Introduction To Modern Planar Transmission Lines: Physical, Analytical, and Circuit Models Approach (Wiley - IEEE) [1 ed.] 1119632277, 9781119632276

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Introduction to Modern Planar Transmission Lines

Introduction to Modern Planar Transmission Lines Physical, Analytical, and Circuit Models Approach

Anand K. Verma Adjunct Professor, School of Engineering Macquarie University, Sydney, Australia Formerly Professor, Department of Electronic Science South Campus, Delhi University New Delhi, India

This edition first published 2021 © 2021 John Wiley & Sons, Inc. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, except as permitted by law. Advice on how to obtain permission to reuse material from this title is available at http://www.wiley.com/go/permission. The right of Anand K. Verma to be identified as the author of this work has been asserted in accordance with law. Registered Office(s) John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, USA John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, UK Editorial Office The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, UK For details of our global editorial offices, customer services, and more information about Wiley products visit us at www.wiley.com. Wiley also publishes its books in a variety of electronic formats and by print-on-demand. Some content that appears in standard print versions of this book may not be available in other formats. Limit of Liability/Disclaimer of Warranty In view of ongoing research, equipment modifications, changes in governmental regulations, and the constant flow of information relating to the use of experimental reagents, equipment, and devices, the reader is urged to review and evaluate the information provided in the package insert or instructions for each chemical, piece of equipment, reagent, or device for, among other things, any changes in the instructions or indication of usage and for added warnings and precautions. While the publisher and authors have used their best efforts in preparing this work, they make no representations or warranties with respect to the accuracy or completeness of the contents of this work and specifically disclaim all warranties, including without limitation any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives, written sales materials or promotional statements for this work. The fact that an organization, website, or product is referred to in this work as a citation and/or potential source of further information does not mean that the publisher and authors endorse the information or services the organization, website, or product may provide or recommendations it may make. This work is sold with the understanding that the publisher is not engaged in rendering professional services. The advice and strategies contained herein may not be suitable for your situation. You should consult with a specialist where appropriate. Further, readers should be aware that websites listed in this work may have changed or disappeared between when this work was written and when it is read. Neither the publisher nor authors shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages. Library of Congress Cataloging-in-Publication Data: Names: Verma, Anand K., 1948– author. Title: Introduction to modern planar transmission lines : physical, analytical, and circuit models approach / Anand K. Verma. Description: Hoboken, New Jersey : Wiley-IEEE Press, [2021] | Includes bibliographical references and index. Identifiers: LCCN 2020050198 (print) | LCCN 2020050199 (ebook) | ISBN 9781119632276 (cloth) | ISBN 9781119632450 (adobe pdf) | ISBN 9781119632474 (epub) Subjects: LCSH: Electric lines. Classification: LCC TK3221 .V47 2021 (print) | LCC TK3221 (ebook) | DDC 621.3815–dc23 LC record available at https://lccn.loc.gov/2020050198 LC ebook record available at https://lccn.loc.gov/2020050199 Cover Design: Wiley Cover Image: © DamienGeso/Getty Images Set in 10/12.5pt STIXTwoText by Straive, Pondicherry, India 10 9 8 7 6 5 4 3 2 1

To my loving and caring wife Kamini Verma and ever-smiling grandchildren Naina, Tinu, and Nupur.

vii

Contents Preface xxi Acknowledgments xxiii Author Biography xxv 1

Overview of Transmission Lines: (Historical Perspective, Overview of Present Book)

1.1 1.1.1 1.1.2 1.1.3 1.1.4

Introduction 1 Overview of the Classical Transmission Lines 1 Telegraph Line 1 Development of Theoretical Concepts in EM-Theory 2 Development of the Transmission Line Equations 6 Waveguides as Propagation Medium 8

1.2 1.2.1 1.2.2

Planar Transmission Lines 8 Development of Planar Transmission Lines 8 Analytical Methods Applied to Planar Transmission Lines

1.3 1.3.1 1.3.2

Overview of Present Book 10 The Organization of Chapters in This Book 11 Key Features, Intended Audience, and Some Suggestions References 15

2

Waves on Transmission Lines – I: (Basic Equations, Multisection Transmission Lines)

2.1 2.1.1 2.1.2 2.1.3 2.1.4 2.1.5 2.1.6 2.1.7 2.1.8 2.1.9

Introduction 19 Uniform Transmission Lines 19 Wave Motion 19 Circuit Model of Transmission Line 21 Kelvin–Heaviside Transmission Line Equations in Time Domain 23 Kelvin–Heaviside Transmission Line Equations in Frequency-Domain 24 Characteristic of Lossy Transmission Line 26 Wave Equation with Source 27 Solution of Voltage and Current-Wave Equation 28 Application of Thevenin’s Theorem to Transmission Line 33 Power Relation on Transmission Line 34

2.2 2.2.1 2.2.2

Multisection Transmission Lines and Source Excitation Multisection Transmission Lines 36 Location of Sources 38

2.3

Nonuniform Transmission Lines

40

1

10

14

36

19

viii

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40

2.3.1 2.3.2

Wave Equation for Nonuniform Transmission Line Lossless Exponential Transmission Line 42 References 43

3

Waves on Transmission Lines – II: (Network Parameters, Wave Velocities, Loaded Lines)

3.1 3.1.1 3.1.2 3.1.3 3.1.4

Introduction 45 Matrix Description of Microwave Network 45 [Z] Parameters 46 Admittance Matrix 48 Transmission [ABCD] Parameter 49 Scattering [S] Parameters 51

3.2 3.2.1 3.2.2 3.2.3

Conversion and Extraction of Parameters 59 Relation Between Matrix Parameters 59 De-Embedding of True S-Parameters 61 Extraction of Propagation Characteristics 62

3.3 3.3.1 3.3.2

Wave Velocity on Transmission Line Phase Velocity 63 Group Velocity 66

3.4 3.4.1 3.4.2

Linear Dispersive Transmission Lines 69 Wave Equation of Dispersive Transmission Lines 69 Circuit Models of Dispersive Transmission Lines 72 References 75

4

Waves in Material Medium – I: (Waves in Isotropic and Anisotropic Media, Polarization of Waves)

4.1 4.1.1 4.1.2 4.1.3

Introduction 77 Basic Electrical Quantities and Parameters 77 Flux Field and Force Field 77 Constitutive Relations 78 Category of Materials 79

4.2 4.2.1 4.2.2 4.2.3 4.2.4 4.2.5 4.2.6

Electrical Property of Medium 80 Linear and Nonlinear Medium 80 Homogeneous and Nonhomogeneous Medium Isotropic and Anisotropic Medium 81 Nondispersive and Dispersive Medium 85 Non-lossy and Lossy Medium 86 Static Conductivity of Materials 86

4.3 4.3.1 4.3.2

Circuit Model of Medium 87 RC Circuit Model of Lossy Dielectric Medium 88 Circuit Model of Lossy Magnetic Medium 90

4.4 4.4.1 4.4.2

Maxwell Equations and Power Relation 91 Maxwell’s Equations 91 Power and Energy Relation from Maxwell Equations

4.5 4.5.1 4.5.2

EM-waves in Unbounded Isotropic Medium EM-wave Equation 96 1D Wave Equation 97

45

63

96

81

94

77

Contents

98

4.5.3 4.5.4 4.5.5

Uniform Plane Waves in Linear Lossless Homogeneous Isotropic Medium Vector Algebraic Form of Maxwell Equations 101 Uniform Plane Waves in Lossy Conducting Medium 102

4.6 4.6.1 4.6.2 4.6.3 4.6.4

Polarization of EM-waves 104 Linear Polarization 104 Circular Polarization 105 Elliptical Polarization 106 Jones Matrix Description of Polarization States

4.7 4.7.1 4.7.2 4.7.3 4.7.4 4.7.5

EM-waves Propagation in Unbounded Anisotropic Medium 110 Wave Propagation in Uniaxial Medium 111 Wave Propagation in Uniaxial Gyroelectric Medium 113 Dispersion Relations in Biaxial Medium 114 Concept of Isofrequency Contours and Isofrequency Surfaces 115 Dispersion Relations in Uniaxial Medium 116 References 118

5

Waves in Material Medium-II: (Reflection & Transmission of Waves, Introduction to Metamaterials)

5.1 5.1.1 5.1.2 5.1.3

Introduction 121 EM-Waves at Interface of Two Different Media 121 Normal Incidence of Plane Waves 121 The Interface of a Dielectric and Perfect Conductor 124 Transmission Line Model of the Composite Medium 124

5.2 5.2.1 5.2.2 5.2.3 5.2.4

Oblique Incidence of Plane Waves 125 TE (Perpendicular) Polarization Case 125 TM (Parallel) Polarization Case 128 Dispersion Diagrams of Refracted Waves in Isotropic and Uniaxial Anisotropic Media Wave Impedance and Equivalent Transmission Line Model 130

5.3 5.3.1 5.3.2

Special Cases of Angle of Incidence Brewster Angle 132 Critical Angle 133

5.4 5.4.1 5.4.2

EM-Waves Incident at Dielectric Slab Oblique Incidence 136 Normal Incidence 138

5.5 5.5.1 5.5.2 5.5.3 5.5.4 5.5.5 5.5.6 5.5.7 5.5.8

EM-Waves in Metamaterials Medium 139 General Introduction of Metamaterials and Their Classifications EM-Waves in DNG Medium 141 Basic Transmission Line Model of the DNG Medium 144 Lossy DPS and DNG Media 146 Wave Propagation in DNG Slab 146 DNG Flat Lens and Superlens 149 Doppler and Cerenkov Radiation in DNG Medium 151 Metamaterial Perfect Absorber (MPA) 153 References 156

6

Electrical Properties of Dielectric Medium Introduction

159

106

132

136

159

139

129

121

ix

x

Contents

6.1 6.1.1 6.1.2 6.1.3 6.1.4

Modeling of Dielectric-Medium 159 Dielectric Polarization 159 Susceptibility, Relative Permittivity, and Clausius–Mossotti Model Models of Polarizability 164 Magnetization of Materials 165

6.2 6.2.1 6.2.2

Static Dielectric Constants of Materials Natural Dielectric Materials 167 Artificial Dielectric Materials 168

6.3 6.3.1 6.3.2 6.3.3

Dielectric Mixtures 173 General Description of Dielectric Mixture Medium 173 Limiting Values of Equivalent Relative Permittivity 174 Additional Equivalent Permittivity Models of Mixture 175

6.4 6.4.1 6.4.2 6.4.3

Frequency Response of Dielectric Materials 178 Relaxation in Material and Decay Law 179 Polarization Law of Linear Dielectric-Medium 179 Debye Dispersion Relation 181

6.5 6.5.1 6.5.2 6.5.3 6.5.4

Resonance Response of the Dielectric-Medium 183 Lorentz Oscillator Model 183 Drude Model of Conductor and Plasma 188 Dispersion Models of Dielectric Mixture Medium 189 Kramers–Kronig Relation 190

6.6 6.6.1

Interfacial Polarization 190 Interfacial Polarization in Two-Layered Capacitor Medium

6.7 6.7.1 6.7.2 6.7.3 6.7.4 6.7.5

Circuit Models of Dielectric Materials 193 Series RC Circuit Model 193 Parallel RC Circuit Model 194 Parallel Series Combined Circuit Model 195 Series Combination of RC Parallel Circuit 196 Series RLC Resonant Circuit Model 200

6.8 6.8.1 6.8.2 6.8.3

Substrate Materials for Microwave Planar Technology 202 Evaluation of Parameters of Single-Term Debye and Lorentz Models Multiterm and Wideband Debye Models 205 Metasubstrates 207 References 208

7

Waves in Waveguide Medium

7.1 7.1.1 7.1.2 7.1.3 7.1.4

Introduction 213 Classification of EM-Fields 213 Maxwell Equations and Vector Potentials 214 Magnetic Vector Potential 214 Electric Vector Potential 215 Generation of EM-Field by Electric and Magnetic Vector Potentials

7.2 7.2.1 7.2.2

Boundary Surface and Boundary Conditions Perfect Electric Conductor (PEC) 219 Perfect Magnetic Conductor (PMC) 220

162

167

190

202

213

219

216

Contents

221

7.2.3

Interface of Two Media

7.3 7.3.1 7.3.2 7.3.3

TEM-Mode Parallel-Plate Waveguide 222 TEM Field in Parallel-Plate Waveguide 222 Circuit Relations 222 Kelvin–Heaviside Transmission Line Equations from Maxwell Equations

7.4 7.4.1 7.4.2 7.4.3

Rectangular Waveguides 223 Rectangular Waveguide with Four EWs 223 Rectangular Waveguide with Four MWs 235 Rectangular Waveguide with Composite Electric and MWs

7.5 7.5.1 7.5.2

Conductor Backed Dielectric Sheet Surface Wave Waveguide TMz Surface Wave Mode 240 TEz Surface Wave Mode 242

7.6 7.6.1 7.6.2

Equivalent Circuit Model of Waveguide 244 Relation Between Wave Impedance and Characteristic Impedance Transmission Line Model of Waveguide 245

7.7 7.7.1 7.7.2 7.7.3 7.7.4

Transverse Resonance Method (TRM) 247 Standard Rectangular Waveguide 247 Dielectric Loaded Waveguide 248 Slab Waveguide 249 Conductor Backed Multilayer Dielectric Sheet

7.8 7.8.1 7.8.2

Substrate Integrated Waveguide (SIW) 253 Complete Mode Substrate Integrated Waveguide (SIW) 253 Half-Mode Substrate Integrated Waveguide (SIW) 256 References 258

8

Microstrip Line: Basic Characteristics 261

8.1 8.1.1 8.1.2 8.1.3 8.1.4

Introduction 261 General Description 261 Conceptual Evolution of Microstrip Lines 264 Non-TEM Nature of Microstrip Line 264 Quasi-TEM Mode of Microstrip Line 265 Basic Parameters of Microstrip Line 266

8.2 8.2.1 8.2.2 8.2.3 8.2.4 8.2.5 8.2.6

Static Closed-Form Models of Microstrip Line 268 Homogeneous Medium Model of Microstrip Line (Wheeler’s Transformation) Static Characteristic Impedance of Microstrip Line 270 Results on Static Parameters of Microstrip Line 271 Effect of Conductor Thickness on Static Parameters of Microstrip Line 272 Effect of Shield on Static Parameters of Microstrip Line 274 Microstrip Line on Anisotropic Substrate 276

8.3 8.3.1 8.3.2 8.3.3 8.3.4 8.3.5

Dispersion in Microstrip Line 278 Nature of Dispersion in Microstrip 278 Waveguide Model of Microstrip 280 Logistic Dispersion Model of Microstrip (Dispersion Law of Microstrip) 282 Kirschning–Jansen Dispersion Model 284 Improved Model of Frequency-Dependent Characteristic Impedance 284

223

237 240

244

252

268

xi

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285

8.3.6

Synthesis of Microstrip Line

8.4 8.4.1 8.4.2

Losses in Microstrip Line 285 Dielectric Loss in Microstrip 285 Conductor Loss in Microstrip 287

8.5

Circuit Model of Lossy Microstrip Line References 296

9

Coplanar Waveguide and Coplanar Stripline: Basic Characteristics

9.1

Introduction 301 General Description

9.2 9.2.1 9.2.2 9.2.3 9.2.4 9.2.5

Fundamentals of Conformal Mapping Method 302 Complex Variable 302 Analytic Function 303 Properties of Conformal Transformation 304 Schwarz–Christoffel (SC) Transformation 306 Elliptic Sine Function 307

9.3 9.3.1 9.3.2 9.3.3 9.3.4 9.3.5 9.3.6

Conformal Mapping Analysis of Coplanar Waveguide 310 Infinite Extent CPW 310 CPW on Finite Thickness Substrate and Infinite Ground Plane 311 CPW with Finite Ground Planes 313 Static Characteristics of CPW 315 Top-Shielded CPW 316 Conductor-Backed CPW 317

9.4 9.4.1 9.4.2 9.4.3 9.4.4 9.4.5 9.4.6 9.4.7

Coplanar Stripline 319 Symmetrical CPS on Infinitely Thick Substrate 319 Asymmetrical CPS (ACPS) on Infinitely Thick Substrate 320 Symmetrical CPS on Finite Thickness Substrate 321 Asymmetrical CPW (ACPW) and Asymmetrical CPS (ACPS) on Finite Thickness Substrate Asymmetric CPS Line with Infinitely Wide Ground Plane 326 CPS with Coplanar Ground Plane [CPS–CGP] 326 Discussion on Results for CPS 327

9.5 9.5.1 9.5.2

Effect of Conductor Thickness on Characteristics of CPW and CPS Structures CPW Structure 329 CPS Structure 330

9.6 9.6.1 9.6.2 9.6.3 9.6.4

Modal Field and Dispersion of CPW and CPS Structures 330 Modal Field Structure of CPW 330 Modal Field Structure of CPS 332 Closed-Form Dispersion Model of CPW 334 Dispersion in CPS Line 336

9.7 9.7.1 9.7.2 9.7.3

Losses in CPW and CPS Structures Conductor Loss 337 Dielectric Loss 341 Substrate Radiation Loss 342

9.8 9.8.1

Circuit Models and Synthesis of CPW and CPS 345 Circuit Model 345

295

301

301

337

329

324

Contents

9.8.2 9.8.3

Synthesis of CPW 346 Synthesis of CPS 347 References 348

10

Slot Line: Basic Characteristics 353

10.1 10.1.1 10.1.2

Introduction 353 Slot Line Structures 353 Structures of the Open Slot Line 353 Shielded Slot Line Structures 354

10.2 10.2.1

Analysis and Modeling of Slot Line Magnetic Current Model 355

10.3 10.3.1 10.3.2 10.3.3 10.3.4

Waveguide Model 357 Standard Slot Line 357 Sandwich Slot Line 370 Shielded Slot Line 371 Characteristics of Slot Line

10.4 10.4.1 10.4.2 10.4.3

Closed-form Models 378 Conformal Mapping Method Krowne Model 380 Integrated Model 381 References 384

11

Coupled Transmission Lines: Basic Characteristics 391

11.1

Introduction 391 Some Coupled Line Structures

11.2 11.2.1 11.2.2

Basic Concepts of Coupled Transmission Lines 394 Forward and Reverse Directional Coupling 394 Basic Definitions 395

11.3 11.3.1 11.3.2 11.3.3

Circuit Models of Coupling 396 Capacitive Coupling – Even and Odd Mode Basics Forms of Capacitive Coupling 399 Forms of Inductive Coupling 401

11.4 11.4.1 11.4.2

Even–Odd Mode Analysis of Symmetrical Coupled Lines Analysis Method 404 Coupling Coefficients 408

11.5 11.5.1 11.5.2 11.5.3

Wave Equation for Coupled Transmission Lines 409 Kelvin–Heaviside Coupled Transmission Line Equations Solution of Coupled Wave Equation 411 Modal Characteristic Impedance and Admittance 414 References 416

12

Planar Coupled Transmission Lines 419

12.1

Introduction 419 Line Parameters of Symmetric Edge Coupled Microstrips

355

376 379

391

397

402

409

419

xiii

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Contents

12.1.1 12.1.2

Static Models for Even- and Odd-Mode Relative Permittivity and Characteristic Impedances of Edge Coupled Microstrips 419 Frequency-Dependent Models of Edge Coupled Microstrip Lines 423

12.2 12.2.1 12.2.2

Line Parameters of Asymmetric Coupled Microstrips 424 Static Parameters of Asymmetrically Coupled Microstrips 424 Frequency-Dependent Line Parameters of Asymmetrically Coupled Microstrips

12.3 12.3.1 12.3.2

Line Parameters of Coupled CPW 430 Symmetric Edge Coupled CPW 430 Shielded Broadside Coupled CPW 432

12.4 12.4.1 12.4.2 12.4.3

Network Parameters of Coupled Line Section 433 Symmetrical Coupled Line in Homogeneous Medium 434 Symmetrical Coupled Microstrip Line in An Inhomogeneous Medium ABCD Matrix of Symmetrical Coupled Transmission Lines 444

12.5 12.5.1

Asymmetrical Coupled Lines Network Parameters [ABCD] Parameters of the 4-Port Network 447 References 452

13

Fabrication of Planar Transmission Lines

13.1 13.1.1 13.1.2 13.1.3 13.1.4

Introduction 455 Elements of Hybrid MIC (HMIC) Technology 455 Substrates 456 Hybrid MIC Fabrication Process 457 Thin-Film Process 459 Thick-Film Process 460

13.2 13.2.1 13.2.2

Elements of Monolithic MIC (MMIC) Technology Fabrication Process 464 Planar Transmission Lines in MMIC 466

13.3 13.3.1 13.3.2

Micromachined Transmission Line Technology MEMS Fabrication Process 471 MEMS Transmission Line Structures 473

13.4 13.4.1 13.4.2 13.4.3 13.4.4

Elements of LTCC 478 LTCC Materials and Process 480 LTCC Circuit Fabrication 482 LTCC Planar Transmission Line and Some Components LTCC Waveguide and Cavity Resonators 489 References 489

14

Static Variational Methods for Planar Transmission Lines

14.1 14.1.1 14.1.2 14.1.3

Introduction 493 Variational Formulation of Transmission Line 493 Basic Concepts of Variation 493 Energy Method-Based Variational Expression 495 Green’s Function Method-Based Variational Expression

14.2 14.2.1

Variational Expression of Line Capacitance in Fourier Domain Transformation of Poisson Equation in Fourier Domain 498

447

455

462

471

484

493

497 498

438

426

Contents

14.2.2 14.2.3

Transformation of Variational Expression of Line Capacitance in Fourier Domain 499 Fourier Transform of Some Charge Distribution Functions 500

14.3 14.3.1 14.3.2 14.3.3

Analysis of Microstrip Line by Variational Method 503 Boxed Microstrip Line (Green’s Function Method in Space Domain) 503 Open Microstrip Line (Green’s Function Method in Fourier Domain) 508 Open Microstrip Line (Energy Method in Fourier Domain) 510

14.4 14.4.1 14.4.2

Analysis of Multilayer Microstrip Line 512 Space Domain Analysis of Multilayer Microstrip Structure 512 Static Spectral Domain Analysis of Multilayer Microstrip 518

14.5 14.5.1 14.5.2

Analysis of Coupled Microstrip Line in Multilayer Dielectric Medium Space Domain Analysis 520 Spectral Domain Analysis 523

14.6 14.6.1 14.6.2 14.6.3

Discrete Fourier Transform Method Discrete Fourier Transform 525 Boxed Microstrip Line 528 Boxed Coplanar Waveguide 531 References 537

15

Multilayer Planar Transmission Lines: SLR Formulation

15.1 15.1.1 15.1.2 15.1.3 15.1.4 15.1.5

Introduction 541 SLR Process for Multilayer Microstrip Lines 541 SLR-Process for Lossy Multilayer Microstrip Lines 542 Dispersion Model of Multilayer Microstrip Lines 544 Characteristic Impedance and Synthesis of Multilayer Microstrip Lines Models of Losses in Multilayer Microstrip Lines 550 Circuit Model of Multilayer Microstrip Lines 553

15.2 15.2.1 15.2.2 15.2.3 15.2.4

SLR Process for Multilayer Coupled Microstrip Lines 553 Equivalent Single-Layer Substrate 553 Dispersion Model of Multilayer Coupled Microstrip Lines 555 Characteristic Impedance and Synthesis of Multilayer Coupled Microstrips Loss Models of Multilayer Coupled Microstrip Lines 557

15.3 15.3.1 15.3.2 15.3.3 15.3.4

SLR Process for Multilayer ACPW/CPW 559 Single-Layer Reduction (SLR) Process for Multilayer ACPW/CPW 560 Static SDA of Multilayer ACPW/CPW Using Two-Conductor Model 561 Dispersion Models of Multilayer ACPW/CPW 564 Loss Models of Multilayer ACPW/CPW 565

15.4

Further Consideration of SLR Formulation References 567

16

Dynamic Spectral Domain Analysis

16.1

Introduction 571 General Discussion of SDA

16.2 16.2.1

Green’s Function of Single-Layer Planar Line Formulation of Field Problem 574

519

525

566

571

571 574

541

549

556

xv

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Contents

16.2.2 16.2.3

Case #1: CPW and Microstrip Structures 576 Case #II – Sides: MW – EW, Bottom: MW, Top: EW

16.3 16.3.1 16.3.2

Solution of Hybrid Mode Field Equations (Galerkin’s Method in Fourier Domain) 587 Microstrip 587 CPW Structure 589

16.4 16.4.1 16.4.2

Basis Functions for Surface Current Density and Slot Field Nature of the Field and Current Densities 590 Basis Functions and Nature of Hybrid Modes 590

16.5 16.5.1

Coplanar Multistrip Structure 596 Symmetrical Coupled Microstrip Line 597

16.6 16.6.1 16.6.2

Multilayer Planar Transmission Lines 598 Immittance Approach for Single-Level Strip Conductors 600 Immittance Approach for Multilevel Strip Conductors 605 References 610

17

Lumped and Line Resonators: Basic Characteristics 613

17.1

Introduction 613 Basic Resonating Structures

17.2 17.2.1 17.2.2 17.2.3 17.2.4 17.2.5 17.2.6

Zero-Dimensional Lumped Resonator 615 Lumped Series Resonant Circuit 615 Lumped Parallel Resonant Circuit 617 Resonator with External Circuit 619 One-Port Reflection-Type Resonator 620 Two-Port Transmission-Type Resonator 623 Two-Port Reaction-Type Resonator 629

17.3 17.3.1 17.3.2

Transmission Line Resonator 629 Lumped Resonator Modeling of Transmission Line Resonator 630 Modal Description of λg/2 Short-Circuited Line Resonator 634 References 636

18

Planar Resonating Structures

18.1 18.1.1 18.1.2 18.1.3 18.1.4 18.1.5

Introduction 639 Microstrip Line Resonator 639 λg/2 Open-end Microstrip Resonator 640 λg/2 and λg/4 Short-circuited Ends Microstrip Resonator Microstrip Ring Resonator 643 Microstrip Step Impedance Resonator 645 Microstrip Hairpin Resonator 649

18.2

CPW Resonator

18.3

Slot Line Resonator

18.4 18.4.1 18.4.2 18.4.3

Coupling of Line Resonator to Source and Load Direct-coupled Resonator 655 Reactively Coupled Line Resonator 656 Tapped Line Resonator 658

584

590

613

639

651 653 654

643

Contents

659

18.4.4

Feed to Planar Transmission Line Resonator

18.5 18.5.1 18.5.2 18.5.3

Coupled Resonators 659 Coupled Microstrip Line Resonator 659 Circuit Model of Coupled Microstrip Line Resonator 661 Some Structures of Coupled Microstrip Line Resonator 664

18.6 18.6.1 18.6.2 18.6.3 18.6.4 18.6.5

Microstrip Patch Resonators 666 Rectangular Patch 667 Modified Wolff Model (MWM) 667 Circular Patch 670 Ring Patch 671 Equilateral Triangular Patch 672

18.7 18.7.1 18.7.2 18.7.3

2D Fractal Resonators 674 Fractal Geometry 674 Fractal Resonator Antenna 682 Fractal Resonators 683

18.8 18.8.1 18.8.2

Dual-Mode Dual-Mode Dual-Mode References

Resonators 686 Patch Resonators 686 Ring Resonators 689

692 697

19

Planar Periodic Transmission Lines

19.1 19.1.1 19.1.2 19.1.3

Introduction 697 1D and 2D Lattice Structures 697 Bragg’s Law of Diffraction 697 Crystal Lattice Structures 698 Concept of Brillouin Zone 701

19.2 19.2.1

Space Harmonics of Periodic Structures 703 Floquet–Bloch Theorem and Space Harmonics 703

19.3 19.3.1 19.3.2 19.3.3 19.3.4 19.3.5 19.3.6

Circuit Models of 1D Periodic Transmission Line 704 Periodically Loaded Artificial Lines 705 [ABCD] Parameters of Unit Cell 707 Dispersion in Periodic Lines 709 Characteristics of 1D Periodic Lines 712 Some Loading Elements of 1D Periodic Lines 718 Realization of Planar Loading Elements 720

19.4 19.4.1 19.4.2

1D Planar EBG Structures 728 1D Microstrip EBG Line 728 1D CPW EBG Line 742 References 748

20

Planar Periodic Surfaces 753

20.1 20.1.1 20.1.2

Introduction 753 2D Planar EBG Surfaces 753 General Introduction of EBG Surfaces Characteristics of EBG Surface 756

753

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Contents

760

20.1.3

Horizontal Wire Dipole Near EBG Surface

20.2 20.2.1 20.2.2

Circuit Models of Mushroom-Type EBG Basic Circuit Model 763 Dynamic Circuit Model 767

20.3

Uniplanar EBG Structures

20.4 20.4.1 20.4.2

2D Circuit Models of EBG Structures 773 Shunt-Connected 2D Planar EBG Circuit Model 773 Series-Connected 2D Planar EBG Circuit Model 780 References 782

21

Metamaterials Realization and Circuit Models – I: (Basic Structural Elements and Bulk Metamaterials)

21.1 21.1.1 21.1.2 21.1.3

Introduction 785 Artificial Electric Medium 785 Polarization in the Wire Medium 786 Equivalent Parallel Plate Waveguide Model of Wire Medium Reactance Loaded Wire Medium 791

21.2 21.2.1 21.2.2 21.2.3 21.2.4 21.2.5

Artificial Magnetic Medium 793 Characteristics of the SRR 794 Circuit Model of the SRR 795 Computation of Equivalent Circuit Parameters of SRR Bi-anisotropy in the SRR Medium 800 Variations in SRR Structure 801

21.3 21.3.1 21.3.2 21.3.3

Double Negative Metamaterials 803 Composite Permittivity–Permeability Functions 803 Realization of Composite DNG Metamaterials 805 Realization of Single-Structure DNG Metamaterials 809

21.4 21.4.1 21.4.2

Homogenization and Parameters Extraction Nicolson–Ross–Weir (NRW) Method 814 Dynamic Maxwell Garnett Model 821 References 827

22

Metamaterials Realization and Circuit Models – II: (Metalines and Metasurfaces)

22.1 22.1.1 22.1.2 22.1.3 22.1.4 22.1.5

Introduction 833 Circuit Models of 1D-Metamaterials 833 Homogenization of the 1D-medium 834 Circuit Equivalence of Material Medium 834 Single Reactive Loading of Host Medium 836 Single Reactive Loading of Host Medium with Coupling Circuit Models of 1D Metalines 839

22.2 22.2.1 22.2.2 22.2.3

Nonresonant Microstrip Metalines 846 Series–Parallel (CRLH) Metalines 846 Cascaded MNG–ENG (CRLH) Metalines 848 Parallel–Series (D-CRLH) Metalines 849

22.3 22.3.1

Resonant Metalines 850 Resonant Inclusions 851

763

771

789

798

814

838

833

785

Contents

22.3.2 22.3.3

Microstrip Resonant Metalines 852 CPW-Resonant Metalines 854

22.4 22.4.1 22.4.2 22.4.3

Some Applications of Metalines 856 Backfire to Endfire Leaky Wave Antenna 856 Metaline-Based Microstrip Directional Coupler 857 Multiband Metaline-Based Components 859

22.5 22.5.1 22.5.2 22.5.3 22.5.4 22.5.5

Modeling and Characterization of Metsurfaces 859 Characterization of Metasurface 862 Reflection and Transmission Coefficients of Isotropic Metasurfaces Phase Control of Metasurface 867 Generalized Snell’s Laws of Metasurfaces 870 Surface Waves on Metasurface 872

22.6 22.6.1 22.6.2 22.6.3

Applications of Metasurfaces 873 Demonstration of Anomalous Reflection and Refraction of Metasurfaces Reflectionless Transmission of Metasurfaces 877 Polarization Conversion of Incident Plane Wave 880 References 886 Index

893

865

873

xix

xxi

Preface The planar transmission lines form the core of modern high-frequency communication, computer, and other related technology. The subject has come up to the present level of maturity over the past three to four decades. The planar transmission lines are used not only as interconnects on the PCB board and IC chips; these are directly needed for the development of microwave and mm-wave components in the form of microwave integrated circuits (MIC). The types of planar transmission lines, i.e. their physical structures and material medium, have been changing with the growth of technology in many other disciplines. Such efforts during recent years propelled the MIC to move in many exotic directions – MMIC, MEMS, LTCC, use of ferroelectrics, and hightemperature superconductors, optically controlled microwave devices, nonlinear planar transmission lines, DGS, EBG, metamaterials, etc. The researchers with varying backgrounds have contributed much to research activities. Already the divergent planar technology has contributed significantly to the advancement of highfrequency electronics and in the near future, more contribution will be made by it. The exotic planar transmission lines are not covered comprehensively in a single book. The present book is an attempt in this direction. The proposed book aims to provide a comprehensive discussion of planar transmission lines and their applications. It focuses on physical understanding, analytical approach, and circuit models for planar transmission lines and resonators in the complex environment. The present book has evolved from the lecture notes, workshop, seminar presentation, and invited lectures delivered by the author at many universities and R&D centers. Some chapters were also initially written for the Ph.D. students to help them to understand the topics. Finally, it has evolved from notes prepared by the author as a scheme for the self-study. The author started his academic career after 17 years of professional experience in

the field of electrical engineering, broadcast transmitters, and satellite communication. At present, the planar transmission lines are taught as part of the course on RF and microwave packaging, advanced electromagnetic field theory, and microwave design. It is also taught as an independent paper. However, a teacher has to consult divergent sources to prepare the lecture notes, as no single source at the teaching-level is available. Moreover, the classroom teaching of the planar transmission lines is not as systematic as the classical metallic waveguide structures. It is due to the very nature of the subject itself. The available books are usually not classroom oriented. Usually, they can be grouped into two categories – 1. Designoriented books, 2. Monograph kind of books. Once we use the first category of books in the classroom, we end up writing only closed-form expressions without any systematic derivation of the expressions. The systematic approach is important in the classroom environment. The second category of books is suitable for an experienced researcher or specialist. It is difficult to use them in the classroom. Thus, a teacher of this subject has to struggle between these two extremes to balance teaching throughout the semester. Finally, a teacher has to depend on personal experience and lecture notes. There is a need to present, in one cover, the divergent topics of the planar transmission lines in a studentfriendly format. The researchers with varying backgrounds in physics, chemistry, engineering, and other fields have joined activities in the expanding area of the planar technology. The early researchers, R&D professionals in the industry, teachers, and students need a text that could be useful in faster acquisition of the physical modeling process and theoretical formulations used in the classical planar transmission lines. Similar treatment is also needed for the modern engineered EBG and metamaterial lines and surfaces.

xxii

Preface

Therefore, the real motivation for writing this intermediate-level book is to fill the gap for a textbook on the planar transmission line that caters to the need for classroom teaching, early researchers with divergent backgrounds, and designers working in the microwave industry. The book is intended to help students both at the undergraduate and postgraduate levels. It also serves the purpose of a resource book for self-study. The detailed derivations of results and physical modeling of the planar transmission lines are two basic concepts followed through the book. The present book is neither a design-oriented book nor an advanced monograph.

The book correlates the physical process with mathematical treatment. The advanced mathematical methods such as the conformal mapping method, variational method, and spectral domain method applied to planar lines are worked out in adequate details. The book further covers modern topics such as the DGS/EBG, metamaterial-based planar transmission lines, and surfaces. The approach used in writing the book is perhaps less formal than most available texts. This approach is helpful for classroom teaching. It also assists the reader to follow the contemporary developments in planar technology.

xxiii

Acknowledgments The author is thankful to Prof. Karu P. Esselle and Prof. Graham Town, School of Engineering, Macquarie University, Sydney, for supporting Adjunct Professorship at Macquarie University. The author is also thankful to Prof. Enakshi K. Sharma, Department of Electronic Science, South Campus, Delhi University, for continuous discussions on topics related to EM-Theory, Wave propagation, etc. The author sincerely appreciates the help and guidance provided by Prof. Kai Chang, Department of Electrical Engineering, Texas A&M University during the review process of the book. The author also appreciates the active interest taken by Mr. Brett Kurzman and his team of Wiley Publishing for the review and friendly administrative support. The author is particularly grateful to Dr. I.J. Bahl, Editor-in-Chief, Int. J. of RF & Microwave ComputerAided Engineering, John Wiley, USA; Prof. Zhongxiang Shen, School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore; and Prof. Ladislau Matekovits, Department of Electronics and Telecommunications, Politecnico di Torino, Italy, for reading book chapters and providing valuable suggestions to improve the book. The author has benefitted from the comments, suggestions, and corrections of many colleagues, teachers, and students. The author would like to thank the following people for their useful contributions toward corrections and useful discussions: Dr. Koteshwar Rao, Dr. Harsupreet Kaur, Dr. Kamlesh Patel, Mr. Amit Birwal, Dr. Ashwani Kumar, Dr. Paramjeet Singh, Dr. Y.K. Awasthi, and Mr. Prashant Chaudhary from the Department of Electronic Science, Delhi University;

Dr. Raheel Hashmi, Dr. Sudipta Chakraborty, and Dr. Rajas Prakash Khokle of School of Engineering, Macquarie University, Sydney; Dr. Nasimuddin of Institute for Infocomm Research, Singapore; Prof. Asoke De, Dr. Priyanka Jain, Ms. Priyanka Garg of Department of Electrical Engineering, DTU, Delhi; Mr. Shailendra Singh of Product Development and Innovation Center, Bharat Electronics Ltd., Bengaluru, India; Dr. Rajesh Singh, Microwave Radiation Laboratory, University of Pisa, Italy; Dr. Archna Rajput, IIT Jammu, India; Dr. Ravi Kumar Arya, Dept. of ECE, NIT, Delhi. The author is especially thankful to his students, Mr. Shailendra Singh and Mr. Prashant Chaudhary for their continuous help in correction of all chapters. The author expresses his unbounded love and regards to his parents – Late Sh. A.P. Verma and Late Tara Devi, Uncle Late Sh. Hira Prasad, to grandmother Late Radhika Devi, and Uncle and Life-guide Sh. Girija Pd. Srivastava. The author also is grateful to his teacher Prof. M.K.P. Mishra for excellent teaching of Circuit Theory in unique style, and for providing support in many ways. The author is thankful to his family members for their encouragement and support. Finally, I wish to express my heartfelt thanks and deepest appreciation to my wife, Kamini. The smiling faces of my grandchildren, Naina, Tinu, and Nupur have always kept me going on with the tiring work of book writing.

Anand K. Verma New Delhi, Sydney

xxv

Author Biography Anand K. Verma, PhD, is an adjunct professor in the School of Engineering, Macquarie University, Sydney, Australia. Formerly, he was professor and head, Department of Electronic Science, South Campus, University of Delhi, New Delhi, India. He has been a visiting professor at Otto van Guericke University, Magdeburg, Germany (2002–2003), and a Tan Chin Tuan Scholar (2001) at Nanyang Technological University, Singapore. He holds a German patent on a microstrip antenna. He has introduced the concept and design method of surface-mounted compact horn antenna used for high gain, wideband, and

ultra-wideband quasi-planar antenna applicable to both linear and circular polarization. He has organized and attended many international symposia and workshops. He has conducted short-term courses and delivered invited lectures at the research institutes in India and several countries. He was also chairman of the TPC, APMC-2004, New Delhi, India. Professor Verma has published over 250 papers in international journals and the proceedings of international and national symposia. He has introduced the concept of single layer reduction (SLR) formulation for the CAD-oriented modeling of multilayer planar lines.

1

1 Overview of Transmission Lines (Historical Perspective, Overview of Present Book)

Introduction The transmission line is at the core of the communication technology system. It forms a medium for signal transmission, and also helps to develop high-frequency passive components and circuit blocks. Historically, both experimental investigations and analytical theories have played significant roles in the growth of transmission line technology. Each type of distinct line structure is responsible for the development of distinct communication technology. The single-wire transmission line with the Earth as a return conductor is responsible for the operation of Telegraphy. It evolved into the coaxial cable that made the Transatlantic Telegraphy possible. The two-wire open line became a medium for the Telephonic transmission. These two line structures are behind the development of the monopole and dipole antenna that made possible the growth of the high-frequency communication using the medium wave (MW), short wave (SW), very high-frequency (VHF), and ultra-high frequency (UHF) bands. The microwave and mm-wave transmission systems are developed mostly around the metallic waveguides, and subsequently also using the nonmetallic dielectric waveguides. Finally, it has resulted in modern optical fiber technology. The planar transmission lines are behind the modern advanced microwave communication components and systems. The present chapter provides a very brief historical overview of the classical and modern planar transmission lines. The chapter presents a historical survey of the development of the electromagnetic (EM) theory also. Next, a brief overview of the organization of the book is discussed. Objectives

•

To present a survey of the developments of the classical EM-theory.

• • •

To present brief historical notes on the classical transmission lines and development of transmission line theory. To present brief historical notes on planar transmission lines. To present an overview of the contents of the book.

1.1 Overview of the Classical Transmission Lines The classical transmission lines such as a single-wire line with the earth as a return conductor, coaxial cable, two-wire line, multi-conductor lines, and waveguides are reviewed very briefly in this section. The historical development of the Telegrapher’s Equations is also presented. The developments of the theoretical concepts of EM-theory are reviewed below. The data related to the review of the EM-theory and transmission lines are collected from the published books [B.1–B.7] and journal articles referred at the end of the chapter. 1.1.1

Telegraph Line

The telegraph is the first coded point-to-point electrical communication system. As early as 1747, William Watson showed the possibility of transmitting an electrical current on a wire using the earth as a return conductor. Thus, overhead single-wire with the earth as a return conductor is the first transmission line. It is to be noted that even the voltaic pile, i.e. the chemical battery of Volta was nonexistent at that time. The Leiden jar, a capacitor to store the static electrical charges, was invented just two years before in 1745. However, much earlier in the year 1663, Otto Von Guericke studied the phenomenon of static electricity and designed a machine to produce it. Thus, the charged Leiden jar became a source of

Introduction to Modern Planar Transmission Lines: Physical, Analytical, and Circuit Models Approach, First Edition. Anand K. Verma. © 2021 John Wiley & Sons, Inc. Published 2021 by John Wiley & Sons, Inc.

2

1 Overview of Transmission Lines

electricity, and single-wire transmission was the communication medium. These were two important ingredients to establish the telegraph link. The third ingredient of telegraphy, the electroscope, invented by William Gilbert around 1600, acted as the receiver for the coded signals. It is interesting to note that the telegraph was conceived without any theoretical investigations on electricity. Even Coulomb’s law was discovered later in 1785. However, the proper telegraphy could be developed only after the invention of voltaic pile, i.e. a chemical battery by Volta in 1799. Further development of the telegraph has an involved history. In 1837, Morse patented his telegraph in the United States, and on January 6, 1838, the first telegram was sent over 3 km distance. Cooke commercialized the telegraph in England and established the 21 km link on April 9, 1839. Thus, an era of electrical communication heralded. In 1844, long-distance Morse’s telegraph between Washington, DC, and Baltimore, Maryland was established. Before further reviewing the growth of the classical transmission lines, it is useful to have a quick look at the theoretical developments related to electricity [B.5]. 1.1.2 Development of Theoretical Concepts in EM-Theory The ancients unfolded our story of Electromagnetism through careful observations of the phenomena of static electricity and natural magnetism. The developments of basic concepts, analytical modeling, and theoretical formulations used in the EM-Theory are emphasized in the review. The theoretical concepts of the electric and magnetic fields followed the mathematical models developed for the gravitational force by Newton in 1687, and subsequently refined by other investigators. The development of the theoretical models of transmission lines inherited the modeling process, and mathematical method of Fourier developed for the transmission of heat in a rod. Electrostatics and Scalar Potential

Newton published his theory of gravitation in his monograph Philosophiae Naturalis Principia Mathematica. Newton viewed the gravitational interaction between two masses through force. The effect of static electricity was known for a long time, at least since 600 BC. However, only in 1600, Gilbert carried out systematic studies of both magnetism and static electricity. The static electricity was generated by the rubbing of two specific objects. He suggested the word electricus for electricity, and the English word “electricity” was suggested by Thomas Browne in 1646. Gilbert also suggested that

the electrical effect is due to the flow of a small stream of weightless particles called effluvium. This concept helped the formulation of one- and two-fluid model of electricity. He also invented the first electrical measuring instrument, the electroscope, which helped further experimental investigations on electricity. In 1733, Fay proposed that electricity comes in two forms – vitreous and resinous, and on combination, they cancel each other. The flow of the two forms of electricity was explained by the two-fluid model. During this time interval, around up to 1745, the electrical attraction and repulsion were explained using the flow of Gilbert’s particle effluvium. In 1750, Benjamin Franklin proposed the one-fluid model of electricity. The matter containing a very small quantity of electric fluid was treated as negatively charged, and the matter with excess electric fluid was treated positively charged. Thus, the negative charge was resinous electricity, and the positive charge was vitreous electricity. Now, the stage was ready for further theoretical and experimental investigations on electricity. In the year 1773, Lagrange introduced the concept of the gravitational field, now called the scalar potential field, created by a mass. The gravitational force of Newton was conceived as working through the gravitational field. The scalar potential field has appeared as a mechanism to explain the gravitational force interaction between two masses. Thus, a mass located in the potential field, described by a function called the potential function, experiences the gravitational force. In 1777, Lagrange also introduced the divergence theorem for the gravitational field. The nomenclature potential field was introduced by Green in 1828. Subsequently, Gauss in 1840 called it “potential.” Laplace in 1782 showed that the potential function ϕ (x,y,z) satisfies the equation ∇2ϕ = 0. Now the equation is called Laplace’s equation. Following the Law of Gravitation, Coulomb postulated similar inverse square law, now called Coulomb’s law, for the electrically charged bodies. He experimentally demonstrated the inverse square law for the charged bodies in 1785. Thus, the mathematical foundation of the EM-theory was laid by Coulomb. The law was also applicable to magnetic objects. The interaction between charged bodies was described by the electric force. In the year 1812, Poisson extended the concept of potential function from the gravitation to electrostatics. Incorporating the charge distribution function ρ, he obtained the modified Laplace’s equation, written in modern terminology, ∇2ϕ = −ρ/ε0. This equation now called Poisson’s equation is the key equation to describe the potential field due to the charge distribution. In the same year, Gauss

1.1 Overview of the Classical Transmission Lines

rediscovered the divergence theorem originally discovered by Lagrange for the gravitational field. In the year 1828, Green coined the nomenclature – the potential function, for the function of Lagrange and modern concept of the scalar potential field came into existence. Green also showed an important relation between the surface and volume integrals, now known as Green’s Theorem. Green applied his method to the static magnetic field also. Green also introduced a method to solve the 3D inhomogeneous Poisson partial differential equation where the considered source is a point charge. The point charge is described by the Dirac’s delta function. The solution of the Poisson’s partial differential equation, using Dirac’s delta function, is now called Green’s function. Neumann (1832–1925) extended the Green’s function method to solve the 2D potential problem and obtained the eigenfunction expansion of 2D Green’s function [J.1–J.5, B.2, B.7].

Magnetic Effect of Current

So far, we have paid attention to the electrostatics. At this stage, Leiden jar was the only source of static electricity. A source for the continuous electric current was not available. Volta in 1799 invented the voltaic pile, i.e. a chemical battery, and the first time a continuous source of electric current came into existence. On April 21, 1820, it led to the discovery of the magnetic effect of current flowing in a wire. The electric current became the source of the magnetic field, encircling the current-carrying wire. The magnetic effect of current was discovered by Orsted (Oersted). In the same year, Ampere showed that the co-directional parallel currents flowing in two wires attract each other, and the countercurrents repel each other. It was a very significant discovery, i.e. creation of the attractive and repulsive magnetic forces without any physical magnet. It firmly established the relation between electricity and magnetism. Ampere further developed an equation, presently called Ampere’s Circuital Law, to connect the current flowing in a wire to the magnetic field around it and developed the right-hand rule. He called the new field of electricity Electrodynamics and Maxwell recognized him as the Father of Electrodynamics. Ampere further modeled the natural magnetic materials as the materials composed of perpetual tiny circulating electric currents. He demonstrated the validity of his concept using the current-carrying conductor in the helical form called a solenoid. The solenoid worked like an artificial bar magnet. In the year 1820 itself, Biot–Savart obtained the

equation using the line integral to compute the magnetic field at a position in the space due to the current flowing in a wire [J.2, B.6, B.7].

Ohm’s Law

The voltaic pile helped the discovery of the magnetic effect of current; however, surprisingly the relation between the current flowing in resistance and voltage across it, known as the Ohm’s law, remained undiscovered. The primary reason was the unstable voltage supplied by the voltaic pile. The discovery of thermoelectricity by Seebeck in 1822 provided a constant voltage source to supply continuous electric current. Using the thermo-piles in the year 1826, Ohm obtained a simple but powerful relation among voltage, current, and resistance. It was the beginning of the Electric Circuit Theory. However, only in 1850 Kirchhoff published his two circuital laws and opened the path for the development of the Network Theory. Kirchhoff also showed that Ohm’s electroscopic force (voltage) and classical potential of Lagrange, Laplace, and Poisson are identical. Interestingly, Ohm’s law could be viewed as a symbol of the International Scientific unity relating to Italy (Volta), Germany (Ohm), and France (Ampere). Based on the magnetic effect of current, in the same year, Johann Christian Poggendorff invented the galvanometer to detect the current in a wire. Kelvin improved its sensitivity by designing the mirror galvanometer in 1858 [B.1, B.6, B.7].

Electric Effect of the Time-Varying Magnetic Field

On knowing the magnetic effect created by an electric current, Faraday argued that the magnetic field can also produce the electric effect. After some attempts, he realized that such an effect can’t be produced by the stationary magnet. In 1831, he could generate the electric potential (electromotive force) and electric current by the time-varying magnetic field of a moving magnet. The phenomenon is called the induction effect. The voltage induction effect demonstrated that electricity could be generated by a purely mechanical process, converting the mechanical energy into electrical energy via the medium of the moving magnetic field. The first DC generator was demonstrated by Faraday himself, and next year French instrument maker Hippolyte Pixii built the first A.C. generator inaugurating the Electrical Age. Now, the electricity was ready to accelerate the growth of human civilization at an unprecedented rate [B.6, B.7].

3

4

1 Overview of Transmission Lines

Concept of the Magnetic Vector Potential

In the process of discovery of induction, Faraday introduced the concept of fields, and also suggested that the electric energy resides in the field around the charged body and the magnetic energy resides in the field around the magnetized body. Thus, he viewed that the electric and magnetic energies reside in the space around the charged or magnetized body, not in the charge or magnet. The field concept has greatly influenced the further development of EM-theory. The field provided a mechanism of interaction between charged bodies. Using Ampere-Biot–Savart law of magnetic forces, and electromagnetic induction of Faraday, Neumann in 1845 introduced the concept of the magnetic vector potential A to describe the magnetic field. Subsequently, Maxwell showed that the time derivative of A computes the induced electric field E. Kelvin in 1847 further extended the concept of the magnetic vector potential A to compute the magnetic field using the relation B = ∇ × A . This relation comes as a solution of the Gauss divergence equation ∇ B = 0 due to the closed-loop of the magnetic field, showing the nonexistence of a magnetic charge. Kelvin further elaborated on the mathematical theory of magnetism in 1851. It is interesting to note that at any location in the space once time-dependent magnetic vector potential function is known, both the magnetic and electric fields could be computed as, B =∇× A ∇ × E = − ∂ B ∂t = ∇ ×

a − ∂ A ∂t

E = − ∂ A ∂t

111 b

Maxwell shared the views of Neumann and Kelvin. However, time-retardation was not incorporated in the scalar and vector potentials. In 1867, Lorentz introduced the concept of retardation in both the scalar and vector potentials to develop the EM-theory of light, independent of Maxwell. The time-retardation only in the scalar potential was first suggested by Riemann in 1858, but his work was published posthumously in 1867 [J.1, J.2, B.6, B.7].

law, and Ohm’s law. Now Maxwell, Newton of the EM-theory, arrived at the scene to combine all the laws in one harmonious concept, i.e. in the Dynamic Electromagnetic Theory. He introduced the brilliant concept of the displacement current, created not by any new kind of charge but simply by the time-dependent electric field. Unlike the usual electric current supported by a conductor, this new current was predominantly supported by the dielectric medium. However, both currents were in a position to generate the magnetic fields. Thus, Maxwell modified Ampere’s circuital law by incorporating the displacement current in it. The outcome was dramatic; the electromagnetic wave equation. Despite such success, the concept and physical existence of displacement current created a controversy that continues even in our time, and its measurement is a controversial issue [J.6–J.8]. In the year 1856, Maxwell formulated the Faraday’s law of induction mathematically, and modified Ampere’s circuital law in 1861 by adding the displacement current to it. Finally in 1865 after a time lag of nearly 10 years, Maxwell could consolidate all available knowledge of the electric and magnetic phenomena in a set of 20 equations with 20 unknowns. However, he could solve the equations to get the wave equations for the EM-wave with velocity same as the velocity of light. Now, the light became simply an EM-wave. In the year 1884, Heaviside reformulated the Maxwell equations in a modern set of four vector differential equation. The new formulation of Maxwell equations was in terms of the electric and magnetic field quantities and completely removed the concept of potentials, considering them unnecessary and unphysical. Hertz has independently rewritten the Maxwell equation in the scalar form using 12 equations without potential function. Hertz worked out these equations only after Heaviside. In 1884, Poynting computed the power transported by the EM-waves. Recognizing the contributions of both Heaviside and Hertz in reformulating Maxwell’s set of equations, Lorentz called the EM-fields equations Maxwell–Heaviside–Hertz equations. However, in due course of time, the other two names were dropped and the four-vector differential equations are now popularly known as “Maxwell’s Equations” [J.1, J.6, J.9, J.10, B.5–B.7].

Maxwell’s Dynamic Electromagnetic Theory

Generation and Transmission of Electromagnetic Waves

At this stage of developments in the EM-theory, the electric field was described in terms of the scalar electric potential, and the magnetic field was described by the magnetic vector potential. Several laws were in existence, such as Faraday’s law, Ampere’s law, Gauss’s

Maxwell’s EM-theory was a controversial theory, and physicists such as Kelvin never accepted it. Hertz finally generated, transmitted, and detected the EM-waves in 1887 at wavelengths of 5 m and 50 cm. In the process, he invented the loaded dipole as the transmitting

1.1 Overview of the Classical Transmission Lines

antenna, rectangular wire-loop receiving antenna, and spark-gap both as transmitter and detector to detect the propagated EM-waves. Thus, he experimentally confirmed the validity of Maxwell equations and opened the magnificent gateway of wireless communication. In the year 1895, Marconi transmitted and received a coded telegraphic message at a distance of 1.75 miles. Marconi continued his works and finally on December 12, 1901, he succeeded in establishing the 1700 miles long-distance wireless communication link between England and Canada. The transmission took place using the Hertzian spark-gap transmitter operating at the wavelength of 366m. In the year 1895 itself, J.C. Bose generated, transmitted, and detected the 6 mm EMwave. He used circular waveguide and horn antenna in his system. In 1897, Bose reported his microwave and mm-wave researches in the wavelengths ranging from 2.5 cm to 5 mm at Royal Institution, London. Of course, the Hertzian spark-gap transmitter was at the core of his communication system. Bose was much ahead of his time as the commercial communication system grew around the low frequency, and the microwave phase of communication was yet to come in the future. In 1902, Max Abraham introduced the concept of the radiation resistance of an antenna [J.11–J.13, B.1–B6].

Further Information on Potentials

Hertz is known for his outstanding experimental works. However, as a student of Helmholtz, he was a high ranking theoretical physicist. Although, he considered, like Heaviside, electric and magnetic fields as the real physical quantities, still he used the vector potentials, now e

m

called Hertzian potentials π and π , to solve Maxwell’s wave equation for the radiation problem. These potentials are closely related to the electric scalar potential ϕ and magnetic vector potential A. Stratton further used Hertzian potentials in elaborating the EM-theory [B.8]. Collin continued the use of Hertzian potentials for the analysis of the guided waves. He also used the A and ϕ potentials in the radiation problems [B.9, B.10]. The use of Hertzian potentials gradually declined. However, its usefulness in problem-solving has been highlighted [J.1, J.11, J.12]. Gradually, the magnetic vector potential became the problem-solving tool if not the physical reality. Further, by using the retarded scalar and vector potentials and Lorentz gauge condition ∇ A = − με∂ϕ ∂t connecting both the vector and scalar potentials, Lorentz formulated the EM-theory of Maxwell in terms of the magnetic vector potential. In his formulation, a current is the

source of the magnetic vector potential A . So, Lorentz considered the propagation of both the magnetic vector and electric scalar potential with a finite velocity that resulted in the retarded time at the field point. However, Maxwell’s scalar potential was nonpropagating. Maxwell did not write a wave equation for the scalar potential, as his use of Coulomb gauge ∇ A = 0 was inconsistent with it. Later on, even electric vector potential F was introduced in the formulation of EM-theory. The nonphysical magnetic current, introduced in Maxwell’s equations by Heaviside, is the source of potential F . The use of vector potentials simplified the computation of the fields due to radiation from wire antenna and aperture antenna. A component of the magnetic/electric vector potential is a scalar quantity. It has further helped the reformulation of EM-field theory in terms of the electric scalar and magnetic scalar potentials [B.9, B.10]. Such formulations are used in the guided-waves analysis. In recent years, it has been pointed out that the Lorentz gauge condition and retarded potentials were formulated by Lorenz in 1867, much before the formulation of famous H. A. Lorentz [J.14, J.15]. However, most of the textbooks refer to the name of Lorentz. Both Heaviside and Hertz considered only the electric and magnetic fields as real physical quantities, and magnetic vector and electric scalar potentials as merely auxiliary nonphysical mathematical concepts to solve the EM-field problem. Possibly, this was not the attitude of Kelvin and Maxwell. They identified the electrical potential with energy, and magnetic vector potential with momentum. The magnetic vector potential A could be considered as the potential momentum per unit charge, just as the electric scalar potential ϕ is the potential energy per unit charge [J.16]. The potential momentum P is obtained as follows: F =

∂ qA dP = qE = − , dt ∂t

P = qA 112

In the above equation, F is the force acting on the charge q, and E is given by equation (1.1.1b). Lebedev in 1900 experimentally demonstrated the radiation pressure, demonstrating momentum carried by the EM-wave. The energy and momentum carried by the EMwave indicate that the light radiation could be viewed as some kind of particle, not a wave phenomenon. A particle is characterized by energy and momentum. Such a dual nature of light is a quantum mechanical duality phenomenon. Einstein introduced the concept of the light

5

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1 Overview of Transmission Lines

particle, called “photon” to explain the interaction of light with matter, i.e. the photoelectric effect. However, Lorentz retained the classical wave model to explain the interaction between radiation and matter via polarization of dipoles in a material creating its frequency-dependent permittivity. It is to be noted that at a location in the space, even for zero B and E fields, the potentials A and ϕ could exist [B.11]. Aharonov–Bohm predicted that the potential fields A and ϕ, in the absence of B and E , could influence a charged particle. Tonomura and collaborators experimentally confirmed the validity of Aharonov– Bohm prediction. The Aharonov–Bohm effect demonstrates that B and E fields only partly describe the EM-fields in quantum mechanics. The vector potential also has to be retained for a complete description of the EM-field quantum mechanically [J.3, J.16, J.17]. However, to solve the classical electromagnetic problems, such as guided wave propagation and radiation from antenna, Heaviside formulation of Maxwell equations and potential functions as additional tools is adequate. EM-Modeling of Medium

The above brief review omitted developments in the electromagnetic properties of the material medium. A few important developments could be summarized. In 1837, Faraday introduced the concept of the dielectric constant of a material. In 1838, he introduced the concept of electric polarization P in dielectrics under the influence of the external electric field. Soon after the discovery of the electron in 1897 by J.J. Thomson, the models around electrons were developed to describe the electromagnetic properties of a material. Around 1898, John Gaston Leathem obtained an important relation D = E + P , connecting the displacement of charges in a material with polarization. Kelvin, in the year 1850, developed the concept of magnetic permeability and susceptibility with separate concepts of B , M, and H to characterize a magnetic material. In 1900, Drude developed the electrical conduction model, now known as the Drude model, after electron theory. Subsequently, the model was extended to the dielectric medium by Lorentz in 1905. The model called the Drude-Lorentz model explains the dispersive property of dielectrics. In the year 1912, Debye developed the concept of dipole moment and obtained equations relating it to the dielectric constant. These models laid the foundation to study of the electric and magnetic properties of natural and

engineered materials under the influence of external fields [B.4, B.6, B.7, B.12] 1.1.3 Development of the Transmission Line Equations Kelvin’s Cable Theory

During the period 1840–1850, several persons conceived the idea of telegraph across the Atlantic Ocean. Finally, in the year 1850, the first under-sea telegraphy, between Dover (Kent, England) and Calais (France), was made operational. However, no cable theory was available at that time to understand the electrical behavior of signal transmission over the undersea cable. In 1854, Kelvin modeled the under-sea cable as a coaxial cable with an inner conductor of wire surrounded by an insulating dielectric layer, followed by the saline seawater acting as the outer conductor [J.18, B.1]. The coaxial cable was modeled by him as a distributed RC circuit with the series resistance R per unit length (p.u.l.) and shunt capacitance C p.u.l. It was the time of the fluid model of electricity. Kelvin further conceived the flow of electricity similar to the flow of heat in a conductor. Fourier analysis of 1D heat flow was in existence since 1822. Following the analogy of heat equation of Fourier, Kelvin obtained the diffusion type equation for the transmitted voltage signal over the under-sea coaxial cable: ∂2 v ∂v = RC ∂x2 ∂t

113

This is the first Cable Theory; Kelvin called the above equation the equation of electric excitation in a submarine telegraph wire. Kelvin’s model did not account for the inductance L p.u.l. and the conductance G p.u.l. of the cable. The cable inductance L is due to the magnetic effect of current, and G is due to the leakage current between the inner and outer conductors. However, cable theory was a great success. Following the method of Fourier, he solved the equation for both the voltage and current signals. At any distance x on the cable, a definite time-interval was needed to get the maximum current of the received signal. The galvanometer was used to detect the received current. This time-interval called the retardation time of the received current signal also depends on the square of the distance. Moreover, the telegraph signals constituted of several waves of different frequencies, and their propagation velocities were frequency-dependent. It limited the speed of signal transmission for long-distance telegraphy. The conclusions of Kelvin’s analysis were ignored, and 1858 transatlantic cable worked only for three weeks. It failed due to the application of 2000 V potential pulse

1.1 Overview of the Classical Transmission Lines

on the cable. The speed of transmission was just 0.1 words per minute. Finally, following Kelvin’s advice and using a very sensitive mirror galvanometer invented by him, the transatlantic telegraph was successfully completed in 1865 with eight words per minute transmission speed [B.1–B.3]. Heaviside Transmission Line Equation

The limitation of the speed of telegraph signals was not understood at that time. The RC model of the cable, leading to the diffusion equation, and use of the time-domain pulse could not explain it. Moreover, it became obvious that the RC model couldn’t be used to understand the problems related to voice transmission over telephonic channels. The telephony was coming into existence. The modern telephone system is an outcome of the efforts of several innovators. However, Graham Bell got the first patent of a telephone in the year 1874. The transmitted telephonic voice signal was distorted. Therefore, an analytical model was urgently needed to improve the quality of telephonic transmission. Heaviside in 1876 introduced the line inductance L p.u.l. and reformulated the cable theory of Kelvin using Kirchhoff circuital laws [B1, B.3]. The formulation resulted in the wave equation for both the voltage (V) and current (I) waves on the line: ∂2 T ∂T ∂2 T + LC = RC , ∂x2 ∂t ∂t2

where T = V, I 114

In the case of line inductance L = 0, the above equation is reduced to the diffusion type cable equation (1.1.3) of Kelvin. Using the Fourier method, Heaviside solved the aforementioned time-domain equation. Only in 1887, he could introduce the line conductance G p.u.l in his formulation to account for the leakage current in an imperfect insulating layer between two conductors. Finally, Heaviside obtained a set of coupled transmission line equations using all four line constants R, L, C, and G. Subsequently, the coupled transmission line equations were called the Telegrapher’s equations. At the end, Heaviside obtained the following modified wave equation: For lossy line ∂2 T ∂T ∂2 T + LC 2 + RG T = RC + LG 2 ∂x ∂t ∂t

a

For lossless line ∂2 T ∂2 V = LC , where T = V, I ∂x2 ∂t2

b 115

To solve the above time-domain equation, Heaviside developed his own intuitive operational method approach by defining the operator ∂/∂t p. The use of the operator reduced the above partial differential equation to the ordinary second-order differential equation. Finally, he solved the equation under initial and final conditions at the ends of a finite length line. In the process, he obtained the expressions for the characteristic impedance and propagation constant in terms of line parameters. Heaviside could obtain results for the line under different conditions. For a lossless line, R = G = 0, the equation (1.1.5b) is obtained. Conceptually, the characteristic impedance provided a mechanism to explain the phenomenon of wave propagation on an infinite line. At each section of the line, it behaved like a secondary Huygens’s source providing the forward-moving wave motion. Heaviside also obtained the condition for the dispersionless transmission on a real lossy line, and suggested the inductive loading of a line to reduce the distortion in both the telegraph and telephone lines. Afterward, his intuitive operational method approach developed into the formal Laplace transform method, widely used to solve the differential equations [J.19, J.20, B.1–B.3, B.13]. The method of Heaviside was further extended by Pupin in 1899 and 1900. Pupin introduced the harmonic excitation in the wave equation as a real part of the source V0ejpt [J.21, J.22]. This was an indication of the use of the modern phasor solution of the wave equation. Similar analytical works, and also practical inductive loading of the line was done by Campbell at Bell Laboratory. He published the results in 1903 [J.23]. In July 1893, Steinmetz introduced the concept of phasor to solve the AC networks of RLC circuits. In 1893, Kennelly also published the use of complex notation in Ohm’s law for the AC circuits [J.24]. Carson in 1921 applied the method to solve Maxwell’s equations for the wave propagation on closely spaced lines, and also analyzed for the mutual impedances. Carson in 1927 developed the electromagnetic theory of the Electric Circuits, and paved the way for the modeling of the wave phenomena using the circuit models [J.25, J.26]. Peijel in 1918, and Levin in 1927 analyzed the wave propagation on the parallel lines. Levin extended the telegrapher’s equations to the multiconductor transmission lines using Maxwell’s equations [J.27]. In 1931 Bewley presented a set of wave equations on the coupled multiconductor lines. Subsequently, Pipes introduced the matrix method to formulate the wave propagation problem on the multiconductor lines [J.28, J.29]. Thus, the theoretical foundation was laid to deal with the complex

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1 Overview of Transmission Lines

technical problems related to transmission lines. Starting with Marconi wireless in 1895, several improvements took place in the long-range wireless telegraphy. Also, the audio broadcasting was developed between 1905 and 1906 and commercially, around 1920–1923, in the long-wave, medium-wave, and short-wave RF frequency bands [B.5]. Now, the time was ripe for microwave and mm-wave communication. The above discussion shows that the Telegrapher’s equations have come in existence due to the contributions of both Kelvin and Heaviside. To recognize their contributions, we call in this book the Telegrapher’s equations as the Kelvin-Heaviside transmission line equations. Also, as the characteristic impedance behaves as the secondary Huygens’s source, so it can also be viewed as the Huygens’s load. Such Huygens’s load distributed over a surface forms the modern Huygens’s metasurface, discussed in the chapter 22 of this book. 1.1.4

Waveguides as Propagation Medium

Heaviside reformulated Maxwell equation in 1884. He rejected the idea of EM-wave propagation in a hollow metallic cylinder. In his opinion, two conductors, alternatively one conductor and the earth as a ground conductor are essential for the EM-wave propagation. However, in 1893 J.J. Thomson expressed the possibility of the EM-wave propagation in a hollow cylinder [B.12]. Next year, Oliver Lodge verified it experimentally. In the year 1895, J.C. Bose used the waveguide and horn antenna for the mm-wave transmission and reception. In 1897 he reported the work at Royal Institution in London [B.5]. However, it was Rayleigh who carried out a detailed solution of boundary-value problems. He obtained the normal mode solution, showing wave propagation in the form of the distinct discrete modes, i.e. the normal modes. He obtained his solutions for both the TE and TM modes, and introduced the concept of the cutoff frequency for modes. He further examined the EMwave propagation on a dielectric waveguide [J.30]. In 1920 Rayleigh, Sommerfeld and Debye continued the researches in this direction. However, only in 1930 proper experimental investigations of the wave propagation in the waveguides were undertaken by G. C. Southworth at Bell Labs, and W.L. Barrow at MIT. In 1934, microwave commercial link was established, and in 1936, Southworth and Barrow discovered the possibility of using the waveguide as a transmission medium. However, they published their works only in 1936 [J.31, J.32, B.5]. During the same time-period, Brillouin also investigated the wave

propagation in a tube [J.33]. Serious analytical work on waveguides was further undertaken by J.R Carson, S.P. Mead, and S.A. Schelkunoff around 1933 [J.34]. Almost forgotten analytical works of Rayleigh was reinvented. Chu and Barrow further investigated the EMwaves propagation in the elliptical and rectangular hollow metallic pipes [J.35]. During 1934, Schelkunoff extended the concept of impedance to the EM-wave propagation in the coaxial line, and obtained the transmission line equations using the electromagnetic theory [J.36]. In 1937, he further extended the theory to the TE and TM mode guided wave propagations, and obtained the circuit models of mode supporting waveguides. Finally, Schelkunoff generalized the standard telegrapher’s equation, using Maxwell’s EM-theory to represent an infinite set of uncoupled and coupled modes of a waveguide by the system of uncoupled and coupled transmission line equations [J.37–J.39]. Subsequently, his method has been extended to planar lines in an inhomogeneous medium supporting the hybrid modes [B.13]. During the World War-II period, important theoretical and practical works were done in the field of waveguide technology for the development of the waveguide-based components and systems. The development of Radar provided the impetus for such research activities.

1.2

Planar Transmission Lines

A brief review of the development of planar transmission lines, influencing modern microwave technology, is presented below. A review is also given for the analytical methods as applied to the planar line parameters. 1.2.1

Development of Planar Transmission Lines

The waveguide is a low-loss transmission medium capable of handling high power transmission. However, it is a bulky structure with limited bandwidth. The fabrication of waveguide-based components is a complex and expensive machining process. The limitations of the waveguide provided an impetus for the growth of planar lines, and technology based on the planar lines. H.A Wheeler, in 1936, developed a low-loss coplanar stripline, and in 1942 created parallel plate strip transmission line on a high permittivity substrate. The line structure was compact and suitable at low RF frequency from 150 MHz to 1500 MHz. However, properly documented stripline was reported by R.M. Barrett only in 1951. Just next year, i.e. in 1952, Grieg and Engelmann

1.2 Planar Transmission Lines

reported microstrip line. Both structures competed with each other. Initially, the stripline in the homogeneous medium was a preferred line, as it is a dispersionless line with a larger bandwidth. It supports the TEM mode propagation. As it is a shielded line, so it also has a higher Q-factor. Whereas, the microstrip in the inhomogeneous medium is a dispersive line as it supports the dominant hybrid mode. It has a smaller bandwidth and lower Q-factor. During 1960, solid-state components started appearing, and microstrip became the preferred line structure for the MIC environment. The microstrip is an open structure that provided easier access for the interconnections. It led to the development of miniaturized microstrip integrated circuit (MIC) technology. Gradually, the discrete active devices were combined with the planar passive microwave components, and the hybrid MIC (HMIC) came into existence. The sixties were a very creative period for the planar line technology. In 1968, Cohn reported the slot line followed by the coplanar waveguide (CPW) that became the medium of MMIC. C.P. Wen in 1969 developed the CPW. It is an interesting and unusual coincidence that the abbreviation of both the line name and inventor’s name is CPW. The integration of the slot line with waveguide took place in 1972 when Meier reported the quasi-planar fin line [J.40–J.44]. Further compactness in the microwave circuits and systems took place through the development of the monolithic MIC (MMIC) circuit concept in the year 1964. At this stage, the MMIC was based on silicon technology. Unfortunately, the program was not successful due to the very lossy Si-substrate. The semi-insulating Si-substrate deteriorated in the process of the formation of active devices, such as bipolar junction transistors (BJTs) on a Si-substrate. The next phase of MMIC development took place for the GaAs substrate-based technology in 1968. It required nearly 10–12 years for its more meaningful development. The span of 1980–1986 was a period of rapid growth for MMIC technology. In 1990s, SiGe based technology was developed that permitted operation of high-efficiency circuits at higher frequencies. The MMIC technology achieved its maturity for the MMIC based on the silicon and indium–phosphide (InP) substrates apart from the GaAs substrate. At the core of the development were the multilayer planar lines and new varieties of active devices [J.45]. Another kind of Si-based technology, namely the micro-electro-mechanical system (MEMS) gradually came to the fields of RF and microwave. Petersen’s reported the MEMS membrane-based switches in

July 1979. However, after a long gap, Yao and Chang developed the surface MEMS switch for DC-4 GHz operation and high-quality MEMS inductor chip could be realized in 1997. Subsequent years witnessed a reduction in operating voltage of MEMS switches. The operation of MEMS in the microwave and mm-wave ranges expanded their applications in the field of the antenna and other microwave systems [J.46–J.48]. The robust and compact multilayer ceramic tapebased microwave technology, called the low-temperature co-fired ceramics (LTCC) gradually acquired significance for the development of the hybrid integrated circuits. It started in 1950–1960 to develop more robust capacitors. The several layers of different materials are used in a single multilayer laminated package to design multifunctionality circuit-blocks. The planar lines in the LTCC are used in the multilayer and multilevel formats as a medium to develop the components and interconnect [J.49, J.50]. Further innovations in the planar microwave technology were added by incorporating the periodic reactive loading of planar lines and planar surfaces resulting in the electronic band-gap (EBG) lines and EBG surfaces for a wide range of applications. Long ago, the theoretical basis for the analysis of the periodic structures was summarized by L. Brillouin [B.14]. The theoretical concept of the metamaterial as a double negative (negative permittivity and negative permeability) material medium, and its radical impact on behaviors of the electromagnetic phenomena were worked out by Victor Veselago in 1967. The practical development of the metamaterials is an outcome of a long history of artificial dielectrics and mixture medium. However, only in 1996–1999, Pendry and co-workers suggested, and further experimentally demonstrated, the artificial negative permittivity below the controlled plasma frequency. It was realized by using the periodic arrangement of thin conducting wires. Further, in 1999 Pendry and co-workers suggested and experimentally produced resonance type magnetic behavior in the split coaxial conducting cylinders. However, only Smith and co-workers worked out the simultaneous negative permittivity and negative permeability in 2000, and experimentally verified it in 2001. Gradually, the concept of metamaterials was added to the planar lines and surfaces resulting in the realization of metalines and metasurfaces. These artificial structures have significantly influenced the design and development of unique antenna, components, and circuits with new characteristics and multifunctionality. Present researches in these fields are in progress in many directions [J.51–J.55].

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1 Overview of Transmission Lines

1.2.2 Analytical Methods Applied to Planar Transmission Lines Assaudourion and Rimai considered the microstrip in the quasi-static limit. They assumed the TEM mode propagation on it. They applied, in 1952, the wellestablished conformal mapping method to compute the characteristic impedance, dielectric, and conductor losses. Between the years 1954 and 1955, Cohn also used the conformal mapping method to get the designoriented results for the characteristic impedance, dielectric and conductor losses of the stripline. He further used the conformal mapping method to get the odd-even mode impedances of the edge-coupled strip lines in 1955 itself. He further obtained these results for the broadside-coupled strip lines in 1960. Following the conformal mapping method, in 1964 and 1965 Wheeler produced more accurate and design-oriented expressions for the computation of characteristic impedance of microstrip line. He extended his analysis to get further results in 1977 and 1978 [J.56–J.64]. In 1969 Cohn suggested another planar line, i.e. the slot line. It is a complementary structure of the microstrip line. He also presented the equivalent waveguide model of the slot line, and obtained the frequencydependent propagation parameters of the slot line. Next in the group of the planar lines is the coplanar waveguide (CPW) proposed by C.P. Wen. He obtained the initial quasi-static line parameters of CPW using the conformal mapping method. Subsequently, the conformal mapping method was applied to analyze several variants of the planar lines [J.65–J.67]. Other quasi-analytical and numerical methods were also used for the analysis of microstrip lines. For instance, in 1968 Yamashita and Mitra introduced the quasi-analytical variational method in the Fourier domain to obtain the quasi-static line parameters of the microstrip line. It was the prelude to the quasianalytical dynamic spectral domain analysis (SDA) of microstrip and other planar lines. The dynamic SDA is a full-wave analysis method that considers the hybrid mode nature of planar lines. After a gap of nearly 10 years, Itoh used the concept of the discrete Fourier transform and Galerkin’s method to get the static line parameters of suspended coupled microstrip lines, and also extended the method to suspended multiconductor microstrip structures. The Fourier domain method was significantly extended by many investigators to other planar structures such as the CPW [J.68–J.71, B.15, B.16]. In 1973 and 1974, Itoh and Mitra introduced the dynamic SDA to obtain dispersion characteristics of

the slot line, and also microstrip line. Jansen extended the dynamic SDA to analyze the higher order modes in the microstrip. The method is very powerful and analytically elegant. It has been used and improved by other researchers in the field of planar resonators, antenna, and line structures. Other powerful methods, such as the method of moments, finite elements, finite-difference time-domain method, and so on have also been developed to analyze the 2D and 3D complex planar structures. The contemporary EM-Simulators are based on these numerical methods. The closed-form models for faster computation of the static and frequencydependent line parameters of planar lines have also been developed by several investigators. The closed-form models of lines, discontinuities, and so on helped the development of the Circuit Simulators [J.72–J.75, B.15, B.16].

1.3

Overview of Present Book

The book presents a seamless treatment of the classical planar transmission lines and modern engineered planar lines using the concept of the engineered electromagnetic bandgap (EBG) structures and metamaterials. The modern EBG and metamaterials based planar lines are the outcome of the classical researches in the artificial dielectrics and concept of homogenization of mixing of inclusions in the host medium. Gradually, the modern microwave planar transmissions became a complex medium of wave propagations on the 1D lines and 2D surfaces. It demanded serious considerations of wave– matter interactions, especially in the engineered materials by the microwaves researchers and engineers. It demanded a physical understanding of various electromagnetic phenomena taking place in the artificially engineered complex medium. It also required the analytical and circuit modeling of the planar transmission lines under the complex environment. The present book: Introduction to Modern Planar Transmission Lines (Physical, Analytical, and Circuit Models Approach) addresses these problems from the very basics, making it suitable for the early comers to the fields. However, the detailed treatment of topics could be also useful to more experienced professionals and engineers. The numerical methods used in the analysis of the planar structures and basis of the EM-simulators are more specialized topics beyond the scope and line of thought followed in the present book. The key concept used throughout the book is the modeling, physical, analytical, and circuit, of the planar

1.3 Overview of Present Book

structures. However, what is the meaning of modeling itself? Scientific modeling is a process of understanding the unknown with the help of known. The reverse is not possible. The method of analogy is a great tool in such a modeling process. The growth of electromagnetic field theories at different stages has evolved from the previously known results of the gravitational field. Likewise, the gradual development of the transmission line theory has used the analogy of heat flow. These are two important illustrative examples discussed in the previous section. The experimental observation and the experimental verification of the theoretically predicted results are further contributors to the modeling process. The scientific modeling process has been examined in depth by the modern educationists [B.17]. The reader can observe such a modeling process in the development of models for the complex planar medium exhibiting unique properties. 1.3.1

The Organization of Chapters in This Book

The chapters of this book are organized into four distinct groups as follows: i) Introductory transmission line and EM wave theory. ii) Basic planar lines and Resonators: Microstrip, CPW, Slot lines, Coupled lines, and Resonators. iii) Analytical Methods: Conformal mapping method, Variational method, Full- wave SDA, and SLR formulation. iv) Contemporary engineered planar structures: Periodic planar lines and surfaces, Metamaterials – Bulk, 1D metalines, 2D metasurfaces. The group i reviews the transmission line and the EM-theory to assist the reader to follow the rest of the chapters with ease. The groups ii and iii form the classical transmission lines, and the group iv is the modern transmission lines and surfaces. The book presents a seamless treatment of the classical planar transmission lines and the modern engineered planar lines and surfaces using the concept of EBG and metamaterials. The modern EBG and metamaterials based planar lines are the outcome of the classical researches in the artificial dielectrics and concept of homogenization of mixing of inclusions in the host medium. The topics of the chapters are selected to provide comprehensive coverage of the needed background to understand the functioning of both the classical and modern lines and surfaces. Each chapter follows a uniform style. The topics within a chapter start with simple concepts and move to a higher complexity level.

Likewise, the chapters are also arranged from the simpler to complex. The distribution of the chapters among the groups is discussed below. The key features of the chapters are also summarized.

Introductory Transmission Line and EM-Wave Theory

The six chapters, chapters 2–7, on the transmission lines and various aspects of the EM-theory are introduced in the book before even commencing with the microstrip in chapter 8. These topics provide the essential background to follow smoothly the topics covered in this book. It could be useful in understanding the analysis and modeling of the planar line structures, the EBG based lines, and surfaces, and also the metamaterials and metasurfaces. Moreover, the topics discussed may also help to understand the modern publications in these fields. The usual undergraduate textbooks on the EM-theory do not cover all the topics. However, the reader’s familiarity with the transmission line and EM-theory is assumed. The reader interested in a more detailed study of these topics can follow the references given at the end of the chapters. Some contents of the chapters are highlighted below. The chapters 2 and 3 on transmission lines are written as a review. However, it goes beyond a regular review, although it starts with the familiar notion of oscillation and wave propagation on lines. Usually, the available textbooks present the transmission line equations and wave equations for the uniform lines only, without any source. The present book covers the transmission line equations and wave equations with a source, and the analysis of the multisection transmission lines is also introduced. Such formulation is used in the chapters 14 and 16 to obtain the Green’s functions of planar transmission lines used with the variational method and full-wave spectral-domain analysis (SDA) method. The chapter on the transmission line adequately covers the concept of dispersion in the wave supporting medium. Also, the impact of the reactive loading of the line on the nature of wave propagation is discussed. Such treatment prepares a reader for the periodically loaded engineered lines and surfaces, both as the bandgap medium and homogenized metamaterial medium. These topics are discussed in chapters 19–22. Chapter 3 covers various parameters used for the characterization of a line section. Understanding of this topic is essential for understanding the microwave components design, the results obtained from EM-simulation, and to develop the circuit models. Chapters 4 and 5 cover the wave’s propagation in the material medium. Again, primarily it is a review of the

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EM-theory. However, its perspective is very broadly applicable to the topics discussed in other chapters. The chapter 4 commences with a basic review of the electromagnetics and elementary electrical properties of the material medium, such as linearity, nonlinearity, homogeneity, inhomogeneity, anisotropy, and losses. It further covers the topic of the circuit modeling of a medium. The known topics of Maxwell’s equations in the differential form, as well as in the vector-algebraic form are presented. The wave propagation is discussed not only in the isotropic and conducting media but also in the anisotropic, uniaxial, gyroelectric, and biaxial media. The complex media are encountered by the wave propagating in the metamaterials. While reviewing the wave polarizations, the Jones matrix description of polarization states is also discussed. It is needed to follow the contemporary developments in the metasurfaces. Chapter 4 ends with the concepts of isofrequency contours and isofrequency surfaces, and the dispersion relations in the uniaxial medium. Chapter 5 reviews both the normal and polarizationdependent oblique incidence of the waves at the interface of two media. It also presents the equivalent transmission line model of the wave’s incidence at the interface of two media. The model can be extended to more number of layers. The formulation has many applications. The model is used for instance in chapter 20 on the EBG surface. The chapter 5 also presents the basic electrodynamics of the engineered metamaterials and formulate the basic characteristics of the wave propagation in the metamaterials. It also discusses some important directions for applications of metamaterials. However, the realization of the bulk metamaterials, metalines, and metasurfaces are followed up in the chapters 21 and 22. Chapter 6 covers a review of the electrical properties of the natural and artificial dielectric media. It also presents various static and frequency-dependent models of the mixture media. The artificial dielectric medium finds its application in modeling the metamaterials. The Lorentz, Drude, and Debye models applicable to the frequency-dependent permittivity are discussed. Chapter 6 further discusses the interfacial polarization and its circuit model. This important topic is usually not discussed in popular textbooks. The modeling of the substrates, using the single term Debye and Lorentz models, as well as the multi-term and wideband Debye models are elaborated. These models help to get the causal effective permittivity of the planar lines of the substrates, useful in the time-domain analysis of pulse propagation on the planar lines. Finally, the chapter ends with a novel concept of artificial metasubstrate.

Chapter 7 comprehensively treats basic waveguide structures. It begins with the classification of the modal EM-fields, and the sources of their generation. The waveguides are analyzed using the scalar electric and magnetic potentials. The spectral domain analysis (SDA) method discussed in chapter 16 is based on these scalar potentials. The concept of the perfect electric conductor (PEC) and the perfect magnetic conductor (PMC) with boundary conditions are introduced. The analysis of the rectangular geometry of the waveguides formed with these surfaces is presented. Thus, apart from the usual all metallic walls, i.e. the PEC based waveguides, all PMC and two PEC and two PMC walls waveguides are also discussed. The dielectric slab waveguides and surface-waveguides are also presented. The concept of the odd/even mode analysis is introduced. These concepts are used in the book for the analysis of symmetrically coupled planar lines in chapters 11 and 12. The simple and powerful transverse resonance method (TRM) is introduced to get the propagation characteristics of the dielectric-loaded waveguides and the multilayer surface-waveguides. Finally, chapter 7 ends with the contemporary substrate integrated waveguide (SIW) developed in the environment of the planar technology. Basic Planar Lines and Resonators

The planar line structures – microstrip, CPW, and slot line are discussed in chapters 8–10, respectively. The chapters 11 and 12 cover the theory of the coupled transmission lines and their realization and analysis in the planar technology environment. The theory of resonating structures and planar lines version of the resonators are discussed in chapters 17 and 18, respectively. The fabrication technologies – MIC, MMIC, MEMS, and LTCC used in the planar lines and components are reviewed in chapter 13. The microstrip is the most commonly used planar line in planar technology. It is in the inhomogeneous medium supporting the hybrid-mode that is approximated as the dispersive quasi-TEM mode. However, at the lower frequency, it is treated in the nondispersive static condition. Chapter 8 introduces the concept of medium transformation from the inhomogeneous medium to the homogeneous medium using Wheeler’s transformation for the lossy microstrip medium. The results on the static microstrip line parameters are summarized. The dispersion law is discussed to get the dispersion model of microstrip. Some other dispersion models are also summarized. The losses and their computation are presented in detail. Finally, chapter 8 ends with the circuit model of the microstrip line giving the complex frequency-dependent characteristic impedance

1.3 Overview of Present Book

and propagation constant. The circuit model explains the behavior of the low-frequency dispersion due to the finite conductivity of the conductors. Several topics are covered for the first time in a book form. The derivations of some frequently used expressions are provided. The coplanar waveguides (CPW) and the coplanar stripline structures (CPS) and their variations are discussed in chapter 9. The approach used in this chapter is based on the detailed derivation of the results using the conformal mapping method. Usually, the available books only summarize the results of the conformal mapping method. However, chapter 9 briefly presents the conformal mapping method as applied to the CPW and CPS. The characteristics of the modes, dispersion, and losses are presented in detail. The results are also presented for the synthesis of the CPW and CPS line structures. Finally, the circuit models of the lossy CPW and CPS are given to get the frequency-dependent complex characteristics impedance and propagation constant. The modeling of the third important planar line, i.e. the slot line is presented in chapter 10. The modeling process is based on the unique waveguide model of Cohn. The model provides the frequency-dependent characteristic impedance and propagation constant of the slot line, supporting the hybrid mode. The waveguide model of the slot line treats the hybrid-mode as a linear combination of the TE and TM modes. The equivalent waveguide model is further extended to the multilayer and shielded slot line structures. The chapter ends with the closed-form integrated model of the slot line to compute the dispersion and loss parameters. The next two chapters 11 and 12 cover the basic characteristic of the coupled lines theory and implementation of theory in the planar technology. Chapter 11 discusses the coupling mechanism and the analysis of symmetrical and asymmetrical coupled lines. The wave equation of the coupled transmission is obtained and solved in some cases. Chapter 12 summarizes the design expressions for the edge coupled and broadside coupled microstrip lines. Similar expressions are also summarized for the coupled CPW line structures. The network parameters for both symmetrical and asymmetrical coupled line sections are discussed in detail. Such an analysis is useful for the design of the filters. At this stage, the further discussion of the planar line structure is discontinued and the fabrication technologies suitable for the planar lines and components are introduced in chapter 13. The chapter 13 discusses, in brief, the four kinds of fabrication technologies – the hybrid microwave integrated circuit (HMIC) suitable for the PCB board medium, the semiconductor based

monolithic MIC (MMIC) technology, the silicon-based micro-electro-mechanical systems (MEMS) technology, and the ceramic tape-based low temperature co-fired ceramic (LTCC) technology. The typical details of the material and conductor parameters used in these technologies are also summarized. The familiarity of the fabrication process could be useful to the researchers and designers developing planar lines models and circuits. The basic discussion and basic analysis of the resonator circuits, lumped and distributed line type, are presented in chapter 17. The implementation of the theory of transmission line resonators, and also the patch resonators, is the subject matter of chapter 18. The emphasis is placed on the circuit modeling of the resonating structures. Chapter 18 also discusses the fractal resonators and the dual-mode resonators with the illustrative examples of their applications. The resonating structures are important components for the development of the planar EBG and metamaterials.

Analytical Methods

Most of the discussions on the planar transmission lines and resonators are centered on the physical models and closed-form expression using circuit modeling. However, the analytical and quasi-analytical methods have been developed in the literature for more versatile and accurate modeling of the planar line structures in the multilayered medium. The conformal mapping method, as applied to the CPW and CPS line structures, is presented in chapter 9. The chapter 14 presents the static variational methods, both in the space-domain and Fourier domain, for the analysis of the microstrip and coupled microstrip. Using the transverse transmission line (TTL) technique, the variational method is extended to the multilayer microstrip lines. The method is extended to the boxed microstrip line and CPW using the Galerkin’s method. However, the planar lines are both lossy and dispersive medium. The closed-form models are normally not available for the multilayer lossy planar lines. Chapter 15 presents the scheme of the single-layer reduction (SLR) formulation that utilizes the variational method and available single layer closedform expressions to compute the dispersion and losses in the multilayer planar lines. The SLR models could be incorporated in the microwave CAD packages for the synthesis of the planar microwave components in the multilayer environment. Finally, the semi-analytical full-wave method, i.e. the dynamic SDA is elaborated in chapter 16. The treatment is at the introductory level with the detailed derivation of expressions. The method also incorporates the multilayered planar lines.

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Contemporary Engineered Planar Structures

These are the topics of current interest and have wideranging applications in the development of novel devices and antenna structures with controlled functionalities. Chapter 19 is dedicated to the periodically loaded transmission lines and their circuit models. The method of the circuit models based analysis of the electronic band-gap (EBG) line is elaborated in detail and applied to the loaded microstrip and CPW line structures. The method is extended to the planar EBG surfaces in chapter 20. The accurate circuit model of the EBG surfaces is presented, and also the process of the EM-simulation and interpretation of the results are discussed. The analyses of the engineered artificial media called the bulk metamaterials, metalines, and metasurfaces are presented in the chapters 21 and 22. The modeling and parameter extraction of the engineered media are also discussed. The metasurfaces are interesting structures with several possible applications both in the development of the circuits and planar antenna with multidimensional functionalities. The unique generalized Snell’s laws are discussed in chapter 22 that helps to control the wave propagation. The polarization control and conversion of the state of polarization of the incident waves are also achieved with metasurfaces. Several applications of the metasurface are summarized. 1.3.2 Key Features, Intended Audience, and Some Suggestions Key Features

Possibly, it is the first comprehensive book dedicated entirely to the planar transmission lines. The book covers the microstrip lines, CPW, and other classical resonating structures from basic to advanced form in one cover. The book further covers the EBG and metamaterial-based planar transmission lines and surfaces. These are the topics of current interest. The sequence of the chapter is logical and of increasing complexity. The emphasis of the book is on the modeling of the planar lines and engineered surfaces using the physical concepts, circuit-models, closed-form expressions, and the derivation of a large number of expressions. It provides an in-depth review of the classical transmission line theory, electromagnetics, modeling of the material medium, and waveguide structures compactly without sacrificing the clarity of presentation. The advanced mathematical treatment of the topics, such as the variational method, conformal mapping method, and SDA for several kinds of planar line structures is carried out in detail to help the new readers unfamiliar with these topics. A large number of illustrative

examples from the published literature are given to clarify the theory and physical principles involved in the contemporary topics of the EBG lines and surfaces, metamaterials, metalines, and metasurfaces. The multilayer structures useful for the MMIC, MEMS, and LTCC technology are covered. The closedform modeling of dispersion, frequency-dependent characteristic impedance, and losses in the multilayer planar transmission lines is covered for the first time in a bookform. Also, the basic fabrication technology of the planar transmission lines in the MMIC, MEMS, and LTCC technology is described to help the modelers of the EMphenomenon and microwave circuit designers. Thus, the book prepares the reader to follow the modern designs of the planar circuits and also to undertake independent researches in the field of planar microwave technology. Intended Audience

The book is intended to help undergraduate students of third/fourth year and also postgraduate students. It is useful to the teachers of microwave engineering in preparing lectures, assignments, and projects. The new researchers in the field of microwave engineering will find the book useful to improve their skills in the modeling of the planar structures. It is also suitable for the self-study of the RF/Microwave professionals in the industries. The selected chapters could be used in classroom teaching. It could be the main text for conducting elective courses at the university level. Further, the book can serve as a reference book even to more experienced users in the industry. Some Suggestions

The students and new researchers can have a fast review of the transmission line theory, EM-theory, and so on from chapters 2–7. Then it will be much easier to follow the remaining text. It will be also helpful in following other advanced books and published current literature. However, experienced readers can read any chapter or topic. The concept for the forward and backward referencing of chapters, sections, and subsections are provided to help the reader to skip the chapters or to go back to an intended topic inside the book. A large number of closed-form expressions are given for the modeling purpose. A reader is encouraged to write the Matlab codes or codes in any other language familiar to him/her. The teacher can assign to students the code development as homework. Such activities will enhance the learning process and skill development. More efforts will be needed to write the codes for the

References

variational method, Galerkin’s method, and the SDA. Of course, the effort is rewarding. The researchers must look into the journals to follow newer investigations and decide the direction of his/her research activities. Good luck!

B.15 Brillouin, L.: Wave Propagation in Periodic Structures -

Electric Filters and Crystal Lattice, 2nd Edition, Dover Publications, New York, 1953. B.16 Itoh, T. (Editor): Planar Transmission Lines, IEEE Press, Piscataway, NJ, 1987. B.17 Khine, M.S.; Saleh, I.M. (Editors): Models and Modeling in Science Education, Vol. 6, Springer Dordrecht Heidelberg, London, New York, 2011.

References Books B.1 Nahin Paul, J.: Oliver Heaviside, Sage in Solitude: The

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Journals J.1 Elliott, R.S.: The history of electromagnetics as Hertz

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would have known it, IEEE Antennas Propag. Soc. Newsl., Vol. 30, No. 3, pp. 5–18, June 1988 Grattan-Guinness, I.: Why did George Green write his essay of 1828 on electricity and magnetism?, Am. Math. Mon., Vol. 102, No. 5, pp. 387–397, DOI: 10.1080/ 00029890.1995.12004591http://about.jstor.org/terms. Wu, A.C.T.; Yang, C.N.: Evolution of the concept of the vector potential in the description of potential interactions, Int. J. Mod. Phys. A, Vol. 21, No. 16, pp. 3235–3277, 2006. Eliseo Fernández, E.: Concepts and instruments: the potential from Green to Kelvin, The Midwest Junto for the History of Science, March 25–27, 1994. Challis, L.; Sheard, F.: The green of Green’s function, Phys. Today, Vol. 56, No. 12, pp. 41–46, 2003. John Roche, R.: The present status of Maxwell’s displacement current, Eur. J. Phys., Vol. 19, pp. 155–16n, 1998. French, A.P.; Tessman, J.R.: Displacement currents and magnetic fields, Am. J. Phys., Vol. 31, pp. 201–204, 1963. Scheler, G.; Paulus, G.G.: Measurement of Maxwell’s displacement current, Eur. J. Phys., Vol. 36, No. 055048, pp. 1–9, 2015. Selvan, K.T.: Presentation of Maxwell’s equations in historical perspective and the likely desirable outcomes, IEEE Antennas Propag. Mag., Vol. 49, No. 5, pp. 155–160, Oct. 2007. Selvan, K.T.: A revisiting of scientific and philosophical perspectives on Maxwell’s displacement current, IEEE Antennas Propag. Mag., Vol. 51, No.3, pp. 36–46, June 2009. Essex, E.A.: Hertz vector potentials of electromagnetic theory, Am. J. Phys., Vol. 45, pp. 1099–1101, 1977. Kraus, J.D.: Heinrich Hertz-theorist and experimenter, IEEE Trans. Microwave Theory Tech., Vol. 36, No. 5, pp. 824–829, May 1988. Susskind, C.: Heinrich Hertz: A short life, IEEE Trans. Microwave Theory Tech., Vol. 36, No. 5, pp. 802–805, May 1988.

15

16

1 Overview of Transmission Lines

J.14 Bladel, J.V.: Lorenz or Lorentz?, IEEE Antennas Propag.

J.31 Southworth, G.C.: Hyper – frequency waveguides –

Mag., Vol. 33, No. 2, pp. 69, April 1991. J.15 Jackson, J.D.; Okun, L.B.: Historical roots of gauge

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invariance, Rev. Mod. Phys., Vol. 73, pp. 663–680, July 2001. Griffiths, D.J.: Resource letter EM-1: Electromagnetic momentum, Am. J. Phys., Vol. 80, No. 1, pp. 7–18, Jan. 2012. Aharonov, Y.; Bohm, D.: Significance of electromagnetic potentials in quantum theory, Phys. Rev., Vol. 115, pp. 485–491, 1959. Thomson, W. (Kelvin): On the theory of the electric telegraph, Proc. R. Soc. London, Vol. 4, No. 11, pp. 382–399, May 1855. Searle, G.F.C., et al. The Heaviside Centenary Volume, The Institution of Electrical Engineers, London, 1950. Whittaker, E.T.: Oliver Heaviside, In Electromagnetic Theory Vol. 1, Oliver Heaviside, Reprint, Chelsea Publishing Company, New York, 1971. Pupin, M.I.: Wave propagation over non-uniform cables and long-distance airlines, Trans. Am. Inst. Electr. Eng., Vol. 17, pp. 445–507, (discussion on pp. 508–512), May 1900. Pupin, M.I.: Propagation of long electrical waves, Trans. Am. Inst. Electr. Eng., Vol. xv, No. 144, pp. 93–142, March 1899. Campbell, G.A.: On loaded lines in telephonic transmission, London, Edinburg Dublin Philos. Mag. J. Sci., Vol. 5, No. 27, pp. 313–330, 1903. Kennelly, A.E.: Impedance, 76th Meeting of American Institute of Electrical Engineers, pp. 175–232, April 1893. Carson, J.R.: Wave propagation over parallel wires: The proximity effect, London, Edinburgh Dublin Philos. Mag. J. Sci., Vol. 41, No. 244, pp. 607–633, 1921 Carson, J.R.: Electromagnetic theory and the foundations of electrical circuit theory, Bell Syst. Tech. J., Vol. 6, pp. 1–17, Jan. 1927. Levin, A.: Electromagnetic waves guided by parallel wires with particular reference to the effect of the earth, Trans. Am. Inst. Electr. Eng., Vol. 46, pp. 983–989, June 1927. Bewley, L.V.: Traveling waves on transmission systems, Trans. Am. Inst. Electr. Eng., Vol. 50, No. 2, pp. 532– 550, June 1931. Pipes, L.A.: Matrix theory of multiconductor transmission lines, London, Edinburgh Dublin Philos. Mag. J. Sci., Vol. 24, No. 159, pp. 97–113, July 1937. Rayleigh, J.W.S.: On the passage of electric waves through tubes, or vibrations of dielectric cylinders, Philos. Mag., Vol. 43, pp. 125–132, Feb. 1897.

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General consideration and experimental results, Bell Sys. Tech. J., Vol. 15, pp. 284–309, 1936. Barrow, W.L.: Transmission of electromagnetic waves in hollow tubes of metal, Proc. IRE, Vol. 24, pp. 1298– 1328, Oct. 1936. Brillouin, L.: Propagation of electromagnetic waves in a tube, Rev. GMn. de l’Elec., Vol. 40, pp. 227–239, Aug. 1936. Carson, J.R.; Mead, S.P.; Schelkunoff, S.A.: Hyperfrequency wave guides – mathematical theory, Bell Sys. Tech. J., Vol. 15, pp. 310–333, 1936. Chu, L.I.; Barrow, W.L.: Electromagnetic waves in hollow metal tubes of rectangular cross- section, Proc. IRE, Vol. 26, 1520, 1938. Schelkunoff, S.A.: The electromagnetic theory of coaxial transmission lines and cylindrical shields, Bell Sys. Tech. J., Vol. 13, No. 4, pp. 532–579, Oct. 1934. Schelkunoff, S.A.: Transmission theory of plane electromagnetic waves, Proc. IRE, Vol. 25, p. 1437, 1937. Schelkunoff, S.A.: Generalized telegraphist’s equations for waveguides, Bell Sys. Tech. J., Vol. 31, No. 4, pp. 784– 801, July 1952. Schelkunoff, S.A.: Conversion of Maxwell’s equations into generalized telegraphist’s equations, Bell Sys. Tech. J., Vol. 34, No. 5, pp. 995–1043, Sept. 1955. Niehenke, E.C.; Pucel, R.A.; Bahl, I.J.: Microwave and millimeter-wave integrated circuits, IEEE Trans. Microwave Theory Tech., Vol. 50, No. 3, pp. 846–857, March 2002. Greig D.D.; Engelmann, H.: Microstrips – A new transmission technique for the kilomegacycle range, Proc. IRE, Vol. 40, pp. 1644–1650, Dec. 1952. Barrett, R.M.: Microwave printed circuits-A historical survey, IEEE Trans. Microwave Theory Tech., Vol. 3, No. 2, pp. 1–9, Mar. 1955. Barrett, R.M.: Microwave printed circuits - The early years, IEEE Trans. Microwave Theory Tech., Vol. MTT-32, No. 9, pp. 983–990, Sept. 1984. Cohn, S.B.: Characteristic impedance of the shieldedstrip transmission line, IRE Trans. Microwave Theory Tech., Vol. MTT-2, pp. 52–57, July 1954. McQuiddy, D.N.; Wassel, J.W.; LaGrange, J.B.; Wisseman, W.R.: Monolithic microwave integrated circuits: An historical perspective, IEEE Trans. Microwave Theory Tech., Vol. 32, No. 9, pp. 997–1008, 1984. Petersen, K.E.: Micro-mechanical membrane switches on Silicon, IBM J. Res. Dev., Vol. 23, No. 4, pp. 376–385, July 1979. Yao, J.; Chang, M.F.: A surface micro-machined miniature switch for telecommunication application

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with single freq. from DC up to 4 GHz, Proc. Transducer’95, pp. 384–387, June 1995. Goldsmith, C.; Lin, T.H.; Powers, B.; Wu, W.R.; Norvel1, B.: Micro-mechanical membrane switches for microwave application, IEEE Microwave Theory Tech. Symp., MTT-S Digest, pp. 91–94, May 1995. Rodriguez, A.R.; Wallace, A.B.: Ceramic capacitor and method of making it, Patent US 3004197, issued 10/ 10/1961. Hajian, A.; Müftüoglu, D.; Konegger, T.; Schneider, M.; Schmid, U.: On the porosification of LTCC substrates with sodium hydroxide, Compos. Part B: Eng., Vol. 157, pp. 14–23, 2019. Veselago, V.: The electrodynamics of substances with simultaneously negative values of ε and μ, Soviet Physics Uspekhi, Vol. 10, No. 4, pp. 509–514, Jan., Feb. 1968. Pendry, J.B.; Holden, A.J.; Stewart, W.J.; Youngs, I.: Extremely low-frequency plasmons in metallic mesostructures, Phys. Rev. Lett., Vol. 76, No. 25, pp. 4773–4776, June 1996. Pendry, J.B.; Holden, A.J.; Robbins, D.J.; Stewart, W.J.: Magnetism from conductors and enhanced nonlinear phenomena, IEEE Trans. Microwave Theory Tech., Vol. 47, No. 11, pp. 2075–2084, Nov. 1999. Shelby, R.A.; Smith, D.R.; Schultz, S.: Experimental verification of a negative index of refraction, Science, Vol. 292, No. 5514, pp. 77–79, April 2001. Smith, D.R.; Padilla, W.J.; Vier, D.C.; Nemat-Nasser, S. C.; Schultz, S.: Composite medium with simultaneously negative permeability and permittivity, Phys. Rev. Lett., Vol. 84, No. 18, pp. 4184–4187, May 2000. Assaudourion, F.; Rimai, E.: Simplified theory of microstrip transmission systems, Proc. IRE, Vol. 40, pp. 1651–1657, Dec. 1952. Cohn, S.B.: Characteristic impedance of the shieldedstrip transmission line, IRE Trans. Microwave Theory Tech., Vol. MTT-2, pp. 52–57, July 1954. Cohn, S.B.: Problems in strip transmission lines, IEEE Trans. Microwave Theory Tech., Vol. MTT-3, No. 2, pp. 119–126, March 1955. Cohn, S.B.: Shielded coupled-strip transmission line, IRE Trans. Microwave Theory Tech., Vol. MTT-3, pp. 29–38, Oct. 1955. Cohn, S.B.: Characteristic impedances of broadsidecoupled strip transmission lines, IRE Trans. Microwave Theory Tech., Vol. MTT-8, pp. 633–637, Nov. 1960. Wheeler, H.A.: Transmission-line properties of parallel wide strips by a conformal-mapping approximation, IEEE Trans. Microwave Theory Tech., Vol. MTT-12, pp. 280–289, May 1964. Wheeler, H.A.: Transmission-line properties of parallel strips separated by a dielectric sheet, IEEE Trans.

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Microwave Theory Tech., Vol. MTT-13, pp. 172–185, March 1965. Wheeler, H.A.: Transmission-line properties of a strip on a dielectric sheet on a plane, IEEE Trans. Microwave Theory Tech., Vol. MTT-25, pp. 631–647, Aug. 1977. Wheeler, H.A.: Transmission line properties of a stripline between parallel planes, IEEE Trans. Microwave Theory Tech., Vol. MTT-26, pp. 866–876, Nov. 1978. Cohn, S.B.: Slotline on a dielectric substrate, IEEE Trans. Microwave Theory Tech., Vol. MTT-17, pp. 768–778, Oct. 1969. Wen, C.P.: Coplanar waveguide: a surface strip transmission line suitable for nonreciprocal gyromagnetic device application, IEEE Trans. Microwave Theory Tech., Vol. MTT-17, pp. 1087–1090, Dec. 1969. Meier, P.J.: Two new integrated-circuit media with special advantages at millimeter-wave lengths, IEEE MTT-S Int. Microwave Symp. Dig., pp. 221–223, 1972. Yamashita, E.; Mitra, R.: Variational method for analysis of microstrip lines, IEEE Trans. Microwave Theory Tech., Vol. MTT-16, No. 4, pp. 251–256, April 1968. Yamashita, E.: Variational method for analysis of microstrip – like transmission lines, IEEE Trans. Microwave Theory Tech., Vol. MTT-16, No. 8, pp. 529–535, Aug. 1968. Itoh, T.; Herbert, A.S.: A generalized spectral domain analysis for coupled suspended microstrip lines with tuning septum, IEEE Trans. Microwave Theory Tech., Vol. MTT-26, No. 10, pp. 820–826, Oct. 1978. Itoh, T.: A generalized spectral domain analysis for multiconductor printed lines and its application to tunable suspended microstrips, IEEE Trans. Microwave Theory Tech., Vol. MTT-26, No. 12, pp. 983–987, Dec. 1978. Itoh, T.; Mittra, R.: Spectral-domain approach for calculating the dispersion characteristics of microstrip lines, IEEE Trans. Microwave Theory Tech., Vol. MTT-21, pp. 496–499, July 1973. Itoh, T.; Mittra, R.: Dispersion characteristics of slot lines, Electron. Lett., Vol. 7, pp. 364–365, 1971. Itoh, T.: Spectral-domain immittance approach for dispersion characteristics of generalized printed transmission lines, IEEE Trans. Microwave Theory Tech., Vol. MTT-28, pp. 733–736, 1980. Jansen, R.H.: High-speed computation of single and coupled microstrip parameters including dispersion, high-order modes, loss and finite strip thickness, IEEE Trans. Microwave Theory Tech., Vol. MTT-26, pp. 75–82, 1978.

17

19

2 Waves on Transmission Lines – I (Basic Equations, Multisection Transmission Lines)

Introduction The two-wire transmission line is a useful medium for the propagation of the voltage and current waves. It is also useful in the modeling of planar transmission lines. The EM-wave propagating on multilayered planar transmission lines could also be analyzed with the help of the multisection transmission lines. The primary purpose of this chapter is to review in detail the wave propagation on a transmission line without and with sources. Several other important topics such as the characterization of a line section, nature of wave velocities, dispersion, and reactively loaded lines are further discussed in chapter 3. Objectives

• •• • • •

To formulate Kelvin–Heaviside transmission line equations. To obtain the solution of the wave equation. To compute the power flow on a transmission line. To use Thevenin’s theorem on a transmission line section to obtain its transfer function. To consider the wave propagation on a multi-section transmission line with the voltage and current sources. To understand the nature of the wave propagation on a nonuniform transmission line.

2.1

Uniform Transmission Lines

This section presents the basic understanding of wave propagation on a uniform transmission line. The Kelvin–Heaviside transmission line equations are formulated using the lumped circuit elements model of the uniform transmission line. The voltage and current wave equations are obtained and solved for a terminated line section. The phenomenon of the standing wave is discussed. The Thevenin theorem of the transmission line network is discussed and the transfer function of a line section is obtained.

2.1.1

Wave Motion

The wave motion could be treated as the transfer of oscillation from one location to another location. A harmonic oscillation is described either by a sine or by a cosine function. A periodic oscillation has a fixed period. An oscillation repeats itself after the periodic time (T). In general, an oscillation, i.e. an oscillatory motion can have any shape such as square, triangular, and so on. However, with the help of the Fourier series, such periodic oscillations can be decomposed to the harmonic functions. Likewise, wave motion can also acquire an arbitrary shape. The arbitrary periodic shape of a wave can be decomposed into the harmonic waves. Figure (2.1) shows the instantaneous amplitude v(t) of the wave generated by an oscillating quantity, such as an oscillating ball (particle) at the location A. It has an angular frequency of oscillation ω radian/sec and maximum amplitude Vmax. The ball is attached to a spring and immersed in a water tank. It generates a wave motion at the surface of the water. The water wave travels with velocity vp and causes another delayed oscillation, at the location B, in a similar ball attached to a spring. Through the mechanism of wave motion, the oscillation is transferred from the location A to the location B separated by distance x. Likewise, the oscillating quantity could be a charge, electric field, magnetic field, or voltage obtained from an oscillator connected to a cable. The equation of the harmonic oscillation at location A is described by the cosine function, va t = Vmax cos ωt

211

The equation of harmonic oscillation that appears at location B after a delayed time t is vb t = Vmax cos ω t − t

212

The wave motion, created by the oscillation at the location A, travels with velocity vp. The time delay in

Introduction to Modern Planar Transmission Lines: Physical, Analytical, and Circuit Models Approach, First Edition. Anand K. Verma. © 2021 John Wiley & Sons, Inc. Published 2021 by John Wiley & Sons, Inc.

2 Waves on Transmission Lines – I

T= A

B Vp

Oscillating particle Transmitted wave

Medium x

Received wave

Oscillating particle

Delayed oscillation as a wave motion-initial oscillation v(t) at location A, and delayed oscillation v(t − x/vp) at location B. The waves propagate from the locations A to B.

setting up the oscillation at the location B is t = x/vp. So the equation of oscillation at the location B is vb t = Vmax cos ω t − x vp

213

Equation (2.1.3) describes the propagating wave through the medium between locations A and B. It is called the wave equation. The phase constant β of the propagating wave is defined as β = ω/vp. On dropping the subscript “b,” equation (2.1.3) is rewritten in the usual form, v t, x = Vmax cos ωt − βx

Figure 2.2

T

216

The wave motion or velocity of a monochromatic wave (the wave of a single frequency) is the motion of its wavefront. For the 3D wave motion in an isotropic space, the wavefront is a surface of the constant phase. In the case of the 1D wave motion, the wavefront is a line, whereas, for the 2D wave motion on an isotropic surface, the wavefront is a circle. Figure (2.3) shows the peak point P at the wavefront of the ID wave. The motion of a constant phase surface is known as its phase velocity vp. Thus, the peak point P at

Wavefront P

*

P′

Voltage

* Distance X

P″ Figure 2.3

Vmax

Wave motion as a motion of constant phase surface. It is shown as the moving point P.

λ

Voltage amplitude V(x, t = 0)

Vmax

2π β

214

In equation (2.1.4), the lagging phase ϕ = βx is caused by the delay in oscillation at the receiving end. The medium supporting the wave motion is assumed to be lossless. The wave variable v(t, x) is a doubly periodic function of both time and space coordinates as shown in Fig (2.2a and b), respectively. The temporal period, i.e. time-period T is related to the angular frequency ω as follows:

215

where f is the frequency in Hz (Hertz), i.e. the number of cycles per sec. The wavelength (λ) describes the spatial period, i.e. space periodicity. It is related to the phaseshift constant (or phase constant), i.e. the propagation constant β, as follows: λ=

Figure 2.1

2π 1 = , ω f

Voltage amplitude V(x = 0, t)

20

Vmin

Vmin

Time, t (a) Harmonic variation of the wave with respect to time, at the location x = 0.

Distance, d (b) Harmonic variation of the wave with respect to the distance, at time t = 0.

Double periodic variations of wave motion.

2.1 Uniform Transmission Lines

the wavefront has moved to a new location P in such a way that phase of the propagating wave remains constant, i.e. ωt − βx = constant. On differentiating the constant phase with respect to time t, the expression for the phase velocity is obtained: vp =

dx ω = dt β

ω β

218

Thus, for a frequency-independent nondispersive wave, the phase constant is a linear function of angular frequency. β = Constant × ω

Angular frequency ω

x vp

2 1 10

On taking the second-order partial derivative of the wave function ψ(t, x) with respect to space-time coordinate x and t, the 1D second-order partial differential equation (PDE) of the wave equation is obtained. Similarly, wave functions ψ(t, x, y) and ψ(t, x, y, z) are solutions of two-dimensional (2D) and three-dimensional (3D) wave equations supported by the surface and free space medium, respectively. These PDEs are summarized below: ∂2 ψ 1 ∂2 ψ = ∂x2 v2p ∂t2

a,

∂2 ψ ∂2 ψ ∂2 ψ 1 ∂2 ψ + + = ∂x2 ∂y2 ∂z2 v2p ∂t2

c,

∂2 ψ ∂2 ψ 1 ∂2 ψ + = ∂x2 ∂y2 v2p ∂t2 ∇2 ψ =

1 ∂2 ψ v2p ∂t2

β

ω

Q

θ Propagation constant β

Figure 2.4 The ω-β dispersion diagram of nondispersive wave.

b d

2 1 11

219

The dispersive nature of the 1D wave motion is further discussed in sections (3.3) and (3.4) of the chapter 3. The phase velocity in a dispersive medium is usually frequency-dependent. It is known as the temporal dispersion. In some cases, the phase velocity could also depend on wavevector (β, or k). It is known as spatial dispersion. The spatial dispersion is discussed in the subsection (21.1.1) of the chapter 21. The dispersion diagrams of the wave propagation in the isotropic and anisotropic media are further discussed in the section (4.7) of the chapter 4, and also in the section (5.2) of chapter 5.

o

ψ t, x = ψ t

217

In a nondispersive medium, the phase velocity of a wave is independent of frequency, i.e. the waves of all frequencies travel at the same velocity. Figure (2.4) shows the nondispersive wave motion on the (ω − β) diagram. It is a linear graph. The slope of the point Q on the dispersion diagram subtends an angle θ at the origin that corresponds to the phase velocity of a propagating wave, vp = tan θ =

The one-dimensional (1D) wave travels both in the forward and in the backward directions. It can have any arbitrary shape. In general, the 1D wave propagation can be described by the following wave function, known as the general wave equation:

The dispersion diagram of the 2D wave propagation over the (x, y) surface is obtained by revolving the slant line of Fig (2.4) around the ω-axis with propagation constants βx, βy in the x- and y-direction. It is discussed in subsection (4.7.4) of chapter 4. 2.1.2

Circuit Model of Transmission Line

A physical transmission line, supporting the voltage/ current wave, can be modeled by the lumped R, L, C, G components, i.e. the resistance, inductance, capacitance, and conductance per unit length (p.u.l.), respectively. The two-conductor transmission line can acquire many physical forms. A few of these forms are shown in Fig (2.5). The lines as shown in Fig (2.5a–c) support the wave propagation in the transverse electromagnetic mode, i.e. in the TEM-mode; while Fig (2.5c) shows the quasi-TEM mode-supporting microstrip. For TEM mode wave propagation, the electric field and magnetic field are normal to each other and also normal to the direction of wave propagation. For the TEM mode, there is no field component along the direction of propagation. However, the quasi-TEM mode also has component of weak fields along the longitudinal direction of wave propagation. The quasi-TEM mode is a hybrid mode discussed in the subsection (7.1.4) of chapter 7. All TEM mode supporting transmission lines can be represented by a parallel two-wire transmission line

21

22

2 Waves on Transmission Lines – I

Strip conductor

Substrate Ground plane (a) Two-wire line. Figure 2.5

(b) Co-axial line.

(c) Microstrip line.

Cross-section of a few two-conductor transmission lines.

(k + 1)th node

kth node L

L

R

R

Δi

i(t)

ik + 1 (x,t)

ik (x,t) Vk (x,t)

C

v(t)

G

C

G

Vk + 1 (x,t) Δx (a) Physical two-wire transmission line.

(b) Equivalent circuit of the two-wire transmission line.

Figure 2.6 RLCG lumped circuit model of a transmission line.

shown in Fig (2.6a). A transmission line is a 1D wave supporting structure. Its cross-sectional dimension is much less than λ/4; otherwise, its TEM nature is changed. The longitudinal dimension can have any value, from a fraction of a wavelength to several wavelengths. The mode theory of the electromagnetic (EM) wave propagation is further discussed in chapter 7. A two-conductor transmission line, or any other line supporting the TEM mode, is modeled as a chain of discrete passive RLCG components. As a matter of fact, by cascading several sections of discrete L-network of the series L and shunt C elements, or even discrete Lnetwork of the series C and shunt L elements, an artificial transmission line can also be constructed. The artificial transmission line is discussed in section (3.4) of chapter 3, and also in the chapters 19 and 22. It plays a very important role in modern microwave planar technology. The behavior of a transmission line is determined in terms of the resistor (R), inductor (L), capacitor (C), and conductance (G); all line elements are in per unit length, i.e. p.u.l. Kelvin introduced the modeling of the telegraph cable laid in the ocean using the RC circuit model. Heaviside further introduced L and G components in the circuit model to improve the modeling of the lossy transmission line [B.1, B.2, J.1, J.2].

Kelvin RC-circuit model of the transmission line leads to the diffusion equation, not to the wave equation. Whereas using the RLGC circuit model, Heaviside finally obtained the wave equation for the voltage/current on a transmission line. Using the RLCG circuit model, shown in Fig (2.6b), the voltage, and current equations are obtained for the transmission line. The set of the coupled voltage and current equations are normally called the telegrapher’s equations; as it was originally developed for the telegraph cables. However, the set of coupled transmission line equations can be called the Kelvin–Heaviside transmission line equations to recognize their contributions.

The Resistance of a Line

The electrical loss in a transmission line, known as the conductor loss, is due to the finite conductivity of the line. It is modeled as the resistance R p.u.l. It is also influenced by the skin effect phenomena at a higher frequency. The instantaneous current i(t) flowing through a lumped resistance Rlum is related to the instantaneous voltage drop v(t) by Ohm’s law: v t = Rlum i t

2 1 12

2.1 Uniform Transmission Lines

The Inductance of a Line

The current flowing in a conductor creates the magnetic field around itself. So the magnetic energy stored in the space around the transmission line, i.e. the time-varying current supporting line section, is modeled by a series inductor L p.u.l. The inductance of a line is not lumped at one point, i.e. it is not confined at one point. It is distributed over the whole length of a line. The instantaneous voltage across a lumped inductor Llum is related to the current flowing through it by v t = Llum

di t dt

2 1 13

In a two-conductor transmission line, the conductors separated by a dielectric medium form a distributed system of capacitance. The electric field energy stored in a line is modeled by the shunt capacitance C p.u.l. The instantaneous shunt current through a lumped capacitor Clum is related to the instantaneous voltage across it by dv t dt

2 1 14

The Loop Equation

vk − vk + 1 = Δv = − LΔx

∂i − RΔx i ∂t 2 1 16

The instantaneous line voltage v(t) and the instantaneous line current i(t) are functions of the space–time variables x and t. In the limiting case of Δx 0, equation (2.1.16) is written as ∂v ∂i = − Ri + L ∂x dt

2 1 17

The Node Equation

The Conductance of a Line

If the medium between two conductors of the transmission line is not a perfect dielectric, i.e. if it has finite conductivity, then a part of the line current shunts through the medium causing the dielectric loss. The dielectric loss of the line is modeled by shunt conductance G p.u.l. The instantaneous shunt current is related to the instantaneous voltage across a lumped conductance by ish t = Glum v t

Figure (2.6b) shows the lumped element equivalent circuit model of a section Δx of the transmission line. The primary line constants R, L, C, G on the per unit length (p.u.l.) basis are related to the lumped circuit elements as Rlum = RΔx, Llum = LΔx, Clum = CΔx, and Glum = GΔx. Figure (2.6b) shows the voltage loop equation and the current node equation for a small line section Δx. These are written as follows:

Differential voltage change across line length Δx = Voltage drop across inductor + Voltage drop across resistance

The Capacitance of a Line

ish t = Clum

2.1.3 Kelvin–Heaviside Transmission Line Equations in Time Domain

2 1 15

Figure (2.6a) shows a physical transmission line that supports the TEM mode wave propagation. Figure (2.6b) shows that this line could be modeled as a chain of the lumped RLCG structure. More numbers of RLCG sections per wavelength are needed to model a transmission line. The RLCG model is the modeling of a transmission line by the lossy series inductor and lossy shunt capacitor. The transmission line supports the wave propagation from power AC to RF and above. Likewise, the corresponding lumped components model of a transmission line also supports such waves. The transmission line structure behaves like a low-pass filter.

Differential shunt current at the node = Current through conductance + Current through capacitor ik − ik +1 = Δi = − GΔx v − CΔx In the limiting case, Δx written as

∂v ∂t

2 1 18

0; and the above equation is

∂i ∂v = − Gv + C ∂x ∂t

2 1 19

The pair of coupled voltage and current transmission line equations in the time-domain summarized below is known as “the time domain telegrapher’s equations”: ∂v ∂i = − Ri + L ∂x ∂t

a,

∂i ∂v = − Gv + C ∂x ∂t

b 2 1 20

The above pair of equations can also be called the Kelvin–Heaviside transmission line equations. The coupled Kelvin–Heaviside transmission line equations relate the voltage and current on a transmission line through the line parameters, RLCG. These parameters

23

24

2 Waves on Transmission Lines – I

are known as the primary constants of a line. The RLCG parameters depend on the physical configuration of a line, i.e. on its physical shapes, dimensions, and the electrical properties of the medium. They could be frequency-dependent parameters also. For simple transmission lines such as a pair of wires and coaxial lines, the closed-form formulas are available to compute them. However, these parameters could also be obtained through measurements. The empirical expressions for the RLCG parameters of a microstrip line are also available [J.3]. The transmission line equations are the complementary parts of Maxwell’s field equations that relate the time-varying electric and magnetic fields in a physical medium through the primary constants of a medium-conductivity (σ), permittivity (ε) and permeability (μ). It is discussed in chapter 4. The transmission line equations can be obtained from Maxwell’s equations. It is discussed in section (7.3) of chapter 7. The voltage and current coupled variables of equation (2.1.20) can be separated. The separation of the variables leads to the wave equations for the voltage and current waves on a transmission line. On differentiating equation (2.1.20a) with respect to the variable x, the voltage wave equation is obtained: ∂2 v ∂i ∂ ∂i +L = − R ∂x2 ∂x ∂x ∂t

= − R

∂i ∂ ∂i +L ∂x ∂t ∂x 2 1 21

∂i from equation (2.1.20b) in equa∂x tion (2.1.21), the voltage wave equation is On substituting

2

2

∂ v ∂v ∂ v + LC 2 = RGv + RC + LG 2 ∂x ∂t ∂t

On comparing the above equations with equation (2.1.11a), the velocity of propagation, for both the current and voltage waves, is vp =

1 LC

2 1 26

It is like the velocity of propagation of an electromagnetic wave in a dielectric medium obtained from Maxwell’s equations, where the primary constant of the line L and C are replaced by the medium constants permeability μ and permittivity ε. The EM-wave is discussed in chapter 4.

2.1.4 Kelvin–Heaviside Transmission Line Equations in Frequency-Domain The time-harmonic instantaneous voltage in the frequency domain, i.e. in the phasor form, is written as v t = Vmax cos ωt + ϕ = Re Vejωt

2 1 27

where “Re” stands for the real part of the voltage phasor V . The voltage phasor V is given by the following expression: V = Vmax ejϕ

2 1 28

The phasor is nothing but a polar form of a complex quantity. Likewise, the instantaneous current in the phasor form is i t = Imax cos ωt + φ = Re Iejωt ,

2 1 29

where current phasor is 2 1 22

The above partial differential equation describes the time-domain voltage wave on a lossy transmission line. Likewise, an equation could be written to describe the current wave on a transmission line: ∂2 i ∂i ∂2 i + LC = RGi + RC + LG ∂x2 ∂t ∂t2 2 1 23 A lossless transmission line has, R = G = 0. The voltage and current waves on a lossless line are given by the following 1D PDEs: ∂2 v ∂2 v = LC ∂x2 ∂t2

2 1 24

∂2 i ∂2 i = LC ∂x2 ∂t2

2 1 25

I = Imax ejφ

2 1 30

The phasor is either a constant or a function of only the space variable. It is not a function of time t. The phasor is shown with a tilde (~) sign in this chapter. However, in the subsequent chapters, the tilde (~) sign is dropped. The phasor is used at a single frequency. Using the phasor notation, the voltage across R, L, and the current through C, G; given by equations (2.1.12)–(2.1.15) in the time domain, can be rewritten in the frequencydomain: V = RI

a,

V = jωLI

b

I = jωCV

c,

I = GV

d

2 1 31

In the above equations, the time derivative ∂/∂t is replaced by jω converting the expression from the time-domain to frequency-domain. Following the conversion process, the time-domain coupled voltage-current

2.1 Uniform Transmission Lines

transmission line is known as the dispersive transmission line. A complex wave traveling on a lossy dispersive line gets distorted as each component of the complex wave travels with different phase velocity. Using the complex propagation constant, the wave equation (2.1.33a and b) are rewritten as

transmission line equation (2.1.20), is rewritten in the frequency-domain: dV = − R + jωL I dx dI = − G + jωC V dx

a 2 1 32 b

d2 V − γ2 V = 0 dx2 d2 I − γ2 I = 0 dx2

On separation of the voltage and current variables, the following voltage and current wave equations are obtained in the frequency-domain: d2 V = R + jωL G + jωC V dx2 d2 I = R + jωL G + jωC I dx2

2 1 33 b

It is noted that the second-order partial differential wave equations in the time-domain are converted to the second-order ordinary differential equations in the frequency-domain. The factors at the right-hand side of the above equations help to define a secondary parameter γ, known as the complex propagation constant of a transmission line: γ = R + jωL G + jωC ,

V I

2 1 34

where γ = α + jβ. The real part α(Np/m) of the complex propagation constant γ is called the attenuation constant and the imaginary part β (rad/sec) is the propagation constant of a lossy transmission line. The parameter β is also known as the phase-shift constant or phase constant. On separating the real and imaginary parts of the above equation, the following expressions are obtained: α = Re γ =

RG − ω2 LC +

RG − ω2 LC

2

+ ω2 LG + RC

2

1 2

2 2 1 35

β = Im γ =

ω2 LC − RG +

RG − ω2 LC

2

+ ω2 LG + RC

2

2 2 1 36

The attenuation constant α, and propagation constant β are given in terms of the primary line constants, R, L, C, G. Normally α and β are frequency-dependent. Thus, the phase velocity of both the current and voltage waves, given by vp = ω/β is frequency-dependent. This kind of

2 1 37 b

The above homogeneous wave equations on a transmission line could be treated as a boundary value problem to get the voltage and current at any location on the line. If a transmission line is infinitely long and excited from one end, then the voltage and current waves on the line always move in the forward direction without any reflection. At any location on the line, the voltage and current are related by another secondary parameter called “the characteristic impedance” of a transmission line:

a

1 2

a

1 2

= Z0

2 1 38

The characteristic impedance of a transmission line could be viewed as a mechanism that explains the wave propagation on a line. It recasts the Huygens’ Principle of the secondary wave formation in terms of the characteristic impedance. The characteristic impedance could be called the Huygens’ load. It is an unusual load impedance with a special property. It absorbs power from the source and itself becomes a secondary source for the further transmission of power in the form of wave motion. In this manner, the wave on a transmission line moves; as the characteristic impedance, i.e. Huygens’ load, acts both as a load and also as a source of the wave motion. The process is similar to the Huygens’ secondary source for the wavefront propagation. The concept of the Huygens’ load is further extended to engineer the Huygens’ metasurface with unique characteristics to control the reflected and transmitted (refracted) EM-waves. It is discussed in subsection (22.5.2) of chapter 22. The characteristic impedance, i.e. Huygens’ load of a lossy line is a complex quantity. Its real part does not dissipate energy like the real part of a normal complex load. The imaginary part of Huygens’s load indicates the presence of losses in a transmission line, whereas in the case of a normal complex load its imaginary part shows the storage of the reactive energy. For a lossless transmission line, Huygens’ load is a real quantity L C that is nondissipative. Huygens adopted the secondary source

25

26

2 Waves on Transmission Lines – I

model to explain the propagation of the light wave in the space [B.3, B.4]. The expression for the characteristic impedance of a line is obtained from equation (2.1.32). Z20 =

ΔV I

×

V ΔI

=

R + jωL , G + jωC

Z0 =

R + jωL G + jωC

Characteristic Impedance

The characteristic impedance Z0 of a uniform lossy transmission line is given by equation (2.1.39). In the limiting case, ω 0, i.e. at a lower frequency, it is reduced to a real quantity that is dominated by the lossy elements of a line:

2 1 39 In general, the characteristic impedance of a lossy transmission line is a complex quantity. However, for a lossless line, the lossy elements are zero, i.e. R = G = 0. It leads to the following expressions: α=0

a,

β = ω LC

b,

Z0 =

L C

Z0 =

Z0 =

2 1 40

2.1.5

Characteristic of Lossy Transmission Line

A transmission line, such as a coaxial cable, a flat cable, used in the computer, or a feeder to TV receiver, is embedded in a lossy dielectric medium. The loss in a dielectric medium is known as the dielectric loss of a transmission line. Of course, when the line is in the air medium, the dielectric loss could be neglected. Likewise, there is another kind of loss, known as the conductor loss, on a transmission line. It is due to the finite conductivity of the conducting wires or the strip conductors forming a line. All open transmission lines tend to radiate some power, leading to radiation loss. In the case of a planar transmission line, there are also other mechanisms of losses. However, this section considers only the conductor and the dielectric losses of a line and their effect on the propagation characteristics of the line.

2 1 41

The characteristic impedance Z0 at very high frequency, i.e. for ω ∞, is also reduced to a real quantity. However, now it is dominated by the lossless reactive elements:

c

There is no attenuation in the propagating wave on a lossless line. If the line inductance L and the line capacitance C are frequency-independent, the transmission line is nondispersive. The characteristic impedance is a real quantity. The line parameters such as the attenuation constant (α), propagation constant (β), and characteristic impedance (Z0) are known as the secondary parameters of a transmission line. These secondary parameters are finally expressed in terms of the physical geometry and the electrical characteristics of material medium of a line. A microwave circuit designer is more interested in these secondary parameters as compared to the primary line constants, RLCG, of a transmission line. The secondary parameters are more conveniently measured and numerically computed for a large class of the transmission line structures. For any practical transmission line, the losses are always present on a line.

R G

L C

2 1 42

At higher frequency, we have ωL >> R and ωC >> G. Therefore, the R and G are normally ignored for the computation of characteristic impedance of a low-loss microwave transmission line. In the intermediate frequency range, the characteristic impedance of a line is a complex quantity. Its imaginary part indicates the presence of the loss in a line. Equation (2.1.42) is also applicable to a lossless line. The characteristic impedance of transmission line in the lossless dielectric medium, or a moderately lossy medium where G could be neglected, is obtained in equation (2.1.43). However, the conductor loss is present on the line: Z0 =

R + jωL G + jωC

1 2

=

L R 1+ C jωL

1 2

≈

L R 1−j C 2ωL 2 1 43

The measured or computed complex characteristic impedance of a line, over a certain frequency range, with a negative imaginary part, indicates that the loss in the line is mainly due to the conductor loss [J.4]. The alternative case of a lossy line, with G 0, R = 0, could be also considered. In this case, the conductor loss is ignored; however, the dielectric loss is dominant. The characteristic impedance of such line is approximated as follows: 1

2 jωL Z0 = ≈ G + jωC

L G 1+j C 2ωC

2 1 44

If the imaginary part of the characteristic impedance of a line is positive over some frequency range, then the dielectric loss dominates the loss in the line. However, if both R and G are moderately present, with ωL >> R and ωC >> G, the real and imaginary parts of

2.1 Uniform Transmission Lines

the characteristic impedance could be approximated by using the binomial expansion as follows: 1

1 ± x 2≈1 ±

x x2 − 8 2

1−j

Z0 =

L C

R2 G2 RG 1+ + 2 2 + 2 2 4ω2 LC 8ω L 8ω C

1+j

G ωC

1 2

L C

2 1 47

In the above expression, ω3 and ω4 terms are neglected. If we neglect ω2 terms and also take G = 0 or R = 0, equation (2.1.47) reduces to either equation (2.1.43) or equation (2.1.44), respectively. It is also possible that with the change of frequency, the imaginary part of characteristic impedance changes from a positive to a negative value indicating that the dominant loss can change from the dielectric loss to the conductor loss. For such cases, R and G are usually frequency-dependent [J.4]. Over a band of frequencies, the imaginary part of the characteristic impedance could be zero leading to R G = L C

2 1 48

It is well known as Heaviside’s condition. On meeting it, a lossy line becomes dispersion-less and the propagation constant β becomes a linear function of frequency, while the attenuation constant becomes frequency-independent. Following the above equation (2.1.48), a lossy line could be made dispersionless by the inductive loading [B.5, B.6]. Propagation Constant

The propagation constant γ of a uniform lossy transmission line is given by equation (2.1.34). It could be approximated under the low-loss condition. Its real and imaginary parts are separated to get the frequently used approximate expressions for the attenuation and phase constants of a line:

γ ≈ jω LC

R G 1−j ωL ωC 2 R R + 1−j 2ωL 8ω2 L2 1−j

1−j

G G2 + 2ωC 8ω2 C2

1 2

R 2

L C

β ≈ ω LC

2 1 46

G R − 2ωC 2ωL

γ = jω LC

α≈

2 1 45

Z0 =

+j

R ωL

constant, whereas the imaginary part gives the propagation constant: +

G L C R GZ0 + = 2 2Z0 2

b

a 2 1 50

The first term of the above equation (2.1.50a) shows the conductor loss of a line, while the second term shows its dielectric loss. If R and G are frequency-independent, the attenuation in a line would be frequencyindependent under ωL >> R and ωC >> G conditions. However, usually, R is frequency-dependent due to the skin effect. In some cases, G could also be frequencydependent [B.7]. The dispersive phase constant β is obtained from the imaginary part of equation (2.1.49): β ≈ ω LC

1−

RG R2 G2 + + 4ω2 LC 8ω2 L2 8ω2 C2 2 1 51

On neglecting the second-order term, β becomes a linear function of frequency and the line is dispersionless. In that case, its phase velocity is also independent of frequency. A lossy line is dispersive. However, it also becomes dispersionless under the Heaviside’s condition – (2.1.48). A transmission line, such as a microstrip in the inhomogeneous medium, can have dispersion even without losses. 2.1.6

Wave Equation with Source

In the above discussion, the development of the voltage and current wave equations has ignored the voltage or current source. However, a voltage or current source is always required to launch the voltage and current waves on a line. Therefore, it is appropriate to develop the transmission line equation with a source [B.8]. The consideration of a voltage/current source is important to solve the electromagnetic field problems of the layered medium planar lines, discussed in chapters 14 and 16. Shunt Current Source

2 1 49

On neglecting ω2, ω3, and ω4 terms, the real part of the propagation constant γ provides the attenuation

Figure (2.7) shows the lumped element model of a transmission line section of length Δx with a shunt current source Is Δx located at x = x0. It is expressed through Dirac’s delta function as Is Δxδ x − x0 . The lumped line constants R, L, C, G are given for p.u.l. The loop and node equations are written below to develop the Kelvin–Heaviside transmission line equations with a current source:

27

28

2 Waves on Transmission Lines – I

∼ I

+

jωL/2

R/2

jωL/2

R/2 ∼

∼ I + ΔI

∼ Ish

∼ V

jωC

+ ∼ ∼ V + ΔV

G

IsΔx δ(x-x0)

–

– x0 Δx

Figure 2.7

Equivalent lumped circuit of a transmission line with a shunt current source.

The Loop Equation

V − R + jωL Δx 2 I − R + jωL Δx 2 I + ΔI − V + ΔV = 0 − ΔV − R + jωL Δx I −

R + jωL ΔxΔI 2 = 0,

By taking ΔxΔI 2 ≈ 0 ΔV = − R + jωL I Δx

The Node Equation

On solving the above equations for the voltage, the following inhomogeneous voltage wave equation, with a current source, is obtained: d2 V − γ2 V = − γZ0 Is δ x − x0 dx2

2 1 54

Away from the location of the current source, i.e. for x x0, equation (2.1.54) reduces to the homogeneous equation (2.1.37a). The wave equation for the current wave, with a shunt current source, could also be rewritten. 2.1.7 Solution of Voltage and Current-Wave Equation The voltage and current wave equations in the phasor form are given in equation (2.1.37). The solution of a wave equation is written either in terms of the hyperbolic functions or in terms of the exponential functions. The first form is suitable for a line terminated in an arbitrary load. A section of the line transforms the load impedance into the input impedance at any location on the line. The impedance transformation takes place due to the standing wave formation. The hyperbolic form of the solution also provides the voltage and current distributions along the line. The exponential form of the solution demonstrates the traveling waves on a line, both in the forward and in the backward directions. A combination of the forward-moving and the backward-moving waves produces the standing wave on a transmission line.

I + Is Δxδ x − x0 − I + ΔI − Ish = 0 The Hyperbolic Form of a Solution

I + Is Δxδ x − x0 − I + ΔI − G + jωC Δx V − I R + jωL Δx 2 = 0 ΔI = − G + jωC V − I R + jωL Δx 2 + Is δ x − x0 Δx

For Δx

V x = A1 cosh γx + B1 sinh γx I x = A2 cosh γx + B2 sinh γx

0, the above equations are reduced to

dV = − R + jωL I dx dI = − G + jωC V + Is δ x − x0 dx

a b

The above equations are rewritten below in term of the characteristic impedance (Z0) and propagation constant (γ) of a transmission line: a,

dI γ = − V + Is δ x − x0 dx Z0

a b 2 1 55

2 1 52

dV = − γZ0 I dx

Figure (2.8a) shows a section of the transmission line having a length ℓ. It is fed by a voltage source, Vg with Zg internal impedance. The general solutions for the line voltage V x and line current I x of the wave equation (2.1.37) are

b 2 1 53

At any section on the line, its characteristic impedance Z0 relates the line voltage V x and line current I x . So the constants A2, B2 are related to the constants A1 and B1. In Fig (2.8a), the point P on the line is located at a distance x from the source end, i.e. at a distance d = (ℓ − x) from the load end. The load is located at d = 0, and the source is located at d = − ℓ. The Vs , and Is are the input voltage and the input current at the port-aa, i.e at x = 0. At x = ℓ, i.e. at the port-bb, VR and IR are the load end voltage and current, respectively.

2.1 Uniform Transmission Lines

Zg

∼ IS a

P

+ ∼ VS

∼ Vg

Zg

∼ VR

Z0, Y0

ZL

∼ Vg

∼ VS

a

b

X

a

d = (ℓ – X)

X

X=ℓ

(a) Finite length transmission line with a voltage source and a load termination.

(b) Input equivalent circuit at plane a-a.

|Vmax| Line voltage

ZTH

|Vmin|

+ ∼ VTh

∼ IS

b ∼ IL ZL

–

Distance from load to source

d=0 b (d) Output Thevenin equivalent circuit at plane b-b.

(c) A standing wave on the line. Figure 2.8

Zin

–

Zin

X=0

∼ IS a

+

∼ V(x)

–

∼ IR

b

Transmission line circuit. The distance x is measured from the source end; whereas the distance d is measured from the load.

The ideal voltage generator Vg has the internal impedance, Zg = 0, i.e. VS = Vg . The phasor form of the line current, from equations (2.1.32b) and (2.1.55a), is written below: I x = − G + jωC Ix = −

On comparing the coefficients of sinh(γx) and cosh (γx), of equations (2.1.55b) and (2.1.56), two constants A2 and B2 are determined: A2 = −

B1 Z0

2 1 57

The phasor line voltage and line current are written as follows: V x = A1 cosh γx + B1 sinh γx 1 A1 sinh γx + B1 cosh γx Ix = − Z0

At x = 0, the line input voltage is VS, giving the value of A1: V x = 0 = A1 = VS

2 1 59

At the receiving end, x = ℓ, the load end voltage and current are 2 1 56

A1 , Z0

•

A1 cosh γx + B1 sinh γx dx

1 A1 sinh γx + B1 cosh γx Z0

B2 = −

The constants A1 and B1 are determined by using the boundary conditions at input x = 0 and output x = ℓ.

a b 2 1 58

V x = ℓ = VR = VS cosh γℓ + B1 sinh γℓ 1 VS sinh γℓ + B1 cosh γℓ I x = ℓ = IR = − Z0

•

a b 2 1 60

At x = ℓ, i.e. at the receiving end, the voltage across load ZL is VR = ZL IR

2 1 61

The constant B1 is evaluated on substituting VR and IR, from equation (2.1.60), in the above equation: B1 = −

VS Z0 cosh γℓ + ZL sinh γℓ ZL cosh γℓ + Z0 sinh γℓ 2 1 62

29

30

2 Waves on Transmission Lines – I

On substituting constants A1 and B1 in equation (2.1.58a), the expression for the line voltage at location P, from the source or load end, is ZL cosh γ ℓ − x + Z0 sinh γ ℓ − x ZL cosh γℓ + Z0 sinh γℓ ZL cosh γd + Z0 sinh γd V d = VS ZL cosh γℓ + Z0 sinh γℓ

V x = Vg

a

V x = VS

The line voltage, in terms of Vg , Zg, and ZL, is obtained on substituting equation (2.1.66) in equation (2.1.63):

2 1 67

b

2 1 63

Likewise, from equations (2.1.64) and (2.1.66), an expression for the line current is obtained:

Similarly, the line current at the location P is obtained as follows: I x = Vg

VS ZL sinh γ ℓ − x + Z0 cosh γ ℓ − x Z0 ZL cosh γℓ + Z0 sinh γℓ VS ZL sinh γd + Z0 cosh γd Id = Z0 ZL cosh γℓ + Z0 sinh γℓ

a

Ix =

b 2 1 64

At any location P on the line, the load impedance is transformed as input impedance by the line length d = (ℓ − x): Zin x = Zin d =

Vx Ix Vd Id

= Z0

ZL + Z0 tanh γ ℓ − x Z0 + ZL tanh γ ℓ − x

a

= Z0

ZL + Z0 tanh γd Z0 + ZL tanh γd

b

ZL + jZ0 tan βd Z0 + jZL tan βd

c

for α = 0, Zin d = Z0

2 1 65 Equations (2.1.65a,b) take care of the losses in a line through the complex propagation constant, γ = α + jβ. However, for a lossless line α = 0, γ = jβ and the hyperbolic functions are replaced by the trigonometric functions shown in equation (2.1.65c). It shows the impedance transformation characteristics of d = λ/4 transmission line section. Equations (2.1.63) and (2.1.64) could be further written in terms of the generator voltage Vg for the case, Zg 0. Figure (2.8b) shows that at the source end x = 0, the load appears as the input impedance Zin. The sending end voltage is obtained as follows: V Vs = Vg − Zg I,swhere Is = Zg +gZin , and Zin x = 0 = Z0 Vs = Vg − Vg Vs = Vg

Z0 ZL cosh γ ℓ − x + Z0 sinh γ ℓ − x Z0 ZL + Z0 Zg cosh γℓ + Z20 + ZL Zg sinh γℓ

ZL + Z0 tanh γℓ . Z0 + ZL tanh γℓ

ZL sinh γ ℓ − x + Z0 cosh γ ℓ − x Z0 ZL + Z0 Zg cosh γℓ + Z20 + ZL Zg sinh γℓ

2 1 68 The above equations could be reduced to the following equations for a lossless line, i.e. for α = 0, γ = jβ, cosh(jβ) = cos β and sinh(jβ) = j sin β: V x = Vg

Z0 ZL cos β ℓ − x + jZ0 sin β ℓ − x Z0 ZL + Z0 Zg cos βℓ + j Z20 + ZL Zg sin βℓ

2 1 69 I x = Vg

Z0 cos β ℓ − x + jZL sin β ℓ − x Z0 ZL + Z0 Zg cos βℓ + j Z20 + ZL Zg sin βℓ

2 1 70 Equation (2.1.65c), for the input impedance, could be obtained from the above two equations. The sending end voltage and current are obtained at the input port – aa, x = 0: Vs = V x = 0 = Vg

Z0 ZL cos βℓ + jZ0 sin βℓ Z0 ZL + Z0 Zg cos βℓ + j Z20 + ZL Zg sin βℓ 2 1 71

Is = I x = 0 = Vg

Z0 cos βℓ + jZL sin βℓ Z0 ZL + Z0 Zg cos βℓ + j Z20 + ZL Zg sin βℓ 2 1 72

Likewise, the expressions for the voltage and current at the output port – bb, i.e. at the receiving end for x = ℓ, are obtained:

Zg Zin = Vg Zin + Zg Zin + Zg

VR = V x = ℓ

Zo ZL + tanh γℓ Z0 ZL + Zo Zg + Z20 + ZL Zg tanh γℓ Z20

2 1 66

= Vg

Z0 ZL + Z0 Zg

Z0 ZL cos βℓ + j Z20 + ZL Zg sin βℓ 2 1 73

2.1 Uniform Transmission Lines

IR = I x = ℓ = Vg

Z0 ZL + Z0 Zg

Z0 cos βℓ + j Z20 + ZL Zg sin βℓ 2 1 74

Two special cases of the load termination, i.e. the short-circuited load and the open-circuited load, are discussed below. The voltage and current distributions on a transmission line for both the cases are also obtained.

For the short-circuited load ZL = 0, the line voltage at the load end is zero V x = ℓ = 0. However, the voltage on the line is not zero. Equations (2.1.63) and (2.1.64) provide the voltage and current distributions on a shortcircuited line:

Ix =

a

Zin d = Z0 coth γd

b

for α = 0, Zin d = − jZ0 cot βd

c

2 1 78

The ℓ < λ/4 open-circuited line section behaves as a capacitive element. The electrical nature of the line section can be controlled by changing its electrical length.

Matched and Mismatched Termination

Short-Circuited Receiving End

sinh γ ℓ − x sinh γℓ

a

VS cosh γ ℓ − x Z0 sinh γℓ

b

V x = VS

Zin x = Z0 coth γ ℓ − x

2 1 75

The input impedance at any distance (ℓ − x) from the source end is Zin x = Z0 tanh γ ℓ − x

a

Zin d = Z0 tanh γd

b

for α = 0, Zin d = jZ0 tan βd

c

The input impedance at any location on a line is Zin (x) = Z0 if it is matched terminated in its characteristic impedance, i.e. ZL = Z0. Normally, the characteristics impedance of a microwave line is a real quantity. The line terminated in Z0 does not create any reflected wave on a transmission line. However, for the mismatched termination ZL Z0, there is a reflected wave on a transmission line, traveling from the load end to the source end.

2 1 76

Exponential Form of Solution

The wave nature of the line voltage and line current becomes more obvious from the exponential form of solutions of the wave equations. The solution of wave equation (2.1.37), for the phasor line voltage and line current, can also be written in the exponential form: V x = V + e − γx + V − eγx

a

At the load end, the voltage is zero. However, the line current is not infinite like the lumped element circuit with a short-circuited termination at output. A shortcircuited transmission line draws only a finite current from the source. The ℓ < λ/4 short-circuited line section behaves as an inductive element. The electrical nature of the line section can be controlled by changing its electrical length [B.9–B.15].

The distance x is measured from the source end. The time-dependent harmonic form of the voltage wave is v x, t = Re V x exp jωt . Finally, it is written as follows:

Open-Circuited Receiving End

v x, t = V + e − αx cos ωt − βx + V − eαx cos ωt + βx

The load impedance is ZL ∞ for an open-circuited transmission line and the load current I x = ℓ = 0 . Again, equations (2.1.63) and (2.1.64) provide the voltage and current distributions on an open-circuited transmission line. The voltage and current waves and the input impedance at location P from the load end can be computed for the open-circuited load as follows: cosh γ ℓ − x cosh γℓ

a

VS sinh γ ℓ − x Z0 cosh γℓ

b

V x = VS Ix =

2 1 77

Ix =

1 V + e − γx − V − eγx Z0

Forward traveling wave

b

2 1 79

Backward traveling wave 2 1 80

For an outgoing wave on a lossy line, the wave amplitude decays and its phase lags; whether the distance is measured from source end or load end. It is accounted for by the proper sign of distance x. The amplitude of a wave is exponentially decaying due to the line losses. It is expressed by the attenuation constant α (Np/m). The expression of the traveling current wave on a line could be written as follows:

31

32

2 Waves on Transmission Lines – I

i x, t =

1 V + e − αx cos ωt − βx − V − eαx cos ωt + βx Z0

Forward traveling wave

Backward traveling wave 2 1 81

If the source at x = 0 is connected to a line of infinite extent, there is no reflection from the load end, as the wave will never reach to the load end to get reflected. Therefore, for the forward traveling voltage and current waves on an infinite line, V− = 0 and the above solutions of the wave equations are written as follows: V x = V + e − γx

a

V + e − γx Z0

b

Ix =

v x, t = V + e − αx cos ωt − βx

c

V + − αx e cos ωt − βx Z0

d

i x, t =

At the load end, the total line voltage is a sum of the forward and reflected waves, and it is equal to the voltage drop VR across the load impedance ZL. It is shown in Fig (2.8a). To simplify expressions for the line voltage and current, the origin is shifted from the source end to the load end. In that case, expressions for both the line voltage and line current, given by equation (2.1.79), have to be modified for the new distance variable x < 0. Now the source and load are located at x = − ℓ and at x = 0, respectively. For the origin at the load end, the reflected wave V−e(ωt + γx) appears as the “forward-traveling wave” from load to the source, whereas the wave incident at the load V+e(ωt − γx) appears as the “backwardtraveling” wave. The voltage drop across the load is VR = ZL IR

2 1 83

Keeping in view that the origin of the distance (x = 0) is at the load, the line voltage and current at the load end are written, from equation (2.1.79), as follows: VR = V x = 0 = V + + V − 2 1 82

The input impedance of an infinite line at any location x is Zin (x) = Z0. Thus, a finite line terminated in its characteristic impedance, i.e. ZL = Z0 behaves as an infinite extent transmission line without any reflection. The characteristic impedance shows a specific feature of a line that is determined by the geometry and physical medium of the line. Once a finite extent line is terminated in any other load impedance, i.e. ZL Z0, the voltage and current waves are reflected from the load. The wave reflection is expressed by the reflection coefficient, Γ(x). The reflection coefficient is a complex quantity and its phase changes over the length of a line. However, its magnitude remains constant over the length of a lossless line. The interference of the forward-moving and backward-moving reflected waves produces the standing wave with maxima and minima of the voltage and current on a line. Figure (2.8c) shows the voltage standing wave. The position and magnitude of the voltage maxima Vmax and voltage minima Vmin are measurable quantities. Their positions are measured from the load end. The reflection coefficient, and also the voltage standing wave ratio, VSWR, i.e. S, is defined using the Vmax and Vmin. The VSWR could be measured with a VSWR meter. Therefore, the reflection on a transmission line is expressed by both the reflection coefficient and the VSWR. The reflection at the input terminal of a line is also expressed as the return-loss, RL = − 20 log10|Γin|. The wave behavior in terms of the reflection parameter is an important quantity in the design of circuits and components in the microwave and RF engineering.

IR = I x = 0 =

1 V + − V− Z0

a b

2 1 84

The voltage reflection coefficient at the load end is defined as follows: ΓL =

Amplitude of reflected voltage V− = + V Amplitude of forward incident voltage 2 1 85

The expression to compute the reflection coefficient at the load end is obtained from the above equations (2.1.83) – (2.1.85): equations: ZL 1 − ΓL Z0 ZL − Z0 ΓL = ZL + Z0

1 + ΓL =

a b

2 1 86

The mismatch of a load impedance ZL with the characteristic impedance Z0 of a line is the cause of the reflection at the load end. For the condition ZL = Z0, the matched load terminated line avoids the reflection on a line, as ΓL = 0. At any distance x, the reflection coefficient is a complex quantity with both the magnitude and phase expressed as follows: Γ x = Γ x ejϕ = Γr x + jΓi x Γx =

Γr x

2

+ Γi x

ϕ = tan − 1 Γi x Γr x

a

2

b

2 1 87

A lossless line has |Γ(x)| = |ΓL|, i.e. on a lossless line magnitude of the reflection is the same at all locations

2.1 Uniform Transmission Lines

on a line. However, the lagging phase ϕ changes with distance. It has 180 periodicity, i.e. for an inductive load, the range of phase is 0 < ϕ < π, and a capacitive load has the phase in the range −π < ϕ < 0. Using equation (2.1.79), the line voltage and line current are written in term of the load reflection coefficient: V x = V + e − γx 1 + ΓL e2γx

a

V + e − γx 1 − ΓL e2γx Z0

b

Ix =

2 1 88

For a lossless transmission line, the above equations are written as follows: V x = V + 1 + ΓL ej ϕ + 2βx V+ Ix = Z0

1 − ΓL ej ϕ + 2βx

a b

2 1 89

Vmax x = V + 1 + ΓL

a

Vmin x = V + 1 − ΓL

b

For a lossy line

Γ x = ΓL e2γx

a

for a lossless line

Γ x = ΓL ej2βx

b

Vmin x

1+ Γ x 1− Γ x

V e Z0

+ γl

1 − ΓL e − 2γℓ

a b

The amplitude factor V+ is determined by the reflections at both the source and load ends. Figure (2.8b) shows that the port voltage Vs and the line current Is , at the input port – aa, are related to the source voltage Vg and its internal impedance Zg by Is =

Vg − Vs Zg

2 1 94

On substitution of equation (2.1.93) in (2.1.94), the voltage wave amplitude V+ is obtained as follows: Zg + Z0 e + γl − Zg − Z0 e − γl ΓL = Z0 Vg 2 1 95 However, the reflection coefficient at the source end is Γg =

Zg − Z0 Zg + Z0

2 1 96

Therefore, the amplitude of the voltage wave launched by the source is Z0 e − γℓ Zg + Z0 1 − Γg ΓL e − 2γℓ

2 1 97

2 1 90

2 1 91

The measurable quantity voltage standing wave ratio (VSWR) is defined as follows: =

+

2 1 93

V + = Vg

The reflection coefficient Γ(x) at any location x from the load end is related to the reflection coefficient at the load ΓL by

Vmax x

Is x = I x = − ℓ =

V+

In the above equations, the origin is at the load end, i.e. x < 0. The maxima and minima of the voltage and current waves along the line occur due to the phase variation along the line. The voltage maximum occurs at ej(ϕ + 2βx) = + 1. In this case, both the forward and reflected waves are in-phase. The voltage minimum occurs at ej(ϕ + 2βx) = − 1. In this case, both the forward and reflected waves are out of phase. Finally, the maxima and minima of the voltage on a line are given as follows:

S = VSWR =

Vs x = V x = − ℓ = V + e + γl 1 + ΓL e − 2γℓ

2 1 92

For a lossless line, the VSWR is constant along the length of a line. Likewise, the current standing wave ratio is also defined. The wave reflection also takes place at the sending end when the source impedance Zg is not matched to the characteristic impedance of a line. The reflection coefficient at x = − ℓ, i.e. at the generator (source) end is defined as Γ(x = − ℓ) = Γg. The voltage and current at the generator end are obtained from equation (2.1.88),

Equation (2.1.88a and b) give the voltage and current waves on a transmission line with the amplitude factor V+. The amplitude factor V+ is given by equation (2.1.97).

2.1.8 Application of Thevenin’s Theorem to Transmission Line Thevenin’s theorem is a very popular concept used in the analysis of the low-frequency lumped element circuits. It is equally applicable to a transmission line network. At the output end of the line, the input source voltage Vg and the line section are replaced by the equivalent Thevenin’s voltage, VTH x = 0 with internal impedance, i.e. the Thevenin’s impedance ZTH [B.12]. Figure (2.8d) shows it. The distance is measured from the load end. Thevenin’s voltage is an open-circuit voltage at the load end. In the case of the open-circuited load, ZL ∞, equation (2.1.86) provides a reflection coefficient ΓL = 1. Thevenin’s voltage is obtained from equations (2.1.88a) and (2.1.97):

33

34

2 Waves on Transmission Lines – I

2Vg Zo e − γℓ Zg + Zo 1 − Γg e − 2γℓ

VTH x = 0 =

On replacing Γg from equation (2.1.96), Thevenin’s voltage is Vg Zo Zo cosh γℓ + Zg sinh γℓ

VTH x = 0 =

Hω =

2 1 98

e − γℓ 2

2 1 105

However, if the transfer function is defined by the ratio of the input voltage Vs at the port – aa to the output voltage VR , H(ω) = e−γℓ. It is obtained from equation (2.1.104) for Zg = 0.

2 1 99 Thevenin’s impedance ZTH is obtained from equation (2.1.88b) by computing Norton current, i.e. the short-circuit current at x = 0. Under the short-circuited load condition at x = 0, ΓL = − 1, and the Norton current is 2Vg e − γℓ Zg + Zo 1 + Γg e − 2γℓ

IN x = 0 =

2 1 100

Thevenin’s impedance is obtained as follows: ZTH =

VTH IN

,

ZTH = Z0

1 + Γg e − 2γℓ 1 − Γg e − 2γℓ

2.1.9

Power Relation on Transmission Line

The average power over a time-period T in any timeharmonic periodic signal is [J.5, B.10] Pav =

1 T

T

v t i t dt

2 1 106

0

where the time-harmonic instantaneous voltage and current waveforms are v t = V0 cos ωt + ϕ

a

i t = I0 cos ωt + ϕ

b 2 1 107

2 1 101

The voltage and current in the phasor form are written as follows: V = V0 ejϕ

a,

I = I0 ejϕ

b

Transfer Function

2 1 108

The transmission line section could be treated as a circuit element. Its transfer function is obtained either with respect to the source voltage Vg or with respect to the input voltage Vs at the port- aa, as shown in Fig (2.8a). The load current is obtained from Fig (2.8d): IL =

VTH ZTH + ZL

2 1 102

A complex number X = a + jb has its complex conjugate, X∗ = a − jb. Thus, the real (Re) and imaginary (Im) parts of a complex number are written as follows: 1 X + X∗ 2 1 Im X = X − X∗ 2

Re X =

a b 2 1 109

The voltage across the load is VR = ZL IL =

2Z0 ZL Vg ZTH + ZL Zg + Z0

− γℓ

e 1 − Γg e − 2γℓ 2 1 103

The transfer function of a transmission line with respect to the source voltage Vg is Hω =

VR Vg

=

ZTH

2Z0 ZL + ZL Zg + Z0

e − γℓ 1 − Γg e − 2γℓ 2 1 104

For a lossless transmission line connected to a matched source and a matched load, i.e.γℓ = jβℓ, Zg = Z0, ZTH = Z0, ZL = Z0, Γg = 0 the transfer function is

On using the above property, the instantaneous voltage and current are written as follows: 1 ∗ Vejωt + V e − jωt a v t = 2 1 ∗ Iejωt + I e − jωt it = b 2 2 1 110 The average power in phasor form is obtained from equations (2.1.106) and (2.1.110), 1 ∗ Re VI 2 It can be expressed in the usual AC form, Pav =

Pav =

2 1 111

1 1 Re V0 ejϕ I∗0 e − jϕ = V0 I∗0 cos ϕ − ϕ 2 2 2 1 112

2.1 Uniform Transmission Lines

The average power on the line is

Available Power from Generator

Figure (2.9a) shows that the maximum available power from a source is computed by directly connecting the load to it. The average power supplied to the load is Pav =

1 1 1 Re VL I∗ = Re I 2 ZL = I 2 RL 2 2 2 2 1 113

The load current is I=

Vg Vg = Zg + ZL Rg + RL + j Xg + XL Vg

2

I =

Rg + RL

2

2

Pav

1 ∗ = Re V x I x 2

On a lossless line, the average power is independent of the distance x from a source. Physically it makes a sense, as the same amount of power flows at any location on the line. Under the matched load termination, ZL = Z0, the input impedance at the source end is Z0 itself. It is shown in Fig (2.9b). The sending end voltage at the input port – aa of a transmission line is VS =

Therefore, the average power supplied to the load is Pav

1 = 2

2

Vg RL Rg + RL

2

+ Xg + XL

2 1 114

2

Under the conjugate matching, XL = −Xg and RL = Rg, the average power supplied to load is maximum: Pav =

1 Vg 8 Rg

2

2 1 115

This is the maximum power available from a generator under the matching condition and delivered to a load RL. At this stage, the maximum power delivered to a load is computed in the absence of the transmission line. For a matched terminated lossless line, the maximum available power from the source is delivered to the load. It is examined below. The voltage and current waves on a line under no reflection case are V x = VS e − jβx

a,

Ix =

VS e − jβx Z0

b 2 1 116

∼

Zg

I

V x = V − e + jβx

a,

Ix = −

Pav =

1 ∗ Re V x I x 2

= −

∼

Zg

ZL

– b (a) Load directly connected to a source.

V − e + jβx b Z0 2 1 119

V− 2 2Z0

2 1 120

However, at the load end amplitude of the reflected voltage wave is V− = ΓLV+; where V + = Vs. Therefore, the average reflected power on the line is

I a

+ VL

Figure 2.9 Load connections to a source.

2 1 118

The average power in the reflected wave is

b

∼

Vg

V g Z0 Zg + Z0

The maximum power is transferred from a generator to the transmission line under the matching condition, Rg = Z0. The maximum available power from the generator to feed the line is given by equation (2.1.115). It is identical to the power determined from equation (2.1.117), as Vs = Vg 2. If the line is not terminated in its characteristic impedance, then a reflection takes place at the load end. The reflected wave travels from the load toward the generator given by

+ ∼

2

2 1 117

2

+ Xg + XL

∗

VS V 1 = Re VS e − jβx S ejβx = 2 Z0 2Z0

∼

∼

Vg

VS

Zin = Z0

– a (b) Line terminated in a matched load.

35

36

2 Waves on Transmission Lines – I

Pav = −

2

2

ΓL 2 V + 2Z0

= −

ΓL 2 V s 2Z0

by a source. The output power Pout is the power supplied to the load.

2 1 121

The negative sign (−) shows that the reflected power travels from the load toward the source. Finally, under the mismatched load, the power delivered to the load is obtained from equations (2.1.117) and (2.1.121)

2.2 Multisection Transmission Lines and Source Excitation This section extends the solution of the voltage wave equation to the multisection transmission line [B.8, B.16]. Next, the voltage responses are obtained for the shunt connected current source, and also the seriesconnected voltage source, at any location on a line. This treatment is used in chapters 14 and 16 for the spectral domain analysis of the multilayer planar transmission lines.

2

V PL = S 1 − ΓL 2Z0

2

2 1 122

For a lossless line, the power balance is written as follows: Pin =

1 Vs 2Z0

2

1 − ΓL

2

= Pout 2 1 123

Incident power

2.2.1

Reflected power

Figure (2.10a) shows a multisection transmission line, consisting of the N number of line sections. Each line section has different lengths (d1, d2,…,dN), different

In equation (2.1.123), input power Pin at the input port – aa of the line enters into the line. It is supplied

Line section

#1 Z01 =

∼ Vg

#2

1 Y01

β1

x=0 Location of junction Load at junction

X0

Voltage at junction

VS

Z03 =

β2

d1

#n 1 Y03

Z0n =

β3 d2

1 Y0n

Z0N =

X3

XN–1

Ynin

XN YN 1O = YL

V1

V1

V3

Vn–1

Vn

VN–1

VN

Γ1

Γ1

Γ3

Γn–1

Γn

ΓN–1

ΓN

(a) Analysis of multi-section line structure. Input

Output

VS

Yʹin =

Γ1

Z01, β1

d1 X=0

X0

X1 (b) Isolated first line section.

Figure 2.10

ZL =

dN Xn

Xn–1

Y3in

1 Y0N

βN dn

X2 Y2in

N

βn d3

X1 Y1in

Reflection coeff. at junction

#3

1 Y02

Z02 =

Multisection Transmission Lines

The multisection transmission line.

1 Zʹin

1 YL

2.2 Multisection Transmission Lines and Source Excitation

characteristic impedances (Z01, Z02,…,Z0N), or different characteristic admittances (Y01, Y02,…,Y0N) and different propagation constants (β1, β2, …, βN). At each junction of two dissimilar lines, the voltage wave reflection occurs with the reflection coefficient Γ1, Γ2, …, ΓN. At each junction, the input admittance of all succeeding line sections appears as a load. The input admittances at junctions (x1, x2, …, xN) are Y1in , Y2in , …, YN in . The distances of the junctions are measured from the origin that is located on the left-hand side. The input terminals of line sections 1,2,…, N − 1 are located at the junctions (x0, x1, x2,…,xN). The voltage source Vs is located at the input of the first line section. The last line section is terminated in the load ZL = 1/YL. The objective is to find the voltage at each junction of the multisection line. Further, the voltage distribution on each line section is determined due to the input voltage Vs . The solutions for the voltage and current wave equations involve four constants. The constants of the current wave are related to two constants of the voltage wave through the characteristic impedance of a line. Out of two constants of the voltage wave, one is expressed in terms of the reflection coefficient at the load end; that itself is expressed by the characteristic impedance and the terminated load impedance. The reflection coefficient can also be expressed by the characteristic admittance and the terminated load admittance. The second constant is evaluated by the source condition at the input end. Figure (2.10b) shows the first isolated line section. The voltage and current waves, with respect to the origin at the load end x1, on the line section (x0 ≤ x ≤ x1) are written from equation (2.1.88):

V x0 = VS = V + e − jβ1 V+ =

e − jβ1

x0 − x1

x0 − x 1

VS + Γ1 ejβ1

+ Γ1 ejβ1

x 0 − x1

223

x 0 − x1

The voltage wave on the transmission line section #1 is

V1 x =

VS e − jβ1 x − x1 + Γ1 ejβ1 x − x1 e − jβ1 x0 − x1 + Γ1 ejβ1 x0 − x1

224

The above expression is valid over the range x0 ≤ x ≤ x1. The voltage at the output of the line section #1 (x = x1), that is at the junction of line #1 and line #2, is V1 x 1 =

e − jβ1

VS x0 − x 1

1 + Γ1 + Γ1 ejβ1

x0 − x1

=

VS 1 + Γ1 , e + jβ1 d1 + Γ1 e − jβ1 d1

225 where d1 = x1 − x0 is the length of the line section #1. The above voltage is input to the line section #2. Equations (2.2.4) and (2.2.5) apply to any line section and at any junction. The voltage Vn − 1 xn − 1 is treated as the input voltage of the nth line section. It is the same as the output voltage of the (n − 1)th line section. The line length d1 and reflection coefficient Γ1 are replaced by dn and Γn, respectively. The cascaded line sections to the right-hand side of any junction act as a load at the junction and the reflection coefficient at the junction is At load end of the line section#1 Γ1 =

Y01 − Y1in Y01 + Y1in

a

At load end of the line section#n V x =V

+ +

e

− jβ1 x − x1

V e − jβ1 Ix = Z01

x − x1

+ Γ1 e − Γ1 e

jβ1 x − x1

j x − x1

, x0 ≤ x ≤ x1

a b

The reflection coefficient Γ1 at the load end, i.e. at x = x1 is given by Z1in − Z01 Y01 − Y1in = 1 Zin + Z01 Y01 + Y1in

Y0n − Ynin Y0n + Ynin

b

Length of the line section#n 221

Γ1 =

Γn =

222

The load at the x = x1 end is formed by the cascaded line sections after location x = x1. The voltage amplitude V+ is evaluated by the boundary condition at the input, x = x0, of the first line section. At x = x0, shown in Fig (2.10b), the source voltage V x0 is Vs and V+ is evaluated as follows:

dn = x n − x n − 1 ,

n = 1, 2, …N

c 226

Equation (2.2.4) is applied to Fig (2.10a) to compute the voltage distribution on any line section. The voltage on line section #2 is V2 x =

V1 e − jβ2 x − x2 + Γ2 ejβ2 x − x2 , x1 ≤ x ≤ x2 e − jβ2 x1 − x2 + Γ2 ejβ2 x1 − x2 227

The voltage at the output of the line section #2, i.e. the junction voltage of the line sections #2 and # 3 at x = x2, is obtained from the above equation:

37

38

2 Waves on Transmission Lines – I

V 1 1 + Γ2 + jβ e 2 d2 + Γ2 e − jβ2 d2

V2 x 2 =

N

228

n=1

Using equation (2.2.5) and above equations, the voltage distribution on the line section #2, and also the junction voltage at x = x2, are obtained: V2 x =

VS e + jβ1 d1

1 + Γ1 + Γ1 e − jβ1 d1

e − jβ2 x − x2 + Γ2 ejβ2 x − x2 e + jβ2 d2 + Γ2 e − jβ2 d2 229

V2 x2 =

VS + jβ e 1 d1

1 + Γ1 + Γ1 e − jβ1 d1

Vn xn =

1 + Γ2 + Γ2 e − jβ2 d2

e + jβ2 d2

2.2.2

VS 1 + Γn ejdn βn + Γn e − jdn βn

2 2 12

Location of Sources

The shunt voltage Vs could be located at any junction and the voltage distribution is computed on any line section due to it. However, it is also interesting to consider a shunt current source and a series voltage source located anywhere on a multisection transmission line. Both kinds of sources create the voltage wave on a line.

2 2 10 th

Finally, the voltage distribution on the n line section and the voltage at the nth line junction can be written as follows: N

Vn x = n=1

V S 1 + Γn − 1 ejβn − 1 dn − 1 + Γn − 1 e − jβn − 1 dn − 1

e − jβn x − xn + Γn ejβn x − xn e + jβn dn + Γn e − jβn dn

2 2 11

Line #1

Current Source at the Junction of Finite Length Line and Infinite Length Line

Figure (2.11a) shows a transmission line circuit with a current source IS located at x = 0 that is the junction of two lines of different electrical characteristics. The open-circuited line #1, with length x = −d1, is located at the left-hand side of the current source. Its characteristics impedance/admittance is (Z01/Y01) and its propagation constant is β1. The infinite length line #2, with

β1,Y01 (Z01)

β2,Y02 (Z02)

β1,Y01 (Z01)

∼ IS

Open

Line #2

Line #1

Line #2

∼ IS

Open

Y– x = –d1

x

x=0

β2,Y02 (Z02)

∞

Y+

x=0

x = –d1

(b) Equivalent circuit.

(a) Transmission line circuit with a shunt current source. ∼ IS

+ Yin = Y+ + Y–

∼ VS –

(c) Replacement of shunt current source by the equivalent voltage source. Figure 2.11

A shunt current source at the junction of two-line sections.

YL = Y02

x = +d2

2.2 Multisection Transmission Lines and Source Excitation

characteristics impedance/admittance (Z02/Y02) and the propagation constant β2, is located at the right-hand side of the current source. It can be replaced by a load admittance YL = Y02 at a distance x = d2, shown in Fig (2.11b). The objective is to find out the voltage waves on both the lines as excited by the current source. The current source IS can be replaced by an equivalent voltage source Vs, shown in Fig (2.11c), at x = 0: VS =

IS IS = − , Yin Y + Y+

2 2 13

where Yin is the total load admittance at the plane containing the current source IS. Y− and Y+ are left-hand and right-hand side admittances at x = 0 given by = jY01 tan β1 d1 For a losslessline , Y + = Y02 2 2 14 The general solution of a voltage wave is given by equation (2.1.79a). The constants V+ and V− are evaluated for the left-hand side of a lossless transmission line. At x = 0, V(x = 0) = Vs. On using this boundary condition in equation (2.1.79a): VS = V+ + V−. At x = −d1 the line is open-circuited with I (x = −d1) = 0. On using this boundary condition in equation (2.1.79b): V + ejβ1 d1 − V − e − jβ1 d1 = 0 . The constants V+ and V−, from these two equations, are obtained as V+

V x = VS e − jβ2 x ,

x≥0

The method can be easily extended to a multisection line structure. For this purpose, the left-hand and right-hand side admittances Y− and Y+ are determined at the plane containing the current source.

a,

V − = V + ej2β1 d1

b

Figure (2.12a) shows the series-connected voltage source VS at x = 0. The location x = 0 is a junction of two transmission lines – line #1 open-circuited finite-length line and line #2 infinite length line. The lines at the left-hand and right-hand sides of the voltage source can be replaced by the equivalent impedances Z− and Z+, respectively. It is shown in the equivalent circuit, Fig (2.12b). Again, the voltage waves on both lines, excited by a series voltage source, could be determined. The voltages across loads Z− (Z1) and Z+ (Z2), shown in Fig (2.12b), are obtained as follows: Line current

I=

VS Z− + Z +

The voltage wave on the left-hand line #1 is obtained by substituting equation (2.2.15) in equation (2.1.79a): VS cos β1 x + d1 , cos β1 d1

− d1 ≤ x ≤ 0 2 2 16 ∼ VS

Line #1

Y01 = Open

β1

1 Z01

β2

b

Voltage across Z2

V x = 0+

c 2 2 18

Line #1 is open-circuited and line #2 is of infinite extent. Therefore, their input impedances at x = 0− and x = 0+ are ∼ VS

1 Z02

∼ I +

–

Z2

Z1

–

+ Z– x = –d1

Z+ x=0

V S Z1 Z1 + Z2 V S Z2 = Z1 + Z2

V x = 0− = −

Line #2

Y02 =

a

Voltage across Z1

2 2 15

Vx =

2 2 17

Series Voltage Source

Y − = Y01 tanh γ1 d1

VS = 1 + ej2β1 d1

The line at the right-hand side of the current source is an infinite length line that supports a traveling wave without any reflection. Therefore, at x = 0, V− = 0 and V+ = VS. The voltage wave on line #2 at the right-hand side is

x=∞

(a) Transmission line circuit with a series voltage source. Figure 2.12 A series voltage source at the junction of two-line sections.

x=0 (b) Equivalent circuit.

39

40

2 Waves on Transmission Lines – I

Z1 = − jZ01 cot β1 d1

a,

Z2 = Z02

b 2 2 19

+

The voltage at x = 0 from equations (2.2.18c) and (2.2.19) is V x = 0 + = VS

= VS

1 × Y02

1 1 1 + jY01 tan β1 d1 Y02

jY01 tan β1 d1 Y02 + jY01 tan β1 d1

2 2 20

For a lossy transmission line, the above equation could be written as follows: V x = 0 + = VS

Y01 tanh γ1 d1 Y02 + Y01 tanh γ1 d1

2 1 21

The voltage wave on the infinite extent lossy line #2 is V x = V x = 0 + e − γ2 x = VS

Y01 tanh γ1 d1 e − γ2 x , Y02 + Y01 tanh γ1 d1

x≥0 2 2 22

The voltage at x = 0− on the lossless line #1 is V x = 0 − = − VS

Y02

Y02 + jY01 tan β1 d1 2 2 23

However, the voltage at x = 0− on a lossy line #1 is V x = 0 − = − VS

Y02 Y02 + Y01 tanh γ1 d1 2 2 24

The voltage wave on the open-circuited lossless line #1 is obtained from equations (2.2.16) and (2.2.23): V x = 0− cos β1 x + d1 cos β1 d1 Y02 V x = − VS Y02 + jY01 tan β1 d1

Vx =

×

2.3

cos β1 x + d1 , cos β1 d1

− d1 ≤ x ≤ 0

permittivity and relative permeability also do not change along the line. For such a uniform lossless or low-loss transmission line, the voltage and current waves travel at a definite velocity from low frequency to high frequency. The uniform transmission line behaves as a low-pass filter (LPF) section. The propagation constant for such a uniform transmission line is the same at any section of the line. The line also has a unique characteristic impedance that is independent of the location on a line. However, if the geometry of a transmission line or an electrical property of the medium of a line changes in the direction of propagation, such a line is no longer a uniform transmission line. It is a nonuniform transmission line. Its electrical properties, such as the RLCG primary constants, propagation constant, phase velocity, and characteristic impedance, become a function of the space coordinate along the direction of propagation. The characteristic impedance of a nonuniform transmission line changes from one end to another end; therefore, it finds application in the broadband impedance matching [B.9, B.10, B.12]. It is also used for the design of delay equalizers, filters, wave-shaping circuits, etc. [J.6–J.9]. It is an essential section of the on-chip measurement system [J.10]. Unlike a uniform transmission line, it shows a cutoff phenomenon, i.e. the wave propagates on the line only above the cut-off frequency. Below the cut-off frequency, the wave only attenuates with distance. Thus, the nonuniform transmission line behaves like a high-pass filter (HPF) section [J.11–J.13, B.17]. The present section obtains the wave equations for a nonuniform transmission line. However, like the uniform transmission lines, the nonuniform transmission lines do not have closed-form solutions for the voltage and current waves. The numerical methods have been used to determine the response of an arbitrarily shaped nonuniform transmission line [J.10, J.11]. However, this section discusses only the exponential nonuniform transmission line to understand its characteristics.

2 2 25

Nonuniform Transmission Lines

The previous sections have presented the voltage and current waves on the uniform transmission line that has no change in the geometry along the direction of propagation. For a uniform line, the relative

2.3.1 Wave Equation for Nonuniform Transmission Line Figure (2.13) shows a nonuniform transmission line. The line parameters (primary line constants) R(x), L(x), C(x), G(x) are distance-dependent. It results in the distance-dependent characteristic impedance, Z0(x), and propagation constant, γ(x). Using equation (2.1.20), the voltage and current equations for a nonuniform transmission line are written as follows:

2.3 Nonuniform Transmission Lines

∂2 i ∂2 i = L x C x + ∂x2 ∂t2

1 ∂C x C x ∂x

∂i ∂x 236

x

If L(x) and C(x) are not a function of x, then equations (2.3.5) and (2.3.6) reduce to the familiar wave equations (2.1.24) and (2.1.25) on a uniform transmission line. For a lossy nonuniform transmission line, it is not possible to get separate voltage and current wave equations in the time domain. However, separate voltage and current wave equations can be obtained in the frequency domain by using the phasor form of voltage and current. The transmission line equations in the phasor form are

x=ℓ

x=0 Figure 2.13 Nonuniform transmission line.

∂v ∂i = − R x i+L x ∂x ∂t

a

∂i ∂v = − G x v+C x ∂x ∂t

b

231

The following expressions are obtained on differentiating equation (2.3.1a) with x and equation (2.3.1b) with t: ∂2 v ∂i ∂R x ∂2 i ∂i ∂L x + i + L x + = − R x ∂x2 ∂x ∂x ∂x∂t ∂t ∂x 232 ∂2 i ∂v ∂2 v = − Gx +C x 2 ∂t∂x ∂t ∂t

233

On substituting equations (2.3.1b) and (2.3.3) in equation (2.3.2): −

∂2 v = −R x ∂x2

−L x

Gx

G x v+C x

∂v ∂2 v +C x 2 ∂t ∂t

+

∂v ∂t

+i

∂R x ∂x

∂i ∂L x ∂t ∂x 234

This equation has both the voltage and current variables v(x, t) and i(x, t). However, most of the transmission lines are low-loss lines. Thus, using R(x) 0 G (x) 0 in equations (2.3.1a) and (2.3.4), the following voltage wave equation is obtained for a lossless nonuniform transmission line:

dV x = −Z x I x dx dI x = −Y x V x dx

a 237 b,

where the line series impedance and shunt admittance p.u.l. are given by Z x = R x + jωL x

a

Y x = G x + jωC x

b

238

The following wave equations for the nonuniform transmission line are obtained: d2 V x 1 dZ x − Z x dx dx2

dV x −Z x Y x V x = 0 dx 239

2

dIx 1 dY x 2 − Y x dx dx

dI x −Z x Y x I x = 0 dx 2 3 10

If Z(x) and Y(x) are not a function of x, the above wave equations reduce to wave equation (2.1.37a and b) for a uniform transmission line. For a lossless nonuniform line, the series impedance and shunt admittance per unit length are Z(x) = jωL(x), Y(x) = jωC(x). The voltage wave equation (2.3.9) could be written as d2 V x 1 dL x − 2 L x dx dx

dV x + dx

ω vp x

2

V x = 0, 2 3 11

2

2

∂ v ∂ v =L x C x 2 + ∂x2 ∂t

1 ∂L x L x ∂x

∂v ∂x

where position-dependent nominal phase velocity of a nonuniform transmission line is given by 235

Likewise, the current wave equation is obtained as,

vp =

1 Lx Cx

2 3 12

41

42

2 Waves on Transmission Lines – I

It is difficult to get a general solution for the above wave equations. However, under the case of no reflection on a line, and the line with a small fractional change in L(x) and C(x) over a wavelength, Lewis and Wells, and Wohler [B.17, J.11] have given the following solution of wave equation (2.3.11): Z0 x V x =V 0 Z0 0

1 2

exp − jω

x

dx v 0 p x 2 3 13

In this expression Z0(x) is the nominal characteristic impedance at any location x on the nonuniform transmission line, whereas characteristic impedance Z0(0) is the nominal characteristic impedance at x = 0. For a uniform line, the phase velocity vp(x) is constant and dL x = 0. The wave equation (2.3.11) Z0 x = Z0 0 , dx is reduced to the wave equation of a uniform transmission line. The solution (2.3.13) is also reduced to the standard solution, V x = V 0 e − jβx . Equation (2.3.13) shows that for increasing characteristic impedance Z0(x) along the line length, the voltage amplitude also increases as the square root of nominal characteristic impedance. Lewis and Wells [B.17] have also given an expression for the reflection coefficient of the nonuniform transmission line terminated in the load ZL at x = ℓ: 1 vp 1− j2ω Z0 Γx = 1 vp 1+ j2ω Z0

x=ℓ x=ℓ x=ℓ x=ℓ

dZ0 x dx dZ0 x dx

Lossless Exponential Transmission Line

The general solution of the wave equation for a nonuniform transmission line is not available. However, the closed-form solution is obtained for an exponential transmission line [J.11, J.13]. This case demonstrates the properties of a nonuniform line. The following exponential variation is assumed for the line inductance and capacitance of a nonuniform transmission line: L x = L0 e2px , C x = C0 e − 2px ,

x=ℓ

x=ℓ

ZL − Z0 x = ℓ ZL + Z0 x = ℓ

For a uniform transmission line Z0(x = ℓ) = Z0, and equation (2.3.14) is reduced to the nominal reflection coefficient, ZL − Z0 x = ℓ ZL − Z0 = ZL + Z0 ZL + Z0 x = ℓ

2 3 15

At higher operating frequency ω, the reflection coefficient for any termination, given by equation (2.3.14), is also reduced to equation (2.3.15). However, reflection occurs at a lower frequency ω on a nonuniform transmission line, even if the nominal reflection coefficient Γnom(x = ℓ) zero, i.e. even if the line is matched at the load end. This behavior is different from that of a uniform transmission line.

2 3 16

where L0 and C0 are primary line constants at x = 0 and p is a parameter controlling the propagation characteristics. The above choice of line inductance and capacitance maintains a constant phase velocity that is independent of the location along the line length. The characteristic impedance of a lossless exponential transmission line changes exponentially along the line length. Its propagation constant is also frequency-dependent. Therefore, a lossless nonuniform line is dispersive. The nominal characteristic impedance at any location x on the line is Z0 x =

Lx = Cx

L0 2px e = Z0 x = 0 e2px C0 2 3 17

The parameter p, defined below, could be determined from the characteristic impedance at the input and output ends of the line: p=

2 3 14

Γnom x =

2.3.2

1 Z0 ℓ log e 2ℓ Z0 x = 0

2 3 18

If the impedances at both ends of a line are fixed, changing the line length, ℓ, can change the parameter p. The parameter p also determines the propagation characteristics of a nonuniform transmission line. The series impedance and shunt admittance p.u.l. of the exponential line can be written as follows: Z x = jωL0 e2px

a,

Y x = jωC0 e − 2px

b 2 3 19

In case of an exponential line, the voltage and current wave equations (2.3.9) and (2.3.10) reduce to d2 V x dV + ω2 L0 C0 V x = 0 − 2p 2 dx dx

2 3 20

d2 I x dI + ω2 L0 C0 I x = 0 + 2p dx dx2

2 3 21

Let us assume the following exponential form of the solution for the above wave equations with separate propagation constants for the voltage and current waves:

References

V x = V0 e − γ1 x

a,

I x = I0 e − γ2 x

b 2 3 22

The above differential equations provide the following characteristic equations: γ21 + 2pγ1 + ω2 L0 C0 = 0

a

γ22 − 2pγ2 + ω2 L0 C0 = 0

b 2 3 23

On solving the above equations, the following expressions are obtained for the complex propagation constants: γ1 = α1 ± jβ1 = − p ± γ2 = α2 ± jβ2 = p ±

p2 − ω2 L0 C0 p2 − ω2 L0 C0

a b 2 3 24

In the case of a uniform transmission line (p = 0), the propagation constants for the voltage and current waves are identical. The parameter p determines the attenuation constant, i.e. α of a nonuniform line. It is positive for the condition Z0(x = ℓ) > Z0(x = 0). Thus, there is an attenuation factor even for a lossless nonuniform line. The factor under the radical sign provides the propagation constant, i.e. the phase-shift constant β. At the cut-off frequency, ω = ωc, β is zero. The cut-off frequency is given by ωc =

p , L0 C 0

Px =

1 1 ∗ V x I x = V0 I0 e − 2 2

γ1 +γ∗2 x

,

2 3 28

where γ1 + γ∗2 = 0. Unlike a lossless uniform transmission line, the phase velocity of the voltage and current waves on a lossless nonuniform transmission line is dispersive as given below: vp =

ω = β

ω p

ω ωc

2 3 29

2

−1

The phase velocity shows singularity at the cut-off frequency. After the cut-off frequency, i.e. for ω > > ωc, it decreases, with an increase in frequency, to a value given by expression (2.3.26).

2 3 25

References

where phase velocity of the voltage and current waves on the line at x = 0 is Vp x = 0 =

However, real parts of the complex propagation constants γ1 and γ2 are nonzero. For p > 0, the voltage wave gets attenuated while the current wave is increased in the positive direction of wave propagation. In the backward direction, the reflected voltage and current waves have opposite behavior. The attenuation in the signal is not due to any ohmic loss of a line. It is due to the continuous reflection of the wave as it progresses on the line. The opposite behavior of the voltage and current waves maintains the constant flow of power (P) at any location on a line:

1 L0 C0

2 3 26

The complex propagation constants can be rewritten as follows: ω ωc

γ1 = α1 ± jβ1 = − p ± jp γ2 = α2 ± jβ2 = p ± jp

ω ωc

Books B.1 Nahin Paul, J.: Oliver Heaviside, Sage in Solitude: The

B.2

2

−1

a

B.3

2

−1

b 2 3 27

The propagation constants β1 and β2 are imaginary quantities for the signal below the cut-off frequency ω < ωc. Under such conditions, no wave propagates on the nonuniform line. The initial signal only gets attenuated. It is called the evanescent mode. The wave propagation takes place only for ω > ωc. Therefore, a nonuniform transmission line behaves like a high-pass filter (HPF).

B.4 B.5 B.6 B.7

B.8

Life, Work, and Times of an Electrical Genius of the Victorian Age, IEEE Press, New York, 1988. MacCluer, C.R.: Boundary Value Problems and Fourier Expansions, Dover Publications, Mineola, NY, 2004. Sears, F.W.; Zemansky, M.W.: University Physics, Addition-Wesley, Boston, MA, 1973. Huygens, C.: Treatise on Light, Macmillan, London, 1912. Karakash, J.J.: Transmission Lines and Filter Networks, Macmillan, New York, 1950. Johnson, W.C. Transmission Lines and Networks, McGraw-Hill, Inc., New York, 1950. Mattick, R.E.: Transmission Lines for Digital and Communication Networks, IEEE Press, New York, 1995. Weeks, W.L.: Electromagnetic Theory for Engineering Applications, John Wiley & Sons, New York, 1964.

43

44

2 Waves on Transmission Lines – I

B.9 Rizzi, P.A.: Microwave Engineering- Passive Circuits,

B.10 B.11

B.12 B.13 B.14 B.15 B.16

B.17

Prentice-Hall International Edition, Englewood Cliff, NJ, 1988. Pozar, D.M.: Microwave Engineering, 2nd Edition, John Wiley & Sons, Singapore, 1999. Ramo, S.; Whinnery, J.R.; Van Duzer, T.: Fields, and Waves in Communication Electronics, 3rd Edition, John Wiley & Sons, Singapore, 1994. Collin, R.E.: Foundations for Microwave Engineering, 2nd Edition, McGraw-Hill, Inc., New York, 1992. Rao, N.N.: Elements of Engineering Electromagnetics, 3rd Edition, Prentice-Hall, Englewood Cliff, NJ, 1991. Sadiku, M.N.O.: Elements of Electromagnetics, 3rd Edition, Oxford University Press, New York, 2001. Cheng, D.K.: Fields and Wave Electromagnetics, 2nd Edition, Pearson Education, Singapore, 1089. Bhattacharyya, A.K.: Electromagnetic Fields in Multilayered Structures, Artech House, Norwood, MA, 1994. Lewis, I.A.D.; Wells, F.H.: Millimicrosecond Pulse Techniques, 2nd Edition, Pergamon Press, London, 1939.

J.5

J.6

J.7

J.8

J.9

J.10

Journals J.1 Searle, G.F.C., et al. The Heaviside Centenary Volume,

The Institution of Electrical Engineers, London, 1950. J.2 Whittaker, E.T.: Oliver Heaviside, In Electromagnetic Theory Vol. 1, Oliver Heaviside, Reprint, Chelsea Publishing Company, New York, 1971. J.3 Verma, A.K.; Nasimuddin: Quasistatic RLCG parameters of lossy microstrip line for CAD application, Microwave Opt. Tech. Lett., Vol. 28, No. 3, pp. 209–212, Feb. 2001 J.4 Hasegawa, H.; Furukawa, M.; Yanai, H.: Properties of microstrip line on Si-SiO2 system, IEEE Trans.

J.11

J.12

J.13

Microwave Theory Tech., Vol. MTT-19, pp. 869–881, 1971. Kurokawa, K.: Power waves and the scattering matrix, IEEE Trans. Microwave Theory Tech., Vol. 13, No. 2, pp. 607–610, 1965. Tang, C.C.H.: Delay equalization by tapered cutoff waveguides, IEEE Trans. Microwave Theory Tech., Vol. 12, No. 6, pp. 608–615, Nov. 1964. Roberts, P.P.; Town, G.E.: Design of microwave filters by inverse scattering, IEEE Trans. Microwave Theory Tech., Vol. 7, pp. 39–743, April 1995. Burkhart, S.C.; Wilcox R.B.: Arbitrary pulse shape synthesis via nonuniform transmission lines, IEEE Trans. Microwave Theory Tech., Vol. 38, No. 10, pp.1514–1518, Oct. 1990. Hayden, L.A.; Tripathi, V.K.: Nonuniform coupled microstrip transversal filters for analog signal processing, IEEE Trans. Microwave Theory Tech., Vol. 39, No. 1, pp. 47–53, Jan. 1991. Young, P.R.; McPherson, D.S.; Chrisostomidis, C.; Elgaid, K.; Thayne, I.G.; Lucyszyn, S.; Robertson I.D.: Accurate non-uniform transmission line model and its application to the de-embedding of on-wafer measurements, IEEE Proc. Microwave Antennas Propag., Vol. 148, No. 1, pp. 153–156, June 2001. Wohlers, M.R.: Approximate analysis of lossless tapered transmission lines with arbitrary terminations, Proc. IRE, Vol. 52, No. 11, 1365, Dec. 1964. Khalaj-Amirhosseini, M.: Analysis of periodic and aperiodic coupled nonuniform transmission lines using the Fourier series expansion, Prog. Electromagn. Res., PIER, Vol. 65, pp. 15–26, 2006. Ghose, R.N.: Exponential transmission lines as resonators and transformers, IRE Trans. Microwave Theory Tech., Vol. 5, No. 3, pp. 213–217, July 1957.

45

3 Waves on Transmission Lines – II (Network Parameters, Wave Velocities, Loaded Lines)

Introduction The transmission line sections are used to develop various passive components. These are characterized by several kinds of matrix parameters. This chapter discusses the matrix parameters and their conversion among themselves. It also discusses various kinds of dispersion and wave propagation encountered on transmission lines. The transmission lines could be loaded by the reactive elements and resonating circuits to modify the nature of the wave propagation on the lines. Such loaded lines are important in modern planar microwave technology. Such loaded lines are introduced in this chapter. The primary purpose of this chapter is to review in detail the matrix description of lines and wave propagations on the dispersive transmission line that supports various kinds of wave phenomena.

Objectives

• • • • • •

To review the matrix representations of the two-port networks using the [Z], [Y], and [ABCD] parameters. To discuss the basic properties and use of the scattering [S] parameters. To understand the process of de-embedding of true [S] parameters of a device. To understand the process of extraction of the propagation constant from the [S] parameters. To understand the phase and group velocities in a dispersive medium. To discuss the circuit modeling of the reactively loaded line supporting both the forward and backward waves.

3.1 Matrix Description of Microwave Network At low frequency, the circuit is described in terms of several kinds of matrices that relate the port voltages to the port currents. These matrices could be the impedance

matrix [Z], admittance matrix [Y], and hybrid matrix [H]. The transmission matrix is defined as the [ABCD] matrix. It is useful in cascading of two or more networks or transmission line sections. At low radio frequency, the voltage and current are measurable parameters. Therefore, the matrix elements of a network and device could be experimentally determined. Normally, the microwave passive components, circuits, and networks are constructed around the transmission lines supporting the TEM or the quasi-TEM mode. Sometimes, the lumped elements are also used. The waveguide sections supporting non-TEM mode are also used to develop components and circuits. As a matter of fact, the voltage and current can be uniquely defined only for the TEM mode supporting structures. However, for non-TEM line structures, only the equivalent voltage and current, based on the power equivalence principle, is defined [B.1, B.2]. The abovementioned parameters are discussed in this section, as these are important for the analysis of the line networks and the networks involving both the line sections and lumped circuit elements. The results of the analysis and measurement are also presented using these parameters. The reader can study these parameters in detail from any of the excellent textbooks [B.1, B.3–B.7]. One basic difference could be seen between the lumped elements based lowfrequency circuits and the transmission line sections based on high-frequency circuits. The low-frequency circuits are the oscillation type circuits, whereas the high-frequency microwave circuits are the wave type circuits. In the case of the low-frequency oscillation type circuits, the port voltage and port current are described by a single voltage or current. In general at any port, for the high-frequency wave-type circuits, the port voltage is described by a sum of the incident and reflected voltages, also the port current is a sum of the incident and reflected currents. It is illustrated in the discussions on the evaluation of the parameters.

Introduction to Modern Planar Transmission Lines: Physical, Analytical, and Circuit Models Approach, First Edition. Anand K. Verma. © 2021 John Wiley & Sons, Inc. Published 2021 by John Wiley & Sons, Inc.

46

3 Waves on Transmission Lines – II

How to characterize the components, circuits, and network made of the transmission line sections and waveguide sections? At the microwave frequency, port voltage and port current are not the measurable quantities. However, from the analysis point of view, the networks can be characterized by the [Z], [Y], and [ABCD] parameters. But these are not the measurable parameters at microwave frequencies. A different kind of matrix parameter, called the scattering or [S]-parameters, is used for the practical characterization of the microwave network and the transmission line structures [J.1]. The [S] parameter is a measurable quantity. A Scalar Network Analyzer is used to measure the magnitude |S| of S-parameter of any microwave circuit and network. For the measurement of the complex [S] parameters, i.e. both the magnitude and phase response of a network, a Vector Network Analyzer (VNA) is used. The Circuit Simulators and the EM-Simulators (Electromagnetic field simulators) are also used to get the frequency-dependent [S] parameters response of the microwave circuits.

3.1.1

[Z] Parameters

The [Z] matrix defines the impedance parameter of a two-port or a multiport network. The matrix elements are evaluated by open circuiting the ports. Therefore, the [Z] parameters or the impedance parameters are also called the open-circuit parameters. The port voltage (VN) and the port current (IN) are the sums of the reflected and incident voltages and currents, respectively. The port current is an independent variable, whereas the port voltage is the dependent variable. Therefore, the port currents are the excitation sources creating the port voltages as the response. The response voltage is proportional to the excitation current and the proportionality constant has the dimension of impedance. The impedance matrix could be obtained for a general linear two-port network, shown in Fig (3.1). The wave entering the port is the incident voltage (Vn+ ) or the incident current (In+ ) wave. The reflected voltage (Vn− ) and reflected current (In− ) waves are also present at the ports. The total port voltage (Vn) or current (In) is the sum of the incident and reflected voltage or current: Vn = Vn+ + Vn− a , In = In+ − In− b ; n = 1, 2 311 In equation (3.1.1), n = 1, 2 is the port number, i.e. port-1, port-2. The power entering the network is taken as a positive quantity for the incident wave, so the power

+

+

+

(V1 , I1 ) Incident wave

Incident wave

[Z] Or [Y]

Reflected wave

–

–

–

–

(V1, I1)

(V2, I2)

Port #1

Port #2

Figure 3.1

+

(V2 , I2 )

Reflected wave

Two-port network to determine [Z] and [Y] parameters.

coming out of the network, i.e. the power of the reflected wave, is taken as a negative quantity. The reflected voltage (Vn− ) is positive. To maintain the negative direction of power flow, the reflected current wave (In− ) is taken as negative in the above equation. At each port, the current entering the port from outside is positive, whereas the current leaving the port is negative. For the linear networks, the voltage at any port is a combined response of the currents applied to all ports. On using the superposition of the voltage responses, the following set of equations is written: V1 = Z11 I1 + Z12 I2 V2 = Z21 I1 + Z22 I2

312

Equation (3.1.2) is written in a more compact matrix form: V = Z I,

313

where [V] and [I] are the column matrices. The two-port impedance matrix is Z =

Z11

Z12

Z21

Z22

314

The [Z] parameter can be easily extended to the N-port networks [B.1, B.3–B.5]. The Z-parameters are the opencircuited parameters. The coefficient of the matrix can be defined in terms of the open circuit condition at the ports: Zii =

Vi Ii

,

i = 1, 2…, and k

i

315

Ik = 0

All the ports are open-circuited, except the ith port at which the matrix element Zii is defined. For instance, in the case of a two-port network, Z11 is obtained when current I1 is applied to port-1 and the voltage response is also obtained at the port-1, while keeping the port-2 open-circuited, i.e. I2 = 0. The coefficient, Z11, is known as the self-impedance of the network. These are the

3.1 Matrix Description of Microwave Network

diagonal elements of a [Z] matrix. The off-diagonal elements of a [Z] matrix are defined as follows: Zij =

Vi Ij

,

k

j

The given circuit is asymmetrical. However, it is a reciprocal circuit. It becomes symmetrical for ZA = ZB.

316

Ik = 0

Example 3.2

In this case, the current excitation is applied at the port-j and the voltage response is obtained at the port-i. All other ports are kept open-circuited allowing Ik = 0, except at the port-j. For instance, in the case of a two-port network to evaluate Z12, the current source is applied at the port-2, and the voltage response is obtained at the port-1, while keeping the port-1 open-circuited. The coefficient Z12 is the mutual impedance that describes the coupling of port-2 with the port-1. A network can have Z11 = Z22, i.e. both of the ports are electrically identical. Such a network is known as the symmetrical network. Furthermore, the voltage response of a network at the port-1 due to the current at the port-2 can be identical to the voltage response at the port-2, due to the current at the port-1. This kind of network is a reciprocal network. It has a Z12 = Z21. If Z12 = Z21 = 0, the ports are isolated one.

Determine the [Z]-parameter of a section of the transmission line of length ℓ shown in Fig (3.3).

Solution Let the port-2 be open-circuited and an incident voltage Vinc = V1+ is applied at the port-1. The voltage wave reaches to the port-2 and reflects from there. It reaches the port-1 as the voltage wave V1+ e − 2 γℓ. The maximum of the voltage wave occurs at the port-2. The total port voltages are given below:

At the port-1 At the port-2

Thus, the [Z] parameters are obtained as follows:

Example 3.1

Z11 =

Figure (3.2) shows lumped elements T-network. Determine the [Z] parameter of the network.

Z21 =

Solution For the port-2 open-circuited, I2 = 0. The voltage at the port-1 is V1 = ZA + ZC I1 , Z11 = V2 = I1 ZC ,

Z21

V1 I1

V2 = I1

= ZA + ZC

+ I1 V1

ZA + ZC

ZC

ZC

ZB + ZC

ZA

= ZC

= Z0

1 + e − 2γℓ 1 − e − 2γℓ

= Z0 coth γℓ

= Z0

2 e − γℓ 1 − e − 2γℓ

= Z0 cos ech γℓ

I2 = 0

I2 = 0

Z0 coth γℓ

Zo cosech γℓ

Z0 cosech γℓ

Z0 coth γℓ

For the lossless transmission line, γ = j β, α = 0 and [Z] is Z = −

j Z0 cot βℓ

j Zo cosec βℓ

j Z0 cosec βℓ

j Z0 cot βℓ

Figure 3.2 Lumped T-network.

b 318

I2 V2

ℓ

+

+

– Port #2

+

I1

I2 Z0, γ

Vinc V 1 Vref

– Port #1

a

317

ZB

ZC

V2 I1

Z =

Likewise, for the port-1 open-circuited, I1 = 0, and the parameters are Z22 = ZB + ZC, Z12 = ZC. The [Z] matrix description of a T-network is Z =

V1 I1

The following [Z] matrix of a line section is obtained by keeping in view that the uniform transmission line is a symmetrical and reciprocal network:

I2 = 0

I2 = 0

V1 = Vinc + Vref = V1+ + V1+ e − 2 γℓ V2 = Vinc + Vref = 2V1+ e − γℓ V+ V+ I1 = Iinc − Iref = 1 − 1 e − 2 γℓ Z0 Z0 I2 = 0

At the port-1 At the port-2

– Port #1 Figure 3.3 A transmission line section.

Vinc V2 Vref – Port #2

47

48

3 Waves on Transmission Lines – II

3.1.2

Solution

Admittance Matrix

To define the [Y] parameters, the voltage is taken as an independent variable and current as of the dependent one for a two-port network shown in Fig (3.1). In this case, the voltage is a source of excitation, and current at the port is the response. Thus, for a linear network, the total port current is a superposition of currents due to the voltages applied at both the ports: I1 = Y11 V1 + Y12 V2 I2 = Y21 V1 + Y22 V2

a,

I1 V1

Y12 Y22

,

3 1 10

Y21 =

V2 = 0

I2 V1

I2 V2

, V1 = 0

Ii Vi

,

Y12 =

I1 V2

3 1 12 V1 = 0

i = 1, 2…, and k

i

3 1 13

Vk = 0

Equation (3.1.13) shows that to get Yii, i.e. the diagonal elements of the [Y] matrix, all the ports are shortcircuited, except the ith port. The current is evaluated at the ith port for the voltage applied at the ith port itself. To get Yij, i.e. the off-diagonal elements of the [Y] matrix, the voltage is applied at the jth port. Yij is the mutual admittance describing the coupling between the jth port and the ith port. The current at the ith port is evaluated or measured, while all other ports are short-circuited. The admittance element Yij is evaluated as Yij =

Ii Vj

,

I1 ZB + ZC = V1 ZA + ZC ZB + Zc − Z2C I2 ZC = = − V1 ZA + ZC ZB + Zc − Z2C

Likewise, the expressions for Y22 and Y12 could be computed by short-circuiting the port-1, V1 = 0. Final [Y] matrix of the T-network is Y =

1 ZB + ZC ΔZ − ZC

ΔZ = ZA + ZC

− ZC ZA + ZC

ZB +

,

where

3 1 15

ZC − Z2C

The above matrix is a reciprocal of the [Z] matrix, given in of length ℓ equation (3.1.7).

3 1 11

The [Y] parameters are extended to a multiport network by defining its matrix elements as follows: Yii =

I1.

V2 = 0

Likewise, for the short-circuited port-1, the Yparameters are Y22 =

ZC ZB + Zc

From the above equations:

Y21

The Y-parameters are defined as the short-circuited parameters. For the short-circuited port-2, V2 = 0, and Y11 and Y21 are defined from equation (3.1.9): Y11 =

For the short-circuited port-2, V2 = 0: I2 = −

I = Y V b,

where [V] and [I] are the voltage and current column matrices. The admittance matrix of the two-port network is Y11 Y21

V1 = ZA + ZC I1 + ZC I2 , V2 = ZC I1 + ZB + ZC I2

Y11 =

319

Y =

The loop equations for the circuit are written as

i = 1, 2…, and k

j

3 1 14

Vk = 0

Example 3.3 Fig (3.2) shows the T-network. Determine the [Y] parameter of the network.

Example 3.4 Determine the [Y] parameter of a section of the transmission line of length ℓ shown in Fig (3.3).

Solution The incident voltage Vinc = V1+ excites the port-1, and it reaches the port-2 as V1+ e − γℓ The port-2 is shortcircuited to determine the [Y] parameter. Under the short-circuit condition, the reflected voltage at the port-2 is − V1+ e − γℓ such that the total voltage at the port-2 is zero. The reflected voltage at the port-1 is − V1+ e − 2γℓ . The total voltage and the total current at the port-1 are V1 = Vinc + Vref , I1 = Iinc − Iref ,

V1 = V1+ − V1+ e − 2γℓ I1 =

V1+ V+ + 1 e − 2γℓ Z0 Z0

At the port-1, the incident current Iinc enters the port, so it is positive, whereas at the port-1, the reflected current Iref leaves the port, so it is negative. At the port-2, the incident current Iinc enters the port-2 from the port-1 side and leaves the port-2, so it is negative, whereas at the port-2, the reflected current Iref from the terminated load, enters the port-2, so it is positive. The total voltage and the total current at the port-2 are

3.1 Matrix Description of Microwave Network

V2 = V1+ e − γℓ + − V1+ e − γℓ = 0, I2 = − Iinc + Iref V+ V+ 2 V1+ − γℓ I2 = − 1 e − γℓ + − 1 e − γℓ = − e Z0 Z0 Z0 Y11

I1 = V1

Y21

I2 = V1

V2 = 0

1 1 + e − 2γℓ = = Y0 coth γℓ Z0 1 − e − 2γℓ = −

V2 = 0

2e − γℓ = − Y0 cos ech γℓ Z0 1 − e − 2γℓ

The line section is symmetrical and reciprocal giving the [Y] parameter: for the lossy line section Y =

Y0 coth γℓ

− Yo cos ech γℓ

– Y0 cos ech γℓ

Y0

a

coth γℓ

and for the lossless line section Y =

– j Y0 cot βℓ

j Yo cos ec βℓ

jY0 cos ec βℓ

− j Y0 cot βℓ

b

These expressions can be written in the matrix form, V1 I1

=

A B C D

V2 I2

3 1 18

In the case of the [Z] and [Y] parameters, the positive current I2 enters the port, while in the above network defining the [ABCD] parameter in Fig (3.4), the output current I2 leaving the port is taken as positive [B.1, B.3]. It is an input to the next circuit block, as shown in Fig (3.5). However, like defining the [Z] and [Y] parameters, to define the [ABCD] parameter current, I2 could be taken as the current entering the output port. In this case, I2 in equation (3.1.18) is replaced by (−I2) [B.1, B.4]. The matrix elements A, B, C, D can be determined from the open and short circuit conditions at the output port. When the output is open-circuited, I2 = 0. Equation (3.1.17) provides the parameter-A and C:

3 1 16

A= 3.1.3

Transmission [ABCD] Parameter

3 1 17

I1 = C V2 + D I2 I1 + V1 – Port #1

I2 A B C D

a,

C=

I2 = 0

I1 V2

b I2 = 0

3 1 19

On many occasions, two or more circuit elements or circuit blocks are interconnected in such a way that the output voltage and current of the first circuit block become the input to the next circuit block. To facilitate such combination or cascading, the circuit elements and blocks are characterized using the transmission parameters, i.e. the [ABCD] matrix, instead of [Z] or [Y] matrix. The great strength of the transmission parameter, i.e. the [ABCD] parameter, is due to its ability to provide [ABCD] matrix of the complete cascaded network, as a multiplication of the [ABCD] matrices of the individual circuit element or circuit block. The [ABCD] parameter, different from the T-matrix, is applicable to a two-port network only. To obtain the transmission matrix description of a two-port network, the output voltage and current are treated as the independent variables. The following expressions relate to the input and output voltage and current of the two-port network shown in Fig (3.4): V1 = A V2 + B I2

V1 V2

+ V2 – Port #2

Figure 3.4 Two-port network for transmission parameter.

The parameter A is the voltage ratio that is a reciprocal of the voltage gain. The parameter C is the trans-admittance of a network. It relates the output voltage of a network to its input current source. When the output is short-circuited, V2 = 0. Equation (3.1.17) again provides the parameters-B and D: B=

V1 I2

,

D=

V2 = 0

I1 I2

3 1 20 V2 = 0

The parameter B is the trans-impedance of a network. It provides the output current when the input of a network is excited by the voltage source. The parameter D is the current ratio giving a reciprocal of the current gain of a network. Fig (3.5) demonstrates the usefulness of the transmission parameters to obtain an equivalent [ABCD] parameter of the cascaded networks. The [ABCD] parameters for the first and the second network are written as V1 I1

=

A1 C1

B1 D1

V2 V3 , I2 I3

=

A2 C2

B2 D2

V4 I4

At the junction of two networks, I2 = I3 and V2 = V3. Therefore, from the above equations, the following expression is obtained: V1 I1

=

A1

B1

A2

B2

V4

C1

D1

C2

D2

I4

3 1 21

49

50

3 Waves on Transmission Lines – II

I1

I3

I2

+ V1 –

+ + V2 V3 – –

A1 B1 C1 D1

A2 B2 C2 D2

I1

Figure 3.5 Cascading of two networks to get one equivalent network.

I4 + –

V4

I4

+ V1 –

+

A B C D

–

V4

Finally, two cascaded networks can be replaced by one equivalent 2-port network having equivalent [ABCD] parameter. It is given by the following expression: A C

B A1 = D C1

A2 C2

B1 D1

B2 D2

3 1 22

Expression (3.1.22) can be extended to the cascading of N-networks by multiplying the individual matrix of each network. Example 3.5 Determine the [ABCD] parameters of the series impedance as shown in Fig (3.6).

Solution The output port is open-circuited, I2 = 0. Therefore, equation (3.1.17) provides V1 = A V2 and I1 = CV2. For the port 2 of Fig (3.6) open-circuited, I2 = 0, V1 = V2 and I2 = I1 = 0. On comparing these equations, the computed parameters are A = 1 and C = 0. For the output port is short-circuited, V2 = 0. Therefore, equation (3.1.17) helps to get, V1 = BI2 and I1 = DI2. Using Fig (3.6) shows, V2 = 0, V1 = ZI2 and I1 = I2. The comparison of these equations provide B = Z and D = 1. Thus, the [ABCD] matrix of series impedance is written as

I1

I2

A C

Solution The output port-2 is open-circuited, I2 = 0. Therefore, from matrix equation (3.1.17): V1 = A V2 and I1 = CV2.At the open-circuited output port 2: I2 = 0, V1 = V2 and I1 = Y V2. On comparing these equations: A = 1 and C = Y. At the short-circuited output port 2: V2 = 0, V1 = BI2 and I1 = DI2.Using Fig (3.7), for V2 = 0, V1 = 0 and I1 = I2. On comparing these equations: B = 0 and D = 1. Finally, the [ABCD] matrix of shunt admittance can be written as B

C

D

=

I1

V2

V1

– Port #1

– Port #2

– Port #1

Series impedance.

A

1

0

Y

1

3 1 24

The [ABCD] matrix could be easily evaluated for the L, T, and π networks, shown in Fig (3.8). The [ABCD] matrix of each element is known and the complete circuit is a cascading of the elements.

V1

Figure 3.6

3 1 23

Determine the [ABCD] parameters of a shunt admittance shown in Fig (3.7).

+

Z

Z 1

Example 3.6

+

+

B 1 = D 0

Figure 3.7

I2

+

V2

Y

– Port #2 Shunt admittance.

3.1 Matrix Description of Microwave Network

ZA

ZA

ZB

ZB

ZC

(a) L-Network.

ZC

ZA

(b) T-Network.

ZB

(c) π-Network.

Figure 3.8 Basic networks.

Example 3.7

A

Determine the [ABCD] parameters of a section of transmission line shown in Fig (3.3).

Solution

cosh γℓ

B =

C

D

A

B

cos βℓ =

Equations (2.1.79) of chapter 2 provide the voltage and current waves on a transmission line: V x = V + e − γx + V − eγx , I x =

V + − γx V − γx e − e Z0 Z0

The V+ and V− are the amplitudes of the forward and reflected waves, respectively. For convenience, the distance x is measured from the port-2. The voltage and current at the port-2 are V x = 0 = V2 = V + + V − , I x = 0 = Z0 I2 = V + − V −

The amplitudes of the forward and reflected voltages in terms of the port voltage and port current are V2 + Z0 I2 , 2

V+ =

V− =

V2 − Z0 I2 2

The voltage and current on a transmission line can be written as e γx + e − γx e − γx −e γx V2 + Z0 I2 2 2 1 e − γx − e γx e − γx + e γx Ix = V2 + I2 Z0 2 2

C

D

Y0 sinh γℓ

j Y0 sin βℓ

Zo sinh γℓ a

cosh γℓ j Zo sin βℓ cos βℓ

; α = 0, γ = jβ

b

3 1 25

The above example can be further extended to a network of several cascaded transmission line sections having different ℓ, Z0, and γ. The overall [ABCD] parameter of the multisection transmission line can be obtained by a multiplication of the [ABCD] matrix of each line section. The line sections can be attached to the series and the shunt lumped elements. Even in such cases, one can find the overall [ABCD] parameter of a complete network. The input impedance, output impedance, Thevenin and Norton equivalent circuits, and power transfer relation, etc. of a complete circuit can be written in terms of the [ABCD] matrix. However, a detailed discussion of these aspects is out of the scope of this book. The reader can follow many available texts for this purpose [B.1, B.2–B.5, B.7, B.8].

Vx =

The voltage and current at the input port-1 are obtained for x = −ℓ: V x = − ℓ = V1 = cosh γℓ V2 + Z0 sinh γℓ I2 sinh γℓ I x = − ℓ = I1 = V2 + cosh γℓ I2 Z0

Above equations can be written in the matrix form: V1 I1

=

cosh γℓ Y0 sinh γℓ

Zo sinh γℓ cosh γℓ

V2 I2

The [ABCD] parameters of the lossy and lossless transmission line sections are given by equation (3.1.25a) and equation (3.1.25b), respectively:

3.1.4

Scattering [S] Parameters

The [Z], [Y], and [ABCD] matrix descriptions of any microwave network or component are based on the port voltage and port current relations. The evaluation of these parameters requires the short-circuiting and open-circuiting of the ports. At the microwave frequency, usually, it is difficult to measure the voltage and current. Similarly, the short-circuiting and opencircuiting of the ports may not be always possible at the microwave frequency. Thus, these parameters are normally not measurable quantities. However, these parameters are useful for the analysis of the microwave circuits built around the lumped and distributed circuit elements. At this stage, another kind of measurable parameters is needed to characterize the microwave circuits. At the microwave frequency, the power could be

51

52

3 Waves on Transmission Lines – II

measured, and also the forward and reflected power waves could be obtained. The frequency and phase of a microwave signal are also measurable quantities. The scattering parameters, also called the S-parameters, are defined for any two-port network, or even the multiport microwave network, in terms of the measurable incident and reflected power waves [J.1].

Using equations (3.1.27) and (3.1.28), the power variable ai is written in terms of the forward RMS voltage Vi+ at the ith port ai =

P1+ =

2 V1+

Z0

a,

P1+ =

2 V1+max

b

2Z0

3 1 26

In equation (3.1.26b), V1+ = V1+max 2 is the RMS voltage of the voltage wave. In general for the two-port or N-port network, the forward power entering the ith port is written as 2

Pi+ =

Vi+ , Z0i

i = 1, 2, …, N

3 1 27

The incident power variable ai at the ith port is defined in a way that the power entering the port is given by the square of the power variable: Pi+ = a2i

Vi+ =

The power variable ai is simply a normalized forward voltage wave, incident on the ith port. The normalization is done with respect to the square root of the characteristic impedance at the port. The forward power variable can also be viewed as the incident normalized current. The power entering the ith port, in terms of the incident RMS current I1+ , is given below: 2

Pi+ = Ii+ Z0i = a2i

ai = Ii+

Z0i ,

i = 1, 2, …, N 3 1 31

The forward port current in terms of the forward power variable is ai , Z0i

Ii+ =

i = 1, 2, …, N

3 1 32

The multiplication of the voltage and current of equations (3.1.30) and (3.1.32), again provides the forward power, Pi+ = Vi+ Ii+ = a2i . Thus, the definitions of the power variable both as the normalized voltage wave and as the normalized current wave are consistent. However, one must be careful about the presence of the square root of the characteristic impedance in the numerator and denominator for two definitions. Consider the reflected power wave at the ith port with characteristic impedance Z0i. Figure (3.10) shows that the ith port is connected to a source with impedance Z0. For the sake of clarity, the port is taken out of the

ℓ=0

+

+

V2

Vi

Z0i Network

Z0

Z02

S

bi

– – V1

– V2

ith Port

Port #2 Figure 3.10

Figure 3.9

3 1 30

3 1 28

V1

Port #1

3 1 29

Z0i ai

ai

Z01

i = 1, 2, …, N

The forward voltage can also be written in term of the power variable as

Basic Concept

A commonly used two-port network is suitable to develop the concept of the S-parameter. Even the Sparameters of a multiport network are measured as the two-port parameters, while other ports are terminated in the matched loads. Figure (3.9) shows the two-port network. It is to be characterized by the S-parameters. The port-1 and port-2 are terminated with the line sections of characteristic impedance Z01 and Z02, respectively. However, most of the two-port networks have Z01 = Z02 = Z0, i.e. the identical transmission line sections at both the ports. The reference impedance Z0 is normally 50Ω. The incident voltage waves at both the ports-V1+ and V2+ , enter the ports and the reflected voltage waves at both the ports-V1− and V2− , come out of the ports. The forward power, i.e. the incident power entering the port-1, is

Vi+ , Z0i

Two-port network for evaluation of S-parameter.

A section of the multiport network. Port is shown extended with length ℓ = 0.

3.1 Matrix Description of Microwave Network

network using an interconnect line of characteristic impedance Z0 with zero length, ℓ = 0. The total power available from the source does not enter the network. A part of it gets reflected. The reflected power in terms of the reflected power variable bi is Pi− = b2i =

2 Vi–

Z0i

i = 1, 2, …, N

,

1 2 1 bi = 2

ai =

bi =

Vi− , Z0i Zoi Ii− ,

Z0i bi

3 1 34

Ii− =

bi Z0i

3 1 35

a

Z0i

b

The definition of the power wave variables given by the equations (3.1.30), (3.1.32), (3.1.34), and (3.1.35) are valid for the special cases of the forward wave and reflected waves. The definition, given in equation (3.1.38), is valid for the general case. It is applicable at any port for any kind of termination. The power-variables ai and bi are complex quantities. The incident and reflected power are

3 1 33

Vi− =

Z0i

3 1 38

The reflected power variable is related to the reflected port voltage and the reflected port current as follows: bi =

Vi + Ii Z0i Vi − Ii Z0i

Pi+ = ai 2 = ai a∗i ,

Pi− = bi 2 = bi b∗i 3 1 39

The power entering the ith port is Pi,in = Pi+ − Pi− = a2i − b2i = ai 1 − Γi

2

, Scattering [S] Matrix

3 1 36

Figure (3.11) shows the N-port network. The power entering the ith port is given in terms of the forward voltage wave Vi+ or the forward power variable ai and the power leaving the ith port is given in terms of the reflected (backward) voltage wave Vi− or the reflected (backward) power variable bi. A part of the microwave power is reflected at the ith port itself and remaining power entering the network comes out of all other ports as ( V1− , V2− , …, VN− ) or as ( b1− , b2− , …, bN− ). The outcoming power from any port is a linear combination of the transmitted power from all other ports. Using either the voltage variables or the power variables, the incident power and reflected power at various ports are correlated as follows:

where the reflection coefficient at the ith port is Γi = bi/ai. The total port voltage and the total port current in term of the power variables can be written as Vi = Vi+ + Vi− = Ii = Ii+ − Ii− =

Z0i ai + bi 1 a i − bi Z0i

a b 3 1 37

The reflected port current Ii− is negative, such that the reflected power Pi− = Vi− Ii− travels in the opposite direction, i.e. it travels away from the port. Using equation (3.1.37), the power variables can be written in terms of the total port voltage and the total port current: V1− V2−

S11 S12

S1N

S21 S22

S2N

V1+ V2+

= VN− Response

b1

S11 S12

S1N

a1

b2

S21 S22

S2N

a2

a, SN1 SN2

SNN

Network

VN+ Excitation

= bN

Response

The incident power at the port is treated as the excitation, and reflected/transmitted power at the port is considered as the response. The network is characterized by the S-parameters.

b SN1 SN2

SNN

Network

(3.1.40)

aN

Excitation

Therefore, the matrix elements Sij relating the excitation (Vi+ or ai) to the response (Vi− or bi) are described as follows:

53

54

3 Waves on Transmission Lines – II

Sjj = + V1 (a1)

N port network

Port1

Port ith

–

V1 (b1) –

+

VN (bN)

VN (aN) Port Nth

Figure 3.11 N-port network showing power variables (ai, bi) in terms of voltage variables Vi+ , Vi− .

bj aj

= ai = 0

Vj− Vj +

,

i

j

3 1 41

Vi+ = 0

Therefore, Sjj is the reflection coefficient (Γj) at the jth port, provided all other ports are terminated in their characteristic impedances. However, if other ports are not terminated in their characteristic impedances, then Sjj is not a measure of the true reflection coefficient of the network or a device at the jth port. The true reflection coefficient at the jth port, under the unmatched load condition, is more than Sjj that is defined under the matched load condition. Transmission Coefficient Sij

Si j Column of the matrix (Excitation at jth port)

Row of the matrix (Response at ith port)

The Sij is defined with the help of the matched termination. The matched termination also helps to measure the matrix elements Sij.

Reflection Coefficient Sii

Figure (3.12) shows that the ith port of a multiport network is terminated in a load equal to the characteristic impedance of the port. The power wave bi coming out of the port is incident on the load ZL, whereas the incident power wave ai is the reflected wave from the load. If a port is terminated in its characteristic impedance, i.e. ZL = Z0, then the reflection from the load at the port is zero, i.e. ai = 0. Thus, for the excitation aj applied at the jth port, while all other ports are terminated in their characteristic impedances, the ports have V1+ a1 = V2+ a2 = = Vi+ ai = 0, i j . However, the power bj is reflected from the jth port. The reflection coefficient at the jth port is obtained from equation (3.1.40):

ith Port

Network

ai = 0

Z0

Terminated load ZL = Z0

If the excitation source is connected only to the jth port and the response is seen at the ith port, while all other ports are terminated in their characteristic impedances, it leads to V1+ = V2+ = = Vk+ = 0, i e a1 = a2 = = ak = 0, k j . It shows that the excitation is zero at all ports, except at the jth port. The transmitted power, i.e. the scattered power, from the jth port is available at all ports, i = 1, 2, …, k. However, at the jth port, a part of the incident power appears as the reflected power. The transmission coefficient, Sij for the power transfer from the jth port to the ith port is defined as Sij =

bi aj

= ak = 0,k j

Vi− Vj +

Z0j Z0i

= Vk+ = 0,k j

Ii− Ij +

Z0i Z0j

Vk+ = 0,k j

3 1 42 Normally, the network has identical port impedances and equal to the system impedance, i.e. Z0i = Z0j = Z0. Equation (3.1.40) is written in compact form as V− = S V + ,

b = S a

3 1 43

The elements of the [S] matrix are determined using equations (3.1.41) and (3.1.42). Properties of [S] Matrix

Some important properties of the [S] matrix description of the network are summarized below, without going for the formal proof of these statements. Usually, elements of the [S] matrix are complex quantities. The detailed discussion is available in the well-known textbooks [B.1–B.5, B.7]. Reciprocity Property

bi

Figure 3.12 At the ith port, the load is terminated in port characteristic impedance.

The [S] matrix of a reciprocal network is a symmetric matrix, i.e. the transpose [S]T of the [S] matrix is equal to the [S] matrix itself: S = ST

3 1 44

3.1 Matrix Description of Microwave Network

Unitary Property

The [S] matrix of a lossless network is a unitary one. However, if the network is not lossless, then it is not unitary. The definition of the unitary matrix provides the following relation for the given [S] matrix: ST S∗= I,

3 1 45

where [S]T is the transpose of the [S] matrix, [S]∗ is a complex conjugate of the complex [S] matrix and [I] is the identity matrix. Thus, for a given 2-port [S] matrix, we have S11 S12 , S21 S22

S =

S∗=

S∗11 S∗12 , S∗21 S∗22

S11 S21 S12 S22

ST=

On substituting these expressions in the unitary relation (3.1.45), the following result is obtained: S11 S21

S∗11 S∗12

=

S∗22

1

0

0

1

S∗21 S12 S22 S11 S∗11 + S21 S∗21

S11 S∗12

S12 S∗11 + S22 S∗21

S12 S∗12 + S22 S∗22

+

S21 S∗22

=

1

0

0

1

• • •

The dot product of any column vector with its complex ∗

conjugate is unity, p p = 1 The dot product of any column vector with the complex conjugate of any other column vector is zero, ∗

p q =0 The [S] matrix forms an orthogonal set of the vectors.

The following equation (3.1.47):

S11 S∗12 + S21 S∗22 = 0 S11 S∗11 + S21 S∗21 = 1 b a , S12 S∗12 + S22 S∗22 = 1 S12 S∗11 + S22 S∗21 = 0 3 1 47 Equations (3.1.47) are generalized for the N-port network: N

Sik S∗ik = 1, k = 1, 2, …, N

3 1 48

Sik S∗ij

3 1 49

i=1 N

= 0, j

k

j, k = 1, 2, …, N

i=1

Equation (3.1.48) shows that both elements have identical columns, whereas in equation (3.1.49) column are not identical. The [S] matrix is formed by the column vector as follows: S = p , q , where p =

S11 S21

and q =

p = i S11 + j S21 q = i S12 + j S22

Hence, for a lossless network the following statements, based on equations (3.1.48) and (3.1.49) are made:

S11 2 + S21

Pd = 1 −

from

2

3 1 53

Phase Shift Property

The [S] parameter is a complex quantity. It has both magnitude and phase. Thus, the [S]-parameter is always defined with respect to a reference plane. In Fig (3.13) [S]-parameter of the N-port network is known at the location x = 0. It is determined at the new location, x = −ℓn. Alternatively, once the [S] parameters are known at x = −ℓn, these are determined at x = 0, i.e. at the port of the network. The location ℓn shows the length of the line connected to each port of an N-port network. Normally, it is the point of measurement of the [S] parameters of the network or device. The interconnecting transmission line is lossless and has propagation constant βn. Thus, the electrical length of the connecting line is θn = βn ℓn. V′1

+

+

V1

Port1 – V1′ : : : + Vn′

S22 3 1 50

3 1 51

written

Equation (3.1.52) is the power balance equations for the lossless two-port networks. The unit input power fed to the port-1 is a sum of the reflected power ( |S11|2 ) at the port-1 and the transmitted power |S21|2 to the port-2. In the case |S11|2 + |S21|2 is less than unity, some power is lost in the network through the mechanism of conductor, dielectric, and radiation losses. The lost power, i.e. the power dissipation in the network, is

S12

Therefore, in the usual vector notation we have

are

S11 2 + S21 2 = 1 a , S12 2 + S22 2 = 1 b 3 1 52

3 1 46 On equating each element of matrix equation (3.1.46), the following relations are obtained:

expressions

–

V1 +

Vn

N-port netwotk [S]

Port N – – : V′n Vn : : x = –ℓn x=0

Figure 3.13

N-port network showing phase-shifting property.

55

56

3 Waves on Transmission Lines – II

For an N-port network, the incident wave at the nth port, x = − ℓn, after reflection from the port at x = 0, returns to x = − ℓn. In the process, it travels the electrical length 2θn. Similarly, if the wave is incident at port #1, located at x = − ℓ1 and arrives at the port-2, located at x = − ℓ2; the electrical length traveled by the wave is θ1 + θ2 = β1 ℓ1 + β2 ℓ2, or 2θ1, on the assumption that β1 = β2, and ℓ1 = ℓ2, i.e. the transmission lines connected at both the ports are identical. The measured or simulated scattering matrix [S ] at the location x = − ℓn is related to the [S] parameters of the network by the following expression S = S e − j 2θn

3 1 54

The [S]-parameter of the network is extracted from equation (3.1.54), as 3 1 55 S = S' e + j 2θn For reducing the cascaded network to a single equivalent network, the [S] parameters cannot be cascaded like the [ABCD] parameters. The [ABCD] matrix is suitable for this purpose. However, it is not defined in terms of the power variables. Therefore, another suitable transmission matrix, called [T] matrix has been defined in terms of the power variables to cascade the microwave networks. The [S] matrix is easily converted to the [T] parameters [B.1, B.2–B.5, B.7, B.9]. The concept of the [S] matrix is used below to some simple, but useful circuits. These examples would help to appreciate the applications of the [S] parameters.

Zg

Solution The 2-port network (device) is connected to a source at the port-1 and a load ZL at the port-2. The source has voltage Vg with internal impedance Zg. The network scattering parameters-[S] are computed under the matched condition. The characteristic impedance of the connecting line between the port-1 and the source is Z01, whereas the characteristic impedance of the connecting line between the port-2 and the load is Z02. The lengths of the connecting lines are zero. The reflection and transmission coefficients are to be determined at the input and output terminals. This is a practical problem for the measurement and simulation of the 2-port network: b1 = S11 a1 + S12 a2 b2 = S21 a1 + S22 a2

a b

i

+

+ S11 S12

a1

Vg

V1 b1

S21 S22

–

Figure 3.14

I2 a2 ZL

V2 b2 –

Port1 Ch. impedance Z01

Port2 Ch. impedance Z02

A two-port network with arbitrary termination.

Figure (3.14) shows that the power variable b2 is the incident wave at the load ZL and the power variable a2 is the reflected wave from the load. Thus, the reflection coefficient at the load is ΓL =

ZL − Z02 a2 = ZL + Z02 b2

ii

From above equations (i) and (ii): b1 = S11 a1 + S12 ΓL b2 b2 = S21 a1 + S22 ΓL b2 S21 b2 = a1 1 − S22 ΓL

a iii b

On substituting b2 from equation (b) in equation (a): b1 = S11 +

S12 S21 ΓL a1 1 − S22 ΓL

iv

The input reflection coefficient at the port-1 is Γin = Γ1 =

Example 3.8 Determine the S-parameters and return loss of a 2-port network with arbitrary termination shown in Fig (3.14).

I1

Reflected wave from port − 1 b1 = Incident wave at port − 1 a1

Γ1 = S11 +

S12 S21 ΓL 1 − S22 ΓL

3 1 56

The reflection coefficient Γ1 is more than S11 of the network. The mismatch at the load degrades the return loss (RL) of the network. It is given by RL = − 20log 10 Γ1

3 1 57

For the port 2 open-circuited (ZL ∞), the waves get reflected in-phase, i.e. ΓL = 1, and for a short-circuited load (ZL = 0) the total reflection is out of phase, i.e. ΓL = −1. If the network is terminated in a matched load (ZL = Z02), the incident waves are absorbed with ΓL = 0 and Γ1 = S11. Likewise, the source reflection coefficient Γg could be defined at the input port-1. Figure (3.14) again shows that b1 is the incident wave on the internal impedance of the source Zg and a1 is the reflected wave from Zg. Thus, Γg =

Zg − Z01 a1 = Zg + Z01 b1

v

3.1 Matrix Description of Microwave Network

The output reflection coefficient Γ2 at the port-2 is obtained from equations (i) and (v): Γ2 =

S21 S12 Γg b2 = S22 + a2 1 − S11 Γg

I1

3 1 58

Again under the matched condition (Zg = Z01) at the input port, Γg = 0. For most of the applications, 50 Ω system impedance is used, i.e. Z01 = Z02 = Z0 = 50 Ω. For a 2-port lossless network, we have the following expressions: S11 S∗11 + S21 S∗21 = 1 a , S12 S∗12 + S22 S∗22 = 1 b

Zin

V1

V2

Port1

Port2

a,

S12 =

1 − S11

2

b 3 1 59

S11 S∗12

S21 S∗22

The network also follows + = 0 . The S-parameters are complex quantities. The S-parameters are written in the phasor form: S11 = S11 ej θ1 , S22 = S22 ej θ2 and S12 = |S12| ej ϕ. From the above equation, the phase relation is obtained: 1 − S11

S11 ϕ =

2

e

θ1 + θ 2 π + 2 2

j θ1 − ϕ

+e

nπ

j ϕ − θ2

=0

Likewise, to compute S22, the port-1 is terminated in Z0. It gives Zout = Z + Z0 at the port-2. The S22 is S22 =

V1 = V1+ + V1− = S11 + 1 V1+

Determine the S-parameters of the series impedance shown in Fig (3.15). Also, compute the attenuation and the phase shift offered by the series impedance.

i

Note S11 =

Therefore, from equations (i) and (ii): S21 =

V2− V1+

= V2+

=0

V2 S11 + 1 V1

I1 =

V1 Z0 V1 and V2 = Z0 I1 = Z + Z0 Z + Z0

iv

Finally, S21 is obtained from equations (iii), (iv) and (3.1.61): V2 Z0 Z S11 + 1 = 1+ V1 Z + Z0 Z + 2Z0

=

2 Z0 Z + 2 Z0 3 1 63

Equations (3.1.61) and (3.1.63) provide the following relation: 3 1 64

The [S] matrix of the series impedance is

To compute S11 that is the reflection coefficient of a network under the matched condition, the port-2 is terminated in Z0. Thus, Zin = Z + Z0 and the reflection coefficient at port-1 is S11 =

iii

However, the port voltage V2 computed from the port current is

S11 + S21 = 1

Solution

V1− V1+

V2 = V2+ + V2− = V2− ii Note V2+ = 0 due to the matched termination

S21 =

Example 3.9

3 1 62

To compute S21, i.e. the transmission coefficient from the port-1 to the port-2 under the matched termination, at first, the total port voltage at the port-2 is obtained:

3 1 60

Therefore, once the complex S11 and S22 are measured, both the magnitude and phase of the S21 are determined. However, usually, both S11 and S21 are obtained from a VNA and also from the circuit simulator or EM-simulator. The magnitude of S21 provides the insertion-loss of the network and ϕ is the phase shift at the output of the network.

Zout − Z0 Z = Zout + Z0 Z + 2 Z0

The total port voltage at the port-1 is a sum of the forward and reflected voltages:

a b

Z0

Figure 3.15 Network of series impedance.

However, for a reciprocal network S12 = S21. Thus, the above equations provide S11 = S22

I2

Z

Zin − Z0 Z = Zin + Z0 Z + 2 Z0

3 1 61

S =

S11 1 − S11

1 − S11 S11

3 1 65

The attenuation and phase shift of a signal, applied at the input port-1 of series impedance Z = R + jX, are computed below.

57

58

3 Waves on Transmission Lines – II

Using S21 from equation (3.1.63), the attenuation offered by the series impedance is α dB = 20 log 10

1 R + jX = 20 log 10 1 + S21 2Z0

α dB = 10 log

1+

R 2Z0

2

+

X 2Z0

The [S] matrix of the shunt admittance is

S =

2

3 1 66

X R + 2Z0

2 2 + YZ0 − YZ0 2 + YZ0

3 1 71

The attenuation of the input signal due to the shunt admittance is

The lagging phase shift of the signal at the output port2, due to the series element, is ϕ = − tan − 1

− YZ0 2 + YZ0 2 2 + YZ0

3 1 67

α dB = 20 log 10

1 Y = 20 log 10 1 + S21 2Y0

α dB = 10 log

1+

2

G 2Y0

+

B 2Y0

2

3 1 72

Example 3.10 Determine the S-parameter of a shunt admittance shown in Fig (3.16). Also, compute the attenuation and the phase shift offered by the shunt admittance.

The lagging phase shift of the signal at the output port-2, due to the shunt admittance, is B G + 2Y0

ϕ = − tan − 1

3 1 73

Solution The shunt admittance is Y = G + jB. To compute S11, the port-2 is terminated in Z0 (=1/Y0) giving Yin = Y + Y0. The reflection coefficient of the shunt admittance under matched termination is S11 =

Y0 − Yin −Y = Yo + Yin 2Y0 + Y

3 1 68

Likewise, to compute S22 of the shunt admittance, the port-1 is terminated in Z0: S22 =

−Y 2Y0 + Y

3 1 69

Following the previous case of the series impedance, the S21 is computed: V2 S21 = S11 + 1 Fig (3.16) shows V1 = V2; V1 therefore, S21 = 1 + S11 =

2 2 + YZ0

Example 3.11 Determine the S-parameters of a transmission line section, shown in Fig (3.17), with an arbitrary characteristic impedance.

Solution The line has an arbitrary characteristic impedance nZ0 and propagation constant β. The Z0 is taken as the reference impedance to define the S-parameter. The reflection coefficient at the load end is ΓL x = 0 =

Z0 − nZ0 1−n = Z0 + nZ0 1+n

Using equation (2.1.88) of chapter 2, the input impedance at the port-1 of the transmission line having characteristic impedance nZ0 is

3 1 70 +

V1

+

V2

+ V1

V1

Y

V2

–

Z0 – V1

Zin Port 1 Figure 3.16

3 1 74

Port 2 Network of shunt admittance.

X = –ℓ Port1

Figure 3.17

nZ0

Z0

V2 –

V2 x=0 Port2 A transmission line circuit with an arbitrary characteristic impedance.

3.2 Conversion and Extraction of Parameters

Zin = nZ0 Zin = nZ0

1 + ΓL x = 0 e − j 2βℓ On substituting ΓL , 1 − ΓL x = 0 e − j 2βℓ 1 + n + 1 − n e − j 2βℓ 1 + n − 1 − n e − j 2βℓ

3 1 75

Thus, the reflection coefficient at the port-1 is S11 =

n2 − 1 1 − e − j 2βℓ Zin − Z0 = Zin + Z0 n + 1 2 + n − 1 2 e − j 2βℓ 3 1 76

The transmission parameter S21 is computed in terms of S11. If the amplitude of the forward traveling voltage wave on the transmission line is V1+ , the total voltage on the transmission line is given by V x = V1+ e − j βx + ΓL x = 0 ej βx ,

3 1 77

where x is measured from the load end, as shown in Fig (3.17). The input port-1 is located at x = − ℓ. The voltage at the port-1 is V1 = V1+ ej βℓ + ΓL x = 0 e − j βℓ

3 1 78

The port voltage V1 is obtained as a sum of the incident and reflected voltages at the port-1: V1 = V1+ + V1− = V1+ 1 + S11 , S11 = V1− V1+ 3 1 79

At the port-2, under the matched termination, ZL = nZ0 giving V2+ = 0 and V2 = V2− . Equation (3.1.77) shows that the voltage at the port-2, i.e. at x = 0 is V2 = V1+ 1 + ΓL x = 0

3 1 80

Using equation (3.1.79), the transmission coefficient, S21 of the circuit shown in Fig (3.17) is obtained as S21 =

V2− V2 = S11 + 1 V1+ V1

3 1 81

On substituting V1 from equation (3.1.78) and V2 from equation (3.1.80) in the above equation S21 is obtained: 1 + ΓL x = 0 1 + S11 ej βℓ + ΓL x = 0 e − j βℓ 2 1 + S11 = 1 + n ej βℓ + 1 − n e − j βℓ

3.2 Conversion and Extraction of Parameters Sometimes, the conversion of one kind of network parameter to another kind is needed for the analysis of a circuit. For instance, if several circuit blocks comprising of the lumped elements and the transmission line sections are cascaded, each circuit block could be expressed by its [ABCD] matrix. It helps to get an overall [ABCD] matrix of the cascaded network. However, the final [ABCD] matrix, describing the cascaded network is further converted to the [S] matrix. Similarly, the [S] matrix of each building block of the cascaded network has to be converted to the [ABCD] matrix to get the overall [ABCD] matrix of the cascaded network. Finally, the overall [ABCD] matrix is converted to the [S] matrix of the cascaded network. The S-parameters are measurable quantities. The performance of a network is measured in the [S] matrix using a VNA. On several occasions, the S-parameters of a line section or a network are known either from the simulations or from the measurements. The S-parameters are used to get the characteristic impedance and the propagation constant of a line, or a network. However, the true S-parameters of a network are needed for this purpose. The true S-parameters are normally embedded in the measured or the simulated S-parameters at the ports of measurement, or the ports of simulation. The true S-parameters of a line or a network are extracted, i.e. de-embedded, from the measured, or simulated, Sparameters at the ports. This is known as the de-embedding process [B.10]. The EM-Simulators have provision to de-embed the true S-parameters from the Sparameters obtained at the measurement or simulation ports. This section presents the conversion of matrix parameters, de-embedding of the S-parameters, and extraction of the propagation characteristics. 3.2.1

S21 =

3 1 82

The present line network is symmetrical and reciprocal. It has S11 = S22 and S21 = S12. The above expressions are checked for n = 1, i.e. for a transmission line of characteristic impedance Z0. For this case, S11 = S22 = 0 and S21 = S12 = e−j βℓ. These are expressions of the Sparameters for a line having characteristic impedance Z0.

Relation Between Matrix Parameters

[Z] and [ABCD] Parameters

Figure (3.18) shows a network with its known [Z] parameters. It requires conversion to the [ABCD] parameters. The [ABCD] and [Z] parameters of the network are summarized below: V1 = A V2 + B I2

a

I1 = C V2 + D I2

b

V1 = Z11 I1 − Z12 I2

a

V2 = Z21 I1 − Z22 I2

b

321

322

59

60

3 Waves on Transmission Lines – II

I1

V = Z I

I2

+

+ Z

V1

V– =

V2

– Port 1

where [I] is a unit or identity matrix. Keeping in view the definition of the [S] matrix, the following relations, between the [S] matrix and [Z] matrix, are obtained:

Network for Z-parameter.

The current (I2) entering the port-2 is taken positively. However, the Z-parameter is defined in Fig (3.18) for the output current leaving the port. In this case, I2 is negative. Equation (3.2.2) is rearranged to get the port voltage and current V1 and I1 at the port-1 in terms of the V2 and I2 at the port-2: Z11 Z11 Z22 V2 + − Z12 I2 Z21 Z21 1 Z22 V2 + I2 I1 = Z21 Z21 V1 =

Z11 Z22 B= − Z12 , Z21

1 C= , Z21

Z22 D= Z21

−1 Y21 − Y11 D= Y21 B=

325

The N-port network, having normalized reference port impedance, Z0n = 1, is considered. The port voltage and port current in terms of the incident and reflected voltage can be written as Vn =

+

a,

In =

In+

I − Y I + Y

323

[S] and [Z] Parameters

Vn−

a,

Y =

− In−

=

Vn+

− Vn−

V = V

+ V

a,

I − S I + S

I = V

+

− V

−

b

[ABCD] and [S] Parameters

Figure (3.19) shows a 2-port network. The known [ABCD] parameters of the network are to be converted to the [S] parameters. The voltage pair V1+ , V2+ , and V1− , V2− are the incident voltage and reflected voltage at both port-1 and port-2. Figure (3.19) also shows the total port voltage (V1, V2) and total port current (I1, I2). The port currents, at both the ports, enter into the network. Therefore, the current I2 is negative. The input port voltage and port current are related to the output port voltage and port current through the [ABCD] parameters as follows: V1 = A V2 + B − I2 I1 = C V2 + D − I2

a b

3 2 11

The port voltage and current are a linear combination of the incident and reflected voltages and currents: Vn = Vn+ + Vn− ;

In =

n = 1, 2

a

1 V + − Vn− Z0 n

b

3 2 12

b

The above equations are written in the column matrix form: −

b

3 2 10

326

+

I + S I − S

Similarly, the following expressions, relating [S] and [Y]-parameters are obtained:

A complete set of the conversion table of parameters is available in textbooks [B.1, B.5, B.7].

Vn+

Z =

b

Likewise, the relations between [Y] and [ABCD] parameters are obtained: − Y22 , Y21 Y11 Y22 C= − − Y12 , Y21

a,

329

S =

324

A=

Z − I Z + I

S =

a

On comparing equations (3.2.1) and (3.2.3), the following conversion expressions are obtained: Z11 A= , Z21

V+ − V–

328

– Port 2

Figure 3.18

V + + V− = Z Z − I V+ , Z + I

+

+ V1 I1

The port voltage is related to the port current through the [Z]-matrix:

+

V1

b 327

I2

–

V1

A

B

C

D

–

+ –

Network for [ABCD] parameter.

Z0

V2 –

Port1 Figure 3.19

V2

Port 2

–

V2

3.2 Conversion and Extraction of Parameters

On substituting equation (3.2.11) in equation (3.2.12): V1 = V1+ + V1− = A V2+ + V2− − B 1 I1 = V + − V1− = C V2+ + V2− Z0 1

V2+ − V2− Z0 V + − V2− −D 2 Z0

a b

3 2 13 To define the [S] parameters, port-2 is terminated in the reference impedance Z0 giving V2+ = 0 , i.e. the reflection from the matched terminated load is zero. The voltage V2− is the incident wave on the load (ZL = Z0), whereas the voltage V2+ is the reflected wave from the load. The above equations are reduced to the following expressions: B V− Z0 2

a

V1+ − V1− = C Z0 + D V2−

b

V1+ + V1− =

A+

3 2 14 On adding the above equations, the following expression is obtained: 2 V1+ = A + B Z0 + CZ0 + D V2−

3 2 15

Equation (3.2.15) provides the transmission coefficient S21, defined as follows: Response at port-2 V− S21 = = 2+ Excitation at port-1 V + = 0 V1 V + = 0 2 2 2 S21 = A + B Z0 + CZ0 + D 3 2 16 The following expression equation (3.2.14): V1+ + V1− A + B Z0 = V1+ − V1− CZ0 + D

is

obtained

from 3 2 17

Equation (3.2.17) gives the following reflection coefficient S11 at the port-1, while port-2 is terminated in the matched load Z0: Response at port-1 V− S11 = = 1+ Excitation at port-1 V + = 0 V1 V + = 0 2

S11

On eliminating V1− the S22 is obtained, whereas S12 is obtained eliminating V2− :

2

A + B Z0 − CZ0 − D = A + B Z0 + CZ0 + D 3 2 18

Similarly, the following expressions are obtained from equation (3.2.13), where the port-1 is terminated in the matched load Z0, i.e. V1+ = 0: V1− = A − B Z0 V2+ + A + B Z0 V2− − V1− = CZo − D V2+ + CZ0 + D V2−

a b 3 2 19

S22 = S12 =

V2− V2+ V1− V2+

=

− A + B Z0 − CZ0 + D A + B Z0 + CZ0 + D

a

=

2 AD − BC A + B Z0 + CZ0 + D

b

V1+ = 0

V1+

=0

3 2 20 If the network is reciprocal, AD − BC = 1, i.e. S12 = S21. For the symmetrical network, S11 = S22 leading to A = D. The known [S] parameters can also be converted to the [A, B, C, D] parameters. Similarly, the [Z], [Y], [ABCD] and [S] parameters are also converted among themselves [B.1, B.3, B.5].

3.2.2

De-Embedding of True S-Parameters

A transmission line section could be treated as a device and its performance can be evaluated by using a VNA. The device is connected to the VNA through connecting cables and connectors. The S-parameters of a device is measured at the external circuit ports that include the effect of the cables and connectors on the S-parameters of the device. Thus, the true S-parameter of a device is embedded in the measured S-parameters of the device. However, it is desired to obtain the true S-parameters of the device under test (DUT). The line section could be the DUT. The process of extracting the S-parameters of the device at the internal device ports (1in, 2in), from the measured S-parameters, at the external circuit ports (1ex, 2ex) is known as the de-embedding process. It is achieved through a calibration process in which the Sparameters of two error boxes are quantified. The error box represents errors in the S-parameters due to cables and connectors connecting the device to the external circuit ports [B.1]. The S or [ABCD] parameter representation of the device at internal ports (1in, 2in) along with the error boxes is shown in Fig (3.20a). The location of the measurement ports, i.e. the external ports (1ex, 2ex) and the device internal ports (1in, 2in), are further shown in Fig (3.20b). Once the error boxes are characterized through their S-parameters, it can be converted to the [Ae Be Ce De] parameters. Similarly, the measured S-parameters of the device and the error box combined are available at the external ports. These can be converted to the measured [Am Bm Cm Dm] parameters. The device [Ad Bd Cd Dd] parameters are related to the other two parameters by the following equation:

61

62

3 Waves on Transmission Lines – II

1in

lex

2in

2ex

DUT (line section)

Error box 1

1in

Error box 2

1ex

DUT (line section)

2in 2ex

Connecting lines to external and internal ports

lex

1in

2ex

2in

(b) Location of ports.

(a) Representation of measurement process with external measurement ports and internal device ports. Figure 3.20

Calibration process in the measurement of S-parameters of a device.

Am

Bm

Cm

Dm

=

Ae

Be

Ad

Bd

Ae

Be

Ce

De

Cd

Dd

Ce

De

−1

3 2 21 The error box 2 is the mirror image of the error box 1 with respect to the DUT. So in the above-given matrix sequence, the third matrix is inverse of the first matrix [B.11]. At the internal device ports, the device [Ad Bd Cd Dd] parameters are de-embedding as follows: Ad

Bd

d

d

C

D

=

Ae Ce

Be De

−1

Am Cm

Bm Dm

Ae Ce

Be De 3 2 22

d

d

d

d

The de-embedded device [A B C D ] parameters are converted to the de-embedded S-parameters of the device. The de-embedded S-parameters could be further converted to the Z and Y-parameters. Thus, any two-port device can be characterized through measurements using suitable parameters- S, Z, or Y. In the case of a transmission line section, the de-embedded Sparameters can be converted to the propagation parameters and the characteristic impedance of the line.

A

B

C

D

=

B C

3.2.3

Extraction of Propagation Characteristics

The true S-parameters of the line section, i.e. its deembedded S-parameters over a range of frequencies are known either through the measurement or through the EM-simulation. This information can be converted to the [ABCD] parameters of a line section as follows [B.1]:

1 + S11 1 − S22 + S12 S21 1 + S11 1 + S22 − S12 S21 Zo 2S21 2S21 1 1 − S11 1 − S22 − S12 S21 1 − S11 1 + S22 + S12 S21 2S21 2S21 Zo 3 2 23

Usually, S11 = S22 and S12 = S21 because a section of the transmission line is treated as the symmetrical and reciprocal network. The frequency-dependent [ABCD]

Z0l =

The above-mentioned concept of de-embedding of the device S-parameters at the internal port of a device is equally applicable to the EM-Simulators – both 2.5D and 3D simulators [B.10]. In EM-simulators, the deltagap voltage source could be used to launch the wave on a line section or a device. It also generates the nonpropagating evanescent modes at the ports. They cause a discontinuity at the external circuit ports, i.e. at the ports of simulation. The port discontinuity affects the S-parameters of the device that is removed by the process of de-embedding [J.2]. The EM-simulators could be used to extract the propagation parameters and the characteristic impedance of a line.

a,

γ = α + jβ =

(3.2.23)

parameters of a lossy line of length ℓ are used to compute the characteristic impedance Z0 and propagation constant γ = α + j β over a range of frequencies [B.1]:

1 1 cosh − 1 A = ln A ± ℓ ℓ 3 2 24

A2 − 1

b

(3.2.24)

3.3 Wave Velocity on Transmission Line

The frequency-dependent secondary line parameters are extracted to get the dispersion characteristics of a line. Once the secondary line parameters of a line section are known, their frequency-dependent primary constants are computed as follows: R = Re γ Z0

a,

γ Z0

c,

G = Re

Im γ Z0 ω Im γ Z0 C= ω

L=

b d 3 2 25

3.3 Wave Velocity on Transmission Line In a communication network, several kinds of electrical signals propagate on a transmission line. The signal could be a modulated or unmodulated carrier wave, the baseband analog signal, or the digital pulses. The TEM mode transmission lines, and also various kinds of non-TEM waveguide structures support wave propagation. The parameters defining these transmitting media could be either frequency-independent or frequency-dependent. The property of the medium has a significant impact on the nature of wave propagation through a medium. The wave velocity has no simple or unique meaning, like the meaning of the velocity of a particle. There are several kinds of wave velocities – phase velocity, group velocity, energy velocity, signal velocity, etc., applied to wave propagation. The significance of several types of wave velocities is inherent both in the complexity of a signal and also in the complexity of the wave supporting medium. This section focuses attention on the meaning of the phase and group velocities only. Section (3.4) demonstrates these two wave velocities as applied to several kinds of the artificial linear dispersive transmission lines. 3.3.1

The concept of phase velocity is applicable to a single frequency wave, i.e. to a monochromatic wave discussed in Section (2.1) of chapter 2. The phase velocity is just the movement of the wavefront. The wavefront is a surface of constant phase, like maximum, minimum, or zero-level points shown in Fig (2.3). It is given by equation (2.1.8) of chapter 2 and reproduced below: ω β

331

The propagation constant β is influenced by the wavesupporting medium. For a lossless TEM transmission line and lossless unbounded space, β is given by

332

where ε and μ are permittivity and permeability of a medium. Thus, pairs (L, C) and (ε, μ) are the parameters that characterize the electrical property of the wave supporting-media. The unbounded medium supports the plane wave propagation. If these parameters are not frequency-dependent, the medium is known as nondispersive. In such a medium, the phase velocity remains constant at every frequency. However, if any of these parameters are frequency-dependent, the propagation constant β is frequency-dependent and consequently, the phase velocity is frequency-dependent. The medium that supports the frequency-dependent phase velocity is known as the dispersive medium. Normally, the characteristic impedance or intrinsic impedance of a dispersive medium is also frequency-dependent. The parameters (L, C) and (ε, μ) are usually independent of signal strength. Such a medium is called a linear medium, whereas the signal strength dependent medium is a nonlinear medium. The characteristics of the medium are discussed in Section (4.2) of chapter 4. The present discussion is only about the linear and dispersive transmission lines. Why a medium becomes dispersive? One reason for dispersion is the loss associated with a medium. The geometry of a wave supporting inhomogeneous structures, commonly encountered in the planar technology, is another source of the dispersion. In the case of a transmission line, the parameters R and G are associated with losses and they make propagation constant β frequencydependent. Likewise, losses make permittivity ε and permeability μ of material medium frequency-dependent complex quantities. However, a low-loss dielectric medium can be nondispersive in the useful frequency band. For such cases, the attenuation and propagation constants are given by α=

Phase Velocity

vp =

β = ω LC a , β = ω με b ,

ωε''r

μr ε'r 2c

a,

β=

ω c

μr ε'r 1 +

1 ε''r 8 ε'r

2

b

333 In equation (3.3.3), c is the velocity of EM-wave in free space. The above expressions are obtained from equations (4.5.16) and (4.5.19) of chapter 4. The complex permittivity is given by ε = ε − jε . The imaginary part, showing a loss in a medium, is related to the conductivity of a medium through relation σ = ωε . If ε and ε are not frequency-dependent, the propagation constant β is not frequency-dependent and the phase velocity is also not frequency-dependent. However, if they are frequency dependent, the phase velocity is also frequency-dependent. The phase velocity in the low-loss dielectric medium is

63

64

3 Waves on Transmission Lines – II

vp =

ω = β

1 1 ε 1− 8 ε με

2

334

Therefore, the presence of loss decreases the phase velocity of EM-wave. This kind of wave is known as the slow-wave. The slow-wave can be dispersive or nondispersive. However, it is associated with a loss. This aspect is further illustrated through the EM-wave propagation in a high conductivity medium. The conducting medium is discussed in subsection (4.5.5) of chapter 4. The attenuation (α), phase constant (β), and phase velocity (vp,con) of a highly conducting medium are given by equation (4.5.35b) of chapter 4 [B.3]: α=β=

πfμσ

vp,con =

ω = vp β

335 2ωε σ

336

The above expressions are obtained for a highly conducting medium, σ/ωε >> 1. It does not apply to the lossless medium with σ = 0. The wave propagation is associated with significant loss (α) given by equation (3.3.5). Moreover, both the attenuation constant and phase velocity are frequency dependent, so a conducting medium is highly dispersive. However, the periodic structures and other mechanisms give slow-wave structures with a small loss. Such structures are useful for the development of compact microwave components and devices [B.1, B.3–B.5, B7, B.12, B.13]. The slow-wave periodic transmission line structures are discussed in chapter 19. Some EM-wave supporting media have cut-off property. They support the wave propagation only above the certain characteristic frequency of a medium or a structure. These media and structures are also dispersive. For instance, the nonmagnetic plasma medium has such cut-off property [B.4, B.14]. The plasma medium is discussed in the subsection (6.5.2) of chapter 6. The permittivity of a plasma medium is given by equation (6.5.16): εp = ε0 1 −

fp f

2

a,

fp =

1 2π

Ne2 me ε 0

b

337 In the above expression, fp is the plasma frequency that is a characteristic cut-off frequency of the plasma medium [B.4, B.14]. The permeability of nonmagnetized plasma is μ = μ0. Other parameters are as follows-ε0: permittivity of free space, N: electron density, e: electron charge, and me: electron mass. The propagation

constant, phase velocity, and plasma wavelength λplasma of the EM-wave wave in a plasma medium are given below: 1 2

fp 1− f

2

2

vp =

fp 1 1− f με0

λplasma

fp 2π v = 1− = f β f

vp =

λplasma v λ

β = ω με0

a − 12

fp = v 1− f 2

2

− 12

b

− 12

c d 338

In equation (3.3.8), v = 1 με0 is the velocity of EMwave in the homogeneous medium with parameters ε0 and μ. The wavelength in the homogeneous medium is λ = v/f. However, the nonmagnetized plasma medium has the parameters ε0, μo supporting the wavelength λ0 = c/f. The nonmagnetized plasma medium behaves as free space. The phase velocity of the EM-waves in a plasma medium is frequency-dependent. Therefore, it is a dispersive medium that supports a fast-wave. It is fast in the sense that the phase velocity is higher than the phase velocity of the EM-wave in free space given by c = 1 μ0 ε0 . The plasma medium exhibits the cut-off phenomenon, similar to the cut-off behavior of the waveguide medium. The waveguide medium is discussed in the section (7.4) of chapter 7. There is no wave propagation at the plasma frequency f = fp. The plasma frequency fp behaves like the cut-off frequency fc of a waveguide. Thus, the waveguide can be used to simulate the electrical behavior of plasma. For f < fp, no wave propagation takes place, as the propagation constant β becomes an imaginary quantity. Such a wave is known as an evanescent wave. It is an exponentially decaying nonpropagating wave (E = E0 e−αz). The standard metallic waveguide also supports the cut-off phenomenon and has a frequency-dependent phase velocity [B.1, B.5, B.7, B.8, B.15–B.17]. The dispersion is a property of the wave-supporting medium. The phase velocity of a wave in a dispersive medium can either decrease or increase with the increase in frequency. Thus, all dispersive media could be put into two groups – (i) normal dispersive medium and (ii) abnormal or anomalous dispersive medium.

Phase velocity (VP)

Figure 3.21 Nature of normal (positive) dispersion.

Relative permitivity (εr)

3.3 Wave Velocity on Transmission Line

Frequency (f) (a) Frequency-dependent relative permittivity dεr / df >0.

Phase velocity (VP)

Figure 3.22 Nature of anomalous (negative) dispersion.

frequency is positive, i.e. dεr/df > 0. Similarly, the anomalous dispersion is called the negative dispersion with dεr/df < 0. The relative permittivity of material undergoes both kinds of dispersion depending upon the physical cause of dispersion. The dispersion is caused by several kinds of material polarizations – dipolar, ionic, electronic, and interfacial polarization. Once the frequency is varied from low-frequency to the optical frequency, the material medium undergoes these polarization changes, and the propagating wave experiences both the normal and anomalous dispersion at different frequencies [B.17, B.18]. It is discussed in chapter 6. The concept of phase velocity applies to a single frequency signal. Now the question is to apply it to a complex baseband signal and a modulated signal. It is possible to use the phase velocity concept to such waveforms through the Fourier series of a periodic signal and using the Fourier integral for a nonperiodic signal. Any signal, periodic or nonperiodic, is composed of a large number of sinusoidal signals. They have a definite amplitude and phase relationship with the fundamental frequency of the signal. A combination of all

Relative permitivity (εr)

Figure (3.21a and b) show the general behavior of a medium having normal dispersion. The relative permittivity of such a medium increases with frequency, i.e. dεr/df is positive, and the phase velocity decreases with frequency, i.e. dvp/df is negative. A microstrip line provides such a medium for the normal dispersion. The effective relative permittivity of a microstrip line increases with frequency leading to a decrease in the phase velocity with an increase in frequency. The microstrip is discussed in chapter 8. Figure (3.22a and b) show the general behavior of an anomalous dispersive medium. The relative permittivity of such a medium decreases with an increase in frequency, i.e. dεr/df < 0 (negative). It leads to an increase in the phase velocity with an increase in frequency, i.e. dvp/df > 0 (positive). A microstrip line on a semiconductor substrate having the Metal, Insulator, Semiconductor (MIS) or the Schottky structure, in the transition region, is an anomalous dispersive medium [J.3, J.4]. It is emphasized that there is nothing abnormal with the anomalous dispersion. Both kinds of dispersions exist in reality. The normal dispersion is also called the positive dispersion as the gradient of εr with

Frequency (f) (b) Frequency-dependent phase velocity dvp / df 0.

65

3 Waves on Transmission Lines – II

sinusoidal components gives a complex signal of definite wave-shape. If the complex waveshape travels through a dispersive medium having frequency-dependent attenuation constant α(f ), the amplitude of each signal component changes differently. Similarly, in a dispersive medium having a frequency-dependent propagation constant β(f ), each signal component travels with a different velocity. It results in different phase-change for each frequency component of the complex wave; so the shape of the wave changes while traveling on a line or through the medium. The numerical inverse Fourier transform provides the wave-shape of a signal in the time-domain at any location in the medium. Thus, the Fourier method helps to apply the concept of phase velocity to complex waveform propagation [J.5, J.6]. Such investigations are important to maintain the signal integrity on the IC and MMIC chips. 3.3.2

Group Velocity

A complex signal composed of two or more frequency components forms a wave-packet. However, the frequency components should not be much different from each other like an amplitude modulated signal. Figure (3.23) shows the wave-packet formed by a group of a narrowband complex signal. The wave-packet has a central or a carrier frequency of higher value, superimposed with a low-frequency envelope. The carrier wave travels with the phase velocity vp, whereas the envelope, i.e. the wave shape travels with the group velocity vg. In a nondispersive medium, the carrier wave and envelope

both travel with the same velocity without a change in the waveshape. However, in the case of a dispersive medium, velocities of the carrier and envelope are different. Depending on the nature of dispersion, the wave could be the forward wave or the backward wave. If the medium has normal (positive) dispersion, the phase velocity, i.e. the velocity of the carrier wave, and the velocity of the envelope, i.e. the group velocity, are in the same direction, as shown in Fig (3.23). The wave is known as the forward wave. However, in the case of a highly anomalous (negative) dispersive medium, under a certain condition, the phase and group velocities are in the opposite directions, forming the backward wave. The carrier and envelope are combined to form a unified wave structure called the wave-packet. In the case of normal dispersion, the group velocity is the energy velocity of a signal and the information travels with the group velocity [B.1, B.4, B.5, B.7, B.14, B.16]. However, in the case of anomalous dispersion, the energy velocity and group velocity are different. In this case, group velocity is not velocity of information. Moreover, the concept of group velocity applies only to a narrow-band wavepacket, not to the wideband signal. The controversy exists at present on the travel of information with a velocity more than the velocity of light [J.7]. Formation of Two-Frequency Wave-Packet

A wave-packet is formed by a linear combination of two signals of equal magnitude with a small difference in angular frequency and phase constant. It is shown in Fig (3.24). The composite voltage wave is given by

Velocity Vg of low frequency envelope Velocity Vp of high frequency carrier Amplitude

66

Figure 3.23

Time

Description of phase and group velocities of a forward-moving modulated wave.

3.3 Wave Velocity on Transmission Line

Angular frequency (ω)

Wave 1 Frequency: ω0 + Δω Phase const: β0 + Δβ

Wave - packet +

Wave 2 Frequency: ω0 – Δω Phase const: β0 – Δβ

ωp

Light line with wave valocity ϕ

ψ

Slow wave region

c=

ω β

β0 Phase constant (β)

v x, t = V0 cos

Figure 3.25 ω − β diagram to get phase and group velocities.

ω0 + Δω t − β0 + Δβ x

+ V0 cos

Wave#1 2V0 cos Δωt − Δβx Envelope

ω0 t − β0 x = Constant,

0,

(3.3.9)

cos ω0 t − β0 x Carrier wave 339

dx ω0 = vp = β0 dt 3 3 10

The velocity of the envelope, i.e. the group velocity, is obtained from the constant phase point on the envelope Δω t − Δβ x = Const ,

ω0 – Δω t − β0 − Δβ x

Wave#2

The carrier wave has frequency ω0 and propagation constant β0. The above expression applies to a narrowband signal, Δω 0, i e Case-II: dω dω i.e. the positive dispersion. Figure (3.26b) indicates that the slope of the propagation constant β in the normal dispersive medium increases nonlinearly with Case-I:

No dispersion

Anomalous dispersion

Angular frequency (ω) (b) Phase constant in a dispersive medium.

Nature of dispersion on (ω − β) the diagram.

dn c dβ β = − dω ω dω ω

3 3 16

Normal dispersion Anomalous dispersion

3 3 15

Figure (3.26a and b) show the dispersive behaviors of the phase velocity (vp) and propagation constant (β). The expression for the group velocity in a dispersive medium, i.e. a medium with frequency-dependent refractive index n ω = εr ω , can be rewritten as follows: n=

c 3 3 13

Phase velocity (Vp)

68

3.4 Linear Dispersive Transmission Lines

angular frequency ω. Therefore, the phase velocity decreases with angular frequency, i.e. dvp/dω is negative. It is shown in Fig (3.26a). For this case, equations (3.3.15) and (3.3.16) show that vg < vp. It is the case applicable to a dispersive microstrip line. dvp dn > 0 . It is the case of the < 0, i e Case-III: dω dω anomalous (abnormal), i.e. the negative dispersion. The propagation constant β of such an anomalous dispersive medium increases nonlinearly with angular frequency ω. However, on the (ω − β) diagram, shown in Fig (3.26b), its value is below the dispersionless medium of the case-I. The slope of the propagation constant β, in the nonlinear region, decreases with an increase in frequency ω, i.e. dn/dω < 0. Figure (3.26a) shows that the phase velocity increases with angular frequency, i.e. dvp/dω is positive. Equations (3.3.15) and (3.3.16) show that vg > vp. However, the group velocity in an anomalous dispersive medium is not the velocity of energy transportation. This case is applicable to the dispersive MIS or Schottky microstrip lines in the transition region [J.3, J.4]. The equations (3.3.15) and (3.3.16) further indicate a possibility of backward wave propagation (vg negative) in an anomalous dispersive medium, if dn/dω is significantly negative, i.e. the medium has a very large negative dispersion. This is the special condition for the existence of backward wave in the anomalous dispersive medium. However, such a medium is very lossy. Lorentz model, discussed in subsection (6.5.1) of chapter 6, explains the phenomenon. The relations between two velocities are obtained below by using the propagation constant (β) and wavelength (λ), instead of the angular frequency (ω): vg = vp + β

dvp a, dβ

vg = vp − λ

dvp b dλ 3 3 17

The condition for the normal dispersion is dvp/dβ < 0, dvp/dλ > 0 and for the anomalous dispersion, it is dvp/dβ > 0, dvp/dλ < 0.

3.4 Linear Dispersive Transmission Lines The velocity of EM-wave propagation in a dispersive medium has been discussed in the previous section. The transmission line is a 1D wave-supporting medium.

The lossy line is a dispersive medium, whereas a lossless line, modeled through the line constants L and C, is a nondispersive medium. It acts as a low-pass filter (LPF). This section shows that a reactively loaded lossless line could be a dispersive medium. A variety of transmission line structures with interesting properties can be developed by using several additional combinations of C and L. Such line structures can be realized with the lumped elements and also by the modification of planar transmission lines. The transmission medium with negative relative permittivity and negative relative permeability has been synthesized with the help of the modified, i.e. reactively loaded line structures. These wave supporting media form a new class of materials known as the metamaterials. They do not exist in nature. However, these novel media have been developed with the defects in the transmission line and using the embedded resonators in the transmission line [B.19, J.8]. The present section considers only an infinitesimal section of the transmission line, modeled as a lumped circuit elements network. The modeled network is reactively loaded to get the loaded line. However, in practice, a finite length of the line is periodically loaded with reactance. Such reactively loaded lines offer novel and improved designs of microwave components and circuits. The periodically loaded line, creating the 1D – EBG and metalines are discussed in the chapters 19 and 22.

3.4.1 Wave Equation of Dispersive Transmission Lines Figure (3.27a) shows the shunt inductor loaded line structure. An infinitesimally small Δx line length is considered. The cascading of a large number line sections, under the condition Δx 0, forms a continuous line. The standard Δx long host transmission line section is modeled by the distributed line constants L and C p.u. l. The host line section is loaded with the inclusion, i.e. with additional shunt inductance Lsh. The loading element, i.e. inclusion is shown in the gray box, so the periodic embedding of reactive inclusion in the host line medium creates an artificial 1D mixture medium. The concept of the mixture material is further discussed in subsection (6.3.1) of chapter 6. The inclusion lumped shunt inductance Lsh distributed over the length Δx is (Lsh/Δx) p.u.l. The total series inductance, (L Δx), gives the total series impedance Z = j ωL Δx and a combination of shunt capacitance (C Δx) and shunt inductance (Lsh/Δx) gives the total shunt admittance, Y = j (ωCΔx − Δx/(ωLsh)). The

69

70

3 Waves on Transmission Lines – II

transmission line equations for the shunt inductor loaded line are obtained as follows: ∗Drop in the line voltage

dispersionless transmission line equation (2.1.24) of chapter 2. On assuming the harmonic solution for the forward-moving voltage, the voltage on the loaded line is

= Voltage drop across the series inductor Δi Δt ∗Drop in the line current = Δv = − LΔx

v x, t = Vmax cos ωt − βx a

v x, t = Vmax Re

Currents through the shunt capacitor and shunt inductor Δi = − CΔx

Δv − i2 Δt

ω2 = ω2c + v20 β2 ,

341 0, Δt

∂v ∂i = −L ∂x ∂t ∂i ∂v i2 = −C − ∂x ∂t dx

a b

The voltage v across the shunt inductor is Lsh ∂i2 dx ∂t

343

From equation (3.4.2a,b), the following voltage wave equation is obtained: ∂2 v ∂2 i = −L 2 ∂x ∂x∂t

∂2 v ∂ ∂v i2 C + =L dx ∂x2 ∂t ∂t

∂2 v ∂2 v L ∂i2 = LC + ∂x2 ∂t2 dx ∂t

344

On substituting ∂i2/∂t from equation (3.4.3) in the above equation, the above wave equation is reduced to ∂2 v ∂2 v L = LC 2 + v, 2 ∂x ∂t Lsh

∂2 v 1 ∂2 v 1 − + v = 0, 2 ∂t LC ∂x2 CLsh 345

where v0 = 1 LC is the phase velocity of the standard nondispersive line without shunt loaded inductance, i.e. for the case Lsh ∞. For a line in the air medium, the phase velocity is the velocity of the EM-wave in a vacuum, i.e. v0 = c. The cut-off frequency is defined as ωc = 1 Lsh C. Finally, the above voltage wave equation of the dispersive transmission line is rewritten as 2 ∂2 v 2∂ v − v + ω2c v = 0 0 ∂t2 ∂x2

347

β=

ω v0

1−

ωc ω

2

348

0, the following

342

v=

ωt − βx

On substituting equation (3.4.7) in equation (3.4.6), the following dispersion relation is obtained:

b

For the limiting case, Δx expressions are obtained:

ej

346

Equation (3.4.6) is known as the Klein–Gordon equation [B.20]. For a line without cut-off frequency, i.e. for ωc = 0, the above equation is reduced to the standard

Equation (3.4.8) is identical to equation (3.3.8a) for the plasma medium. The cut-off frequency ωc of the loaded line corresponds to the plasma frequency ωp. Thus, a plasma medium can be modeled by the shunt inductor loaded transmission line. Further, Fig (3.27a) shows that below the cut-off frequency, i.e. for ω < ωc the circuit is reduced to the L-L circuit presenting inductors in both the series and shunt arm. Such a medium is called the epsilon negative (ENG) medium, discussed in section (5.5) of chapter 5. For frequency ω < ωc, the propagation constant β is the imaginary quantity and the voltage wave cannot propagate on the shunt inductor loaded line. The voltage wave propagates only for ω > ωc. Thus, the shunt inductor loaded line behaves like a high-pass filter (HPF). It is like a metallic waveguide discussed in subsection (7.4.1) of chapter 7. For ω < ωc, the loaded line supports the nonpropagating attenuated wave, called the evanescent wave. Its attenuation constant α is obtained by the following expression: α=

ω v0

ωc ω

2

−1

349

The phase and group velocities of the wave on the loaded dispersive line are vp =

ω = β

v0 1−

ωc ω

2

a,

vg = v 0

1−

ωc ω

2

b

3 4 10 Figure (3.27b) shows the dispersion behavior, i.e. the function ω = f(β), on the (ω − β) diagram for the shunt inductor loaded line. It shows the frequency-dependent behavior of the propagation constant β and attenuation constant α. The dashed line is the light line. Its slope ψ at the origin is the phase velocity of the EM-wave in free space, i.e. vp = v0 = c. The propagation constant β exists

3.4 Linear Dispersive Transmission Lines

LΔx

i + Δi

i

i2 CΔx

Lsh/Δx v + Δv

V

Angualr frequency (ω)

y β

Fast wave region ω = f(β)

P ϕ

ωc

Slow wave region α

ψ

Δx (a) Shunt inductor loaded line.

Light line (V0 = c)

o

x Propagation/ attenuation costants (α,β)

Y Cut-off region

Vp

Time

V0

t2

Vg 0

ωc Angular frequency ω

t

t3

Amplitude

Phase/group velocity (Vp/Vg)

(b) Dispersion nature of shunt inductor loaded line.

t1 O

(c) Nature of phase and group velocities.

X2 X3 Distance (d) Pulse width spreading. X1

X

Figure 3.27 Shunt inductor loaded line and its characteristics.

above the light line only for ω > ωc and wave attenuation occurs for ω < ωc. At location P, the local slope ϕ provides the group velocity vg, whereas the slope ψ of the point P at the origin O provides the phase velocity vp of the propagating EM-wave on the loaded transmission line. Figure (3.27c) shows the variations in the phase velocity and the group velocity with frequency. For ω ωc, vp ∞ , vg 0, and for ω ∞, both wave velocities move toward the light line, i.e. toward the phase velocity (v0) of the unloaded host transmission line. From equation (3.4.10a) dvp/dω is negative, i.e. the phase velocity decreases with frequency, whereas from the equation (3.4.10b), dvg/dω is positive, i.e. the group velocity increases with frequency. However, both velocities tend toward v0, i.e. phase velocity in the unloaded host medium. Figure (3.27b) also shows that the propagation constant β increases with frequency. Thus, the shunt inductor loaded line has normal dispersion. The wave

considered is a forward-moving wave with both the phase and group velocities in the same direction, as both anti-clockwise gradients ψ and ϕ defining the phase and group velocities respectively are positive. Above the cut-off frequency, i.e. above the light line in the fast-wave region, the phase velocity is higher than the phase velocity of the unloaded L-C type transmission line. The shunt inductor loaded line supports the fast-wave. The wave is fast as compared to the wave velocity, shown as the slope of the light line, on an unloaded transmission line. The group velocity vg is also frequency-dependent, causing dispersion in the envelope of a wave-packet. Therefore, if a voltage wave-packet, say a Gaussian wave-packet, propagates on a shunt inductor loaded line, it disperses while moving forward. Figure (3.27d) shows that its amplitude decreases and the pulse width increases. This is known as the group dispersion. The time delay of the envelope over the distance d is

71

72

3 Waves on Transmission Lines – II

τd =

d dβ =d vg dω

3 4 11

In the present subsection, the propagation constant of the loaded line is obtained by solving the wave equation. However, the dispersion relation and also the characteristics impedance of the loaded line could be obtained from the circuit analysis. This simple method is applicable to several interesting cases of loaded lines forming the basis for the modern Electromagnetic bandgap (EBG) materials and metamaterials. 3.4.2 Circuit Models of Dispersive Transmission Lines The above discussion demonstrates that the reactive loading of a line modifies the electrical characteristics of an unloaded host line. This section considers a few such modifications. Shunt Inductor Loaded Line

The total series impedance and total shunt admittance of the shunt inductor loaded dispersive line, shown in Fig (3.27a), are given by Z = j ωL Δx

Y = j ωC −

a,

1 Δx ωLsh

b

3 4 12 The series impedance and shunt admittance per unit length (p.u.l.) are z = j ωL

1 y = j ωC − ωLsh

a,

b 3 4 13

The complex propagation constant of the wave on the shunt inductor loaded line is

L = C 1 − ωc ω

γ = α + jβ = =j

zy =

ω ωc 1− ω v0

2

1 2

2

Z0

=

1 − ωc ω

2

3 4 15 For the case, ω < ωc, the characteristic impedance Zod is an imaginary quantity that stops the signal transmission through the line. The line behaves like a high-pass filter (HPF). For the case ω > ωc, the characteristic impedance Zod is a real quantity that allows the signal propagation on the line. However, the characteristic impedance Zod in the pass-band is frequency-dependent. Backward Wave Supporting Line

Figure (3.28a) shows the circuit model of the standard low-pass filter (LPF) type LC transmission line that can also be realized by cascading several lumped elements LC unit cells. However, the distance between two units cells should be a fraction of wavelength, i.e. number of LC unit cells should 10 or more per wavelength. Figure (3.28b) shows the dual of the LC line realized again by cascading several lumped elements CL unit cells. It is called the CL transmission line. It is a high-pass filter (HPF) type transmission line. Its propagation characteristic is obtained using the circuit analysis: γ = jβ = ± β=

zy = ±

1 j ωC

1 j ωL

1 ω LC

1 2

3 4 16

Equation (3.4.16) gives the rectangular hyperbolic relationship for ω and β in the first and second quadrants of the (ω − β) diagram shown in Fig (3.28c). The phase velocity is vp =

1 − ωL ωC − ωLsh

1 2

z ωL = y ωC − 1 ωLsh

Z0d ω =

ω = β

ω2

LC

3 4 17

The group velocity is obtained separately for both the negative and positive propagation constants:

1 2

3 4 14

The propagation constant β for ω > ωc is identical to equation (3.4.8). The attenuation constant α is obtained for ω < ωc. The expressions for the phase velocity and group velocity follow from the expression of β as discussed previously. The characteristic impedance of the loaded dispersive transmission line Zod(ω) is given by

dβ 1 1 = ± 2 leading to dω ω LC 1 = + ω2 LC for β − tive vg = dβ dω vg = − ω2

LC

for β + tive

a b 3 4 18

The phase velocity and group velocity are in opposite directions to each other, i.e. vg > 0, vp < 0 or vg < 0, vp > 0. This kind of wave is called the backward wave.

3.4 Linear Dispersive Transmission Lines

L

L

L

C

C

C

C

C

C L

L

Δx

L

Δx

(a) Standard LPF- type LC line.

(b) Backward wave HPF- type CL line. Light line

Light line ω

Vg > 0, Vp < 0 Forward direction wave propagation region

Q

I- Quadrant Light cone

II - Quadrant

Vg < 0, Vp > 0

P

ϕ ψ

Backward direction wave propagation region

ϕ ψ

–β

β

(c) Positive and negative gradients of CL line supporting forward and backward propagating backward waves. Figure 3.28 Lumped elements models of short transmission line sections and dispersion diagram of CL line.

If the power is moving from the source to load in the positive direction, the group velocity is in the positive direction from the source to the load, i.e. from the left side to the right side of Fig (3.28b). The power always flows from the source to load. Therefore, the direction of the power flow decides the direction of the EM-wave propagation. Normally, the direction of the group velocity is in the direction of the power flow. However, the phase velocity appears to be from load to the source. It means at the end of a CL line section the leading phase shift is obtained. It is unlike the lagging phase shift for an LC line section. Figure (3.28c) shows the above results on the (ω − β) diagram. In both the quadrants, the slope ψ at the origin determines the phase velocity, and the local slopes ϕ at the points P and Q determine the group velocities. Both these slopes rapidly increase with an increase in frequency, showing the fast increase in both velocities as given by equations (3.4.17) and (3.4.18). Thus, the lossless CL-line is a highly dispersive medium. The slope at the origin and the local slope have opposite sense, i.e. one is clockwise and another is anti-clockwise, showing the antiparallel nature of the phase and group velocities. Such waves are the backward waves. The second quadrant with vg > 0, vp < 0 shows the positive direction of backward wave propagation, while the first quadrant

with vg < 0, vp > 0 shows the negative direction of backward wave propagation, i.e. the reflected wave on the CL-line. The wave in the shaded light-cone region is fast-wave, whereas outside the light-cone it is the slow-wave. Equations (3.4.16) and (3.4.17) show that the propagation constant β decreases with an increase in frequency, whereas the phase velocity increases with an increase in frequency. Therefore, the CL-line has an anomalous dispersion. It is a highly dispersive transmission line. In the case of the material medium, the backward wave with the anomalous dispersion is associated with the strong absorption region. It is discussed in the subsection (6.5.1) of chapter-6 using the Lorentz model. The standard LPF type transmission line is known as the right hand, i.e. the RH-transmission line and the HPF type transmission line supporting the backward wave is known as the left hand, i.e. the LH-transmission line. The RH-line corresponds to the double-positive (DPS), i.e. both μ, ε positive medium, whereas the LH-line corresponds to the double negative (DNG), i.e. both μ, ε negative medium. These media are discussed in section (5.5) of chapter 5. The composite RH-LH (CRLH)-transmission line, giving some unique transmission characteristics, models the so-called metamaterials with negative permittivity and negative

73

74

3 Waves on Transmission Lines – II

Lsh

Lsh ωc L ω

β= ±

C

Lsh L

= ± L

β= ±

v0 ω

Δx Figure 3.29

Inductor loaded CL-line.

permeability [B.19, J.8]. This section considers only one isolated LC unit. The cascading of several such units forms the backward wave supporting LH-transmission line known as the metalines. It is further discussed in chapter 22.

Series Connected Parallel Lsh-C Type Line

The backward wave supporting CL-line, discussed above, has no cut-off frequency. Figure (3.29) shows the modified CL- line by adding a shunt inductor Lsh, inside the gray box, across the series-connected capacitor C. It is an HPF type CL-line that supports the backward wave with a cut-off frequency. The propagation characteristics of this line are obtained from the series impedance and shunt admittance p.u.l.: z=

1 − j ωLsh

ωLsh = , 2 + j ωC j ω Lsh C − 1

1 y= j ωL 3 4 19

The propagation constant of the line is

γ = jβ = ± = β=

j

ωLsh 1 zy = ± j ω2 CLsh − 1 jωL

Lsh L ω2 CLsh − 1 Lsh L

1 2

1 2

1 ω ωc 2 − 1

3 4 20

The cut-off frequency is ωc = 1 CLsh . The phase velocity of the usual LC-line is v0 = 1 LC. Therefore, the propagation constant of the shunt inductor Lsh in the series arm loaded CL-line is

1 1 − ωc ω

1 1 Lsh C ω

2

1 1 − ωc ω

2

1 1 − ωc ω

2

3 4 21

The inductor loaded CL-line behaves like a high-pass filter. The wave propagates for ω > ωc. For the frequency below cut-off, i.e. for ω < ωc, the wave is in the evanescent mode. The (ω − β) diagram of the inductor loaded CL-line is similar to the (ω − β) diagram of Fig (3.28c). However, the cut-off frequency is not shown in Fig (3.28c). A reader can easily add the cut-off frequency ωc in the dispersion diagram of Fig (3.28c). Unlike the unloaded CL, the present loaded CL line shows the cut-off frequency behavior. The present HPF type loaded CL line also supports the dispersive backward wave with phase velocity and group velocity opposite to each other. The propagation constant β decreases with frequency, whereas the phase velocity increases with frequency. It shows that the loaded CL-line has anomalous dispersion. The phase and group velocities of the backward wave are vp = vg =

ω = β

ω2 v0

1 = dβ dω

1−

ωc ω

2

vCL p

=

ω2 ωc 1− v0 ω

2

3 2

=

1−

ωc ω

2

vCL 1− p

a ωc ω

2

3 2

b

3 4 22 In summary, the inductor loaded CL-line shown in Fig (3.29) supports the backward wave. Above the cutoff frequency, i.e. for ω > ωc, and for the limiting case ω ∞, equations (3.4.22a) and (3.4.22b) reduce to equations (3.4.17) and (3.4.18), respectively. The phase velocity 2 of the unloaded CL-line is vCL LC = ω2 v0 . For p =ω ωC = 0 also, the above equations are reduced to equations (3.4.17) and (3.1.18). Therefore, above the cut-off frequency, i.e. for ω > ωc, the inductive loading of the transmission line shown in Fig (3.29) supports the backward slow-wave outside the light cone and the backward fast-wave within the light cone. Its dual structure, shown in Fig (3.27a), always supports the forward fast-wave. Series Capacitor Loaded LC-Line

The normal LC-line can also be loaded with a series capacitor Cs in the series arm. Figure (3.30) shows the

References

L

Cs

C

Δx Figure 3.30 Series capacitor loaded LC-line.

series capacitor loaded LC line. The propagation parameters of the loaded line are computed using the circuit analysis. The series arm impedance and the shunt arm admittance p.u.l. are given below: z = j ωL −

1 ωCs

a,

y = j ωC

b 3 4 23

The propagation constant of the capacitor loaded LCline is γ = jβ = ±

1 zy = ± jωC × jωL 1 − 2 ω LCs

ωc = ± jω LC 1 − ω

2

1 2

1 2

ω ωc β= ± 1− ω v0

2

1 2

,

3 4 24 where the phase velocity of the unloaded LC-line is v0 = 1 LC. The cut-off frequency of the loaded line is ωc = 1 LCs . For ω > ωc, the line behaves as the HPF. In absence of the series capacitance Cs loading, the line behaves like the LPF. The series-arm impedance is capacitive at a frequency below cut-off, i.e. for ω < ωc. The circuit of Fig (3.30) is reduced to the C-C line, i.e. a line with capacitive elements in both the series and shunt arms. It corresponds to the mu-negative (MNG) medium discussed in the section (5.5) of chapter 5. The C-C line blocks the low-frequency signal. Therefore, for the frequency ω < ωc, the propagation is in the evanescent mode with high attenuation. The phase and group velocities of the propagating waves are obtained as vp = v0 1 −

1 ωc 2 − 2

ω

a , vg = v0 1 −

ωc 2 ω

capacitor loaded LC-line. This line supports the forward wave and it is a normal dispersive transmission line. It supports the fast-wave above the cut-off frequency. This line is the dual structure of the line shown in Fig (3.29) that supports the backward wave propagation. Additional numbers of configurations for the loaded transmission lines could be obtained. For instance, the L-C section of a line, supporting the forward wave, could be cascaded with the C-L section of a line, supporting the backward wave. The composite line forms an interesting kind of the transmission line structure [B.19, J.8]. Both the series and parallel reactive loading of the lines can be done. Such loaded line structures have been realized in the planar technology to obtain novel properties useful for the development of novel microwave devices. They form the so-called metamaterials. The concept of metamaterials has been introduced in chapter 5 and elaborated in chapter 21. Chapter 22 considers the planar 1D-metalines and 2Dplanar metasurfaces, and chapter 19 discusses the planar periodic transmission lines.

References Books B.1 Pozar, D.M.: Microwave Engineering, 2 B.2

B.3

B.4

B.5 B.6

1 2

b

B.7

3 4 25 Equations (3.4.24) and (3.4.25) for β, vp, and vg are identical to equations (3.4.10a) and (3.4.10b) for the transmission line shown in Fig (3.27a). Thus, Fig (3.27b and c) also show the behavior of the present series

B.8 B.9

nd

Edition, John Wiley & Sons, Singapore, 1999. Fache, N.; Olyslager, F.; De Zutter, D.: Electromagnetic and Circuit Modeling of Multiconductor Transmission Lines, Clarendon Press, Oxford, NY, 1993. Rizzi, P.A.: Microwave Engineering-Passive Circuits, Prentice-Hall International Edition, Englewood Cliff, NJ, 1988. Ramo Simon, W.J.R.: Van Duzer Theodore, Fields, and Waves in Communication Electronics, 3rd Edition, John Wiley & Sons, Singapore, 1994. Collin, R.E.: Foundations for Microwave Engineering, 2nd Edition, McGraw-Hill, Inc., New York, 1992 Carson, R.S.: High-Frequency Amplifiers, 2nd Edition, John Wiley & Sons, New York, 1982. Elliott, R.S.: An Introduction to Guided-Waves and Microwave Circuits, Prentice-Hall, Englewood Cliff, NJ, 1993. Gardial, F.E.: Lossy Transmission Lines, Artech House, Boston, MA, 1987. Freeman, J.C.: Fundamentals of Microwave Transmission Lines, John Wiley, New York., 1996.

75

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3 Waves on Transmission Lines – II

B.10 Swanson, D.G.; Hoefer, W.J.R.: Microwave Circuit

B.11

B.12 B.13 B.14

B.15

B.16 B.17 B.18

B.19

B.20

Modeling Using Electromagnetic Field Simulation, Artech House, Boston, MA, 2003. Weber, R.J.: Introduction to Microwave Circuits, Radio Frequency and Design Applications, IEEE Press, New York, 2001. Collin, R.E: Field Theory of Guided Waves, IEEE Press, New York, 1991. Orfanidis, S.J.: Electromagnetic Waves and Antenna, Free Book on Web. Staelin, D.H.; Morgenthaler, A.W.; Kong, J.A.: Electromagnetic Waves, Prentice-Hall, Englewood Cliff, NJ, 1994. Sadiku, M.N.O.: Elements of Electromagnetics, 3rd Edition, Oxford University Press, New York, 2001. Cheng, D.K.: Fields and Wave Electromagnetics, 2nd Edition, Pearson Education, Singapore, 1089. Balanis, C.A.: Advanced Engineering Electromagnetics, John Wiley & Sons, New York, 1989. Mattick, R.E.: Transmission Lines for Digital and Communication Networks. IEEE Press, New York, 1995. Engheta, N.; Ziolkowski, R.W.: Metamaterials: Physics and Engineering Explorations, John Wiley & Sons, Inc., New York, 2006. Remoissenet, M.: Waves Called Solitons: Concepts and Experiments, Springer, New York, 1996.

Journals J.1 Kurokawa, K.: Power waves and the scattering matrix,

J.2

J.3

J.4

J.5

J.6

J.7

J.8

IEEE Trans. Microwave Theory Tech. Vol. 13. No. 2, pp. 607–610, 1965. Lei, Z.; Wu, K.: Short-open calibration technique for field theory-based parameter extraction of lumped elements of planar integrated circuits, IEEE Trans. Microwave Theory Tech., Vol. 50, No. 8, pp. 1861–1869, Aug. 2002. Hasegawa, H.; Furukawa, M.; Yanai, H.: Properties of microstrip line on Si-SiO2 system, IEEE Trans. Microwave Theory Tech., Vol. MTT-19, pp. 869–881, 1971. Jager, D.; Rabus, W.: Bias-dependent phase delay of Schottky contact microstrip line, Electron. Lett., Vol. 9, pp. 201–202, 1973. Veghte, R.L.; Balanis, C.A.: Dispersion of transient signal in microstrip transmission lines, IEEE Trans. Microwave Theory Tech. Vol. 34, pp. 1427–1436, Dec. 1986. Verma, A.K.; Kumar, R.: Distortion in gaussian pulse on microstrip-like transmission lines, Microw. Opt. Technol. Lett., Vol. 17, No. 4, pp. 253–255, March 1998. Hua, C.; Dogariu, A.; Wang, L.J.: Negative group delay and pulse compression in superluminal pulse propagation, IEEE J. Sel. Top. Quantum Electron., Vol. 9, No. 1, pp. 52–58, Jan–Feb 2003. Lai, A.; Caloz, C.; Itoh, T.: Transmission line based metamaterials and their microwave applications, Microwave Mag., Vol. 5, No. 3, pp. 34–50, Sept. 2004.

77

4 Waves in Material Medium – I (Waves in Isotropic and Anisotropic Media, Polarization of Waves)

Introduction The characteristics of EM-wave propagating on a planar line are strongly dependent on the nature of the materials used in planar technology. The familiarity with the characteristics of the medium and EM-wave propagation in the unbounded medium is important to understand the working of the planar transmission lines. These topics are extensively covered in several books [B.1–B.15]. Broadly speaking, the present chapter covers basic electrical characteristics of the material media and the EM-waves propagation in the unbounded dielectric media – both isotropic and anisotropic. In the first part of the present chapter, and also in chapter 6, attention is paid to the physical processes and the circuit models to understand the electrical properties of the material medium. The electrical and magnetic properties of the materials appear as the responses to the electric and magnetic excitations. Such excitations could be in the form of the circuit sources, such as the voltage and a current source. It could also be in the form of the field sources, such as the electric field intensity (E) and magnetic field intensity (H). The excitation could be any of three forms, namely (i) time-independent, i.e. the static or DC type; (ii) frequency-dependent, i.e. the time-harmonic dependent, or AC (phasor) type; and (iii) arbitrary time-dependent, i.e. the transient type. The discussion is limited to the static and time-harmonic type of responses of the materials, i.e. the material response and behavior in the frequency-domain. Objectives

• •• •

To review the EM-field quantities and medium parameters. To review the basic electrical properties of media. To obtain elementary circuit models of media. To review Maxwell’s equations.

• •• •

To present the wave equation in the unbounded lossless and lossy isotropic dielectric medium. To review wave polarizations. Jones matrix description of polarization states. To present the wave equation in the unbounded lossless anisotropic dielectric medium.

4.1 Basic Electrical Quantities and Parameters The electrical charge and the electric current are the primary electrical sources for the creation of the electric field and the magnetic field, respectively. The charge, also current (displacement current), is described by the flux field, i.e. the flux density ( D, B ). Two electrically charged bodies or two current-carrying conductors interact through the force fields, i.e. the field intensity ( E , H). The static charge creates the static electric field around itself, whereas the electric current creates the magnetic field around itself. The magnetic charge does not exist in nature. Sometimes, we talk about the magnetic charge, only as a mathematical source for the magnetic field. It is a hypothetical creation to maintain the symmetry of the field equations. The charge and current, i.e. flux fields are not determined by a medium, whereas the electric and magnetic interactions, i.e. force fields, between two separate bodies in a medium, are influenced by the electromagnetic parameters of the medium. 4.1.1

Flux Field and Force Field

The electric and magnetic fields are visualized through the line of flux. Thus, the electric charge (Qe) is described by the electric flux (Ψe). The total charge (QT) on a physical body and the corresponding electric flux are related by Gauss’s law:

Introduction to Modern Planar Transmission Lines: Physical, Analytical, and Circuit Models Approach, First Edition. Anand K. Verma. © 2021 John Wiley & Sons, Inc. Published 2021 by John Wiley & Sons, Inc.

78

4 Waves in Material Medium – I

QT = Ψe

411

If the charge is distributed throughout the volume of a body, in the form of volume charge density ρe, the elemental charge in the volume element dv is ρedv. The flux is further expressed as the electric flux density (D), i.e. the flux per unit area. The flux through the elemental surface area is dΨe = D d s It is shown in Fig. (4.1). The flux density is a vector quantity. It is also called the electric displacement vector. Gauss’s Law for Electric Flux

Total electric flux coming out of a closed surface = Total charge enclosed inside the volume of a closed surface, i.e. ρe dv

D ds = sur

412

vol

The above expression is the integral form of Gauss’s law. It can be converted to the differential form by using Gauss’s vector integral identity, ∇ D dv = vol

ρe dv,

∇ D = ρe

vol

413 Gauss’s Law for Magnetic Flux

Similar to the electric charge distribution, the magnetic charge distribution can be assumed in a volume of the body. The magnetic charge density is expressed as ρm. The magnetic charge creates a magnetic flux Ψm. Similar to the case of the electric charge, the elemental magnetic charge in the volume dv is ρmdv, and the elemental mag-

Again, by using Gauss’s vector integral identity, the above expression is written below in the differential form: ∇ B = ρm

415

However, the magnetic charges are not found in nature, i.e. ρm = 0. Therefore, ∇ B =0

416

The amount of charge, or current, is an absolute quantity. It does not dependent on the material medium. Thus, the corresponding flux or the flux density is also not dependent on the surrounding medium. In brief, the charge and current create the electric and magnetic flux field, i.e. the flux densities

D , B ; and these are

not influenced by a material medium. Experiments demonstrate that the electrically charged body, or a current-carrying conductor, interacts with other charged body, or another current-carrying conductor. Such interaction, i.e. the mutual force, is influenced by the medium surrounding these bodies. Therefore, medium-independent flux densities

D, B

cannot

explain the interaction between two charged bodies or current-carrying conductors. The interactions between the charges and current-carrying conductors take place through the force fields, expressed by the electric field intensity E and magnetic field intensity H . The field intensities

E , H are also responsible for the electro-

magnetic (EM)-power transportation through a medium. However, the field intensities are influenced by the electrical and magnetic properties of a medium.

netic flux coming out of the surface is dΨm = B d s , where B is the magnetic flux density as ds is the elemental surface of the enclosed volume v. It is also called the magnetic displacement vector. The Gauss’s law for the magnetic charge and magnetic flux can be written in the integral form as follows: ρm dv

B ds = sur

4.1.2

Constitutive Relations

It is noted above that the charge and current create the electric and magnetic flux fields, described by the flux densities

D, B

They also create the force field

described by the electric and magnetic field intensities 414

E , H . The flux density parameters D, B are related

vol

to the force field intensity parameters E , H by the following constitutive relations:

∧ n dψe D ds

Figure 4.1

The unit vector n is in the direction normal to the surface.

D = ε0 εr E = ε E

a

B = μ0 μr H = μH

b, 417

where ε0 and μ0 are the permittivity and permeability of the free space. The permittivity of any medium is its ability to store electric energy, such as a capacitor.

4.1 Basic Electrical Quantities and Parameters

Therefore, it is identified as the capacitance of the free space. Any dielectric material medium can store more electric energy through the mechanism of electric polarization. The electric dipole is created during the process of polarization and the total induced charge is shown as electric flux density D. Chapter 6 presents a detailed discussion of material polarization. The electric polarization of material under the influence of an external electric field gives a higher value of permittivity as compared to the permittivity of the free space. This is known as the relative permittivity εr of a medium. It is also known as the dielectric constant of a medium. The permittivity of a medium is ε = ε0εr. For an isotropic dielectric medium, εr is a scalar quantity. However, for an anisotropic medium, it is a tensor quantity. Likewise, the free space has also an ability to store magnetic energy. It is expressed as its permeability. Magnetic material is magnetized by the process of magnetization under the influence of an external magnetic field. Thus, magnetic material stores more magnetic energy as compared to the free space. The ability of a magnetic material to store magnetic energy is expressed through its relative permeability. The permeability of the medium is μ = μ0μr. The permittivity ε0 and permeability μ0 of the free space are the primary physical constants ε0 = 8.854 × 10−12 F/m, μ0 = 4π × 10−7 H/m. Again, for the isotropic magnetic medium, the relative permeability μr is a scalar quantity, and for an anisotropic medium, it is a tensor quantity. It is interesting to note that the velocity of EM-wave and the characteristic (intrinsic) impedance of the free space are given in terms of these primary constants, c=

1 ≈ 3 × 108 m sec, μ0 ε0

η=

μ0 = 377 Ω ε0 418

A material medium with a finite conductivity dissipates energy in the form of heat. The finite conductivity of a medium is due to the presence of free charge carriers. The free space is considered as a lossless medium because it has no free charge carrier. The conduction current flows through a medium under the influence of an external electric field. The conduction current density ( J c) in a medium is related to the electric field intensity

tensor. As free space has no conductivity, there is nothing like the relative conductivity of a medium. However, sometimes the conductivity of a medium is expressed in terms of the conductivity of copper. In summary, the electrical properties of a material are described by the relative permittivity (εr), relative permeability (μr), and conductivity (σ). A material can have all three properties at a time, or it can have one predominant property at a time. Assuming the case of one predominant property at a time, all materials are classified into three basic categories. 4.1.3

Category of Materials

Dielectric Materials

The dielectric materials support electric polarization of bound charges and electric displacement current through it. At the micro-level, the electric polarization creates dipole moments that appear as the permittivity of the dielectric material at the macro-level. The permittivity is frequency-dependent and lossy. So, permittivity is a complex quantity showing the lossy nature of dielectric materials. This is discussed in chapter 6. The relative permittivity of natural dielectric material is always positive and more than unity. However, engineered artificial dielectrics can have relative permittivity 0 < εr < 1. Such materials are known as the epsilon near zero (ENZ) materials. Under certain conditions, it can also acquire negative permittivity, creating the epsilon negative (ENG) medium [B.16]. It is discussed in section (5.5) of chapter 5. The electric displacement current flowing through a dielectric medium does not involve the flow of current through free charges. It is associated with timedependent electric fields. The electric displacement current density is a vector quantity. It expressed as follows: Jd=

∂D ∂E =ε ∂t ∂t

4 1 10

Magnetic Materials

The magnetic materials support magnetization, i.e. magnetic polarization by the magnetic field. It also supports magnetic displacement current due to the timedependent magnetic field:

( E ) by Ohm’s law: J c = σE,

419

where σ is the conductivity of a conducting medium. The conductivity (σ) of the isotropic conducting medium is a scalar and for an anisotropic conducting medium it is a

Md =

∂B ∂H =μ ∂t ∂t

4 1 11

A diamagnetic material has permeability μ ≤ μ0 and a paramagnetic material has permeability μ > μ0. The diamagnetic behavior is caused by induced magnetic

79

80

4 Waves in Material Medium – I

dipole moments, creating a magnetic field that opposes the externally applied magnetic field. However, the paramagnetic property comes into existence due to the alignment of already randomly existing magnetic dipole moments in the material. The conducting ferromagnetic materials have much larger permeability due to domain formation. However, the engineered artificial magnetic materials can have relative permeability less than unity, 0 < μr < 1. These are known as the mu near-zero (MNZ) materials [B.16]. Again, the engineered materials can also have mu negative, giving the mu negative (MNG) materials. It is discussed in section (5.5) of chapter 5. Conductors

It supports the flow of the free charge carrier, i.e. electrons. The free charge carriers are responsible for the conduction current density. It is given by equation (4.1.9). The free electrons confined within a neutral conductor form the plasma medium. The artificial dielectrics, including the wire-medium (i.e. rodded medium), are also modeled as the plasma medium. The ionosphere also supports the plasma medium [B.2]. The plasma medium is modeled using the Drude model, discussed in subsection (6.5.2) of chapter 6. The plasma medium provides a means to engineer materials with negative permittivity. These materials are called the epsilon negative materials, i.e. the ENG materials.

4.2

Electrical Property of Medium

Any material can be electrically characterized by the relative permittivity (εr), relative permeability (μr) and conductivity (σ), or resistivity (ρ). However, these electrical parameters are not constants for any given material. For instance, these are both temperature and frequencydependent. Also, these may not be uniform throughout the volume of a material. Further, the characterizing parameters may depend on the field intensity, the direction of the applied field, operating frequency, working temperature, and pressure. The parameters can also

depend on the history of a medium. However, the static value of these parameters, at room temperature, is treated as constant. A special kind of material, called chiral material, requires another parameter called chirality, i.e. the handedness of materials for its characterization [B.13, B.17]. Several properties of the medium are described briefly in this section. 4.2.1

Linear and Nonlinear Medium

The relative permittivity (εr) and the relative permeability (μr) of a linear material do not depend on the magnitude of the electric or magnetic field intensity, respectively. However, for nonlinear materials, the relative permittivity and relative permeability are functions of the electric and magnetic field intensity, respectively, and expressed as εr(E) and μr(H). Thus, for an isotropic nonlinear medium, equation (4.1.7a) is written as D = ε0 εr E E ,

421

and so forth. The where εr(E) = εr1 + εr2E + εr3E + coefficients εr2, εr3, … indicate the order of nonlinearity in the nonlinear relative permittivity. In the case of a time-harmonic electric field, i.e. E = E0ejωt, equation (4.2.1) is written as, 2

D = ε0 εr1 E0 ejωt + ε0 εr2 E20 ej2ωt + ε0 εr3 E30 ej3ωt + 422 It is obvious that while the input signal has only one frequency ω, shown in Fig. (4.2), the output of a nonlinear medium has several harmonically related frequency components, ω, 2ω, 3ω, … and so forth. Thus, a sinusoidal input signal gets distorted, once it passes through a nonlinear medium. Such distortion also occurs in an amplifier in the nonlinear region. Similarly, the relative permeability of a nonlinear magnetic medium is a function of the amplitude of the magnetic field. The constitutive relation, given by equation (4.1.7b), is written as B = μ0 μr H H

ω, 2ω, 3ω E

ω Input t

Figure 4.2

Nonlinear medium (εr)

Output

E

Response of nonlinear medium showing generation of harmonics.

t

423

4.2 Electrical Property of Medium

εr3 εr3

h3

εr2

h2

εr1

X Y

εr2 εr1

h1 Z

(a) Nonuniform medium.

O

h1

h2

h3 X

(b) Step variation of relative permittivity with substrate height.

Figure 4.3 Inhomogeneous medium showing a step variation of relative permittivity with substrate height.

where μr(H) = μr1 + μr2H + μr3H2 + and so forth. The coefficients μr2, μr3, … indicate the order of nonlinearity in the nonlinear relative permeability of a magnetic medium. 4.2.2 Homogeneous and Nonhomogeneous Medium The relative permittivity (εr) and the permeability (μr) are not necessarily uniform throughout the volume of a medium. These parameters could also be positiondependent. The variation in εr and μr could be in discrete steps, or they could be continuous functions of the position. Likewise, the conductivity of a medium can also be a function of position. If the parameters εr, μr, and σ are uniform throughout the medium, the medium is called homogeneous; otherwise, it is a nonhomogeneous medium or an inhomogeneous medium. A multilayer dielectric medium, forming a parallel capacitor, as shown in Fig. (4.3), is a nonhomogeneous medium, where the relative permittivity εr(x) is a function of position x in discrete steps. The conductivity of a doped Si substrate is a function of the depth of penetration of the charged carrier, forming a continuously variable nonhomogeneous medium. 4.2.3

Isotropic and Anisotropic Medium

Inside the isotropic dielectric medium, the electric displacement vector D and the electric field intensity E are parallel to each other, i.e. the applied electric field views the same relative permittivity of a medium in all directions. Likewise, the magnetic displacement vector B is parallel to the magnetic field intensity H within the isotropic medium. These properties are expressed through constitutive relations (4.1.7a) and (4.1.7b). For the isotropic media, permittivity and permeability are scalar quantities. However, there are dielectrics, such as quartz, sapphire, alumina, MgO, and so forth, where D and E are not

parallel to each other, i.e. they are not in the same direction. Such dielectrics form the anisotropic medium. In such a medium, the relative permittivity viewed by the applied electric field is direction-dependent. For instance, Fig. (4.3a) forms a composite anisotropic medium as the effective permittivity along the x-axis is different from the effective permittivity along the z-axis. Similarly, magnetic materials such as ferrite, garnet, and so forth are also anisotropic because B and H vectors are not in the same direction. Several authors have treated the properties of anisotropic medium and EM-wave propagation through such media in detail [B.1–B.4, B.9, B.11, B.13–B.15, B.17–B.23]. This subsection reviews basic concepts related to anisotropic media. The relative permittivity and relative permeability of these anisotropic media are not scalar quantities. They are tensor quantity, εr , μr described by 3 × 3 matrices. The constitutive relations of such electric and magnetic media are written as follows:

D = ε0 εr E ,

Dx

εr,xx

εr,xy

εr,xz

Ex

Dy

= ε0 εr,yx

εr,yy

εr,yz

Ey

Dz Bx B = μ0 μr H,

εr,zx εr,zy εr,zz μr,xx μr,xy μr,xz

a

Ez Hx

By

= μ0 μr,yx

μr,yy

μr,yz

Hy

Bz

μr,zx

μr,zy

μr,zz

Hz

b

424 In general, elements of permittivity and permeability matrices could be complex quantities and also frequency-dependent, accounting for the losses and dispersion in the material medium. Equation (4.2.4a) shows that for the anisotropic dielectric medium, the electric flux density D is not parallel to electric field intensity E . Likewise, equation (4.2.4b) shows that the vector B is not parallel to the vector H . For instance, if the incident field on an anisotropic dielectric medium has only Ex component, i.e. x-polarized incident E-field,

81

82

4 Waves in Material Medium – I

it generates all three components of electric flux density – Dx, Dy, and Dz. The same applies to the anisotropic magnetic medium. The above equations can be written in a more compact form as D = ε0 εr

E

a

B = μ0 μr

H

b 425

The above permittivity and permeability matrices could be either symmetric or anti-symmetric. Thus, the anisotropic materials could be divided into two broad groups: (i) symmetric anisotropic materials and (ii) anti-symmetric anisotropic materials. The symmetric anisotropic materials support linearly polarized EMwaves propagating as the normal modes of the homogeneous unbounded medium. However, circularly polarized EM-waves are the normal modes of the antisymmetric anisotropic medium. The normal modes of media travel without any change in polarization.

The above relation also holds for a real permittivity tensor of symmetric anisotropic dielectric material. In a dielectric material case, the off-diagonal elements of the matrix are symmetrical, i.e. εr, xy = εr, yx, and so forth. Similar expression can also be obtained for the symmetric anisotropic magnetic material. However, some media do not follow this symmetry rule. To describe an anisotropic medium, two sets of the coordinate systems are used: one set is for the crystal structure axes of the anisotropic medium; and another set is used for the physical axes of the line structure. Figure (4.4) shows the crystal axes (ξ, η, ς), i.e. (xi, eta, zeta), and the physical axes (x, y, z) of a microstrip line on an anisotropic substrate. The crystal axes (ξ, η, ς) are rotated with respect to the physical axes (x, y, z) by the angles θ1, θ2, and θ3. The offdiagonal elements of [εr] in equation (4.2.4) are present due to the nonalignment of two coordinate systems. However, if they are aligned, i.e. θ1 = θ2 = θ3 = 0, then the off-diagonal elements of [εr] are zero; and the constitutive relation (4.2.4a) reduces to Dx

Symmetric Anisotropic Materials

The complex permittivity matrix ε∗r , showing the permittivity tensor, of symmetric anisotropic dielectric material is a Hermitian symmetric matrix, i.e. the following relation holds: ε∗r = ε∗r

T

426

In the above equation, the matrix elements ε∗r are the complex conjugate of the matrix elements ε∗r . The superscript T shows the transpose of the permittivity matrix.

Dy Dz

= ε0

εrξ

0

0

Ex

0 0

εrη 0

0 εrζ

Ey Ez

The crystal axes (ξ, η, ς) are also known as the principal coordinate system of a material medium; and the diagonal relative permittivity components εrξ, εrη and εrζ are known as the principal relative permittivity components [B.1, B.3, B.10, B.12–B.14, B.24]. Normally, all relative permittivity components are positive quantities. However, it is possible to get one component as a

Crystal axis Y

ξ

w η Strip conductor

εrη

y εr|| z εr⟂

h εr⟂

Figure 4.4

X Structural axis

θ3 εrζ

Uniaxial substrate

Ey

θ1

Anisotropic substrate

x Z

εrξ

t=0

θ2

427

ζ

The crystal axes (ξ, η, ς) and the physical axes (x, y, z) of a planar anisotropic sheet.

4.2 Electrical Property of Medium

negative quantity in the engineered composites that provide unique EM-wave characteristics [J.1, J.2, B.16]. Along the principal axes, the components of the vector D are parallel to vector E . The dielectric materials are further classified into three categories: Type I: Isotropic materials. The relative permittivity components of these materials are identical, i.e. εrζ = εrη = εrζ. Thus, the relative permittivity of isotropic material is a scalar quantity. Type II: Uniaxial materials. These are anisotropic materials with relative permittivity components εrζ = εrξ = εr⊥, and εrη = εr . It is shown within a box in Fig. (4.4). In the case of alignment of crystal axes along the physical axes, permittivity components are expressed as εxx = εzz = εr⊥, εyy = εr . Thus, the permittivity tensor of the uniaxial anisotropic substrate is expressed as follows εr = ε0

εr,xx εr,yx

εr,xy εr,yy

428

For a uniaxial substrate, shown in Fig. (4.4), the applied external electric field Ey faces the relative permittivity component εr . The permittivity component εr is parallel (||) to the normal (y-axis) of an anisotropic substrate surface located in the (x-z) plane. In the (x-z) plane, the remaining two relative permittivity components have identical values εr⊥. The relative permittivity components εr⊥ are in the plane normal (⊥) to the y-axis, and also normal to the external electric field Ey. The yaxis is the main axis. It is also known as the C-axis, or the optic-axis, or the extraordinary axis. In the direction of the optic axis, the permittivity is different. The other two x- and z-axes are known as the ordinary axes. The x- and z-polarized EM-waves, known as the ordinary waves, in the (x-z)-plane travel with the same velocity, whereas y-polarized EM-wave, known as the extraordinary wave, travels with another velocity. The nonhomogeneous medium as shown in Fig. (4.3) is also a uniaxial medium with the x-axis as the optic axis, supporting the extraordinary wave propagation. The y and z-axes are the ordinary axes, supporting the ordinary wave propagation. Figure (4.4) shows a microstrip line of width w on an anisotropic substrate of thickness h. It forms a parallelplate capacitor placed in the (x-z)-plane. It views εr component of the uniaxial relative permittivity. Whereas, if the parallel plates of the capacitor are placed either in the (x-y)-plane or the (y-z)-plane, it will view the εr⊥ component of a uniaxial substrate medium. Thus, a uniaxial dielectric medium offers two different values

of capacitance, depending on the placement of the parallel plates, connected to a voltage source. Normally, manufacturers provide data for εr and εr⊥ of the uniaxial substrates. The constitutive relation for the uniaxial medium aligned to the physical axes is Dx Dy

= ε0

Dz

εr⊥ 0

0 εr

0 0

Ex Ey

0

0

εr⊥

Ez

429

In the case of the crystal axes ξ, η of an anisotropic substrate, shown in Fig. (4.4), have an angle θ = θ1 = θ2 with respect to the physical axes x, y; the relative permittivity components could be computed by the following expressions [B.24]: εrxx = εrξ cos 2 θ + εrη sin 2 θ εryy = εrξ sin 2 θ + εrη cos 2 θ

4 2 10

εrxy = εryx = εrξ − εrη sin θ cos θ Type III: Biaxial materials. For such an anisotropic medium, all three principal relative permittivity components are different, i.e. εrξ εrη εrζ. Such a medium is known as the biaxial medium [B.3]. The above nomenclatures are also applicable to the permeability of a magnetic material.

Anti-symmetric Anisotropic Materials

The plasma medium, i.e. the electron gas model of metal or ionosphere, is treated as an isotropic medium. However, in the presence of a static biasing magnetic field, it becomes a uniaxial dielectric material with off-diagonal elements for the permittivity matrix. This permittivity matrix is anti-symmetric. It does not support linearly polarized characteristic waves as the normal modes. It forms a gyroelectric medium, i.e. electrically gyrotropic medium. Similarly, in the presence of a static biased magnetic field in the z-direction, ferrite medium becomes a gyromagnetic medium, i.e. magnetically gyrotropic medium. Both plasma and ferrite media, under the magnetic biasing, support propagation of circularly polarized characteristic waves as the normal modes [B.2–B.4, B.21, B.22]. The permittivity and permeability tensors in matrix form for these media are summarized below.

Gyroelectric Medium

The relative permittivity matrix for gyroelectric medium, under the applied magnetic field in the z-direction, is expressed as follows:

83

84

4 Waves in Material Medium – I

εr = εr =

εr

jκ

− jκ

εr

0

0

0

εr,zz

where, εr = 1 + ωp =

Ne2 , mε0

0

ω2p ω2c

− ω2

, κ=

a ω2p ωc ω2c

− ω2

ω

, εr,zz = 1 −

ω2p

b

ω2

eB0 ωc = m

In the above expressions, ωp and ωc are the plasma frequency and cyclotron frequency. The cyclotron frequency is also called the gyrofrequency. The jκ is a cross-coupling factor responsible for the gyration property of the relative permittivity of a medium. Other parameters are N: electron density, e: electron charge, m: electron mass, and B0: DC magnetic field. In the case of the extremely high magnetic field, ωc ∞ and εr = 1, κ = 0. Also, in the absence of magnetic field, B0 = 0, κ = 0, and the gyrotropic medium is reduced to a uniaxial dielectric medium. In some cases, there is no plasma medium in the z-direction and εr, zz = 1.

Gyromagnetic Medium

The relative permeability matrix of a gyromagnetic ferrite medium is given below:

μr = μr =

εr =

c

4 2 11

μr

permeability tensors. These tensors for uniaxial MD materials are expressed as follows: 0 εr,0 0

jκ

0

− jκ μr

0

0 1 ω0 ωM ωωM where, μr = 1 + 2 , κ= 2 ω0 − ω2 ω0 − ω2 and ωM = μ0 γMs , ω0 = μ0 γH0

a

0

b c 4 2 12

In the above equations, H0 is the biasing magnetic field in the z-direction, Ms is the saturation magnetization, and the parameter γ is the gyromagnetic ratio. Again the cross-coupling factor jκ is responsible for the gyration property of the relative permeability of a medium. The permeability function shows a singularity at the frequency ω = ω0. It is suppressed in the presence of losses. In the case of unbiased and demagnetized ferrite, H0 = 0 and Ms = 0 leading to k = 0 and μr = 1. The ferrite, in this case, acts as a dielectric medium with permeability μ = μ0.

Magneto-dielectric Composite Materials

The magneto-dielectric (MD) materials have both electric and magnetic properties. These materials are characterized by the simultaneous presence of permittivity and

μr =

0

εr,0

0

0 μr,0

0 0

εr,e 0

0

μr,0

0

0

0

μr,e

a

b 4 2 13

The above-given expressions for the uncoupled independent relative permittivity and permeability tensors show that there is no coupling between the electric and magnetic fields. Ferrites have such property. However, ferrites are very lossy at microwave frequency. The low-loss MD materials are synthesized by mixing ferrites/hexaferrite and their composites with a polymer as a host medium [J.3]. The periodic structures are embedded in the host medium to engineer MD materials for antenna applications in VHF and UHF bands [J.4]. The metamaterial composites are also MD materials with simultaneously negative permittivity and permeability in a certain frequency band. Magnetoelectric Materials

The magnetoelectric materials are general bianisotropic electromagnetic materials with cross-coupling of electric and magnetic fields. These materials have anisotropy for both the permittivity and permeability with additional cross-coupling of the electric and magnetic field. In such materials, the electric flux density vector D and also magnetic flux density B depends on both applied E and H . It shows that in the bianisotropic materials, the E fields not only generate an electric polarization but also create the magnetic polarization, i.e. magnetization. Similarly, H fields applied to such materials create both magnetization and electric polarization. The constitutive relation relating four flux and field vectors for a linear magneto-electric medium is expressed as follows [B.21–B.23]: D =ε E +ξ H a B =ζ E +μ H

b 4 2 14

The above given four material parameter tensors ε, μ, ξ, and ζ describe the bianisotropic magnetoelectric materials. The tensors ε and μ are usual permittivity and permeability tensors, whereas ξ and ζ are magneto-electric coupling tensors. In general, 36 complex material parameters are required to characterize

4.2 Electrical Property of Medium

chirality parameter κ (kappa) and the cross-coupling parameter χ (chi). The chirality parameter κ measures the degree of the handedness of the medium. The parameter χ is due to the cross-coupling of fields. It decides the reciprocity (χ = 0) and nonreciprocity (χ 0) of the medium, giving the reciprocal and nonreciprocal material medium, respectively. In absence of cross-coupling, i.e. χ = 0, the parameters ξ and ζ are reduced to imaginary quantities, and the bi-isotropic medium is reduced to a nonchiral simple isotropic medium for κ = 0 and to a chiral medium for κ 0. It is also known as Pasteur medium. It supports the left-hand and righthand circularly polarized waves as the normal modes of propagation. It is a reciprocal medium. For κ = 0, χ 0 another medium, called Tellengen medium, is obtained. It is a nonreciprocal medium. The general bi-isotropic medium has χ 0, κ 0. It is a nonreciprocal medium. The gyrotropic medium and bianisotropic medium support left-hand and right-hand circularly polarized EM-waves. However, there is a difference. The gyrotropic medium supports the Faraday rotation, i.e. rotation of linearly polarized wave while propagating in the medium, whereas bianisotropic medium does not support it [B.21].

bianisotropic materials. However, the material parameter matrix could be diagonalized. So, for the uniaxial bianisotropic, i.e. the magneto-electric, materials, constitutive relations are written as follows [B.15, B.21]:

D=

B =

ε

0

0

0

ε

0

0

0

εz

ζ

0

0

0

ζ

0

0

0

ζz

E +

ξ

0

0

0

ξ

0

H

a

H

b

0 0 ξz μ 0 0 E +

0

μ

0

0

0

μz 4 2 15

In the absence of cross-coupling, i.e. for ξ = ζ = 0, the bianisotropic medium is reduced to an MD medium. Figure (4.5a) shows groups of general bianisotropic medium- isotropic, bi-isotropic, biaxial anisotropic, and bianisotropic media [B.25]. Two cases have already been discussed. The bi-isotropic materials are isotropic materials also showing cross-coupling of electric and magnetic fields. However, Fig. (4.5b) shows that the general bi-isotropic medium has special forms – isotropic, Pasture, Tellegen, and bi-isotropic. For general bi-isotropic medium, the medium tensors are reduced to scalars, and the constitutive relations given by equation (4.2.14) are reduced to the following simpler form [B.23]: D = εE + ξH where, ξ = χ + jκ

a , B = ζE + μH μ0 ε0

c,

ζ = χ − jκ

4.2.4

The medium in which the EM-wave propagates with equal phase velocity at all frequencies is called the nondispersive medium. For such a medium, the relative permittivity and relative permeability are real quantities, and they are independent of frequency. However, practically every material medium has losses; and thus their relative permittivity and relative permeability are

b μ0 ε0

d

4 2 16 where ξ /με is nearly unity. The magnetoelectric coupling parameters ξ and ζ have two components: the 2

Biaxial anisotropic ([ε], [μ]) (Direction dependence. No magneto – electric coupling) Bi-isotropic (ε, μ, ξ, ζ) (No-direction dependence. Magneto – electric coupling)

Isotropic (ε, μ) (No-direction dependence. No magneto – electric coupling) Bi-anisotropic ([ε], [μ], [ξ], [ζ]) (Direction dependence. Magneto – electric coupling)

(a) Groups of bi-anisotropic materials.

Nondispersive and Dispersive Medium

Pasture (Reciprocal, χ = 0 Chiral, k ≠ 0)

Isotropic (Reciprocal, χ = 0 Non-chiral, k = 0)

Tellengen (Non-reciprocal, χ ≠ 0 Non-chiral, k = 0)

Bi-isotropic (Non-reciprocal, χ ≠ 0 Chiral, k ≠ 0)

(b) Groups of bi-isotropic materials.

Figure 4.5 Classification of bianisotropic and bi-isotropic materials.

85

86

4 Waves in Material Medium – I

complex quantities, and these are a function of frequency. Thus, normally a material medium supports frequency-dependent phase velocity. Such a medium is called the dispersive medium. This kind of dispersion is known as material dispersion [B.2, B.3, B.9, B.10]. The material dispersion in the microwave and the mm-wave ranges is normally negligible for the substrates used in the planar technology. However, a composite substrate, made of the layered dielectric media, is dispersive. Likewise, the artificial metamaterial substrate is also dispersive. The plasma and conducting media are also dispersive in the microwave range. The material dispersion is an important phenomenon near the optical frequency. The constitutive relation for the dispersive medium is D ω =ε ω E ω

4 2 17

Lorentz oscillator model of a dielectric material, discussed in section (6.5) of chapter 6, helps to understand the frequency-dependent origin of the ε(ω). When an EM-pulse, like a Gaussian pulse, passes through a dispersive medium, its pulse-width widens due to the separation of different frequency components, as each frequency component travels at a different velocity in the dispersive medium. This is known as the pulsespreading phenomenon. It limits the speed of digital data transmission through the dispersive medium, as the digital bits can overlap each other. However, a dispersive medium can be nonlinear also, where the pulse spreading can be balanced by the nonlinearity. Under such combined dispersion, and nonlinearity, a pulse can propagate without changing its shape. Such robust EM-waves are called the solitons. The solitons are useful in optical communication. However, the solitons have also been generated in the microwaves frequency range 4.2.5

4.2.6

Static Conductivity of Materials

Figure (4.6a) shows a cylindrical section of the lossy material of conductivity (σ), i.e. resistivity ρ = 1/σ. Its cross-sectional area and length are A and h, respectively. The lossy material can be modeled as a resistor R, also shown in Fig. (4.6a). The conduction current density Jc flowing through the conductor, due to the free mobile charges, is Jc = Ic/A, where conduction current is Ic = V/R. The electric field intensity in the material is E = V h. On combining these expressions, the following relation is obtained: Jc=

All physical dielectric and magnetic material media have some amount of loss. Normally, the substrates used in the microwave planar technology do not have a high loss, except the doped semiconducting substrates. The lossy dielectrics are known as the imperfect dielectrics. The losses in the dielectric and magnetic materials change the relative permittivity and permeability into complex quantities: a b 4 2 18 The imaginary parts of the relative permittivity and permeability are taken as negative quantities due to

h E RA

J c = σE h where, σ = RA

a b c 4 2 19

The above expression (4.2.19b) is Ohm’s law for a lossy medium. The following expression of the equivalent resistance of a lossy material, in terms of its resistivity, follows from the above equation (4.2.19c): R=

Non-lossy and Lossy Medium

εr ∗ = εr − jεr , μr ∗ = μr − jμr

our choice of time-harmonic function ejωt. However, εr and μr are always positive quantities in a passive lossy medium. The lossy capacitor and the lossy inductor can model the complex relative permittivity and permeability of a medium. This is called the circuit modeling of a material medium. The modeling of the relative permittivity of a medium by the capacitor shows that both the medium and capacitor can store electric energy. Likewise, the relative permeability of medium and the corresponding inductor model both are the magnetic energy storage components.

ρh A

4 2 20

In general, the Ohm’s law for the anisotropic medium is written in the vector form as J c = σ E . Further, the expression is also obtained below for the conductivity σ of the material in terms of the basic parameters of mobile charges in a lossy material, i.e. in terms of electron charge and its mobility. The expression is also applicable to semiconductors. In the case of a semiconductor, the conduction current is due to both the electrons (negative charges) and holes (positive charges). Figure (4.6a) considers h = Δx length of a cylindrical section of conducting material with a cross-sectional area ΔA. The free charges move in the direction x with a velocity vx on the application of electric field intensity Ex. The conduction current in the x-direction is the rate

4.3 Circuit Model of Medium

Id

Id

A

Icap

A

Vx V

R

͠

E

Dielectric, ε*r

d

V ͠

Ir

C

R=1/G

V

Ic

h = Δx

(b) RC circuit model of a dielectric medium.

ε′r ε′r / ε″r

Icap = ωCV = ε0 ε′r E

E (a) A cylindrical section of conducting material.

I δ

ε″r

θ Ir = V/R = ε0 ε″r E

ω (d) Frequency response of a dielectric material.

(c) Defining loss-tangent.

Figure 4.6 Circuit model, parameters of a dielectric medium.

of flow of total charge ΔQe contained in a volume, ΔV = ΔA × Δx. Ic =

ΔQc ΔV dx = ρe = ρe ΔA Δt Δt dt

4 2 21

If N is the number of free charges per unit volume, with charge q on each carrier, the charge density is ρe = Nq. The equation (4.2.24) of the conductivity is changed to σ = N q μm

In the limiting case, v x = Lt Δx Δt = dx dt . Δt

Thus, I c = ρe ΔA v x , density is Jc=

0

and the conduction current

Ic = ρe v x ΔA

4 2 22

In a material, the charge movement is random due to the scattering and so forth. However, an average motion is assumed, giving the drift of charges in the x-direction with a drift velocity v x. The drift velocity is proportional to the electric field intensity that provides another expression for Jc: v x = μm E

a

J c = ρe μm E

b

4 2 25

In the case of a conductor, the charge carrier is electron, i.e. q = qe (the electron charge) and μ = μem (electron mobility). However, for a semiconductor, its conductivity σs is due to both electrons and holes leading to the following expression: σs = σe + σh = Ne qe μem + Nh qh μhm ,

4 2 26

where Ne and Nh are numbers of electrons and holes per unit volume. The charges on electron and holes are equal qe = qh = e = 1.6 × 10−19 Coulombs. The electron and hole mobilities, in a semiconductor, are μem and μhm, respectively. The signs of qe and μem are negative, whereas the signs of qh and μhm are positive. However, the conductivity σs of a semiconductor is always positive.

4 2 23 where constant μm is called the mobility of a charge. It is noted that μ is also used as a symbol for permeability. On comparing equations (4.2.19b) and (4.2.23b), the following expression for conductivity is obtained: σ = ρe μm

4 2 24

4.3

Circuit Model of Medium

The circuit model helps to understand the electrical property of a medium. It is further useful for simulating the electrical responses of a medium. The electrical

87

88

4 Waves in Material Medium – I

property of a dielectric medium is expressed through relative permittivity that shows the electric energy storage ability of the medium. The capacitor also stores electric energy. Therefore, a dielectric medium is modeled as a capacitor. The loss in a medium is due to the dissipation of energy that is modeled as a resistor. Similarly, the permeability of a medium, such as an inductor, shows its ability to store magnetic energy. Therefore, the permeability of a medium is modeled as an inductor [B.10–B.12].

Both current components are shown in Fig. (4.6b). The loss-tangent, showing the dissipation factor of the RC circuit, is defined using Fig. (4.6c): Loss current Ir Dissipation factor, tan δ = = Icap Reactive current 1 tan δ = ω RC 434

4.3.1 RC Circuit Model of Lossy Dielectric Medium

Therefore, the dissipation factor, i.e. the loss tangent (tan δ), of lossy dielectric material is defined using equation (4.3.2) as follows: Ir Jr A ε = r, = 435 tan δ = Icap εr Jcap A

Figure (4.6b) shows a parallel-plate capacitor, containing a lossy dielectric medium with complex relative permittivity ε∗r = εr − jεr . It further shows its RC circuit model that is a parallel combination of the capacitor (C) and resistor (R). It is connected to a time-harmonic voltage source v = v0ejωt that produces a time-harmonic electric field, E = E0ejωt in the dielectric medium. The displacement current density in the dielectric medium is dD d Jd = = ε0 ε∗r E0 e jωt dt dt Jd = jωε0 ε∗r E = jωε0

εr − jεr E

431

Jd = jωεr + ωεr ε0 E Id = Jd × A = ωεr + jωεr ε0

a

Cd ε0 A d εr = ω ε0 R A εr =

a b 436

AV d

b

The displacement current density has two components with the quadrature phase: Jd = Jr + j Jcap Jr = ωεr ε0 E

where A is the area of the parallel-plate capacitor. The loss current is zero, i.e. Ir = 0 for a lossless capacitor, and also for a lossless dielectric medium. It leads to R ∞ , and εr = 0, tanδ = 0. In Fig. (4.6c), the angle θ is the power-factor angle. On comparing equations (4.3.2) and (4.3.3), an equivalence is obtained between the lossy dielectric medium and a lossy capacitor

Jcap = ωεr ε0 E

a b

432

The reactive current density (Jcap) flows through εr, i.e. through the capacitor. It is shown by the presence of “j.” Thus, the real part of the relative permittivity εr is modeled as a capacitor C. The resistive current density (Jr), causing a loss in the dielectric medium, flows through εr , i.e. through resistor R. The imaginary part of relative permittivity εr is modeled as a resistor R, parallel to the capacitor C. It is obvious from equation (4.3.1) that the loss caused by εr is a positive quantity only when the complex relative permittivity ε∗r is defined as a subtractive combination of its real and imaginary parts mentioned in equation (4.2.18a). The current through the equivalent circuit is I = Ir + j Icap = 1 R + j ωC V = G + j ωC V Ir = V R

a

Icap = ωC V

b 433

On replacing the dielectric medium of Fig. (4.6b) by the air medium, i.e. εr = 1, capacitance C0 is obtained: ε0 A 437 C0 = d On using the above equations, the real and imaginary parts of a complex relative permittivity and loss tangent are defined in terms of the circuit elements: C εr = a C0 1 εr = b ω RC0 ε 1 c tan δ = r = εr ω RC 438 The above equations provide a practical means to measure the dielectric constant of any dielectric material with the help of a parallel-plate capacitor. Equation (4.3.8a) also gives a practical definition of the relative permittivity of a homogeneous dielectric medium. The relative permittivity is a ratio of capacitances of a parallel-plate capacitor, with a material medium and with the air medium; while keeping the geometry of the parallel-plate capacitor unchanged. We get a

4.3 Circuit Model of Medium

homogenized dielectric medium even if the parallelplate capacitor, as shown in Fig. (4.3a), is made of layered dielectric sheets. The measurement of capacitance C ignores the layered medium and views it as a homogeneous medium. So, the relative permittivity of a material is the macrolevel homogenization concept that ignores the microlevel discrete composition of a medium. The concept of homogenization is important to design the engineered metamaterials using the discrete metallic and nonmetallic structures embedded in a host medium. It is discussed in section (21.4) of chapter 21. Figure (4.6d) shows the frequency response of a lossy dielectric medium, as predicted by the RC circuit model. The real part of the permittivity εr is frequency independent, whereas the imaginary part of the permittivity εr decreases hyperbolically with frequency. Some dielectric materials may not exhibit this kind of frequency response. More realistic circuit models may be needed for such a dielectric medium. Chapter 6 discusses a few more circuit models of the dielectric media. The loss-tangent of a dielectric is also a measurable quantity. Manufacturers provide data on it. However, the loss of a semiconducting substrate is characterized by the conductivity (σ) of a substrate. Even a dielectric material can have some amount of free charge carriers, contributing to its conductivity (σ). The finite conductivity causes a dielectric loss in the material. The imaginary part of the complex relative permittivity εr arises due to the damping of oscillation during the polarization process of a dielectric material, under the influence of an externally applied AC electric field discussed in chapter 6. However, it is difficult to distinguish between two sources of the dielectric loss; the contribution of the free charge carriers (conduction current) and the contribution of the dielectric polarization (polarization current). Therefore, both could be grouped in the total loss-tangent. The parallel-plate capacitor, shown in Fig. (4.6b), supports two kinds of current densities – the conduction current density, Jc given by equation (4.1.9), and the displacement current density, Jd given by equation (4.3.1a). The total current density is Jt = σ E + jωε0 εr + ε0 εr E Jt = σ + ωε0 εr E + jωε0 εr E 439 Loss current Reactive current The total loss-tangent, from equations (4.3.4) and (4.3.9) of dielectric material is tan δt =

σ + ωε0 εr ωε0 εr

Equation (4.3.9), in a changed form, is rewritten as follows: σ Jt = jωε0 εr − j εr + E a ωε0 Jt = jωε0 εr − j εreq E

4 3 11 The equivalent εreq of lossy dielectrics, due to the combined effect of polarization and finite conductivity, is σ 4 3 12 εreq = εr + ωε0 The lossy dielectric medium is also described by the concept of the complex equivalent conductivity σ∗eq = σeq + j σeq . Using the expression Jt = σ∗eq E and equation (4.3.9), the complex equivalent conductivity is expressed as follows: σ∗eq = σeq + jσeq = σ + ωε0 εr + jωε0 εr

4 3 13

The real part of a complex equivalent conductivity causes the dielectric loss in a medium, whereas its imaginary part stores the electric energy of the dielectric medium. Therefore, the imaginary part of a complex equivalent conductivity is related to the relative permittivity of a medium, and its real part is associated with the imaginary part of the complex relative permittivity: σeq = ω ε0 εr a σeq = σ + ω ε0 εr

b 4 3 14

Sometimes, loss due to the polarization causing εr is ignored, say in a semiconductor, if the loss due to the conduction current is more significant. The loss characteristic of a semiconducting substrate is given by its conductivity. In that case, the loss-tangent is expressed in terms of the conductivity of a medium: ε σ tan δ = r = 4 3 15 εr ω ε0 εr The above expression helps to convert the conductivity of a substrate to its loss-tangent at each frequency of the required frequency band. It can also be used to convert the loss-tangent to the conductivity at each frequency. The equivalence between the relative permittivity and capacitance has been obtained by treating both the dielectric and capacitor as electric energy storage devices. Thus, the complex relative permittivity is equivalent to a complex capacitance. Figure (4.6b) provides the admittance of a lossy capacitor: Y = G + jωC,

4 3 10

b

Y = jω C − j

G = jωC∗ , ω 4 3 16

89

4 Waves in Material Medium – I

where the complex capacitance C∗ is given by ∗

C = C −jC

4 3 17

The real and imaginary parts of a complex capacitance, and also loss-tangent, are given by the following expressions: C =C G C = ω C G 1 = = tan δ = C ωC ω RC

a b c

transmission line by the variational method. It is discussed in chapter 14.

4.3.2

The complex relative permittivity is also expressed as follows: ε∗r

C −jC = εr − j εr = C0

4 3 19

The above expression is helpful in the computation of the real and imaginary parts of the effective relative permittivity of a lossy planar transmission line. The complex line capacitance of a lossy planar transmission line can be numerically evaluated. It also helps to compute the effective loss-tangent of a multilayered planar

μ∗r = μr − jμr =

Ir R

I

I

V (a) Circuit containing magnetic material.

O

V (b) Equivalent circuit.

μ′r μ″r

ω

(c) Frequency response of real and imaginary parts of permeability. Figure 4.7

L∗ L0

4 3 20

A magnetic material placed inside a coil, shown in Fig. (4.7a), gets magnetized in the process of magnetization. The current causing the magnetization is known as the magnetization current (Im). However, during the magnetization process, some power is lost. It is viewed through the magnetic loss-current (Ir). The magnetization of a magnetic material involves the storage of magnetic energy. So, a lossy magnetic material is modeled through the RL equivalent circuit, shown in Fig. (4.7b).

L

Im

Magnetic material

Circuit Model of Lossy Magnetic Medium

The magnetic loss in magnetic materials is due to the process of magnetization that results in a complex relative permeability, μ∗r = μr − jμr . The relative permeability is defined as a ratio of two inductances; inductance (L∗) of the coil with a magnetic material, and inductance (L0) of the same coil with the air-core,

4 3 18

μ′r = L/L0 , μ″r

90

Circuit model and frequency response of lossy magnetic material.

4.4 Maxwell Equations and Power Relation

The inductor models the stored magnetic energy in a magnetic material, whereas the resistor models its loss. A time-harmonic current I = I0 ejωt flows through the coil containing the magnetic material. The voltage across the coil and current through it are given below: dI = jωμ∗r L0 I, dt μr + j μr V I= = −jV ∗ j ωμr L0 ωL0 μ 2r + μ V = L∗

a 2 r

b 4 3 21

Figure (4.7b) shows the current flowing in the parallel RL equivalent circuit. By comparing the above current, with the circuit model current; the expressions for the circuit elements, in terms of the material parameters, are obtained as follows: I = Im + Ir = − j L = L0

μ 2r + μ μr

R = ωL0

V V + ωL R

2 r

μ 2r + μ μr

a b

2 r

c 4 3 22

The magnetic loss-tangent of the lossy inductor is defined below that helps to get the magnetic loss-tangent of magnetic material: Ir ωL = Im R μr tan δm = μr

tan δm =

a b 4 3 23

A lossless magnetic material has μr = 0, i.e. L = μr L0 , R ∞. It means an inductor, without the resistor, models the lossless magnetic material. It shows that the permeability is an ability of a magnetic material to store magnetic energy, just as the permittivity of dielectric material is its ability to store electric energy. For the case of the low-loss magnetic materials, μr μr , the following expressions are obtained from equations (4.3.22b) and (4.3.22c): L L0 ωL μ μr = R r μr =

It is observed that for a low-loss magnetic material, μr is independent of frequency, whereas μr increases linearly with frequency, as shown in Fig. (4.7c). The above equations show that inductor models the real part of the complex permeability, whereas resistance models its imaginary part. This kind of frequency response may not be a realistic description of a hysteretic magnetic material.

a

4.4 Maxwell Equations and Power Relation The experimental observations on the time-dependent electromagnetic field are described by a set of the vector differential equations, known as the Maxwell equations. These sets of equations form the fundamental laws of nature, governing behaviors of the EM-field in free space, and also in material media. Each part of Maxwell equations shows a definite nature of the EM-field, explored by different investigators; Faraday, Ampere, and Gauss. However, Maxwell has put them in the form of a consistent set of equations using a mathematical format called the quaternions. He has also introduced the concept of displacement current. On solving the equations, either for the electric field or the magnetic field, the wave equations are obtained for the electric and magnetic fields traveling at a velocity of light in free space. Poynting has determined the power carried by the EM-wave. Finally, Heaviside replaced the quaternions by the vector notations and presented the Maxwell equations in the present form, i.e. in the convenient form of the vector differential equations [B.26–B.28]. Such a formulation of the electromagnetic theory has opened the grand path of modern research and investigation. So, the modern form of Maxwell equations can be truly called Maxwell–Heaviside equations. However, in this text, we use normal terminology to follow the current practice.

4.4.1

Maxwell’s Equations

The set of Maxwell’s equations, given below, consists of four time-dependent vector equations; relating the sources such as conduction current density ( J c ), electric displacement current density

J d = ∂ D ∂t , and mag-

netic displacement current density b 4 3 24

the magnetic field (H respectively:

J d,m = ∂ B ∂t

to

and the electric field ( E ,

91

92

4 Waves in Material Medium – I

∇× E = −

∂B − Mext ∂t

∇ × H = Jc +

∂D + J ext ∂t

∇ D =ρ ∇ B =0

a b c d 441

All quantities in the above equations are space-time dependent. The current densities J c , J d , and J d,m are not the power supplying sources to the propagating EM-wave in a medium. These current densities are created by the externally applied magnetic and electric current densities Mext and J ext supplying power to the EM-wave and partly getting absorbed as loss in a conducting medium. The right-hand sides of the above equations could be treated as the sources (excitations) and the left-hand fields as the responses. The force field quantities ( E ,H), flux field quantities ( D, B ), and current densities are functions of both the space variables and time variable. Further, in any material medium, the flux field quantities are related to the force field quantities by the constitutive relations, given in equation (4.1.7). The conduction current density in a lossy medium is related to the electric field, as given in equation (4.1.9). In general, εr, μr, σ are the tensors quantities for an anisotropic medium. However, these are scalar quantities for an isotropic medium. They are also treated as complex quantities to include the losses of a medium. In the case of a complex conductivity σ∗, its real part is responsible for the loss in a medium, whereas its imaginary part accounts for the energy storage. In a dispersive medium, εr, μr and σ are also frequency-dependent. The characteristics of various kinds of media, such as dielectrics, conductors, plasma, semiconductors, ferrites, and so forth are accounted for in Maxwell’s equations through the constitutive relations applicable to these physical media. Maxwell’s equations, along with the constitutive relations, are the field equations, not the force equations, i.e. these equations do not express the forces exerted by the fields on stationary or moving charges. This is achieved through Lorentz’s force equation: F =q E + v × B ,

442

where q is the charge on a mass m that is moving with velocity v. In this case, Lorentz’s force acting on a moving charge is equal to Newtonian force: m

dv = F =q E + v × B dt

443

Lorentz’s force has two components: (i) electric force component and (ii) magnetic force component. The electric force F e = qE is exerted either on a moving or on a stationary charge in the static, or in the time-varying electric field. The magnetic force F m = qv × B is exerted only on a moving charge in the static, or in the time-varying, magnetic field. In the case of a time-varying electric, or the time-varying magnetic field, both fields are always present and are related through Maxwell’s equations (4.4.1a) and (4.4.1b). Thus, both components of Lorentz’s force are present on a moving charge in a time-varying EM-field. It is observed that in the absence of the external sources, in a lossless medium Maxwell’s equation (4.4.1a) states that a time-varying magnetic field creates a time-varying electric field; and Maxwell’s equation (4.4.1b) states that a time-varying electric field ( D = ε0 εr E ) creates a time-varying magnetic field. Thus, Maxwell’s equations (4.4.1a) and (4.4.1b) form a set of the coupled equations, showing an interdependence of the time-varying electric and magnetic fields. It is like the two-variable simultaneous equations that occur in ordinary algebra. However, in the present case, the field variables ( E ,H are vector quantities. The coupled partial differential equations are solved either for the E or H by following the rules of the vector algebra. The solutions provide the wave equation either for the electric ( E ) or for the magnetic (H) field. Equation (4.4.1c) of Maxwell’s equations is a divergence relation. It shows that the electric field originates from a charge, and ends on another charge. Equation (4.4.1d) shows that the divergence of the magnetic field is zero; meaning thereby that a magnetic charge does not exist in nature. The magnetic field exists as a closed-loop around a current-carrying conductor. However, sometimes a presence of the hypothetical magnetic charge is assumed, and the divergence equation (4.4.1d) is modified as ∇ B = ρm. This assumption maintains the symmetry of Maxwell’s equations. Likewise, to maintain the symmetry of Maxwell’s equations, a hypothetical magnetic current density (−Jm) term is also to be added to equation (4.4.1a). The modified Maxwell equations in the symmetrical form are given below: ∇ × E = − Jm − ∇ × H = Jc +

∂B − Mext ∂t

∂D + J ext ∂t

a b

∇ D =ρ

c

∇ B = ρm

d 444

4.4 Maxwell Equations and Power Relation

ε, μ, σ

n^

E, H

Jext, Mext

S

ds

S

A

n^

V A

Material media

C

d1 (a) External surface electric and magnetic (b) A closed-surface S currents creating a field in outer enclosure. formed by perimeter C.

(c) A volume V enclosed by the surface S.

Figure 4.8 Surfaces and volume used in the integral form of Maxwell equations.

In a real material medium, a current can flow due to the time-dependent electric polarization (P), and also due to the time-dependent magnetic polarization (M). These are known as the electric and magnetic polarization currents and are incorporated in Maxwell’s equations as the electric and magnetic displacement current densities. Sometimes, the external sources, magnetic current density − Mext and electric current density J ext are further added to Maxwell equations (4.4.1a) and (4.4.1b). The external current sources are retained in the modified form of Maxwell equations. Figure (4.8a) shows externally impressed current sources, creating fields in an enclosure. These externally applied current sources supply power to create the electric field and magnetic field in a material medium. They also cause the flow of current in a lossy medium. It is examined in the next section while discussing the energy balance of the electromagnetic field [B.8, B.9]. The lossless free space is treated as a charge-free and source-free medium, i.e. Mext = J ext = J c = − J m = ρ = ρm = 0. A divergence of Maxwell’s equations (4.4.1b) in absence of J ext , and using equation (4.4.1d), leads to the following continuity equation: ∇ Jc = −

∂ρ ∂t

445

The above equation shows the conservation of charge. In any enclosed volume, the decrease of charge is associated with a flow of current out of the volume. Normally, the time-harmonic fields are used in most of the applications. The time-dependent electric field and other field quantities are written in the phasor form for the time-harmonic field: E x, y, z, t =

Re E x, y, z

ejωt

Instantaneous time Space coordinate Time harmonic dependent field dependent phasor variation field

446

H x, y, z, t = Re H x, y, z ejωt

a

D x, y, z, t = Re D x, y, z ejωt

b

B x, y, z, t = Re B x, y, z ejωt

c

J x, y, z, t = Re J x, y, z ejωt

d

q x, y, z, t = Re q x, y, z ejωt

e 447

It is noted that the same notation is used for both the and only time-space dependent fields [ E x, y, z, t space-dependent fields [ E x, y, z in the phasor form. The context of the discussion can clarify the situation. In general, the space coordinates based [ E x, y, z etc.] phasor fields are complex quantities, with both the magnitude and phase angle associated with them. The real part Re E x, y, z ejωt gives actual sinusoidal field quantity with phase relation. E x, y, z, t is the instantaneous field quantity without any phase term. Further, the following relations are useful: d E x, y, z ejωt = jωE x, y, z ejωt dt j E x, y, z ejωt dt = − E x, y, z ejωt ω

a b 448

Using the above equation for the electric field and similar equation for the magnetic field, Maxwell’s equations (4.4.1), in the external source free medium, are rewritten below in the phasor form for the timeharmonic field: ∇ × E = − jω B

a

∇ × H = J c + jω D

b

∇ D =ρ

c

∇ B =0

d 449

93

94

4 Waves in Material Medium – I

where flux density vectors are related to field intensity vectors in an anisotropic medium by the following tensor constitutive relations: D =ε E

a

B =μH

b

Jc = σ E

c

∇× E S

a

∇ × H = σ + jωε0 εr E

b

∇ E =ρ ε

c

∇H=0

d,

E dl = −

4 4 11 B = μ0 μr H, J c = σ E. The differential form of the above given Maxwell’s equations provide relations between the electric field and the magnetic field at any location in the medium. Also, the sources are specified at a point in the medium. The Maxwell equations (4.4.9) involving the flux densities D , B and field intensities E , H apply to both the isotropic and anisotropic media, whereas Maxwell’s equations (4.4.11) involving the field intensities E , H apply to the isotropic medium only. The differential form of Maxwell equations does not account for the creation of the fields in the space due to the sources such as the charge or current distributed over a line, surface, or volume. This case is incorporated in Maxwell’s equations by converting them to the integral forms. It is achieved with the help of two vector identities: ∇ × A ds S

4 4 12 ∇ A dv

A ds =

Gauss divergence theorem S

∂ ∂t

B ds

= −

S

∂ψm , ∂t 4 4 15

where ψm is the magnetic flux. It is the Faraday law of induction that gives the induced voltage V, i.e. the emf, on a conducting loop containing the time-varying magnetic flux, ψm:

where the scalar constitutive relations are D = ε0 εr E ,

A dl =

B ds S

It is assumed that a surface enclosing the magnetic field does not change with time. By using equation (4.4.12), equation (4.4.14) is changed in the following form:

C

∇ × E = − jωμ0 μr H

c

S

∂B ∂ ds = − ∂t ∂t

4 4 14

4 4 10 In the case of an external source-free homogeneous isotropic medium, Maxwell’s equations are written in terms of field intensities only:

Stokes theorem

ds = −

emf = V =

E dl = −

∂ψm ∂t

4 4 16

Likewise, using Maxwell’s equation (4.4.1b) and (4.4.12) for J ext =0, the second Maxwell’s equation is written in the integral form, giving the modified Ampere’s law: Hdl = C

JC d s + S

∂ ∂t

D ds = S

Jc + Jd d s S

4 4 17 In the above equation, Jc and Jd are conduction and displacement current densities creating the magnetic field H . The above expression is generalized Ampere’s law due to Maxwell. The magnetomotive force, mmf, is obtained as follows: mmf = H d l =

Jc + Jd d s

a

S

C

H dl =

Jc d s + S

∂ψe ∂t

b 4 4 18

V

4 4 13 Figure (4.8b) shows the existence of a vector A over the surface S. Its boundary is enclosed by the perimeter C. Stoke theorem is defined with respect to Fig. (4.8b) and Gauss divergence theorem with respect to Fig. (4.8c). The unit vector n shows the direction of a normal to the surface S. Figure (4.8c) shows a vector A existing in the whole of the volume V that is enclosed by the surface S. Maxwell’s equations in the integral form, for Mext =0, are obtained by taking the surface integral of Maxwell’s equation (4.4.1a):

where ψe is the time-dependent electric flux. Equation (4.4.18b) is Maxwell’s induction law giving the induced mmf due to the time-varying electric field. For the source free medium with Jc = 0, it is the complementary induction law of Faraday’s law of induction.

4.4.2 Power and Energy Relation from Maxwell Equations A medium supporting the electromagnetic fields also stores the EM-energy and supports the power flow. The EM-power is supplied to the enclosure by the

4.4 Maxwell Equations and Power Relation

time-dependent external electric current density Jext and the time-dependent external magnetic current density Mext. They create the time-dependent electric field ( E and the time-dependent magnetic field (H ) shown in Fig. (4.8a). The external power, supplied by the source to the medium, is Pext =

H Mext + E J ext ds

4 4 19

where the fields E and H are normal to each other. The power density S is defined by a vector product of E and H, known as the Poynting vector: S = E ×H

4 4 20

The divergence of the above equation provides the power entering, or leaving, a location in the space: ∇ S =∇

E ×H =H∇× E−E ∇×H 4 4 21

S

The field and source quantities have RMS values, and these are also time-dependent. The power on a transmission line, carrying the voltage and current wave, is P = VI cos ϕ, i.e. a scalar product of the voltage and current. The EM-wave is a transverse electromagnetic wave,

The energy contained per unit volume, i.e. the energy density, in a dispersive and a nondispersive medium, in the form of the electric and magnetic energy, is given by the following expressions:

1 1 ε0 εr E E + μ0 μr H H 2 2 1 ∂ 1 ∂ εr ω E E + μ0 μ ω H H W = ε0 2 ∂ω 2 ∂ω r

Nondispersive medium W =

a

Dispersive medium

b

A physical medium is dispersive. For a dispersive medium, the modified equation (4.4.22b) is valid [B.2, B.8, B.11–B.15]. The energy density W is a positive quantity, i.e. the following relations must be satisfied: ∂ εr ω ∂ω

∂ μr ω ∂ω

> 0,

>0

4 4 23

The medium having finite conductivity σ dissipates the EM-energy in the form of heat given by the Joule’s law: P d = E Jc = σ E

2

4 4 24

The total power carried in a medium, in the form of the EM-wave, is

(4.4.22)

σ E 2 dv

Pdis =

4 4 27

V

Thus, power (Pext) supplied by the external source is balanced by the following equation: Pext = Pwave + Pdis +

∂ W energy ∂t stored

4 4 28

Using Maxwell equations, the power balance equation (4.4.28) is evaluated in terms of the field quantities and external sources. The resulting power balance equation identifies the above-mentioned expressions for the power in a wave, energy dissipated in the medium, energy stored in the medium, and also the energy supplied by the external electric and magnetic currents. The dot product of equation (4.4.1a) is taken with H,

Pwave =

E × H ds

4 4 25

S

The integration is carried over the cross-section of the medium carrying the EM-power. The total energy stored in volume V is W energy = stored

1 2

and the dot product of equation (4.4.1b) with E, and subtract one from another to get the following expression: H ∇ × E − E ∇ × H = − H Mext + E J ext − σ E E − μ0 H

∂ μH ∂t r

− ε0 E

∂ εr E ∂t

4 4 29

ε0 εr E 2 + μ0 μr H 2 dv V

4 4 26 The total power dissipated in volume V of the medium is

In the above equation, J c = σ E is used. In the case of a time-invariant medium, the relative permittivity and relative permeability of a medium are constants. By using an equation (4.4.21), the above equation is rewritten as follows:

95

96

4 Waves in Material Medium – I

∇

E × H = − H Mext + E J ext −σ E E −

∂1 ε0 εr E 2 + μ0 μr H 2 ∂t 2 4 4 30

H Mext + E J ext dv = − v

σ E 2 dv +

E × H ds + V

S

Pext

Pwave

∂ W energy ∂t stored

is the oscillating electric

and magnetic energy in the medium.

medium with Mext = J ext = 0 to obtain separate wave

∇× ∇× E

= − μ0 μr

(4.4.31)

V

equations for the electric and magnetic fields. It is like getting the voltage and current wave equations on a transmission line. Maxwell equations are further presented in the vector algebraic form. The present section is concerned with the uniform 1D wave propagation in an unbounded isotropic dielectric medium, also in a conducting medium. The EM-wave propagation in the anisotropic medium is discussed in section (4.7).

EM-wave Equation

source-free medium Mext = J ext = 0, are solved, using the rule of vector algebra, for the electric field intensity E by substituting H equation (4.4.1a):

∇× ∇× E

∂ ∂E , σ E + ε0 εr ∂t ∂t

In the above equation, the identity ∇ × ∇ × E

stored

Maxwell’s coupled equations (4.4.1), in the external

Maxwell’s coupled vector differential equations (4.4.1a) and (4.4.1b) are solved in an external source-free

∂ ∇× H ∂t

ε0 εr E2 + μ0 μr H 2 dv

Wenergy

4.5.1

4.5 EM-waves in Unbounded Isotropic Medium

∇ × ∇ × E = − μ0 μr

∂1 ∂t 2

Pdis

The power supplied by the external sources is positive. So, the total power in a medium is negative. Out of the total power supplied to a medium, Pwave is power carried away by the EM-wave, Pdis is power loss in the medium due to the finite conductivity of the medium, and

On taking volume integral of the above equation and further using Gauss divergence theorem (4.4.13), the following expression is obtained:

=

= − μ0 μr

∂ ∂t

J c + ε0 εr

from equation (4.4.1b) to

∂E ∂t

(4.5.1)

2

∇2 E = μ0 μr σ

∂E ∂ E + μ0 ε0 εr μr 2 ∂t ∂t

∇ 2 H = μ0 μr σ

∂H ∂2 H + μ0 ε0 εr μr 2 ∂t ∂t

452

∇ ∇ E − ∇2 E is used. The above wave equation for

To get the time-harmonic field, i.e. E x, y, z, t =

the electric field is valid in an isotropic, homogeneous, and lossy medium. In a homogeneous medium, the (μr, εr) are not a function of position. Further, the

E x, y, z exp jωt , the time differential variable is replaced as follows: ∂/∂t jω and ∂ 2/∂t2 − ω2. The above wave equations for the time-harmonic fields are written as,

medium is taken as charge-free, i.e. ρ = 0, ∇ E = 0. Likewise, the following wave equation is obtained for the magnetic field:

∇2 E − jωμ0 μr σ + jε0 εr ω E = 0

a

∇2 H − jωμ0 μr σ + jε0 εr ω H = 0

b

453

4.5 EM-waves in Unbounded Isotropic Medium

The complex propagation constant γ = α + jβ is defined as follows: γ2 = jωμ0 μr σ + jε0 εr ω

454

For a lossless medium σ = 0, and the propagation constant is a real quantity: γ = jω μ0 ε0 μr εr

β = ω μ0 ε0 μr εr = ω

α=0

με

In the above equation, the real part of the complex relative permittivity is εr = εr . On separating the real and imaginary parts, the attenuation constant (α) and propagation constant (β) are obtained: μ0 μr ε0 εr 2 μ0 μr ε0 εr β=ω 2

α=ω

a b

455 In a homogeneous medium, propagation constant β is also expressed as the wavenumber k. In free space, μr = εr = 1. The velocity of the EM-wave is equal to the velocity of light (c) in free space: c=

ω = β0

1 , μ0 ε 0

ω = β c vp = = εr

vp =

c μr εr c n

a b

1 + tan 2 δ + 1

12

a

12

b 4 5 11

The wave equation (4.5.3a) and (4.5.3b) for the ( E , H) fields in a lossy and lossless (α = 0) media are rewritten below:

456

where β0 = ω μ0 ε0 is the propagation constant, i.e. the wavenumber (k0) in free space. A lossless material medium is electrically characterized by (εr, μr). However, it is also characterized by the refractive index n = μr εr . In the case of a dielectric medium, it is n = εr . The velocity of the EM-wave propagation in a medium is

1 + tan 2 δ − 1

∇2 E − γ2 E = 0

a

∇2 E + β2 E = 0

b

∇2 H − γ2 H = 0

c

∇2 H + β2 H = 0

d 4 5 12

The propagation constant β is also expressed as the wavenumber k of the wavevector k . Sometimes in place of the complex propagation constant γ, the complex wavevector k is used as a complex propagation constant, i.e. k∗ = β − jα. On using the complex k, the field is written as E0 e−jkx = E0 e−j(β − jα)x = (E0e−αx) e−jβx.

457 For a lossy medium, the complex propagation constant can be further written as: γ2 = − ω2 μ0 μr ε0 εr − j

σ ωε0

458

For a lossy dielectric medium, εr is defined as a complex quantity: εr ∗ = εr − jεr where, εr =

a σ ωε0

b

4.5.2

1D Wave Equation

For the wave propagating only in the x-direction, equations (4.5.12a) and (4.5.12b) are reduced to the 1D wave equations: d2 E − γ2 E = 0 dx2

a

d2 H − γ2 H = 0 dx2

b

459 It is like the previous discussion on the complex relative permittivity ε∗r in a lossy dielectric medium, with the following expressions for the loss-tangent and propagation constant: tan δ =

εr σ = εr ωε0 εr

γ2 = − ω2 μ0 μr ε0 εr 1 − j tan δ α2 − β2 + j2αβ = − ω2 μ0 μr ε0 εr 1 − j tan δ

a b c 4 5 10

4 5 13 Equation − γx

(4.5.13a)

has

the

solution,

E x =

− αx − jβx

= E 0e e . The time-harmonic wave proE 0e pagating in the x-direction is E x, t = Re E x ejωt = Re E 0 e − αx ej ωt − βx

a

Likewise, H x, t = Re H x ejωt = Re H 0 e − αx ej ωt − βx

b

4 5 14

97

98

4 Waves in Material Medium – I

The field equations in the time-domain are also written as follows: E x, t = E 0 e − αx cos ωt − βx

a

H x, t = H 0 e − αx cos ωt − βx

b 4 5 15

In the case of a lossy medium, equation (4.5.11) shows that both α and β depend on the loss-tangent of a medium. For a lossless medium, tan δ = 0, leading to α = 0, and β = ω c μr εr. In the case of low conductivity, i.e. the low-loss medium, the approximation tan δ 1, or (σ/ωε0εr) 1, can be used. In such a medium σ ωε0εr, the contribution of the conduction current is small as compared to that of the displacement current. Such a medium is a dielectric medium with a small loss. 1 On approximating, 1 + tan 2 δ 2 ≈ 1 + 1 2 tan 2 δ ; the following expression is obtained from equation (4.5.11a): α=

ω μ0 ε0 μr εr tan δ = 2

ω μr εr tan δ 2c 4 5 16

The above equation computes the dielectric loss of a low-loss dielectric medium. The approximation 1 + tan 2 δ + 1 ≈ 2 + tan 2 δ 2 ≈ 2 is used to get an approximate value of β for such medium from equation (4.5.11b): ω β = ω μ0 ε0 μr εr = μr εr = β0 μr εr c 4 5 17 For a low-loss dielectric medium the dielectric loss, due to tan δ, increases linearly with frequency ω. However, the propagation constant β is dispersionless, giving the frequency-independent phase velocity. The above approximation can also be carried out in a little different way: 1 + tan 2 δ ≈ 1 +

σ2 2 ω2 ε2o ε2r 12 2

μ0 μr ε0 εr σ × 2 2 ω2 ε2o ε2r σ μ0 μr σ μr α= = η εr 2 ε0 εr 2 α≈ω

For a dielectric medium

α=

a σ η 2 εr

b 4 5 18

In the above equation, η = μ0 ε0 = 377Ω is the characteristic (intrinsic) impedance of free space. A low-loss medium is a mildly dispersive medium, with the frequency-dependent phase velocity:

σ2 2 ω2 ε2o ε2r σ2 = ω μ0 ε0 μr εr 1 + 8 ω2 ε2o ε2r σ2 β ≈ β0 μr εr 1 + 8 ω2 ε2o ε2r ω σ2 vp = ≈ v0 1 − β 8 ω2 ε2o ε2r β≈ω

μ0 μr ε0 εr 2

1+

12

+1

a b 4 5 19

In the above equation, v0 = c μr εr is the velocity of the EM-wave in the lossless dielectric medium. The presence of the loss has decreased the phase velocity, i.e. a lossy medium supports the dispersive slow-wave propagation. The use of α and β from equations (4.5.11a) and (4.5.11b) provide more accurate results. The dispersion in a medium is always associated with loss. This fundamental property is further discussed in chapter 6.

4.5.3 Uniform Plane Waves in Linear Lossless Homogeneous Isotropic Medium Figure (4.9a) shows the propagation of the TEM wave in the x-direction of an unbounded medium. Figure (4.9b) further shows that for the TEM wave, electric and magnetic field components, i.e. the pairs (Ey, Hz), or (Ez, Hy), are transverse to the direction of propagation, i.e. ± x - direction. For the field pair (Ey, Hz), the wave is y-polarized; and for the field pair(Ez, Hy), it is the z-polarized. The polarization of an EM-wave is determined by the direction of the E vector. The propagating wave is called the uniform plane wave, as amplitudes of electric and magnetic fields are constant over the equiphase surfaces. Figure (4.9c) shows that the phases of the Ey field at any instant of time, over the equiphase surfaces, are either 0 or 180 . The (y-z)-plane is the equiphase surface. The field components of a uniform plane wave do not change with y and z coordinates, i.e. ∂/∂y(field E or H) = ∂/∂z(field E or H) = 0. The field components are a function of x only. So, the field components of the EM-wave propagating in the x-direction can be written as follows: Ei = E0i ej ωt Hi = H0i e

βx x

j ωt βx x

a b

4 5 20

where, i = y, z In the above expressions, the (−) sign shows the wave propagation in the positive x-direction, whereas

4.5 EM-waves in Unbounded Isotropic Medium

Direction of propagation y

y

Ey-field

Wave propagation

Ey Hy x

z

E Hz z

x

z

Hz-field (a) TEM mode wave propagating in the x-direction.

(b) Field components of TEM mode wave propagating in the ± x-direction. Equiphase surface y

Eo

n^

y

y

k r

Ey Eo t

O

x

z Eo z (c) y-Polarized EM-wave showing phase reversal.

(d) Wave propagation in an arbitrary direction.

Figure 4.9 TEM mode wave in an unbounded medium.

the (+) sign is for the wave propagation in the negative xdirection. The wave propagation in the positive xdirection is discussed below. The above expressions related to a uniform plane wave, in an external source-free Mext = J ext = 0 lossless J c = 0 medium, can be applied to the Maxwell equations (4.4.1). In the present case, the del operator is replaced by a derivative with respect to x, i.e. ∇ x ∂ ∂x as a derivative with respect to y and z are zero. Maxwell first curl equation is reduced to a simpler form: ∂ ∂ × xEx + yEy + zEz = − μ0 μr x xHx + yHy + zHz ∂x ∂t −y

∂Ey ∂Hy ∂Ez ∂Hx ∂Hz +z = − μ0 μr x +y +z ∂x ∂x ∂t ∂t ∂t

4 5 21 On separating each component of the fields, the following expressions are obtained: ∂Hx =0 ∂t ∂Hy ∂Ez = μ0 μr ∂x ∂t ∂Ey ∂Hz = − μ0 μr ∂x ∂t

a b c 4 5 22

Likewise, the following expressions are obtained from Maxwell’s second curl expression (4.4.1b): ∂Ex =0 ∂t ∂Ey ∂Hz = ε0 εr − ∂x ∂t ∂Hy ∂Ez = ε0 εr ∂x ∂t

a b c 4 5 23

It is seen from the above equations that the Ex and Hx components, in the direction of propagation, are timeindependent, i.e. constant. They do not play any role in the wave propagation and can be assumed to be zero, without affecting the wave propagation [B.3]. Only transverse field components play a role in wave propagation. The time-varying Hy component generates the Ez, whereas the time-varying Ey component generates the Hz. It is also true for another time-varying pair (Hz, Ez). Maxwell divergence relations also show ∂Ex/ ∂x = ∂Hx/∂x = 0. Again, Ex and Hx components do not show any variation along the direction of propagation that is essential for wave propagation. So, in the TEM mode propagation, the longitudinal field components are zero, i.e. Ex = Hx = 0. By using equation (4.5.20) with the above equations, the time-harmonic fields are rewritten as follows:

99

100

4 Waves in Material Medium – I

∂Hz ∂x ∂Hy ∂x ∂Ez ∂x ∂Ey ∂x

= − jωε0 εr Ey

a

= jωε0 εr Ez

b

= jωμ0 μr Hy

c

= − jωμ0 μr Hz

d

Ei = E0ie−px ejωt and Hi = H0i e−px ejωt for the wave propagation in the positive x-direction. These are decaying nonpropagating evanescent mode waves. These are only decaying oscillations. The wave equations (4.5.13a) and (4.5.13b), for the Ey and Hz field components, are solved to get the total solution as a superposition of two waves traveling in opposite directions: 4 5 24

On using equation (4.5.20) with the above equations, the following algebraic expressions are obtained for the EM-wave propagating in the positive x-direction: βx Hz = ωε0 εr Ey

a

βx Hy = − ωε0 εr Ez βx Ez = − ωμ0 μr Hy

b c

βx Ey = ωμ0 μr Hz

d

+ j ωt − βx Ey x,t = Eoy e

− j ωt + βx Eoy e

+

Forward moving wave Backward moving wave + j ωt − βx Hz x, t = Hoz e

Hz x, t =

−

1 + j ωt − βx E e − η oy

− j ωt + βx Hoz e − j ωt + βx Eoy e

4 5 25 The wave impedance in free space, or in homogeneous unbounded medium, is defined in a plane normal to the direction of propagation [B.2]. For instance, Fig. (4.9a) shows the direction of propagation is along the x-axis, and wave impedance is defined in the (y-z) plane. It is also called the intrinsic impedance η0 of free space and intrinsic impedance η of material filled homogeneous space. The following expression is obtained for the wave impedance η, for y-polarized waves, from equation (4.5.25a): ω μ0 ε0 μr εr Ey β =x x =x Hz ωε0 εr ωε0 εr μ0 μr μr a η =x = x η0 ε0 εr εr μr b η = + x η0 εr η =

4 5 26 Equation (4.5.25b) provides the following wave impedance for the z-polarized waves: − Ez β η = =x x a Hy ωε0 εr 4 5 27 − Ez μr b η =x = xη0 Hy εr The positive wave impedance of the (Ey, Hz) or (−Ez, Hy) fields corresponds to a wave traveling in the +x direction. However, for the (Ez, Hy) fields, the wave impedance is negative showing the wave propagation in the negative x-direction. Under certain conditions, a medium can have an imaginary value of propagation constant, i.e. βx = − jp. In this case, the wave impedance becomes reactive, and there is no wave propagation. Again, the wave equation (4.5.20) is reduced to

a

b 4 5 28

In the above expression, μ0 ε0 = η0 = 377 Ω is the intrinsic or characteristic impedance, of the uniform plane in free space. The power movement is obtained from the Poynting vector S = E × H = yEy × zHz = y × z Ey Hz = x Ey Hz . The power of the forward wave travels in the positive x-direction. The direction of the power movement is the direction of the group velocity. In the x-direction, the direction of the phase velocity is associated with the direction of the propagation vector, i.e. the wavenumber β = k . Figure (4.9d) shows the propagating EM-wave in an arbitrary direction of the wavevector k = xkx + yky + zkz The wavevector k is normal to the equiphase surface. The position vector of a point P at the equiphase surface is r = xx + yy + zz. The following expressions describe the propagating wave as a solution to the wave equations (4.5.12a) and (4.5.12c): r , t = E or ej

ωt − k r

H r , t = H or ej

ωt − k r

E

a b

where, k r = kx x + ky y + kz z

c

k k = k2 = k2x + k2y + k2z

d, 4 5 29

where k = ω με. Equation (4.5.29d) is the dispersion equation. In the above equations, field quantities show time dependence, i.e. temporal dependence, through factor ejωt and space dependence, i.e. spatial dependence, through factor e − j k r . It also shows the lagging phase of the propagating wave in the positive direction.

4.5 EM-waves in Unbounded Isotropic Medium

This is the convention adopted by engineers. On several occasions, physicist prefers an alternative sign convention, i.e. e−jωt and ej k r . It leads to a leading phase for propagating waves in a positive direction. A reader must be careful while reading literature from several sources. For y-polarized wave propagating in the x-direction with ky = kz = 0 and kx = βx − jα, the solution of the wave equation could be written as follows: + − αx j ωt − βx x − αx j ωt + βx x Ey x,t = Eoy e e + Eoy e e

Hz x, t =

1 + − αx j ωt − βx x − αx j ωt + βx x E e e − Eoy e e η oy

b

The second terms of the above equations show wave propagation in the negative x-direction. The wave equations show the decaying propagating waves. For the case α = 0, the above equations are the same as that of equations (4.5.28). 4.5.4 Vector Algebraic Form of Maxwell Equations Maxwell’s equations in the unbounded medium could be also written in the vector algebraic form. The del operator can be replaced as follows: ∇ − j k . Using equation (4.4.9), for the charge-free lossless medium ρ = σ = 0, two sets of Maxwell equations, for the isotropic and anisotropic media, are written in the following algebraic forms: Set #I for the isotropic medium: k × E = ωμ H

a

∇ × H = − j k × H = jωε E

k × H = − ωε E

b

∇ E =0

k E =0

c

∇H=0

k H=0

k × k × E = ωμ k × H = − ω2 με E

k × E = ωB

e

k × H = − ωD

f

k D =0

g

k B =0

h 4 5 31

a

4 5 30

∇ × E = − j k × E = − jωμ H

Set #II for the anisotropic medium:

d

Equations (4.5.31a) and (4.5.31b) show that for the positive values of μ and ε, the triplet E , H, k follows the right-handed orthogonal coordinate system. Equations (4.5.31c) and (4.5.31d) show that in an isotropic medium, the field vectors E and H are orthogonal to the wavevector k . Equations (4.5.31c) and (4.5.31d) directly follow from the first two Maxwell equations by taking their dot product with the wavevector k . Equation (4.5.31a) further shows that H is normal to both E , k vectors, and equation (4.5.31b) shows that E is normal to both k , H vectors. In brief, the vectors E , H, k are orthogonal to each other, and there is no field component along the wavevector k , i.e. the wave is a transverse electromagnetic (TEM) type. Also, in an isotropic medium, E is parallel to vector D and H is parallel to vector B. This statement does not hold for the anisotropic medium. Maxwell equations (4.5.31e)–(4.5.31h) apply to an anisotropic medium. In an anisotropic medium, equations (4.5.31e) and (4.5.31f) show that B is normal to vectors k , E and D is normal to vectors k , H . However, B is not parallel to H. Also, D is not parallel to E. It is discussed in subsection (4.2.3). Equations (4.5.31a) and (4.5.31b) are solved E to get the vector algebraic form of wave equation as follows:

k × k × E + ω2 με E = 0

On using identity, k × k × E = k k E − E k k ,

Likewise, the wave equation for D could be written that is useful for the EM-wave propagation in an anisotropic medium. Equation (4.5.32b) is an eigenvalue equation, and the nontrivial solution for E 0 provides the eigenvalue of k belonging to the propagating waves in the unbounded isotropic medium. The medium supports two numbers of linearly y-polarized waves known

k k − ω2 με E = 0

a

(4.5.32)

b

as normal modes propagating in ±x-directions with same phase velocity (vp) given below: k = ± ω με = ± k0 μr εr ω c vp = ± = ± k μr εr

a b

4 5 33

101

102

4 Waves in Material Medium – I

In the above equation, k0 = ω μ0 ε0 is the wavenumber in free space. Using the intrinsic impedance with

k × E = ωμ H

H=

kk × E = ωμ

equation (4.5.31a), the magnetic field vector and Poynting vector are obtained below:

εμ 1 k× E = k× E η μ

a

εμ kk × H = − k × H = − ηk × H ωε ε 1 1 E2 E ×k× E = E E k− E k E = S= E ×H = k η η η ∗ 1 E2 Time averaged Sav = Re E × H = 2η 2 k × H = − ωε E

E = −

In the case of the propagation of waves in an isotropic medium, the wavevector k and Poynting vector S both are in the same direction. It provides the phase and group velocities in the same direction. 4.5.5 Uniform Plane Waves in Lossy Conducting Medium The loss-tangent (tan δ), given in equation (4.5.10), is much greater than unity, i.e. tan δ 1 for a highly conducting medium. It means a contribution of the conduction current is much more than that of the displacement current 1 . However, in a conducting medium, i.e. σ ωε0 εr the approximation for a low-loss medium is taken differently. The propagation constants of the EM-wave in a highly conducting and also in a low-loss medium are obtained from equation (4.5.4) as follows: Highly conducting medium σ ≈ jωμ0 μr σ γ2 = − ω2 μ0 μr ε0 εr 1 − j a ωε0 εr 1+j ωμσ γ = α + jβ = j ωμσ = 2 ωμσ b α=β= 2 Low − loss medium ω σ 12 ω σ γ=j μr εr 1 − j μr ε r 1 − j ≈j c ωε0 εr c 2ωε0 εr ση ω γ = α + jβ = c +j μr ε r 2 c 4 5 35

In a lossy medium, the plane wave propagates in the xdirection with the uniform field components in the (y-z)plane as shown in Fig. (4.9a). The field components are given by equation (4.5.24), incorporating the conductivity σ of a medium. They are modified as,

∂Hz ∂x ∂Hy ∂x ∂Ez ∂x ∂Ey ∂x

b c

(4.5.34)

d

= − σEy − jωε0 εr Ey

a

= σEz + jωε0 εr Ez

b

= jωμ0 μr Hy

c

= − jωμ0 μr Hz

d

4 5 36

Using the field solutions from equations (4.5.20), the above equations are reduced to the following forms: γ Hz = σ + jωε0 εr Ey

a

− γ Hy = σ + jωε0 εr Ez

b

γ Ez = − jωμ0 μr Hy

c

γ Ey = ωμ0 μr Hz

d

4 5 37

In the above equations, the complex propagation constant γ is given by equation (4.5.4). The conducting medium is highly dispersive, whereas the low-loss medium is nondispersive. Using equations (4.5.35a,b) with equation (4.5.12a), the wave equation and the phase velocity in a conducting medium are given below: d2 E − jωμσ E = 0 dx2

a

ω vp = = β

b

2ω μσ

4 5 38

It shows that the conducting medium is dispersive, and the phase velocity increases with an increase in frequency. The characteristic impedance (intrinsic impedance) of the low-loss and highly conducting media are obtained from equations (4.5.37) and (4.5.35a) as follows:

4.5 EM-waves in Unbounded Isotropic Medium

Ey Ez γ = − = η∗c = Hz Hy σ + jωε0 εr

Low − loss η∗c =

jωμ0 μr σ + jωε0 εr

High − loss η∗c =

1 2

=

jωμ0 μr σ + jωε0 εr

η 1 − j σ ωε0 εr 1 2

≈

jωμ0 μr σ

≈η 1 + j

1 2

1 2

= 1+j

The characteristic impedance, i.e. the intrinsic impedance ηc, of a high-loss conducting medium is a complex quantity, with an equal magnitude of real and inductive imaginary parts. The real part of η∗c is known as the surface resistance, Rs incurring an Ohmic loss in the conducting medium; and its imaginary gives the internal inductance Li of a conducting medium: πμf σ πμf ωLi = σ

a

Rs =

4 5 40 b

For the unbounded medium, |Rs| = |ω Li|, and the internal inductance Li is due to the penetration of the magnetic field in the medium. It is further discussed in subsection (8.4.2) of chapter 8. The expressions for the magnetic and electric fields and Poynting vector in a conducting medium are given below: Ey = E0 e − γ x ejωt = E0 e − α x ej ωt − βx E0 Hz = ∗ e − α x ej ωt − βx ηc

a

Hz = H 0 e − α x ej

c

b

ωt − βx 2

Sav = Re Ey × Hz x =

E0 − 2 α x 1 e Re ∗ x 2 ηc

d

πf μ0 μr σ

Power after traveling distance x

α,

b

1

π μ f 2 − 2α x e σ 4 5 42

The power transmitted through the conducting lossy medium is a complex quantity. Its real part gives the power that comes out from the medium of length x, whereas the imaginary part gives stored energy due to the field penetration in the conductor. The input power density available at x = 0 is S0 = H2o 2 π μ f σ . The power density after traveling distance x in a highly conducting medium is 1

Sout =

H2 0 π μ f 2 − 2αx e 2 σ

4 5 43

The field decreases by a factor e−αx, whereas the wave travels through a lossy medium. If the wave travels a distance x = δ = 1/α, known as the skin depth, the field is decreased by 1/e of its initial strength, i.e. approximately 37% of its initial field strength. However, the power decreases at a faster rate, i.e. by the factor e−2αx. If the initial power density is S0, the power density at distance x is

1 Skin – depth, δ = = α

a 1

b

πμσf

4 5 44 The attenuation constant α in the above equation is used from equation (4.5.35b). The power loss of wave traveling a distance x is computed after computing the power loss at unit distance x = 1m:

αdB = − 8 686 α dB m

αdB = − 8 686

(4.5.39)

1 2

S x = S0 e − 2αx

The α, β, and ηc for a highly conducting medium are given by equations (4.5.35b) and (4.5.39b). The uniform plane wave in an unbounded conducting medium is still TEM-type. However, field components Hz and Ey are not in-phase. These are in-phase in a dielectric medium shown in Fig. (4.9a). The power transported per unit area, in a conducting medium, in the x-direction is also given by the following expression: S x=1 = − 20 log 10 e S0

a

Ey H∗z η∗ Hz H∗z H2 0 Sx = = c = 1+j 2 2 2

4 5 41

αdB = 10 log 10

σ 2ω ε0 εr

x δ

a (4.5.45)

dB

b

103

104

4 Waves in Material Medium – I

polarization. It is normal to the direction of propagation, i.e. the x-axis. Both these polarizations are linear polarization. Figure (4.9a and c) show that for the EM-wave propagating in the x-direction, the tip of the Ey field component moves along the y-axis from +E0 to 0 to −E0. The movement and rotation of the tip of the

In the above equation, the value of e is 2.71828. The power loss is about 9 dB per skin-depth. The attenuation constant α of a lossy medium is defined by equation (4.5.44a) as follows: dS x = − 2 α S0 e − 2αx = − 2α S x dx dS x dx Power loss per unit length = α = 2S x 2 × Power avialable at lenght x

E -vector could be seen using the instantaneous E x, t -vector of the wave propagating in the x-direction. In Fig. (4.9c), the path of the tip of the Ey-field vector, in the plane of polarization, traces a line with respect to time. The linear trace demonstrates the vertical linear polarization. However, if both Ey and Ez field components are present, any of the three kinds of wave polarizations can be obtained: (i) linear polarization, (ii) circular polarization, and (iii) elliptical polarization. These polarizations are shown in Fig. (4.10a–c). The type of wave polarization depends upon the magnitude and the relative phase of the orthogonal Ey and Ez field components. The polarization states are briefly discussed below. Finally, the Jones vector and Jones matrix descriptions are summarized to describe elegantly the polarization states and their control by the polarizing devices.

4 5 46

4.6

Polarization of EM-waves

The uniform plane wave in the unbounded medium is the TEM-type wave. The monochromatic EM-wave is characterized by amplitude, phase, and polarization states. The microwave to optical wave devices can appropriately manipulate these characteristics to steer the EM-waves in the desired direction with shaped wavefront. The modern metasurfaces, discussed in sections (22.5) and (22.6) of chapter 22, can achieve such controls on the reflected and transmitted waves. In general, both E and H fields have two orthogonal field components in a plane normal to the direction of propagation, the x-direction, as shown in Fig. (4.9a and b). The field components are in the (y-z)-plane. For the TEM mode, it is possible to get either (Ey, Hz) or (Ez, Hy) pair of fields. Both pairs of field components can also exist. The orientation of the electric field component and the movement of the tip of the resultant Efield determine the polarization of the EM-wave. Thus, the (Ey, Hz) pair of the EM-wave is called a y-polarized wave, as only the Ey component of wave exists. The (Ez, Hy) pair of the EM-wave is called the z-polarized wave. The (y-z)-plane is known as the plane of z A′ Ez

│E│

A

B

Figure (4.10a) shows Ey and Ez field components of the EM-wave propagating in the x-direction. The E-electric field vector in the (y-z)-plane could be written in the phasor form as follows: E x = y E0y + z E0z e − jk0 x

θ = tan − 1

y

Ez

Ez

x

c

B ω ωt

y

O

x

A Ey

B′

Type of polarizations.

(b) RH-circular polarization.

b

Eoz E0y

│E│

ω Ey

a

461

ωt

Ey

E20y + E20z

for x = 0, E =

z

E0

(a) Linear polarization. Figure 4.10

Linear Polarization

z

θ O

4.6.1

(c) RH-elliptical polarization.

y

4.6 Polarization of EM-waves

The ejωt time-harmonic factor is suppressed in the above equation (4.6.1a). Equation (4.6.1b) shows the magnitude of the E-field, and equation (4.6.1c) computes its inclination with respect to the y-axis. For y-polarized wave, E0z = 0, and for the z-polarized wave, E0y = 0. In general, the field components E0y and E0z are complex quantities. For the in-phase field components, these are expressed as E0y=|E0y| ejϕ and E0z=|E0z| ejϕ. The instantaneous field components are considered to trace the movement of the tip of the E x, t -vector: E x, t = Re

y E0y ejϕ + z E0z ejϕ ejωt

E y = E0y cos ωt + ϕ ,

a

E z = E0z cos ωt + ϕ

In the above equations, both field components have an identical phase (ωt + ϕ). Figure (4.10a) shows the slant or inclined linearly polarized wave with the in-phase Ey and Ez components. The tip of the electric vector ( E ) moves along line A-O-B with respect to time. However, the slant angle θ does not change with time. If both the E-field components are either in-phase (A − O − B) or out of phase (A/ − O − B/) and have the same magnitude, i.e. E0y = E0z = E0, the corresponding inclination angle of the linear polarization trace, with respect to the y-axis, is θ = 45 and 135 , respectively. For the linear polarization, the total E-field given by equation (4.6.1a) could also be written as follows: a

E x = E0 z − y e − jk0 x

b 463

4.6.2

Circular Polarization

The circular polarization, shown in Fig. (4.10b), is obtained for two orthogonal field components of equal magnitude, and phase in quadrature. So, to get the circular polarization, two electric field components oscillate at the same frequency and meet the following conditions:

• • •

RHCP

Equal amplitude: The magnitudes of Ey and Ez are equal, i.e. |E0y| = |E0z| = E0. Space quadrature: The Ey and Ez field components are normal to each other. Time (phase) quadrature: The phase difference between the Ey and Ez field components are (ϕ = ± 90 ), i.e. E0y = E0, and E0z = E0e±π/2 = ± j E0.

The phasor form of the E-field vector of the circularly polarized waves, meeting the above conditions, could be written from equation (4.6.1a) as follows:

E x = E0 y + jz e − j k0 x E x = E0 y − jz e

a

− j k0 x

b 464

The ejωt time-harmonic factor is suppressed in the above equations. The time-domain forms of the circularly polarized waves, using the instantaneous E x, t -field components, at any location in the positive xdirection and also at the x = 0, i.e. in the (y-z)-plane, are expressed as follows:

b

462

E x = E0 y + z e − jk0 x

LHCP

LHCP

E x, t = Re E0 y + jz ej ωt − k0 x

a

RHCP

E x, t = Re E0 y − jz ej ωt − k0 x

b

LHCP

At x = 0, E t = E0 y cos ωt − z sin ωt

c

RHCP

At x = 0, E t = E0 y cos ωt + z sin ωt

d

−1

LHCP

E x, t

= E0 , θ = tan

− tan ωt = − ωt

RHCP

E x, t

= E0 , θ = tan − 1 tan ωt =

ωt

e f

465 The handedness, i.e. the sense of rotation of the E -vector of the circularly polarized wave is determined by the direction of rotation of the E x, t -vector in the transverse (y-z)-plane with respect to time, whereas the EM-wave propagates in the x-direction. Therefore, using equation (4.6.5f), Fig. (4.10b) shows the anti-clock rotation of the E-field vector tracing a circle of radius E0 in the (y-z)-plane. The sense of rotation is considered with respect to cos(ωt), taking t = 0. It shows the wave propagating in the positive x-direction, i.e. the wave coming toward an observer standing on the positive xaxis is the right-hand circularly polarized wave (RHCP). Its field vector is given by equation (4.6.4b). Similarly, equation (4.6.5e) shows the clockwise rotation, i.e. the left-hand circularly polarized wave (LHCP) propagating in the positive x-direction. The magnetic fields of the RHCP and LHCP wave are obtained using equation (4.6.4) with equation (4.5.34a): RHCP LHCP

E0 E0 x × y − jz e − jk0 x = z + j y e − jk0 x a η0 η0 E0 E0 x × y + jz e − jk0 x = z − j y e − jk0 x b H x = η0 η0 H x =

466 In the case of the wave propagation in the negative x-direction, i.e. the wave moves away from the observer standing on the positive x-axis, the role of RHCP and LHCP gets interchanged. Further, the handedness of circular polarization can be reversed by applying 180

105

106

4 Waves in Material Medium – I

phase-shift to either the y or z component of the E-field vector. The circular polarization could be further considered as a linear combination of two linearly polarized waves. Alternatively, the linear polarization can also be considered as a linear combination of the left-hand and right-hand circularly polarized waves [B.9]. 4.6.3

Using the above equation and identity cos ωt = Ey t E0y , sin ωt =

Ez t = E0z cos ωt cos ϕ

Elliptical Polarization

E y t = y E0y cos ωt,

E z t = z E0z cos ωt ± ϕ

the following

equation of the ellipse is obtained:

Ey t Ez t = cos ϕ E0z E0y

Two orthogonal field components, in the space quadrature, of unequal magnitude and arbitrary phase angle (ϕ) between them, generate the elliptical polarization, i.e. the resultant E-field vector rotates in the plane of polarization such that its tip traces an elliptical path as shown in Fig. (4.10c). The instantaneous orthogonal E-field components in the x = 0 plane are given below:

2

1 − Ey t E0y ,

sin ωt sin ϕ 1−

Ey t E0y

2

1 − cos 2 ϕ

P E2y − Q Ey Ez + R E2z = 1 where, P =

E20y

1 2 cos ϕ 1 , Q= , R= 2 2 2 sin ϕ E0y E0z sin ϕ E0z sin 2 ϕ

468 The semi-major axis OA, the semi-minor axis OB of the ellipse shown in Fig. (4.10c), and the axial ratio (AR) of the polarization ellipse are given below [B.9, B.29]:

E t = y E0y cos ωt + z E0z cos ωt ± ϕ 467

OA =

1 E2y + E2z + E4y + E4z + 2E2y E2z cos 2ϕ 2

OB =

1 E2y + E2z − E4y + E4z + 2E2y E2z cos 2ϕ 2

AR =

Emax OA = = Emin OB

1 2

1 2

a 1 2

b

E2y + E2z + E4y + E4z + 2E2y E2z cos 2ϕ E2y + E2z − E4y + E4z + 2E2y E2z cos 2ϕ

The tilt angle θ of the polarization ellipse, i.e. inclination of the major axis OA with y-axis is [B.9, B.29]: θ=

1 2

2Ey Ez 1 tan − 1 2 cos ϕ 2 Ey + E2z

4 6 10

For ϕ π/2, the polarization ellipse is inclined with respect to the y-axis. The linear and circular polarizations are obtained as special cases from the elliptical polarization. For instance, for Ez(t) = 0 the wave is horizontally polarized in the y-direction. For E0y = E0z = E0 and ϕ = ± π/2, the LHCP/RHCP wave is obtained as equation (4.6.9) is reduced to an equation of a circle with OA = OB. For the linear polarization, AR is infinity. However, for the circular polarization, AR is unity. In the case E0y = E0z = E0 and ϕ π/2, the wave is not circularly polarized and its AR is cotϕ/2. For a practical circularly polarized antenna, the axial ratio is frequencydependent and its axial ratio bandwidth is defined as the frequency band over which AR ≤ 3dB.

1 2 1 2

(4.6.9)

1 2

, 1 ≤ AR ≤ ∞

c

4.6.4 Jones Matrix Description of Polarization States The polarizing devices change the state of polarization. For instance, the polarizing devices could change the rotation of the linear polarization or convert the linear polarization into circular polarization. The Jones matrix method describes and manipulates the polarization states of the EM-wave using a 2 × 1 column vector, known as the Jones vector and transfer matrix of the polarizing device, known as the Jones matrix [B.30, B.31]. The Jones matrix is used in chapter-22 with metasurfaces. Jones Vector

The polarization state of the EM-wave propagation in the x-direction is given by equation (4.6.1). The orthogonal E-field components can be expressed in the form of the following column vector, known as the Jones vector:

4.6 Polarization of EM-waves

E0y

E x = y E0y ejϕy + z E0z ejϕz e − jk0 x =

In the above expression, the common phase angle has been absorbed in the propagation factor e − jk0 x . The Jones vector describes polarization states of any plane wave field. Some common polarization states are summarized below with respect to Fig. (4.10):

a

jΔϕ

E0z e

where, differential – phase Δϕ = ϕz − ϕy

b

4 6 11

1

Horizontal polarization HP along the y-axis

a

0

0 1

Vertical polarization VP along the z-axis

b

Linear polarization LP at an angle θ with y-axis

sin θ

=

1 1 2 1

c d

1 1 2 j

Left − hand circular polarization LHCP

(4.6.12)

θ = 45

1 −j

1 2

Right − hand circular polarization RHCP

cos θ

e

In the above expressions, the normalized magnitude of the E-field components are |Eoy| = |Eoz| = 1.

The Jones matrix elements are interpreted in the terms of the co-polarized and cross-polarized outgoing waves after transmission/reflection from a slab/surface:

Jones Matrix

Jones matrix

Figure (4.11) shows that the polarizing device could be described by a 2 × 2 transfer matrix, i.e. the Jones matrix [J]. It relates the output of the device to its input. The input could be the incident wave at certain slab/surface, acting as a polarizing device, and the output could be the transmission or reflection of the waves. out

E Eout y

=

in

= J E , E out = J E in J11 J12

Ein y

J21 J22

Ein z

=

Eout z

a b

Outgoing

Polarizing Incoming

waves

device

waves

where E out and E in are output and input Jones vectors, and J is Jones matrix 4 6 13

J =

J11 J12 J21 J22

Port #2 Polarizing device [J]

Input

Jones Matrix of Linear Polarizer

A linear polarizer allows the transmission of the incoming wave only along the transmission axis of the polarizer and blocks the transmission of the orthogonal polarizations. Jones matrices of the linear polarizers are summarized below:

Vertical Polarizer VP

Figure 4.11

1 0

a

0 0 0 0

b

0 1

Output

Linear Polarizer (Incident waves)

4 6 14

where Jyy and Jzz are responsible for the co-polarized outgoing waves, and Jyz and Jzy account for the presence of cross-polarized waves at the output. The co-polarized output waves have the same polarization as that of the incident input waves. Whereas, the cross-polarized output waves have orthogonal polarization with respect to the polarization of the incident input waves.

Horizontal Polarizer HP Port #1

Jyy Jyz , Jzy Jzz

=

(Outgoing waves)

Polarizing device described by Jones matrix.

where 0 ≤ py ≤ 1; 0 ≤ pz ≤ 1

py 0 0

pz

, c 4 6 15

107

4 Waves in Material Medium – I

For instance, at an inclination angle θ = 45 , the linearly polarized wave, given by Jones vector of equation (4.6.12c), is incident on the horizontal polarizer. The polarizer will provide a horizontally polarized wave given by equation (4.6.12a). It could be examined with the help of equation (4.6.13). Jones Matrix of a Linear Polarizer Rotated at Angle θ with the y-Axis

Figure (4.12) shows the (y-z)-coordinate system and also the (e1-e2) -coordinate system rotated at an angle θ with respect to the y-axis. The original polarizer, located in the (y-z)-coordinate system is described by the Jones matrix [J]; and the rotated polarizer at an angle θ is described as the rotated Jones matrix [Jrot(θ)]. The rotated polarizer is located in the (e1-e2) coordinate system. The relation between the Jones matrix [J] of unrotated polarizer and Jones matrix [Jrot(θ)] of the rotated polarizer is obtained by the coordinate transformation through the rotation Jones matrix [Rθ(θ)] and the inverse rotation Jones matrix [Rθ(θ)]−1 = [Rθ(−θ)]. Figure (4.12) shows that the unit vectors of two coordinate systems are related through the following transformations: y = cos θ e1 − sin θ e2 z = sin θ e1 + cos θ e2 y z e1 e2

= =

cos θ

− sin θ

e1

sin θ

cos θ

e2

cos θ

sin θ

y

− sin θ

cos θ

z

a b 4 6 16

The following rotation Jones matrix [Rθ(θ)] and its inverse [Rθ(−θ)] can be defined from the above relations that could be useful to transform the vectors from one co-ordinated system to another:

Rθ θ = −1

Rθ θ

cos θ

sin θ

− sin θ

cos θ

= Rθ − θ =

e2-axis

− sin θ

sin θ

cos θ

b

Transformation of E-vector Components

The rotation Jones matrix [Rθ(θ)] transforms the vector components from the rotated (e1-e2) coordinate system to the (y-z) Cartesian coordinate system. Whereas the inverse rotation Jones matrix [Rθ(−θ)] transforms the vector components from the (y-z) Cartesian coordinate system back to the rotated (e1-e2) coordinate system. Transformation of Jones Matrix of Polarizer

To perform the transformation of the E-vector components from one to another coordinate system using a polarizer, the Jones matrix describing the polarizer has to be transformed from one to another coordinated system. Thus, the above relations transform the Jones matrix [J] = [J]Car of a linear polarizer, given by equation (4.6.14) in the Cartesian (y-z) coordinate system, to the Jones matrix [Jrot(θ)] = [J]e of the polarizer in the rotated (e1-e2) coordinate system. The input/output relations of the E-field in both coordinate systems are expressed in terms of their respective Jones matrices: Cartesian coordinate system out

in

E

= J Car E Car

a Car

e1 − e2 coordinate system in

= Je E

E e

b e

4 6 18

θ θ y-axis

Figure 4.12

cos θ

The above rotation matrices are used to transform both the Jones vector and the Jones matrix from one to another coordinate system. The Jones matrix concept is applied to the polarizing system in two steps:

out

e1-axis

a

4 6 17

Rotated z-axis

108

The (y–z) and rotated (e1–e2) Coordinate systems.

In the above expression, subscript “Car.” with Jones matric stands for the (y-z) Cartesian coordinate system; and the subscript “e” with Jones matric stands for the general (e1-e2) coordinate system. In the present case, it is the anticlockwise rotated Cartesian system, as shown in Fig. (4.12).

4.6 Polarization of EM-waves

Using the rotation Jones matrices of equation (4.6.17a), the input and output E-field vectors could be transformed from the rotated (e1-e2) coordinate system to the Cartesian (y-z) coordinate system: in

= Rθ θ

E out

E out

b e

On substituting the above equations in equation (4.6.18a), the following expression is obtained: out

= J Car Rθ θ

E e

in

E

=

Rθ θ

−1

J Car Rθ θ

e

in

E e

4 6 20 On comparing the above equations against equation (4.6.18b), we get the transformed Jones matrix [J]e of the rotated polarizer in the (e1-e2) coordinate system from the Jones matrix [J]Car of the original polarizer in the Cartesian (y-z) coordinate system: J e = Rθ θ

−1

J e = Rθ − θ

sin θ

cos θ

JLP θ

e

py

0

cos θ

sin θ

0

pz

− sin θ

cos θ

py cos θ + pz sin θ

py − pz sin θ cos θ

py − pz sin θ cos θ

py sin 2 θ + pz cos 2 θ

2

2

=

It is noted that the original linear polarizer in the Cartesian system has no cross-polarization element. However, the rotated linear polarizer has a crosspolarization element. The above transformation can be applied to the horizontal polarizer (py = 1, pz = 0) rotated at an angle θ, and also to the linear polarizer rotated at an angle θ = 45 , to get the following rotated Jones matrices: Horizontal polarizer cos 2 θ JHP θ e = sin θ cos θ JHP θ = 45

e

=

J Car Rθ θ

sin θ cos θ

a

sin 2 θ

1 1 2 1

1

b

1

Linear polarizer JLP θ = 45

e

=

1 2

py + pz

py − pz

py − pz

py + pz

c

J Car Rθ θ

Alternatively, J Car = Rθ θ

4 6 23

a J e Rθ − θ

b 4 6 21

The transformation (4.6.21b) transforms the Jones matrix [J]e describing a polarizer in the rotated (e1-e2) coordinate system to the Jones matrix [J]Car in the Cartesian coordinate system. The transformation equation (4.6.21b) is obtained by matrix manipulation. However, it could also be obtained independently, as it is done for equation (4.6.21a). The use of the coordinate transformation for the polarizer is illustrated below by a few illustrative simple examples. Application of Jones matrix to the more complex polarizing system is available in the reference [B.30].

•

− sin θ

e

out

E

cos θ

4 6 22 4 6 19

E

Car

Rθ θ

=

a e

= Rθ θ

e

in

Car

E

JLP θ

2) The anisotropic polarizer with cross-coupling, in the Cartesian system, is given by equation (4.6.14). The polarizer is rotated by an angle θ. The rotated Jones matrix of the anisotropic polarizer is obtained as follows: Janiso θ Janiso θ

e

e

=

cos θ

− sin θ

Jyy

Jyz

cos θ

sin θ

sin θ

cos θ

Jzy

Jzz

− sin θ

cos θ

=

Jyy cos 2 θ + Jzz sin 2 θ

Jyy − Jzz sin θ cos θ

− Jyz + Jzy sin θ cos θ

+ Jyz cos 2 θ − Jzy sin 2 θ

Jyy − Jzz sin θ cos θ

Jyy sin 2 θ + Jzz cos 2 θ

+ Jzy cos 2 θ − Jyz sin 2 θ

+ Jyz + Jzy sin θ cos θ

4 6 24

Examples:

Jones Matrix for Retarder (Phase Shifter)

1) The Jones matrix of a linear polarizer given by equation (4.6.15c) is transformed below to the rotated Jones matrix [J]e = [JLP(θ)] of a linear polarizer that is rotated at an angle θ:

The wave retarder also called the waveplate alters the relative phase between two orthogonal field components passing through it. In this respect, it is acting as a phase shifter. The waveplates are designed using the birefringent, i.e. anisotropic material with orthogonal fast-axis

109

110

4 Waves in Material Medium – I

and slow-axis. The relative permittivity, also the refractive index, of the anisotropic material, has lower value along the fast-axis and higher value along the slow-axis, causing relatively slower phase velocity of the EM-wave propagation along the slow-axis. The half-wavelength thick slab, called the half-wave plate, changes the direction of the linear polarization at its output. Whereas, the quarter-waveplate, i.e. a quarter-wavelength thick slab, converts the linearly polarized incident waves into the circularly polarized waves at its output. The waveplates, i.e. the wave retarders, are characterized by the Jones matrices as discussed below. For the EM-wave propagating in the x-direction, the field components at the output of a wave retarder could be written from equation (4.6.11), by normalizing the magnitude of field components to the unity, as follows: Eout x y x Eout z

= x=d

1

0

Ein 0y x

0

ejΔϕ

Ein 0z x

Ein 0y

=

x a

ejΔϕ Ein 0z x JR Δϕ =

1

0

0

ejΔϕ

x=0

x=d

b

4 6 25 The JR(Δϕ) is the Jones matrix of a wave retarder (waveplate) given by equation (4.6.25b). At the input, the incident wave is linearly polarized and at the output of the retarder slab of thickness d, the differential phase Δϕ = ϕz − ϕy is the relative phase between the field components Eout d and Eout d. y z

Jones Matrix of Half-waveplate

The Jones matrix of a half-waveplate and the field components at its output are obtained by taking the relative phase Δϕ = π: JR Δϕ = π = x Eout y Eout z

x

1

0

0

jπ

= x=d

=

e

=

0

0

−1

0

0

−1

Ein 0z

x

The Jones matrix of a quarter-wave retarder and also the field components at the output are obtained by taking the relative phase Δϕ = − π/2: 1

JR Δϕ = − π 2 = x Eout y Eout z

x

= x=d

0

0 e − jπ

2

1

0

E0

0

−j

E0

=

1

0

0

−j =

x=0

a 1

−j

E0

b

4 6 27 In the above equation, both field components are equal to E0 = 1. It is noted that at the output of the quarter-waveplate, the wave is a right-hand circularly polarized wave. In the case, input wave components are in Ein y x = 0 = m cos ϕy and Ez x = 0 = n cos ϕy, the field components at the output of the quarter-waveplate, using equation (4.6.25a) are Eout y = m cos ϕy

a

Eout z = n cos ϕz = n cos ϕy − π 2 = n sin ϕy

b

Eout y m

2

+

Eout z n

2

=1

c 4 6 28

Equation (4.6.28c) shows that an ellipse is traced by the E-field in the (y-z)-plane. In this case, the quarterwaveplate produces an elliptically polarized wave. However, for the case m = n, it degenerates into the circularly polarized wave. Further, for the rotated retarders the rotated Jones matrices could be obtained, similar to the case of the rotated polarizer, using equation (4.6.21a).

4.7 EM-waves Propagation in Unbounded Anisotropic Medium

x=0

x

− Ein 0z x

Jones Matrix of Quarter-waveplate

a

Ein 0y x

1

Ein 0y

1

Two field components are in-phase at the input of the half-waveplate.

b 4 6 26

It is noted that at the output of the half-wave plate the phase difference between two field components is 180 .

Two cases of wave propagations in the uniaxial anisotropic media – without off-diagonal elements and with off-diagonal elements, are considered in this section. The dispersion relation is also discussed leading to the concept of hypermedia useful for the realization of hyperlens [J.1, J.5–J.7].

4.7 EM-waves Propagation in Unbounded Anisotropic Medium

4.7.1

On eliminating Hz and Hy from the above equations, wave equations for the electric field transverse components are obtained. Likewise, the wave equations for magnetic field transverse components are obtained on eliminating Ez and Ey:

Wave Propagation in Uniaxial Medium

The unbounded lossless homogeneous uniaxial medium is considered. The y-axis is the optical axis, i.e. the extraordinary axis. In the direction of the optic axis, the permittivity is different as compared to the other two directions. The medium is described by a diagonalized matrix with all off-diagonal elements zero. The permeability of the medium is μ0 and its permittivity tensor is expressed as follows: =

ε = ε = ε0

εr⊥

0

0

0 0

εr 0

0 εr⊥

d2 Ey dx2 d2 Ez dx2 d2 H y dx2 d 2 Hz dx2

471

Figure (4.13a) shows the TEM plane wave propagation in the x-direction. The TEM waves have Ex = 0, Hx = 0; Ey 0, Hy 0; Ez 0, Hz 0. Also, the uniform field components in y and z-directions do not vary, i.e. ∂/∂y (field) = ∂/∂z(field) = 0. Under these conditions, the following Maxwell equations provide the transverse field components of electric and magnetic fields: ∇ × E = − jωμ0 H

a

∇ × H = jωε0 εr E

b

a 473 b a b

Slow wave axis (Optic-axis)

474

x

Direction of wave propagation

(a) Wave propagation in the uniaxial medium. Figure 4.13 Wave propagation uniaxial media.

b

+ k20 εr⊥ Hy = 0

c

+ k20 εr Hz = 0

d

Direction of bias (H0)

z

x z

+ k20 εr⊥ Ez = 0

Equation (4.7.5a) is a 1D wave equation of the ypolarized electric field Ey 0 and Ez = 0. The magnetic field component Hz is along the z-axis. The Ey field component causes polarization in the dielectric medium creating relative permittivity εr along the y-axis. Further, the electric field Ey is transverse to the (x-z)-plane containing the direction of propagation x. Such waves are called the transverse electric (TE) waves. The zpolarized electric field with Ez 0 and Ey = 0 follows the wave equation (4.7.5b). The Ez component generates another polarization in the dielectric medium creating relative permittivity εr⊥ along the z-axis. In this case, Hy-component is transverse to the (x-z)-plane. These waves are called the transverse magnetic (TM) waves. In the present case, also in the case of the oblique incident of the plane waves, the waves are still TEM only. However, the TE and TM terminology is normally used for the non-TEM mode waves supported by the waveguides. It is discussed in chapter 7. A reader should be careful in the dual use of terminology TE/TM to show the mode of propagation, and also the polarization of

On expansion, the above equations provide the following sets of transverse field components:

y

a

475

472

∂Ez = jωμ0 Hy ∂x ∂Ey = − jωμ0 Hz ∂x ∂Hz − = jωε0 εr Ey ∂x ∂Hy = jωε0 εr⊥ Ez ∂x

+ k20 εr Ey = 0

y

Direction of wave propagation

(b) Wave propagation in the gyroelectric medium.

111

112

4 Waves in Material Medium – I

propagation. For the guided waves, TE/TM is modes of propagation; and in the open medium, it indicates the polarization of the propagating waves. The second use in the context of obliquely incident EM-waves at the interface of two medias is further discussed in section (5.2) of chapter 5. On solving the above wave equations, the timeharmonic wave propagating in the positive x-direction is obtained as,

E x, t = y Ey x + z Ez x

ejωt

e o = y E0y e − jk x + z E0z e − jk x ejωt

a

where, ke = k0 εr

b

ko = k0 εr⊥ c vpe = εr c vpo = εr⊥

c

H x, t = y Hy x + z Hz x

d e

f 476

In the above equations, k0 and c are wavenumber and velocity of EM-waves in free space. Also, ke and ko are wavenumbers of the extraordinary waves and ordinary waves, respectively, traveling in the x-direction with phase velocities vpe and vpo. The extraordinary waves are y-polarized, i.e. TE-polarized wave viewing the permittivity component εr . The ordinary waves are z-polarized, i.e. TM-polarized wave viewing the permittivity component εr⊥. Thus, an obliquely incident linearly polarized EM-waves, with Ey and Ez components, entering the slab of the anisotropic medium is split into two distinct normal mode waves and travel with two different phase velocities. They come out from the slab with a phase difference. This phenomenon is known as double refraction or birefringence. The dispersion relation for both normal waves is discussed in subsection (4.7.5). The wave impedances ηe and ηo of the extraordinary waves and ordinary waves propagating in the x-direction are obtained by substituting the field solutions of equation (4.7.6) in equations (4.7.3) and (4.7.4):

TM − waves Ez 0, Ey = 0 E0z ωμ η0 ηo = = − o0 = x εr⊥ H0y k

Waveplates and Phase Shifters

The incident linearly polarized wave on a slab at 45 has two in-phase E-field components. After traveling a distance d, the field components develop a phase difference Δϕ. So, for the case Eoy = Eoz = E0 and a phase difference Δϕ = 90 at the output of the slab of thickness x = d, the uniaxial anisotropic dielectric slab converts linearly polarized incident waves into the circularly polarized waves. It is shown below:

ejωt

e o = y H0y e − jk x + z H0z e − jk x ejωt

TE − waves Ey 0, Ez = 0 E0y ωμ η = e0 = 0 x ηe = H0z k εr

The plasma medium could be taken as an example. It is a uniaxial anisotropic medium with εr⊥ = 1 and εr = εr. In this case, y-polarized extraordinary waves travel with a slower phase velocity vpe as compared to a phase velocity vpo of the z-polarized ordinary waves. So, the extraordinary waves are also known as the slow-waves with εr > εr⊥. The ordinary waves are called fast-waves. The optic axis of the uniaxial medium is called the slow-wave axis and ordinary axis as the fast-wave axis.

a

b 477

Incidence wave

E x = 0 = y + z E0

π Phase difference Δϕ at output k − k d = 2 λ0 d= 4 εr − εr⊥ e

a

o

b 478

The electric components of the extraordinary and ordinary waves and also the total E-field at the output of the slab are Eey x = d = E0 e − jk d = E0 e − j e

Eey x = d = − j E0 e − jk Eoz x = d = E0 e − jk

o

o

d

a

d

E x = d = − j y + z E0 e E x = d, t = Re

k0 d + π 2

b − jko d

c

− j y + z E0 e − jk d ejωt o

d 479

The wave at the output of the slab is a left-hand circularly polarized wave. Such a slab converting the incoming linearly polarized wave to the circularly polarized waves is called the quarter-waveplate. A waveplate d = λ0 2 εr − εr⊥ , i.e. with ke − ko d = π, the half-waveplate rotates the 45 linearly polarized waves by 90 and output electric field component is E x = d = − y + z E0 . The phase of the outgoing waves could be controlled by changing the thickness of a slab. The waveplate acts as a phase shifter. In the case of uniaxial anisotropic plasma medium, the permittivity components are εr = εr and εr⊥ = 1.

4.7 EM-waves Propagation in Unbounded Anisotropic Medium

The above characteristics of a slab are also realized by thin metasurfaces discussed in subsections (22.5) and (22.6) of chapter 22. 4.7.2 Wave Propagation in Uniaxial Gyroelectric Medium Figure (4.13b) shows uniform TEM-waves propagation in the z-direction in an unbounded uniaxial gyroelectric medium created by the magnetized plasma on the application of the DC magnetic field H0 in the z-direction. The permittivity tensor [εr] of the medium is given equation (4.2.11). Due to the presence of off-diagonal matrix elements ±jκ in the permittivity matrix of the gyroelectric medium, the Ex component of the linearly polarized incident wave also generates the Ey component with a time quadrature. It is due to the presence of factor “j.” Similarly, the Ey component of an incident wave generates Ex component also with a time quadrature. The presence of two orthogonal E-field components with a time quadrature in a gyroelectric medium creates the left-hand circularly polarized (LHCP) and right-hand circularly polarized (RHCP) waves as the normal modes in the uniaxial gyroelectric medium. Both circularly polarized waves travel with two different phase velocities. Thus, the gyro medium with the cross-coupling gyrotropic factor ±jκ has the ability of polarization conversion. Maxwell equation (4.7.2a) is expanded in the usual way to get the transverse field components Ex, Ey and H x, H y : ∂Ey = jωμ0 Hx , ∂z ∂Ex = − jωμ0 Hy ∂z

a

However, the Maxwell equation (4.7.2b) in the present case is expanded differently: y

z

∂ ∂x

∂ ∂y

∂ ∂z

Hx

Hy

0

−

0

0

εr

jκ

0

Ex

= jωε0 0 y

0

− jκ

εr

0

Ey

z

0

0

εr,zz

0

x

0

0

∂Hy = jωε0 εr Ex + jκEy ∂z ∂Hx = jωε0 − jκEx + εr Ey ∂z

E z = E 0 e − jβz z

a

H z = H 0 e − jβz z

b

4 7 12

On substituting the above equations in equations (4.7.10) and (4.7.11), the following sets of equations are obtained: βz Ey = − ωμ0 Hx βz Ex = ωμ0 Hy

a b

βz Hy = ωε0 εr Ex + jκEy

c

βz Hx = − ωε0 − jκEx + εr Ey

d

4 7 13

On solving the above equations for Ex and Ey, the following characteristics equation is obtained: β2z − εr k20 jκ k20

− jκ k20 β2z − εr k20

Ex Ey

= 0,

4 7 14

where wavenumber in free space is k20 = ω2 μ0 ε0. The det [ ] = 0 of the above homogeneous equation provides the nontrivial solutions giving the following two eigenvalues of the propagation constant βz: β2z − εr k20

b 4 7 10

x

propagation in the z-direction. However, for the wave propagation in the x-direction Ex = 0, Ez 0 and εr, zz permittivity component occurs in the wave propagation. Similar is the case for the wave propagation in the ydirection. Further, due to the cross-coupling between Ex and Ey components in the above equations, it is not possible to obtain a single second-order wave equation for either Ex or Ey. However, the solution could be assumed for the field vectors E z and H z as follows:

− jκ k20

= 0,

jκ k20 β2z − εr k20 2 β2z = k20 εr ± κ = βz± βz+ = k0 εr + κ

βz− = k0 εr − κ

β2z − εr k20

2

− κ k20

2

=0

a b 4 7 15

a b 4 7 11

It is shown below that the eigenvalue βz+ and βz− are the propagation constant of two circularly polarized normal mode waves propagating in the z-direction. The wave with propagation constant βz+ travels at slower velocity compared to the wave traveling in an isotropic medium with relative permittivity εr. The wave with propagation constant βz− is a faster traveling wave. +

For the uniform plane wave propagating in the positive z-direction, ∂Hy/∂x = ∂Hx/∂y = 0. In the above equations, it is noted that the εr, zz component of permittivity does not play any role in the TEM mode wave

The −electric fields, i.e. the eigenvectors E z and E z for both the normal mode waves are obtained by substituting the eigenvalue βz = βz+ and βz = βz− in equation (4.7.14), and using equation (4.7.12a):

113

114

4 Waves in Material Medium – I

For βz+ normal mode wave +

Ey = − jEx = − jE0 The E-field of wave is, +

Likewise, for βz− normal mode wave E +

Using E

+

z = xEox + yEoy e − jβz z ,

E

+

z, t = Re E

E −

z = E0 x − jy e − jβz

+

z = E0 x + jy e

z ejωt , equation (4.7.16a)

shows the RHCP waves coming toward an observer. Likewise, equation (4.7.16b) is for the LHCP waves coming toward an observer. Suppose the x-polarized wave with E z = 0 = xE0 is incident on the gyroelectric slab of thickness d. At the

z

− jβz− z

E0 E0 x−jy + x + jy 2 2 + − E0 E0 x − j y e − j βz z + x + j y e − j βz z At distance z inside the slab E z = 2 2 βz+ + βz− βz+ − βz− βz+ − βz− z 2 E z = E0 x cos − y sin e−j 2 2

Ey β + − βz− = − z z Ex 2 β + − βz− ϕ z=d = − z d 2 ϕ z = tan − 1

a b 4 7 18

The above equation shows that the E-field polarization vector rotates while the wave travels in the medium. For the wave reflected at the end of the slab, the total rotation at the input is 2ϕ. This is known as Faraday rotation. It is the characteristic of a gyrotropic medium – gyroelectric, as well as gyromagnetic [B.2–B.4]. The wave propagation in the gyromagnetic medium is obtained similarly [B.3]. Similar to the gyroelectric medium, the gyromagnetic

4.7.3

+ω μ

εxx

0

0

Ex

0

εyy

0

Ey

0

0

εzz

Ez

=0

(4.7.17)

b

Dispersion Relations in Biaxial Medium

A biaxial medium could be considered with scalar permeability μ and permittivity tensor [ε]. The off-diagonal elements of the matrix equation (4.2.4a) are zero. Maxwell equations (4.5.31a) and (4.5.31b) are used in the present case with permittivity tensor [ε] in place of a scalar ε. The wave equation (4.5.32a) is suitably modified to incorporate the tensor [ε]:

xkx + yky + zkz kx Ex + ky Ey + kz Ez − k2x + k2y + k2z 2

a

medium also supports the circularly polarized normal modes. The word gyro indicates rotation and the gyro media supports circularly polarized normal mode wave propagation. They do not support the linearly polarized EM-waves. The analysis of the wave propagation in other complex media- bi-isotropic and bianisotropic is cumbersome. However, it can be followed by consulting more advanced textbooks [B.13, B.17, B.21–B.23].

+ ω2 μ ε E = 0, where k k = k2x + k2y + k2z

k k E −E k k

(4.7.16)

b

plane of entry, the linearly polarized electric field can be decomposed into the RHCP and LHCP waves traveling in the positive z-direction. The electric field at any distance inside the slab is a sum of two circularly polarized waves:

At input of the slab E z = 0 = xE0 =

However, the wave is still linearly polarized with a rotation of ϕ with respect to the x-axis. The angle of rotation ϕ at the output of the slab is

a

xEx + yEy + zEz

4.7 EM-waves Propagation in Unbounded Anisotropic Medium

− k2y + k2z

kx ky −

kx ky kx kz

Let

k2x

kx kz k2z

+

ky kz − k2x + k2y

ky kz

εxx

0

0

Ex

0 0

εyy 0

0 εzz

Ey Ez

kx ky

kx kz

kx ky

ω2 μεyy − k2x + k2z

ky kz

kx kz

ky kz

ω2 μεzz − k2x + k2y

4 7 20

k2 − ω2 μεyy = n2 ; k2 − ω2 μεzz = p2

Using equation (4.7.20), equation (4.7.19) is rewritten as, k2x − m2 kx ky kx kz

kx ky k2y

−n

ky kz

2

kx kz

Ex

ky kz

Ey

k2z − p2

Ez

=0 4 7 21

The nontrivial solution for Ei (i = x, y, z) of the above homogeneous equation is det[ ] = 0, i.e. 2

+ mpky

2

+ mnkz

2

= mnp

2

k2y k2x k2z + + =1 m2 n2 p2 4 7 22 The above dispersion relation is a quadratic equation of any component of k2. So, there are two solutions for any component of k. Two solutions correspond to two normal modes of propagation in the anisotropic medium. At a fixed frequency, equation (4.7.22) is the equation of an ellipsoid surface in the k-space (wavevector space), i.e. the normal space. For an isotropic medium, it is reduced to a sphere. Further, on knowing the wavenumber, the field components Ei (i = x, y, z) can be determined from equation (4.7.21). 4.7.4 Concept of Isofrequency Contours and Isofrequency Surfaces The discussion of dispersion in the uniaxial medium requires an understanding of the concept of isofrequency contours and isofrequency surfaces in the 2D and 3D kspace. Figure (4.14a–d) explain the concept of isofrequency contours. The wave propagation is considered in the isotropic (x-y)-plane. The dispersion relations for the 2D waves propagation in the isotropic medium and also in air medium are expressed as

k2x k2x

=0

(4.7.19)

ω2 μεxx − k2y + k2z

k2 − ω2 μεxx = m2 ;

npkx

+ ω2 μ

Ex Ey Ez

2

ω vp

+

k2y

=

+

k2y

ω = c

=0

a 4 7 23 2

b

At a fixed frequency ω, the above equations are equations of circles in the (kx-ky)-plane. Figure (4.14a) shows that the radius of the circle increases with an increase in frequency. It forms a light cone. Figure (4.14b) further shows the increase in the 2D wavevector at the increasing order of frequencies ω1 < ω2 < ω3 < ω4. The concentric contours of the wavevector are known as the isofrequency contours, displaying the dispersion relation. The light cone of the 2D dispersion diagram is generated by revolving Fig. (2.4) of the 1D dispersion diagram, shown in chapter 2, around the ω-axis. Likewise, 3D isofrequency surfaces are obtained. It is discussed in the next section. The propagating wave in the z-direction is described by equation (4.7.12). The propagation constant βz = kz =

k2 − k2x + k2y

of the waves must be real.

So, the propagating waves are obtained only for k2x + k2y < k2 , i.e. within the light cone. Outside the light cone, i.e. for k2x + k2y > k2, the waves are nonpropagating evanescent waves. Thus, the light cone divides the k-space into the propagating and evanescent wave regions shown in Fig. (4.14a,c and d). Further, any point P on the isofrequency contour surface, shown in Fig. (4.14b), connected to the origin O shows the direction of the wavevector. It is also the direction of the phase velocity vp. The direction of the normal at point P shows the direction of the Poynting vector, i.e. the direction of the group velocity vg. For an isotropic medium, both the vp and vg are in the same direction. It is noted that the wavefront is always normal to the wavevector k . However, for the anisotropic medium, the phase and group velocities may not in the same direction. It is discussed below.

115

4 Waves in Material Medium – I

ω ky

Vg P

Propagating waves

kz = 0 Evanescent waves

Vp O

ω1 < ω2 < ω3 < ω4

(a) Light cone in the k-space.

(b) Isofrequency contours of waves at increasing frequency. Light line

ky Propagating waves kx

ω

Propagating waves

kx

Evanescent waves (c) Regions for propagating and evanescent waves. Figure 4.14

Light line Evanescent waves

ky

4.7.5

kx

ω1 ω2 ω3 ω4

kx

Evanescent waves

116

(d) Light cone in the (ω – kx) plane.

Dispersion diagrams of the wave propagating in the z-direction in the isotropic medium.

Dispersion Relations in Uniaxial Medium

This section considers the dispersion relation of a uniaxial anisotropic permittivity medium as a special case of the dispersion relation (4.7.19) of the biaxial medium. The permittivity along the optic axis, i.e. the z-axis is εzz = ε and permeability of the medium is μ. The (xy)-plane, with permittivity tensor elements εxx = εyy = ε⊥, is a transverse plane. So, the medium is isotropic in the (x-y)-plane. The propagation constant along the z-axis is kz and in the transverse plane, it is kt, satisfying the relation k2 = k2x + k2y + k2z = k2t + k2z . To simplify equation (4.7.19) for the uniaxial medium, the transverse wavevector k t in the (x-y)-plane is aligned such that the wave propagates only along the x-axis, i.e. ky = 0 and kx = kt. Under such alignment, equation (4.7.19) is reduced to the following dispersion relation: ω2 με⊥ − k2z

0

0

ω2 με⊥ − k2

kt kz 0

kt kz

0

ω2 με − k2t

The above equation provides the following characteristic equation:

4 7 24

ω2 με − k2t − kz kt

2

=0

4 7 25 The above expression is also obtained from equation (4.7.19) even for the alignment of the wavevector k t to the y-axis, i.e. kx = 0 and ky = kt. So, the characteristic equation (4.7.25) gives the following two independent dispersion relations: k2 = ω2 με⊥

a

k2z

b

ε⊥

+

k2t ε

= ω2 μ

k2 = k2y + k2z = k20 εr⊥ k2z

εr⊥ =0

ω2 με⊥ − k2z

k2 − ω2 με⊥

+

k2y εr

= k20

c d 4 7 26

The above equations (4.7.26a) and (4.7.26b) demonstrate the presence of two normal modes of wave propagation in the 3D (kx, ky, kz) – space. Next, the 3D

4.7 EM-waves Propagation in Unbounded Anisotropic Medium

Likewise, equation (4.7.26d) is the dispersion relation of the extraordinary waves, giving the isofrequency ellipse in the (ky-kz)-plane, as shown in Fig. (4.15b). In this case, the wavenumber is direction-dependent because the relative permittivity for this case is direction-dependent, ke = εr θ k0 . The wavevector ke subtends an angle θ with respect to the z-optic axis. In this case, wavevectors components are kz = ke cos θ = εr θ k0 cos θ, and ky = ke sin θ = εr θ k0 sin θ . Using, these relations with equation (4.7.26d), the following equation of ellipse of permittivity, supporting the extraordinary wave, is obtained:

dispersion relation is reduced to the 2D dispersion relation given by equations (4.7.26c) and (4.7.26d). Equation (4.7.26a) is an equation of sphere in the 3D k-space at a fixed frequency, i.e. at ω = constant. It shows the dispersion relation of the ordinary waves with the wavevector ko = k0 εr⊥. At increasing order of frequencies, equation (4.7.26a) provides isofrequency spherical surfaces. Likewise, equation (4.7.26b) is an equation of ellipsoid in the 3D k-space at a fixed frequency. It provides the dispersion relation of the extraordinary waves in the form of the concentric isofrequency ellipsoidal surfaces. The wavevector of extraordinary wave ke is rotation dependent. The 2D dispersion diagram of both the ordinary and extraordinary waves is considered in the (y-z)-plane with the help of equations (4.7.26c) and (4.7.26d). Equation (4.7.26c) is the equation of an isofrequency dispersion circle, shown in Fig. (4.15a) in the (ky-kz) plane. It is the dispersion relation of the ordinary waves in the

kz

Vg

Vg

θ Vp

Vp

ky

ky

(a) Dispersion of ordinary waves.

kz

(b) Dispersion of extraordinary waves in positive uniaxial medium.

kz Vg Vg Vp Vp Vp

ky

ky

Vg

(c) Dispersion of extraordinary waves in the negative uniaxial medium.

cos 2 θ sin 2 θ 1 + = 2 2 2 n0 ne n θ

b

The dispersion relation (4.7.27b) is in the term of refractive indices. The refractive index n(θ) is the direction-dependent, and n0 and ne are refractive indices for the ordinary and extraordinary waves. The wave

of the ordi-

nary wave is independent of permittivity ε along the z-axis.

kz

a

4 7 27

o

isotropic (y-z)- plane. The wavenumber k

cos 2 θ sin 2 θ 1 + = εr⊥ εr εr θ

(d) Dispersion of extraordinary waves in the hyperbolic uniaxial medium.

Figure 4.15 Dispersion diagrams in the uniaxial anisotropic medium.

117

118

4 Waves in Material Medium – I

analysis using the refractive index, in place of permittivity and permeability, is commonly used for the optical wave propagation in the uniaxial medium [B.18–B.20]. Figure (4.15a) shows that the ordinary waves have both the phase and group velocities, on the isofrequency contour, in the same direction. Figure (4.15b) shows the elliptical dispersion diagram for the extraordinary waves. In this case, there is a deviation of the direction of the group velocity from the direction of the phase velocity. The inner circle of Fig. (4.15b) shows the dispersion diagram of the ordinary waves with relative permittivity εr⊥ = n⊥ refractive index . Along the ky, the extraordinary waves have relative permittivity εr = n refractive index , εr > εr⊥ . Only along the z-optic axis, i.e. along the kz-axis, the identical relative permittivity for both kinds of waves is obtained. In all other directions, the ordinary wave is a fast-wave as compared to the extraordinary wave. The medium is known as the positive uniaxial medium. Figure (4.15c) shows the case of the negative uniaxial medium. In this case, εr < εr⊥ and the ordinary wave is a slow-wave as compared to the extraordinary wave in all directions except along the y-axis. The phase velocities vp1 and vp2 of the ordinary and extraordinary waves using equations (4.7.26a) and (4.7.27a) are expressed through the following relations [B.3]: v2p1 = vo

2

=

1 με⊥

v2p2 = vo 2 cos 2 θ + ve 2 sin 2 θ c where ve = εr

other. So, the hypermedium supports the backward wave propagation, and it is a metamaterial medium. The hyperlens is discussed in the subsection (5.5.6) of chapter 5.

References Books B.1 Baden-Fuller, A.J.: Microwaves, 3rd Edition, Pergamon

Press, New York, 1990. B.2 Jordan, E.C.; Balmain Keith, G.E.: Electromagnetic

B.3

B.4 B.5

B.6

B.7

a B.8

b c 4 7 28

The phase velocity vp1 of the ordinary wave is independent of the angle θ. However, the phase velocity vp2 of the extraordinary wave is dependent on the angle θ. In Figure (4.15b), both velocities are identical only for the wave propagation along the z-optic axis. Finally, it is possible to artificially realize a uniaxial anisotropic material with one of the permittivity components as a negative quantity, say εr⊥ = − |εr⊥|. In this case, equation (4.7.26d) shows the hyperbolic dispersion relation. This medium is known as the hypermedium. It supports the wave propagation with a larger value of wavenumbers and can convert an incident evanescent wave to the propagating waves. Such engineered materials are needed by the hyperlens [J.1, J.7]. Figure (4.15d) shows the dispersion relation of the hypermedium with phase and group velocities. In this case, y-components of vp and vg are opposite to each

B.9 B.10 B.11 B.12 B.13 B.14

B.15

B.16

Wave and Radiating System, Prentice-Hall India, New Delhi, 1989. Ramo, S.; Whinnery, J.R.; Van Duzer, T.: Fields, and Waves in Communication Electronics, 3rd Edition, John Wiley & Sons, Singapore, 1994. Collin, R.E.: Foundations for Microwave Engineering, 2nd Edition, McGraw-Hill, Inc., New York, 1992. Rao, N.N.: Elements of Engineering Electromagnetics, 3rd Edition, Prentice-Hall, Englewood Cliff, NJ, 1991. Sadiku, M.N.O.: Elements of Electromagnetics, 3rd Edition, Oxford University Press, New York, 2001. Cheng, D.K.: Fields and Wave Electromagnetics, 2nd Edition, Pearson Education, Singapore, 1989. Weeks, W.L.: Electromagnetic Theory for Engineering Applications, John Wiley & Sons, New York, 1964. Balanis, C.A.: Advanced Engineering Electromagnetics, John Wiley & Sons, New York, 1989. Hipple, A. Von: Dielectrics, and Waves, Artech House, Norwood, MA, 1995. Plonsey, R.; Collin, R.E.: Principle and Applications of Electromagnetic Fields, TMH, New Delhi, 1973. Polivanov, K.: The Theory of Electromagnetic Fields, Mir Publisher, Moscow, 1975. Orfanidis, S.J.: Electromagnetic Waves and Antenna, Free Book on the Web. Staelin, D.H.; Morgenthaler, A.W.; Kong, J.A.: Electromagnetic Waves, Prentice-Hall, Englewood Cliffs, NJ, 1994. Landau, L.D.; Lifshitz, E.M.; Pitaevskii, L.P.: Electrodynamics of Continuous Media, Pergamon Press, New York, 1984. Engheta, N.; Ziolkowski, R.W. (Editors): Metamaterials: Physics and Engineering Explorations, Wiley-Interscience, John Wiley & Sons, Inc., Hoboken, NJ, 2006.

References

B.17 Lakhtakia, A.; Varadan, V.K.; Varadan, V.V.: Time-

B.18 B.19 B.20 B.21 B.22

B.23

B.24 B.25 B.26 B.27

B.28

B.29

Harmonic Electromagnetic Fields in Chiral Media, Springer-Verlag, Berlin, Heidelberg, 1989. Born, M.; Wolf, E.: Principles of Optics, 6th Edition, Pergamon Press, New York, 1980. Haus, H.A.: Waves and Fields in Optoelectronics, Prentice-Hall, Englewood Cliffs, NJ, 1984. Yariv, A.; Yeh, P.: Optical Waves in Crystals, Wiley, New York, 1984. Kong, J.A.: Electromagnetic Wave Theory, Wiley, New York, 1986. Mackay, T.G.; Lakhtakia, A.: Electromagnetic Anisotropy and Bianisotropy: A Field Guide, World Scientific, Singapore, 2010. Lindell, I.V.; Sihvola, A.H.; Tretyakov, S.A.; Viitanen, A.J.: Electromagnetic Waves in Chiral and Bi-anisotropic Media, Artech House, Boston, MA, 1994. Hoffmann, R.: Microwave Integrated Circuit Handbook, Artech House, Boston, MA, 1985. Capolino, F. (Editor): Theory and Phenomena of Metamaterials, CRC Press, Boca Raton, FL, 2009. The Heaviside Centenary Volume, The Institution of Electrical Engineers, London, 1950. Whittaker, E.T.: Oliver Heaviside, In Electromagnetic Theory, Vol. 1, Oliver Heaviside, Reprint, Chelsea Pub. Co., New York, 1971. Behrend, B.A.: The work of Oliver Heaviside, In Electromagnetic Theory, Vol. 1, Oliver Heaviside, Reprint, Chelsea Pub. Co., New York, 1971. Kraus, J.D.: Antenna, 2nd Edition, McGrawHill, 1988.

B.30 Collett, E.: Polarized Light: Fundamentals and

Applications, Marcel Dekker, Inc., 1993. B.31 Saleh, B.E.A.; Teich, M.C.: Fundamental of Photonics,

Wiley, New York, 1991.

Journals J.1 Jacob, Z.; Alekseyev, L.V.; Narimanov, E.: Optical

J.2

J.3

J.4

J.5

J.6

J.7

hyperlens: far-field imaging beyond the diffraction limit, Opt. Express, Vol. 14, No. 18, pp. 8247–8256, 2006. Lee, H. et al.: Development of optical hyperlens for imaging below the diffraction limit, Opt. Express, Vol. 15, pp. 15886–15891, 2007. Seddon, N.; Bearpark, T.: Observation of the inverse Doppler effect, Science, Vol. 302, pp. 1537–1539, Nov. 2003. Mosallaei, H.; Sarabandi, K.: Magneto-dielectrics in electromagnetics: concept and applications, IEEE Trans. Ant. Propagat., Vol. 52, No. 6, pp. 1558–1567, June 2004. Alu, A.; Engheta, N.: Pairing an epsilon–negative slab with a mu-negative slab: anomalous tunneling and transparency, IEEE Trans. Antenna Proag., Special Issue on Metamaterials, Vol. 51, No. 10, pp. 2558–2570, Oct. 2003. Lee, H.; Xiong, Y.; Fang, N.; Srituravanich, N.; Durant, S.; Ambati, M.; Sun, C.; Zhang, X.: Realization of optical superlens imaging below the diffraction limit, New Journal of Physics, Vol. 7, No. 255, pp. 1–16, 2005. Kim, M.; Rho, J.: Metamaterials and imaging, Nano Convergence, Vol. 2, No. 22, pp. 1–16, 2015.

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5 Waves in Material Medium-II (Reflection & Transmission of Waves, Introduction to Metamaterials)

Introduction The characteristics of material media and EM-waves propagation in unbounded media are discussed in the previous chapter 4. Continuing the topics, this chapter is about the normal and oblique incidence of EM-waves at the interface of two media. The characteristics of both the normal and oblique incident EM-waves are obtained using an analytical method and convenient equivalent transmission line models. Under certain conditions, the interface surface of two media could acquire the property of a perfect electric conductor (PEC), or perfect magnetic conductor (PMC), or even a reactive impedance surface (RIS). These surfaces play a significant role in the modern microwave and antenna technologies. Some electromagnetic characteristics of the engineered composite materials, with negative permittivity and permeability, are also presented in the present chapter. These artificially structured materials are known as metamaterials. The realization, circuit modeling, and some applications of the metamaterials and metasurfaces are further presented in chapters 21 and 22, respectively. The properties of natural and artificial dielectrics are further examined in chapter 6. Objectives

• • • •

To discuss the normal and oblique incidence of the EMwaves at the interface of two media. To present the transmission line model of the normal and oblique incidence of the EM-waves. To obtain the dispersion diagrams of refracted waves in the isotropic and anisotropic media. To obtain characteristics of Brewster and the critical angles of incidence.

• • •• • •

To discuss the general properties of metamaterials media and their classifications. To obtain some characteristics of the EM-wave propagation in the metamaterials. To obtain the circuit models of metamaterials. To discuss the possibility of obtaining the flat lens, superlens, and hyperlens beyond the diffraction limit. To discuss the nature of the Doppler effect and Cerenkov radiation in the metamaterials. To discuss metamaterials as thin microwave absorbers.

5.1 EM-Waves at Interface of Two Different Media The EM-waves can strike the interface of two media with different electrical characteristics, either normally or obliquely. If the second medium is a dielectric medium, the waves undergo both reflection and transmission, whereas if the second medium is a perfect conductor, the reflection occurs. The obliquely incident plane waves follow the well-known Snell’s law (also called Snell-Descartes law) of reflection and refraction. There are important applications of obliquely incident EM-waves.

5.1.1

Normal Incidence of Plane Waves

Figure (5.1a) shows an interface of two media #1 and #2. The media are electrically characterized by the primary parameters εri, μri, σi; i = 1, 2 and also by the secondary parameters, such as the refractive index ni, and intrinsic or wave impedance ηi. Initially, both media are considered lossless dielectric media. Next, the second medium is treated as a PEC.

Introduction to Modern Planar Transmission Lines: Physical, Analytical, and Circuit Models Approach, First Edition. Anand K. Verma. © 2021 John Wiley & Sons, Inc. Published 2021 by John Wiley & Sons, Inc.

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5 Waves in Material Medium-II

The incident wave in the medium #1, propagating in the x-direction with propagation constant k1, is y-polarized The direction-x is with an electric field component Einc y normal to the interface PQ along the y-axis; so the incident magnetic field component Hinc z is +z directed. This is a special case of the TM-polarized obliquely incident wave, discussed in section (5.2.2). The incident ray strikes at the location O and gets partly reflected and refracted (i.e. transmitted) at O with the field components

− Href z

Eref y ,

tra Etra y , Hz

, and

x = 0−

+ Eref y x

inc

Hz

x = 0−

= Etra y x

ref

x

i e, 1 + 1 1− η1

x = 0− Eref Etra m m = inc , inc Em Em Eref 1 Etra m m = η2 Einc Einc m m

x = 0+

x = 0−

= Hz

x

, x = 0+

,

1−Γ =

Etra m Einc m

x=0

x=0

η2 Htra z η1 Hinc z

ref

ref

+ jk1 x , Hz = − z E y = yEref m e

x=0

Eref m e + jk1 x η1

b

Transmitted plane wave tra

tra

− jk2 x , Hz = z E y = yEtra me

Etra m − jk2 x e η2

c, 511

where η1 and η2 are the intrinsic impedance of medium #1 and medium #2, respectively. The total tangential components of the electric and magnetic fields in both media are continuous across the interface at x = 0:

a

Einc Eref Etra m − m = m η1 η1 η2

b 512

η1 τ η2

d

a =τ=

a

c

η − η1 =Γ= 2 η2 + η 1 =

Einc m e − jk1 x η1

Reflected plane wave

1+Γ=τ

On solving the above last two equations, the reflection coefficient Γ, and the transmission coefficient τ are computed at the interface: Eref m Einc m

inc

ref tra Einc m + Em = Em

,

tan

+ Hz x

inc

− jk1 x E y = yEinc , Hz = z m e

, respectively. In

the medium #2, the propagation constant is k2. In general, the k is a wavevector with three components. Figure (5.1a) shows that the direction of propagation is decided by the direction of the Poynting vector. The field components are summarized in equation (5.1.1) on suppressing the time-harmonic dependence ejωt: Einc y x

Incident plane wave

2η2 η2 + η1

b 513

For a lossy medium, the intrinsic impedance is a complex quantity, and reflection and transmission coefficients are also complex quantities. However, the magnitude of the reflection coefficient of a passive medium is always equal to or less than unity, i.e. |Γ| ≤ 1. The magnitude and phase of the reflected and transmitted waves are different from that of the incident waves. For a lossless composite medium, the matching is obtained, i.e. no reflection Γ = 0, for an incident wave

at the interface with η1 = η2. It could be realized in two ways: for μ1 = μ2 = μ0 for μ1

μ2 , εr1

εr1 = εr2 μr1 μ εr2 = r2 εr1 εr2

a b 514

In equations (5.1.4a,b), the first case is the usual one, as both media are identical. However, the second case is interesting; as the matching is possible even for dissimilar media. It may not be a practical one with natural materials. However, for the artificially engineered metamaterials, both μr and εr can be tuned independently to achieve the matching condition, It is discussed in subsection (5.5.8). The impedance-matched Huygens’ metasurface has also been developed by tuning of electric and magnetic inclusions discussed in subsection (22.5.2) of chapter 22.

5.1 EM-Waves at Interface of Two Different Media

Sometimes, the concept of the reflectance (reflectivity) R and transmittance (transmissivity) T are used to show the portion of the reflected and transmitted powers: Eref Pref m = inc P Einc m

2

Etra Ptra m = Pinc Einc m

2

R= T=

Power balance

2

2

2η1 2η1 2η2 2η1

R= Γ2=

,

η2 − η1 η 2 + η1

total

2

a

R + T = 1,

E total

η1 4η1 η2 = η2 η1 + η2 2 2 η1 Γ + τ =1 η2

T= τ2

,

waves. The field is uniform in the x-direction. However, it creates partly a standing wave, and partly the traveling wave moving in the x-direction:

H b

2

− jk1 x x = yEinc + Γe + jk1 x m e 1 x = z Einc e − jk1 x − Γe + jk1 x η1 m

a b 516

c

515

For Γ > 0, i.e. for η2 > η1, the above expression can be decomposed as follows:

The total field in medium #1 is a combination of the incident and reflected waves causing interference of

In space

total

− jk1 x x = yEinc − Γe − jk1 x + Γe − jk1 x + Γe + jk1 x m e

total

x = yEinc 1 + Γ e − jk1 x + j2Γ sin k1 x m

E E total

In space-time E

a

x, t = yEinc m 1 + Γ cos ωt − k1 x − 2Γ sin k1 x sin ωt Traveling wave

b

Standing wave 517

total

In equation (5.1.7), E

total

x, t = y Re E

x e

jωt

is taken. Likewise, for Γ < 0, i.e. for η2 < η1, the wave in the medium can be written as follows: In space – time total

E

x, t = yEinc m 1 + Γ cos ωt − k1 x − 2Γ cos k1 x cos ωt Traveling wave

Standing wave

1 S∗ x = x Etotal x H∗total x z 2 y 1 1 − Γ 2 + j2 Γ sin 2k1 x 2η1 =x 1 2 inc 2 τ Em , x > 0 2η2

Emin = 1 − Γ Emax = 1 + Γ Emax 1+ Γ VSWR = = Emin 1− Γ Einc m ,

Einc m

a b 519

The power densities in medium #1 (x0) are obtained using Poynting vector relation:

x θc

a

for dielectric media sin θc =

In this case, equation (5.3.3c) provides the real value of the angle θ2 as sinθ2 < 1 corresponding to an angle of incidence θ1. Figure (5.5a) shows the reflection and transmission of the obliquely incident plane wave at the interface for the case εr1 > εr2.

b

=0

for magneto − dielectric media sin θc =

Case #1: θ1 < θc

Figure (5.5b) shows that the refracted wave travels along the interface x = 0+ in the y-direction and no component of the refracted (transmitted) wave propagates in the medium #2. Equation (5.2.28a) also shows that for the TM-polarized incident wave, the reflection coefficient is ΓTM = − 1 at the angle of refraction θ2 = π/2. It demonstrates that under the total reflection condition, occurring at the critical angle θ1 = θc, the interface behaves as a PEC for the TM-polarized incident wave. However, equation (5.2.28b) provides also a transmission coefficient τTM = 2η2 η1. It shows that the transmitted wave in the medium #2, confined at the interface, travels along the y-axis at the interface x = 0+. The transmitted field components at an angle θ2 = π/2 in the medium #2 are given in equation (5.2.18). In the case of the TE polarization of the obliquely incident wave, equation (5.2.27a,b) shows the reflection coefficient ΓTE and transmission coefficient ⊥ = +1 TM = τ . In this case, the interface acts like a PMC. τTE ⊥ Again, the transmitted wave in the medium #2 propagates in the y-direction at the interface x = 0+. It is seen from the field equation (5.2.10) for θ2 = π/2.

1 2

1 − μr1 εr1 μr2 εr2 2 sin 2 θc

For the real value of the critical angle θc, permittivities of media are taken as εr1 > εr2. Thus, the critical angle of the interface is a fixed quantity. The angle of refraction θ2 deciding the direction of wave propagation in medium #2 is a function of angle incidence θ1. Figure (5.5a–c), applicable to for both the TM and TE polarizations, consider three cases, θ1 < θc, θ1 = θc, θ1 > θc, of propagation for the obliquely incident plane wave.

533

Figure (5.5c) shows the case for θ1 > θc. Equation (5.3.3c) of Snell’s refraction law shows that sinθ2 > 1. Therefore,

133

134

5 Waves in Material Medium-II

Reflected wave

y

Medium#1 εr1 θ1

y

Reflected wave

Transmitted wave

Transmitted wave

Medium#1 εr1 θ1

θ2

θ1 = θc

x

θ1

θ2 = π / 2 z εr2

εr2 Incident wave

x

Incident wave

Medium#2

Medium#2

(θ1 < θc)

(θ1 = θc)

(a) Partial reflection.

(b) Complete reflection.

Reflected wave

y

Transmitted wave e–jk2yy

Medium#1 εr1 θ1 θc

e–αx z

θ1 Incident wave

x εr2

Medium#2 (θ1 > θc)

(c) Complete reflection and Surface-wave propagation.

Figure 5.5

Oblique incidence of plane wave at three different angles of incidence.

there is no real solution to the angle θ2. However, the wave propagation in medium #2 requires sinθ2 and cosθ2, not the value of the angle of refraction θ2. The sinθ2 and cosθ2 could be obtained from Snell’s Law given by equation (5.2.7c), also from the wavevector k2 and its x and y-directed components k2x and k2y using equation (5.2.1): εr1 sin θ1 > 1, for εr1 > εr2 sin θ2 = εr2 εr1 cos θ2 = ± j sin 2 θ1 − 1 εr2 k2y εr1 sin θ2 = , k2y = k2 sin θ1 k2 εr2 k2x cos θ2 = , k2x = ± jα; k2 εr1 sin 2 θ1 − 1 where, α = k2 εr2

a b c d

in a usual passive medium. The expressions (5.3.5b,d, e) are used for both the TE and TM polarization to compute the reflection and transmission coefficients, and also the refracted fields in medium #2 for the case θ1 > θc TE Polarization

Equation (5.2.8c,d) for the reflection and transmission coefficients of the obliquely incident TE-polarized wave, using an equation (5.3.5a) are reduced to the following expression: ΓTE ⊥ =

η2 η1 cos θ1 − j εr1 εr2 sin 2 θ1 − 1

1 2

η2 η1 cos θ1 + j εr1 εr2 sin 2 θ1 − 1

1 2

= + 1ejϕ⊥ At

θ1 = θc , sin θ1 =

e

and

ΓTE ⊥

535

τTE ⊥

The negative (−) value is selected in further computation as it provides exponentially decaying fields with distance from the interface in the medium #2. The choice of positive (+) value results in the exponentially growing field with the distance that is physically not possible

a εr2 εr21

= + 1, θ2 = π 2 2 cos θ1 = cos θ1 + η1 η2 cos θ2 2 cos θ1 = cos θ1 + j η1 η2 εr1 εr2 sin 2 θ1 − 1

At θ1 = θc , sin θ1 =

εr2 εr1 , and τTE ⊥ =2

b

1 2

c d 536

5.3 Special Cases of Angle of Incidence

Fig (5.5c), the interface supports the slow-wave type surface wave propagation. It is demonstrated below. For the case θ1 > θc, the electric and magnetic field component and power flow of the TE-polarized wave in the medium #2 is obtained from using equation (5.3.5) with equation (5.2.10a):

The interface surface acts as a PMC surface for the TEpolarized incident wave under the case θ1 ≥ θc because ΓTE ⊥ = + 1. Over a certain frequency band, it is realized as the artificial magnetic conductor (AMC) surface. At the critical angle, θ2 = π/2, the transmitted wave propagates along the y-direction at the interface x = 0+, as shown in Fig (5.5b). For the case θ1 > θc, shown in TE inc − xk2 Etra ⊥z = zτ⊥ Em e

εr1 εr2 sin 2 θ1 − 1

1 2

× e − jk2 y sin θ1

εr1 εr2

TE inc − αx − jk2y y Etra e ⊥z = z τ⊥ Em e

Htra ⊥ =

inc τTE ⊥ Em

η2

∗

cos θ1 x sin θ2 − y cos θ2 e − αx e − jk2y y

∗

τTE ⊥

2

2

S =E×H = ∗

S =

a

τTE Einc ⊥ m η2 k2

2

Einc m

537

2

η2

cos θ1 y sin θ2 + x cos θ2

cos θ1 yk2y + xjα cos θ2

ω ω = k2y k2 sin θ1

vp εr2 = εr1 sin θ1

In equation (5.3.8), as εr1 > εr2; so vp < vp, i.e. the phase velocity of the surface wave is less than that of in the medium #2. Therefore, the surface wave is a slow-wave. tra

=

Htra,z ∗

τTM Einc m

= z

k2 τTM Einc m η2 ∗

S = E ×H =

c

TM Polarization

All three cases of the angle of incidence apply to the TMpolarized obliquely incident plane wave. For θ1 > θc, the reflection and transmission coefficients of the TM-polarization, given by equation (5.2.28) are reduced to ΓTM = −

η1 η2 cos θ1 − j εr1 εr2 sin 2 θ1 − 1

1 2

η1 η2 cos θ1 + j εr1 εr2 sin 2 θ1 − 1

1 2

= − 1ejϕ τTM = At

TM

a 2 cos θ1

η1 η2 cos θ1 + j εr1 εr2 sin 2 θ1 − 1

θ1 = θc , θ2 = π 2, sin θ1 =

and Γ

TM

= −1 τ

TM

a

e − αx e − jk2y y

b

2

Einc m

η2 k2

b

εr2 εr1 c

The electric and magnetic field components of the TM polarization, also the complex Poynting vector in the medium #2 under θ1 > θc, could be obtained using equation (5.3.5), from equation (5.2.18). The results are summarized below:

− x k2y + y jα e − αx e − jk2y y

τTM

= 2η2 η1

1 2

539

εr2 εr1 538

E

e

− 2αx

e − 2αx

In equation (5.3.7a,b,c), the y-directed wavevector k2y and attenuation factor α are given by equation (5.3.5). Equation (5.3.7a,b,c) shows that along the direction normal to the interface, i.e. + x-direction, the field is exponentially decaying in medium #2; showing the confinement of field near the interface. However, the wave propagates in the y-direction. It is also evident from the complex Poynting vector giving real power transportation in the y-direction, while imaginary power shows storages of energy in the x-directed evanescent field. So the interface supports a surface wave, excited at the interface of two media by the obliquely TEpolarized incident plane wave at the angle of incidence θ1 ≥ θc. The surface wave, in more detail, is discussed in chapter 7. Its phase velocity along the y-axis is vp =

b

2

y k2y + x jα

e − 2αx

c

5 3 10

135

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5 Waves in Material Medium-II

5.4 EM-Waves Incident at Dielectric Slab

The expression (5.3.9a) shows that there is a total reflection Γ

TM

= − 1 at the interface for θ1 ≥ θc, with-

out any transmission of power from the medium #1 to medium #2. The transmission coefficient shows the presence of the x-directed evanescent field in the medium #2. Under such conditions, the interface acts as a perfect electrical conductor (PEC); so the interface surface of two media can act as the artificial electric conductor (AEC). However, there is an exponentially decaying field in the medium #2, and the interface supports the surface wave propagation along the interface at x = 0+ in the y-direction. The real power carried in the y-direction along the surface is given by the real part of equation (5.3.10). The imaginary part of the Poynting vector shows the stored energy in the evanescent field. Figure (5.5c) shows the surface wave propagation in the y-direction that also occurs in the case of the obliquely incident TE-polarized waves. This subsection shows the existence of a surface wave at the interface of natural media. However, artificially engineered metasurfaces discussed in subsection (22.5.5) of chapter 2 has additional ability to control the surface wave in the desired manner, and also reradiate it as the leaky wave.

The EM-waves can strike the slab embedded in a homogeneous medium both normally and obliquely. Both cases of wave incidence and their transmission line models are discussed below. The analysis can be extended to the multilayer medium. 5.4.1

Oblique Incidence

Figure (5.6a) illustrates the oblique incidence of the TEpolarized waves on a three-layered dielectric medium. It is desired to find overall reflection and transmission coefficients of a dielectric slab of thickness d embedded in a homogeneous medium, while the TE-polarized wave is obliquely incident at the first interface located at the y-axis. The forward and reflected waves are present in both the media #1 and #2 and finally transmitted to the medium #3. The media #1 and #3 have identical electrical properties. Extending the process given in equations (5.2.1)– (5.2.7), the total E and H-fields in three media are written as follows:

E − field in medium #1 total

− jk1 E m1 x, y = zEinc m e

x cos θ1 + y sin θ1

− jk1 + zEref m e

− x cos θ1 + y sin θ1

a

E − field in medium #2 total

− jk2 E m2 x, y = zEm2f m e

x cos θ2 + y sin θ2

− jk2 + zEm2b m e

541

− x cos θ2 + y sin θ2

b

E − field in medium #3 total

− jk1 E m3 x, y = zEtra me

x − d cos θ1 + y sin θ1

c

y Medium #2

Medium #1

η1

Href k1

Eref

η2

k3

k2

θ1

z

θ1

o

Medium #3 Etran η3 = η1 θ1

Htran

k2

Medium #1 Medium #2 Medium #3 η1

θ2

Γ23

Γ12

Einc x d

k1

Einc

d Hinc (a) TE-polarized oblique incident. Figure 5.6

η1

η2

Eref

Plane-wave incident on a dielectric slab.

0

y Ezinc

Etran

z

x

(b) Transmission line model.

x

5.4 EM-Waves Incident at Dielectric Slab

H − field in medium #1 total

H m1 x, y =

Einc m x sin θ1 − y cos θ1 e − jk1 η1

x cos θ1 + y sin θ1

Eref m x sin θ1 + y cos θ1 e − jk1 η1 H − field in medium #2 total

H m2 x, y =

− x cos θ1 + y sin θ1

Em2f m x sin θ2 − y cos θ2 e − jk2 η2

x cos θ2 + y sin θ2

Em2b m x sin θ2 + y cos θ2 e − jk2 η2 H − field in medium #3 total

H m3 x, y =

Etra m x sin θ1 − y cos θ1 e − jk1 η1

εi μi = k0 εri μri

ki also, ηi = ωεi

a, c,

μi μri = η0 b εi εri ωμi ηi = i = 1, 2 d ki 543

Further, in equations (5.4.1b) and (5.4.2b), the superscripts m2f and m2b show the forward and backward moving waves in the medium #2. The tangential Ez and Hy field components at the first interface located at x = 0 are continuous: Ez − field

=

total

E m1 x, y

= E m2 x = 0− − jk1 sin θ1 y − jk1 sin θ1 y Einc + Eref m e m e − jk2 sin θ2 y − jk2 sin θ2 y Em2f + Em2b m e m e

x, y

a x = 0+

−

Einc m η1

Htotal m1 x, y cos θ1 e

x = 0−

− jk1 sin θ1 y

+

= Htotal m2 x, y Eref m η1

cos θ1 e

x = 0+

a

− jk1 sin θ1 y

Em2f Em2b = − m cos θ2 e − jk2 sin θ2 y + m cos θ2 e − jk2 sin θ2 y b η2 η2 545 The continuity of the y-components of the field across the interface also provides the phase matching giving the following result: k1y = k2y ,

k1 sin θ1 = k2 sin θ2

b

c

The dispersion relation (4.5.29d) of chapter 4 provides the following expressions for the propagation constant of propagating wave, in the medium #2 and medium #3, in the x-axis direction: Medium#2 k22x + k22y = ω2 μ2 ε2 ,

k2x =

ω2 μ2 ε2 − k21 sin 2 θ1

a

ω2 μ1 ε1 − k21y =

ω2 μ1 ε1 − k21 sin 2 θ1

b

Medium#3 k3x = k1x =

547 After canceling the phase-matching factors in equations (5.4.4) and (5.4.5), the amplitude matching at the interface #1 provides the following expressions: ref m2f m2b Einc m + Em = Em + Em

cos θ1 inc cos θ2 m2f Em − Eref Em − Em2b m = m η1 η2

a b 548

b 544

Hy − field

542

+

x − d cos θ1 + y sin θ1

ηi =

total

a

− x cos θ2 + y sin θ2

In equations (5.4.2a,b,c) superscripts m1, m2, and m3 correspond to medium #1, #2, and #3, respectively. The propagations constant and intrinsic impedance in media are ki =

+

546

The above equations are solved to get the following set m2b of expressions for Em2f m and Em : 1 ref 1 + P Einc m + 1 − P Em 2 1 ref 1 − P Einc Em2b m = m + 1 + P Em 2 η cos θ1 where, P = 2 η1 cos θ2 Em2f m =

a b c 549

After cancellation of the phase-matching factors at the interface #2 (at x = d), the continuity of tangential components Ez and Hy provide the following expressions:

137

138

5 Waves in Material Medium-II

Ez − field

total

− jϕ + Em2f m e

Hy − field

total

E m2 x, y

total

= E m2 = E m3 x, y

x = d− m2b + jϕ Em e = Etran m total

H m1 x, y

x = d−

a x = d+

b , where ϕ = k2 d cos θ2

c

total

= H m3 x, y

5 4 10 x = d+

cos θ2 m2f − jϕ cos θ1 tran + jϕ = Em e − Em2b Em , m e η2 η1 − jϕ + jϕ − Em2b = PEtran Em2f m e m e m

d

Equations (5.4.10a,b,c) are again solved for Em2f m and Em2b m :

1 + jϕ 1 + P Etran m e 2 1 − jϕ = 1 − P Etran m e 2

Em2f m =

a

Em2b m

b 5 4 11

Equations (5.4.9) and (5.4.11) are equated to get a pair of expressions for the reflection (ΓTE) and transmission (τTE) coefficients: 1 + P + 1 − P ΓTE = 1 + P e + jϕ τTE 1−P + 1 + P Γ where, ΓTE = τTE =

TE

= 1−P e

− jϕ TE

τ

Eref m Einc m Etra m Einc m

a b

The reflection and transmission coefficients of both the TE and TM-polarized incident waves are identical in form, except that parameters P and Q are different. 5.4.2

Normal Incidence

For the case of the normally incident TE-polarized wave, the angles of incidence and refraction are θ1 = θ2 = 0∘. Furthermore, ϕ = k2d, (k2x = k2), and P = η2/η1. Under, this condition, equation (5.4.13a,b) for the reflection and transmission coefficients are reduced to the following expressions:

c

ΓNor =

d

τNor =

η2 − η1 η2 + η1

1−

4η2 η1 η 2 + η1

2

1 − e − j2k2 d η2 − η 1 η2 + η1

1−

e − jk2 d η2 − η1 η2 + η1

2 − j2k2 d e

5 4 12 The above equations are solved to get the following expressions for the reflection and transmission coefficients of the obliquely incident TE-polarized wave: 1 − P2 e − jϕ − e + jϕ 1 + P2 e + jϕ − 1 − P2 e − jϕ 4P τTE = 2 + jϕ − 1 − P2 e − jϕ 1+P e η cos θ1 where, P = 2 η1 cos θ2 ΓTE =

a b c 5 4 13

For the obliquely incident TM-polarized wave on the three-layered medium, shown in Fig (5.6a), the process can be repeated to get the following expressions, similar to expressions (5.4.14): ΓTM =

jϕ

a b c d 5 4 14

b

5 4 15 The above expressions are valid for the normally incident TM-polarized waves also. Figure (5.6b) shows the equivalent transmission line model of the three-layered medium. The reflection coefficients at both interfaces can be obtained from the expression, Γ = (ZL − Z0)/(ZL + Z0). At the first interface, corresponding medium impedances are ZL η2 and Z0 η1, while at the second interface, these are ZL η1 and Z0 η2. The reflection coefficients Γ12 and Γ23 are defined at the first and second interfaces. The above equations are rewritten as follows: ΓNor = τNor =

− jϕ

1−Q e −e 1 + Q2 e + jϕ − 1 − Q2 e − jϕ 4Q τTM = 2 + jϕ 1+Q e − 1 − Q2 e − jϕ η cos θ1 where, Q = 1 η2 cos θ2 ϕ = k2 d cos θ2 2

a

2 − j2k2 d e

Γ12 1 − e − j2k2 d 1 + Γ12 Γ23 e − j2k2 d 4η2 η1 η2 + η1

2

e − j2k2 d 1 + Γ12 Γ23 e − j2k2 d

ΓNor ≈ Γ12 1 − e − j2k2 d η2 − η1 , η2 + η 1 η − η2 = − Γ12 = 1 η2 + η1

where, Γ12 = Γ23

a b c d e 5 4 16

5.5 EM-Waves in Metamaterials Medium

The approximate expression (5.4.16c) is used for the small reflection at the interfaces, i.e. for |Γ12Γ23| < < 1. The above expressions for total reflection and transmission of a slab can also be obtained from the theory of multiple reflections [B.5, J.2]. Using the relation (5.4.3), equation (5.4.16) can be recast in the following format also: ΓNor = τNor

μ2 k 1 2 − μ1 k 2

2

1 − e − j2k2 d

a

2

μ2 k1 + μ1 k2 − μ2 k1 − μ1 k2 2 e − j2k2 d 4μ1 μ2 k1 k2 e − jk2 d = μ2 k1 + μ1 k2 2 − μ2 k1 − μ1 k2 2 e − j2k2 d

b 5 4 17

In the case of the obliquely incident TE waves, in the above equations, we replace the wavevectors k1 and k2 as k1 k1x k1 cos θ1 and k2 k2x k2 cos θ1 respectively; where the waves propagate in the x-direction. The waves are confined in the y-direction. As a special limiting case, if the electrical parameters of the slab of thickness d and the host medium are identical, the k2 = k1, μ2 = μ1. It leads to ΓNor = 0, i.e. no reflection at the interface, and τNor = e − jk2 d , i.e. the slab provides a complete transmission with a lagging phase k2d. This is the case of the matched layer seen in equation (5.4.15a) for η1 = η2. The matching applies even to the interface of electrically dissimilar media given by the condition in equation (5.1.4b). The expressions obtained in the present subsection are useful to characterize the metamaterials slab. It is discussed in subsection (21.4.1) of chapter 21.

5.5 EM-Waves in Metamaterials Medium The electromagnetic properties of metamaterials, propagation of EM – waves in the metamaterials media, and circuit models of metamaterials are presented in this section. Some applications of the metamaterials are also discussed. The realization of engineered metamaterials and their further modeling are discussed in chapter 21. 5.5.1 General Introduction of Metamaterials and Their Classifications The materials existing in nature, ignoring losses, are characterized electromagnetically by the relative permittivity and relative permeability. Normally, these are positive quantities. Such a medium is called the double-positive (DPS) medium. It has a positive real refractive index n. It is placed in the first quadrant of the (μr, εr)-plane as shown in Fig (5.7). The conductivity or resistivity is added to a medium to account for the losses. Some noble metals, such

as silver and gold, have negative permittivity in the visible frequency bands. Drude model, based on the plasma model of metal, is used to model such behavior. It is discussed in chapter 6, section (6.5.2). An identical phenomenon exists in the ionospheres. Below the so-called plasma frequency mentioned briefly in section (3.3.1) of chapter 3, the permittivity of a plasma medium becomes negative. In this case, the medium does not support wave propagation. The waves become oscillatory decaying evanescent mode. The permeability of the plasma medium remains a positive quantity. Under the evanescent mode condition, the plasma medium below the cut-off frequency is called the epsilon-negative (ENG) medium. It is shown in the second quadrant in Fig (5.7). Thus, below the plasma frequency, the medium reflects the wave incident on it from the DPS medium; while inside the ENG medium only decaying evanescent mode exists. However, above the plasma frequency, the ENG plasma medium becomes a normal DPS medium. Likewise, a gyrotropic medium, such as a ferrite medium, has negative permeability over a lower microwave frequency range. However, its permittivity is a positive quantity. Such a medium is called the munegative (MNG) medium. It is shown in the fourth quadrant of Fig (5.7). Both the ENG and MNG media have imaginary refractive index n and support only decaying evanescent mode, as shown in Fig (5.7). As a single negative medium, both the ENG and MNG belong to the group of the single negative, i.e. (SNG) medium. Both the ENG and MNG are described by the following forms of the Drude model: ENG MNG

εr = 1 − ωep ω

2

μr = 1 − ωmp ω

a 2

b, 551

where ωep and ωmp are the electric plasma and the magnetic plasma frequency respectively. The magnetic plasma is conceptual as no magnetic charge is available in nature. The natural dielectric and magneto-dielectric media support the slow-wave propagation as for these media εr > 1, μr > 1. However, it is possible to engineer a DPS metamaterial for 0 < εr < 1, μr ≥ 1 and 0 < μr < 1, εr ≥ 1. The first medium is called epsilon near zero (ENZ) medium, while the second one is called mu near-zero (MNZ) medium. They belong to the group called near-zero (NZ) medium. These media support the fast-wave. The DNG medium also supports the ENZ and MNZ cases. The NZ medium is shown as a dark circle at the origin in Fig (5.7). These media are obtained just above the plasma frequency. The refractive index of the ENZ and MNZ is numerically less than unity, supporting the fast-wave. Even a zero refractive index could be obtained. It is known as the “nihility” [B.6].

139

5 Waves in Material Medium-II

μr

ENG (Plasma)

DPS (Dielectrics) (εr > 0, μr > 0) n: +tive Vp

Amp

Amp

(εr < 0, μr > 0) n: Img.

t Vg Forward - wave propagation

t Evanescent wave

εr

Vp Amp

Amp

140

t Vg

t

(εr < 0, μr < 0) n: –tive

(εr > 0, μr < 0) n: Img.

Backward - wave propagation

Evanescent wave

DNG (Metamaterials)

MNG (Gyrotropic medium)

Figure 5.7

Electrical grouping of the materials media in the (μr, εr)-plane.

Following the above-discussed nomenclature scheme of a material medium, it is natural to envisage an artificial medium with both the permittivity and permeability as negative quantities. Such an engineered medium could be called a double negative medium (DNG) medium. It is shown in the third quadrant of Fig (5.7). It has a negative refractive index (−n). The DPS medium supports the wave propagation such that both the phase velocity vp and group velocity vg are in the same direction, whereas the DNG medium supports wave propagation with phase velocity vp and group velocity vg in opposite directions. Such EM-wave is known as the backward wave. However, no DNG material is known to exist in nature. The DNG material medium has been synthesized over certain frequency bands, from the microwave to optical frequency ranges. The DNG materials are commonly known as metamaterials. However, artificially engineered ENG, MNG, ENZ, and MNZ media also belong to the metamaterials. Even the DPS could be engineered, and called a metamaterial, to realize the controlled values of permittivity and permeability to meet the specific design requirement. One such requirement is to develop the microwave absorbers discussed in subsection (5.5.8). Several dedicated books are available on the topics of the metamaterials [B.6–B.10]. Depending on their characteristics, several other names have also been proposed for the metamaterials [B.7, B.10]. Veselago proposed the concept and

electrodynamics of the metamaterials [J.3]. The metamaterials have been synthesized by embedding either the resonating type inclusions or the C-L type artificial transmission line inclusions in a host medium. The metamaterials could be either resonant metamaterials or nonresonant metamaterials, so the metamaterials are structured composite materials. Starting with J.C. Bose, such composite materials have a long history [J.4–J.6]. The concept of composite materials and their modeling is discussed in section (6.3) of chapter 6. The C-L line supports the backward wave propagation. It is discussed in section (3.4) of chapter 3. A metamaterial, i.e. the DNG medium, is also called the backward wave (BW) medium, or BW material [J.7]. The normal forward wave supporting DPS medium could be called forward wave (FW) medium, or FW material. The physical dimensions of the resonating inclusions, or the C-L line inclusions, are much smaller than the operating wavelengths. The metamaterials could be treated as the homogenized medium described by the parameters permittivity and permeability. The inclusions are arranged in the host medium either periodically, or nonperiodically. The periodically arranged inclusions, with a period λg/2, form another group of metamaterials known as the electronic bandgap (EBG) materials. It is discussed in chapters 19 and 20. Even the ENG and MNG materials have been artificially developed using such inclusions.

5.5 EM-Waves in Metamaterials Medium

The (μr, εr)-plane of Fig (5.7) summarizes the abovediscussed four-groups of material media with their basic electrical characteristics. The circuit models and further wave characteristics of these media are shown in Fig (5.10). The synthesis, modeling, and some illustrative applications of the metamaterials are discussed in chapter 21. This section only presents some basic electrodynamics properties of the DNG medium. 5.5.2

EM-Waves in DNG Medium

This section considers the EM-wave propagation in the lossless DNG medium, the concept of the negative refractive index, the basic circuit model of a DNG medium, and the lossy DNG medium.

Maxwell’s equation (4.4.11a,b) of chapter 4 could be written for a lossless (σ = 0) DNG medium by using (−|εr|) and (−|μr|) in place of their usual positive values: ∇ × E = − jωμ0 − μr H

a

∇ × H = jωε0 − εr E

b

∇×

For a DPS medium, the directions of the vectors k and S are identical, i.e. they are parallel vectors giving the relation k S > 0. Therefore, in a DPS medium, both the phase and group velocities are in the same direction giving v p v g > 0. It is shown in the first quadrant of Fig (5.7). The phase of the propagating EM-wave in the DPS medium lags while traveling in the S -direction. In the case of a DNG medium, both permittivity and permeability are negative. Using Maxwell’s equations (5.5.2c,d), and replacing ∇ − jk the wavevector triplet- E , H, k relations for the DNG medium are written as follows: k × E =ωμ

Lossless DNG Medium

∇ × E = − jωμ0 μr

are combined into one diagram as shown in Fig (5.8c).

−H

− H = jωε0 εr E

c d 552

Equations (4.5.31a) and (4.2.3b) of chapter 4 show Maxwell’s equations for a DPS medium in terms of the wavevector k and the field vectors

−H

a

k × H =ωε E

b 553

However, in the above expression reversal of the direction of the magnetic field involves the reversal of the direction of the power flow toward the source. Physically, it is not possible, so the above equations are rearranged as follows by associating the negative sign with the wavevector k : −k

× E =ωμ H

a

−k

× H = −ω ε E

b 554

E , H . They form

the wavevector triplet ( E , H, k ) relations in the righthand (RH) coordinate system, shown in Fig (5.8a) for a DPS medium. Normally, the direction of the wavevector k determines the direction of the wave propagation, i.e. the direction of the phase velocity vp. However, the Poynting vector ( S ) given by equation (4.4.20) provides another power-vector triplet ( E , H, S ) that determines the true direction of wave propagation. It is the direction of the energy flow from the source to a load. So the corresponding group velocity vg defines the direction of the wave propagation. Figure (5.8c) shows both the sets of the vector triplets of a DPS medium. The field vectors are rotated in cyclic order so that the power flow is maintained from left to right-hand side, i.e. from a source located at the origin O, power flows outwardly in a positive direction. For both the phase and group velocities, the DPS medium follows the right-hand (RH) coordinate system, so a normal DPS medium is called the right-handed, i.e. the RHmedium. As the RH-system holds for both triplets, they

The above wavevector triplet-

E , H, k

is shown in

Fig (5.8b), i.e. in the left-hand (LH) coordinate system, so the DNG medium is also called the left-handed, LH-medium. Both the power-vector and wavevector triplets and their combination are further shown in Fig (5.8d). The field vectors are rotated to maintain the power flow in a positive direction. Figure (5.8d) shows that the phase and group velocities are opposite to each other ( v p v g < 0) as for a DNG medium the vectors k and S are antiparallel, i.e. k S < 0 . The DNG occupies the third quadrant in the (μr, εr)-plane as shown in Fig (5.7). In conclusion, a DNG medium supports the backward wave propagation, whereas a forward wave is supported by the DPS medium. As the phase velocity travels toward the source, while energy is traveling from the source to a load, a propagating EM-wave in a DNG medium, in the direction of the vector S , has a leading phase. This is a unique property of the DNG medium. It significantly influences the EM-wave characteristics of the DNG medium [J.8, B.6, B.10].

141

142

5 Waves in Material Medium-II

→ H

→ H

→ k

→ –k ×

→ E

(b) (E,H,k) triplet in (LH)-coordinate system.

(a) (E,H,k) triplet in (RH)-coordinate system.

→ E → Vp

→ E → Vg O

O

→ S

→ H

→ E

→ H

→ E → → Vp,Vg

→ k

O → H

→ → k S

→→ → ( E, H, S ) – Triplet

→→ → ( E, H, k ) – Triplet

→→ →→ ( E, H, S, k) – Composite triplet

(R-H) System

(R-H) System

(R-H) + (R-H) System

→ → (c) DPS medium (RH-medium,Vp,Vg > 0). → E → Vg O → H

→ E

→ Vp → –k

→ S

→ E

→ Vp O → H

→ –k

O → H

→ Vg → S

→→ → ( E, H, S ) – Triplet

→→ → ( E, H, k ) – Triplet

→→ →→ ( E, H, S, k) – Composite triplet

(R-H) System

(L-H) System

(R-H) + (L-H) System

→ → (d) DNS medium (LH-medium,Vp,Vg < 0). Figure 5.8

RH and LH-coordinate systems for the DPS and DNG media.

Refractive Index of DNG Medium

The above discussion shows that Maxwell’s equations in the DNG medium are written in the LH-coordinate system. However, the wave equation (4.5.32) of chapter 4 for the DPS medium remains valid for a lossless (σ = 0) DNG medium. It provides the following expressions for the propagation constant β = kDPS and refraction index of a DPS medium: DPS

k

= k 0 μr ε r k

a,

nDPS =

μr εr

b

555

The evaluation of the square root of negative permeability and negative permittivity is a critical issue in the DNG medium. The negative number (−1) is exp

(±jπ). However, to meet the physical condition, discussed in subsection (5.5.3), we take {−1 = exp(−jπ)} [J.8, J.9]. Therefore, the square roots of negative permeability and negative permittivity are obtained as follows: − μr =

e − jπ μr = e − jπ

− εr = e − jπ

2

εr = − j

2

μr = − j

μr , and

εr

Using the above relations, the refractive index of a DNG medium, and also the propagation constant, are obtained as follows:

5.5 EM-Waves in Metamaterials Medium

nDNG = nDNG = − DNG

k

− εr =

− μr ×

μr e − jπ

2

εr e − jπ

2

= e − jπ

μr ε r

= β = −

μr ε r a

μr εr k 0

b

It is interesting to note that the refractive index for a DPS medium nDPS is a positive quantity, whereas for a DNG medium nDNG is a negative quantity. So the metamaterials are also known as the negative refractive index materials, i.e. the NIM. Snell’s law of refraction for a DNG medium is also modified accordingly. The negative refractive index also shows the reversal of the direction of the phase velocity of the EM-wave. However, first let us discuss the intrinsic impedance, i.e. the wave impedance for the DNG and SNG media.

556

The propagation constants of EM-waves in the ENG and MNG media are obtained below: βENG = ± β0

μr − εr =

jβ0

μr εr

a

β

εr − μr =

jβ0

μr εr

b

MNG

= ± β0

559 Expressions (5.5.9 a,b) show that the ENG and MNG media do not support wave propagation. Figure (5.7) shows that these media are placed in the second and fourth quadrants in the (μr, εr)-plane. They only support the decaying evanescent mode.

Wave Impedance of DNG and SNG Media

Following equation (4.5.26b) of chapter 4, the wave impedance ηDNG in a DNG medium is written below: ηDNG = η0

− μr e − jπ = η0 − jπ − εr e

2 2

μr = η0 εr

μr = ηDPS εr 557

Like the wave impedance in a DPS medium η , the wave impedance in the DNG medium ηDNG is a positive quantity; showing the outward power flow from the source into a DNG medium. However, the wave impedances of the ENG and MNG media are reactive due to nonpropagating evanescent mode: DPS

ηENG = η0

μr = η0 ejπ − εr

ηMNG = η0

− μr = e − jπ 2 η0 εr

2

μr = jηDPS εr μr = − jηDPS εr

a

b 558

The inductive/capacitive reactive wave impedances of the ENG and MNG media create the reflecting surfaces. The circuit model of the metamaterials, discussed in section (5.5.3), elaborates on the nature of the RIS. Further details of the artificial RIS surface is discussed in section (20.2) of chapter 20. It is noted that the ENG/MNG medium is realized through the nonpropagating evanescent wave. Such an environment is provided by a rectangular waveguide below the cutoff region. It is commented in subsection (7.4.1) of chapter 7.

Negative Refraction in DPS-DNG Composite Medium

Figure (5.9a) shows the DPS-DNG composite medium. The incident ray is in the third quadrant of the DPS medium #1. For the DPS medium #2, the refracted ray comes out in the first quadrant; as the angle of refraction (θ2) is positive. The refraction in the DPS-DPS composite medium follows Snell’s law given by equation (5.2.7c). However, if the medium #2 is DNG-type then the angle of refraction (−θ2) is negative due to the negative refractive index. It follows from Snell’s law: n1 sin θ1 = − n2 sin θ2 = n2 sin − θ2 5 5 10 Figure (5.9a) shows that due to the negative angle of refraction (−θ2), the refracted ray in the DNG medium #2 emerges from the fourth quadrant. It shows the reversal of Snell’s law in the DPS-DNG composite medium, as compared to Snell’s law in the DPS-DPS composite medium. Figure (5.9b) shows that the wavevector DNG

k2 in medium #2 must be in the reverse direction DNG to meet the phase-matching condition, kDPS 1y = k2y , at the interface of the composite medium. In the case of the TE-polarized incident waves, the wavevectors of the incident, reflected, and refracted rays are given by equations (5.2.1a–d), whereas their Poynting vectors are given in equations (5.2.5a–c). For the DPS-DPS composite medium, both vectors are in the same direction after refraction, showing the presence of the forward-wave in medium #2. However, for the DPS-DNG medium, these vectors of the transmitted wave in the DNG medium #2 are given as

143

5 Waves in Material Medium-II

y Medium #2 (DPS/DNG) → I S → DPS k2 θ Forward-wave

II

Normal

y II Medium #1 (DPS)

k2

2

→ S

→ k2

k1t k2t

III

→ S

x

k2n

DNG

Backward-wave k1n

IV

DNG

Refraction of the obliquely incident EM-wave at the interface of the DPS-DPS and DPS-DNG composite medium.

tra DNG

=

2 Etra m

2η2

a

+ x cos θ2 − y sin θ2

b 5 5 11

The wavevector k 2 = k 2 , satisfying the phasematching at the interface, of the above expressions, is obtained from Fig (5.9b). The Poynting vector is written from the wavevector diagram shown in Fig (5.9a). It is also obtained by using equation (5.2.4) to compute the Poynting vector in the DNG medium for the angle of refraction (−θ2). The above expressions further show that the DNG medium #2 supports the backward-wave DNG

propagation because the vectors k 2 are anti-parallel.

tra DNG

μ =

βDPS = ω με, βLine = ω LC

ε = Y ω jω

b 5 5 13

− j ωC 1 = − 2 jω ωC

a,

ε =

− j ωL 1 = − 2 b jω ωL

5 5 14

The unbounded DPS medium and a transmission line both support the forward wave propagation in the TEM mode, so Fig (3.28a) of chapter 3 models a DPS medium by the LC transmission line. The equivalence between the material parameters ε, μ and the circuit parameters C, L is discussed in subsection (3.4.2) of chapter 3. The characteristics impedance and propagation constant of the DPS medium and equivalent transmission lines are summarized below: L C

a,

In equation (5.5.13), Z(ω) and Y(ω) are the series impedance and shunt admittance of the equivalent line. Following the above discussion, a DNG medium, supporting the backward-wave propagation, could be modeled through the CL-line, shown in Fig (3.28b) of chapter 3. Using the above expressions, the material parameters of a DNG medium could be modeled as follows:

and S av

5.5.3 Basic Transmission Line Model of the DNG Medium

μ , ZLine = o ε

μ and ε could be treated through the analogous series inductance L, and shunt capacitance C, i.e. L μ, C ε. The expressions L = Z(ω)/jω and C = Y(ω)/jω for series inductance and shunt capacitance, respectively, model the DPS medium parameters as follows: μ = Z ω jω

DNG

ZDPS = o

DNG

(b) Wavevector diagrams.

= xk2x + yk2y = − nk2n + tk2t = k2 − x cos θ2 + y sin θ2

S av

k2 k2n

(a) Oblique incidence of EM-wave at DPS-DPS.

k2

DPS x

k1

k2t

–θ2

k1t

θ1

→ k1

Figure 5.9

I Medium #2 (DPS/DNG) k2t

Medium #1 (DPS)

Interface

144

a b 5 5 12

The equivalent LC transmission line is an analog of the DPS medium. Therefore, the medium parameters

The above expressions show that the CL-line model provides negative values for the permeability and permittivity, as needed for a DNG medium. The characteristic impedance and the propagation constant of the EM-wave in a DNG medium follows from the above equations: ZDNG = o

μ = ε

βDNG = ± ω

1 × ω2 L = ω2 C

μ ε =

L C

1 1 × 2 = ω2 C ωL

a 1 ω2

LC

b

5 5 15 Expression (5.5.15b) shows that even a lossless DNG medium is highly dispersive due to the presence of a term ω2 in the denominator. It is unlike a DPS medium which is nondispersive for the lossless case. However, a dispersive medium is always associated with loss to meet the

5.5 EM-Waves in Metamaterials Medium

DPS (Dielectrics)

μr (L)

ENG ( Plasma )

βDPS = β0

εr μr, ηDPS = η0

μr εr

βENG = jβ0 │εr││μr│ , ηENG= jηDPS ±

Evanescent wave

Forward - wave propagation εr (C) βMNG = jβ0 │εr ││μr│, ηMNG = – jηDPS ±

ηDNG = ηDPS, βDNG = – β0

│εr││μr│

Backward - wave propagation DNG (Metamaterials)

Evanescent wave MNG (Gyrotropic medium)

Figure 5.10 Circuit models of four kinds of the medium on the (μ, ε)-plane.

Kramer–Kronig condition of causality. It is discussed in subsection (6.5.4) of chapter 6. The causality fails for a fictitious nondispersive DNG medium. The discussion shows that the DNG medium parameters and wavenumber, i.e. εr, μr, n, k, are complex quantities [J.8]. The DPS transmission line LC-model can also be extended to the ENG and MNG media. Both these media do not support any EM-wave propagation. Only the decaying evanescent mode is supported by them, as these media are reactive. The ENG medium is obtained for −εr and +μr that correspond to negative shunt capacitance and positive series inductance. Likewise, the MNG medium is obtained for −μr and +εr, i.e. for negative series inductance and positive shunt capacitance. However, the negative capacitance and negative inductance in reality correspond to equivalent positive inductance and equivalent positive capacitance, respectively. It is shown below from their reactance: For negative capacitance −j jXc = = jωLeq , ω −C For negative inductance j , jXL = jω − L = − ωCeq

Leq =

1 ω2 C

Ceq =

1 ω2 L

a

b 5 5 16

Figure (5.10) shows the equivalent T-network unit cell for all four kinds of media in the (εr, μr)-plane, i.e. in the corresponding (C, L)-plane. The εr(C)-axis and μr(L)axis show the capacitive shunt element and the

inductive series element of the equivalent T-network of material media. The DPS medium in the first quadrant is changed to the ENG medium by taking a negative value for shunt capacitance that corresponds to a shunt inductor. The L-L network in the second quadrant corresponds to the ENG medium. The MNG medium is obtained by taking a negative value for the series inductance of the DPS medium. It results in the C-C network in the fourth quadrant. The characteristics impedance of the ENG and MNG equivalent lines are inductive and capacitive respectively. Of course, by taking negative values for both the series inductance and shunt capacitance of the DPS medium, the DNG medium is created. It is shown as a CL unit cell in the third quadrant of Fig (5.10). Individually, the ENG and MNG media do not support EM-wave propagation. However, jointly they form a transparent medium, and EM-wave propagates through the joint medium. It is known as the tunneling phenomenon [J.10, J.11]. The above description of circuit modeling is also applicable to a waveguide below the cut-off frequency. The TE-mode waveguide below cut-off frequency provides the inductive load. So it could be viewed as an ENG medium, while the TM-mode waveguide below the cut-off frequency provides the capacitive load and it could be viewed as the MNG medium [B.6, B.8]. The behavior of the modal wave impedance of a rectangular waveguide below the cut-off frequency is discussed in subsection (7.4.1) of chapter 7.

145

146

5 Waves in Material Medium-II

5.5.4

Lossy DPS and DNG Media

A lossy DPS medium is characterized by the complex permittivity and complex permeability ε∗DPS = εr − jεr r and μ∗DPS = μ − jμ , also ε ε and μ μ . The comr r r r r r r plex wavenumber for the EM-wave in a lossy DPS medium is obtained as follows: k∗DPS = k0

ε∗DPS ≈ r

where

εr − j

=

μ∗DPS ε∗DPS r r

a

εr 1 − j

εr 2 εr

εr 2εr μ∗DPS ≈ r

and,

μr − j

μr

b

2 μr

5 5 17

However, in the case of a complex DNG medium, the complex refractive index is n∗DNG = − (n + jn ); as Re(n∗DNG) is a negative quantity and Im(n∗DNG) is still a positive quantity. The above discussion is for the propagating waves in the DPS and DNG media. However, in case the waves are nonpropagating (evanescent) in both media, the real part of the wavenumber is an imaginary quantity i.e. k = − jα . The evanescent wave behaves differently in the DPS and DNG media. It is examined below. The electric fields of the x-directed propagating and nonpropagating EM-waves in the unbounded lossy DPS and DNG media could be expressed as follows: Propagating wave ∗DPS

k

∗DPS

≈ k0 ≈ k0

EDPS = E0 e + j ωt − k

ε μ 1 εr μr + μr εr εr μr − r r −j 2 4 εr μr εr μr

DPS

E E

1 εr μr + μr εr 2 ε r μr

εr μr − j

5 5 18

DNG

= E0 e

= E0 e + j ωt −

x

k − jk x

,

− k x j ωt − k x

e

+ j ωt − k∗DNG x

+ j ωt + k + jk x

= E0 e = E0 e DNG − k x j ωt + k x = E0 e e E

a , b

Evanescent wave k = − jα EDPS = E0 e −

k + α x jωt

e

α − k x jωt

c α >k

On separating the real and imaginary parts of a complex wavenumber in the DPS medium, k∗DPS = k − jk , the following expressions are obtained:

E

k = Re k∗DPS = k0

The propagating EM-wave is attenuated while traveling in both the DPS and DNG lossy media k 0. However, the DPS medium offers a lagging phase, whereas the DNG medium offers a leading phase to the wave traveling in the positive x-direction. Poynting vector decides the direction of the EM-wave propagation. The λg/2-line resonator could be designed in the DPS-DNG composite, with a length λg/4 in each medium, without any phaseshift at the output. The classical λg/2-line resonator, in a DPS medium, has 180 phase at the output. It is further seen from the above equation that the evanescent wave is decaying with distance x while traveling in the DPS medium. The enhancement of amplitude by the DNG medium could be viewed as the step-up transformer action, whereas it is increasing in amplitude while traveling in the DNG medium. This property is more clearly seen in a lossless medium with k = 0.

k = Im k∗DPS =

k0 2

εr μr

a

εr μr + μr εr εr μr

b 5 5 19

The DPS medium has εr > 0, εr > 0, μr > 0, μr > 0. So using the above equations, it is obvious that k’ > 0, k > 0. The above expressions are also valid for the DNG medium with some modifications. In the case of a DNG medium, the medium parameters are εr < 0, μr < 0. However, even in the DNG medium, the electric and magnetic losses are positive quantities, i.e. εr > 0, μr > 0. Using equation (5.5.19), the complex wavenumber in the DNG is given as follows: k∗DNG = − k − jk = − k + jk

n∗DPS = k∗DPS

c k = k0 ω

≈

εr μr −

j εr μr + μr εr 2 εr μr

a

n∗DPS = n − jn , where n =

ε r μr , n =

1 εr μr + μr εr 2 εr μr

= E0 e

e ,

d 5 5 22

5 5 20

The complex DPS medium is also described by the complex refractive index: ∗DPS

DNG

b 5 5 21

5.5.5

Wave Propagation in DNG Slab

The EM-wave propagating through a normal DPS slab provides the lagging phase of the propagating part of the field at the output end of the slab. It is noted that the nonpropagating decaying evanescent wave also exists, along with the propagating wavefield components, inside the slab. However, for a propagating wave, the DNG slab provides the leading phase at its output. The increasing evanescent wave exists inside a DNG slab, along with the propagating wavefield components.

5.5 EM-Waves in Metamaterials Medium

Ezinc

DPS η1

y

y

DNG η2

k1

Hyinc

k2

DPS η1

Free space

Ezinc

k1

+│μ1│

–│μ2│ +│μ1│

+│ε1│

–│ε2│ +│ε1│

Hyinc

DPS

DNG

η0

η1DPS

ηDNG 2

η0

k0

k1DPS

k2DNG

k0 τtotal

Γtotal= 0

x o d (a) DNG slab in the host DPS medium.

Free space

o

x dDPS dDNG (b) Composite DPS - DNG slab in the host DPS medium.

Figure 5.11 EM-wave propagation through the DNG and composite DPS-DNG slabs.

Thus, the DNG slab can act both as (i) a phasecompensator and (ii) as an amplitude-compensator, i.e. as a field amplifier, without any power amplification. The field amplifier is like a voltage step-up transformer without any power amplification. The first property, in the form of the DGS-DNG composite slab, is helpful in the design of compact sub-wavelength resonator [J.12, B.6]. A normal DPS resonator is half-wavelength long. The second property offers an opportunity to design subwavelength resolution superlens [J.2, J.13–J.21]. This subsection explains both properties of a DNG slab. The superlens is discussed briefly in the next subsection.

Figure (5.11a) shows a DNG slab, μ2 = − |μ2|, ε2 = − |ε2|, η2, of thickness d embedded in the DPS host medium, μ1 = + |μ1|, ε1 = + |ε1|, η1. The TE-polarized wave is normally incident at the first interface. The DNG slab supports the backward wave with the wavevector k2 in the opposite direction, as power flows from the left to right. This has important consequences. Let us assume that in Fig (5.11a), the slab is a DPS type, and it is impedance matched with the host DPS medium, i.e. η1 = η2. The reflection and transmission coefficients are obtained from equation (5.4.15): a,

− jk2 τNor DPS = e

DPS

d

= e − j k2 d

b 5 5 23

However, if the DPS slab is replaced by a matched DNG slab, then the following expressions are obtained, from equation = − kDPS = − k2 on using kDNG 2 2 (5.5.6b): ΓNor DNG = 0

a,

− jk2 τNor DNG = e

DPS

d

= e + j k2 d

− jk1 τNor toal = e

DPS DPS d1

DNG DNG d2

e + jk2

= e − j k1

DPS DPS d1

− kDNG dDNG 2 2

5 5 25

Phase – Compensation in the DPS-DNG Slab

ΓNor DPS = 0

(ϕ = + |k2|d). This property of the DNG slab is useful in compensating the lagging phase of a DPS slab in a DPS-DNG composite slab. Figure (5.11b) of the composite DPS-DNG slab illustrates the application of a DNG slab as a phase compensator. Again, the impedance matching of both slabs with the host medium is assumed, i.e. η1 = η2 = η0, such that the total reflection coefficient is zero, ΓNor total = 0. The total transmission coefficient at the out of the DNG slab is given as follows:

To get the compensated phase, i.e. the zero phase, at the output of the DPS-DNG slab, the following condition, obtained from the above equation, must be met: DPS − kDNG dDNG =0 kDPS 1 d1 2 2

= k0 k0 μr1 εr1 dDPS 1

μr2 εr2 dDNG 2

dDPS nDNG 2 1 = , nDPS dDNG 1 2

5 5 26

and nDNG are refractive indices of the DPS where nDPS 1 2 and DNG slabs. The thicknesses of the slabs need not be equal. The dimension of thicknesses could be even in the sub-wavelength. It is useful in designing of very compact sub-wavelength cavity resonators and parallel plate waveguides [J.12]. Such resonance can be developed even in the compact composite ENG-MNG slab [B.6]. The present concept also finds application in designing properly matched electrically small dipole antenna [B.6].

b 5 5 24

A complete transmission of waves occurs through the DPS/DNG slab. However, the DPS slab provides the lagging phase (ϕ = − |k2|d) at the output of the slab, whereas the DNG slab provides the leading phase

Amplitude-Compensation in the DNG Slab

The resolving power of an optical lens is restricted by the wavelength of a source. This is known as the diffraction limit of the lens. Fourier optics, i.e. the wave optics adequately explain it in terms of the propagating waves and decaying evanescent waves generated by the object

147

5 Waves in Material Medium-II

kz

y Propagating wave

Evanescent region

Object

Image

(ii)

o

kt < ko

(i)

Propagation region

x

Lens

z Evanescent wave

ky

kt > ko (a) Wavefield of an object in DPS medium. y Propagating wave

(b) Grouping of waves in k-space.

Evanescent wave in near-field

DNG

Object Image

y

Image

Figure 5.12

Antisotropic DNG

z Evanescent wave

x

d (c) Wavefield of an object DNG slab. (Near –field superlens).

x Evanescent wave coverted to propagating wave

(d) Wavefield of an object in hyperbolic DNG. slab(Far-field hyperlens).

Creation of image using wave optics.

source shown in Fig (5.12a). The object source located at x = 0 generates an arbitrary wave field that can be decomposed in terms of the plane wave spectrum. Their superposition reconstructs the wave field. Thus, the wavefield in the real DPS space, propagating in the xdirection, is composed of 2D Fourier plane wave components in the Fourier domain, i.e. in the k-space (propaDPS

gation vector space). The wave field E x, y, z, t is expressed through the following 2D Fourier integral: DPS

y, z, x, t =

1 4π2

DPS

E

Image formation in far-field

o z Evanescent wave

E

Propagating wave

Object

o Amplified evanescent wave

148

y, z, x, t = E ky , kz e − j

= where kDPS x

∞

∞

E ky , kz ej

ωt − k r

−∞ −∞ ∞ ∞

1 4π2

−∞

−∞

dky dkz

×

ej ωt − kx x dky dkz

a

k20 − k2y + k2z , ko = ω c

b

ky y + kz z

5 5 27 The wave propagates in the x-direction and the spectral field components are in the (ky − kz)-plane. The spectral field amplitudes E ky , kz with phase distribution within

the bracket [ ] are Fourier components of the electric DPS

field E y, z, x, t . The propagation constant kDPS of x the plane wave, given by equation (5.5.27b), is obtained from the dispersion relation (4.5.29d) of chapter 4. The propagation constant kDPS of propagating plane waves x must be a real quantity, whereas imaginary values of kDPS provide decaying evanescent waves in the x-direcx tion. Thus, Fourier components of the wave originated by an object could be divided into two groups: propagating waves and nonpropagating evanescent waves. The grouping of waves is shown in Fig (5.12b). In a DPS medium, the propagation constants for both groups of waves, from equation (5.5.27b), could be written as follows: Propagating waves kDPS = x

k20 − k2y + k2z = βx ,

k20 > k2t = k2y + k2z

a

Evanescent waves kDPS = −j x

k2y + k2z − k20 = − jαx , k20 < k2t = k2y + k2z

b

5 5 28

5.5 EM-Waves in Metamaterials Medium

The electric fields of both the propagating and evanescent waves in the DPS medium could be written from the above relations as follows: Propagating waves DPS

E

y, z, x, t =

1 4π2

∞

∞

−∞

−∞

E ky , kz e − j

ky y + kz z

ej ωt − βx x dky dkz a

Evanescent waves DPS

E

y, z, x, t =

E ky , kz ej ωt −

1 4π2

∞

∞

−∞

−∞

ky y + kz z

e − αx x dky dkz

b

5 5 29 Figure (5.12a) shows both the propagating and exponentially decaying evanescent waves in the DPS medium, generated by an object. The evanescent waves decay fast within a distance under λ. The higher value of transverse components of the wavevector (ky, kz) corresponds to the finer spatial details of the object. So at the image-plane, finer spatial details of an image are lost. The DPS based lens cannot recover the lost spatial details in the evanescent waves. The maximum value of the transverse wavevector k t, max = yky + zkz at the cut-off wavenumber kx = 0 determines the limit of the finer spatial details, i.e. the diffraction limit:

5.5.6

DNG Flat Lens and Superlens

The focal length (f ) of a thin lens is related to its radius of curvature (R) by the expression f = R/(n − 1), where n is the refractive index of a lens. For n = +1, the lens does not refract, and the focusing of the EM-wave at the image plane does not occur. However, for a DNG lens with n = −1, refraction occurs. Veselago [J.3] has shown that due to negative refraction in a DNG slab, even a slab acts as a flat lens. Pendry has further shown that such a lens is a perfect lens in the near-field region as it enables recovery of the decaying evanescent field at the image plane [J.2]. Such recovery of the evanescent waves is not possible with a normal lens, irrespective of the size of its aperture. The perfect lens, also called the superlens, has a subwavelength resolution, breaking the diffraction limit barrier. The superlens creates an image in the near-field very close to the lens, so it is difficult to use it in practice. However, anisotropic hyperbolic DNG lens creates a perfect image in the farfield region, by converting the evanescent waves into the propagating waves. Such a lens is called the hyperlens. The proof of concept has been demonstrated experimentally for both the superlens and hyperlens in optical and microwave range. The proper functioning of these lenses is limited by the losses associated with a DNG medium. A brief theory of three lenses is presented in this section. Veselago Flat Lens

kDPS = x

k20 − k2y + k2z =

k20 − k2t, max = 0

kt, max = ω c Maximum resolution Δ =

a 2π kt, max

=λ

b 5 5 30

The maximum resolution Δ is the minimum adjacent distance of the finer spatial details of the object. It called the diffraction limit. The relation Δ × kt,max = 2π is obtained in subsection (19.1.1) of chapter 19 from the relation between the direct space period (Δ) and the propagation vector in the k-space. The diffraction limit for a very wide lens aperture could be reduced to λ/2 [J.13]. The DNG planar slab acts as a flat lens [J.2, J.13– J.15, J.22]. The DNG slab provides exponentially increasing evanescent waves inside the slab. So the DNG can compensate for the decaying evanescent waves that reach the image plane creating a highresolution image with finer details. It is shown in Fig (5.12c). It is also examined in the next subsection (5.5.6).

Figure (5.13) shows the ray diagram of a DNG slab acting as a lens due to the negative refraction [J.3]. It has two foci N and Q. The focal length f1 of focus N is located inside the DNG slab of thickness d. The focal length f2 of focus Q is located outside the slab. To avoid mismatch at the interface, the DNG slab has = η0 . Figure (5.13) shows the μr = εr = − 1 and ηDNG 1 angles of incidence θi and refraction θr and other geometrical dimensions. Snell’s law for the DNG slab is written below: 1 sin θr = sin θi 5 5 31 n The first focal length f1 is determined as follows: H1 , H1 = d1 tan θi d1 H1 tan θr = , H1 = f 1 tan θr f1 d1 tan θi f1 = tan θr tan θi =

ie

a b c 5 5 32

The incident ray OM and the output ray P1 Q are parallel, i.e. θi = θo.

149

5 Waves in Material Medium-II

Medium #I ε0, μ0, n0 = +1 M θi

O

θi

O

Medium #II

Medium #III I ε0, μ0, n0 = +1

–εr, –μr, n = –1 P

θr

H1

θr

N

θr

n22 k20 − k2y + k2z

= − βx ,

n22 k20 > k2y + k2z

a

Evanescent waves x

kDNG = − x

H2

θʹr

f1

kDNG = − x

Q

θr M1

Propagating waves Image plane

y

Object plane

150

θ0

−j

k2y + k2z − n22 k20

= jαx , k20 < k2y + k2z

DNG P1

b 5 5 36

Oʹ

d1

Figure 5.13

d D

f2

The electric fields of both the propagating and evanescent waves, inside the DNG slab, are obtained using the above equations with equation (5.5.29a,b): Propagating waves

Iʹ

Ray diagram of a DNG flat lens.

DNG

The second focal length f2 is computed as follows: H2 , d − f1 H2 , tan θ0 = tan θi = f2 d − f 1 tan θr i e f2 = tan θi tan θr = tan θr =

H2 = d − f 1 tan θr H2 = f 2 tan θi

y, z, x =

E a b c

5 5 33

E ky , kz e − j

d f2 − =1 f 1 d1 5 5 34

Equation (5.5.31) shows a matched DNG slab has θi = θr. For this case, both focal lengths, and also distance D between the object and image at the second focus are given below: f 1 = d1

a,

f 2 = d − d1

b,

D = 2d c 5 5 35

Pendry Superlens Lens

In the case of a DNG-based matched flat superlens, the image is reconstructed at the second focus by amplification of the evanescent electric fields. Thus, the DNG slab provides an opportunity to design subdiffraction lens, i.e. practically a lens without any diffraction limit. The refractive index of the DNG, matched to air medium, is nDNG = − 1 . The longitudinal wavevector kx, given 2 by equation (5.5.28), is rewritten for the backward wave supporting DNG slab as follows [J.2, J.13, J.22]:

∞

∞

−∞

−∞

ej ωt + βx x dky dkz

ky y + kz z

a

Evanescent waves DNG

E

y, z, x =

E ky , kz ej ωt −

For d > f1, equations (5.5.32c) and (5.5.33c) provide a relation between two focal lengths: d − f 1 d1 tan θi , f2 = f 1 tan θi

1 4π2

1 4π2

∞

∞

−∞

−∞

ky y + kz z

eαx x dky dkz

b 5 5 37

Equation (5.5.37a) shows that the propagating waves acquire the leading phase (βxd) at the end of the DNG slab of thickness x = d. The evanescent waves are amplified exponentially by αxd at the end of the slab. Figure (5.12c) shows such behavior of the DNG based superlens. At the output of the DNG slab, the wave is still evanescent waves in the near-field region; although at the image plane its magnitude level is the same as that of at the object plane. So at the second focus, the image has high resolution beyond the diffraction limit. It is noted that the evanescent waves still propagate at the interface along the y-axis in the (y − z)-plane as the surface waves. The surface wave is bounded in the x-direction. The amplification of the evanescent wave is also treated through the concept of coupled quasi-particles called plasmon-polariton, especially in the optical frequency range [J.13, B.11]. HyperLens

The anisotropic DNG medium-based hyperlens converts the evanescent waves to the propagating waves by reducing the values of the transverse wavenumber below the wavenumber in the medium. It is realized by the hyperbolic dispersion relation of the uniaxial

5.5 EM-Waves in Metamaterials Medium

anisotropic medium discussed in section (4.7.5) of chapter 4. Thus, a hyperlens projects the highresolution image in the far-field region. It is shown in Fig (5.12d). The hyperlens has been realized in the cylindrical geometry using very thin alternate curved layers of conducting and dielectric films. The object is placed at the center and the propagating waves are available at the flat end of the cylindrical medium in which the hyperlens is embedded. The normal optical lens, attached with the hyperlens, carries out the optical processing to create the high-resolution image in the far-field region [J.17, J.18]. 5.5.7 Doppler and Cerenkov Radiation in DNG Medium The DNG medium acts inversely on Doppler and Cerenkov radiations. It is examined below.

Doppler Effect

Doppler effect is related to the change in frequency of a source due to the relative motion of the source and receiver. If a source is moving away, i.e. receding, from the receiver in a DPS medium, the received frequency is less than the stationary frequency of the source. However, if the source is moving toward the receiver, the received frequency is increased. Let us examine the receding case of a moving source. Figure (5.14a) shows that a source #S is moving away from receiver #R with velocity Vs along the positive xaxis. The source occupies positions P1 (x = d1) and P2(x = d2) at time t1 and t2 (t2 > t1), respectively. The source S radiates frequency fs and the radiated spherical wave propagates with the phase velocity Vp in the DPS medium. Figure (5.14a) shows the spherical wavefronts at time t1,…,t4. The phase of the received wave at time t1 and t2 could be written as

DPS medium 1

Wavefront motion (Vp)

2 Source #(S) motion (Vs) P1

Receiver# R

3 4

P2 P3 P4

X

d1 Low frequency

High frequency

(a) Doppler effect in DPS medium.

DNG medium

Wavefront motion (Vp)

Source #(S) motion (Vs)

Receiver# R

High frequency

Low frequency

(b) Inverse Doppler effect in DNG medium. Figure 5.14 Doppler effect in the DPS and DNG media. The source receding the receiver.

151

152

5 Waves in Material Medium-II

Phase at time t1 ϕ1 = 2πf s t1 − βd1

a

Phase at time t2 ϕ2 = 2πf s t2 − βd1 + βVs t2 − t1 Change in phase during the interval Δt = t2 − t1

b

Δϕ = 2πf s − βVs Δt

c 5 5 38

The received frequency fR of the wave at the stationary receiver is the rate of change in phase, i.e. ωR = 2πfR = LtΔt 0(Δϕ/Δt). The received frequency, from the above expression (5.5.38c) is obtained as LtΔt

0

Δϕ Δt

f DPS = fs 1 − R

= 2πf R = 2πf s −

2πf s Vs Vp

Vs , Vs < Vp Vp

5 5 39

Inverse Doppler Effect

The DNG medium, shown in Fig (5.14b), supports the backward wave propagation with opposite phase velocity. In this case, the spherical wavefronts move in the backward direction. Equation (5.5.39) is used to get the frequency of the received waves from a source moving away from the receiver in the DNG medium: Source moving away from receiver f DNG = fs 1 + R

Vs , Vs < Vp Vp

a

Source moving toward receiver = fs 1 − f DNG R

Vs , Vs < Vp Vp

the planar transmission lines radiate the Cerenkov type radiation within a substrate resulting in high substrate loss [B.13]. It is discussed in subsection (9.7.3) of chapter 9. In a DPS medium the directions of the radiated power, i.e. the direction of the Poynting vector, and the wavevector are in the same direction. However, in a DNG medium, these directions are opposite to each other, giving inverse Cerenkov radiation [J.3]. Both Cerenkov radiation and inverse Cerenkov radiation are similar to the shock waves of supersonics generating the radiation cone. Figure (5.15a) shows the generation of the radiation cone of the spherical wavefronts by a moving charge, with velocity Vc, in a DPS medium. The charged particle velocity Vc is greater than the phase velocity Vp of the EM-wave in a DPS medium, i.e. Vc > Vp( = c/n). The positive refractive index is n, and c is the velocity of EM-wave in free space. The locations P0–P4, on the positive x-axis, are successive positions of the moving charge at time t0–t4. During the time interval T = (t4 − t0), the spherical wavefront acquires radius P0N = VpT = (c/n)T, while the charged particle travels a distance P0P4 = VcT. Figure (5.15a) also shows other spherical wavefronts for the time sequence for t < T. Following the Huygens principle, the inclined tangential wavefront of the secondary radiation is NP4. The wavevector (k), i.e. P0N, is normal to the wavefront NP4. It forms a Cerenkov angle ϕc at P0 or angle θc at P4. It is obtained from the triangle P0N P4: sin θc = cos ϕc =

b

5 5 40 The received frequency of the radiated waves from a source, moving towards the receiver, is reduced at the stationary receiver. In the case, the source is moving away from the receiver, the received frequency increases = f s 1 + Vs Vp . as f DPS R Due to the reversal of the change in the received frequency, the DNG medium supports the inverse Doppler Effect [J.6]. It has been experimentally confirmed both in the microwave and optical frequency ranges. The inverse Doppler Effect could be used to design tunable and multifrequency radiation sources [J.23–J.25]. Cerenkov Radiation

The Cerenkov radiation, also called the Cherenkov (or Cherenkov) radiation, in the DPS medium is generated from the charged particles traveling with a velocity faster than the velocity of the EM-wave in that medium [B.12]. The radiation from a leaky-wave antenna closely follows the radiation mechanism of Cerenkov radiation. Likewise,

Vp T Po N c = = Vc T Po P4 nVc 5 5 41

Figure (5.15b) shows the directions of the wavevectors DPS DPS kDPS and Poynting vectors SDPS of the 1 , k2 1 , S2 upper and lower radiation from the charged particle moving in the positive x-direction. Cerenkov radiation is a linearly polarized EM-wave with an electric field component parallel to the path of the charged particle, i.e. the x-axis. Cerenkov radiation is directive in the forward direction within the cone angle 2ϕc. Similarly, a leaky-wave antenna is highly directive. The beam becomes narrower for the higher value of the refractive index n. Inverse Cerenkov Radiation

Figure (5.15c) shows that the charged particle moving with a velocity Vc (Vc > Vp) in a DNG medium with a negative refractive index −|n|. It creates backward radiation with the negative Cerenkov angle, sin θc =

c , − n Vc

sin − θc =

c n Vc 5 5 42

5.5 EM-Waves in Metamaterials Medium

Wavefront

DPS medium

Wave vector Vp Power flow Vg 90˚

N DPS medium ϕc P0

ϕc

θc

Vc P1

P2 P3

t=0

k1DPS Vp x

P4

P0

t=T

ϕc ϕc

ϕc

P0 P1 P2 t=0 Vp Wave vector

DNG medium

P3

Vc

P4

P4 Wavefront

Power flow Vg S1DNG DNG medium

ϕc

θc

P2 Vc P3

(b) Wavevector and Poynting vector of Cerenkov radiation cone in DPS medium.

Wavefront N 90˚

Wavefront

x P1

k2DPS S2DPS N

(a) Cerenkov radiation cone in DPS medium.

Vg

Cerenkov radiation

Vp

Wavefront

Power flow

S1DPS N

x

Wavefront (c) Inverse Cerenkov radiation cone in DNG medium.

Inverse Cerenkov θc radiation θ Inverse Cerenkov c radiation S2DNG Vg Power flow

ϕc

k2DNG

P0 – ϕc

Wave vector Vp

Vc

x

Vp k1DNG Wave vector

(d) Wavevector and Poynting vector of Cerenkov radiation cone in DNG medium.

Figure 5.15 Cerenkov radiation in DPS and DNG media.

Figure (5.15d) shows that the wavevectors kDNG and Poynting vectors SDNG , for , kDNG , SDNG 1 2 1 2 both the upper and lower radiations, are anti-parallel that causes the backward radiation. Thus, a DNG medium supports the reverse, i.e. inverse Cerenkov radiation. The metamaterial medium could be continuously varied from the DPS to DNG forming the forward endfire to broadside to backward end-fire antenna [J.26, J.27, B.6, B.7]. The inverse Cerenkov radiation has been experimentally confirmed in the microwave frequency range [J.28]. The realization of such a microstrip antenna is illustrated in the subsection (22.4.1) of chapter 22.

5.5.8

Metamaterial Perfect Absorber (MPA)

The absorbing materials are needed to absorb undesired reflected EM-waves and interfering signals. The perfect absorbing materials absorb 100% of incident RF power

without any reflection, scattering, and transmission. The frequency-dependent absorbed power A(ω), i.e. attenuation, of an absorber, in absence of scattering and diffraction, is expressed as A ω = 1−R ω −T ω or A ω = 1 − S11 ω

2

− S21 ω

a 2

b, 5 5 43

where R(ω) is the reflectivity (reflectance) related to the reflection coefficient (S11) and T(ω) is the transmissivity (transmittance) of the absorbing sheet [B.14]. The attenuation A(ω) is also called absorptivity (absorptance). For 100% absorption, both R(ω) and T(ω) must be zero. To get R(ω) = 0, the absorbing sheet must be matched to the free space impedance η0 = 377 Ω over a frequency band and for getting T(ω) = 0 the absorbed power must be dissipated in the absorbing sheet. The transmissivity can be made zero even by placing a conducting sheet

153

5 Waves in Material Medium-II

behind the absorbing sheet. However, it may result in multiple reflections degrading R(ω) [J.29]. Salisbury Absorber

The classical Salisbury absorber, shown in Fig (5.16a), is a narrow band resonant type absorber. The absorbing resistive screen has resistive 377Ω/sq impedance so that it is matched to free space. A metallic sheet (electric-wall) placed behind the absorbing screen at a distance λ0/4 creates the magnetic-wall, i.e. the high impedance surface (HIS) at the plane of the resistive screen. The equivalent transmission line model of free space shows the absorbing screen as a 377Ω load followed by an open-circuited termination corresponding to the magnetic-wall. The concept of magnetic-wall is discussed in the subsection (7.2.2) of chapter 7. At the magnetic-wall, the total tangential components of the incident and reflected waves provide a high-intensity electric field, i.e. Etotal = 2Einc t 0 . However, the total tangential component of the magnetic field is zero, i.e. Htotal = 0. The total elect tric field induces a current on the resistive screen placed at the magnetic-wall. The absorbed incident RF power is dissipated. Salisbury absorbing sheet is narrowband,

thick, and bulky. By placing several absorbing resistive screens above the conducting plane, over a distance λ0, a broadband absorber, known as Jaumann absorber, is obtained [B.15]. Again, it is a thick and bulky absorber. The metasurface provides an alternate arrangement for creating the high impedance surface to develop a thin and compact absorber. The realization of the metasurface is discussed in section (22.5) of chapter 22.

Metasurface Absorber

Figure (5.16b) shows the composition of the metasurface-based absorber [J.30, J.31]. A thin dielectric sheet d < < λ0 is backed by a conducting surface. An inductive surface is created at the dielectric surface. The capacitive grid of lines or patches is constructed on the dielectric surface such that the surface resonance creates the metasurface, i.e. the high impedance surface at the plane of the air-dielectric interface. Like a Salisbury absorber, again 377 Ω/sq resistive screen is placed at the interface to get the impedance matching with free space. The absorbed RF power is dissipated as heat.

Air medium

Dielectric medium

Z0 = 377Ω

d= λ Resistive 4 screen Resistive screen absorber (a) Salisbury absorber. Incident wave

Open circuit

Magnetic wall (open circuit)

Electric wall (short circuit)

Z0 = 377Ω

Equivalent transmission line

Reflected wave

Air medium Resistive screen Metasurface Zd Kd

DPS medium Thin dielectric sheet

Rs d 1 [B.11, J.7]. They are also engineered to get the fractional relative permittivity 0 < |εr| < 1 [J.6, J.8]. Such materials are known as the epsilon-near-zero (ENZ) materials. Even negative permittivity could be obtained over a certain frequency band that is discussed in chapter 21. In this section, we discuss the dielectric constant of both the natural and artificial dielectric materials.

a

Nαtotal εo 1 − Nγαtotal εo

H

me T

b

Several polarizations εr = 1 +

6 1 31 The bi-isotropic and bi-anisotropic materials are discussed in section (4.2.3) of chapter 4. The above concept of the polarization and cross-polarization, developing bi-anisotropy in the split ring resonator (SRR), is discussed in subsection (21.2.4) of chapter 21 on metamaterials.

6.2 Static Dielectric Constants of Materials The natural dielectric materials are bulky for many applications in microwave and antenna technology. They have relative permittivity |εr| > 1. The lightweight

n

1−γ

a

Nn αn εo n

Nn αn εo

b 621

Clausius–Mossotti model could be obtained by taking γ = 1/3 in the above equations. The Onsager model could be obtained for γ = 1/(2εr + 1). Like a low-density gaseous dielectric-medium, the solid dielectrics also have three kinds of polarization conditions:

•• •

Only electronic polarization. Electronic + ionic polarization. Electronic + ionic + orientational polarizations.

(dipolar)

Normally, the artificial dielectrics could also be treated as a low-density medium with induced dipole kind

167

168

6 Electrical Properties of Dielectric Medium

of polarization [B.11]. However, like many natural polar dielectrics, the induced dipole polarization-based artificial dielectrics are not temperature-dependent. The induced dipole polarization is more akin to the electronic polarization of natural dielectrics. The artificial dielectrics are discussed briefly in subsection (6.2.2). A solid material could be nonpolar or polar material. 6.2.1.1 Nonpolar Solid

The nonpolar solids do not have permanent dipolar polarization. They can acquire either electronic polarization or a combination of electronic and ionic polarization. A suitable value of the interaction parameter γ can be selected to match the computed relative permittivity with the experimentally obtained value. Lorenz and Onsager model can provide an approximate value of permittivity. The measurements of the dielectric constants, both at low frequency and high frequency, are used to know the electronic and ionic polarization ratio that gives information on the relative strength of ionic and electronic polarization. The susceptibility, and also relative permittivity, of this kind of material is also temperature-independent [B.1, B.2]. 6.2.1.2 Polar Solid

No theory exists to compute the dipolar polarization Pd of a polar solid. The dipolar polarizability, αd = μ2/3kT, is valid for the polar gases and polar liquids. The permanently polarized molecules of gases and liquids can rotate freely. However, the molecules (dipoles) of a solid are restricted by the structural arrangement of a crystal and have the directional character of bonding. It varies from material to material. The rotation of the permanent dipole in a solid is not random under the thermal agitation. Therefore, a factor 3kT associated with αd is not correct. However, some rotation of polar molecule is available showing the temperature dependence of the relative permittivity, i.e. there is a decrease in εr with an increase in the temperature. At lower temperatures, the dipolar polarizability αd = 0 is showing freezing, i.e. decrease of εr. Anyway, the total polarization of a natural polar solid is due to all three kinds of polarization. 6.2.2

Artificial Dielectric Materials

Several dielectric materials used in the microwave applications are heterogeneous, i.e. they are mixtures of two or more numbers of individual dielectric materials. Such mixtures are also called two and multiphase mixtures. Normally, materials of some regular or irregular geometrical shapes, called the inclusions, are embedded in

another material, called the host. The dielectric or metallic inclusions, i.e. the metallic particles of several geometrical shapes, can be treated as mega molecules. They get polarized under the influence of an external electric field. Traditionally, the mixing formulae have been developed for the dielectric constants of such mixtures for the geophysical, remote sensing, and hydrological research. However, recently a wide range of artificial dielectric materials are developed by using a dielectric mixture, dielectric-metal mixture, and dielectricferroelectric mixture [B.17, B.18]. Even a porous material can be treated as a mixture of dielectrics and air pockets. For instance, 99.6% alumina (Al2O3) ceramic could be treated as a mixture of aluminum oxide as a host material and the air pockets of diameter 1. However, due to diamagnetism, permeability is μr < 1, so magnetization reduces the value of the refractive index. On suppressing magnetization in inclusions, the refractive index can be enhanced. Sometimes, one of the electrical properties could be predominant, while others can be ignored. In some cases, both properties can be tailored for the realization of engineered metamaterials. Normally such materials are anisotropic. Over years, several closed-form expressions have been developed to model the heterogeneous materials, both natural and artificial, by treating them as equivalent homogeneous material with equivalent relative permittivity. This modeling process is called the homogenization of the heterogeneous mixture in which the host and inclusions ignore their electric behaviors in a range of operating frequency bands. The inclusions could be randomly

6.2 Static Dielectric Constants of Materials

distributed in a volume of a host material. Developing such artificial material is cumbersome. However, if inclusions form a regular structure, such as the crystal structure of a solid, or periodically arranged metallic inclusion in the artificial dielectrics, the material could be treated as a homogeneous medium only at the lower frequency. In lower frequency band, the size of the inclusion is much smaller than the half wavelength of the exciting field; and also lattice constant (separation between two adjacent inclusion elements) is also smaller than the half wavelength. In this case, the EM-waves interacting with the material undergo refraction. The EM-waves view the material as a homogeneous medium described by the permittivity, permeability, or refractive index. In the case, the separation between inclusions is halfwavelength, the homogenization does not work, and the Bragg’s diffraction patterns are obtained. The EM waves view the material medium as an arrangement of discrete inclusions. Such a material mixture, called band-gap materials, could be 1D, 2D, and 3D-medium. They exhibit interesting electromagnetic properties and are known as the photonic bandgap (PBG) or electromagnetic band-gap (EBG) materials. The 1D and 2D EBG materials are discussed in chapters 19 and 20. Similarly, artificial inclusions can also create so-called metamaterials. These can be subjected to the process of homogenization with both their equivalent relative permittivity and relative permeability as the negative quantities. The metamaterials are introduced in section (5.5) of chapter 5 and further discussed in the chapter 21. The equivalent permeability is obtained only if either the host or the inclusion material is magnetic [J.9]. Further, it is also possible to get the equivalent complex relative permittivity that includes the losses in a mixture. In brief, the engineered artificial dielectrics have a long history – starting with a composite mixture of natural dielectric inclusions in a host dielectric-medium, to the metallic inclusion of several metallic shapes in a host air-medium or a host low permittivity (foam) medium, to the EBG material, and metamaterials. A few classical illustrative examples of artificial dielectrics and the process of their modeling, using the concept of polarization are presented below. Also, their transmission line modeling is discussed to get the relative permittivity of these media. It helps to understand the realization of 1D and 2D EBG materials and metamaterials. 6.2.2.1 Classical Examples of Artificial Dielectrics

Figure (6.2a–d) shows embedded metallic spheres or metallic disks as inclusions in the host low permittivity dielectric-medium, or the host air-medium. A linear peri-

odic arrangement of inclusions creates the 1D artificial dielectric-medium, whereas the 2D lattice structures create the 2D artificial dielectric-medium. Similarly, the 3D lattice structures with such inclusions create artificial dielectrics. If the lattice constant d and diameters of inclusion are much smaller than the wavelength of external field incident on the material, then artificial dielectrics are treated as the homogeneous medium. In the present discussion, the interaction between inclusions is ignored. However, mutual coupling occurs among closely spaced inclusions [B.11]. Otherwise, they form EBG materials. These are discussed in chapter 19. The spherical and disk inclusions are normally embedded, shown in Figure (6.2a and b), in a host foam-medium. However, disks could be either printed or pasted over a dielectric sheet. Figure (6.2c and d) show that metallic spheres are supported by the insulated rods in the air-medium. Due to the symmetry, the spherical inclusions create the isotropic dielectric-medium, whereas the disk inclusions create an anisotropic dielectric-medium. Figure (6.3a) shows that on the application of the external electric field, the metallic sphere gets polarized, creating electric dipole moment p. However, the external magnetic fields on the surface of the sphere are distorted and induced eddy current creates magnetic dipole with fields opposing the external magnetic field. The diamagnetic effect decreases the permeability of the host medium below unity. The computation of electric and magnetic polarizability is discussed further in this section. Figure (6.3b) shows that the magnetic field on the surface of the disk could be undistorted, and there is no induced eddy current. Therefore, the magnetization is absent in this case. Figure (6.4a and b) shows the 1D periodic arrangement of the thin metallic strip and metallic rods inclusions in a low permittivity host dielectric-medium creating the 1D artificial dielectrics. Figure (6.4c and d) shows the 2D periodic arrangement of these inclusions creating the 2D artificial dielectrics. Figure (6.4a–d) shows the external electric field is normal to the strips and rods. The polarized inclusions create the capacitive effect, and relative permittivity is developed in the artificial dielectrics. However, the created permittivity is small. Although under such polarization, the refractive index is more than unity, and phase velocity is reduced causing a delay in propagation as compared to the propagation in free space. Such dielectrics are known as the phase delay dielectrics. However, in case, the electric field is parallel to the strips and rods, it creates the magnetic effect, i.e. inductive loading, in the opposite direction reducing permeability less than unity. The magnetic

169

170

6 Electrical Properties of Dielectric Medium

+++

+++

---

---

---

+++

+++

+++

---

---

---

d

d

Ey

P y +++ r θ pa O --d

Ey

+++

+++

d

+++ X

--d

Hx

Hx

--d

d

(a) 1D-spherical / disk inclusions embedded in low permittivity dielectrics.

(b) 2D-spherical / disk inclusions embedded in low permittivity dielectrics.

Conducting spheres

Conducting spheres

Hx

Host air medium

Host air medium Ey Hx

Ey Hx Insulating rods

Insulating rods

(c) 1D-spherical inclusions embedded in the air medium. Figure 6.2

(d) 2D-spherical inclusions embedded in the air-medium.

1D and 2D artificial media created by spherical inclusions in low permittivity dielectric host medium and host air-medium.

Ey + + +

Ey

Metallic sphere

Ey

+ ++

Ey

Ey

Metallic disk

Hx

Ey Hx

---

Hx

--Hx

Propagation

Z

(a) Distortion in the magnetic field on the surface of a sphere. Figure 6.3

Z Propagation

(b) Undistorted magnetic field on the surface of the disk.

Metallic spherical and disk inclusions in low permittivity dielectric host/air host medium.

6.2 Static Dielectric Constants of Materials Host dielectric medium Hx

Host dielectric medium

Ey Strip conductors

(a) 1D-metallic strips in low permittivity dielectrics.

Hx Ey

Host dielectric medium Hx

Ey

Conducting rods

(b) 1D-metallic rods in low permittivity dielectrics.

Strip conductors

(c) 2D-metallic strips in low permittivity electrics.

Host dielectric medium

TE01 - Mode Conducting rods Hx

Ey

a Ey Conducting rods

(d) 2D-metallic rods in low permittivity dielectrics.

(e) 1D-parallel plate waveguide medium.

Figure 6.4 Metallic strips, conducting rods and parallel-plate waveguide inclusions forming the 1D and 2D media.

effect could depend on the separation of strip conductors. It increases the phase velocity causing a phase advance. Such dielectrics are known as the phase advance dielectrics [B.11, J.6, J.7]. In the case of closely spaced rodded or wire medium, with incident electric field parallel to the wire stronger polarization can occur and the wire medium behaves like an electric plasma medium. It is discussed in section (21.1) of chapter 21. Again, this medium provides positive permittivity less than unity. The wire medium acts as the phase advance dielectrics. However, below the plasma frequency, the wire medium has negative permittivity. Figure (6.4e) shows the 1D periodic arrangement of parallel conducting plates supporting the TE01 kind of wave propagation. It is like the TE10 mode of a rectangular waveguide, discussed in chapter 7. Using expression (7.4.27) of chapter 7, the refractive index and relative permittivity of the parallel-plate artificial dielectrics are computed as follows:

c βTE10 π n= = = 1− β0 vp aβ0 λ0 = 1− λc εr = 1 −

λ0 λc

2 1 2

λ0 = 1− 2a

2 1 2

2 1 2

a 2

= 1−

fc f

2

b 622

For f > fc, the refractive index n and relative permittivity are less than unity; so the present medium supports fast-wave propagation. For f < fc, it behaves as an ENG medium, discussed in section (5.5.3) of chapter 5. By comparing the above expression for the relative permittivity against expression (6.5.16d), it is observed that the parallel-plate medium behaves like a plasma medium. Below the cut-off frequency, the medium does not support the wave propagation. The rodded medium, called

171

172

6 Electrical Properties of Dielectric Medium

the wire medium, also behaves like a plasma medium. It is further discussed in chapter 21 as an essential component for the realization of metamaterials. 6.2.2.2 Computation of Dielectric Constant of Artificial Dielectrics

The polarizability of a metallic sphere, shown in the unit cell of Fig (6.2a), is computed under the assumption that neighboring spherical inclusions do not interact and influence the local field of the dipole created at the center of the sphere. The dipole moment p is y-directed due to the static or low-frequency external field Ey. The total electric potential Vtot at any location P at the distance r is a summation of the potential Vext due to external E-field and potential Vdip due to the dipole moment p [J.7]: a,

Vtot = Vext + Vdip ,

Vtot = − Eext r cos θ +

Vdip =

p cos θ 4πεr2

c

Using equation (6.2.3c) at r = a, total potential on the surface of the metallic sphere of radius “a” is zero, giving an expression for the electric dipole moment p. It is further used with equation (6.1.3) to obtain the expression for the static or low-frequency electric polarizability αee (we have used this symbol for the electronic polarization also) of the metallic sphere embedded in low permittivity dielectric (εr ≈ 1, i. e. ε ≈ ε0) medium: p = 4πε0 a3 Eext a ,

p = αee Eext

On comparing equations − a and b

αee

= 4πε0 a

b 3

c

624 Equation (6.2.4c) is identical to equation (6.1.19b) of electronic polarization as both equations are due to the induced dipoles under the assumption Eloc = Eext. The effective relative permittivity of the homogeneous medium, i.e. the static relative permittivity of the artificial dielectrics is obtained by substituting equation (6.2.4c) in equation (6.1.8): 625

The above expression is improved by using an equation (6.2.4c) with Lorentz model given in equation (6.1.14a): εr = 1 +

4πNa3 1 − 4π 3 Na3

a,

μr = 1 − 2πNa3

b

b

623

εr = 1 + 4πNa3

αm = − 2πμ0 a3

627

p cos θ 4πεr2

Vext = − Eext r cos θ

γ is obtained by solving the Laplace equation for the 3D lattice of spherical inclusion. For the 2D lattice, modified result for the parameter γ is obtained [B.11]. It is noted that only one inclusion is located within a unit of the 3D lattice. For the cubic lattice of lattice constant d, the density of the inclusion is N = 1/Δv = 1/d3. The time-varying magnetic field of the incident TEM wave causes the magnetic polarization, i.e. the magnetization in the spherical inclusions. On substituting magnetic polarizability αm of the metallic sphere, given below, in equation (6.1.27b), the relative permeability of the artificial magnetic medium is obtained [J.7]:

626

In equation (6.2.6) the interaction constant (γ) is taken as 1/3. A little different value of the interaction constant

It is noted that the electric polarizability is a positive quantity giving the increased value of relative permittivity, more than unity. However, the magnetic polarizability is a negative quantity, giving permeability less than unity, 0 < μr < 1. The Lorentz model can also be used to get an improved expression for μr, similar to equation (6.2.6). Once the effective values for εr and μr of the artificial dielectrics with the magnetic property are computed, the refractive index can be obtained from n = εr μr . The reduced value of relative permeability μr lowers the refractive index. It is not a desirable feature [J.6, J.7]. The resonating inclusions can also provide a negative value of permittivity and permeability. The Lorentz and Drude dispersion models predict such negative values over a certain range of frequencies. For the metallic sphere periodically arranged in a dielectric host medium, the effective permittivity and permeability of the artificial medium could be computed with the help of MG mixing rule, discussed in section (6.3.3). A modern treatment of frequency-dependent polarizability of some geometrical shapes of inclusion, Lorentz, and Drude dispersion models, and also MG models, from the field theory point of view, is discussed by Bohren and Huffman [B.20]. The polarization response of the inclusions under the high-frequency incident EM-wave is frequencydependent. It results in the material dispersion, i.e. the frequency dependence of εr and refractive index. Lorentz’s oscillator model, discussed in section (6.5.1) explains such dispersion. The artificial dielectricmedium could be treated as a loss-less plasma medium. Thus, using equation (6.5.5b) the frequency-dependent relative permittivity of the artificial dielectric-medium is written as follows:

6.3 Dielectric Mixtures

ε∗rc f = 1 + for

f εrh and εrh > εri cases.

f 1 1−f + εri − εrh 2εrh 1−f 1 f + εrh − εri 2εri

εr eq min = εrh +

εr eq max = εri +

a

b

6 3 12 The variational expressions are more accurate as compared to the capacitor models. The variational expressions provide a narrower margin between the minimum and maximum values of εr eq. The average value of the lower and upper bounds provides the more accurate value of the equivalent relative permittivity of a mixture [J.16]. Figure (6.6a) compares the lower and upper bounds, obtained from the capacitor model and also from the variational method, against the finitedifference time-domain (FDTD) results of the equivalent relative permittivity of a mixture of host medium εrh = 1 and inclusion with εri = 16. All FDTD results are within the bounds of both methods. However, bounds of the variational methods are closer to the FDTD results. 6.3.3 Additional Equivalent Permittivity Models of Mixture Several models of the equivalent relative permittivity of a mixture medium, apart from the capacitor models discussed above, have been reported with a varying degree of accuracy. Three models presented below are valid for the isotropic host (εrh) and isotropic inclusion (εri) materials. Both materials are individually homogeneous and their mixture is dilute; i.e. the contrast K = εri/εrh is small, with normally εri > εrh. 6.3.3.1 Maxwell Garnett Formula

6.3.2.3 Variational Model of Bounds on the Equivalent Relative Permittivity of the Mixture

Hashin and Shtrikman have provided the following expressions, using the variational method, for minimum and maximum values of the equivalent relative permittivity εr eq of the 3D mixture [J.15, J.16]: εr eq min = εrh +

εr eq max = εri +

f 1 1−f + εri − εrh 3εrh 1−f 1 f + εrh − εri 3εri

a

b

J.C Maxwell Garnett who provided the mixing rule [J.19] was another researcher, not the famous J.C. Maxwell [B.24, J.20]. The effective or equivalent relative permittivity εr eq of a mixture of the 3D spherical shape, and also the circular cylinder shape, i.e. the 2D cylindrical dielectric inclusion material, in the host dielectric-medium is computed by the following phenomenological MG formulae [B.17, J.16, J.19, J.20]: 3D spherical inclusion εr eq = εrh 1 +

3f εri − εrh εri + 2εrh − f εri − εrh

a

2D cylindrical inclusion 6 3 11

The above expressions are valid for εri > εrh. For a 2D mixture, the bounds could be written as follows:

εreq = εrh 1 +

2f εri − εrh εri + εrh − f εri − εrh

b, 6 3 13

175

176

6 Electrical Properties of Dielectric Medium

16

Equivalent relative permittivity

Real equivalent relative permittivity

FDTD Results

14

• Variational method bound

12 10

• Capacitor model bound

8 6 4

Looyenga 103

102

0

0.1

0.2

0.3 0.4 0.5 0.6 0.7 0.8 0.9 Inclusion volume fraction (a) Host medium εrh = 1, inclusion εri = 16 [J.13].

M-G

S-BG

Maxwell

101 100

2

A-BG

Experimental results

104

0

1.0

0.2

0.4 0.6 0.8 Inclusion volume fraction (c) Comparison of computed results of the real part of the equivalent relative permittivity of the rubber-CIP mixture against exp.[J.15].

1

Image equivalent relative permittivity

4

Equivalent relative permittivity

Bruggeman

3 Sihvola FDTD results 2 Maxwell Garnet

105 Looyenga

104

A-BG

103 Experimental results

102 101

S-BG M-G

100 0-1

Maxwell 0

1

0

0.1 0.2 Inclusion volume fraction (b) Host medium εrh = 1, inclusion εri = 51 [J.19].

0.3

0.2

0.4 0.6 0.8 Inclusion volume fraction (d) Comparison of computed results of the Img. part of the equivalent relative permittivity of the rubber-CIP mixture against exp.[J.15].

1

Figure 6.6 Comparisons of several mixture formulae against the FDTD and experimental results. Source: Kärkkäinen et al. [J.16], Merrill et al. [J.17], and Kärkkäinen and Sihvola [J.18].

where f is the volume fraction of the inclusion material. In case, the factor f (εri − εrh) in the denominator of the above equation is dropped, the Maxwell model of a mixture is obtained [J.17]. The MG model is more suitable for the well-separated noninteracting inclusion of lowinclusion density. For the higher density of inclusion, the polarized field of inclusions, forming dipoles, interacts with each other. In this case, the MG model underestimates the permittivity of the mixture as shown in Fig (6.6b). The MG model is also suitable to obtain homogenization of the cubic-centered inclusion of artificial dielectrics and metamaterials. This aspect is further discussed in subsection (21.4.2) of chapter 21. The

ellipsoidal inclusions can also be incorporated in the mixing rule [J.21]. The phenomenological MG formula can be obtained from the Clausius–Mossotti expression (6.1.14a). To get the MG mixing formula, the air is taken as the host medium and the spherical dielectric inclusion of radius ‘a’ is replaced by the electric dipole of electric polarizability αe obtained from the EM Theory [B.20, B.25]: αe = 4πε0 a3

ε − ε0 , ε + 2 ε0

6 3 14

where ε is the permittivity of the dielectric sphere. In the case of a metallic sphere ε ∞, the above expression is

6.3 Dielectric Mixtures

reduced to equation (6.2.4c). Each unit cell of the volume Δv of the artificial material contains one spherical inclusion, i.e. the density (concentration) of inclusion is N = 1/Δv. Using equation (6.3.14), density N can be obtained in terms of the volume fraction f. The expression for the factor Nαe/ε0 used in equation (6.1.14a) is obtained as follows: vi 4π 3 f= = a N, Δv 3 Nαe 3f ε − ε0 = ε0 ε + 2ε0

3f N= 4πa3

a b 6 3 15

On substituting equation (6.3.15b) in equation (6.1.14a), the following MG model for the effective relative permittivity εr of the artificial medium is obtained: εr = 1 +

3f ε − ε0 ε + 2 ε0 − f ε − ε0

6 3 16

Equation (6.3.16) is valid for the spherical dielectric inclusions of permittivity ε and volume fraction f in the air host medium. The above expression is reduced to the MG equation (6.3.13a) for the host medium with relative permittivity εrh and inclusion with relative permittivity εri. To achieve this end, the parameters are replaced as follows: ε0 ε0εrh, ε ε0εri, and εr ε0εreq in the above equation and multiply the total expression by the factor εrh. The MG equation (6.3.13) can also be used for the magnetic inclusion μri in the host magnetic medium μrh: 3D cylindrical inclusion μr eq = μrh 1 +

3f μri − μrh μri + 2μrh − f μri − μrh

a

concept of Mie magnetic and electric resonances. It is discussed in section 21.4.2 of chapter 21. The 3D and 2D MG formula can also be extended to a host medium with two inclusion materials, with relative permittivity εri1 and εri2, and volumetric fractions f1 and f2 [J.22]: εr eq = εrh 1 +

3 f 1 k1 + f 2 k2 1 − f 1 k1 + f 2 k2

a

εr eq = εrh 1 +

2 f 1 k1 + f 2 k2 1 − f 1 k1 + f 2 k2

b

where, k1 =

2f μri − μrh μri + μrh − f μri − μrh

6 3 17 The above discussion shows that the Clausius– Mossotti model of homogenization is based on the concept of the polarizability of the inclusion as a dipole, whereas the classical MG model computes the effective (equivalent) permittivity/permeability of the homogenized heterogeneous mixture in terms of the permittivity/permeability of the inclusion and host materials. Both forms of mixing models have been used to develop only dielectric-based metamaterials useful in the THz, infrared, and optical frequency ranges by using the

k2 =

εri2 − εrh c εri2 + 2 εrh 6 3 18

3D case εr eq = εrh

1 + 2f 1−f

a

2D case εr eq = εrh

1+f 1−f

b 6 3 19

6.3.3.2 Sihvola Formula

Sihvola has suggested the following more general formulae for the equivalent relative permittivity of a mixture material containing the 3D spherical, and also 2D circular cylindrical, inclusions [B.17, J.16, J.18, J.23–J.25]: 3D case

εr eq − εrh εr eq + 2 εrh + a εreq − εrh =f

b

and

If the inclusion material is a conducting sphere εri ∞, both MG formulae of equation (6.3.13), are reduced to

2D cylindrical inclusion μr eq = μrh 1 +

εri1 − εrh εri1 + 2 εrh

2D case

εri − εrh εri + 2εrh + a εr eq − εrh

a

εreq − εrh εr eq + εrh + a εr eq − εrh =f

εri − εrh εri + εrh + a εr eq − εrh

b 6 3 20

On changing the value of the empirical parameter ‘a’, various classical mixing formulae could be obtained. Thus, for the 3D spherical inclusion case; a = 0 gives Maxwell Garnett formula, a = 2 gives the 3D symmetrical Bruggeman formula [J.26] and a = 3 gives the coherent potential rule [J.27]. Likewise, for the 2D cylindrical inclusion case a = 1 gives the 2D symmetrical

177

178

6 Electrical Properties of Dielectric Medium

Bruggeman formula. Further, a = 2 gives the 2D coherent potential rule. Both symmetrical Bruggeman formulae (S-BG) are summarized below [B.17]: 3D case

1−f

εrh − εr eq εri − εr, eq +f =0 εrh + 2εr eq εri + 2εr eq

a

2D case

1−f

εrh − εr eq εri − εr, eq +f =0 εrh + εr eq εri + εr eq

b 6 3 21

Sihvola formula for a = 1.2 and contrast ratio K = (εri/εrh) > 1 provides better results for the equivalent relative permittivity against the numerically simulated results. It also gives a more accurate formula for the contrast ratio K < 1. The parameter “a” is a function of contrast ratio (K) and volumetric function (f ). It can be computed using the following empirical expression [J.18] that is used with Sihvola formula to get the equivalent relative permittivity of a mixture: a f, K = 1 27 + 1 43e − 0 048K f 2 + − 2 76 − 0 09e − 0 043K f + 2 35, for K > 1

a

a f, K = 1 06f 2 + − 1 23 + 0 44e − 0 5 95K f + 1 7, for K < 1

b 6 3 22

The accuracy of MG, Bruggeman, and Sihvola’s empirical models are compared in Fig (6.6b) for the host medium εrh = 1, inclusion, with K = 51, against the FDTD results. Additional comparison against the results of FDTD show that Sihvola’s empirical formula has higher accuracy, and the next accurate model is the Bruggeman model. There are power-law [J.16] and logarithmic–law [J.28] also, given below, to compute the equivalent relative permittivity εr, eq of mixtures: Power law εβr,eq = fεβri + 1 − f εβrh Logarithmic law

a

ln εr,eq = f ln εri + 1 − f ln εrh b 6 3 23

The logarithmic law is a special case of the power-law. Likewise, the Looyenga rule could be obtained for β = 1/3 [J.29] and Birchak rule for β = 1/2[J.30]. The popular Bruggeman model is also put in the symmetrical and asymmetrical formats summarized below.

6.3.3.3 Bruggeman Formula

The asymmetric Bruggeman (A-BG) formulae is written as follows [J.17, J.23]:

A − BG

εri − εr eq εr eq = 1−f εri − εrh εrh

13

6 3 24

The expression (6.3.21) is a symmetric one, i.e. εri and εrh, also f and (1-f), can be interchanged without affecting equivalent relative permittivity of mixture. This model is more suitable for a random mixture where clustering is allowed [B.17]. The asymmetrical Bruggeman formula (6.3.24) shows better agreement with the available experimental data of Doyle and Jacob [J.17]. If the inclusion material is a solid conducting sphere, then εri ∞. The equations (6.3.21a) and (6.3.24) provide the following expressions: εrh 1 − 3f εrh = 1−f

S − BG εr eq = A − BG εr eq

a b

3

6 3 25 The above mixture models can be extended to compute the complex equivalent relative permittivity of a mixture of materials by replacing εrh = εrh − jεrh and εri = εri − jεri [J.17]. In case of a conducting inclusion of finite conductivity, we take ε∗ri = 1 − j

σ , ε0 ω

6 3 26

where σ is the conductivity of an inclusion conductor. The result on a Re(εr eq) and Img(εr eq) is shown in Fig (6.6c and d) respectively, for a mixture of carbon iron power (CIP) inclusion, with σ = 104 at 4GHz, in the host rubber medium with εrh = 2.43 − j0.029. The asymmetrical Bruggeman model shows better agreement with the measured data. Hoffman [B.19] has given the following expression to compute the equivalent relative permittivity εreq of a porous material: εr eq = εrh

1 − 3 P + 2 εrh 1 + 2 − 3 P εrh

6 3 27

where the porosity (P) in a host material is taken as the inclusion. It is determined by equation (6.3.3). For the alumina as a host, the relative permittivity is εrh = 10.22.

6.4 Frequency Response of Dielectric Materials The full building up of any kind of polarization, giving the static permittivity of a dielectric material, takes a finite amount of time. Thus, the polarization and

6.4 Frequency Response of Dielectric Materials

permittivity are time-dependent, or a frequencydependent, phenomena causing material dispersion and loss. The dielectric materials show two kinds of responses to the applied external field (i) relaxation response (ii) resonance response. The dipolar (orientational) polarization is described by Debye relaxation model. It also describes the relaxation response of the interfacial polarization. Normally, the relaxation response is operative up to microwave frequency. The atomic and ionic polarizations take place due to the displacement of bound charges under the influence of the external field. These polarizations are described by Lorentz type harmonic oscillator model. Lorentz model is also reduced to Drude model, applicable to conductors. Both Debye and Lorentz harmonic oscillator models provide frequency-dependent complex relative permittivity. The real and imaginary parts of the complex permittivity of dielectric material are related to each other by the well-known Kramers–Krönig relation [B.5, B.10]. This section first presents the decay law of the charges inside a conductor, then the polarization law of a linear dielectric-medium, and finally Debye dispersion relation for the complex relative permittivity. Lorentz and Drude harmonic oscillator models are presented in section (6.5). In brief, the present and next sections model the dielectric responses using the microlevel polarization concepts – both the nonresonant and resonant types. Section (6.7) presents the circuit models of Debye and Lorentz polarization processes also.

6.4.1

Relaxation in Material and Decay Law

6.4.1.1 Relaxation in Conducting Medium

Figure (6.7a) shows a conductor of volume V and conductivity σ that contains charge density ρv. Under the influence of the electric field E, current density J flows out of volume V so that the charges appear at the surface, satisfying the following condition: Rate of flow of charges out of surface S = Rate of decrease of charges in volume V bounded by surface S I=

dQ d = − ρ dv dt dt v v

s

dρv dv J ds = − dt v

On using the divergence theorem, we get the following continuity equation:

∇ J dv = − v

v

∇ J = −

dρv dt

dρv dv dt dρv σ + ρ =0 dt ε0 εr v 641

To get the final form of the above equation, we have used Ohm’s law J = σ E and relations D = ε0 εr E and ∇ D = ρv . The above equation is the decay law, describing the decay of charges, i.e. the migration of the charges from the inside of volume V to the surface of a conductor. The above equation is solved as follows, to get the exponential decay law under the initial condition ρv(t) = ρ0 at time t = 0: dρv = − ρv

σ dt, ε0 εr

ln ρv = −

σ t + ln ρ0 ε0 εr

t

ρv = ρ0 e − τ , 642 where τ = (ε0εr/σ) is the relaxation time. The charges inside the volume exponentially decrease toward the stable condition until all charges migrate to the surface, i.e. ρv 0, t ∞. On inserting external charges in volume V, the system becomes unstable and it is relaxed after reaching the stable, i.e. steady-state condition. During relaxation time duration t = τ, the charges decrease to the value ρv = ρ0/e, i.e. 36.8% of its original quantity. The relaxation time is the material-dependent parameter. For instance, the relaxation time of copper is τ=

εr ε0 1 × 8 856 × 10 − 12 F m = σ 5 8 × 107 S m

= 1 527 × 10 − 19 Sec For the seawater with σ = 4S/m and εr = 80, the relaxation time is τ = 1.77 × 10−10 Sec. It means for a conductor, the relaxation time is very short, and for dielectrics, it is large. Therefore, the dielectrics hold the charges for a longer duration. Say for the glass, with σ = 10−12S/m and εr = 6, the relaxation time is τ = 53.14 Sec. ≈ 1 min . 6.4.2 Polarization Law of Linear DielectricMedium The polarization law of a dielectric-medium demonstrates its relaxation response, resulting from the applied external electric field. Under the influence of

179

6 Electrical Properties of Dielectric Medium

– J

σ ρv

– E

(a) Flow of charges from inside of a conductor to its surface.

τ

P∞ 63.2%

Polarization (P)

P∞ Polarization (P)

180

P

(P∞ – P)

τ

Time (t)

Time (t)

Response

Response E

E

E0

E0

Time (t)

Time (t) (b) Depolarization on removal of the electric field. Figure 6.7

(c) Building up of polarization under electric field excitation.

Polarization and depolarization process in a dielectric material.

an external field, the atoms, molecules, and dipoles of a linear dielectric-medium start aligning themselves along the direction of the field. Equation (6.1.5) provides a steady-state,

i.e.

the

static

polarization,

P

∞

=

ε0 χr E ext . The χr is the static susceptibility of a material. For a nonpolar material, the initial polarization is zero, so the polarization increases from P = 0 to P = P∞. The time-domain polarization law, for a linear dielectric-medium, could be stated as follows: The rate of increase of polarization dP dt is proportional to the difference between the steady − state value P ∞ and polarization P at any time t, i e

dP dt

Remaining polarization at time t,

dP =C P∞−P dt τ

dP + P = ε0 χr E ext dt

a b, 643

where τ = 1/C is the proportionality constant in the time unit. Equation (6.4.3b) is the Debye dispersion law in the time-domain. It is similar to the RC circuit model of material response. The above equation is solved to get the polarization decay law and also its rise law.

6.4 Frequency Response of Dielectric Materials

6.4.2.1 Polarization Decay Law

Let us assume that the medium is initially fully polarized with polarization P∞ under the influence of the external field of magnitude E0. At time t = 0, the external field E0 is removed, as shown in Fig (6.7b). The depolarization process, i.e. the decay of the polarization, starts. It is described by dP + P = 0, τ dt

dP dt = − , P τ

P = P∞ e

− tτ

Phasor form Eext ω = E0 ejωt , P ω = P0 ejωt

a

P0 ejω t + jω τ P0 ejω t = ε0 χr E0 ejω t P ω = ε0

χr Eext ω 1 + jω τ

b 646

By using the above equation with equation (6.1.6a) in the frequency-domain, the following expression is obtained:

644 The above expression is known as the polarization decay law. During the time t = τ, the polarization P = P∞/e is decreased by 36.8% of P∞.

Figure (6.7c) shows that under the application of the step type electric field, the polarization reaches the maximum P∞. Equation (6.4.3b) is solved under condition t = 0, P = 0, i.e. no initial polarization, as follows: dt , τ

P = ε0 χr E0 − Ce

647

χr Eext ω 1 + jωτ

ε∗r ω = εr − jεr = 1 + χ∗r ω =

− tτ

χr 1 + j ωτ

χr 1 + j ωτ

a b, 648

P = ε0 χr E0 1 − e − t τ , 645 where constant, C = ε0χrE0 = P∞. The final form of the above equation is called the polarization rise law. During the relaxation time, t = τ, 63.2% polarization builds up. Therefore, the relaxation is a process under which the delayed polarization response, after a finite time interval, appears on the application of excitation. The charging of the capacitor through a series resistor is also described similarly. Therefore, the RC circuit could be used to model the polarization process.

6.4.3

D ω = ε0 1 +

Therefore, the complex relative permittivity of a dielectric medium is

6.4.2.2 Polarization Rise Law

dP = ε0 χr E0 − P

D ω = ε0 Eext ω + P ω ,

Debye Dispersion Relation

The first-order phenomenological differential equation (6.4.3) is a statement of the time-domain Debye dispersion relation. To obtain the frequency-domain Debye dispersion relation, instead of the step type field, the dielectric-medium is subjected to the sinusoidal timeharmonic field. The polarization, as a response, also follows the sinusoidal time behavior. Equation (6.4.3) can be rewritten in the frequency-domain using the phasor forms as follows:

where χ∗r ω is the complex frequency-dependent susceptibility of a material. The equation (6.4.8a) is Debye’s dispersion relation for a frequency-dependent dielectricmedium. The real and imaginary parts of the complex relative permittivity ε∗r ω are given below: εr =

1 + χr + ωτ 1 + ωτ 2

2

a,

εr =

χr ωτ 1 + ωτ

2

b 649

For ω 0, the static value, or the low-frequency limit of the relative permittivity, is obtained, as εr = 1 + χr , εr = 0 . For ω 0, the frequency is very low and the material medium gets sufficient time to build up to full polarization. At ω ∞,εr = 1, εr = χr ωτ ≈ 0 , i.e. at the extremely high frequency, the medium is not able to polarize at all. In this case, the medium behaves as free space. Such behavior is because the expression (6.4.9) does not take into account the presence of other kinds of polarization. However, it accounts for the role of one relaxation in the modeling of the frequency response of the relative permittivity. Therefore, the present model is improved by accounting for various kinds of polarization. At any other frequency ω, the dielectric loss is obtained through εr .

181

182

6 Electrical Properties of Dielectric Medium

6.4.3.1 Debye’s Dispersion Equation for Three Kinds of Polarizations

Three kinds of static polarizations considered in equation (6.1.23) have three relaxation times. However, the electronic and ionic polarization at the optical and infrared frequency are so fast that their relaxation times are taken as zero, i.e. polarizations take place instantaneously without any time lag. The delayed response, leading to Debye dispersion relation, is obtained only due to the permanent dipolar polarization, with relaxation time τd. Following the dispersion model of equation (6.4.7), the static expression (6.1.23) could be modified to get the following frequency-dependent total polarization: P ω = Nε0

αee + αei +

μ2 1 3kT 1 + j ω τd

Eloc ω 6 4 10

On the assumption Eloc ω = Eext ω , and using equation (6.1.5), equation (6.4.10) provides the frequencydependent complex relative susceptibility and complex relative permittivity of a low-density medium: Nμ2 3 kT 1 + j ω τd Nμ2 ε∗r ω = 1 + N αee + αei + 3 kT 1 + j ω τd

χ∗r ω = N αee + αei +

ω

ε∗r ω

∞ =1+N

+

αei

0 = 1 + N αee + αei +

2

b

It is noted that there is no dielectric loss at very high and very low frequency, as the limiting values of the relative permittivity are real quantities, with εr = 0. The contribution of the dipolar (orientational) polarization to the static relative permittivity is obtained from the above expressions: 0 − εr ω

∞

6 4 13

The εr is the same as εr. Using equations (6.4.12a) and (6.4.13), the frequency-dependent complex relative permittivity, given by equation (6.4.11b), is rewritten as

εr ω

0 − εr ω 1 + jωτd

∞

Δεr 1 + jωτd

a b

6 4 14 The above equation is the Debye dispersion equation. The difference relative permittivity Δεr is a change in the relative permittivity at two steady-state polarization levels. The local field could be replaced with the Lorentz field to improve the above model. The local field can also be replaced by the Onsager field to avoid the Mossotti catastrophe. Even experimentally determined empirical parameter γ can be used to improve the accuracy of the Debye model. However, the nature of the Debye dispersion equation remains unchanged. Only the relaxation time τd gets modified. For instance, using the modified relaxation time τm for Mossotti field, the real and imaginary parts of the Debye dispersion equation, and also loss-tangent of the medium, are given by the following expressions [B.1]:

εr ω =

6 4 12

Nμ2 = εr ω 3kT

∞ +

b

a Nμ 3 kT

ε∗r ω = εr ω

εr ω = εr ω

The above Debye’s dispersion relation accounts for three kinds of polarization. The limiting values of the relative permittivity at a very high frequency (optical frequency, ω ∞), and at low frequency (static, ω 0) are obtained as αee

∞ +

a

6 4 11

ε∗r

ε∗r ω = εr ω

tan δ = where,

εr ω

∞ +

0 − εr ω 1 + ω2 τ2m

ε ω εr ω = r εr ω εr ω τm =

εr ω

εr ω εr ω

0 − εr ω 1 + ω2 τ2m

∞

a

∞ ωτm

0 − εr ω 0 + εr ω 0 +2 τd ∞ +2

b ∞ ωτm ∞ ω2 τ2m

c d

6 4 15 Figure (6.8) shows Debye dispersion behaviors of a dielectric material. Beyond the dielectric relaxation frequency, fre = 1/τm, the contribution of the dipolar polarization is reduced and finally, its contribution vanishes. The remaining high-frequency relative permittivity is due to ionic and electronic polarization. In the transition region, the Debye dispersion model provides the anomalous, i.e. negative, dispersion in the medium, as in this region, the phase velocity increases with the increase in frequency. The anomalous dispersion is discussed in chapter 3, section (3.3). Furthermore, as there is no relaxation behavior at low and at a very high frequency, the imaginary part of the relative permittivity εr vanishes at both ends of frequency band giving the maximum value of εr max , i.e. the absorption peak, near the middle of the transition frequency. The frequency fmax, at which the maximum εr max occurs, is obtained by differentiating equation (6.4.15b) with respect to ω and equating the expression to zero. εr max is computed at frequency fmax:

6.5 Resonance Response of the Dielectric-Medium

2

ε′r(ω = 0) – ε′r(ω = ∞)

ε′r(ω = 0)

ε″r

ε′r(ω = ∞) ω τm

1 Frequency

(Log scale)

Figure 6.8 Relaxation type Debye dispersion behaviors of a dielectric material.

ωmax τm = 1 εr max =

εr ω

f max =

1 2πτm

0 − εr ω 2

a ∞

b 6 4 16

The εr max is determined in terms of the limiting values of the real parts of the complex relative permittivity. The frequency-dependent relative permittivity εr ω and tan (δ) are measured experimentally. In practice, the Debye model is curve-fitted to the experimental results of εr ω ∞ , Δεr and εr ω by evaluating its parameters εr ω , and τm. It is discussed in section (6.8).

6.5 Resonance Response of the Dielectric-Medium In general, the polarization of material could be due to a combination of several polarizations. The oscillator model discussed in this section, along with the relaxation model, provides a realistic description of dispersion in both the normal and artificial materials. Debye model, applicable to a relaxation type response of the dipolar polarization, is discussed in the previous section. Above the relaxation frequency, the material gets depolarized and its permittivity is decreased. However, most of the practical dielectrics used in the microwave range, say up to 100 GHz, do not show a meaningful change in dielectric constant. It means most of the materials used in

the microwave range are nonpolar and the relative permittivity of these materials is due to the ionic and electronic polarization. However, materials show the resonance peaks in relative permittivity and also the absorption peaks in the infrared and visible frequency ranges. At the molecular level, such polarization resonances are modeled by the 1D linear harmonic oscillator, known as the Lorentz model. This model also is reduced to the conductivity model of Drude. Therefore, it is also called the Lorentz–Drude model [B.1, B.10]. The harmonic oscillator model can be reduced to the relaxation type model for the overdamped case. The RLC circuit also provides Lorentz–Drude kind of response of dielectric materials presented in subsection (6.7.5). The Lorentz model is also applicable to the artificial magnetic material. It is expressed by equation (21.2.4) of chapter 21. The artificial dielectrics developed by introducing the inclusions, such as an array of electrically small conducting scatters – spheres, discs, thin wires, metallic strips, etc., in a host dielectric-medium, also show resonance peaks. The metallic inclusions of artificial dielectric materials act as macromolecules and exhibit frequency-dependent relative permittivity, even at a lower frequency. They show Lorentz type resonance and anomalous dispersion. The insertion material, i.e. the inclusion, could also be a high permittivity material, different from the host medium. Such an artificial dielectrics medium provides interesting dielectric properties for many engineering applications. The dispersion models – both Debye and Lorentz types are important for these materials [B.11, J.31–J.33]. 6.5.1

Lorentz Oscillator Model

Each of the polarization in a dielectric material, natural or artificial, discussed above is modeled separately by the harmonic oscillator. Figure (6.9a) shows the displaced bound charges, under the influence of external field Eext, and the spring model of Hook’s type restoring force for the c-kind harmonic oscillator with the charge qc and mass mc. The generic symbol “c” could be the electron, ion, dipole, or inclusion. The external electric field E ext displaces the charge by a distance x. The attractive force, i.e. the restoring force, due to the local electric field E loc between the charges, is modeled by the spring. The motion of a charge, like the motion of a mass attached to the stretched spring, can be written as d2 x dx q Eloc + ω2c x = c , 2 + γc mc dt dt

651

183

6 Electrical Properties of Dielectric Medium

ε′r max (ω)

Eext +

Normal dispersion

ε′r(ω = 0)

– E1oc

Negative (Anomalous) dispersion

ε′r(ωe) = εrT Normal dispersion

ε′r min (ω)

Spring model

+

ε″r

–

ωL ωe ωU

0

(a) Harmonic oscillator model of resonance type Lorentz polarization.

Real and imaginary parts of complex relative permittivity

184

Frequency (ω)

(b) Normal and anomalous Dispersion.

ε″r (ω)

1 0

ε′r (ω) 0

ωp

Frequency (ω)

(c) Response of plasma medium. Figure 6.9

Harmonic oscillator model of resonance type Lorentz polarization and its frequency response, showing both the positive and negative dispersions. Also the response of the Drude model of the plasma medium.

where γc is the damping coefficient, in the frequency unit, of the oscillation indicating losses in materials and ωc is the characteristic resonance frequency. In absence of losses, the system performs a simple harmonic motion with frequency ωc. For Nc numbers of the c-type oscillators per unit volume with the dipole moment pc = qcx, the polarization, using an equation (6.1.4), due to all oscillators is Pc = Nc pc = Nc qc x

Mossotti field Eloc = Eext + Polarization Eq

where plasma frequency ωp = ωc = Polarization Eq

a

d2 Pc dPc + + γc dt dt2 Nc q2c = Eext mc

652

Using Mossotti–Lorentz field, i.e. the Lorentz field, from equation (6.1.9), the following equation of motion for the polarization can be written from the equations (6.5.1) and (6.5.2):

Pc 3ε0

ω2c

ω2c −

b

Nc q2c , ε0 mc

Nc q2c − 3ε0 mc

Nc q2c Pc 3 ε0 mc

c

1 2

d

d2 Pc dPc 2 + ω c Pc = ε0 ω2p Eext e 2 + γc dt dt 653

6.5 Resonance Response of the Dielectric-Medium

The characteristics resonance frequency ωc is reduced to ωc , due to the Mossotti field. However, for the lowdensity artificial dielectric material and gases, it could be ignored. Thus, the subsequent discussion assumes ωc = ωc . For the time-harmonic excitation, polarization (Pc) is also time-harmonic; expressed as Eext = E0ejωt, Pc = P0ejωt. Under the time-harmonic condition, the above-given polarization equation (6.5.3e) is reduced to the following expression: ω2c − ω2 + jω γc Pc =

Nc q2c

Eext

mc

654

On comparing the above equation with equation (6.1.5), i.e. P c = ε0 χr E ext , and also using equation (6.1.7a), the following expressions, related to three dispersion models, are obtained for the relative susceptibility and complex relative permittivity of the c-type oscillator: χrc =

ω2p Nc q2c ε0 mc = ω2c − ω2 + jωγc ω2c − ω2 + jωγc

Lorentz model ε∗rc = 1 + =1+ Drude model Debye model

ε∗rc = 1 + ε∗rc = 1 +

a

Nc q2c ε0 mc − ω2 + jωγc

ω2c

ω2p ω2c − ω2 + jωγc ω2p − ω2 + jωγc

, for ωc = 0 c

ω2p ω2c 1 + jω

b

γc ω2c

inclusions. However, in the real material, or a mixture of materials, several kinds of resonating elements are present. The relative permittivity of such a material is due to a linear combination of all noninteracting polarizations:

c

Δεr =

Nc q2c 1 ε0 mc ω2c

ω2c − ω2 + jωγc

a,

εrT = 1 + c

Nc q2c 1 ε0 mc ω2c

b 657

To obtain the relative permittivity around the generic resonance frequency ωe of the e-type resonating inclusions, the total relative permittivity includes the contribution of all nonresonating polarizations below ωe i.e. for ωc < ωe. The total complex relative permittivity due to nonresonating c-type polarizations and resonating e-type polarization is given below, along with its real and imaginary parts: ε∗r ω = 1 +

= εrT +

655 The c-type oscillators, constituting a dielectric material, are located in the vacuum with εr = 1. In the absence of the restoring force, there is no resonance, i.e. ωc = 0, and the Lorentz model is reduced to the Drude model. Drude model, discussed in the next subsection (6.5.2), applies to the modeling of the conducting medium of metal, and also to the wire medium discussed in chapter 21. Thus, the Lorentz and Drude models help to synthesize the artificial resonant and nonresonant types’ dispersive metamaterials. Further, the Debye model, in a form little different from equation (6.4.8a), is obtained from the Lorentz model under the condition ω2c + jωγc >> ω2 . In equation (6.5.5b), only one kind of resonating element is present. It applies to the artificial dielectric materials and metamaterials with one kind of resonating

c

At the frequency much below the resonance, ω < < ωc, i.e. ω 0, each kind of polarized element adds the nonresonating factor Δεr to the total static, i.e. low-frequency relative permittivity (εrT). It is obtained as follows:

c e c> ω2

Nc q2c ε0 mc =1+ ω2c − ω2 + jωγc

ε∗rc = 1 +

εr ω = εrT + εr ω =

Ne q2e ε0 me

Nc q2c 1 Ne q2e 1 + 2 2 2 ε0 mc ωc ε0 me ωe − ω + jωγe

Ne q2e 1 2 2 ε0 me ωe − ω + jωγe Ne q2e ε0 me

ω2e − ω2 ω2e − ω2

2

+ ω2 γ2e

ωγe ω2e

− ω2

2

+ ω2 γ2e

a b c 658

In case only one kind of polarizing element is present in the host vacuum medium, εrT = 1. The above relation is examined below at (i) low frequency, (ii) around resonance, and (iii) high frequency above the resonance.

6.5.1.1 Low-Frequency Relative Permittivity

On taking the limiting case, ω 0, of equation (6.5.8b), the quasistatic low-frequency value is obtained for relative permittivity:

185

186

6 Electrical Properties of Dielectric Medium

εr ω = 0 = εrT + 2 εr ω = 0 = εrT + where

2εrE ωe ω2p

a

Ne q2e = 2ε0 me ωe ωe ω2p εr ω = 0 = εrT + 2 ωe

εrE =

dεr ω γ = 0, giving Δω = ± e 2 dω γe Upper frequency, ωU = ωe + 2 γe Lower frequency, ωL = ωe − 2

Ne q2e 1 2ε0 me ωe ωe

b c

In the case of the artificial dielectrics and metamaterials, static relative permittivity εr ω = 0 could be the relative permittivity of a host medium. The host medium could be the vacuum itself with εr ω = 0 = 1, i.e. εrT = 1, εrE = 0. In this case, no polarizing element is present. In the frequency range below the resonance frequency, 0 < ω < ωe, the relative permittivity is highly dispersive and εr ω increases to a peak value. 6.5.1.2 Relative Permittivity at and Near the Resonance Frequency

Near resonance, on using approximations, the approximate complex relative permittivity is obtained from equation (6.5.8a) along with its real and imaginary parts: ω ≈1 ωe

Relative permittivity near resonance ε∗r ω εrE = εrT + γ Δω + j e 2 εrE Δω εr ω = εrT + γ 2 Δω 2 + e 2 εrE ωe − ω = εrT + γ 2 ωe − ω 2 + e 2 γe γe εrE εrE 2 2 εr ω = = 2 γ γ 2 Δω 2 + e ωe − ω 2 + e 2 2

a

b

c

d 6 5 10

The relative permittivity and absorption peak, i.e. the maximum value of εr(ω), are obtained at the resonance frequency, ω = ωe, Δω = 0: εr ω = ωe = εrT

a,

εr max ω = ωe =

b 6 5 12

659

Approximation Δω = ωe − ω, ωe + ω ≈ 2ωe ,

a

2εrE b γe 6 5 11

The maximum and minimum values of the real part of the relative permittivity εr ω , given by equation (6.5.8b), are obtained by taking

On substituting the ωL and ωU in equation (6.5.10c), the following values of the maximum εr max ω and minimum εr min ω are obtained: εrE γe εrE = εrT − γe

εrmax ω = ωL = εrT +

a

εr min ω = ωU

b 6 5 13

6.5.1.3 High-Frequency Relative Permittivity Above Resonance

At high frequency above the resonance, ω ∞, the steady relative permittivity is obtained from equation (6.5.8b) as, εr ∞ = εrT

6 5 14

Figure (6.9b) shows the frequency response of both εr ω and εr ω corresponding to material dispersion and loss, respectively. The resonance type Lorentz dispersion model exhibits both the normal and anomalous dispersion in a material. The εr ω rises to the peak value εr max ω , from the quasistatic value εr ω = 0 , with an increase in frequency. It is given by equation (6.5.13a). The resonance occurs at ω = ωc. Further, an increase in frequency causes a decrease in relative permittivity εr ω from its peak εr max ω to the minimum εr min ω The decreasing slope of relative permittivity εr ω is a linear function of frequency. Further increase in frequency, above εr min ω , increases the relative permittivity εr ω from εr min ω to εrT ω . So for an increase in the frequency from (ω = 0) to (ω = ωL=ωe − Δω), and also from (ω = ωU = ωe + Δω) to (ω ∞), the slope of εr ω is positive, i.e. dεr dω > 0. Therefore, both in the lower and upper frequency ranges the dispersion is normal. However, between the frequency range from (ω = ωL = ωe − Δω) to (ω = ωU = ωe + Δω), the slope is negative, i.e. dε /dω < 0. It shows that over a narrow frequency band around the resonance, the dispersion is anomalous. Further, at the frequency, ω = ωe + Δω, the value of relative permittivity is less than the steady-state relative permittivity εrT contributed by the nonresonating elements of the medium. It is interesting to note that if the steady-state

6.5 Resonance Response of the Dielectric-Medium

overall material dispersion is anomalous. It is obtained from Debye’s relaxation model. Figure (6.10a) shows two extended Debye responses I and II for both Maxwell–Wagner and dipolar polarizations with a noticeable difference in the magnitude of their relative permittivity. However, in practice, such difference may not be visible for two Debye type polarizations. The Maxwell–Wagner polarization provides the extended Debye-I response, and the dipole polarization provides the extended Debye-II response. The magnitude of the relative permittivity of both polarizations tends to the same value εr ω = ∞ at high frequency, when polarizations are lost. The minima and maxima in εr ω for two Debye type polarizations could be visible in the measurements [B.21]. The Lorentz oscillator model is reduced to the

Extended debye-I

Electronic polarization

Dipolar polarization Ionic Polarization

Maxwell-Wagner polarization

ε′rdipol

Real relative permitivity

ε′rmw

relative permittivity of a material is εrT = 1, then the highly dispersive relative permittivity can become negative at a frequency ω > ωe over a narrow band. In chapter 3, subsection (3.3.2), it is discussed that the anomalous dispersive medium, under special condition, supports the backward wave propagation. However, losses are very high as shown in Fig (6.9b). There could be polarization also in heterogeneous dielectrics due to the embedded inclusions in a host medium. This kind of polarization arises due to Maxwell–Wagner interfacial polarization, which is discussed in the next section (6.6). Figure (6.10a and b) shows dispersion and loss behaviors of a hypothetical material with several polarizations. In general, the εr(ω) decreases with an increase in frequency, i.e. the

Infra-red

Ultraviolet

Microwave

ε′relee

ε′rion Extended debye-II

1

Resonance (lorentz) type polarization

Relaxation (debye) type polarization Frequency

log scale

ω1

ω2 Frequency log-scale

Ultraviolet Electronic polarization

Dipolar polarization

Infra-red Ionic polarization

Microwave

Maxwell wagner polarization

Img. relative permitivity

(a) Dispersions due to several kinds of polarization.

ω3

ω4

(b) Losses due to several kinds of polarizations. Figure 6.10

Dispersion behavior of a hypothetical material showing two relaxation type dispersion and two resonance type dispersion.

187

188

6 Electrical Properties of Dielectric Medium

Debye relaxation model by ignoring d2Pc/dt2 term in the polarization differential equation (6.5.3e). It can be written as γc dPc Nc q2c + Pc = Eext 2 ωc dt mc ω2c

6 5 15

A comparison of equation (6.5.15) with equation (6.4.4) shows that it is a relaxation type dispersion model with relaxation time τ = γc ω2c. It is further noted from equation (6.5.8c) that if εr ω = 0, i.e. there is no loss in the material, εr ω becomes nondispersive. Thus, dispersion in a material is associated with material loss. The material dispersion and loss are related through Kramers–Kronig relations [B.5]. It is summarized in the subsection (6.5.4). If we measure the frequencydependent real part of relative permittivity εr ω , the measured frequency-dependent loss εr ω could be recovered using Kramers–Kronig relations. 6.5.2

Drude Model of Conductor and Plasma

Drude model provides the DC and frequency-dependent conductivity of metal under the time-harmonic field excitation. It shows Debye-type relaxation at a frequency below the plasma frequency, as shown in Fig (6.9c). Drude model is a special case of the Lorentz model given by equation (6.5.5c). The metal has free electrons; so in absence of the restoring force, there is no characteristic resonance, i.e. ωc = 0. The following Drude model expressions can be written from equation (6.5.5c) for the complex susceptibility and complex relative permittivity of the plasma medium of a metal: χ∗r =

ω2p

a,

− ω2 + jωγ

ε∗r = 1 + χ∗r = 1 −

ω2p

= εr =

ω

γc ω2p ω2 + γ2c

J =

d

ω2 ω2p ω2

+

γ2

plasma medium of the conductor as an epsilon negative (ENG) medium, discussed in section (5.5) of chapter 5. Even in the case of a lossless plasma medium, the ENG medium is obtained for the frequency ω < ωp. Figure (6.9c) shows that the Drude model provides wideband negative permittivity below the plasma frequency. Equation (6.5.16f) shows that in the region of negative permittivity, dielectric losses increase with a decrease in frequency. The ENG medium is highly dispersive and highly lossy. However, the Drude model provides negative permittivity over a wider bandwidth, whereas the resonant Lorentz model provides the negative permittivity over a very narrow bandwidth. In the ENG medium, for ω < ωp, the incident EM-wave cannot propagate through the plasma medium. The wave propagation occurs only for ω > ωp. Thus, the plasma frequency ωp acts like the cut-off frequency of a waveguide. Therefore, the plasma medium could be modeled as a waveguide discussed in chapter 7. Alternatively, the waveguide can be treated as a plasma medium. In the case of metals, the plasma frequency occurs in the optical and UV frequency ranges. However, the plasma medium has been also created artificially using the wire-medium. It brings down the plasma frequency in the GHz frequency range. It is discussed in section (21.1.1) of chapter 21. The current density is the polarization current density due to the time-varying electric field. Using equation (6.1.5), the polarization current is computed. On using equation (6.5.16), the frequency-dependent conductivity is further computed:

b c

ω2 − jωγ

Re ε∗r = εr = εr = 1 − Im

Ne2 ε0 me

ω2p

for γc = 0, εr = 1 −

ε∗r

ω2p =

incident, with frequency ω2 < ω2p − γ2c , views the

e

dP = jω P = jωε0 χr E ext dt

J = σ ω E ext jε0 ω2p σ∗ ω = jγ − ω

6 5 16 where ωp is the plasma frequency of the free electrons of charge e and mass me. N is the electron density. The free electrons are confined within a thin layer of conductor form the plasma medium. The propagation constant in the lossy plasma medium is β = β0 εr . The EM-wave

b

also

c

Static conductivity σ ω = 0 = σ∗ ω =

f ,

a

σ ω=0 1 + jωτ

where relaxation time, τ =

ε0 ω2p γ

d e

1 γc

f 6 5 17

Equations (6.5.17d) and (6.5.17e) are the static and frequency-dependent Drude models, respectively, for the conductivity of a metal. It shows the Debye-type relaxation, similar to equation (6.4.8b).

6.5 Resonance Response of the Dielectric-Medium

6.5.3 Dispersion Models of Dielectric Mixture Medium

6.5.3.1 Maxwell Garnett–Debye Model

Several expressions have been considered in section (6.3) to evaluate the static equivalent relative permittivity of the dielectric mixture medium formed by scattered inclusion material, with relative permittivity εri, in a host continuum with relative permittivity εrh. We use the terminology equivalent relative permittivity, in place of usual effective relative permittivity for the corresponding equivalent (effective) homogeneous medium of the mixture, to avoid confusion of similar terminology used in the microstrip description. The inclusion and host materials can have either Debye type or the Lorentz type dispersion behavior. In the case of the metallic inclusion, the plasmonic model for the permittivity could be used to get the Drude type dispersion. The MG mixing formula has been combined with these dispersion models to obtain the dispersion model of a mixture medium [J.34–J.38]. In particular, either the inclusion or host material could be frequency-dependent. In general, both could be frequency-dependent. Maxwell Garnett formula is summarized below incorporating Debye, Lorentz, and Drude models, for the dispersive inclusion, εri(ω) and the static relative permittivity εrh of the host continuum [J.34].

εreq ω εreq ω

0 = εrh 1 +

εri ω

∞ = εrh 1 +

εri ω

Debye model of equation (6.4.14) has been modified for the mixing medium using the MG formula (6.3.13) for the 3D spherical dispersive inclusions. The inclusion itself follows the Debye model with relaxation time τdi. In the case of a mixture, it is modified to τeq; as the mixture is treated as an equivalent homogeneous medium. Following equation (6.4.14a), Maxwell Garnett–Debye model giving the dispersive complex equivalent relative permittivity of a twophase mixture is summarized below [J.34].: ε∗req ω = εreq ω

∞

εreq ω

+

0 − εreq ω 1 + j ω τeq

6 5 18 The homogenized mixture could be also treated with the help of the circuit model, discussed in subsection 6.7.3. The low- and high-frequency equivalent relative permittivity of a mixture, containing Debye type frequency-dependent inclusion, are obtained by modifying the MG formula (6.3.13) as follows:

3f εri ω 0 − εrh 0 + 2εrh − f εri ω 0 − εrh 3f εri ω ∞ − εrh ∞ + 2εrh − f εri ω ∞ − εrh

a 6 5 19 b

Finally, the relaxation time of the mixture is given by τeq = τdi

1 − f εri ω 1 − f εri ω

ε∗req ω = εreq ω

∞ + 2 + f εrh 0 + 2 + f εrh

∞

∞ +

ω2p,eq ω20,eq − ω2 + jω γeq 6 5 21

6 5 20 In equation (6.5.20), f is the volume fraction of a mixture. The real and imaginary parts of the complex equivalent relative permittivity of the mixture could be separated. It helps to obtain its loss-tangent of the mixture.

6.5.3.2 Maxwell Garnett–Lorentz Model

Similarly, Lorentz resonance type model (6.5.5b) could be modified for the two-phase mixture and get the following Maxwell Garnett–Lorentz model [J.34]:

In the present model, inclusion is dispersive and the host-medium is frequency independent. The εreq(ω ∞) of the mixture is given by equation (6.5.19b). The plasma frequency ωp, eq and resonance frequency ω0, eq of the mixture are expressed in terms of the plasma frequency ωp, i and resonance frequency ω0, i of the inclusion: ωp,eq =

f

1 − f εri ω

ω20,eq = ω20,i +

3εrh ωp,i ∞ + 2 + f εrh

1 − f εri ω

a

1−f ω2 b ∞ + 2 + f εrh p,i

6 5 22

189

190

6 Electrical Properties of Dielectric Medium

In equation (6.5.22), f is the volume fraction of a mixture. The damping factor of the inclusion remains unchanged in the mixture, i.e. γeq = γi. The resonance frequency of a mixture is higher than that of the resonance frequency of inclusion, and it increases with a low-density mixture. The plasma frequency of a mixture increases for a denser mixture.

6.6

6.5.3.3 Maxwell Garnett–Drude Model

Drude model has been discussed above as a special case of the Lorentz model, as applied to the metals with free electrons showing no resonance. However, the metallic inclusions are used in a host medium. The mixture behaves like a dielectric-medium, showing the Lorentz type resonance with resonance frequency ωmetal 0,eq given below. For the resonance frequency ωmetal 0,eq , the MG– Lorentz model (6.5.21) could be treated as the MG– Drude model applicable to the metallic inclusions: ωmetal 0,eq =

1 − f εri ω

1−f ∞ + 2 + f εrh

1 2

ωp,i , 6 5 23

where for a metallic inclusion εri(ω ∞) ≈ 1. It shows that the mixture behaves like the nonmetallic type Drude model. However, for the usual metallic Drude model ω0, eq = 0 in equation (6.5.21) to get the plasma-medium type response. If the measurement results for the real and imaginary parts of any mixture medium are known at a few frequencies, the multiterms Debye models could be empirically obtained to get the curve-fit expression of experimental results for the wideband applications. It is discussed in section (6.8). 6.5.4

Kramers–Kronig Relation

The general relation between the frequency-dependent real and imaginary parts of complex relative permittivity εr ω and εr ω is expressed by the Kramers–Kronig relation given below [B.5, B.10]: 2 εr ω = εr ω = ∞ + π εr ω = −

2 π

∞

0

∞

0

ω εr ω dω ω 2 − ω2

ω εr ω − εr ∞ ω

2

− ω2

not the part of Kramers–Kronig causal relation. If the imaginary part of a dispersive lossy dielectric is measured at one frequency, the first expression of Kramers–Kronig can provide frequency-dependent real relative permittivity at all other frequencies.

dω

a

b 6 5 24

It can be shown that Debye’s equation follows the Kramers–Kronig relation. However, the D.C. conductivity could be the part of the Debye model, although it is

Interfacial Polarization

The interfacial polarization, i.e. Maxwell–Wagner polarization, occurs in the layered inhomogeneous dielectric and semiconductor media, such as the metal insulatorsemiconductor (MIS) structure, Schottky structure, PN-junction, etc. [B.1, J.2–J.5]. It also occurs in heterogeneous mixing materials, where the contact between the inclusion and host also forms the boundary layer [J.39, J.40]. The interfacial polarization provides Debye kind of polarization. Normally, it occurs in the lower range of microwaves. This section presents the polarization equation of Maxwell–Wagner polarization for a two-layered structure. The next section discusses circuit modeling. 6.6.1 Interfacial Polarization in Two-Layered Capacitor Medium Figure (6.11a) shows the two-layered parallel-plate capacitor to examine the interfacial polarization. The lossy dielectric layers have relative permittivity εr i, conductivity σi, and substrate thickness hi (i = 1,2). On the application of a voltage V across the capacitor, the layers support conduction current density, Ji. Once the switch is closed at t = 0, the interfacial polarization process starts with the flow of the transient polarization current. The migrated charges, from the interior of the medium, accumulate at the interface, as shown in Fig (6.1d). Finally, the steady-state condition is reached. However, initially, there is no accumulated charge at the interface. The time taken for the accumulation of charges is described by the relaxation time of the media given by τ1 =

ε1 ε2 , τ2 = , where εi = ε0 εr i , i = 1, 2 σ1 σ2 661

The relaxation time is defined in equation (6.4.2). 6.6.1.1 Boundary Conditions

Using Ohm’s law, the conduction current densities, normal to the interface of two media, are expressed as Jn1 = σ1 En1

a,

Jn2 = σ2 En2

b,

662

6.6 Interfacial Polarization

t=0 h2 V

h1

εr2 Medium #2 σ2

J2

σ1

J1

H

ε1

ρ1

ρ2

ε2

Medium #1 εr1

(a) Two-layered lossy capacitor medium exhibiting interfacial polarization at the interface.

(b) Equivalent circuit of two-layered lossy inhomogeneous medium to compute time constant.

Figure 6.11 Interfacial polarization at the interface of two-layered lossy dielectrics.

where En1 and En2 are the normal components of the electric field in the medium #1 and medium #2, respectively. Initially, the current follows across the interface, and there is no accumulation of charges: Current entering the interface = Current leaving the interface En1 σ2 Jn1 = Jn2 a , = En2 σ1

b 663

However, during the transient process, the charges build up at the interface, giving Jn1 Jn2. The acculturated negative surface charge ρs provides discontinuity to the normal components of the electric displacement (electric flux density) vectors: Dn1 − Dn2 = ρs

εr1 En1 − εr2 En2 =

ρs ε0

664

On using equations (6.6.1) and (6.6.3b), the accumulated charges at the surface are obtained in terms of the relaxation times: ρs σ1 ρs σ2 En1 , = εr1 − εr2 = εr1 − εr2 En2 ε0 σ2 ε0 σ1 ρs = τ1 − τ2 σ1 En1 For no accumulation of charge at the interface, εr1 σ1 = ρs = 0 εr2 σ2

same material having different conductivity (σ1 σ2). The latter case is obtained in a semiconducting material through the doping or by the formation of a depletion layer at the interface. The case h1 = h2 = h is considered below. The case h1 h2 is considered in subsection (6.7.4) using the circuit model. The double-layered structure of Fig (6.11a) is analyzed under both the steady-state and transient conditions. Under the initial condition, at time t = 0, there is no surface charge at the interface of two media, ρs(0) = 0. Under the transient condition, the time-dependent surface charge ρs(t) accumulates at the interface, and finally, under the steady-state condition for t ∞, the static charges ρs(∞) accumulate at the interface.

6.6.1.2 Steady-State Condition

Under the steady-state condition, the applied voltage V at the capacitor is En1 h + En2 h = V En2 1 +

a b c 665

If τ1 = τ2, ρs = 0, i.e. there is no interfacial polarization in a homogeneous medium. The interfacial polarization occurs only when the relaxation times of both media are different, τ1 τ2. Different relaxation times are obtained for two different materials (εr1 εr2), or even for the

En2 =

En1 En2

=

a V , h

σ1 V σ1 + σ2 h

b,

En2 1 +

σ2 σ1

or En1 =

=

V h

σ2 V c σ1 + σ2 h 666

The surface charge density at the interface is obtained by substituting En1 from equation (6.6.6c) in equation (6.6.5a): ρs = ε1 −

ε2 σ1 σ2 V σ2 σ1 + σ2 h

ε1 σ2 − ε2 σ1 V ρs = σ1 + σ2 h

667

191

192

6 Electrical Properties of Dielectric Medium

6.6.1.3 Transient Condition

During the transient process Jn1 Jn2, and it results in a flow of the interfacial polarization current: ∂ρs = Jn2 − Jn1 ∂t

∂ρs = En2 σ2 − En1 σ1 ∂t

a,

b

The final interfacial polarization, showing the Debye type response, in the time-domain could be written as follows: ρs t =

V σ2 ε1 − σ1 ε2 h σ1 + σ2

where

Tr =

668 Both field quantities, En1 and En2, are computed by using equations (6.6.4) and (6.6.6a): V b h On adding the above equations (a) and (b), En1 is obtained: En1 ε1 ε2 − En2 = ρs ε2

a,

En1 + En2 =

ε1 V ρ + s + 1 En1 = ε2 ε2 h ε2 V ρs + En1 = ε1 + ε2 h ε1 + ε2 V also, En2 = − En1 h ε1 V ρs − En2 = ε1 + ε2 h ε1 + ε2

a

b 669

The following interfacial polarization equation is obtained by substituting En1 and En2 from equation (6.6.9) in equation (6.6.8b): ∂ρs σ1 + σ2 σ2 ε1 − σ1 ε2 V ρs + = − ∂t ε1 + ε2 ε1 + ε2 h ∂ρs = − Aρs + B ∂t σ1 + σ2 σ2 ε1 − σ1 ε2 V where, A = ,B = ε1 + ε2 ε1 + ε2 h

Tr =

ρs t = Ke

−

σ1 + σ2 ε1 + ε2

t

+

ε1 + ε2 σ1 + σ2

b

ε1 + ε2 1 ρ1 + 1 ρ2

c

The lossy dielectric layers could be viewed as the complex capacitors represented by the RC-circuit model, as shown in Fig (6.11b). The relaxation time constant Tr of the inhomogeneous layered medium could also be obtained from Fig (6.11b), showing the parallel combination of two complex capacitors corresponding to two layers of lossy dielectrics: 1 1 1 = + R ρ1 ρ2

a

C = ε1 + ε2

b ε1 + ε2 σ1 + σ2

Tr = RC =

c 6 6 14

b c

The circuit model can be extended to get the relaxation time constant for n numbers of interfaces, corresponding to (n+1) numbers of lossy dielectric layers. It also provides the relaxation frequency of the Debye type polarization response: n+1

Tr = D A

i=1 n+1 i=1

σ2 ε1 − σ1 ε2 V σ1 + σ2 h 6 6 11

The constant K is evaluated from the initial condition, at t = 0, ρs(0) = 0: σ2 ε1 − σ1 ε2 V =0 σ1 + σ2 h σ2 ε1 − σ1 ε2 V K= − σ1 + σ2 h

a

6 6 13

On solving the above equation, the transient surface charge density is obtained at the interface:

K= −

Tr

a

6 6 10

− Aρs t + B = De − At , B ρs t = Ke − At + , where, A

1 − e−t

ρs 0 = K +

6 6 12

εi

1 ρi

a,

fr =

1 Tr

b

6 6 15

The above analysis can be extended to get the Debye type polarization of the interfacial polarization in the frequency-domain. In subsection (6.7.4), the circuit model is used to get such results. In the case of interfacial polarization, the polarization is not achieved using the concept of polarizability as no dipole formation is involved. The interfacial polarization is achieved by the process of charge accumulation at the interface of heterogeneous media. However, even for the interfacial polarization, the polarizability could be extracted, as done in equation (6.7.36).

6.7 Circuit Models of Dielectric Materials

6.7 Circuit Models of Dielectric Materials

the expression is obtained for the complex capacitance of the circuit:

The dielectric dispersion behavior of a homogeneous material, at the molecular level, is described either by the relaxation type response or by the resonance type response. The homogeneous dielectric materials can be divided into two broad groups: (i) Debye materials and (ii) Lorentz materials [J.41]. The heterogeneous dielectrics with the interfacial Maxwell–Wagner polarization can be called Maxwell–Wagner materials. However, they also show the relaxation response [B.1]. In previous sections, these responses are considered using the polarization models. The polarization works at the microlevel and is finally responsible for the creation of relative permittivity in a material at the macrolevel, ignoring details of each polarized molecules. It is known as the process of homogenization. Likewise, the circuit model elements depict the microlevel polarization response, relaxation, or resonance type, to the applied field. In the end, the circuit models of materials – natural or artificial, help to compute their complex relative permittivity. Thus, the circuit models help to obtain homogenized media. The circuit models are useful in the development of a new kind of artificial dielectric materials with interesting properties. This section presents the modeling of polarization responses, and complex relative permittivity of materials, using the circuit models.

jωτQ + Q = Cs V,

Series RC Circuit Model

The relaxation in a dielectric material is related to its delayed polarization response, under the excitation of an external field. The growth of the polarization is described by the first-order differential equation, given by equation (6.4.3). The delayed polarization, with relaxation time, can also be modeled by the RC series circuit shown in Fig (6.12a). On closing of the switch, the capacitor Cs is charged to Q. The voltage around the circuit is V = Vc + VR , Rs

IRs +

dQ Q + = V, dt Cs

τ

Q =V Cs dQ + Q = Cs V, dt 671

where the time constant of the circuit is τ = RsCs. Equation compares well with the polarization equation (6.4.3). Let the solution of equation (6.7.1) is Q = Q0ejω t on the application of a sinusoidal voltage V = V0ejω t. On substituting these in the above equation,

Q Cs = 1 + jωτ V 672

The above relation shows that the RC circuit can model the Debye type of polarization of a dielectric material giving the following expressions for complex relative permittivity, along with its limiting values: C ω − jC ω C∗ ω Cs C0 = = 1 + jωτ C0 C0 Cs 0, ε∗r 0 = C0 ∞ , ε∗r ∞ = 0

ε∗r = εr − jεr =

a

For

ω

b

For

ω

c

673 The complex relative permittivity can be separated into the real and imaginary parts: εr 0 1 + ω2 τ2 ε = ωτ tan δ = ε

εr ω =

a,

εr ω =

ωτεr 0 1 + ω2 τ2

b

c 674

The above relation can be obtained directly from the admittance of the RC circuit: Y=

6.7.1

C∗ ω =

jω Cs jω Cs = 1 + jω Cs Rs 1 + jω τ

675

On using τ = RsCs and Y = j ω C∗(ω), the complex capacitance of the circuit is C∗ ω =

Cs 1 + jωτ

676

Equation (6.7.6) is identical to equation (6.7.2). Figure (6.12b) shows the Debye type frequency response of both the εr ω and εr ω , obtained from the RC circuit model. It is seen that at the optical frequency, i.e. at the 0. It is unrealistic, as higher frequency, ω ∞, εr ω εr of even the vacuum is unity, not zero. The model could be improved by assuming that the capacitance relation (6.7.2) provides the susceptibility, in place of relative permittivity. Alternatively, a capacitance can be added in parallel to the RC circuit. It is discussed in subsection (6.7.3). The imaginary part of the complex relative permittivity εr ω is zero, both at ω = 0, and ω ∞. Further, an absorption peak is obtained at a frequency, ω = ωmax = 1/τ giving εr ωmax = εr 0 2. On an increase in the frequency of the applied external electric field, polarization is not fully developed, as

193

6 Electrical Properties of Dielectric Medium

ε′r (ω = 0) C = S C0 ε′r (ω)

I

VR

t=0

ω

ε″r (ω)

CS

V

0

ε″r (ωmax)

RS VC

ωmax

0 (a) RC series circuit for modeling of relaxation time.

ω

(b) Frequency-dependent ε′r and ε″r as Debye type RC circuit response.

ε′r (ω = 0) C = P C0 ε′r (ω)

I

RP

V

0

CP

Frequency

ω

Frequency

ω

ε″r (ω)

194

0

Y (c) RC parallel circuit to model nonrelaxation type material. Figure 6.12

(d) Frequency response of nonrelaxation type of material.

Series and parallel equivalent circuits of the relaxation and nonrelaxation type materials.

sufficient time is not available for the full charging of the capacitor. To polarize the material completely, the electric field must be applied for a duration more than the relaxation time τ of the dielectrics, or its circuit model. Once the experimental results on the εr ω and εr ω are known, the numerical value of Cs and Rs can be determined to model the material by the Rs Cs series circuit.

shown in Fig (6.12c). However, it is a noncausal model. It is analyzed by using the input admittance Y: Y = Gp + jωCp C∗ ω =

Gp + jωCp jω

a,

Y = jωC∗ ω

b

c 677

The complex capacitance provides the complex relative permittivity ε∗r ω of a medium: 6.7.2

Parallel RC Circuit Model

Some dielectrics do not show noticeable variation in the relative permittivity εr . However, the imaginary part of their relative permittivity εr decreases with frequency. Such material can be modeled by the RC parallel circuit,

ε∗r ω = εr − jεr = εr =

Cp C0

a,

C∗ ω C0 εr ω =

1 ω C 0 Rp

b 678

6.7 Circuit Models of Dielectric Materials

Figure (6.12d) shows the frequency response of real εr ω and imaginary εr ω parts of complex relative permittivity. The parallel RC circuit does not show any relaxation and the real εr ω is frequency independent, while imaginary εr ω is given by a hyperbolic curve. The material response modeled around simple series and parallel RC circuits may not correspond to any real material. Still, such simple models have been used to model dispersive materials. For instance, the RC parallel circuit has been used to model the diamond film deposited on the Si substrate [J.42]. The combination of the RC networks can provide more realistic material modeling. 6.7.3

Parallel Series Combined Circuit Model

RS Y

C* (ω)

(a) Circuit model of homogeneous material, showing two kinds of polarization.

ε′r (ω = ∞) =

CS

ω 2 Cs τs ω Cs + j ω Cp + 2 2 1 + ω τs 1 + ω2 τ2s

6 7 10

The admittance can be further written using the equivalent complex capacitance, C∗ ω = εr − jεr C0

6 7 11

Y = jωC∗ ω = ε r + jεr ωC0

On comparing the above equations, the expressions are obtained for real εr ω , imaginary εr ω , complex ε∗r ω and tanδ of a lossy material in terms of the circuit elements: Cp Cs 1 + C0 C0 1 + ω2 τ2s Cs ω τs εr ω = C0 1 + ω2 τs 2 εr ω =

a b

ε∗r ω = εr − jεr , ε∗r ω =

Cp Cs 1 + C0 C0 1 + jω τs

tan δs = ω τs , tan δ =

also, tan δ =

tan δs Cp 1+ 1 + ω2 τ2s Cs

c εr εr d 6 7 12

Cp C0

ε″r (ω = ∞) = ε″r (ω = 0) = 0 ε″r (ω)

Cp

Y=

ε′r (ω = 0) = (Cp +Cs) ε′r (ω) C0

Figure (6.13a) shows a parallel combination of a capacitor Cp and RsCs circuit to model a homogeneous material medium with two kinds of polarizations (i) showing no dispersion, like the electronic and ionic polarizations up to very high frequency; and (ii) showing Debye type dispersion in the lower frequency range. The first kind of polarization of host medium is modeled by the capacitance Cp, while the RsCs series circuit, corresponding to lossy inclusion, models the second kind of polarization, like the dipolar polarization. The circuit could also model the artificial dielectric medium. The impedance of the series arm and also input admittance of the circuit are given below: 1 1 Z s = Rs + a , Y = jω Cp + b jω Cs Zs 679

The time constant τs = RsCs, with relaxation frequency ωs = 1/τs = 1/(Rs Cs), controls the relaxation response of the circuit. Its input admittance is

0 0.01

0.1 1.0 10 100 ωτs Frequency log scale

0 0.01

0.1 1.0 10 100 ωτs Frequency log scale

(b) Frequency response of Debye material.

Figure 6.13 Circuit model and relaxation response of Debye material.

195

6 Electrical Properties of Dielectric Medium

The above equations show that the circuit of Fig (6.13a) models Debye-type relaxation response of a material. For the limiting case ω 0, the static relative permittivity is due to both kinds of polarizations, whereas at the optical frequency, ω ∞, the dipolar kind of polarization or polarization in inclusions of a mixture, does not exist. The following expressions are obtained for both the limiting cases and also the contribution of the dipolar polarization to the relative permittivity: Cp + Cs 0 εr ω = 0 = C0

For ω

εr ω

1 + ω ωs

The above equations could be combined to write the following expression: εr ω = 0 − εr ω = ∞ 1 + j ω τs

The above equation is identical to the MG–Debye model, given by equation (6.5.18). Figure (6.13b) shows the usual frequency response for a Debye-type material. The component’s values can be computed for the circuit model from the experimental results. The circuit model is useful for the design of components and also to study distortion in a signal propagating through the medium. More general circuit model, useful for the SPICE circuit simulator, are developed for a Debye-type dispersive material [J.43].

c

∞ =0

d

Contribution of dipole polarization Cs = εr ω = 0 − εreff ω = ∞ C0

e 6 7 13

The contribution of the dipolar polarization to the overall polarization of a material decreases at the frequency above the relaxation frequency, i.e. for ω > ωs (ωs τs = 1). From equations (6.7.12) and (6.7.13), the dispersion expression of the material could be written as

6.7.4

Series Combination of RC Parallel Circuit

Figure (6.11a) shows an inhomogeneous lossy medium formed with a two-layered lossy dielectric capacitor. Figure (6.14a) shows its circuit model, using the series

ε′req (ω)

ε′req (ω = 0)

t=0

ε′req (ω = ∞)

Rp2

v

0 0.01

Cp2

Cp2

0.1 1.0 10 100 ωτs Frequency log scale

v Rp1

Rp1 Cp1

Cp1

ε″req (ωmax)

ε″r (ω = 0) = 0

C*(ω)

0 0.01 0.1 0 0.01

1.0 0.1

100 ωτs

10 1

10

100

Frequency log scale (a) Circuit model of two-layered inhomogeneous medium. Figure 6.14

b

2

6 7 15

Cp ∞ = , C0

t=0

a

6 7 14

ε∗r ω = εr ω = ∞ +

b

∞ εr ω

εr ω =

a

εr ω = 0 = 0 For ω

εr ω = 0 − εr ω = ∞ 1 + ω ωs 2 εr ω = 0 − εr ω = ∞ ω ωs

εr ω = εr ω = ∞ +

ε″req (ω)

196

(b) Relaxation type response of interfacial polarization.

Circuit model of the two-layered inhomogeneous medium under the interfacial polarization.

6.7 Circuit Models of Dielectric Materials

combination of two parallel RC circuits. Each parallel circuit is the analog of each layer of the lossy dielectric-medium. Such an inhomogeneous dielectricmedium is encountered in the microstrip line structures based on Si-SiO2 and Schottky-contact substrates [J.2–J.5]. It involves the Maxwell–Wagner type interfacial polarization. In the case of the Si-SiO2 structure, the upper SiO2 layer is a lossless passivation layer with Rp2 ∞ and the lower Si- layer is a high conductivity doped silicon layer. Thus, two cases are considered for the two-layered medium shown in Fig (6.11a): (i) upper layer lossless and (ii) upper layer lossy. The first circuit model of Fig (6.14a) shows the case (ii), and it is reduced to another circuit model for the case (i), that is a series combination of Rp1 Cp1 circuit and C p2. The interfacial

Y1 = Z=

polarization response of the inhomogeneous medium could be obtained from the response of the circuit models. The Maxwell–Wagner type polarization also occurs in the polycrystalline materials – semiconductor and ceramic [J.40]. The low-resistance grains are molded by one parallel RC circuit and narrow high-resistance grain boundaries are modeled by another parallel RC circuit.

6.7.4.1 Lossless Upper Layer

The input impedance Z and input admittance (Y = 1/Z) of the lossless upper layer, shown as the reduced circuit in Fig (6.14a), is determined below

1 1 1 + jωCp1 , Z = + Rp1 jωCp2 Y1 6 7 16

1 + jωRp1 Cp1 + jωRp1 Cp2 jωCp2 1 + jωRp1 Cp1

Y=

ω2 Cp2 τ1 + Cp2 Rp1 − Cp2 τ1 + jωCp2 1 + ω2 τ1 τ1 + Rp1 Cp2 1 + ω2 τ1 + Rp1 Cp2

2

where the time constant of the circuit is τ1 = Rp1 Cp1. The whole circuit is treated as an equivalent complex capacitor with a single lossy dielectric-medium: ε∗req = εreq − jεreq , Y = j εreq − jεreq ωC0 = εreq + jεreq ωC0 , 6 7 18 where C0 is the capacitance of the structure with airmedium. On comparing equations (6.7.17) and (6.7.18), the expressions for the real and imaginary parts of ε∗req are obtained in terms of the circuit elements: εreq ω = εreq

Cp2 C0

1 + ω2 τ1 τ1 + Rp1 Cp2 1 + ω2 τ1 + Rp1 Cp2

Cp2 ωRp1 Cp2 ω = 2 C0 1 + ω τ1 + Rp1 Cp2

2

a

,

6 7 17

6.7.4.2 Limiting Cases

The frequency ω 0 gives the static value of equivalent relative permittivity of the two-layered inhomogeneous medium. Whereas the ω ∞ gives its equivalent relative permittivity when interfacial polarization is lost, and relative permittivity is due to the remaining polarizations at a higher frequency. The limiting values of the equivalent relative permittivity are obtained from the equations (6.7.19a,b): εreq ω = 0 =

Cp2 C0 Cp2 C0

εreq ω

∞ =

εreq ω

∞ =0

a, 1 Rp1 Cp2 1+ τ1

εreq = 0

=

b

εreq ω = 0 Rp1 Cp2 1+ τ1

c d

6 7 20 2

b 6 7 19

The responses of Fig (6.14b) show that the εreq ω ∞ , with an decreases from εreq ω = 0 to εreq ω increase in frequency, whereas the εr eq ω exhibits a

197

198

6 Electrical Properties of Dielectric Medium

maximum absorption peak in the transition region of εreq ω . The frequency ωmax at which εreq, max ω occurs is computed on differentiating equation (6.7.19b) with respect to frequency ω and equating it to zero. The εreq, max ω is obtained at ω = ωmax: ωmax = εreq max εreq max

1 τ1 + Rp1 Cp2 Cp2 Rp1 1 ωmax = , 2 C0 τ1 + Rp1 Cp2 Cp2 1 ω = 2 C0 Cp1 + Cp2

a

τeq =

Rp1 Rp2 Rp1 + Rp2

τeq =

Cp1 + Cp2 1 1 + Rp1 Rp2

τeq =

also,

τeq =

Cp1 + Cp2 a

εr1 h2 + εr2 h1 σ 1 h2 + σ 2 h1

εr1 h1 + εr2 h2 σ1 h1 + σ2 h2

b

b 6 7 21

The equivalent circuit model of Fig (6.14a) can be identified with the two-layered medium of Fig (6.11a), that has σ2 = 0, as follows:

6 7 24 Equation (6.7.24b) is obtained with the help of equation (6.7.22). For the multilayer lossy dielectric layers medium, the equivalent relaxation time could be computed as follows: εri hi

Cpi

A A h1 Cp1 = ε , Cp2 = ε , Rp1 = , Aσ1 h1 1 h2 2 ε A Rp2 = ∞ , τ1 = r1 , C0 = ε0 , h = h1 + h2 , σ1 h

i

τeq =

1 Rpi

=

i

6 7 22 where A is the area of the parallel-plate capacitor.

The first circuit model of Fig (6.14a) corresponds to both lossy layers of Fig (6.11a). In this case, the steady-state complex relative permittivity can also be obtained, without ignoring Rp2. The admittance function for each parallel circuit is given below: 1 1 + jωτ1 Y1 = + jωCp1 = Rp1 R1

a

1 1 + jωτ2 + jωCp2 = Y2 = R2 Rp2

b

with the time constants; τ1 = Rp1 Cp1

c

τ2 = Rp2 Cp2

d

6 7 25

σi hi i

The expressions (6.7.24) and (6.7.25) for the equivalent relaxation time of the two-layered medium and the multilayered medium could be rewritten as, τeq =

6.7.4.3 Lossy Upper Layer

i

τ1 Rp1 + τ2 Rp2 Rp1 τ2 + Rp2 τ1 = 1 Rp1 + 1 Rp2 Rp1 + Rp2

a

τi Rpi τeq =

i

b

1 Rpi i

6 7 26 6.7.4.4 Equivalent Relative Permittivity

6 7 23 The inhomogeneous medium of Fig (6.11a) could be replaced by a homogeneous medium with an equivalent complex relative permittivity and an equivalent relaxation time. The equivalent relaxation time could be computed following the process discussed in the subsection (6.6.1). In this process, two parallel circuits shown in Fig (6.14a) are connected in parallel as shown in Fig (6.11b). Two parallel resistances are combined, and also parallel capacitors are combined to provide the equivalent relaxation time:

To obtain the equivalent relative permittivity of the composite medium, the input admittance of the circuit shown in Fig (6.14a) is extended to the multilayer case: Y1 Y2 , Y1 + Y2 1 Y= Rp1 + Rp2

Two layers Y =

1 + jωτ1 1 + jωτ2

a

1 + jωτeq

Π 1 + jωτi Multilayer Y =

i

1 + jωτeq

b Rpi i

6 7 27 The admittance, Y = j ω ε0 ε∗req C0 , of the equivalent single-layer complex capacitance is equated to the above two-layered admittance equation to obtain the complex equivalent relative permittivity:

6.7 Circuit Models of Dielectric Materials

Y = jωε0 ε∗req C0 = jωε0 εreq − jεreq C0

ε0 εreq ω = 0 =

= ωε0 εreq + jεreq C0 jωε0 ε∗req C0 =

a

a

1 + Rp2

Rp1

τ1 + τ2 − τeq C0 Rp1 + Rp2 τi − τeq

1 + jωτ1 1 + jωτ2

b,

i

ε0 εreq ω = 0 =

1 + jωτeq

C0

b

Rpi i

6 7 28

6 7 30

where C0 = Aε0/(h1 + h2) is the capacitance of the parallel-plate capacitor with air-medium. On separating the real and imaginary parts, the following expressions are obtained: ε0 εreq ω =

τ1 + τ2 − τeq + ω2 τ1 τ2 τeq C0 Rp1 + Rp2

ε0 ε req ω =

1 + ω2 τ2eq

1 − ω2 τ1 τ2 + ω2 τeq τ1 + τ2 ωC0 Rp1 + Rp2

1 + ω2 τ2eq

In the case of ω ∞, we get the high-frequency relative permittivity of two-layered and multilayered media: ε0 εreq ω

∞ =

τ1 τ2 C0 Rp1 + Rp2 τeq

a

ε0 εreq ω

∞ =

1 C0 1 Cp1 + 1 Cp2

b

Multilayer ε0 εreq ω

∞ =

ε0 εreq ω = εreq ω =

τ1 + τ2 − τeq ε0 εreq ω = 0 1 + ω2 τ2eq

1+ +

b

6 7 31

In the case of ω 0, the static relative permittivity is computed for both the two-layered and multilayer dielectric media:

C0 Rp1 + Rp2

1 1 Cpi

C0 i

6 7 29

ε0 εreq ω =

a

ω2 τ2eq

The dispersion relation, given by equations (6.7.29a), can be recast in terms of the limiting values of equivalent relative permittivity:

ω2 τ1 τ2 τeq

+

C0 Rp1 + Rp2

1+

ω2 τ2eq

×

τeq τeq

ω2 ε0 εreq ω = ∞ τ2eq 1 + ω2 τ2eq

εreq ω = 0 + εreq ω = ∞ τ2eq ω2 + εreq ω = ∞ − εreq ω = ∞

6 7 32

1 + ω2 τ2eq

εreq ω = εreq ω = ∞ +

εreq ω = 0 − εreq ω = ∞ 1 + ω2 τ2eq

Likewise, the εr eq ω , given by equation (6.7.29b), can be written as follows:

ε0 εreq ω = ε0 εreq ω =

εreq ω =

1 ω C0 Rp1 + Rp2

1+

ω2

τ2eq

1 ω C0 Rp1 + Rp2

εreq ω = ∞ τeq ωτ1 τ2

+

1 + ω2 τ2eq

+

ω τeq τ1 + τ2 − τ1 τ2 C0 Rp1 + Rp2

+

1 + ω2 τ2eq

ω ε0 εreq ω = ∞ τeq τeq τ1 + τ2 − τ1 τ2

εreq ω = 0 − εreq ω = ∞ 1 + ω2 τ2eq

τ1 τ2 1 + ω2 τ2eq ωτeq

6 7 33

199

200

6 Electrical Properties of Dielectric Medium

Equations (6.7.19a,b) are special case of equations (6.7.32) and (6.7.33) for the lossless upper layer, with Rp2 0. On comparing equations (6.7.32) and (6.7.33) with equation (6.4.15), it is seen that the εreq ω follows the Debye relaxation response. However, εreq ω has an additional term εreq ω = ∞ τeq

ω τ1 τ2 due to the

DC conductivity of the lossy dielectric layers. The equivalent DC conductivity σeq is given by εreq ω = ∞ τeq ε0

σeq =

6 7 34

τ1 τ2

The σeq is taken as σ in further discussion. The real and imaginary parts of the complex equivalent permittivity ε∗req ω can be combined to provide the following dispersion equation of Maxwell–Wagner polarization: ε∗req ω

= εreq ω = ∞ +

εreq ω = 0 − εreq ω = ∞ 1 + j ω τeq

−j

σ ω ε0 6 7 35

The Maxwell–Wagner polarization follows the Debye relaxation response. The above relation with DC conductivity does not satisfy Kramers–Kronig causal relations [J.44]. However, it can be further extended to more numbers of dielectric layers. The polarizability αint of the interfacial polarization can also be extracted as follows:

6.7.5

Series RLC Resonant Circuit Model

Lorentz type dielectric materials can be modeled by the RLC series resonant circuit, shown in Fig (6.15a). The RLC circuit can also be used to model the capacitor loaded wire-medium and split-ring resonator (SRR) for the realization of the generic model of dispersive effective relative permittivity and permeability. These are discussed in subsections (21.1.1) and (21.2.1) of chapter 21. One type of resonating polarization is modeled by one kind of RLC series circuit. Therefore, the multiple kinds of resonating polarizations in a material are modeled by a series combination of multiple series resonant circuits, shown in Fig (6.15b). At the frequency below their resonance frequency, the resonating elements contribute to the fixed value of relative permittivity. The following integrodifferential equation can be written from Kirchhoff’ law:

L

di 1 + Ri + idt = v t dt C

dQ d2 Q dQ Q , L 2 +R + =v t dt dt C dt For open switch at t = 0, v t = 0, As i =

Pinterfacial = Nαint Eext = ε0 ε∗req ω − εreq ω = ∞ αint =

and εr = 1 εreq ω ∞ . Equations (6.1.4) and (6.1.6) are valid for the polarization in a natural material, not for the polarization in the composite heterogeneous considered presently. The interfacial polarizability can be combined with another polarizability to get the total polarizability of an inhomogeneous dielectric-medium.

ε∗req ω = 0 − εreq ω = ∞ 1 + jωτeq

Eext

a

σ ε0 −j ωε0 N

for

b,

t>0 L

d2 Q dQ Q + =0 2 +R dt C dt

a b

c 6 7 37

6 7 36 where N is the density of mobile charge carrier. Equation (6.7.36a) is obtained by using equations (6.1.4), (6.1.6), and (6.1.7). In the process, the following replacement is also carried out: εr ε∗req ω = 0 ,

R

L

I

t=0

C

R1

C1

R2

L2

C2

R3

L3

I

V (a) One resonating circuit corresponding to one type of polarization. Figure 6.15

L1

Equation (6.7.37) describes the oscillating behavior of the circuit. Let its solution be Q(t) = Q0eKt. On substituting it in equation (6.7.37), the following expressions are obtained:

t=0

V (b) Three resonating circuits corresponding to three types of polarization.

Series resonant circuit for modeling of Lorentz type material.

C3

6.7 Circuit Models of Dielectric Materials

1 =0 K2 + γK + ω20 = 0 C where damping factor γ = R L LK2 + RK +

ω0 = 1

b

LC

Two roots of solutions K1,2 = −

c

γ γ ± j ω20 − d, 4 2 6 7 38 2

where, and, ωd =

K1,2 = −

j 2L Δω + jγ 2 γ 2 Y∗ = 2L Δω 2 + γ 2 2 j Δω + 2L Δω 2 + γ 2 2

ω20 −

b

a

b 6 7 43

γ2 4

c 6 7 39

In equation (6.7.39), ωd is the frequency of the damped oscillator. It is less than the natural frequency ωo of the undamped oscillation. Three cases of oscillation are discussed below: Case #I: If ωo > γ/2, i.e.1 LC > R 2L, the coefficients K1 and K2 are complex quantities, and the oscillatory charge decays as Q t = 2A e

Y∗ =

a

γ ± jωd 2

6 7 42

where ω and γ are given by equation (6.7.38). An approximation can be used near resonance Δω = ω0 − ω, ω ≈ ω0, ω + ω0 ≈ 2ω to get equation (6.7.43):

where ω0 is the natural frequency of oscillation. Thus, the solution of equation (6.7.37) is Q t = Ae + K1 t + Ae + K2 t

1 1 = Z∗ R + j ωL− 1 ωC jω L , Y∗ = 2 ω0 − ω2 + jω γ

Y∗ =

a

− 2γt

cos ωd t

6 7 40

Case #II: For ω0 < γ/2, there is no oscillation and the charge decays with two relaxations. Case #III: For ω0 = γ/2, i.e. K1 = K2, the charge decay is critically damped for ω0 = γ/2, giving the following expression: Q t = Ae − αt ,

α=

R 2 = 2L RC

6 7 41

The above behavior is similar to that of an RC series circuit, giving Debye kind of dispersion response. So the critically damped Lorentz material behaves as Debye material. However, its relaxation time is τ = 1/α = RC/2, instead of RC. On comparing equation (6.7.37) with equation (6.5.3), it is obvious that the charge Q on the capacitor is an analog of the polarization P. The damping factor and the natural frequency of the circuit are analog of the damping factor and the natural frequency of the oscillating polarizing element. Thus, the response of the circuit model could provide the relative permittivity of a material. The complex admittance of the circuit shown in Fig (6.15a) is

The admittance of the parallel-plate capacitor, containing the material with relative permittivity ε∗r = εr − jεr is Y∗ = jω ε0 εr − jεr C0 = ω C0 ε0 εr + jεr , where C0 is the capacitance of the parallel-plate capacitor without material. On comparing admittances of the circuit and material, the following expressions are obtained for the relative permittivity in terms of Lorentz resonator circuit: Δω ωε0 εr C0 = , 2L Δω 2 + γ 2 2 using K = εr ω = εr ω = Q=

1 2ωε0 C0 L Δω

2 Δω

KΔω + γ 2

2

Kγ + γ 2

2

a 2

2

ω0 L , for Lorentz oscillator R

b c d

6 7 44 The εr ω and εr ω obtained from the resonator circuit model is the same as that of obtained from the onedimensional molecular Lorentz oscillator given by equation (6.5.10). The circuit parameter K corresponds to the material parameter εrE. The εrT is added to equation (6.7.44) to get the relative permittivity due to all kinds of polarization processes, as shown in Fig (6.15b). The frequency responses of the real εr ω and imaginary εr ω parts of the complex relative permittivity are obtained from the measurements. Using these results, the values of R, L, C, and ω0 parameters of the resonator circuit model can be extracted. The circuit model also provides the frequency response of εr ω and εr ω , as shown in Fig (6.9b).

201

202

6 Electrical Properties of Dielectric Medium

A more general form of Debye and Lorentz dispersion models, using the time derivative of the external field, i.e. the polarizing field, has been developed using the small Hertzian dipoles as the inclusions in the host medium. By further loading, the Hertzian dipoles inclusions with simple resistance, RC and RL circuits, etc., and by loading it with a diode and operational amplifiers new kinds of artificial absorbing materials and smart materials have been reported. These materials are developed using the circuit models for more general polarization of the macromolecules of passive and active inclusions [J.33, J.45, J.46].

6.8 Substrate Materials for Microwave Planar Technology The static and frequency-dependent relative permittivities are discussed in the previous sections, for both the natural and artificial dielectric materials. The frequency dependence of relative permittivity of natural materials has been perceived using Debye and Lorentz type phenomenological polarization models. Further, these models have also been viewed through the equivalent circuit models. However, the substrate materials used in the microwave do not follow a single relaxation frequency of the Debye model; nor do they follow a single resonance frequency of Lorentz model; a real-life substrate material, say FR-4 of complex composition, can have several relaxations due to the presence of several polarizations. Even in the case of a single relaxation frequency, or a single resonance frequency, it is not easy to evaluate the characteristic parameters of both the models for a specific substrate material, whose frequency-dependent real or complex relative permittivity is known experimentally at few frequencies. Normally, to design and simulate components and circuits in the microwave planar technology, relative permittivity and loss tangent of a substrate are taken as constant, i.e. frequency-independent. However, it results in the noncausal response in the time-domain. It adversely influences the computations of the pulse propagating on interconnect in such a medium. Further, the noncausal constant permittivity model does not provide the correct group delay response over the broadband. The Debye type causal dispersion models are needed not only for the substrate materials but also for the composites and artificial dielectrics, for their application to the time-domain EM-solvers. The following two kinds of extended Debye models have been developed for their use in the EM-solvers:

••

Multiterm Debye models Wideband Debye model.

A summary of both models is presented below. The characteristic parameters of the single-term Debye and Lorentz models are also evaluated using experimental results. The first step to develop a Debye model for any substrate material is to get accurate experimental data on the frequency-dependent relative permittivity. It is normally obtained by using the cavity resonator method or an air-coaxial line. The microstrip line and patch resonator are also used for this purpose with some limitations [J.47–J.51]. The modeling process can be started from the experimental data of manufacturers of substrates. The engineered artificial dielectric materials, using nonresonant metallic inclusions in a host dielectricmedium, are discussed in subsection (6.2.2). However, resonant metallic inclusions embedded in the dielectric host can create high permittivity and high permeability artificial materials depending upon the type of the resonant inclusions [J.52–J.57]. Such magneto-dielectric materials are useful for the synthesis of engineered substrates, called the metasubstrates, useful for the development of compact microstrip based circuits and antenna [J.58]. A brief discussion of metasubstrates is presented at the end of the present section.

6.8.1 Evaluation of Parameters of Single-Term Debye and Lorentz Models 6.8.1.1 Single-Term Debye Model

The Debye model of a lossy dielectric material with single relaxation frequency fre can be written from equation (6.7.35), with some changes, as follows: Δεr σ −j 1 + j f f re 2πfε0 Δεr = εr ω = 0 − εr ω = ∞

ε∗r f = εr f = ∞ + where,

a b 681

In equation (6.8.1), relaxation frequency is fre = 1/τd and σ is DC conductivity of the dielectric-medium. So a lossy and dispersive dielectric material modeled by the Debye model is characterized by four parameters Δεr , f re , εr ω = ∞ , and σ . For known Δεr and εr ω = ∞ , the static relative permittivity εr ω = 0 can be easily obtained. The real and imaginary parts of a complex relative permittivity can be separated. These are evaluated at two frequencies f1 and f2 from the known experimental results. This arrangement provides the following four numbers of equations for

6.8 Substrate Materials for Microwave Planar Technology

4.35

Wideband Lorentzian Debye Manufacturer’s data

4.3 4.25 ε′r

Wideband Lorentzian with DC conductivity Debye without DC conductivity Debye with DC conductivity Manufacturer’s data

0.4

0.3 ε″r

4.2

0.2 4.15 0.1

4.1 4.05

1

0.1

10

100

0

1000

0.1

1

10

100

Frequency (GHz)

Frequency (GHz)

(a) Frequency-dependent real relative permittivity.

(b) Frequency-dependent imaginary relative permittivity.

1000

Figure 6.16 Response of Debye model for FR-4 substrate [J.59]. Source: Koledintseva et al. [J.59].

In equation (6.8.2), if f2 > f1, then εr1 f 1 > εr2 f 2 as the relative permittivity decreases with increasing frequency. Also, as the dielectric loss increases with frequency, so tan δ f 1 < tan δ f 2 , i e εr1 f 1 εr2 f 2 < εr2 f 2 εr1 f 1 . The experimental data must fulfill these physical conditions. The following expressions are obtained from the above equations, for the characteristics parameters of the Debye model; in terms of the experimentally known real and imaginary parts of the relative permittivity at f1 and f2 [J.48, J.49]:

evaluations of four numbers of characteristics parameters [J.48, J.49]: Δεr 1 + f 1 f re Δεr = εr f = ∞ + 1 + f 2 f re f 1 f re Δεr σ = 2 − 2πf 1 + f 1 f re 1 ε0

εr1 f 1 = εr f = ∞ +

2

a

εr2 f 2

2

b

εr1 f 1

εr2 f 2 =

c

f 2 f re Δεr σ 2 − 2πf 1 + f 2 f re 2 ε0

d 682

f re =

εr2 f 2 f 2 − εr1 f 1 f 1 εr1 f 1 − εr2 f 2

σ = 2πf 1 ε0 εr1 f 1 + εr f = ∞ =

f 1 f re ε f 1 − εr2 f 2 2 f 1 − f 22 r1

a

1+

f2 f re

εr1 f 1 − εr2 f 2 + εr1 f 1 f 1 f re 2 − εr2 f 2 f 1 f re 2 − f 2 f re 2

εr f = 0 = εr1 f 1 + εr1 f 1 − εr f = ∞

f 1 f re

The above parameters in equation (6.8.3a–d) are evaluated for the FR-4 substrate, using the manufacturer’s data. Figure (6.16a) shows Debye’s response for both

2

2

b 683 f 2 f re

2

c d

the real and imaginary parts of the FR-4 with one relaxation time. It also shows the manufacture’s data. The evaluated Debye model parameters are obtained as follows:

203

204

6 Electrical Properties of Dielectric Medium

εreq f = 0 = 4 301, εreq f = ∞ = 4 096, σ = 2.294 × 10−3S/m and τd = 2.294 × 10−11s. At the lower end of frequency, a large increase in εr ω is seen due to the presence of σ. If DC conductivity is ignored σ = 0, then only three parameters are evaluated for the single-term Debye model, namely Δεr , f re , εr f = ∞ . For both the measured value of relative permittivity and tanδ at frequency f1, and the known value of static relative permittivity at f = 0, Debye parameters could be evaluated to get the Debye model [J.60]. In the case of slowly varying relative permittivity in the relaxation region, the Cole– Cole and other versions of the Debye model can provide a better fit against the experimental results [B.21]. These variations can be attempted for the substrates and artificial dielectric materials.

In general, five numbers of characteristics parameters, namely εr f = ∞ , εr f = 0 , f 0 , fre, and σ, are to be determined. The following five numbers of equations are used with experimental results for three numbers of imaginary parts of the susceptibility and two numbers of real parts of the susceptibility at three frequencies: f1, f2, and f3: χr f 1 = χr f 2 = χr f 1 = χr f 2 =

6.8.1.2 Single-term Lorentz Model

Normally, the Lorentz model is applicable over a narrow band for dispersive dielectrics, such as a mixture of aluminum inclusions of 10–15 μm diameters in the polymer host with εr = 2.15. However, it could apply to the wideband dielectrics also, such as FR-4 [J.59], if the resonance frequency is very high and the Q-factor is less than unity. The single-term Lorentz model, with the single resonance frequency f0 (= ω0/2π), can be written, in the modified form, using an equation (6.5.8) and other expressions of section (6.5). The DC conductivity σ of the lossy dielectric material can also be included [J.48, J.49]. Δεr σ −j 2 2πfε0 1 − f f 0 + j f f re Δεr = εr ω = 0 − εr ω = ∞

ε∗r f = εr f = ∞ +

a

where,

b

f re = f 20 Δf

c 684

In equation (6.8.4), the relaxation frequency parameter fre is defined in terms of 3dB bandwidth Δf of Lorentz resonance at f0. The above expression of complex relative permittivity can be written in terms of the frequencydependent complex susceptibility; its real and imaginary parts can be separated: ε∗r f = εr f = ∞ + χ∗r f

χ∗r

a

f = χr f − jχr f

χr f = χr f =

b

Δεr 1 − f f 0

2

2

2

2 2

2

1 − f f 0 2 + f f re Δεr f f re 1 − f f0

+ f f re

c −j

σ 2πfε0

d 685

χr f 3 =

Δεr 1 − f 1 f 0 1 − f1 f0

2 2

2

+ f 1 f re

Δεr 1 − f 2 f 0

a

2

2

b

2

2

2 2

+ f 1 f re 1 − f1 f0 Δεr f 2 f re

2

−j

σ 2πf 1 ε0

c

2 2

+ f 2 f re 1 − f2 f0 Δεr f 3 f re

2

−j

σ 2πf 2 ε0

d

2 2

2

−j

σ 2πf 3 ε0

e

1 − f 2 f 0 2 + f 2 f re Δεr f 1 f re

1 − f3 f0

+ f 3 f re

686 In case the DC conductivity is ignored, i.e. σ = 0, equation (6.8.6e) could be dropped. The measured susceptibility, both real and imaginary, are needed only at two frequencies. In case, only the susceptibility parameters are needed, three characteristic parameters Δεr , f 0 , and f re could be determined to get the single-term Lorentz model. A set of three equations for the measured imaginary parts of the susceptibility, at three frequencies f1, f2, and f3, are needed to evaluate these parameters. In this case again σ = 0, and equations (6.8.6c) and (6.8.6d) could be dropped. The system of nonlinear equations can be solved using the MATLAB program. While evaluating the parameters, these parameters are taken as the positive real quantities and measurement frequencies must be both above and below the resonance frequency, i.e. f1 < f0 < f2, or f2 < f0 < f3. If the experimental data are χ1 5GHz = 1 2, χ2 8GHz = 2 0, and χ3 15GHz = 0 8 , then the evaluated parameters are Δεr = 3.82, f0 = 10.1 GHz, and fre = 99.03 GHz, which corresponds to Δf = 1.03 GHz. For three numbers of experimental data on FR-4 given in Fig (6.16b), the evaluated parameters are εreq f = 0 = 4 301, εreq f = ∞ = 4.096, σ = 2.294 ∗ 10−3S/m, f0 = 39.5GHz, Δf = 200GHz. These parameters are used to develop the single-term Lorentz model for both the real and imaginary parts of the complex relative permittivity. Figure (6.16) shows the results of the Lorentz model which are almost identical to that of the Debye model. In the present case, due to high resonance frequency and very wide bandwidth, the Q-factor of

ε′r

8

5

10

6

4

8

3

6

2

4

1

2

4 2

10.4

tanδ (×10–3)

10.5

ε′r

0 1

6

11 16 21 26 Frequency (GHz)

31

0

36

6

7

(a) Debye model for Duriod-6010.

9 8 Frequency log10(f)

10

tanδ (×10–2)

6.8 Substrate Materials for Microwave Planar Technology

0

(b) Debye model for FR-4.

Figure 6.17 8 terms Debye models for relative permittivity and loss-tangent. Source: Zhou and Melde [J.61].

a

In equation (6.8.7), εr ω = ∞ is the relative permittivity at high frequency. The incremental relative permittivity Δεri and ωrei are the polarization strength and relaxation frequency of the ith Debye type polarization. Further, σ is DC conductivity that influences very low frequency εr ω . It is dropped for many cases or is known from the low-frequency measurement. N = 2 to 8 numbers of Debye terms are normally used to accurately curve-fit the experimental results. For the assumed value of N, the system of nonlinear equations is solved in MATLAB to get the parameters Δεri and ωrei [J.61]. However, the empirical approach has also been applied to the FR-4 substrate by taking N = 8 [J.37]. The real εr ω of FR-4 shows a linear dependence of frequency, on the logarithm scale. One-term per decade of frequency could be taken. The total experimentally obtained real part of relative permittivity is divided by N to get the uniform value of Δεri at each of i=1 to 8. Next, each of eight numbers of Debye polarization strength Δεri could be fine-tuned to curve-fit the model to experimental results. However, the genetic algorithmbased optimization process appears to be more efficient and systematic for the evaluation of these parameters. Figure (6.17) shows the results for the eight-term Debye model for both RT–Duriod 6010 and FR-4 substrates, using the genetic algorithm approach. At 10 GHz, these results follow closely the manufacture’s results [J.48]. Both dielectric substrates have a substantial difference in their frequency response due to the different nature of polarizations involved.

b

6.8.2.2 Wideband Debye Model

resonance is less than unity and the Lorentz model applies to the wideband dispersion [J.59]. 6.8.2

Multiterm and Wideband Debye Models

The single-term Debye model is not sufficient to model the frequency-dependent real and imaginary parts, or the loss-tangent, accurately over a wider frequency range, as the substrate materials have several polarizations with different relaxation times. The multiterm or the wideband Debye models are needed to improve the accuracy of the curve-fitting of the measured dispersion results [B.21]. It involves several more characterizing phenomenological parameters and several approaches have been used to evaluate these parameters from the experimental results [J.47–J.49, J.59– J.63]. Such models are also accommodated in the modern EM-Simulators that require only a few experimental data to generate Debye models. A brief introduction to such Debye models is given below.

6.8.2.1 Multiterm Debye Model

Debye model of expression (6.8.1) can be extended to the multirelaxation frequency materials. It provides frequency-dependent relative permittivity; as a superposition of Debye type relaxations due to N number of polarizations: ε∗r ω = εr ω = ∞ +

N

Δεri σ −j 1 + j ω ω ωε rei 0 i=1 N

Real part εr ω = εr ω = ∞ +

Δεri i = 1 1 + ω ωrei

2

N

ω ωrei Δεri σ Imaginary part εr ω = 2 − ωε0 i = 1 1 + ω ωrei

c

687

Both the relative permittivity and loss-tangent, say for FR4, show slow variations over a very large frequency range. This makes discrete multiterm Debye modeling a laborious process. The relaxation can be treated as a distributed function and replace the summation in the above

205

206

6 Electrical Properties of Dielectric Medium

formulation by an integral in terms of the lower (τ2) and upper (τ1) relaxation time [B.21]. In the case of the logarithmic variation, Frohlich distribution is used to get the εr ω = 0 − εr ω = ∞ ln τ1 τ2 ε ω = 0 − εr ω = ∞ εr ω = εr ω = ∞ + r ln τ1 τ2 ε ω = 0 − εr ω = ∞ 1 + jωτ1 arg εr ω = r 1 + jωτ2 ln τ1 τ2 ε∗r ω = εr ω = ∞ +

ln ln

1 + jωτ1 1 + jωτ2 1 + jωτ1 1 + jωτ2

= εr ω = ∞ + a

τ2 1 + ω2 τ21 1 ln 22 τ1 1 + ω2 τ22 2

Δεr m2 − m1

ln

, τ1 > τ2

a b

688

c

In the place of Frochlich distribution function, hyperbolically decaying relaxation function has been used to get the following Debye model [J.64]:

− j tan − 1 ωτ2 − tan − 1 ωτ1 , τ2 > τ1

The four characteristic parameters,(τ1), (τ2), a, and εr ω = ∞ are evaluated by the trial and error process to curve-fit the measured results, for both the real and imaginary parts of the complex permittivity. In the case of the FR-4 substrate, the curve-fitted parameters are εr ω = ∞ = 4 2, a = 0.06, τ1 = 1.6 ps, and τ2 = 1.6 ms. The model also helps to compute the frequency-dependent loss-tangent with an increasing trend as a function of frequency. The above expressions are valid for wideband applications. However, the evaluation of four numbers of characteristic parameters from the experimental is not a direct process. Another closed-form wideband Debye model for FR-4 substrate has also been suggested that is simple to use. This model also considers continuously varying relaxation frequency. The polarization strength Δεri is replaced by the average value of the change in the relative permittivity between two assumed frequency limits in the logarithmic scale. The change in relative ε∗r ω = εr ω = ∞ +

τ2 τ1 τ2 τ1

τ2 τ1

Unlike the multiterms Debye model, the above model evaluates only four characteristic parameters, τ1, τ2, εr ω = 0 , and εr ω = ∞ , from the experimental results, using the genetic algorithm-based optimization [J.61].

ε∗r ω

following Debye expression for the frequency-dependent complex relative permittivity of material with continuously variable relaxation time [J.61]:

ω2 + jω ω1 + jω ln 10

689

permittivity at two frequencies is known from the experimental results. The lower and upper frequencies are taken as ω1 = 10m1 and ω2 = 10m2 , m2 > m1 . The average value is Δεri = Δεr m2 − m1 . So the summation over the N relaxations of equation (6.8.7a) can be replaced by an integral, where relaxation frequency is substituted by another frequency variable [J.47]: On taking ωrei = 10x , m1 ≤ x ≤ m2 , ω1 fc 7 4 27

fc f

2

− 1,

f < f c,

where k = β is the propagation constant of the EM-wave in the homogeneous medium. Under the condition f > fc, the propagation constant βmn z = +β

Modal Propagation Constant

2

2

f = fc jβ

The propagation constant and the wave impedance of modes are examined below for understanding the behavior of modes.

fc f

0,

Characteristics of Modes

mπ a

1−

mπ 2 nπ 2 + = ωc με a b mπ 2 nπ 2 + a b mπ a

2

nπ + b

a

1 − fc f

2

is used for the propagating wave in the forward z-direction, whereas under the condition f < fc, the βmn z = − jβ

fc f

2

− 1 acts an attenuation constant

and provides attenuation to the decaying evanescent wave in the forward z-direction. The cut-off occurs at the frequency f = fc. The wavelength of mnth mode λmn g , known as the guided-wavelength, is given by λmn g =

λ=

2π = βmn z λ0 εr μr

2π β

fc 1− f

2

=

λ fc 1− f

2

a

b, 7 4 28

where λ is the wavelength of the EM-wave in a homogeneous medium of the waveguide. The guidedwavelength is a real number for the propagating mode and is an imaginary number for the nonpropagating evanescent mode. It is infinite at the cut-off frequency. Wave Velocities

2

b, 7 4 26

Following the case of a plasma medium with cut-off phenomenon presented in equations (3.3.8) and (3.3.13) of chapter 3, the phase and group velocities of the wave in a waveguide could be obtained as

227

7 Waves in Waveguide Medium

fc vp = v 1 − f fc vg = v 1 − f vg × vp = v

2

−1 2

a,

vp =

c,

vg =

λmn g λ

2 1 2

2

λ λmn g

η v

b

v

d

fc 1− f Zzw TEzmn =

e

In the above equations, the velocity of the EM-wave in the homogeneous medium inside to a waveguide is v = c εr μr , and the corresponding wavelength is λ = λ 0 ε r μr .

Modal Wave Impedance

The concept of the characteristic impedance of a transmission line is discussed in subsection (2.1.3) of chapter 2. It explains the wave phenomenon on a line. It also helps to treat a section of the transmission line as a circuit element. For a similar purpose, the concept of wave impedance of the TEz and TMz modes, i.e. Zzw TEzmn and Zzw TMzmn , respectively, are introduced. The wave impedance is defined as a ratio of the transverse field components. For the TEzmn mode, the wave impedance is Zzw TEzmn =

Ey Ex ωμ = = mn Hy − Hx βz

7 4 30

z Z TE

0

z ZTM

Evanescent mode

η

Reactive region

1

z Z ΓE

f/fc

Propagating mode

Real region

(a) Wave impedance of non-propagating and propagating regions and also at the cut-off frequency. Figure 7.6

,

∞,

f > fc

f = fc fc f

,

2

Wave impedance behavior and inductive surface.

7 4 31

f < fc ,

−1

μ μr is the intrinsic impedance of = η0 εr ε the EM-wave propagating in a homogeneous medium of the waveguide and η0 = 120π is the intrinsic impedance of the free space. The wave impedance of the propagating TEzmn, mode under the condition f > fc, is a real quantity providing the propagating wave. The wave impedance is higher than the intrinsic impedance of the medium inside a waveguide. For the evanescent mode under the condition f < fc, the wave impedance Zzw TEzmn offers an inductive reactance that reflects the power supplied to the waveguide. So the TE mode waveguide, below cut-off frequency, could be viewed as the ENG metamaterial medium discussed in the subsection (5.5.3) of chapter 5. At the cut-off f = fc, the wave impedance is infinitely large creating the artificial magnetic conductor (AMC) surface with in-phase reflection. The AMC surface is discussed in section (20.2) of chapter 20. Figure (7.6a) shows the impedance behavior of the TE mode waveguide. Figure (7.6b) shows the realization of the inductive surface created by an array of TEzmn mode waveguide for the operation below cut-off frequency f ≤ fc. where η =

Real wave impedance

Inductive

Wave reactance

The following expression is written from equations (7.4.27) and (7.4.30):

z Z TE

2

+ jη 7 4 29

Capacitive

228

y x

(b) Realization of an inductive surface using an array of z TEmn mode waveguide below the cut-off frequency.

7.4 Rectangular Waveguides

Likewise, the following wave impedance of the TMzmn mode is obtained: Zzw TMzmn =

βmn z ωε η

Zzw TMzmn =

a 1−

fc f

2

,

0, − jη

f > fc f = fc

fc f

b

2

− 1,

f < fc 7 4 32

It is obvious that below the cut-off frequency, i.e. for f < fc the wave impedance of the TMzmn mode offers capacitive reactance. It reflects the TMzmn mode. So the TM mode waveguide, below cut-off frequency, could be viewed as the MNG metamaterial medium discussed in subsection (5.5.3) of chapter 5. Likewise, at the cut-off frequency f = fc, the wave impedance of TMzmn mode is zero creating the artificial electric conductor (AEC) surface exhibiting out-of-phase reflection. It discussed in section (20.2) of chapter 20. The nature of the TE/TMmode rectangular waveguide, above and below cut-off frequency, is further validated with the help of the circuit model in subsection (7.6.2). The TEzmn and TMzmn modes show the high pass filtering property. The wave impedance of both modes has opposite behaviors as shown in Fig (7.6a). At the cutoff frequency, the wave impedance for the TEzmn mode is very large, showing the open circuit behavior, whereas for the TMzmn mode it is zero, showing the short-circuit behavior. At high frequency, the wave impedances move toward η, i.e. toward the intrinsic impedance of the homogeneous medium inside the waveguide. For the TMzmn modes, the arrangement of Fig (7.6b) creates a capacitive surface.

Modal Fields

Figure (7.7) shows the modal field for a few TEzmn modes. The integers “m” and “n” denote half guidedwavelength variations of the field along the x and y-axes of a waveguide, i.e. along the broad-face (width) and narrow-face (height) of the waveguide. The modal field pattern repeats itself along the direction of propagation, i.e. in the z-direction, at every λg/2 distance. The upper part of Fig (7.7a) shows the E and H-fields distribution of the fundamental TEz10 mode. The E-field lines are shifted by λ/4 (90 ) with respect to the center of the closed-loop of the H-field lines.

The conduction current and its continuity to the displacement is shown in the lower part of Fig (7.7a). The conduction current flows on the inner surface of the broad-face of a waveguide. Further, electric displacement current, entering toward the lower broad-face of the waveguide, completes the current loop. The electric displacement current, generated by the time-varying electric field, is at the center D of the loop of the magnetic field. It shows λg/4 shifting from the location of maxima of the electric field, because of the presence of “j” with the displacement current. It indicates the π/2 phase-shift of the displacement current. Figure (7.7b) shows the E and H-fields distribution of the TEz20 and TEz30 modes. The width “a” accommodates 2 and 3 numbers of TEz10 to form the TEz20 and TEz30 mode, respectively. Each TEz10 mode, constituting TEz20 and TEz30 modes, is separated by the EWs. It is noted that at the plane of the EW, the total electric field is zero and the magnetic field is maximum. Figure (7.7c–e) show the electric variations of three TEzm0 modes (m = 1,2,3). The location of the maximum of the electric field, along the broad-face, is mode dependent and repeats itself in the z-direction. The maximum of the E-field for the TE10 mode is at the center of the waveguide. Also, the cut-off frequency of TEz30 mode is equal to the cut-off frequency of TEz10 mode of the width a/3 of the original waveguide. The TEz10 mode is the fundamental mode, with the lowest cut-off frequency. There is no field variation for the TEzm0 mode along the height (y-direction) of a waveguide. Figure (7.8) shows field lines of the TMz21 mode. The broad-face view shows the presence of a closed-loop of the electric field that is created by the magnetic displacement current at the center of the electric field loop. The E-field lines start at the positive charges on the waveguide wall and after bending end at the negative charges. The pattern is repeated along the z-axis. The magnetic displacement current is generated by the time-varying magnetic field. Surface Current on the Waveguide

The EM-wave in a waveguide induces a current on its inner surfaces. The surface current density is related to the tangential component of the magnetic field on a surface, J s = n × H t,

7 4 33

where n is the unit vector, normal to the conducting surface. The current density is useful for the computation of conductor loss on the walls of a waveguide. It is computed for a perfectly conducting surface. However, an

229

230

7 Waves in Waveguide Medium

Ey

b a

a Y Ey X (c) TE10 mode E- field variation in cross-section. Conduction current D

a

D

X

D

Ey

b

X

Ey a

Z 0

Y λg/4

λg/2

3 λg/4

λg

Ey

5 λg/2

Ey X (d) TE20 mode E-field variation in cross-section.

(a) TE10 mode- upper figure --- H-field, (⚫, ×) E-field arrow. lower figure conduction current, (⚫, ×) displacement current arrow.

Ey

a/2 b EW

Ey

Ey

EW

a/2

a Y Ey

a/3

Ey Ey

EW

EW (e) TE30 mode E- field variation in cross-section.

a/3 EW

EW a/3

(b) TE20 and TE30 modes. Figure 7.7

Field and current distribution of TEzm0 modes (m = 1,2,3).

X

7.4 Rectangular Waveguides

–

× -H lines

+

–

+

×

× ×

×

×

× ×

×

×

×

×

×

×

×

×

×

×

×

×

×

×

×

×

×

×

×

×

×

×

×

×

×

×

× × × ×

X × ×

X

λg/4

0

Y

λg/2

Rightside figure,

Leftside figure --- H-field lines × , : Electric displacement current

×

× ×

×

× ×

×

λg Z

3 λg/4

(•, × ) H-field arrow,

E-field lines

: Magnetic displacement current.

,

Figure 7.8 Field lines of TMz21 mode.

actual conducting surface has finite conductivity that influences the computation of conductor loss. Let us consider the surface current distribution of the TEz10 fundamental mode, for m = 1, n = 0 on inner surfaces of a perfectly conducting waveguide. The following field components for TEz10 mode are obtained from equation (7.4.19): m Em x = Ez = 0,

Em y =

−π π B10 sin x e − jβz z a a

π B10 βz π sin x e − jβz z a ωμ a B10 2 π m k − β2z cos x e − jβz z Hm y = 0, Hz = − j ωμ a Hm x =

where k2 − β2z = k2c = π a

2

The surface current density is easily obtained at all four inner conducting surfaces of the waveguide shown in Fig (7.5a). Left hand sidewall at x=0: Hm y

= − y Hm z Js=yj

m J s x = a, 0 ≤ y ≤ b = − x × y Hm y + z Hz

= y

Hm z

x=a

x=a

B10 2 k − β2z e − jβz z J s = −y j ωμ = −y j

B10 π 2 − jβz z e ωμ a

b

The bottom wall at y=0: m J s 0 ≤ x ≤ a, y = 0 = y × x Hm x + z Hz

− z Hm x

=

+x

Hm z

y=0 y=0

j B10 π π x e − jβz z J s = −x cos a ωμ a B10 βz π π sin x e − jβz z −z ωμ a a 2

7 4 34

J s x = 0, 0 ≤ y ≤ b = x × y

Right-hand sidewall at x=a:

+

z Hm z

The upper wall at y=b: J s 0 ≤ x ≤ a, y = b = − J s 0 ≤ x ≤ a, y = 0

d

7 4 35

x=0

The current distribution on the upper wall of the inner surface is sketched in Fig (7.7a).

x=0

B10 2 k − β2z e − jβz z ωμ B10 π 2 − jβz z =yj e ωμ a

c

EM-wave Power Transfer in Waveguide

a

The EM-wave power in a waveguide is transported in forms of the power associated with the TEzmn and

231

232

7 Waves in Waveguide Medium

TMzmn modes. This is computed by using the Poynting vector relation,

a

cos

2

0 ∗

S = E ×H ,

7 4 36

a

sin 2

∗

where H is the complex conjugate of the complex vector H . The powers contained in the TEzmn and TMzmn modes are computed separately, as these modes exist independently of each other.

0

ε0p =

a , 2 a, a , 2 0,

mπ x dx = a mπ x dx = a 1, p = 0 2, p

0

,

m

0

a

m=0 m

0

b

m=0

p = m or n

c 7 4 40 2

TEzmn Mode

The power density of TEzmn mode transported in the zdirection is ∗ 1 Re E × H z TEzmn 2 1 m∗ m m∗ = Re Em x Hy − E y Hx 2

PTE mn = Bmn

2

βz a 2 ωμ ε0m

b β2 ε0n c

= Bmn

2

β a 2 ωμ ε0m

b f mn β2c 1 − c f ε0n

PTE mn

= Bmn

2

ω με a × 2 ωμ ε0m

Bmn = 2η

2

PTE mn

STE z =

7 4 37

On using an equation (7.4.19) with equation (7.4.37), the following expression is obtained: nπ mπ nπ Bmn cos x sin y e − jβz z b a b

STE z =

+

mπ mπ nπ Bmn sin x cos y e − jβz z a a b

×

mπ ∗ βz mπ nπ sin B x cos y ejβz z a mn ωμ a b

Bmn 2 βz 2 ωμ mπ + a

nπ b 2

sin 2

2

cos 2

mπ nπ x sin 2 y a b

7 4 38 The mode power crossing the cross-section of the waveguide is b a

= Bmn

0 0 2

ε0m

b ε0n

β2c

β2c

STE z dxdy

βz 2 ωμ

PTE mn = Bmn

nπ b 2

2

a ε0m

βz a 2 ωμ ε0m

b ε0n b ε0n

+

mπ a mπ a

2

a ε0m

2

+

b ε0n

nπ b

12

f mn 1− c f

f mn 1− c f

2

12

12

2

7 4 41

TMzmn Mode

The power carried by the waveguide in the TMzmn mode is also computed similarly [B.5], PTM mn

Amn 2 ηβ2c a = 2μ2 2

b 2

f mn 1− c f

2

12

7 4 42

mπ nπ x cos 2 y a b

TEzmn

PTE mn =

a

b ε0n

2

nπ ∗ βz mπ nπ B x sin y ejβz z × cos b mn ωμ a b

1 Re 2

STE z =

2

+ nπ and Using the cut-off wavenumber, β2c = mπ a b equation (7.4.27), equation (7.4.39) is written as

2

,

7 4 39 where the following expressions are used in equation (7.4.39):

The total power transmitted in the waveguide is a sum of power carried by both the TEzmn and TMzmn modes. Attenuation

The conducting walls of a waveguide are not the perfect ones. They have finite conductivity. The modal EMfields induce the modal surface currents that cause the conductor loss in a waveguide. Once the conductor loss is computed on the four walls of a waveguide, its attenuation constant αc, due to the conductor loss, is computed easily. Only the case of the TEz10 fundamental mode is considered below. Conductor Loss

The total conductor loss per unit length is twice the sum of conductor loss on one broad-face and one narrow-face of the waveguide:

7.4 Rectangular Waveguides

10 PTE z c

= 2 Pc z

+

x=0

Loss at narrow-face

2 Pc z

y=0

Loss at broad-face 7 4 43

The conductor loss of the fundamental mode TE10 at the left-hand side narrow-face of the waveguide, shown in Fig (7.5a), is computed using equation (7.4.35a):

π 2π 2π f 10 c = 2π = βc = = v a λc 1 a με = 2 f 10 c

1 Rs Js x = 0 2

=

x=0

2

10 PTE z = c

dy

Rs B210 π ω2 μ2 a

4

Rs B210 π ω2 μ2 a

=

0 b

Rs B210 π 4 dy = 2 ω2 μ2 a 10 PTE c

z

x=0

=

0 Rs B210 b 2 ω2 μ2

π a

10 PTE c

a 4 π2 f 2 1 2 π2 4 f 10 c

b+

=

Rs B2 π z = 2 10 ω μ2 a

a 10 z = PTE c

Rs B210 π π2 f 2 μ a

10 z = Rs B210 PTE c

1 Rs Js y = 0 2

2

4

2

f

b+

a 2

f 10 c 2

a f 2 f 10 c

2

f 10 c f

1+

2

2b a 7 4 47

a y=0

4

4

The conductor loss at bottom broad-face, using equation (7.4.35c) is 10 PTE z c

7 4 46

On using equation (7.4.46) with equation (7.4.45), the following expression is obtained:

b 10 PTE z c

με f 10 c

4

ε π μ a

a 2 2 f a ε 1+ 2 2

a 1+ 2

fc f

fc f 2

dx

2

2b a

2b a 7 4 48

0 a

Rs = 2

The power P0 at the location z = 0 decreases to the Pmn (z) at the location z by the following relation:

Js y = 0 J∗s y = 0 dx 0

a

Rs = 2 0

B210 π ω2 μ2 a

cos

2

π a

2

a

Rs B210 π = 2 ω2 μ 2 a 10 z PTE c

4

2

− 2αc z PTE , mn z = P0 e

π B2 π 2 2 π x + 2102 β sin 2 x a a ωμ a z cos 2

π π x + β2z sin 2 x a a

dx

dx

0 y=0

=

Rs B210 π 2 ω2 μ 2 a

2

π a

a 2

2

+ β2z

b

Rs B 2 π = 2 10 ω μ2 a 10 PTE c

2

π 2 a π b+ a 2 a

π 2 a π b+ a 2 a

2

z =

Rs B210 ω 2 μ2

=

Rs B210 ω 2 μ2

π a π a

2

4

2

2

+

PTE 10 z = 2αc PTE 10 z dz PTE10 z αc = c TE 2P10 z

7 4 50

The attenuation constant for the TEz10 mode is obtained by substituting equations (7.4.41) and (7.4.48) in equation (7.4.50):

The total conductor loss p.u.l. is Rs B210 π ω2 μ2 a

where αc is the attenuation constant due to conductor loss. The loss of transmitted power p.u.l. is equal to the conductor loss p.u.l. for the TEz10 mode, 10 PTE z = − c

7 4 44

10 PTE z = c

7 4 49

a 2 β 2 z

a π ω2 με − + 2 a

Rs B210

2

π 2 a b + ω2 με a 2 a 2 a 2 b + ω με 2 π

10 αTE = c

ε π μ a

B210 a π b η 2 a 1+

7 4 45 The following expressions are used to simplify equation (7.4.45) for the conductor loss:

=

Rs bη

2

a 2b f 10 c 1+ f 2 a 2

1−

2b f 10 c f a

f 10 1− c f

f 10 c f

2

2 1 2

2

2 1 2

Np m, 7 4 51

233

Conductor loss (αc) (Np/m)

7 Waves in Waveguide Medium

0.5 0.2 0.1 TM11

0.05 0.02

TE11

TE20

0.01

TE10

5

10

50 20 Frequency (GHz)

100

200

(a) Conductor loss of fundamental and higher-order modes. 0.050 0.045 0.040 0.035 0.030 0.025 0.020 0.015 0.010 0.005 0.000

Conductor loss (αc) (Np/m)

Conductor loss (αc) (Np/m)

234

0

20

40 60 Frequency (GHz)

80

100

(b) Conductor loss of TE10 mode in the propagation region using two methods [J.1]. Figure 7.9

29.5

Purturbation method

24.5 19.5

Exp

14.5 9.5

Boundary matching method

4.5 –0.5 11.45 11.47 11.49 11.51 11.53 11.55 11.57 Frequency (GHz) (c) Conductor loss of TE10 mode in cut-off region using two methods [J.1].

Conductor loss for the TEz10 and higher-order modes. Source: From Yeap et al. [J.1]. Public domain.

where the surface impedance of conductor is Rs = π f μcond σcond (for a conductor μcond = 1). The cutoff frequency of the TEz10 mode is given by equation (7.4.26). The intrinsic impedance of the material medium in a waveguide is η = η0 εr . Likewise, the conductor loss of the TM11 mode is obtained:

11 αTM c

34.5

2 Rs = ηab

b a + 2 a2 b f 11 1− c f

2 1 2

Np m 1 1 + 2 a2 b

Figure (7.9b) confirms the accuracy of the conductor loss computation of the TEz10 -mode of the air-filled waveguide with a = 1.3 cm and b = 0.64 cm, in the propagation region, using the present approximate perturbation method, as against the measurement results and the results of the boundary matching method. However, Fig (7.9c) shows that the perturbation method does not work in the cut-off region. The boundary matching method shows more accurate loss computation in the cut-off region also [J.1–J.3]. Dielectric Loss

7 4 52 The general expression for the conductor loss is obtained for both the TEmn and TMmn modes [B.5]. Figure (7.9a) shows the plot of conductor loss for the TEz10 and TMz11, and other higher-order modes for a waveguide with a cut-off frequency of the fundamental mode TEz10 about 5 GHz. The fundamental mode TEz10 is least lossy, as compared to other higher-order modes. At the cut-off, all modes show a sharp increase in the loss.

If the waveguide is loaded with a lossy dielectric material, with the complex relative permittivity ε∗r = εr − j εr , the propagating EM-wave modes suffer the dielectric loss. So the dielectric loss is modedependent and shows the cut-off behavior, like the conductor loss. The dielectric loss can be computed from the complex propagation constant using the dispersion relation (7.4.18b). It is reproduced below βz =

k2 − β2c ,

7 4 53

7.4 Rectangular Waveguides

where the lossy dielectric medium has the wavenumber k: μr ε∗r = k0

k = k0

μr εr − jεr

7 4 54

k20 μr εr − jεr − β2c =

βz =

βz =

k20 μr εr − β2c − jk20 μr εr

k 2 − β2c − jk20 μr εr ;

k20 μr εr k 2 − β2c

1−j

βz =

k 2 − β2c

βz ≈

k 2 − β2c − j

k20 μr εr k 2 − β2c

2

μr εr

Where, k = k0 1 2

= βd − jαd 7 4 55

The propagation constant in the dielectric medium of the waveguide is βd = Re βz = = k0

μr εr

k 2 − β2c 1−

fc 2 f

7 4 56

rad m

The attenuation constant, due to the lossy dielectric, in the dielectric medium of the waveguide is k20 μr εr

αd = Im βz = 2 αd =

k

2

− β2c

ω μ0 ε0 εr μr 2

εr

=

k20 μr εr 2k0 μr εr

1 1−

fc f

fc 2 f

μ ε

1 1−

fc 2 f

=

η 2

σd 1−

fc 2 f

Hz = 0, i e ϕm = 0 at x = 0, a; 0 ≤ y ≤ b 7 4 59

Hz = 0, i e ϕm = 0 at y = 0, b; 0 ≤ x ≤ a 7 4 60

Np m 7 4 58

where σd is the conductivity of the imperfect dielectric medium. 7.4.2

The TEz mode has Hz component with Ez = 0, and all other field components are generated by scalar magnetic potential ϕm x, y, z = ϕm x, y e − j βz z given by equation (7.4.1).Wave equation (7.4.2) for the potential function ϕm(x, y) is solved under the following boundary conditions on the MWs, obtained from equation (7.1.35).

ii) Boundary condition on the bottom and top walls: 7 4 57

ωε αd = 2

TEz Mode

i) Boundary conditions on left and right-hand sidewalls: 2

1 1−

not to discuss the detailed field solutions with the decaying field in the air-medium. We are interested in treating the rectangular dielectric slab waveguide as a complementary structure of the metallic waveguide, shown in Fig (7.5a). The electromagnetic field is assumed to be confined inside the dielectric waveguide and there is no exponentially decaying field in the air region at the surface of the dielectric waveguide. Such a simplistic field confinement model of the dielectric waveguide is achieved by assuming that the surface of the rectangular waveguides is formed by the MWs, i.e. by the perfect magnetic conductors (PMC), as against the EW (PEC) surfaces of the metallic rectangular waveguide. The TEz and TMz modes for the dielectric slab waveguide are obtained below under the condition of four MWs.

Rectangular Waveguide with Four MWs

Figure (7.5b) shows a rectangular waveguide made of high permittivity perfect dielectric slabs. Normally, this waveguide has the field confinement in two transverse directions, as the fields show exponential decay in the air-medium adjacent to the dielectric slab. Marcatili [J.4] has obtained the approximate solution of the field problem related to the dielectric slab structure. The slab structure supports the hybrid mode with respect to the z-direction of propagation. However, our concern is

Following the method of separation of variables discussed for the metallic rectangular waveguide, the following modal solution is obtained from wave equation (7.4.2) for the scalar magnetic potential ϕm(x, y): mπ nπ x sin y ; a b m = 1, 2, 3…; n = 1, 2, 3 …

ϕm mn x, y = Amn sin

7 4 61 ϕm mn

x, y , the single superwhere for the eigenfunction script “m” indicates the magnetic scalar potential and the paired subscripts (“m, n”) show the integral modal numbers, indicating the numbers of half-wavelength variations of the fields along the x- and y-axes, shown in Fig (7.5b). The modal field components are obtained by using an equation (7.4.61) with equation (7.1.35):

235

236

7 Waves in Waveguide Medium

nπ mπ nπ sin x cos y b a b mπ mπ nπ x sin y Em cos y = Amn a a b − βz ∂φm mn Hm x = ωμ ∂x Em x = − Amn

a b

Hm y = = Hm z =

c

− βz ∂φm mn ωμ ∂y d

Amn 2 mπ nπ k − β2z sin x sin y j ωμ a b

e, 7 4 62

where the z-directed propagation constant is given by mπ a

2 βmn z = ± k −

nπ + b

2

2

mπ a

mπ a

nπ b

2

+

π a

2

+

m=n

7 4 69

0

2

7 4 64

2

nπ ; b 2

+

7 4 65

where c is the velocity of the EM wave in the free space. The dielectric medium has μr = 1. The fundamental mode of all MWs waveguide for width a > b is obtained by taking m = n = 1. The cut-off frequency for the TE11 mode is c 2π εr

mπ nπ x cos y ; a b m = 0, 1, 2, …; n = 0, 1, 2, 3…

The field components of the TMz modes are obtained on substituting equation (7.4.69) in equation (7.1.28)

m = 1, 2, 3 …; , n = 1, 2, 3…,

f 11 c =

∂ϕe x, y = 0 at y = 0, b; 0 ≤ x ≤ a ∂y 7 4 68

ϕemn x, y = Bmn cos

7 4 63

The cut-off frequency of the mnth mode is obtained mn from kmn με c = 2π f c c 2π μr εr

0≤y≤b

The following eigenfunction ϕemn x, y for the TMz mode is obtained as a solution of equation (7.4.20) under the above boundary conditions:

12

The cut-off wave number is mn βmn c = kc =

∂ϕe x, y = 0 at x = 0, a; ∂x

ii) Boundary condition on the bottom and top walls:

Hx = 0, i e

− Amn βz nπ mπ nπ sin x cos y ωμ b a b

f mn c =

Hey = 0, i e

7 4 67

− Amn βz mπ mπ nπ cos x sin y ωμ a a b

=

i) Boundary condition on the left and right-hand sidewalls:

π b

2

7 4 66

The field components for the TE11 fundamental mode are obtained from equation (7.4.62). TMz Mode

The TMz mode has Ez component and no longitudinal magnetic field component, i.e. Hz = 0. All other field components, given by equation (7.1.28), are generated by the electric scalar potential ϕe x, y, z = ϕe x, y e − jβz z . The modal function ϕe(x, y) is a solution of wave equation (7.4.20), under the boundary conditions on the MWs obtained from equation (7.1.28).

Eex =

− jβz ∂ϕemn x, y jωε ∂x

Bmn βz mπ mπ nπ x cos y sin ωε a a b − jβz ∂ϕemn x, y Eey = jωε ∂y =

Bmn βz nπ mπ nπ cos x sin y ωε b a b Bmn 2 mπ nπ Eez = k − β2z cos x cos y jωε a b nπ mπ nπ Hex = − Bmn cos x sin y b a b mπ mπ nπ Hey = Bmn sin x cos y a a b =

a

b c d e 7 4 70

In equation (7.4.70), both m and n cannot be simultaneously zero, as it leads to zero fields. All field components should be multiplied by the factor e − jβz z to get the complete space-dependent field components. The time-harmonic field components are obtained by multiplying the above field components by the wave-factor ej ω t − βz z . The modal propagation constant is 2 βmn z = ± k −

mπ a

2

+

nπ b

2

12

7 4 71

7.4 Rectangular Waveguides

i) Boundary conditions at the bottom and top EWs:

The cut-off wave number is βmn c

=

kmn c

=

mπ a

2

nπ + b

2

7 4 72

mn με, the cut-off frequency of On using kmn c = 2π f c th the mn mode is

f mn c =

c 2π μr εr

mπ a

2

+

nπ 2 ; b

m = 0, 1, 2…; n = 0, 1, 2 m=n

7 4 74

The TM10 is the fundamental mode of all MWs based waveguide. The TE10 mode does not exist for this waveguide. The cut-off frequency of the TM10 mode is lower than the cut-off frequency of the TE11 mode. It is opposite to the case of a metallic waveguide. The field components for the fundamental TM10 mode are obtained from equation (7.4.70). 7.4.3 Rectangular Waveguide with Composite Electric and MWs Figure (7.5c) shows a rectangular waveguide with top and bottom perfect conductors, forming the EWs, and the open sides, forming the MWs. This structure is a perfect parallel-plate waveguide structure that has no fringing field at the sidewalls. The electromagnetic energy is confined within the waveguide structure. This section discusses the modal EM-field configuration supported by the structure. The structure supports both the TEz and TMz modes. However, its fundamental mode is the static TEM mode, static in the (x–y)-plane, obtained as a solution of the Laplace’s equation discussed in subsection (7.3.1). A microstrip line could be modeled by such a waveguide [B.13]. This aspect is discussed in chapter 8. The side MWs, over a limited frequency band, can also be realized using the electronic/photonic bandgap (EBG/PBG) structures [J.5]. The EBG/PBG is discussed in chapters 19 and 20. TEz Mode

ii) Boundary conditions at both magnetic sidewalls: at x = 0, a; 0 ≤ y ≤ b

7 4 73

0

c = 2 a εr

∂ϕm x, y = 0 at y = 0, b; 0 ≤ x ≤ a ∂y 7 4 75

m Hm z = 0, i e ϕ x, y = 0,

The dielectric medium has μr = 1. The fundamental mode for width a > b is obtained by taking m=1, n=0. The cut-off frequency for the TM10 mode is f 10 c

Em x = 0, i e

The ϕm x, y, z = ϕm x, y e − j βz z , generating the TEz mode with Ez=0 and Hz 0, is obtained as a solution of wave equation (7.4.2) under the boundary conditions of section (7.2) and field equation (7.1.35).

7 4 76 The following modal eigenfunction is obtained by solving wave equation (7.4.2) under the above boundary conditions: mπ nπ x cos y ; a b m = 1, 2, 3 ; n = 0, 1, 2, 3…

ϕm mn = Amn sin

7 4 77

The following field components are obtained on sub− jβz z in equation (7.1.34): stituting ϕm mn e nπ mπ nπ sin x sin y b a b mπ mπ nπ Em x cos y cos y = Amn a a b m − jβz ∂ϕ Hm x = jωμ ∂x − Amn βz mπ mπ nπ = cos x cos y ωμ a a b − jβz ∂ϕm Hm y = jωμ ∂y Amn βz nπ mπ nπ = sin x sin y ωμ b a b Amn 2 mπ nπ Hm k − β2z sin x cos y z = jωμ a b Em x = Amn

a b

c

d e 7 4 78

In equations (7.4.78), the modal propagation constant βmn is used βz z . The modal propagation constant is given by βmn z

= ± k − 2

mπ a

2

nπ + b

2

12

7 4 79

The cut-off wave number is βmn c =

mπ 2 nπ 2 ; + a b m = 1, 2, 3…; n = 0, 1, 2, 3…

7 4 80

The cut-off frequency of the mnth mode is f mn c =

c 2π μr εr

mπ a

2

+

nπ b

2

7 4 81

237

238

7 Waves in Waveguide Medium

In the case of dielectric medium, μr = 1 and for a > b, the first higher-order mode is TEz10 mode with a cut-off frequency f 10 c =

c 2a εr

,

7 4 82

The propagation constant βz of the TMzmn mode is given by 2 βmn z = ± k −

mn βmn c = kc =

+

2

12

7 4 87

The scalar electric potential function ϕe x, y, z = ϕe x, y e − jβz z generating the TMz modes in the waveguide of Fig (7.5c) is obtained as a solution of the wave equation (7.4.20) under the following boundary conditions, obtained from equation (7.1.28). i) Boundary conditions at the bottom and top EWs at y = 0, b; 0 ≤ x ≤ a 7 4 83 ii) Boundary conditions at both magnetic sidewalls: ∂ϕe x, y = 0 at x = 0, a; 0 ≤ y ≤ b ∂x 7 4 84

mπ a

2

+

nπ b

2

7 4 88

mn με, the cut-off frequency of On using kmn c = 2π f c z the TMmn mode is

TMz Mode

Hey = 0 i e

nπ b

2

The cut-off wave number is

where c is the velocity of the EM-wave in the free space. The guided-wavelength of TEzmn mode can be computed. Its wave impedance is ωμ/βz.

Eez = 0, i e ϕe x, y = 0

mπ a

f mn c =

c 2π μr εr

mπ a

2

+

nπ 2 ; b

7 4 89

m = 0, 1, 2, 3 …; n = 1, 2, 3, … For the dielectric medium of the waveguide μr = 1, the cut-off frequency of the TMz01 mode is c f 01 7 4 90 c = 2b εr z As a > b, the cutoff frequency f 10 c of the TE10 mode is 01 z less than f c of the TM01 mode. However, this structure also supports the TEM mode with no cut-off frequency. Therefore, the TEM mode is the dominant mode of a parallel-plate waveguide and the TEz10 mode is the first higher-order mode.

The modal scalar electric potential function ϕemn x, y is obtained by solving wave equation (7.4.20) under the above boundary conditions: mπ nπ ϕemn x, y = Bmn cos x sin y ; 7 4 85 a b m = 0, 1, 2, 3…; n = 1, 2, 3…

Current Distribution on the Bottom and Top Conductors

The field components of the TMzmn mode are obtained by substituting ϕe x, y, z = ϕe x, y e − jβz z in equation (7.1.28):

where ± n is the direction of a unit vector normal to the conducting surface. The surface current flows on the inner surfaces of the upper and lower conductors. The current density is useful for the computation of conductor loss on each plane of the conducting surfaces.

− jβz ∂ϕemn x, y jωε ∂x Bmn βz mπ mπ nπ = sin x sin y ωε a a b e − jβz ∂ϕmn x, y Eey = jωε ∂y − Bmn βz nπ mπ nπ = cos x cos y ωε b a b Bmn 2 mπ nπ k − β2z cos x sin y Eez = jωε a b nπ mπ nπ cos x cos y Hex = Bmn b a b mπ mπ nπ Hey = Bmn sin x sin y a a b

Eex =

a

b c d e 7 4 86

The modal surface current densities, on the upper and lower conducting planes of the composite surface waveguide, are determined from the surface magnetic fields: mn

J

= n × H mn ,

7 4 91

TEz Mode

At the upper conducting plane, i.e. at y = b, the magnetic field components from equation (7.4.78) are − Amn βz mπ mπ cos x cos nπ ωμ a a − Amn βz mπ mπ = cos x −1 n ωμ a a Amn βz nπ mπ Hm sin x sin nπ = 0 y = ωμ b a Amn 2 mπ Hm k − β2z sin x −1 n z = jωμ a Hm x =

a b c 7 4 92

7.4 Rectangular Waveguides

the longitudinal and transverse current distributions of the TE10 mode. To sketch the symmetrical current distribution, the origin is shifted by x = a/2. The longitudinal current shows edge singularities. The Ey, Hx, Hz field components, in the composite waveguide for the TEzm0 modes, are proportional to mπ x mπ x mπ x cos , cos and sin , respectively. a a a

Using equation (7.4.91) and (7.4.92), the surface current density at the upper conductor, y = b, is mn

J

m m m = − y × xHm x + zHz = zHx − xHz

Therefore, the longitudinal current density Jmn and z transverse current density Jmn x are Amn βz mπ mπ cos x −1 n a ωμ a a Amn 2 mπ k − β2z sin x −1 n b = − Hm z = − jωμ a 7 4 93

m = − Jmn z = Hx

Jmn x

TMz Mode

At the upper conducting plane, i.e. at y=b, the magnetic field components of the TMzmn modes are obtained from equation (7.4.86),

The current densities for a thin parallel-plate waveguide b ωc, the characteristic impedance ZVI 0 is a real quantity. Therefore, both the waveguide and its equivalent transmission line behave as the high-pass filters. Let us also compute the characteristic impedance of the equivalent transmission line of the TE10 mode waveguide by using the power-current definition. It is achieved with the help of equations (7.6.5) and (7.6.6),

7.6.2

Transmission Line Model of Waveguide

The equivalent transmission line model of a waveguide can be obtained by using the equivalence of power carried by the original waveguide and the equivalent transmission line. Furthermore, characteristic impedance and propagation constant of the equivalent transmission line can be taken the same as the wave impedance and propagation constant of the waveguide modes. The equivalent modal voltage and modal current can be obtained from Maxwell’s equation [B.1, B.3, B.10]. The lumped circuit model of the equivalent transmission line could be obtained from the results of the equivalent transmission line with respect to Fig (7.13). In chapter 2, the following expressions are obtained for the characteristic impedance and propagation constant of the TEM mode transmission line, shown in Fig (7.13a):

245

246

7 Waves in Waveguide Medium

μ

μ

μ

μ

ε

ε

ε

(a) TEM-mode line. Figure 7.13

μ

μ ε/k2c

μ

μ/k2c ε

μ ε/k2c μ ε/k2c

ε

μ/k2c

(b) TE-mode line.

ε

ε

(c) TM-mode line.

Lumped element circuit model of the waveguide modes.

Z = ± Y

Zc = ±

2

X B

γ= ±j

a,

XB

b 7 6 10

Equations (7.6.10) (+) sign is used for the forwardmoving wave, whereas (−) sign is used for the backward-moving wave. Equations (7.3.13a,b) show correspondence of series L, and shunt C to permittivity and permeability of medium respectively, For the TE-mode waveguide, the characteristic impedance of the equivalent transmission line is equal to the wave impedance (ωμ/βTE) and its propagation constant is βTE. The series reactance XTE p.u.l. and shunt susceptance BTE p.u.l. of the equivalent TE-mode transmission line are obtained as follows: XTE BTE

TE = j ZTE c γ

ωμ = j TE βTE β

XTE BTE = j XTE

,

line

Z = j XTE = j ωμ

ZTE 0 =

XTE = BTE

jωμ =η j ωε + k2c jωμ

7 6 11 Also, BTE

γ = − TE = − X TE

Y = jB

k2c

− ω με ωμ 2

γTE = ± j ωμ ωε − 7 6 12

k2 = j ωε − c ωμ

ZTM 0 =

Equations (7.6.11) and (7.6.12) provide the series impedance Z p.u.l. and shunt admittance Y p.u.l. of the equivalent transmission line model for the TE mode as shown in Fig (7.13b). The series inductance of the equivalent transmission line is L = μ p.u.l. and the shunt connected parallel resonant circuit components p.u.l. are L = μ k2c and C = ε. For the TM-mode supporting rectangular waveguide, the series reactance XTM p.u.l. and shunt susceptance BTM p.u.l. of its equivalent TM-mode transmission line are shown in Fig (7.13c). The XTM and BTM are obtained similarly: TM

TM = ZTM 0 γ

βTM TM β2 jβ = j ωε ωε

Z = j XTM =

7 6 14

Equation (7.6.14) provide the equivalent TM-mode transmission line shown in Fig (7.13c). The series arm has series resonant circuit elements L = μ and C = ε k2c p.u.l. whereas the shunt arm has capacitance C = ε p.u.l. The reactance and susceptance loaded equivalent transmission lines for both the TE and TM modes behave as the high-pass filters. The cut-off frequency for both cases is ωc = kc με . The characteristic impedance and propagation constant for both the equivalent lines for the propagating waves are obtained as follows:

waveguide

TE 2

TM

γTM β2 Also, B = − TM = 2TM = ωε X β ωε Y = j BTM = j ωε TM

= j XTM

line

waveguide

j ω2 με − k2c k2 = j ωμ + c ωε j ωε 7 6 13

XTM = BTM

k2c ωμ

k2c ωε

a

1 − fc f

2

1 2

= ± j k2 − k2c

j ωμ + k2c jωε =η j ωε

γTM = ± j ωε ωμ −

1

1 2

1 − fc f

b

2

c

1 2

= ± j k2 − k2c

1 2

d 7 6 15

Expressions (7.6.15) are the same as those given in equations (7.4.31) and (7.4.32). Figure (7.6) shows that at the cut-off frequency ZTE ∞ and ZTM 0, i.e. 0 0 at the cut-off frequency, there is no wave propagation. The equivalent transmission line of TE-mode, shown in Fig (7.13b), at a frequency below cut-off frequency ω < ωc becomes all inductive networks as it is an L–L line. It shows that below the cut-off frequency, the TEmode waveguide acts as a 1D ENG medium. Above the cut-off frequency ω > ωc, the waveguide acts as a 1D DPS medium. Thus, the below cut-off frequency, the TE mode waveguide, axially loaded with negative permeability (−|μr|), acts as the 1D DNG medium.

7.7 Transverse Resonance Method (TRM)

The negative permittivity (−|εr|) is offered by the TE mode waveguide below the cut-off frequency. The negative permeability could be realized with the help of printed splitring resonators (SRR) [J.10]. The SRR is discussed in the section (21.2) of chapter 21. Likewise, the equivalent transmission line of TM-mode, shown in Fig (7.13c), becomes all capacitive C–C line below the cut-off frequency. Thus, the TM-mode waveguide below cut-off frequency acts as a 1D MNG medium. Such TM – mode waveguide axially loaded with the negative permittivity wire medium can act as a 1D DNG medium. The wire medium is discussed in section (21.1) of chapter 21. The behavior of the L–L line as the ENG plasma type medium and behavior of the C–C line as the MNG type medium are presented in the section (5.5) of chapter 5. The evanescent mode has propagation constant βz = − jα and equations (7.4.31b) and (7.4.32b) provide inductive and capacitive wave impedances for the TE and TM modes, respectively, below the cut-off frequency. The waveguide in the TE/TM mode acts as 1D medium – DPS, ENG, and MNG medium. So the equivTM alent permittivity εTE and equivalent permeabileq , εeq TE TM ity μeq , μeq of both modes of waveguide medium could be also computed. The propagation constants (βTE, βTM) for both modes are obtained from equations (7.6.15b and d) to compute equivalent permittivity and equivalent permeability: βTE = ωμ ωε − =ω

k2c ωμ

μ ε 1−

1 2

k2c ω2 με

=ω

TE where, μTE eq = μ, εeq = ε 1 −

βTM = ωε ωμ − =ω

k2c ωε

TE μTE eq εeq

k2c ω2 με

ε μ 1−

TM μTM eq εeq

TM where, εTM eq = ε, μeq = μ 1 −

= μ 1−

ω2c ω2

= ε 1−

1 2

=ω

a ω2c ω2

b

k2c ω2 με

7.7 Transverse Resonance Method (TRM) The transverse resonance method (TRM) is a powerful and simple method to determine the propagation constant of a layered medium waveguide, including the surface wave supporting dielectric slab waveguides [B.2, B.3, B.5, B.9]. The method is also applicable to the traveling wave antenna [B.5, B.14]. The guided wave structures have EM – field confinement in the lateral direction, i.e. in the transverse direction, either due to the presence of metallic walls, as in the case of a standard waveguide, or due to the exponentially decaying field, i.e. the evanescent mode field, as in the case of a planar open surface waveguide. In the transverse plane, transverse to the direction of propagation, these structures could be treated as a resonating section of the equivalent transmission line. The resonating line avoids the traveling wave and confines the field in the form of a standing wave. The waveguide in the transverse direction could be treated as the equivalent transmission supporting the resonance. It is assumed that the characteristic impedance of the equivalent transmission line is equal to the wave impedance of the mode supported by the waveguide. The field is confined, i.e. trapped, in the transverse direction. So a condition of resonance is assumed to exist in the transverse plane of a waveguide. Therefore, the method is called the transverse resonance method (TRM). At a suitable location in the transverse plane, the total impedance or total admittance looking toward the left and right-hand sides of the location is either zero or infinity. Alternatively, at any location, under the resonance condition, the total reactance or susceptance is zero. This section uses the concept of TRM to determine the propagation constants of several waveguide structures. 7.7.1

c k2c ω2 με d 7 6 16

Standard Rectangular Waveguide

Figure (7.14a) shows the cross-section of a standard rectangular waveguide supporting TE10 mode. The field is confined along the x-axis due to the conducting walls (PEC) at x = 0 and x = a. Using equation (7.6.1a), the TE mode wave impedance of the wave propagating in the z-direction is z

ZTE w = Equations (7.6.16) demonstrate the plasma-like behavior of the waveguide. Below cut-off frequency, the negative permittivity and negative permeability are realized for the TE and TM mode, respectively. They result in the 1D ENG and MNG medium.

Ey ωμ = 100 − Hx βz

771

To use the TRM, a plane wave is assumed with components Ey and Hz, propagating in the x-direction, such that the equivalent transmission is along the x-axis. Its characteristics impedance is

247

248

7 Waves in Waveguide Medium

p

Y

p

Y Ey Hz

TEx

Ey Hx

p

Hx

X

Z

x=0

p

X

Zin

(a) Cross-section of the waveguide. Figure 7.14

10

Zw –

Y

βx

(b) Equivalent transverse transmission line.

TRM applied to the rectangular waveguide.

x

Z0 = ZTE w =

Ey ωμ = 100 Hz βx

772

Figure (7.14b) shows both ends shorted equivalent transverse transmission line. It is due to the presence of the conducting walls (PEC) at both ends of a waveguide. At x = 0, the input impedance of the shortcircuited line of length x = a is zero,

it is desired to determine the propagation constant βz for both modes by using the TRM. The equivalent transverse transmission line approach applies to both TEy and TMy modes. The propagation constant for the hybrid mode satisfies the following dispersion relations in two media: Medium #1

x

10 Zin = ZTE w tan βx a = 0 mπ m = 1, 2, 3…… β10 x = a

Medium #2 a

mπ a mπ a

2

2

+ β2y1 + β2z = ω2 ε1 μ1

a

+ β2y2 + β2z = ω2 ε2 μ2

b,

(7.7.3)

Therefore, the longitudinal propagation constant βz of the TEm0 mode of the rectangular waveguide in the z-direction is mπ 2 b, 773 a where k is the wavenumber of EM-wave in the homogeneous medium of a waveguide and βx is the transverse propagation constant in the x-direction. This result has been obtained in equation (7.4.71) by the field analysis. However, the TRM computes the results with much less effort. β2z = k2 − β2x = k2o −

7.7.2

x=a

+

Y Y

Dielectric Loaded Waveguide

Figure (7.15) shows the cross-sections of horizontally and vertically layered dielectric-loaded rectangular waveguides. The EM-waves propagate in the z-direction. The inhomogeneous medium waveguide supports the hybrid mode that is described by TE and TM modes with respect to a normal to the interface of two dielectric media [J.11–J.18]. Figure (7.15a) supports the TEy, i.e. the longitudinal sectional electric (LSEy) mode, and also the TMy, i.e. the longitudinal sectional magnetic (LSMy) mode. Normally the field components and waveguide parameters are determined by solving the wave equation in each medium [B.1, B.5, J.11–J.13]. However,

774 where longitudinal propagation constant βz could be y y y y βLSE βLSM either βTE or βTM for the LSEy or LSMy z z z z mode respectively. The y-directed transverse propagation constants βy1 and βy2 in both the media are to be computed with the help of the TRM, to finally determine the propagation constant of both modes. Let us apply the TRM to the horizontally layered waveguide shown in Figure (7.15a) that also shows the equivalent transmission line section for two media in the transverse plane. Both the upper and lower transmission line sections are short-circuited at the ends. Their input impedances at the plane p–p are Zin+ = j Z02 tan βy2 b − b1

a

Zin− = j Z01 tan βy1 b1

b,

775

where Z01 and Z02 are the characteristic impedances and βy1, βy2 are the propagation constants of the lower and upper equivalent transmission lines, respectively. Under the resonance condition in the transverse plane of the waveguide, the total input reactance at the plane p–p is zero, Zin+ + Zin− = 0 Z02 tan βy2 b − b1

= − Z01 tan βy1 b1 776

7.7 Transverse Resonance Method (TRM)

y=b Y b n b1

Z02 βy2

b – b1 ε2 μ2

+

p Z– Z

p

ε1 μ1

βy1 Z01

b1

x=0

x=a

m

X

y=0

(a) Horizontally loaded waveguide.

Equivalent circuit

Y b ε1 μ1

n

p

a1

ε2 μ2

Z01, βy1

(a–a1) βy2, Z02

p x=a X (b) Vertically loaded waveguide.

x=0

m

a1

x=0

x=a Equivalent circuit

Figure 7.15 Dielectric loaded rectangular waveguide.

TEy (LSEy) Hybrid Mode

For the TEy hybrid mode, characteristic impedances of the upper and lower line sections are obtained from equation (7.6.1a), y

Z02 = ZTE w2 =

ω μ2 βmn y2

a,

y

Z01 = ZTE w1 =

ω μ1 βmn y1

In superscript “mn”, m is an modal integer, and modal number n is the order of roots of equation (7.7.10). Again, under the transverse resonance condition, the following dispersion relation of the TMy mode is obtained:

b

βmn y1

In superscript “mn”, m is an modal integer, and modal number n is the order of roots of equation (7.7.8). Thus, under the transverse resonance condition, the following dispersion relation is obtained from equations (7.7.6) and (7.7.7): ωμ2 ωμ tan βy2 b2 = − mn1 tan βy1 b1 βmn βy1 y2 βmn y1 μ1

cot βy1 b1 = −

βmn y2 μ2

a

cot βy2 b2

b, 778

where b2 = b − b1. On substituting the propagation constants βy1 and βy2 from equation (7.7.4), equation (7.7.8) y could be solved to obtain the propagation constant βTE z of the TEy mode, i.e the LSEy mode. TMy (LSMy) Hybrid Mode

For the TMy mode, characteristic impedances of the upper and lower line sections are obtained from equation (7.6.1b), y

Z02 = ZTM w2 =

βmn y2 ωε2

a,

y

Z01 = ZTM w1 =

βmn y1 ωε1

b 779

βmn y2

tan βy2 b2 ωε1 ωε2 ωε1 ωε2 cot βy1 b1 = − mn cot βy2 b2 βmn β y1 y2

777

tan βy1 b1 = −

a b 7 7 10

Again, equation (7.7.10), along with equation (7.7.4), y for can be solved to get the propagation constant βTM z the mode TMy, i.e. the LSMy mode. Normally equations (7.7.8) and (7.7.10) are obtained by the field matching approach. However, that is a time-consuming process. Similarly, the propagation constants for LSEx and LSMx modes are determined for the vertically loaded waveguide shown in Fig (7.15b), using the TRM method as applied to the transmission line equivalent circuit [B.1, B.5, J.14–J.18].

7.7.3

Slab Waveguide

Figure (7.16a) shows the slab waveguide that is infinite in the y-direction, so that there is no field variation in the ydirection. The z-axis is the direction of propagation of the guided EM-wave. On the symmetry consideration with respect to the plane p–p, the propagating EM-field could be decomposed into the even and odd modes for both the TE and TM modes. Figure (7.16) shows the field components and the direction of propagation for both modes.

249

250

7 Waves in Waveguide Medium

x

x

q p

X

q

Ey

Ey z

q p

q

p

Z Ey

Y

(b) Field distribution.

X

X Ex

Hz Z

Ez

Direction of propagation

Y

Ey

Figure 7.16

Z

Hy

(c) TEz-mode.

Slab waveguide supporting even and odd surface wave modes.

εo

∞ βox q

0

ε μ

d

Z0a

p P Z

Magnetic wall (a)

TEz-

d

0

Z0d βdx

Z in –

Zin p

Open

even-mode.

∞ βox

X εo

+

q–

q d

P

q μo

Direction of propagation

(d) TMz-mode.

X q μo

Odd mode

Even mode

(a) Slab waveguide. Hx

Y

p

Z0a

q

q

q

ε μ

+

Z in –

d

p

p Electric wall

p Z

Z0d βdx

Zin p

Short

(b) TEz- odd-mode. Figure 7.17

TEz Even and odd modes for slab waveguide.

Figure (7.16b) shows the even and odd mode excitations. The propagation constants for four modes are normally determined by solving the wave equation [B.5, B.9, B.16]. However, the TRM is applied to accomplish this

task. Figure (7.17a and b) show that the plane of symmetry p–p can be treated as the MW or EW corresponding to the even and odd mode excitations. The magnetic and EWs arrangement bifurcates the slab waveguide.

7.7 Transverse Resonance Method (TRM)

TEz Even Mode

Figure (7.17a) shows one-half of the dielectric slab waveguide (μ = μ0), with the MW, supporting the even mode. It also shows the equivalent transmission line in the transverse plane of the waveguide. This structure supports the TEz surface wave mode. The transverse resonance method is used in the (x–z)-plane as the field is bounded by the MW, along the lower part of the x-axis. For analysis, the x-axis is shifted to the top surface of the slab waveguide. The even mode Ey field is maximum at the MW plane p–p, showing that the plane p–p is open-circuited. The plane q–q is the interface of the air-dielectric media. The upper side of the plane q–q is air-medium in the cut-off region that provides the field confinement in the x-direction by the evanescent mode. The infinitely long upper section of equivalent TEx mode transmission under the cut-off condition offers an inductive load, as shown in equation (7.4.31). Following equation (7.6.1), the characteristic impedance of the lower dielectric and upper airline sections of the TEx modes are x

Z0d = ZTE wd =

ωμ0 βdx

a,

x

Z0a = ZTE wa = j

ωμ0 β0x

b, 7 7 11

where βdx and β0x are the propagation constant of these equivalent lines in the transverse x-direction. The input impedance of the lower and upper sections of the transmission lines at the plane q–q are Zin− = − j Z0d cot βdx d ωμ = − j 0 cot βdx d βdx

a , Zin+ = j

ωμ0 β0x

b 7 7 12

The transverse resonance condition, at the plane-qq, satisfies the following relation: Zin+ + Zin− = 0,

β0x cot βdx d = βdx

7 7 13

The dispersion relations, using equation (7.5.2), in the slab and air-medium are Lower Slab medium Upper Air medium

β2dx + β2z = β2d = ω2 εr ε0 μ0 − β20x

+

β2z

=

β20

= ω ε0 μ0 2

a b

7 7 14 In equation (7.7.14) air-medium, acting as the upper line-section, is in the cut-off mode with x-directed imaginary propagation constant jβ0x. On subtracting the above dispersion equations, the following expression is obtained: β2dx + β20x = εr − 1 k20 where,

k20 = ω2 ε0 μ0 7 7 15

The eigenvalue equation (7.7.13) and the dispersion relation (7.7.15) are rewritten as u tan u = w

u2 + w 2 = v 2

a,

where, u = βdx d, w = β0x d, v =

b

1 − εr k0 d

c 7 7 16

The parameter v is known for a given slab-waveguide structure at the assumed frequency. Equations (7.7.16a and b) can be solved for u and w simultaneously. Finally, the propagation constant βz, for TEz even mode, is obtained from equation (7.7.14). Depending upon the frequency, a given structure can support several TEn (even) modes, where n numbers of the solutions of equation (7.7.16) give n numbers of modes. The graphical method is also used for the solution of propagation constant βz [B.5, B.9, B.12].

TEz Odd Mode

Figure (7.17b) shows that for the TEz odd mode excitation, the Ey field component is zero at the plane of symmetry p–p, i.e. at the EW. The equivalent transverse line shows the short circuit at the plane p–p. The plane q–q is the interface of two media. The upper air-medium is again in the cut-off region, supporting the evanescent mode in the transverse x-direction for the field confinement. Therefore, the input impedance of the lower and upper sections of the transmission line at the plane q–q are Zin− = j Z0d tan βdx d = j Zin+ = j

ωμ0 β0x

ωμ0 tan βdx d βdx

a b 7 7 17

Under the transverse resonance condition, we get the following equation: Zin+ + Zin− = 0

a

βdx d cot βdx d = − β0x d

b

7 7 18

Equation (7.7.18), with the help of equation (7.7.16), is rewritten as u cot u = − w

a,

u2 + w 2 = v 2

where, u = βdx d, w = β0x d, v =

b

1 − εr k0 d

c

7 7 19 Again, the above equations can be solved for u and w, once the parameter v is known for the given structure at the assumed frequency. Finally, the propagation constant βz for the TEz odd mode surface wave is computed at the assumed frequency.

251

252

7 Waves in Waveguide Medium

TMz Even Mode

Figure (7.16d) shows that this mode has even symmetry for the Hy-field, giving maximum Hy-field component at the plane p–p. Therefore, the plane p–p could be replaced by an EW. It shows that the TEz odd mode equivalent circuit shown in Fig (7.17b) could be used in this case also. However, for this case of TMz even mode, the characteristic impedance of the equivalent lower and upper transmission line sections correspond to the wave impedance of the TMx mode. Again, the TMx mode in the transverse x-direction of air-medium is in the cut-off region. Following expression (7.4.32), it offers capacitive wave impedance. Therefore, on using equations (7.6.1b) and (7.7.17), the characteristic impedances of equivalent TMx mode transmission lines and their input impedances at the plane q–q are written as Z0d =

x ZTM wd

Zin− = j

β = dx ωε

β a , Z0a = = − j 0x b ωε0 β c , Zin+ = − j 0x d ωε0 7 7 20

The transverse resonance condition is satisfied as follows: a,

βdx d tan βdx d = β0x d b εr 7 7 21

Equation (7.7.21) could be rewritten as follows along with the dispersion relation: u tan u = εr w

a,

u2 + w 2 = v 2

where, u = βdx d, w = β0x d, v =

b

1 − εr k0 d

c

7 7 22 Again, the propagation constant βz of the TMz even mode at an assumed frequency is obtained from the above equations and equation (7.7.14).

TMz Odd Mode z

a , Zin+ = − j

β0x ωε0

The TM odd mode is excited for zero Hy-field component at the plane of symmetry p–p shown in Fig (7.16d). It shows that the plane of symmetry p–p could be replaced by the MW, as is the case for the TEz even mode. Therefore, the equivalent transmission line shown in Fig (7.17a) is again applicable to this case by using the characteristic impedances of TM mode. The air-medium, in the x-direction, is in the cut-off region offering the capacitive wave impedance. Therefore, the input impedance of the lower and upper transmission lines at the plane q–q is

b 7 7 23

Under the transverse resonance condition, the following relations are obtained: Zin+ + Zin− = 0

βdx d cot βdx d = − β0x d b εr 7 7 24

a,

Equation (7.7.24) and the dispersion relation could be rewritten as u cot u = − εr w

u2 + w2 = v2

a,

where, u = βdx d, w = β0x d, v =

b

1 − εr k0 d

c

7 7 25

x ZTM wa

βdx tan βdx d ωε

Zin+ + Zin− = 0,

βdx cot βdx d ωε

Zin− = − j

For a given structure and frequency, v is a known parameter and equation (7.7.25) can be solved for u and w simultaneously. Finally, the propagation constant βz is computed for the odd mode of the TMz surface wave from the dispersion relation (7.7.14). 7.7.4 Conductor Backed Multilayer Dielectric Sheet Figure (7.18a) shows the double-layer dielectric sheet backed by a conductor. It is a multilayer slow-wave structure. The propagation constants for the TEz and TMz modes could be easily obtained by using the TRM. Let us do it for the TM mode. The TRM is applied through the equivalent transmission line shown in Fig (7.18b). The wave propagation in the transverse x-direction in the air-medium is in the cutoff region, offering a capacitive wave impedance and imaginary propagation constant. The characteristic impedance for three-line sections supporting TMx are given below: Z0d1 =

βd1x ωε1

a , Z0d2 =

βd2x ωε2

b , Z0a = − j

β0x c ωε0 7 7 26

The propagation constants in three media meet the following dispersion relations: β2d1x = k21 − β2z where,

k2i

=

a , β2d2x = k22 − β2z εri k20

i = 1, 2

b , β20x = β2z − k20

c d

7 7 27 The input impedance at the plane q–q shown in Fig (7.18b) are given as follows:

7.8 Substrate Integrated Waveguide (SIW)

∞ μo

X q

ε2

d2

εo

Z0a q

μ2

Z0d2

ε1 d1

+

q

q

Z in –

Zin2 –

Z0d1

μ1

p

p Z

Zin1

Short

Figure 7.18 Double-layer slow-wave structure.

− Zin1 =j

βd1x tan βd1x d1 ωε1

− = Zod2 Zin2

a

− Zin1 + j Z0d2 tan βd2x d2 − tan βd2x d2 Z0d2 + j Zin1

Zin+ = Z0a = − j

β0x ωε0

b c 7 7 28

On applying the transverse resonance condition, Zin+ + Zin− = 0 at the plane q–q, the following dispersion expression is obtained: b tan βd1x d1 + a tan βd2x d2 a − b tan βd1x d1 tan βd2x d2

=b

a

where, a=

b=

εr1 β0x = εr1 βd1x εr2 β0x = εr2 βd2x

βz k0 2 − 1 εr − βz k0 2 β z k0 2 − 1 εr2 − βz k0 2

b

7.8 Substrate Integrated Waveguide (SIW) The metallic rectangular waveguide, discussed in section (7.4) is popular in the mm-wave frequency range due to its low-loss. In mm-wave and THz ranges the planar lines are lossier. It is difficult to integrate the standard rectangular waveguide with planar technology. However, the synthetic rectangular waveguide has been developed in the PCB and LTCC technology that has been integrated with other planar lines, say microstrip. This structure is known as the substrate integrated waveguide (SIW), also called the laminated waveguide. Several components and leaky antenna have been developed using SIW [J.19–J.26]. There are several formats for the SIW. This section briefly presents the basic characteristics and simple modeling of two important structures. 7.8.1 Complete Mode Substrate Integrated Waveguide (SIW)

c

βd1x d1 =

2πd1 λ0

εr1 − βz k0

2

d

βd2x d2 =

2πd2 λ0

εr2 − βz k0

2

e 7 7 29

Elliott [B.15] has obtained equation (7.7.29) by a long process of the field matching at the interface. However, the TRM is a direct method. For d2 = 0, the above equation is reduced to equation (7.7.21). It is the case for the conductor backed single layer dielectric sheet. The TRM could be also applied to compute the propagation constant βz of the TEz mode. The method is easily extended to the N layers of the dielectric sheets.

Figure (7.19a) shows the synthetic rectangular substrate integrated waveguide (SIW) embedded in a substrate of the thickness h and relative permittivity εr. The top and bottom walls of the SIW are conducting surfaces, i.e. the EWs, whereas the sidewalls of height h are created by two rows of arrays of the metalized vias of diameter d drilled in the substrate. The vias are separated by the distance S. The center-to-center distance between two rows is W. The structure behaves as a rectangular waveguide filled in with dielectrics of relative permittivity εr, although there are some important differences from the standard metallic waveguide. Figure (7.19b) shows that the SIW can be excited by the microstrip and matching is achieved using a linear transition [J.26]. The width Wt and length Lt of the feed microstrip linear taper are optimized on the EM-simulator for wideband matching. Figure (7.19b) shows both the input and output ports.

253

254

7 Waves in Waveguide Medium

W

d s weff

Metallized surface s

Wt

h

w

X

Lt

Array of metallized via holes

Y Z

Substrate (εr, h)

Wm

Metallized ground plane

(a) SIW structure.

(b) Microstrip-fed SIW.

Surface current

Substrate

Array of metallized via holes

Y X

(c) CPW fed SIW. Figure 7.19

Z

(d) Surface current on SIW.

Synthetic SIW created through two rows of metalized via holes.

Figure (7.19c) shows that the SIW can also be fed through the CPW [J.26]. The mode supported by a waveguide is related to the surface current of the waveguide. Figure (7.19d) shows the surface currents for the TE10 mode in the SIW, it is similar to the standard metallic waveguide. There is no longitudinal surface current on the side-walls. Therefore, any narrow gap, created by the periodic arrangement of the metalized vias holes, will not disturb the TE10, or even TEmo modes. However, TEmn (n 0) and TMmn modes have longitudinal surface currents on the sidewalls and they cannot flow on the SIW sidewalls. Therefore, unlike the standard waveguide, the SIW does not support the TEmn (n 0) and TMmn modes. The SIW is primarily a periodic structure, due to the periodic array of the metallic post, created through the metalized vias. However, the size of via (d) and its separation (S) are adjusted so that the stopband characteristic is avoided in the useful frequency band. This process is known as the homogenization that replaces the periodic discontinuous side-walls by the continuous equivalent metallic walls. The concept of homogenization has been used for other kinds of periodic surfaces too [B.6]. The SIW structures have been analyzed using several types of numerical methods, including the commercial EM-Simulators. However, for design, the equivalent waveguide model could also be used.

The waveguide model uses the closed-form expressions for the effective width weff of the SIW shown in Fig (7.19a) that is less than W. On replacing the width W of the standard rectangular waveguide by Weff, the expressions presented in section (7.4) is used to compute the cut-off frequency, dispersion, losses and wave impedance of the modes of the SIW. The results of the waveguide model are compared, with satisfactory accuracy, against the results of the numerical methods and also against the experimental results. The equivalent width of the SIW is dependent on W, S, and d of the SIW. Any of the following expressions can be used [J.6, J.20, J.25]: d2 0 95 S d2 d2 +01 = W − 1 08 S W W = 1 + 2W − d S

Weff = W −

a

Weff

b

Weff

− 4W 5S4 Weff = W −

d W−d

d2 W − d

d2 d3 − 1 1 s 6 6 S2

2

c

3

d, 781

where d < λg/5, S ≤ 2d and λg = λ0 εr . Equation (7.8.1b) [J.22] and equation (7.8.1c) [J.25] are more

7.8 Substrate Integrated Waveguide (SIW)

0.35

300 250 200 150 Multimode calibration Rectangular waveguide Measured

100 50

Attenuation constant (m–1)

Propagation constant (m–1)

350

0.3 0.25

Multimode calibration FDTD

0.2 0.15 0.1 0.05 0

0 14

14.5

15

15.5 16

16.5 17

17.5

18

15

20

25

30

Frequency (GHz)

Frequency (GHz)

(a) Dispersion of TE10 -mode of SIW.

(b) Attenuation constant of SIW.

Figure 7.20 Dispersion and loss characteristic of the SIW. Source: Xu and Wu [J.22].

accurate expressions. The first one does not show d/w dependence and is acceptable provided (S-d) is small, giving closely spaced via holes. It gives very small leakage loss from the sidewall. The equivalent width is in between W and (W-d). The outer width of the equivalent rectangular waveguide is W and its inner width is (W-d). The above expression can be used to compute the cut-off frequency of the fundamental mode TE10, or even the cut-off frequency of the higher-order mode. Equation (7.8.1d) has been used to determine the cut-off frequency of the TE20 mode [J.20]. The cut-off frequency of the TE10 mode, using equation (7.8.1a), has accuracy within ±5% and the TE20 mode has the accuracy +4 % / − 9%, using an equation (7.8.1d) for the SIW on a substrate with εr = 2.2, h = 20 mil. Figure (7.20a) compares favorably the dispersion characteristic of the TE10 mode of the SIW, on the substrate εr = 2.33, h = 0.508 mm, using the equivalent waveguide model against the results of both the numerical method and experimental results [J.22]. The accuracy of the above expressions can be tested from the dispersion results of the SIW obtained from the EM-Simulator [J.24]. The EM-simulated S21 parameters of two SIW of lengths ℓ1 and ℓ2 (ℓ1 > ℓ2) provide the phase difference. The propagation constant is evaluated from the Sℓ212 and Sℓ211 . The dispersion relation is used to extract the effective width of the SIW: Sℓ212 − Sℓ211 rad m ℓ1 − ℓ2

βSIW = π Weff Weff =

+ βSIW

= kd

π ω2 μ0 ε0 εr − β2SIW

αd = αc =

k20 εr tan δ Np m 2 βSIW

a

Rs Weff η

2

1 − kc k0

where, Rs =

Weff kc +2 h k0

ωμ0 2σ, and

η=

2

Np m b

μ0 ε0 εr

c 783

In equation (7.8.3) conductivity σ of all walls is taken the same, whereas due to gaps between vias holes in the side-walls the conductivity of the equivalent homogeneous surface gets reduced. There is a leakage loss from the side-walls, as the RF power is coupled to the adjacent TEM mode supporting the parallel-plates guide. The total loss of the SIW can be evaluated from the S21 of SIW obtained from the EMSimulator. The total attenuation coefficient, αtotal, due to conductor, dielectric and leakage losses, can also be evaluated from the S21 of two unequal line lengths of the SIW: Total loss in SIW

a

= 10 log 10 1 −

2 2

Once the propagation constant βSIW is evaluated using an EM-Simulator, equation (7.8.2b) computes the equivalent width of the SIW. It is used to verify the accuracy of the expressions of equation (7.8.1). The dielectric loss and conductor losses of the TE10 mode SIW can be computed from the standard expressions of the metallic waveguide [J.23, B.2, B.3]:

2

αtotal =

b, 782

where kd is the wavenumber in the substrate.

Sℓ111

2

S11 2 + S21

+ Sℓ211

2

−

Sℓ112

2

dB

2

+ Sℓ212

a 2

ℓ1 − ℓ2 ℓ1 > ℓ2

Np m, b 784

255

7 Waves in Waveguide Medium

a

1.0 0.8 0.6

Region-II

Region-I

ℓ

X

Input w taper feed

S

d X

W

Ouput taper feed

Ey

256

0.4 Radiation model at 10 GHz EM-Simulation at 10 GHz Radiation model at 40 GHz EM-Simulation at 40 GHz

0.2

a Z

Y

Region-II

h

0.0

–0.4

–0.2

0.0 a 0.2

Air region (a) HM-SIW broad-face and end view.

II

I

1.0

II

εr

εr

Electric wall

Array of vias holes

0.8

ωʼeff,HMT

ωʼeff,HMT

εr

0.6

(b) Electric field variation inside and outside HM-SIW[J.27].

ω I

0.4

x/w

Electric wall

Magnetic wall

(c) Development of equivalent HM-SIW model. Figure 7.21

Modeling of HM-SIW. Source: Modified from Lai et al. [J.27].

The above EM-Simulation process can also be used to evaluate the conductor, dielectric and leakage losses and their corresponding attenuation coefficients separately. The attenuation in the SIW can be also computed using the FDTD and multimode calibration. Figure (7.20b) illustrates the attenuation of the SIW on the substrate εr = 2.33, h = 0.508 mm. It is noted that the leakage loss is high at the lower end of the frequency band [J.22]. The leakage loss of the SIW can be effectively controlled by controlling the sizes of the vias holes and their separation. Further, it is also desirable to avoid any stopband appearance in the useful bandwidth of SIW. These are achieved through the following conditions [J.26]: d < 0 25λc a ,

d < S ≤ 2d

b

785

7.8.2 Half-Mode Substrate Integrated Waveguide (SIW) The SIW discussed is a low-profile and low-cost structure, easily integrated with the planar technology. However, the SIW is a wide width structure, occupying more areas of the substrate. The half-mode SIW (HM-SIW), shown in Fig (7.21a), with half of the conducting plane width of the SIW, offers a more compact waveguide structure that has been used for the development of

filters, 3 dB coupler, directional couplers, etc. [J.28– J.31]. The HM-SIW can be simulated and analyzed using the EM-Simulators. However, in this section design equations are presented, as these are useful for the development of components [J.27]. Again, the HM-SIW is fed through a microstrip line and a microstrip linear taper, optimized by the EM-Simulator. Figure (7.21a) shows that one sidewall of the HM-SIW has an EW, realized with help of an array of metalized vias holes, while the other sidewall is open to the dielectric substrate with εr, h. Again, beyond the substrate edge, there is an infinite air-medium. The HMSIW structure has two distinct regions – region-I, inside the waveguide supporting TEmo type mode, called quasiTEm o half mode (HM), and region-II supporting the evanescent radiation mode. In the case of the quasiHM, the open sidewall is approximately located at the center of the standard rectangular waveguide. It results in the cosine type Ey-field distribution. So m = 0.5, 1.5, 2.5,…. The HM-SIW supports a bound mode, propagating in the Z-direction with propagation constant βz, with one side field confinement by the EW and another side field confinement by the evanescent Ey-field. Figure (7.21b) shows EM-Simulated cosine type Ey-field distribution across the cross-section of the HM-SIW. At x = w, Ey-field is zero as usual. However,

7.8 Substrate Integrated Waveguide (SIW)

the maximum of Ey-field is shifted inside the HM-SIW by the distance “a.” Figure (7.21b) shows the location of the maximum and location of distance “a.” The HMSIW has been modeled using two models – model-I, the radiation model, and model-II, the equivalent waveguide model [J.27].

Model-I, Radiation Model

The model-I helps to compute the evanescent Ey-field outside the HM-SIW, i.e. the radiation field. It is computed by obtaining the magnetic current on the open sidewall. The magnetic current is computed through the Ey-field component of the quasi-TEm o half mode by following the standard TEmn mode discussed in section 7.4. The magnetic scalar potential and the field components of the quasiTEm o, following equations (7.4.17) and (7.4.34), can be written as follows for fundamental mode (m = 0.5): e − jβz z , 0 ≤ x ≤ w

x ϕm m 0 x, y, z = Bm 0 cos βm 0 w − x

Em y

=

− Bm 0 βxm 0

sin

βxm 0

w−x

e

− jβz z

x x Hm x = Bm 0 βm 0 βz ωμ sin βm 0 w − x

e − jβz z

βz = k2 − βx2 m0

1 2

a

c

Weff,HM

b

d,

a , k = k0 εr

b

ωμ βz

d 787

In equation (7.8.7), the origin has been shifted to the location of Ey maximum by the distance a, as shown in Fig. (7.21b). The z-directed magnetic current is obtained from the Ey component, Mz = − 2 − x × y Ey Bm o π π sin W−x =z W−a 2 W−a

ΔW 03 = 0 05 + h εr 2 W 104Weff,HM − 261 38 + 2 77 c + + ℓn 0 79 eff,HM 3 2 h h h 789 where,

e − jβz z

c , ZTE w =

1 d2 d2 +01 2W − 1 08 S 2W 2 = Weff,HM + ΔW

b

where the directional propagation constant, wavenumber, dispersion relation, and wave impedance are given by the following expression for the fundamental quasiTEm o half mode, (m = 0.5): π 2 W−a

The equivalent waveguide model of the HM-SIW is based on the similar model of the SIW discussed previously. However, it has one sidewall as the EW, whereas its other sidewall is the MW. Figure (7.21c) shows the stepwise development of the composite walls equivalent waveguide model of the HM-SIW. If the width of a SIW is 2W and width of the HW-SIW w, the effective width (Weff,HM) of the HM-SIW, using an equation (7.8.1b), can be computed. However, there is a fringing field at the open side of the HM-SIW that can be accounted by further extending the width by ΔW and placing a MW, giving the final equivalent waveguide mode of width Weff, HM [J.27]: Weff,HM =

786

βxm 0 =

Model-II, Equivalent Waveguide Model

a

2 x x 2 Hm z = − Bm 0 βm 0 βm 0 ωμ k − βz

× cos βxm 0 w − x

[J.27]. Fig (7.21b) shows that the evanescent mode Ey-field, computed by the radiation model, agrees satisfactory with results obtained from the EM-simulation.

e − jβz z 788

In expression (7.8.8) image current has been accounted for. Using the standard radiation integral, the Ey-field component can be evaluated in the homogeneous open region-II that has effective relative permittivity between 1 and εr. However, for a thin substrate, one can take the effective relative permittivity ≈1

Equation (7.8.9) is valid for the range 2.2 ≤ εr ≤ 15, 0.254 mm ≤ h ≤ 2.54 mm, 2.5 mm ≤ w ≤10 mm. It covers the frequency range from 2 GHz to 60 GHz. On considering the width of HM-SIW as an λg/4 open-end resonator, the cut-off frequency and the dispersion relations are obtained: f c,TE0 5,0 =

3c 4 εr Weff,HM

βZ,TE0 5,0 = εr k20 −

a π

2Weff,HM

2

1 2

b 7 8 10

Figure (7.22a) compares satisfactorily dispersion of the fundamental TE0.5,0 mode of the HM-SIW, using the equivalent waveguide model-II, against the EM-Simulation and experimental results [J.27]. Figure (7.22b) compares the attenuation of the quasiTE0.5,0 mode HM-SIW against the attenuations of the SIW and 50 Ω microstrip line. The width of 40 mm, 80 mm, and 100 mm long SIW is 2W = 5.25 on a substrate εr = 2.2, tan δ= 0.001, h = 0.254 mm and strip conductivity σ = 5 × 107 S/m. The diameter of via d = 0.05 mm and the separation S between two via is

257

7 Waves in Waveguide Medium

10 Attenuation constant (Np/m)

1800 Propagation constant (βz), rad/m

258

1500 1200 900 EN-Simulation

600

HM-SIW mode Exp

300 0

20

Figure 7.22

25

30

35

40

45

50

55

60

9 8

MS SIW

7

HMSIW

6 5 4 3 2 1 0 20

25

30

35

40

45

50

55

Frequency (GHz)

Frequency (GHz)

(a) Dispersion of quasi-TE0.5,0-mode.

(b) Attenuation of microstrip, SIW, HM-SIW.

60

Dispersion and attenuation characteristics of -TE0.5, 0-mode HM-SIW. Source: Lai et al. [J.27].

0.6 mm. The attenuation of the microstrip increases almost linearly with frequency from 20 GHz to 60 GHz. The attenuation in microstrip is less than the attenuation of both SIW and HM-SIW at frequencies below 35 GHz. The SIW shows a sudden increase in attenuation at 58 GHz, due to the bandstop of the periodic nature of the SIW. The HM-SIW shows a higher cutoff frequency. Its loss is decreased below both these structures above 35 GHz and no bandstop occurs up to 60 GHz. However, as compared to a microstrip line, both the SIW and HM-SIW occupy more area on a PCB. The HM-SIW has wider single-mode operation bandwidth, as compared to that of the SIW. In the case of the HM-SIW, the cut-off frequency of next higher quasiTE1.5,0 mode occurs at a frequency that is three times of the cut-off frequency of the fundamental quasiTE0.5,0 mode. The single-mode operation bandwidth of the HM-SIW is two times the bandwidth of the SIW.

B.4 Collin, Robert E.: Antenna and Radio Wave

Propagation, McGraw-Hill, Inc., New York, 1985. B.5 Balanis, C.A.: Advanced Engineering Electromagnetics,

John Wiley & Sons, New York, 1989 B.6 Ramo Simon, Whinnery John R., Van Duzer

B.7 B.8 B.9 B.10

B.11

B.12

References Books

B.13

B.1 Harrington, R.F.: Time-Harmonic Electromagnetic

Fields, McGraw-Hill Book Company, New York, 1961 B.2 Collin, Robert E.: Field Theory of Guided Waves, IEEE

B.14

Press, New York, 1991. B.3 Collin, Robert E.: Foundations for Microwave

Engineering, 2 York, 1992

nd

Edition, McGraw-Hill, Inc., New

B.15

Theodore: Fields and Waves in Communication Electronics, 3rd Edition, John Wiley & Sons, Singapore, 1994. Sadiku, N.O.: Elements of Electromagnetics, 3rd Edition, Oxford University Press, UK,2004. Chang, D.K.: Field and Waves Electromagnetics, Pearson Education Asia, Delhi, 2001. Rozzi, T.; Mongiardo, M.: Open Electromagnetic Waveguides, IEE Press, U.K., 1997. Jordan, Edward C., Balmain Keith G. E.: Electromagnetic Wave and Radiating System, PrenticeHall, New Delhi India, 1989. Plonsey Robert, Collin, Robert E.: Principle and Applications of Electromagnetic Fields, Tata McGrawHill, New Delhi, 1973. Elliott, R.S.: An Introduction to Guided Waves and Microwave Circuits, Englewood Cliffs, NJ: PrenticeHall, 1993 Itoh Tatsuo (Editor): Numerical Techniques for Microwave and Millimetre-Wave Passive Structures, Wiley-Interscience, New York, 1989. Walter, C. H.: Traveling Wave Antennas, McGrawHill, 1965, Dover, 1970. Mirshekar-Syahkal, D.: Spectral Domain Method for Microwave Integrated Circuits, ResearchStudies Press Ltd., John Wiley & Sons, N.Y., 1990.

References

B.16 Koul, S. K.: Millimeter Wave and Optical Dielectric

J.12 Weeks, W.L.: Propagation constants in rectangular

Integrated Guides and Circuits, Wiley-Interscience, New York, 1997. J.13

Journals J.1 Yeap, K.K.; Tham, C.Y.; Yassin, G.; Yeong, K.C.:

J.2

J.3

J.4

J.5

J.6

J.7

J.8

J.9

J.10

J.11

Attenuation in rectangular waveguides with finite conductivity walls, Radioengineering, Vol. 20, No. 2, pp. 472–478, June 2011. Yeap, K.K.; Tham, C.Y.; Yassin, G.; Yeong, K.C.; Propagation in lossy in rectangular waveguides, Electromagnetic Waves Propagation in Complex Matter, Ed. InTech Pub., 2011 Lee, C.S.; Lee, S.W.; Chuang, S. L.: Plot of modal field distribution in rectangular and circular waveguides, IEEE Trans. Microwave Theory Tech., Vol. 33, No. 3, pp. 271–274, 1985. Marcatili, E.A.J.: Dielectric rectangular waveguide and directional coupler for integrated optics, Bell Syst. Tech. J., BSTJ, Vol. 48, No. 7, pp. 2071–2102, Sept. 1969. Chang, C.C.; Qian, Y.; Itoh, T.: Analysis, and applications of uniplanar compact photonic bandgap structures, Prog. Electromagn.Res., PIER, Vol. 41, pp. 211–235, 2003. Krischnining, M.; Jansen, R.H.; Koster, N.H.L.: New aspects concerning the definition of microstrip characteristic impedance as a function of frequency, IEEE., MTT-S, Int MicrowaveSymp. Dig., pp. 305–307, 1982. Bianco, B.; Panini, L.; Parodi, M.; Ridella, S.: Some considerations about the frequency dependence of the characteristic impedance of uniform microstrip, IEEE Trans. Microwave Theory Tech., Vol. MTT- 26, No. 3, pp. 182–185, Mar. 1978. Knorr, Jeffrey B.; Kuchler, Klaus-Dieter: Analysis of coupled slots and coplanar strips on dielectric substrate, IEEE Trans. Microwave Theory Tech., Vol. MTT- 23, pp. 541–548, July 1975. Cohn, S.B.: Slot line on a dielectric substrate, IEEE Trans. Microwave Theory Tech., Vol. MTT- 17, No. 10, pp. 768–778, Oct. 1969. Marque, R.; Martel, J.; Mesa, F.; Medina, F.: Lefthanded-media simulation and transmission of EM waves in subwavelength split-ring-resonator-loaded metallic waveguides, Phys. Rev. Lett., Vol. 89, No. 18, pp. 183901-1–183901-4, Oct. 2002. Pincherle, L.: Electromagnetic waves in metal tubes filled longitudinally with two dielectrics, Phys. Rev. Vol. 66, No. 5&6, pp. 118–130, Sept. 1944.

J.14

J.15

J.16

J.17

J.18

J.19

J.20

J.21

J.22

J.23

J.24

waveguide partially filled with dielectric, IRE Trans. Microwave Theory Tech., Vol. 7, No. 2, p. 294, April 1959. Gardial, F.E.; Vander Vorst, A.S.: Wave propagation in a rectangular waveguide loaded with an H-plane dielectric slab, IEEE Trans. Microwave Theory Tech., Vol. MTT, No.1, p. 56, Jan. 1969. Vartanian, P. H.; Ayres, W. P.; Efelgesson, A. L.: Propagation in dielectric slab loaded rectangular waveguide, IRS Trans. Microwave Theory Tech., Vol. MTT- 15, No.5, pp. 215–222, April 1958. Eberhardt, N.: Propagation in the off center E-plane dielectrically loaded waveguide, IEEE Trans. Microwave Theory Tech., Vol. MTT, No.15,pp. 282–289, May 1967. Seckelmann, R.: Propagation of TE modes in dielectric loaded waveguides, IEEE Trans. Microwave Theory Tech., Vol. MTT, 4, No.11, pp. 518–527, Nov. 1966. Witt, H.R.; Biss, R.E.; Price, E.L.: Propagation constants of a waveguide containing parallel sheets of finite conductivity, IEEE Trans. Microwave Theory Tech., Vol. MTT-15, No.4, pp. 232–239, April 1967. Gardiol, F.E.: Higher-order modes in dielectrically loaded rectangular waveguides, IEEE Trans. Microwave Theory Tech., Vol. MTT- 16, No.11, pp. 919–924, Nov. 1968. Deslandes, D.; Wu, K.: Integrated microstrip and rectangular waveguide in planar form, IEEE Microwave Wireless Compon. Lett., Vol.11, No.2, pp. 68–70, Feb. 2001. Cassivi, Y.; Perregrini, L.; Arcioni, P.; Bressan, M.; Conciauro, G.: Dispersion characteristics of substrate integrated rectangular waveguide, IEEE Microwave Wireless Compon. Lett., Vol.12, No.8, pp. 333–335, Sept. 2002. Deslandes, D.; Wu, K.: Single-substrate integration technique of planar circuits and waveguide filters, IEEE Trans. Microwave Theory Tech., Vol. 51, No. 2, pp. 593– 596, Feb. 2003. Xu, F.; Wu, K.: Guided wave and leakage characteristics of substrate integrated waveguide, IEEE Trans. Microwave Theory Tech., Vol. 53, No. 1, pp. 66–73, Jan. 2005. Deslandes, D.; Wu, K.: Accurate modeling, wave mechanisms, and design considerations of a substrate integrated waveguide, IEEE Trans. Microwave Theory Tech., Vol. 54, No. 6, pp. 2516–2526, June 2006. Deslandes, D.; Wu, K.: Design consideration and performance analysis of substrate integrated waveguide components, Proc. 32th Eur. Microw. Conf., Milan, Italy, vol. 2, pp. 881–884, Sept. 2002.

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260

7 Waves in Waveguide Medium

J.25 Salehi, M.; Mehrshahi, E.: A closed-form formula for

dispersion characteristics of fundamental SIW mode, IEEE Microwave Wireless Compon. Lett., Vol. 21, No. 1, pp. 4–6, Jan. 2011. J.26 Wu, K.; Deslandes, D.; Cassivi, Y.: The substrate integrated circuits – a new concept for high-frequency electronics and optoelectronics, TELSIKS 2003 6th Int. Conf., vol. 1, pp. P-III–P-X, October 2003. J.27 Lai, Q.; Fumeaux, C.; Hong, W.; Vahldieck, R.: Characterization of the propagation properties of the half-mode substrate integrated waveguide, IEEE Trans. Microwave Theory Tech., Vol. 57, No. 8, pp. 1996–2004, Aug. 2009. J.28 Hong, W.; Liu, B.; Wang, Y. Q.; Lai, Q. H.; Wu, K.: Half mode substrate integrated waveguide: A new guided wave structure for microwave and millimeter-wave

application, Proc. Joint 31st Int. Infrared Millimeter Wave Conf./14th Int. Terahertz Electron. Conf., pp. 18–22, Shanghai, China, Sep. 2006. J.29 Liu, B.; Hong, W.; Wang, Y. Q.; Lai, Q. H.; Wu, K.: Half mode substrate integrated waveguide (HMSIW) 3 dB coupler, IEEE Microw. Wireless Compon. Lett., Vol. 17, No. 1, pp. 22–24, Jan. 2007. J.30 Wang, Y.; Hong, W.; Dong, Y. D.; Liu, B. et al.: Half mode substrate integrated waveguide (HMSIW) bandpass filter, IEEE Microw. Wireless Compon. Lett., Vol. 17, No. 4, pp. 256–267, Apr. 2007. J.31 Liu, B.; Hong, W.; Zhang, Y.; Tang, H. J.; Yin, X. X.; Wu, K.: Half mode substrate integrated waveguide 180 3-dB directional couplers, IEEE Trans. Microw. Theory Tech., Vol. 55, No. 12, pp. 2586–2592, Dec. 2007.

261

8 Microstrip Line Basic Characteristics

Introduction

8.1

The microstrip line contributes about ninety percent toward the MIC and MMIC technology, as against other planar transmission lines, such as coplanar waveguide (CPW), slot lines, etc. It has found its applications in the frequency range 1–100 GHz and beyond up to a few THz. Unlike a coaxial cable and a two-wire transmission line, embedded in a homogeneous medium, the microstrip line is constructed in an inhomogeneous medium. The basic line parameters; such as propagation constant, attenuation factor, and characteristic impedance, of a coaxial cable and a two-wire transmission line are frequency independent, i.e. these lines are nondispersive. However, the basic line parameters of microstrip and several other planar transmission lines are frequency-dependent, i.e. the planar transmission lines are dispersive. The main emphasis in this chapter is to understand the basic parameters and characteristics of the microstrip line. It can help us to compute its propagation characteristics, characteristic impedance, and losses in terms of the physical parameters of a microstrip line.

Figure (8.1) shows the cross-sectional views of a few basic microstrip structures. Figure (8.1a) shows the standard form of an open microstrip line. It is also called just the microstrip or the microstrip line. The strip conductor width of the microstrip is w and the thickness of the strip conductor is t. The microstrip is printed on a dielectric sheet, called the substrate, with relative permittivity εr1 and thickness h1. The open space above the microstrip is unbounded medium. Therefore, the microstrip forms an inhomogeneous transmission medium for the EM-wave propagation, as the strip conductor is located at the interface of two dielectrics media – the substrate with relative permittivity εr1 and the air-medium with εr2 = 1. For simplicity, relative permittivity and thickness of a substrate are taken as εr and h, respectively. The real-life microstrip-based components are always enclosed in a metallic enclosure, shown in Fig (8.1b), to protect the circuits against the environment. The metallic enclosure also provides the EM-shielding to the microstrip circuits and also the mechanical strength to the substrate. It further acts as a heat-sink to the high-power microwave devices. The presence of a metallic enclosure, i.e. the top shield and two sidewalls, has a significant influence on the propagation characteristics and the characteristic impedance of an open microstrip line. Therefore, either their presence has to be accounted for at the design stage itself, or the top shield, and sidewalls should be kept far away from the strip conductor, so that they do not alter the characteristics of an open microstrip line. The topshielded microstrip line, shown in Fig (8.1c), has the sidewalls far from the strip conductor. Therefore, the sidewalls are not shown in this case. The upper dielectric layer of relative permittivity εr2 and thickness h2, i.e. the dielectric cover to the microstrip, is called the superstrate. Figure (8.1a–c) do not show any superstrate. The upper dielectric is just an air-medium. In the case of the shielded and the enclosed microstrip lines, both the lower (substrate) and the upper (superstrate)

Objectives

• • • • • •

To explain the general nature of the mode of propagation of the EM-wave on a microstrip line, including the quasi-static nature of wave propagation. To understand Wheeler’s transformation that transforms the inhomogeneous medium microstrip to the homogenous medium microstrip through the concept of the effective relative permittivity. To present closed-form static models, to compute effective relative permittivity and the characteristic impedance of a microstrip under several kinds of physical conditions. To understand the nature of microstrip dispersion and to develop dispersion models. To model the dielectric and conductor losses of a microstrip line. To present a circuit model of a lossy microstrip line.

General Description

Introduction to Modern Planar Transmission Lines: Physical, Analytical, and Circuit Models Approach, First Edition. Anand K. Verma. © 2021 John Wiley & Sons, Inc. Published 2021 by John Wiley & Sons, Inc.

262

8 Microstrip Line

Strip conductor of thickness t Open space W

εr2 = 1

t

εr1

Dielectric substrate

Open space

W

Open space

H

εr1

h1

Ground plane (c) Top-shielded microstrip.

h1

Top shield Ground Open space h2 conductor εr2 = 1 W t εr1

Semiconductor

L

Figure 8.1

H

L

Oxide/ polyimide layer εr1

H

h1

L

Ground plane (d) Stripline.

t

h1

Strip conductor of thickness t εr2 = 1 W

Top shield

h2 εr2 = 1

W

H

(b) Enclosed microstrip.

(a) Open standard microstrip.

εr1

t εr1

Ground plane

t

h2 εr2 = 1

h1

Ground plane

W

Top shield

Top shield

Ground plane (e) Thin-film microstrip (TFMS).

Ground plane (f) Micro-coplanar strip (MCS).

Cross-sections of some microstrip structures.

dielectric layers are present. However, layers can have identical relative permittivity and thickness, i.e. εr2 = εr1 with h2 = h1. This special structure, shown in Fig (8.1d), is the strip transmission line. The stripline is developed in a homogeneous dielectric-medium. The stripline can also be asymmetrical. This line is also used in MIC technology. However, it is not much popular like a microstrip; as the strip conductor of a strip transmission line is not approachable for mounting the components. There are many more interesting and useful variations in the basic microstrip line [B.1–B.7, J.1, J.2]. One such variation, shown in Fig (8.1e), is known as the thin film microstrip (TFMS) line. The TFMS is constructed on a semiconductor substrate. The ground plane conductor is deposited on a supporting semiconductor that need not be expensive semi-insulating type. The substrate is a thin layer (a few micron thicknesses) of the oxide or polyimide on which the strip conductor of microstrip is deposited. The thin-film microstrip line is a useful structure for MMIC technology. Normally, it is difficult to ground any device in the microstrip, as it may be brittle substrates such as alumina and semiconducting substrates. Drilling a hole in these substrates is a difficult and critical task. Figure (8.1f ) shows the micro-coplanar strip (MCS) structure. It is a microstrip line with an additional coplanar ground conductor strip that helps to ground a device in the microstrip technology [J.3]. Figure (8.2a) shows the 3D view of λg/2-section of an infinite-length microstrip line. The microwave signal is

applied between the strip conductor and the ground plane conductor. It also shows two sets of transverse electric and magnetic field lines, separated by half wavelength, in the direction of propagation. In general, the substrate of a microstrip could be made of dielectric material, magnetic material, semiconductor, ferroelectric material, etc. However, a dielectric sheet is commonly used as the substrate. Normally, the substrates are lossy materials and they cause the dielectric loss in a microstrip line. The strip and ground conductors can be made of copper, aluminum, gold, and even superconductors, depending upon the particular technological requirement. The conducting strip and ground plane are usually several skin-depth thick conductors. The ground plane is taken as an infinitely thick conductor; however, the thickness of the strip conductor is either neglected or taken as finite. We have to account for the influence of the strip conductor thickness on the propagation characteristics and the characteristic impedance of a microstrip line. Figure (8.2b) demonstrates the crosssection of a thick conductor (thickness t) microstrip with more field lines in the air-region. The strip conductor is printed on a substrate either through the process of etching on a premetallized substrate or by the process of deposition of a conducting layer on a substrate. In the process of etching, the unwanted parts of the pre-coated conducting sheet are removed either by the chemical etching or by the mechanical milling. The fabrication processes of the microstrip, and other planar lines

8.1 General Description

Magnetic field w Electric field Direction of wave propagation, Jz : Current density

Jz

Air y

λg/2

t

Microwave source

εr

h

x y Ground plane Dielectric substrate

x z

Strip conductor of thickness t (a) Microstrip supporting wave propagation. Electric field Magnetic field W W t

Cf

Cp

Cf

y x Ground plane (b) Microstrip cross-section showing more field lines in air-medium due to conductor thickness t.

(c) Partial capacitances of microstrip.

Figure 8.2 The fields and wave of a microstrip.

also, are discussed in chapter 13. The strip and ground conductors are not perfect conductors. They have a finite conductivity that causes loss to the EM-wave propagating on a microstrip line. This loss is called the conductor loss. The microstrip and its several variants are primarily twoconductor transmission lines. A two-conductor transmission line, in a homogeneous dielectric-medium, supports the TEM-mode wave propagation. Figure (8.2a) shows that in the case of a microstrip line, the electric and magnetic fields are in two different dielectric media. Thus, a microstrip line is located in an inhomogeneous dielectricmedium. Figure (8.2b) further shows the field distribution of a microstrip line in the transverse plane. The electric field is not totally inside the dielectric-medium, between the

strip and ground conductors. A part of the total electric field, forming the fringing field, is away from the central region of the strip conductor. Figure (8.2c) shows that the central capacitance Cp p.u.l. of microstrip is due to the main electric field, whereas both side fringe capacitance Cf p.u.l. is due to the fringing field. The total line capacitance of microstrip is due to both Cp and Cf. The next section demonstrates, with the help of Maxwell’s equations, that a microstrip line in an inhomogeneous medium cannot support the pure TEM mode wave propagation, as there are always electric and magnetic field components along the direction of propagation. In general, a microstrip supports the so-called hybrid mode, discussed in chapter 7. However, at the lower frequency, the hybrid mode could be

263

264

8 Microstrip Line

approximated by the TEM kind of mode that is known as the quasi-TEM mode. For the quasi-TEM mode, the magnitude of the longitudinal field components can be neglected, as compared to the magnitude of the transverse field components. Figure (8.3) exhibits the conceptual evolution of a microstrip from the coaxial line. It helps to appreciate the TEM nature of a microstrip line.

8.1.1

Non-TEM Nature of Microstrip Line

Maxwell’s equations are used below to demonstrate that technically a microstrip line cannot support the TEM mode propagation, i.e. a mode with no electric and magnetic field components along the direction of propagation. Figure (8.2a) shows that the electric field E undergoes the refraction, i.e. bending of its path, when it enters from the air-medium to the dielectric substrate medium. At the

Coaxial line

Open microstrip Figure 8.3

Deformed coaxial line

Ex

dielectric

= Ex

811

air

As there is no free charge at the interface of two media, Maxwell’s equation could be written as ∇ × H = jε0 εr ω E

812

On using the above expressions in both media, the following expression is obtained at the interface:

Conceptual Evolution of Microstrip Lines

Figure (8.3) demonstrates the evolution of a microstrip line from a coaxial line. The circular coaxial line supports the pure TEM mode propagation as the fundamental mode. At higher frequency, the higher-order modes can also appear. However, these can be ignored, as normally, the single fundamental mode propagation is more important. The circular coaxial line is gradually deformed into the elongated elliptical coaxial line that too supports the pure TEM mode propagation. The elongated sides can be replaced by the vertical sidewalls. Thus, a stripline structure is obtained that again supports the pure TEM mode propagation. Both sidewalls and the materials above the strip conductor can be removed. It results in a microstrip structure. The microstrip appears to support the TEM mode, as it has evolved from the TEM mode supporting the coaxial line. However, due to the presence of fields in two different dielectric media, the microstrip does not support the pure TEM mode. It supports the quasi-TEM mode.

8.1.2

interface of two media, the tangential components of electric field Ex in both media must be continuous:

1 ∇×H jε0 εr ω ∂Hz ∂Hy − ∂y ∂z

x dielectric

=

= εr dielectric

1 ∇× H jε0 ω ∂Hz ∂Hy − ∂y ∂z

x air

air

813 The normal component of the magnetic field Hy is also continuous at the interface of two media: ∂Hy ∂z

= dielectric

∂Hy ∂z

814 air

The following expression is obtained from the above two equations: εr

∂Hz ∂y

− air

∂Hz ∂y

= εr − 1 dielectric

∂Hy ∂z

815

For an inhomogeneous microstrip medium, εr 1. Therefore, the Hz component of the magnetic field exists, as the right-hand side of equation (8.1.5) is nonzero. However, for a homogeneous medium, say for the air dielectricmedium, εr = 1 and the right-hand side of the above equation is zero, leading to Hz = 0. Likewise, the second Maxwell’s equation can be used to demonstrate that for an inhomogeneous microstrip medium, the Ez component of the electric field also exists. Thus, a microstrip line supports the hybrid mode with Ez 0 and Hz 0. However, the longitudinal field components, Ez and Hz, are significant only at a higher frequency. At the lower end of the microwave,

Elliptical coaxial line

Stripline

Evolution of stripline and microstrip line from the TEM-mode coaxial line.

8.1 General Description

say at a frequency below 4 GHz, these field components can be neglected as compared to the dominant transverse field components. It leads to approximately TEM mode. The approximate TEM mode is called the quasi-TEM mode. Even at the low frequency, the metallic ground plane backed dielectric substrate supports the fundamental TM0 surface wave mode. The surface wave mode is discussed in chapter 7. The TEM mode couples with the TM0 surface wave mode producing the dispersive nature of a microstrip line. The quasi-TEM mode can be regarded as a combination of the TEM mode and the TM0 surface wave mode [J.4]. Thus, the pure TEM mode supporting line is nondispersive one, whereas the quasi-TEM mode supporting line is dispersive. The material dispersion in a substrate is negligible in the microwave and mm-wave ranges.

8.1.3

Quasi-TEM Mode of Microstrip Line

The separation of variable method, applicable to the metallic waveguides, is not applicable to solve the wave equation for a microstrip line in an inhomogeneous medium. For the case of a microstrip line, and other planar transmission lines, several numerical methods, called the full-wave methods have been developed to obtain the frequencydependent hybrid mode field solutions. We present below the limiting case at low frequency. In this case, the wave equation could be reduced to Laplace’s equation, showing the static nature of microstrip fields in the transverse plane [B.8]. Thus, the quasi-TEM mode of a microstrip could be dealt with the Laplace’s equation. Its solution is much simpler as compared to the solution of Helmholtz’s wave equation. The wave equation for the electric and magnetic field is given below: ∇2

E H

+ ω2 εμ

E H

=0

816

Figure (8.2a) shows that the strip conductor of a microstrip line supports the time-varying current density J z h, z located at the distance h from the ground plane. The retarded magnetic vector potential at any location below the strip conductor in the transverse (x-y)-plane is [B.9]: A r =

μ 4π

J z h, z e − jω r−r

με r − r

ds

817

In the above equation, ds is the length of the current element and r, r are radial vectors. For the field just below the conducting strip, i.e. at |r − r | = h, the magnetic vector potential is A r =

μ 4π

J x h, z e − j2πh λ ds r−r

818

If the substrate thickness h is much smaller than the wavelength λ, (2πh/λ 1), the exponential factor showing the phase variation of magnetic vector potential can be neglected and it is given by A r =

μ 4π

J z h, z ds r−r

819

The above magnetic vector potential of a magnetostatic field is valid for the substrate thickness, h λ/2π. In this case, the time retardation in the magnetic vector potential is ignored. It shows that the transverse field of a microstrip at a longer wavelength, with limiting case of ω 0, could be treated as the static field in the transverse plane. Although, the time-dependent wave still propagates along the microstrip line as the quasi-TEM mode. Thus, the magnetostatic field expressed as below, models the quasi-TEM mode of a microstrip line: ∇×H =0

8 1 10

The electric field of the TEM mode is related to the magnetic field through Maxwell’s equation: ∇ × E = − jμ0 μr ωH

8 1 11

The following Laplace equation is obtained from the above two equations: ∇2 E = 0

8 1 12

In the transverse (x-y)-plane, Laplace equation could be further written in terms of the electric potential V: ∇2 V = 0,

8 1 13

where E = − yV h. Therefore, the quasi-TEM mode assumption requires the solution of Laplace’s equation in the transverse (x-y)-plane of a microstrip line. The static field solution is useful only in the lower microwave frequency range, say around 4 GHz. The static microstrip line parameters are improved empirically to incorporate the frequency-dependent behavior of a microstrip line. In summary, a microstrip and other planar transmission lines are analyzed by following three approaches:

•

Full-wave methods: These methods, such as spectral domain analysis (SDA), integral equation method, method of lines, finite element method (FEM), finite difference time domain (FDTD), mode-matching method, etc., numerically solve the wave equation. They provide frequency-dependent line parameters. The SDA is discussed in chapter 16. The SDA is a semi-analytical method involving more analytical processing and less computational time.

265

266

8 Microstrip Line

• •

Static methods: There are several static methods, such as the variational method in the Fourier domain and the space domain, conformal mapping methods, method of moment (MoM), finite difference method, etc. The conformal mapping method and the variational method are discussed in chapters 9 and 14, respectively. Closed-form models: Several static closed-form models, for the frequency-independent microstrip line parameters, and several dynamic closed-form models, for the frequency-dependent microstrip line parameters, have been reported. A few of these models are discussed in this chapter.

The above methods and models have been summarized in several books [B.1–B.7, B.10]. The full-wave methods involve the rigorous solution of the wave equation (8.1.6) that is analytically demanding and numerically inefficient, i.e. it takes a long time on a computer to get the results. Therefore, these methods are not suitable for the computer-aided design (CAD) that is the core of the modern circuit simulators, such as ADS, Microwave Office, etc. However, the full-wave methods can provide accurate results on the propagation behavior and also for the field and current distributions of a microstrip line and other planar transmission lines. The full-wave methods are suitable for the detailed investigations of the planar transmission lines and passive components. These methods are the foundation block of the modern 2D and 3D EM-field simulators – Momentum, Microwave Studio, HFSS, LINMIC, Sonnet, Empire, etc. The EM-Simulators are useful for the numerical experimentations without conducting the real physical experimentations on the planar transmission lines and components. The real-life experimental investigations are expensive and time-consuming. These methods are also used to generate numerical data for the multidimensional curve-fitting to obtain numerically efficient CAD-oriented closed-form models. The closed-form models are the foundation blocks of the modern microwave circuit simulators mentioned above. Most of the practical design works are carried out on these circuit simulators. The static methods are also numerical methods, but they are analytically much simpler and computationally much faster. The static methods are combined with the closedform dynamic models, discussed in chapter 15, to provide the design-oriented dispersion and loss models for the multilayer microstrip and other planar lines.

8.1.4

Basic Parameters of Microstrip Line

The above discussion shows that a microstrip line supports the hybrid mode of propagation that can be approximated by the TEM kind of wave propagation. Thus, a microstrip

line could be modeled as a parallel conductor transmission line that we have discussed in chapter 2. The microstrip and other planar transmission lines, such as the TEM mode supporting two-conductor transmission lines, are described by the secondary line parameter – the complex propagation constant γ and the characteristic impedance Z0. These secondary line parameters are described by the primary line constants – R, L, C, G, i.e. the resistance, inductance, capacitance, and conductance per unit length ( p.u.l.) of a microstrip line. The secondary parameters are directly measurable quantities. These are useful in the design of microwave components. Therefore, a microstrip and other planar transmission lines are characterized in terms of the secondary line parameters. Careful consideration is needed for the modeling of a microstrip line, supporting the hybrid mode wave propagation, by the TEM mode. The static solution of Laplace’s equation provides the frequency-independent secondary line parameters: characteristic impedance Z(f = 0), propagation constant β(f = 0), and attenuation constant α(f = 0). Finally, the frequency-independent phase velocity vp(f = 0) is obtained. The microstrip line is described as a line without any dispersion. However, this is not the correct description of microstrip and other planar transmission lines. The microstrip and other planar transmission lines are primarily dispersive transmission lines. Their characteristic impedance Z(f ), propagation constant β(f ), and attenuation constant α(f ) are frequency-dependent parameters. Moreover, the fundamental TEM mode is supplemented by the appearance of higher-order guided modes. Further, the radiation modes, viewed as the leaky modes, also exist. Figure (8.4) shows the results of the full-wave analysis, using the eigenmode solution, on frequency-dependent effective relative permittivity, with respect to the normalized wavelength h/λ, of a top-shielded microstrip line on an alumina substrate (w/h1 = 4.72, εr = 9.8, h2 = 5h1) for the quasi-TEM mode [B.1, J.5, J.6]. In the hybrid mode format, the quasi-TEM mode is the HE0 hybrid mode. It also shows the appearance of higher-order hybrid modes, i.e. the HE1 and HE2 hybrid modes, with an increase in frequency; as the normalized substrate thickness (h/λ) increases with frequency. The higher-order modes appear above their cutoff frequencies. These are the bound guided modes. Other full-wave methods are also used to get higherorder modes [J.7]. Figure (8.4) further shows the presence of the unbound radiation mode at all frequencies. It is significant at a higher frequency. Thus, in the case of an open microstrip line operating in the higher-order mode, the leaky mode could be obtained. It is useful to design the leaky-wave antenna. The useful frequency range of the microstrip line, as a wave-propagating medium, is limited by the appearance of the first higher-order mode HE1.

8.1 General Description

Eeffective relative permittivity εreff (f)

10 9.8 8 εreff (f = 0)

HE0(Quasi-TEM mode) HE1 - mode

6

4 HE2 - mode

2

Operating frequency range Radiation mode

0 0

0.02

0.04

0.06

0.08

0.1

Normalized substrate thickness (h/λ) Figure 8.4 The quasi-TEM mode and higher-order modes of a microstrip line on the alumina substrate [J.5]. Source: Modified from Ermert [J.5].

The cutoff frequency of the first higher-order mode should be known, to estimate the frequency range of the quasiTEM operation. The microstrip line can tolerate, without any damage, only a finite amount of microwave power. Therefore, the power handling capability of a microstrip line should also be estimated. It is important for the high-power application of a microstrip line [B.2]. Of course, for the usual low-power application, this information is normally not needed. In brief, our primary concern is to model a microstrip line by the static closed-form models. Next, the models are improved by incorporating the frequency-dependent behavior in it. Finally, the closed-form frequencydependent models should be able to accommodate several variations in the microstrip line, shown in Fig (8.1). It is considered in chapter 15. At present, the microstrip is treated as a lossless transmission line with R = G = 0 that is embedded in an inhomogeneous dielectric-medium. The process of obtaining the equivalent homogeneous dielectric-medium for a primarily inhomogeneous medium microstrip line is discussed in the next section. From chapter 2, the characteristic impedance and phase constant of a lossless line are given below: Z0 =

L C

a,

β = ω LC

b

8 1 14

The phase velocity of the TEM-wave propagating on a transmission line is vp =

ω = β

1 LC

8 1 15

The phase velocity of a microstrip line in the air-medium is the same as that of the velocity of light in the free space.

The phase velocity of the EM-wave in the nonmagnetic dielectric-medium with μr = 1 and εr 1 is vp =

ω = β

1 1 = μ0 ε0 εr

c εr

8 1 16

The wavelength λ0 of the signal in the free space is decreased in a dielectric-medium. The guided wavelength (λg) of the TEM-wave in a dielectric-medium is λg =

λ0 εr

8 1 17

The guided-wave propagation constant (βg) of the EMwave propagating in a dielectric-medium is related to the propagation constant (β0) in the free space by βg = β0 εr

8 1 18

The characteristic impedance can be written in terms of the line capacitance C p.u.l.: Z0 εr =

1 Z0 εr = 1 = εr vp C

8 1 19

If the relative permittivity of a medium is frequencydependent, i.e. if the medium is dispersive, then the phase velocity, propagation constant, guided wavelength, and characteristic impedance associated with the transmission line are also frequency-dependent. A parallel-plate transmission line with the homogeneous dielectric-medium inside it also represents a microstrip line. The line capacitances p.u.l. for a parallel-plate transmission line in a homogeneous dielectric-medium and the air-medium are given by ε0 εr w ε0 w a , Co = C εr = 1 = b, C εr = h h 8 1 20

267

268

8 Microstrip Line

where w is the width of the parallel-plate line, and h is the separation between two parallel-plates. The relative permittivity of a homogeneous medium is taken as a ratio of the above given two capacitances: εr =

C εr C εr = 1

dropped for the sake of simplicity, as it has been done above for the line capacitance.

8.2 Static Closed-Form Models of Microstrip Line

8 1 21

The above expression is very useful both from the experimental and theoretical points of view. The relative permittivity of a dielectric sheet is measured at low frequency with help of a capacitance meter, by measuring the capacitances with and without a dielectric sheet. During the process, the same separation between the plates, for both the measurements, is maintained. Likewise, once an expression is available to determine the line capacitance of a microstrip, the relative permittivity of a microstrip medium is computed by calculating the line capacitances with and without dielectric substrate, while keeping the w/h-ratio unchanged. The relative permittivity obtained is called the effective relative permittivity of an equivalent homogeneous medium of microstrip, as all field lines are not confined in the dielectric-medium, some fringing field is always present in the air-medium. The value of the effective relative permittivity is always more than unity and less than the relative permittivity of a substrate. The line capacitance can be obtained by solving Laplace’s equation discussed above. The characteristic impedance of a TEM-mode transmission line can also be expressed in terms of these two line capacitances Z0 εr =

1 c

=

C εr C εr = 1

1 , c C C0

This section introduces the concept of Wheeler’s transformation, filling-factor, and effective relative permittivity. Wheeler’s transformation is used to convert the inhomogeneous medium microstrip line to an equivalent homogeneous medium microstrip. The closed-form models are also presented to compute the static effective relative permittivity and the static characteristic impedance of a microstrip line with zero strip conductor thickness. Subsequently, the effects of finite conductor thickness, top shield, and the sidewalls are accounted for on the effective relative permittivity and the static characteristic impedance. These closed-form models are important for the MIC design.

8.2.1 Homogeneous Medium Model of Microstrip Line (Wheeler’s Transformation) Figure (8.5) shows the steps involved to obtain Wheeler’s transformation that provides the equivalent homogeneous medium microstrip line from the original microstrip line structure in an inhomogeneous medium. The concept of partial capacitance is used in the process.

•

8 1 22

where c is the velocity of EM-wave in the air-medium. Sometimes, the parametric description of quantities is Original microstrip in inhomogeneous medium w

Step #(a)

Step #(b)

εr1

Step #(c)

C (εr)

w h

h

ε reff

wʹ = w1+ w2

w2 h

C (εr = 1)

Inhomogeneous medium. Figure 8.5

Equivalent microstrip in homogeneous medium

εr2 = 1 εr1

wʹ w1

Step (a): The microstrip line is on the substrate of thickness h with relative permittivity εr1 = εr. The uppermedium is an infinite block of air with relative permittivity εr2 = 1. The strip conductor has width w, with zero

εreff

C (εreff) Homogeneous medium.

Equivalent homogeneous dielectric-medium for a microstrip line.

h

8.2 Static Closed-Form Models of Microstrip Line

thickness. The equivalent homogeneous medium microstrip line is located in the dielectric-medium of the infinite extent with a relative permittivity called the effective relative permittivity (εr = εreff). The effective relative permittivity depends on the relative permittivity (εr) and w/h ratio of the original microstrip line. It also depends, to some extent, on the thickness of the strip conductor that is neglected at present. Following the definition of the relative permittivity of a homogeneous dielectric-medium given by equation (8.1.21), an expression can be written for the effective relative permittivity of a microstrip line in terms of its line capacitances: εreff

w C εr , w h = εr , w h C εr = 1, h

821

The line capacitance of a microstrip line is obtained by solving Laplace’s equation. Once the effective relative permittivity of microstrip is known, the guided-wavelength and the characteristic impedance are obtained by replacing εr for εreff equations (8.1.17) and (8.1.19): λg =

λ0 εreff

a,

Z0 εr ,

w Z0 εr = 1 = h εreff

b 822

• •

Step (b): The microstrip is treated as an equivalent parallel-plate capacitor with two parallel dielectric media. The material dielectric with relative permittivity εr1 = εr occupies the width w1 and the air-medium occupies the width w2. The total equivalent strip width w = w1 + w2 of a microstrip remains unchanged. For the equivalent homogeneous medium, the strip width w is the same and relative permittivity is εreff. Step (c): The microstrip line is described by two partial parallel-plate capacitors C1(εr) and C2(εr = 1) corresponding to the dielectric and air media, respectively. The equivalent homogeneous medium microstrip line is described by the effective line capacitance Ceff (εreff). Wheeler’s transformation is a static process in which the electrostatic energy remains unchanged. It is expressed as 1 1 1 Ceff V2 = C1 V2 + C2 V2 , 2 2 2

q=

C1 C1 + C2 825

The partial capacitances p.u.l. could be written as C1 =

ε0 εr w 1 ε0 w 2 ε0 εreff w 1 + w 2 , C2 = , and Ceff = h h h 826

On substituting expressions of equation (8.2.6) in equation (8.2.4), the following expression is obtained: εreff = εr

w1 w2 + w1 + w2 w1 + w2

827

Likewise, from equations (8.2.5) and (8.2.6), the fillingfactor is expressed as q=

εr w 1 εr = w2 εr w 1 + w 2 εr + w1

828

The above expression shows that the filling-factor q (>1, i.e. in the case of a very wide width microstrip, total electric field lines are confined in the dielectric substrate region w1, giving w2 = 0 and q = 1 as the limiting case. For the case of equal division of electrostatic energy between the air and dielectric-media, C1=C2 and the ratio of widths is w2/w1= εr. Under this limiting condition, the filling-factor is q = 1/2. It applies to a microstrip with a very narrow strip width. On substituting equation (8.2.9) in equation (8.2.7), the following expression is obtained [J.8, J.9]: εreff f = 0

1 + q εr − 1

8 2 10

823

where V is the potential between the strip conductor and the ground plane. The above equation provides Ceff = C1 + C2

1 C1 V 2 2 , q= 1 1 C1 V 2 + C2 V 2 2 2

824

The filling-factor (q) is defined as a ratio of the electrostatic energy p.u.l. in the dielectric-medium to the total electrostatic energy p.u.l. in the microstrip line:

The above expression is called Wheeler’s transformation. The transformation is bi-directional. If the right-hand side of equation (8.2.10) is known, then it transforms the known inhomogeneous medium microstrip line to a homogeneous medium microstrip. It is assumed that the expression for the filling-factor in terms of w/h-ratio is known. Wheeler has obtained such expression for the filling-factor from the conformal mapping method. The result is given below. If

269

270

8 Microstrip Line

microstrip is embedded in a homogeneous dielectricmedium of relative permittivity εreff(f = 0), i.e. the left-hand side of equation (8.2.10) is known, the equation (8.2.10) transforms it to the standard open microstrip line in an inhomogeneous medium. The reverse medium transformation could be called inverse Wheeler’s transformation. This concept is useful for the modeling of multilayer microstrip and multilayer CPW structures. It is discussed in chapter 15. The filling-factor has a range 0.5 ≤ q ≤ 1. Under limiting cases, the effective relative permittivity of the microstrip line could be separated into two groups:

The following more accurate expression for the function F(w/h, εr) is obtained from the expressions given by Hammerstad–Jensen [J.12]: F w h, εr = 1 +

a=1+

+ For q = 0 5, w

h, very narrow strip conductor εr + 1 εreff f = 0 = a 2 For q = 1 0, w h very wide strip conductor εreff f = 0 = εr

b 8 2 11

On knowing the expression for the filling-factor, equation (8.2.10) computes the effective relative permittivity of a microstrip line for any w/h-ratio. In general, the filling-factor is a function of both w/h-ratio and εr of the substrate. It is expressed in the functional form using the structure-dependent function F(w/h, εr): q w h, εr =

F w h, εr + 1 2

8 2 12

The dependence of the filling-factor on the relative permittivity εr is estimated, from the expression (8.2.8), as follows [J.10, J.11]: 1 1 ≈ 1+ 1 W2 εr 1+ εr W1 1 qW h ≈ 1− εr

q W h, εr =

F

w 12 , εr = 1 + h w h

− 12

+ w h

εr − 0 9 εr + 3

2

1 w 52 h

4

1 ln 1 + 18 7

b = 0 564

a

+ 0 432 1 w 18 1 h

3

b

0 053

c 8 2 15

In the above expressions, the parameters “a” and “b” are functions of w/h ratio and εr, respectively.

8.2.2 Static Characteristic Impedance of Microstrip Line The static characteristic impedance of a lossless microstrip line, with zero conductor thickness, is obtained below, in terms of the characteristic impedance of a microstrip line with zero conductor thickness on the air-substrate. The static characteristic impedance is the dc characteristic impedance, i.e. for the case frequency f 0. The equation (8.1.14) helps to write the following expression: Z0 w h, εr1 = εr , εr2 = 1 = Z0 w h, εreff =

L w h, εreff C w h, εreff 8 2 16

qW h

In the above expression, we first assumed a special case for w1 = w2 and then multiplied it by the w/h dependent filling-factor, q(w/h). For the air-substrate, q(w/h, εr) = 0 leading to εreff(f = 0) = 1. For a very high permittivity substrate, the dependence of the filling-factor on permittivity εr can be ignored. Wheeler [B.1] has given the following approximate expression for the structure-dependent function F(w/h, εr) that is independent of εr: − 12

4

−a b

−1

8 2 13

12 F w h, εr = 1 + w h

1 ln 49

w h

10 w h

However, for a nonmagnetic dielectric-medium, the line inductance does not depend on the relative permittivity. For a microstrip in an equivalent homogeneous dielectric-medium, the expressions for its line inductance and line capacitance are a L w h, εreff = L w h, εreff = 1 ε0 εreff w C w h, εreff = = εreff C w h, εreff = 1 b h 8 2 17 The expression for the static characteristic impedance of a lossless microstrip line is obtained from the above two equations: Z0 w h, εr1 = εr , εr2 = 1, f = 0 =

w , for > 1 h

a

+ 0 04 1 −

w w , for ≤ 1 b h h

Z0 w h, εr1 = εr2 = 1 εreff f = 0 8 2 18

2

8 2 14

The static nature of the line characteristics is shown by writing frequency f = 0 as the parameter. It is dropped subsequently, unless it is needed to emphasize the static

8.2 Static Closed-Form Models of Microstrip Line

nature. Several closed-form expressions, with different accuracy, have also been proposed [B.1]. The closed-form expressions of Hammerstad–Jensen are summarized below for the characteristic impedance Z0(w/h, εr1 = εr2 = 1) of a microstrip line on the air-substrate [J.12]: Z0 w h, εr1 = εr2 = 1 = 60 ln

F + w h

where, F = 6 0 + 0 2832 exp −

30 666 w h

1+

2 w h

2

a

0 7528

b

8 2 19 The above expression has error about 0.01% for w/h < 1 and error 0.03% for w/h < 1000 with respect to the exact conformal mapping method. The static effective relative permittivity of a microstrip line with conductor thickness t = 0 is computed from expressions (8.2.10), (8.2.12), and (8.2.15).

8.2.3 Results on Static Parameters of Microstrip Line The filling-factor (q) is primarily a function of the w/h ratio. However, it also depends to some extent on εr [J.10, J.11]. It is incorporated in equation (8.2.15). Figure (8.6a) shows its variation from 0.5 to 1 corresponding to the narrow- and wide-strip conductors for permittivity εr = 2.5. The effective relative permittivity varies from (εr + 1)/2, for a very narrow-strip conductor, to permittivity εr, for a very wide-strip conductor. Figure (8.6b) shows an increase in εreff f = 0 with an increase in the w/h-ratio. It is due to the confinement of more electric field in the dielectric region below the strip conductor with an increase in w/ h-ratio. Table (8.1a) shows some computed results for εreff(f = 0) on the plastic substrate (εr = 2.3) and alumina substrate (εr = 9.8) using Hammerstad–Jensen (H-J) and Wheeler’s (Wh.) closed-form expressions. Results are also

εr = 20

4.0

Sqrt (εreff)

0.9 0.8

0.6

εr = 2.5 0.1

Characteristic impedance Ω

1.0

0.5

140

12 10 8 6 5 4 3 2

3.0

2.0

0.7

1

10 w h (a) Variation in filling-factor.

εr = 5

7

110

10 80

16

50 20

1.5 0.1

100

0.2

0.4 0.6 0.81.0 2.0 W/h-Ratio

4.0

(b) Variation in the square root of effective relative permittivity.

Characteristic impedance Ω

Effective filling fraction, q

15

1.0

100

εr = 1

2 4 10

0.1

0.2

0.3 0.4 0.50.6

1.0

W/h-Ratio (c) Variation in the characteristic impedance of narrow line. Figure 8.6 Characteristics of microstrip lines. Source: Maloratsky [J.1].

0.1

1.0 10.0 W/h-Ratio (d) Variation in the characteristic impedance of wide line.

271

272

8 Microstrip Line

Table 8.1a Comparison of computed εreff(f = 0) by several numerical methods and closed-form models. εr

2.3

MOL

SDA

Var

H-J

Wh.

εr

w/h

εreff

Z0 Ω

0.100

1.738

1.758

1.738

1.737

1.700

2.2

0.50

1.93

120

1.000

1.838

1.836

1.835

1.836

1.822

1.00

1.99

89.4

2.098

2.096

2.034

2.100

2.092

1.50

2.03

73.2 54.1

w/h

10.00 9.8

Table 8.1b The εreff(f = 0) and characteristic impedance of microstrip lines.

0.100

5.926

5.986

6.150

5.929

5.739

2.50

2.10

1.000

6.574

6.563

6.555

6.579

6.563

4.00

2.60

8.382

8.406

8.345

8.389

8.395

0.50

3.28

92.1

1.00

3.43

68.1

1.50

3.53

55.5

2.50

3.69

40.8

4.00

3.85

29.5

10.00

Comparison of εreff(f = 0) against experimental results w/h

Exp.

SDA

Var

H-J

Wh.

εr⊥ = 9 4

0.100

6.465

6.459

6.431

6.448

6.405

εrII=11.6

1.259

7.614

7.601

7.606

7.658

7.607

h=05±

9.140

9.787

9.775

9.758

9.725

9.702

4.8

9.30

Source: Edwards and Owens [J.13] and Hammerstad`and Jensen [J.12].

0 05 mm Source: Edwards and Owens [J.13] and Hammerstad`and Jensen [J.12].

presented using the numerical methods; such as the method of the line (MOL), variational (var.) method, and SDA at 1 GHz. As compared to the MOL, the results of the SDA have a maximum deviation of 1.5%, the variational method and Wheeler’s expression have above 3% deviation for the narrow microstrip; whereas the H-J model has deviation within 0.5%. Table (8.1a) also shows experimental results on the anisotropic substrate (Sapphire) to compare the accuracy of models against them [J.13, J.14]. Table (8.1b) shows a few computed results for both static effective permittivity εreff(f = 0) and characteristic impedance of a microstrip line. Figure (8.6c and d) further shows a decrease in the characteristic impedance of a microstrip line with the increase in relative permittivity of the substrate and also with an increase in w/h-ratio. It is because the characteristic impedance of a microstrip line is inversely proportional to the line capacitance that increases with the w/hratio and also with the relative permittivity of a substrate.

8.2.4 Effect of Conductor Thickness on Static Parameters of Microstrip Line Depending upon the substrate and the fabrication process, a microstrip line has strip conductor thickness in the range, t = 1 μm–35 μm, which is usually t/h < 0.3. Several numerical methods, including the conformal mapping methods, have been used to account for the effect of conductor thickness on microstrip line parameters, including losses. Figure (8.2b) shows that the conductor thickness t increases the content of the electric field lines in the air-medium. It results in a decrease in the effective relative permittivity of a microstrip line. It further decreases the characteristic

impedance of a microstrip due to the increase in the effective width of the strip conductor. Wheeler presented a method to compute the equivalent strip conductor width, weq(t/h, w/h, εr), of microstrip with finite conductor thickness [J.8, J.9]. The equivalent conductor width, with zero conductor thickness, is more than the physical width of the strip conductor. It helps to obtain expressions for the effective relative permittivity and the characteristic impedance of a microstrip line. In general, the equivalent strip conductor width is a function of t/h, w/h, and εr. It could be written in the form of the differential increase in the width of a strip conductor: w eq t h, w h, εr = w + Δw t h, w h, εr

8 2 20

Wheeler [J.8], using the conformal mapping method, has obtained the following equation for the differential increase in the width for a microstrip line on the air-substrate:

Δw t h, w h, εr = 1 =

t 2 1 + ln h t h t 4π 1 + ln π t h

, for w h > 0 16 , for w h ≤ 0 16

8 2 21 The above expression applies to a conductor of small thickness, i.e. for t/h < 0.1. Wheeler [J.9] has also obtained another expression that applies to all widths of thick conductor microstrip line on the air-substrate, in the range 0 < w/h < ∞, t/w < 1, and t/h < 1:

Δw t h, w h, εr = 1 =

10 872

t ln π t h

2

+

1 π w h + 1 10

2

8 2 22

8.2 Static Closed-Form Models of Microstrip Line

The equation (8.2.22) is to be modified for a microstrip line on a dielectric substrate. While replacing the original microstrip of a finite conductor thickness t by an equivalent microstrip with t = 0, the characteristic impedance of the original microstrip, with known w/h and t/h, and the characteristic impedance of the equivalent microstrip, with equivalent strip conductor width weq(t/h, w/h, εr) and t = 0, should remain unchanged. The line inductance and line capacitance of a microstrip view the differential conductor width in different ways. The line inductance does not depend on the relative permittivity of a substrate. Therefore, the differential conductor width for the line inductance remains unchanged for a microstrip on the air and the dielectric substrate: w eq t h, w h, εr

Induc

However, with the increasing εr of a substrate, more electric field lines concentrate in the dielectric region just below the width of the strip. It means the line capacitance experiences linearly decrease in the differential conductor width with increasing εr. It is assumed that the differential width due to the fringe field of a strip conductor decreases with an increase in relative permittivity εr: Cap

=

Δw t h, w h, εr = 1 εr

8 2 24

The strip conductor thickness has extended the width and increased the line capacitance that decreases the characteristic impedance. To maintain the constant characteristic impedance, the line inductance must be increased with differential width. Therefore, the relative changes in line capacitance, impedance, and inductance can be written as follows: Δw t h, w h, εr Cap ΔC Δw t h, w h, εr = 1 = = w C wεr Δw t h, w h, εr Z0 ΔZ0 = Z0 w ΔL Δw t h, w h, εr = 1 Indc Δw t h, w h, εr = 1 = = w L w

a b c

8 2 25 The characteristic impedance of microstrip is a function of L and C, i.e. Z0 = Z0 (L, C). Keeping in view decrease in Z0 with line capacitance, the differential change in Z0 with respect to differentials of L and C is dZ0 =

∂Z0 ∂Z0 dL − dC ∂L ∂C

Additionally, using Z0 =

a

∂Z0 = ∂C

L

d 1 1 C−2 = − dC 2

b L1 CC

d 8 2 26

On substituting equations (8.2.25a)–(8.2.25c) in the above equation, the following expression is obtained for the differential width Δw t h, w h, εr Z0 , as viewed by the characteristic impedance of a microstrip line on the dielectric substrate: Δw t h, w h, εr

Z0

=

1 1 Δw t h, w h, εr = 1 1+ 2 εr 8 2 27

Hammerstad–Jensen [J.12] have given the following set of closed-form expressions for the differential width of microstrip over air-substrate: Δw t h, w h, εr = 1 =

t 10 872 ln π t h coth2 6 517 w h

a

Δw t h, w h, εr 1 1 Δw t h, w h, εr = 1 = 2 coth εr − 1

b 8 2 28

The next task is to use the expression (8.2.20) to compute the characteristic impedance Z0(t/h, w/h, εr) and effective relative permittivity εreff(t/h, w/h, εr) of a microstrip line with strip conductor thickness. Two equivalent schemes are used for the computation of the characteristic impedance of a microstrip. In the first scheme, the extension in width weq(t/h, w/h, εr = 1) is obtained on the air-substrate, and then the characteristic impedance with conductor thickness is computed from the following equation: Z0 t h, w h, εr =

Z0 t = 0, εr = 1, w eq t h, w h, εr = 1 εreff t h, w h, εr 8 2 29

The effective relative permittivity εreff(t/h, w/h, εr) in the above equation is unknown. It could be computed from the following expression due to Bahl and Garg [J.17]: εreff t h, w h, εr = εreff t = 0, w h, εr −

εr − 1 t h 46 w h 8 2 30

L C, we have following:

∂Z0 1 d 1 1 1 = L2 = ∂L 2 LC C dL and

1 L ΔL ΔC + 2 C L C dZ0 1 ΔL ΔC + = Z0 2 L C

ΔZ0 =

= w + Δw t h, w h, εr = 1 8 2 23

Δw t h, w h, εr

Therefore, differential change in Z0 is

c

In the second scheme, we first obtain the extension in width on the dielectric substrate, i.e. weq(t/h, w/h, εr), and use previously discussed expressions to compute characteristic impedance on air-substrate with t = 0 and

273

8 Microstrip Line

effective relative permittivity with t = 0. Finally, the characteristic impedance of microstrip with finite conductor thickness, i.e. t 0, is computed from the expression given below: Z0 t h, w h, εr =

Z0 t = 0, εr = 1, w eq t h, w h, εr εreff t = 0, w eq t h, w h, εr , εr 8 2 31

The above expression does not require equation (8.2.30) for the computation of characteristic impedance. On equating equations (8.2.29) and (8.2.31), the following expression is obtained to compute effective relative permittivity of microstrip with finite conductor thickness: εreff t h, w h, εr = εreff t = 0, w eq t h, w h, εr , εr Z0 t = 0, εr = 1, w eq t h, w h, εr = 1 Z0 t = 0, εr = 1, w eq t h, w h, εr

×

2

h2 and the sidewall is at the distance “a” from the edge of the strip. The shield is used to avoid radiation from the circuit and also to protect the circuit from outside interference and the environment. Sometimes it is also used for the postfabrication adjustment of the designed circuit. For the far placed top shield and side shielding walls, i.e. h2/h1 > 10 and a/w > 20, the influence of shield on the characteristic impedance and effective relative permittivity could be disregarded. If the sidewalls are far away, the enclosed microstrip is reduced to the top-shielded microstrip, shown in Fig (8.8b). The line capacitance of an open microstrip Cmic increases due to additional shunt capacitance Csh introduced by the shield. These capacitances are shown in the circuit model of Fig (8.8c). Following equations (8.1.21) and (8.1.22), the effective relative permittivity and characteristic impedance of the unshielded and shielded microstrip lines could be obtained:

8 2 32 The above expression is more accurate as compared to equation (8.2.30). Figure (8.7a and b) illustrates variations in the effective relative permittivity [J.15] and characteristic impedance [B.11, J.16] with the conductor thickness. Both the effective relative permittivity and characteristic impedance decrease with an increase in the conductor thickness. The characteristic impedance also decreases with an increase in the w/h-ratio.

Unshielded line εreff = Z0 =

Cmic εr Cmic εr = 1

a

1

b

Cmic εr Cmic εr = 1 Cmic εr + Csh εr = 1 = Shielded line εShield reff Cmic εr = 1 + Csh εr = 1 c

c

ZShield = 0 1 c

d

Cmic εr + Csh εr = 1 Cmic εr = 1 + Csh εr = 1

8 2 33

8.2.5 Effect of Shield on Static Parameters of Microstrip Line Figure (8.8a) shows a shielded microstrip line in a metallic box. The substrate thickness is h1 (original h) and its relative permittivity is εr1 (original εr). The top shield height is

The above equations show that the characteristic impedance of an open microstrip is reduced in the presence of the shield, ZShield < Z0 . The effective relative permittivity of 0 microstrip is also reduced due to the presence of a shield,

8.2

t = 0.5 μm

8.1

t = 1.0 μm

8.0

t = 1.5 μm

7.9

1

2

3

4

5 6 7 Frequency (GHz)

8

9

(a) Effect of conductor thickness on εreff (f) [J.15]. Figure 8.7

10

65

63

On Scale #1

w/h = 1·0 On Scale #2 w/h=2·0 49 On Scale #3

48

Alunina (εr = 9.8)

34

33.5 Scale #3

t = 0.2 μm

Scale #2

8.3

50 w/h = 1·5

Scale #1

8.4

w = 200 μm, d = 200 μm

Characteristic impedance (Z0, Ω)

GaAs (εr = 12.9)

Characteristic impedance (Z0, Ω)

67 Effective relative permittivity

274

33 0.01 0.1 (t/h) Normalized characteristic impedance

0.001

(b) Effect of conductor thickness on Z0 [J.16].

Variation in microstrip parameters with conductor thickness. Source: Farina and Rozzi [J.15] and Gunsto and Weale [J.16].

t

εr2 = 1

h2 t

a εr1 h1

(a) Enclosed microstrip.

W

εr2 = 1 εr1

h2

Cmic (εr)

a

W

Csh (εr = 1)

8.2 Static Closed-Form Models of Microstrip Line

h1

(b) Top shielded microstrip.

(c) Circuit model.

Figure 8.8 Shielded microstrip line and its circuit model. Source: Hoffmann [B.1] and March [J.18].

εShield < εreff . The electric field of the microstrip also moves reff from the dielectric of the substrate to the air-region in the presence of a shield that reduces the effective relative permittivity of an open microstrip. The characteristic impedance and effective relative permittivity of a shielded microstrip can be determined by the static and dynamic SDA methods presented in chapters 14 and 16, respectively. Bahl [J.17] and March

[J.18] have reported the closed-form models for this purpose with higher error as compared to the results of the SDA. The modified March’s model [J.19] is summarized below. It has a deviation of 1

8 2 46 The w/h-dependent equivalent isotropic relative permittivity εeq r,iso w h with substrate thickness h and the stripwidth w can be used to compute the correct static effective relative permittivity, and the characteristic impedance of the original microstrip on the anisotropic substrate by using the closed-form expression of Hammerstad–Jensen. The process of obtaining the equivalent isotropic substrate has a 0.5% deviation against the SDA results.

8.3

Dispersion in Microstrip Line

The microstrip line is a dispersive transmission line with frequency-dependent phase velocity. Thus, its effective relative permittivity εr eff(f ) is also frequency-dependent which is computed by the full-wave methods [B.6, B.8,

B.9, J.5, J.6, J.29–J.36]. Normally, the frequency dependence is significant above 4 GHz. The frequency-dependent effective relative permittivity εr eff(f ) of a microstrip increases with frequency and its phase velocity decreases with the increasing frequency. It has significant implications for both the digital and RF analog signals on a microstrip. On a dispersive interconnect, frequency components of digital signal travel with different velocities. It results in a distorted pulse due to pulse spreading, ringing, etc. The distorted pulse limits the rate of high-speed data transmission. Likewise, a complex RF analog signal traveling on a microstrip also gets distorted due to dispersion. The guided waveεr eff f is also frequency-dependent. length λg = λ0 Thus, microstrip-based filters, couplers, and matching networks become frequency sensitive and these devices are frequency band limited. The characteristic impedance of microstrip is also frequency-dependent that creates a problem for broadband matching. A microwave circuit designer cannot work only with the static models of the microstrip transmission line parameters. Several closed-form models are reported in the literature [J.10, J.11, J.37–J.46]. Some closed-form dispersion models are based on the curve-fitting of dispersion data, obtained from the rigorous field-theoretic formulations. A few models are based on the coupling between the TEM mode and the lowest order surface wave TM0 mode. Some authors have considered the dispersion due to the coupling between TE1 surface wave mode and TEM mode. The experimental and numerical investigations have shown dispersion in a microstrip at a frequency below the cutoff frequency of the TE1 surface wave mode. Therefore, models based on the coupling between the TM0 surface wave mode and TEM mode appear to be more appropriate, as the TM0 surface wave mode has no cutoff frequency. This section presents a discussion on the nature of dispersion and closed-form models to compute the frequency-dependent effective relative permittivity and the characteristic impedance of a microstrip line.

8.3.1

Nature of Dispersion in Microstrip

The increase in the effective relative permittivity εreff(f ) with frequency is due to the coupling between the TEM parallel-plate mode and the TM0 surface wave mode [B.2]. During the process of field coupling, the electric field lines move from the air-region to the dielectric-region increasing the effective relative permittivity. The surface wave mode is supported by a metal-coated dielectric sheet. It is discussed in the section (7.5) of chapter 7. Thus, the quasi-TEM mode is viewed as a combination of these two modes: Quasi-TEM

TEM parallel-plate mode + TM 0 surface wave mode 831

8.3 Dispersion in Microstrip Line

εr

w εr εr

h

h

Parallel plate waveguide (w/h→∞)

εreff (f)

w

Original microstrip εr

Point of inflection

h

Metal backed dielectric sheet (w/h→0)

εreff ( f = 0) 0

f1

fi

f2

Frequency (b) Microstrip dispersion behavior.

(a) Microstrip limiting case with respect to w/h-ratio.

Figure 8.12 Limiting cases of microstrip with respect to w/h-ratio and frequency.

The above description of the quasi-TEM mode of a microstrip line leads to two limiting cases shown in Fig (8.12a). For w/h ∞, microstrip tends to a parallel-plate waveguide supporting the TEM fundamental mode. This case gives εr eff(f = 0) εr, whereas for w/h 0, the microstrip tends to metal-supported dielectric waveguide that supports the TM0 surface wave mode. This case leads to εreff(f = 0) (εr + 1)/2. Through the process of mode coupling, the electric field lines of a dispersive microstrip move from the air-medium to the dielectric-medium, so that the εreff(f ) increases from its lower limiting value εr eff(f = 0) to the upper limiting value εr for f ∞. The frequencydependent nature of εreff(f ) of a microstrip line in the frequency range 0 ≤ f < ∞ is shown in Fig (8.12b). Figure (8.12b) illustrates that εreff(f ) is a monotonically increasing function of frequency, i.e. ∂εr eff(f )/∂f > 0 in the range εr eff(f = 0) ≤ εr eff(f ) < εr. However, the frequency-dependent effective relative permittivity εreff(f ) has an inflection at the frequency fi, known as the inflection frequency. Schneider [J.39] states the following physical conditions that should be satisfied by the frequencydependent function of the effective relative permittivity: i Initial conditions εreff f εreff f = 0 , as f ∂εreff f 0, as f 0 ∂f ii Final condition εreff f εr , as f ∞ ∂εreff f 0, as f ∞ ∂f ∂ 2 εreff f iii Inflection condition = 0, at f = f i ∂f 2

0

ia ib ii a ii b iii

832 The first initial condition states that with decreasing frequency, εreff(f ) tends to a static value of the effective relative

permittivity of microstrip and the second initial condition states that there is no initial jump (discontinuity) in εreff(f ) at f = 0. The first final condition states that εreff(f ) saturates to the value εr with f ∞ and the second final condition states about the smooth saturation of εreff(f ) with increasing frequency. The presence of the inflection frequency fi shows that below it the change in εreff(f ) is accelerated one, and above it the change in εreff(f ) is de-accelerated, leading to its saturation to relative permittivity εr. All dispersion models must follow these physical conditions. However, the dispersion models following Schneider’s condition may not have identical accuracy. Figure (8.12b) shows that a microstrip line is less dispersive, both in the lower frequency range from 0 to f1 and in the upperfrequency range from f2 to (f ∞). It shows a rapid change in εreff(f ) during the frequency range f1–f2, around the inflection frequency, leading to higher dispersion [J.10, J.11]. Therefore, if a complex signal has all its frequency components, fundamental and several higher harmonics, below f1, the signal may not be much distorted due to dispersion. Similar is the case for the distortion in a signal if its fundamental and higher harmonics occupy the frequency spectrum f2 < f < ∞. However, if frequency components of a complex signal occupy the frequency spectrum f1 < f < f2, the signal undergoes a severe distortion due to the significant dispersion in the microstrip. The confinement of the electric field between the strip conductor and the ground plane suggests that a microstrip line can be treated with the help of the waveguide model. The waveguide model of a microstrip has been developed through the efforts of many investigators [J.46–J.48]. The model has limited accuracy. However; it is a simple one and it provides several characteristics of a microstrip line. Moreover, it has been used to study the behavior of several

279

8 Microstrip Line

kinds of discontinuities in a microstrip [B.12, J.49]. A consolidated formulation of the waveguide model is presented below.

and the characteristic impedance of a microstrip line. It is also used to obtain the cutoff frequency of the higherorder mode of a microstrip line. The cutoff frequency of the waveguide model is

Waveguide Model of Microstrip

Figure (8.13a) illustrates the equivalent parallel-plate waveguide model with the equivalent width weq = w + Δw of a microstrip so that it contains all the electric field lines between two conducting strips [B.12, J.47, J.49]. The top and bottom are the electric walls (EW), whereas the sidewalls are the magnetic walls (MW). This waveguide is discussed in chapter 7. It supports the fundamental TEM mode with zero cutoff frequency, which is also supported by the original microstrip line. The next higher-order mode is TE10. The waveguide model is used to obtain expressions for the frequency-dependent effective relative permittivity

C w h, εr =

ε0 εr w eq f = 0 h

1 fc = 2 με

m w eq

2

n + h

2

12

833

For the TE10 mode, the modal numbers are m = 1, n = 0. The cutoff frequency fT of the TE10 mode is fc

10

= fT =

c 2w eq μr εr

,

834

where c is the velocity of the EM-wave in the free space. A dielectric substrate has μr = 1. For equivalence, the line capacitances of the original microstrip line and the parallelplate waveguide should be identical:

εreff 0 1 = cZ w h, εr vp Z w h, εr

Waveguide medium

(8.3.5)

Microstrip line

weq = w + Δw εr

MW

w

h

Ground conductor

MW

8.3.2

εr

h

Ground conductor

(a) Parallel-plate waveguide model of a microstrip. 60 Characteristic impedance Ω

58 Characteristic impedance Ω

280

Waveguide model 54

SDA 50

V-P

I-P

50

4

8 12 Frequency (GHz)

16

20

(b) Comparison of the waveguide model with SDA. Figure 8.13

Rautio 3-D

Zhu & Wu 3-D

45

0

V-I

55

0

5

10

15

Frequency (GHz)

(c) Z0 using several definitions [J.50].

Parallel-plate waveguide model of a microstrip. Source: Zhu and Wu [J.50].

20

8.3 Dispersion in Microstrip Line

The above equation gives an expression of the static equivalent width of the parallel-plate waveguide: w eq f = 0 =

h εreff 0 hεreff 0 = ε0 εr cZ0 w h, εr ε0 εr cZ0 w h, εr = 1 836

The static effective relative permittivity and characteristic impedance, used in the above equation, are obtained from the closed-form expressions of Hammerstad–Jensen, or Wheeler, presented in section (8.2). The following empirical relation can be written for the frequency-dependent effective relative permittivity that is known as Getsinger’s model [J.45]: εreff f = εr −

εr − εreff 0 1 + f fT 2

837

The above expression meets Schneider’s physical conditions mentioned in the previous subsection. It is noted that the operating frequency f is normalized by the cutoff frequency of the TE10 waveguide mode. The cutoff frequency fT is modified by taking into account the fringe field through a parameter G, called the Getsinger’s parameter. The modified expression of cutoff frequency fT of a microstrip on a dielectric substrate (μr = 1) is obtained in terms of the Getsinger’s parameter G. For this purpose, the equivalent width weq, from equation (8.3.6), is substituted in equation (8.3.4), to get the following expression: fT = ft

εr εreff 0

= ft

1 G

838

The normalized frequency ft and parameter G are expressed as follows: Z0 w h, εr ft = 2μ0 h

a,

εreff 0 G= εr

b 839

On substituting equation (8.3.8) in equation (8.3.7), the Getsinger’s dispersion model is obtained: εreff f = εr −

εr − εreff 0 1 + G f ft 2

8 3 10

The accuracy of the above model compares well against several available models [J.4]. The waveguide model is also used to get an expression for the frequency-dependent characteristic impedance of a microstrip line [J.47, J.48, J.51]. To do so, we have to obtain an expression for the frequencydependent equivalent width of the waveguide model. Frequency-Dependent Equivalent Width

The physical width of a microstrip strip conductor is enlarged due to the fringing electric field. The electric field lines move below the strip conductor in the dielectric

region with an increase in frequency. Such enhanced confinement of the electric field in the dielectric region is due to an increase in the coupling, with increasing frequency, between the TEM mode and the TM0 surface wave mode. It results in the monotonic decrease in width, i.e. ∂weqf(f )/∂f < 0, of the waveguide model toward the physical width w of a microstrip. Schneider’s physical conditions are adapted to the frequency-dependent equivalent width as follows [J.46]: i Initial condition w eq f = 0 = w + Δw, as f w eq f ∂w eq f 0, as f ∂f ii Final condition w eq f w, as f ∂w eq f 0, as f ∂f ∂ 2 w eq f iii Inflection condition = 0, at ∂f 2

0

ia

0

ib ∞

∞ f = fi

ii a ii b iii 8 3 11

The dispersion model of Getsinger’s can be empirically modified to model the frequency-dependent behavior of the equivalent width: w eq f = w +

w eq f = 0 − w 1 + G f ft

8 3 12

2

Frequency-Dependent Characteristic Impedance

The following replacements are done in equation (8.3.6) to get an expression for the frequency-dependent characteristic impedance of a microstrip line: εr

εreff f = 0

c=

1 ,η = μ0 ε0 0

Z0 w h, εr , f =

εreff f , w eq f = 0 μ0 = 377 Ω ε0 377h

w eq f εreff f alternatively, Z0 w h, εr , f = Z0 f = 0

εreff f − 1 εreff 0 − 1

εreff 0 εreff f

w eq f ,

a

b, 8 3 13

where εreff(f ) and weq(f ) are obtained from equations (8.3.10) and (8.3.12), respectively. The above expression shows an increase in the characteristic impedance of a microstrip line with frequency. Figure (8.13b) compares the frequencydependent characteristic impedance of a 50Ω microstrip line on a substrate εr = 10.0, h = 0.635 mm, w/h = 1.0, as computed by the waveguide model, against the results of the SDA [B.1, J.52]. The computed results from the waveguide model are not accurate. However, the deviation in the computed characteristic impedance is

281

282

8 Microstrip Line

within 2% for the frequency up to 10 GHz. It is interesting to note that this simple model shows that the dispersion behavior, i.e. the increase in characteristic impedance with frequency, follows the power-current definition of characteristic impedance. Figure (8.13c) shows the full-wave results, using the P-V, V-I, P-I, and 3D port definitions, of the characteristic impedance of microstrip on an alumina substrate, εr = 9.7, h = 0.635 mm, w/h = 1.0 [J.36, J.50]. There is no unique definition of hybrid-mode characteristic impedance of a microstrip line [J.51]. However, power-current (PI) is more acceptable and even the 3D port definitions from both sources are close to the P-I definition; although these definitions are noncasual. A causal definition of the characteristic impedance has also been proposed [J.53]. Krischning–Jansen model provides a set of closed-form expressions to compute the frequency-dependent characteristic impedance of a microstrip line that follows the powercurrent definition [J.54]. Subsection (8.3.5) summarizes the expressions. At this stage, it is also noted that the Zo εr , f = εreff f definition is popular among microZo εr , f = 0 wave designers, although it shows a decrease in the characteristic impedance of a microstrip line with frequency. However, equation (8.3.13b) suggests alternate empirically obtained expression for the characteristic impedance that exhibits increasing characteristic impedance, similar to the P-I-based model, with an increase in the frequency [J.14]. The effect of the conductor thickness can be incorporated in this expression by using weq(f ) in place of the strip width w, and by modifying the expression for the εreff(0) and εreff(f ) suitably. Higher-Order Mode

Using expression (8.3.4) and the replacement discussed above, the cutoff frequency of higher-order modes of a microstrip can be estimated by the following expression: mc 8 3 14 f c m, 0 = 2w eq f εreff f It has been demonstrated that m = 0, 1, 2… approximately corresponds to the hybrid modes HE0 (HE0,0), HE1 (HE1,0), HE2 (HE2,0), etc. [B.12, J.49]. The fundamental mode HE0 is the quasi-static mode. The effective relative permittivities of the higher-order modes HE1, HE2, etc., are dispersive and can be computed using the waveguide model. The guided wave propagation constant of the HEm mode obtained from the waveguide model is βm g where

f c m, 0 = k 1− f

βm g = β0 k = β0

In the above equation, k is the wavenumber in the equivalent homogeneous medium. The following dispersion equation for the higher-order modes are obtained from the above equations: εm reff f = εreff f = 0 1 −

a

εm r eff f

b

εr eff f = 0

c

2

8 3 16

8.3.3 Logistic Dispersion Model of Microstrip (Dispersion Law of Microstrip) Figure (8.12b) shows the phenomenological dispersion behavior of a microstrip. It is used to develop the logistic dispersion model (LDM) [J.4]. The LDM is based on the coupling of the TEM mode and TM0 surface wave mode to determine the inflection frequency fi. The dispersion in a microstrip line is due to the growth of the effective relative permittivity εr eff(f ) from its static value εr eff(f = 0) to the relative permittivity εr of a substrate with the increase in frequency, in the range 0 ≤ f ≤ ∞. The following statement can be made on the dispersion law for the growth of the effective relative permittivity of a microstrip line with increasing frequency: “The rate of increase of the effective relative permittivity with frequency α [Effective relative permittivity at given frequency] × [Remaining fractional relative permittivity of the substrate]” The above statement of the dispersion law can be written in the form of the first-order differential equation: dεreff f εr − εreff f = Kεreff f , df εr

8 3 17

where K is the proportionality constant. The constant K is structure-dependent, i.e. it depends on the w/h-ratio and relative permittivity εr of a substrate. The K could be estimated and expressed in terms of the static effective relative permittivity εreff(f = 0) and εr. For the sake of convenience, the K is taken on the natural logarithms as follows: K = ln

3 εreff 0 − εr εreff 0

8 3 18

The solution of the differential equation (8.3.17) leads to the following logistic dispersion equation: εreff f =

2

f c m, 0 f

εr , 1 + e− K f + C

8 3 19

where C is the constant of integration. Using Schneider’s initial condition (8.3.2), equation (8.3.18), and e−c = M, constant M is obtained from equation (8.3.19): 8 3 15

M=

εr − εreff 0 εreff 0

8 3 20

8.3 Dispersion in Microstrip Line

The operating frequency in equation (8.3.19) is normalized by the inflection frequency fi. The final form of the logistic dispersion equation is εr εr eff f = 8 3 21 1 + M e − K f fi The above equation meets Schneider’s conditions (8.3.2). However, it does not meet the condition (8.3.2–i b). The inflection frequency is determined from the coupling frequency fk, TM [B.2, J.10]: εr eff 0 − 1 εr − εr eff 0

tan − 1 εr f k,TM = c

2πh

εr − εr eff 0

f k,TM w 3 1+B h

A = ax + b, 8 3 24

For − 1 ≤ x ≤ 0, i e 0 1 ≤ w h ≤ 1 a = − 0 1122εr + 1 428, b = 0 0649εr + 0 3136 For 0 < x ≤ 0 7, i e 1 < w h ≤ 5 a = − 0 0927εr + 0 9081, b = 0 0648εr + 0 3142 For 0 7 < x ≤ 1, i e w h ≥ 5 A = 0 95 For 10 ≤ εr ≤ 20, x ≥ − 1, i e w h ≥ 0 1 A = 0 95 8 3 25 For 20 ≤ εr ≤ 30, w h ≥ 0 6 w h

− 0 1435

01≤w h≤1

B=05 08+e For x > 0 3, i e

1≤w h≤2

8 3 29

− 1 7047x

w h>2

B=08

8 3 30

The LDM expression (8.3.21) does not meet Schneider’s condition (8.3.2-i b) that results in higher value for the effective relative permittivity, as compared to results of the SDA. A correction factor is incorporated into the expression of the LDM. The form of correction term is obtained from the first derivative of equation (8.3.21) in which εr is replaced by (εr − εreff(0)) to avoid a significant change in the nature of the LDM. The correction factor Δεr eff(f ) is, M K εr − εr eff 0 P

−Q

f , fi

f ≤ fi

M K εr − εr eff 0 − Q f i , f > fi P f fi f 2 , P = 1 + M e−R where Q = k , R = K fi f

8 3 31

The corrected LDM is εr

εr eff f =

− EΔεr f 1 + M e − K f fi 0, 0 1 ≤ w h < 5 where E = , for 1 ≤ εr ≤ 20 1, w h≥5 and

E=

1,

05≤w h > j ωε, the wave equation (4.5.3) of chapter 4 is written as the diffusion equation in three forms: ∇2 E = jωμσ E

a,

∇2 H = jωμσH

c

∇2 J = jωμσ J

b

8 4 27

d2 J z = jωμσJz = Γ2 Jz dy2 The propagation constant Γ is Γ=

j 2πfμσ = 1 + j

y

Z

1

δ=

πfμσ

L

(a) Longitudinal current flowing on the surface of the conductor. Figure 8.15

V

y

Jz y = J0 e

b

8 4 29

,

8 4 30

− Γy

y y −j = J0 e δ × e δ −

Amplitude Phase of current density 8 4 31

Longitudinal direction Jz

(b) Penetration of surface current inside the metal strip.

Surface current penetration in the thick metal strip.

a

where inside a conductor μ = μ0. If J0 is the surface current density at the top surface of a conductor, then the solution of equation (8.4.28) is written as

x

Zseries

I

πfμσ

The parameter δ is known as the skin-depth inside a planar thick conductor. It is given by

Jz z

8 4 28

Γ= 1+j δ

Figure (8.15a) shows that the longitudinal current density (Amp/m2) Jz, associated with the EM-wave, flowing in the z-direction on the surface of a conductor. The current I is supplied by the voltage source V, connected across the line of length L. The conductor width in the x-direction is infinite, leading to the uniform current flow on a strip. The strip conductor section offers a series of impedance Zseries to the source. It is evaluated below. Figure (8.15b) shows that while the surface current density Jz, at the top of the conductor, flows in the z-direction, it also penetrates inside the conductor in the y-direction. Such penetration of the surface current is responsible for the conductor loss in a microstrip line. The penetration of longitudinal Jz in the y-direction is described by the 1D diffusion equation:

Depth

288

8.4 Losses in Microstrip Line

The lineal current density per unit width (Amp/m), flowing in the z-direction, is ∞

∞

Jzw = 0

=

J0 e −

Jz y dy =

1+j δ

y

dy =

J0 δ 1+j

R=

0

J0 δ − 45∘ 2

Therefore, the z-directed lineal current density is 45 out of phase, with respect to the surface current density at the surface of the conductor. The current decays exponentially inside a conductor. At the surface of the finite conductivity conductor, the electric field intensity E0 is related to the voltage drop, J0 = σ E0. The lineal current density per unit width is δσ E0 8 4 33 Jzw = 1+j The above equation gives an expression for the surface impedance: E0 1+j = σδ Jzw 1 δ Rs + jωLi = + jωμ σδ 2

Xi = ωLi =

Rs w

b

1 1 Re V I∗ = Re E0 L Jzw w ∗ 2 2 1 1 = Re Rs Jzw L Jzw w ∗ = Rs Jzw 2 Lw 2 2

P=

The power loss per unit surface area is p = (1/2)Rs|Jzw|2 = (1/2) Rs |Js|2, where Jzw = Js is the linear surface density per unit width. It is related to the surface magnetic field by the expression |Js| = |Ht|. The power loss over the conducting surface is P=

8 4 34 b,

w

Jzw x dx = Jzw w

8 4 35

0

In the above equation, Jzw(x) is assumed to be uniform with respect to x. It is an unrealistic assumption, as the current distribution across the width of a conducting strip has an edge singularity. However, at present the voltage drop across a length of the conductor L is V = E0 L

8 4 36

Therefore, the series impedance Zseries offered by the block of a strip conductor connected to a source is obtained as follows: =

1 2

Rs Js 2 ds

8 4 39

S

where equation (8.4.34b) is obtained using equation (8.4.30). The real part Rs of the above equation is known as the surface resistance. It is responsible for the conductor loss. Its imaginary part forms the internal inductance of a conductor, due to the penetration of the magnetic field. At the field penetration y = δ/2, the surface resistance and surface reactance are numerically equal, i.e. ω Ls = Rs. Figure (8.15a) shows the line current I (Amp) passing through the conductor of width w. It is computed as follows:

V E0 L L = = Zs I Jzw w w

a,

The power loss in a conductor of length L and width w, carrying the uniform surface current Jzw per unit width, is

a

Zs = Rs + jωLi =

Zseries =

1 Rs = w σδw

8 4 38 8 4 32

I=

of the series impedance is resistance Rseries or simply R, i.e. the line resistance. The line resistance and the internal reactance p.u.l., i.e. for L = 1, are

1+j σδ

L w 8 4 37

For L = w, Zseies = Zs, i.e. any size of a square conductor offers the same series impedance to the source. The real part

To compute the conductor loss of a microstrip line, the above equation is applied to both the strip and ground conductors. This method is known as the perturbation method [B.9]. It requires determination of the surface current density Js over both the strip and ground conductors. It has been done by numerical methods [J.60]. The closed-form Maxwell’s distribution function takes care of the edge singularity [J.61, J.62]. The closed-form expressions for the current distribution over the ground conductor have also been obtained [J.63]. However, for simplicity, the uniform current density is assumed over the strip conductor and uses the parallel-plate waveguide model to compute the conductor loss of a microstrip line. It is used to compute the conductor loss of a waveguide discussed in chapter 7. Next, the results are summarized for the perturbation method that takes care of Maxwell’s distribution function of surface current distribution showing the edge singularity. Waveguide Model for Computation of Conductor Loss of Microstrip Line

The waveguide model of a microstrip line is a parallel-plate transmission line of conductor width w and conductor thickness t. Two conducting plates are separated by a distance h, i.e. the thickness of the substrate. The parallel-plate line contains a dielectric of relative permittivity εr eff(f ). Using equation (8.4.38a), the RF line resistance p.u.l. is R=

2 σδs w

8 4 40

The factor 2 in the above equation is due to two strip conductors, lower and upper ones, forming the parallel-plate

289

290

8 Microstrip Line

line. The δs is the skin-depth of a strip conductor. However, if the strip and ground conductors are of different materials with different conductivities, the line resistance p.u.l is 1 1 R = Rss + Rsg = + , σδs w σδg w

8 4 41

where δg is the skin-depth of the ground conductor. In the above equation, Rss and Rsg are the surface resistance of strip and ground conductors, respectively. The conductor loss p.u.l. is given by equation (8.4.18): αC =

R 2Z0 εr

=

Rss + Rsg 2Z0 εr

Np length

8 4 42

R=

2 σtw

8 4 45

For the above case, the current is uniformly distributed throughout the strip thickness and conductor loss is frequency independent. At high frequency, above fc2, corresponding to t ≥ 2δs, the conductor loss follows the square root law given by (8.4.43c). These frequency limits are given by the following expressions [J.64, J.65]: a

Upper frequency limit f c2

b, 8 4 46

As the characteristic impedance of a parallel-plate waveguide is Zo εr = 120π h w 1 εr = h w μ0 ε0 εr, the conductor loss of a microstrip line using equations (8.4.18), (8.4.30), and (8.4.40) is πδs

εr eff f Np length λ0 h 27 3δs εr eff f αC = dB length λ0 h ε0 εr eff f f dB length αC = 27 3 h πσ

αC =

a b c 8 4 43

The waveguide model of microstrip helps to replace εr εr eff(f ) in the above expressions. For the strip and ground plane conductors, with different conductivities, the conductor loss is αC =

πμr μ0 2Z0 εr w

1 + σs

1 σg

f

Np length 8 4 44

The nonmagnetic conductors have μr = 1. The various forms of the above expressions to compute the conductor loss are valid only for a wide microstrip with w/h ∞. For w/h ≤2, these expressions show about 80% higher conductor loss, as compared to the experimental results [J.61, J.62]. The conductor loss increases with the square root of frequency, provided εr eff(f ) does not change much with frequency. Although the expression for the surface resistance is obtained for an infinitely thick conductor, however it is valid for thickness t ≥ 3δs. The above expressions show that at nearly zero frequency, f 0, the conductor loss is zero. In practice, this not the case. The skin-depth becomes larger than the conductor thickness with decreasing frequency. For the case of the conductor thickness less than the skin-depth, i.e. for t < δs corresponding to lower frequency limit fc1, the RF resistance is replaced by the dc resistance given below:

R 2πL 4 = πμσt2

Lower frequency limit f c1 =

where R = 1 σ w t and L are resistance and inductance p.u.l. of a microstrip. It is also obvious from equation (8.4.43c) that the conductor loss increases with increasing permittivity of the substrate. Equation (8.4.44) shows that the conductor loss decreases with increasing strip width w. In the above equations, conductivity of a conductor is treated as constant, i.e. independent of the frequency. However, the computed conductor loss deviates from the experimental results with increasing frequency. The following empirical expression for the frequencydependent conductivity improves the computed frequency-dependent conductor loss [J.66]: σ f =σ 0

1 + C0 f,

8 4 47

where σ(0) is the given or measured dc conductivity of a conductor, f is the frequency in GHz and for a microstrip line the constant C0 = 0.045. Drude model, discussed in chapter 6, section (6.5), shows a decrease in the conductivity with frequency. However, the experimentally obtained conductor loss follows the above relation. Possibly the conductor–dielectric interface morphology is responsible for such behavior. Figure (8.16a) compares the computed conductor loss by the present method against the experimental results of a microstrip [J.66]. It shows good agreement of the computed and measured conductor loss. For the conductor thickness much more than the skin-depth, t > > δs, the conductor loss shows a linear behavior on the logarithmic frequency scale. In the lower frequency range, t < δs, the conductor loss deviates from it. However, if the expression for the surface impedance, given by equation (8.4.34), is replaced by the thickness-dependent surface impedance, more accurate computed results are obtained. It is shown in Fig (8.16a). Such expression is presented later on. The minimum conductor loss, i.e. the loss smaller than the loss of the infinitely thick conductor, is obtained for a conductor thickness in the range 1.142 δs < t < 2.851 δs.

8.4 Losses in Microstrip Line

Measured Calculated

Finite thickness conductor Infinitely thick conductor

102 10–1

11 10 9 8 7 6 5 4 3 2 1

Conductor loss αc (dB/m)

Conductor loss αc (dB/m)

104

10–2

Zo = 55 ohm w = 12.5 um h = 7.4 um t = 1.3 um εr = 3.12 w/h = 1.69,t/h = 0.17 σ(0) = 3.9e7 (Au) Co = 0.00445

0

Frequency (GHz) (a) w = 20 μm, t = 0.65 μm, h = 2.8 μm, εr = 3.0, tanδ = 0.01, σ = 3.6 × 107 s/m.[J.66].

20

Exp t CST HK, σ(0) HK, σ(f)

40 60 80 Frequency (GHz)

100

(b) Conductor loss on TFML(55 Ω) [J.67].

Figure 8.16 Conductor of the microstrip line. Source: Konno [J.66] and Bansal et al. [J.67].

Perturbation Method Using Surface Current with Edge Singularity

A narrow microstrip line has a concentration of current at the edges, resulting in higher conductor loss. The waveguide model does not treat the edge singularity of the current distribution on a strip conductor and the nonuniform current distribution on a ground plane. A summary of two methods that take care of the edge singularity of the surface current is given below. Conformal Mapping Method

Collin [B.8] has corrected the expression for a surface resistance of both the strip and ground conductors given in equation (8.4.34) by using the conformal mapping method. The results are summarized below: For strip conductor Rss = Rs

1 1 4πw + 2 ln π π t

LR

where for w h ≤ 0 5 LR = 1; 0 5 < w h ≤ 10 LR = 0 94 + 0 132w h − 0 0062 w h For ground conductor w h , 0 1 ≤ w h ≤ 10 Rsg = Rs w h + 5 8 + 0 03h w

a 2

b

8 4 48 where Rss and Rsg are surface resistances of the strip and ground conductors, respectively. In the above expression, the effective width of the ground plane is w + 5.8h of the uniform current density. Collins has taken surface

y=

ωμ σ . Liou [J.68] has suggested another resistance Rs = multiplying factor for correct expression of the surface impedance that accounts for w and t of a conductor: Zs Improved = Zs

2t 1+ w

w δ 1−e s −

t δ 1−e s −

8 4 49 It is seen that for w

∞ , t/w

0, Zs(Improved)

Zs.

Perturbation Method

The perturbation method is used to compute the conductor loss of both the thin and thick strip conductors. The conductor thickness of a thin conductor is less than the skindepth. However, the method is applied separately to the strip conductor and the ground conductor. In the case, the strip current distribution is expressed through Maxwell’s function, Jz = I

w 2 2 − x2 , equation (8.4.39)

shows the logarithmic divergence. To avoid it, the integration omits the endpoints of the width and stops at ±(w/2 − Δ), where Δ is known as the stopping distance [J.69]. For an isolated strip, Holloway and Kuester have given the values of the stopping distance in a tabular form [J.70]. However, the following closed-form expressions for the reciprocal of normalized stopping distance Δ, as a function of normalized strip conductor thickness, are also available for the CAD application [J.71]:

− 60 89 x4 + 282 17 x3 − 27 764 x2 + 0 5103 x + 9 1907 711 01 x6 − 2120 2 x5 − 1038 3 x4 + 9881 7 x3 − 12798 x2 + 6935 8 x − 1372 7

for 0 03 ≤ x < 0 64 for 0 64 ≤ x < 1 5

− 183 05 x4 + 1583 x3 − 4981 9 x2 + 6618 6 x − 2807 6 0 2953 x4 − 7 7733 x3 + 73 991 x2 − 293 88 x + 581 98 − 0 0198 x3 + 0 9848 x2 − 12 104 x + 240 01

for 1 5 ≤ x < 2 76 for 2 76 ≤ x < 8 0 for 8 0 ≤ x < 16

8 4 50

291

292

8 Microstrip Line

The normalized parameters used in the above expressions are given below: Reciprocal normalized stopping distance y = t Δ t Normalized strip thickness x = 2 0δs

a b 8 4 51

Using the concept of stopping distance, Holloway and Kuester (HK) have computed the conductor loss of a microstrip line as a summation of the conductor loss on the strip (αcs) and conductor loss on the ground plane (αcg), i.e. αc = αcs + αcg [J.69, J.72]. The expressions are summarized below: Loss on strip Rsm w−Δ αcs = 2 ln Np m 2π wZ0 w eff , h, t, εr , f Δ cot kc t + csc kc t where, Rsm = μ0 ωt Im kc t σ 12 ωε0 Loss on ground plane Rsm αcg = π w Z0 w eff , h, t, εr , f w h w − ln 1 + tan − 1 2h w 2h kc = ω μ0 ε0 1 − j

a b

Surface Impedance of Composite Layers Conductor

Figure (8.17) shows a two-layered conductor of an infinite extent. It has uniform current distribution at the upper conducting layer of thickness t, conductivity σ1, and permeability μ1. The lower conductor is infinitely thick. Its conductivity and permeability are σ2 and μ2, respectively. The 1D diffusion equation (8.4.28) is solved for the current density in both the conducting media. The more appropriate solution involves the solution of the 2D diffusion equation that also accounts for the width of the conducting strip [J.73]. Medium – I Jz1 = A sinh Γ1 y + B cosh Γ1 y , 0 < y ≤ t Medium – II Jz2 = C e − Γ2 y − t , y ≥ t 1+j = 1 + j πfμ1 σ1 and where Γ1 = δs1 1+j Γ2 = = 1 + j πfμ2 σ2 δs2

a b

c 8 4 53

c

In both media, the longitudinal current density obeys Ohm’s law: Ezi =

2

d 8 4 52

In the above equation, “Im” stands for imaginary part, Rsm is the surface impedance and kc is the wavenumber in a conductor. In the above expression, Holloway and Kuester have taken the DC bulk conductivity of the conductor. However, equation (8.4.47) suggests frequency-dependent conductivity. Furthermore, for a thin film microstrip line (TFML), shown in Fig (8.1e), the conductivity is thickness dependent and is less than the conductivity of a bulk metal. Figure (8.16b) shows the conductor loss, up to 100 GHz, of a 55Ω TFML constructed with the gold film, where thin-film conductivity of gold is taken as 3.9e7 S/m, instead of the bulk conductivity of gold that is 4.1e7 S/m. The Holloway and Kuester model with frequency-dependent, conductivity σ(f ) follows the experimental results more closely, as compared to their model with σ(0). Figure (8.16b) also shows the results of the commercial EM-Simulator CST, with more deviation against the experimental results [J.67]. In the above discussion, a single layer of metal for a strip conductor is considered. However, a practical microstrip line has composite layers of metals to improve its adhesion to the dielectric substrate. The fabrication process is discussed in chapter 13. A more general expression for the surface impedance, accounting for the composite layer of metals, is obtained below.

Jzi σi

i = 1, 2 showing medium

8 4 54

The tangential magnetic field components are obtained from Maxwell’s equation: Hxi =

1 dEzi σi dEzi = 2 jωμi dy Γi dy

i = 1, 2

8 4 55

Equation (8.4.29) is used to get the final form of the above equations. The electric and magnetic fields in both media are obtained from the above equations: 1 A sinh Γ1 y + B cosh Γ1 y σ1 1 = A cosh Γ1 y + B sinh Γ1 y Γ1

Ez1 =

a

Hx1

b

Medium I

8 4 56

Jz z t

μ1 μ2

y

x σ1 σ2

Figure 8.17 EM-wave propagating in Z-direction of the two-layered metallic conductor.

8.4 Losses in Microstrip Line

C − Γ2 e σ2 C − Γ2 = e Γ2

Ez2 = Medium II Hx2

y−t

Zs y = 0 = 1 + j Rs1 tanh Γ1 t

a 8 4 57

y−t

b Rs t = Re Zs y = 0

At the interface of both conductors, the following continuity conditions are used: Ez1 = Ez2 ,

Hx1 = Hx2

8 4 58

The above expressions provide a ratio of two constants: B Γ1 σ2 Γ2 σ1 = − Γ1 σ2 Γ2 σ1 A

= 1 + j Rs1 tanh

Γ2 σ1 cosh Γ1 t Γ1 σ2 Γ2 σ1 cosh Γ1 t + sinh Γ1 t Γ1 σ2

sinh Γ1 t +

•

= Rs1

t δs1 tanh 1 + j t δs1 1+j

a b 8 4 65

Medium II is a perfect dielectric insulator, i.e. σ2 leading to Rs2 ∞. Equation (8.4.64) is reduced to

Zs y = 0 = 1 + j Rs1 coth 1 + j Rs t = Re Zs y = 0

t δs1

= Rs1 coth 1 + j

a t δs1

b

8 4 59 Γ1 1+j = = 1 + j Rs1 δs1 σ1 σ1 Γ2 1+j = = 1 + j Rs2 σ2 δs2 σ2

where,

B = − A

a 8 4 60 b

Rs2 cosh Γ1 t Rs1 Rs2 cosh Γ1 t + sinh Γ1 t Rs1

sinh Γ1 t +

8 4 61

8 4 66 The above equation provides Leontovich surface impedance of a conductor of finite thickness [J.63]. This is an important case involving either a thin ground conductor over an air-medium or a thin ground conductor over the dielectric substrate. These conditions arise in a microstrip. The above expression is also used for different conductors of the strip and the ground plane. The surface resistance obtained from the above equation is

At the top surface of the conductor I, i.e. at y = 0, the surface current density is Jsz = − Hx

8 4 62

y=0

Ez

y=0

Jsz

= −

Ez1 Hx1

Rs t = Rs1

RF = y=0

Using the above two equations, an expression is obtained for the surface impedance of conductor I having finite thickness t. The conductor I is supported by another conductor of an infinite extent. Zs y = 0

where,

sinh y + sin y , where cosh y − cos y

a

y = 2t δs1

8 4 63

Γ1 B B = − 1 + j Rs1 = − σ1 A A

sinh Γ1 t + Rs2 Rs1 cosh Γ1 t cosh Γ1 t + Rs2 Rs1 sinh Γ1 t Zs = Rs + jωLis

= 1 + j Rs1

2t 2t + sin δs1 δs1 2t 2t cosh − cos δs1 δs1

sinh

The surface impedance of the conductor I is Zs y = 0 =

a

0,

b 8 4 67

The above expression is used in equation (8.4.42) to compute the conductor loss of microstrip with conductor thickness less than, and also more than, the skin-depth. The finite conductor thickness, through the resistance factor (RF), modifies the surface impedance Rs1 = Rs of the infinitely thick conductor. For a minimum loss, the RF of equation (8.4.67b) must be minimized with respect to the thickness t. It gives the optimum conductor thickness for the minimum conductor loss [J.66].

b

8 4 64 In the above equation, Rs and Lis are the surface resistance and internal surface inductance of the composite conductor’s system. The following special cases are considered:

•

Medium II is a perfect conductor, i.e. σ2 Rs2 0. Equation (8.4.64) is reduced to

∞, leading to

Wheeler’s Incremental Inductance Rule

Wheeler’s incremental inductance rule determines the line resistance from the internal inductance of a conductor to compute the conductor loss of a microstrip line for the strip conductor thickness more than the skin-depth, t ≥ 1.1 δs [J.61, J.62, J.74, J.75]. Equation (8.4.38b) provides the

293

8 Microstrip Line

0.2

Strip conductor δ /2 thickness t, width w s

δs/2 t

δs/2 Substrate

δs/2

h δs/2

Conductor loss αc (dB/cm)

294

0.15

(

,

) Exp. [J.72]

εr1 = 12.9, h1 = 100 μm, t = 3 μm σ = 4.1×107 S/m, tanδ1 = 0 W = 10μm

W = 130μm

0

5

10

15 20 25 30 35 Frequency (GHz) (b) Comparison of Wheeler's method against experimental results [J.75].

40

Application Wheeler’s incremental rule to compute the conductor loss of a microstrip line. Source: From Verma and Bhupal [J.75].

following expression for the line resistance in terms of the internal inductance: R = Xi =

Rs = ω Li w

8 4 68

Equation (8.4.34b) shows that the internal inductance Li = μδs/2 is due to the half skin-depth (δs/2) penetration of the magnetic field in a conductor. Figure (8.18a) shows the magnetic field penetration in a conducting strip from all directions, whereas in the ground conductor, it is in one direction. After the field penetration, the effective width and thickness of the strip conductor are reduced to w − δs and t − δs, respectively. The substrate thickness is increased to the thickness h + δs. The dielectric substrate does not influence the line inductance. Therefore, the internal inductance due to the field penetration is equal to the incremental change in the line inductance, Li = ΔL. The incremental line inductance can be easily computed from the total and external line inductances. The external line inductance Lext of microstrip is computed without any field penetration, i.e. from w, t, and h of a microstrip line. Whereas the total line inductance Ltot is computed from w − δs, t − δs and h + δs that takes care of the field penetration in the strip and ground plane conductors. The line inductance of a microstrip line is obtained from L = Z0(air)/c, where c is the velocity of EM-wave in the free space. Thus, the internal inductance is

=

) Wheeler

0.05

Ground conductor (a) δs/2 -δ penetration of magnetic field/current around strip conductor and in the ground.

Li = ΔL =

,

0.1

0

Figure 8.18

(

ΔZ0 c

1 Z0 w − δs , h + δs , t − δs , εr = 1 − Z0 w, h, t, εr = 1 c 8 4 69

The computation of characteristic impedance of a microstrip line with conductor thickness using the expressions of Hammerstad–Jensen has already been discussed in subsection (8.2.2). The line resistance is obtained from equations (8.4.68) and (8.4.69): R = ωLi ω Z0 w − δs , h + δs , t − δs , εr = 1 − Z0 w, h, t, εr = 1 = c = βΔZ0 8 4 70 The conductor loss for a microstrip line is computed by the following expression, obtained by substituting the above equation in equation (8.4.42): βΔZ0 π ΔZ0 = 2Z0 w, h, t, εr λ0 Z0 w, h, t, εr 27 3ΔZ0 αC = dB length λ0 Z0 w, h, t, εr

αC =

Np length

a b,

8 4 71 where ΔZ0 is the differential change in the characteristic impedance of a microstrip line, due to the penetration of the magnetic field. The above method is accurate for the conductor thickness t ≥ 1.1 δs against the experimental results [J.75, J.76]. Figure (8.18b) shows the accuracy of the present formulation of Wheeler’s inductance rule against the experimental results. The classical formulation of Wheeler’s inductance rule is not as accurate [J.61, J.62]. Moreover, the present formulation applies to a microstrip in a layered medium, discussed in chapter 15. It is also applicable to a coupled microstrip in the layered dielectric-medium [J.77]. Normally, the conductor surface is

8.5 Circuit Model of Lossy Microstrip Line

not smooth. The uneven conductor surface causes an increase in the conductor loss; as the surface current has to travel a longer path. The surface roughness can increase the conductor loss by 50%–60% [J.1]. The conductor loss models discussed in this section normally work up to 100 GHz. However, the working range of the models could be extended up to 1 THz, and beyond, considering the empirically obtained frequency-dependent model of the strip and ground conductors’ conductivity [J.55].

computes the complex characteristic impedance. However, it does not provide low-frequency dispersion [J.78]. The alternate computation of the dynamic RLCG parameters is discussed below. The frequency-dependent primary line parameters R(f ), L(f ), C(f ), G(f ) can be approximately determined from the dispersion relation, frequency-dependent characteristic impedance, dielectric loss, and conductor loss models discussed in this chapter. Several models working in the different frequency ranges have been combined to get the integrated model of each of these parameters of a lossy microstrip working up to THz frequency ranges [J.55]. These models can be used to obtain the R(f ), L(f ), C(f ), G(f ) parameters from the following set of equations:

8.5 Circuit Model of Lossy Microstrip Line The characteristic impedance of the lossy microstrip is a complex quantity. Moreover, the microstrip line also shows low-frequency dispersion, say below 1 GHz, due to the field penetration in the conducting strip and ground planes with finite conductivity [J.15]. The above-discussed models ignore such important aspects. The quasi-static RLCG circuit model of the lossy microstrip, shown in Fig (8.19a),

R f = 2 Z0 w eff , h, t, εr , f αC

a

L f = Z0 w eff , h, t, εr , f

b

εreff w eff , h, t, εr , f c

C f = εreff w eff , h, t, εr , f c0 Z0 w eff , h, t, εr , f G f = 2αd Z0 w eff , h, t, εr , f

L(f )

R(f )

C(f )

G(f )

(a) Circuit model of a lossy microstrip. 700

400 HFSS Direct model Circuit model

300 250 Substrate: ρs = 10 Ω-cm Conductor: Aluminum (σ0 = 3.7 × 107 S/m)

200 150

HFSS Direct model Circuit model

600 Conductor loss (Np/m)

Effective relative permittivity

350

100

500 400 300 200 Substrate: ρs = 10 Ω-cm Conductor: Aluminum (σ0 = 3.7 × 107 S/m)

100

50 0

0 0.1

1

10 100 Frequency (GHz)

(b) Effective permittivity of a lossy microstrip.

1000

0.1

1

10 100 Frequency (GHz)

(c) Total loss of a lossy microstrip.

Figure 8.19 Dispersion and total loss of lossy microstrip using the circuit model. Source: Awasthi et al. [J.55].

1000

c d, 851

295

296

8 Microstrip Line

where c is the velocity of the EM-wave. Using the above expression, the complex characteristic impedance and complex propagation constant of a lossy quasi-TEM microstrip are computed from the following well-known equations:

Z∗0 f =

R f + jωL f G f + jωC f

γf =

R f + jωL f

a,

γ f = αT f + jβ f

G f + jωC f

b c, 852

where αT(f ) = αd(f ) + αc(f ) is the total loss as a combination of the dielectric and conductor losses. The circuit model provides interaction between individual line parameters and improves the computation of characteristic impedance, dispersion, and losses. The dispersion and loss results are presented in Fig (8.19b and c), respectively, for a microstrip on the Si-substrate: εr = 11.9, h = 450 μm, t = 0.5 μm, w/h = 0.67. The lossy Si-substrate has resistivity ρs = 10 Ω − cm and aluminum conductor has conductivity σ0 = 3.7 107S/m [J.55]. Figures (8.19b,c) compare the results of the circuit model against the results of the EM-Simulator, HFSS. Both results are almost identical. The strong low-frequency dispersion is noted.

B.10 Gunter, K.: Practical Microstrip Design and Applications,

Artech House, Boston, 2005. B.11 Frey, J.; Bhasin, K.B.: Microwave Integrated Circuits, Artech House, Dedham, MA, 1985. B.12 Wolff, I.: The waveguide model for the analysis of microstrip discontinuities, In Numerical Techniques for Microwave and Millimeter-Wave Passive Structures, Itoh Tatsuo, John Wiley & Sons, New York., 1989. B.13 Rozzi, T.; Mongiardo, M.: Open Electromagnetic Waveguides, IEE, London, 1997.

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Measurements on 10-100 Ω Microstrip Lines on Sapphire, IEEE Trans. Microw. Theory Tech., Vol. MTT-25, pp. 172–185, March 1965. Hammerstad, E.O.; Jensen Ø.: Accurate models for microstrip computer-aided design, IEEE MTT-S Int. Microwave. Symp. Dig., pp. 407–409, 1980. Farina, M.; Rozzi, T.: Spectral domain approach to 2D-modelling of open planar structures with thick lossy conductors, IEE Proc. -Microw. Antenna Propagat., Vol. 147, No. 5, pp. 321–324, Oct. 2000. Gunsto, M.A.R.; Weale, J.R.: Variation of microstrip impedance with strip thickness, Electron. Lett., Vol. 5, No. 26, pp. 697–698, Dec. 1969. Bahl, I.J.; Garg, R.: Simple and accurate formulas for a microstrip with finite conductor thickness, Proc. IEEE, Vol. 65, No. 11, pp. 1611–1612, 1977. March, S.: Microstrip packaging. Watch the last step, Microwaves, Vol. 20, No. 12, pp. 83–94, 1981. Verma, A.K.; Kumar, R.: New empirical unified dispersion model for shielded, suspended, and composite-substrate microstrip line for microwave and mm-wave applications, IEEE Trans. Microw. Theory Tech., Vol. MTT-46, No. 8, pp. 1187–1192, Aug. 1998. Yamashita, E.; Atsuki, K.; Mori, T.: Application of MIC formula to a class of integrated – optics modulator analysis, IEEE Trans. Microw. Theory Tech., Vol. MTT-25, pp. 146–150, 1977. Szentkuti, B.T.: Simple analysis of anisotropic microstrip lines by a transform method, Electron. Lett., Vol. 12, No. 25, pp. 672–673, 1976. Szentkuti, B.T.: Corrections to simple analysis of anisotropic microstrip lines by a transform method, Electron. Lett., Vol. 13, No. 3, pp. 92, 1977. Kusase, S.; Terakado, R.: Mapping of two-dimensional anisotropic regions, Proc. IEEE, Vol. 67, No. 1, pp. 171–172, 1979. Kobayashi, M.; Terakado, R: New view on an anisotropic medium and its application to the transformation from anisotropic to isotropic problems, IEEE Trans. Microw. Theory Tech., Vol. MTT-27, No. 9. pp. 769–775, 1978. Owens, R.P.; Aitken, J.E.; Edwards T.C.: Quasi-static characteristics of microstrip on an anisotropic sapphire substrate, IEEE Trans. Microw. Theory Tech., Vol. MTT-24, pp. 499–505, Aug. 1976. Fritsch U. and Wolff, I.: Characterization of anisotropic substrate materials for microwave applications, IEEE. MTT-S Digest, pp. 1131–1134, 1992. Verma, A.K.; Awasthi, Y.K.; Singh, H.: Equivalent isotropic relative permittivity of microstrip on multilayer anisotropic substrate, Int. J. Electron., Vol. 96, No. 8, pp. 865–875, Aug. 2009.

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J.73 Faraji-Dana, R.; Chow, Y.L.: The current distribution and

J.76 Goldfarb, M.E.; PLatzker, A.: Losses in GaAs microstrip,

AC resistance of a microstrip structure, IEEE Trans. Microw. Theory Tech., Vol. MTT-38, No. 9, pp. 1268–1277, Sept. 1990. J.74 Wheeler H.A.: Formula for the skin effect, Proc. IRE, Vol. 30, No. 9, pp. 412–424, Sept. 1942. J.75 Verma, A.K.; Bhupal, A.: Conductor loss of multilayer microstrip line using the SLR formulation, Microw. Opt. Technol. Lett., Vol. 19, No. 1, pp. 20–24, Sept. 1998.

IEEE Trans. Microw. Theory Tech., Vol. MTT-38, pp. 1957–1963, 1990. J.77 Verma, A.K.; Nasimuddin: Determination of conductor loss of multilayer-coupled microstrip lines for CAD application, Microw. Opt. Technol. Lett., Vol. 38, No. 5, pp. 409–415, Sept. 2003. J.78 Verma, A.K.; Nasimuddin: Quasi-static RLCG parameters of lossy microstrip lines for CAD application, Microw. Opt. Technol. Lett., Vol. 28, No. 3, pp. 209–212, Feb. 2001.

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9 Coplanar Waveguide and Coplanar Stripline Basic Characteristics

Introduction The coplanar waveguide (CPW), coplanar stripline (CPS), and the slot line are uniplanar transmission lines with all conductors, the signal and ground conductors, on one side of the substrate [J.1]. A microstrip requires via-holes for the shunt connections of the components. It is difficult to drill a semiconductor or a ceramic substrate. The vias also add parasitic inductance to the FET or HEMT devices. A thin semiconducting substrate, down to 100 μm or less, is used to get a suitable microstrip line width, even for a 50 Ω line. The backside of a wafer requires thick conductor plating to provide mechanical strength to it. Furthermore, the line characteristics of a microstrip are influenced by the variation in thickness of a substrate. However, the microstrip-based circuit has a common ground to all components and devices. The CPW provides both the shunt and series connections of the components, without making any via-hole in the substrate. However, the bond-wires are used in a CPW to connect two ground planes. These are inconvenient in an MMIC fabrication. The CPW supports the elliptically or circularly polarized magnetic field in the slot region that is needed to develop the nonreciprocal magnetic devices. The CPS does not need the bonding wire. Also, the CPS takes much less space, as compared to the CPW. The relatively large range of characteristic impedance can be obtained with the CPS. Both the CPS and CPW are less dispersive as compared to a microstrip line. These lines are also less sensitive to the substrate thickness and no backside processing of the substrate is required. However, the main drawback of a CPS is the lack of shielding that causes stray coupling to other lines. This drawback could be avoided by adding the coplanar ground planes on both sides of the CPS line [J.2].

Objectives

• • • • • •

To present a general description of the uniplanar transmission lines. To review the conformal mapping method as applicable to the CPW and CPS. To get analytical expressions of several CPW and CPS structures using the conformal mapping method. To understand the nature of modes and present dispersion relations for the CPW and CPS. To model dielectric and conductor losses of the CPW and CPS. To present synthesis expressions for the CPW and CPS.

9.1

General Description

The CPW and CPS have been proposed by C.P. Wen [J.1]. S. B. Cohn has proposed the slot line that is discussed in chapter 10. Figure (9.1a–c) shows these uniplanar transmission lines, both in the 3-D and in the crosssectional view. The ground conductors (G) of a CPW, shown in Fig (9.1a), are infinite in lateral dimension. Normally, the central strip (S) is symmetrically placed with respect to the pair of ground conductors. The slot width (W) separates the central strip conductor (S) from the ground conductor (G). The signal conductor can also be placed asymmetrically, giving the asymmetrical CPW (ACPW). Figure (9.1b) shows the CPS. It is a complementary line to the CPW in the sense that conductor strips and dielectric gaps of the CPW are interchanged by the dielectric gaps and conductor strips in a CPS. In the case of a slot line, shown in Fig (9.1c), both the ground conductors have infinite lateral dimensions. Once their widths are finite, the slot line is reduced to a CPS. Similarly, the slot line could be viewed as a limiting case of a CPW with zero central strip width.

Introduction to Modern Planar Transmission Lines: Physical, Analytical, and Circuit Models Approach, First Edition. Anand K. Verma. © 2021 John Wiley & Sons, Inc. Published 2021 by John Wiley & Sons, Inc.

302

9 Coplanar Waveguide and Coplanar Stripline

G

W S W

G

(a) CPW. Figure 9.1

S

G

(b) CPS.

S

(c) Slot line.

Some uniplanar transmission lines with cross-sectional views.

G

W S W

G

G

G W S W G

(a) Ground conductors infinite, substrate infinite/ finite.

G

W S W

G

εr (d) Top-shielded CPW.

W S W

G

εr

εr

εr

Figure 9.2

G

(b) Ground conductors (c) Conductor-backed finite, substrate infinite/ finite. CPW, i.e. CB-CPW.

G

W S W

G

G

εr

W S W

G

εr

(e) Top-shielded CB-CPW.

(f) Enclosed CPW.

Some CPW structures.

There are several forms of the standard CPW structure, such as infinite extent CPW, finite substrate CPW, finite ground plane CPW, top-shielded CPW, conductor-backed CPW (CB-CPW), enclosed CPW, etc. Figure (9.2a–f ) shows these structures. A CPW structure supports the quasi-TEM mode that has no cutoff frequency for the fundamental quasi-TEM mode. The CPW structures are analyzed by the static methods, like the conformal mapping method [B.1, B.2, B.3, J.1, J.3– J.8], the variational method [J.9], etc. They are also analyzed by the full-wave methods [B.1, B.4, B.5]. However, the static conformal mapping method is very popular, as it provides simple CAD-oriented closed-form expressions for the characteristic impedance, propagation constant, and losses. Under the usual environment of the MIC, the static expressions can be used up to 20 GHz. This chapter presents the conformal mapping method for the analysis of the CPW and CPS structures. The variational method and the spectral domain method are discussed in chapters 14 and 16, respectively.

9.2 Fundamentals of Conformal Mapping Method This section briefly reviews the basics of the complex variable that leads to the conformal mapping method,

especially with the help of Schwarz–Christoffel (SC)transformation. Once applied to the CPW and CPS structures, the SC-transformation requires the elliptic sine function that is less familiar. An elementary discussion on the elliptic sine function is presented. Several mathematical books cover this topic [B.2, B.3, B.6–B.9]. 9.2.1

Complex Variable

A dependent complex variable w = u + jv is a function of another independent complex variable z = x + jy. Figure (9.3a,b) jointly defines these functions using two Argand diagrams. w = F z = u x, y + jv x, y

921

The complex variable z is defined in the complex zplane, also called (x–y plane), shown in Fig (9.3a) and the w is defined in the complex w-plane, also called (u–v plane), shown in Fig (9.3b). In the case of a single-valued complex variable function, for each value of a complex number, say P(x, y) in the z-plane, there is only one corresponding complex number, say Q(u, v) in the w-plane. The variables u and v, i.e. the real and imaginary parts of the complex variable w, are functions of independent variables x and y. If a curve C is traced by the point P(x, y) in the z-plane, then due to the functional dependence, the point Q(u,v) also traces another curve C in the w-plane. The curve C is called the

9.2 Fundamentals of Conformal Mapping Method

z-plane

jy

jv

Curve - C

Curve - Cʹ Q(u, v) o u

P(x, y)

o

w-plane

x

(a) Z, i.e. (x, y)-plane.

(b) W, i.e. (u,v)-plane.

Figure 9.3 Mapping of point P from Z-plane to W-plane as point Q.

z-plane jv

jy

w-plane Q

P

Z

2π

r θ x

0 C (a) Circular cylinder in z-plane.

u

(b) Parallel plane in w-plane.

Figure 9.4 Transformation of the circular cylinder to a parallel plane.

mapping or the image of the curve C. The function F(z) is called the mapping function. In the process of mapping, the geometrical shape can get changed. Let us see it through an example. Figure (9.4) shows that the mapping function w(z) = ln(z) transforms a circular cylinder located in the z-plane to a parallel plate in the w-plane. The independent variable z can be written in the polar coordinate as z = rejθ. It describes a circular cross-section of radius r, while argument θ changes from 0 to 2π. The mapping function in the polar coordinate is w z = ln z = ln r + j θ + 2πk , k = 0, 1, 2, 3… Therefore, 1 ln x2 + y2 = C 2 y Im w = v x, y = θ + 2πk = tan − 1 + 2πk x Re w = u x, y = ln r =

a b, 922

where k = 0, 1, 2… For the principal value, k = 0. Let us interpret the above transformation (mapping) graphically, as shown in Fig (9.4). The point P(x, y) in the z-plane traces a circle of radius r, while argument θ changes from 0 to 2π radians. The point Q(u, v), in the w-plane, is located at distance u = ln(r) = C moves along a line CQ parallel to the v-axis from θ = 0 to 2π. In this process, a circle in the z-plane is transformed into a

straight line in the w-plane, i.e. the circular cylinder is transformed into a parallel plate as there is no change in the z-direction after the conformal mapping.

9.2.2

Analytic Function

The above discussion is related to the mapping of points, lines, and the enclosed area. However, attention is to be paid to the transformation of an angle between two lines, or between two curves. It involves the derivative of a complex function, i.e. dw/dz. The uniqueness of the derivative is satisfied only by a special class of the complex functions, known as the analytic function. An analytic function satisfies the following Cauchy–Riemann conditions [B.8, B.10]: ∂u ∂v = ∂x ∂y ∂u ∂v = − ∂y ∂x

a 923 b

The following Laplace’s equations could be obtained from the above equations: ∂2 u ∂2 u + =0 ∂x2 ∂y2

a

∂2 v ∂2 v + =0 ∂x2 ∂y2

b

924

303

304

9 Coplanar Waveguide and Coplanar Stripline

It shows that an electrostatic potential function is adequately described by the analytic function. The functions u(x, y) and v(x, y) are known as the conjugate functions. If u (x, y) represents the electrostatic potential, then v(x, y) would correspond to the electrostatic flux. The reverse representation is also applicable. In summary, the complex analytic function, w = F(z), involves two coordinate systems – the z-plane, i.e. [(x-y)system] and the w-plane, i.e. (u-v)-system. The analytic function can be used for the mapping of points, lines, and enclosed regions from the (x-y)-system to the (u-v)-coordinate system with a unique derivative of the function, i.e. dw/dz is independent of direction. The change in the angle between two curves, located in the (x-y)-system, is further examined for their transformation in the (u-v)-system, with the help of an analytic function. Figure (9.5a) shows that a point P in the z-plane moving along the curve C from z0 to z0 + Δz. Figure (9.5b) shows the corresponding movement of the point Q, from the location w0 to w0 + Δw, in the w-plane along the path C . Let the curves be described by the parameter t. The path C of the curve in the z-plane is z=z t ie

x=x t , y=y t

925

The path C of the curve in the w-plane is w=w t ie

u=u t , v=v t

926

The derivative dz/dt is tangent to the curve C at z0 in the z-plane. The derivative dw/dt is tangent to the curve C at wo in the w-plane. Two derivatives are related as follows: dw dw dz dz = =F z dt dz dt dt

927

The function F(z) is analytic; therefore, its derivative at z0 is unique: dw dt

jy

w = w0

= F z0

z-plane Curve - C zo + ∆z P θo zo

928

z = z0

jv

w-plane

Curve - Cʹ

wo + ∆w Q

ϕo = θo + α wo

x (a) Tangent at P in Z-plane.

Figure 9.5

dz dt

u (b) Rotation of tangent at Q in W-plane.

Rotation of curve during conformal transformation.

The complex derivatives could be expressed in the polar forms, dw dz jϕ0 w = w 0 = ρ0 e , dt dt and F z0 = Re jα ,

z = z0

= r0 ejθ0

929

where r0, ρ0; θ0, and ϕ0 correspond to z0P and w0Q. The R and α are related to dw/dz. The substitution of the above equation (9.2.9) in equation (9.2.8) provides ρ0 ejϕ0 = r0 R ej θ0 + α . On separating the real and imaginary parts, the following expressions are obtained: ϕ0 = θ0 + α = θ0 + arg F' z0

a

ρ0 = r0 R

b

9 2 10

The above equations show that on transformation (mapping), the curve in the z-plane is rotated by an angle α in the w-plane. The angle of rotation α is equal to the argument of the derivative dw/dz at z0. The magnitude of the derivative dz/dt, i.e. the length z0P in the z-plane, is modified by the factor R on mapping. Figure (9.5b) demonstrates the rotation of the curve in the w-plane. If two curves in the z-plane intersect at z0, both curves rotate by the angle α, as both curves have identical dw z = z0 . Therefore, the angle between them remains dt unchanged on their mapping from the z-plane to the w-plane. As the angle between two curves in the w-plane confirms the angle in the z-plane on the transformation (mapping), such geometrical transformation using the analytic function is called the conformal mapping. 9.2.3

Properties of Conformal Transformation

Some more properties of the conformal mapping transformations are summarized below. A reader can consult references for their proofs [B.3, B.7, B.11, B.12]. Under the conformal mapping transformation, from the z-plane to the w-plane, the angle between two intersecting curves remains unchanged. The z-plane, i.e. the (x,y) coordinate system, is an orthogonal one. The (u,v) coordinate system forming the w-plane for an analytical function is also an orthogonal system. This property can be demonstrated using Cauchy–Riemann conditions. Figure (9.6a) shows two orthogonal surfaces, u(x, y) = constant and v(x, y) = constant, in the z-plane. As the plane, u(x,y) = constant and plane v(x,y) = constant are described by their normal, is obtained from grad (u) = ∇u and grad (v) = ∇v, respectively, the following expression is obtained: ∇u ∇v = xux + yuy

xvx + yvy = ux vx + uy vy 9 2 11

9.2 Fundamentals of Conformal Mapping Method

The Cauchy–Riemann condition (9.2.3) in the above equation provides ∇u ∇v = 0

9 2 12

The above expression shows that the normal of surfaces u = constant and v = constant are orthogonal to each other. Therefore, the (u–v) coordinate system forms an orthogonal coordinate system. Figure (9.6b) shows that u = constant and v = constant curves, in the z-plane, are transformed to the parallel planes in the (u − v) coordinate system. In the z-plane, the constant u and constant v curves could have any geometrical form such as circle, parabola, hyperbola, ellipse, straight line, etc. All of them can be conformally transformed to the parallel lines in the (u − v) coordinate system, on using a suitable analytic mapping function. Thus, if the system of curved surfaces, in the z-plane, are equipotential surfaces, then they are transformed to a parallel-plate capacitor in the w-plane, where capacitance can be easily computed. Normally, the conformal mapping process is used to solve the electrostatic problems. The electrostatic potential (ϕ) is expressed through the variable u; and the electric flux (ψ), corresponding to the charges on a conductor, z-plane

u = constant

v = const.

u = const.

jv

∇u

v = constant

jy

90˚ ∇v w-plane

x u (a) u = constant and v = constant (b) Transformation into parallel orthogonal lines in W-plane. orthogonal curves in Z-plane.

is expressed by the variable v. The variables v and u can be interchanged for the description of the potential and electric flux. Laplace’s equation remains invariant on the transformation from the z-plane to the w-plane. Further, the electrostatic energy in both the z-plane and w-plane remains unchanged. Therefore, under the conformal transformation of the arrangement of conductor and enclosed space, the capacitance remains invariant in both the z-plane and the w-plane. In conclusion, we state that in the process of the conformal mapping, only the geometry of the arrangement of conductors is changed to an analytically convenient geometry, while the capacitance remains invariant in both the z- and w-planes. It helps to evaluate the line capacitances of several complicated transmission lines. The conformal mapping is just a geometrical transformation process. It is not a physical transformation process. The physical properties and composition of the enclosed space, such as the electrical (permittivity, conductivity), magnetic (permeability), mechanical properties, etc., remain invariant under the conformal mapping transformation. The physical nature of the boundaries, such as the electric wall (EW) and magnetic wall (MW), also remain invariant under the conformal mapping transformation from the z-plane to the w-plane. An application of the conformal mapping is demonstrated below, by computing the line capacitance of a coaxial line of inner conductor radius r = a and radius of outer conductor r = b. Figure (9.7a) shows the coaxial line in the z-plane. Let us take the following mapping function that transforms the coaxial line to a parallel plate line, shown in Fig (9.7b): w z = ln z = u + jv

Figure 9.6 Transformation of curves into parallel lines.

z-plane

jy z

jv

w-plane

θ = 2π b

z a

x

εr (a) Coaxial line in z-plane.

θ=0

o ℓn(a)

εr ℓn(b)

u

(b) Transformed parallel plate line in w-plane.

Figure 9.7 Conformal mapping of the coaxial line. Dark surfaces are conductors.

9 2 13

305

306

9 Coplanar Waveguide and Coplanar Stripline

space within the polygon in the w-plane, of course without changing the relative permittivity of the space. The opposite can be carried out with the help of the SC-transformation, i.e. a polygon with enclosed space can be transformed into the line segments with open half-space. Figure (9.8) illustrates the SC-transformation. The SC-transformation is written in the form of the derivative of the complex function w, as follows:

The complex variable z of the above equation, in the polar coordinate, describes a circle in the (x–y)-plane: z = x + jy = rejθ , y θ = tan − 1 x

r2 = x2 + y2 9 2 14

The transformation function ln(z) in the polar coordinate is w z = ln rejθ = ln r + jθ 9 2 15

F w =

Thus, the transformed variables in the (u–v)-plane are

dw = A z − x1 dz N

u = ln r v=θ

a b

9 2 16

2πε0 εr 2πε0 εr = ln b − ln a ln b a

z − x2

The points x1 x2 x3… xN, forming line segments in the z-plane, are located on the x-axis. We move on the x-axis from the left to right in such a manner that points w1, w2, w3, …, wN create an anticlockwise polygon in the wplane. In the above equation, z is a real variable x, as y = 0. The constant A is a complex quantity. Its magnitude determines the size and its angle (argument), i.e. ∠A, determines the orientation of a polygon in the wplane. The parameters k1, k2,…,kN are determined by the exterior angles θ1, θ2, …, θN of the polygon. The angle of the SC-transformation is Γ0 =

dw = F Z = A− dz

N

x1 x2 , x2 o, o, x3 …, xN , etc., along the real axis in the z-plane. The SC-transformation transforms them in the form of a polygon in the w-plane, shown in Fig (9.8b) [B.3, B.6, B.12]. In the process, the upper half of the open space in the z-plane is also transformed into an enclosed

w-plane W4

jv z-plane

ki z − xi is the total external angle

x1

x2

o

x3

θ3

W3

θ4

xN

θ2 Enclosed space

W5

SC - transformation

εr

θ5

εr

x WN

Direction of points Left to right

θN

Figure 9.8

W2 θ1

Path

W1

o (a) Location of points in z-plane.

ki z − xi , i=1

of the polygon. The angle ∠z − xi along the line acquires the value π or 0, depending on the position of z(=x) with respect to points xi. Figure (9.9a,b) illustrates the process. Figure (9.9a) shows that the variable x crossing a fixed point xi from the left to right undergoes the change of angle by π radian. Figure (9.9b) shows the movement of the variable x after crossing the point xi does not involve any change in angle.

Figure (9.8a) shows an arrangement of the line segments,

space

N

9 2 19

9 2 17

Schwarz–Christoffel (SC) Transformation

Upper half

− kN

− ki

i=1

jy

… z − xn

9 2 18

where the angle 9.2.4

− k2

i=1

The u = constant and v = constant planes (lines) in the (u − v)-system form a parallel plate transmission line. The r = constant, i.e. r = a, b, are the circular conducting surfaces. Therefore, u1 = ln(a) and u2 = ln(b) are also the conducting surfaces in the (u–v)-plane. In the z-plane, θ = 0, 2π forms two circles with the dielectric medium between them. Therefore, in the (u–v)-plane, v1 = θ = 0 and v2 = θ = 2π; further u = ℓn(a) and ℓn(b) enclose dielectric medium between two conducting parallel planes. Figure (9.7b) shows that the width of the parallel plate conductor in the w-plane is w = 2π and the separation of the parallel plates is h = ln(b) − ln(a). Thus, the line capacitance p.u.l. of the original coaxial line obtained from the transformed parallel plate transmission line is C=

z − xi

=A

− k1

u

(b) Formation of the polygon in w-plane.

Formation of a polygon in w-plane by transforming collinear points in z-plane.

9.2 Fundamentals of Conformal Mapping Method

xi

x

xi

X-axis

(a) x < xi

x

X-axis

terms of the trigonometric and algebraic functions. The special kind of function, known as the elliptic sine function, is used for such cases [B.3, B.10, B.13].

(b) x > xi 1st Form of the 1st Kind of the Elliptic Integral

Figure 9.9 Angle change while crossing a fixed point.

x − xi = π, for x xi x − xi = 0, for x xi

; ;

i = 1, 2, …N i = 1, 2, …N

a b

The following definite integral of the variable t is known as the 1st kind of the elliptic integral: w

z= 0

9 2 20 The crossing points x1, x2, x3, …, xN in the z-plane appear as the vertices of a polygon in the w-plane. The external angle at each vertex of the polygon is given by θi θi = ki π, i = 1, 2, …N ki = ; π 9 2 21 To determine the parameter ki, we assume the shape of the polygon in which line segments of the z-plane is to be transformed. The following relation holds for the interior angles α1, α2, …, αN of a polygon: 9 2 22

For the interior angle of the SC-transformation, equation (9.2.18) is modified into dw = A z− x1 β1 z− x2 β2 … z −xn βN F w = dz α1 α2 αN − 1 , β2 = − 1 , βN = −1 where, β1 = π π π 9 2 23 Thus, the SC-transformation is available in two forms – using the external angle or using the internal angle of a polygon. The sum of external angles of any polygon is ki π = 2π

k ≤1 w ≤ 1, 9 2 25

where the upper limit w is a real quantity with 0 ≤ t ≤ w and k is a parameter (modulus) such that |k| ≤ 1. Further, the quantity w is located on the real axis Re(t) of the t-plane such that |w| ≤ 1. The value z of the above elliptic integral, for the known w and k, could be obtained using the numerical methods. The above elliptic integral also defines the inverse sine function as a special case: For k = 0,

z = sin − 1 w =

w

dt

0

1 − t2

The above function z is the sine inverse function, sin−1(w) with |w| ≤ 1. At present, the sine inverse function is defined with the help of a definite integral. Normally, sin−1(w) is defined through geometrical means. For k 0, the integral given by equation (9.2.25) defines the elliptical sine inverse function that is a generalized form of the sine inverse function. It also helps to get the elliptic sine function. Elliptic sine inverse function w 0

dt 1 − t2 1 − k2 t2

Elliptic sine function

,

z = sn − 1 w, k = k ≤1 w ≤1

sn z, k = w, k

N

i=1

ki = 2

9 2 24

i=1

The above condition imposed on ki can be used as a check of its correct determination for a polygon. 9.2.5

1 − t2 1 − k2 t2

, modulus

9 2 26

αi + θi = π, αi + ki π = π π − αi , i = 1, 2, …N ki = π

N

dt

a

b 9 2 27

For k = 0, inverse sin e function, sn − 1 w, 0 = sin − 1 w = z

a

For k 0 elliptic sine function, sn z, k = w, k For k = 0, sine function, sn z, 0 = sin z = w

Elliptic Sine Function

b c

9 2 28

The analytic functions involved in the SC-transformation are generally expressed with the help of the definite integrals. Sometimes these integrals are not solved in

The elliptic sine inverse function and thus the elliptic sine function sn(z, k) are obtained with the help of equation (9.2.27) using the following process: Assumed parameter

sn z, k

=

The numerically calculated value of z in z – plane using elliptic integrals

w, k Known value of w located in t – plane

9 2 29

307

308

9 Coplanar Waveguide and Coplanar Stripline

For the change in a variable t = (1 − k 2t 2)−1/2 with the complementary modulus (k ) as k 2 = 1 − k2, the above complementary integral is written as

The elliptic cosine function cn(z) and another function dn(z) are defined as follows: 1 − sn2 z

cn z =

1−k

dn z =

2

a

1

9 2 30

sn2

z

b

0

The parameter “k” is dropped in the above functions cn(z) and dn(z).

dθ

ϕ=

The complete elliptic integral, in the whole range of variable t, is defined by combining equations (9.2.32) and (9.2.33): 1 k

dt

z=

1 − t2 1 − k2 t2

π 2

π , 2

z=

dθ

9 2 35 For the case 0 ≤ t ≤ 1, z is a real quantity K(k). For the case, t > 1, i.e. for 1 ≤ t ≤ 1/k, z is an imaginary quantity jK(k). However, for 0 ≤ t ≤ 1/k, z is a complex quantity given by the following expression:

b

1 − k sin θ 2

0

2

9 2 31 As w = sin (9.2.27) is

π 2

1

= 1, the integral shown in equation

z = ± K k + jK k = ±

1

z = sn − 1 1, k = K k =

dt 1 − t2

0

1 − k2 t2

1 k

,

+j 1

for 0 ≤ t ≤ 1

t2 − 1 1 − k2 t2

1

,

1≤t≤

0 1 0

1−t 2

z-Plane Kʹ (w-plane) Aʹ (–K + jKʹ) Fʹ Eʹ Dʹ (K + jKʹ)

(z-Plane) SC - transformation F (–∞)

A B

–1/k –1

C

D

O +1 +1/k

E(∞) x (t)

(a) Location of points along x (t)-axis. Figure 9.10

1 − k2 t 2

2

1−k t 2

9 2 36 To define the elliptic function geometrically, two complex planes are used: the t-plane (or the z-plane) with t as a real number on the x-axis, i.e. x = t, y = 0, and the zplane (or the w-plane). Figure (9.10) shows the correlation, i.e. the transformation between these planes.

1 k

t-Plane

dt 1 − t2

dt

+j

9 2 33

jy

t2 − 1 1 − k2 t2 1

In the above equation, K(k) is the value of the elliptic inverse sine function for the parameter k (modulus). The K(k) can be computed only for 0 ≤ t ≤ 1. However, the complimentary elliptic sine inverse function can be obtained for 1 ≤ t ≤ 1/k as follows: dt

dt

± K k + jK k = ±

9 2 32

1 k

dt 1 − t2 1 − k2 t2

0

K k =

0≤t≤1 k

,

a

1 − k2 sin 2 θ

0

1−k t

Complete Elliptic Integral

0

ϕ

0≤t ≤1

It is noted that K(k ) = K (k).

The 1st kind of the elliptic integral could be arranged in a different format. Let t = sin θ and w = sin ϕ. We can change the limits t = 0 and t = w = ϕ of the integral (9.2.25), and the following integral expression could be obtained:

For

1−t

, 2 2

2

9 2 34

2nd Form of the 1st Kind of the Elliptic Integral

z=

dt

Kk =

Formation of the rectangle using SC-transformation.

Bʹ (–K)

Oʹ

Cʹ (K)

K

(b) Formation of the rectangle after mapping.

9.2 Fundamentals of Conformal Mapping Method

is achieved with the help of the 1st kind of elliptic integral. The transformation process is explained below

The transformation of points located on the real x-axis of the z-plane to the points located in the complex w-plane The parameter of a fixed value sn z, k

=

t Point on the x – axis of z – plane t − plane

a point on the z i e w – plane

appears as an upper limit of the integral

Figure (9.10) shows the SC-transformation using the elliptic function sn(z, k) = t that transforms the points located on the real axis in the t-plane (i.e. the z-plane) to the vertices of a rectangle in the complex z-plane (i.e. the complex w-plane). There are two important aspects of the SC-transformation:

• •

9 2 37

Value of integral shown as

The line segments along the x-axis are transformed into the sides of a rectangle in the complex w-plane. In the t-plane, movement is from the left to right, whereas in the complex w-plane, the rectangle formation follows the anticlockwise path.

K k =K k =

2

1 1+ k ln 2 π 1− k

9 2 38

k 21 4 k , for + 168 4

k ≥ 0 65

3rd Form of the 1st Kind of Elliptic Integral

−1

, for

1 1+ k , for ln 2 π 1− k

k =

The following approximate expressions are used to evaluate the elliptic integral [B.3, B.14, J.6]: For the parameter k γ > δ. Sometimes the role of the modulus k and complimentary modulus k could be interchanged [J.12]. Thus, α = t3 = c2, β = t2 = b2, γ = t1 = a2, δ = t0 = 0. Thus, the aspect-ratio of the CPW on an infinite substrate with thick finite extent coplanar ground planes is k = k3 = kG 1 =

rectangular structure in the w-plane by using the following SC-transformation: t dt 9 3 20 u + jv = w t = t t − t1 t − t2 t − t3 t0

t

u

w-plane

4 EW x1

MW

(c) Parallel plate line.

x-plane

( εr – 1 )

2

a b

c2 − b2 c2 − a2

9 3 22

The aspect-ratio, k1 = a/b, applies to the infinite width ground planes, and the correction factor, F1 =

c2 − b2

c2 − a2 , mentioned in equa-

tion (9.3.17b), is due to the finite width of ground planes. Following the discussion of subsection (9.3.1), the line capacitance of a CPW on the air-substrate is C1 εr = 1 = 2ε0

K k3 K k3

9 3 23

a 1 k

=N 0

dp 1 − p2

1 − k2 p2

b

c d,

9 3 21

Step #II

Using Figure (9.15a), step II considers the numbered vertically half of the original CPW in the lower half-plane that is a finite thickness substrate backed by the air-substrate. To compute the line capacitance for this case, the dielectric–air interface is replaced by a MW, and relative permittivity (εr − 1) is used in the infinite extent lower half of the x-plane, shown in Fig (9.15d). The

9.3 Conformal Mapping Analysis of Coplanar Waveguide

transformation from the z-plane of Fig (9.15a) to the x-plane of Fig (9.15d) is carried out using the following mapping function: πz 2h

x = cosh 2

The above equation provides the correction factor F2 of equation (9.3.18b). The line capacitance of the lower half of the CPW with the dielectric layer is

9 3 24

Using the above equation, the transformed locations are summarized in Table (9.4). The following SC-transformation maps structure of Fig (9.15d) to a parallel-plate capacitor in the w-plane, shown Fig (9.15e): x

C2 = 2ε0 εr − 1

C2 εr = 2C1 + C2 = 4ε0

x − x1 x − x2 x − x3 x − x4

x1

9 3 25 The above integral is transformed into a standard elliptic integral of the 1st kind with the help of equation (9.3.21): x1 = δ = 1, x2 = γ = cosh 2

πa , 2h

9 3 26

q=

cosh πc 2h − cosh πb 2h k2 = k24 = cosh 2 πc 2h − cosh 2 πa 2h × k4 =

kG 2

2

Zo εr =

9.3.4

Points in the z-plane

Points in the x-plane

1) z = 0

x = x1 = 1

2) z = a

x = x2 = cosh 2

3) z = b

x = x3 = cosh 2

4) z = c

x = x4 = cosh 2 ∞)

x

7) z = (x − jh)(x

∞

x=0 ∞)

x

−∞

K k3 K k4 + 2ε0 εr − 1 K k3 K k4 K k3 4ε0 K k3

1 K k4 K k3 εr − 1 2 K k4 K k3

1 K k4 K k3 2 K k4 K k3

a b

30π K k3 εr eff K k3

9 3 31

Static Characteristics of CPW

Infinite Dielectric Thickness

Table 9.4 Transformation using x = cosh2(π z/2h).

6) z = −jh

4ε0

1 2

9 3 27

5) z = x(x

9 3 29

9 3 30

cosh 2 πa 2h − 1 cosh 2 πb 2h − 1

sinh 2 πc 2h − sinh 2 πb 2h = sinh 2 πc 2h − sinh 2 πa 2h sinh πa 2h × sinh πb 2h

C εr = C εr = 1

εr eff = 1 +

Using equation (9.3.21c), the aspect-ratio is 2

K k3 K k4 + 2ε0 εr − 1 K k3 K k4

The effective relative permittivity, filling-factor, and characteristic impedance of the considered CPW are given below:

εr eff =

πb πc , x4 = α = cosh 2 2h 2h

x3 = β = cosh 2

9 3 28

Using equation (9.3.14), the total line capacitance p.u.l. of CPW with the finite substrate and finite ground conductors is

dx

u + jv = w =

K k4 K k4

πa 2h πb 2h πc 2h

The effective relative permittivity of a CPW on an infinitely thick substrate, shown in Fig (9.12), does not depend on the a/b aspect-ratio. However, its characteristic impedance depends on a/b-ratio. For a fixed value of b, the separation between two grounds is fixed and an increase in the width of the central strip conductor (s = 2a) narrows the gap w. It results in an increase in the line capacitance that decreases the characteristic impedance of a CPW. Figure (9.16) shows such a decrease in the characteristic impedance with an increase in a/b-ratio [B.16]. Several values of the strip width (2a) and separation of the ground planes (2b) can provide identical value for the characteristic impedance. Thus, the dimensions (2a) and (2b) are selected for the compactness of the CPW, and also for the minimum conductor loss.

315

9 Coplanar Waveguide and Coplanar Stripline

90

100 7.00 3 6.98

80

6.96

70 100

60

6.94

c/b = 3

c/b =

50 40

6.92

c/b = 1.5

1.5

6.90 6.88

εr = 13 , h = 300 μm 2b = 200 μm

30 20 0.0

1.5,3,100 [B.16, J.3]. For a thick substrate, h/b ≥ 3, and for the wide ground planes c/b ≥ 3 the effect of finite substrate thickness and effect of the finite width of ground planes on the characteristic impedance and effective relative permittivity could be ignored.

Effective rel. permittivity (εr e f f )

100

Characteristic impedance (Ω)

6.86

9.3.5

6.84 1.0

0.5 Aspect ratio - a/b

Figure 9.16

Changes in CPW characteristics with aspect ratio-(a/b) and c/b-ratio. Source: Gupta et al. [B.16]. © 1996, Horizon House Publications, Inc.

Finite Dielectric Thickness

The characteristic impedance and effective relative permittivity of a CPW are influenced to some extent by the finite size of the substrate thickness and finite width of the ground planes. At a fixed aspect-ratio a/b, the decrease in h/b results in more electric fields in the air medium of the lower half-plane and less electric fields in the dielectric medium. This causes a decrease in line capacitance for the lower part of CPW. It increases its characteristic impedance and also decreases its effective relative permittivity εr eff. The finite ground conductor width c decreases the line capacitance, as some of the electric field lines move to the air regions. The characteristic impedance of CPW increases and its effective relative permittivity decreases with a decrease in c/b ratio. These changes are shown in Fig (9.16) for c/b-ratio jy

Top-Shielded CPW

Figure (9.17a) shows a top-shielded CPW structure on a finite thickness (h) substrate with relative permittivity εr. However, the ground conductors have an infinite width and the top shield is located at the height h1. The medium between the CPW strip conductors and the top shield is air, i.e. εr = 1. The top shield is used to protect a CPW against the environment. It is also useful for the post-fabrication adjustment of the line parameters. The conformal mapping method is applied separately to the upper and lower spaces of the structure. Both slots are modeled by the MW and the conducting strips are treated as the EW. The line capacitance p.u.l. of the present CPW is C5(εr) = line capacitance C3(εr = 1) of the upper half of shielded space + line capacitance C2(εr) of the lower half of finite thickness substrate, i.e. C5 εr = C3 εr = 1 + C2 εr C2 εr = C1L εr = 1 + C2

where C1L is given by equation (9.3.12) and C2 is given by equation (9.3.13). We have to obtain the expression only for C3(εr = 1). The first transformation, using the function given below, maps all conductors of the upper half-space, shown in Fig (9.17a), on the real t-axis, shown in Fig (9.17b):

z-plane 5

6 h1 εr1 = 1 o a b 1 2 3 h εr

4

(∞ + jh1) ∞ z

εr = 1 (a) Original structure. Im(t) Air-medium εr = 1 t1 t2 t3

t5 –∞

5

6

1 2

3

–K(k2) + jK(kʹ2) t4 4 Re(x)

(b) Intermediate transformation. Figure 9.17

jv

K(k2) + jK(kʹ2)

2

EW

1

εr = 1

MW

t-plane

a b, 9 3 32

MW

316

3 4,5 EW 6 u (c) Parallel plate line.

Conformal mapping of the upper half of the top-shielded CPW on the finite thickness.

9.3 Conformal Mapping Analysis of Coplanar Waveguide

Table 9.5 Transformation using x = cosh2(πz/2h1). Points in the z-plane

Points in the t-plane

1) x = 0

t1 = 1

2) x = a

t2 = cosh 2

3) x = b

t3 = cosh 2 ∞)

4) x = x(x

5) x = (x + jh1)(x 6) z = jh1

t = cosh

2

C5 εr = 1 = 2ε0

t4

∞

εr eff

q=

πx 2h1

9 3 33

Table (9.5) shows the transformed locations of the strip edges in the t-plane. Next, the SC-transformation converts the strips and gaps located on Re(t)-axis into a rectangle in the wplane, shown in Fig (9.17c). The mapping function for this transformation is written as follows: t

W= t0

dt

9 3 34

t t − t1 t − t2 t − t3

The above integral can be put in the standard form of the elliptical integral of the 1st kind by the change of the variable expressed through equation (9.3.21). The aspect ratio K5 is given by α−δ β−γ , α−γ β−δ

K25 = 2

K5 =

K5 =

tanh πa 2h1 tanh πb 2h1

1 − K25

a b

πb πa where α = cosh 2 , β = cosh 2 , γ = 1, δ = 0 2h 2h

c

9 3 35 Figure (9.17c) shows the transformation of the locations from the t-plane to w-plane forming the parallelplate capacitor. The line capacitance C3(εr = 1) of the upper half of the shielded CPW is C3 εr = 1 = 2ε0

K k5 K k5

9 3 36

The line capacitance p.u.l. of the shielded CPW on a finitely thick substrate with the top shield is

C5 ε r = 1 + εr − 1 = C5 ε r = 1

Lower half – space 9 3 37

The line capacitance p.u.l. of the shielded CPW in the air medium is

K k4 K k4 K k5 K k3 + K k5 K k3

K k4 K k4

a

b

K k5 K k3 + K k5 K k3

9 3 39 The following aspect-ratios, from equations (9.3.5a), (9.3.11a), and (9.3.35a), are used in the above equations for the infinitely wide ground planes: a sinh πa 2h , K4 = b sinh πb 2h tanh πa 2h1 2 , K j = 1 − K2j K5 = tanh πb 2h1

K3 =

9 3 40

The characteristic impedance of the top-shielded CPW on the air and dielectric substrates are 60π K k5 K k3 + K k5 K k3 Z0 ε r = 1 Z0 εr = εr eff

Z0 εr = 1 =

a 9 3 41 b

Only the infinitely wide ground planes are considered above. However, for the finite width of the ground planes separated by distance 2c, shown in Fig (9.14b), the aspect-ratios K3, K4, K5 should be modified according to expressions (9.3.17) and (9.3.18) as follows: 2

K3 mod = K3

c2 − b2 c2 − a2

K4 mod = K4

sinh 2 πc 2h − sinh 2 πb 2h sinh 2 πc 2h − sinh 2 πa 2h

b

K5 mod = K5

sinh 2 πc 2h − sinh 2 πb 2h sinh 2 πc 2h − sinh 2 πa 2h

c

2

K k5 K k3 K k4 C5 εr = 2ε0 + 2ε0 + 2ε0 εr − 1 K k5 K k3 K k4 Upper half – space

9 3 38

The effective relative permittivity and the filling-factor q of the shielded CPW are

πa 2h1 πb 2h1

t5 −∞ t6 = 0

∞)

K k5 K k3 + 2ε0 K k5 K k3

K i mod = 1 − K2i mod ,

a

i = 3, 4, 5,

d 9 3 42

9.3.6

Conductor-Backed CPW

Figure (9.2c) shows the CB-CPW structure. The conductor backing provides mechanical support to the thin and fragile semiconductor and quartz substrates. It also acts

317

9 Coplanar Waveguide and Coplanar Stripline

as a heat sink to the active microwave devices. The CBCPW is a mixed structure of the CPW and microstrip structures. This structure can support the unwanted microstrip mode that is the dominant mode of a thin substrate and wide slot width, w/h > 2, CB-CPW. For a moderate aspect-ratio, such as s/2h = 1/3 and w/h = 2/3, the structure is less dispersive, as compared to a microstrip line of w/h = 2/3. In this section, the conformal mapping method is applied to obtain the static effective relative permittivity and the characteristic impedance of a top-shielded CB-CPW, shown in Fig (9.2e). The line capacitance of the top-shielded CB-CPW is a sum of the line capacitances of the upper half C3(εr = 1) and the lower half C2(εr) of the structure: C εr = C3 εr = 1 + C2 εr ,

9 3 43

where C3(εr = 1) is given by equation (9.3.36) for height h1 of the top shield. Following the same process, C2(εr) is given by C2 εr = 2ε0 εr where aspect ratio

K k6 K k6 K6 =

tanh πa 2h tanh πb 2h

b

2

K k5 K k6 + 2ε0 εr K k5 K k6 K k5 K k6 C εr = 1 = 2ε0 + 2ε0 K k5 K k6 C εr = 2ε0

a b 9 3 45

The effective relative permittivity and filling-factor q of the shielded CB-CPW structure are obtained from the above equations: K k5 K k6 + εr K k5 K k6 C εr = εr eff = K k5 K k6 C εr = 1 + K k5 K k6 K k6 K k6 q= K k5 K k6 + K k5 K k6

a

b

9 3 46 The characteristic impedance of the structure is 60π 1 9 3 47 Z εr = K k6 εr eff K k5 + K k5 K k6

a

K6 = 1 − K26

c

9 3 44 In the above expression, h is the thickness of a conductor-backed substrate. The line capacitance of the top shield CB-CPW on the dielectric and the air substrates are

For h1 = h, εr eff = (εr + 1)/2. For the shield height ∞ , K5 a/b, the expressions for simple CBh1 CPW are obtained. The presence of a top or a bottom shield reduces the characteristic impedance of a CPW. The characteristic impedance also decreases with an increase in a/b-ratio, shown in Fig (9.18a). The nearness of the conductor backing to the strip conductors increases εr eff [J.4]. It is shown in Fig (9.18b).

60 εr = 10, h+h1 = 2b 50 40 h = b 10

30 20

05

10

02

Effective relative permittivity

Impedance (Ohms)

318

εr = 10, h+h1 = 2b 8 7 6 5 4

h = b 02 05 10

0.2 0.3 0.4 0.5 0.6 0.7 0.8 Aspect ratio - a/b

0.2 0.3 0.4 0.5 0.6 0.7 0.8

(a) Variation in characteristic impedance.

(b) Variation in the effective relative permittivity.

Figure 9.18

Aspect ratio - a/b

Variation in the line parameters of top-shielded conductor-backed CPW with aspect-ratio. Substrate thickness is a parameter. Source: Ghione and Naldi [J.4].

9.4 Coplanar Stripline

w –∞

s

w

w1 –∞

εr

(a) Symmetrical CPS on infinitely thick substrate. w1

s

s

w2

w

(b) Asymmetrical CPS on infinitely thick substrate. w

h . εr

s

Ground ∞

(c) Symmetrical CPS on finitely thick substrate. Ground –∞

h . εr

(d) Asymmetrical CPS on finitely thick substrate.

w

h . εr

εr

w2

s

h . εr

2c 2a 2b

Ground ∞

(f) CPS with coplanar ground planes.

(e) Asymmetrical CPS with infinitely wide ground strip.

Figure 9.19 Some CPS structures.

9.4

Coplanar Stripline

Figure (9.19) shows several CPS structures. It consists of two strip conductors of width w separated by a slot-gap of width s. The CPS line also supports the quasi-TEM mode and is less dispersive, as compared to a microstrip line. To reduce the losses on the CPS line, normally a wider strip width, i.e. w ≥ 4s, is maintained [J.13]. However, the strip width (w) has to be kept small to exclude the parasitic non-TEM modes and also to reduce the MMIC chip area. The CPS line further supports the TM0 and TE0 dielectric slab parasitic modes. The TM0 surface wave mode has no cutoff frequency [J.14, J.15]. This section discusses the conformal mapping method to get the static line parameters of the CPS structures. The dispersion and losses are also considered in sections (9.6) and (9.7).

opposite polarity on two conductors attract each other to produce the increased current density, i.e. the edge singularity, at the inner edges of a CPS line. The CPS line is a complementary structure of the CPW structure shown in Fig (9.12a), i.e. the slots and conductor strips of the CPW are interchanged by the conductor strips (w) and slot (s) to get the CPS structure. The conformal mapping, shown in Fig (9.12b), is also applicable to a CPS by interchanging the MW to the EW, and also EW to the MW. The CPS, after mapping in the w-plane, is shown in Fig (9.20c). It forms a parallel-plate capacitor. The line capacitance C1 for the upper half of the CPS structure with air medium (εr = 1) and the line capacitance C2 for the lower half of the CPS with dielectric medium (εr) are computed as follows: ε0 K k1 2K k1 εr ε0 K k1 C2 = 2K k1

C1 =

9.4.1 Symmetrical CPS on Infinitely Thick Substrate Figure (9.20a) shows the zero conductor thickness symmetrical CPS on a finitely thick substrate. It also shows the TEM field lines. Figure (9.20b) further shows the current distributions across both the strips. The charges of

w

Current

εr = 1

w s

εr h (a) Electric and magnetic field lines of CPS supporting TEM mode.

w

s

EW w

(b) Current distribution on CPS.

Figure 9.20 Field and current distribution on the CPS line.

941 b

The modulus k1, giving the aspect-ratio and complementary modulus k1 , is given by equation (9.3.5). The total line capacitance p.u.l. of symmetrical CPS on an infinitely thick dielectric substrate and air substrate is

jv w-plane [–K(k1) + jK(kʹ1)] [K(k1) + jK(kʹ1)] MW

E-field H-field

a

εr

EW MW

u

(c) Parallel plate line in w-plane.

319

320

9 Coplanar Waveguide and Coplanar Stripline

C εr = C1 + C2 =

εr + 1 ε0 K k1 2 K k1

ε0 K k1 K k1

C εr = 1 =

the way the SC-transformation (9.3.21a) is applied to get the rectangular figure in the w-plane. Figure (9.21a) shows the ACPS with dimensions δ = − w1, γ = 0, β = s, α = s + w2. The SC-transformation from equation (9.3.21) is

a b 942

z

w=A

The effective dielectric constant and the filling-factor of symmetrical CPS line on an infinitely thick substrate are obtained from the above equations: C εr εr + 1 = εr eff = C εr = 1 2 1 q= 2

a b

Like a CPW on an infinitely thick substrate, the above equation shows the equal division of field in the air and dielectric regions for a CPS structure also. The characteristic impedances of the CPS on the air and dielectric substrates are

Z εr =

1 = vp C

μ0 ε0 ×

K k1 K k1 = 120π ε0 K k 1 K k1

120π K k1 × εr eff K k1

k7 = k7 =

a b

jv 5 ∞

2′ x

EW

MW εr

946

1 2

b

a 947 b

The total line capacitances p.u.l. of the ACPS line with dielectric and air-substrates are C εr = C1 εr = 1 + C2 εr = εr + 1 ε0 C εr = 1 = 2ε0

K k7 K k7

K k7 K k7

a b 948

[K(k7) + jK(k′7)] 3′ EW

1′

(a) Asymmetrical CPS on infinitely thick substrate. Figure 9.21

=

s w1 + w2 + s s + w1 s + w2

w-plane

z-plane

Path

εr

1 2

a

K k7 K k7 K k7 C2 = ε0 εr K k7

Figure (9.21a) shows an asymmetrical CPS (ACPS) structure in the z-plane [J.14]. It has two strip conductors of width w1 and w2 separated by a slot gap s. The SCtransformation, given by equation (9.3.21b), is used to get the parallel plate line in the w-plane shown in Fig (9.21b). Locations of conducting strips and slot edges in the z-plane are marked by points α, β, γ, δ, such that α > β > γ > δ. The modulus, given by equation (9.3.21c), could be either k or complementary modulus k depending upon

εr = 1 W1 S W2 γ α β δ 4 3 1 2

1 − k72

1 2

C1 = ε0

9.4.2 Asymmetrical CPS (ACPS) on Infinitely Thick Substrate

–∞ 6

w1 w2 s + w1 s + w2

The line capacitances C1 for ACPW with the air-filled upper half-space and the line capacitance C2 for the dielectric-filled lower half-space is given as follows:

944

jy

945

z + w1 z z − s z − s − w2

The above function maps the strip conductor line segments (1)–(2) and (3)–(4) in the z-plane as the line segments (1 )–(2 ) and (3 )–(4 ) (EW) along the jv-axis in the w-plane. The slot segments (2)–(3) and (4–5–6–1) are mapped as (2 )–(3 ) and (4 , 5 , 6 , 1 ) MW. The modulus k given by equation (9.3.21) is the complementary modulus for the present case:

943

Z εr = 1 =

δ

dt

u MW (5′,6′) 4′ (b) Parallel plate line in w-plane.

The conformal mapping of the asymmetrical CPS line on the infinitely thick substrate.

9.4 Coplanar Stripline

The effective relative permittivity, filling-factor, and characteristic impedances of the ACPS on the air and infinitely thick substrates are C εr εr + 1 = C εr = 1 2 1 q= 2 1 = Z εr = 1 = vp C εr = 1

εr eff =

a b μ0 ε0

K k7 2ε0 K k7

K k7 K k7 60π K k7 Z εr = εr eff K k7 = 60π

The capacitance C2 is obtained by following the process given in section (9.3.2) and shown in Fig (9.22). It is noted that Fig (9.22a) is a complementary structure of Fig (9.13a). In this process, the strip conductors of Fig (9.13a) are replaced by the MW, and the slots are replaced by the strip conductors, i.e. by the EW. Thus, as shown in Fig (9.22a), the slot gap is s = 2a and the strip width (w) is related to 2b = s + 2w. The mapped Fig (9.22b) provides us the line capacitance C2 : ε0 εr − 1 K k2 2K k2

C2 εr − 1 =

c

Modulus k2 , from equation − 9 3 11 πs sinh 4h k2 = π s + 2w sinh 4h

d 949

The SC-transformation process presently discussed is equally applicable to the ACPW structure with two different slot gaps.

1 − k22

k2 =

a

b

c 9 4 11

9.4.3 Symmetrical CPS on Finite Thickness Substrate

Using Fig (9.22b), the line capacitance C1 of the upper region in the air medium could be written as:

The symmetrical CPS on a finite thickness substrate, shown in Fig (9.22a), can be treated as a complementary structure of the CPW on a finite thickness substrate [J.16]. However, the assumption of a complementary structure is valid only for a thick substrate, or more correctly only for a CPS line on the air-substrate [J.17]. In any case, it is easier to implement the concept. The line capacitance C1 of a CPS for the upper halfspace, with air medium, is given by equation (9.4.1a). Following the discussion in section (9.3.2), the line capacitance of a CPS for the lower half-space, with an inhomogeneous medium, is

ε0 K k1 2K k1 2a s where, k1 = = 2b s + 2w

C2 εr = C1 εr = 1 + C2 εr − 1 ,

C1 =

k1 =

jy

C εr = ε0

9 4 10

w

h

εr

ε0 K k1 K k1

w

MW

2a

(a) Symmetrical CPS on finitely thick substrate.

w-plane

[–K(k2) + jK(k′2)] MW

[K(k2) + jK(k′2)]

x EW

εr = 1

a b 9 4 13

(εr –1)

MW Air-medium

K k1 ε0 εr − 1 K k2 + K k1 2K k2

C εr = 1 =

jv

MW

c

The total line capacitance p.u.l. of a CPS on the finite thickness substrate and the air substrates are C εr = 2C1 εr = 1 + C2 εr − 1 ,

z-plane 2b s

b

9 4 12

where C1 is the line capacitance of a CPS for the upper half-space and C2 is the line capacitance of a CPS with the lower half-space filled in with infinitely thick dielectric medium having relative permittivity (εr − 1). Air-medium εr = 1

1 − k21

a

MW (b) Parallel plate line in w-plane.

Figure 9.22 The symmetrical CPS line on the finite thickness substrate.

EW u

321

322

9 Coplanar Waveguide and Coplanar Stripline

For a CPS line under the condition h 0, the factor K k2 K k2 decreases giving a decrease in the value of εr eff and an increase in the value of the characteristic impedance. However, this solution to the problem is artificial [B.14, J.19, J.20]. The failure of the complementary model of the CPS, with respect to a CPW structure on the finitely thick substrate, suggests the need for a new conformal mapping transformation. It is discussed below.

The effective relative permittivity and the filling-factor of a CPS line on the finite thickness substrate are εr eff = q=

K k2 C εr 1 = 1 + εr − 1 C εr = 1 2 K k2

1 K k2 2 K k2

K k1 K k1

a

K k1 K k1

b 9 4 14

The characteristic impedance of the CPS on the finite thickness substrate is obtained from equation (9.4.4) with εr eff taken from the above equation (9.4.14). Let us examine the limiting case, h 0. The physical condition suggests that the substrate is gradually replaced by the air-substrate. Therefore, for h 0, εreff should decrease and the characteristic impedance should increase. However, as h 0, (πs/4h) ∞ and also [π(s + 2w)/4h] ∞ that gives k2 0. It means for h 0, K k2 K k2 increases gradually [B.14] and equation (9.4.14) provides the contradictory results with the increasing value of εr eff and the decreasing value of the characteristic impedance. Such nonphysical results are due to the assumption of the CPS as a complementary structure of the CPW [J.16]. To remove the inconsistency, Ghione and Naldi [J.18] have assumed that the phase velocities of the CPW on the infinitely thick substrate and its complementary CPS structure are identical. Under this assumption, the effective relative permittivity εr eff of the CPS can be written from equation (9.3.15a) as K k1 1 εr − 1 K k1 2

εr eff = 1 +

w εr

jy

z-plane

s

w

K k2 K k2

The total line capacitance C(εr) of a CPS on a finitely thick substrate is still given by equation (9.4.13a). However, C2, i.e. the line capacitance on the (εr − 1) substrate, has to be recomputed. For this purpose, the CPS is redrawn in Fig (9.23a) and an EW of length (1)–(6) is placed along the y-axis at the center of the slot. It is assumed that the capacitance C2 is due to the series combination of two identical sections of capacitance C2 in the third and fourth quadrants, i.e. C2 = C2 2. The conformal mapping to compute the line capacitance C2 is achieved in the following two steps: Step #1

The EW (1)–(6) of the z-plane is transformed as a coplanar strip (t6 − t1) in the t-plane, shown in Fig (9.23b), on an infinitely thick substrate with relative permittivity (εr − 1) by using the following mapping function:

9 4 15

t = cosh 2

t-plane Re (t)

x

1 2 3 4→∞ Path –jh 5→∞ 6 Air-medium EW εr = 1 (a) CPS in z-plane.

h

Alternate Conformal Mapping of the CPS Line on Finitely Thick Substrate

(–∞←) t5 t6

t1

t2

(εr – 1)

t3 t4 (→∞)

Path

(b) Strip and gap located in t-plane.

jv

w-plane MW

EW

K(k3) + jK(k′3)]

(εr – 1)

EW

u MW (c) Parallel plate in w-plane. Figure 9.23

Lower half-space mapping of symmetrical CPS on finite substrate thickness.

πz 2h

9 4 16

9.4 Coplanar Stripline

Table 9.6 Transformation using t = cosh2[πz/2h]. Points in z-plane

C2 = ε0 εr − 1

Points in the t-plane

1) z = 0 s 2) z = 2 s 2w + s 3) z= w + = 2 2 4) z ∞ 5) z = (−jh + x), x ∞ 6) z = − jh

t1 = 1 t2 = cosh 2 t3 = cosh 2 t4 = ∞ t5 = − ∞ t6 = 0

C2 = πs 4h π 2w + s 4h

K k3 K k3

C2 ε0 εr − 1 K k3 = K k3 2 2

a b 9 4 19

The total line capacitance of a CPS on a finitely thick dielectric and air substrates are C εr = 2C1 + C2 , K k1 ε0 εr − 1 K k3 + 2 K k1 K k3 K k1 C εr = 1 = ε0 K k1 C εr = ε0

The path of transformation is shown on both the zand t-planes. The locations (1)–(6) on the z-plane are transferred as the corresponding locations t1, t2, t3, t4, t5, t6 on the t-plane, as shown in Table (9.6).

The coplanar strip structure from the t-plane is transformed as a rectangle in the w-plane. It is shown in Fig (9.23c). The mapping is achieved through the following SC-transformation: t

w = u + jv = t6

dt t − t6 t − t1 t − t2 t − t3

= K k3 + jK k3 9 4 17 The modulus k3 is obtained from equation (9.3.21d) [J.12, J.20]: α−δ β−γ ; where δ = t6 = 0, α−γ β−δ πs γ = t1 , β = t2 = cosh 2 4h π α = t3 = cosh 2 s + 2w 4h πs π − 1 cosh 2 s + 2w cosh 2 2 4h 4h k3 = π πs s + 2w − 1 cosh 2 cosh 2 4h 4h 2 πs 2 π cosh s + 2w sinh 4h 4h = π πs s + 2w cosh 2 sinh 2 4h 4h πs tanh 4h k3 = π s + 2w tanh 4h k23 =

9 4 18 The partial line capacitance C2 for the one-half of the CPS, and also C2 for the complete CPS, on the finitely thick substrate is

b

9 4 20 Finally, the effective relative permittivity and the filling-factor of the CPS are εr eff =

Step #2

a

q=

K k3 C εr 1 = 1 + εr − 1 K k3 C εr = 1 2 1 K k3 2 K k3

K k1 K k1

K k1 K k1

a b 9 4 21

The characteristic impedance of the CPS is obtained from equation (9.4.4) on using equation (9.4.21a) for εr eff. Thepresence of tanh in place of sinh in the modulus alters the limiting conditions. For the case h ∞, k3 k1 = s/(2w + s) and for h 0, k3 1. The decreasing value of K k3 K k3 is obtained for the substrate thickness h 0. Therefore, for h 0, εr eff decreases and the characteristic impedance increases for a CPS line. It is the physically correct description. It is noted that the above expression is obtained without using the complementary condition for the CPS line. Figure (9.24) shows the effective relative permittivity computed using the CPW-based expression (9.4.15) and also using the expression (9.4.21). The complementary condition holds only for the thick substrates, i.e. for 2h/(2w + s) > 2 and low permittivity substrate. The CPS structure could be provided with the additional floating ground plane at the backside of the substrate [J.21]. The backside metallization of a CPS provides mechanical strength and also heat sink facility for the active devices. The presence of the backside conductor increases the effective relative permittivity of a CPS. The CPS with the bottom conductor supports two fundamental modes – the CPS mode and the parasitic microstrip mode. The parasitic microstrip mode is significant for a thin substrate with a wide strip conductor and wide slot width. Both the dielectric and conductor losses of the line structure are high, as compared to the normal CPS line without conductor backing.

323

9 Coplanar Waveguide and Coplanar Stripline

9.4.4 Asymmetrical CPW (ACPW) and Asymmetrical CPS (ACPS) on Finite Thickness Substrate

6 2.0

2h/(s + 2w) 5

1.0

Figure (9.25) shows the ACPW and ACPS as the complementary pairs [J.12, J.22]. It is seen in the previous section that the complementary pair’s concept does not hold for a thin and high permittivity substrate. It gives an error in the computation of the line parameters.

1.0 Effective relative permittivity

324

0.5

4 0.5 3

0.2 0.2

2

ACPW Line

Figure (9.26) shows that the conformal mapping of the ACPW structure is carried out in two steps. Step #1 transforms the ACPW with lower half-space of the inhomogeneous medium to the ACPW with lower half-space filled in with the infinitely thick homogeneous medium having assumed relative permittivity (εr − 1). Step #2 transforms the CPW structure, on the infinitely thick substrate, of the step #1 to a parallel-plate capacitor in the w-plane. The line capacitance of the ACPW structure is

0.1

0.1

1 Eq.(9.4.21) Eq.(9.4.15) 0 0.0

0.2

Figure 9.24

0.4 0.6 s/(s + 2w)

s

C εr = 2C1 εr = 1 + C2 εr − 1 ,

where C1 is the line capacitance of the upper/lower halfspace of the ACPW on the air-substrate. C2 is the line capacitance of the ACPW with lower half-space filled with infinite extent dielectric having relative permittivity (εr − 1).

ACPW Step #1 w1

w2

s

εr

The following mapping function transforms the lower half-space of the ACPW located in the z-plane to an ACPW on the infinite extent substrate in the t-plane:

ACPS

Figure 9.25

ACPW and ACPS as complementary pairs.

t = sinh

jY

–∞ 10 9

9 4 22

w2

εr

h

1.0

Validation of concept of complementary structure (εr = 9.9). Source: Zhu and Wang [J.20]

w1 h

0.8

z-Plane z1 w1 z2 z3 z4w2 z5 x 1

h,εr

2

34

5

6 7

–jh 8

(a) ACPW on finite thickness substrate.

–∞←

t2

t3

t-plane t4 Re(t)

6′ →∞ –j 8′ –j ∞↓ 7′,9′ (b) ACPW in t-plane with infinite thickness substrate. (εr – 1) 1′ 2′ 3′4′ 5′

w-Plane

jv w(t3) 4′

MW

EW

(εr – 1)

w(t4)

t1

10′

5′

MW

2′ w(t2) EW 1′w(t ) 1

(d) ACPS in w-plane. Figure 9.26

u

π z 2h

w-Plane

jv w(t3) 4′

EW

MW

(εr – 1)

w(t4) 5′

EW

2′ w(t2) MW u 1′w(t1)

(c) ACPW in w-plane.

Mapping of asymmetrical CPW/CPS of finite thickness substrate.

9 4 23

9.4 Coplanar Stripline

The transformation is shown in Fig (9.26a,b). The transformation rotates lines (8)–(7) and (8)–(9) about jy-axis such that the closed finite space of the substrate thickness h in the z-plane, opens to the infinite lower half-space in the t-plane. The t-plane transformation is not needed for an ACPW in the air medium. The above mapping function maps the locations z1, z2, z3, z4, etc. in the z-plane to the corresponding locations in the t-plane. These are summarized in Table (9.7). Step #2

Figure (9.26c) shows that the following SCtransformation converts the line segments of the lower half-space in the t-plane to a rectangle in the w-plane: w = u + jv = K k + jK k dt =A t − t1 t − t2 t − t4 t − t5 t where, k =

t4 − t2 t5 − t1 t5 − t2 t4 − t1

k =

t5 − t4 t2 − t1 t5 − t2 t4 − t1

a

1 2

b 1 2

Table 9.7 Transformation using t = sinh(πz/2h). Points in the z-plane

Points in the t-plane

π s + 2w 1 4h π t2 = − sinh s 4h t3 = 0 π s t4 = sinh 4h π t5 = sinh s + 2w 2 4h t6 ∞ t7 = − j∞ t8 = − j t9 = − j∞ −∞ t10

s z1 = − w1 + 2 s z2 = − 2 z3 = 0 s z4 = 2 s z5 = w2 + 2 z6 ∞ z7 − jh + ∞ − jh z8 − jh − ∞ z9 −∞ z10

t1 = − sinh

The role of the modulus k and complimentary modulus k gets interchanged. The modulus k and k are obtained from equation (9.3.21). On substituting the variable t from Table (9.7) in the above equation, the modulus k = k8 , k = k8 is obtained.

c

9 4 24 πs π π sinh s + 2w 1 + sinh s + 2w 2 2 sinh 4h 4h 4h k28 = π πs πs π s + 2w 1 + sinh sinh + sinh s + 2w 2 sinh 4h 4h 4h 4h π πs πs πs s + 2w 1 − sinh sinh + 2w 2 − sinh sinh 2 4h 4h 4h 4h k8 = π πs πs π s + 2w 1 + sinh sinh + sinh s + 2w 2 sinh 4h 4h 4h 4h

In the limiting case of infinite extent substrate, i.e. for h ∞, the above equations for k8 and k8 reduce to equation (9.4.6a,b), applicable to k7 and k7 . The line capacitance of the lower/upper space of the ACPW on the air-substrate, total line capacitance of the ACPW on the air-substrate, and total line capacitance on the dielectric substrate are C1 = ε 0

K k7 K k7

C εr = 1 = 2C1 = 2ε0 C εr = 2ε0

a K k7 K k7

K k7 K k8 + ε0 εr − 1 K k7 K k8

b c 9 4 26

The effective relative permittivity and filling-factor of the ACPW are

εr eff = q=

a 9 4 25 b

K k7 C εr 1 = 1 + εr − 1 K k7 C εr = 1 2 1 K k7 2 K k7

K k8 K k8

K k8 K k8

a b 9 4 27

The characteristic impedance of the ACPW on the air and finite extent dielectric substrates are Z εr = 1 =

1 = vp C εr = 1

K k7 K k7 60π K k7 Z εr = εr eff K k7

Z εr = 1 = 60π

μ0 ε 0

K k7 2ε0 K k7 a b 9 4 28

The effective relative permittivity εr eff could be examined under the limiting cases, h ∞ and h 0. For

325

326

9 Coplanar Waveguide and Coplanar Stripline

h ∞, the ACPW line is on an infinite extent substrate k7 , k8 k7 , q = 1 2. For this case, leading to k8 εr eff = (εr + 1)/2 that shows an equal distribution of the field in the upper and lower media. For h 0, k8 0, i.e. K k8 K k8 0 , i.e. q 0, so εr eff 1. It is physically correct, as h 0 removes the substrate and the lower half-space is filled with air-medium. ACPS Line

The ACPS could be treated through complementary Fig (9.26d). However, it is better to assume identical phase velocity for both the ACPW structure and its complementary ACPS structure [J.18]. Thus, equation (9.4.27) for εr eff and q are also applicable to an ACPS. On using the effective relative permittivity εr eff with equation (9.4.9), the characteristic impedance of the ACPS line on the finite extent substrate is computed. The alternate expressions are also available in the literature for the modulus k8 and k8 [B.14, B.16, J.12]. 9.4.5 Asymmetric CPS Line with Infinitely Wide Ground Plane

coupling of the CPS structure is reduced in the MCL. The TM0 dielectric slab waveguide mode that exists with the CPS is eliminated by the ground plane of the MCL. The structure could be treated as a limiting case of the ACPS structure with w2 ∞. Thus, for an infinitely thick substrate (h ∞), the modulus k9 is obtained from equation (9.4.6) for w2 ∞, w1 = w: s k9 = s+w k9 =

w s+w

1 2

a 1 2

9 4 29 The characteristic impedance of the MCL obtained from equation (9.4.9d) is Z εr = where,

60π K k9 εr eff K k9 εr + 1 εr eff = 2

b

However, for a finite substrate thickness, there is another modulus also. For the lower half-space, the strip widths are w1 = w, w2 ∞. The modulus k10 is obtained from equation (9.4.25):

πs π π sinh s + 2w + exp s + 2w 2 2 sinh 4h 4h 4h k210 ≈ π πs πs π s + 2w + sinh sinh + exp s + 2w 2 sinh 4h 4h 4h 4h πs π exp s + 2w 2 2 sinh 4h 4h ≈ π πs π sinh s + 2w + sinh exp s + 2w 2 4h 4h 4h πs 2 sinh 2 4h k10 = π πs s + 2w + sinh sinh 4h 4h 2

a

9 4 30

The asymmetric CPS line with an infinitely wide ground plane, shown in Fig (9.19e), is commonly used in the electro-optics modulators [J.23, J.24]. It is also known as the micro coplanar line (MCL). The line to line

k 10 =

b

1 − k210

a

b 9 4 31

Another expression for modulus k10 is also available [J.12, B.14, B.16]. The effective relative permittivity and filling-factor from equation (9.4.27) are K k9 K k10 1 εr eff = 1 + εr − 1 a 2 K k9 K k10 q=

K k9 K k9

K k10 K k10

b 9 4 32

The MCL could be provided with another ground plane at the back of the substrate. Such MCL could be

viewed as a microstrip line with an additional coplanar ground plane that helps the mounting of shunt components in the microstrip technology. This structure, also called micro-coplanar strip (MCS), was proposed and analyzed by Yamashita et al. [J.25].

9.4.6 CPS with Coplanar Ground Plane [CPS–CGP] The CPS with two infinitely wide coplanar ground planes forms the (CPS–CGP) line. It is shown in Fig (9.27) [J.2, J.20, J.26]. The presence of ground planes reduces the

9.4 Coplanar Stripline

2c 2b w h,εr

d

s w

w

CPW

line to line coupling that is a drawback of the conventional CPS line. The TM0 dielectric slab mode, supported by the CPS line, is eliminated. The spacing between the signal lines and ground planes is changed to modify the characteristic impedance and effective relative permittivity of the CPS–CGP line. Mclean and Itoh have carried out both the full-wave and quasi-static analysis of the structure [J.2]. The CPS–CGP structure can also be treated as a complementary structure of the finite ground plane CPW on a finitely thick substrate. It is shown in Fig (9.27). The characteristic impedance of the CPS–CGP structure on the air-substrate is given by equation (9.4.4) for the standard CPS. However, the modulus has to be modified to account for the lateral ground planes. The CPS–CGS line is treated as a complementary structure of the CPW with the finite ground plane. Following the discussion of previous sections, the phase velocities of both structures are taken as identical [J.18]. The modulus k11 and k11 are obtained from equation (9.3.17): a c 2 − b2 b c 2 − a2

k11 =

1 − k211

1 2

=

s 2w + s

d + 2w + s d d+w+s d+w

a b

where, 2a = s, 2b = 2w + s, 2c = 2d + 2w + s

9 4 33 Likewise, the modulus k12 and k12 in the presence of the finitely thick substrate are obtained from equation (9.3.18):

k12

πa sinh 2h = πb sinh 2h

k12 =

K k11 K k11 120π K k11 Z εr = εr eff K k11

d

Figure 9.27 The CPS with the coplanar ground plane and its complimentary CPW.

1 − k212

sinh

2

sinh 2

πc − sinh 2 2h πc − sinh 2 2h

2πb 2h 2πa 2h

1 2

a

b

9 4 34 The effective relative permittivity and the characteristic impedance of the CPS–CGP structure are

K k12 K k12

Z εr = 1 = 120π

CPS-CGP

h,εr

k11 =

K k11 1 εr − 1 2 K k11

w

2a s

d

d

εr eff = 1 +

9 4 35

a 9 4 36 b

Zhu and Wang [J.20] have shown that the concept of the complementary pair is satisfactory for h/b ≥ 0.5. For the thinner substrates, on using the direct conformal mapping transformation, they have obtained the following improved expression for the modulus k13: πc π b+a π b−a − tanh 2 tanh 2 4h 4h 4h πc 2 π b + a 2 π b−a 1 − tanh tanh 4h 4h 4h

tanh 2 k13 = tanh 2 k13 =

1 − k213

1 2

a

b

9 4 37 The k12 is replaced by k13 in equation (9.4.35) to compute the improved value of the effective relative permittivity for the CPS–CGP structure. 9.4.7

Discussion on Results for CPS

Figure (9.28) shows the variation in the effective relative permittivity and the characteristic impedance of the symmetrical CPW and symmetrical CPS on alumina (εr = 10) substrate [J.18]. The CPS is treated as the complementary structure of the CPW and their phase velocities are taken as identical, i.e. their effective relative permittivities are identical. The normalized substrate thickness (h/b) is taken as the parameter. Figure (9.28a) shows a decrease in the effective relative permittivity of CPW/CPS for a thinner substrate. It causes an increase in characteristic impedance for both the CPS and CPW for a thin substrate. These results on the characteristics impedance for the CPS and CPW are shown in Fig (9.28b,c), respectively. However, the CPS line provides a much higher value of the characteristic impedance. The characteristic impedance of the CPS increases, while that of the CPW decreases with an increasing aspect-ratio (a/ b). Therefore, the characteristic impedance can be controlled both by varying the substrate thickness and the slot width for a fixed width of the strip. The strip width could be decided by the space consideration on a substrate and by considering the conductor loss [J.12]. The conformal mapping results are compared against the results of the variational method for both the upper and lower bounds. In conclusion, it is seen that the characteristic impedance of

327

300

Conformal mapping method Variational method

6 h/b =

Characteristic impedance (Ω)

Effective relative permittivity

9 Coplanar Waveguide and Coplanar Stripline

∞

5 0.5

4 3 2 0.1 0.1

0.2

0.3 0.4 0.5 0.6 Aspect - ratio (a/b)

0.7

Conformal mapping method Variational method

250 200 150 100

h/b = 0.1 0.5 ∞

0.1

0.8

Characteristic impedance (Ω)

(a) Effective relative permittivity of CPS and CPW. 160

h/b =

140

0.1

0.2

0.3 0.4 0.5 0.6 Aspect - ratio (a/b)

0.7

0.8

(b) Characteristic impedance of CPS.

Conformal mapping method Variational method

120 100 80

0.5 ∞

60 40

0.1

0.2

0.3 0.4 0.5 0.6 Aspect - ratio (a/b)

0.7

0.8

(c) Characteristic impedance of CPW.

6

Characteristics of CPS and CPW on the finitely thick substrate. Source: Ghione and Naldi [J.18]. © IET.

Ch. impedance (Ω)

Figure 9.28

Eff. realtive permittivity

328

h increasing h=

5 4 0

CPS Conventional 10

30 20 Frequency (GHz)

0.3125 mm 0.635 mm 1.270 mm

∞

40

115

Conventional CPS

105 95

CPS-CGP

85

50

(a) Effective relative permittivity of CPS – CGP line [J.26]. Figure 9.29

125

0.2 0.6 1.0 1.4 1.8 Outer slot-width (mm) (b) Characteristic impedance of CPS – CGP line [J.2].

Characteristic of CPS–CPG line on an alumina substrate (εr = 9.9, h = s = w = 0.635 mm). Source: McLean and Itoh [J.2]. © IEEE. Svacina [J.26]. © IEEE.

the asymmetric CPS and CPW are significantly controlled by varying the strip width ratio and the ratio of the slot gaps, respectively [J.22]. However, impedance control in the case of ACPS is more as compared to the control in ACPW. In the case of a CPS–CPG line on a finitely thick substrate, Fig 9.29a shows that the addition of coplanar ground increases the effective relative permittivity of

the CPS, as more field lines are confined in the dielectric region. Furthermore, the CPS–CPG line is less dispersive, as compared to the standard CPS line. Figure (9.29b) shows that the coplanar ground plane strips reduce the characteristic impedance of CPS significantly. The coplanar ground plane strips can be used as an additional means to control the characteristic impedance of a CPS structure.

5.5

170

5.4

150

Ch. impedance (Ω)

Eff. relative permittivity

9.5 Effect of Conductor Thickness on Characteristics of CPW and CPS Structures

5.3 5.2 5.1 5.0 4.9 0.1 0.2 0.5 1 2 5 10 Strip width (w1)/Slot gap (s)

(a) `Effective relative permittivity of MCL.

130 110 90

∞

w2/w1 = 1 CPS 52

MCL 70 50 0.1 0.2 0.5 1 2 5 10 Strip width (w1)/Slot gap (s) (b) Characteristic impedance of MCL.

Figure 9.30 Characteristic of MCL structure on a finitely thick substrate (εr = 9.9, s/h = 0.1). Source: Svacina [J.26]. © IEEE.

Figure (9.30a,b) shows the variation in the effective relative permittivity and characteristic impedance of the MCL structure on a finitely thick alumina substrate (εr = 9.7) [J.26]. The nearness of the large ground plane, i.e. a decrease in slot width, reduces the effective relative permittivity. It also reduces the characteristic impedance. Figure (9.30b) also shows the gradual approach of the CPS line to the MCL, as one of the stripconductors increases to infinity. It shows that the characteristic impedance of the MCL is less than that of the CPS for the same slot width.

9.5 Effect of Conductor Thickness on Characteristics of CPW and CPS Structures The conductor thickness of the CPW and CPS structures, like a microstrip line, has a noticeable influence on their effective relative permittivity, characteristic impedance, and conductor loss. Such influences are also frequencydependent and are obtained using the full-wave methods and also incorporated empirically in the results of the conformal mapping method [B.16, B.17, J.27–J.29]. 9.5.1

normalized conductor thickness [J.28]. The increase in conductor thickness increases the equivalent widths of the strip conductors that reduce the slot gap width causing an increase in the CPW line capacitance. The enhanced value of the line capacitance reduces the characteristics impedance Z0 = 1 c C εr C εr = 1 of the CPW. For t/w = 0.02, an increase in the guided wavelength is about 1% and a decrease in the characteristic impedance is about 1.5% with respect to the results of zero conductor thickness. To account for the effect of conductor thickness on the CPW line parameters, the physical width(s) of the central signal strip is replaced by the equivalent width [B.16]. Two empirical relations are used for this purpose. Effectively, conductor thickness decreases the slot width w: seq = s + Δs

a

weq = w − Δs

b

where, Δs =

t 8πw 1 + ln 2πεr t

Alternatively, Δs =

t 1 + ln 4 − 0 5 ln π

CPW Structure

The conductor thickness (t) causes more confinement of the electric fields in the air-medium of two slot regions causing a decrease in the effective relative permittivity. Figure (9.31a) shows the full-wave method results of a CPW for the frequency-dependent slowing-factor 1 εr eff f, t . It also demonstrates an increase in the slowing-factor with the increase in normalized conductor thickness (t/w) strip conductors. Figure (9.31b) further shows the decrease in the frequency-dependent characteristic impedance with an increase in the

c

t h

2

+

t πs

2

8πw t

d 951

The equation (9.5.1d) is taken from the reference [B.17]. The aspect-ratio parameter k1,t of the CPW, on infinite thickness substrate, with conductor thickness is obtained from equation (9.3.5): seq k1,t = a seq + 2w eq k1,t =

1 − k21,t

b 952

329

9 Coplanar Waveguide and Coplanar Stripline

5.0 0.10

0.33

Characteristic impedance (Ω)

Slowing factor [1/Sqrt (εreff (f))]

330

0.04 0.32 0.02 t/h = 0.00 0.31

t/h = 0.00 0.02 0.04

4.8

0.10 4.6 εr = 20.0, h = 1.0 mm

εr = 20.0, h = 1.0 mm 0.00

2

4

6 8 10 Frequency (GHz)

12

2

(a) Slowing factor. Figure 9.31

4

12

(b) Characteristic impedance.

Effect of conductor thickness on the line parameters of CPW. Source: Kitazawa et al. [J.28]. © IEEE.

Likewise, the aspect-ratio parameter k2,t of the CPW, on finite thickness substrate, with conductor thickness could be obtained from equation (9.3.11) on using seq and weq. The effective relative permittivity and characteristic impedance are computed by the following empirical relation [B.16]: 0 7 εr eff t = 0 − 1 t w εr eff t = εr eff t = 0 − K k K k + 0 7t w 30π K kt Zt = εr eff t K kt

a b 953

The above expressions are approximate and do not provide good results for a narrow slot width CPW.

9.5.2

6 8 10 Frequency (GHz)

effective relative permittivity. For a change in the normalized strip thickness from 0 to 0.1, the εr eff(f ) is changed by 6%, whereas the Z0(f ) is changed by 9 and 22% at h/λ = 0.01 and h/λ = 0.1, respectively.

9.6 Modal Field and Dispersion of CPW and CPS Structures The quasi-static descriptions of the CPW and CPS structures are not sufficient in the mm-wave range. The realistic descriptions of the fields, modes, and dispersion are only possible with the help of the full-wave analysis [B.1, J.30]. This section discusses the results obtained from the full-wave analysis to appreciate the modal fields and dispersion. The closed-form expressions for dispersion are also presented.

CPS Structure

The full-wave results, shown in Fig (9.32), demonstrate that the effective relative permittivity and the characteristic impedance of a CPS decrease with an increase in the conductor thickness [J.29]. Results are shown for a CPS on GaAs (εr = 12.9) substrate with slot width (b − a)/h = w/h = 0.5. The normalized conductor thickness t/(b − a) = t/w is a parameter. The characteristic impedance, using the power– voltage definition, decreases, and the effective relative permittivity increases with increasing frequency. The changes, due to the conductor thickness, in the characteristic impedance, especially in the higher frequency range, are more as compared to the changes in the

9.6.1

Modal Field Structure of CPW

Figure (9.2) presents several structural forms of the CPW line. Each structure has a different field configuration and supports a separate modal field. Thus, the dispersion behavior of the CPW line is different for each structure. The finite thickness substrate CPW, and also the CBCPW, is discussed below. CPW on a Finite Substrate

Figure (9.33a) shows the electric and magnetic field lines of a CPW for the quasi-static case, supporting the TEMtype mode. Both the ground planes are at zero potential. The identical potential on both ground conductors is

9.6 Modal Field and Dispersion of CPW and CPS Structures

100 t/w = 0.0, 0.025, 0.05

80 70

Eff. relative permittivity

0.1 8.0

GaAs εr = 12.9, w/h = 0.5

7.5

t/w = 0.0, 0.025, 0.05

6.5

60

Ch. imp (Z0)

90

50

0.1

5.5 0.1

Figure 9.32

0.3

0.5 0.7 Normalized frequency (h/λ)

0.9

1.0

Effect of conductor thickness on effective relative permittivity and the characteristic impedance of a CPS. Source: Rahman and Nguyen [J.29]. © IEEE.

W

εr

S

Current

Magnetic field lines

W Increasing frequency

h

w εr h

Electric field lines

1.0 0.8

TM0

TM1

TM2

0.6 0.4

CPW-mode

0.2 0.0 0.0

0.25 0.5 0.75 1.0 1.25 Normalized frequency (h/λd) (c) TM-surface wave modes [J.15].

w

(b) Current distribution on strip conductors.

1.5

Normalized propagation const. (β/β0)

Normalized propagation const. (β/β0)

(a) TEM-mode field distribution.

s

Increasing frequency

1.0 0.8

TE0

TE1

0.6 0.4

CPW-mode

0.2 0.0 0.0

0.25 0.5 0.75 1.0 1.25 Normalized frequency (h/λd)

1.5

(d) TE-surface wave modes [J.15].

Figure 9.33 Fields, current, and modes on finite thickness substrate CPW. Source: Riaziat et al. [J.15]. © IEEE.

331

332

9 Coplanar Waveguide and Coplanar Stripline

maintained with the help of the air-bridge, (not shown in the figure), connecting them. Such CPW supports the even mode [J.4]. If both ground conductors are at different potentials, the CPW supports the odd mode and a slot line mode could be excited. In the case of the evenmode operation, electric field lines originate at the central conductor and terminate at the ground planes. The magnetic fields surround the central conductor. One can place any other material in the slot region, or just below the central strip, for their interaction with CPW fields. It helps to characterize the material using CPW. Figure (9.33b) demonstrates the longitudinal current distribution across the width of strips, showing maximum value, i.e. the singularity, at the edges. The current decreases away from the slot edges. The decrease is faster at a higher frequency. The edge singularity of the longitudinal current is associated with the excess charge concentration at the edges of strip conductors. A CPW without the conductor backing supports the quasi-static mode that is a combination of the TEMmode (static CPW mode) and the surface wave mode. The surface wave modes are supported by a conductor-backed dielectric sheet. It is discussed in section (7.5) of chapter 7. In the case of a CPW with a finite thickness substrate, the structure supports both the TM- and TE-type surface wave modes shown in Fig (9.33c,d) [J.15]. These modes are excited by the slot fields of a CPW. Figure (9.33c) shows the range of normalized propagation constant for the CPW mode (dashed line), corresponding to (0.48 ≤ εr eff/εr ≤ 0.64). The TM0 surface wave mode has no cutoff frequency. It is similar to the CPW mode (TEM-mode). At lower frequency, phase velocities of the TM0 surface mode and CPW mode are different. So coupling between these two modes is weak, leading to a small dispersion. However, at high frequency, close to the point of intersection, both modes have the same phase velocity leading to a strong coupling that results in a large dispersion. Thus, in the case of a dispersive CPW, without conductors backing, the dispersion is due to the coupling between the TEM mode (CPW mode) and the surface wave mode. The CPW should be used below the first point of intersection of modes, shown in Fig (9.33c). The operating frequency should meet the condition h ≤ 0.15 λd, where λd is the wavelength in a dielectric medium. Above the point of mode intersection, the loss in a CPW increases with frequency due to the surface wave loss. The TE0 surface wave mode is excited at the substrate thickness 2h ≈ 0.25 λd. Figure (9.33d) shows that near 2h ≈ 0.35 λd, the phase velocities of TE0 surface wave mode and CPW mode are almost identical. For h ≈ 0.12 λd, the

TE0 mode can be avoided, and also dispersion due to coupling between the TM0 mode and CPW mode can be reduced. Conductor-Backed CPW (CB-CPW)

Figure (9.34) shows the electric field configurations of different modes of the CB-CPW structure with the finite width coplanar ground planes. It is noted that the CBCPW structure can support normal CPW even mode, microstrip mode, first higher-order microstrip mode, and image guide mode, depending on the structural parameters and the operating frequency. However, in practice, the structural parameters are selected such that only the CPW and microstrip modes are supported by the structure. The back conductor is at the floating potential; still, it supports the undesired microstrip mode. The CB-CPW also behaves like a parallel plate waveguide. Figure (9.34e) demonstrates that its fundamental mode is the TEM parallel-plate mode and it also supports other TE/TM higher-order modes. The phase velocity of the TEM parallel-plate mode is less than the phase velocity of the TEM-type CPW mode. It causes a loss of power in a CPW in the absence of lateral confinement of fields. The TE1/TM1 modes appear at a higher frequency, above h = 0.12 λd. The coupling between the CPW mode and TE1/TM1 parallel plate modes causes power loss and dispersion. The loss and dispersion can be controlled, if h ≤ 0.12 λd. The lateral confinement of fields in a CB-CPW is achieved by connecting the ground planes of the upper layer, using an array of vias, to the conductor backing. In this case, the structure also supports the rectangular waveguide modes. The coupling between the waveguide mode and the CPW mode also causes dispersion. This is again avoided by operating the CPW below the cutoff frequency of the waveguide modes. 9.6.2

Modal Field Structure of CPS

A CPS line also supports the quasi-TEM mode. Figure (9.35a) shows its field lines. At higher frequency, it also supports the higher-order modes. Figure (9.35b) presents the propagation constants of the fundamental TEM mode and the first higher-order mode [J.31]. The first higher-order modes appear around 9 GHz and the next higher-order mode appears at 40 GHz not shown in Fig (9.35b). The CPS line on an infinitely thick substrate does not support any parasitic surface wave mode or the dielectric slab mode. However, it supports the leaky-wave

9.6 Modal Field and Dispersion of CPW and CPS Structures

Electric field lines w

s

w

s

w

w

Electric field lines

εr h

εr h

(b) Microstrip quasi-static mode.

(a) CPW mode. s

w

w

w

Electric field lines

εr h

Normalized propagation const. (β/β0)

w Electric field lines

εr h

(c) Microstrip first higher-order mode.

s

(d) Image guide-like mode.

1.0 TEM

0.8 TE1 / TM1

0.6 0.4

TE2 / TM2 TE3 / TM3

0.2 0.0 0.0

0.25 0.5 0.75 1.0 1.25 Normalized frequency (h/λd)

1.5

(e) Parallel plate modes [J.15].

Magnetic field lines w εr

Electric field lines

s

w

h

(a) TEM field of CPS line.

Normalized propagation const. (β/β0)

Figure 9.34 The modes on finite width conductor-backed CPW. Source: Riaziat et al. [J.15]. © IEEE.

3.0 ε = 10, h = 0.03″ r strip - width w = 0.065″ , slotwidth s = 0.004″

2.5

Fundamental CPS mode

2.0 1st higher order mode

1.5 1.0

0

5

15 10 20 Frequency (GHz)

25

(b) The quasi-TEM mode and appearance of 1st higher-order mode on CPS [J.31].

Figure 9.35 Field and modes of CPS line. Source: Goverdhan et al. [J.31].

radiation mode that causes the power loss to the CPS. This aspect of power loss, both in the CPW and CPS structures, is further discussed in the next section [J.32]. The CPS on a finite thickness substrate can excite the TM- and TE-type dielectric waveguide modes both in the even and odd mode formats. The

TM0 and TE0 modes have no cutoff frequency. Again the power of a CPS is lost to these parasitic guided modes, as these modes become the leaky modes. A narrow strip width CPS excites the TM-type substrate mode, whereas the wide strip excites both the TM- and TE-type substrate modes.

333

334

9 Coplanar Waveguide and Coplanar Stripline

The conductor-backed CPS, i.e. CB-CPS, supports two fundamental modes – the CPS mode and the parasitic microstrip mode [J.33]. It also supports parallel-plate modes. The parasitic microstrip mode is significant for a thin substrate and wide strip dimension. The CBCPS further supports the surface wave mode due to the conductor-backed dielectric. The higher-order modes appear above the critical frequency given by

fc =

1 h

961

2μ0 ε0 εr − 1

By using a thin substrate, the critical frequency could be pushed to a very high value, so that the CPS line could be used without the appearance of higher-order modes. 9.6.3

Closed-Form Dispersion Model of CPW

The closed-form dispersion model of the CPW, on a finitely thick substrate, is presented below. However, coplanar ground conductors are infinitely wide. The conductors are treated as the perfect conductors. Next, the dispersion model is improved to take into account the finite conductivity and finite thickness of the strip conductor. CPW on a Finitely Thick Substrate

Hasnain et al. [J.34] have adopted the dispersion expression of a microstrip line proposed by Yamashita et al. [J.35] to model the dispersion behavior of a CPW on a finitely thick substrate. The dispersion model is given below: εr eff f =

εr eff 0 +

εr − εr eff 0 1 + aF − b

2

962

In the above equation, F is the normalized frequency with respect to the cutoff frequency of the TE1 surface wave mode: F = f f TE c c TE fc = 4h εr − 1

a b 963

The expression for f TE c is taken from equation (7.5.28) of chapter 7. The static effective relative permittivity εr eff(0) of a CPW is obtained from equation (9.3.15). Empirically obtained constants a and b are dependent on the CPW structure:

b=18 s a ≈ exp u log +v w u = 0 54 − 0 64q + 0 015q2 v = 0 43 − 0 86q + 0 54q2 s q = log h

a b c d e 964

The model has 5% accuracy for the ranges 0.1 < s/w < 5, 0.1 < s/h < 5, 1.5 < εr < 50, 0 < f/fTE < 10. Figure (9.36a) compares the dispersion in a 50Ω CPW, on the GaAs substrate (εr = 13, h = 100 μm) with central conductor width s = 85 μm, slot width w = 50 μm, against a 50 Ω microstrip on the same substrate with strip width w = 73 μm. The static εr eff(0) of a CPW is lower compared to εr eff(0) of a microstrip line. At very high frequency, for both lines, εr eff(f ) approaches to εr. Therefore, for a very wide bandwidth signal, a CPW is more dispersive than a microstrip. However, below f TE c , a CPW is less dispersive as compared to a microstrip line. Jackson et al. [J.37] have shown that the static εr eff(0) can be used for the design of the GaAs-based MMIC up to 40 GHz. The small dispersion in a CPW is due to the confinement of fields in the slot region that is to a large extent independent of frequency. The nature of dispersion in the CPW, shown in Fig (9.36a), suggests that the CPW dispersion can also be modeled using the LDM of chapter 8. However, a set of parameters applicable to the CPW has to determined from the data obtained using an EM-Simulator. Improved Dispersion Model

The above model ignores the thickness and conductivity of the strip conductors. The finite conductivity of a strip conductor permits penetration of the magnetic field inside the strip that creates the internal inductance [J.38]. The line capacitance increases with an increase in the finite thickness of the strip conductor. Both parameters are combined to modify the static effective relative permittivity, εr eff(f = 0). So, the εr eff(f = 0) used in equation (9.6.2) could be replaced by the conductor thickness and skin-depth-dependent effective relative permittivity εr eff(t, δ, f = 0). The εr eff(t, δ, f = 0) is computed below. The field penetrates through all sides of the strip and ground conductors by the skin-depth, δ = 1 μ0 πfσ , where σ is the conductivity of conductors. It is shown in Fig (9.38). In the process, the strip width s is reduced and gap width w is increased. Finally, the aspect-ratio k1δ due to the skin-depth is changed. Using the equality

9.6 Modal Field and Dispersion of CPW and CPS Structures

52 50

CST

Analytical

CM

48 46 44 0

1.0

(a) Comparison of dispersion behaviors of microstrip and CPW [J.34].

80

40 60 Frequency (GHz)

20

100

(c) Low-frequency behaviors of Real part of Z0 of CPW with finite conductivity [J.36].

7.4

0.0

7.2 7.0 6.8

HFSS

CST

Analytical

CM

Im. impedance (ohm)

Effective dielectric constant

HFSS

Normalized propagation const. ( β/β0 )

4.0 3.8 εr = 13 , h = 100 μm Microstrip 3.6 3.4 Z = 50 Ω, f TE = 216 GHz 0 c 3.2 CPW 3.0 2.8 2.6 2.4 2.2 2.0 –1.0 –.5 0.0 .5 Normalized frequency log (f/fcTE)

Re. impedance (ohm)

εreff (f) β/β0 =

Nor. propagation const.

54

6.6 6.4

–1.0 –2.0 HFSS

CST

CM

–3.0

6.2 –4.0

6.0 0

20

40 60 Frequency (GHz)

80

100

0

40 60 Frequency (GHz)

20

80

100

(d) Low-frequency behaviors of Imag. part of Z0 of CPW with finite conductivity [J.36].

(b) Low-frequency dispersion behaviors of CPW with finite conductivity [J.36].

Figure 9.36 Dispersion behaviors of CPW. CM: Circuit model. Source: Hasnai et al. [J.34]. © IEEE. Verma et al. [J.36]. © John Wiley & Sons.

of the phase velocity of the EM-wave on the CPW considered on the air-substrate and the EM-wave in the airmedium and also using equation (9.3.7), the total line inductance p.u.l. is obtained as follows: L δ C εr = 1, δ = ε0 μ0 , where, k1,δ = k1,δ =

c εr eff δ, t, f = 0

μ0 K k1,δ 4 K k1δ

K k1,δ K k1δ

εr eff δ, t, f = 0 =

c

εr eff δ, t, f = 0 = P εr eff t, f = 0

Using equation (9.3.7b), line capacitance p.u.l. of a CPW, on the air-substrate, with conductor thickness t is given below: 966

The modified aspect-ratio k1,t is given in equation (9.5.2). The conductor thickness-dependent total line capacitance p.u.l. can be written as follows: 967

The phase velocity of the EM-wave on the CPW is

where P =

K k1,δ K k1δ

4ε0

K k1,t K k1,t

b

965

C εr , t = C εr = 1, t εr eff t, f = 0

=

εr eff δ, t, f = 0 = c2 L δ C t, εr

a

1 − K21δ

K k1,t C εr = 1, t = 4ε0 K k1,t

1 L δ C εr , t

εr eff δ, t, f = 0 = c2

μ K k1,δ Lδ = 0 4 K k1,δ

s−δ s + 2w + 2δ

vp =

K k1,t K k1,t

εr eff t, f = 0

εr eff t, f = 0 a

K k1,t K k1,t

b

968 The above equation gives the conductor thickness and skin-depth-dependent relative permittivity εr eff(t, δ, f = 0) in terms of factor P due to the skin-depth and conductor thickness. On replacing the εr eff(f = 0) used in equation (9.6.2) by εr eff(δ, t, f = 0), the final dispersion expression is obtained for the CPW with finite strip conductivity and finite strip thickness: εr eff f, t, δ =

Pεr eff t, 0 +

εr − Pεr eff t, 0 1 + aF − b 969

2

335

336

9 Coplanar Waveguide and Coplanar Stripline

The above model is valid for frequency, f > 1/(μ0σ t s). Below this frequency, the current distribution is almost uniform across the strip and the line inductance has to be computed as the DC inductance [J.39].

shows such strong dispersion. The present analytical models show similar behavior. The complex impedance is obtained using the circuit model (CM), discussed in the section (9.8.1).

Frequency-Dependent Characteristic Impedance

9.6.4

The characteristic impedance for a CPW can be computed from Z0 f, εr , t, δ =

Z0 f = 0, εr = 1, t, δ

9 6 10

εr eff f, t

The characteristic impedance, Z0(f = 0, εr = 1, t, δ), of a finite conductivity CPW on the air-substrate is obtained as follows: Z 0 f = 0, εr = 1, t, δ = μ K k1,δ = 0 4 K k1,δ

Lδ C εr = 1, t

1 K k1,t K k1,t × 4ε0 K k1,t K k1,t

Z0 f = 0, εr = 1, t, δ = 30π = Z0 f = 0, εr = 1, t

K k1,t K k1,t

K k1,t K k1,t

1 2

P

Using the above equations, we can compute the characteristic impedance of a CPW of strip conductors with finite conductivity and finite thickness from the following expression: Z0 f = 0, εr = 1, t εr eff f, t, P = 1

P

The full-wave methods, such as the SDA [J.40], variational [J.13], and mode matching [J.29], have been used to obtain the frequency-dependent characteristic impedance and effective relative permittivity of a CPS line. In a lower frequency range, the CPS line behaves as a quasi-TEM mode line. However, in a higher frequency range, its hybrid nature appears. The voltage–power definition [J.29, J.40] shows an increase in the characteristic impedance of a CPS line with increasing frequency. The power– current definition shows an initial decrease and then an increase in characteristic impedance with increasing frequency [J.13]. The effective relative permittivity εr, eff(f ) shows an increase with an increase in frequency. Closed-Form Dispersion Model of CPS Line

P

9 6 11

Z 0 f, εr , t, δ =

Dispersion in CPS Line

9 6 12

The characteristic impedance Z0(f = 0, εr = 1, t), frequency-dependent effective relative permittivity, and factor P are obtained for various CPW structures from our previous discussions. The effect of penetration of the magnetic field inside the strip conductor of finite conductivity can also be viewed as the creation of effective permeability in a CPW and the factor P is related to it. This aspect is accommodated in the frequency-dependent modeling of the multilayer CPW structure using the SLR process [J.36]. It is discussed in the section (15.3) of chapter 15. Figure (9.36b–d) shows the effect of finite conductivity on a CPW designed on GaAs substrate, εr = 12.9, tan δ2, h = 380 μm, s = 16 μm, w = 12 μm, t = 3 μm, σ = 4.1 × 107 S/m over the frequency range 1 –100 GHz. There is a significant increase in the relative permittivity and real part of characteristic impedance at the lower end of frequency due to the dispersive behavior. It is due to frequency-dependent magnetic field penetration. The imaginary part of the characteristic impedance also

The closed-form expression, discussed in this section, to compute the frequency-dependent effective relative permittivity εr eff(f ) of the CPW structure, has also been used for the CPS line up to mm-wave range [J.32, J.34, J.41, J.42]. The effect of conductor thickness on the static effective relative permittivity and characteristic impedance should be accounted for. The following expressions, for the CPS on an infinitely thickness substrate, compute the thickness-dependent line parameters [B.16, J.42]: εr eff t = εr eff t = 0 − Zt =

120 π εr eff

K k1,t t K k1,t

1 4 εr eff t = 0 − 1 t s K k1 K k1 + 1 4t s

a b

9 6 13 For a finite thickness substrate, k1, t is replaced by k2, t. While calculating the aspect-ratios k1, t and k2, t, using equations (9.3.5) and (9.3.11), the modified dimensions seq = s − Δw, weq = w + Δw should be used in place of the physical dimensions s and w. Equation (9.5.1c) is used to compute Δw. The finite conductivity can also be accommodated in the computation. However, its effect, showing an increase in both the effective relative permittivity and characteristic impedance at the lower end of frequency, is also accounted for using the CM discussed in section (9.8.1). Figure (9.37a) presents the dispersive behavior of a CPS on GaAs substrate, εr = 12.9, tan δ2, h = 670 μm, s = 7 μm, w = 7 μm, σ = 4.1 × 107 S/m, over the

9.7 Losses in CPW and CPS Structures

25

HFSS

Sonnet

Analytical

[J.43]

Effective relative permittivity, εreff (f)

Phase constant β (Rad/cm)

30 t = 0.5 μm

20 t = 3 μm

10

t = 9 μm

5 0 0

10

20

30

40

50

60

21 18

HFSS Sonnet [J.43] Analytical Circuit model

15 12 9 6 0

5

Frequency (GHz)

(a) Dispersive phase constant. Img. characteristic impedance Z0 (Ohm)

Re. characteristic impedance Z0 (Ohm)

200 180

Sonnet [J.43] Circuit model

140 120

0 –20 –40

5

10 Frequency (GHz)

15

Sonnet Circuit model

–80 –100

20

(c) Low-frequency dispersion in Real Z0.

HFSS [J.43]

–60

–120

100 0

20

(b) Low-frequency dispersion.

220

HFSS Analytical

10 15 Frequency (GHz)

0

5

10 15 Frequency (GHz)

20

(d) Low-frequency dispersion in Img. Z0.

Figure 9.37 Dispersion behavior of CPS. Source: Majumdar and Verma [J.42]. © Taylor & Francis Ltd.

frequency range 1–60 GHz. Figure (9.37b–d) shows the dispersion of CPS for s = 8 μm, w = 4 μm, σ = 4.1 × 107 S/m over 1–20 GHz. The present analytical model does not account for the finite conductivity. So, it does not show low-frequency dispersion. The circuit model (CM) accounts for the effect of both the finite conductivity and loses on dispersion. It shows the low-frequency dispersion for effective relative permittivity and also for the real and imaginary parts of characteristic impedance. The results follow closely the results of two EMSimulators, HFSS and Sonnet [J.42], and also the experimental results [J.43].

9.7

Losses in CPW and CPS Structures

There are three prime mechanisms for the power loss in both the CPW and CPS structures – conductor loss, dielectric loss, and radiation loss. Under normal circumstances, conductor loss, due to the finite conductivity of conducting strip conductors, is the main source of power loss. The dielectric loss is small due to low loss-tangent of the substrate materials. The radiation loss is also small at the microwave and the lower end of mm-wave. However, under certain circumstances, it becomes a major source of loss. This section discusses the mechanism of losses and presents closed-form expressions to compute them.

9.7.1

Conductor Loss

The conductor loss in the CPW and CPS is decided by the finite conductivity of the strip conductors and geometry of the structures. It also strongly depends on the thickness of strip conductors. It increases with a decrease in the normalized conductor thickness t/δs, where δs is the skin-depth. Several quasi-static and full-wave methods have been reported for the computation of conductor loss [J.44–J.46]. The quasi-static methods, such as the conformal mapping methods [B.3, J.12, J.17, J.47], and numerical methods, like MOM, [B.18, J.48] are based on the perturbation approach. This method assumes that the magnetic fields tangential to the conductor surfaces do not change significantly when the conductivity is changed from infinite to a finite practical value. However, the assumption is not correct at the low frequency; where magnetic field penetration in a practical conductor is significant, leading to an increase in the internal inductance of a line [J.39]. The penetration of the magnetic field influences not only the conductor loss but also the dispersion characteristics and characteristic impedance [J.49, J.50]. It is discussed in subsection (9.6.3). The current density on the strip conductor shows the edge-singularity that causes the integral, used in the perturbation method, to diverge. However, the problem of divergence has been overcome by the concept of stopping

337

338

9 Coplanar Waveguide and Coplanar Stripline

distance while integrating the current density on the strip conductor [J.51, J.52]. The quasi-static Wheeler’s incremental inductance method discussed below, can be used to compute the conductor loss in both the CPW and CPS structures [B.16, J.25, J.53]. However, this method is applicable only for the conductor thickness greater than the skin-depth, i.e. for t > 1.1 δs [J.54]. The perturbation method with stopping distance concept is also presented. It applies to a conductor thickness both greater and less than the skin-depth. Wheeler’s Incremental Inductance Method

Wheeler’s incremental inductance method normally uses the partial derivative of the characteristic impedance of a CPW [B.16]. However, this process is not very accurate [J.12]. A more accurate and general formulation of Wheeler’s inductance rule, in terms of the fractional characteristic impedance, is discussed in subsection (8.4.2) of chapter 8 [J.53, J.55]. It is also applicable to the CB-CPW structures and the CPW with top shield, shown in Fig (9.38). The expression presented here is also applicable to the multilayer case. The expression (8.4.71) of chapter 8, applicable to a microstrip line, is given below for the CPW case: αc =

27 3 εCPW r eff εr , w, s, h, t × λ0 ΔZ0 εr = 1, w, s, h, t, δs Z0 s + Δs, w − Δs, h, εr = 1

971 dB m

The above expression is written for a CPW on the finite thickness substrate. Figure (9.38b) shows that the conductor thickness increases the strip width (s) of the central conductor to the enlarged width s + Δs and decreases the slot width (w) to (w − Δs). The characteristic impedance Z0(s + Δs, w − Δs, h, εr = 1) with conductor thickness t is for the CPW on the air-substrate. It is, along with εCPW r eff εr , w, s, h, t , obtained from equation (9.5.3). Figure (9.38c) shows that the strip width and conductor thickness decrease due to skin-depth while the slot width increases. The incremental change in width (Δs) due to the conductor thickness and skindepth is computed using equation (9.5.1d). The

w s w

H

εr h (a) Top-shielded CPW. Figure 9.38

Δs/2 Δs/2 t

Δs/2 t

w

s

w

(b) Extension of strip width due to conductor thickness.

difference characteristic impedance, due to the skindepth, ΔZ0, is defined as follows: ΔZ0 εr , w, s, h, t, δs = Z0 s + Δ s − δs , w − Δ s + δs , h +

δs , εr = 1 2

− Z0 s + Δs, w − Δs, h, εr = 1 972 On the right-hand side, the first characteristic impedance takes care of both the conductor thickness and the skin-depth δs in the strip conductor. Figure (9.38c) shows the field penetration δs/2 all around the strip conductor. The effective thickness of the strip conductor is reduced to t = t − δs, where δs = 1 μ0 πfσ. The substrate thickness is increased to the thickness h + δs/2. The incremental increase in the strip width (Δ s) due to the effective conductor thickness t is given by Δ s = t π 1 + ln 4 − 0 5 ln

t h

2

+ t πs

2

973 The effective strip conductor width due to both effective conductor thickness t and skin-depth δs is (s + Δ s − δs) and the effective slot width is (w − Δ s + δs). The characteristic impedance corresponding to the first term of equation (9.7.2) is computed for the effective strip width (s + Δ s − δs), effective slot width (w − Δ s + δs), and the effective substrate thickness (h + δs/2) with relative permittivity εr, using expression (9.3.16). The characteristic impedance, corresponding to the second term, is also computed without skin-depth using the same expression. The above-discussed process is also applicable to a CBCPW. In this case, εCPW r eff t = 0 has to be computed from equation (9.3.46). However, in this case, the substrate thickness with skin-depth is (h + δs). The difference characteristic impedance in for CB-CPW is ΔZ = Z0 s + Δ s − δs , w − Δ s + δs , h + δs , εr = 1 − Z0 s + Δs, w − Δs, h, εr = 1 974 The characteristic impedance of the CB-CPW is computed from equation (9.3.47).

δ2/2

δ2/2 w

s

δ2/2 w

(c) δ/2 - field penetration around strip-conductor.

Application of Wheeler’s inductance rule to compute conductor loss.

9.7 Losses in CPW and CPS Structures

In the above equations, Im[ ] stands for the imaginary part of the expression under the bracket [ ]. For a conductor, μc = μ0, the wavenumber in a conductor is given by

The conductor loss of a CB-CPW, with the top shield, could be also computed. The suitable expression for εCPW reff t = 0 is used for the CB-CPW with a top shield. The shield height with skin-depth is (H + δs). The difference characteristic impedance for this case is

kc = ω μc εc = ω

ΔZ = Z0 s + Δ s − δs , w − Δ s + δs , h + δs , H + δs , εr = 1 − Z0 s + Δs, w − Δs, h, H, t, εr = 1

μ0 ε 0 − j

σ σ = ω μ0 ε0 1 − j ω ωε0

1 2

978

975 Modified Perturbation Method

CPW Structure

Wheeler’s incremental inductance method presented above does not apply to the case of a strip conductor thickness less than the skin-depth, i.e. for t < 1.1 δs [J.53]. Even for the conductor thickness slightly more than the skin-depth, it is not accurate. The modified perturbation method, using the concept of the stopping distance (Δ), computes conductor loss of the CPW and CPS structures without any restriction on the conductor thickness [J.56, J.57]. The closed-form expression of the conductor loss is obtained by modifying the classical perturbation method in which the integration limit excludes the strip edge singularity by excluding a small distance, called the stopping distance (Δ), from both edges of the strip conductor. The conductor losses for both the CPW and CPS are computed by the following modified perturbation method [J.56]:

Figure (9.39a) shows the CPW structure on a finite thickness substrate and with infinite extent ground conductors. However, to avoid the edge singularity, the stopping distance Δ is considered at all conductor edges. The strip can acquire a trapezoidal shape in the fabrication process. The current distribution (J) on the central strip conductor is taken as follows, showing singularity at x = ± a, b [J.17]:

αc = αc =

For CPW

Rsm 4ZCPW 0

Rsm CPW CPS Δ 4Z0 a−Δ −a + Δ

J I

J I

J=

dx + 2

J I

b+Δ

b

J=

The modified surface impedance Rsm, that takes care of both the upper and lower surfaces of the strip conductor, is given by the following expression [J.58, J.59]:

∆

cot kc t + cosec kc t kc t

∆ ∆ ∆ θ w w s a a b b

(a) CPW structure.

∆ w b

∆ a

b a b

c

A x 2 − a 2 x 2 − b2

;

x >b

9 7 10

Equation (9.7.6b) for a CPW is written below by taking into account the stopping distance Δ and symmetry of the CPW while evaluating the integral for the conductor loss computation:

977

∆ θ

a

In the above equation, I is the current on the strip conductor. The current density on the ground plane of CPW is

976

Rsm = ωμc t Im

x > (b − a), εCPS reff = εr + 1 2 The conductor loss αc for the CPS line is obtained as follows, on substituting equation (9.7.16) in equation (9.7.6) and carrying integration with respect to Fig (9.39b):

The expressions (9.7.13) and (9.7.14) for the stopping distance Δ are valid for the CPS also. However, the presence of a ground plane disturbs its value. Figure (9.40b) shows that the conductor loss of the CPS computed by the present method is almost identical to the results obtained using the mode-matching method [J.45, J.62, J.63]. The CPS is on the GaAs substrate εr = 12.9, a = 10 μm, b = 25 μm, t = 1.5 μm, and σ = 3 × 107 S/m. 9.7.2

Dielectric Loss

The CPW and CPS lines are fabricated on the low-loss substrates. Therefore, the dielectric loss is much smaller than the conductor loss. The dielectric loss for a CPW on a finite thickness substrate, with and without conductor backing, is computed by the single-layer reduction (SLR) formulation. The method also includes the top shield. Figure (9.41) shows the shielded finite thickness substrate CPW with the steps involved in the SLR process.

341

342

9 Coplanar Waveguide and Coplanar Stripline

This method is also applicable to the CPW with a multilayer dielectric medium [J.64]. It is discussed in section (15.3) of chapter 15. The dielectric loss of the CPW structure on the finite thickness substrate of Fig (9.40a) is computed by the following expression [J.64, J.65]:

αCPW d

π εr,eq λ0 εr eff

εr eff − 1 tan δeq Np m εr,eq − 1 εr,eq εr eff − 1 tan δeq = 27 29 dB m εr eff εr,eq − 1 λ0

αCPW = d

a b, 9 7 19

where εreff of the CPW structure, without conductor backing, with conductor backing, and with/without a top shield, is computed from expressions presented previously. The effect of the conductor thickness and frequency, in the computation of εr eff could also be included in the computation. The real permittivity εr used in these expressions is replaced by the complex relative permittivity ε∗r = εr − jεr , tan δ = εr εr of a lossy substrate. The complex effective relative permittivity ε∗r eff is computed to place the CPW in a homogeneous lossy dielectric medium, as shown in Fig (9.41b). Figure (9.41c) shows the next step of the SLR process that reduces the CPW structure, located in the homogeneous medium, to a CPW on the infinitely thick substrate with equivalent relative permittivity εr, eq and equivalent losstangent tanδeq. The εr, eq and tanδeq, i.e. the complex equivalent relative permittivity ε∗r,eq , take care of finite thickness of a substrate, conductor backing, and also top shield. The complex equivalent relative permittivity ε∗r,eq is computed as follows: ε∗r,eff = 1 + q ε∗r,eq − 1

9 7 20

for, q = 1 2, ε∗r,eq = 2ε∗r,eff − 1

The real part of the above equation gives εr, eq of the equivalent infinitely thick substrate. The tanδeq of the equivalent substrate is

h2, εr1, tanδ2 = 1 εr2 = 1 w s w h1, εr1, tanδ1 (a) Shielded CPW. Figure 9.41

εʹreff

tan δeq =

Im ε∗r,eq Re ε∗r,eq

9 7 21

Figure (9.42a) presents the dielectric loss of a CPW, using the SLR and MOM-based software LINPAR [B.18], on a GaAs substrate with and without conductor backing for the slot width w = 0.02 mm, 0.2 mm. The presence of a conductor backing increases the dielectric loss, as the field lines are confined in the lossy dielectric region. For the same reason, the dielectric loss increases with a strip width of a CB-CPW, whereas it decreases with strip width for the CPW without conductor backing. Both methods show good agreement. Figure (9.42b) further shows the effect of the top shield on the CB-CPW. The proximity of the top shield increases the dielectric loss. However, for the shield height more than the substrate thickness, its influence can be neglected. Figure (9.42b) also shows that for the same loss-tangent and the same CPW structure, the dielectric loss increases with an increase in the relative permittivity of a substrate. Again, it is due to more field confinement in the dielectric region of a high permittivity substrate. 9.7.3

Substrate Radiation Loss

The fundamental modes of the CPW and CPS, under the certain condition, can become the leaky mode and radiate power into the substrate. This kind of radiation loss is very strong, as compared to both the dielectric and conductor losses. The mechanism of such substrate radiation is related to Cherenkov-type radiation [B.19]. Cherenkov has shown experimentally that if a charged particle is moving in a dielectric medium with a uniform velocity vq greater than the EM-wave velocity vm in that medium, it radiates the EM shock wave forming a radiation cone, shown in Fig (9.43a). For the Cherenkov radiation, also called the Cerenkov radiation, power increases with f3 and is sensed by a photographic film. The mechanism of the Cherenkov radiation is discussed in subsection (5.5.7) of chapter 5.

w s w w s w εreq

(b) CPW in open homogeneous medium.

tanδeq

(c) CPW on equivalent substrate.

The SLR process to compute the dielectric loss of conductor-backed CPW with a top shield. Source: Verma et al. [J.64]. Verma and Bhupal [J.65].

9.7 Losses in CPW and CPS Structures

0.3

0.21

w = 0.02 mm w = 0.2 mm

0.2

Dielectric loss (dB/m)

Dielectric loss (dB/m)

0.25

w = 0.02 mm

0.15

(

0.1

, (

,

,

,

) SLR

No conductor backing ) MOM

,

εr1 = 9.8

0.19 0.17

h1 = 1.59 mm, S in mm, tanδ1 = 0.001, s = 0.2 mm, f = 1 GHz

0.15 0.13 0.11 0.09

εr1 = 12.9, tanδ1 = 0.001, h1 = 0.2 mm, f = 1 GHz 0.05 10

εr1 = 12.9

0.23

Conductor backing w = 0.2 mm

εr1 = 2.32

0.07 0.314

610 210 410 Strip width (S) (micrometer)

(a) Dielectric loss with/without conductor backing.

w= 0.02 0.02 0.02

1.314 0.814 Shield height (h2/h1)

w= 0.2 0.2 0.2

1.814

(b) Dielectric loss of CPW with a top shield.

Figure 9.42 Comparison of computation of dielectric loss CPW structures. Source: Verma et al. [J.64].

Figure (9.43b) shows the wave propagating on a CPS structure. The propagating wave induces the dipolar charges on the CPS. The dipolar charges move on the conductor faster than the phase velocity of the EM-wave

in the surrounding dielectric substrate medium. Thus, a signal traveling on the CPS structure radiates power in the substrate in the form of Cherenkov-type radiation, causing a serious loss in the signal. The radiation in

Signal on CPS

EM-wave from charged particle ψ

Motion of charged particle

Photographic plate

Radiation loss / mm

(a) Radiation from moving charged particle.

Radiation cone in substrate (b) Radiation in a substrate in the form of radiation cone.

0.6 0.4 0.2 0 0

0.2

0.4 0.6 Frequency (THz)

0.8

1.0

(c) Radiation loss for CPS [J.66]. Figure 9.43 Cherenkov-type radiation loss. Source: Grischkowsk et al. [J.66]. © APS.

343

344

9 Coplanar Waveguide and Coplanar Stripline

the substrate is at an angle ψ forming the radiation cone. Figure (9.43c) experimentally shows the strong f3dependent radiation loss from the 150 Ω CPS taken on silicon on the sapphire (SOS) substrate. The CPS has strip width (s) 5 μm, strip gap (w) 15 μm, and aluminum conductor thickness (t) 0.5 μm [B.20, J.66–J.68]. The dots show the experimental loss of the CPS line from DC to 1 THz that is approximately described by the following curve-fitted expression: αrad = 0 2 f + 0 65f 3 Np mm

9 7 22

The first term of the above expression is the conductor loss of the aluminum conductor-based CPS that is small. It increases slowly as the square root of frequency in the THz range. The second term is the strong radiation loss due to Cherenkov-type leaky mode radiation. Rutledge et al. have obtained analytic expressions to model this unique radiation loss on both the CPW and CPS structures [B.20]. Their expressions have been modified by other investigators [J.68]. A summary of their expressions is presented below. Figure (9.43a) shows the cone angle of Cherenkovtype radiation of a moving charge by the following expression: cos ψ =

vm ; vq

vq > vm ,

9 7 23

where vm is the phase velocity of the EM-wave in the surrounding medium and vq is a uniform velocity of the moving charge. In the case of a CPS, shown in Fig (9.43b), the charge velocity vq is identified to the movement of the dipoles on a strip conductor that is related to the velocity of the fundamental mode with propagation constant kz. The EM-wave, radiated in the substrate, has the substrate mode velocity vm, with the propagation constant kd. Thus, the velocities are vm = c/kd and vq = c/kz. The radiation cone angle for the coplanar line is In dielectric medium In air medium

cos ψ = kz kd =

cos ψ = kz k0

εr eff εr

1 2

a b

9 7 24 The above relation also shows the phase matching, at the interface, of the radiated wave in a dielectric medium. The substrate leaky-mode radiation loss occurs only for kz < kd. Similarly, the radiation in the airmedium occurs, if kz < k0. This condition is needed for

working of a leaky-wave antenna [J.69, J.70]. The bound-mode wave propagation on a transmission line, without any radiation, occurs if kz > kd > k0. The radiation in the substrate occurs for the condition kd > kz > k0. However, radiation in air-medium occurs for the condition k0 > kz. For the CPS and CPW on a thick substrate, wavenumber is kd = εr k0 for the radiated wave inside the infinitely thick substrate. The wavenumber of the fundamental quasi-static mode, on both the line strucεr + 1 2k0. For kd > kz, the fundamentures, is kz = tal quasi-static mode becomes leaky, and radiates in the substrate. The CPS and CPW structures, on the infinitely thick substrate, are unconditionally leaky with radiation in the substrate. The expression for the radiation loss follows the cubic frequency dependence under the quasistatic approximation [J.15, J.41, J.66, J.67]: αCPS sw = = αCPW sw

π 2 π 2

5

5

3 2

2 3 − 8 1 − 1 εr 2 w 2 c3 K k K k

1 + 1 εr εr

f3 a

3 2

1 1 − 1 εr 2 s + 2w εr 3 f 2 1 + 1 εr c3 K k K k

b

9 7 25 where aspect-ratio is k = s s + 2w , k = 1 − k2 and c is the velocity of the EM-wave in the free space. In the case of a CPW, s is the width of central strip and w is the slot width, whereas for a CPS, w is the strip width and s is the slot gap. The CB-CPW line is also unconditionally leaky [B.16]. For the CPS and CPW lines on a finitely thick substrate, Frankel et al. [J.41] have observed that substrate radiation loss does not follow the cubic frequency dependence. It follows frequency dependence less than the exponent factor 3. This can be accommodated by taking cos ψ = εr eff f εr and by modifying the above loss expression as follows: 5 αCPS sw = π

αCPW sw

3− 8 2

π 5 = 2 2

εr eff f εr

1−

ε

f εr

εr eff f εr

1−

εr eff f εr

2

3 2

s + 2w 2 εr 3 f c3 K k K k

a

2 3 2

s + 2w 2 εr 3 f c3 K k K k

b

9 7 26 The finitely thick substrate supports the TE/TM surface waves. Therefore, the surface wave loss for the TE/TM polarization could be computed separately

9.8 Circuit Models and Synthesis of CPW and CPS

1.0

1.0 TM0 TM1

TM2

0.6 0.4 0.2 0

TE1

Range of (kz / kd) Mode crossing point

0

TE0

0.8 kdTE/k0

kdTM/k0

0.8

0.25 0.5 0.75 1.0 1.25 1.5 Normalized frequency (h/λg) (a) TM-surface wave mode.

0.6

TE2

0.4 0.2 0

Mode crossing point Range of (kz / kd)

0

0.25 0.5 0.75 1.0 1.25 1.5 Normalized frequency (h/λg) (b) TE-surface wave mode.

Figure 9.44 Mode crossing of surface wave modes in a conductor-backed dielectric slab. Source: Riaziat et al. [J.15]. © IEEE.

[J.68]. The above expressions ignore this aspect. The computation of frequency-dependent effective relative permittivity of both the CPW and CPS lines, on the finitely thick substrate, has already been presented. The expressions are valid for 0.1 < s/w < 10 and h > 3w and for wavelength λ0 > s + 2w. The CB-CPW supports the TM and TE surface wave modes on the conductor-backed dielectric substrate of thickness h. If the wavenumbers of the TM/TE surface wave modes exceed the wavenumber of the CPW line, the quasi-TEM mode of CPW becomes leaky and radiates power in that mode in a substrate. Figure (9.44a, b) shows the mode crossings of both the TM and TE surface wave modes [J.15]. The broken lines show the εr eff(f )/εr of the CPW line in the range 0.48–0.64. It is determined by the CPW structure. The surface wave propagation constant is normalized by kd = εr k0 . For the wideband operation of the CPW, without excessive substrate radiation loss, the substrate thickness should be such that the fundamental surface wave mode does not cross the normalized wave number of the quasiTEM mode of a CPW. This is achieved by taking substrate thickness h < 0 15 λd , where λd = λ0 εr . This condition is equally applicable to a microstrip line that also supports the spurious TM0-mode surface wave.

9.8 Circuit Models and Synthesis of CPW and CPS The computation of the characteristic impedance, effective relative permittivity, conductor loss, and dielectric

loss of the CPW and CPS structures is discussed in the previous sections. In these computations, the structural parameters on a given substrate are known quantities. This is known as the analysis of the CPW or CPS. However, in designing RF circuits, it is required to determine the structural dimensions of the CPW/CPS on a specified substrate for the assigned characteristic impedance. This process is known as the synthesis of CPW or CPS. This section presents the final expressions of the synthesis of the CPW and CPS structures. Even in the case of the analysis, the losses also influence Z0 and εr eff(f, t). Such influence can be accommodated through the RLCG-circuit model discussed in this section.

9.8.1

Circuit Model

The lossy CPW, CPS, and other quasi-TEM planar lines are described either by the secondary line parameters, such as the complex propagation constant γ and the complex characteristic impedance Z∗0, or by the primary line constants R, L, C, G p.u.l. In the case of the quasiTEM planar lines, the primary line constants are frequency dependent. Figure (9.45) shows the equivalent TEM-type transmission line model of the lossy quasiTEM planar lines. The frequency-dependent and also conductor thickness-dependent primary line constants, R(f,t), L(f,t), C(f,t), G(f,t), shown in Fig (9.45), can be evaluated from the known frequency and conductor thicknessdependent secondary line parameters, Z0(f, t), εr eff(f, t), αc(f, t), and αd(f, t) as follows [J.50]:

345

346

9 Coplanar Waveguide and Coplanar Stripline

G(f)

Figure 9.45

9.8.2

L(f)

R(f)

C(f)

Equivalent transmission line model of lossy quasiTEM planar lines.

R f, t = 2Z0 f, t αc f

a

εr eff f, t c εr eff f, t C f, t = cZ0 f, t 2αd f, t G f, t = Z0 f, t

Lf =

Z0 f, t

b c d, 981

where c is the velocity of the EM-wave in the free space. The secondary line parameters are computed, for the CPW and CPS structures, using the analytical expressions presented in this chapter and for the microstrip line, using expressions presented in chapter 8. The complex propagation constant γ and the complex characteristic impedance Z∗0 of the planar lines are computed from the following expressions: Z∗0 f, t = Z0 f, t + jZ0 f, t =

R f, t + jωL f, t G f, t + jωC f, t

1 2

a

γ f, t = αT f, t + jβ f, t = R f, t + jωL f, t

G f, t + jωC f, t

1 2

The characteristic impedance of a microstrip line depends on a single structural parameter, i.e. the w/hratio. However, the characteristic impedance of a CPW depends on two structural parameters, i.e. on the w/hratio and s/h-ratio. Thus, the CPW of a given value of characteristic impedance can be synthesized by several combinations of w/h-ratio and s/h-ratio [J.72–J.75]. There is no unique choice of the w/h-ratio and s/h-ratio to achieve the given characteristic impedance of a CPW. Therefore, two sets of synthesis expressions are given below – one for the assumed s/h (s: strip width) ratio and another for the assumed w/h (w: slot width) ratio. Usually, the slot width ratio (w/h) is less than 1 for a compact CPW. This choice also gives small dispersion, as the electric field is confined to the slot region and the so-called microstrip mode is suppressed. The CPW has negligible dispersion up to 60 GHz. Thus, the slot width in the range 0.1 ≤ w/h ≤ 1 and corresponding to the strip width in the range 0.1 ≤ s/h ≤ 8 are used to achieve characteristic impedance of a CPW between 25 and 100 Ω on GaAs substrate, εr = 12.9, h = 0.2 mm [J.72].

Synthesis Expression Set - I (Known w/h-Ratio)

The synthesis expressions, given by Deng [J.72], are summarized below. These are valid for a substrate with any relative permittivity. This set of expressions assumes w/h (slot width/substrate thickness)-ratio. For the given characteristic impedance Z0(εr), on a known substrate (εr, h), it computes the s/h (strip width/substrate thickness)-ratio:

b

s w = G1 εr , Z0 , w h h h

982 The characteristic impedance Z0(f, t) and εr eff(f, t) are evaluated from the above expressions. At this stage, effect of the losses and the magnetic field penetration on Z0(f, t) and εr eff(f, t) are included. Besides, the imaginary part of Z∗0 f, t is also computed. These are not possible with analytical expressions presented in this chapter. Figures (9.36) and (9.37) show the results of circuit model (CM) for the CPW and CPS. These results show the low-frequency dispersion in the line parameters. The above-computed R(f,t), L(f,t), C(f,t), G(f,t), can be further used to develop closed-form models of the primary line constants. Such expressions are available for a standard microstrip line [J.71].

Synthesis of CPW

For Z0 < =

η0 2 εr + 1

1 exp 4 4

For Z0 ≥

π εr eff η0

983

G 1 ε r , Z0 , w h

η0 Z 0 o

+ exp

−

π 4

εr eff

η0 −1 Z 0 0

a

2 εr + 1

G1 εr , Z0 , w h =

1 exp 2π 8

εr eff 0

Z0 1 − η0 2

−1

b

984 The εr eff(0) is the static effective dielectric constant that is a function of εr, Z0 and the assumed w/h-ratio. It can be computed by the following expression:

9.8 Circuit Models and Synthesis of CPW and CPS

εr eff 0 = εr eff εr , Z0 , w h = A Z0 , εr B εr , Z0 , w h Z0 K k A Z0 , εr = 1 + 2 εr − 1 εr + 1 η0 K k B εr , Z0 , w h = sech

ε5r 4π εr + 1

6

exp where aspect − ratio k

k=

η0 Z0

a b

2

exp

1 + 0 0016εr Z0

π 1+g w πw − exp 2 h 2h π 2+g w max −1 2 h

w w ln 0 6 + h h

c 985

1 − k2

k =

d,

e

εr + 1 2 η0 = 120π

f

g=

g

The approximate expression for K(k)/K(k ) is given by equation (9.2.39). The above expressions are valid for w/h ≤ 10/[3(1 + ln εr)] and s/h ≤ 80/[3(1 + ln εr)]. The above expressions also compute the static effective relative permittivity in the process of synthesis. Usually, its value is less than the value computed by the static method discussed previously or by the full-wave methods. For a practical range of parameters, its accuracy is 1.5% that gets degraded for the extreme case of the w/h- and s/h-ratios. The above expressions are complicated. However, once w/h- and s/h-ratios of a CPW are known, εr eff(0) can be accurately computed using equation (9.3.15). Another set of expressions to compute G1, that does not involve computation of εr eff(0), is also available [J.75]. It is valid for the substrates with relative permittivity range 6 ≤ εr ≤ 100.

Synthesis Expression Set - II (Known s/h-Ratio)

The slot width (w/h) ratio of a CPW, for known, εr, Z0, h can be computed if s/h-ratio is assumed [J.72]: w s h = h G2 εr , Z0 , s h

εr eff 0 = εr eff εr , Z0 , s h = A B B = 1 + tanh Q1 = exp

ε7r 4π2

εr + 1

1 + 0 0004 εr

8

η0 Z0

a 2

Q1

Z0 s s ln g h hg

b c 987

The parameter A of the above equation is computed from the expression (9.8.5b). The aspect-ratio k is obtained from the following expression: 1+1 g s π s − exp 2 h 2g h π 1+2 g s −1 exp 2 h

exp π k=

988

The above set of equations also computes εr eff(0). However, its value is more than the results obtained from the analysis formula. It can be computed accurately using the expression (9.3.15). An alternate set of expression, for 6 ≤ εr ≤ 100, is also available to compute G2 without computing εr eff(0) [J.75].

986

On replacing w/h-ratio by s/h-ratio in the expression (9.8.4), the expression of G1 is used for G2. The w/h-ratio is replaced by the s/h-ratio. However, εr eff(0) has to be computed as follows:

9.8.3

Synthesis of CPS

The expressions for the analysis of the CPS can be taken in the iterative process to synthesize the CPS. The synthesis expression for the CPS on the infinitely thick substrate is available [B.14]. Three different sets of the

347

348

9 Coplanar Waveguide and Coplanar Stripline

Table 9.8 The values for the coefficient α1 − α9. α1

α2

α3

α4

α5

α6

α7

α8

α9

−1.0083 −0.4394 −1.0050 −0.9255

0.4261 0.1753 0.6229 0.4156

2.3613 −0.0848 4.7557 3.1556

1.1966 −0.0524 2.3716 1.3762

9.7854 1.8679 8.4331 1.6588

4.0479 1.8822 4.2339 2.0995

3.7066 1.2015 3.5531 2.2560

2.0667 2.1294 2.1547 1.8277

4.2467 4.2530 4.4647 3.8805

Set

Set #1: Low impedance High impedance Set#2: Low impedance High impedance

closed-form expressions have also been proposed to synthesize the symmetrical CPS on a finitely thick substrate [J.76–J.78]. Two sets of accurate expressions are given below [J.76]. Set #1 (Known s/h-ratio): In this case, the substrate details (εr,h) and characteristic impedance Z0(εr) of CPS are specified. Further, the slot width (s) is assumed appropriately. The strip width (w) is computed as follows: s a w= G ε r , Z0 ε r , X X=s h

b 989

Set #2 (Known w/h-ratio): In this case also, the substrate details (εr,h) and characteristic impedance Z0(εr) of CPS are specified. However, the strip width (w) is assumed appropriately and we calculate the slot width (s) as follows: s = w G εr , Z0 εr , X

a

X=w h

b

References Books B.1 Simon, R.N.: Coplanar Waveguide Circuits

B.2

B.3 B.4 B.5 B.6

B.7

9 8 10 The curve-fitted expression for the factor G[εr, Z0(εr), X] is given below: G εr , Z0 εr , X = G1 + G2 + G3 where G1 = α1 exp Xα2 εαr 3 G2 = exp α5 G3 = α7 εαr 8

a α4

B.9

b

α6

Z0 εr 120π

Z0 εr 120π

Z0 εr 120π

B.8

c

B.10

α9

d 9 8 11

The expression for the factor G(εr, Z0(εr), X) is common for both the sets and the value of the X is suitably assumed. The values for the coefficient α1 − α9, given in Table (9.8) for both sets of expressions, apply to the low impedance and high impedance CPS. For the low impedance lines, Z0 εr < 120π 2 εr + 1 , and for the high impedance line, the relation is Z0 εr > 120π 2 εr + 1 . The formulas are valid for the range s/h ≤ {10/[3(1 + ln(εr))]}, w/h ≤ {10/(1 + ln(εr))} and 2.2 ≤ εr ≤ 50. It has accuracy 1.56% against the results of the analysis formula.

B.11 B.12 B.13

B.14 B.15

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B.16 Gupta, K.C.; Garg, R.; Bahl, I.; Bhartia, P.: Microstrip nd

B.17

B.18 B.19

B.20

Lines and Slot Lines, 2 Edition, Artech House, Boston, MA, 1996. Kneppo, I.; Fabian, J.; Bezousek P.; Hrnicko, P.; Pavel, M.: Microwave Integrated Circuits, Chapman & Hall, London, 1994. Sarkar, T.K.; Harrington, R.F.: LINPAR – Software, Artech House, Boston, 1999. Smith, G.S.: An Introduction to Classical Electromagnetic Radiation, Cambridge University Press, NewYork, 1997. Rutledge, D.B., et al.: Millimeter Components and Techniques, Infrared and Millimeter Waves Part II, Chapter-1, In Planar Integrated Circuit Antennas Vol. 10, K.J. Button, Academic Press, New York, NY, 1983

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J.15

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J.21

J.22

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J.23 Chung, H.; Chang, W.S.C.; Adler, E.J.: Modelling and

J.24

J.25

J.26

J.27

J.28

J.29

J.30

J.31

J.32

J.33

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coplanar strip line, Microw. Opt. Technol. Lett., Vol. 44, No. 2, pp. 199–202, Jan. 2005. J.78 Deng, T.Q.; Leong, M.S.; Kooi, P.S.; Yeo, T.S.: Synthesis formulas for coplanar lines in hybrid and monolithic MIC, Electron. Lett., Vol. 32, pp. 2253–2254, Nov. 1996.

353

10 Slot Line Basic Characteristics

Introduction S.B. Cohn proposed the basic slot line structure and, along with others, also demonstrated its application [B.1, B.2, J.1– J.6]. It is truly a uniplanar transmission line, as there is no backside metallization. Normally, it is used along with the microstrip and CPW structures. At the mm-wave frequency, the microstrip conductor losses are high and its dimensions are very small. So it requires specialized fabrication technology. The slot line and coplanar line structures are useful in the mm-wave and sub-mm-wave ranges [B.3], [J.7, J.8]. To achieve the EM-field confinement in the slot region, normally a substrate of high permittivity, i.e. εr ≥ 9, is used. A significant part of the electric field is in the air region, above the conducting plane. The effective relative permittivity of a slot line is about 15% lower than that of a microstrip. Its characteristic impedance is between 40 Ω and 200 Ω. A slot line mounted in a rectangular waveguide forms the fin line that is another useful structure for the applications in the mm-wave ranges [B.4]. The slot line is primarily a non-TEM mode structure that can be accurately analyzed only by using the full-wave methods, such as the SDA discussed in chapter 14 [B.5– B.7]. However, Cohn has modeled it as a rectangular waveguide supporting the quasi-TE10 mode [J.1]. This model shows a cutoff frequency that is an unrealistic description of a practical slot line. Still, it is interesting to view the 2D geometry of a slot line, as a 3D rectangular waveguide. Furthermore, the static conformal mapping method has been also used with some modifications to compute the line parameters of a slot line [J.9–J.11].

Objectives

•• •

To present several geometrical forms of the slot line. To discuss the magnetic current model of a slot line. To present the equivalent waveguide model of standard slot line.

••

To extend the waveguide model to a multilayer slot line. To present the closed-form integrated model of slot line to compute its line parameters.

10.1

Slot Line Structures

The slot line could be used on single layer and multilayer substrates. It could be a conductor-backed structure that could be also covered with a shield. A brief description of some of these structures is presented below.

10.1.1

Structures of the Open Slot Line

A few configurations of the open slot line structure are shown in Fig (10.1). The lateral conducting planes are infinitely large. Figure (10.1a) is a basic standard slot line. The effective relative permittivity, or the slowing-factor given by λg/λ0, and characteristic impedance are determined by the relative permittivity of the substrate, slot-width, and operating frequency. The slot line could be considered as a complementary structure of a microstrip line. The central strip conductor of microstrip is replaced by a slot and two lateral very large bare substrate portions are replaced by the infinitely large two lateral conducting planes of a slot line. Further, the ground conductor of the microstrip is replaced by the bare face of its dielectric substrate. However, even though a slot line is a two-conductor line, like a CPS, it does not support the TEM-mode. For infinitely wide conductors of a slot line, its line capacitance is C ∞. Therefore, the TEM-mode characteristic impedance of a slot line vanishes, Z0 = (1/vpC) 0. However, a practical slot line is fabricated with large, but finitely wide strip conductors. In that case, the slot line mode could be approximated by the quasi-TEM mode with zero cutoff frequency. However, Cohn has modeled it as a rectangular waveguide supporting the quasiTE10 mode with a cutoff frequency.

Introduction to Modern Planar Transmission Lines: Physical, Analytical, and Circuit Models Approach, First Edition. Anand K. Verma. © 2021 John Wiley & Sons, Inc. Published 2021 by John Wiley & Sons, Inc.

354

10 Slot Line

Air medium Air medium

h2, εr2

S

h1, εr1

h, εr

Air medium (a) Standard slot line. Figure 10.1

Air medium h2, εr2

S

Air medium (b) Sandwich slot line.

Air medium S

S

2h1, εr1

h1, εr1 Air medium (c) Composite substrate slot line.

EW/MW

S Air medium (d) Bilateral slot line.

Some open slot-line structures.

Figure (10.1b) shows that the sandwich slot line is located at the interface of two dielectric media [J.2, J.9, J.12–J.14]. For the identical lower and upper dielectric layers, it reduces to an embedded (buried) slot line [J.13]. The sandwich slot line has a higher effective relative permittivity and lower characteristic impedance compared to a standard slot line on the same substrate with an identical s/h ratio. Therefore, an overlay, i.e. the superstrate, serves the purpose of controlling the line parameters. It further acts as a protective cover that also reduces radiation loss. The composite substrate slot line, shown in Fig (10.1c), is another useful multilayer structure [J.7, J.9] that also controls the line characteristics and distortion in a signal. For the case εr1 > εr2, it diverts the EM-wave energy from the conducting planes to the dielectric region that reduces the conductor loss of a slot line [J.7]. The substrate #1 also provides mechanical support to substrate #2. The composite substrate slot line has higher characteristic impedance, as compared to a standard slot line. It is again due to the diversion of the EM-field away from the surface conductors [J.15]. Figure (10.1d) shows the double-sided slot line, also called the bilateral slot line [J.15–J.17]. The structure is a broadside coupled slot lines. Due to the symmetry, it supports both the even and odd modes. In the case of the odd-mode excitation, the tangential components of electric fields in both the slot regions are equal and opposite that form the electric wall (EW) at the plane of symmetry. The structure could be reduced to a conductor-backed slot line (CBSL), shown in Fig (10.2a). However, in the case of the even-mode excitation, the tangential components of the electric field in the slot regions are equal and in the same direction that forms the magnetic wall (MW) at the plane of symmetry. The even-mode excitation provides nearly a standard slot line. The characteristic impedance of the odd-mode excited bilateral slot line is always much less than that of a standard slot line on a substrate of identical thickness and identical slot-width. On several occasions, the characteristic impedance of a bilateral slot line is one-half to that of the characteristic impedance of a slot line. It means to get fixed characteristic impedance for a

given substrate; wider slot-width could be used in case of a bilateral slot line. Thus, a bilateral slot line overcomes the fabrication difficulties associated with the narrow slot of a standard slot line. Due to the symmetrical field distribution, a bilateral slot line is less dispersive than a standard slot line. The bilateral slot line is more suitable as a feed line to a tapered slot-radiator for the wideband antenna application [J.5, J.15, J.18].

10.1.2

Shielded Slot Line Structures

The slot line structures are also used with a conductorbacking and top shield [J.15]. The slot line could be fully enclosed in a metallic shield too. Fig (10.2) shows a few of such structures. The CBSL appears to be a useful structure, as it permits mounting of the circuits on a metallic ground plane. The conductor backing adds mechanical strength to a slot line. It also provides a good heat-sink and dc biasing to the active devices. However, the structure has a serious problem of power-leakage, due to the leakymode, in the substrate region. It gets coupled to the neighboring circuits on the same substrate, degrading their performances. The CBSL structure is normally not suitable as a transmission medium. The wide-slot CBSL is commonly used as the leaky-wave antenna [J.19–J.24]. Figure (10.2b) shows that the CBSL could be modified by adding a superstrate to the structure. For the case εr1 > εr2, it has identical behavior to that of CBSL, shown in Fig (10.2a). However, for the case εr2 > εr1, known as the inverted conductor-backed slot line (ICBSL), the structure has more useful features – both with respect to the signal transmission and upward radiation in the outer space [J.21]. In the lower frequency range, it is possible to obtain a frequency band that is suitable for the guided wave propagation without the leaky-mode and radiation. However, radiation can be observed in some frequency ranges. The standard slot line is an open structure that causes excessive radiation at the discontinuity and also crosscoupling to the neighboring circuits. Figure (10.2c) shows added top shield to form the shielded suspended slot line. The top shield is also added to other slot lines. At higher

10.2 Analysis and Modeling of Slot Line

Air medium Air medium εr, h

S

h2, εr2 h1, εr1

(a) Conductot backed slot line (CBSL).

h3, εr3 =1 S

h2, εr2 S

(b) Inverted CBSL εr2 > εr1.

h1, εr1 (c) Shielded suspended slot line.

Figure 10.2 Some conductor-backed shielded slot lines.

frequency, a shielded slot line could be as efficient as a finline and much easier to integrate with circuits [J.8]. The shielded suspended slot line structure permits propagation of a band of frequency without radiation. The lower end of frequency is limited by the cutoff phenomenon and the higher frequency is limited by the radiation. The lower ground plane has a stronger influence on its performance. The practical planar circuits are always mounted in the shielded enclosures, to protect them from the environment and RF interferences. For conducting walls very near to the slot region, the slot mode moves to the cutoff, where field confinement in the slot is less [J.13].

10.2 Analysis and Modeling of Slot Line Cohn suggested two approximate models for a slot line – the magnetic current model and the equivalent waveguide model [J.1]. The magnetic current model, discussed in this section, demonstrates the EM-field confinement in the slot region, due to the exponential decay of the field away from the slot. It also demonstrates the existence of the elliptically polarized wave in the slot region. However, the model does not compute the frequency-dependent slot line parameters. It is achieved with the help of the equivalent waveguide model, discussed in the next section. The equivalent waveguide model has been further extended to the multilayer shielded slot lines and the coupled slot lines [J.13, J.25, J.26]. The spectral-domain analysis (SDA) of the slot line is discussed in chapter 16. A summary of the results of the conformal mapping method as well as the closed-form models of the slot line to compute the losses and dispersion are also presented subsequently.

H-field λg/2

Z Jm

Y Eϕ

Substrate

Magnetic Current Model

Figure (10.3) illustrates an approximate 3D-view of the field lines of a slot line in the cylindrical coordinate, with Eϕ, Hr, and Hz field components. The electric and magnetic fields are shown separately for better clarity, although they are orthogonal to each other. The magnetic field pattern along

X

E-field Figure 10.3

Approximate 3D view of the dominant mode of a slot line with a cut to show field lines.

the slot line repeats itself at every λg/2 distance, where λg is the guided wavelength of a slot line. The current flowing on the surface of the strip conductors causes the conductor loss. The fictitious magnetic current Jm in the slot region is assumed. It is related to the slot electric field through expression J m = E × n. The magnetic current model of the slot field is valid for a narrow slot, i.e. s/λ0 1. Therefore, kc is an imaginary quantity. For a narrow slot, the field component Hz does not change across the slot-width, i.e. ∂Hz/∂ϕ = 0. Thus, the transverse field components of the TE-mode, from equation (10.2.1), are Er = Hϕ = 0,

Eϕ = j

ωμ ∂Hz , k2c ∂r

Hr = − j

βz ∂Hz k2c ∂r 10 2 4

The Hz-field component satisfies the wave equation in the cylindrical coordinate system: ∂ 2 Hz 1 ∂Hz 1 ∂ 2 Hz + + + k2c Hz = 0 r ∂r r2 ∂ϕ2 ∂r2

10 2 5

On using the method of separation of variable, the solution of the above equation is written as follows: Hz = R r F ϕ 2

a 2

dR F dR 1 dF + 2 R 2 + k2c RF = 0 2 + r dr r dϕ dr r 2 d2 R r dR 1 d2 F 2 2 + k + r = − = n2 c R dr2 R dr F dϕ2

F

b 10 2 6

On separating equations for R(r) and F (ϕ), the following two equations are obtained: d2 F r2 d2 R r dR 2 + + r2 k2c − n2 = 0 2 + n F=0 a , R dr2 R dr dϕ d2 R 1 dR n2 b + k2c − 2 R = 0 2 + r dr r dr 10 2 7

The solution of equation (10.2.7a) is the linear combination of harmonic functions. The solution of the Bessel equation of nth order (10.2.7b) is written using a linear combination of the first and second kinds of Hankel function [B.9], as it provides radial outgoing wave from the magnetic current source. F ϕ = A cos nϕ + B sin nϕ Rr =

C Hn1

kc r +

D Hn2

a

kc r

10 2 8

b

The approximate uniform field distribution with respect to ϕ shows the zeroth-order solution by taking n = 0. The first and the second kind of Hankel functions for the zeroth-order are approximated by the following expressions [B.9, B.10]: H

H

1

0 r

2 π exp j kc r − πkc r 4

kc r =

2 exp πkc r

∞

2

0 r

kc r =

∞

− j kc r −

a π 4

b 10 2 9

On multiplying the above equations by ejωt, equation (10.2.9a) shows the inward moving wave and equation (10.2.9b) shows the outward moving wave from the slot region. Thus, equation (10.2.9b) shows the radiation from the slot that is always present in an open structure. The radiation could be both in the free space, forming the space-wave and also in the dielectric substrate forming the surface wave. However, the present discussion is not about radiation. The present discussion is concerned with the field confinement in the slot region, forming the guided wave on a slot line. The radiation term of the solution (10.2.8) is discarded by taking D = 0, giving the zeroth-order (n = 0) Hz-field for the guided wave on the slot line: 1

Hz = A H0 kc r

10 2 10

For getting the above expression, the constant A × C is again denoted by A. It does not influence the solution. Equation (10.2.3b) shows that the argument (kcr) of the above Hankel function is an imaginary quantity. On using the identity given below [B.9], the azimuthal electric field Eϕ and the radial magnetic field Hr components are written from equations (10.2.4) and (10.2.10):

10.3 Waveguide Model

d 1 H x dx 0

1

= − H1 x 1

λg λ0

Eφ = − η A

a

H 1 kc r 1 − λg λ 0

2

b

10 2 11

1

A H1 kc r

Hr =

1 − λ g λ0

c

2

As the argument of the Hankel function is an imaginary 1 quantity, Hankel function Hn jx is given by the modified Bessel function Kn [B.9, B.10]: Kn x =

π j 2

n+1

Hn1 jx ,

10 2 12

where n is the order of functions. There is a 90 phase difference between Hz and Hr magnetic field components. Therefore, these orthogonal magnetic field components may produce the circular polarization at a certain distance r = rcp, where their magnitudes must be the same: 1

H1

Hr =1 Hz

kc rcp

1

kc rcp

H0

=

1−

λg λ0

2

10 2 13 The right-hand side of the above expression is 4, or r λ0 > 1 εr − 1, Hankel function used in equation (10.2.14) is approximated as 1

H1 j kc r = −

2 π

π e− 2 kc r

kc r

10 2 15

The voltage ratio, in this case, is approximated by the following expression on using equation (10.2.3): Vx r εr − 1 =π V λ0 2

1 4

e

−

π r λ0

2 εr − 1

10 2 16 The above expression has error 0 air WG #B

C slot O

–b/2 A x=0

#1

air A

EW/MW Width

a

D S

z 0, has two sections – the WG #A and WG #B. The WG #A is the irissupported dielectric section of thickness h and the WG #B is an air-filled waveguide of infinite length in the cutoff region. The waveguide #2, in the direction of z < 0, is the air-filled waveguide of infinite length, also in the cutoff region. The waveguide #2 and WG #B, with the width a < λ0/2, load the junction reactively. The nature of the reactive loading, i.e. the inductive or capacitive, depends on the TE/TM modes in the cutoff regions. Summary of Expressions used in Waveguide

The analysis of the rectangular waveguide with all-EW is discussed in chapter 7. A summary of expressions is presented below for immediate reference. The expressions for the TE and TM modes are given with some changes. The constant B = − j Bmn ωμ β2c is used in the expression (7.4.19), and the constant A = − j Amn ωε β2c in the expression (7.4.24), to get the following expressions of the TE/TM modes [B.9]:

359

10 Slot Line

Y

EW/MW

Waveguide #2 Waveguide #1 EW +b/2 h Air medium WG WG #B +S/2 #A Air medium Gap 0 Z –S/2 Metallic Bd plane –b/2 εr

EW

+b/2

–b/2 x=0

a/2 EW/MW

(b) Cross-section of the waveguide showing the formation of the iris from a section of slot line.

(a) Longitudinal section of waveguide model of three sections.

P

P

β1h

+b/2

θ

Waveguide #1

Z –S/2 Ba εr –b/2

Bd

Z=0

β1,Y01 Open

+S/2 Waveguide Gap WG #A # 1a

β1,Y01 εr Yio P

Z=0

Z

Z = h, p

(d) Transmission line model of a waveguide section.

(c) Waveguide model of slot region. Plane PP is the reactive plane, either EW/MW. Figure 10.5

Hx

–S/2

Metallic plane

EW

Waveguide model of iris

Ey

εr

X

x=a

EW

+S/2

EW

Slot -gap

Ba Bd

Y

Y

Metallic plane

WG #B Air medium

360

Analysis of the slot line as a waveguide.

The waveguide model considers the EM-wave propagation in the z-direction with the propagation factor exp{j(ωt − γz)}. The wave impedances for the TE and TM modes, i.e. ZTE and ZTM from equations (7.4.31) and (7.4.32) of chapter 7, are given below:

TE-modes:

Hz = B cos βx x cos βy y η βy f B cos βx x sin βy y βc f c ηβ f Ey = − j x B sin βx x cos βy y βc f c Ey Ex , Hy = Hx = − ZTE ZTE

Ex = j

10 3 1

ZTE

TM-modes:

ZTM = η 1 −

Ez = A sin βx x sin βy y Hx = j

βy f A sin βx x cos βy y βc η f c

βx f A cos βx x sin βy y βc η f c Ey = − ZTM Hx Ex = ZTM Hy ,

mπ , a

βy =

nπ , b

also, γ = jβz = j k2 − β2c

1 2

βc =

fc f

2

−1 2

a 10 3 3

2 1 2

b

The intrinsic impedance of the waveguide medium is 10 3 2

η = μ ε and for the free space it is η0 = 377 Ω, such that η = η0 εr for a dielectric medium. The transverse wavenumbers βx, βy, and the cutoff wavenumber are

Hy = − j

βx =

fc = η 1− f

β2x + β2y =

, k = k0 εr , λ = λ0

mπ a

2

+

nπ 2 , b

βz = k2 − β2c

εr , λg = 2π βz , λ0 = 2π k0

1 2

10 3 4

10.3 Waveguide Model

In the above equation, a is the dimension of broad-face, and b is the dimension of the narrow-face of the equivalent waveguide. For the TE-mode, either of two modal numbers m and n could be zero. However, for the TM-mode neither of two modal numbers m and n is zero, i.e. for TM-mode m, n ≥ 1. If either m or n is zero for the TM-modes, all field components are zero. The modal numbers m, n are integers. The cutoff wavelength and cutoff frequency for both the TE and TM modes are given as follows: λc,mn =

f c,mn

2π = βc

2 m a

2

+

βc 1 = = 2π με 2 με

n b

a

2

m a

2

n + b

in waveguide #1. The field expressions (10.3.1) of the TEmodes in the transverse x–y plane show that the Ey and Hx field components have the following proportionality relations: Ey Hx

πx 2π n y cos a b πx 2π n y cos sin a b

Rn cos n = 1, 2, …

b 10 3 5

10 3 6 b

At z = 0 and x = a/2, the total Ey and Hx are given as the summation of fields of all modes: E y = R0 +

2

a

sin

Hx = − yi,0 R0 −

2π n y b

yi,n Rn cos

n = 1, 2, …

a 2π n y b

10 3 7

Determination of Susceptances Bd and Ba for Case-A

Figure (10.5a and b) shows the longitudinal and crosssectional views of the (EW–EW) waveguide model. It is assumed that both the waveguides #1 and #2 are excited by the aperture field of an iris, located at the junction z = 0 of the waveguides. At the plane of the iris at the right-hand side of z = 0, the susceptance Bd is present due to WG #A and WG #B in the cutoff region for all modes. At the same iris plane, from the left-hand side of z = 0, the susceptance Ba is present due to the WG #2 again in the cutoff region. At first, the susceptance Bd at z = 0, offered by the waveguide #1, is computed. The iris slot-gap s is modeled by the air-filled waveguide #1a with a cross-section (a × s) that excites waveguide #1. It is shown in Fig (10.5c). The waveguide #1 has width “a” and height “b” and is divided into two waveguides – WG#A filled with strip-slot-loaded dielectric substrate of thickness h and WG#B filled with air medium and is under the cutoff region. The WG#A could be treated as terminated in either the inductive reactance or capacitive reactance created by the air-filled waveguide WG#B in the cutoff region. The TE mode WG#B in the cutoff region provides the inductive reactance at x = h and, whereas the TM mode in cutoff provides the capacitive reactance. Further, the capacitive reactance could be replaced by the extended length of WG#A terminated in the open-circuit, and the inductive reactance is replaced by the extended length of WG#A terminated in the short-circuit. The input admittance Yi at plane z = 0, looking toward waveguide #1 containing dielectric of relative permittivity εr, is a susceptance Bd, as the dielectric is to be taken as lossless. To get an expression for the Bd (Yi = j Bd), let us consider the waveguide #1 excited by the aperture field of the iris. It has been noted earlier that, due to symmetry, only TE1,2n (n = 0,1,2,…) and TM1,2n (n = 1,2,…) modes are present

b,

where R0 and Rn are constants. The input wave admittance for the TE10 mode is yi,0 and for the hybrid mode, it is yi,n. The linear combination of other remaining modes TE1,2n (n ≥ 1) and TM1,2n (n ≥ 1) with input wave admittance yi,n constitute the hybrid modes. The wave admittances of the TE10 mode and hybrid modes are expressed as follows: yi,0 = − yi,n = −

Hx Ey

a TE10

10 3 8

Hx

TE1,2n

+ Hx

TM1,2n

Ey

TE1,2n

+ Ey

TM1,2n

b

A constant field Ey in the capacitive iris slot is assumed to excite the TE10 and other higher-order modes, mentioned above, in the waveguide #1. The iris slot is modeled as the waveguide #1a, shown in Fig (10.5c), and its assumed field components at z = 0 are written as follows: C0 , for y ≤ Ey =

s 2

s b 0, for < y ≤ 2 2 s Hx = − C0 yi , for y ≤ 2

a 10 3 9 b

In the above equation, yi is the input wave admittance of waveguide #1a of height s supporting only TE10 mode. As the length of the exciting waveguide #1a, i.e. the thickness of iris is zero, therefore yi is also the input wave admittance of the waveguide #1 that is related to its input admittance Yi. Let us define the fractional slot gap δ as s δ= , s b 10 3 10 b The assumed slot-field of equation (10.3.9) ignores higher-order modes and is valid for a narrow slot of

361

362

10 Slot Line

fractional width δ ≤ 0.15. The constant slot field Ey could be expressed as follows, in the form of Fourier series, giving its modal composition: E y = C 0 = R0 +

Rn cos n = 1, 2, …

2π n y b 10 3 11

The Fourier coefficients R0 and Rn are determined by using the orthogonality relation. Thus, the coefficient R0 is obtained by integrating both sides of the equation over −s/2 ≤ y ≤ s/2 and coefficient Rn is obtained by the same integration process after multiplying both sides of the above equation by cos{(2πm/b)y}:

−s 2

s C0 dy = C0 = C0 δ b

b 2

C0 cos −b 2

1 2

Ey Hx dydx −s 2 0

10 3 15

s 2 a

1 Pi = 2

−s 2 0

1 C20 yi dydx = C20 yi s a 2 s 2

10 3 12

Ey dy = s Ey −s 2

10 3 16

b 2

2π m y dy = R0 b

s 2 a

Pi =

The voltage over the slot is V =

s 2

1 R0 = b

needed, as the slot line is described through its characteristic impedance and not through the wave impedance. The input power at z = 0 to the waveguide #1 is obtained using the assumed field expressions in the slot by equation (10.3.9):

C0 cos

2π m y dy b

2π n y cos b

2πm y dy b

−b 2

Using the above given two equations, the characteristic admittance Yi of the waveguide #1a, in terms of its wave admittance, is obtained as follows:

b 2

+

Rn n

C0 cos −b 2

Yi =

sin nπδ Rn = 2C0 δ nπδ

For m = n

= Y i C0 s

2

=

1 2 C y sa 2 0 i

a 10 3 17

Likewise, the input waveguide admittances Yi0 and Yin corresponding to the TE10 and higher-order modes of the waveguide #1 for the height b are Yi0 =

a y 2b i,0

b,

Yin =

a y 2b i,n

c 10 3 18

s 2

C0 yi dy = −s 2

a y 2s i

2

10 3 13

Next, it is desired to obtain an expression for the input wave admittance yi of waveguide #1a at z = 0 in terms of the modal input wave admittances yi, 0 and yi, n of waveguide #1. To achieve it, equation (10.3.7b) is substituted in equation (10.3.9b) and integrated over slot height ±s/2: s 2

Pi = Yi V2 = Yi Ey s

−s 2

2π n yi,0 R0 dy + yi,n Rn cos y dy b n = 1, 2, …

C0 yi = yi,0 R0 +

n = 1, 2, …

yi,n Rn

sin nπδ nπδ

On substitution of R0 and Rn from equations (10.3.12) and (10.3.13) in the above equation, the following expression is obtained: b sin nπδ yi,n y = yi,0 + 2 s i nπδ n = 1, 2, …

2

10 3 14

So far the wave admittance has been used in the discussion. It is defined as a ratio of the transverse magnetic field component and transverse electric field component given by equation (10.3.1). It can be replaced by the waveguide characteristic admittance defined for the TE10-mode on the voltage-power basis [B.8]. It is discussed in section (7.6) of chapter 7. The change of admittance is

On substituting the yi from equation (10.3.17), and yi0 and yin from equation (10.3.18) in equation (10.3.14), the following expression for the input admittance of the waveguide #1 at z = 0 is obtained: Yi = j Bd = Yi0 + 2

Yin n = 1, 2, …

sin π n δ πnδ

2

10 3 19 The input admittance is capacitive. It has two components. The first component Yi,0 is due to the TE10 mode in the waveguide #1 containing dielectric substrate, followed by the air-filled infinite waveguide. The second component of the input admittance, under the sign, is contributed by the higher-order TE1,2n and TM1,2n n ≥ 1 modes in the waveguide #1. The next task is the computation of the Yi,0, and Yi,n.

10.3 Waveguide Model

The intrinsic impedance of free space is η0 =

Computation of Input Admittance Yi,0 due to the TE10 Mode

The input admittance Yi,0 due to the TE10 mode is obtained by using the transmission line model of the waveguide #1 with two waveguide sections WG #A and WG #B. Figure (10.5d) shows the process to compute the input admittance Yi0 due to the TE10 mode in waveguide #1. At z = h, shown in Fig (10.5c), the TE10 mode in the air-filled waveguide section WG #B is evanescent. Further, the waveguide section WG #B could be treated as a finite length line terminated in an open-circuit, offering capacitive load admittance Y0 to the waveguide section WG #A. In the case a slot line is enclosed in a metallic box, the air-filled equivalent waveguide section WG #B could be treated as a line terminated in the short-circuit. This case is discussed in subsection (10.3.3). The capacitive load Y0 terminated at the plane PP can be further replaced by the extended open line section of the length θ. It is noted that the infinite length WG #B of characteristics admittance Y0 is replaced by the extended length θ of the WG #A having characteristic admittance Y01 and propagation constant β1. Hence, the total electrical length of the waveguide section WG #A is (β1h + θ) that is terminated in the open-circuited load ZL = ∞ or YL = 0. The TE10 mode in the WG #A is a lossless propagating mode. Therefore, the input admittance of the extended line length θ at the plane PP is Yin

PP

= Y0 = jY01 tan θ,

θ = tan− 1

Y0 γ 2π , β1 = 1 = jY01 j λg

10 3 20

The guided wavelength λg1 is for the TE10 mode in the dielectric-filled WG #A. At z = 0, using equation (2.1.78) of chapter 2, the input admittance of the waveguide #1 is computed by replacing it with the extended WG #A of finite length (β1h + θ) terminated in the open. The input admittance of the waveguide #1 seen by the TE10 mode is

Y0 jY01

β β = ωμ η 0 k0

a TE Y 2b w

Y0 = YPV 0 =

aγ j 2bη0 k0

10 3 24

The parameter γ = jβ is the z-directed propagation constant of the TE10 mode air-filled waveguide section WG #B. Likewise, the characteristic admittance Y01 of the dielectric-filled waveguide section WG #A for the TE10 mode is Y01 = YPV 0 = −j where, η1 = η0

aγ1 a λ0 = 2bη1 k1 2 bη0 λg1 εr , k0 = 2π λ0 , k1 =

2π εr λ0

10 3 25

γ1 = jβ1 , β1 = 2π λg1

Computation of Input Admittance Yi,n Due to Higher-order Modes

At z = 0, the input admittance Yi,n, for n > 0, of the combined TE1,2n and TM1,2n modes, i.e. the hybrid mode is obtained from equation (10.3.8b) as follows: Hx Yi,n = −

Ey

TE1,2n

+

TM1,2n

Ey

TE1,2n

Ey

TM1,2n

n

where,

10 3 22

10 3 23

Therefore, using the above two equations, the characteristic admittance Y0 of the air-filled waveguide section WG #B for the TE10 mode is

Hx

TM1,2n

Ey

TM1,2n

+1

Dn =

10 3 26

n

YTM + YTE Dn i = i 1 + Dn

10 3 21

The characteristic admittances Y0 and Y01, defined on the power-voltage basis, are related to the wave admittance of the TE mode. The wave admittance is given in terms of the physical parameters of a waveguide. It is discussed in chapter 7. Using equation (7.4.30) of chapter 7, the wave admittance of the TE mode supporting air-filled waveguide is given below [B.8]: YTE w =

YPV 0 =

Yi,n

Yi,0 = jY01 tan β1 h + θ , Yi,0 = jY01 tan β1 h + tan − 1

μ0 ε0 = 377 Ω. Using equation (7.6.9) of chapter 7, the characteristic admittance on the power-voltage basis is related to the TE mode wave admittance YTE w by the following expression:

Ey

TE1,2n

Ey

TM1,2n

a b

The expressions for the field components of both the TE1,2n and TM1,2n modes, in waveguide #1, are needed to evaluate Dn. At z = 0, i.e. at the plane of the iris, shown in Fig (10.4a), only TE10 mode with constant Ey field in the slot region is considered. The slot field is given by equation (10.3.9a). Therefore, the total Ex field component of both the TE and TM higher-order modes must be zero at z = 0. So, only the TE10 mode field is retained at z = 0:

363

364

10 Slot Line

Ex

TE1,2n

+ Ex

TM1,2n

=0

10 3 27

On substituting Ex field components from equations (10.3.1) and (10.3.2) in the above equation, the following expression is obtained: 10 3 28

Again, on substituting Ey, for both the TE and TM modes, from equations (10.3.1) and (10.3.2) in equation (10.3.26b); and using the resulted expression with equation (10.3.28), the following relation is obtained: β2x = β2y

mπ a nπ b

2

10 3 29

As discussed previously, due to the symmetry of the structure, the waveguide #1 has m = 1 and n 2n. Hence, the above equation is written as follows: Dn =

γn1 =

+

2

2nπ b

2π n b 1− = b n

n YTM 0 TMn Y01

b,

TE Mode

For dielectric – filled waveguide a a β a × YTE × n1 = w = 2b 2b ωμ 2b

n For air – filled waveguide YTE = 0

2

a

TMn n where YTE are the characteristic admittance of 01 and Y01 the dielectric-filled waveguide section WG #A for the n n and YTM are TE1,2n and TM1,2n modes, respectively. YTE 0 0 the characteristic admittances for the air-filled waveguide section WG #B. The characteristic admittances for the TE and TM modes defined with respect to power–voltage ratio are related to their respective wave admittances [B.8]. It is discussed in chapter 7.

10 3 30

Following the method adopted above to obtain expression (10.3.21) for the input admittance of the TE10 mode, n n and YTM the expressions for the input admittance YTE i i of TE1,2n and TM1,2n (n ≥ 1) modes could be obtained. The TE1,2n mode of the WG #B in the cutoff region offers an inductive load to the WG #A at x = h, whereas the TM1,2n mode offers a capacitive load. These considerations help to compute the input admittance for both the higherorder modes. The derivation of the results could be followed from the steps given for the shielded slot line on the suspended composite substrate in Fig (10.7). The results summarized below, and also the expression (10.3.21), are limiting cases of the relevant equations, for h3 = 0, h1 ∞, in subsection (10.3.3). The equations to compute n n the input admittance YTE and YTM of the present structure i i are given below [J.1]:

π a

−1 n n YTM = YTM 01 tanh γn1 h + tanh i

n YTE 01 =

2

b 2na

n YTE 0 YTEn 01

10 3 31

β η0 βy B = A ZTM x η0

Dn =

−1 n n = YTE YTE 01 coth γn1 h + coth i

2

2π − λ0

1 2

2

εr 1 2 − 2a λ0

εr 2

1 2

γn1

βn η 0 k0

a 2b

a b 10 3 32

TM Mode

For dielectric – filled waveguide a a ωε × YTM × w = 2b 2b β

=

a 2b

k1 η1 βn1

a

n For air – filled waveguide YTM = 0

a 2b

k0 η0 βn

b

n YTM 01 =

10 3 33 In the above equations, the propagation constants γn1 = j βn1 and γn = j βn are for both modes in the dielectric-filled WG #A and air-filled WG #B, respectively. The propagation constants for the dielectric-filled waveguide section A and the air-filled waveguide section B are obtained from equation (10.3.4) on using m = 1, n 2n:

2π λ0

2

2π n b 1− = b n

2

2π n b 1− = b 2π n

βn1 η 1 k1

2

π εr − a

2

1 1 2 − 2 λ λc

1 2

1 2

10 3 34

10.3 Waveguide Model

where λ = λ0 εr. In the above equation, the cutoff wavelength of the TE1,2n mode is λc = 2a. It is related to the following expression [B.8]: 1 1 1 = 2− 2 λ2g λ λc

10 3 35

The above equation follows from equation (10.3.4). Using equations (10.3.34) and (10.3.35), the propagation constant γn1 for n ≥ 1 mode, in the dielectric-filled waveguide section WG #A, is obtained. Likewise, the propagation constant γn for the air-filled waveguide section WG #B is obtained using γ = jβ, λg = 2π/β. Both expressions are given below: γn1

γn γn1 γ rn = γn1 h + tanh − 1 n1 εr γn

qn = γn1 h + coth − 1

2

2πn b 1− = b nλg1

2π n γn = 1+ b

1 2

a 10 3 36

1 2

2

bγ 2π n

b

Two more parameters qn and rn are defined for equation (10.3.31) as n YTE 0 TEn Y01

qn = γn1 h + coth − 1 rn = γn1 h + tanh − 1

a

n n YTE = YTE 01 coth qn i

n YTM γ η k0 γ 0 = n1 1 = n1 TMn γ η k ε Y01 r γn n 0 1

b

b

On substituting the characteristic admittances from equations (10.3.32a) and (10.3.33a) for the TE1,2n and TM1,2n (n ≥ 1) modes, respectively, the above equations are written as a γn1 n YTE = coth qn a i j 2bη1 k1 10 3 41 ja k1 n = tanh r b YTM n i 2 bη1 γn1 On substituting equations (10.3.41) and (10.3.30) in equation (10.3.26a), the input admittance of combined TE1,2n and TM1,2n modes, i.e. the hybrid mode, at z = 0 is

Yi,n

On using equations (10.3.32) and (10.3.33) for the TE1,2n and TM1,2n modes, the ratio of characteristic admittances can be evaluated for both modes, in terms of their propagation constants: a

n n YTM = YTM 01 tanh rn i

a,

10 3 40

b

n YTE γ η k1 γ 0 = n 1 = n TEn γn1 η0 k0 γn1 Y01

10 3 39 b

On using the above equations for qn and rn in equation (10.3.31), the input admittances of the TE1,2n and TM1,2n modes can be expressed as

10 3 37

n YTM 0 TMn Y01

a

ja k1 a γn1 tanh rn + coth qn 2 bη1 γn1 j 2bη1 k1 = b 2 1+ 2an

=

ja k0 2 bη0 γn1

b γn1 2a nk0 b 1+ 2a n

b 2an

2

2

εr tanh rn −

coth qn 2

10 3 42 10 3 38

A few more parameters are defined below to recast the above equation in the more convenient form:

Thus,

p= 1 λg1

λ0 λ0 λ0 = , also λg = λc = 2a, λ = εr 2a λc 2

So, u =

=

1 λ

λ0 = λg1

2

−

1 λc

2

= εr

εr − p2 =

γ1 jk0

1 λ0

2

−

1 λc

d, v=

a, 2

,

u=

λ0 2π β γ = × 1 = 1 λg1 k0 2π j k0

1 1 = 2 εr − p2 λ2g1 λ0

γ = k0

where γ = γ0 is the propagation constant of the air-filled waveguide. The propagation constants, given by

p2 − 1, u =

v j

b c

10 3 43

e,

equations (10.3.36a) and (10.3.36b), can be modified as follows, by using parameters p, u, and v:

365

366

10 Slot Line

∗ For dielectric – filled waveguide section WG#A Fn1

1 2

2

b b γn1 = 1 − = 2π n nλg1

bu = 1− nλ0

1 2

2

bu = 1− 2a np

2

1 2

a (10.3.44)

∗ For air – filled waveguide section#B Fn =

bγ 2π n

b γ = 1+ 2π n n

2

1 2

2

b v k0 2π n

= 1+

= 1+

Let us change two factors involved in equation (10.3.42) in terms of parameters Fn1 and p: bγn1 bγ π 2π λ0 = Fn1 p = n1 = Fn1 = Fn1 2ank0 2πn ak0 2ak0 2a ak0 aπ aπ 1 = = = bγn1 2bγn1 λ0 bγn1 2apbγn1 2p π 1 1 = = bγn1 4p nFn1 2n 2p 2π n

1 2

Yi,0 = j

b

b 2an

η 0 Bd =

n = 1, 2 …

=j

1 2pη0

sin π nδ π nδ

2

εr tanh rn − pFn1 2 coth qn 1+

b 2an

au π uh v tan − tan− 1 2b ap u

1 2p n = 1, 2, …

1+

b 2 2an

Fn1

sin 2 π nδ n π nδ 2

2

Fn1

sin π nδ π nδ

2

1 n

The above capacitive susceptance is due to the discontinuity offered by the waveguide #1 that has an iris at z = 0 supported on the dielectric-loaded WG #A section followed by the WG Section B in the cutoff mode. The first term of the above equation is the susceptance due to TE10 mode and the second term is the susceptance due to combined higher-order modes TE1,2n and TM1,2n (n = 1,2,…) in a section of WG #B. The rate of convergence of the series shown in the above equation is slow. It requires improvement in convergence. To do so, the limiting value of the term under the sign is obtained.

10 3 47 The above capacitive input susceptance is due to the nonpropagation higher-order modes. At z = 0, the input admittance Yi0 given by equation (10.3.21), could be written in terms of the variables p, u, and v. To do so, the characteristic admittance Y01, Y0/jY01 and electrical length (β1h) are expressed in terms of p, u, and v. Equations (10.3.24), (10.3.25), and (10.3.43) are used to get the following expressions: a λ0 au × = 2bη0 λg1 2bη0 2π h 2π π uh = u k0 h = uh = β1 h = λg1 λ0 ap Y0 γ λg1 v = − = − jY01 k 0 λ0 u Y01 =

εr tanh rn − p2 F2n1 coth qn

10 3 50

The input admittances at z = 0, due to all higher modes, TE1,2n and TM1,2n for n ≥ 1, are obtained from equations (10.3.19) and (10.3.46) as follows: Yi,n

10 3 49

2

10 3 46

2

au π uh v − tan−1 tan 2bη0 ap u

Finally, the input capacitive susceptance Bd, given by equation (10.3.19), at z = 0 is rewritten with the help of equations (10.3.47) and (10.3.49) as

+

εr tanh rn − pFn1 2 coth qn 1+

b

Using variables of equation (10.3.48) in equation (10.3.21), the input admittance of TE10 mode is obtained as

Equation (10.3.42) could be rewritten as 1 η0 4pnFn1

2

a

10 3 45

Yi,n = j

bv 2 anp

1 2

Let Sn =

b 2an

1+

2

Fn1

sin 2 π nδ n π nδ 2 10 3 51

Under the limiting condition, h ∞ and f c i.e. λ0 = ∞, λg ∞, a ∞, also Lt Fn1 f λg λg1 ∞ Lt tanh rn

a

εr tanh rn − p2 F2n1 coth qn

λg λg1

∞

1, Lt cothqn λg λg1

∞

0, 1,

1 the following expres-

sion is obtained:

b

Sn

c 10 3 48

Sn = εr − p2

2 sin 2 π nδ 2 sin π nδ 2 = u n π nδ n π nδ 2

10 3 52

10.3 Waveguide Model

The series summation of equation (10.3.50) is rewritten as follows: Sn = n = 1, 2, …

n = 1, 2, …

n = 1, 2, …

Sn = u 2 = u2

Sn − Sn +

n = 1, 2, …

sin 2 π nδ n π nδ 2

n = 1, 2, …

Sn

At z = 0, the susceptance Bd looking into the waveguide #1 given by equation (10.3.51) is rewritten by using equations (10.3.51), (10.3.52), and (10.3.57):

10 3 53 η0 B d =

1 2πδ

≈ u2 ln

+

3 2 +

1 − 0 3379 ln δ

au πuh v tan − tan − 1 2b ap u

+

u2 2 ln πδ 2p

εr tanh rn − p2 F2n1 coth qn

1 2p n = 1, 2, …

2

b 2a n

1+

− u2

Fn1

10 3 54

10 3 58

The above summation has been evaluated by expressing the sine function in the form of exponentials and also using the identity given by Collin in Appendix-A.2 [B.11]. Thus, under the condition δ 0, the field is confined in the slot region providing the capacitive susceptance Bc that exists on both sides of the iris. Hence, from equations (10.3.54) and (10.3.50), the following expression is obtained: u2 1 ln − 0 3379 p δ

η0 B c ≈

The first term of the above susceptance is due to the TE10 mode in waveguide #1 with WG sections – WG #A and WG #B. The second term is the capacitive reactance of the slot iris. The third term is the susceptance contributed by the higher-order modes in waveguide #1. At z = 0, the susceptance Ba offered by the air-filled waveguide #2 in the region z < 0 is obtained from the above equation by using εr = 1 or h = 0:

10 3 55

η 0 Ba = −

However, a more accurate formula in the limiting case λg1 ∞ is [J.1, B.13], η 0 Bc ≈

u2 2 ln p πδ

=

u2 1 ln − 0 452 p δ

+

10 3 56 The difference in Bc is due to our assumption of the constant Ey and Hx fields accross the iris slot. The slot field is a function of y, not a constant as assumed in equation (10.3.9). Using the above expression, equation (10.3.53) is improved as Sn = n = 1, 2, …

n = 1, 2, …

Sn − Sn

sin 2 π nδ n π nδ 2

av v2 2 − ln 2b 2p πδ 1 − p2 F2n1

1 2p n > 1

1+

b 2a n

− u2

2

Fn1

sin 2 πnδ n π nδ 2

10 3 59 It is seen that for εr = 1, γn1= γn, so tanh (rn) = coth (qn) = 1, also Fn1 =

2 + u2 ln πδ

the

1−

bu 2anp

2

Thus, the expression under

sign is

10 3 57

1 − p2 1 −

1 − p2 F2n1 1+

b 2an

−u = 2

2

b 2an

1+

b 2an

2

2

u 1+

b 2an

Fn 2

= 1+

b 2an

2

u2 p2

1 − p2 + −u = 2

Fn

Fn

b 2an

2

u2 − u2

2

Fn 10 3 60

− u2 = u2

2

1+

b 2an

1 −1 Fn

= v2 1 −

1 Fn

367

368

10 Slot Line

At z = 0, the susceptance Ba, looking toward the air-filled waveguide #2 is η0 Ba = −

av v2 2 − ln 2b 2p πδ

1 1 + v2 1 − 2p n = 1, 2, … Fn

sin 2 π nδ n π nδ 2

the MW and symmetry, TE11, TE13, TE15,… and TM11,TM13, TM15,…, i.e. TE1,2n and TM1,2n modes are excited, where n = 1/2, 3/2, 5/2, etc. Under this value for n, the following limiting value is obtained: δ

Lt

0 n = 12, 32, …

sin 2 π nδ 4 u2 8 − 0 3379 ≈ ln 2 = ln δ p πδ n π nδ 10 3 65

10 3 61 At z = 0, the total susceptance Bt = Bd + Ba is obtained, for the case-A of the (EW–EW) waveguide, from equations (10.3.58) and (10.3.61): η 0 Bt =

a πuh v − v + u tan − tan − 1 2b ap u 1 εr + 1 2 + − p2 ln p 2 πδ +

1 2 n = 1, 2,

v2 1 − …

1 Fn

+ Mn

η0 Bd =

sin 2 π nδ n π nδ 2 10 3 62

where Mn for real Fn1 is; εr tanh rn − p2 F2n1 coth qn Mn = − u2 b 2 Fn1 1+ 2an 2π nhFn1 rn = + tanh − 1 b 2π nhFn1 + coth − 1 qn = b

Fn1 εr Fn Fn Fn1

εr tan rn − p2 Fn1 2 cot qn 1+

b 2an

2

a

2πnh Fn1 Fn1 + tan − 1 b εr Fn 2πnh Fn1 Fn qn = + cot − 1 b Fn1

+

1 2p

εr tanh rn − p2 F2n1 coth qn n = 12, 32, …

1+

η0 Ba = − 10 3 63

+

b

b 2a n

2

− u2

Fn1

sin 2 π nδ n π nδ 2

2

v 8 ln 2p πδ 1 2p

v2 1 − n = 12, 32, …

1 Fn

sin 2 π nδ n π nδ 2 10 3 67

c

The first term of the above equations shows the capacitive susceptance offered by the iris in the (EW–MW) waveguide. The second term is the capacitive susceptance due to the higher-order modes. At z = 0, the total susceptance Bt = Bd + Ba, for case B of the (EW–MW) waveguide, is obtained by adding the above two equations:

− u2

a

Fn1

rn =

u2 8 ln 2p πδ

10 3 66

Mn for Fn1 imaginary is; Mn =

The above expression provides the capacitive susceptance offered by the iris in the (EW–MW) waveguide. Therefore, the susceptance Bd and Ba for the case of MW at y = ±b/2 is written, from equations (10.3.58), (10.3.61), and (10.3.65), as follows:

b

η0 Bt =

c 10 3 64

The above expression for the total susceptance Bt is valid for the narrow slot, δ = s b ≤ 0 15, s ≤ λ0 4 εr .

Case B: Magnetic Wall at y = ±b/2

In this case, two broad-faces, separated by height b in Fig (10.4) and (10.5), are treated as the MW. The equivalent waveguide model, with the EW–MW surfaces, is formed. The rest of the analysis follows case A. However, this waveguide does not support the propagating TE10 mode. Therefore, its contribution to the susceptance Bd, through the first term of equation (10.3.50), is dropped. Also keeping in view

εr + 1 8 − p2 ln 2 πδ 1 1 sin 2 π nδ + Mn v2 1 − + 2 13 Fn n π nδ 2 n = , ,… 1 p

2 2

10 3 68 The above expression is a modification of equation (10.3.62). The first term of equation (10.3.62), due to the TE10 mode, is dropped. The parameters Mn for the real and imaginary Fn1 are given by equations (10.3.63) and (10.3.64), respectively. Slowing-Factor

The slowing-factor 1 εreff f is determined by using the transverse resonance condition, i.e. Bt = 0 at z = 0. For a given slot line, i.e. known substrate thickness h and relative permittivity εr and assumed b, equations (10.3.62) and (10.3.68) are solved for the parameter p at a given

10.3 Waveguide Model

frequency. At the transverse resonance, a = λg/2, i.e. p = λ0/λg, where λg is the guided wavelength of the slot line. The value of the parameter p is in the range 1 < p < εr corresponding to λg1 λ0, λg1 ∞. The solution provides 1 p = λg λ0 = 1 εreff f , i.e. the dispersive slowing-factor. For the correct choice of dimension b, both equations (8.3.62) and (8.3.68) provide almost the same values for the slowing-factor. Thus, at each assumed frequency, the effective relative permittivity εreff(f ) of a slot line is determined, once the parameter p is evaluated. It should be independent of parameter b. Usually, b is several times of the slot-width, b > 10 s.

The characteristic impedance of a slot line is defined based on voltage–power ratio, as follows: V2+ , 2P +

10 3 69

where P+ is the average power flow of the wave traveling in + x-direction on a slot line, shown in Fig (10.4a, d) and (10.5b) and V+ is the peak amplitude of the voltage across the slot. The energy U p.u.l. travels with the group velocity vg, i.e. U=

2P+ vg

10 3 70

In the case of the resonant line length x = λg/2, two oppositely traveling waves form the standing wave, such that the total power on the line is 2P+. The stored energy in the λg/2 line length is Wt = (2P+/vg) × (λg/2) = P+λg/vg. Using phase velocity vp = λgc/λ0 from equation (7.4.29b) of chapter 7, the stored energy is Wt =

P+ λ0 2π P + vp = × vp vg c ω vg

10 3 71

At the center of the slot (x = λg/2), the voltage is maximum: V0 = 2V +

10 3 72

The characteristic impedance of the slot line, from equations (10.3.69) to (10.3.71), is Z0 =

vp π V20 × 4 ωWt vg

10 3 73

The stored energy in a cavity, at resonance, is given by [J.1]: Wt =

V20 dB , 4 dω

vp π 1 ω d η0 Bt dω vg

10 3 75

However, ω = 2πc/λ0, at the transverse resonance p = λ0/2a = λ0/λg, ω = 2π c/2a p = π c/a p, dω = − (π c dp)/ap2 = − ω(dp/p). Therefore, the above characteristic impedance is written as Z0 = 377

π p

Δp − Δ η 0 Bt

vp vg

10 3 76

An expression for vp/vg, i.e. ratio of the phase and group velocities for a slot line, is needed to compute its characteristic impedance. The group velocity is related to the propagation constant βx of the slot line by dω 1 2π = , where, βx = 10 3 77 dβx dβx λg dω λ2g 1 1 1 = − vg = = = − dλg 1 dλg d 2π 2π d 1 df dω λg 2π df λg λ2g df vg =

Characteristic Impedance

Z0 =

Z0 =

10 3 74

where B is the susceptance, at the port of the cavity. In the present case, the susceptance is Bt at the port z = 0 giving the following expression:

λg 1 dλg = − vg λg df 10 3 78 Using equation (7.4.29b) of chapter 7, the phase velocity is also expressed as follows: vp =

λg c = f λg λ0

10 3 79

λ0 dp d λ0 1 dλ0 λ0 dλg = − 2 . , = λg df df λg λg df λg df c dλ0 c λ0 = − 2 = − However, λ = i e f df f f Further, p =

Therefore,

d λ0 λ g λ0 1 1 dλg = − + df λg f λg df dλg λg d λ0 λg 1 = − λg + df df f λ0

10 3 80

From equations (10.3.78) and (10.3.79), Δ λ0 λ g λg 1 = + λ g λ0 Δf vg f

10 3 81

On substituting λg from equation (10.3.79) in the above equation, the following relation is obtained: Δ λ0 λg vp f =1+ Δf vg λ0 λg

10 3 82

The Δ(λ0/λg), i.e. Δp and Δf, is computed from two separate solutions of η0Bt = 0 for the fixed values of εr, s, h, and b and for two slightly different values of a = λg/2 = λ0/2p, incremented plus and minus from the defined parameter “a” at the frequency f. The frequency f is taken midway in the Δf interval. Once vp/vg is obtained, it can be substituted in equation (10.3.76) to compute the characteristic impedance of a slot line.

369

10 Slot Line

10.3.2

Sandwich Slot Line

The waveguide model discussed above is extended to a sandwich slot line, shown in Fig (10.1b) [J.12–J.14]. The method is also applicable to the coupled slot lines [J.25, J.26]. Following the discussion of the previous section, the waveguide model is developed for a sandwich slot line, shown in Fig (10.6a). Figure (10.6b) shows its equivalent waveguide of width a = λg/2, in the direction of the x-axis and height ±b/2, in the direction of the y-axis. The width is not shown in Fig (10.6b). Similar to Fig (10.4c), the broadface of the present equivalent waveguide could be either an EW or a MW. A pair of strip conductors of a slot line appears as the iris located at the interface of two dielectric sheets. The z = 0 plane of Fig (10.6b) is a junction of two modeled waveguides – the waveguide #1 for z ≥ 0 and the waveguide #2 for z ≤ 0. For the case of EW at y = ±b/2 and on using equation (10.3.58), the susceptance Bd1 and Bd2, i.e. Bdi (i = 1,2) for the z > 0 and z < 0 sides are written below: η0 Bdi =

a ui π ui hi vi tan − tan − 1 2b ap ui

+

εri tanh rni − p2 F2ni coth qni n = 1, 2, 3

1+

b 2an

εri tanh rni − p2 F2ni coth qni

Let, Mni =

u2i 2 ln 2p πδ

− u2i

2

Fni

+

1+

b 2an

2

− u2i

i = 1, 2

Fni 10 3 84

For waveguide #1: au1 π u1 h1 v1 tan − tan − 1 2b ap u1

η0 Bd1 =

+

u21 2 ln 2p πδ

1 sin 2 πnδ Mn1 2p n = 1, 2, … n πnδ 2

+

10 3 85 Likewise, for waveguide #2: η0 Bd2 =

au2 π u 2 h2 v2 tan − tan − 1 2b ap u2 +

1 2p

+

u22 2 ln 2p πδ

1 sin 2 π nδ Mn2 2p n = 1, 2, … n π nδ 2 10 3 86

The total susceptance at plane z = 0 Bt = Bd1 + Bd2

sin 2 π nδ n π nδ 2

10 3 87

10 3 83

a π u 1 h1 v1 − tan − 1 u1 tan ap u1 2b +

1 2 2 u + u22 ln 2p 1 πδ

+

+ u2 tan

1 2p n = 1, 2, 3

πu2 h2 v2 − tan − 1 ap u2

Mn1 + Mn2

10 3 88

sin 2 π nδ , n π nδ 2

Y Air

Air

Z h2 S

εr2

Y X

εr1

h1 Air

EW/MW

+b/2 εr2 +S/2

Air εr1 Gap 0

–S/2 Metallic plane –b/2 h2

h1

Z

Waveguide #1

η0 Bt =

Waveguide #2

370

EW/MW

Bd2 Bd1 (a) Sandwich slot line. Figure 10.6

Sandwich slot line and its equivalent waveguide model.

(b) Equivalent waveguide model.

10.3 Waveguide Model

where

ui =

1−

bui 2anp

2

Fni =

1+

bv 2anp

2

p2 − 1 c , F n =

εri − p2

v1 = v2 = v =

a,

The parameter Mni for the real Fni and imaginary Fni (i = 1, 2) are obtained from equations (10.3.63) and

η 0 Bt = +

a πu1 h1 v − tan − 1 u1 tan ap 2b u1 1 2 εr1 + εr2 − 2p2 ln 2p πδ

+

+ u2 tan 1 2p n = 1, 2, 3

Likewise, for the case of the MW at y = ±b/2, using equation (10.3.66), the total susceptance Bt at z = 0 is written as follows: 1 8 η 0 Bt = εr1 + εr2 − 2p2 ln 2p πδ +

1 2p

Mn1 + Mn2 n = 12, 32, …

sin 2 π nδ n π nδ 2

In the case of the MW at y = ±b/2, the first two terms of equation (10.3.90), due to the TE10 mode, are dropped in the above equation. In the present case, the equivalent waveguides #1 and #2 are partly occupied with dielectric slabs corresponding to the substrate and superstrate of the sandwiched slot line. If the two substrates are widely different, either u1 or u2 can be imaginary. In that case, u2i = − ui 2 and ui = j|ui|. For the ui imaginary, the first term of equation (10.3.90) is replaced by another factor as follows: ui tan

v ui

i = 1, 2 10 3 92

At the assumed frequency, the parameter p = λ0 λg = εreff f is obtained from the solution of either equation (10.3.90) or equation (10.3.91), for a large value of b, b > 10 s. The characteristic impedance is obtained from equation (10.3.76). The zeroth-order solution for the guided wavelength is λg = λ0

2 εr1 + εr2

10 3 93

b

d

(10.3.89)

(10.3.64), respectively. Using equation (10.3.58), the total susceptance Bt, at z = 0, is πu2 h2 v − tan − 1 ap u2 Mn1 + Mn2

(10.3.90)

sin 2 π nδ n π nδ 2

The above analysis is used for the dielectric substrate– superstrate. However, the permeability of the substrate– superstrate can be easily accounted for in the analysis [J.12].

10.3.3 10 3 91

π hi ui v − tan − 1 ap ui π hi u i − tanh − 1 ui tanh ap

, i = 1, 2

Shielded Slot Line

Figure (10.7a) shows a suspended composite substrate slot line, totally enclosed inside a metallic box [J.13]. Figure (10.7b) shows its equivalent waveguide model. Figure (10.7c) shows the cross-section of the equivalent waveguide. The EM-wave on the slot line propagates in the x-direction. In the present case, the EW, due to the side shield conductors, are placed at y = ±b/2. The lower region, below the strip conductors, comprising of three dielectric media, is converted to the right-hand side waveguide #1 shown in Fig (10.7b). The WG #1 is terminated in the conducting wall at z = (h1 + h2 + h3). The upper region of the structure, with air medium, appears to the left-hand side waveguide #2 terminated in the conducting wall at z = −h4. Hence, at the plane z = 0, a junction is created by two waveguides – waveguide #1 and waveguide #2. The metallic iris is located at the junction of two waveguides. Fig (10.7c) shows the cross-sectional view of the iris corresponding to the a = λg/2 section of a slot line. The structural details are taken into account, while computing the capacitive susceptance Bd at z = 0, using the expressions of subsection (10.3.1). The total Ey and Hx fields at z = 0 plane and x = a/2 are still given by equation (10.3.7). The TE10 mode and all higher-order modes are at the cutoff or non-propagating in the air region of thickness h1 and h4. However, in the dielectric regions, the TE10 mode propagates. The first few higher-order modes may be in the cutoff, depending upon height “b” of the equivalent

371

10 Slot Line

Y

H

Y

εr2 h2 εr1 h1

Z

Air

0 –S/2

z

h4

(a) Suspended composite slot -line.

εr2

εr3

x

εr

+b/2

–b/2 A x=0

EW (c) Cross-section of equivalent waveguide.

Multilayer shielded slot line and its equivalent waveguide model.

and TM1,2n. The following expressions are obtained for them:

Input Admittance Yi,0 of the TE10 Mode

The input admittance of waveguide #1 is computed in three steps with reference to Fig (10.8). The input admittance of the equivalent transmission line of the waveguide section of length h terminated in load YL is

Computation of Bd at z = 0 for Waveguide #1

Yin = Y0

The input admittance Yi, related to the capacitive susceptance Bd, at the plane z = 0 for the waveguide section #1, is given by equation (10.3.19). The Yi has two components – (i) input admittance Yi,0 of the TE10 mode, and (ii) input admittance Yi,n contributed by the higher-order modes TE1,2n

h2

εr = 1

εr2

Y02

Y02

εr2

εr2

Yin2 P

Z=0

P

β3 h3

P

Y′in2

Y03 εr3

Y′in2

P

Z

(b) Step-II.

(a) Step-I. h3

θ2

θ3

Y03

Y03

εr3

εr3

Open

γ

P

β2 h2

P

Y02

Yin3 P

Z =0 (c) Step-III.

Computation of the input admittance.

10 3 94

Figure (10.8a) shows the 1st step. The last section of the waveguide #1 is terminated in the EW forming the shortcircuited load YL ∞.

Yin1

Y01

YL + Y0 tanh γ h Y0 + YL tanh γ h

Open

h1

Figure 10.8

s

B x = a = λg/2

–b/2

h1

(b) Longitudinal section of equivalent waveguide model.

waveguide. The input wave admittance of the fundamental TE10 mode yi,0, and the input wave admittance, yi,n of the combined nth TE1,2n, and TM1,2n modes, i.e. the hybrid modes, are given by equation (10.3.8). The slot field Ey, given by equation (10.3.9), is assumed to be constant in the slot region. The analysis is valid for the narrow slot δ = s/b ≤ 0.15 [J.1].

Yin1

Metallic plane

Metallic plane

εr1 = 1

h2

h3

EW

Waveguide #1

εr4 = 1

b

Figure 10.7

Air

+S/2

Y A

B

EW

εr3 h3

Waveguide #2

S

EW

+b/2

Air

εr4 h4

Air

Short circuit

372

Y′in2

P

Z

10.3 Waveguide Model

Yin1 = Y01 coth γh1 ,

10 3 95

where Y01 is the characteristic admittance of the waveguide section h1. For the TE10 mode, Y01 is given by equation (10.3.24) and γ is the propagation constant of the waveguide. The 2nd step of Fig (10.8b) shows that Yin1 acts as a load to dielectric (εr2)-filled waveguide section h2 with characteristic admittance Y02. This load is replaced by the extended open-end line of length θ2. Its input admittance is Yin1 = jY02 tan θ2 = Yin1 = Y01 coth γ h1 Y01 coth γ h1 tan θ2 = j Y02 10 3 96 The following equation (10.3.48c):

expression

is

obtained

Y0 v = − jY02 u2

from

10 3 97

The input admittance Yin,2 of the waveguide section h2 with extended length θ2 is Yin,2 = jY02 tan β2 h2 + θ2

10 3 98

The 3rd step of Fig (10.8c) shows that Yin,2 acts as a load to the dielectric (εr3)-loaded waveguide section h3 with the characteristic admittance Y03. The load Yin,2 is again replaced by an open circuit line section of length θ3. Its input admittance is Yin,2 = jY03 tan θ3 = jY02 tan β2 h2 + θ2 Y02 tan β2 h2 + θ2 , tan θ3 = Y03

Input Admittance Yi,n of Higher-order Modes

To compute Yin for the waveguide #1, the input admitn n and YTM , of the TE1,2n and TM1,2n (n ≥ 1) tances YTE i i modes, are obtained. The process is identical to the previous case for the TE10 mode. n YTE for the TE1,2n Mode i

n Figure (10.9) shows the three-step process of obtaining YTE i at z = 0 for waveguide #1. In the air-filled waveguide section of length h1, the TE1,2n higher-order modes are in the cutoff region, giving the imaginary inductive characteristic impedance. Hence, the waveguide #1 is terminated in the inductive load at z = (h2 + h3). The inductive load at z = (h2 + h3) is also achieved if the TE1,2n mode supporting the waveguide section of length h1 is terminated in a shortcircuit [J.1]. Figure (10.9a) shows the 1st step to compute the input admittance of section h1 for load YL ∞:

TEn YTEn in1 = Y01 coth γn1 h1

Figure (10.9b) of the 2nd step shows that the YTEn in1 is connected as a load to the output of section h2. It can be expressed as a short-circuited line of extension θ2. The input TEn admittance Y in1 due to line length θ2 is

cothθ2 =

Y02 u2 = Y03 u3

u2 v tan β2 h2 − tan− 1 coth γ h1 u3 u2

v = p2 − 1

1 2

u2 = εr2 − p2

However, using equations (10.3.38) and (10.3.44), the ratio of the characteristic admittances are YTEn Yn Fn 01 = TEn = Y F Y02 n2 n2 2πn Fn h 1 γn h1 = γn1 h1 = b

, γ = v k0 , k = 2π λ0 , 1 2

, u3 = εr3 − p2

1 2

a b 10 3 107

10 3 102 The input admittance of the section h3 with line extension θ3 is Yin,3 = jY03 tan β3 h3 + θ3

YTEn 01 coth γn1 h1 YTEn 02 10 3 106

10 3 101 where

10 3 105

TEn

10 3 100 tan θ3 =

10 3 104

TEn Y in1 = YTEn 02 cothθ2 = Y01 coth γn1 h1

10 3 99 where using an equation 10 3 48a

a u3 tan 2bη0 π u3 h 3 u2 π u2 h2 v + tan − 1 − tan − 1 tan coth γ h1 u2 ap u3 ap

Yi,0 = Yin,3 = j

10 3 103

On using equations (10.3.48a) and (10.3.48b), the above equation is written as the input admittance Yi,0 of waveguide #1 for the TE10 mode:

The input admittance YTEn in2 for the line length (γn2h2 + θ2) is TEn YTEn in 2 = Y02 coth γn2 h2 + θ2 2π n Fn2 h2 γn2 h2 = b

where,

a b 10 3 108

rd

The 3 step nected to line

of Fig (10.9c) shows that YTEn in2 is a load section h3. The load YTEn in2 is replaced

conby a

373

10 Slot Line

h1

TEn

Yin1

εr = 1

Y02 εr2

TEn

TEn

Y′in1

(a) Step-I. γn3 h3

P

TEn

TEn

Yi P

=

θ3 TEn

Y03 εr3

TEn

Y03 εr3

Z

P

(b) Step-II.

P

TEn

Y02 εr2

Yin2 P

h3

TEn

Y02 εr2

Short circuit

γn1

TEn

TEn

TEn Yin1

Y01

θ2

P

Short circuit

Short circuit

TEn

γn2 h2

P

h2

Y′in2

374

Y03 εr3

TEn

Yin3

z=0

Y

TEn

Z

P

Y′in2

(c) Step-III. Figure 10.9

n Computation of the input admittance YTE of the equivalent transmission line of TE1,2n mode. i

short-circuited line of length θ3. The input admittance of line length θ3 is

Figure (10.9c) shows that finally the input admittance is

YTEn in3

TEn YTEn in 3 = Y03 coth γn3 h3 + θ3 aγn3 where YTEn 03 = j 2bη3 k3

TEn TEn YinTEn 2 = Y03 coth θ3 = Y02 coth γn2 h2 + θ2

cothθ3 = where,

YTEn 02 YTEn 03

coth γn2 h2 + θ2

a

YTEn Fn2 02 = Fn3 YTEn 03

10 3 109

where,

Fn1 = Fn =

2π nFn3 h3 Fn2 2πnFn2 h2 Fn 2π nFn coth + coth − 1 + coth − 1 h1 cot h b Fn3 b Fn2 b

1+

bv 2anp

2

, Fn2 =

1−

bu2 2anp

The variables v, u2, and u3 are given in equation (10.3.102). Input Admittance YTMn i

The similar three-step process, shown in Fig (10.10), is followed to obtain the input admittance of higher-order of the waveguide #1. Again, TM1,2n modes, i.e. YTMn in1 TM1,2n higher-order modes are in cutoff for the waveguide #1 section h1 that is short-circuited at the end. It is the 1st step and shown in Fig (10.10a). However, the characteristic admittance of the line section h1, supporting the evanescent

b

10 3 110

The input admittance YTEn is obtained from equa03 tion (10.3.32a). The θ2 and θ3 are replaced to get the expresof the waveguide #1 for sion for the input admittance YTEn i the TE1,2n mode at z = 0.

b

TEn = YTEn YTEn i in 3 = Y03 coth

a

(10.3.111)

2

2

, Fn3 =

1−

bu3 2anp

(10.3.112)

TM1,2n modes, is capacitive. Hence, the line section h1 offers a capacitive load, YTMn in1 , to the line section h2, instead of the inductive load, as in the case of the evanescent TE1,2n modes discussed previously. The capacitive load YTMn is in1 given by TMn YTMn in1 = Y01 coth γn1 h1

10 3 113

Figure (10.10b) of the 2nd step shows that the input admittance YTMn in1 is connected as a load to the line section h2. The capacitive load YTMn in1 is further treated as an open-ended line of length θ2, i.e. YL 0 using equation (10.3.94).

10.3 Waveguide Model

h2 Short circuit

εr = 1

Y02 εr2 TMn

Yin2

TMn

P

γn3 h3

P

Y′in2

TMn

Yi

TMn

Y03 εr3

Yin3

Z =0

P

TMn

Y′in2

θ3 TMn

TMn

Y03 εr3

TMn

TMn

Z

(b) Step-II.

P

Y03 εr3

P

Y′in1

(a) Step-I. h3

Y02 εr2

P

Open-circuit

Yin1

TMn

Y02 εr2

TMn

γn1 TMn

θ2 TMn

TMn

Yin1

TMn

Y01

P

γn2 h2

P

Open-circuit

h1

Z

(c) Step-III. Figure 10.10

Computation of the input admittance of the equivalent transmission line of TMn mode.

TMn

TMn TMn n Y in1 = YTM 02 tanh θ2 = Yin1 = Y01 coth γn1 h1

tanh θ2 =

YTMn 01 coth γn1 h1 YTMn 02

TMn

TMn Y in2 = YTMn 03 tanh θ3 = Y02 tanh γn2 h2 + θ2

tanh θ3 =

YTMn 02 tanh γn2 h2 + θ2 , YTMn 03

10 3 114 Following equations (10.3.33) and (10.3.44), the following expression is obtained: n YTMn YTM For air − filled waveguide 1 Fn2 0 01 = = TMn TMn εr2 Fn Y02 Y02

10 3 115 1 Fn2 tanh θ2 = coth γn1 h1 , εr2 Fn

10 3 116

where γn1h1 is given by equation (10.3.107b). Figure (10.10b) shows the input admittance YTMn of the in2 open-circuited line of length (γn2h2 + θ2): TMn YTMn in2 = Y02 tanh γn2 h2 + θ2

10 3 117

Again, Fig (10.10c) of the 3rd step shows that admittance TMn YTMn in2 is connected as a load to the line section h3. The Yin2 is replaced by the open-circuited line section θ3. Its input TMn admittance Yin2 is

YTMn = in

10 3 118 where following equation (10.3.115), the below expression is written: YTMn εr2 Fn3 02 = εr3 Fn2 YTMn 03 tanh θ3 =

εr2 Fn3 tanh γn2 h2 + θ2 εr3 Fn2

10 3 119

10 3 120

The electrical length of line section h2, γn2h2, is given by equation (10.3.108b). Finally, the input admittance of line section h3 at z = 0 is given below. The characteristic admitis obtained from equations (10.3.33) and tance YTMn 03 (10.3.45b). TMn YTMn = YTMn in in3 = Y03 tanh γn3 h3 + θ3 ε r3 =j YTMn 03 4pnFn3 η0

a b 10 3 121

jεr3 2πnFn3 h3 εr2 Fn3 2π nFn2 h2 1 Fn2 2πnFn h1 + tanh− 1 tanh− 1 tanh tanh coth 4pnη0 Fn3 b εr3 Fn2 b εr2 Fn b 10 3 122

375

376

10 Slot Line

The expressions (10.3.111) and (10.3.122) are rewritten below in terms of arguments qn and rn for nth TE1,2n and TM1,2n modes.

aγn3 coth qn j2bη0 k0 jεr3 = tanh rn 4pnFn3 η0

YTEn = i

a

YTMn i

b

10 3 123

where, 2πnFn3 h3 + coth − 1 b 2πnFn3 h3 rn = + tanh − 1 b

qn =

Fn2 2πn Fn 2πnFn h Fn2 h2 + coth − 1 coth coth Fn1 Fn2 b b εr2 Fn3 2πnFn2 h2 1 Fn2 2πn + tanh − 1 F n h1 tanh coth εr3 Fn2 b εr2 Fn b

The capacitive susceptance Bd of the iris, supported on a dielectric substrate, due to the TE10, TE1,2n, and TM1,2n modes looking into the waveguide #1, is obtained by

η0 Bd = Yi,0 +

u23 2 ln 2p πδ

+

1 2p n = 1, 2 …

modifying equation (10.3.58) tions (10.3.114) and (10.3.123):

εr3 tanh rn − p2 F2n3 cothqn 1+

Due to TE10 mode, Due to iris

b 2π n

2

− u23 ×

Fn3

a (10.3.124) b

with

help

of

equa-

sin 2 πnδ n π nδ (10.3.125)

Due to TE1,2n + TM1,2n modes

The input admittance Yi,0 of the waveguide #1 for the TE10 mode is given by equation (10.3.104).

η0 Ba = −

Computation Ba at z = 0 for Waveguide #2

+

Further, the capacitive susceptance Ba of iris, in the airfilled waveguide #2, is obtained by taking h2 = h3 = 0, and h1 = h4, and by modifying equation (10.3.58):

av v2 2 coth vk0 h4 − ln 2b πδ 2p 1 2 n = 1, 2, …

v2 1 −

coth 2π nFn h4 b Fn

sin 2 π nδ n π nδ 2

10 3 126

At z = p, the total susceptance Bt = Bd + Ba is ηBt =

a 2b

− v coth vk0 h4 + u3 tan 1 p

where,

Mn =

εr

1 2

− p2 ln

2 πδ

π h3 u 3 u2 π h2 u2 v + tan − 1 − tan − 1 tan coth vk0 h1 ap u3 ap u2 1 , 2

v

v2 1 − n 1, 2, …

coth 2πnFn h4 b Fn

Mn

sin 2 π nδ n π nδ 2

εr3 tanh rn − p2 F2n3 coth qn − u23 1 + b 2an 2 Fn3

Under the transverse resonance condition, Bt = 0. At assumed frequency, the variable p is determined for a slot line. The value of the parameter “p” leads to the determination of frequency-dependent effective relative permittivity. The frequency-dependent characteristic impedance of the present slot line structure is determined by using equations (10.3.76) and (10.3.82).

(10.3.127)

(10.3.128)

10.3.4

Characteristics of Slot Line

The slowing-factor λg λ0 = 1 εreff f , or the effective relative permittivity εreff(f ), and the characteristic impedance of a slot line are dependent on their physical parameters: relative permittivity (εr), a substrate thickness (h), slot-width (s), and frequency, even if the strip conductor

10.3 Waveguide Model

+++

4.0

+

+

+

+

+

+

+ + +

+

+ + +

+

+ +

+

+ +

+

3.5

0.62

b = 4.5 mm

Full-wave method Waveguide model Experimental

+

+

+

1.5

0.54

+

+

λg/λ0

+

εreff

0.58 0.56

b = 10 mm

+

0.60

3.0 t 2.5

s/h = 2.0

0.52

1.5

εr

h

180 160

1.0 1.0

.8 .6

140

0.8

.4

0.6

120

0.4

100

.2

0.50 0.48

S

220 200

+

+

+

εr = 9.6 s/h = 2.0

.1

0.2

.02

80 0.1

0.46

Z0 (Ω)

4.5

60

.02

b

0.44

2.0 0

3

6

9

12

15

18

Frequency (GHz) (a) Comparison of results of the full-wave method and waveguide model against experimental results [J.27].

Figure 10.11

λg/λ0 Z0

40

.02 .04 .05 .06 .07 .08 .09 h/λ0 (b) Computation of slowing-factor and Z0 using waveguide model [J.2].

Nature of dispersion in the slot line. Source: Boremann [J.27]. © IEEE. Mariani et al. [J.2]. © IEEE.

thickness and losses are ignored. The waveguide model provides information on the line parameters. The modal behavior of a slot line using the waveguide model could be compared against the results obtained from the fullwave analysis of Boremann, using the scattering-type TRM [J.27]. Figure (10.11a) compares such results of εreff(f ) for a slot line, on the substrate εr = 9.7, h = 0.635, t = 7.03 μm, slot-width s = 0.5 mm, and line width b, against the experimental results. The equivalent waveguide model does not correctly describe the dispersion behavior of a slot line at the lower frequency, due to the presence of the cutoff phenomenon. The full-wave results show no cutoff phenomenon and follow the experimental results. It demonstrates the quasi-TEM-type mode of a finite width slot line, as the εreff(f ) tends toward the static εreff(f 0). Therefore, the quasi-TE10 mode, shown by the waveguide model, is not a correct physical description of the finite width practical slot line. However, much above the cutoff frequency, the results of the waveguide model follow both the full-wave and experimental results. Figure (10.11b) shows the results for the slowingfactor (λg/λ0) and the characteristic impedance of a slot line on the substrate with εr = 9.6 using Cohn’s equivalent waveguide model [J.1, J.2]. The λg/λ0 curves for various slot-widths, i.e. s/h-ratio as a parameter, converge εr − 1 The normalized substrate toward h λ0 ≈ 1 4 thickness, h λ0 ≈ 1 4 εr − 1 , corresponds to the cutoff

frequency of the TE1 surface wave mode that occurs nearly at h = λg/4. Thus, the slot line model is useful for the subεr − 1 . It shows that effecstrate thickness h λ0 < 1 4 tive relative permittivity is εreff < (εr − 1)/2. The behavior of characteristic impedance with respect to h/λ0 is more complicated. It firstly increases, next remains flat, and finally decreases with frequency. However, its increase is obvious with an increase in the slot-width. Figure (10.12a and b), on the substrate with εr = 16, demonstrates that a slot line forms the complementary pair with a microstrip line. It is obvious that the microstrip characteristic impedance decreases with an increase in strip width (s = w), i.e. w/h-ratio, whereas for a slot line, it increases with an increase in the slot-width ratio, i.e. s/h. Thus, the slot line is a reciprocal structure of a microstrip. An increase in the slot-width results in the slot field confinement in the air medium that increases the characteristic impedance. Figure (10.12b) shows that the effective relative permittivity of a slot line decreases with s/h-ratio. However, as a complementary pair, the effective relative permittivity of a microstrip increases with the increase of s(=w)/h-ratio. The effective relative permittivity of a slot line is always less than that of a microstrip line for the same s/h-ratio and it increases with frequency. Kitazawa and Itoh have further compared, using the SDA, the εreff(f ) and the characteristic impedance of the slot line against the symmetrical CPW, on a substrate with εr = 12.8, h = 0.1 mm, tan δ = 0.0006, at 60 GHz [J.28].

377

10 Slot Line

4.0 – 6.4 GHz 5.4 GHz 2.5 – 8.0 GHz

εr = 16

11 10

Microstrip

εr = 16

80 GHz

90 8

Slot line Microstrip

50

Slot line

54 GHz

70 εreff

Z0 (Ω)

1.7 GHz

30 GHz

6

17 GHz

4

30 0.1

0.3

0.5

0.1

0.7 0.9 1.0

0.3

s/h - ratio

Slot line and microstrip as the complementary pairs. Source: Mariani et al. [J.2]. © IEEE.

Slot line

7.5

S = 0.04 mm

εreff

Zpv

70

6.5 CPW

W = 0.04 mm S = 0.08 mm

0

20 40 60 Frequency (GHz)

80

(a) Frequency-dependent Z0 of slot line and CPW. Figure 10.13

CPW

150

Slot line W (mm) CPW

0.02

0.02

100

0.10

CPW 0.40 εreff

3.5 0.0

Closed-form Models

A slot line of finite width conductors could be viewed as a CPS structure, supporting the quasi-static TEM mode. The quasi-static nature of a slot line is mentioned by Cohn and others [J.1, J.2, J.9, J.29]. So the method of moment (MOM) has been used to obtain the characteristic impedance of a slot line of finite width [J.29]. Likewise, the variational method is used to get the line parameters of the CPS [J.30]. The conformal mapping method, along with empirical expressions for the virtual heights, is also reported to compute the frequency-dependent effective relative permittivity and

4.5

CPW (W = 0.10 mm, S = 0.05 mm)

40

10.4

W (mm)

5.5

Quasistatic

εreff

Zpv Zpi Zvi

dielectric region that increases its effective relative permittivity and decreases its characteristic impedance. Boremann has obtained the frequency-dependent effective relative permittivity and characteristic impedance of the CBSL, along with the appearance of high-order mode [J.27].

0.10

60 50

0.9 1.0

(b) εreff of a slot line and microstrip.

Figure (10.13a) shows frequency-dependent characteristic impedance of CPW using three definitions – power–voltage (pv), power–current (pi), and voltage–current (vi). The power–voltage definition, applicable to a slot line also, shows a small increase in the Z0(f ) of the CPW with frequency. In the case of a slot line, the increase in the Z0(f ) is very significant, indicating the highly dispersive nature of a slot line. Figure (10.13b) further compares the effective relative permittivity and Z0 of a CPW and slot line for increasing slot-width. The characteristic impedance of CPW is always less than that of a slot line. As a matter of the fact, for a wide-slot (s = 0.4 mm) CPW, the characteristic impedance is one-half of the characteristic impedance of a slot line. Likewise, εreff of a slot line is always less than that of the CPW structure. Simon [J.13] has shown that in the case of the suspended slot line, the nearness of the top shield to the slot line decreases its effective relative permittivity and increases its characteristic impedance. The presence of the conductor backing increases the field concentration of a slot line in the

80

0.7

s/h - ratio

(a) Zo of a slot line and microstrip. Figure 10.12

0.5

Zpv

50

Ch. impedance (Zpv) (Ω)

120 110

Z0 (Ω)

378

Slot line

0.1 S (mm)

0 0.2

(b) εreff and Z0 of slot line and CPW.

Comparison of slot line against the CPW. Source: Kitazawa and Itoh [J.28]. © IEEE.

10.4 Closed-form Models

the characteristic impedance of three kinds of slot lines: the standard, sandwich, and composite substrate slot lines, shown in Fig (10.1) [J.9]. Krowne has given a set of closed-form expressions to compute both the frequency-dependent slowing-factor (λg/λ0) and the characteristic impedance of a slot line over the range 2.2 ≤ εr ≤ 20 [J.29]. Garg and Gupta have reported the curve-fitted expression to compute the slowingfactor (λg/λ0) and frequency-dependent characteristic impedance of a narrow gap slot line on high permittivity substrate, over the range 9.7 ≤ εr ≤ 20.0, 0.02 ≤ s/h ≤ 2.0, as 0 01 ≤ h λ0 ≤ 1 4 εr − 1 [J.31]. This model has about 2% accuracy against the results of Cohn’s waveguide model. Janaswamy and Schaubert [J.32, J.33] have provided another set of the closed-form expressions of a wide slotwidth slot line on the low permittivity substrate, over the range 2.2 ≤ εr ≤ 9.8, 0.006 ≤ h/λ0 ≤ 0.06 and 0.005 ≤ s/λ0 ≤ 2.0. This model also has about 2% accuracy against the results of SDA. The above-mentioned closed-form models do not account for the strip conductor thickness. Moreover, the low-frequency dispersion occurs in the slot line due to the finite conductivity of the strip conductor. Such lowfrequency dispersion is also not accounted for by these models. An integrated model of the slot line has also been reported, over the range 2.2 ≤ εr ≤20 [J.10, J.11, J.33, J.34]. This model is a combination of the models of Garg and Gupta, and Janaswamy and Schaubert. Further, the integrated model incorporates the effect of the conductor thickness on the slot line parameters. It computes the dielectric loss, as well as the conductor loss. Next, the integrated model helps to get the circuit model with R(f ), L(f ), C(f ), G(f ) parameters. These line parameters compute the low-frequency dispersion of a slot line, due to the finite conductivity of strip conductors. A summary of the closed-form models is given below.

10.4.1

Conformal Mapping Method

The characteristic impedance of the slot line structures, shown in Fig (10.1a–c), is computed by using the following result of the conformal mapping [J.9]: 60π K k0 Z0 = εreff K k0

10 4 1

In the above equation, K(k0) and K(k0 ) are the complete elliptic integrals of the first kind and their modulus k0 and complimentary modulus k0 are given by α0 1 + α0 πs α0 = tanh 2h0

k20 = 2

a,

2

k0 = 1 − k20

b

c 10 4 2

The h0 is the hypothetical “virtual height” below and above the slot, forming a region in which most of the EM-energy is confined. The virtual height h0 is obtained empirically for three cases of the slot line. Standard Slot Line

Figure (10.1a) shows a finite substrate thickness slot line. The virtual height is proportional to the substrate thickness h and inversely proportional to the relative permittivity εr of the substrate. An empirical expression for the frequencyindependent virtual height is h0 = h 1 +

133 εr + 2

10 4 3

The effective relative permittivity of a standard slot line is obtained from the following expression: εreff = 1 +

εr − 1 K k1 K k0 2 K k1 K k0

10 4 4

The complete elliptical integral of the 1st kind is computed by the approximate expression discussed in chapter 9, section (9.2). The modulus is given by α1 1 + α1 πs α1 = tanh 2h k21 = 2

2

k 1 = 1 − k21

a,

b

c 10 4 5

This method has accuracy within 2% for the computation of effective relative permittivity over the following range: substrate thickness h/λ0 ≤ 0.01, slot-width 0.02 ≤ s/h ≤ 1, and relative permittivity 2.22 ≤ εr ≤ 20. The frequency-dependent effective relative permittivity is obtained, using the above equations, by taking the virtual height h0 as a frequency-dependent parameter through the following empirical relations: h0 = h 1 +

0 0133 λ0 εr + 1 h

2

,

10 4 6

where λ0 is the free space wavelength and the slowingεeff f . Once εreff is obtained, factor is λg λ0 = 1 the static or frequency-dependent characteristic impedance is computed using equation (10.4.1). The model is valid for 9.7 ≤ εr ≤ 20, 0.02 ≤ s/h ≤ 1.0, and 0 01 ≤ h λ0 ≤ 0 25 εr − 1. Sandwich Slot Line

Figure (10.1b) shows the sandwich slot line structure. In this case, there are two frequency-dependent virtual heights: h01 below the slot and h02 above the slot. These are given by the following empirical relations:

379

380

10 Slot Line

h01 = h1 1 +

0 0133 λ0 εr1 + 2 h1

2

0 0133 εr2 + εr1 − εr2 h1 h2 + 2

h02 = h2 1 +

λ0 h2

b

The h1 and h2 are the thicknesses of the substrate and the superstrate. The frequency-dependent effective relative permittivity of the sandwich slot line is obtained as follows: εr1 − 1

K k1 K k2 + εr2 − 1 K k1 K k2

K k01 K k02 + K k01 K k02 Where, for i = 1, 2 k2i = 2

αi 1 + αi

2

a , k i = 1 − k2i

b , αi = tanh

πs 2hi

c

10 4 9 also for i = 1, 2 k20i = 2

αoi 1 + αoi

2

a , k oi = 1 − k2oi b , αoi = tanh

πs 2hoi

c

10 4 10

K k1 K k0 K k1 K k0

10 4 12

K k2 K k0 K k2 K k0

,

where ki and ki (i = 1,2) are given by equation (10.4.9). The k0 and k0 are computed from equation (10.4.2). However, for computing α0, equation (10.4.11) is used in equation (10.4.2c). The above expression has accuracy of 2%– 3% for the sandwich slot line against the results of the waveguide model [J.1].

10.4.2

10 4 8

−1

εr2 − εr1 2

εr2 − 1 + 2

2

10 4 7

εreff = 1 +

εreff = 1 +

a

Krowne Model

Krowne [J.35] has given the closed-form expressions to εreff f and compute the slowing-factor λg λ0 = 1 the characteristic impedance of a slot line shown in Fig (10.1a). The model has been tested for 9.6 ≤ εr ≤ 20. However, it is also suitable for the substrate of lower permittivity, such as εr = 2.22. For the wider slot-width, s/h ≥ 0.2, the error in the computation of characteristic impedance is within 4% and for the narrow slot-width, the error goes up to 14.5%. For the computation of λg/λ0, the error is within ±3.7% against the results given by Mariani et al. [J.2] for 2.22 ≤ εr ≤ 20, 0.02 ≤ s/h ≤2.0, and 0.015 ≤ h/λ0 ≤ 0.08.

The Composite Substrate Slot Line

Slowing-Factor λg/λ0

Figure (10.1c) shows the structure. The frequencydependent virtual height in this case is

The slowing-factor is computed by the following expression:

h0 = h2 1 +

0 0133 λ0 εr1 + εr2 − εr1 h1 H + 2 h1

2

λg = f 1 εr λ0

f2

H = h 1 + h2

h λ0

f4

s h

+ f5

s h 10 4 13

10 4 11 The effective relative permittivity for this case is computed as follows:

f 1 εr = 3 549εr − 0 56

s s + f3 h h

a,

f 2 s h = 0 5632 s h

The parameters f1 to f5 of the above equation are given as follows:

0 104 s h

0 266

f 3 s h = − 0 8777 s h 0 81 + 0 4233 s h − 0 2492 f 4 s h = − 1 269 × 10 − 2 ln 50 s h 1 7 + 0 0674 ln 50 s h + 0 2 f 5 s h = 1 906 × 10 − 3 ln 50 s h 2 9 − 7 203 × 10 − 3 ln 50 s h + 0 1223

b c d e

10 4 14

10.4 Closed-form Models

λ0 2 λg λg = λ0 A1 + B1 f 1 + C1 f 2 + D1 f 3 + E1 + F1 + G1

Characteristic Impedance

εreff f, t = 0 =

The characteristic impedance of a slot line is computed by the following expression: Z0 = g1 εr , s h, h λ0 g2 s h, h λ0

10 4 15

10 4 18

The parameters of the above equation are given as follows: g1 εr , s h, h λ0 = 11 εr T , where T = p1 p1 a1 a2 a3

s s s s

s h , h λ0

The set of parameters involved in the above expressions, in terms of the physical parameters of the slot line, have been obtained from the models of Garg-Gupta and Janaswamy and Schaubert. These are available in the Appendices I and II at the end of chapter 10 [J.10, J.11]. The combined model is valid over a wider range of parameters, as compared to individual models, i.e. over the range 2.22 ≤ εr ≤ 20, 0.02 ≤ w/h ≤ 1.0 (i.e. 2 × 10 − 4 ≤ w λo ≤

a

h, h λ0 = a1 s h h λ0 2 + a2 s h h λ0 + a3 s h h = − 30 21 ln s h − 46 03 h = 0 5073 ln s h + 3 358 s h + 6 442 h = − 2 013 × 10 − 2 ln s h − 0 1374 s h + 0 2365

b c d e

10 4 16 g2 s h, h λ0 = p2 s h

h λ0

2

+ p3 s h

h λ 0 + p4 s h

0 25 εr − 1 ), 0 01 ≤ h λ0 ≤ 0 25 εr − 1 . The combined model has an average accuracy of 2% against the full-wave results. Figure (10.14a and b) illustrates one such comparison, for three closed-form models, against the fullwave results [J.11, J.37]. The integrated model follows more closely the full-wave results. The model of Svacina for the characteristic impedance does not follow the full-wave results. At the second stage of development, the integrated model accounts for the effect of the conductor thickness on frequency-dependent effective relative permittivity and characteristic impedance. The strip conductor thickness reduces both the effective relative permittivity and the characteristic impedance of a slot line. However, its effect, in the integrated model, is accounted for in different ways. The frequency- and thickness-dependent effective relative permittivity, i.e. εreff(f, t), is computed as follows:

a

p2 s h = − 1 176 × 104 s h 0 502 − 6 311 × 103 s h − 162 7 b p3 s h = 900 5 s h 0 28 + 1262 s h − 123 8 c 0 46 + 30 96 d p4 s h = 1 637 ln 50 s h + 40 99 s h

10 4 17

10.4.3

Integrated Model

Characteristic impedance, Z0 (Ω)

Effective relative permittivity, √εeff

The integrated model of a slot line is a closed-form model with four stages of development [J.10, J.11, J.33, J.34]. At the first stage, it has combined the closed-form models of Garg-Gupta [J.31], and Janaswamy and Schaubert [J.32, J.36] to compute the frequency-dependent effective relative permittivity and characteristic impedance. At this stage, conductor thickness of strips and losses are not accounted for. The combined model is written in the following parametric form:

3.4 3.2 3.0 Kitazawa [J.37]

2.8

IM

2.6

Krowne [J.35] Svacina [J.9]

2.4 2.2

2

4

6

8 10 12 14 Frequency,f (GHz)

16

(a) Frequency-dependent εeff (f). Figure 10.14

18

20

a

Z0 f, t = 0 = A2 + B2 g1 + C2 g2 + D2 g3 + E2 + F2 b

80 75 70 65

Kitazawa [J.37] IM

60

Krowne [J.35] Svacina [J.9]

55 50

2

4

6

8 10 12 14 Frequency,f (GHz)

16

18

20

(b) Frequency-dependent Z0 (f).

Comparison of three models of slot line, εr=20, s/h=0.5, t=0, against the full-wave results. Source: Majumdar and Verma [J.11]. © Springer.

381

382

10 Slot Line

εreff f, t = 0 − εreff f, t =

εr − 1 t h t , for 4 × 10 − 5 ≤ < 6 67 × 10 − 4 46 λ0 s h εr − 1 t h t − 46 λ s h 0

εreff f, t = 0 −

a

p0

, for

6 67 × 10 − 4

where, p0 = 0 0006f 2 − 0 0369f + 0 7714

b

In the above expression, the frequency-dependent effective relative permittivity εreff(f, t = 0) is computed from equation (10.4.18). To compute the frequency- and thickness-dependent characteristic impedance, i.e. Z0(f, t), reduced size of the equivalent slot gap seq = s − Δs is used, in place of the

t 1 1+ × ln 2π εr Δs =

10 872 t λ0

q0

1 π s h +11

+

t 1 1+ 2π cosh εr − 1

× ln

t 1 + ln 4 − 0 5 ln π

t λ0

10 4 19

t ≤ < 3 3 × 10 − 3 λ0

2

physical gap s in equation (10.4.18a). The same process can be used with the models of Krowne and also Svacina discussed it previously. The incremental reduction in the slot-gap can be computed from the following empirical relations:

for 4 × 10 − 5 ≤

,

10 872 t λ0 coth2

q0

+

t πs

6 517 s h

2

,

t < 3 6 × 10 − 4 λ0

, for 3 6 × 10 − 4 ≤

for 6 × 10 − 4 ≤

t < 6 × 10 − 4 λ0

a

t ≤ 3 33 × 10 − 3 λ0

2, for f < 18 GHz where, q0 =

b

0 092 f 0 4184 , f ≥ 18 GHz

10 4 20

Figure (10.15a–d) compares results of the integrated model, for the frequency- and thickness-dependent effective relative permittivity and the characteristic impedance of a slot line, against the full-wave results from the EMsimulators (HFSS, Sonnet). The HFSS excites a slot line through the wave-port (WP). However, Fig (10.15a and c) demonstrates that the results of HFSS (WP) do not follow the full-wave results of the SDA [J.37]. So, the slot line is considered as a limiting case of the CPW with a central strip conductor w 0; i. e. w = 0.1 μm for minimum slot-width s = 10 μm. The HFSS (CPW) provides a correct result for the slot line [J.10]. Normally, the results of the integrated model are in between the results of two EM-Simulators. At the third stage of the development of the integrated model, computations of the dielectric and conductor losses are incorporated in the integrated model. These are due to the imperfect dielectric substrate and imperfect conductors. At the mm-wave range, the radiation loss and leakage loss through surface wave mode also become important [J.38]. However, it is not included in the integrated model. The dielectric loss of a slot line is computed using the following expression, obtained in chapter 8:

αd =

π εr εreff f, t − 1 tan δ Np m λ0 εreff f, t εr − 1

10 4 21

The εreff(f, t) is computed from equation (10.4.19). Figure (10.16a) compares the dielectric loss of a slot line between 2 GHz and 30 GHz, computed by the integrated model against two full-wave results [J.38, J.39]. The average and maximum deviations of the integrated model are 0.011 Np/m and 0.041 Np/m, respectively. The integrated model computes the conductor loss of a slot line using the perturbation method with the concept of stopping distance Δ. The method applies to the microstrip and the CPW also. It is discussed in chapter 9. A slot line could be treated as a limiting case of the CPW structure by reducing the width of its central conductor to zero. Under this limiting case, the conductor loss of a slot line is computed from the following expression [J.10]: αc ≈

Rsm ln 16 Z0 f K2 k0 s

2s +1 Δ

s+Δ s−Δ

Np m, 10 4 22

where modulus k0 is given by equation (10.4.2).

10.4 Closed-form Models

5 HFSS (WP),

Effective relative permittivity εreff

Effective relative permittivity εreff

4.5 HFSS (CPW)

4 Kitazawa [J.37],

3.5

Integrated model

3 2.5 2

4

5

15 10 Frequency (GHz)

3 f = 2 GHz

2.5 2

20

70 65 60

Kitazawa [J.37],

HFSS (CPW) Integrated model

30

f = 10 GHz

100

40

50

60

70

Integrated model

80

90

HFSS SONNET

95 90

f = 20 GHz

85 80

f = 2 GHz

75 70

0 5 10 15 20 (c) Frequency-dependent Z0 (f, t) at t = 20 μm, εr = 20, s/h = 0.5.

Figure 10.15

20

(b) Thickness-dependent εreff (f, t),at 2 GHz, 10GHz, 20 GHz; εr = 20,s/h = 0.5. Characteristics impedance Z0 (Ohms)

Characteristics impedance Z0 (Ohms)

10

105

75

0

10

20

30 40 50 60 70 Conductor thickness, t (μm)

80

90

(d) Thickness-dependent Z0 (f, t) at 2 GHz, 10 GHz, 20 GHz; εr = 9.8, s/h = 0.5.

Comparison of the integrated model against HFSS and Sonnet. Source: Majumdar and Verma [J.10]. © IET.

The characteristic impedance Z0(f ) is computed using equation (10.4.18b). The surface impedance Rsm of the strip conductor of thickness t and conductivity σc in S/m is computed from the following expression: Rsm = ωμc t Im

cot kc t + csc kc t kc t

kc = ω μ0 ε0 1 − j

σc ωε0

a

1 2

b 10 4 23

Finally, the stopping distance for a slot line is computed from the following empirical relation: y = T 1 eT 2 x a , where, T1 = 12 093e − 25 101w 4 T2 = − 50w + 30w 3 + 30w 2 − 0 691w + 1 0109 where, y

0

Integrated model

Conductor thickness, t (μm)

80

HFSS (WP),

SONNET

HFSS

(a) Frequency-dependent √εreff (f,t) at t =10 μm , εr = 20, s/h = 0.5.

55

f = 10 GHz

3.5

1.5 0

f = 20 GHz

4.5

t Δ and x

b c

t 2δs ; 0 816 < x < 7 3 d 10 4 24

In the above equation, the slot-width s is in mm.

Wheeler’s incremental inductance rule, discussed in chapter 8, can also be used to compute the conductor loss of a slot line [J.11]. However, it does not apply to the conductor thickness near and less than the skin-depth. Figure (10.16b) compares the conductor loss computations, using the integrated model and Wheeler’s model, against the full-wave results [J.39]. The results of two EM-simulators are also used in comparison. Results of the integrated model, full-wave method, and Sonnet have average and maximum deviations (4.91%, 7.37%), (15.1%, 27.1%), and (2.07%, 4.49%), respectively, against the results of HFSS. At the final stage of the integrated model, the secondary line parameters are used to obtain the R(f ), L(f ), C(f ), G(f ) parameters of a circuit model of the slot line. The process is discussed in section (9.8) of chapter 9 for a CPW. The circuit model helps us to get low-frequency dispersion for both the effective relative permittivity and characteristic impedance. It also provides the imaginary part of the complex characteristic impedance. The accuracy of the circuit model is

383

10 Slot Line

0.2 Rozzi [J.38] Kitazawa [J.39]

0.16

Conductor loss αc (Np/m)

Dielectric loss,αd Np/m

W = 0.25 mm W = 0.5 mm 0.7mm

Integrated model W = 0.5 mm

0.12

W = 0.25 mm

W = 0.7 mm

0.08 0.04 0

2

6

10

14 18 22 Frequency (GHz)

26

30

(a) Comparison of computed dielectric loss for εr = 9.8, t = 6 μm, h = 0.635 mm[J.11].

Integrate model HFSS

50

Kitazawa [J.39],

1.25

Integrated model,

1

f = 60 GHz

0.75

f = 30 GHz

Circuit model SONNET α

40 30 20 εreff

10 0 0.01

1 0.1 Frequency (GHz)

HFSS

0.5 f = 10 GHz

0.25 0

f = 2 GHz

0

9 8 7 6 5 4 3 2 1 0 10

10

20 30 40 Conductor thickness t (μm)

50

120 100 Re(Z0)

80 60

Integrate model

40

Circuit model SONNET

HFSS

20 Img(Z0)

0 –20 –40 0.01

0.1

1

10

Frequency (GHz)

(a) εreff (f, t) and total attenuation. Figure 10.17

Sonnet, Wheeler

(b) Comparison of computed conductor loss for εr = 9.8, t = 6 μm, h = 0.635 mm[J.10].

Characteristic impedance, Z0 (Ohms)

70 60

1.5

Comparison of the integrated model, for loss computation, against full-wave methods. Source: Majumdar and Verma [J.11] © Springer. Majumdar and Verma [J.10]. © IET.

Attenuation constant, α (Np/cm)

Figure 10.16

Effective relative permittivity

384

(b) ReZ0 (f, t) and total Img Z0 (f, t).

Low-frequency dispersion in a slot line (εr = 12.9, t = 6 μm, s/h = 0.5).

higher for computation of line parameters, including losses, compared to the accuracy of individual components of the integrated model. Figure 10.17a and b compares the results of the slot line parameters obtained from both the integrated model and circuit model against two EM-Simulators.

B.3 Di Paolo, F.: Network, and Devices Using Planar

Transmission Lines, CRC Press, New York, 2000. B.4 Bhat, B.; Koul, S.K.: Analysis Design and Applications of B.5 B.6

References B.7

Books B.1 Hoffmann, R.: Microwave Integrated Circuit Handbook,

Artech House, Boston, 1985. B.2 Gupta, K.C.; Garg, R.; Bahl, I.; Bhartia, P.: Microstrip Lines and Slot Lines, 2nd Edition, Artech House, Boston, 1996.

B.8 B.9

Fin-Lines, Artech House, Boston, 1987. Rozzi, T.; Mongiardo, M.: Open Electromagnetic Waveguides, IEE Press, UK, 1997. Mirshekar-Syahkal, D.: Spectral Domain Method for Microwave Integrated Circuits, Research Studies Press Ltd., John Wiley & Sons, New York., 1990. Itoh, T. (Editor): Numerical Techniques for Microwave and Millimetre-Wave Passive Structures, John Wiley & Sons, New York, 1989. Jordan, E. C.; Balmain, K.G.: Electromagnetic Wave and Radiating System, Prentice-Hall India, New Delhi, 1989. Ramo, S., Whinnery, J. R., Van Duzer, T.: Fields, and Waves in Communication Electronics, 3rd Edition, John Wiley & Sons, Singapore, 1994.

References

B.10 Balanis, C.A.: Advanced Engineering Electromagnetics,

J.12 Cohn, S.B. Sandwich slot line, IEEE Trans. Microwave

John Wiley & Sons, New York, 1989. B.11 Collin, R.E.: Field Theory of Guided Waves, IEEE Press,

Theory Tech., Vol. MTT- 19, No. 9, pp. 773–774, Sept. 1971. J.13 Simons, R.: Suspended slot line using double-layer

New York, 1991. B.12 Abramowitz, M.; Stegun, I.A.: Handbook of Mathematical

Functions with Formula, Graphs, and Mathematical Tabless, Martino Publishing, Eastford, CT, 2014. B.13 Marcuvitz, N.: Waveguide Handbook, M.I.T. Radiation Laboratory Series, Vol. 10, pp. 218–219, McGraw-Hill, New York, 1951.

J.14

J.15

J.16

Journals J.1 Cohn, S.B.: Slot line on a dielectric substrate, IEEE Trans.

J.2

J.3

J.4

J.5

J.6

J.7

J.8

J.9

J.10

J.11

Microwave Theory Tech., Vol. MTT- 17, No. 10, pp. 768– 778, Oct. 1969. Mariani, E.O.; Heinzman, C.P.; Agrios, J.P.; Cohn, S.B.: Slot line characteristics, IEEE Trans. Microwave Theory Tech., Vol. MTT- 17, No. 12, pp. 1091–1096, Dec. 1969. Hunton, J.K.: Novel contributions to microwave circuit design, IEEE, MTT-S, Int. Microwave Symp. Dig., pp. 753–755, 1989. Donn, C.: A miniature, circularly polarized cross-slot line antenna, IEEE AP-S Int. Symp. Dig., (San Jose, CA), pp. 1328–1331, June 1989. Povinelli, M.J.; Johnson, J.A.: Design and performance of wideband dual-polarized stripline notch arrays, IEEE AP-S Int. Symp. Dig., (Syracuse, NY), pp. 200–203, June 1988. Robinson, G.H.; Allen, J.I.: Slot line application to miniature ferrite devices, IEEE Trans. Microwave Theory Tech., Vol. Vol. MTT- 17, No. 12, pp. 1097–1101, Dec. 1969. Samardzija, N.; Itoh, T.: Double-layered slot line for millimeter-wave integrated circuits, IEEE Trans. Microwave Theory Tech., Vol. MTT- 24, No. 11, pp. 827– 831, Nov. 1976. Abdel-Moniem, A.; El-Sherbiny: Millimeter-wave performance of shielded slot-lines, IEEE Trans. Microwave Theory Tech., Vol. MTT- 30, No. 5, pp. 750–756, May 1982. Svacina, J.: Dispersion characteristics of multilayered slot lines—a simple approach, IEEE Trans. Microwave Theory Tech., Vol. MTT- 20, No. 2, pp. 1826–1829, Sept. 1999. Majumdar, P.; Verma, A.K.: Integrated closed-form model and circuit model of lossy slot line, IET Microw. Antennas Propag Vol. 5, No. 14, pp. 1763–1772, 2011. Majumdar, P.; Verma, A.K.: Modified closed-form dispersion and loss models of slot-line with conductor thickness, J Infrared Milli Terahz Waves, Vol. 31, pp. 271– 287, 2010.

J.17

J.18

J.19

J.20

J.21

J.22

J.23

J.24

J.25

J.26

dielectric, IEEE Trans. Microwave Theory Tech., Vol. MTT- 29, No. 10, pp. 1102–1107, Oct. 1981. Han, K.J.; Kim, J.P.: Guide wavelength estimation of sandwich-type double-layer slot line, Microwave Opt. Technol. Lett., Vol. 43, No. 3, pp. 261–264, Nov. 2004. Coetzee, J.C.; Malherbe, J.A.G.: Characteristic impedance for double-sided slot lines, Microwave Opt Technol Lett., Vol. 3, No. 3, pp. 85–88, Mar. 1990. Coetzee, J.C.; Malherbe, J.A.G.: Dispersion characteristics for double-sided slot lines, Electron. Lett., Vol. 25, No. 20, pp. 1383–1385, Sept. 1989. Janaswamy, R.: Even-mode characteristics of the bilateral slot line, IEEE Trans. Microwave Theory Tech., Vol. MTT- 38, No. 6, pp. 760–765, June 1990. Simons, R.N.; Taub, S.; Lee, R.Q.; Young, P.G.: Microwave characterization of slot line and coplanar silicon for a slot antenna feed network, Microwave Opt Technol. Lett., Vol. 7, No. 11, pp. 489–494, Aug. 1994. Shigesawa, H.; Tsuji, M.; Oliner, A.A.: Conductor backed slot line and coplanar waveguide: dangers and full-wave analysis, IEEE MTT-S Int. Microwave Symp. Dig., vol. 1, pp. 199–202, June 1988. Zehentner, J.; Machac, J.; Mrkvica, J: Novel selected modes on the conductor-backed slot line, IEEE MTT-S Int. Microwave Symp. Dig., pp. 961–964, Seattle, June 2002. Zehentner, J.; Machac, J.; Mrkvica, J.; Tuzi, C.: The inverted conductor-backed slot line – a challenge to the antenna and circuit design, Proc. 33rd European Microwave Conf., (Munich), pp. 73–76, 2003. Zehentner, J.; Machac, J.; Mrkvica, J.; Tuzi, C.: Modes on the standard and inverted conductor-backed slot line, IEEE MTT-S Int. Microwave Symp. Dig., vol. 2, pp. 677–680, June 2003. Zehentner, J.; Machac, J.; Mrkvica, J.: Even and odd modes on a conductor backed slot line, Proc. of 32nd European Microwave Conf., vol. 2, pp. 609–612, Sept. 2002. Das, N.K.; Pozar, D.M.: Full-wave spectral-domain computation of material, radiation, and guided wave losses in infinite multilayered printed transmission lines, IEEE Trans. Microwave Theory Tech., Vol. MTT- 39, No.1, pp. 54–63, Jan. 1991. Simons, R.: Suspended coupled slot line using double-layer dielectric, IEEE Trans. Microwave Theory Tech., Vol. MTT29, No. 2, pp. 162–165, Feb. 1981. Simons, R.: Suspended broadside- coupled slot line with overlay, IEEE Trans. Microwave Theory Tech., Vol. MTT- 30, No. 1, pp. 76–81, Jan. 1982.

385

386

10 Slot Line

J.27 Boremann, J.: A scattering-type transverse resonance

J.28

J.29

J.30

J.31

J.32

J.33

J.34 Majumdar, P.; Verma, A.K.: Conductor thickness based

formulation and its application to open, conductor-backed and shielded slot line (M) MIC structures, IEEE MTT-S Int. Microwave Symp. Digest, pp. 695–698, June 1991. Kitazawa, T; Itoh, T.: Propagation characteristics of co-planar type transmission lines with lossy media, IEEE Trans. Microwave Theory Tech., Vol. MTT- 39, No. 10, pp. 1694–1700, Oct. 1991. Lee, J.J.: Slot line impedance, IEEE Trans. Microwave Theory Tech., Vol. MTT- 39, No. 4, pp. 666–672, April 1991. Yamashita, E.; Yamazaki, S.: Parallel-strip line embedded in or printed on a dielectric sheet, IEEE Trans. Microwave Theory Tech., Vol. MTT- 16, No. 11, pp. 792–793, Nov. 1968. Garg, R.; Gupta, K.C.: Expression for wavelength and impedance of slot line, IEEE Trans. Microwave Theory Tech., Vol. MTT- 24, No. 8, pp. 532, Aug. 1976. Janaswamy, R.; Schaubert, D.H.: Characteristic impedance of a wide slot line on low-permittivity substrates, IEEE Trans. Microwave Theory Tech., Vol. MTT- 34, No. 8, pp. 900–902, Aug. 1986. Majumdar, P; Verma, A K.: Propagation characteristics of slot – line in lossy media with conductor thickness, Proc. Int. Symp. Antenna & Propagat. ISAP, Taiwan, Oct. 2008.

J.35

J.36

J.37

J.38

J.39

propagation characteristics and losses of slot-line, Proc. Asia-Pacific Microwave Conf., APMC, Hong-Kong, China, Dec. 2008. Krowne, C.M.: Approximations to hybrid mode slot line behavior, Electron. Lett., Vol. 14, No. 8, pp. 258–259, April 1978. Janaswamy, R.; Schaubert, D.H.: Dispersion characteristics for wide slot lines on low-permittivity substrates, IEEE Trans. Microwave Theory Tech., Vol. MTT- 33, No. 8, pp. 723–726, Aug. 1985. Kitazawa, T.; Fujiki, Y.; Hayashi, Y.; Suzuki, M.: Slot line with a thick metal coating, IEEE Trans. Microwave Theory Tech., Vol. MTT- 21, No. 9, pp. 580–582, Sept. 1973. Rozzi, T.; Moglie, F.; Morini, A.; Marchionna, E.; Politi, M.: Hybrid modes, substrate leakage, and losses of slotline at millimeter-wave frequencies, IEEE Trans. Microwave Theory Tech., Vol. MTT- 38, No. 8, pp. 1069–1078, Aug. 1990. Kitazawa, T.; Kuo, C.W.; Kong, K.S.; Itoh, T.: Planar transmission lines with finitely thick conductors and lossy substrates, IEEE MTT-S Int. Microwave Symp. Digest, pp. 769–772, June, 1991.

Appendix – I The coefficients A1 – G1 and parameters f1 – f3 for equation (8.4.18a). 1) 2.22 ≤ εr ≤ 3.8 A) 0.0015 ≤ w/λ0 ≤ 0.075 f 1 = ln εr C1 =

i,

f2 =

w h

ii ,

6 3εr 0 945 238 64 + 100w h

h λ0

f 3 = ln

iii , A1 = 1 045

vi , D1 = 0 0599 −

8 3695 100εr

iv , B1 = − 0 365

vii , E1 = 0 viii , F1 = 0

v ix , G1 = 0

B) 0.075 ≤ w/λ0 ≤ 1.0 f 1 = ln εr

i , f2 =

D1 = 0 0561 −

w h

h λ0

ii , f 3 = ln

0 1234 εr

vii , E1 =

iii , A1 = 1 194

iv , B1 = − 0 24

v , C1 = 0

0 621εr 0 835 w λ0 0 48 viii , F1 = 0 ix , G1 = 0 1 344 + w h

x

2) 3.8 ≤ εr ≤ 9.8 A) 0.0015 ≤ w/λ0 ≤ 0.075 f 1 = ln εr

i,

f2 =

w ii , h

1 03684 × 10 − 3 εr C1 = w h + 0 435 F1 = 0

ix , G1 = 0

1 2

f 3 = ln

h λ0

vi , D1 = 0 046 − x

iii , A1 = 0 9217

iv , B1 = − 0 277

0 0365 ε2r

w λ0 9 06 − 100w λ0

v ,

vii , E1 = 0 viii ,

vi ,

x

Appendix – II

B) 0.075 ≤ w/λ0 ≤ 1.0 f 1 = ln εr

i,

f2 =

w h

ii ,

f 3 = ln

h λ0

h λ0

D1 = 0 139 1 + 0 52εr ln 14 7 − εr

iii , A1 = 1 05

iv , B1 = 0

vii ,

E1 = 1 411 × 10 − 2 εr − 1 421 ln w h − 2 012 1 − 0 146εr

viii ,

F1 = − 0 04 + 0 04063

x

w λ0 εr

v , C1 = 0 vi ,

w λ0

ix , G1 = 0 111

3) 9.7 ≤ εr ≤ 20 A) 0.02 ≤ w/h ≤ 0.2 f 1 = log 10 εr

i,

f2 =

w h

ii ,

B1 = − 0 448 v , C1 = − 0 38

h λ0

f 3 = log 10

iii ,

vi , D1 = − 0 29

A1 = 0 829

w + 0 047 h

iv ,

vii , E1 = 0 viii ,

F1 = 0 ix , G1 = 0 x B) 0.2 ≤ w/h ≤ 1 w h

f 1 = log 10 εr

i , f2 =

C1 = − 0 077

vi , D1 = − 0 094

E1 =

− 0 0022

ii ,

f 3 = log 10

w + 0 0032 log 10 h

h λ0

iii , A1 = 0 745

iv , B1 = − 0 483

w + 0 121 vii , h h + 0 0064 εr viii , F1 = 0 ix , λ0

v ,

G1 = 0 x

Appendix – II The coefficients A2 – F2 and parameters g1 – g3 for equation (8.4.18b). 1) 2.22 ≤ εr ≤ 3.8 A) 0.0015 ≤ w/λ0 ≤ 0.075 w h

g1 = ln εr

i,

B2 = 133 5

w λ0 − 0 1423 εr − 131 1

g2 =

C2 = 12 94 − 0 6378εr +

ii ,

h λ0

g3 = ln

εr − 2 06 + 0 85 w h

12 48 εr − 2 06 + 0 85 w h

h λ0 + 307 39

iv ,

2 246w h

h λ0 +

D2 = 2 89w h − εr 0 1385w h + 0 0309 ln εr E2 = 134 77

iii , A2 = 60

2

2

v ,

vi ,

vii ,

w λ0 viii , F2 = 3 69 sin

εr − 2 22 π 2 36

ix

B) 0.075 ≤ w/λ0 ≤ 1.0 g1 = εr

i , g2 =

B2 = 10 34εr +

w h w λ0

ii , g3 =

iii , A2 = 166 5

2

13 34εr − 45 56 −

1 2 32εr − 0 56 1+ w h w h w 2 h D2 = − 1247 4 + 39 23 h λ0

C2 =

h λ0

h λ0

iv ,

684 48εr + 1848 − 37 224

v ,

1 2

32 5 − 6 67εr 100h λ0

2

−1

vii , E2 = 0 viii , F2 = 0

2 87ε2r − 9 08εr + 15 675 ix

vi ,

387

388

10 Slot Line

2) 3.8 ≤ εr ≤ 9.8 A) 0.0015 ≤ w/λ0 ≤ 0.075 g1 = εr

i , g2 =

w h

B2 = − 2 15 + 31 37 w λ0

06

06

iii , A2 = 73 6 iv ,

+ 0 753 h λ0

36 23 ε2r + 41 − 225 w h + 0 876εr − 2

C2 = 4 9789 + 2 3485εr + E2 = 638 9 w λ0

h λ0

ii , g3 = ln

viii ,

w λ0

v ,

vi , D2 = 0 51εr

w w + 1 0812 h h

F2 = 0 ix

B) 0.075 ≤ w/λ0 ≤ 1.0 g1 = ln εr i , g2 =

w h

w λ0

B2 = 7 892 − 3 6144εr + C2 = w h

p−1

h iii , A2 = 100 479 iv , λ0 w + 0 66 72 471 − 33 1908εr λ0

ii , g3 =

50 tan − 1 2εr − 0 8 ln 100h λ0 +

v

100h λ0

2

+ 1 vi

where, p = 1 11 + 0 132 εr − 27 7 100h λ0 + 5 , w w + 0 66 1421 − 650 8εr vii , D2 = 154 74 − 70 872εr + λ0 λ0 w w E2 = + 0 66 − 186 151 + 85 25εr + 5 543εr viii , F2 = 0 λ0 λ0

ix

3) 9.7 ≤ εr ≤ 20 A) 0.02 ≤ w/h ≤ 0.2 g1 = log 10 εr

i , g2 =

w h

B2 = − 131 16 − 19 58 log 10

h λ0

iii , A2 = 169 475

iv ,

w − 11 78 log 10 εr − 48 33 log 10 εr h

h − 388 48 log 10 εr + 863 14 v , λ0 01 h − 100h λ0 2 1 07 log 10 εr + 1 44 C2 = − 137 14 + 2 + log 10 εr 60 23 + 691 4 λ w h 0 h + log 10 εr 2 − 39 41 log 10 εr − 1298 9 + 95 02 vi , λ0 h w − 250 8 + 3283 2 vii , E2 = 44 28 log 10 w h D2 = 1100 viii , F2 = 0 ix λ0 h +

h λ0

ii , g3 =

− 3200

B) 0.2 ≤ w/h ≤ 1 w h ii , g3 = iii , A2 = 93 431 iv , h λ0 w w 2 + 19 83 log 10 εr + 116 01 log 10 εr B2 = − 62 335 − 365 69 h h h w 2 w 3 h + 321 5 − 13 841 − 2300 − 5 75 log 10 εr 2 − 128 81 λ0 h h λ0 g1 = log 10 εr

i , g2 =

C2 = 243 762 + 13 35 − 55 01 log 10 εr

w h − 1556 5 h λ0 3

20700 log 10 εr

+ 241 31 log 10 εr h λ0

2

vi ,

2

+ 3 484

w h

2

v ,

2

+ 7900

h λ0

2

vii ,

Appendix – II

w w 2 w − 230 log 10 εr 2 − 331 8 log 10 εr − 2135 32 h h h w 2 w 2 − 587 88 log 10 εr 2 + 1093 76 log 10 εr vii , h h w 3 w 3 w 2 E2 = − 4 174 log 10 εr 3 + 13 932 log 10 εr 2 − 29 858 log 10 εr viii , F2 = 0 ix h h h

D2 = 307 5 − 1500

h λ0

+ 5087 5 log 10 εr

2

389

391

11 Coupled Transmission Lines Basic Characteristics

Introduction The coupled transmission line structures are important for the design of several microwave components such as directional couplers, filters, phase shifters, delay lines, 3dB-hybrids, matching networks, etc. The coupling between lines is a desirable feature of the coupled lines. However, the multiple pairs of telephone lines, multiconductor strips on the PCB boards and IC chips have an undesired coupling, called the cross-talk. It leads to the distortion and degradation in the signals and malfunctioning of the digital circuits and networks. Such undesirable couplings should be either avoided or at least controlled to the minimum acceptable limits. This chapter covers the basic theoretical backgrounds needed for the design of the coupled lines [B.1–B.25]. The theoretical formulations are used to analyze the planar coupled transmission line structures in chapter 12. Objectives

•• •• • • •

To review some coupled line structures. To introduce basic concepts of coupled lines. To develop the circuit models of coupling. To present even–odd mode analysis of symmetrical coupled lines. To discuss capacitance and inductance matrices of multiconductor lines. To obtain the wave equations for asymmetrical and symmetrical coupled lines. To discuss the C and π-modes of asymmetrical coupled lines.

11.1

Some Coupled Line Structures

The proximity of two transmission lines with a common ground conductor, shown in Fig (11.1), provides an electromagnetic coupling between them. It results in

the transfer of RF power from one line to another line. The coupling of the RF power is viewed as the coupling of the electric and magnetic fields of the TEM or nonTEM mode, supported by the line structures. The coupling is also viewed through the mutual capacitance and mutual inductance between the coupled lines. Thus, the coupling is described by the capacitive (electric) coupling and the inductive (magnetic) coupling. The coupling can also exist on the multiconductor lines, and is described through Maxwell’s capacitance matrix and inductance matrix [B.1–B.3, J.1–J.5]. The planar transmission lines – the strip lines, microstrip lines, CPW structures, etc. – are used to form the coupled transmission lines. These lines are either the edge coupled lines as shown in Fig (11.2a–c), or the broadside coupled lines, shown in Fig (11.3a–c). In the case of the coupled strip lines, both upper and lower dielectric layers are identical. Normally, the edge-coupled lines have a weak coupling (15–25 dB), or a medium coupling (6–15 dB), whereas the broadside coupled lines provide a strong coupling (3–6 dB) [B.4–B.15]. The coupling between two coupled transmission lines, mentioned above, is a continuous coupling throughout the proximity range of two lines. There is another group of coupling known as the discrete coupling between two lines, or between two waveguides. It is achieved through the coupling slots, known as the aperture coupling between two transmission lines or between two waveguide structures. Fig (11.4a–c) shows three cases of the aperture coupling: the broadside coupling of two waveguides [B.10, B.11], the aperture coupled two microstrip lines located on the opposite sides of a common ground plane [J.6], and the microstrip line-fed image dielectric waveguide [J.7]. The discrete coupling is also achieved by interconnecting two transmission lines at discrete locations, using the stubs [B.5–B.10, B.13, B.14]. Figure (11.5a and b) show the stub coupled transmission lines (hybrids) for the waveguide and microstrip structures. Presently, we are only interested in the

Introduction to Modern Planar Transmission Lines: Physical, Analytical, and Circuit Models Approach, First Edition. Anand K. Verma. © 2021 John Wiley & Sons, Inc. Published 2021 by John Wiley & Sons, Inc.

392

11 Coupled Transmission Lines

continuously coupled planar lines: strip lines, microstrips, and CPW. However, the reader can consult available textbooks and published works for the aperture coupling, and the stub coupling forming the hybrids. The planar coupled line structures, shown in Figs (11.2 and 11.3), could also be fabricated in the multilayered media: dielectric, ferrite, semiconducting, and other media. The coupling between the lines is strongly influenced by the medium and dimensions of the strip conductors. In the case of the coupled strip lines, the medium is homogeneous. However, two strip widths could be different. Such coupled line structure is called the asymmetrical coupled lines. The coupled lines with identical widths form the symmetrical coupled lines. This chapter is concerned with only the uniformly coupled transmission lines, giving the uniform coupling between the lines. The nonuniform transmission lines, say exponential coupled

Line #1 Line #1

Line #2

Common ground conductor

Common

Figure 11.1

Line #2

Ground

Two-wire transmission lines.

microstrips, could be used to get the nonuniform coupling [J.8, J.9]. Such a structure provides stronger coupling and wider bandwidth. The microstrip and CPW coupled lines are the inhomogeneous media coupled lines, as the medium is inhomogeneous. However, they are either symmetrical or asymmetrical coupled lines. The symmetrical edge coupled microstrip structure is more popular for the design of several microwave components [B.4–B.10, B.15]. The asymmetrical coupled microstrip structure is the less used one. However, they provide both the coupling and impedance transformation simultaneously [B.16, B.17, J.10–J.13]. The planar coupled transmission lines are analyzed by three methods:

•• •

Even–odd mode analysis. Coupled transmission line equations. Coupled-mode analysis.

The first method is simple and direct one; however, applicable to only symmetrical lines [B.4, B.6–B.10, B.15]. The method decouples the symmetrical coupled transmission lines into two independent transmission lines, supporting two independent modes called the even-mode and the odd-mode. These individual modes are the normal modes of the coupled lines. The normal modes of the coupled lines are further discussed in

εr W1 Line #1

W2 Line #2

W1 Line #1

εr

(a) Coupled strip lines in a homogeneous medium.

Ground conductor W1 Line #1

W2 Line #2

W2 Line #2 εr

(b) Coupled microstrip lines in inhomogeneous medium.

Ground conductor εr

(c) Coupled CPW lines in an inhomogeneous medium. Figure 11.2

Edge coupled planar transmission lines.

11.1 Some Coupled Line Structures

εr 3 = 1

Line #1

Line #2 h2 εr 2

h Line #2 ε

Line #1

h1 εr 1

r

Conducting shield (a) Coupled strip line in a homogeneous medium.

(b) Coupled microstrip line in an inhomogeneous medium.

εr = 1 Line #2 εr Line #1 h

εr

(c) Coupled CPW line in an inhomogeneous medium. Figure 11.3 Broadside coupled planar transmission lines.

Coupling slots mon in common ground plane

Coupling slots

Waveguide #2

εr h

Line #2

εr h

Waveguide #1

Line #1 (b) Aperture coupled microstrip.

(a) Aperture coupled waveguides.

Coupling slots in ground plane

Dielectric waveguide

Microstrip

(c) Aperture coupled dielectric waveguide and microstrip. Figure 11.4 Apertures (slot) coupled transmission line structures. Source: Collin [B.10, B.11].

393

Waveguide #1

11 Coupled Transmission Lines

Substrate Line #1

Stub

Stub

Stub

Stub

Waveguide #2

394

Line #2

(a) Stubs coupled waveguide. Figure 11.5

(b) Stubs coupled microstrip/strip lines.

Stubs coupled transmission line structures.

Port #3

Port #4

Port #3

Port #4

Port #1

Port #2

Port #1

Port #2

(a) Co directional coupler. Figure 11.6

(b) Counter-directional coupler.

Schematic diagrams of couplers.

subsection (11.5.2). The second method applies to both the symmetrical and asymmetrical coupled lines, supporting the TEM and quasi-TEM modes [B.1–B.4, B.15 J.10, J.14]. The C and π-modes are the normal modes of the asymmetrical coupled lines. The coupled transmission line equations are obtained through the equivalent coupled lumped circuit model. The third method is more general and helps to understand the nature of coupling [B.18, J.1, J.9, J.15]. However, the method is more involved. The last two methods are also applicable to the multiconductor coupled line structures. The present chapter discusses only the first two methods applicable to a pair of coupled transmission lines.

11.2 Basic Concepts of Coupled Transmission Lines The basic nature of coupling and the basic definition of parameters to characterize the coupled transmission lines are discussed in this section. The next section presents the circuit model of the coupled lines. 11.2.1 Forward and Reverse Directional Coupling A pair of coupled transmission lines forms the 4-port network. Ideally, the coupling between two lines

Table 11.1 Designation of the ports of a 4-port coupler.

Name of ports

Codirectional coupler

Counter-directional coupler

Input port Coupled port Through port Isolated port

Port Port Port Port

Port Port Port Port

#1 #4 #2 #3

#1 #3 #2 #4

could be either the codirectional, i.e. the forward directional coupling or the counter directional, i.e. the reverse directional coupling. Figure (11.6a and b) shows the schematic diagrams of both types of couplings. The upper pairs of ports, ports-(3) and (4), are associated with the coupled line, while the lower pairs of ports, ports-(1) and (2), are associated with the mainline. In the case of a codirectional (forward) coupling, the direction of the coupled power at port #4 is the same as that of the direction of the input power flowing in the mainline, from port #1 to port #2. Port #3 is the isolated port, and ideally, no coupled power is available there. In the case of a counter-directional (reverse) coupling, the direction of the coupled power, at port # 3, is in the direction opposite to that of the direction of input power in the mainline. Table (11.1) shows the designation of the ports of a 4-port coupler.

11.2 Basic Concepts of Coupled Transmission Lines

Plane of symmetry

Dielectric waveguide #1

Dielectric waveguide #2 EZ1 EZ2 HZ1 HZ2

(a) Co directional coupled dielectric waveguide.

Plane of symmetry

I1

I2

Hϕ1 Hϕ2 Coupled line #2

Main line #1

(b) Counter-directional coupled two-wire lines.

Figure 11.7 Coupled lines structure.

The coupled transmission lines supporting the nonTEM modes, i.e. the TE, TM, and hybrid modes, provide the codirectional coupling. Such coupled transmission line structures are formed by the metallic waveguide, dielectric waveguide, etc. The codirectional coupling is physically explained by the mode coupling, shown in Fig (11.7a), of the hybrid mode coupled dielectric waveguides. The longitudinal field components EZ and Hz of the main guide #1 are in the direction of the wave prorogation. The direction of field components, Ez and Hz, determines the direction of the power flow in a waveguide. At the plane of symmetry, the tangential field components of both the waveguides #1 and #2 are con-

and non-TEM (TM0 surface wave) mode [J.16]. The main mode is the TEM one. Therefore, the coupled microstrip lines, or even the coupled CPW, have the counterdirectional coupling property. However, the presence of the TM0 surface wave modes also gives some codirectional coupling at the isolated port that degrades the isolation and directivity of a microstrip coupler. Further, the strength of the TM0 mode increases with an increase in frequency. It causes more degradation in the directivity of a microstrip coupler at a higher frequency. However, a wideband high directivity codirectional coupler has been developed in microstrip [J.17, J.18].

tinuous, i.e. E z1 = E z2 and H z1 = H z2 . Therefore, the directions of the field components in both the coupled waveguide #2 and the main waveguide #1 are the same, i.e. the powers in both the waveguides flow in the same direction. So, a non-TEM mode coupler is a codirectional (forward) coupler. A pair of transmission lines in a homogenous medium supporting a pure TEM mode, i.e. a mode with no longitudinal field component, forms a counter-directional (reverse) coupler, shown in Fig (11.7b). The current (I1), in the mainline #1, comes out of the plane of the paper, showing the direction of power flow in the mainline #1. The magnetic field HΦ1 around mainline #1 is in the anticlockwise direction. At the plane of symmetry, the magnetic fields of both the lines are continuous,

11.2.2

i.e. H ϕ1 = H ϕ2 . Consequently, the magnetic field H ϕ2 around the coupled line #2 is in the clockwise direction. The induced current I2 in line #2 enters the plane of the paper. It is in the direction opposite to the current I1 in line #1. So, the direction of the coupled power in the coupled line #2 is opposite to that of the power flow in the mainline #1. In conclusion, the TEM-type coupled transmission lines create a counter-directional (reverse) coupler. Strictly speaking, a microstrip line supports the hybrid mode. The hybrid mode, i.e. quasi-TEM mode, of a microstrip line is a linear combination of the TEM mode

Basic Definitions

The basic characteristics of the symmetrical directional coupler are defined using the scattering parameters, discussed in subsection (3.1.4) of chapter 3 [B.3, B.9, B.10, B.15, B.19]. Figure (11.8) shows the schematic diagram of a 4-port codirectional coupler that is used to define its scattering parameters. It is a coupler showing the coupling from port #1 to port #4 or from port #2 to port #3. Ideally, there is no coupling between port #1 and port #3, and vice versa. Also, there is no coupling between port #2 and port #4 and vice-versa. The general scattering matrix of the 4-port network is given below:

S =

S11 S21

S12 S22

S13 S23

S14 S24

S31 S41

S32 S42

S33 S43

S34 S44

11 2 1

Port #3

Port #4

Port #1

Port #2

Figure 11.8 Symmetrical codirectional coupler.

395

396

11 Coupled Transmission Lines

If all ports are matched terminated, normally to the 50 Ω, then S11 = S22 = S33 = S44 = 0, as there is no reflection at these ports. Also, due to the ideal isolation of ports, discussed above, S13 = S31 = S24 = S42 = 0. The voltage coupling coefficient kc is defined as S14 = S32 = kc

11 2 2

The through transmission coefficient is given as 1 − k2c

S12 = S34 = kt =

11 2 3

For no dissipation in a coupler, the power balance for the unit input power is written as follows: k2c + k2t = 1,

11 2 4

where k2c and k2t are the coupled and through powers, respectively. For a loose, i.e. weak, coupling, kc ≈ 0.1 and kt ≈ 1. For a quadrature coupler, two output signals are 900 phase apart, i.e. S14 = S32 = kc and S12 = S34 = j kt. The scattering matrix for the matched reciprocal quadrature coupler could be written as follows:

S =

0

jkt

0

kc

jkt

0

kc

0

0 kc

kc 0

0 jkt

jkt 0

Pc = 10 log 10 Kc Pi = − 20 log 10 S41 = − 20 log 10 kc , 11 2 6

where Pi is the input power at port #1 and Pc is the coupled power at port #4. The coupling is a measure of the power transferred from the mainline (waveguide) to the coupled line (waveguide) under the matched condition. This is also known as the insertion loss between port #1 and port #4. The transmission factor (Kt = Pt/Pi), i.e. the insertion loss (IL), in dB is a measure of the power appearing at the through (direct) port #2 of the mainline: = 10 log 10 Kt

= − 20 log 10 S21 = − 20 log 10

1 − k2c ,

11 2 7 where Pt is the power available at the through port #2. The reflection factor in dB, defined below, also known

11 2 8

The return loss is a measure of the reflected power at the input port. Ideally, no power should be transferred to the isolated (decoupled) port #3. However, in a practical coupler, some undesired power is always transferred to the isolated port #3. Thus, out of the coupled power, from the mainline to the coupled line, a major portion of the power is transferred to the coupled port #4. However, some undesired power is also available at the isolated (decoupled) port #3. The isolation factor (Kd = Pd/Pi) of port #3 in dB is defined as Isolation dB = 10 log 10

Pd Pi

= − 20 log 10 S31 ,

11 2 9 where Pd is the power at the isolated port #3. Out of the coupled power, the maximum power is desired at the coupled port #4 and minimum power at the isolated port #3. This characteristic is defined by the directivity of a coupler: Directivity dB = 10 log 10

Coupling dB = 10 log 10

Pt Pi

Return loss = − 20 log 10 S11

11 2 5

The coupling factor (Kc = Pc/Pi) in dB is defined as

IL dB = 10 log 10

as the return-loss, is measured at the input port #1, when all other ports are terminated in the matched loads:

Pd , Pc

11 2 10

where Kdir = Pd/Pc is the directivity factor. Thus, the directivity of a coupler is a measure of the isolation of a coupled port from the isolated port. It is also seen that the isolation (dB) is a summation of directivity (dB) and coupling (dB), i.e. Kd(dB) = Kdir(dB) + Kc(dB). In the above definitions, the S-parameters in dB are negative quantities, as the power available at all ports is less than the incident (input) power.

11.3

Circuit Models of Coupling

The electromagnetic coupling between two transmission lines takes place through the process of electric and magnetic inductions [B.4, B.10]. Electrostatic induction is a familiar concept. Once uncharged conductor #A is brought near a charged conductor #B, the charges of opposite polarity are induced on the conductor #A and capacitance is formed between two conductors. Thus, two originally isolated conductors are coupled through the mutual capacitance between them. Likewise, the law of dynamic induction (Faraday law of induction) is another familiar concept. A time-varying electric current in the conductor #A creates a time-varying magnetic field

11.3 Circuit Models of Coupling

could be obtained from its line capacitance. In the case of the coupled TEM transmission lines, such direct computation is applicable. The symmetrically coupled TEM modes are treated as a superposition of two normal TEM modes. These normal modes are called the even mode and odd mode. However, for the asymmetrical coupled lines, the normal modes are the C-mode and π-mode. These normal modes could be excited on the coupled transmission lines. The even–odd mode analysis method, applicable to the symmetrical coupled lines only, separates the coupled lines into two uncoupled lines. Likewise, the coupled asymmetrical lines are separated into the C-mode supporting line and the π-mode supporting line.

around it that couples with the conductor #B to induce a potential on it. The magnetic field coupling is represented by the mutual inductance between two conductors. In brief, the coupling between the two transmission lines is due to two mechanisms:

••

Capacitive coupling, i.e. the electric coupling. Inductive coupling, i.e. the magnetic coupling.

In this section, the circuit models of both the couplings are obtained. The models are based on direct capacitances and direct inductances. These lumped elements are evaluated in terms of the elements of Maxwell’s capacitance matrix, and inductance matrix. In the case of the symmetrical coupled planar lines, the concept of even–odd modes capacitance and inductance is useful.

Even-Mode Excitation

Figure (11.10a) shows the even-mode excitation on the symmetrical coupled lines by applying identical voltages of the same polarity, i.e. V1 = V2 = +1V, on both the conductors. The electric field lines of both the conductors are terminated in the same direction on the ground conductor. At the plane of symmetry-AA , the magnetic fields cancel each other forming a magnetic wall with Ht = 0, i.e. there is no surface-current on the magnetic wall. Therefore, from the circuit point of view, the magnetic wall is an open-circuited structure. Figure (11.10b) shows the rearranged original π-equivalent circuit of Fig (11.9b) with a plane of symmetry. Figure (11.10c) shows the separated even-mode equivalent circuit with the open-circuited plane of symmetry. The even-mode line capacitance is obtained as follows:

11.3.1 Capacitive Coupling – Even and Odd Mode Basics Figure (11.9a) shows the cross-section of a 3-conductor coupled transmission lines. A positive potential is applied on the conductor #1 that terminates its electric field lines partly on the ground conductor #3, and partly on the conductor #2. In the process, it induces the negative charges on the conductor #2 and establishes the capacitive coupling through the direct mutual coupling capacitance Cd12 p.u.l. between two conductors. The potentials at the conductors #1 and #2 further create direct self-capacitances Cd11 and Cd22 p.u.l. with respect to the ground conductor #3. Figure (11.9b) shows that the capacitive coupling is modeled by the π-network of the direct capacitances. For the asymmetrical coupled lines, shown in Fig (11.9a), both conductors have different sizes, giving unequal self-capacitance, i.e. Cd11 Cd22 between the conductors and ground. However, for symmetrical coupled lines, both conductors are physically identical, giving identical self-capacitance, i.e. Cd11 = Cd22 . The characteristic impedance of a single conductor TEM line

Conductor #1

Conductor #2

V1

Conductor #1 I1

Ce = Cd11 Odd-Mode Excitation

Figure (11.11a) shows the odd-mode excitation of the symmetrical coupled lines by applying equal voltages of opposite polarity on two coupled conductors. At the plane of symmetry-AA , the electric field lines cancel each other; forming an electric wall with zero tangential Cd12

Conductor #2

V2

Conductor #3

Cd11

V1

V2

Cd22

Common ground (a) Cross-section of two-conductor lines.

(b) Equivalent direct capacitances π-network.

Figure 11.9 The capacitive coupling between two conductors.

11 3 1

397

11 Coupled Transmission Lines

Hϕ-field

+

A

2Cd12

1

+ Cd11

E-field 1′

A′ Ground conductor (a) Even-mode fields.

2Cd12

Magnetic wall

Plane of symmetry (Magnetic wall) A

2

Cd22

2′

A′

(b) Equivalent circuit with magnetic wall A-A′. A 1

Open-circuit

2Cd12

Cd11

1′ A′ (c) Equivalent circuit of even-mode line. The even-mode excitation of the symmetrical coupled lines.

Plane of symmetry (Electric wall) A

Hϕ-field

+

A 2Cd12

2Cd12

1

– Cd11

Electric wall

Figure 11.10

2

Cd22

E-field A′ Ground conductor (a) Odd-mode fields.

2′

1′ A′

(b) Equivalent circuit with electric wall A-A′. 2Cd12

1

Cd11

A Short-circuit

398

A′ 1′ (c) Equivalent circuit of the odd-mode line. Figure 11.11

The odd-mode excitation of the symmetrical coupled lines.

11.3 Circuit Models of Coupling

electric field component, i.e. Et = 0. However, the magnetic fields of both conductors add together. Thus, from the circuit point of view, the electric wall is a shortcircuited structure. Figure (11.11b) shows the rearranged original π-equivalent circuit with the plane of symmetryAA treated as an electric wall. Finally, Fig (11.11c) shows the separated odd-mode supporting line. The odd-mode line capacitance is obtained by shortcircuiting the output terminals: C0 =

Cd11

+

2Cd12

11 3 2

It is possible to compute the even- and odd-mode line capacitances of the symmetrical coupled strip lines, microstrip lines, and CPW, using either the closed-form models or numerical methods [B.4–B.9, B.12, B.13, B.16, B.17, B.20, J.10, J.11, J.19–J.21]. The equations (11.3.1) and (11.3.2) are used to obtain the direct capacitances Cd11 and Cd12 of the symmetrical circuit model from the even- and odd mode-line capacitances. To excite the C-mode on the asymmetrical coupled lines, two unequal voltages with zero (cipher) phase difference are applied to both the conductors. It appears that the nomenclature C-mode is derived from the word-Cipher. To excite the π-mode, both the conductors are excited with out of phase unequal voltages, i.e. with 180 (π) phase difference that gives the nomenclature π-mode. The above-given description is about the capacitive coupling in the terms of the direct capacitances, forming the π-model. However, the direct capacitances appear due to the electrostatic induction. Therefore, the electric coupling is also described by Maxwell’s capacitance matrix of the multiconductor transmission lines [B.21, J.2, J.3, J.22]. The elements of the capacitance matrix are used to determine the direct capacitances of the π-model. 11.3.2

Forms of Capacitive Coupling

The capacitive coupling of the symmetrical coupled transmission lines is expressed in either of three ways: (i) the coefficients of capacitance matrix, (ii) the direct capacitances of the π-circuit model, and (iii) the even– odd mode capacitances. The choice depends on the ease of their computations. These three forms of capacitive coupling are examined below. Capacitance Matrix

Figure (11.9a) of the (2 + 1) conductors can be replaced by a system of the (N + 1) conductor transmission lines. Where N is the number of transmission lines and the additional line is the common ground conductor. If charges q1, q2, …qN are placed on the line conductors

#1, 2, 3, …, N, apart from retaining self-charges, these charges on respective conductors induce opposite polarity charges on all other conductors. Let us consider the induced charge on the conductor #1. However, if the original charge (self-charge) on the conductor #1 is q11, the total charge (q1) on it is a summation of self-charge and all other induced charges from the conductors 2 to N: q1 = q11 + q12 + q13 + …q1N

11 3 3

The self-charge on conductor #1 is proportional to the potential on it, i.e. q11 = C11V1. The induced charge on conductor #1 is proportional to the potential Vj on the jth conductor, i.e. the induced charge on conductor #1 is q1j = C1jVj. Therefore, if the conductors #1 to N are charged with the positive potentials V1…Vj…VN, the following expression is obtained for the total charges q1…qN on the N numbers of conductors: q1 = C11 V1 + C12 V2 + C13 V3 – – – C1N VN q2 = C21 V1 + C22 V2 + C23 V3 – – – C2N VN ––

––––

–––––

––––

––

a

–––

qN = CN1 V1 + CN2 V2 + CN3 V3 – – CNN VN q = C V

b 11 3 4

The [C] is Maxwell’s capacitance matrix of the coefficients of capacitance p.u.l:

C =

C11

C12

C1N

C21

C22

C2N

CN1

CN2

CNN

11 3 5

The coefficients of the capacitance matrix are the shortcircuit parameters. The Cii is the self-coefficient of the capacitance of the ith conductor due to a potential Vi on it, while all other conductors are grounded. Cij is the coefficient of capacitance matrix showing the mutual capacitance between the ith and jth conductors. The potential Vj is applied on the jth conductor, while all other conductors are grounded. The mutual capacitances (Cij), i.e. the off-diagonal elements of the capacitance matrix, are always negative as the induced charges have opposite polarity. Thus, Cii and Cij are obtained as follows: Cii =

qi Vi

a, Vj = 0

Cij = −

qi Vj

b;

j

i

Vi = 0

11 3 6 The coupled lines of Fig (11.9a) illustrate an application of the capacitance matrix. The potentials V1 and

399

400

11 Coupled Transmission Lines

On comparing the above equation (11.3.8) against equation (11.3.5) for N = 2 conductors, the following relations are obtained between the coefficients of capacitance matrix and the direct capacitances:

V2 are applied to two conductors. The charges on the conductors, in terms of the coefficients of capacitance matrix, are: q1

=

C11

− C12

V1

a

− C21 C22 V2 q1 q C11 V2 = 0 = , C21 V2 = 0 = − 2 V1 V1 q q C22 V1 = 0 = 2 , C12 V1 = 0 = − 1 V2 V2 q2

Mutual capacitance between lines C12 = − Cd12

b

In the above equation, the mutual capacitance C12 is negative and the reciprocity relation C21 = C12 holds for the coupled lines within an isotropic medium. Direct Capacitances

Figure (11.9b) shows the direct capacitance, Cd12 = Cd21 , between conductors, and the direct capacitances, Cd11 and Cd22, between conductors and the ground. Following expressions could be written for the total charges on each conductor of the coupled transmission lines: q1 = Cd11 V1 + Cd12 V1 − V2

q1 = Cd11 + Cd12 V1 − Cd12 V2

q2 = Cd12 V2 − V1 + Cd22 V2

q2 = − Cd12 V1 + Cd22 + Cd12 V2

= q2

Cd11 + Cd12

− Cd12

V1

− Cd12

Cd22 + Cd12

V2

c

11 3 9

11 3 7

q1

C11 = Cd11 + Cd12 a C22 = Cd22 + Cd12 b

Self capacitance of line#1 Self capacitance of line#2

However, the off-diagonal coefficients of the capacitance matrix given by equation (11.3.7a) are negative giving C12 = Cd12 . In the case of symmetrical structure (C11 = C22), the even mode is excited for V1 = V2 and the odd mode is excited for V1 = − V2. Using these conditions in equation (11.3.8), the even- and odd-mode line capacitances are obtained in terms of the direct capacitances. Using equation (11.3.9), they can be further expressed in terms of the coefficients of the capacitance matrix. On several occasions, such as the use of the variational method discussed in chapter 14, the even- and odd-mode line capacitances could be easily computed for the symmetrical coupled microstrip lines. The coefficients of capacitance matrix are computed from the even- and odd-mode line capacitances:

11 3 8

Even–mode line capacitance Ce = q1 V1 = Cd11 Cd11

Odd–mode line capacitance

C 0 = q1 V 1 =

Coefficient of capacitance

C11 = Ce + C0 2

Capacitive Coupling Coefficient

The voltage coupling is due to electric field coupling. It is defined by the capacitive coupling coefficient kc, [0 ≤ kc ≤ 1]. For kc = 0, there is no electric field coupling between lines #1 and #2. So, kc = 1 provides full coupling between two lines. However, it is not possible. In Fig (11.9b), the time-varying voltage V1 is applied to the input port #1. The current flowing in the circuit is j ω Cd Cd I1 = d 22 d12 V1 C22 + C12 The coupled voltage V2 across the output port of the conductor #2 is V2 =

I1 Cd = d 12 d V1 d j ω C22 C22 + C12

+

2Cd12

a,

Ce = C11 − C12

b

c,

Co = C11 + C12

d

e,

C12 = C0 − Ce 2

f

11 3 10

On using equation (11.3.9), the voltage coupling kc2 from the port #1 to port #2 and also the coupling kc1 from the conductor #2 to conductor #1 are obtained as follows: kc2 =

V2 Cd C12 = d 12 d = V1 C22 C22 + C12

a

kc1 =

V1 Cd C12 = d 12 d = V2 C11 C11 + C12

b 11 3 11

The voltage coupling factor kc between the asymmetric coupled lines is defined as a geometric mean of kc1 and kc2: kc =

Kc1 Kc2 =

C12 C11 C22

11 3 12

11.3 Circuit Models of Coupling

As for the symmetrical coupled lines C11 = C22, so the voltage coupling kc, using equation (11.3.10), is C12 Co − Ce = C11 Co + Ce

11.3.3

11 3 13

ψ = L I

Forms of Inductive Coupling

The inductive coupling of the symmetrical coupled transmission lines is also expressed in either of three ways: (i) the coefficients of the inductance matrix, (ii) the direct inductances of the T-circuit model, and (iii) the even–odd mode inductances. Again the choice depends on the ease of their computations. These three forms of inductive coupling are examined below.

Inductance Matrix

The inductive coupling, among N numbers of multiconductor coupled lines, is properly described in terms of the coefficients of inductance matrix Lij p.u.l [B.10, B.22, J.2, J.3, J.22]. The flux linkage ψj in the jth line is proportional to the current in the ith line, i.e.ψj Ii or ψj = LjiIi. So, the current I1 in the conductor #1 creates its self-inductance L11, while currents in all other conductors are zero. The currents I2, I3, … IN in the neighboring conductors also provide the flux linkage from these conductors to the conductor #1 through mutual inductances. The total flux linkage to conductor #1 is ψ1 = L11I1 + L12I2 + … + L1NIN. The flux linkages in each one of N conductors, due to the currents in all N conductors, are given by the following set of linear equations:

11 3 14 where the inductance matrix p.u.l. is given as

Ld22

Line #2

Port #1

L11 – L12

L22– L12

Port #2

I1

I1

L12 = M

Ld12

(a) Equivalent T - circuit model of asymmetrical coupled line. Figure 11.12

L1N

L21

L22

L2N 11 3 15

LN2

LNN

The coefficients of the inductance matrix are defined as the open-circuited parameters. Thus, the self-inductance Lii at the port #i, due to the current Ii, is obtained, while all ports are open-circuited, i.e. Ij=0 (i j). Likewise, the mutual inductance Lij between the ith and jth ports, when current flows only in the jth line, is obtained, while all other ports are open-circuited: Lii =

ψi Ij = 0 Ii

a , Lij =

11.3.3.1

b; i

j

Direct Inductances

Figure (11.12a) shows the T-type equivalent circuit, in terms of the direct inductances of the inductively coupled asymmetrical coupled lines. A relation exists between the direct inductances and coefficients of the inductance matrix. The total flux linkages in line #1 and #2, due to

+

V21

–

Coupling

I2 = 0

L11 Load

(b) Inductively coupled lines with induced voltage.

Inductive coupling of the coupled lines.

ψi Ii = 0 Ij

11 3 16

Source V1 Line #1

I2

L12

LN1

V2 Line #2 Ld11

L11 L =

L22

Line #1

a

b

Open - circuit

kc =

ψ1 = L11 I1 + L12 I2 + … + L1N IN ψ2 = L21 I1 + L22 I2 + … + L2N IN – –– –– –– – –– –– –– ψN = LN1 I1 + L2N I2 + … + LNN IN

401

402

11 Coupled Transmission Lines

currents I1 and I2 in the conductors #1 and #2, are given as follows: ψ1 = Ld11 I1 + Ld12 I1 + I2 ,

ψ1 = Ld11 + Ld12 I1 + Ld12 I2

ψ2 = Ld22 I2 + Ld21 I1 + I2 ,

ψ2 = Ld21 I1 + Ld22 + Ld21 I2

ψ1

Ld11 + Ld12

Ld12

ψ2

Ld22

+

Ld21

I2 11 3 17

Due to the reciprocity relation, in the above equation, we have Ld21 = Ld12. The flux linkages in terms of the coefficients of inductance matrix are written from equation (11.3.14) as follows: ψ1 ψ2

L11 = L12

L12 L22

I1 I2

11 3 18

On comparing the above two matrix equations, the following relations are obtained between the direct inductances of the equivalent T-circuit model and coefficients of inductance matrix: L11 = Ld11 + Ld12

a

L22 = Ld22 + Ld12

b

L12 = Ld21 = M

c

L12

b

11 3 22

Figure (11.12b) shows a pair of inductively coupled transmission lines with the induced voltage V21 in the line #2, due to the current I1 in the line #1, while the line #2 is open-circuited. The open-circuited voltage at terminals of line #2 is V2 = V21. The inductive voltage coefficients, kL2 from line #1 to line #2 and kL1 from line #2 to line #1, are obtained as follows: For line #1 with short- circuited load, V1 = j ωL11I1, and the induced voltage in line #2 is V2 = V21 = j ωL12I1, i.e. V2 = V21 = jωL12 × similarly,

kL1 =

V1 L12 = V1 , jωL11 L11

kL2 =

V2 L12 = V1 L11

L21 L22

a b

11 3 23

11 3 19 The symmetrical coupled lines have Ld11 = Ld22 . The matrix equation (11.3.17) provides the following evenand odd-mode inductances, in terms of the direct inductances, under the even-mode (I1 = I2) and odd-mode excitations (I1 = −I2): ψ1 = Ld11 + 2Ld12 I1 ,

Le = ψ1 I1 = Ld11 + 2Ld12

a

ψ1 = Ld11 I1 ,

L0 = ψ1 I1 = Ld11

b

11 3 20 Likewise, the matrix equation (11.3.18) provides the coefficients of inductance matrix for the even-mode and odd-mode line inductances: Le = L11 + L12 L0 = L11 − L12

a

Inductive Coupling Coefficient

I1

= Ld21

Le + L0 2 Le − L0 = 2

L11 =

a b

The reciprocity relation provides L12 = L21. The inductive voltage coupling coefficient kL for the asymmetrical coupled lines is defined as a geometric mean of kL1 and kL2. It also provides an expression of kL for symmetrical coupled lines (L22 = L11): kL = kL =

L12 L11 L22 Le − Lo Ld = = d 12 d Le + Lo L11 + L12

kL1 kL2 = L12 L11

a b 11 3 24

If the transmission lines are coupled both electrically and magnetically, the coupling coefficient C is defined as [J.15, J.23], C=

kC kL

11 3 25

11 3 21 The above equations can also be obtained from Fig (11.12a) on using the open- and short-circuit conditions at the plane of symmetry under the even- and oddmode excitations. To create the plane of symmetry, the shunt-connected mutual inductance Ld12 is replaced with a parallel combination of two inductances of value 2Ld12. The plane of symmetry (magnetic/electric wall) is placed in the middle of two inductances. The coefficients of inductance matrix for the symmetrical coupled lines are computed in terms of the evenand odd-mode inductances:

11.4 Even–Odd Mode Analysis of Symmetrical Coupled Lines Levy [B.15, J.24] developed the even–odd mode analysis method to obtain the reflection coefficient, transmission coefficient, coupling, and directivity of the TEM mode supporting symmetrical coupled transmission lines (strip lines) structure in a homogeneous medium. However, the method is also applied to the analysis of coupled microstrip and CPW lines, supporting the quasi-TEM mode [B.4–B.15]. The method has also

+

Port #4

Line #2 Z0

V3

Port #1

+

Z0

V4

V2

Input port

V

+ Port #2

Line #1

V1

Z0

Z0

L

Through port

Coupled port Port #3 +

Isolated port

11.4 Even–Odd Mode Analysis of Symmetrical Coupled Lines

(a) Original symmetrical coupled lines.

+ V4e Port #4

Line #2

I4e

I3e

+V 2

Z0

Z0e Port #1 +V1e Z0 +V 2

I1e

Input port

+ V2e Port #2

Line #1

I2e

Z0

Plane of symmetry (Magnetic wall)

Isolated port

Z0

θe = βeL

Through port

Coupled port Port #3 +V3e

Z0

θo = βoL

–V4o

Line #2 I4o

I3o

–V 2

Port #4 Z0

Z0o Port #1 +V1o Z0 +V 2

I1o

Input port

+V2o Port #2

Line #1

I2o Plane of symmetry (Electric wall)

Z0

Through port

Coupled port Port #3 –V3o

Isolated port

(b) Even-mode excited coupled lines.

(c) Odd-mode excited coupled lines. Figure 11.13

Coupled lines resolved in two separate lines supporting the even and odd modes.

been used to the coupled waveguide structure [B.15]. Figure (11.13) shows symbolically the 2-conductor coupled lines. The RF input voltage V is applied to the input port #1, whereas port #2, port #3, and port #4 are the through, coupled, and isolated ports, respectively. The coupled symmetrical lines, shown in Fig (11.13a), support two independent normal modes, called the even

mode and odd mode discussed in the previous section. The phase velocities of the even and odd modes of the coupled microstrip and CPW structures are different. Their characteristic impedances are also different. In the case of coupled strip lines, the even and odd modes have identical phase velocities, due to homogeneous medium. However, the even and odd mode

403

404

11 Coupled Transmission Lines

Port #1

Port #2

Port #3 Port #4 (a) Two-conductor 4-ports uniformly coupled symmetrical lines. Axis of odd symmetry (Ground potential)

Axis of even symmetry

(b) Field distributions of even-odd modes of a coupled strip lines.

εr

εr (c) Field distributions of even-odd modes of coupled microstrip lines. E-field H-field

E-field H-field

Magnetic wall

Electric wall (d) Field distributions of even-odd modes of a coupled CPW.

Figure 11.14

Cross-sections of coupled strip lines, microstrip, and CPW showing even–odd mode field distributions.

characteristic impedances of a coupled strip line are still different, due to different field distributions for both the modes. Figure (11.14a–d) shows the coupled lines section, the field distributions of both modes for the edge coupled strip lines, microstrip lines, and CPW. The even and odd modes analysis is equally valid to the symmetrical broadside coupled transmission lines. 11.4.1

Analysis Method

The even–odd mode analysis is applied to the coupled microstrip lines. The coupled strip line structure is treated as its special case. Figure (11.13a) shows the original coupled strip lines excited by the voltage source V at its input port #1. The source has an internal impedance Z0. The port #2, port #3, and port #4 are terminated in the load Z0. Under proper excitations, the twin conductors support two normal modes – the even and odd modes [J.25]. Figure (11.13b) shows that the even mode is

excited by the application of identical polarity voltages +V/2 at port #1 and port #3. The plane of symmetry, between two conductors, is a magnetic wall. Likewise, the opposite polarity voltages ±V/2 applied at these ports excite the odd mode. In this case, the plane of symmetry is an electric wall. The currents and voltages of the even and odd modes, at all the 4-ports, are given below: Even mode

Odd mode

I1e = I3e , I2e = I4e V1e = V3e , V2e = V4e

I1o = − I3o , I2o = − I4o V1o = − V3o , V2o = − V4o

a b 11 4 1 a b 11 4 2

In summary, the concept of the even and odd modes decouples the original pair of coupled lines into two

11.4 Even–Odd Mode Analysis of Symmetrical Coupled Lines

Z0

I2

I1 [A,B,C,D]

V1

T

Γ

V2

Z0

Z0

Zin Port #2

Port #1

The above expressions are used for both the even and odd modes giving (Γe) and (Te), and also (Γo) and (To). Following equation (11.4.3), the ABCD matrix descriptions for both the even- and odd-mode lines could be written as follows: V1e I1e

Figure 11.15

2-Port network for even- or odd-mode line section.

V1o isolated single lines, supporting the even and odd modes. So, the even-mode supporting line is just a single conductor TEM line in presence of the magnetic wall, whereas the odd-mode supporting line is another single conductor TEM line in the presence of the electric wall. The Zoe and Zoo are the even- and odd-mode characteristic impedances of these single lines. Thus, a section of the 4-port coupled lines structure is partitioned into two 2-port line sections – the even-mode and odd-mode line sections, shown in Fig (11.13). Figure (11.15) shows that the 2-port line section could be treated as the 2-port network. It is characterized by the ABCD matrix discussed in subsection (3.1.3) of chapter 3. The input voltage–current (V1,I1) are related to the output voltage–current (V2,I2) as follows [B.10, B.23]: V1 I1

=

A B C D

V2 I2

=

cos θ sin θ j Zo

jZo sin θ

V2 I2 ,

cos θ

11 4 3 where the characteristic impedance Zo could be either Zoe or Zoo for the even- or odd-mode line section, respectively. The above expression is given in equation (3.1.25) of chapter 3. Likewise, the electrical length θe = βeL, θo = β0 L corresponds to the even and odd modes. However, for the coupled strip line propagation constants βe = βo = β, giving θe = θo = θ. However, the coupled microstrip lines have θe θo. The physical length of the coupled lines section is L. The input impedance of the line section, terminated in a matched load, is computed. It helps to obtain the reflection coefficient (Γ), showing a mismatch at the input port, and the transmission coefficient (T), showing the available signal at the output port [B.23]: AZo + B CZo + D A + B Zo − CZo − D Γ= A + B Zo + CZo + D 2 T= A + B Zo + CZo + D

Zin =

=

I1o

=

cos θe

j Zoe sin θe

sin θe j Zoe

cos θe

cos θo

j Zoo sin θo

sin θo j Zoo

cos θo

V2e I2e V2o I2o

a

b 11 4 5

S-Parameters of Coupled Microstrip Lines

To get the S-parameters, the port voltages are computed. For the input voltage V = 1 at port #1, shown in Fig (11.13a), the even- and odd-mode voltages at port #1 are ½. The even- and odd-mode reflected signals, A1e and A1o, at port #1 are obtained as follows: Reflected signal = Reflection coefficient × Input signal 1 For the even mode A1e = Γe × a 2 1 For the odd mode A1o = Γo × b 2 11 4 6

The total reflected signal A1 at port #1 of the original coupled lines is a combination of the even and odd modes reflected signals. It is written in terms of the S-parameter giving the return-loss: 1 Γe + Γo 2 Return–loss RL dB = − 20 log 10 S11 S11 = A1 = A1e + A1o =

a b 11 4 7

Likewise, the even- and odd-mode transmitted signals A1e and A1o at the output port#2 are obtained as Transmitted signal = Transmission coefficient × Input signal 1 a For the even mode Ale = Te × 2 1 b For the odd mode Alo = To × 2

11 4 8 The total transmitted signal A2, at the through port #2 of the original coupled lines, helps to obtain the S21 and insertion loss:

a b

1 Te + To 2 Insertion loss, I L dB = − 20 log 10 S21

S21 = A2 =

c 11 4 4

a b 11 4 9

405

406

11 Coupled Transmission Lines

Likewise, the coupled signal A3 at the coupled port #3 computes the coupling: 1 a S31 = A3 = Γe − Γo 2 b Coupling, C dB = − 20 log 10 S31

Likewise, using equations (11.4.4c) and (11.4.5), the following expressions are obtained for the transmission coefficients of the even and odd modes: Te =

11 4 11 The directivity of a coupler is D dB = − 20 log 10

A4 A3

εr effe f L

a

θo = βo L = β0

εr effo f L

b

11 4 16

o

The port voltages and port currents (V1, I1) and (V2, I2) at the port #1 and port #2, respectively, are computed as a linear combination of the even and odd voltages:

At port#2

V1 I1 V2 I2

= =

V1e I1e V2e I2e

+ +

V1o

a

I1o V2o

b

I2o

11 4 17 Equations (11.4.5) and (11.4.17) provide V1 I1

,

b

Input Impedance Matching

At port#1

θe = βe L = β0

2

To =

The input port #1 must be matched to the source impedance Z0. The condition for input impedance matching is obtained below.

11 4 12

The scattering parameters are complex quantities. Therefore, their magnitudes are taken for the determination of the return-loss, insertion-loss, coupling, isolation, and directivity. The even and odd modes’ electrical lengths of the coupled microstrip lines, in an inhomogeneous medium, are different:

a,

e

11 4 10 Finally, the signal at the isolated port #4 provides the isolation: 1 S41 = A4 = Te − To a 2 b Isolation, I dB = − 20 log 10 S41

2

=

cos θe + cos θo sin θe sin θo j + Zoe Zoo

j Zoe sin θe + Zoo sin θo cos θe + cos θo

Γe =

Zoe Zo sin θe j − Zo Zoe

Γo =

b, 11 4 14

where

o

Zoe Zo sin θe + Zo Zoe Zoo Zo sin θo = 2 cos θo + j + Zo Zoo = 2 cos θe + j

Γ=

Zin − Z0 = 0, Zin + Z0

Zin = Z0 =

AZ0 + B CZ0 + D 11 4 19

For a reciprocal network with A = D, the above matching condition is reduced to B = Zo C Z0 11 4 20

a

o

e

The input impedance of coupled lines under matching at port #1 is

CZ20 + Z0 A = AZO + B

e

Zoo Zo j − sin θo Zo Zoo

I2

11 4 18

11 4 13 where β0 is the propagation constant in the free space. The effective even- and odd-mode relative permittivity, εr effe(f ) and εr effo(f ), are frequency-dependent. The above S-parameters are obtained in terms of the reflection and transmission coefficients for the even and odd modes. A section of the coupled line in an isotropic medium forms the reciprocal network with A = D. Using equations (11.4.4) and (11.4.5), the even- and odd-mode reflection and transmission coefficients are computed as

V2

On substituting the matrix elements B and C of equation (11.4.18) in the above equation, the following matching condition is obtained: Zoe Zoo Zo Zo sin θe + sin θo = sin θe + sin θo Zo Zo Zoe Zoo Zoe Zoo Zoo sin θe + Zoe sin θo = 2 Zoe sin θe + Zoo sin θo Zo

11 4 21

a b 11 4 15

The perfect matching, at the input port, is satisfied at only one center frequency with a narrow frequency band, as θe,θo and Zoe, Zoo are frequency-dependent, due to the frequency-dependent nature of εr effe(f ) and εr effo(f ).

11.4 Even–Odd Mode Analysis of Symmetrical Coupled Lines

Isolation Between Ports

The parameter S41 determines the isolation between ports #1 and #4. However, the ports #2 and #4 should also be isolated from each other. It is needed to improve

where, r =

2 cos θo + cos θe + j

θe + θo 1 −j r + 2 r θe + θo 1 2 cos +j r+ 2 r 2 sin

Zoe , and Z0 ≈ Zoo

11 4 22 a

b

Z0e Z0o

It is noted that for the coupled strip lines, in a homogeneous medium, θe = θo, giving the perfect isolation as A4 = 0. However, in the case of coupled microstrip lines θe θo; therefore, the perfect isolation is not possible for a microstrip-based coupler. For a loose coupling case, Zoe ≈ Zoo i e r + 1r ≈ 2, giving A4 θe − θo = tan A2 2

1 sin θo − sin θe r 1 sin θo + sin θe r+ r θe + θo cos 2 θe + θo sin 2

2 cos θo − cos θe + j r +

A4 Te − To Isolation = = = A2 Te + To

A4 θe − θo = tan A2 2

directivity. Using equations (11.4.9a) and (11.4.11a), the isolation between the ports #2 and #4 is obtained as follows [J.24]:

A4 ≈

θe − θo 2

L = (π/2) × (vp/ω0). The electrical length of the coupled section at any other frequency is θ = βL =

ω π vp , vp 2 ωo

θ=

π f × 2 fo

11 4 25

Using the input matching condition (11.4.24), the even–odd modes reflection and transmission coefficients of the coupled strip lines are obtained from equations (11.4.14)–(11.4.16):

11 4 23 For a loose coupling |A2| ≈ 1, and |(θe − θo)/2 | is small. Both the directivity and isolation degrade for loose coupling (≤10 dB). The isolation also degrades with an increase in frequency. Coupled Strip Lines

The symmetric coupled strip lines are in a homogeneous medium with equal εr effe(f ) = εr effo(f ) = εr, giving θe = θo = θ. Therefore, for coupled strip lines the input matching condition, given by equation (11.4.21), is reduced to a simpler form: Zo =

Zoe Zoo

11 4 24

The above expression is also used for the design of the microstrip-based coupler. To obtain the frequency response of a coupler, the variable θ is expressed in terms of the normalized frequency f/f0, where f0 is the central design frequency of a coupler for maximum coupling. Let the length of the coupled strip lines section is L = (λg/4). It gives the electrical length as θe = θo = θ = βgL = (2π/λg) × (λg/4) = (π/2). The length of the coupled section at the center frequency ω0 is

j Γe = − Γo = Te = To = where,

Zoe − Zoo

Zoo sin θ Zoe

11 4 26

2

= 2 cos θ + j

a Zoe + Zoo

Zoo sin θ Zoe

b

11 4 27 Thus, using the above relation and equations (11.4.7)– (11.4.11), the output signals, i.e. the S-parameters, at all the four ports are given below: Input matched A1 = 0, Perfect isolation A4 = 0,

Coupling A3 = Γe Through port A2 = Te

11 4 28 The coupled signal at port #3 is

A3 = j

Zoe − Zoo Zoe + Zoo

sin θ 2Zo cos θ + j sin θ Zoe + Zoo 11 4 29

407

408

11 Coupled Transmission Lines

The maximum A3, showing the maximum coupling π (C), is obtained for the electrical length θ = : 2 C = A3 =

Zoe − Zoo Zoe + Zoo

a,

1 − C2 =

2Zo Zoe + Zoo

b

11 4 30 Finally, under the matched condition, the following expressions are obtained for the coupled and through signals at the ports #3 and 2 of a stripline coupler: A3 = Γ e = j A2 = T e =

C sin θ

a

1 − C cos θ + j sin θ 2

1 − C2

b

1 − C2 cos θ + j sin θ

11 4 31 The two output signals, at port #2 and port #3, are in the quadrature-phase. The expression for θ, given by equation (11.4.25), is used to obtain the frequency response at all ports, i.e. the frequency-dependent return-loss, insertion loss, coupling, isolation, and directivity of a microstrip and strip lines couplers. The expressions for the S-parameters are also applicable to the coupled CPW lines. To compute the frequency response

Zoe =

Le Ld + 2Ld = 11 d 12 Ce C11

1 2

Alternatively, εr effe 1 = = Zoe = vp1 Ce cCe c c 1 vpe = = = εr effe Le C e

=

L11 + L12 C11 − C12

at all ports, expressions are needed for εr effe(f ), εr effo(f ), Zoe, and Zoo of the coupled microstrip lines, coupled strip lines, and coupled CPW. The expressions are presented in chapter 12. Chapter 14 also presents the variational method to determine the static εr effe(f = 0) and εr effo(f = 0). The present method is valid only for the symmetrical coupled lines. The next section (11.5) discusses the coupled wave equations approach that is also used for the asymmetrical coupled lines.

11.4.2

Coupling Coefficients

Equation (11.4.30) for the coupling is strictly valid only for the symmetrical coupled strip lines in a homogeneous medium. More accurate expressions for the symmetrical coupled microstrip lines, involving using both the capacitive and inductive couplings, are obtained below. These expressions for coupling, in terms of the evenand odd-mode line parameters, are also applicable to the symmetrical coupled CPW. Using equations (11.3.1) and (11.3.20a), the evenmode characteristic impedance Zoe, phase velocity vpe, and even-mode effective relative permittivity εr effe are obtained as follows:

1 2

a

1

,

Ce εr Ce εr = 1 1

=

Ld11 + 2Ld12 Cd11

Zoe =

ηo εo Ce εr Ce εr = 1 1

L11 + L12 C11 − C12

εr effe = c2 L11 + L12 C11 − C12

Zoo =

1 2

=

L11 − L12 C11 + C12

c

phase velocity vpo, and effective relative permittivity εr effo are also given below:

1 2

Alternatively, εr effo 1 ηo εo = , Zoo = Zoo = cCo c Co εr Co εr = 1 Co εr Co εr = 1 c 1 1 1 = = = vpo = εr effo Lo C o L11 − L12 C11 + C12 Ld11 Cd11 + 2Cd12 εr effo = c2 L11 − L12 C11 + C12

11 4 32

d,

where c is the velocity of EM-wave in free space and ηo = 120π is the intrinsic impedance of the free space. Likewise, the odd-mode characteristic impedance Zoo, Lo Ld11 = Co Cd11 + 2Cd12

b

a

b c d

11 4 33

11.5 Wave Equation for Coupled Transmission Lines

The closed-form expressions for the characteristic impedance and effective relative permittivity are presented in section (12.1) of chapter 12. Therefore, the direct circuit constants Ld11, Ld12, Cd11, and Cd12 are expressed in terms of the even- and odd-mode secondary parameters as follows: Zoe vpe = Zoo vpo =

1 , Cd11 Cd11

Cd11 = 1 , + 2Cd12

Cd12

εr effo − Zoo

a,

Zoo = Ld11 vpo

b

11 4 35

1 Zoe Zoo 1 = + Zoe εr effe + Zoo εr effo vpo 2 vpe 2c 1 Zoe Zoo 1 Zoe εr effe − Zoo εr effo = − = 2 vpe 2c vpo

Ld11 + Ld12 =

a

Ld12

b

11 4 36

εr effe 1 = cZoe vpe Zoe

1 = 2c

Zoe = Ld11 + 2Ld12 vpe

a

εr effe Zoe

The capacitive (kC) and inductive (kL) coupling coefficients are obtained as follows:

b

11 4 34

kC =

Co − Ce Cd = d 12 d , Co + Ce C11 + C12

Lo − Le Ld = d 12 d , kL = Lo + Le L11 + L12

kC =

Zoe εr effo − Zoo εr effe Zoe εr effo + Zoo εr effe

Zoe εr effe − Zoo εr effo kL = Zoe εr effe + Zoo εr effo

In the case of a homogeneous medium coupled strip lines εr effe = εr effo = εr, and the inductive and capacitive coupling coefficients kC = kL = C are given by equation (11.4.30a). The coupling coefficient of the coupled microstrip lines, or the coupled CPW, in the inhomogeneous media is computed from the following expression [J.15, J.23]: C=

kC kL

11 4 38

For the symmetrical coupled strip lines, kC = kL and the above equations are reduced to equation (11.4.30a).

11.5 Wave Equation for Coupled Transmission Lines A pair of coupled asymmetrical transmission lines of length dx, located in an isotropic homogeneous/ inhomogeneous medium, is shown in Fig (11.16a). Figure (11.16b) shows its equivalent circuit using the coefficients of capacitance and inductance matrices. The coupling of power from one line to another line takes places through the following two mechanisms:

• •

Coupling due to the time-varying magnetic field, i.e. through the inductive coupling. Coupling due to the time-varying electric field, i.e. through the capacitive coupling.

The equivalent circuit, shown in Fig (11.16b), takes into account both coupling mechanisms. The

a 11 4 37 b

coupling capacitance, i.e. the mutual capacitance, is a negative quantity, C12 = − Cm, whereas the inductive coupling, i.e. the mutual inductance, is a positive quantity, L12 = M. The self-inductance, L11 and L22, and the self-capacitance, C11, and C22, of line #1 and line #2 are the elements of the inductance and capacitance matrices discussed in section (11.3). On the application of the basic induction law of Faraday for the magnetic coupling and induced displacement current law of Maxwell for the electric coupling, a wave equation is obtained for the asymmetrical coupled transmission lines. It can be easily extended to N numbers of the coupled transmission lines. The matrix form of the formulation is a compact one, and easier to manipulate. The time-harmonic variation is assumed for the phasor voltage and phasor current [B.4–B.6, B.21, B.22, J.5, J.10, J.17, J.23, J.26]. 11.5.1 Kelvin–Heaviside Coupled Transmission Line Equations The Kelvin–Heaviside transmission line equations of subsection (2.1.3) of chapter 2 can be extended to the coupled transmission line also. In view of two coupling mechanisms, the equivalent circuit of the coupled lines is obtained in two steps. Figure (11.17a) shows the inductive coupling through the induced emf in line #1, L12 dx (∂ i2/∂ t), caused by the time-varying current i2 in line #2. The polarity of the induced voltage follows Lenz’s law. Figure (11.17b) shows the capacitive coupling, through the induced displacement current, C12dx(∂v2/∂t), due to

409

11 Coupled Transmission Lines

dx

V1 + dV1 Line #1

i1

C11 L 11

V2+ dV2 C12 = –Cm

V1

Line #2 Line #2

V2

(a) Coupled asymmetrical lines. Figure 11.16

i1

C22

L22

C11 L

11

C11 L 11

C11

C12 L12

C12 L12

C12

L22 C22

C22

L22

C22

(b) Equivalent circuit of coupled asymmetrical lines.

Coupled asymmetrical lines in an inhomogeneous medium.

(L12dx)

(L11dx)

+

di2 dt –

i1

i1 + di1

i1 + di1 dv2 dt

i2

L12

Region

Line #1

Coupling

(C11dx) Line #1 V1

V1 + dV1

(C12dx)

410

Line #1 V1

dx

dx

(a) Inductive coupling through induced emf.

i1

(b) Capacitive coupling through induced displacement current. (L12dx)

(L11dx)

Line #1 V1

V1 + dV1

+ (C11dx)

(C12dx)

di2 dt –

dv2 dt

i1 + di1

V1 + dV1

dx (c) Combined capacitive and inductive couplings. Figure 11.17

The coupling mechanisms in the asymmetrical coupled transmission lines.

the time-varying electric field between lines #1 and #2. Figure (11.17c) shows both couplings simultaneously. The following loop and node equations are written, from Fig (11.17c), for line #1 due to the couplings from the line #2: ∂i1 ∂i2 − L12 dx − v1 + dv1 = 0 ∂t ∂t ∂v1 ∂i1 ∂i2 − = L11 + L12 a ∂x ∂t ∂t ∂v1 ∂v2 Node equation i1 − C11 dx − C12 dx − i1 + di1 = 0 ∂t ∂t ∂i1 ∂v1 ∂v2 − = C11 + C12 b ∂x ∂t ∂t Loop equation v1 − L11 dx

11 5 1

Likewise, the voltage and current equations for the line #2, due to the coupling from line #1 to line #2, are written: ∂v2 ∂i2 ∂i1 = L22 + L21 ∂x ∂t ∂t ∂i2 ∂v2 ∂v1 − = C22 dx + C21 ∂t ∂t ∂t −

a b 11 5 2

Using the phasor notation, the above Kelvin– Heaviside coupled equations are rewritten below in terms of the impedances and admittances: −

dv1 = Z11 i1 + Z12 i2 dx

a,

−

dv2 = Z21 i1 + Z22 i2 dx

b

11 5 3

11.5 Wave Equation for Coupled Transmission Lines

−

di1 = Y11 v1 + Y12 v2 dx

di2 = Y21 v1 + Y22 v2 dx

−

a,

The L11, C11, and L22, C22 pairs are the self-inductance and self-capacitance of the conductors #1 and #2. The pairs R11, G11, and R22, G22 account for the conductor loss and the leakage loss of the conductors #1 and #2. The G12 is the leakage loss between the conductors #1 and #2 through the leakage current in the lossy dielectric medium. The R12 is caused by the proximity effect of two conductors [J.10, J.26]. For an isotropic and linear medium, the following reciprocity relations hold [B.22, B.23]:

b

11 5 4 The above equations, in the phasor form, are generalized to include the dielectric and conductor losses through the conductance G and resistance R, respectively. The phasor voltages and currents are written without the tilde sign v1 , i1 , etc. The self-impedance, mutual impedance, self-admittance, and mutual admittance of the coupled lines p.u.l. are given by the following expressions: Z11 = R11 + jωL11

a,

Y11 = G11 + jωC11

b

Z12 = R12 + jωL12

c,

Y12 = G12 + jωC12

d

Z22 = R22 + jωL22

e,

Y22 = G22 + jωC22 f 11 5 5

−

d v1 dx v2

d i1 − dx i2

−

= =

Z11

Z12

i1

Z12

Z22

i2

Y11

Y12

v1

Y12

Y22

v2

d V = Z I dx

a,

−

= jω = jω

Z12 = Z21

L12

i1

L12

L22

i2

C11

C12

v1

C12

C22

v2

where [V] and [I] are the column matrices for the voltage and current on the coupled transmission lines. The matrix form easily accommodates N numbers of conductors [J.2, J.25, J.26]. The [Z] and [Y] are the n × n order impedance and admittance matrices for N numbers of coupled lines. The above matrix equations are solved to obtain the following voltage wave equation for coupled transmission lines:

+

R11

R12

i1

R12

R22

i2

+

−

d d I = − Z Y V 2 V = Z dx dx d2 V = Z Y V dx2

Z Y =

11 5 9

Z11 Y11 + Z12 Y12

Z11 Y12 + Z12 Y22

Z12 Y11 + Z22 Y12

Z12 Y12 + Z22 Y22 11 5 10

b

11 5 6

a

G11

G12

v1

G12

G22

v2

Z Y =

a1 b2

11 5 7 b

b1 a2

11 5 11

where, a1 = Z11 Y11 + Z12 Y12 ,

a2 = Z12 Y12 + Z22 Y22

b1 = Z11 Y12 + Z12 Y22 ,

b2 = Z12 Y11 + Z22 Y12 11 5 12

Similarly, the current wave equation is also obtained for the coupled transmission lines.

11.5.2 2

Y12 = Y21

The coupled line equations for the isotropic and linear medium are written below, in more compact matrix notation:

L11

d I = Y V b, dx 11 5 8

a,

Solution of Coupled Wave Equation

The following form of the solution is assumed for the voltage wave propagation in the x-direction on the coupled lines: V = V0 e − γ x ,

11 5 13

where γ is the complex propagation constant. On substituting the above solution in the wave equation (11.5.9), the following expression is obtained:

411

412

11 Coupled Transmission Lines

γ2 V = Z Y V ,

Z Y − γ2 U

V

= 0, 11 5 14

where [U] is the unit matrix. The eigenvalues of the above equation are obtained from the following characteristics equation: Det Det

a1

b1

b2

a2

−

γ2

0

0

γ2

a1 − γ2

b1

b2

a2 − γ2

=0 11 5 15

=0

γ4 − γ2 a1 + a2 + a1 a2 − b1 b2 = 0 a1 + a2 1 ± a1 − a2 2 + 4b1 b2 γ2 = 2 2

Equation (11.5.16) provides two solutions for the propagation constants, indicating the existence of two linearly independent modes known as the normal modes. Two normal modes are two independent solutions of wave equations. In the present case, two normal modes are supported by a pair of coupled lines. Each normal mode has ± propagation constants, indicating the presence of the forward and backward traveling modal voltage waves. Likewise, the structure of the N-coupled line supports N numbers of normal modes. Each normal voltage mode has a distinct propagation constant. The phase velocity of the N normal modes of the N-coupled lines is given by vp,i = 1 L C = c εreff,i i = 1, 2, …, N . The coupled lines could be either an asymmetrical or symmetrical structure. The medium could also be either an inhomogeneous medium or a homogeneous medium. Further, a medium could be isotropic or anisotropic. However, this subsection only considers the coupled transmission lines in the isotropic medium with four cases: Asymmetrical coupled lines in an inhomogeneous medium. Asymmetrical coupled lines in a homogeneous medium. Symmetrical coupled lines in an inhomogeneous medium. Symmetrical coupled lines in a homogeneous medium.

βc =

ω L11 C11 + L22 C22 − 2L12 C12 + F 1 2

where, F =

L11 C11 − L22 C22

2

In the case of the asymmetrical coupled lines in an inhomogeneous medium, such as the asymmetrically coupled microstrips and CPW, the self-impedance and the self-admittance of two conductors are unequal, i.e. Z11 Z22 and Y11 Y22. The C-mode and π-mode are the distinct normal modes of asymmetrical coupled lines. The normal modes have distinct propagations constants and distinct characteristic impedances. The propagation constants for the C and π-modes are obtained from equation (11.5.16): γ1,2 = ± γc a ,

1 2

11 5 16

• • • •

Case # 1: Asymmetrical Coupled Lines in the Inhomogeneous Medium

γ3,4 = ± γπ b ,

11 5 17

where (+) indicates the forward-moving C and π-modes and (−) indicates the backward moving C and π-modes. The propagation constants of these modes are given below: 1 a1 + a2 1 + a1 − a2 2 + 4b1 b2 2 2 2 1 a1 + a2 1 − a1 − a2 2 + 4b1 b2 2 = 2 2

γc 2 =

a

γπ 2

b 11 5 18

The propagation constants of a lossless asymmetrical coupled transmission lines, in the isotropic inhomogeneous medium, can be obtained in terms of the primary line constants. For this purpose, the following expressions are used: Z11 = jωL11 , Z12 = jωL12 ,

Z22 = jωL22

a

Y11 = jωC11 , Y12 = − jωC12 , Y22 = jωC22 a1 + a2 j2 ω2 = L11 C11 + L22 C22 − 2 L12 C12 , a1 − a2 2 2 = j4 ω4 L11 C11 − L22 C22

2

4 b1 b2 = 4j4 ω4 L12 C11 − L22 C12 L12 C22 − L11 C12

2

b c

11 5 19 In the above equation, the negative value is used for the mutual capacitance (C12). Normally, the line constants of microstrip and CPW kind of structures are frequency-dependent [J.4]. The following expressions for the propagation constants βc and βπ of the lossless C and π-modes, γc, π = j βc,π, are obtained on using the above equations with equation (11.5.18):

2

+ 4 L12 C11 − L22 C12 L12 C22 − L11 C12

a 11 5 20 b

11.5 Wave Equation for Coupled Transmission Lines

βπ =

ω L11 C11 + L22 C22 − 2 L12 C12 − F 1 2 2 11 5 21

The phase velocities and effective relative permittivities for C and π-modes are given below: ω c ω c a , vπ = vc = = = b βc εreff c βπ εreff π cβ cβ c, εreff π = π d εreff c = c ω ω 11 5 22 For L11 = L22 and C11 = C22, the asymmetrical coupled lines structure is reduced to the symmetrical coupled lines. Also, the C and π-mode effective relative permittivities εreff c and εreff π are reduced to the even- and oddmode effective relative permittivities εreff e and εreff o given by equations (11.4.32d) and (11.4.33d). Case #2: Asymmetrical Coupled Lines in a Homogeneous Medium

In a homogeneous medium, the phase velocities of the EM-wave for both the transmission lines are identical: 1 1 = 11 5 23 L11 C11 L22 C22 ω2 L11 C11 = ω2 L22 C22

Z11 Y11 = Z22 Y22

11 5 24

Using the above expressions, the following expressions are obtained from equations (11.5.12): a = a1 = a2 = Z11 Y11 + Z12 Y12

a

b1 = Z11 Y12 + Z12 Y22

b

b2 = Z12 Y11 + Z22 Y12

c

The condition b1 b2, (Z11 Z22, Y11 Y22 ), is valid for the asymmetrical coupled lines in a homogeneous medium. The following expressions for the propagation constants of the C and π-modes in a homogeneous medium are obtained, on using the above equations with equation (11.5.18): γc 2 = Z11 Y11 + Z12 Y12 1 2

11 5 26 γπ 2 = Z11 Y11 + Z12 Y12 − Z11 Y12 + Z12 Y22 Z12 Y11 + Z22 Y12

Case # 3: Symmetrical Coupled Lines in an Inhomogeneous Medium

Both lines are identical for the symmetrical coupled lines in an inhomogeneous medium, i.e. Z11 = Z22 and Y11 = Y22. Equation (11.5.12) provides a1 = a2 = a and b1 = b2 = b. Equation (11.5.18) gives the following expressions of the propagation constants for the C and π-modes: γc 2 = a + b γπ 2 = a − b

γc = ± a + b γπ = ± a − b

1 2

11 5 27

a b 11 5 28

In the case of the symmetrical coupled lines, the γc corresponds to a sum mode, also known as the even mode, i.e. γc = γe, while the γπ corresponds to a difference mode, i.e. the odd mode leading to γπ = γo. Thus, the even and odd modes are the normal modes of the symmetrical coupled lines. The propagation constants of these modes are further written in terms of the line parameters of the symmetrical coupled lines:

•

The propagation constant of the even mode (sum mode): γe = ± Z11 Y11 + Y12 + Z12 Y11 + Y12 1 2

γe = ± Z11 + Z12 Y11 + Y12

•

1 2

11 5 29

The propagation constant of the odd mode (difference mode): γo = ± Z11 Y11 − Y12 − Z12 Y11 − Y12

11 5 25

+ Z11 Y12 + Z12 Y22 Z12 Y11 + Z22 Y12

For lossless asymmetrical coupled lines in a homogeneous medium, the propagation constants βc and βπ are computed from the above equations.

γo = ± Z11 − Z12 Y11 − Y12

1 2

1 2

11 5 30 For lossless symmetrical coupled lines, the even- and odd-mode propagation constants are reduced to the following expressions: βe = ± ω

L11 + L12 C11 − C12

βe = ± ω

Ld11 + 2Ld12 Cd11

βo = ± ω

L11 − L12 C11 + C12

βo = ± ω

Cd11 + 2Cd12 Ld11

a

b

11 5 31 The above equations have already been obtained in the form of equations (11.4.32c) and (11.4.33c) by using the even–odd mode analysis method. It is noted that

413

414

11 Coupled Transmission Lines

and βo = βo εr effo = βe = βo εr effe = ω c εr effe ω c εr effo . For an isolated line, the above equations are reduced to βe = βo = β = ω L11 C11 . Case #4: Symmetrical Coupled Lines in the Homogeneous Medium

The even-mode and odd-mode phase velocities are equal in a homogeneous medium, their propagation constants are also equal, i.e. γe = γo. Using equations (11.5.29) and (11.5.30), the following condition is obtained: Z12 Y11 = − Z11 Y12 ,

L12 C11 = − L11 C12 L12 C11 = L11 Cm , 11 5 32

d2 v 1 dx2 v2

=

a1

b1

v1

b2

a2

v2

d2 v1 − a 1 v 1 − b1 v 2 = 0 dx2 d2 v 2 − a2 v2 − b2 v1 = 0 dx2

The characteristic impedances of both the C and π-modes, computed in this section, are normally expressed in terms of the voltage-ratios (Rc, Rπ) of the C and the π-modes: a, c

b π

11 5 33 The C or π-mode phasor voltage traveling waves in the x-direction, on each of two conductors, is written as: v1 = v01 e − γ x

a,

v2 = v02 e − γ x

b

11 5 34

The coupled voltage waves, on the coupled transmission lines, are described by equation (11.5.9). It is rewritten below:

Rc =

L22 C22 − L11 C11 + F 2 L12 C22 − L11 C12

where, F =

L11 C11 − L22 C22

a, 2

b

On substituting equations (11.5.34) in the above equations, the following characteristic equations are obtained: γ2 v1 − a1 v1 − b1 v2 = 0

a

γ v2 − a2 v2 − b2 v1 = 0

b

2

11 5 37 Two values of the propagation constants, corresponding to the C or π-mode, are obtained. The voltage-ratio of either of the modes is

11.5.3 Modal Characteristic Impedance and Admittance

v2 Rπ = v1

a

11 5 36

where Cm is the capacitive mutual coupling, Cm = −C12.

v2 Rc = v1

11 5 35

Rπ =

v2 γ2 − a1 b2 = = 2 v1 b1 γ − a2

The following expressions, for the voltage ratios of the C or π−mode, i.e. for Rc and Rπ, are written by using equation (11.5.18) with the above equation: 1 1 a Rc = a2 − a1 + a2 − a1 2 + 4b1 b2 2 2b1 1 1 b Rπ = a2 − a1 − a2 − a1 2 + 4b1 b2 2 2b1 11 5 39 Using equations (11.5.12) and (11.5.19), the above equations could be rewritten in terms of the equivalent circuit elements as follows:

L22 C22 − L11 C11 − F 2 L12 C22 − L11 C12

+ 4 L12 C11 − L22 C12

The circuit elements of the asymmetrical coupled lines could be determined from the static analysis. The closedform expressions for these circuit elements of the asymmetrically coupled microstrips are presented in subsection (12.2.1) of chapter 12. The symmetrical coupled lines have a1 = a2 and b1 = b2. In this case, the above voltage-ratios are reduced to Rc = + 1, corresponding to the even mode

a

b Rπ = − 1, corresponding to the odd mode 11 5 41

11 5 38

L12 C22 − L11 C12

b 11 5 40 c

Therefore, to excite the C-mode with Rc + 1, the voltages v1 and v2 must have the same polarity; and to excite the π−mode with Rπ − 1, the voltages v1 and v2 must have opposite polarity. The coupled lines phasor equations, in terms of the Z-matrix, are given by equation (11.5.7a). Substitution of the assumed solutions (11.5.34) in equation (11.5.7a), using a similar solution for the current waves, and taking γ = γc for the C mode, provides the following expressions:

11.5 Wave Equation for Coupled Transmission Lines

γc v1 = Z11 i1 + Z12 i2

a

γc v2 = Z22 i2 + Z12 i1

b,

γ v2 − Z12 i1 i2 = c Z22 11 5 42

On substituting i2 in equation (11.5.42a), γc Z22 v1 = Z11 Z22 i1 + Z12 γc v2 − Z12 i1 γc

v2 Z22 − Z12 v1

= Z11 Z22 − Z212

i1 v1

11 5 43

The characteristics admittance of the line #1 for the C-mode is Yc1 = i1/v1 and its voltage-ratio is Rc = v2/v1. The Yc1 for line #1 follows from the above equation: Yc1 = γc

Z22 − Z12 Rc 1 , = Zc1 Z11 Z22 − Z212

γe 1 = Z11 + Z12 Ze

11 5 45

Using equation (11.5.29), the characteristics impedance of the even mode is further written below in terms of the primary line constants of the symmetrical coupled lines: Ze =

Z11 + Z12 = γe

Z0 =

Z11 + Z12 Y11 + Y12

a

Ze =

L11 + L12 C11 − C12

b

11 5 46 The above relation is already incorporated in equation (11.4.32a) by using the even–odd mode analysis method. In the above equation, a negative sign is taken for C12. Further, the characteristics admittance for the π-mode is Z22 − Z12 Rπ 1 = 2 Zπ1 Z11 Z22 − Z12

Z11 − Z12 = γ0

Z11 − Z12 Z11 − Z12 Y11 − Y12

=

Z11 − Z12 Y11 − Y12

a

Zo =

L11 − L12 C11 + C12

b 11 5 49

Again, the above relation is incorporated in equation (11.4.33a) by using the even–odd mode analysis method. The C and π-modes on line #2 of the asymmetrical coupled lines could be considered to obtain its characteristic admittance for both modes. The current i1 is solved from equation (11.5.42a) to get the relation i1 = (γc v1 − Z12 i2)/Z11. On substituting i1 in equation (11.5.42b), the following expression is obtained: γc Z11 v2 = Z11 Z22 i2 + Z12 γc v1 − Z12 i2 γc Z11 v2 − Z12 v1 = Z11 Z22 − Z212 i2 γc Z11 − Z12

v1 v2

= Z11 Z22 − Z212

11 5 47

The characteristics admittance of the odd mode follows from the above equations for Rπ = −1, Z11 = Z22 and γπ = γo:

i2 v2 11 5 50

Y11 + Y12

=

Yπ1 = γπ

11 5 48

Using equation (11.5.30), the characteristics impedance for the odd mode can be further written in terms of the primary line constants of the symmetrical coupled lines:

Z11 + Z12 Z11 + Z12

γ0 1 = Z11 − Z12 Z0

11 5 44

where Zc1 is the characteristics impedance of the C-mode supporting line #1. The symmetrical coupled lines, supporting the even mode, has Z11 = Z22, Rc = +1, and γc = γe. The even-mode characteristics admittance follows from the above equation: Ye =

Yo =

The characteristics admittance of line #2 of a coupled transmission lines for the C-mode is Yc2 = i2/v2 and its voltage-ratio is Rc = v2/v1. The Yc2 for the line #2, from the above equation, is obtained as: Yc2 =

γc Z11 Rc − Z12 1 = Rc Z11 Z22 − Z212 Zc2

11 5 51

Likewise, the characteristics admittance of line #2 of the π-mode is Yπ2 =

γπ Z11 Rπ − Z12 1 = 2 Rπ Z11 Z22 − Z12 Zπ2

11 5 52

The characteristic impedance/admittance of both modes is related through the voltage-ratio of both modes given by equation (11.5.33). The C and π-modes are excited by the identical pair of unequal voltages. To excite the C-mode, both voltages have the same polarity. However, in the case, the π-mode one voltage on the

415

416

11 Coupled Transmission Lines

strip conductor has the opposite polarity. It helps to define the voltage-ratio of one mode in terms of current-ratio of another mode as follows [B.17, J.10]: Rc =

V2c I1π = − V1c I2π

a,

Rπ =

V2π I1c = − V1π I2c

b

B.5 Gupta, K.C.; Garg R.; Bahl I.; Bhartia P.: Microstrip B.6 B.7

11 5 53 Using the relations Vπ i = Zπ iIπ i(i = 1, 2) and Vc i = Zc iIc i(i = 1, 2), the following expressions are obtained: I1π V1π Z1π Z2π 1 = − = − I2π V2π Z2π Z1π Rπ I1c V1c Z1c Z2c 1 = − = − Rπ = − I2c V2c Z2c Z1c Rc Yc1 Yπ1 = = − Rc Rπ Therefore, Yc2 Yπ2

Rc = −

a b

B.11

c

B.12

For the symmetrical coupled lines, even-mode characteristics admittances for both the lines are equal. So is the case of the odd-mode characteristics admittances. However, for asymmetrical coupled lines, the C and π-mode characteristics admittances, for line #1 and line #2, are unequal: For symmetrical coupled lines Ye1 = Ye2 , Yo1 = Yo2 Yc2 , Yπ1

Yπ2

B.9

B.10

11 5 54

For asymmetrical coupled lines Yc1

B.8

B.13 B.14 B.15 B.16

a b

B.17

11 5 55 The capacitive and inductive coupling coefficients, kC and kL, and the final coupling coefficient C of the asymmetrical coupled lines are obtained from equations (11.3.12), (11.3.24), and (11.3.25). The concepts and formulations of this chapter are used further in Chapter 12 to analyze the coupled microstrips and coupled CPW structures.

B.18

B.19 B.20

B.21

References Books

B.22

B.1 Fache, N.; Frank, O.; Daniel, D.Z.: Electromagnetic

and Circuit Modelling of Multiconductor Transmission Lines, Clarendon Press, Oxford, NY, 1993. B.2 Bewley, L.V.: Travelling Waves on Transmission Systems, Wiley, New York, 1951. B.3 Noyan, K.; Aksun M.I.: Modern Microwave Circuits, Artech House, Boston, 2003. B.4 Hoffmann, R.: Microwave Integrated Circuit Handbook, Artech House, Boston, 1985.

B.23 B.24

B.25

Lines and Slot Lines, 2nd Edition, Artech House, 1996. Edward, T.C.: Foundations for Microstrip Circuit Design, John Wiley & Sons, New York, 1987. Fooks, E.H.; Zakarevicius, R.A.: Microwave Engineering Using Microstrip Circuits, Prentice-Hall PTR, New Jersey, 1989. Di Paolo, F.: Network, and Devices Using Planar Transmission Lines, CRC Press, New York, 2000. Edward, T.C.; Steer, M.B.: Foundations of Interconnects and Microstrip Circuit Design, 3rd Edition, John Wiley & Sons, New York, 2000. Collin, R.E.: Foundations for Microwave Engineering, 2nd Edition, McGraw-Hill, Inc., New York, 1992. Collin, R.E.: Field Theory of Guided Waves, IEEE Press, New York, 1991. Simon, R.N.: Coplanar Waveguide Circuits Components and Systems, John Wiley, New York, 2001. Wolff, I.: Coplanar Microwave Integrated Circuits, John Wiley & Sons, New York, 2006. Young, L. (Editor): Advances in Microwaves, Vol. 8, Academic Press, 1974. Young, L. (Editor): Advances in Microwaves, Vol. 1, Academic Press, 1966. Kneppo, I.; Fabian J.; Bezousek P.; Hrnicko P.; Pavel M.: Microwave Integrated Circuits, Chapman & Hall, London, 1994. Mongia, R.; Bhartia, P.; Bahl, I.J.: R. F. and Microwave Coupled-Line Circuits, 2nd Edition, Artech-House, Boston, 1999. Hung-Chia, H.: Coupled Mode Theory- As Applied to Microwaves and Optical Transmission, VNU Science Press, the Netherland, 1984. Dworsky, L.N.: Modern Transmission Line Theory and Applications, John Wiley, New York, 1979. Costa, L.; Valtonen, M.: Implementation of Single and Coupled Microstrip Lines in APLAC, CT-33, ISBN 951222-3881-0, Helsinki University, Finland, Dec. 1997. Djordjevic, A.R; Harrington, R.F.; Sarkar, T.K.; Bazdar, M.B.: Matrix Parameters for Multiconductor Transmission Lines – Software and User’s Manual, Norwood Artech House, 1989. Mattick, R.E.: Transmission Lines For Digital and Communication Networks. IEEE Press, New York, 1995 Rizzi, P.A.: Microwave Engineering- Passive Circuits, Prentice-Hall Int. Ed., New Jersey, 1988. Orfanidis, S.J.: Electromagnetic Waves and Antenna, Free Book on Web, ECE Department, RutgersUniversity, Piscataway, NJ, 2016. Kuznetsov, P.I.; Stratonovich, R.L.: The Propagation of Electromagnetic Waves in Multiconductor Transmission Lines, Pergamon Press, London, 1964.

References

Journals J.1 Yasumoto, K.: Coupled – mode formulation of

J.2

J.3

J.4

J.5

J.6

J.7

J.8

J.9

J.10

J.11

J.12

J.13

multilayered and multiconductor transmission lines, IEEE Trans. Microw. Theory Tech., Vol. MTT-44, No. 4, pp. 585–590, April. 1996. Pipes, L.A.: Matrix theory of multiconductor transmission lines, Philos. Mag., Vol. 24, No. 159, pp. 97–11, 1937. Bazdar, M.B.; Djordjevic, A.R.; Harrington, R.F.; Mautz, J.R.; Sarkar, T.K.: Evaluation of quasi- static matrix parameters for multiconductor transmission lines using Galerkin’s method, IEEE Trans. Microw. Theory Tech., Vol. MTT- 42, No. 7, pp. 1223–1228, July 1984. Tripathi, V.K.: A dispersion model for coupled microstrips, IEEE Trans. Microw. Theory Tech., Vol. MTT-34, No. 1, pp. 66–67, 1986. Marx, K.D.: Propagation modes, equivalent circuits, and characteristic terminations for multiconductor transmission lines with inhomogeneous dielectrics, IEEE Trans. Microw. Theory Tech., Vol. MTT-21, No. 7, pp. 450–457, July 1973. Rao, J.S.; Joshi, K.K.; Das, B.N.: Analysis of small aperture coupling between the rectangular waveguide and microstrip line, IEEE Trans. Microw. Theory Tech., Vol. MTT-29, No. 2, pp. 150–154, Feb 1981. Miao, J.-F.; Itoh, T.: Coupling between microstrip and image guide through a small aperture in the common ground plane, IEEE Trans. Microw. Theory Tech., Vol. MTT-31, No. 4, pp. 361–363, April 1983. Sobhy, M.I.; Hosny, E.A.: The design of directional couplers using exponential lines in inhomogeneous media, IEEE Trans. Microw. Theory Tech., Vol. MTT-30, No. 1, pp. 71–76, Jan 1982. Adair, J.E.; Haddad, G.I.: Coupled-mode analysis of non-uniform coupled transmission lines, IEEE Trans. Microw. Theory Tech., Vol. MTT-17, No. 10, pp. 746–752, Oct. 1969. Tripathi, V.K.: Asymmetric coupled transmission lines in an inhomogeneous medium, IEEE Trans. Microw. Theory Tech., Vol. MTT-23, No. 9, pp. 734–739, 1975. Allan, J.L.: Non-symmetrical coupled lines in an inhomogeneous dielectric medium, Int. J. Electron., Vol. 38, pp. 337–347, 1975. Tripathi, V.K.; Chang, C.L.: Quasi-TEM parameters of non-symmetrical coupled microstrip lines, Int. J. Electron., Vol. 45, No. 2, pp. 215–223, 1978. Tripathi, V.K.: Equivalent circuits and characteristics of inhomogeneous non-symmetrical coupled line two-port

J.14

J.15

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J.18

J.19

J.20

J.21

J.22

J.23

J.24

J.25

J.26

circuits, IEEE Trans. Microw. Theory Tech., Vol. MTT25, No. 2, pp. 140–142, Feb. 1977. Jones, E.M.T.; Bolljahn, T.: Coupled strip transmission line filters and directional couplers, IEEE Trans. Microw. Theory Tech., Vol. MTT-4, No. 2, pp. 75–81, April., 1956. Krage, M.K; Haddad, G.I.: Characteristics of coupled microstrip transmission lines-I: Coupled mode formulation of inhomogeneous lines, IEEE Trans. Microw. Theory Tech., Vol. MTT-18, No. 4, pp. 217–222, 1970. Verma, A.K.; Kumar, R.: A new dispersion model for microstrip line, IEEE Trans. Microw. Theory Tech., Vol. MTT-46, No. 8, pp. 1183–1187, 1998. Ikalainen, P.K.; Matthaei, G.L.: Wide-band, forward – coupling microstrip hybrid with high directivity, IEEE Trans. Microw. Theory Tech., Vol. MTT-35, No. 8, pp. 719–725, Aug. 1987. Verma, A. K.; Nasimuddin, N.; Ojha, R.K.: Synthesis and analysis of optimum directivity suspended microstrip coupler, Asia-Pacific Microwave Conference, APMC -Dec. 2000, Sydney, Australia. Bryant, T.G.; Weiss, J.A.: Parameters of microstrip transmission lines and coupled pairs of microstrip lines, IEEE Trans. Microw. Theory Tech., Vol. MTT-16, No. 12, pp. 1021–1027, 1968. Hammerstad, E.; Jensen, O.: Accurate models for Microstrip Computer-Aided Design, IEEE., MTT- S, Int. Microwave Symposium Digest, pp. 407–409, 1980. Garg, R.; Bahl, I.J.: Characteristics of coupled microstrip lines, IEEE Trans. Microw. Theory Tech., Vol. MTT-27, No. 7, pp. 700–705, 1979. Also corrections in, Vol. MTT- 28, No., pp. 272, 1980. Wei, C.; Harrington, R.F.; Mautz, J.R.; Sarkar, T.K.: Multiconductor transmission lines multilayered dielectric media, IEEE Trans. Microw. Theory Tech., Vol. MTT-32, No. 4, pp. 439–450, April 1984. Krage, M.K; Haddad, G.I.: Characteristics of coupled microstrip transmission lines-II: Evaluation of coupled lines parameters, IEEE Trans. Microw. Theory Tech., Vol. MTT-18, No. 4, pp. 222–228, 1970. Levy, R.: Transmission line directional couplers for very broad-band operation, Proc. IEE, Vol. 112, No. 3, March 1965. Carson, J.R.; Hoyt, R.S.: Propagation of periodic currents over a system of parallel wires, Bell Syst. Tech. J., Vol. 6, pp.495–545, 1927. Rice, S.O.: Steady-state solutions of transmission line equations, Bell Syst. Tech. J., Vol. 20, No. 2, pp. 131– 178, April 1941.

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12 Planar Coupled Transmission Lines Introduction The basic theoretical concepts and formulations of the coupled lines are presented in chapter 11. This chapter is concerned with their applications to the planar coupled transmission lines, such as the coupled microstrips and CPW. These planar coupled lines are needed for the design of important microwave components such as directional couplers, filters, phase shifters, delay lines, 3 dB-hybrids, matching networks, etc. The circuits and components based on the coupled lines are discussed in several books [B.1–B.11]. Objectives

• • •• • •

To compute static and dynamic parameters of symmetrical coupled microstrip lines. To compute static and dynamic parameters of asymmetrical coupled microstrip lines. To compute static parameters of coupled CPW. To obtain the 4-ports and 2-ports [Z], [Y], and [ABCD] parameters of coupled strip lines. To obtain the 4-ports and 2-ports [Z], [Y], and [ABCD] parameters of coupled microstrip lines. To obtain the [ABCD] parameters of the C-mode and π-mode of asymmetrical coupled microstrip lines.

12.1 Line Parameters of Symmetric Edge Coupled Microstrips Figure (12.1) shows coupled microstrips in the inhomogeneous media. The medium could be homogeneous also. This section presents both the static and frequencydependent closed-form models to compute the line parameters of symmetrical edge coupled microstrips in terms of physical dimensions. It also accounts for the effects of the top shield, and the finite conductor thickness in the models [B.1]. Bryant and Weiss [J.1] and also Krage and Haddad [J.21] have provided the static numerical results on

the parameters of the symmetrically coupled microstrips. The method of moment-based software is available for such computation [B.2, J.2, J.3]. A summary of the frequencydependent models, developed by Kirschning and Jansen is also presented [J.4], for the symmetrical coupled microstrip lines. The methods are also available for computation of line parameters for the broadside coupled microstrips [J.5, J.6]. In chapter 14, the variational method, applicable to the microstrip lines in the layered dielectric medium, is presented to compute the static coupled line parameters more systematically. 12.1.1 Static Models for Even- and Odd-Mode Relative Permittivity and Characteristic Impedances of Edge Coupled Microstrips Figure (12.1) shows shielded asymmetrical coupled microstrips of strip widths (w1,w2), separation s, and strip conductor thickness t. The substrate is having relative permittivity εr, thickness h1, and is covered with a shield height h2. However, in the case of symmetrically coupled microstrips, strip conductors have width w1 = w2 = w. The presence of a top shield creates additional capacitance in the air medium, between the top shield and strip conductors, thereby reducing the effective relative permittivity and characteristics impedance of coupled microstrips. The physical dimensions of the coupled microstrip are normalized with respect to the substrate thickness: u = w h1 , g = s h1 , th = t h1 , hs = h2 h1 12 1 1 The expressions summarized below have accuracy within 1% in the range 0.1≤ u ≤10, 0.1≤ g ≤10, and 1≤ εr ≤ 18. These expressions are used in the commercial circuit simulator for the microwave design applications [B.1, B.3, J.4, J.7]. Even-Mode Effective Relative Permittivity

Following the process used in subsection (8.2.1) of chapter 8 applicable to the single conductor microstrip, an

Introduction to Modern Planar Transmission Lines: Physical, Analytical, and Circuit Models Approach, First Edition. Anand K. Verma. © 2021 John Wiley & Sons, Inc. Published 2021 by John Wiley & Sons, Inc.

420

12 Planar Coupled Transmission Lines

h2

w1

s

w2

h1 Figure 12.1

structure-dependent variable p is given below. It determines the parameter ae(p) of equation (12.1.3b). The parameter be(εr) is permittivity dependent. The closedform expressions for parameters ae(p) and be(εr) are summarized below:

t εr

Asymmetrical coupled microstrip lines with a top shield. (For the symmetrical coupled lines, w1 = w2 = w.)

2

ae p = 1 +

expression is written below for the even-mode effective relative permittivity of the symmetrically coupled microstrips, in terms of the even-mode filling-factor qe [J.7]: εr effe f = 0 =

εr + 1 2

+

Where even mode filling – factor Fe + 1 qe = 2

Ft,e =

Fs,e =

1 733 , for hs ≤ 39 hs

1

d

for, hs > 39

12 1 3 where F∞, e is the even-mode-related function of open coupled microstrips, i.e. for the case h2 ∞. The conductor thickness is accounted for by using a correction factor Ft,e, and the shield height is accounted for by another correction factor Fs,e of March [J.8]. The more accurate expression is also available [J.9]. The

qo = q ∞ ,o − qt,o qs,o

a , where, qt,o =

0 053

b c

b

c

tanh 1 626 + 0 107hs −

εr − 0 9 εr + 3 0

u 20 + g2 + g exp − g 10 + g2

12 1 4

Odd-Mode Effective Relative Permittivity

The improved Hammerstad and Jensen model [B.1, J.4, J.7], to compute the static odd-mode effective relative permittivity, in terms of the odd-mode filling-factor, is given below: εr effo f = 0 =

b

2 ln 2 th π u

a

The above equations are a modification of equation (8.2.15) given in chapter 8.

a

− ae p be εr

3

p=

The even-mode function Fe is dependent on the structural parameters u, g, and εr. The even-mode function Fe, taking care of the conductor thickness and the top shield height, is Fe = F ∞ ,e − Ft,e Fs,e

p 18 1

a

12 1 2

10 p

ln 1 + be εr = 0 564

εr − 1 Fe 2

εr effe f = 0 = 1 + qe εr − 1

F ∞ ,e = 1 +

p p4 + 52 1 1 + ln 4 p + 0 432 49 18 7

εr + 1 + ao u, εr − εr eff f = 0 qo 2 + εr eff f = 0 , 12 1 5

where εr eff(f = 0) is the static effective relative permittivity of a single isolated microstrip line, given in subsection (8.2.1) of chapter 8. The parameter ao (u,εr) is given below: εr + 1 × 2 1 − exp − 0 179 u

ao u, εr = 0 7287 εr eff f = 0 −

12 1 6

The following modified odd-mode filling-factor qo takes into account the correction factors qt, o and qs,o, due to the conductor thickness and shield height, respectively:

2ℓn 2 th π u

b

q ∞ ,o = exp − co gdo , co = bo εr − b0 εr − 0 207 exp − 0 414 u bo εr =

qs,o =

0 747 εr , d0 = 0 593 + 0 694 exp − 0 562 u 0 15 + εr tanh

9 575 − 2 965 + 1 68hs − 0 311h2s , 7 − hs 1

for hs < 7 for hs ≥ 7

c

d,

12 1 7

12.1 Line Parameters of Symmetric Edge Coupled Microstrips

where q∞, o is the filling-factor of the open structure, i.e. for the case h2 ∞. In the case of the odd mode, the electric field confinement in the line structure is stronger. So, the odd mode case has hs>7, whereas for the even mode, due to weak field confinement, hs> 39 for taking qs,e = qs,o.

The closed-form expressions to compute the static evenmode characteristic impedance of open coupled microstrips of zero conductor thickness, due to Kirschning and Jansen [J.4], are summarized below: Zo f = 0

εr,eff f = 0 εreff,e f = 0

1 − Z o f = 0 η0

Q2

a

Q9 = ln Q7 Q8 + 1 16 5

b

Q8 = exp − 6 5 − 0 95 ln g − g 0 15 Q7 = 10 + 190g

2

1 + 82 3g

5

c

3

d

Q6 = 0 2305 + 1 281 3 ln g10 1 + g 5 8

10

+ 1 5 1 ln 1 + 0 598 g1 154

Even-Mode Characteristic Impedance

Zoe, ∞ εr , f = 0 =

Q10 = Q2 Q4 − Q5 exp ln u Q6 u − Q9

εr,eff f = 0 Q4

e

Q5 = 1 794 + 1 14 ln 1 + 0 638 g + 0 517g2 43

f

12 1 11 The effect of the top shield on the even/odd mode characteristics impedance is accounted below [B.4, J.8].

,

12 1 8 where the static characteristic impedance and effective relative permittivity, Zo(f = 0) and εr, eff(f = 0), of a single isolated microstrip, are computed in subsection (8.2.2) of chapter 8. The even mode εreff,e(f = 0) is determined from equation (12.1.2). The effect of the top shield and conductor thickness on the even-mode characteristic impedance Zoe, ∞(f = 0, εr) is accounted for separately. The parameter Q4 of equation (12.1.8) is evaluated from the following set of equations:

Effect of the Top Shield on Even-Mode Characteristic Impedance

The proximity of the top shield to strip conductors reduces the characteristic impedance of open coupled microstrips. It is accounted for by using the following even-mode impedance correction factor ΔZoe : Zoe f = 0, εr , u, hs = Zoe, ∞ f = 0, εr , u, hs where, ΔZoe

∞

a

− ΔZoe f = 0, εr , u, hs ΔZoe f = 0, εr = 1, u, hs f = 0, εr , u, hs = εr,effe f = 0, εr , u

b

12 1 12 2Q1 Q2 exp − g uQ3 + 2 − exp − g u − Q3 1 − 0 387 Q3 = 0 1975 + 16 6 + 8 4 g 6 + 241

Q4 =

ln

g10 1+ g 34

a

b

10

Q2 = 1 + 0 7519 g + 0 189 g2 31 Q1 = 0 8695 u0 194

c d 12 1 9

For the static even mode, εr,effe(f = 0, εr, u) and Zoe,∞(f = 0, εr, u, hs ∞) are computed using equations (12.1.2) and (12.1.8). The even-mode impedance correction factor on the air-substrate is obtained from the following set of equations [B.4]: ΔZoe f = 0,εr = 1,u,hs = f e u,hs ge g,hs f e u,hs = 1− tanh −1 A + B + C u u A = −4 351 1 + hs

1 842

C = −2 291 1 + hs

1 90

, B = 6 639 1 + hs

ge g,hs = 270 1− tanh D + E

1 861

1 + hs −F 1 + hs

Odd-Mode Characteristic Impedance

D = 0 747 sin 0 5 π x , E = 0 725 sin 0 5 π y

The expressions to compute the static odd-mode characteristic impedance are summarized below [J.4]:

log 10 F = 0 11−0 0947 g

Zoo, ∞ εr , f = 0 =

Zo f = 0

εreff f = 0 εreff,0 f = 0

1 − Z o f = 0 η0

log 10 x = 0 103 g−0 159, log 10 y = 0 0492 g−0 073 12 1 13

εreff f = 0 Q10 12 1 10

The odd mode εr,effo(f = 0) is determined from equation (12.1.5), and the parameter Q10 is computed as follows:

Effect of the Top Shield on Odd-Mode Characteristic Impedance

The expression for the odd-mode characteristic impedance of the top-shielded coupled microstrips, using the correction factor ΔZoo, is written below:

421

12 Planar Coupled Transmission Lines

Zoo f = 0,εr ,u,hs = Zoo,∞ f = 0,εr ,u,hs where, ΔZoo

∞

−ΔZoo f = 0,εr ,u,hs ΔZoo f = 0,εr = 1,u,hs f = 0,εr ,u,hs = εr,effo f = 0,εr ,u

a b

12 1 14 For the static odd mode, εr, effo(f = 0, εr, u) and Zoo(f = 0, εr, u, hs ∞) are computed using equations (12.1.5) and (12.1.10). The odd-mode impedance correction factor, on the air-substrate, is obtained from the following set of equations [B.4]: ΔZoo f = 0, εr = 1, u, hs = f o u, hs go g, hs f o u, hs = uJ , J = tanh 1 + hs

1 585

go g, hs = 270 1 − tanh G + K

6

1 + hs − L 1 + hs

The Δu is a change in the strip width of a single conductor [J.10]. To account for the effect of the strip conductor thickness, the normalized strip-width u is replaced by the thickness-dependent strip-width ut,e, and ut,o, for the even mode and odd mode, in the expressions of even and odd modes effective relative permittivity and characteristic impedance.

Another Simple Closed-Form Expression

The even- and odd-mode line capacitances can be computed from the following expressions obtained using the conformal mapping method [J.11]: Ce = 2ε0 εr

K k1 K k2 + K k1 K k2

a

Co = 2ε0 εr

K k3 K k4 + K k3 K k4

b

G = 2 178 − 0 796 g, log 10 20 429 g0 174 , g > 0 858

K=

g ≤ 0 858

1 30,

where, k1 = tanh πw 4h tanh π w + s

2 51 g − 0 462 , g > 0 873

L=

k2 = tanh πw 4 h + πw

g ≤ 0 873

2 674,

tanh π w + s

k3 = tanh πw 4h coth π w + s 12 1 15

4h 4 h + πw

4h , k4 = w w + s 12 1 17

The K(k) is the first kind elliptical integral and Effect of Conductor Thickness on Microstrip

The conductor thickness increases the normalized effective even- and odd-mode strip conductor widths as follows: ut,e =u+Δu 1−0 5 exp −0 69Δu Δt

a

ut,o =ut,e +Δt

b

where, Δt= Δu=

th g εr

c

1 25 2 4π u−2 th th 1+ ln + π th 1+ exp −100 u−1 2π

d

s/h = 0.1

90

Present method Kirschning–Jensen Galerkin method HFSS

0.6

70

1.1 2.1

50 30

Odd mode ch. impedance (Ω)

12 1 16 Even mode ch. impedance (Ω)

422

K k =K

teristic impedance are computed using equations (11.4.32) and (11.4.33) of chapter 11. Likewise, the even- and oddmode effective relative permittivities are computed as εr,eff, e/o = Ce, o(εr)/Ce, o(εr = 1). Figure (12.2a and b) compares the results of the even- and odd-mode characteristic impedance computed by the present method against other closed-form models [J.4], Galerkin’s method, and results of the EM-simulator, HFSS. The method has accuracy within 5% even for a very tight coupling

90 2.1 70

1 − k2 . The even- and odd-mode charac-

Present method Kirschning–Jensen Galerkin method HFSS

1.1 0.6

50

s/h = 0.1

30 10

10 0.0

0.5

1.0

1.5

2.0

2.5

w/h-Ratio (a) Even-mode characteristic impedance. Figure 12.2

0.0

0.5

1.0

1.5

2.0

2.5

w/h-Ratio (b) Odd-mode characteristic impedance.

Even-and odd-mode characteristic impedance of symmetrical couple microstrips on an alumina substrate with εr = 9.8. Source: Abbosh [J.11]. © 2009 The Institution of Engineering and Technology.

12.1 Line Parameters of Symmetric Edge Coupled Microstrips

(s/h = 0.0001), whereas the Kirschning–Jansen model has error 20% for s/h = 0.1. The effect of a conductor thickness on the line parameters can be accounted for by computing the normalized effective line widths for both the modes [B.1]: w eff,e w Δw Δ w h1 = + 1 − 0 5 exp − 0 69 h1 h1 h1 Δ t h1 weff,o w eff,e Δt = + h1 h1 h1 t 1 t h1 = where h 1 ε r s h1

a

Fe f n = P1 P2 P3 P4 + 0 1844P7 f n

1 5763

P1 = 0 27488 + u 0 6315 + 0 525 1 + 0 0157 f n − 0 065683 exp − 8 7513u P2 = 0 33622 1 − exp − 0 03442εr P3 = 0 0363 exp − 4 6u 1 − exp − f n 38 7 P4 = 1 + 2 751 1 − exp − εr 15 916

b c 12 1 18

20

P5 = 0 334 exp − 3 3 f n 15 P6 = P5 exp − f n 18

3

4 97

8

+ 0 746

0 368

P7 = 1 + 4 069P6 g0 479 exp − 1 347g0 595 − 0 17g2 5 12 1 20

Bedair has also provided the closed-form expressions to compute the odd-mode capacitance [J.42]. Even-Mode Characteristics Impedance

It is computed from the following expression [J.4]:

12.1.2 Frequency-Dependent Models of Edge Coupled Microstrip Lines

Zo,e εr , f n = Zo,e εr , f n = 0

At high frequency, normally above 4 GHz, the even- and odd-mode effective relative permittivity and characteristic impedance are frequency-dependent. Both modes have different field distributions, resulting in different frequency dependence. Getsinger has extended his model to the symmetrically coupled microstrip [J.12]. However, it has lower accuracy. Garg and Bahl [J.13] have also given simpler but less accurate expressions for the frequency-dependent characteristic impedance of both modes. The Getsinger’s model has the advantage of its applicability to the asymmetrically coupled microstrips also. It is discussed in the next section. Kirschning and Jansen [J.4] have provided a set of more accurate closed-form expressions to compute the frequencydependent line parameters of edge-coupled symmetrical microstrip, shown in Fig (12.1). These expressions are summarized below. The dispersion model has an accuracy better than 2.5% over the normalized frequency fn up to 20; and better than 1.5% for fn ≤ 15, and εr ≤ 12.9, where fn = f. h GHz − mm.

The even-mode dispersion is given by the following expression, in the format of Getsinger’s model [J.4]: εreff,e f n = εr −

εr − εreff,e f n = 0 1 + Fe f n

0 9408 − de

12 1 19

The static even-mode effective relative permittivity is computed using equation (12.1.2), and the parameter Fe(f ) is evaluated using the following set of equations:

εr,eff f = 0

− 0 9603 Ce

Q0

− 0 9603

Q0

,

12 1 21 where the static even-mode characteristic impedance Z0,e(εr, fn = 0) is given by equation (12.1.8). The effective relative permittivity εr, eff(fn = 0) for the single isolated microstrip has been computed in subsection (8.2.1) of chapter 8. The parameters Ce and de are computed from the following set of expressions: Ce = 1 + 1 275 1− exp −0 004625pe εr

1 674

f n 18 365

2 745

−Q12 + Q16 −Q17 + Q18 + Q20 exp −22 2u1 92

de = 5 086 qe re 0 3838 + 0 386qe 6

1 + 1 299re × 1−εr 1 + 10 1−εr 6 pe = 4 766 exp −3 228u0 641 , qe = 0 016 + 0 0514εr Q21

4 524

12

, re = f n 28 843

12 1 22

The expressions for the variables Q11–Q21 are further given below: Q11 = 0 893 1 − 0 3

Even-Mode Effective Relative Permittivity

Ce

0 9408 εr,eff f = 0

Q12 = 2 121 f n 20

1 + 0 7 εr − 1 4 91

1 + Q11 f n 20

4 91

exp − 2 87g g0 902 Q13 = 1 + 0 038 εr 8 Q14 = 1 + 1 203

51

εr 15

4

1 + εr 15

4

Q15 = 1 887 exp − 1 5 g0 84 gQ14 1 + 0 41 f n 15

3

u2

Q13

0 125 + u1 626

Q13

423

424

12 Planar Coupled Transmission Lines

1 + 0 403 εr − 1

Q16 = 1 + 9

2

Q17 = 0 394 1 − exp − 1 47 u 7 1 − exp − 4 25 f n 20

0 672

×

1 87

1 − exp − 2 13 u 8

Q18 = 0 61

where Zo(εr, fn), the frequency-dependent characteristic impedance of a single isolated microstrip, is computed by equation (8.3.36) of chapter 8. Equation (12.1.10) computes the odd-mode characteristic impedance Zo,o(εr, fn = 0). The Q-variables are obtained from the following set of expressions:

Q15

1 593

1 + 6 544g4 17 4

49

Q19 = 0 21g

1 + 0 18g

1 + 0 1u

1 + 0 1 εr − 1

Q20 = 0 09 + 1

Q21 = 1 − 42 54 g

0 133

27

2

1 + f n 24

3

Q19

exp − 0 812g u

Q22 = 0 925 f n Q26

Q23 = 1 + 0 005f n Q27

12 1 23

Q25 = 0 3f 2n 10 + f 2n Q26 = 30 − 22 2 Q27 = 0 4g

Odd-Mode Effective Relative Permittivity

The odd-mode dispersion expression, in the format of Getsinger’s model, is given below [J.4]:

tan

−1

2 481 εr 8

P10 = 0 242 εr − 1

1 5763

1 + 0 0576 εr − 1

P9 = P8 − 0 7913 1 − exp − f n 20

1 424

×

0 946

0 55

P11 = 0 6366 exp − 0 3401f n − 1 tan − 1 1 263 u 3 P12 = P9 +

1 − P9

P13 = 1 695P10

1 629

1 + 1 183u1 376

0 414 + 1 605P10

P14 = 0 8928 + 0 1072 1 − exp − 0 42 f n 20

3 215

P15 = 1 − 0 8928 P14 1 + P11 P12 exp − P13 g

1 092

12 1 25 The coefficients P1–P4 are taken from equation (12.1.20). The absolute value is taken in the last expression of equation (12.1.25). Odd-Mode Characteristics Impedance

The frequency-dependent odd-mode characteristics impedance is computed from the following expression: Zo,o εr , f n = Zo εr ,f n +

Q28 = 0 149 εr − 1

12

2 5 εr − 1 3

2

3 575 + u0 894 5 + εr − 1

1 + 3 εr − 1 13 15

5 + εr − 1

94 5 + 0 038 εr − 1

Q29 = 15 16 1 + 0 196 εr − 1

1 + 0 025u2

4 29

1 + 2 333 εr − 1

εr − 1 13

1+

19

12

2

− Q29

15

3

2

12 1 24

The static odd-mode effective relative permittivity is computed using equation (12.1.5), and the parameter Fo(f ) is evaluated using the following set of equations:

P8 = 0 7168 1 + 1 076

1 + 0 812 f n 15

12 1 27

εr − εr,effo f = 0 = εr − 1 + Fo f

Fo f n = P1 P2 P3 P4 + 0 1844 f n P15

0 84

1 536

1 + 0 3 f n 30

Q24 = 2 506Q28 u0 894 1 + 1 3u f n 99 25

25

1 + 0 033u2 5

εr,effo f n

1 536

Zo,o εr , f = 0 εr,eff,o f εr,eff,o f = 0 Q22 −Zo εr , f Q23 , 1 + Q24 + 0 46g 2 2 Q25

12 1 26

12.2 Line Parameters of Asymmetric Coupled Microstrips This section presents the closed-form models to compute the static and frequency-dependent parameters of the asymmetrically coupled microstrips. The static parameters are computed by two closed-form models – one computes the direct capacitances of the π-circuit model, shown in Fig (11.9b) of chapter 11, using the results of the symmetrically coupled microstrips [J.14], while another computes these capacitances by the partial capacitance method [J.15]. The more accurate former model is used in a CAD-package also [B.1]. The network analog method is also used to compute the capacitance matrix of the asymmetrically coupled microstrips [J.16]. The dispersion in the asymmetrically coupled microstrips is accounted for by extending the Getsinger’s model [J.16, J.17]. 12.2.1 Static Parameters of Asymmetrically Coupled Microstrips The expressions for the propagation constants and characteristic impedances of two normal modes, i.e. the C and π-modes, of an asymmetrically coupled microstrips have been obtained in section (11.5), chapter 11. These are also obtained in terms of the elements of the capacitance and inductance matrices. The inductance of the microstrips on a dielectric substrate is independent of the relative permittivity. Therefore, the inductance matrix is determined from the capacitance matrix on the air-substrate:

12.2 Line Parameters of Asymmetric Coupled Microstrips

C εr = 1 = L = μ0 εo

C11 εr = 1

− C12 εr = 1

− C12 εr = 1

C22 εr = 1

C11 εr = 1

− C12 εr = 1

− C12 εr = 1

C22 εr = 1

a

−1

b 12 2 1

The following expressions are obtained, from equations (11.5.20)–(11.5.22) of chapter 11, on using (εr, eff, i = (βi/β0)2, i = c, π), for the static effective relative permittivity of the C and π-modes: c2 L11 C11 + L22 C22 − 2L12 C12 + F 2 c2 L11 C11 + L22 C22 − 2L12 C12 − F εr,eff,π f = 0 = 2 where, F = 4 L12 C11 − L22 C12 L12 C22 − L11 C12 εr,eff,c f = 0 =

+ L11 C11 − L22 C22

2 1 2

a b

c 12 2 2

where c is the velocity of EM-wave in free space. The static modal characteristics impedance Zc1 and Zπ1 of the line #1 for the C and π-modes are given by equations (11.5.44) and (11.5.47) of chapter 11. These equations could be rewritten in terms of the circuit parameters. The expression for the Zc1 of the C-mode is obtained below. The elements of the [L]-matrix are related to the elements of the [C]-matrix through the phase velocity of the C-mode: L C =

εreff c , c2

L11 L12 L12 L22 L11 =

L11 L12

=

L12 L22

εreff c C11 − C12 c2 − C12 C22

εreff C = 2 c C11 C22 − C212

εreff c c2 C11 C22 − C212

εreff c L12 = 2 c C11 C22 − C212 εreff c L22 = 2 c C11 C22 − C212

C22

−1

C22 C12 C12 C11 a

εreff c 1 1 = c C11 − C12 Rc YC1

Zc1 =

C12

Rc = − Zc2 Zc1

1 Zc2 Rπ Zc1 Z22 − Z12 Rc Z11 Rc − Z12

= Rc

L11 L22 − L212

c εreff,c L22 − L12 Rc

Z12 Rπ − Z22 L12 Rπ − L22 = Z11 Rπ − Z12 L11 Rπ − L12 Z12 Rc − Z22 L12 Rc − L22 = Rπ = Z11 Rc − Z12 L11 Rc − L12

12 2 6

a b 12 2 7

Using equation (12.2.7a), equation (12.2.4) for Zc1 could be rewritten, in terms of the voltage ratio Rπ, as follows: Zc1 =

c L12 L11 − Rπ εreff,c

12 2 8

Following the above process, expressions for the static characteristic impedance Zπ1 of the π-mode of line #1 can also be obtained. The static characteristic impedance of C and π-modes for both line #1 and line #2 are summarized below [B.5, J.18–J.20]:

c

12 2 4

b

Rc =

b

On using Z11 = jωL11, Z22 = jωL22, Z12 = jωL12, equation (11.5.44) for the characteristics impedance of C-mode on line #1 is written as

a

The above equations provide the following identities:

= C11

12 2 5

The voltage ratio Rc of C-mode is given by equation (11.5.40a) of chapter 11. The above equation of Zc1 can also be expressed in terms of the elements of [L]-matrix of asymmetrical coupled lines and the voltage ratio Rπ of the π-mode. To do so, the relation between Rc and Rπ in terms of the circuit elements is needed. Using equations (11.5.44), (11.5.51), and (11.5.53) of chapter 11, the following relations are obtained:

Conductor #1 Zc1 =

12 2 3

Zc1 =

Using equation (12.2.3) with the above equations, the following expression for the Zc1 of the C-mode is obtained:

Zπ1 = =

εr,eff,c f = 0 c

c εr,eff,c f = 0 εr,eff,π f = 0 c c εr,eff,π f = 0

L11 − L12

1 C11 − C12 Rc 1 Rπ = Yc1

a

1 C11 − C12 Rπ L11 − L12 Rc =

1 Yπ1

b

Conductor #2 Zc2 = − Rc Rπ Zc1

c

Zπ2 = − Rc Rπ Zπ1

d 12 2 9

425

426

12 Planar Coupled Transmission Lines

To use the above expressions, the elements of the capacitance matrix are computed. At first, the direct capacitances are computed, using the closed-form expressions; and next, the capacitance matrix is obtained. The method of moments computes all elements of a capacitance matrix more effectively [B.2, J.2, J.3].

Figure (11.9b) shows the direct capacitances of the π-circuit model of the asymmetrically coupled microstrips. The direct capacitances of the asymmetrically coupled microstrips are obtained below on modifying the direct capacitances of the symmetrically coupled microstrips [J.14]. The following normalized dimensions are used: u2 = w 2 h,

g=s h

12 2 10

Equation (11.4.34) of chapter 11 compute the direct capacitances Cd11,Sym u, g, εr and Cd12,Sym u, g, εr for the symmetrically coupled microstrips of strip widths, w1 = w2 = w. The mutual coupling capacitance, Cd12,asym u1 , u2 , g, εr , of the asymmetrically coupled microstrips is obtained from the following average value of two symmetric coupled microstrips, each having strip widths u = u1 and u = u2, respectively: Cd12,asym u1 , u2 , g, εr 1 Km

2

= Cd12,sym u1 , g, εr

− Km

+ Cd12,sym u2 , g, εr

− Km

12 2 11 The exponent Km is determined from the empirical expression: Km = 0 95 + 0 33 ln u1 u2 − 0 4 ln g

ΔC1 = 1−Ks εr

Cd12,sym u1 ,g,εr −Cd12,asym u1 ,u2 ,g,εr

a

ΔC2 = 1−Ks εr

Cd12,sym u2 ,g,εr −Cd12,asym u1 ,u2 ,g,εr

b

12 2 14 The relative permittivity-dependent parameter Ks(εr) is given by

The Method Based on Symmetrically Coupled Microstrips

u1 = w 1 h,

The capacitance correction factors ΔC1 and ΔC2 are computed from the following expressions:

12 2 12

Two direct capacitances, Cd11,asym u1 , u2 , g, εr , Cd22,asym u1 , u2 , g, εr of the asymmetrically coupled microstrips, between the strip conductors and ground, are obtained by modifying the direct capacitances of two symmetric coupled microstrips; Cd11,Sym u1 , g, εr , Cd11,Sym u2 , g, εr corresponding to the strip widths u=u1 and u=u2, respectively. The direct capacitances of the asymmetrically coupled microstrips are obtained as follows: Cd11,asym u1 , u2 , g, εr = Cd11,sym u1 , g, εr + ΔC1

a

Cd22,asym u1 , u2 , g, εr = Cd22,sym u2 , g, εr + ΔC2

b 12 2 13

Ks εr = 1 1 + 0 58 1 − 1 εr Ks εr = 1 = 0 21 − 0 023 ln u1 u2

Ks εr =

exp − g 1 56 + 0 22 ln u1 u2

a

b

12 2 15

The elements of the following capacitance matrix of the asymmetrically coupled microstrips are obtained, using equation (11.3.9) of chapter 11; from the direct capacitances Cd11,asym u1 , u2 , g, εr , Cd22,asym u1 , u2 , g, εr , and Cd12,asym u1 , u2 , g, εr : C εr =

C11 εr − C12 εr

− C12 εr C22 εr

12 2 16

To compute the inductance matrix, using equation (12.2.1b), the capacitance matrix [C (εr = 1)] , on air-substrate is evaluated by taking εr = 1 in the above equation. Next, the matrix [L (εr = 1)] is evaluated from the expression L εr = 1 = μ0 ε0 C εr = 1 − 1 . Finally, elements of the capacitance and inductance matrices, [C (εr)] and [L (εr = 1)] , are substituted in equations (12.2.2) and (12.2.9) to compute the static modal effective relative permittivity and modal characteristic impedances of the C and π-modes. 12.2.2 Frequency-Dependent Line Parameters of Asymmetrically Coupled Microstrips Tripathi [J.17] developed the waveguide dispersion model to compute the frequency-dependent modal effective relative permittivity and the modal characteristic impedance of the asymmetrically coupled microstrips, by obtaining the frequency-dependent capacitance and inductance matrices. The model has been adapted in a CAD package [B.1]. The waveguide model of the coupled microstrip lines could be called the “Tripathi model.” Simpler but less accurate dispersion model, by extending Getsinger’s model of a single microstrip, is also reported [J.12, J.16]. Tripathi Waveguides Dispersion Model

Figure (12.3a and b) shows the asymmetrically coupled microstrips and its equivalent waveguide model. The

12.2 Line Parameters of Asymmetric Coupled Microstrips

w1

εr,eq3 weq3 g εr,eq2 h

w2 g h

εr

εr,eq1 weq1

(a) Asymmetric coupled microstrips.

Cd12

V1

V2

Cd11

Cd22

weq2

(b) Waveguide model.

(c) π-circuit model.

Figure 12.3 Asymmetric coupled microstrips and its equivalent circuit model.

waveguide model treats two strip conductors of the widths w1 and w2 as the parallel plate waveguides of height h. It also treats the coupling gap region g as another waveguide of height g. Figure (12.3c) further shows the π-capacitance model of the coupled lines structure. The waveguides have static equivalent relative permittivity, εr, eq1(εr, f = 0), εr, eq2(εr, f = 0), and εr, eq3(εr, f = 0). Their static equivalent widths are weq1(f = 0), weq2(f = 0), and weq3(f = 0). Equation (11.3.9) of chapter 11 relates the static direct capacitances, Cd11 , Cd22 ,Cd12 of the π-model to the capacitance matrix elements, C11, C12, C22. The static equivalent relative permittivities of three equivalent waveguides are computed as follows: εr,eq1 εr ,f = 0 = εr,eq2 εr ,f = 0 = εr,eq3 εr ,f = 0 =

Cd11 εr C11 εr −C12 εr = C11 εr = 1 −C12 εr = 1 εr = 1

a

Cd22 εr C22 εr −C12 εr = d C C22 εr = 1 22 εr = 1 −C12 εr = 1

b

Cd11

Cd12 εr C12 εr = C12 εr = 1 εr = 1

c

Cd12

12 2 17 The above equations take into account the negative value of the coupling capacitance. The static equivalent widths of the waveguides, for the air-substrate, are also computed as follows from the direct capacitances: h Cd11 εr = 1 εo h Cd22 εr = 1 f=0 = εo g Cd12 εr = 1 f=0 = εo

w eq1 f = 0 =

a

w eq2

b

w eq3

presented in chapter 14. In practice, the equivalent widths and equivalent relative permittivity are frequency-dependent parameters. Getsinger’s model is used to estimate its frequency dispersion: εr,eq,j εr , f = εr − w eq,j f = w j − where,

εr − εr,eq,j εr , f = 0 1 + G f fp

w j − w eq,j f = 0 1 + f fp

2

2

, j = 1, 2, 3

a

,

j = 1, 2, 3

b c

G = 0 6 + 0 009 2μ0 hf p

12 2 19 The above expression for G is optimized for the alumina substrate. It could be corrected for other substrates to get improved results. Another expression for the parameter G = εreq, j/εr, (j = 1, 2, 3) given in equation (8.3.9) of chapter 8 can also be used. At this stage, the task is to compute the inflection frequency fp. It is obtained by using the transverse resonance method, shown in Fig (12.4a and b). Three lossless interconnected equivalent waveguides are treated as the open-end line sections of lengths wj, j = 1, 2, 3. Their characteristic impedances and propagation constants are obtained from the following expressions: Zj = βj = ω

μ0 ε0 hj

εr,eq,j f = 0 ,

μ0 εr,eq,j f = 0 ,

j = 1,2,3 h1 = h2 = h,h3 = g j = 1,2,3

a b

12 2 20

c

Under the transverse resonance condition, the total admittance at the plane X–X is zero: 12 2 18

Subsection (12.2.1) presents the closed-form expressions for the static elements of the capacitance matrix and direct capacitances of π-model of asymmetrical coupled microstrip lines. The method of moments (MOM) is also used to compute the static capacitance matrix [B.2, J.2, J.3]. In the case of the symmetrically coupled microstrips, Cd11 = Cd22 and these capacitances can also be computed by the variational method

Y1 + Y2 = 0;

1 Z1 tan β1 w eq,1 f = 0 + Y2 = 0

a

in Zin 2 = Z2 cot β2 w eq,2 f = 0 ,Z3 = Z3 cot β3 w eq,3 f = 0

b

1 1 Y2 = = in Z2 Z2 + Zin 3

c

12 2 21 Usually, weq,3(f = 0) is much smaller than the strip conductor widths. So, the coupling capacitance C12 can be treated as frequency independent. The open stub

427

12 Planar Coupled Transmission Lines

Open

X

Z3 β3

Weq3

Z3in

X

Stub#3

428

Yʹ1 Z1 β1

Open

Z3in

Weq1

Z2 β2

Yʹ1

Weq2 Z1in Z2in Stub#2

(b) Equivalent circuit of stubs at plane X–X.

Application of the transverse resonance method.

#3 is replaced by the impedance Zin 3 = 1 ωC12 . Thus, transverse resonance condition at the plane X–X is modified as follows: in in Zin 1 + Z2 + Z3 = 0

Z1 cot β1 w eq,1 f = 0 + Z2 cot β2 w eq,2 f = 0 + 1 ωC12 = 0

12 2 22

The above equations (12.2.21) or (12.2.22) is solved numerically for the first transverse resonance frequency, i.e. for the first root ω1. It is the inflection frequency fp of the coupled microstrips. It helps to compute the frequency-dependent equivalent widths weq,j ( f ), and equivalent relative permittivity, εr,eq,j (εr, f ), j = 1, 2, 3. The frequency-dependent capacitance matrix elements are computed as follows: C11 εr , f = Cd11 εr , f + Cd12 εr , f

a

C22 εr , f = Cd22 εr , f + Cd12 εr , f

b

also C εr , f

Yʹ2

X

Stub#1 X (a) Stub equivalent circuit. Figure 12.4

Z2in

Z1in

Open

=

C11 εr , f

− C12 εr , f

− C12 εr , f

c

C22 εr , f

12 2 23 where C11 εr ,f = εr,eq,1 f weq,1 f h + εr,eq,3 f w eq,3 f g

a

C22 εr ,f = εr,eq,2 f weq,2 f h + εr,eq,3 f w eq,3 f g

b

C12 εr ,f = εr,eq,3 f weq,3 f g

c

Equation (12.2.2) normally computes the static C and π- effective relative permittivity, as the capacitance and inductance matrix elements are frequency independent. On using the frequency-dependent elements of both [C(εr, f )] and [L(εr = 1, f )] matrices, equation (12.2.2) computes the dispersion, εr, eff, c(f ), εr, eff, π(f ), of both the C and π-modes, whereas equation (11.5.31), or equations (11.4.32d) and (11.4.33d), computes εr, eff, e(f ), εr, eff, o(f ) of the symmetrically coupled microstrips. Figure (12.5a and b) shows the illustrative results using the dispersion model of Tripathi against the results of Getsinger’s model for both the symmetrical and asymmetrical coupled microstrips [J.17]. On using the frequency-dependent elements of both [C(εr, f )] and [L(εr = 1, f )] matrices, equation (12.2.9) computes the dispersive characteristic impedances, Zc1(f ), Zπ1(f ), Zc2(f ), Zπ2(f ) of both the C and π-modes for two asymmetric strip conductors. Likewise, on using the frequency-dependent elements of both [C(εr, f )] and [L(εr = 1, f )] matrices, equations (11.5.46) and (11.5.49) of chapter 11 compute the Zoe(f ), Zoo(f ) for the even and odd modes of the symmetrically coupled microstrips. Figure (12.6) shows the illustrative results using Tripathi’s model for the even- and odd-mode characteristics impedance, Zoe(f ), Zoo(f ), of the symmetrically coupled microstrips [J.17]. It also compares the results against the results of Jansen [J.4].

12 2 24 The frequency-dependent inductance matrix elements are determined by evaluating the above capacitances on the air-substrate: Lf

C εr = 1, f 1 = 2 c

1 , c2 C11 f

L εr = 1, f

− C12 f

C22 f

=

− C12 f

−1

12 2 25

Extended Getsinger’s Dispersion Models for C and π-Modes

Getsinger’s dispersion model applicable to a single microstrip, discussed in subsection (8.3.2) of chapter 8, is further extended to get the dispersion relations to compute the εr, eff, c (εr, f ) and εr, eff, π (εr, f ) of both the C and π-modes of the asymmetrically coupled microstrips, shown in Fig (12.7a and b) [J.16]:

12.2 Line Parameters of Asymmetric Coupled Microstrips

Even 7

0.30, 0.19

6

0.86, 1.12 Odd

8

εr,eff,e(f), εr,eff,o(f)

8 εr,eff,e(f), εr,eff,o(f)

w/h, s/h 0.86, 1.12

εr = 10

w1 = 0.6 mm, g = 0.4 mm h = 0.635 mm, εr = 9.7 w2 = 1.2 mm 0.3 mm

7

C-mode π-mode

6

1.2 mm 0.3 mm

0.30, 0.19 5 0

2

5 0

4 6 8 10 12 Frequency (GHz)

Tripathi

Getsinger

2

4 6 8 10 12 Frequency (GHz)

Tripathi

Getsinger

(b) Dispersion of C and π-modes.

(a) Dispersion of even and odd modes.

Figure 12.5 Dispersion of symmetrical and asymmetrical coupled microstrip. Source: From Tripathi [J.17]. © 1986 IEEE.

For C − mode εr,eff,c εr , f = εr − g mm 0.06

Even–odd mode impedance

w1= 0.6 mm 70

f cp = ZC0,eq εr , f = 0

0.4

40 Tripathi,

0.06 0

9 6 12 Frequency (GHz)

3

15

Figure 12.6 Dispersive characteristic impedances of even and odd modes. Source: From Tripathi [J.17]. © 1986 IEEE.

Z01

Z01

Z02

Zc1 Zc2

+ –

Z02 RcV1

(a) C-mode excitations.

Z02

Zπ1

Z01 + V2 –

εr − εr,eff,π εr , f = 0 1 + Gπ f f πp

2

c

d

12 2 26

Jansen [J.4]

30

a

b

where, Gπ = 0 6 + 0 009 Zπ0,eq εr , f = 0 f πp = Zπ0,eq εr , f = 0 2μ0 h

50 Odd mode

2μ0 h

For π − mode εr,eff,π εr , f = εr −

1.8 1.8

Even mode

2

where, Gc = 0 6 + 0 009Zc0,eq εr , f = 0

0.4

60

Z01 + V1 –

1 + Gc f f Cp

h = 0.635 mm, εr = 9.7

80

20

εr − εr,eff,c εr , f = 0

Zπ2

– +

Z02 RπV2

(b) π-mode excitations.

Figure 12.7 C- and π-mode excitations of the asymmetrically coupled microstrips.

The static modal effective relative permittivities, εr, eff, c (εr, f = 0) and εr, eff, π(εr, f = 0), are computed using equation (12.2.2). The above waveguide dispersion model requires numerical evaluation of the inflection frequency fp of the coupled microstrips. However, Getsinger’s model computes the inflection frequencies, f cp and f πp , separately for both C and π-modes in terms of the equivalent static mode impedances, Zc0,eq εr , f = 0 and Zπ0,eq εr , f = 0 . The equivalent mode impedances of the C and π-modes excited in the coupled microstrips, shown in Fig (12.7a and b), are to be evaluated. Figure (12.7a and b) shows that the voltage pair (+V1, +RcV1) excites the C-mode, and the voltage pair (+V2, −RπV2) excites the π-mode. Figure (12.8a and b) further shows that the mode voltage ratios Rc and Rπ modify the corresponding source, characteristic impedance, and load impedance of each of the C and π-modes. Figure (12.8a) demonstrates the parallel combination of two conductors of the C-mode, as both conductors are at the same potential. It provides the equivalent C-mode characteristic impedance. Likewise, Fig (12.8b) demonstrates the series connection of two conductors of the π-mode, due to the

429

430

12 Planar Coupled Transmission Lines

Z02 / |Rc|

Z01

Z01|| Z02/ Rc Z01

Zc1

Z

Zc2/Rc

Z01 + V1 –

+ –

Zc1|| Zc2/ Rc

Z02 / |Rc| V1

Z01|| Z02/ Rc

+ V1

–

Zπ1

Z01 + V2 –

(a) C-mode equivalent circuit. Figure 12.8

Zπ1+ Zπ2/ |Rπ|

Zπ2/ |Rπ|

– +

Z01+Z02/ |Rπ|

Z02/ |Rπ|

+ 2V2

V2

–

Equivalent C- and π-mode impedances.

Zo,c1 Zo,c2 Rc Zo,c1 + Zo,c2 Rc

a

Zπ0,eq εr , f = 0 = Zo,π1 + Zo,π2 Rπ

b

Zc0,eq εr , f = 0 =

However, using the conformal mapping method, Keung and Cheng [J.22] have obtained the closed-form expressions for the even- and odd-mode capacitances of the symmetrical edge coupled CPW. It is shown in Fig (12.9), on a finite thickness substrate, with thickness h and relative permittivity εr. Its coupling gap is 2a. The 2b is a summation of the coupling gap and two strip widths, whereas the separation between two coplanar ground planes is 2c.

12 2 27 The characteristic impedance of the C and π-modes are computed from equation (12.2.9a and b). The voltage ratios Rc and Rπ are computed from equation (11.5.40) of chapter 11. The above model is also applicable to the symmetrically coupled microstrip giving dispersion model for the even and odd modes [J.12, J.16].

Even-Mode CPW Capacitance

The even-mode excitation has a magnetic wall (MW) at the plane of symmetry of the edge-coupled CPW. It separates the coupled lines into two quasi-TEM even-mode supporting lines. The even-mode line capacitance, Ce(εr) p.u.l. is given by the following expression:

Line Parameters of Coupled CPW

Ce εr = 2ε0

This section summarizes results on the line parameters of the symmetric edge coupled CPW and the line parameters of a shielded broadside coupled CPW. Normally, the closed-form results are obtained using the conformal mapping methods [J.22–J.24]. 12.3.1

Z01+Z02/ |Rπ|

(b) π-mode equivalent circuit.

opposite potential on both conductors. It gives the equivalent π-mode characteristic impedance. Both equivalent modal impedances are given below:

12.3

Z02/ |Rπ|

where k2e1 = sinh 2 k2e2 =

Symmetric Edge Coupled CPW

Wen [J.25] presented an analysis of the symmetrical edge coupled CPW on an infinite thickness substrate. Magnetic wall for even mode Electric wall for odd mode

K ke1 K ke2 + εr − 1 ε0 K ke1 K ke2

sinh 2

b2 − a2 c2 − a2 πb − sinh 2 2h πc − sinh 2 2h

a b

πa 2h πa 2h

c 12 3 1

c b a

h εr

Figure 12.9

Symmetrical edge-coupled CPW on finite thickness substrate. Source: From Keung and Cheng [J.22]. © 1996 IEEE.

12.3 Line Parameters of Coupled CPW

In the above expressions, the K(k) and K (k) are the complete elliptic integrals of the first kind and its complement. The ratio K(k)/K (k) is evaluated by the closedform expression [B.7, J.26]. It is also summarized in subsection (9.2.4) of chapter 9.

The parameters kc2 and kc3 are further computed using the following expressions: K kc2 = αβ3 K kc2

a

K kc3 = α 1 − β3 K kc3

b

Odd-Mode CPW Capacitance

The odd-mode excitation has an electric wall (EW) at the plane of symmetry. It separates the symmetrical coupled CPW in two quasi-TEM odd-mode supporting lines. The odd-mode line capacitance, Co(εr) p.u.l., is given by the following expression: Co εr = 2ε0 where,

K ko1 K ko2 K ko3 + εr −1 ε0 + K ko1 K ko2 K ko3

k2o1 =

b 12 3 2

The parameters ko2 and ko3 are computed in terms of the incomplete elliptic integral of the first kind in Jacobian notation F(ϕ, k) using the following expressions: ko2 , kc2 kc2 K kc2

ko3 , kc3 kc3 K kc3 β1 + β2 β3 = 2

=

β1 β3

a

=

1 − β2 1 − β3

b

F sin − 1

c

12 3 3

The parameters β1,β2, and another parameter α, in terms of structural dimensions, are given by the following expressions: α=

K kc1 K kc1

K2c1 =

a

sinh 2 sinh 2

F sin

πc 2h πb 2h

−1

β1 = F sin − 1 β2 = F

sinh 2 sinh 2

πb − sinh 2 2h πc − sinh 2 2h

πc πa − sinh 2 2h 2h πc πa cosh 2 sinh 2 2h 2h K kc1 πc πa − sinh 2 sinh 2 2h 2h πc sinh 2 2h K kc1

Kk = p; K k eπp − 2

eπp + 2

2

1115

104–12 000

12.4

0.001 (107)

0.35, 0.60, 0.625

Used at THz

Si (high resistivity)

= 30μm)

350 μm

Air

High resistivity Si-wafer

350 μm

Ground conductor

Micromachined wafer supporting ground plane

Strip width W Silicon Aluminium thickness t = 1μm

–Air

gap

100μm)

Si

Ground plane Width ℓ = 1200μm (e) Inverted microstrip line. Figure 13.17

Microstrip

Glass

Membran

Front view (d) Membrane-supported coupled microstrips.

(c) Signal and ground conductor on membrane.

Circuit wafer

474

Si

Si-wafer

Metalized carrier wafer

(f) Suspended membrane-supported microstrip with a top shield.

Some membrane supported microstrip type of MEMS lines.

provide the ground plane is more suitable for the on-chip integration of the MEMS structure. Figure (13.17c) shows another membrane-supported microstrip. The ground plane is fabricated on another side of the membrane. The microstrip on the membrane is a low impedance line. Figure (13.17d) shows the coplanar strip lines fabricated on the membrane. By adding the ground plane, it has been converted to a suspended coupled MEMS microstrip lines on the membrane. Figure (13.17e) shows the fabrication of an inverted microstrip on the glass substrate. Of course, the signal

strip conductor is fabricated on the silicon wafer itself. The suspended and inverted microstrip lines in the MEMS format can also be provided with the top shield as shown in Fig (13.17f ) [J.39]. Figure (13.18) shows some more variants of MEMS strip lines. The line structure in Fig (13.18a) is on lowresistivity Si-substrate. However, a passivation layer of SiO2 improves, the loss performance of the line. The Spin-On-Glass (SOG) layer is used for another strip. The thickness of the SOG layer is in a range of 20–100 μm. Figure (13.18b) shows the same line

13.3 Micromachined Transmission Line Technology

LR Si Air Cavity

SOG SiO2

SOG membrane

LR Si

Air Cavity LR Si (a) Strip lines on SiO2 coated low resistivity Si-substrate.

Dielectric layer

(b) Strip lines on SOG membrane with low resistivity Si-cavity. LR Si Air Cavity

SOG SOG

HR Si

HR Si

(c) Strip lines on high resistivity Si-substrate. Figure 13.18

(d) Strip lines on SOG with low resistivity Si-cover.

Some variants of MEMS strip lines on low and high resistivity substrates.

structure on the SOG membrane that is supported inside a cavity of the low-resistivity Si-substrate. Figure (13.18c) shows the same line structure, where the stripline is directly printed on the high-resistivity Si-substrate. The second line is printed on the SOG layer that is covered with a dielectric layer. In Fig (13.18d), the dielectric cover is replaced by the low-resistivity Sisubstrate cavity. The metallic ground planes can be added in both the silicon cavities of Fig (13.18b, d) such that the structure work as an off-set broadside coupled MEMS microstrip lines. The microstrip line is also constructed just on the air-substrate by removing even the thin dielectric membrane. Figure (13.19a) shows the metallic membranebased microstrip line on the air-substrate. Of course, dielectric pillars are created on the semi-insulating GaAs-substrate to support the strip conductor. The steps involved in the fabrication process are shown in Fig (13.19b) [J.41]. The CPW has also been constructed in the MEMS configuration. Figure (13.20a) shows a finite ground CPW (FGC) on a high-resistivity Si-substrate with a SiO2 passivation layer. Figure (13.20b) further shows that the

MEMS technology itself can provide the Si-packaging with a cavity to the CPW. Figure (13.20c) shows that after etching out the Si-substrate, the SiO2 membrane supported FGC, i.e. CPW with a cavity backing, is obtained [J.42]. Figure (13.20d) further shows that the structure can be backed by the metalized cavity on another substrate [J.40]. Microshield Technology

In the membrane-based MEMS transmission lines, discussed above, the ground plane is normally created on another substrate that is bonded to the dielectric membrane and signal conductor supporting substrate. However, the ground plane can also be fabricated inside the cavity of a substrate. Such a cavity-backed line structure is called the micro-shield structure that is available for both the microstrip and a CPW. The micro-shielded microstrip structures are shown in Fig (13.21). Figure (13.21a) shows the membrane-supported, suspended microstrip line. The metalized cavity in the substrate, Si/GaAs, provides the ground plane [J.43]. Figure (13.21b) shows the substrate-supported, inverted microstrip line. Again, the metalized cavity in the

475

476

13 Fabrication of Planar Transmission Lines

X

Semi-insulating GaAs Ground metal deposition on GaAs substrate

Y

Dielectric post

Strip conductor

Semi-insulating GaAs Dielelctric post formation X Y Top view of air-supported microsrip line Signal strip conductor Ground conductor

Semi-insulating GaAs Sacrifical layer of photoresist: patterning and backing

Dielectric post

GaAs substrate

GaAs substrate

Semi-insulating GaAs Thin Au evaporation and second metal patterning

Line at X–X cross-section Line at Y–Y cross-section. Strip conductor

t

Dielectric post

t

GaAs substrate

Semi-insulating GaAs

Front longitudinal view of MEMS line

Sacrificial layer lift-off and line formation

(a) MEMS microstrip at air substrate fabricated on GaAs-substrate. Figure 13.19

(b) Fabrication process of air substrate microstrip on semi-insulating GaAs-substrate.

Microstrip on air-substrate. Source: From Lee et al. [J.41]. © 2004 John Wiley & Sons.

Si

SiO2

Cavity SiO2

Si

Si (a) CPW on Si-substrate.

(b) CPW on Si substrate with Si cavity cover. Membrane

Air

Si

SiO2 membrane

Si Si

(c) CPW on SiO2 membrane. Figure 13.20

Cavity

(d) Cavity-backed CPW.

MEMS CPW on high resistivity Si-substrate.

substrate provides the ground plane. The metalized cavity in the substrate also minimizes the signal crosstalk between the adjacent lines, as it isolates the substrate from the line by controlling the radiation to the substrate. The ground plane is available at the top, so there is no need for via hole in the micro-shield microstrip lines. Finally, Fig (13.21c) shows the micro-shielded microstrip line with both the upper and lower ground cavities [J.42]. It is a three-wafer structure. The microstrip is supported on a membrane.

The CPW structure has also been fabricated in the micro-shield format. Figure (13.22) shows the microshielded cavity-backed CPW structures [J.38, J.45– J.47]. Figure (13.22a) shows a metalized cavity in the upper wafer. The ground plane is constructed on another wafer that is bonded to the upper wafer. The coplanar ground conductors on the membrane are connected to the bottom ground plane through the walls of the metalized cavity. Thus, the present CPW structure avoids the air-bridge. The structure looks like a

13.3 Micromachined Transmission Line Technology

Ground plane Membrane

Ground plane

Air cavity

Air cavity

Si-substrate (a) Cavity-backed membrane-based suspended microstrip line.

(b) Cavity-backed inverted microstrip line. Micromachined ground plane wafer

Upper cavity membrane Lower cavity

Micromachined circuit wafer Metallized shield wafer

(c) Membrane supported suspended microstrip line with both the upper and lower metalized Cavities[J.42]. (Reproduce from [J.42], ©1998 IEEE). Figure 13.21

MEMS microshielded microstrip line. Source: From Herrick et al. [J.42]. © 1998 IEEE.

conductor-backed suspended CPW. The self-packaging is added to the upper cavity, as shown in Fig (13.22b). In the case of Fig (13.22b), the lower cavity is filled in with the substrate material. The white strips are three coplanar conductors of a CPW. V-Groove CMOS Technology

The above-discussed fabrication of the MEMS transmission lines is mostly for the semi-insulating GaAssubstrate and the high-insulating Si-substrate. It is achieved by several photolithographic steps, wafer bonding, and backside etching. This process is not compatible with the CMOS process. Another bulk micromachining

process from the front-side of the CPW line structure is suggested by Milanovic et al. [J.44]. It is suitable for the CMOS process. Two cases are considered below. On the CMOS-grade Si-substrate, the SiO2 layer is deposited that acts as the supporting membrane for the CPW line structure. For front etching, two openings in SiO2 are provided. In the first case, shown in Fig (13.23a, b) for the front-view and top view, the openings are at the end of the finite ground CPW (FGC). In the second case, shown in Fig (13.23c, d), two openings are in between the ground conductor and central strip conductor. The CPW structure is patterned over the SiO2 membrane and the Si-substrate etching is carried out from the

Air filled upper cavity Top side metallization Si

Membrane Cavity

Si Metalized lower wafer (a) Cavity-backed MEMS CPW. Figure 13.22

Substrate filled lower cavity (b) Self-packaged MEMS CPW.

Micro-shielded MEMS CPW. Source: Based on Milanovic et al. [J.44].

477

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13 Fabrication of Planar Transmission Lines

CPW G

G

Sio2 Metal OpeningS4P Opening

S

Air

a = 24 μm b = 30 μm c = 130 μm

G Si-substrate

G

Metal 1 [A1]

(a) Front-side of the first case.

S

Open area

Metal 2 [Pad]

(b) Top-view of the first case.

G

CPW

G

Sio2 S4P Metal Opening

Air

S

a = 20 μm b = 50 μm

c = 140 μm

G

S

G

Si-substrate

Metal 1 [A1] (c): Front-side of the second case. Figure 13.23

Open area

Metal 2 [Pad]

(d) Top-view of the first case.

MEMS CPW in CMOS technology. Source: Based on Milanovic et al. [J.44]. © 1997 IEEE.

front openings. The bulk micromachining creates the Vgroove in the substrate due to the anisotropic etching. This structure is used up to 40 GHz. The V-groove microstrip line can be extended to the coupled microstrip structures also [J.48].

LIGA Technology

This is a 3D molding process to create the micromechanical microwave and mm-wave transmission lines, filters, etc. This process provides the large aspect-ratio metal structure of 1 μm–1 mm thickness that is needed for the high-power MMIC applications. The microstrip and CPW structures fabricated by the LIGA process are shown in Fig (13.24a, b), respectively [J.49]. Figure (13.24c) further shows the coupled microstrip lines fabricated on a fused quartz substrate by the LIGA process. The LIGA is a German acronym for the Lithographie, Galvanoformung, Abformung (Lithography, Electroplating, and Molding) fabrication technology. The LIGA process controls the thickness of a conductor that is an additional design parameter. Figure (13.25) shows the LIGA fabrication process for the coupled microstrip. The process is started with the metalized quartz substrate of 120 μm thickness on which 300 μm sheet resist is deposited. The sheet resist is exposed to the X-ray through the Si3N4/SiO2 mask to create the

coupled microstrip that is electroplated with 200 μm Ni or Cu. Waveguide Technology

The MEMS technology has been also used to develop THz range metallic waveguide structures. Its process is shown in Fig (13.26). A Si-substrate is a double-side coated with the silicon nitride layer of 1000 Å. At the step-I, an opening is created by removing the silicon nitride through which the etchant KOH is applied to create half of the waveguide. At step-II, a channel is etched out in the Sisubstrate. At the step-III, the waveguide-containing wafer is bonded to the unetched wafer. At the step-IV, the channel is coated with the chrome (250 Å) and gold (5000 Å) metal composite to create 1/2 waveguide. Finally, at the step-V, two such channel sections are combined to form a rectangular waveguide [J.50] in MEMS technology.

13.4

Elements of LTCC

The substrate technology could be categorized into three groups – laminates based PCB technology, ceramic-based technology, and semiconductor-based technology. The PCB and semiconductor-based technologies are discussed in the previous sections. The ceramic-based interconnect technology has the following forms – a thick

13.4 Elements of LTCC

Electroplated metal

Electroplated metal

Substrate

Substrate

Thin film ground plane

Thin film ground plane (a) Thick conductor microstrip line.

(b) Thick conductor CPW line. Electroplated metal

Substrate

(fused quartz)

Thin film ground plane (c) Thick conductor coupled microstrip line. Figure 13.24

LIGA planar transmission lines.

300 μm-thick sheet PMMA

˚ Ti 600 A ˚ Cu 3600 A ˚ Ti 600 A 120 μm-thick quartz (a) Application of plating base.

1–2 μm-thick spun on PMMA

(b) Application of sheet resist. X-ray Si3N4/SiO2 membrane mask

200 μm-thick Electroplated Ni or Cu

X-ray stop material (3–4 μm Au)

(c) X-rays exposure of the resist.

(d) Conductor electroplating.

˚ Ti 250 A ˚ Au 7500 A (e) Final coupled microstrip line. Figure 13.25

LIGA fabrication process.

479

13 Fabrication of Planar Transmission Lines

(110) silicon (0.05 inches)

(110) silicon (0.05 inches)

KOH etch

Silicon nitride (1000 Å)

Waveguide channel

(100) silicon (0.05 inches)

Step-II: Wafer etched to create a channel. b

Step-III: Wafer with channel bonded to wafer. Figure 13.26

Waveguide channel

Silicon nitride (1000 Å)

Chrome (250 Å) Gold (5000 Å)

Step-I: Si3N4 defines waveguide height.

Polyimide bonding layer

480

a/2 (100) silicon (0.05 inches)

Step-IV: Gold plating of the channel to create ½-waveguide.

Fabrication process of MEMS waveguide.

film on the ceramic substrate, copper plating on the ceramic substrate, thin film on the ceramic substrate, high-temperature cofired ceramic, i.e. HTCC technology and the low-temperature cofired ceramic, i.e. LTCC technology. The thick and thin-film technologies have already been considered for the fabrication of transmission lines and the related structures. Primarily, these are single-layer technologies and are adapted to the multilayer structures. However, each ceramic layer has to be processed separately, and they are bonded together. This circuit configuration is not very compact and also not cost-effective. Thus, there is a need for the vertical integration of passive components and interconnects, as well as an integration of the active components in a modular form. The multilayer ceramic technology offers a solution to high-density circuit integration. In the case of the HTCC technology, the embedded passive components and interconnects are fabricated on several layers of the ceramic substrates. Finally, the assembled layered structure, i.e. a bundle of all ceramic layers is burnt ( fired) at a temperature over 1500 C in an oven/furnace. However, the HTCC process requires high-temperature melting point conductors such as tungsten and other refractory metals. These metals are poor conductors and are also expensive. Thus, the HTCC packaging technology is not a popular one. The ceramic tape-based LTCC has become a more popular technology, as the firing temperature of the LTCC is in the range 800–900 C [J.51]. Gold and silver are used as conductors in the LTCC process. These conductors

have good electrical conductivity. The LTCC offers significant advantages over other technologies for the MCM used in the RF and microwave applications. It also accommodates both the digital and analog circuits in the module. The LTCC has become an important RF technology that accommodates antenna, filters, couplers, and other passive components along with active devices mounted on the surface and in the cavities. A typical LTCC multilayer structure, with embedded passive components, such as capacitor and inductor, and strip lines, and also surface-mounted SMD component, resistor, and RFIC is shown in Fig (13.27a) [J.52]. Figure (13.27b) shows a stack of multilayer dielectric tapes, called the green-tape, on which these components are fabricated/mounted. This section discusses the LTCC fabrication process and its layered medium environment for the development of the planar transmission lines, including waveguide. The detailed description of the LTCC technology could be read from the books and reports [B.26, B.27, J.52]. 13.4.1

LTCC Materials and Process

The LTCC technology is developed around the thin ceramic dielectric tapes (0.005 − 0.008 ). These are cut into equal pieces and stacked together to form a multilayer structure, as shown in Fig (13.27b). It shows seven dielectric layers with eight metallic surfaces for the construction of components and line structures. The

13.4 Elements of LTCC

Printed conductor SMD component

Photopatterned conductor

Metallic surface #1

Resistor Dielectric layer #1 RF IC 0.5775 mm Capacitor Inductor Stacked via Buried Strip line via

(a) A typical LTCC structure. Figure 13.27

Dielectric layer #7

BGA bumping

Metallic surface #8 (b) A stacked layer of green dielectric tape.

A typical LTCC structure with embedded and surface mounted components.

number of layers could be more, say up to 70–80. The last layer is the metal layer that is fabricated by thick film technology. The unfired thickness of the tape is in the range of 50 –250 μm. There is about 10% shrinkage after firing. The typical dielectric constant of tape is about 7.4 and loss-tangent 0.002. More information can be obtained from the Dupont tape selection guide. The photolithography is adapted to print the conductors for constructing the line structures, electrodes, etc. The dielectric tapes, i.e. the LTCC tapes, are made of glass-ceramic composite materials – alumina ceramic filler 40%, glass (silica) 45%, and organic binder, i.e. plasticizer and dispersant (deflocculant) 15%. The tape casting process, i.e. the manufacturing of the LTCC tape is adapted from the plastic and paper industries. The base LTCC tape is called “the green-tape,” as it is soft before baking. It is formed by spreading slurry consisting of ceramic powders mixed in an organic (or water) vehicle, together with appropriate binder and dispersant. The flowchart of the tape casting process is shown in

Ceramic slurry

Powder Solvents

Fig (13.28a). The binder, dispersant (deflocculant), plasticizer, and organic vehicle determine the property of a slurry. The dispersant breaks apart the agglomerates, into individual particles. The binder creates the polymer matrix that holds the chemicals for further processing. The binder determines the mechanical property of dielectric tape. Finally, the plasticizer makes the green-tape bendable (soft) without breaking during the process of punching, cutting, rolling, and laminating. The solvents should have good solubility for the binder and plasticizer. These should also have a moderate evaporation rate after the tape casting [J.53–J.55]. The homogenous slurry is poured on a Mylar (ceramic) tape carrier that is passed under the doctor blade shown in Fig (13.28b) to produce uniform tapes of desired thickness. After drying, the tapes become the ceramic-filled “green-tapes.” It is kept in rolls. The “green-tape” is the trade name of Dupont for the dielectric tapes. It is also manufactured by other manufacturers. The tape characteristics from different suppliers

Hot air

Dispersant Doctor blade

Mixing Plasticizer

Binder Mixing

Viscosity control

Base Coil movement system Mylar coil

Green ceramic coil

Tape casting (a) Flow-diagram of LTCC green-tape casting. Figure 13.28

Casting process of green-tape.

(b) Green-tape casing on the conveyor belt.

481

482

13 Fabrication of Planar Transmission Lines

are different. The conductor strips are made of the paste of gold, silver, silver–copper, Pt–Au or Pd–Au, etc. The metallic paste is also used to fill in via holes. 13.4.2

LTCC Circuit Fabrication

The passive components such as resistor, inductor, and capacitor in the LTCC fabrication process could be embedded components, i.e. they are located at the inner layers of the LTCC dielectric green-tape. However, the discrete and the surface-mount components are located at the uppermost layers and attached to the LTCC at the end of the cofiring process known as the postcofiring process. The microwave passive components designed around the planar transmission lines could be located partially on more than one dielectric layer. The sectionalized vertical arrangement of circuits makes the arrangement very compact. The interconnections between parts of the transmission lines and embedded components and ground planes are achieved by vias. The vias are holes in the dielectric layers. The vias are also used to bring terminals of a circuit to the surface of an LTCC. The arrays of vias are used to create the imperfect walls of a waveguide and also a cavity embedded in the dielectric layers. It is similar to the substrate integrated waveguide (SIW) discussed in section (7.8) of chapter 7. The ground plane could be located at the end of the dielectric stack, called LTCC metal (LTCC-M). The ground planes, or grounded conductors, can exist at several layers, and they are interconnected by using vias. Figure (13.29) shows a typical stack of six dielectric layers with several arrangements of vias, buried resistors, capacitors, and conductors. It also shows the postfired (attached after the cofired process) resistor. The vias are terminated on a cover pad and are filled in with

Via covered or catch pad

Surface resistor

Via stagger

metallic paste before the cofiring process. The important steps of the LTCC manufacturing process are shown in Fig (13.30) [J.54–J.56]. The process assumes that the circuit design has already been completed using the design tools like a Circuit Simulator that contains libraries of LTCC embedded components, via and many transmission lines (microstrip, stripline, and CPW) circuits. The complete LTCC circuit can also be simulated on the 3D EM-simulators. The green dielectric tape roll (i.e. unfired dielectric tape) is cut into pieces and preconditioned in the furnace. The registration holes, used for the alignment of all layers in the vertical stacking, are punched or burnt. The Mylar-tape is removed. The blanking die is used to create the orientation marks and the final working dimension of the green-tape. Next, the vias are made by the laser tools, CNC or punching tool. The vias are filled with metal paste, using the thick film screen printer with a stencil metal mask. The screen printing process of via filling is shown in Fig (13.31a). A CNC machine is used to create a cavity in the dielectric layer, or to cut a partial depth in the dielectric layer. Figure (13.31b) shows the process. It is filled with a high permittivity dielectric paste for fabricating small size-embedded capacitance. The next stage is the printing of the conductors for fabricating the transmission line-based circuits and interconnects. It is shown in Fig (13.30). The thick film screen printing process is used at the surface of the green-tape. The circuit and interconnect printing is partially done on several dielectric layers. The conductors of different layers forming the circuit are connected by the metal-filled vias. At this stage, the circuit, with embedded components in several dielectric layers of the green-tapes, is ready for cofiring. A stack is made of the green-tape containing

Top Layer 1

Via

Layer 2

Buried resistor

Layer 3

Baried capacitor Buried via

Thermal via

Layer 5

Inner layer Stacked via conductor

Via patch Figure 13.29

Outer layer conductor

Via diameter

Layer 4

Layer 6 Bottom

A typical arrangement of vias, passive components, and conductors in the dielectric stack.

13.4 Elements of LTCC

Banking

Via punching

Via filling

Conductor printing

Green tape roll Collating

Final inspection

Figure 13.30

Electrical testing

Co-firing

Pre-cutting

Laminating

Steps of LTCC circuit fabrication process.

partial circuits before the cofiring. It is known as the collating shown in Figs (13.30) and (13.31c). The green-tape layers collated and stacked with special tools are fixed with collating pins through the registration holes on the dielectric layers. The whole stack of the green-tapes is laminated by a thermal process at 100 C and 3000 PSI pressure. It is known as the thermo-compression process that sticks all the dielectric tape layers to form a composite, i.e. the monolithic single layer. Figure (13.31d) shows the process. The laminated layers are cut to the proper size. Finally, the laminated layers are burnt in the sintering furnace at the peak temperature of 850 C. The temperature control for the sintering process is shown in Fig (13.31e). The typical cycle time is three hours. As all the layers are sintered at the same time, the process is called cofiring. During the sintering process, the glass filler of the green-tape melts and spreads to form the host medium for the alumina particles. The sintering process model is shown in Fig (13.31f ). The cofired LTCC circuit is electrically tested for the open and short-circuit tests. The SMT components, like chip capacitor, resistor or IC chips are mounted on the outer surface of the LTCC as shown in Figs (13.27a) and (13.29). The microstrip-matching network is also screen printed on the outer surface. These are postcofired processes. Familiarity with the basic LTCC process helps to appreciate the advantage offered by it, as against the

traditional thick film process and HTCC. Some characteristics of the LTCC process are summarized below:

• • • • • • •

The green-tape dielectrics are produced in parallel that results in the cost-effective process and increased yield. It is one single-firing operation to develop the complete circuit. It reduces production time and cost. The firing operation takes about three hours. The partial circuit on each dielectric layer can be inspected separately before stacking. Thus, the defects can be repaired that increases the reliability and yield. Multiple dielectric layers permit innovative compact circuits of filter, balun, resonator, patch antenna, etc. The ferroelectric layer can be also incorporated for the voltage-controlled phase shifter, filter, delay line, etc. The LTCC circuit can effectively operate under the harsh environment. It can withstand high temperatures even at 500 C. The LTCC is having a better coefficient of thermal expansion (CTE); so it is better matched to the silicon substrate, as compared to organic or thick-film ceramic (alumina) substrate. Thus, it improves the reliability of the flip-chip components in the thermal cycle testing. The LTCC has thermal conductivity 3 W/m K that is of course less than thermal conductivity (24.7 W/m K) for

483

13 Fabrication of Planar Transmission Lines

Tool

Conductor paste Stencil

Squeegee

CNC milling Machined edge

Machined structure

LTCC Porous Nest

LTCC tape

Vacuum

(b) Cavity making in the dielectric tape.

(a) Screen printing process of via filling. Location pin

Registration hole Substrate layer

Before processing Aligning fixture

After processing

Tape #1 Tape #2

18 mm (c) Collating and stacking process.

(d) Thermo compression process. Glass

Glass

Ramp = 10 °C 850 Temp °C

484

30 min 350 45 min

Time (e) LTCC temp. profile for sintering. Figure 13.31

Alumina

Alumina

(f) A sintering process model.

Some LTCC circuit fabrication processes.

96% alumina. However, it is better than the thermal conductivity of FR-4 (1.7 W/m K). The thermal management of the LTCC is improved by their thermal via shown in Fig (13.29). Normally, several thermal vias are provided from the bottom of the heat-generating active circuits to the metallic ground, acting as the heat spreader. Sometimes, the vias and channels below the active components carry fluid for heat removal. The real difficulty with the LTCC process is associated with the tape shrinkage (12–16%) in the planar X–Y direction. The shrinkage of the tape to be accounted for at the design stage itself. However, zero shrinkage tape has been developed as the LTCC-M with the metal

ground that also removes the tape shrinking in the planar X–Y direction, but not in the Z-direction. 13.4.3 LTCC Planar Transmission Line and Some Components The planar transmission lines in the LTCC structure are used both as interconnects and as a medium for the development of distributed circuit elements. The common planar transmission line structures – microstrip, CPW, stripline, slot line, are easily accommodated in the LTCC. These can also be printed on the outer layer of the LTCC after the postfiring process. In this case, both thick and thin-film printing technologies are used.

13.4 Elements of LTCC

Microstrip line LTCC cavity

Ground plane CPW probe pad

LTCC tape

Microstrip line

LTCC cavity Cavity edge (a) The ground plane on the 2nd layer. Two pads for the CPW probe.

Vias

Ground plane

(b) 3-D view of LTCC microstrip line.

Microstrip line

Microstrip line

Layer #3 Dielectric layer Via Layer #8 External ground plane (c) Side -view of LTCC microstrip line. Figure 13.32

Ground conductor

Arrays of via

(d) Laminated cavity-backed microstrip.

Microstrip lines in LTCC.

However, thin-film printing requires a very smooth surface that is available only after polishing the outer surface. It is an expensive exercise, although very narrow lines with narrow spacing could be printed on the LTCC substrate. Normally, thick-film printing is used at the outer and inner dielectric layers, with a minimum line width of about 3–4 mils. The line structures operate at the frequency 1–40 GHz and above 100 GHz [J.55– J.57]. Several distributed components such as filter, coupler, and balun have been fabricated with the planar transmission lines on one dielectric layer or the multiple dielectric layers.

can be approached from the top for measurement purposes. Figure (13.32a) also shows two pads for the CPW probe. The buried (embedded) microstrip line is embedded in a homogeneous medium. Therefore, it is less dispersive than the surface-mounted microstrip line. Figure (13.32d) further shows that the microstrip line can also be located in a cavity formed within the LTCC [J.58]. Such microstrip lines have better isolation. The sidewalls of the cavity are fabricated by the double column of vias. A top ground plane can also be added to form the TEM mode supporting the stripline. The microstrip on the green-tapes can have low-loss, 0.08 dB/m even at 35 GHz [J.59].

Microstrip

The microstrip in the LTCC format has a more flexible characteristic impedance. The characteristic impedance of a surface-mounted microstrip can be controlled by changing the location of a ground plane at any dielectric layer, as shown in Fig (13.32a, b). In Fig (13.32a), the ground conductor is located on the second layer and next could be located at the third layer. The ground plane of the microstrip is connected, using vias, to the external ground of the LTCC located at the eighth layer. It is shown in Fig (13.32c). The embedded microstrip line

CPW to Microstrip Transition Structure

The shielded microstrip in a homogenous medium is a stripline. This line is a preferred structure in the LTCC technology, because it is a broadband nondispersion line. However, the wafer probe uses the CPW launcher. In the LTCC technology, a transition is needed. A basic form of the transition is shown in Fig (13.33) [J.60–J.62]. The microstrip is located at the upper dielectric and the ground plane is located at the next dielectric layer. The CPW is printed at the lower dielectric layer and its two

485

486

13 Fabrication of Planar Transmission Lines

Microstrip

Ground plane

Pads for CPW

CPW line

Figure 13.33

Microstrip CPW transition in the LTCC.

Input

Transmitted

εr εr

Isolated Coupled (a) LTCC broadside coupler. Figure 13.34

W2 S0 S

W3

W2 S S0

W3

W2 h1

εr

h2

εr

S

W1

(b) Conductor backed CPW coupler.

W2

S W1

h2 h1

(c) Microstrip coupler.

Coupler in LTCC technology.

coplanar ground conductors are connected through via to the ground plane of microstrip. A transition, on 5 mils “green-tape” works from 10 MHz to 100 GHz.

transmission line section is replaced by the equivalent T-network shown in Fig (13.35b). Its LTCC realization is shown in Fig (13.35c) [J.65].

Coupled Microstrip Line

Bandpass Filter

Figure (13.34a) shows that the broadside microstrip coupler with very tight coupling, i.e. 3 dB coupling. The broadside coupler can be easily accommodated in the LTCC. Two microstrip conductors are located at two different dielectric layers. The thickness of the dielectric layers between two strip conductors could be adjusted to meet the design requirement. Moreover, as the structure is buried inside a homogeneous medium, the even and odd mode phase velocities are equalized to achieve a high directivity and less dispersion. The structure has been used as the power combiner and splitter in the frequency range 8–14 GHz [J.63]. Figure (13.34b, c) show other coupler structures in LTCC using the CPW and microstrip line. A wide range of coupling from −10 to −2.6 dB in the frequency range of 0.4–4.0 GHz has been reported [J.64].

Figure (13.36a) shows a stripline resonator in the LTCC environment. It is useful in developing the LTCC bandpass filter. Figure (13.36b) shows another type of bandpass filter using line resonators in LTCC. The passive components such as inductors, capacitors, and resonator are realized in the LTCC to develop compact filters [J.66–J.69]. Figure (13.37a) further shows a hybrid π-bandpass filter. Figure (13.37b) shows its corresponding LTCC structure in the multilayer format [J.70]. The filter is designed at 2.5 GHz. Four sections are cascaded to achieve four transmissions zero, at 1.90, 1.70, 4.25, 5.25 GHz for getting high rejection at both the lower and upper bands. The passive lumped components, L and C, and the step-impedance resonator are all constructed using the planar line structures at different levels of the LTCC. The partial structures are connected through vias to fabricate a compact filter. The LTCC structures are complex 3D structures. This kind of structure can be analyzed using the 3D/2.5D EM

Branch-line Coupler

A compact branch-line coupler, shown in Fig (13.35a), has been realized in the LTCC environment. The

13.4 Elements of LTCC

8 Vias connecting layers 1and 6 Port 1 Port 3

Port 4 Z1,θ

Port #1

Port 2 Layer 1

Port #2 0.92 Ls

Z2,θ

1.17 Layer 2

0.92

Z2,θ

Z1,θ

Port #4

Port #3

0.3

0.92

0.25

CP LM

(a) Circuit of branch line coupler.

Ls 1.17 0.92

0.92

Layer 3

CP

LM

≈

Z2,θ

Ls

1.22

Ls

0.76 1.27

0.86 Layer 4

Cs2

Cs

1.17

Cs

(b) T - Equivalent circuit of a line section.

1.22

LTCC substrate each layer thickness:0.09 mm Dielectric constant: 7.8, Loss tangent: 0.0015

1.27

0.97

Layer 5

Cs1 Layer 6

(c) LTCC structure of branch line coupler. Figure 13.35

Branch line coupler in the LTCC. Source: Kuo et al. [J.65]. © 2006 IEEE.

9.7 mm 2.2 mm

Ground conductor Strip conductor

w

Air cavity

Embedded microstrip resonators Opening for probing Bottom ground plane 0.42 mm

Ground conductor (a) Stripline resonator with 2 embedded air-cavity. Figure 13.36

(b) Embedded microstrip BPF in LTCC [J.66].

The embedded microstrip bandpass filter in LTCC. Source: Modified from Choi et al. [J.66]. © 2003 IEEE.

487

488

13 Fabrication of Planar Transmission Lines

Layer 1 Cs L1a C1a

L1b

θ2b Z2b

θ2a Z2a θ1a

θ1b

Z1a

Z1b

Layer 2

Via

Layer 3 C1b Layer 4 Ground plane

Cell #a

Layer 5

Cell #b

Layer 6

(a) Hybrid π-bandpass filter.

Layer 7 Dielectric constant: 7.8 Loss-tangent: 0.0043 Layer thickness: 90-90-135-90-90-90-180 μm Circuit size: 5.4 mm × 3.9 mm × 0.765 mm Figure 13.37

Layer 7b (b) LTCC structure for hybrid π-bandpass filter.

Bandpass filter in LTCC. Source: From Jeng et al. [J.70]. © 2006 IEEE.

Via holes

Via holes

Conductor layer

LTCC substrate Conductor layer (b) Rectangular coaxial line.

(a) Laminated LTCC -waveguide. Microstrip feedlines L Metal 1

W

Microstrip feedlines

Metal 2

External slot

External slot

External slots

OSL

Cavity 1

H

Cavity 2

Metal 5 Internal slots Cavity 3

Via walls Via walls (c) Microstrip line fed cavity[J.71]. Figure 13.38

Via walls Via walls (d) Microstrip line fed 3-poles cavity based bandpass filter [J.71].

Waveguide and cavity structures in the LTCC. Source: From Lee et al. [J.71]. © 2005 IEEE.

Metal 1 Substrate 1 Metal 2 Substrate 2-6 Metal 7 Substrate 7-11 Metal 12

References

simulators. However, most of the structures could be broken in parts and could be modeled separately as the lumped elements or as the transmission line sections. The final results could also be obtained by a combination of these blocks. Such modeling is approximate, and the performance of a model should be compared and improved by comparing its results against the results of the EM-Simulator. An analytical model is helpful for a fast design process.

B.10 Mattox, D.M.: Handbook of Physical Vapor Deposition

B.11

B.12 B.13

13.4.4

LTCC Waveguide and Cavity Resonators

The multilayer LTCC is a very versatile medium. The low-loss waveguide structure, air-filled or the dielectricfilled could be developed in the LTCC. The sidewalls are fabricated by the double column of the closely spaced vias. The top and bottom conductors of the rectangular waveguide could be deposited on two dielectric layers. Figure (13.38a) shows the rectangular laminated waveguide. The rectangular coaxial line (stripline) is also created in the LTCC. It is shown in Fig (13.38b) [J.58]. Similarly, a cavity resonator is also created by using vias for the sidewalls. Figure (13.38c) shows a cavity fed by the aperture-coupled microstrip line. The cavity has a Q factor 372 at 60 GHz. It is further used to develop three poles LTCC bandpass filters as shown in Fig (13.8d) [J.71].

B.14 B.15 B.16 B.17

B.18 B.19 B.20 B.21

References

B.22

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(PVD) Processing: Film Formation, Adhesion, Surface Preparation, and Contamination Control, Noyes Publications, New Jersey, 1998. Haskard, M.; Pitt, K.: Thick-film Technology and Applications, Electrochemical Publications Ltd, UK, 1997. Robertson, I.D.; Lucyszyn, S. (Editors): RFIC and MMIC Design and Technology, IEE, London, 2001. Goyal, R.: Monolithic Microwave Integrated Circuits: Technology & Design, Artech House, Norwood, MA, 1989. Marsh, S.: Practical MMIC Design, Artech House, Inc., USA, 2006. Nguyen, C.: Radio-Frequency Integrated-Circuit Engineering, John Wiley & Sons Inc., 2015. Steer, M.: Microwave and RF Design: A Systems Approach, SciTech Publishing, 2010, USA. Minin, I. (Editor): Microwave and millimeter-wave Technologies from Photonic Bandgap Devices to Antenna and Applications, InTech Pub., Mar. 2010. Lee, Yun-Shik: Principles of Terahertz Science and Technology, Springer, 2009. Kuh, Ernest S.: Multichip Modules, World Scientific Press, Singapore, 1992 Medjdoub, F. (Editor): Gallium Nitride (GaN): Physics, Devices, and Technology, CRC Press, 2016. Rossi, T. and M. Farina: Advanced Electromagnetic Analysis of Passive and Active Planar Structures, IEE Pub., UK, 1999. Santos Hector J. De Los: Introduction to Micromechanical Microwave System, 2, Artech House, 2004. Varadan V.K.; Jose, K.A.: RF MEMS, and Their Applications, John Willey Ltd., 2003. Rebeiz, G.M.: RF MEMS: Theory, Design, and Technology, Wiley-Interscience, 2003. Johnstone R.W.; Parameswaran, A.: An Introduction to Surface-Micromachining, Springer, New York, 2012. ISBN:9781402080203 Ivanka, Y.: Multilayered Low-Temperature Cofired Ceramics (LTCC) Technology, Springer, USA, 2005. Kulke, R.: LTCC – An Introduction and Overview, IMST GmbH, Germany, 2001.

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on sapphire (SOS) device technology, Oki Tech. Rev., Vol. 71, No. 4, Issue 200, pp. 66–69, Oct. 2004. Johnson, R.A.; De la Houssaye, P.R.; Chang, C.E.; Chen, P.-F.; Wood, M.E.; Garcia, G.A.; Lagnado, I.; Asbeck, P.M.; Advanced thin-film silicon-on-sapphire technology: microwave circuit applications, IEEE Trans. Electron Dev., Vol. ED-45, No. 5, pp. 1047–1053, May 1998. Gardial, F.E.: Design and layout of microstrip structures, IEE Proc., Vol. 135, No. 3, June 1988. Coonrod, J.: Understanding when to use FR-4 or highfrequency laminates, On Board Technol., pp. 26–30, Sept. 2011. High frequency circuit materials product selection guide, Rogers cor., USA. Thin film ceramic design guide, CoorsTek, USA. Product Guide: Applied Thin-Film Products (ATP), USA. RT/duroid® 5870/5880 High Frequency Laminates, Rogers Corporation, USA. Hasegawa, H.; Furukawa, M.; Yanai, H.: Properties of the microstrip line on SiSiO2 system, IEE Trans. Microwave Theory Tech., Vol. MTT-19, No. 11, pp. 869–881, Nov. 1971. Tseng, H.Q.; Saunier, P.; Ketterson, A.; Witkowski, L. C.; Jones, T.: Embedded transmission line (ETL) MMIC for low-cost high-density wireless communication applications, IEEE Trans. Microwave Theory Tech., Vol. MTT-45, No. 12, pp. 2540–2548, Dec. 1997. Fine line thick film solution, API Technologies Corp., USA. Guillaume, B.; Mabire, J.; Maurin, P.; Reffet, D.; Chirol, F.; Polzer, E.K.: Microwave applications using photo-imageable technology; Microwave J., Vol. 40, No. 3, pp. 102–106, Mar. 1997. The Basics of Screen Printing Thick Film Inks, Du Pont, USA. Process of Thick Film Technology, http://www.ett.bme. hu/vlab/lab. Crnojević-Bengin, V.; Budimir, D.: Integrated waveguide bandpass filters using thick-film technology, Microwave Rev., Vol. 10, No. 1, pp. 9–13, June 2004. Pan, J.; Tonkay, G.L.; Quintero, A.: Screen printing process design of experiments for fine line printing of thick film ceramic substrates, Proc. Inter. Sym. Microelectronics, pp. 264–269, Nov. 1998, San Diego, USA. Dziurdzia, D.; Novak, S.; Gregorczyk, W.; CieŜ, M.: Photo patterned thick film resonators, 14th Int. Conf. Microwave Radar Wirel. Commun., MIKON-2002, Vol. 2, pp. 467–471, May 2002.

film microstrip lines on polyimide, IEEE Trans. Comput. Packg. Manfact. Technol. Part-B, Vol. 21, No. 2, pp. 171–176, May 1998. Malnovic, V.; Ozgur, M. Degroot, D.C.; Jargon, J.A.; Gaitan, M; Zaghloul, M.E.: Characterization of broadband transmission for coplanar waveguides on CMOS silicon substrates, IEEE Trans. Microwave Theory Tech., Vol. MTT-46, No. 5, pp. 632–640, May 1998. Williams, D.F.: Metal–insulator–semiconductor transmission lines, IEEE Trans. Microwave Theory Tech. Vol. 47, pp. 176–181, Feb. 1999. Verma, A.K.; Nasimuddin; Sharma, E.K.: Analysis and circuit model of a multilayer semiconductor slow-wave microstrip line, IEE Proc.-Microwave Antennas Propag., Vol. 151, No. 5, pp. 441–449, Oct. 2004. Jäger, D.: Slow-wave propagation along variable Schottky-contact microstrip line, IEEE Trans. Microwave Theory Tech., Vol. MTT-24, No. 9, pp. 566–573, Sept. 1976. Verma, A.K.; Nasimuddin, Sharma, E.K.: Propagation characteristics of Schottky contact suspended slow-wave microstrip line, IEEE Microwave Wirel. Comp. Lett., Vol. 11, No. 9, pp. 385–387, 2001. Konno, M.: Conductor loss in thin-film transmission lines, Electron. Commun. Japan Part-2, Vol. 82, No. 10, pp. 83–91, 1999. Yun, Y.; Jeong, J.H.; Kim, H.S.; Jang, N.: Basic RF characteristics of fishbone-type transmission line employing comb-type ground plane (FTLCGP) on PES substrate for use in flexible passive circuits, Electron. Telecommun. Res. Inst. ETRI J., Vol. 37, No. 1, pp. 128–137, Feb. 2015. Ogawa, H.; Hasegawa, T.; Banba, S.; Nakamoto, H.: MMIC transmission lines for multi-layered MMIC’s, IEEE MTT-S Int. Microwave Symp. Dig., pp. 1067–1070, June 1991. Darwish, A.; Ezzeddine, A.; Huang, H.C.; Mah, M; Analysis of three-dimensional embedded transmission lines (ETL’s), IEEE Microwave Guided Wave Lett., Vol. 9, No. 11, pp. 447–449, Nov. 1999. Ishigawa, T.; Yamashita, E.: Characterization of buried microstrip lines for constructing high- density microwave integrated circuits, IEEE Trans. Microwave Theory Tech., Vol. 44, pp. 840–847, June 1996. Kim, J.; Qian, Y.; Feng, G.; Ma, P.; Judy, J.; Chang, M. F.; Itoh, T.: A novel low-loss low crosstalk interconnect for broad-band mixed-signal silicon MMIC’s, IEEE Trans. Microwave Theory Tech., Vol. 47, No. 9, pp. 1830–1835, Sept. 1999.

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compact LTCC branch line coupler using modified -T equivalent circuit model for transmission line, IEEE Microwave Comput. Lett., Vol. 16, No. 2, pp. 90–93, Feb. 2006. Choi, B.G.; Stubbs, M.G.; Park, C.S.: A Ka-band narrow bandpass filter using LTCC technology, IEEE Microwave Comput. Lett., Vol. 13, No. 3, pp. 388–89, Sept. 2003. Liang, H.; Sutono, A.; Laskar, J.; Smth, W.R.: Material parameter characterization of multilayer LTCC and implementation of high Q-resonators, IEEE MTT(S) Dig., Vol. 4, pp. 1901–1904, 1999. Eurskens, W.; Wersing, W.; Gohlke, S.; Wannemnacher, V.; Hild, P.; Weigel, R.: Design and performance of UHF band inductors, capacitors and resonators using LTCC technology for mobile communication systems, IEEE Int. Microw. Symp. Dig., pp. 1285–1288, Baltimore, MD, 1998. Lee, Y.C.; Park, C.H.: A novel high Q LTCC strip line resonator for millimeter wave applications, IEEE Microwave Comput. Lett., Vol. 13, No. 12, pp. 499–501, Dec. 2003. Jeng, Y.H.; Chang, S.F.R; Lin, H.K.: A high stopbandrejection LTCC filter with multiple transmission zero, IEEE Trans. Microwave Theory Tech., Vol. MTT-54, No. 2, pp. 633–638, Feb. 2006. Lee, J.H.; Pinel, S.; Papapolymerou, J.; Laskar, J.; Tentzeris, M.M.: Low-loss LTCC cavity filters using the system-on-package technology at 60 GHz, IEEE Trans. Microwave Theory Tech., Vol. MTT-53, No. 12, pp. 3817–3823, Dec. 2005.

493

14 Static Variational Methods for Planar Transmission Lines Introduction The variational method is a powerful tool to compute the line capacitance and other line parameters of microstrip and coupled microstrips on a single-layer substrate, as well as, on the multilayer substrate. The Galerkin’s method, a sophisticated version of the variational method, is also discussed. Galerkin’s method computes the line parameters of both the microstrip and CPW structures. The method is used both in the space domain and the Fourier transformed domain, i.e. in the Fourier domain. The latter method is also known as the static spectral domain analysis (SDA). The static results of this chapter are used in chapter 15 to develop the closed-form models for the multilayer microstrip and coupled microstrip lines to compute the frequency-dependent propagation constant, characteristic impedance, conductor and dielectric losses. Objectives

• • • • • •

To present variational formulations of line capacitance, both in space and Fourier domains. To compute line parameters of the boxed single-layer and multilayer microstrip in the space domain. To compute line parameters of the open single-layer and multilayer microstrip in the Fourier domain. To compute line parameters of the coupled multilayer microstrip in space and Fourier domains. To compute line parameters of the multilayer boxed microstrip using the discrete Fourier transform. To compute line parameters of the multilayer boxed CPW using the discrete Fourier transform.

14.1 Variational Formulation of Transmission Line The variational method computes the upper and the lower bounds on the line capacitance. It means the line capacitance can be computed with a relatively small

error for an assumed charge distribution function on a conductor, or an assumed potential distribution around the conductor. The variational method is based on the calculus of variation that can be studied from the standard textbooks [B.1–B.4]. It is like the maximum and minimum problem of ordinary calculus. However, some familiarity with the nomenclature of the calculus of variation is useful to follow the variational method. 14.1.1

Basic Concepts of Variation

The differential equation, with known boundary conditions, is normally used to formulate a physical problem. The physical problem could also be reformulated in the form of a definite integral. Its lower and upper limits are associated with the boundary or range. The integrand of the definite integral is known as the Lagrangian of the variational problem. The variational method is concerned with obtaining the unknown function of the integrand under the stationary conditions, i.e. under the maximum/minimum conditions for the variational integral, known as the functional. The stationary conditions give the lower and upper bounds of the functional. Let us consider the formulation of a physical problem that is constructed through following definite integral: b

Iy =

F x, y, a

dy dx, dx

14 1 1

where integral I(y) is known as the functional. The integrand F(x,y,dy/dx), i.e Lagrangian function, is a function of independent variable x, dependent variable y(x), and derivative dy/dx. We have to determine the function y(x) that provides the maximum/minimum, i.e. the stationary value for the integral i.e. functional I(y), within the range a < x < b. It is noted that in the case of a simple function of differential calculus, the local maximum and minimum of the function are obtained at the discrete stationary points, whereas in the case of the functional of the variational calculus, the extrema of the functional are obtained for the stationary functions. Thus, the

Introduction to Modern Planar Transmission Lines: Physical, Analytical, and Circuit Models Approach, First Edition. Anand K. Verma. © 2021 John Wiley & Sons, Inc. Published 2021 by John Wiley & Sons, Inc.

14 Static Variational Methods for Planar Transmission Lines

domain of a simple function is a region of coordinate space, whereas the domain of a functional is a set of the admissible functions in the functional space. The problem of the variational calculus is dealt with by an operator called variation δ. It works like a differential operator. Thus, the variation δy of a function y(x) is a small change in y for a fixed value of independent variable x, i.e. for no variation in x, i.e. δx = 0. The values of the function y(x) are given at x = a, x = b, i.e. y(x = a) = A, y(x = b) = B. These are the boundary conditions, such that there is no variation in y, δy = 0 at x = a, x = b. To obtain the unknown function y, i.e. y(x) under the stationary condition of the functional I(y), the variation in I(y) must be zero, i.e. δI(y) = 0. If the second variation δ2I(y) is positive, the function y(x) provides the minimum value of the integral functional I(y). However, for δ2 I(y) negative, it provides the maximum value of the integral functional I(y). Figure (14.1a and b) show the minimum and maximum values of the variational functional I(y). Figure (14.1c) shows the nonvariational functional I(y). It is noted that the vertical axis of Fig (14.1) shows the values of the functional for the admissible functions y0, y1, etc., along the horizontal axis. In the case of a variational expression, the minimum functional Imin(y0),

shown in Fig (14.1a), is obtained at y = y0. The value Imin(y0) is known as the true value of the functional for the correct solution y0. However, even for the approximate solution y = y1, the value of the functional does not deviate significantly from its true value Imin(y0), although I(y1) > Imin(y0). Thus, any admissible approximate or assumed solution y(x) will have the value of functional more than its true value. The assumed solution should be optimized so that the value of functional approaches to the true value Imin(y0). This is the minimization of the functional I(y) with respect to the assumed solution. Similar is the case for the maximum of the functional Imax(y0), shown in Fig (14.1b). In this case, the functional value has I(y1) < Imax(y0) for y = y1. Thus, any admissible approximate or assumed solution y(x) will have the value of functional less than its true value. The assumed solution should be maximized so that the value of the functional approaches to the true value Imax(y0). This is the maximization of the functional I(y) with respect to the assumed solution. However, if the functional I(y) is a nonvariational, the value of the functional I(y1) is greatly different from its true value of I(y) at y = y0. It is shown in Fig (14.1c). The stationary conditions for the minimum and maximum of the functional I(y) are summarized below:

First variation δI y = 0 For the minimum condition, the 2nd variation positive nd

For the maximum condition, the 2

I(y0) True value y1 y

(a) Imin(y) soln. giving upper bound. Figure 14.1

True value I(y0)

y0

y1

c

y

(b) Imax(y) soln. giving lower bound.

14 1 2

capacitance. Further, in this variational formulation, the line capacitance C is a functional, and potential function ϕ is an unknown function. The approximate form of the unknown trial function could be assumed based on physical consideration. Likewise, the maximum condition, shown in Fig (14.1b), provides the lower bound of the functional I(y), i.e. the variational value of I(y), for y y0 is always lower than the

I(y1)

True value I(y0)

I(y1) y0

δ I y = − tive 2

I(y)

I(y)

I(y1)

δ I y = + tive

a b

2

variation negative

The minimum condition provides the upper bound of the functional I(y), i.e. the variational value of I(y), for y y0 is always higher than the true value of I(y) at y0. The relation I(y1) > Imin(y0) is shown in Fig (14.1a) for the assumed function of y = y1. This variational formulation is applied in section (14.1.2) to compute the line capacitance C using the energy method based variational expression. It provides the upper bound on the line

I(y)

494

y0

y1

(c) Nonvariational function.

Nature of variational (a and b) and nonvariational (c) functionals.

y

14.1 Variational Formulation of Transmission Line

true value of I(y) at y = y0 i.e. I(y1) < Imax(y0). The assumed function y1(x), the solution of a given problem, is known as the trial function. The variational method of Fig (14.1b) is applied in section (14.1.3), using the Green’s function method-based variational expression, to compute the lower bound of the line capacitance. In this case, the charge density distribution is an unknown function. 14.1.2 Energy Method-Based Variational Expression Collin [B.1] has provided the variational expressions for both the upper and lower bounds of the line capacitance of a transmission line. The expression for the upper bound on the line capacitance, i.e. the lower bound on the characteristics impedance Z0, is based on the electrostatic energy. The variational expression for the lower bound of the line capacitance, i.e. the upper bound on the characteristic impedance, is based on Green’s function approach. The Green’s function method is discussed in subsection (14.1.3). A summary of both expressions is presented below. For a full discussion, the reader should consult Collin. The electrostatic energy p.u.l. stored in the field surrounding a transmission line, shown in Fig (14.2a), is

+

ϕ2 = V0 + +

ρ(x,y) + + S2 Conductor #2 + + + Field region:R y

ε0 εr 2

Wc =

ε0 εr 2

∗

E t E t dxdy = ∂ϕ ∂x

2

ε0 εr 2

∂ϕ ∂y

+

Wc =

1 C ϕ2 − ϕ1 2

2

=

C=

ε0 εr V20

∇t ϕ ∇t ϕ dx dy = Cond#2

where, V0 =

∇t ϕ dℓ

ε0 εr V20

∇t ϕ 2 ds

a

S

b 14 1 5

(b) Lower and upper bounds of line capacitance. Y

Upper bound on Z0 Green function method

P(r,θ)

R

Z0 true a

b

X

εr

Lower bound on Z0

Figure 14.2 Transmission lines and variational bounds.

14 1 4

Cond#1

Green function method (Lower bound on C) CLower

(c) Lower and upper bounds of line characteristic impedance.

dxdy,

Using the equations (14.1.3) and (14.1.4), the expression for the line capacitance is obtained as

ϕ1

Energy method

2

1 CV20 2

Ctrue

(a) Two conductors transmission line.

− ∇t ϕ dxdy

where E t is the transverse electric field in the transverse (x–y)-plane, and ϕ(x, y) is the potential function giving potentials ϕ1 and ϕ2 at conductor #1 and conductor #2 respectively. The potential difference between both the conductors is V0. For a grounded conductor #1, ϕ1 = 0. If C is the line capacitance p.u.l., the electrostatic energy p.u.l. is

(Upper bound on C) CUpper

S1

− ∇t ϕ

14 1 3

Energy method

Conductor #1

x

Wc =

(d) Cross-section of the coaxial line.

495

496

14 Static Variational Methods for Planar Transmission Lines

In equation (14.1.5), the differential length dl is along any path between conductor #1 and conductor #2. The integration is carried out over the cross-sectional region (R) occupied by the electrostatic field. The potential function ϕ(x, y) is unknown. We assume the form of solution for the potential function ϕ(x, y) and compute the line capacitance. However, the accuracy of the computed result is uncertain. If expression (14.1.5), i.e. the energy method is variational, in that case, the first-order error in our assumption of the potential function ϕ(x, y) results in less error, i.e. the second-order error in the computation of the line capacitance. Therefore, it is to be demonstrated that expression (14.1.5) is a variational expression. For this purpose, the variation in the electrostatic energy is zero: δWc = δ

∂ϕ ∂x

S

= −

R

2

+

∂ϕ ∂y

dxdy

∂2 ϕ ∂2 ϕ δϕdxdy = 0 + ∂x2 ∂y2

14 1 6

∇t ϕ ∇t ϕ dx dy μ εV0

2

The potentials on the conductors are ϕ1 = 0, ϕ2 = V0. Equation (14.1.7) is also a variational expression that provides the lower bound on the characteristic impedance, shown in Fig (14.2c). An application of the variational formulation is illustrated below by computing the line capacitance of a coaxial cable. Ex. 14.1 Calculate the line capacitance of a coaxial cable of radii a and b shown in Fig (14.2d). Use the energy method-based variational expression for this purpose.

Solution: Let us assume that the potential at P(r, θ) between the inner and outer conductors is expressed by the following polynomial function: N

ϕ = ϕ0 +

α i ri ,

∇t ϕ = r

∂ϕ ∂ϕ +θ = α1 ∂r r∂θ

i = 1, 2, 3, …

i

i=1

where r is a radial distance and αi is the ith coefficient of the assumed polynomial. For the uniform field, potential is independent of θ. Let us approximate the potential by one term i.e. the order of the polynomial is N = 1:

iii

The variational line capacitance using equation (14.1.5) is ∇t ϕ 2 dxdy

ε0 εr

Cv =

s2 s1

∇t ϕ dl

2

2π b 2 0 a α1 r

ε0 εr

=

drdθ

2 b a α1 dr

iv

1 2 b − a 2 α1 2 b+a 2 = ε0 εr π 2 2 b−a α1 b − a

ε0 εr 2π =

However, the true value of the line capacitance of a coaxial line is 2πε0 εr b+a ≈ πε0 εr ln b a b−a

C0 =

14 1 7

ii

The potential at the inner conductor r = a is ϕ = V0 and so

2

The integral is zero, due as ∇2ϕ = 0. The reader should follow the details in Collin [B.1]. Moreover, Collin shows that δ2Wc > 0. Hence, expression (14.1.5) i.e. the energy method provides the minimum value, i.e. the upper bound for the line capacitance, shown in Fig (14.2b). The characteristic impedance of a line is given by 1 = Z0

ϕ = ϕ0 + α 1 r

where, for b > a; ln

b a

=2

b−a b+a

+

1 b−a 3 b+a

3

+

b−a b+a

≈2

v

The above expression is identical to that of the expression (iv). It shows that the variational expressions are approximate. Thus, error in the variational expression occurs due to neglecting the higher-order terms of the logarithmic expansion. If b = 2a, then Cv ≈ 3πε0εr; however, its true value is C0 ≈ 2.885πε0εr. The variational method has a 3.97% error in Cv. The value of the variational capacitance is higher than the true value of the line capacitance. The error in the computed variational line capacitance is reduced by taking another term of the expression for the potential function, i.e. N = 2. ϕ = α0 + α1 r + α2 r2 giving ∇t ϕ = α1 + 2α2 r ε 0 εr

Cv =

2π b 0 a b a

2πε0 εr =

vi

α1 + 2α2 r 2 rdr dθ

α1 + 2α2 r dr

2

b2 − a2 2 b3 − a3 α1 + 4α1 α2 + b4 − a4 α22 2 3 2

b − a 2 α21 + b2 − a2 b − a 2α1 α2 + b2 − a2 α22

Cv πε0 εr −

2

b − a 2 α21 + b2 − a2 b − a 2α1 α2 + b2 − a2 α22 b2 − a2 α21 +

8 3 b − a3 α1 α2 + 2 b4 − a4 α22 = 0 3

vii

14.1 Variational Formulation of Transmission Line

The variational line capacitance Cv is a function of two parameters, i.e. coefficients α1 and α2. We know that Cv is always higher than the true line capacitance. Thus, Cv has to be minimized with respect to α1 and α2, so that it approaches the true value of the line capacitance.

dCv = 0, dα1

dCv =0 dα2

Using equations (vii) and (viii), the following matrix equation is obtained:

2Cv 2Cv 8 b − a 2 − 2 b2 − a2 b − a b2 − a2 − b3 − a3 3 πε0 εr πε0 εr 2Cv 8 3 2Cv 2 2 2 3 2 2 b−a b −a − b −a b − a − 4 b4 − a4 πε0 εr πε0 εr 3

The variational value of the line capacitance Cv is obtained from the nontrivial solution of the above equation. For the nontrivial solution, the determinant of the above equation is zero. For b=2a, it computes the CV = 2.889πε0εr. The value of Cv for N = 2 is less than that of N = 1. The error in the computed value of Cv is only 0.14% higher than the true value of the line capacitance. In the above example, we have adopted the Rayleigh– Ritz method to minimize the line capacitance to determine its value. Thus, if the variational line capacitance is C = N/D, its minimization with respect to the coefficients αi(i = 1, 2, …) requires the following condition: ∂C 1 ∂N N ∂D =0 = − ∂αi D ∂αi D ∂αi ∂N ∂D −C = 0, i = 1, 2, …, n ∂αi ∂αi

14 1 8

The ratio of numerator function N(αi) and denominator function D(αi) defines the line capacitance C(αi). Condition (14.1.8) provides n number of linear homogeneous equations. The solution of the system of linear equations determines the line capacitance. The coefficients αi(i = 1, 2, …, n) of the assumed polynomial function can also be determined by solving the set of homogeneous linear equations.

14.1.3 Green’s Function Method-Based Variational Expression Green’s function formulation of the variational expression for the line capacitance provides the lower bound of the line capacitance, i.e. the upper bound on its characteristic impedance. It is illustrated in Fig (14.2). The charge distribution on the line conductor is an unknown function. The assumed charge distribution function, treated as the trial function, is based on physical consideration. We summarize below the Green’s

viii

α1 α2

=0

ix

function method. The reader can follow Collin for more details [B.1]. Figure (14.2a) shows that conductor #1 is grounded and conductor #2 is positively charged to the potential ϕ = ϕ2 = V0, such that there is a charge distribution ρ(x , y ) on its surface. Alternatively, a charge distribution ρ(x , y ) placed on the surface of a conductor can create the potential function ϕ(x, y) around the conductor. The potential function ϕ(x, y) describes the electric field around the conductor. It satisfies Poisson’s equation: ∇t 2 ϕ x, y = −

1 ρ x ,y , ε0 εr

14 1 9

where the locations (x , y ) and (x, y) are the source point and field point, respectively. The potential ϕ(x, y) could be taken as a response of the charge density ρ(x , y ) located on a conductor at the source point (x , y ). The Dirac’s delta function δ(x − x )δ(y − y ) describes the unit charge density excitation on the surface of a conductor. The response of the unit charge density at the field point (x, y) is the potential Green’s function G(x, y x , y ). It also satisfies Poisson’s equation and meets the boundary conditions given below: ∇2t G

x, y x , y =

−

1 δ x−x δ y−y εr ε0

a 0, x x, y y where G x, y x , y = 0 at the conductor#1, and also at ∞ b

14 1 10 Therefore, for a given problem, Green’s function is obtained by solving the above homogeneous partial differential equation under the boundary conditions. The potential at any field location is a superposition of the potential responses due to the charges. For a distributed charge, it is given by the following integral: ϕ x, y =

G x, y x , y ρ x , y dℓ C2

14 1 11

497

498

14 Static Variational Methods for Planar Transmission Lines

The potential ϕ(x, y) at the surface of conductor #2, that is charged to potential V0, is G x, y x , y ρ x , y dℓ

ϕ x, y = V0 =

14 1 12

C2

In equation (14.1.12), the charge on the conductor #2 is described by the distribution function ρ(x , y ). Equation (14.1.12) is the integral equation, where V0 is known and the charge distribution function ρ(x , y ) is unknown. By solving it, we can obtain the charge distribution on the conductor. In the space domain, this process is called the method of moments, where the integral equation is converted to a matrix equation [B.5, B.6]. The line capacitance p.u.l. is C=

Q V0

ρ x , y dl

a , where charge p u l , Q =

b

C2

14 1 13 The variables x , y , and dl are dummy variables. Expression (14.1.13a) for the line capacitance is not variational in nature. To make the expression variational, we rewrite it as follows, in terms of Green’s function: 1 V0 Q = CV Q2

1 = CV

G x, y x , y ρ x, y ρ x , y dldl C2 C2 ρ

x , y dl

2

14 1 14 It could be shown that δ(1/Cv) = 0, for a δ{ρ(x , y )} variation in the charge density ρ(x, y), and δ2(1/Cv) > 0 [B.1]. Hence, the inverse variational line capacitance 1/Cv provides the minimum value for the line capacitance. The variational expression for the line capacitance, in term of the potential function, is 1 1 = 2 CV Q

ϕ x, y ρ x , y dl

14 1 15

C2

For any charge distribution, the variational 1/Cv is higher than the true value 1/Ctrue, 1/CV > 1/Ctrue, i.e. CV < Ctrue. Hence, the value of variational line capacitance is always less than the true value of line capacitance. Green’s function method gives the lower bound on a line capacitance, shown in Fig (14.2b). It means Green’s function method, using the unknown charge distribution, gives the upper bound on the characteristic impedance, shown in Fig (14.2c). It is noted that the free space permittivity ε0 is not incorporated in expression (14.1.15) for the line capacitance. It has been subsequently incorporated in the expression for the Green’s function. Two variational expressions for the line capacitance have been summarized above. The energy method, with

an unknown potential function, gives the upper bound on the line capacitance. In this case, the parameters (coefficients) of the trial potential function are optimized to get the minimum value of variational line capacitance to bring it closer to the true value of the line capacitance. Likewise, Green’s function method, with unknown charge distribution function, gives the lower bound on the line capacitance. The parameters (coefficients) of the trial charge distribution function are optimized to get the maximum value of the line capacitance. The maximized line capacitance approaches to its true value. Figure (14.2b and c) show the upper and lower bounds on the line capacitance and the characteristic impedance respectively. The average value of line capacitances, obtained from the lower and upper bounds, is more accurate. However, it is a time-consuming process. Thus, we have to decide one variational expression. The decision depends upon the numerical accuracy and efficiency of the numerical process. For a shielded microstrip line, Green’s function method is more commonly used.

14.2 Variational Expression of Line Capacitance in Fourier Domain The potential Green’s function, in the space domain, for a boxed, i.e. shielded, microstrip is obtained in terms of the slowly convergent infinite series. Thus, the computation of the line capacitance of a shielded microstrip takes a longer time, if the sidewalls and top shield are kept at large distances to simulate an open microstrip structure. The open microstrip and the microstrip with top shield, however, without sidewalls, are analyzed more efficiently in the Fourier domain. In the Fourier domain, the second-order PDE is reduced to the second-order ODE and the closed-form Green’s function is obtained. The usual Fourier domain method does not apply to a shielded microstrip line, as the Fourier transform is not defined for the boxed structure. However, the Fourier method, using the discrete Fourier transform (DFT), applies to a boxed microstrip line. The DFT formulation is known as the static spectral domain analysis (SDA). Moreover, the DFT method is also applicable to the multiple strip planar structures, such as the CPW lines, etc. All three methods are discussed in this chapter. 14.2.1 Transformation of Poisson Equation in Fourier Domain Fourier transform of a function ϕ(x, y), with respect to a continuous variable x in the range (−∞, ∞), and the

14.2 Variational Expression of Line Capacitance in Fourier Domain

inverse Fourier transform ϕ β, y to recover the function ϕ(x, y) from it, are defined as follows: ϕ β, y =

∞ − ∞ϕ ∞

1 ϕ x, y = 2π

x, y ejβx dx

−∞

ϕ β, y e

Poisson’s equation in the Fourier domain is obtained from equation (14.2.7), on using equations (14.2.4) and (14.2.6):

a

− jβx

dβ

d2 ϕ β, y 1 f β δ y−y , − β2 ϕ β, y = − ε0 εr dy2 14 2 9

b, 14 2 1

where β is a Fourier variable. Fourier transform is symmetrical with respect to β if the function ϕ(x, y) is symmetrical with respect to variable x, ϕ x, y = ϕ − x, y

ϕ β, y = ϕ − β, y

a,

b

14 2 2 Fourier transform of a derivative of a function ϕ(x, y) is obtained as follows: ∂ϕ 1 = ∂x 2π ∂2 ϕ 1 = ∂x2 2π

∞ −∞ ∞

− jβ ϕ β, y e − jβx dβ

a

− β2 ϕ β, y e − jβx dβ

b

−∞

where ϕ β, y and f β are the potential and charge distribution function, respectively, in the Fourier domain. Equation (14.2.9) is an ODE not a PDE, like the original Poisson’s equation.

14 2 3 2

∂ϕ is obtained ∂x2 from equations (14.2.1a) and (14.2.3b) as,

14.2.2 Transformation of Variational Expression of Line Capacitance in Fourier Domain To avoid taking the inverse Fourier transform of Green’s function or potential function, the variational expression of the line capacitance is expressed in terms of Fourier transformed variables. Parseval theorem is helpful in this process. It expresses the definite integral of a product of two functions in the range (−∞, ∞) in terms of their Fourier transforms. It is established for integral I as follows: ∞

I=

Therefore, Fourier transform F of

F

∂2 ϕ = ∂x2

∞ −∞

The derivative respect to y is ∂ϕ 1 = ∂y 2π ∂2 ϕ 1 = ∂y2 2π Thus, F

∂2 ϕ = ∂y2

∂2 ϕ jβx e dx = − β2 ϕ β, y ∂x2

of

∞ −∞ ∞

the

function

∂ϕ β, y − jβx e dx ∂y

−∞ ∞ −∞

ϕ(x,y)

14 2 4

14 2 5 b

Using equations (14.1.1a), Fourier transform of Poisson’s equation (14.1.9) is −∞

∞ −∞

f x δ y − y ejβx dx

14 2 7 On a thin strip conductor, located at distance y , the charge distribution f(x) is a function of x only. Hence, the charge distribution function ρ(x , y ) has been written, in the above equation, as ρ x ,y = f x δ y−y

−∞

−∞

ϕ β, y dβ

∞

−∞

−∞

f x ϕ x, y dx =

f x dx

f x ej

1 2π

∞ −∞

−β x

1 2π

∞ −∞

ϕ β, y e − jβx dβ

dx

ϕ β, y f β dβ

Using equation (14.2.10), the variational expression of the line capacitance, given by equation (14.1.15), in the Fourier domain is ∞ 1 1 1 ϕ β, y f β dβ = = 14 2 11 2 C Cv 2πQ − ∞ In the above equation, y is the location of strip conductor containing charge distribution. Using Parseval theorem, the potential function, given by equation (14.1.11), in the Fourier domain, is

∂2 ϕ jβx d2 ϕ β, y e dx = 2 ∂y ∂y2

∂2 ϕ ∂2 ϕ jβx 1 + e dx = − 2 2 ∂x ∂y ε0 εr

1 2π

∞

14 2 10 with

14 2 6

∞

−∞ ∞

∞

a

∂2 ϕ β, y − jβx e dx ∂y2

=

f x ϕ x, y dx =

14 2 8

ϕ β, y = G β, y f β ,

14 2 12

where G β, b is Green’s function in the Fourier domain. Using the symmetrical charge distribution, i.e. f − β = f β on a conductor, expression (14.2.12) is rewritten as 1 1 = CV π

∞ 0

f β Q

2

G β, y dβ

14 2 13

Again, it is noted that ε0 is not incorporated in equation (14.2.13). However, the permittivity of medium is normally used with the expression for the Green’s function.

499

14 Static Variational Methods for Planar Transmission Lines

Expression (14.2.13) requires assumed trial charge dis-

f x =

tribution function, f β Q , in the Fourier domain. Therefore, before preceding further, some popular charge distribution functions on the strip conductor, and their Fourier transforms, are present below.

1,

−w 2 ≤ x ≤ w 2

0,

x >w 2

f β = =

∞ − ∞f

w 2 jβx − w 2 e dx

x ejβx dx =

2 ejβw 2 − e − jβw j2 β

2

=w

Constant Charge Distribution

The total charge p.u.l on the strip conductor is Q=

w 2 − w 2 dx

=w

Therefore, the normalized charge

distribution function in the Fourier domain is sin βw 2 f β = βw 2 Q

14 2 16

Linear Charge Distribution

Figure (14.3c) shows the linear charge distribution on a strip conductor [J.2]. It is a better description of the charge distribution on a strip conductor, although it does not show the edge singularity. However, it shows an increase in charge accumulation at the edges of the strip conductor: f x =

x,

−w 2 ≤ x ≤ w 2

0,

x >w 2

Y

Y

1

1

f(x) 0

+w/2

X

+w/2

X

0 –w/2 +w/2 (c) Linear charge distribution.

(b) Constant charge distribution.

Y

Y f(x)

(a) Microstrip conductor of width w.

0

–w/2

X

X –w/2

0

+w/2

(d) Cubic charge distribution. Figure 14.3

X

–w/2

0

+w/2

(e) Maxwell charge distribution.

X

14 2 17

Fourier transform of function (14.2.17) is obtained as follows:

f(x)

Figure (14.3a) shows a symmetrically placed strip conductor of width w. Figure (14.3b) shows the constant charge distribution f(x) on a strip conductor of width w [J.2]. It does not account for the edge singularity.

–w/2

sin βw 2 βw 2 14 2 15

The charge on a strip conductor distributes itself symmetrically in such a way that it has a singularity (sudden large increase) at the edges of a strip conductor. Maxwell’s distribution function, obtained using the conformal mapping method, describes such a charge distribution on the strip conductor. Mitra and Itoh have also computed the charge distribution, showing singularity, on a strip conductor using the function theoretic method [J.1]. However, simpler functions are also used in practice, as the accuracy of the variational method is not very sensitive toward the assumed charge distribution function. Some symmetrical charge distribution functions on a conducting strip, shown in Fig (14.3), and their Fourier transforms are presented below.

Y

14 2 14

Fourier transform of the above function is

14.2.3 Fourier Transform of Some Charge Distribution Functions

f(x)

500

0

s

w

(f) Off-set location of strip conductor.

A few common charge distribution functions and potential distribution.

X

14.2 Variational Expression of Line Capacitance in Fourier Domain

f β =

∞

w 2

x ej βx dx =

−∞

− w2

w 2

x cos βx + j sin βx dx

Q=

w 2

f β =

sin βw 2 βw 2

w 2

−

w 2 sin βw 4 4 βw 4

2

14 2 18 The function |x| sin βx an odd function; so its integral is zero. The function |x| cos βx is an even function. The total charge p.u.l on the strip conductor and also the normalized charge distribution function in the Fourier domain are given below: w 2

Q=

w 2

−w 2

w2 x dx = 2 xdx = 4 0

sin βw 2 f β =2 βw 2 Q

− w2

w 4+A = 4 f β 4 A+1 = Q A+4

= 2 x cos βxdx 0 2

1+A

a

cos

3

2x w

dx = w +

sin βw 2 βw 2

x3 dx

0

12A A + 4 βw 2

+

βw sin βw 2 −2 2 βw 2

sin βw 4 βw 4

+

2

2

b 14 2 22

Maxwell’s Charge Distribution Function

Figure (14.3e) shows Maxwell’s charge distribution function [J.4]. It shows the edge singularity at x = ± w/2.

2

b

w2

a

1

sin βw 4 − βw 4

16A w3

1−

f x =

2

x w 2

−w 2 ≤ x ≤ w 2

,

0

14 2 19

x >0 14 2 23

Cubic Charge Distribution

Figure (14.3d) shows a more realistic cubic charge distribution on a strip conductor [J.3]:

f x

1+A

=

x w 2

3

−w 2 ≤ x ≤ w 2

,

0,

Rayleigh–Ritz optimization process is used to evaluate constant A. However, normally, we use A = 1. Fourier transform of the function (14.2.20) is obtained as follows: ∞ −∞ w 2

= f β =

1+A jβx

− w2

2x W

e dx + A

3

− w2

2 βw 1 + A sin β 2

cos

ej β sin θ − nθ dθ

1 π2 j e π −π 2 On substitution of sin θ = t, 1 1 ejβt dt J0 β = π − 1 1 − t2

a β sin θ

dθ

b

c 14 2 24

f β = −∞

1

ejβx dx

1 − 2x w

2

14 2 25

w 2

12A wβ2

βw 4 βw 16 − sin + 2 βw 2 βw

−π 2

∞

2x 3 jβ x e dx w +

π 2

1 π

Fourier transform of equation (14.2.23) is

ejβ x dx w2

Jn β =

For n = 0; J0 β =

x >w 2 14 2 20

f β =

Fourier transform of function (14.2.23), in terms of the zeroth-order Bessel’s function, is obtained below from the integral form of Bessel’s function of nth order [B.7]:

1

= 2

sin 2

βw 4 14 2 21

Both the total charge p.u.l on the strip conductor and the normalized charge distribution function in the Fourier domain are given below:

1 − 2x w

−w 2

jβx

2

e dx

On using the substitution 2x/w = t and equation (14.2.24c), we get 1

f β =

w 2

−1

ej

β w 2

t

1 − t2

dt =

πw βw J0 2 2

14 2 26

501

502

14 Static Variational Methods for Planar Transmission Lines

Both the total charge p.u.l on the strip conductor and the normalized charge distribution function are w 2

Q=

1

−w 2

dx =

1 − 2x w

2

πw 2

Let 2sin(t) − 1 = p, cos(t) dt = 0.5 dp. For t = 0, π/2; p = −1, +1.

a

f β = J0 βw 2 Q

b 14 2 27

ρi x =

1− 0,

x−s w

2 x−s −w w

2

, s≤x≤s+w

i = 1, 2, 3, …

s+w s

ρi x dx,

s+w

Qi = s

cos i − 1 π 1−

2 x−s w

x−s w

−1

2 1 2

dx

14 2 29

Qi = 0

cos i − 1 π sin t 1 − 2 sin t − 1

2 1 2

w cos t dt 14 2 30

ρi βn =

wπ j e j8

wπ j e − j8

i−1

π 2

+ βn s + w2

i − 1 π2 − βn s + w2

J0 J0

βn w π + i−1 2 2 βn w π − i−1 2 2

i−1

π , 2

i = 1, 2, 3, …

The discrete Fourier transform of function ρi(x), over the range 0 ≤ x ≤ L, is defined as follows: ρi βn =

L 0

ρi x sin βn x dx,

14 2 33

where βn = nπ/L, n = 1, 2, … is the discrete Fourier variable. Therefore, the discrete Fourier transform of equation (14.2.28) is ρi βn =

cos i − 1 π x w− s

s+w s

1−

2 x−s w

−1

2 1 2

sin βn x dx

14 2 34 Let 2((x − s)/w) − 1 = p. Thus, dx = (w/2)dp and for x = s, (s + w); p = −1, +1. The above integral is written as ρi βn =

1

w 2

cos i − 1

π 2

1 + p sin βn s + 1 − p2

−1

Let (x − s)/w = sin(t); dx = w cos t dt. For x = s, s + w; t = 0, π/2. π 2

14 2 31

Discrete Fourier Transform of ρi(x)

otherwise

The above charge distribution function ρi(x) is used with the DFT method discussed in the section (14.6). The location of the strip conductor is shown in Fig (14.3f ). The strip conductor of width w is located at x = s and the center of the strip is at x = s + w/2. Thus, the charge distribution density function ρi(x) is singular at x = s and s + w. The factor cos[(i − 1)π{(x − s)/w}] helps to generate the trial basis functions for i = 1,2,3, … useful in Galerkin’s method. The total charge p.u.l distributed over the width of the strip conductor is

dp

14 2 32

14 2 28

Qi =

wπ π J0 i − 1 cos 2 2

Qi =

Expression (14.2.23) is modified to obtain the following modified Maxwell’s distribution function that is suitable to generate a sequence of functions, showing the edge singularity [B.8]: i−1 π

−1

1+p 12

The cos [ ] could be expressed in the form of exponentials and then equation (14.2.24c) could be used to evaluate the above integral in term of the zeroth-order Bessel’s function to get the following expression:

Modified Maxwell’s Distribution Function

cos

π 2 1 − p2

cos i − 1

1

w 2

Qi =

1 2

w 2

1+p

dp

14 2 35 Again, cos [ ] and sin [ ] could be expressed in the form of exponentials and then equation (14.2.24c) could be used to evaluate the above integral giving the following expression:

+ e−j +e

−j

i − 1 π2 − βn s + w2

i−1

π 2

+ βn s + w2

J0 J0

βn w π − i−1 2 2 βn w π + i−1 2 2

14 2 36

14.3 Analysis of Microstrip Line by Variational Method

For Mi = i − 1 π2 + βn s + w2 and Ni = − i − 1 βn s + w2 , expression (14.2.36) is written as

π 2

+

wπ sin Mi J0 Mi − βn s + sin Ni J0 Ni − βn s , 4 i = 1, 2, 3, …

ρi βn =

14 2 37 Finally, using equations (14.2.32) and (14.2.37) the normalized charge distribution function is sin Mi J0 Mi − βn s + sin Ni J0 Ni ρi βn , = Qi 2J0 i − 1 π2 cos i − 1 π2

i = 1, 2, …

14 2 38

14.3 Analysis of Microstrip Line by Variational Method This section presents the space domain-based variational method to compute the line capacitance of a shielded microstrip line called the boxed microstrip line, shown in Fig (14.4a) [B.1, B.8, B.10–B.12]. It also presents the Fourier domain variational method to compute the line capacitance of an open microstrip line, shown in Fig (14.5a). Both the variational methods are further applicable to a microstrip line under the multilayer environment [B.13, B.14]. The transverse transmission line (TTL) technique conveniently provides Green’s function for the multilayer microstrip and CPW structures [J.5– J.8]. The TTL method is discussed in section (14.4). 14.3.1 Boxed Microstrip Line (Green’s Function Method in Space Domain)

εr1 and substrate thickness h1 [B.9, J.3]. The height of the top shield, i.e. the electric wall (EW), from the ground conductor is H. The separation between the sidewalls (EW) is L. The region–I, −L/2 ≤ x ≤ L/2, 0 ≤ y ≤ h1, contains the substrate of relative permittivity εr1 and thickness h1. The region-II, −L/2 ≤ x ≤ L/2, h1 ≤ y ≤ h2(H − h1), contains the air medium with εr2 = 1 and thickness h2. The charge density ρ(x , y ) on the strip conductor creates the potential field, ϕi(x, y), in both the regions. It is obtained from the solution of 2D-Poisson’s equation: ∂2 ϕ i ∂2 ϕi ρ x ,y + = − , 2 2 εi ∂x ∂y

The functions ϕ1(x, y) and ϕ2(x, y) are potential functions in the regions-I and –II, respectively. Four walls are the electric walls, such that ϕi(x,y) = 0 on them. Equation (14.3.1), in terms of the Green’s function Gi(x, y|x , y ), is rewritten as −

∂2 G i ∂2 G i + = ∂x2 ∂y2

1 δ x−x δ y−y , εi

0,

EW (Upper ground) H EW (Side–wall)

h2

εr2 W Region #1

εr1

O L EW (Lower ground) (a) Boxed Microstrip Line.

h1

EW (Side–wall)

Region #2

X

x ,y

y,

14 3 2 where (x , y ) is the location of the unit charge on the strip conductor of width w and (x, y) is the location of a potential field point. The Green’s function G(x, y|x , y ) is a potential response of the system, i.e. the microstrip structure under excitation of the unit charge source. For an infinitely thin conductor located at y = h1, it is described by Dirac’s delta function: 14 3 3

The total potential at any point (x, y), due to the ρ(x , y ) charge distribution, is

Y

Y

x = x ,y = y x

ρ x , y = ρ x δ y − h1

Figure (14.4a) shows a boxed microstrip line of width w, printed on the isotropic substrate of relative permittivity

14 3 1

i = 1, 2

εr3 εr2

Current distribution on lower ground

εr1 εr1 2. So Chebyshev type basis functions provide faster convergence, even for one or two-term basis functions [J.20, J.22–J.24]. Their Fourier transforms are summarized below: Jx,p = j − 1 Jz,p = − 1

p−1

p−1

αn w 2 pπ Jp 2 αn pπ αn w 2 Jp − 1 2 2

sin Ex n x =

a b,

cos

Slotline

The microstrip line, shown in Fig (16.1b), can be treated as a boxed slot line also, where strip conductor is replaced by the slot-width w, and both dielectric interfaces are replaced by the conducting strips. The considered slot line structure is symmetrical with respect to the y-axis. For the fundamental mode, an EW is in the direction of the y-axis by treating one conductor at the positive potential and another conductor at the negative potential. The edge conditions also require that the Ez field is a regular function with Ez = 0 at the inner edges of the conductors, and Ex has a singularity (maxima) at the edges. The singular sinusoidal functions are popular as the basis functions for the slot-field components [B.1, J.9]. The electric field components Ex(x) and Ez(x) are the even and odd function, respectively, with respect to x. Such functional behavior could be described by the following basis functions to generate the EH-fundamental and even higher-order modes. Exn x =

Ezn x =

πx w

1 − 2x w 2 πx sin 2n w ; n = 1, 2, … 1 − 2x w 2

a

b 16 4 12

2n − 1

πx w

1 − 2x w

16 4 11

where αn is the discrete Fourier variable for the boxed microstrip. The continuous Fourier variable α is used for the side open microstrip. The basis functions are defined by number p. It also determines the order of the Bessel function Jp(αn). In this case, the computer code becomes more involved, unlike the sinusoidal based basis function mentioned above, that has the zeroth-order Bessel function for all order of basis functions. However, using Chebyshev polynomial, frequency-dependent current distribution has been obtained that provides a more accurate computation of frequency-dependent effective relative permittivity of the microstrip [J.22].

cos 2 n − 1

Fourier transform of the above function (16.4.12) is the same as that of equation (16.4.6). The odd higher-order hybrid mode could be obtained from the following basis functions [J.9]:

Ezn x =

2

πx w ; n = 1, 2, … 1 − 2x w 2

a

2n − 1

b, 16 4 13

where w is the slot-width and the field components. These are zero elsewhere. The solution of the characteristic equation of a slot line can provide both the bound fundamental and higher-order modes. However, due to the complex propagation constant, space leaky mode could be generated. Figure (16.6a) shows the even transverse electric field Ex(x), in relative unit i.e. normalized field, in the slot region of a slot line, on a substrate with εr = 2.6, h = 1.2 mm, w = 60 mm, for the fundamental even mode. It provides the bound fundamental mode, shown in Fig (16.6b). The first odd Ex(x) field generates the first higher-order bound mode and also the first space leaky odd mode at about 1.4 GHz. These are also shown in Fig (16.6b). Space leaky mode decreases at a higher frequency [J.9].

CPW Structure

Figure (16.1a) shows the CPW structure that supports the fundamental, even, and odd hybrid modes. For the even hybrid mode, the Ez(x) field component, with respect to x is even and Hz(x) is odd. The odd hybrid mode has Ez(x) odd and Hz(x) even, as respect to the variable x. The plane of symmetry, located at x = 0, is the MW of the fundamental modes and higher-order even mode with Hz(x) = 0 on it. For the odd mode, the plane of symmetry is an EW, with Ez = 0 at x = 0. Several kinds of basis functions have been used as the basis functions to expand the slot fields [B.2, J.1– J.4, J.7–J.10, J.25, J.26]. However, the singular sinusoidal basis functions appear to be more appropriate. Figure (16.7a) shows the location of slots of width s at x = ± c, where c = w + s/2. The CPW could also be treated as the coupled slot line. The total slot fields, Ex,m(x) and Ez,m(x), in both the slot regions, for the fundamental and even modes, and also for the odd modes are expressed by the following singular sinusoidal basis functions:

593

16 Dynamic Spectral Domain Analysis

10 εr = 2.6

Dominant bound mode

4 2 0

1st space leaky mode

–2

(β/k0)

Normalized prog. const.

h = 1.2 mm 6

–0.5

0.5

0.0

2 k0

1.0 2x/w

0

1

2 3 4 5 6 Frequency (GHz) (b) Existence of bound and leaky modes.

2

−

cos mπ x−c s 1− 2 x−c s

+

1− 2 x + c s 2

Odd mode Exm x =

cos mπ x−c s

1− 2 x + c s

, m = 1,3,… b

+

sin mπ x−c s

2

1− 2 x + c s

−

,

1− 2 x−c s 2 m = 1,3,…

2

a

1− 2 x−c s 2

sin mπ x + c s

sin mπ x + c s

the region x < 0, and the second term shows the center of the slot at x = c in the region x > 0. We obtain below the Fourier transforms of the above field distributions for the SDA application. The characteristic equation provides both the real and complex propagation constants for an open CPW. Figure (16.7b) shows the bound fundamental mode, and also generation of the first surface leaky mode at h/λo = 0.12 for the CPW on a substrate with εr = 10, h = 1 mm, s = 0.6 mm, 2w = 0.25 mm. The propagation constants are normalized by the free space wavenumber k0. Figure (16.7b) also shows the first two surface wave modes, kTM0 and kTE1. To obtain Fourier transform of the slot-field, the following definition is used:

, 2

m = 0,2,4… cos mπ x + c s

sin mπ x−c s

c

, m = 2,4…

d

1− 2 x−c s 2

16 4 14 Both the Ex,m(x), and Ez,m(x) are zero on the perfectly conducting strips. In the above equations (16.4.14a–d), the first term shows the center of the slot at x = −c in

f αn , y =

L −L

f x, y ejαn x dx, where, αn =

0.1

MW

0.08

S β/k0 α/k0

2

2W O n=0

C

X

β/k0

n=0

S

Dominant bound mode

Ex (x) -field

Y

Ez (x) -field

0.06 kTE1

Surface leaky εr = 10 k 0 mode h = 1 mm s = 0.6, w = 0.25 mm

0 0.00 (a) Slot Ex and Ez field distribution.

kTM0

0.04 1

C

2n − 1 π L 2 16 4 15

3

Figure 16.7

0

α/k0

cos mπ x + c s 1− 2 x + c s

Ezm x =

1

1st space leaky mode

Bound and leaky mode on a slot-line. Source: Zehentner et al. [J.9]. © 2004, Microwave Review. Reproduced by permission of Microwave Review.

Even mode Exm x =

Ezm x =

β/k0 α/k0

0.9

(a) Even/odd Ex(x) field in slot region. Figure 16.6

3

1st higher order bound mode

1.0

0.8

–4 –1.0

Dominant bound mode

1.1

α/k0

w = 60 mm 8 Ex (x) (Relative unit)

594

0.05

0.10 0.15 h/λ0 (b) Fundamental and surface leaky mode.

0.02 0

Bound and leaky mode on a CPW. Source: Zehentner et al. [J.9]. © 2004, Microwave Review. Reproduced by permission of Microwave Review.

16.4 Basis Functions for Surface Current Density and Slot Field

The function f(x, y) is an even function, for the x variable, with respect to the y-axis.

s cos mπ x − c ejαn x s 2 II = dx s c− 2 x−c 2 2 c+

1−

s

Fourier Transform of Ezm(x) Even Mode

Let us obtain the Fourier transform of equation (16.4.14b) as follows:

− c−

s 2

cos mπ

a

Ezm αn =

Ezm x dx =

− c+

−a

s 2

1−

x+c s

2 x+c s

2

ejαn x dx

s II = 2

I c+

+

c−

s 2

s 2

x−c cos mπ s 1−

2 x−c s

2 x−c s s =u x = u + c and dx = du s 2 2 s 2 s For x = c − , u = c − − c = − 1, and 2 s 2 s 2 s +c , u= c + −c = + 1 for x = 2 s 2

Let

2

ejαn x dx

=

II 16 4 16

Let

2 x+c = u, s

For x = −

x=

s s u − c, dx = du 2 2

s 2 s +c , u= − c − + c, 2 s 2

s 2 s , u= − c + + c, 2 s 2 mπu cos jα s u − c 2 ×e n 2 du 1 − u2

for x = − c − s I= 2

=

+1 −1

s − jcαn 1 e 2 2

+1 −1

e

j

α s n 2

+ mπ 2 u

+e

j

πs −jcαn αn s mπ αn s mπ J0 e + + J0 − 4 2 2 2 2

αns 2

+1 j

e

+c

+ mπ 2

du u

+e

αns mπ 2 − 2

j

u

du

1 − u2

−1

πS jcαn αn s mπ αn s mπ e + + J0 − , J0 4 2 2 2 2 m = 1, 3, 5… 16 4 18

Ezm αn = I + Π = = J0 Ezm

= +1

J0

sπ e 2

− jαn c

+ e jαn c 2

αn s mπ αn s mπ + − + J0 2 2 2 2 sπ cos αn c αn = 2 αn s mπ αn s mπ + − + J0 2 2 2 2

,

m = 1, 3, 5… 16 4 19

Fourier Transform of Ezm(x) Odd Mode

Let us obtain equation (16.4.14d).

u

the

du sin

On using the Bessel function integral [B.10], 1 1 ejαt dt, the integral-I is evaluated: J0 α = π − 1 1 − t2 I=

−1

mπ u jαn 2s u e 2 1 − u2

Fourier transform of Ezm(x) of equation (16.4.14b) is

= − 1;

αn s mπ 2 − 2

1 − u2

cos

s jcαn 1 e 2 2

II =

In equation (16.4.16), the first integral-I is evaluated over the left-hand side slot shown in Fig (16.7a) and the second integral-II is evaluated over the right-hand side slot.

+1

Ezm αn =

− c − 2s − c + 2s

, m = 1,3,5…

16 4 17 The second integral, over the right-hand side of Fig (16.7a), is evaluated as follows:

−

c + 2s c − 2s

Fourier

transform

mπ x + c jαn x e s 2

2 x+c 1− s

I' mπ x − c jαn x e sin s 2 x−c 1− s

2

of

dx

dx

II' 16 4 20

595

596

16 Dynamic Spectral Domain Analysis

Let us evaluate the above integral in two parts − c − 2s

I =

Let

− c + 2s

1−

2 x+c s

2

Using equations (16.4.17) and (16.4.18), Fourier transform of equation (16.4.14a) is determined: dx Exm αn = I − II =

2 x+c =u s

s I = 2

+1

mπu jαn 2s u − c e 2 du 1 − u2

sin

+1

e

j

αns 2 u

+ mπ 2

−e

j

αn s mπ 2 u− 2

du

1 − u2

−1

πs − jcαn e J0 4j

αn s mπ + 2 2

− J0

αn s mπ − 2 2

− jαn c

− e jαn c 2

×

αn s mπ αn s mπ + − + J0 2 2 2 2 jsπ sin αn c × Exm αn = − 2 αn s mπ αn s mπ J0 + − + J0 , m = 0, 1, 2, 2 2 2 2 16 4 24

Fourier Transform of Exm(x) Odd Mode

16 4 21 Likewise, II =

sπ e 2

J0

−1

s = e − jcαn 4j =

mπ x + c jxαn e s

sin

Fourier Transform of Exm(x) Even Mode

πs jαn c J0 e 4j

αn s mπ + 2 2

− J0

αn s mπ − 2 2

Using equations (16.4.21) and (16.4.22), Fourier transform of equation (16.4.14c) obtained: Exm αn = I + II =

16 4 22 Therefore, Fourier transform of equation (16.4.14d) is Ezm αn = I − II =

sπ e − jαn c − ejαn c × 2j 2 αn s mπ − − J0 2 2

αn s mπ + 2 2 sπ sin αn c × Ezm αn = − 2 αn s mπ αn s mπ + − J0 − J0 2 2 2 2 J0

,

J0 Exm J0

sπ e − jαn c + ejαn c × 2 2j

αn s mπ αn s + − − J0 2 2 2 jsπ cos αn c × αn = − 2 αn s mπ αn s + − − J0 2 2 2

mπ 2

mπ 2

,

m = 1, 3, … 16 4 25

m = 2, 4…

Fourier transforms of equations (16.4.14a–d) are summarized below:

16 4 23

Even mode

sπ αn s mπ αn s mπ sin αn c J0 + − + J0 , 2 2 2 2 2 sπ αn s mπ αn s mπ cos αn c J0 + − + J0 , Ezm αn = 2 2 2 2 2 Odd mode sπ αn s mπ αn s mπ + − − J0 Exm αn = − j cos αn c J0 2 2 2 2 2 sπ αn s mπ αn s mπ sin αn c J0 + − − J0 Ezm αn = − 2 2 2 2 2 Exm αn = − j

16.5

Coplanar Multistrip Structure

The previous sections have discussed the single strip microstrip, slot line, and CPW structures. The CPW structure has two slots in the same plane containing

m = 0, 2, 4…

a

m = 1, 3, 5…

b 16 4 26

m = 1, 3, 5…

c

m = 0, 2, 4…

d

the strip conductors. There are occasions when two, three, four, etc., microstrip conductors are located on the same plane, i.e. at the interface of the substrate and air dielectric media. These structures are useful for the design of coupler, filter, etc. The multiple strip

16.5 Coplanar Multistrip Structure

Y

Y εr2

W1 S1

W2 S2

W3

εr2

h2

1 2W

S3 W4 S4 W5 εr1

εr1

h1 L X

–L

h2

2 2W

2S

h1

–(S+W) (S+W)

0 –L LX (b) Symmetrical coupled microstrip line.

(a) Multistrip and multislot planar line. Figure 16.8 Same level multiconductors and multislots.

conductors are also used as interconnects in the VLSI technology that has pushed the clock-rate in the GHz range. A generic form of such structure is shown in Fig (16.8a). The SDA has been used to obtain the propagation constant of all modes supported by this structure [J.27, J.28]. The symmetrically coupled microstrip line, with identical conductors shown in Fig (16.8b), is a more popular structure. Its physical behavior and equivalent circuit line are discussed in section (11.3) of chapter 11. In this section, the SDA is used to obtain the coupling between the lines. The discussion is limited to the symmetrically coupled microstrip. The SDA has also been used to the asymmetrically coupled microstrip with unequal strip widths [B.2]. The mathematical process used below follows the discussion of section (14.6) of chapter 14. 16.5.1

Symmetrical Coupled Microstrip Line

Jx α n =

M

t=1 m=1

Jz α n =

N

M

t=1 m=1

t ctm Jxm

αn

M

1

c1m Jxm αn + m=1

a

m=1

M

Jz α n =

2

c2m Jxm αn M

1

d1m Jzm αn + m=1

2

d2m Jzm αn

b,

m=1

16 5 2 1 Jxm

1 Jzm

αn and αn are Fourier transformed where basis functions of current density on the conductor #1. 2

The Fourier transformed basis functions Jxm αn and 2

Jzm αn are current densities on conductor #2. Equation (16.5.1) is substituted in equation (16.3.1) to get the following system of equations: M

Z11 αn

t

ctm Jxm αn t=1m=1 N

M

+ Z12 αn

t

= Ex αn

a

t

= Ez αn

b

dtm Jzm αn t=1m=1

N

M

Z21 αn

t

ctm Jxm αn t=1m=1 N

M

+ Z22 αn

dtm Jzm αn t=1m=1

16 5 3 The inner product of equation (16.5.3a) is taken with t

the testing function Jxk αn k = 1, 2, … and of equat

a 16 5 1

t

dtm Jzm αn

M

Jx α n =

N

The impedance kind of Green’s function given by equation (16.3.1) in the Fourier domain is applied to the microstrip structure containing any number of coplanar strips, shown in Fig (16.8a). The surface current density components in the Fourier domain Jx αn and Jz αn , i.e. Jx(x) and Jz(x) in the space domain, are the total current density on all strip conductors in the range, −L to L. Likewise, the electric field components Ex αn and Ez αn , i.e. Ex(x) and Ez(x), are the total electric fields in all slots in the range, −L to L. For the N number of strip conductors, equation (16.3.2), for the current density, can be written as follows: N

for both the transverse and longitudinal current density Jx αn and Jz αn . For the two-conductor symmetrical coupled microstrip line, shown in Fig (16.8b), equation (16.5.1a and b) are reduced to

b,

where t is the number of strip conductors, i.e. t = 1, 2, … N; m = 1, 2, …M is the number of terms of basis function

tion (16.5.3b) with the testing function Jzℓ αn , (ℓ = 1, 2, …) such that the right-hand side of both equations is zero over the range −L to L, due to the complementary nature of perfect strip conductors and slots. Under the inner product process, the above equation (16.5.3) is reduced to the following system of equations:

597

598

16 Dynamic Spectral Domain Analysis N

M

N

M

ctm Ptim αn + t=1m=1

dtm Qtim = 0

a

t=1m=1

t

phase. For this case, the total current density Jz αn is

testing function k = i = 1, 2, …M N

M

N

ctm Rtim

o

M

αn +

t=1m=1

dtm Stim

=0

b

M

t=1m=1 N

M

t=1m=1

N

Ptim αn

M

Qtim

t=1m=1 N

Rtim αn

M

t=1m=1

ctm

0 =

dtm

Stim

c, 0

∞

t∗

t

∞

t∗

t

Jz m αn Z12 αn Jx i αn

n=1

αn =

∞

16 5 5 t∗ Jx m

t αn Z21 αn Jz i

αn

n=1

Sti n αn =

∞

t∗

t

Jz m αn Z22 αn Jz i αn

n=1

In equation (16.5.5), the suffix (∗) is used for the complex conjugate of a function. The above expressions are applied to the symmetrical coupled microstrip line shown in Fig (16.8b). Let us consider the case when transverse current on the strip could be neglected, i.e. Jx = 0, and Jz αn has a one-term basis function. This assumption is valid for the fundamental coupled mode. If two strips of width 2w are symmetrically located, its Fourier transform at the center is Jz1 αn . The Fourier transforms of conductor # 1 and conductor # 2 are obtained from the shifting theorem as follows: 1

Conductor #1

Jz αn = e − jαn

Conductor #2

2 Jz

αn = e

s+w

Jz1 αn

a

+ jαn s + w

Jz2 αn

b 16 5 6

For the even-mode excitation, currents on both strip conductors are equal in magnitude with the same phase. t

For this case, the total current density Jz αn is e

Jz αn = e − jαn

s+w

+ e + jαn

s+w

Jz1 αn = 2 cos αn s + w Jz1 αn

− e − jαn 2j

s+w

Jz1 αn 16 5 8

16 5 7

t

Z22 αn Jz i αn

2

=0

16 5 9

n=1 t

e

In equation (16.5.9), the condition Jz i αn = Jz αn is used for the even mode, and the condition t

Jx m αn Z11 αn Jx i αn

n=1

Qti m αn =

s+w

The expression for Fourier transformed current density Jz1 αn has already been obtained in section (16.4). For the case considered, the characteristic equation, from equation (16.5.5), is ∞

where the number of strip conductors is, t = 1, 2…N and Pti m αn =

2j e + jαn

= 2j sin αn s + w Jz1 αn

16 5 4

Rti n

Jz α n =

t=1m=1

testing function ℓ = i = 1, 2, …M N

For the odd mode excitation, currents on both strip conductors are equal in magnitude and opposite in

o

Jz i αn = Jz αn for the odd mode. Also, the same function is used as the basis and testing function. The numerical solution of the characteristic equation, giving the fundamental even and odd mode propagation constants, for the known physical parameters of the coupled microstrip line at the assumed frequency is obtained from the characteristic equation (16.5.9). In the case of more number of basis function for both the longitudinal and transverse currents, the even and odd mode propagation constants are obtained from equation (16.5.4c) by taking its det = 0. The complex propagation constants could also be obtained, showing the presence of the leaky modes.

16.6 Multilayer Planar Transmission Lines Figure (16.9a–f ) show some multilayer planar transmission line structures used in the MIC, MMIC, MEMS, LTCC technology. The dielectric layers could be isotropic or anisotropic. Generally, these are lossy dielectric layers. Sometimes they could be treated as lossless. Three cases are considered in this section. i) Single strip microstrip and single slot slot-line Figure (16.9a, b) shows the microstrip line and the slot-line under four-layered dielectric medium. The structures could be easily reduced to several useful cases, such as shielded, dielectric covered, suspended, inverted structures, etc. ii) Coplanar multistrip microstrip lines Figure (16.9c,d) shows the edge coupled microstrip lines and CPW structures respectively. Multiple

16.6 Multilayer Planar Transmission Lines

y

y Top wall

εr4 εr3

2W

εr3

h3

εr2

h2

εr1

h1 Bottom wall

z y

εr4

h4

x

h4 W

2S

h3

W

εr2

h2

εr1

h1 x

o

(a) Multilayer microstrip.

(b) Multilayer slot -line. y

p εr4

h4

εr4

h4

h3

εr3

h3

εr2

h2

εr2

h2

εr1

h1

εr1

h1

εr3

o

W

W

S

x

C p (c) Edge coupled microstrip.

o

x (d) Multilayer CPW structure.

. εr4

h4

εr4

h4

εr3

h3

εr3

h3

εr2

h2

εr2

h2

εr1

h1

εr1

h1

(e) Broadside-coupled microstrip.

(f) Broadside-coupled CPW.

Figure 16.9 Multilayer planar transmission line structures.

conducting strips are located at the same level. Again these structures are under four-layered dielectric medium. iii) Multilevel multiconductor microstrip Figure (16.9e,f ) shows the broadside-coupled microstrip and the broadside coupled CPW structure under four-layered dielectric medium. In this case, the conducting strips are located at two levels. Normally, the second and third layers are identical. In that case, it is a three-layered structure. These structures could also be reduced to several technologically useful cases. The above mentioned structures are typical. Several dielectric layers and a number of the strip conductors

can be added at different levels to meet practical requirements. The single or multilayer strips at any one level are considered by the total surface current density at that level. Likewise, the multiple slots at any one level are considered by the total slot fields at that level. The real task before us is to find the impedance kind Green’s function for the microstrip type structures used in the multilayer condition and the admittance kind Green’s function for the slot kind of the multilayer structure. Both kinds of Green’s functions are the inverse of each other. In general, two approaches are used to obtain the Green’s function of a multilayer planar transmission line: (i) immittance approach developed by Itoh [B.1, J.4, J.29–J.32] and transfer matrix approach developed by Davies and Syahkal [B.2, B.4, J.11, J.16–J.19]. However,

599

600

16 Dynamic Spectral Domain Analysis

this section considers only the immittance approach due to its simplicity. There are other approaches also to obtain the Green’s function of the multilayer planar transmission lines. These could be followed in the open literature after having acquaintance with one approach.

TEy Mode

16.6.1 Immittance Approach for Single-Level Strip Conductors

Ez α = jαϕ

m

Ex α = − jβz ϕ

m

α , Hx α = −

Ey α = 0,

Hy α =

m

jα ∂ϕ α α ∂ϕ α =− jωμ ∂y ωμ ∂y

m 1 ∂2 + k2 ϕ α 2 jωμ ∂y m

The immittance approach avoids the solution of the wave equation in each layer of a multilayer dielectric medium for the hybrid mode wave propagation. Each layer involves four unknown constants for the electric and magnetic scalar potentials. It is a very cumbersome process to determine the constants of the multilayer case. This difficulty is faced even while solving Laplace’s equation for the quasi-static case of the planar lines in the layered medium. The transverse transmission line (TTL) method is used in section (14.4) of chapter 14 to obtain Green’s function of the quasi-static multilayer microstrip and CPW structures. The immittance approach is an extension of the TTL technique applied to the dynamic SDA. The quasi-static method involves one Laplace’s equation for one electric potential. Thus, it involves only one equivalent transmission line network to represent the multilayer microstrip line. However, the dynamic analysis of a microstrip line, or a CPW structure, involves the hybrid mode that is a linear combination of the LSE (TEy) and LSM (TMy) modes described by the wave equation (16.2.2) for the electric and magnetic potentials. Therefore, two equivalent transmission lines are needed – one for the LSE (TEy) mode and another for the LSM (TMy) mode. Figure (16.9a) shows that the y-axis is the direction of inhomogeneity of the dielectric layers and the hybrid mode propagates in the z-direction. This section at first obtains the transverse transmission line (TTL) equivalence circuit for the layered medium, shown in Fig (16.9a–f ), for the LSE (TEy) mode and LSM (TMy) mode, respectively. Next, it is applied to the TTL models to obtain Green’s functions of several multilayered planar transmission line structures.

α,

Hz α = −

m

jβz ∂ϕ α β ∂ϕ α =− z jωμ ∂y ωμ ∂y

16 6 1 TMy Mode e

Ex α = − Ey α =

α ∂ϕ α , ωε ∂y

e

Hx α = jβz ϕ α

e 1 ∂2 + k2 ϕ α , jωε ∂y2 e

Ez α = −

Hy α = 0 e

e jβz ∂ϕ α β ∂ϕ α =− z , Hz α = − jαϕ α jωε ∂y ωε ∂y 16 6 2

Fourier transforms of the potential and field components are obtained for the laterally open structure by e,m

ϕ

∞

α, y =

−∞

ϕe,m x, y ejαx dx

16 6 3

For the shielded structure, the continuous Fourier variable α is replaced by the discrete Fourier variable αn. The inverse Fourier transform is ϕe,m x, y =

1 2π

∞ −∞

ϕ

e,m

α, y e − j αx + βz z dα

16 6 4

Equation (16.6.4) shows that the TEy and TMy waves could be treated as a superposition of the spectral plane waves

described

by

the

e

m

ϕ

α, y e − j αx + βz z

and

ϕ α, y e − j αx + βz z . The spectral plane waves travel in the (x–z)-plane, with components of propagation constants βz and α along the z-axis and x-axis, respectively. Figure (16.10) shows the spectral waves travel in the v direction. The direction of propagation of the spectral waves, i.e. v, is θ degree inclined with the z-axis, such that cos θ =

TTL Equivalent Circuits for LSE and LSM Modes

The LSE (TEy) and LSM (TMy) modes of the structure shown in Fig (16.9a) are described by magnetic potential ϕm(x, y)ej(ωt − βz) and electric potential ϕe(x, y)ej(ωt − βz), respectively. These potentials satisfy the wave equay tion (16.2.2) both for the TEy mode (Em y = 0) and TM mode (Hey = 0). The five field components for the TEy mode and TMy mode are given by equations (16.2.3) and (16.2.4), respectively. These field components in the Fourier domain are summarized below:

m

βz α2

+

β2z

,

sin θ =

α α2

+ β2z

16 6 5

The modes TEy and TMy have five field components in the (x, y, z) coordinate system. A new coordinate system (u, v, y) is considered for the propagation of the spectral waves. The variables of both coordinate systems in the transverse plane, with respect to the y-axis, are related by the following coordinate transformation: u v

=

sin θ

− cos θ

z

cos θ

sin θ

x

16 6 6

16.6 Multilayer Planar Transmission Lines

X

V

X

Ev (α)

α

Hu (α) βz

y

α

∿

θ

∿

V

∿

θ βz

y

Z

Z

Hv (α) u

u

∿

Eu (α)

(a) TEy → TEv mode change. Figure 16.10

(b) TMy → TMv mode change.

Spectral wave propagation.

Thus, in the (y, u, v) coordinate system, the transformed TEy mode has three field components

Equivalent Transverse Transmission Line for LSE (TEy) Mode

It is interesting to note that five field components of the TEy mode in the (x, y, z) coordinate system are reduced to three field components of the TEv in the (u, v, y) coordinate system. Using equation (16.6.6), the electric field components

Ez , Ex

and magnetic field components

Hz , Hx , from the (x, y, z) coordinate system are transformed into the

Eu , Ev

Hu , Hv

and

field compo-

nents in the (u, v, y) coordinate system as follows: Eu = sin θEz − cos θEx . Using equations (16.6.1), (16.6.4) and (16.6.5), the field components in the (u, v, y) coordinate system are

Hy , Eu , Hv

and three field components are zero

(Ey = Ev = Hu = 0). In the (y, u, v) coordinate system, the usual TEy mode is transformed into the TEv mode. The TEv mode is defined with respect to the v-axis, Ev = 0. Thus, the original TEy mode with five field components is now TEv mode with only three field components. In the original structure, it is assumed that the modal field propagates in the y-direction to define the TEy mode. However, the actual hybrid mode propagates in the z-direction with the propagation constant βz. The TEv mode could also be treated as the mode propagating in the y-direction with constant propagation γ, i.e. m

α

Eu α =

α2

+ β2z

× jαϕ

Eu α = j α2 + β2z ϕ

m

m

+

α2 + β2z

× − jβz ϕ

−

×

βz dϕ ωμ dy

−

m

+

α α2 + β2z

YTE = − b

−

=

α dϕ ωμ dy

βz α2 + β2z

=0

α, y eγ y

For the ith layer of dielectric

m

m

=

γ jωμ

a

YTEi =

γi jωμi

b 16 6 8

α2 + β2z dϕm ωμ

ϕ

α2 + β2z ϕ

α eγ y

dy m

dϕ dy

c

β dϕ − z ωμ dy

Hv α = −

1 jωμ

α2 + β2z

ωμ j m

m

α dϕ − ωμ dy

= −

Eu α

d ϕ

α2 + β2z ×

−

Hv α

α m

Hv α = cos θHz + sin θHx =

m

× jαϕ

=0

+ β2z

ϕ α, y eγy . Therefore, TEv mode is supported by a transmission line of the propagation constant γ. Its characteristic admittance YTE is the wave admittance of the TEv mode given by the following expression:

m

α2 + β2z

βz α2

× − jβϕ

a βz

Hu α = sin θHz − cos θHx = m

α2 + β2z

α

Ev α = cos θEz + sin θEx = α

βz

−

m

dy

d

16 6 7

The (−) sign of the YTE, with respect to Fig (16.10), is to maintain the Poynting vector in the positive y direction. It is noted that the Hv α is in the negative direction along the v-axis. The Eu α -field component is in the

601

602

16 Dynamic Spectral Domain Analysis

direction of the Ju α component of surface current density on the strip conductor. Thus, the Ju α current component is taken as the source of the TEv mode. Now, the equivalent transverse transmission line (TTL) could be drawn for the TEy mode in the (y, u, v) coordinate system as shown in Fig (16.11a). Each dielectric layer of thickness hi is represented by a transmission line section of length hi. The characteristic admittance of the line section is YTEi, and the propagation constant γi is given by γi =

α2 + β2z − εi k2 ,

i = 1, 2, 3, 4

e

α

Eu =

α2 + β

× − 2

βz dϕ ωε dy

+

Equivalent Transverse Transmission Line for LSM (TMy) Mode

Using equation (16.6.6), the five field components of the LSM, i.e. TMy mode, in the (z, x, y) coordinate system can also be reduced to three field components in the (u, v, y) coordinate system as shown in Fig (16.10b). The expression (16.6.5) is applied to the field components of the LSM (TMy) – mode as given by equation (16.6.2):

Ev α =

−

e

βz

−

α2 + β2z

× −

α dϕ ωε dy

=0 a

e

βz

Ev α = cosθEz + sinθEx =

16 6 9

The surface current Ju α is located at the plane containing the strip conductor. The lower and uppermost transmission lines are short-circuited, as Eu α = 0 at both the bottom and top perfect conductors shown in Fig (16.11a).

y

Eu = sin θEz − cos θEx . Using Ex and Ez from equation (16.6.2), the following transformations are obtained:

α2 + β2z

−

βz dϕ ωε dy

−

α dϕ ωε dy

e

α α2 + β2z

α2 + β2z dϕe α ωε dy

b

16 6 10 Hu α = sin θHz − cos θHx e α = × − jαϕ − α2 + β2z

βz α2 + β2z = −j

Hv = cos θHz + sin θHx e βz = × − jαϕ + α2 + β2z

e

× jβz ϕ

e

α2 + β2z ϕ

α α2 + β2z

× jβz ϕ

e

a

=0

b

16 6 11

y εr4, h4

γ4 ,YTE4 Port #1

∿ J (α)

γ3 ,YTE3

u

εr3, h3

YU

εr2 ,h2

L YTE

εr4, h4

γ4 ,YTM4

YU TE4

∿ J (α)

γ3 ,YTM3

TE

v

εr3, h3

Ze(α) γ2 ,YTE2

0

γ1 ,YTE1

0

∿

∿

y = h1 + h2

L YTM1

∿

Eu (α)

y = h1 + h2

∿ J (α)

u

Figure 16.11

εr1 ,h1

Zm(α)

L YTM

Zm(α)

∿ J (α) (a) Equivalent circuit of

γ1 ,YTM1

U YTM

∿

Ze(α)

Eu (α)

γ2 ,YTM2

L YTE1

εr1 ,h1

εr2 ,h2

U YTM4

v

TEy

-mode.

(b) Equivalent circuit of TMy -mode.

Equivalent transverse transmission line (TTL) for TEy and TMy – modes in (u, v, y) coordinate system and equivalent circuit at y = h1 + h2 plane.

16.6 Multilayer Planar Transmission Lines

Thus, the (TMy)-mode has three field components in the (u, v, y) coordinate system – Ev α , Hu α , Ey α , as shown in Fig (16.10b). It shows that these are the field components of the TMv mode. Hence, the TMy mode, e

ϕ α, y eγy propagating in the y-direction, with five field components in the (z, x, y) coordinate system is transformed in the TMv mode in the (u, v, y) coordinate system with three field components. Figure (16.11b) shows a representation of the TMv mode by the transverse transmission line (TTL) sections, with characteristic admittance YTMi: e

YTMi =

YTMi =

Hu α Ev α jωεi γi

− j α2 + β2z ϕ α

= −

e

α2 + β2z

−

dϕ α dy

16 6 12 Equation (16.6.9) gives us the propagation constant γi in the y-direction. The TMv mode has the Ev α electric field component that is generated by the Jv α surface current density, as the strip is located in the (u, v, y) coordinate system. Figure (16.11b) shows that the current source Jv α is located at y = h1 + h2. The propagation constant γi is common to both TEv and TMv modes, as they are not the independent modes. They are the components of the hybrid mode having propagation constant γi in the y-direction. Figure (16.11b) shows that the bottom and top transmission lines are short-circuited. So at both the bottom and top conducting surfaces, the electric field’s v-component is Ev α = 0. The above-discussed equivalent TTL can be applied to the circuits of the hybrid mode to determine the Green’s function in the Fourier domain at the plane y = h1 + h2 containing the strip conductor. Determination of the [Z] – Type Green’s Function

Figure (16.11) shows that at the plane y = h1 + h2, the current sources, Ju α and Jv α , are present. They generate TEv and TMv mode, respectively. The transmission line sections, above and below current sources, present m

impedances Z α, h1 + h2 and Z α, h1 + h2 at the port #1, i.e. at the plane y = h1 + h2 for TEv and TMv modes respectively. The equivalent circuits of Fig (16.11) provide the corresponding electric fields, Eu α, h1 + h2 and Ev α, h1 + h2 , at the plane y = h1 + h2 as follows: e

TE mode

Eu α = Z α Ju α

TMv mode

Ev α = Z

u

m

α Jv α

a b

m

e

1 α + YU TE α 1 α = L YTM α + YU TM α

Z α = Z

m

a

YLTE

16 6 14 b

Equation (16.6.13) indicates that the hybrid mode of the actual structure in the (z, x, y) coordinate system is decoupled in two independent modes, TEu and TMv, in e

× ωεi

i = 1, 2, 3, 4

e

e

The port impedances Z α and Z α are obtained L U from the port admittances YLTE , YU TE and YTM , YTM of the transmission line sections looking downward (L) and upward (U):

16 6 13

m

the (u, v, y) coordinate system. The Z α and Z α are the impedance kind of Green’s functions in the (u, v, y) system for the TEu and TMv modes, respectively. However, the impedance kind of Green’s function is to be obtained in the (z, x, y) coordinate system, belonging to the hybrid mode propagation. The port admittances L

U

L

U

YTE α and YTE α , also YTM α and YTM α , are obtained on using the following input admittance equation: Yin = Y0

YL + Y0 tanh γd , Y0 + YL tanh γd

16 6 15

where Yin is the input admittance of a line section of length d terminated in the load YL. The short-circuited line #1 of Fig (16.11a) has YL = ∞. Using Fig (16.11a), the L

lower port admittance YTE α is obtained in terms of L

the layer admittance YTE1 of the first dielectric layer #1: L

YTE1 α = YTE1 α coth γ1 h1 L

YTE α = YTE2 α

L YTE1

a

α + YTE2 α tanh γ2 h2

b,

L

YTE2 α + YTE1 α tanh γ2 h2

16 6 16 where YTEi α , i = 1, 2, 3, 4 is the characteristic admittance of the ith section of the equivalent transmission L

lines. In the above equation (16.6.16), YTE1 α is a load to the line section #2. Likewise, the admittance of the U

upper port admittance YTE α is obtained in terms of U

the layer admittance YTE4 α fourth dielectric layer #4:

of the short-circuited

U

YTE4 α = YTE4 α coth γ4 h4 U

YTE α = YTE3 α

U YTE4

a

α + YTE3 α tanh γ3 h3 U

YTE3 α + YTE4 α tanh γ3 h3

b 16 6 17

603

604

16 Dynamic Spectral Domain Analysis

At this stage, equation 16.6.13 is to be transformed from the (u, v, y) coordinate system to the (z, x, y) coordinate system by the change of variables from Eu α , Ev α and Ju α , Jv α to Ez x , Ex x , Jz x , and Jx x . To get the coordinate transformation, equation (16.6.13) could be written in the matrix form:

Similarly, using Fig (16.11b), the following expressions are obtained for the

L YTM

U YTM

α and

α:

L

YTM1 α = YTM1 α coth γ1 h1 L

YTM α = YTM2 α

L YTM1

a

α + YTM2 α tanh γ2 h2

b

L

YTM2 α + YTM1 α tanh γ2 h2

16 6 18 L YTM4

α = YTM4 α coth γ4 h4

U

YTM α = YTM3 α

L YTM4

Eu α Ev α

e

Z α

=

0

0 Z

m

α

Ju α Jv α

16 6 20

a

α + YTM4 α tanh γ3 h3

YTM3 α +

L YTM4

On using the coordinate transformation (16.6.6), the above equation (16.6.20) is written as follows:

b

α tanh γ3 h3

16 6 19

sin θ

− cos θ

cos θ

sin θ

e

m

Z

Since

cos θ

− cos θ

cos θ

sin θ

Ex α

=

Z

sin θ

sin θ

Ez α

m

=

Ez α Ex α

−1

− cos θ

e

m

sin θ

cos θ

− cos θ

sin θ e

− cos θ

sin θ

Z

m

m

m

Z −Z

e

Z

m

Jx α

Jz α Jx α

α sin θ e

− Z α sin θ cos θ + Z

α sin θ cos θ m

Z α cos 2 θ + Z m

Z −Z e

m

e

α sin θ cos θ

α cos 2 θ

sin θ cos θ

Jz α

α sin θ

e

α cos θ m

m

− Z α cos θ

α cos θ

e

=

Z

2

− Z α sin θ cos θ + Z Z α sin 2 θ + Z

Jx α

the above equation (16.6.21) is written as follows:

Z α sin θ

m

sin θ

− Z α cos θ

α cos θ

cos θ

e

cos θ

e

Z α sin θ Z

Z α sin θ + Z

Jz α

16 6 21

sin θ

2

α

− cos θ

Jx α

sin θ e

=

Z

m

sin θ

Jz α

α sin θ

−1

0

0

− Z α cos θ

=

Ev α

=

Ex α

α cos θ

Eu α

Z α

e

Z α sin θ

=

e

Ez α

e

sin θ cos θ m

Z α cos 2 θ + Z

α sin 2 θ

α sin 2 θ

Jz α Jx α

Jz α Jx α 16 6 22

The above equations (16.6.22) give the impedance type Green’s function of the structure, shown in Fig (16.9a), in the Fourier domain with the following components: e

Z11 αn = Z αn sin 2 θ + Z

m

m

Z12 αn = Z21 αn = Z − Z e

Z22 αn = Z αn cos θ + Z 2

m

αn cos 2 θ e

sin θ cos θ αn sin θ 2

a b c

16 6 23

The matrix elements of equation (16.6.23) are defined at the plane y = h1 + h2. For an enclosed structure, a discrete Fourier variable α = αn is used in the above equations. The Green’s function is also applicable to the symmetrical coupled microstrip, separated in the even and odd mode supporting structures.

16.6 Multilayer Planar Transmission Lines

y = h1 + h2. These are obtained from the equivalent TTL network shown in Fig (16.11) as follows:

Determination of [Y] Type Green’s Function for Slot Line and CPW

Figure (16.9b,d) shows the multilayer slot line and CPW. The admittance kind of Green’s function is more suitable for these structures. The equivalent TTL network representation of TEy and TMy modes could be applied to the slot case also. At the plane y = h1 + h2, containing the slots and conducting strips, the following expressions are written from equation (16.6.13): e

Ju α = Y α Eu α

TEu mode v

TM mode

Jv α = Y

m

a

α Ev α

e

Y

e

Y α sin 2 θ + Y

b,

=

m

Y −Y

e

m

m

α cos 2 θ

Y −Y e

e

m

m

Y12 αn = Y21 αn = Y − Y e

Y22 αn = Y αn cos 2 θ + Y

m

sin θ cos θ

αn sin 2 θ

2

a b

16 6 28

c

Equations (16.6.23) and (16.6.28) are independent of the number of strip conductors and slots at one level, so both Green’s functions could be applied to any number of strips and slots at one level. 16.6.2 Immittance Approach for Multilevel Strip Conductors The conducting strips could be located at any level of dielectric layers. Figure (16.9e,f ) shows the broadside coupled microstrip and the broadside coupled CPW structure respectively. Normally, εr2 = εr3 and h2 = h3, i.e. the second and third dielectric layers could be combined into a single layer. Following the discussion of the previous section, the equivalent TTL networks can be

TEy mode

e

Y α cos θ + Y

sin θ cos θ

αn cos 2 θ e

α =

L YTM

α +

U YTM

a α

e

=

Y α 0

0 Y

m

α

Eu α Ev α

sin θ cos θ m

α sin θ 2

Ez α Ex α

drawn. These are shown in Fig (16.12a,b), for the LSE (TEy) and LSM (TMy) modes with two current sources for each of the conductor planes. Figure (16.12a) shows that the planes y = h1 and y = h1 + h2 + h3, containing the strip conductors, could be treated as port #2 and port #1, respectively, where current sources Ju2 α and Ju1 α are located. They generate the TEy mode. Likewise, Fig (16.12b) shows that the current sources Jv2 α and Jv1 α generate the TMy mode. The port-voltages appear as the responses at port #1 and port #2 due to the excitation currents, so the port voltages and currents are related through the equivalent [Z]-matrix. In the case of the TEy mode, the port voltages correspond to the field components, Eu2 α and Eu1 α . The excitation current sources Ju2 α and Ju1 α are applied at port #2 and port #1, respectively. In the case of the TMy mode, the response Ev2 α and Ev1 α are considered as the port voltages at port #2 and port #1, and the excitation current sources Jv2 α and Jv1 α are applied at the ports. Using the concept of the superposition, the following Z-matrix formulations are obtained for both the modes:

a

Eu α, h1 = Eu2 α =

b

α +

16 6 26

16 6 27

Eu α, h1 + h2 + h3 = Eu1 α = Ze11 Ju1 α + Ze12 Ju2 α Ze21 Ju1

16 6 25

b

The above expression (16.6.26) is transformed from the (u, v, y) coordinate system to the (z, x, y) coordinate system:

m

The above equation (16.6.27) could be written directly from equation (16.2.22). The above expression provides us the admittance type Green’s function in the Fourier domain with the following components: Y11 αn = Y αn sin 2 θ + Y

m

Ju α Jv α

where the admittance parameters Y α and Y α are the decoupled Y-kind Green’s functions at the plane

Jz α Jx α

U

The determination of the above admittance parameters has already been discussed. Equation (16.6.24) is written in the matrix form:

16 6 24

e

L

Y α = YTE α + YTE α

Ze22 Ju2

α

16 6 29

605

606

16 Dynamic Spectral Domain Analysis

Short-circuited top

y 4 Port #1

γ4

∿ J (α) u1

γ3

0

εr4, h4 YTE4

U1 YTE

YTE3 εr3, h3

YL1 TE

γ4 Ze11

γ2

εr2, h2 YTE2

YU2 TE

γ1

YTE1 ∿ Ju2(α) εr1, h1

YL2

Port #2

Short-circuited top

y

Port #1

γ3

v1

Ze22

0

Port #1 Eu1(α)

γ1

YTM1 εr1, h1

YL2

Figure 16.12

TMy mode

∿ J

v2

Port #1 Ev1(α)

Eu2(α)

Zm22

TM

Port #2

∿

Ev2

Z

∿

∿

Ju1(α)

Zm11

YU2 TM

∿

∿

∿

YL1 TM

Short-circuited bottom

Port #2 Z

YTM3 εr3, h3 εr2, h2 YTM2

Port #2

TE

U1 YTM

γ2

Short-circuited bottom

∿

∿ J (α)

εr4, h4 YTM4

∿

Jv1(α)

Ju2(α)

Jv2

[Z] -Matrix equivalent circuit

[Z] -Matrix equivalent circuit

(a) LSE (TEy)-mode.

(b) LSM (TMy)-mode.

Equivalent TTL network for two-level strip conductors in (u, v, y) – coordinate system.

m Ev α,h1 + h2 + h3 = Ev1 α = Zm 11 Jv1 α + Z12 Jv2 α a m Ev α,h1 = Ev2 α = Zm 21 Jv1 α + Z22 Jv2 α

b,

16 6 30 where Ze11 and Ze22 are the driving point, i.e. the input impedance, at the port #1 and port #2, respectively, for e

in terms of the upper and lower port admittance YU1 TE and at the port #1, while the port #2 is open-circuited, YL1 TE i.e. Ju2 α = 0: Ze11 =

e

the TEy mode. The Z12 and Z21 are the transfer impedances, i.e. the impedance at port #1 and port #2 due to the current sources at port #2 and port #1 respectively. y m m m Likewise, Zm 11 , Z22 , Z12 and Z21 are defined for the TM e

e

mode. The Ze11 , Ze22 , Z12 and Z21 for the TEy mode are determine below, using the TTL network shown in Fig (16.12a). The upper end of the line section #4 and the lower end of the line section #1 are short-circuited.

Ze22 =

YU1 TE YU2 TE

1 + YL1 TE 1 + YL2 TE

a 16 6 31 b

Therefore, the following expressions are obtained, using the transmission line theory, for the port admitL1 tance YU1 TE and YTE : At y = h1 + h2 + h3

+

YU1 TE = YTE4 coth γ4 h4 16 6 32

Determination of Ze11

It is determined from equation (16.13b). The input port impedance Ze11 is determined, from equation (16.6.29a), At y = h1− , load to line section #2 At y = h1 + h2

−

line section #2

At y = h1 + h2 + h3

−

line section #3

TE YL2 TE = Y1 = YTE1 coth γ1 h1

a

YTE 2 = YTE2

+ YTE2 tanh γ2 h2 YTE2 + YTE 1 tanh γ2 h2

b

YL1 TE = YTE3

+ YTE3 tanh γ3 h3 YTE3 + YTE 2 tanh γ3 h3

c

YTE 1

YTE 2

16 6 33

16.6 Multilayer Planar Transmission Lines L1 Once at the plane y = h1 + h2 + h3, YU1 TE and YTE are e known, we can get the function Z11 from equation (16.6.31a).

due to the line sections #3 and #4. YU1 TE is given by equation (16.6.32). Again, following equation (2.2.5) of chapter 2 the potential at y = h1 + h2 + h3, i.e. at port #1, is V1 = Eu1 α =

Ze12

Determination of

Figure (16.13a) shows that the Ze12 is related to the field response Eu1 α at the open-circuited port #1, due to the current source Ju2 α applied at port # 2. It follows from equation (16.6.29a) for Ju1 α = 0. The potential Eu1 α is obtained in terms of the current source Ju2 α by using the process discussed in section (2.2) of chapter 2. The process starts with replacing the current source Ju2 α by the voltage source at the port #2 as follows: Ju2 α Vs = L2 YTE + YU2 TE

Γh 3 =

Eu1 α = eγ3 h 3 + = =

16 6 35

YTE2 − YU TE2 YTE2 + YU TE2

=

YU1 TE + YTE3 tanh γ3 h3 YTE3 + YU1 TE tanh γ3 h3

Vh2 =

Port #1

γ3

γ2 Port #2 γ1 0

U1 YTE

YTE3 εr3, h3

YL1

Source εr2, h2 YTE2

YU2 TE

YTE1 εr1, h1

YL2 TE

Open

∿ J (α) u2

Figure 16.13

Vh2 YTE3 YTE3 cosh γ3 h3 + YU1 TE sinh γ3 h3 Vh2 YTE3 sinh γ3 h3 YU1 TE + YTE3 coth γ3 h3

Vs YTE2 sinh γ2 h2 YU TE2 + YTE2 coth γ2 h2

εr4, h4 YTE4

U1 YTE

∿ J (α)

YTE3 εr3, h3

YL1

Open

εr2, h2 YTE2

YU2 TE

YTE1 ε Response r1, h1

YL2 TE

γ4 Port #1

γ3

Source

u1

γ2 Port #2

(a) Source at port #2 and response at port #1.

+

2Vh2 YTE3 γ3 h 3 − e − γ3 h 3 + YU1 TE e

e − γ3 h 3

Short - circuited top

TE

Short - circuited bottom

eγ3 h 3

16 6 40

The following expressions are also obtained from equations (16.6.34), (16.6.39), and (16.6.40):

y

Response εr4, h4 ∿ E1u(α) YTE4

U1 Vh2 YTE3 + YU1 TE + YTE3 − YTE U1 U1 YTE3 + YTE + YTE3 − YTE e − γ3 h3

The Vh2 is obtained from equation (16.6.35) as

b

Short - circuited top

YTE3 − YU1 TE YTE3 + YU1 TE YTE3 − YU1 TE e − γ3 h3 YTE3 + YU1 TE

16 6 39

In equation (16.6.36), YTE2 and YTE3 are characteristic admittances of line #2 and line #3, respectively. The − admittance YU TE2 is a load to line #2 at y = (h1 + h2)

γ4

YTE3

Eu1 α =

16 6 36

y

eγ3 h 3

a

where, YU TE2 = YTE3

16 6 38

Vh2 1 +

where Γh2 is the reflection coefficient at the plane y = h1 + h2. The above relation is obtained from equation (2.2.5) of chapter 2. The reflection coefficient is given by Γh 2 =

YTE3 − YU1 TE YTE3 + YU1 TE

Thus, the potential at the port #1 is

16 6 34

Vs 1 + Γh2 , γ 2 e h 2 + Γh 2 e − γ 2 h 2

16 6 37

where Γh3 is the reflection coefficient at y = h1 + h2 + h3 obtained from

The potential, corresponding to the electric field at the interface of h2 and h3, i.e. at y = h2, is Vh2 =

V h 2 1 + Γh 3 eγ3 h3 + Γh3 e − γ3 h3

γ1 0

∿ E

2u(α)

TE

Short - circuited bottom (b) Source at port #1 and response at port #2.

Determination of the transfer (mutual) impedance of the LSE (TEy) – mode.

607

608

16 Dynamic Spectral Domain Analysis

Eu1 α =

YL2 TE

+

Ju2 α YTE2 YTE3 U2 YTE sinh γ2 h2 sinh

γ3 h 3

Z

×

e 21

YTE2 YTE3 × U1 YL1 + Y TE TE sinh γ2 h2 sinh γ3 h3 1 YU + Y coth γ h YL2 TE3 3 3 TE3 TE + YTE2 coth γ2 h2

α =

1 YU TE2 + YTE2 coth γ2 h2

YU1 TE + YTE3 coth γ3 h3

16 6 47

16 6 41 Therefore, the transfer impedance from equation (16.6.41) and equation (16.6.29a) showing the component of Green’s function is Z

e 12

α =

YTE2 YTE3 × + sinh γ2 h2 sinh γ3 h3 1 U + YTE2 coth γ2 h2 YU1 TE + YTE3 coth γ3 h3

YL2 TE

YU TE2

YU2 TE

16 6 42 In equation (16.6.42), YU TE2 is given by equation (16.6.36b). For three-layered structure, h2 = 0, equation (16.6.42) is reduced to Z

e 12

α =

YTE3 × U2 YL2 + Y TE TE sinh γ3 h3 1 U YU + Y TE TE3 coth γ3 h3

e 22

α e

= YTE1 coth γ1 h1

YU1 TE = YTE4 coth γ4 h4 YTE 3 = YTE3

YU1 TE + YTE3 tanh γ3 h3 YTE3 + YU1 TE tanh γ3 h3

+ YTE2 tanh γ2 h2 YTE2 + YTE 3 tanh γ3 h3 e

Ev1 α Eu2 α Ev2 α

e

m

m

m

e

u1 v1

a

v2

b

16 6 46

Likewise, Z 21 α is obtained from Fig (16.13b) by applying Ju1 α current at port #1, while port #2 is open-circuited. It can be written from equation (16.6.42) by using the symmetry:

=

Z11

0

0

e

Z12

0

m Z11

0

m Z12

e Z21

0

e Z22

0

0

Z21

0

Z22

m

m

Ju1 α Jv1 α Ju2 α Jv2 α

16 6 48

The coordinate transformation equation (16.6.6) can be written as follows for two sets of variables (u1, v1) (z1, x1) and (u2, v2) (z2, x2):

u2

Finally at y = h1+ YU2 TE = YTE2

m

16 6 44

16 6 45

YTE 3

e

are used for getting the Z11 , Z22 , Z12 and Z21 on replacing TEy mode by TMy mode everywhere, so for the TMy mode the characteristic admittances are taken from equation (16.6.12) and propagation constants are the same as that of the TEy mode. The real structure is in the (z, x, y) coordinate system, supporting the hybrid mode. However, equations (16.6.29) and (16.6.30) are in the (u, v, y) coordinate system. To convert them in the (z, x, y) coordinate, these equations are arranged in the matrix form: Eu1 α

The input impedance Z 22 α is obtained at port #2, while the port #1 is open-circuited. It is given by equation (16.6.31b). The upper and lower parts of the port U2 admittances YL2 TE and YTE , shown in Fig (16.13b), are determined as follows: YL2 TE

e

16 6 43

The above expression (16.6.43) is the same as the Green’s function of the three-layered structure obtained by Itoh [J.4]. Determination of Z

e

The expressions for Z11 , Z22 , Z12 and Z21 are obtained from equations (16.2.31), (16.6.42), and (16.6.47). In these equations, the characteristic admittances for the TEy mode are taken from equation (16.6.8) and propagation constants from equation (16.6.9). These equations

=

sin θ

− cos θ

0

0

z1

cos θ

sin θ

0

0

x1

0

0

sin θ

− cos θ

z2

0

0

cos θ

sin θ

x2

u 16 6 49 Using the above transformation (16.6.49), the field variables Eu1 α , etc., and current variables Ju1 α , etc., in the (u, v, y) coordinate system are transformed to the field variables Ez1 α … and current variables Jz1 α … in the (z, x, y) coordinate system. The coordinate transformation is applied to equation (16.6.48) to get the following transformed relation:

16.6 Multilayer Planar Transmission Lines e

Ez1 α u

Ex1 α Ez2 α

=

Ex2 α Ez1 α u

Ex1 α Ez2 α

=

Ex2 α Ez 1 α =

Ez 2 α

0

sin θ

− cos θ

0

0

Jz1 α

m Z11

0

m Z12

cos θ

sin θ

0

0

Jx1 α

e Z21

0

e Z22

0

0

0

sin θ

− cos θ

Jz2 α

0

Z21

0

Z22

m

0

0

cos θ

sin θ

Jx2 α

0

0

m

e Z11 sin θ m Z11 cos θ e Z21 sin θ m Z21 cos θ

e

e

Z11 sinθ −Z11 cosθ Z12 sinθ −Z12 cosθ

− cosθ sinθ

0

0

Z11 cosθ Z11 sinθ Z12 cosθ Z12 sinθ

sinθ

cosθ

Z21 sinθ −Z21 cosθ Z22 sinθ −Z22 cosθ

0

0

0

0

e

− cosθ sinθ m

Z11 sin 2 θ + Z11 cos 2 θ

Ex 1 α

−Z11 sinθcosθ + Z11 sinθsinθ

Ex 2 α

e

e − Z12 cos θ m Z12 sin θ e − Z22 cos θ m Z22 sin θ

0

Ez 1 α e

=

e

e Z12 sin θ m Z12 cos θ e Z22 sin θ m Z22 cos θ

0

Ex 2 α

Ez 2 α

e − Z11 cos θ m Z11 sin θ e − Z21 cos θ m Z21 sin θ

cosθ

sinθ

Ex 1 α

e

Z12

Z11

m

e

m

Z21 sin 2 θ + Z21 cos 2 θ e m −Z21 cosθsinθ + Z21 sinθcosθ

m e

m

e

m

Jx1 α Jz2 α Jx2 α Jz 1 α

m

Jx 1 α

e

Jz 2 α

e

16 6 50

Jz1 α

m m m m Jx 2 α Z21 cosθ Z21 sinθ Z22 cosθ Z22 sinθ e m e m −Z11 sinθcosθ + Z11 sinθcosθ Z12 sin 2 θ + Z12 cos 2 θ m e e m Z11 sin 2 θ + Z11 cos 2 θ −Z12 cosθsinθ + Z12 sinθcosθ e m e m −Z21 cosθsinθ + Z21 sinθcosθ Z22 sin 2 θ + Z22 cos 2 θ m e e m Z21 sin 2 θ + Z21 cos 2 θ −Z22 cosθsinθ + Z22 sinθcosθ

e

m

−Z12 cosθsinθ + Z12 sinθcosθ e

Jz 1 α

m

Z12 cos 2 θ + Z12 sin 2 θ e

Jx 1 α

m

−Z22 cosθsinθ + Z22 sinθcosθ

Jz 2 α

e m Z22 cos 2 θ + Z22 sin 2 θ

Jx 2 α

16 6 51

The Z-kind of the Green’s function relates the surface current density Jz α and Jx α to the electric field components Ez α and Ex α at two layers designated as two ports shown in Fig (16.13).

11

e

m

Zzx = Z11 −Z11 sin θ cos θ

12

e

m

Zzx = Z12 −Z12 sin θ cos θ

11

11

m

e

11

e

m

12

12

m

e

12

e

m

21

e

m

Zzx = Z21 −Z21 sin θcos θ

22

e

m

Zzx = Z22 −Z22 sin θcos θ

21

21

m

e

21

e

m

22

22

m

e

22

e

m

Zzz = Z11 sin 2 θ + Z11 cos 2 θ, Zzz = Z12 sin 2 θ + Z12 cos 2 θ,

11

m

e

12

m

e

Zzx = Zxz = Z21 −Z21 sin θcos θ, Zxx = Z11 cos 2 θ + Z11 sin 2 θ Zxz = Zzx = Z12 −Z12 sin θcos θ, Zxx = Z12 cos 2 θ + Z12 sin 2 θ

~ Z11 zz ~ Z11 xz

~ Z11 zx ~ Z11 xx

~ Z12 zz ~ Z12 xz

~ Z12 zx ~ Z12 xx

~ Jz 1 (α ) ~ Jx 1 (α )

Zzz = Z21 sin 2 θ + Z21 cos 2 θ,

−−−− = −− ~ ~ Z 21 E z 2 (α ) zz ~ ~ Z 21 E x 2 (α ) xz

−− ~ Z 21 zx ~ Z 21 xx

−− ~ Z 22 zz ~ Z 22 xz

−− ~ Z 22 zx ~ Z 22 xx

−−− , ~ Jz 2 (α ) ~ Jx 2 (α )

Zxz = Zzx = Z21 −Z21 sin θcos θ, Zxx = Z21 cos 2 θ + Z21 sin 2 θ

~ E z 1 (α ) ~ E x 1 (α )

Zzz = Z22 sin 2 θ + Z22 cos 2 θ,

21

m

e

22

m

e

Zxz = Zzx = Z22 −Z22 sin θcos θ, Zxx = Z22 cos 2 θ + Z22 sin 2 θ

16 6 53 Response port Excitation port

In each block of equation (16.6.52), the cross diagonal 11

Response field component

11

12

12

21

21

elements are equal, i.e. Zzx = Zxz , Zzx = Zxz , Zzx = Zxz

12 Z zx

22

22

and Zzx = Zxz . In the case of the coupling between the slots, the [Y]kind of Green’s function is used:

Excitation current component

(16.6.52)

Jz1 α

where the matrix element Z12 zx is explained above. The x-current component at the port #2, i.e. Jx2, creates the z-electric field component at port #1. From the above two equations, the elements of the [Z]-kind of dyadic Green’s function are given below:

Jx1 α Jz2 α

Ez 1 α = Y

Jx2 α where, Y = Z

Ex 1 α Ez 2 α

a

Ex 2 α −1

b

16 6 54

609

610

16 Dynamic Spectral Domain Analysis

Figure 16.14

εr3 = 1

h3

εr3 = 1

h3

εr2 = εr

h2

εr2 = εr

h2

εr1 = 1

h1

εr1 = 1

h1

(a) Microstrip–slot line coupling.

(b) Coupled microstrip with septum.

The 4 × 4 matrix is to be inverted. Alternatively, starting from equations (16.6.29) and (16.2.30), the admittance form is expressed as follows: TEy mode

TMy mode

Ju1 α = Ju2 α =

e Y11 Eu1 e Y21 Eu1

α + α +

e Y12 Eu2 e Y22 Eu2

m

m

m

m

α

References Books B.1 Itoh, T. (Editor): Numerical Techniques for Microwave

α

Jv1 α = Y11 Ev1 α + Y12 Ev2 α

16 6 55

B.2

Jv2 α = Y21 Ev1 α + Y22 Ev2 α

where, using equation (16.2.31), we have e

U1

L1

m

U1

L1

Y11 = YTE + YTE Y11 = YTM + YTM

Two levels coupled structures.

B.3

e

U2

L2

m

U2

L2

Y22 = YTE + YTE Y22 = YTM + YTM

B.4

16 6 56 The rest of the process for obtaining the admittance type Green’s function of the two-level slot structures could be repeated like the computation of the elements of the [Z]- matrix. The method can be extended to more number of dielectric layers and more levels containing strip conductors and slots [B.2]. However, it is analytically more cumbersome. If the conductors or slots on both the levels are symmetrical, as shown in Fig (16.9e,f ), the present analysis is not needed. The MW/EW, parallel to the x-axis, can be placed in between two levels. It reduces the structure to the single-level case for the even/odd mode excitation. The present method is more general and is useful for the asymmetrical structure, shown in Fig (16.14a, b) [B.9, J.4, J.5]. The dynamic SDA has been used for more involved cases, even with the anisotropic substrates [J.30]. Only computation of the propagation constants has been discussed. However, the method has also been used to compute the frequency-dependent characteristic impedance [B.1, B.2, J.33, J.28]. The method has been further used for the analysis of the patch resonators, and microstrip antenna [B.3, J.32].

B.5

B.6

B.7 B.8 B.9 B.10

and Millimeter-Wave Passive Structures, John Wiley & Sons, New York, NY, 1989. Mirshekar-Syahkal, D.: Spectral Domain Method for Microwave Integrated Circuits, Research Studies Press Ltd., John Wiley & Sons, New York, NY, 1990. Garg, R.; Bhartia, P.; Behl, I.; Ittipiboon, A.: Microstrip Antenna Design Handbook, Artech House, Boston, 2000. Bhat, B.; Koul, S.K.: Stripline-Like Transmission Lines for Microwave Integrated Circuits, Wiley Eastern Ltd., New Delhi, 1989. Goel, A.K.: High-Speed VLSI Interconnects Modelling, Analysis, and Simulation, John Wiley & Sons, New York, NY, 1994. Nguyen, C.: Analysis Methods for RF, Microwave, and Millimetre –Wave Planar Transmission Lines, John Wiley & Sons, New York, NY, 2000. Bhattacharyya, A.K.: Electromagnetic Fields in Multilayered Structures, Artech House, Boston, 1994. Scott, C.R.: The Spectral Domain Method in Electromagnetics, Artech House, Boston, 1989. Kinayman, N.; Aksun, M.I.: Modern Microwave Circuits, Artech House, Boston, 2003. Ryshik, I.M.; Gradstein, I.S.: Tables of Series, Products and Integrals, 4th Edtion, Academic Press, New York, NY, 1965.

Journals J.1 Itoh, T.; Mittra, R.: Spectral-domain approach for

calculating the dispersion characteristics of microstrip lines, IEEE Trans. Microw. Theory Tech., Vol. MTT-21, pp. 496–499, July 1973.

References

J.2 Itoh, T.; Mittra, R.: A technique for computing

J.3 J.4

J.5

J.6

J.7

J.8

J.9

J.10

J.11

J.12

J.13

J.14

J.15

J.16

dispersion characteristics of shielded microstrip lines, IEEE Trans. Microw. Theory Tech., Vol. MTT-22, pp. 896–898, Oct. 1974. Itoh, T.; Mittra, R.: Dispersion characteristics of slotlines, Electron. Lett., Vol. 7, pp. 364–365, 1971. Itoh, T.: Spectral domain immittance approach for dispersion characteristics of generalized printed transmission lines, IEEE Trans. Microw. Theory Tech, Vol. MTT-28, pp. 733–736, 1980. Itoh, T.; Hebert, A.S.: A generalized spectral domain analysis for coupled suspended microstrip lines with tuning septum, IEEE Trans. Microw. Theory Tech., Vol. MTT-26, No. 10, pp. 820–826, Oct. 1978. Itoh, T.: Generalized spectral domain method for multiconductor printed lines and its application to tunable suspended microstrips, IEEE Trans. Microw. Theory Tech., Vol. MTT- 26, pp. 983–987, 1978. Huang, W.-X.; Itoh, T.: Complex modes in lossless shielded microstrip lines, IEEE Trans. Microw. Theory Tech., Vol. MTT-36, No. 1, pp. 163–165, 1988. Railton, C.J.; Rozzi, T.: Complex modes in boxed microstrip, IEEE Trans. Microw. Theory Tech., Vol. MTT-36, No. 5, pp. 865–873, 1988. Zehentner, J.; Mrkvica, J.; Machácˇ, J.: Spectral domain analysis of open planar transmission lines, Mikrotalasna Revija (Microw. Rev.), Vol. 10, pp. 36–46, Nov. 2004. Mittra, R.; Itoh, T.: A new technique for the analysis of the dispersion characteristics of microstrip lines, IEEE Trans. Microw. Theory Tech., Vol. MTT-19, pp. 47–56, Jan. 1971. Davies, J.B.; Mirshekar-Syahkal, D.: Spectral domain solution of arbitrary coplanar transmission line with a multilayer substrate, IEEE Trans. Microw. Theory Tech., Vol. MTT-25, pp. 143–146, 1977. Denlinger, E.J.: A frequency dependent solution for microstrip transmission lines, IEEE Trans. Microw. Theory Tech., Vol. MTT-19, pp. 30–39, Jan. 1971. Jansen, R.H.: High-speed computation of single and coupled microstrip parameters including dispersion, high-order modes, loss and finite strip thickness, IEEE Trans. Microw. Theory Tech., Vol. MTT-26, pp. 75– 82, 1978. Meixner, J.: The behavior of electromagnetic fields at edges, IEEE Trans. Antenna Propagat., Vol. AP-20, No. 4, pp. 442–446, July 1972. Van de Capelle, A.R.; Luypaert, P.J.: Fundamentaland higher order modes in open microstrip lines, Electron. Lett., Vol. 9, pp. 345–346, July 1973. Mirshekar-Syahkal, D.; Davies, J.B.: Accurate solution of microstrip and coplanar structures for dispersion and

J.17

J.18

J.19

J.20

J.21

J.22

J.23

J.24

J.25

J.26

J.27

J.28

J.29

dielectric and conductor loss, IEEE Trans. Microw. Theory Tech., Vol. MTT-23, pp. 694–699, 1979. Mirshekar-Syahkal, D.; Davies, J.B.: Accurate analysis of coupled strip-fin line structure for phase constant, characteristic impedance, dielectric and conductor losses, IEEE Trans. Microw. Theory Tech., Vol. MTT-30, pp. 906–910, 1982. Mirshekar-Syahkal, D.; Davies, J.B.: An accurate unified solution to various fin-line structures, of phase constant, characteristic impedance, and attenuation, IEEE Trans. Microw. Theory Tech., Vol. MTT-30, pp. 1854–1861, 1982. Mirshekar-Syahkal, D.: An accurate determination of dielectric loss effect in MMIC’S including microstrip and coupled microstrip lines, IEEE Trans. Microw. Theory Tech., Vol. MTT-31, pp. 950–954, 1983. Fokas, A.S.; Smitheman, S.A.: The Fourier Transforms of the Chebyshev and Legendre Polynomials, arXiv:1211.4943v1 [math.NA], Nov. 2012. Krage, M.K.; Haddad, G.I.: Frequency-dependent characteristics of microstrip transmission lines, IEEE Trans. Microw. Theory Tech., Vol. MTT-20, pp. 678– 688, Oct. 1972. Kobayashi, M.; Iijima, T.: Frequency dependent characteristics of current distributions on microstrip lines, IEEE Trans. Microw. Theory Tech., Vol. MTT-37, No. 4, pp. 799–801, April 1989. Kitazawa, T.; Hayashi, Y.: Propagation characteristics of striplines with multilayered anisotropic media, IEEE Trans. Microw. Theory Tech., Vol. MTT-31, pp. 429– 433, June 1983. Dixit, A.; Jiu, L.; Moll, V.H.; Vignat, C.: The finite Fourier transform of classical polynomials, J. Aust. Math. Soc., Vol.98, pp. 145–160, 2015. Schmidt, L.P.; Itoh, T.: Spectral domain analysis of dominant and higher order modes in fin lines, IEEE Trans. Microw. Theory Tech, Vol. MTT-28, pp. 981–985, 1980. Jansen, R.H.: Fast accurate hybrid mode computation of nonsymmetrical coupled microstrip characteristics, 7th European Microwave Conference, Copenhagen, Denmark, 5–8 Sept. 1977. Kitazawa, T.; Kao, C.W.; Itoh, T.: Planar transmission lines with finitely thick conductors and lossy substrates, IEEE MTT-S Int. Microwave Sym. Dig., pp. 769–772, Boston, MA, 10–14 July 1991. Knorr, J.B.; Kuchler, K.D.: Analysis of coupled slots and coplanar strips on a dielectric substrate, IEEE Trans. Microw. Theory Tech., Vol. MTT-23, pp. 541–548, 1975. Jansen, R.H.: The spectral-domain approach for microwave integrated circuits, IEEE Trans. Microw. Theory Tech, Vol. MTT-33, No. 10, pp. 1043–1056, Oct. 1985.

611

612

16 Dynamic Spectral Domain Analysis

J.30 Tounsi, M.L.; Touhami, R.; Khodja, A.; Yagoub, M.C.

J.32 Itoh, T.; Menzel, W.: A full-wave analysis method for

E.: Analysis of the mixed coupling in bilateral microwave circuits including anisotropy for MICs and MMICs applications, Prog. Electromagn. Res., PIER, Vol. 62, pp. 281–315, 2006. J.31 Menzel, W.: A new interpretation of the spectral domain immittance matrix approach, IEEE Microw. Guid. Wave Lett., Vol. 3, No. 9, pp. 305–306, Sept. 1993.

open microstrip structures, IEEE Trans. Antenna Propagat., Vol. AP-29, No. 1, pp. 63–68, Jan. 1981. J.33 Knorr, J.B.; Tufekcioglu, A.: Spectral-domain calculation of microstrip characteristic impedance, IEEE Trans. Microw. Theory Tech., Vol. MTT-23, pp. 725–728, Sept. 1975.

613

17 Lumped and Line Resonators Basic Characteristics

Introduction The resonator is one of the most basic circuit elements in the microwave and other communication systems. They have a primary role either to accept or to reject a band of frequencies. It has three basic characteristics parameters – resonance frequency, unloaded Q-factor, and coupling coefficient. These basic characteristics apply to all kinds of resonators. The resonators are used in the design of filters, frequency discriminators, duplexers, wave meters, oscillators, tuned amplifiers, etc. The resonators are also used as the sensors to measure material properties and other physical parameters. They further act as antennas to radiate power in the free space and also in the material bodies. This chapter presents the basic characteristics of lumped and line resonators. They act as the circuit models for the planar resonators discussed in chapter 18.

•• • • •

To review the basic characteristics of lumped resonators. To discuss the one-port, two-port, and reaction-type resonators. To understand the coupling of lumped resonators to the external circuits. To discuss the circuit models of transmission line resonators. To present the modal analysis of transmission line resonators.

17.1

Basic Resonating Structures

A section of the open- or short-circuited transmission line supports the standing wave, not the usual traveling wave. Such a standing wave supporting line section forms the basic distributed line resonator. There

is no transportation of electrical energy from one location on a resonating line to another location. Only there is an exchange of energy from the electric field to the magnetic field. It is like the exchange of energy from the potential energy to the kinetic energy of a vibrating string or an oscillating pendulum [B.1]. The physical dimensions of a resonator are expressed in terms of the wavelength, corresponding to the resonance frequency. It is known as the normalized dimension. The resonating structures, from the dimension point of view, are grouped into the following four categories:

•• ••

Zero-dimensional resonator, One-dimensional resonator, Two-dimensional resonator, Three-dimensional resonator.

All microwave resonator circuits follow the above classification [B.2]. These four resonating structures are shown in Fig (17.1). A zero-dimensional resonator has its all dimension very small ( fr, it behaves as an inductor. The phase of the circuit current at frequencies f1 and f2 are tanϕ = X/R = 1, i.e. ϕ = 45∘. The frequency difference (f2–f1) is known as the half-power or 3-dB bandwidth of the circuit. It depends on the internal resistance R of a circuit and is related to another characteristic parameter called the quality-factor, i.e. Q-factor, of the resonant circuit.

The inductor and capacitor are devices to store the magnetic and electric energy, respectively. In a resonant circuit, the energy is transferred from an inductor to a capacitor and back. At the resonance, the magnetic and electric energies are equal. The average stored magnetic energy, electric energy and average power loss in a circuit, over the time period T, are Wm =

1 2 LI 4 max

a,

Pl =

1 2 RI 2 max

c

Wc =

1 CV2max 4

b 17 2 6

17.2 Zero-Dimensional Lumped Resonator

At the resonance, the total stored energy in a circuit is WT = Wm + Wc and in the inductor, WT = 2Wm or in the capacitor, WT = 2Wc, giving the following expressions for the unloaded Q-factor of a series resonant circuit: ωr WT 2ωr Wm ωr 1 2 LI2max ωr L a = = = Pl Pl R 1 2 RI2max 1 b or, QU = ωr RC 17 2 7 QU =

The input impedance of a series resonant circuit, around its resonance frequency, is further expressed in terms of the Q-factor. The frequency around ωr is defined by the frequency detuning parameter εf, and the input impedance is rearranged as follows:

line resonator as a lumped series resonant circuit. The expression (17.2.10b) shows that the Zin of a lossy resonant circuit can be computed from the Zin of a lossless circuit by replacing the resonance frequency ωr of the lossless circuit with a complex resonance frequency, ωr(1 + j/2 QU). At the half-power frequencies f1 and f2, the real and imaginary parts of Zin are equal, i.e. R = |X|. Under this condition, the 3 dB bandwidth of a resonant circuit is obtained from equation (17.2.10b): At half – power frequencies f 1 and f 2 ; R = 2R Δω ωr QU

R= X,

Bandwidth ω2 − ω1 = 2Δω = BW = f 2 − f 1 =

Zin = R + j ωL − 1 ωC

Zin = R + j where, εf =

a

ω ωr − ω ωr

b 17 2 8

Δω f − fr ω − ωr ω = = = −1 fr ωr ωr ωr

ω =1+δ ωr

17 2 9

1+δ − 1+δ

Zin = R 1 + jQU

−1

≈ R 1 + jQU 1 + δ − 1 − δ Zin = R 1 + j2δQU

alternatively, Zin = j

b

X = ωL − dX dω

= 2L

dX 1 =L+ 2 dω ωC a;

as L =

ω = ωr

QU =

and,

1 , ωC

ωr L , R

QU =

1 ω2r C

1 ωr dX 2 R dω

b ω = ωr

17 2 12

The input impedance around the resonance frequency, using equation (17.2.8), could be expressed in terms of the unloaded Q-factor:

Zin = R 1 + j2QU

fr QU

To get a very narrow bandwidth, the Q-factor must be large, i.e. the internal resistance R of a series resonant circuit, signifying the circuit loss, must be very small. Another useful general expression for the Q-factor is also obtained in terms of the reactance slope of a series resonant circuit:

Another variable, i.e. the normalized deviation of frequency from the resonance frequency, is defined as follows: δ=

a

17 2 11

= R + j L C ω LC − 1 ω LC L εf C

ωr QU

a Δω ωr

b

2RQU j ω − ωr 1 + ωr 2QU

c, 17 2 10

where Δω = ω − ωr and εf ≈ 2δ. The above approximate expression for the input impedance is valid around the resonance frequency. It is useful in the modeling of a

17.2.2

Lumped Parallel Resonant Circuit

Figure (17.4a) shows a parallel resonant circuit for the computation of the resonance frequency ωr. The source internal impedance is ignored, i.e. Rg = 0. The input admittance of the parallel resonant circuit is Yin =

1 1 + j ωC − R ωL

17 2 13

Figure (17.4b) shows the voltage response across R with a voltage maximum at the resonance frequency. The voltage maximum at the resonance, Vmax = {R/(R + Rg)}V, is the terminal voltage for a source with internal impedance Rg. Figure (17.4b) also shows the drop in the terminal voltage to Vmax 2 at the half-power frequencies f1 and f2. The average power loss Pℓ in the resistance

617

17 Lumped and Line Resonators

Rg V Vmax I C

L

R

0.707 Vmax

V

Yin Port #1

f1

(a) Parallel resonant circuit.

fr f2 Frequency

(b) Voltage response of the parallel resonant circuit.

R

Input impedance

0.707 R

Inductive Capacitive reactance reactance f1

fr

f2

Frequency

f1

fr

f2

Frequency

+90° +45° Phase

618

0° –45° –95°

(c) Input impedance magnitude and phase variation with frequency. Figure 17.4

Behavior of the parallel resonant circuit.

R, electric energy wc stored in the capacitor, and magnetic energy wm stored in the inductor are given below: 1 Vmax Pl = 2 R wm =

ωr =

1 LC

a,

1

fr =

2π

b

LC

17 2 15

2

a,

1 wc = Vmax 2 C 4

1 1 Vmax 2 IL, max 2 L = 4 4 ω2r L

b c, 17 2 14

where IL,max is the maximum of current IL through the inductor. At the resonance frequency, the susceptance part of Yin is zero giving,

At the resonance wc = wm, i.e. the total stored energy is wT = 2wm. Following definition of equation (17.2.5), the unloaded Q-factor is obtained as follows: QU = 2 × ωr

1 4 Vmax

2

1 2 Vmax

QU = ωr RC =

ω2r L 2

R

=

R ωr L

ωr C G 17 2 16

17.2 Zero-Dimensional Lumped Resonator

Figure (17.4c) shows a variation of the input impedance, both magnitude and phase, with frequency. Below resonance, the input impedance is inductive, while above the resonance, it is capacitive. The input susceptance and its derivative are 1 B = ωC − ωL

a,

dB dω

of a parallel resonant circuit, around the resonance frequency, and also its 3 dB bandwidth, are obtained: Yin = G 1 + j2QU Zin =

= 2C

b

1 = Yin

ω = ωr

17 2 17

Δω ωr R

1 + j2QU

a b

Δω ωr

3 dB Bandwidth = f 2 − f 1 =

fr QU

An alternate expression for the unloaded Q-factor of a parallel resonant circuit is written from the above expressions: 1 ωr dB QU = 2 G dω

17 2 19 17.2.3

,

17 2 18

ω = ωr

where G = 1/R is the conductance. Following the previous discussion on series resonance, the input admittance

Resonator with External Circuit

A resonant circuit, series or parallel, never works in isolation. It is always connected to an external circuit. Figure (17.5) shows the following three basic configurations of the resonators with external circuits:

Lprobe

L

C

R

Patch resonator

(a′) Equivalent circuit of probe-fed microstrip patch resonator.

(a) One-port reflection-type resonator (Coaxial probe-fed microstrip patch).

L Rg

Cc R

C

(b′) Equivalent circuit of gap-fed microstrip line resonator.

Dielelctric resonator

θ

P

L

Rr Cr

d Z0

Cc

R

Vg

(b) Two-port transmission-type resonator (Gap fed microstrip line resonator).

θ

Z0

Lr

R M

Z0

C1

R1

L1

Z0 Z0

Microstrip line (c) Two-port reaction-type resonator (Microstrip coupled dielectric resonator).

c

(c′) Equivalent circuit of microstrip line coupled dielectric resonator.

Figure 17.5 Basic types of planar resonators with equivalent circuits.

C Z

Z0

619

620

17 Lumped and Line Resonators

•• •

One-port reflection type Two-port transmission type Two-port reaction type.

where, X slope

Figure (17.5) also show the equivalent circuits of these configurations. The one-port resonator reflects back power to the source. In the case of a two-port transmission-type resonator, at resonance, the power is partially coupled (transmitted) from the source to a resonator; partially reflected to the source. Next, it is coupled from the resonator to an external load. In the case of a microstripbased reaction-type resonator, shown in Figure (17.5c), the input and output ports are its external circuits. The dielectric resonator reacts to the coupling and reflects the power. Figure (17.5c) also shows that the resonatorloaded line section acts like a band-reject parallel resonant circuit [J.3]. The coupling between a resonator and an external circuit is a fundamental characteristic of a practical resonator. The coupling is expressed through the concept of the external Q-factor (QE), and also through the coupling coefficient (β) between the resonator and external circuit. In the discussion to follow, the emphasis is placed on three fundamental characteristics of a practical resonator – resonance frequency, unloaded Q-factor, and coupling coefficient. Once these parameters are known, either experimentally or by using an EM-Simulator, the R, L, C elements of the circuit model of a resonating structure can be evaluated. 17.2.4

ωr L Rg

QE,in =

One-Port Reflection-Type Resonator

Figures (17.3a and 17.4a) show the one-port series and parallel resonant circuits act as the reflection-type resonators. This section presents the basic concept of coupling through the external Q-factor, and the coupling coefficient for both types of reflection resonators. Reflection-Type Series Resonator

Figure (17.3a) shows a lossy series resonant circuit connected to a source with internal resistance Rg. This circuit is also obtained from Fig (17.7a) by taking RL = 0. The coupling is treated as the power loss in the external resistance, Rex = Rg. The external Q-factor is defined as follows: Average energy stored in resonant circuit QE = ωr Power loss in external circuit

ω = ωr

a, =

QE,in =

ωr dX 2 dω

X slope Rg

ω = ωr

b c

ω = ωr

17 2 21 The above equation (17.2.21b) expresses the external Q-factor in terms of the reactance slope parameter, X(slope), at the resonance. It is applicable to both the lumped and distributed, i.e. transmission line-type resonator. Equation (17.2.21a) could be obtained from equation (17.2.21b). The total Q-factor of the reflection-type series resonator, with external source resistance Rex = Rg, is known as the loaded Q-factor, QL. It is computed by using the total resistance in series with LC circuit, QL =

ωr L Rg + R

17 2 22

It clearly shows that the loaded Q-factor is less than the unloaded Q-factor given by equation (17.2.12). From the above equation, the following relation, between the loaded, unloaded, and external Q-factors, is obtained: Rg 1 R + = QL ωr L ωr L

1 1 1 = + QL QU QE 17 2 23

Figure (17.3a) is a direct coupled series resonant circuit. It has a low Q-factor. The coupling of a resonator to an external circuit is defined by the coupling coefficient β: β=

Power loss in external circuit Pex Power loss in resonator circuit PL

ω = ωr

17 2 24 In the case of a reflection-type series resonant circuit, the coupling coefficient β is computed as follows: β= β=

1 2 Rg I2max Rg = R 1 2 RI2max QU Unloaded Q – factor = External Q – factor QE

a b 17 2 25

ω = ωr

17 2 20 Following the same method to derive equations (17.2.7) and (17.2.12), the above definition is used to obtain the following expressions for the external Q-factor of a series resonant circuit at the input port #1:

The relation between the Q-factors and the coupling coefficient is obtained from equation (17.2.23): 1 1 β 1+β = + = QL QU QU QU QU = QL 1 + β = QE β

a b 17 2 26

17.2 Zero-Dimensional Lumped Resonator

The input impedance of a series resonator, given by equation (17.2.10b), can be rewritten as follows, in terms of the coupling coefficient and the loaded Q-factor: RQU Δω ωr QL 1 + β Δω Zin = R 1 + j2 ωr Zin = R 1 + j2 QL 1 + β δ Zin = R 1 + j2QE βδ Zin = R + j2

a b

R=2 c d

Normalized input impedance Zin = =

R 1 + j2QE βδ 1 + j2QE βδ = Rg β

Zin Rg e,

where δ = Δω/ωr is the normalized frequency deviation. In the above equation, the system impedance is Z0 = Rg. At resonance δ = 0 and Zin = R leading to β = Rg/Zin, i.e. Zin = Rg/β. The condition β = 1, i.e. the critical coupling between a resonator and the external circuit, is needed for the matching of a resonator at the resonance to the source. Under the condition of critical coupling, the S-parameter at the input port-1, S11 = 0. However, equal power is dissipated in the resonator and the external circuit. The critical coupling is achieved for the case QU = 2QL. The condition β < 1 (QU < 2QL) results in the under-coupling, whereas β > 1(QU > 2QL) provides the over-coupling. It is obvious that in the case of under-coupling, the loaded Q-factor is higher as compared to its value at the critical coupling. It provides reduced 3dB bandwidth for an under-coupled resonator. The coupling is weak with a higher value of external Q-factor, i.e. for QE > QU. Therefore, more power is dissipated in the resonator. In the case of over-coupling, the loaded Q-factor is low, as compared to its value at the critical coupling, thus giving a larger 3 dB bandwidth. Also, the coupling is stronger with a lower value of the QE, i.e. for QE < QU. Therefore, more power is dissipated in the external load. The coupling coefficient of a series resonant circuit to the source (Rg = Z0) is computed from the known value of |S11| for a series resonator: S11 =

S11

R − Z0 = + Z0 R β −1 = β +1

At the resonace, δ = 0, and 1−β S11 ω = ωr = SRes 11 = 1+β

RQU Δω , ωr

BW = ω2 − ω1 = 2Δω =

ωr QU 17 2 29

17 2 27

Zin − Z0 Zin + Z0 R 1 + j2QU δ = R 1 + j2QU δ 1 + j2QU δ = 1 + j2QU δ

Equation (17.2.28a) defines the circles in the Γ-plane of the Smith chart. It provides information on the input impedance given by equation (17.2.27a). The unloaded 3dB bandwidth, at the half-power frequencies, is obtained from the expression (17.2.10b) under the condition R = |X|:

Rg 1 + j2QU δ − 1 Rg 1 + j2QU δ + 1 1 − β + j2 QU δ a 1 + β + j2QU δ

b 17 2 28

The magnitude of the reflection coefficient |S11| gives the return loss (RL) at the resonance. The RL and phase ∠S11 are obtained from equation (17.2.28): 1 − β 2 + 2QU δ 2 1 + β 2 + 2QU δ 2

RL = − 20 log S11 = − 20 log − RL dB SRes 11 = 10

S11 = tan − 1

20

, S11 ≤ 1

a b

2QU δ 2QU δ − tan − 1 1−β 1+β

c 17 2 30

The RL at resonance is measured or the EM-simulated quantity giving SRes 11 . The coupling conditions are summarized below: For critical coupling β = 1, i e

SRes 11 = 0, R = Rg

For under-coupling β < 1, β =

1 − SRes 11 , R > Rg b 1 + SRes 11

For over-coupling β > 1, β =

1 + SRes 11 , R < Rg 1 − SRes 11

a

c

17 2 31 The expression for the over-coupling is an inversion of the expression for the under-coupling. The requirement β > 1 is met, while retaining SRes 11 < 1 for both the under- and over-coupling cases. For an under-coupled series resonant circuit, β < 1, i.e. R > Rg, whereas for a critically coupled case β = 1, i.e. R = Rg. The overcoupled resonator has β > 1, i.e. R < Rg. Figure (17.6a) shows the input impedance variation of a reflection-type series resonant circuit with frequency, for three coupling conditions. At lower frequency f1, i.e. below resonance fr, the input impedance is capacitive. At the resonance Zin = R. It is marked on the horizontal axis of the Smith chart. At frequency f2 above the resonance, the input impedance is inductive. Normally, the system impedance is Z0 = Rg = 50Ω. The nature of coupling, i.e. the critical-, under-, or over-coupling, is determined from the position of the locus of the input impedance on the Smith chart. The points A, B, and C show the under-coupling, critical-coupling, and overcoupling, respectively. The coupling coefficient (β) can

621

622

17 Lumped and Line Resonators

f1 C

R=0

f1

f2

f2 B

fr

fr

A fr

R=1

f1

f2

f1

R=∞

fr

R=0 f2 A

f1

f1

R=∞

f2

(b) Parallel resonant circuit.

Input impedance variation of the one-port reflection-type resonators with frequency. (A) Under-coupling, (B) Critical-coupling, (C) Over-coupling.

be located on the horizontal axis of the Smith chart. Furthermore, Fig (17.6a) also shows the dotted QU-circles, i.e. the unloaded Q-factor circles, at which R = |X|. The intersections of the impedance circles with the QU-circles, for three cases of coupling, provide us the lower and upper 3 dB frequency, f1 and f2. It helps us to determine the unloaded Q-factor: QU =

fr

R=1 C

f2

(a) Series resonant circuit. Figure 17.6

fr B

ωr fr = ω2 − ω1 f2 − f1

For QU = 1 + β QL ;

a fr 1 + β f2 − f1

QL =

b 17 2 32

The loaded Q-factor could be also determined from the 3 dB bandwidth of the return-loss response. In

Rg

L

R

V

RL Port #1

V Port #1

C

L

RL Port #2

(b) Series-connected parallel resonant circuit.

R

Rg

Rg

Figure (17.4a) shows a lossy parallel reflection-type resonator circuit, connected to a source with internal resistance Rg. Figure (17.7b) is also reduced to Fig (17.4a) for

C

Port #2

(a) Series-connected series resonant circuit.

Reflection-Type Parallel Resonator

R L

Rg

C

this case, the QU is further obtained from the loaded Q-factor QL. Once ωr, β, and QU are known for any resonating structure, either from the experiment or from the EM-Simulation, the R, L, C components of a series resonator are determined from the following expressions: Rg RQU R= a , L= b β ωr 1 c C= ωr RQU 17 2 33

R RL

V

V

L

RL

C Port #1

Port #2

(c) Shunt-connected parallel resonant circuit.

Port #1

Port #2

(d) Shunt-connected series resonant circuit.

Figure 17.7 Circuit configurations of series and parallel resonators.

17.2 Zero-Dimensional Lumped Resonator

the output short-circuited, i.e. RL = 0. The Q-factors, coupling coefficient, and input impedance are determined. The external Q-factor is obtained using the definition given in equation (17.2.20): QE,in

ωr C = Gg

where,

B slope a , QE,in = Gg ωr dB B slope ω = ωr = 2 dω ω = ωr

S11 =

ω = ωr

b c

17 2 34 In the above equation, Gex = Gg = 1/Rg is the external conductance. The susceptance slope parameter at resonance, B slope ω = ωr, provides a more general definition of the external Q-factor. It is applicable to both the lumped and distributed line-type parallel resonant circuit. The loaded Q-factor of the parallel resonant circuit is obtained by taking total conductance across the LC circuit: QL =

ωr C Gg + G

17 2 35

Equation (17.2.31), showing the relation between SRes 11 and the coupling coefficient, is valid for a parallel resonator also. Of course, equation (17.2.25a) for the coupling coefficient β is modified as follows: QU R ωr L R = = QE ωr L Rg Rg For the under-coupling β < 1, i e R < Rg

b

For the critical coupling β = 1, i e R = Rg For the over-coupling β > 1, i e R > Rg

c d

β=

a

17 2 36 Equation (17.2.31) for β is valid for a reaction-type parallel resonator also. However, the above-given conditions are taken into account. The input impedance of a parallel resonator, given by equation (17.2.19b), is rewritten as R R = 1 + j2QU δ 1 + j2QL 1 + β δ R Zin = 1 + j2QE βδ Zin =

Once input impedance is known, an expression is obtained to compute the S11 that is related to the coupling coefficient.

a b 17 2 37

The normalized input impedance of a parallel reflection-type resonator is R Rg R Rg Zin = = Rg 1 + j2QU δ 1 + j2QL 1 + β δ β β = Zin = 1 + j2QU δ 1 + j2QL 1 + β δ 17 2 38

Zin − 1 β − 1 − j 2 QU δ = β + 1 + j 2 QU δ Zin + 1

RL = − 20 log S11 = − 20 log S11 = − tan − 1

a

β − 1 2 + 2 QU δ 2 β + 1 2 + 2 QU δ 2

b

2QU δ 2QU δ − tan − 1 1−β 1+β

At the resonace, δ = 0, and S11 Re s Also, S11 = 10 − RL dB

20

ω = ωr

Re s = S11 =

c 1−β 1+β

, S11 ≤ 1

d e

17 2 39 Figure (17.6b) shows the normalized input impedance variation of a parallel resonant circuit, with respect to the frequency deviation δ, on a Smith chart. The impedances of the series and parallel resonant circuits show the inverse relationship. The unloaded Q-factor is computed after obtaining the f1, f2, and fr from the Smith chart. The under-, critical-, and over-couplings are shown on the Smith chart. The coupling coefficient β is also obtained at the horizontal axis of the Smith chart, and QL from the 3dB points of the |S11| response. Once QL and β are known, the unloaded Q-factor of the resonator is determined from equation (17.2.26). On using these parameters, along with known ωr, the circuit elements of a parallel resonator are extracted from the following expressions: R = β Rg G + Gg QL = ωr Rg Rg β = L= ωr QE ωr QU

C=

a G + Gg ωr

QU 1+β

b c 17 2 40

17.2.5

Two-Port Transmission-Type Resonator

The series and parallel resonant circuits can be connected as a two-port network, forming the transmission-type resonators, both in the series configuration and in the shunt configuration. Figure (17.7) shows four circuit configurations. The circuit arrangements exhibit either the bandpass filter (BPF) or the bandstop filter (BSF) response. The resonators are directly coupled to the source and an external load. We first discuss the external Q-factor, and the coupling coefficient. Next, the S-parameter response of the circuit is presented. The measured S-parameter response of a resonator circuit is obtained from the Network Analyzer. It can also be obtained using an EM-Simulator.

623

17 Lumped and Line Resonators

Case-1: Series-Connected Series Resonant Circuit

QE =

Figure (17.7a) shows the transmission-type series resonator connected in series with the source and load. Figure (17.8a) shows that the circuit acts as a BPF. The external Q-factors are computed at both the input and the output ports. Normally in a transmission system, Rg = RL = Z0, where Z0 = 50Ω is the system impedance. The total external load, in the series with the LC circuit, is 2Z0. The total load, including the internal resistance of the resonator, is R + 2Z0. The external and loaded Q-factors are written as follows:

f1 fr

Res IL

Frequency

a,

QL =

ωr L R + 2Z0

b 17 2 41

The series-connected series resonant circuit, shown in Fig (17.7a), is analyzed below using the following matrix expression for the S-parameters [B.9, B.10, B.13]: S =

Z Z + 2Z0

2Z0 Z + 2Z0

2Z0 Z + 2Z0

Z Z + 2Z0

, 17 2 42

Res IL 3 dB

f2

ωr L 2Z0

f1 fr f2 Frequency

S21

dB

S21 dB

3 dB

Res IL +90° +45°

Phase

+45°

Inductive reactance

Capacitive reactance

0°

f1 fr

f2

Phase

+90°

Inductive reactance

0°

fr

Frequency Capacitive reactance

–45°

Frequency

–45°

–90°

–90° (a) Series-connected series resonant circuit. f1 fr

Res IL

f2 Frequency

(b) Series-connected parallel resonant circuit. Res IL 3 dB

S21

S21

Res IL +90°

0°

Capacitive reactance

Inductive reactance f1 fr

f2 Frequency

+45° Phase

+90° +45°

0°

–45°

–45°

–90°

–90°

(c) Shunt-connected parallel resonant circuit. Figure 17.8

f1 fr f2 Frequency

dB

dB

3 dB

Phase

624

Capacitive reactance

fr

Frequency Inductive reactance

(d) Shunt-connected series resonant circuit.

Sketched insertion loss and phase response ∠S21(f ) of series/shunt-connected series and parallel resonant circuits.

17.2 Zero-Dimensional Lumped Resonator

where Z0 is the system impedance and Z is the impedance of the series resonant RLC circuit near resonance frequency, Z = R + j(ωL − 1/ωC) = R + j2RQUδ. The above [S]-matrix is obtained in Ex. 3.9 of chapter 3. The insertion loss (IL) and phase shift from the S21 are obtained below. The source impedance is a real quantity, and the ports are symmetrical, i.e. Z0 = RG = RL. S21 f = S21 f

2

2Z0 2Z0 = Z + 2Z0 R + 2Z0 + j2RQU δ

a

2Z0 + R 2Z0

2

2RQU δ 2Z0 + R

1+

b,

2

17 2 43 where δ = (Δω/ωr). The denominator is simplified as follows:

Deno =

2Z0 R + 2Z0 1+

= 1−

2

2RQU δ 2Z0 + R

1+

2

2RQU δQL ωr L

2

R ωr L R + 2Z0 ωr L

2

=

1 + 2δQL 1−

QL QU

2 2

17 2 44 The condition for the half-power provides the 3dB BW, from which the QL can be determined. Next, QU is also determined using the value of SRes 21 f r . At resonance, the normalized frequency deviation is zero δ = 0. The |S21(f )|-response and 3 dB bandwidth from it are obtained as follows: SRes 21 = 1 − QL QU , S21 f For

2

=

2δQL = 1,

SRes 21

QU =

QL 1 − SRes 21

a

2

1 + 2δQL

2 2

fr QL = f2 − f1

b

c, 17 2 45

− IL dB 20 is IL (dB). It is a measured or where SRes 21 = 10 EM-Simulated positive quantity. The IL and phase shifts are obtained as follows:

2

= − 20 log 10 1 − QL QU 2

+ 10 log 10 1 + 2QL δ

a

At resonance, δ = 0, and IL f = f r = − 20 log 10 1 − QL QU ϕ = − tan − 1

b

ωL − 1 ωC 2RQU δ = − tan − 1 R + 2Z0 R + 2Z0

In limit, for ω

1

=

IL dB = − 10 log 10 S21 f

0, ϕ

ω = ωr , δ = 0, ϕ = 0; ω

c

+ π 2; ∞, ϕ

−π 2

d 17 2 46

Figure (17.8a) shows sketches of the magnitude and phase responses of the series-connected series resonant circuit. The series resonant circuit behaves as a series capacitor at a frequency below the resonance frequency, giving positive phase response. At a frequency above the resonance frequency, it acts as a series inductor, giving negative phase response. The 3 dB frequency points, f1 and f2, are marked on both responses. These responses are obtained either experimentally on a Vector Network Analyzer or an EM-simulator. The QL and QU are determined from equations (17.2.45c) and (17.2.46b), respectively. The coupling coefficient (β) is also computed from SRes 21 . In the present case of the equal coupling at both the input and output, the following relations hold between the Q-factors: 1 1 2 = + QL QU QE β=

QL =

QU QE QE + 2QU

a

2QU QE

b 17 2 47

Because of equal coupling at the input and output, the β mentioned above is twice of the coupling coefficient only at the input. The following relation is obtained between SRes 21 and β at the resonance: QL 1 = 1− QU 1 + 2QU QE 1 β = = 1− 1+β 1+β

SRes 21 = 1 −

alernatively, the coupling coefficient, β =

a SRes 21 1 − SRes 21

b

17 2 48 The RL, and also unloaded Q-factor of the circuit, are computed as follows:

625

626

17 Lumped and Line Resonators

Z R + j ωL − 1 ωC = Z + 2Z0 2Z0 + R + j ωL − 1 ωC R + j2RQU δ = 2Z0 + R + j2RQU δ Q 1 + j2QU δ S11 = L QU 1 + j2QL δ QL At resonace, SRes 11 = QU Phase, S11 = tan − 1 2QU δ − tan − 1 2QL δ

Case- 2: Series-Connected Parallel Resonant Circuit

S11 =

RL dB

= − 20 log 10

ω = ωr

QU = QL 10

QL QU

Figure (17.7b) shows the two-port series-connected parallel resonant circuit. Primarily it is a BSF. Figure (17.8b) shows the stopband response. The analysis is carried out by computing the S-parameters. The admittance of the parallel resonator, around resonance frequency, is

a

1 ωL Δω = G 1 + j2QU = G 1 + j2δQU ωr

Y = G + j ωC −

b c d

RL dB 20

17 2 52

e 17 2 49

Further, using equation (3.1.64) of chapter 3 the coupling coefficient at the resonance is also obtained: Res SRes 11 + S21 = 1

SRes 11 = 1 −

β 1 = 1+β 1+β

β=

1 − SRes 11 SRes 11

The resonator, with admittance Y, is loaded at both ends by the external loads, Rg = RL = Z0, i.e. by the admittance 1/(2Z0) = 0.5 Y0. Accounting for the loss of a resonator, loss in the external circuit, and the total loss both in the resonator and external circuit, the expressions for the unloaded, external and loaded Q-factors are written below: ωr C a, G ωr C QL = G + 0 5Y0

QU =

17 2 50 Res The S-parameters SRes 21 and S11 are obtained either from the measurement or from the EM-simulation. Once the coupling coefficient, Q-factors, and the resonance frequency of a two-port series-connected series resonant circuit are known; the component values of a series resonator are obtained as follows:

R = 2Z0 β

a,

L = RQU ωr

b,

C=1

ω2r L

QE =

ωr C 0 5Y0

b c 17 2 53

The S21 (f ) is obtained using equation (17.2.42): S21 f =

2Z0 2Y = Y0 + 2Y Z + 2Z0

17 2 54

c

17 2 51 1 Y0 0 5Y0 =1+ =1+ S21 f 2Y G 1 + j 2δQU

1 S21 f 1 S21 f

=

G + 0 5Y0 + j 2δGQU G 1 + j 2δQU

=

G + 0 5Y0 ωr C × G ωr C

=

QU 1 + j 2δQU × QL QU QL 1 + j 2δQU

=

QU 1 + j 2δQL QL 1 + j 2δQU

The IL and phase are obtained as follows: S21 f = S21 f

2

QL 1 + j 2δQU QU 1 + j 2δQL =

QL QU

2

1 + 2δQU 1 + 2δQL

17 2 55

1 + j 2δQU G G + 0 5Y0 × ωr C ωr C 1 + j 2δQU

where, a

2 2

ϕ = tan − 1 2δQU − tan − 1 2δQL

b c 17 2 56

For

f

2δ ≈ εf = 0, εf

f = f r , εf = 0, f

∞ , εf

f fr − f fr − ∞,ϕ

0−

17 2 57

ϕ=0 ∞,ϕ

0+

Figure (17.8b) shows the phase plot of S21 (f ). For f < fr and QU > QL, the phase ϕ is negative, i.e. a parallel resonant circuit behaves as an inductor at a frequency

17.2 Zero-Dimensional Lumped Resonator

below the resonance frequency. Likewise, for f > fr and QU > QL, a parallel resonant circuit behaves as a capacitor at a frequency above the resonance frequency. 2

IL = − 10 log 10 S21 f = − 20 log 10

QL 1 + 2δQU − 10 log 10 QU 1 + 2δQL

At resonance, δ = 0, ILRes = − 20 log 10 QU = QL 10

2 2

a

QL QU

ILRes 20

Q 1 + j 2δQU f = 1 − S21 f = 1 − L QU 1 + j 2δQL QU − QL = QU 1 + j 2δQL

SRes 11 f r S11 f

2

2

=

=

QU − QL QU SRes 11 f r

ωr C G

QL =

ωr C G + 2Y0

QE =

ωr C 2Y0

b c

An expression for the IL is obtained from the Sparameter of the circuit, given by equation (3.1.71) of chapter 3: YZ0 2 + YZ0 2 2 + YZ0

2 2 + YZ0 YZ0 − 2 + YZ0

− S =

S21 f = 1 S12 f

a

17 2 62

2 2 + YZ0

=1+

YZ0 G =1+ 1 + j2δQU 2 2Y0

The coupling coefficients at both ports are identical. Equation (17.2.36) gives β1 = β2 = β = Y0/G, and the above expression is rewritten as 1

b

S21 f c

d 17 2 59

The resonance frequency fr and 3 dB lower f1, and upper f2 frequencies are obtained from the experimental or EM-Simulated |S21(f )| response. Finally, the circuit components G, L, C for a parallel resonant circuit connected in series are obtained as follows: QL G ωr C G × = = QU ωr C G + 0 5Y0 G + 0 5Y0 1 b , L= Cω2r

a,

17 2 61

2

GQU a , C= ωr

QU =

17 2 63

2

1 + 2δQL 2 Half − power corresponds to 2δQL = 1, fr QL = f2 − f1

0 5Y0 QL G= QU − QL

Figure (17.7c) shows the shunt-connected parallel resonator that has admittance Y = [G + j(ωC − 1/ωL)] = (G + j2δ QUG), where G = 1/R shows the internal loss of a parallel resonator. The resonator is loaded at both ports with Rg = RL = Z0. Figure (17.8c) shows that it behaves as a BPF. Taking care of total losses in the internal conductance G, and both end external conductance Y0, the Q-factors are written as follows:

b 17 2 58

The second term of the equation (17.2.58a) shows the frequency-dependent bandstop behavior, as shown in Fig (17.8b). As QL < QU, so the IL response S21 (f ) on the logarithmic scale is a positive quantity. equation (17.2.58b) extracts the unloaded Q-factor. Following Fig (17.6b), the QU can also be determined from the corresponding bandwidth obtained from the Smith chart. The Smith chart can also be used to determine the QL [J.3]. Normally, one attempts to determine the QL from the 3 dB bandwidth shown in Fig (17.8b). However, the above IL equation defines the bandwidth relating to QL that is dependent on the coupling (QU=βQE). The 3dB location satisfactorily works only if ILRes (fr) is more than 18 dB [J.3]. However, the QL could be also determined from the return-loss response using S11(f ): S11

Case-3: Shunt-Connected Parallel Resonator

=1+

1 1 + j 2δQU 2β

However, 2β is the total coupling coefficient at both ports that relates QL and QU through equation (17.2.26), i.e. 2β = (QU–QL)/ QL. On substituting 2β in the above equation, the following expressions are obtained: S21 f =

QU − QL QU

1 1 + j2δ QL

At resonace δ = 0; SRes 21 f r S21 f

2

=

SRes 21 f r

2

17 2 60

=

a QU − QL QU

2

b

2

1 + 2δ QL

c

2

At Half − power, 2δ QL = 1, c

17 2 64

QL =

Phase, S21 = ϕ = − tan − 1 2δQL

fr f2 − f1

d e 17 2 65

627

628

17 Lumped and Line Resonators

Again, its S-parameters responses, IL, and phase response are obtained from the S21(f ) of the shuntconnected admittance:

The IL of the circuit is IL dB = − 10 log S21 f = − 20 log

QU − QL QU

2

+ 10 log 1 + 2δQL

2

a S21 f =

At resonance δ = 0, QU − QL QU

ILRes dB = − 20 log

b

=

17 2 66 Figure (17.8c) shows the sketched amplitude and phase responses of S21(f ). The QL is determined from the 3 dB BW. For f < fr, δ is negative that leads to positive φ and the resonator acts a shunt-connected inductor. At f = fr, δ = 0 leading to ϕ = 0. For f > fr, δ is positive that leads to negative ϕ, and the resonator acts as a shunt-connected capacitor. The ILres(dB) at the resonance frequency is a measurable positive quantity. However, it appears as a negative quantity, in Figure (17.8c), with respect to no loss case. It is also available from the EM-simulator. The unloaded Q-factor is determined as follows: QU − QL rres = 10 − IL dB QU QL QU = 1 − S21 f r

2Y0 QL QU − QL

a,

S21 f =

=

1 + j2QU δ 1 + QU QE + j2QU δ QL QU

1 + j2QU δ 1 + j2QL δ

a

Phase, S21 = ϕ = tan − 1 2QU δ − tan − 1 2QL δ

b

IL dB = − 20 log 10 QL QU − 10 log 10 1 + 2QU δ + 10 log 10 1 + 2QL δ

2

2

c

= S21 f r 17 2 67

C=

GQU ωr

1 L= Cω2r

Figure (17.8d) shows the sketched amplitude and phase responses of S21(f ). The 3 dB frequencies, f1 and f2, are indicated on the response. However, the halfpower frequency does not provide correctly the needed bandwidth to get the loaded Q-factor QL, as it depends on the coupling coefficient also. However, the internal resistance R of a resonator is evaluated from the insertion-loss at the resonance:

b c 17 2 68

Case-4: Shunt-Connected Series Resonance

Figure (17.7d) shows the shunt-connected series resonator. The circuit acts as a BSF. Three Q-factors, and also the impedance of the series resonant circuit, are written as follows: ωr L QU = R ωr L QL = R + Z0 2

1 + j2QU δ 1 + Z0 2 R × ωr L ωr L + j2QU δ

17 2 70 20

In the above equation, the insertion loss ILresdB is taken at the resonance frequency. After evaluation of the resonance frequency, QU and QL, finally R, L, C are computed from equation (17.2.61): G=

2 2R 1 + j2QU δ = 2 + YZ0 2R 1 + j2QU δ + Z0

a,

ωr L QE = Z0 2

b

c,

Z = R 1 + j2QU δ

d 17 2 69

Re s S21 fr =

QE 2ωr L Z0 = QU + QE 2ωr L Z0 + ωr L R

Re s S21 fr =

2R 2R + Z0

a

Re s Z0 2 S21 fr Re s 1 − S21 f r

b

R=

17 2 71 The unloaded Q-factor is related to QL, QU = QL/ SRes 21 f r . To compute the unloaded Q-factor QU, the QL can be evaluated from the power response of S11(f ) as follows:

17.3 Transmission Line Resonator

Input

Output

Output

Input

(a) λg /2 - line resonators.

(b) λg/2 - hairpin resonators.

DGS slot in ground plane Microstrip

Input

Output

Slot gap

Output

Input

(c) λg/2 - L resonators.

(d) DGS resonator in the ground plane.

Figure 17.9 Some reaction-type resonators.

YZ0 − Z0 = 2 + YZ0 Z0 + 2R 1 + j2QU δ − Z0 = Z0 + 2R + j4RQU δ ωr L Z0 2 × × S11 f = − ωr L R + Z0 2 1 1 + 2QU δR R + Z0 2 × ωr L ωr L Q 1 S11 f = − L QE 1 + j2QL δ Q − QL 1 = − U QU 1 + j2QL δ

S11 f = −

Re s fr At resonace δ = 0, S11

S11 f

2

=

Re s S11 fr

1 + 2QL δ

2

=

QU − QL QU

a

2

b

2

c

2

fr f2 − f1 2QL δ

For half − power, 2QL δ = 1, QL = Phase, S11 = ϕ = π − tan − 1

d e 17 2 72

Once R, ωr, and QU are evaluated, the L and C of a resonator can be computed.

17.2.6

Two-Port Reaction-Type Resonator

Figure (17.5c) shows the dielectric resonator-based reaction-type resonator, magnetically coupled to a microstrip line. The coupling with the dielectric resonator loads the microstrip line with the series-connected parallel resonant circuit. Thus, a reaction-type resonator acts as a band-reject circuit. It can also be described by

the shunt-connected series resonant circuit. Fig (17.9) shows some more electrically or magnetically coupled reaction-type resonators. These are commonly used to develop BSFs. The resonators in Fig (17.9a and d) are electrically coupled to the microstrip. The defected ground structure (DGS) is a λg/2-slot-type resonator that is modeled by a shunt-connected series resonant circuit. However, due to the coupling capacitor, it appears as a series-connected parallel resonant circuit to the microstrip line, like any other reaction-type resonator [J.4]. Normally it is used to develop high rejection low-pass filters [J.5]. The resonators in Fig (17.9b and c) are magnetically coupled to the microstrip line. However, the nature of the coupling is decided by the predominance of electric or magnetic fields in the coupling region. At the ends of the line resonators, the electric field is dominant, and around the middle region of the line resonators, the magnetic field is dominant. There can be many variations in the physical structures of these resonators. Some of these could provide more compact planar filters. The coupled resonators are further discussed in section (18.5) of chapter 18.

17.3

Transmission Line Resonator

The transmission line resonators are basic building blocks of the RF, microwave, and mm-wave circuits, as the lumped element-based resonators are normally not suitable for the frequency above 3 GHz. However, in some cases, the lumped element-based circuits are reported even up to mm-wave ranges. A section of the transmission line, such as a two-wire line, coaxial, microstrip, CPW, slot line, fin line, etc., acts as a resonator. It is also

629

630

17 Lumped and Line Resonators

Voltage n = 1 n=2

Voltage R

n=1

n=2 C

L

n=2

n=2

Yin

Zin

C

Yin

Zin

n = 1 Current

n = 1 Current

(a) Both ends short-circuited line, ℓ = nλg/2. Voltage n=2

R

L

(b) One end short-circuited line, ℓ = (2n – 1)λg/4. Voltage n = 1

n=1

n=2 R C

Yin

L

R

n=2

n = 1 Current

L

Yin

(c) Both ends the open-circuited line, ℓ = nλg/2. Figure 17.10

n=2

Zin

n = 1 Current

Zin

C

(d) One end open-circuited line, ℓ = (2n-1)λg/4.

Basic transmission line resonators, showing voltage and current distribution for the fundamental and first harmonic and also their lumped equivalent circuits.

used for the realization of capacitors and inductors at microwave frequency. This section presents the modeling of a section of the transmission line as a lumped resonator. The section further presents a complete modal description of the λg/2 line resonators. 17.3.1 Lumped Resonator Modeling of Transmission Line Resonator The output end of a line section is either short-circuited or open-circuited. A fixed line length can support a large number of discrete resonating modes. However, at a fixed frequency, the line lengths could be either even multiple of λg/2 or an odd multiple of λg/4 showing series or parallel type of resonances. Figure (17.10) shows four basic structures of the line resonators with voltage and current distribution. It also shows equivalent lumped resonator models around the resonance frequency. Case-I: Both Ends Short-Circuited λg/2 Line

The λg/2-line section, short-circuited at both ends, is shown in Fig (17.10a). It also shows the magnitude of the voltage and current standing-waves for the first two modes n = 1, 2. As both ends are short-circuited, the voltage is zero at both ends, whereas the current is maximum at both ends. At the fundamental resonance frequency (n = 1), the line length is ℓ = λg/2. In the case of the TEM mode line in the air medium, the guided

wavelength λg = λ0; otherwise, λg is influenced by the material medium, and also by the geometry of the planar line. The left-hand side port is excited by a current loop, creating the magnetic field that induces the exciting current at the input. The input impedance of a short-circuited lossy line section is obtained from subsection (2.1.7) of chapter 2: Zin = Z0 tanh α + jβ ℓ = Z0

sinh αℓ cosh jβℓ + cosh αℓ sinh jβℓ cosh αℓ cosh jβℓ + sinh αℓ sinh jβℓ 17 3 1

On using the identity, cosh(jx) = cos x, sinh(jx) = j sin x, the above expression is reduced to Zin = Z0

tanh αℓ + j tan βℓ 1 + j tanh αℓ tan βℓ

17 3 2

A section of the lossless line, with α = 0, gives the reactive input impedance, or reactive input susceptance. Figure (17.11) shows their behaviors. Zin = jZ0 tan βℓ = jX Yin = − jY0 cot βℓ

a b,

17 3 3

where β is the propagation constant of the line and ℓ is its length. For a short-circuited line resonator, the first resonance occurs at β ℓ = π/2, i. e. ℓ = λg/4. Figure (17.11a) shows

17.3 Transmission Line Resonator

π/2

π

3π/2

2π

βℓ

0

π/2

π

3π/2

βℓ

BL

XC

0

Input susceptance

Input reactance XL

BC

(a) Input reactance w.r.t. electrical length. Figure 17.11

(b) Input susceptance w.r.t. electrical length.

Reactance variation of short-circuited line with frequency βℓ = (2πℓ/c) f.

that at the first resonance, the input reactance of the lossless short-circuited line is infinite, Zin = ∞. Figure (17.11b) shows that the input susceptance of the ℓ = λg/4 line is zero, Yin = 0. The behaviors repeat at the odd multiple of λg/4 line length. It is also obvious from the voltage and current distributions on the λg/4-line, shown in Fig (17.10b). To maintain the Zin = ∞ at the input, the input must be open-circuited, forming the λg/4 line resonator. The λg/4 line, short-circuited at the output, behaves like a parallel resonant circuit, shown in Figure (17.10b). For the line length β ℓ10 dB) at the center frequency, shows a stronger coupling with a higher value of kC. The value of kC for a weakly edge-coupled microstrip resonator is kC < 0.1.

fm r,e = =

a

kC =

f er,e

18 5 6

L11 C11 − C12 fm r,o

1 = 2π Le C11 2π fr

1 L11 + L12 C11 a

1 + kL 1 = = 2π Lo C11 2π fr = 1 − kL

1 L11 − L12 C11 b,

b,

where the self-resonance frequency of the uncoupled resonator fr, and the capacitive coupling coefficients kC are defined as follows: 1 2π L11 C11

2

1

18 5 4

fr =

f er,e f er,0

b 18 5 5

As (1 − kC) < 1 and (1 + kC) > 1, the even-mode frequency is higher than the odd-mode frequency. The electric, i.e. the capacitive, coupling coefficient kC can be expressed in terms of the even- and odd-mode resonance frequencies as follows:

18 5 7 where the magnetic coupling is defined as kL = L12/L11, 0 ≤ kL < 1. The above equations are obtained using equation (11.3.21) of chapter 11. Once the even- and oddmode resonance frequencies of magnetically coupled resonators are known from the EM-Simulation of the S21-frequency response, the magnetic coupling is determined from the following equations: fm r,e fm r,o

2

=

1 − kL , 1 + kL

kL =

fm r,o

2

− fm r,e

2

fm r,o

2

+ fm r,e

2

18 5 8 In this case, the above results show that the odd-mode frequency is higher than the even-mode frequency, as (1 − kL) 1. Figure (18.21b) shows the S21 response of the magnetic coupling of coupled resonators

18.5 Coupled Resonators

Even mode

0

0o

f er,o

fr

1

2 3

0o

f er,e

fm r,e

fm r,o

fr

(b) Magnetic coupling.

(a) Electric coupling.

1

2 3

S21 (dB)

0

Odd mode

S21 (dB)

2 3

Phase

1

S21 (dB)

0

Even mode

Phase

Odd mode

fH r,o

fr

fH r,e

(c) Hybrid coupling. Figure 18.21

S21 response of coupled microstrip resonator.

for three separation gaps of the resonators. Again, a larger separation of two resonance peaks indicates stronger coupling. Also, the separation of peaks, for the same coupling distance, is more for the magnetic coupling, as compared against the electric coupling of Fig (18.21a). It demonstrates that the magnetic coupling is stronger than the electric coupling, i.e. kL > kC. It indicates weak coupling for the edge-coupled microstrip resonators, whereas the broadside-coupled microstrip resonators have stronger coupling. Further, the phases of the electric coupling and magnetic coupling are the reverses of each other. So, if the electric coupling is taken as a positive coupling, then the magnetic coupling is a negative coupling.

Hybrid Coupling – Both Capacitive and Inductive Coupling

In general, the coupled microstrip resonators have both the electric and magnetic couplings. Such mixedcoupling is called a hybrid coupling. The pure electric and pure magnetic couplings are an idealization of the coupling mechanism. The electric coupling and magnetic coupling, discussed above, are the dominant electric coupling and dominant magnetic coupling. The mixed-coupling is due to the presence of both C12 and

L12. The even- and odd-mode resonance frequencies of the hybrid coupling are computed as follows: 1 Even mode f H r,e = 2π Le Ce 1 fr = = a 2π L11 + L12 C11 − C12 1 + kL 1 − kC 1 Odd mode f H r,o = 2π Lo Co 1 fr = = b 2π L11 − L12 C11 + C12 1 − kL 1 + kC 18 5 9 Following the previous discussion, the hybrid coupling coefficient kH is defined as follows: fH r,o

2

=

fH r,e

1 + kL 1 − kC 1 − kL 1 + kC

fH r,o

2

− fH r,e

2

fH r,o

2

+ fH r,e

2

fH r,o

2

− fH r,e

2

fH r,o

2

+ fH r,e

2

Define, kH =

≈ kL − kC

kH = kL − kC

a

b c 18 5 10

663

18 Planar Resonating Structures

The above equation (18.5.10a) is obtained ignoring (kL.kC) < 1. Equation (18.5.10c) shows that magnetic and electric couplings are out-of-phase. In this case, the electric and magnetic couplings cancel each other. However, if the mutual inductance or the mutual capacitance change sign, both couplings enhance the hybrid coupling [B.6]. The above equations also show the relation among three coupling coefficients. The case of only electric and magnetic couplings follows from the case of the combined coupling. Figure (18.21c) shows the S21 frequency response of the edge-coupled microstrip resonators for the combined coupling. For the same separation of resonators, two peaks of the combined coupling are less apart, as compared to that of the magnetic coupling, whereas it is more than that of the electric coupling, i.e. kL > kH > kC. It is possible that at a specific frequency, called the null frequency fnull, the transmission of a coupled resonator could be practically zero. It is related to the uncoupled resonator frequency as follows [J.40]: f null = f r

kL kC

18 5 11

It shows that for kL > kC, the null occurs in at a frequency above fr in the upper stopband, and in the case of kC > kL, the null occurs at the frequency below fr lower stopband. This is very useful for the design of a

Resonator#1

Resonator#2

BPF with the transmission zeros near the lower and upper cutoff frequencies of the filter. However, it requires independent control of the electric and magnetic couplings [J.40]. Such applications are illustrated below.

18.5.3 Some Structures of Coupled Microstrip Line Resonator The λg/2-end coupled microstrip, or CPW resonators, used in the design of a BPF, are electrically coupled resonators. Figure (18.22a–d) shows four cases of the coupling arrangements of the λg/2-microstrip coupled open square resonators [J.37, J.41, B.6]. The first case shows the electric field coupling between the open ends, where the electric field is dominant. The second case shows the magnetic coupling between two arms, where the magnetic field is dominant, due to the maximum of current at the center of the resonators. The electric field decays faster than the magnetic field, providing stronger magnetic coupling. Figure (18.21a and b) shows responses for the first and second cases of Figure (18.22). The third and fourth cases are for the mixed coupling. Figure (18.21c) shows the response of the mixed coupling. The coupling is controlled by changing the separation between the resonators. The zero coupling is obtained at some distance that produces a transmission zero at the null frequency. These

Resonator#1

Resonator#2 Magnetic field coupling region

Electric field coupling region

(a) Electric field coupling.

(b) Magnetic field coupling.

Mixed field coupling region

Resonator#1

Resonator#2

Resonator#1

664

(c) Mixed field coupling- I. Figure 18.22

Coupling arrangements between open square resonators.

Mixed field Coupling region

(d) Mixed field coupling- II

18.5 Coupled Resonators

L1

0

Le

L2

S11

Lm

Lf

S11 , S21 (dB)

Sm

–20

–40 S21 –60

Microstrip L2

Se

Zo –80

L1

1.5

(a) Coupled bent resonators2(L1 + L2) + Lm = λg/2.

2.0

2.5 3.0 Frequency (GHz)

3.5

(b) Frequency response with two transmission zeros [J.40]. 0 Prototype S21 Simulated S21 Simulated S11 Measured S21 Measured S11

–10

Output CPW

S parameters (dB)

Input CPW

–20 –30 –40 –50

–60 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 Frequency (GHz) (c) Slot line-based coupled spiral resonators forming 4-pole BPF. Figure 18.23

(d) Response of BPF showing both sides transmission zeros [42].

Coupling arrangements of bent microstrip resonators and slot line resonator. Source: From Tang et al. [J.40]. © 2009, IEEE. From Azadegan and Sarabandi [J.42]. © 2004, IEEE.

structures have been used to develop the multipole BPF using the coupling matrix for the coupled multiple resonators. In the case of the microstrip coupled open-square resonators, it is difficult to control the electric and magnetic couplings, almost independent of each other. However, Fig (18.23a) shows a bent form of λg/2-coupled resonator [J.40] that provides better independent control of the electric and magnetic couplings. The longer coupled lengths Lm and their separation Sm control the magnetic coupling kL, whereas the bent lengths, near open ends Le, and their separation Se control the electric coupling kC. The coupling-ratio kL/kC can be adjusted to place the transmission zero at the left-hand side of the passband of 2-pole BPF. The input and output of the 2-pole BPF of

Fig (18.23a) are tapped fed. Their location Lf controls another transmission zero of BPF. The resonators are designed around high-impedance (Z0 = 110 Ω) microstrip, as it provides a larger tuning range for the coupling coefficients. The frequency response of Fig (18.23b) demonstrates both transmission zeros. Figure (18.23c) shows the slot-line-based, CPW-fed, coupled resonator 4-pole BPF. To reduce the size of the BPF, the resonators are spiral loaded at both ends [J.42]. It is a series resonant-type structure that involves both the electric and magnetic couplings. By controlling these couplings, the transmission zero can be placed on either left or right of the passband. Figure (18.23d) shows a response to the 4-pole BPF with the transmission zeros on both sides.

665

666

18 Planar Resonating Structures

The inductive and capacitive couplings to the microstrip resonators, for instance, the SIR type, can also be achieved with the help of the short-circuited stubs [J.43, J.44]. These coupling arrangements provide high-performance filters. There are several other coupling arrangements for the microstrip resonators.

18.6

Microstrip Patch Resonators

The two-dimensional microstrip resonators can be arranged in three groups – (i) patch and ring resonators, (ii) 2D space-filling fractal resonators, and (iii) dual-mode resonators. These structures are important for both the microwave circuits and antenna applications. This section summarizes the basic properties of patch and ring resonators. The fractal resonators and dual-mode resonators are discussed in the next sections (18.7) and (18.8), respectively. Figure (17.2) of chapter 17 shows several geometrical shapes of microstrip resonators – the patch and rings. These have been extensively studied and used for both the circuit and antenna applications [B.2–B.6, B.9, B.12, B.19]. Four structures, shown in Fig (18.24), are discussed in this section. Using the full-wave analysis

methods, these resonating structures are analyzed to obtain the resonant frequencies, Q-factors, and field structures. However, the cavity model of these structures has a special attraction, as it helps visualization of the resonating modes [J.45–J.49]. It is a simple and effective method to get the resonant frequencies and Q-factors. The wave equation in the enclosure, made of the EW and MW, is solved like all metallic cavity resonators [B.10–B.12, B.15, B.16]. The cavity resonators are just the enclosed waveguide sections. The metallic waveguide of a rectangular or circular cross-section supports TEZmn and TMZmn modes, defined with respect to the Zdirection of propagation. The reflecting metallic surfaces are placed at two ends of a waveguide section to form a cavity resonator with TEZmnp and TMZmnp modes, where p is an integer, showing the number of half-wavelength variations along the Z-axis. The method of separation of variables, along with boundary conditions, discussed in chapter 7, is used to get the modal solutions. This method has been used to get the cavity model of the patch resonator. Below the results are presented, without going into details of their derivations. The expressions are not presented for the fields inside a cavity. This information is available in the standard books [B.2, B.10–B.12, B.16].

W

ρ

a

θ

n

0.0

0.0

L

m

Substrate (b) Circular patch.

(a) Rectangular patch. εr

Y

εr

Y

b –a , 0

ρ

a

X

X

Substrate

a , a 2√3 2

√3

θ 0.0

X

0.0 a

Substrate (c) Ring resonator. Figure 18.24

εr

Y

εr

Y

Substrate

X a ,– a 2 2√3

(d) Equilateral triangular patch.

Some microstrip patch resonators. Thickness h of the substrate is along Z-axis.

18.6 Microstrip Patch Resonators

To model the patch resonators as the cavity resonators, the bottom and top surfaces are taken as the EWs, whereas the sidewalls are treated as the MWs. The substrate thickness h is the separation between two EWs. The substrate thickness is much less than the half-wavelength, i.e. p = 0. Normally, the integer p is dropped from the resonating-mode description of a patch resonator. As there is an E-field component in the Z-direction, the patch resonators support the TMzmn modes. Following the discussion in chapter 7, these modes could be described by the electric scalar potential eigenfunction ϕe x, y, z = ϕemn x, y e − jβz Z for a rectangular crosssection; and by ϕe x, y, z = ϕemn θ, ρ e − jβz Z for the circular disc and circular ring cross-sections. The wave equation is solved in an appropriate coordinate system to get the eigenfunction of a patch resonator [B.2, B.12]. The case of the equilateral triangular patch resonator is more involved. However, Schelkunoff has provided the eigenfunction of equilateral triangular cross-section waveguide [B.20]. It is used for the cavity model of a triangular patch resonator [J.48, J.49]. The wave equations, in both the rectangular and cylindrical coordinate systems, for the electric scalar potential function, are given below: ∂ 2 ϕe x, y ∂ 2 ϕe x, y + + k2c ϕe x, y = 0 2 ∂x ∂y2

a

1 ∂ 2 ϕe + k2c ϕe ρ, θ = 0 ρ2 ∂θ2

b

1 ∂ ∂ϕe ρ ∂ρ ρ ∂ρ

+

where, k2c = k2 − β2z

k = ω με

c,

d 18 6 1

The MW surfaces are located just at the edges of the conducting patches, so that relative permittivity εr is maintained inside the cavity. Under the MW boundary condition ∂ϕe ∂n = 0 , the electric scalar potential eigenfunction and also the modal cutoff wavenumber, kc = kmn, are computed. Under the resonance constraint, the modal resonance frequencies are obtained as f mn =

18.6.1

kmn ckmn = 2π με 2π εr

18 6 2

Rectangular Patch

Figure (18.24a) shows a rectangular patch resonator. Its eigenfunction and cutoff wavenumber are obtained by solving equation (18.6.1a): mπ nπ cos L W 2 mπ nπ 2 1 + kmn = L W where, m, n = 0, 1, 2, 3, … ϕemn = cos

a 2

b,

18 6 3

Using the above eigenfunction, the electric and magnetic field components are obtained from equation (7.1.28) of chapter 7 with a detailed solution in subsection (7.4.2). Figure (18.25a–c) shows the Ez-field distribution of a few TMmn-modes. The magnetic currents (M), over the MWs, are also shown. At the location of Ez = 0, the input impedance is zero; and at the edge, where Ez is maximum, it is maximum. The feed can be suitably located to achieve the matching. The magnetic currents are determined at the apertures of substrate thickness. They help to compute the radiated power. It is important for antenna applications [B.2, B.5, B.21]. It is also important for evaluating the radiation Q-factor of a resonator. The radiation is less for a patch resonator on a thin and high permittivity substrate, such as on an alumina substrate. Following the process applicable to a rectangular waveguide, the conductor loss and dielectric losses of various modes are computed. Finally, the total Q-factor and the bandwidth ≈1% are obtained. The cavity model, in its simplest form, ignores the fringe fields at the edges. So, the computed resonance frequency of a patch deviates from the measured one by several percent. The fringe fields increase both the length and width of a patch. In that case, the MWs should be placed at the end of the effective length Leff and effective width Weff. It helps to include the total field within the cavity model. Likewise, the fringe fields also decrease the relative permittivity of a patch resonator. The Leff and Weff are computed using the expression of openend discontinuity [B.3], and the effective relative permittivity is obtained from closed-form expression discussed in subsection (8.2.1) of chapter 8. This arrangement improves the computation of resonance frequency. However, improvement is not adequate. Wolff and Knoppik [J.47] have noted that the fringe fields are mode dependent, and the dynamic relative permittivity of a patch resonator could be evaluated for more accurate computation of its resonance frequency. Their model has been generalized as the modified Wolff model (MWM) to improve the results further. The MWM applies to the patch resonators of several shapes under the multilayer conditions. The multilayer substrates could be lossless, lossy, isotropic, and anisotropic [J.50–J.60]. The MWM process, for a rectangular patch resonator, is summarized below. 18.6.2

Modified Wolff Model (MWM)

Figure (18.26a) shows a shielded multilayer microstrip rectangular patch resonator. The structure can be reduced to several important cases, including the single-substrate open-patch resonator. The MWM computes the resonance frequency of the fundamental mode with accuracy 0.5%, and the resonance frequencies

667

668

18 Planar Resonating Structures

Y

Y W

M

Y W

M

Y M

M

Ez

M

Patch

M

Patch

M

M

Ez

X

X

Ez

Ez

L L

(a) TM01- mode. Y

(b) TM10- mode. M

Y

W

X

X

M M

Patch

M

Ez

X

Ez

L X

(c) TM20- mode. Figure 18.25

Ez-field distribution of a few TMmn-modes. (M: Magnetic current at magnetic walls.)

of the higher-order modes with an accuracy within 1.7% [J.50, J.52]. Using the concept of the mode-dependent dynamic relative permittivity εr,dyn, the effective length (Leff), and effective width (Weff), equation (18.6.3) provides the following expression to compute the modal resonance frequency: f mn

kmn c = = 2π με 2 εr,dyn

m Leff

2

+

n Weff

2 1 2

Cej,dyn(εr1, εr2, εr3, W or L), (j = 1, 2) are the fringe capacitance, along the length and width of the patch. These capacitances are evaluated as follows [J.47, J.50]: Co,dyn εr1 =

Co,stat εr1 γn γm

a

Co,stat εr1 =

ε0 εr1 LW h1

b,

γi =

1, for i = 0 2 for i

18 6 6

18 6 4 The mode-dependent dynamic relative permittivity of a complete structure is computed by computing the dynamic capacitance of a structure with dielectric layers, and the dielectric layers replaced by an air medium: Cdyn εr1 , εr2 , εr3 , W Cdyn εr1 = εr2 = εr3 = 1, W where, Cdyn εr1 , εr2 , εr3 , W = C0,dyn εr1 εr,dyn =

+ 2Ce1,dyn εr1 , εr2 , εr3 , W + 2Ce2,dyn εr1 , εr2 , εr3 , L

a

b 18 6 5

In the above equation, C0,dyn(εr1) is the central dynamic patch capacitance that is not influenced by the dielectric cover and shield. The capacitances

0

Ce1,stat εr1 , εr2 , εr3 , W γn Ce2,stat εr1 , εr2 , εr3 , L = γm

Ce1,dyn εr1 , εr2 , εr3 =

a

Ce2,dyn εr1 , εr2 , εr3

b 18 6 7

In the above equation (18.6.6), i = n, m is the modal number. The static fringe capacitance of the length-side of the patch is computed after subtracting the central capacitance from the total capacitance of a patch in the layered medium: 1 Ce1,stat εr1 , εr2 , εr3 , W = Ct L − Co,stat εr1 , 18 6 8 2

18.6 Microstrip Patch Resonators

L

Normalized freq. [fr / fro]

1.0

Patch W

Dielectric layers

h3

εr3

h2

εr2

h1

εr1

y w

εr = 9.6

0.8

6.8

0.6

2.33

0.4 EXP (

0.2

) MWM, t = 0

MWM, t = 0.03556 mm

Conducting ground plane

x

James

0.04 0.08 0.12 0.16 0.20 Normalized thickness [h / λd]

Three layers shielded patch

(a) Multilayer shielded microstrip patch.

0.24

(b) Effect of conductor and substrate thickness on resonance frequency. 4

4.25 MWM Combined TTL & TLM Variational Exp

εr2 = 2.32

10 01

3.75 3.50

εr2 = 10.0 w = 1.9 cm, L = 2.29 cm, h = 0.159 cm εr1 = 2.32

3.25 3.00

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

Superstrate thickness (h2) cm

30 22

0

–2

–4

–8

Dearnley - Barel Mod. wolff Ham - Jen

2

4

6

8

Average freq. (GHz)

(c) Effect of superstrate on the resonance frequency. Figure 18.26

20 02 2112

Modes

–6

2.75 0.0

11

2 Average deviation (%) of fr

4.00 Resonant freq. (GHz)

Zeroth order

(d) Accuracy of models for higher-order modes.

Shielded multilayer microstrip rectangular patch resonator. Source: From Verma and Rostamy [J.52]. © 1993, IEEE.

where the total line capacitance (Ct) p.u.l. of the layered medium patch is computed using the variational method in the Fourier domain, discussed in subsection (14.4.2) of chapter 14. The result, applicable to the present structure, is given below: 1 1 = Ct πε0

∞

0

f β Q βh1

2

1 d βh1 Y

where, Y = εr1 coth βh1 εr3 coth βh3 + εr2 tanh βh2 + εr2 εr2 + εr3 tanh βh2 coth βh3

a

b 18 6 9

A layer can be eliminated by taking the thickness of a substrate to zero. Further, h3 ∞ and εr3 = 1 eliminate the top shield. So, the structure can be reduced to several interesting cases. Several layers can be accommodated by getting additional terms in the admittance parameter Y. In the above equation, the function f β is assumed to be Fourier domain cubic charge density function along the width W of the patch, and Q is the total charge p.u.l. on the patch. The details are presented in chapter 14. The process is repeated to determine the static fringe capacitance Ce2,stat(εr1, εr2, εr3) of the width side of a patch.

669

670

18 Planar Resonating Structures

Next, the Leff is computed, using the equivalent width (Weq), from the following expressions: Leff = L + Weq =

Weq − W 2

εreff W + 0 3 εreff W − 0 258

120πh1 Z εr1 , εr2 , εr3 , W εreff W

a b 18 6 10

The effective relative permittivity εreff(W) and the characteristic impedance Z(εr1, εr2, εr3) are determined, from the width-side of the patch, using the variational expression of the line capacitance: C εr1 , εr2 , εr3 , W C εr1 = εr2 = εr3 = 1 Z εr1 , εr2 , εr3 , W 1 = c C εr1 , εr2 , εr3 , W C εr1 = εr2 = εr3 = 1, W εreff W =

a

b 18 6 11

The accuracy of the evaluation of the integral, used in equation (18.6.9a), determines the accuracy of characteristic impedance Z(εr1, εr2, εr3, W). Similarly, the Weff is determined by exchanging W by L and L by W in the above expressions. The effect of the conductor thickness on the resonance frequency could also be accounted for [J.52]. However, it is not significant. Figure (18.26b) compares the results, using MWM, and other methods, for the normalized resonance frequency of a standard patch resonator against the experimental results. It shows a monotonic decrease in the resonance frequency with increasing normalized substrate thickness. The resonance frequency fr is normalized with respect to the zeroth-order resonance frequency, f ro = 15 L εr1 GHz. The patch length L is in cm. Figure (18.26c) compares the results of a dielectric covered patch. Figure (18.26d) shows the accuracy of the MWM for the computations of higher-order modes. In general, the MWM has an accuracy of 0.7% and 1.7% for the fundamental and higher-order modes, respectively. The MWM has also been used to compute the Q-factors due to the dielectric, conductor, radiation, and surface-wave losses [J.54, J.55]. It helps to determine the bandwidth and input impedance of a multilayer patch resonator. The patches could be the circular disc, annular ring, and triangle also. The anisotropy in the substrate is also accounted for in the MWM [J.53,

f mn = aeff

J.54]. Due to its high accuracy, the MWM is used to develop a sensor to measure the complex relative permittivity of material in the sheet, paste, or liquid form [J.56].

18.6.3

Circular Patch

Figure (18.24b) shows the geometry of a circular patch resonator of the radius a, on a substrate with relative permittivity εr, and thickness h. The circular patch could be under the multilayer environment also [J.51, J.55, J.57]. The cavity is formed with the MW just at the circumference of the patch. Equation (18.6.1b) is solved for the potential function ϕemn , under the boundary condition ∂ϕe/∂ρ = 0, at ρ = a, where “a” is the radius of a circular patch. The solution for the potential function ϕemn is expressed as a linear combination of two solutions in terms of Bessel functions of the first and second kinds. However, the solution using the second kind of Bessel function is infinite at ρ = 0. So, it is dropped as a nonphysical solution, giving the following expression of the eigenfunction and wavenumber of the TMznm mode [B.11, J.45]: ϕemn = Jm kmn ρ ejmφ Jm kmn a = 0,

b

where, the first 4 roots of Bessel function χ11 = 1 8412, χ21 = 3 0542, χ01 = 3 8318, χ31 = 4 2012

c 18 6 12

The subscript n is the radial modal number. It shows the radial variation of the modal field of the TMzmn mode. Another modal number m shows the angular (ϕ) variation, over the range 0 ≤ ϕ ≤ 2π, of the modal field. The modal number n is also the order of Bessel function, giving the decaying cosine radial field, along ρ, 0 ≤ ρ ≤ a, with n roots. The subscript n shows the number of roots of the derivative of Bessel function with respect to ρ. For the modal number n = 1, the first root of m = 0–3 order Bessel functions provides the TMz01 , TMz11 , TMz21 , TMz31 resonating modes. Once roots χ'mn are known, the resonance frequency is computed by using kmn = χmn/a in equation (18.6.2). However, for accurate computation of the resonance frequency, the MWM is used; in that case, the radius “a” is replaced by the effective radius aeff and εr is replaced by εr,dyn:

cχmn 2πaeff εr,dyn

2h =a 1+ πaεr

a kmn = χmn a

a a h + 1 41εr + 1 77 + 0 268εr + 1 65 ln 2h a

18 6 13

1 2

b

18.6 Microstrip Patch Resonators

In the above equation, the dynamic relative permittivity is computed by converting the circular patch to an equivalent rectangular patch, on taking the width of the rectangle 2a. Its length is computed to maintain the area of the rectangular patch the same as that of the circular patch [J.51, J.55]. In place of equation (18.6.13b), another more accurate expression is also available to compute the aeff for a wide range of parameters [J.61]. The MWM is also applicable for computing resonance frequency, various Q-factors, input impedance, and bandwidth of a multilayer lossy circular patch [J.55]. The values of roots χmn show that the TMz11 -mode is the fundamental mode. Once ϕemn is known, the field equations are obtained in the cylindrical coordinate [B.11]. Figure (18.27) shows the field and current

patterns of four modes [J.46]. The entering and leaving displacement current provide continuity to the conduction current on the patch. The symmetric mode TMz01 has no angular (azimuthal) field variation. Whereas for the other three modes, the electric field, i.e. the displacement currents, in pairs are obtained, corresponding to m = 1, 2, 3. 18.6.4

Ring Patch

Figure (18.24c) shows the ring-type patch resonator, with inner radius a and outer radius b. The concentric cylindrical MWs are located at ρ = a and ρ = b. The potential function, in terms of the first and second kind Bessel functions, within the MWs, is again a solution of

2a

2a

m=0

m=1

(a) TM01- mode.

(b) TM11- mode.

2a

2a m=2 (c) TM21- mode. Current on top plane,

m=3 (d) TM31- mode. Magnetic field

Displacement current (Electric field) Figure 18.27

Field and current patterns of a few TMmn modes of microstrip circular patch resonator. Source: Watkins [J.46]. © 1969, Institution of Engineering and Technology.

671

672

18 Planar Resonating Structures

equation (18.6.1b). It also provides a constraint condition. Its solutions give the n number of roots χmn for the mth-order Bessel functions. These results are shown below [B.2, B.11, B.19, J.46]:

Once the potential function is known, the field components can be computed. Figure (18.28a–d) shows the field and current distributions on the patch surface of a ring resonator for a few TMzmn modes [B.19].

ϕemn = Jm kmn ρ Ym kmn ρ − Jm kmn ρ Ym kmn ρ ejmϕ a Jm kmn a Ym kmn b − Jm kmn b Ym kmn a = 0 kmn = χmn a

For C = b a ratio, and χmn = kmn a,

Jm Cχmn Ym χmn − Jm χmn Ym Cχmn = 0

b

18.6.5

c

Figure (18.24d) shows the geometry of an equilateral triangular patch resonator with side length a, on a substrate with thickness h and relative permittivity εr. Other triangular shapes have also been considered [B.19]. The TMzm,n,ℓ field components have been obtained from the following eigenfunction [J.48, J.49]:

d

For C = 2, the first 3 roots are, χ11 = 0 6773, χ21 = 1 3406, χ31 = 1 9789 additional root, χ02 = 3 1966, χ12 = 3 2825

e

18 6 14 The calculation of resonance frequency requires replacement of parameters: a aeff, b beff and εr εreff. The required expressions are summarized below: εreff =

1 1 εr + 1 + εr − 1 2 2

1+

10 W h

−1 2

Equilateral Triangular Patch

2πℓ u π m + n v−w + b cos 3b 2 9b 2πm u π n−ℓ v−w + cos + b cos 3b 2 9b 2πn u π ℓ−m v−w + b cos + cos 3b 2 9b

ϕemn = cos

where W = b − a

a

where, ℓ + m + n = 0, u =

beff = b + We f − W 2

b

v−w = −

aeff = a − We f − W 2

c

where, We f = W +

We f = 0 =

120πh , Zo εreff

We f = 0 − W 1 + f fp fp =

d

2

k2mn =

4π 3a

Z0 2μh

e

The factor [We(f ) − W]/2 in the above equation is an increase in one side width of the strip conductor. It reduces the inner radius a, and increases the outer radius b. The following simpler equations for aeff, and beff are also used: beff = b + 3 4 h

b 18 6 16

Using χmn = kmn/a, the resonance frequency is calculated from equation (18.6.2): f mn =

cχmn 2π aeff εreff

3 a 3 x + y, b = 2 2 2 3

b

2

m2 + n2 + mn

c

The integers m, n, ℓ do not represent the halfwavelength variation along the arms of a triangle. However, the boundary value problem is solved under the constraint m + n + ℓ = 0. The resonance frequency is obtained on substituting kmn in equation (18.6.2) [J.49, B.20]: f mn =

2c m2 + n2 + mn 3aeff εr

aeff = a + a,

1 3 x + y, 2 2

18 6 18

18 6 15

aeff = a − 3 4 h

a

18 6 17

In the place of the above equation, again the MWM can be used. It applies to the multilayer ring resonator also [J.59].

h εr

1 2

a b 18 6 19

TMz1,0, − 1

is the fundamental mode of an The mode equilateral triangular patch resonator. The modes TMz1,0, − 1 , TMz− 1,0,1 , and TMz0, − 1,1 are the degenerate modes. Figure (18.29a and b) shows the field patterns of the fundamental and next higher-order modes, i.e. the TMz1,0, − 1 and TMz1,1,2 [J.49]. The conduction currents on the patch are at a right angle to the magnetic field lines. These are terminated in the displacement currents, shown as the dot and cross. The fringe field is accounted through the empirically obtained effective side length aeff, while keeping εr unchanged. It is

2a

2a

2b

2b

(a) TM02 - mode.

(b) TM11 - mode.

2a

2a

2b

2b

(c) TM21 - mode.

(d) TM31 - mode. Magnetic field

Current on top plate, Displacement current (Electric field) Figure 18.28

Field and current patterns of a few TMnm-modes of microstrip ring patch resonator. Source: Sharma et al. [B.19]. © 1989, John Wiley & Sons.

Electric field Magnetic wall

Magnetic field

(a) TM1,0,–1 - mode. Figure 18.29

Electric field Magnetic wall

Magnetic field

(b) TM1,1,–2 - mode.

Field patterns of two TMnml-modes of microstrip equilateral triangle patch resonator. Source: Helszajn and James [J.49]. © 1978, IEEE.

side-arm a = 6.35 mm εr = 24.5 – 25

0

h = 05 mm TM10 6.316 GHz

–5 –10

TM11 10.939 GHz

–15 0

2

4

TM03 TM02 18.948 GHz 12.632 TM12 GHz 16.710 GHz

6 8 10 12 14 16 18 20 Frequency (GHz)

(a) First five modes of an equilateral triangular patch. Figure 18.30

Reasonant frequency (GHz)

18 Planar Resonating Structures

S11 (dB)

674

3.5

εr = 2.32 εr = 6.8 εr = 10.2

3

Ensemble

2.5 2

h = 1.59 mm, tanδ = 0.001

1.5 1 0.5 0

40

60

80

100 a (mm)

120

140

(b) Fundamental mode resonance frequency using MWM.

Resonance frequency of the equilateral triangular patch.

unrealistic. Again, the MWM computes the resonance frequency more accurately through the effective length and εr,dyn [J.58]. Moreover, it applies to the lossy multilayer cases also, giving the Q-factors and bandwidth [J.60]. Figure (18.30a) shows the EM-simulated resonance frequencies of the first five modes of an equilateral triangular patch resonator on the LaAl2O3 substrate (εr = 24.5 − 25) of thickness 0.5 mm, and side arm a = 6.35 mm. Figure (18.30b) shows a variation of the resonance frequency of the fundamental mode, with respect to the side-arm length [J.62]. It further compares the accuracy of the MWM against the results of the SDA [J.63], and also against the full-wave EM-simulator Ensemble.

18.7

SDA

MWM

geometrical arrangement, the planar metallic surface of a microstrip resonator is created not by the 2D metallic patch but by filling the required 2D-space with an infinitely long line, arranged in a specific manner. Several kinds of curves are available to achieve the objective of the space-filling that creates a plane sheet with a very long line. In the place of a metallic line, one can equally take the slot line and can create a slot area in the metallic ground plane of a microstrip line. The slots are also used repeatedly to defect a metallic patch or a line resonator. Before discussing the fractal resonator, it is useful to appreciate some important curves of the fractal geometry. These curves do not follow the integral dimensional arrangements, like 0, 1, 2, and 3 dimensions, of the Euclidean space. The fractal curves have the fractional dimension between 1 and 2.

2D Fractal Resonators

The miniaturization of the planar antenna, filters, and other microwave components are very important for the front-ends of the modern wireless and satellite communication systems. The antenna itself is a narrow band resonating element. The physical size of the microstrip resonator can be reduced by folding the line or increasing the relative permittivity of a substrate. The size reduction of a line is also achieved by the inductive or capacitive loading. The loading is realized through the stubs or by cutting slots in the line conductor, such as the defected microstrip structure (DMS), or by the slots in the ground plane, forming the DGS. These loadings also reduce the size of a resonating patch. This section introduces a different approach, i.e. the planar fractal technology, to the problem of miniaturization of the planar resonators. It has also been extensively used with the antenna and filter technologies. The fractal technology is based on fractal geometry. In this

18.7.1

Fractal Geometry

The fractal object is defined through the concept of the non-Euclidean dimension, called the similarity dimension. There are also other kinds of non-Euclidean fractal dimensions [B.22]. The fractal dimension could be viewed as the space-filling ability of the fractal shapes. Mandelbrot has initiated the present interest in the fractal geometry [B.23]. He emphasized that shapes of the objects in nature are more effectively described by the fractal geometry, i.e. the geometry of broken lines. However, the natural shapes of objects are created by the complex forces of nature. By just adopting the fractal geometrical description of an object, we cannot modify or control the natural forces. So, just by adopting the fractal geometry, we cannot get the optimum or unique electrodynamic properties of an object, although useful changes can be derived. Yet, the effort has been made to achieve something like the fractal electrodynamics

18.7 2D Fractal Resonators

[B.24]. Despite its widespread innovative use in the microwave and antenna technologies, some limitations will be obvious by the end of this section. The fractal curves, used in the planar microwave technology, are normally described with the help of the similarity dimension. In the case of the Euclidean 2D plane, all shapes, such as a square, circle, etc., have a fixed dimensional number, i.e. 2. These Euclidean shapes completely fill in the 2D Euclidean space. Likewise, the line and cube fill in the 1D and 3D Euclidean spaces, respectively. Even though several 2D Euclidean objects have the ability to fill in the 2D-space, their electrical characteristics as resonators, filters, and antennas are different. Different 2D conventional geometrical shapes, even for the same area, or the same length of the perimeter, can provide different performances of the resonators, filters, and antennas. A similar situation holds for various fractal curves. The fractal curve with a higher similarity dimension, or with identical curve length, or identical enclosed area, may not provide identical electrical performances, when used as the resonators, filters, and antenna. The space-filling curves, creating the non-

Original line section

Euclidean planes, have their own similarity dimensions. Before proceeding further, it is important to define the similarity dimension, so that it could be applied to the fractal curves. Similarity Dimension and Basic Characteristics of Fractal Structures

A larger object can be broken into smaller similar objects, called the scaled copies of the original large object. On assembly of several numbers of the scaled copies, in a particular sequence or format, a larger object can be created. The copies and the large object are similar in shape; they are different only in the scale. Let us apply this concept to 1, 2, and 3-dimensional objects of the Euclidean geometry, i.e. to a geometrical line, square, and cube, shown in Fig (18.31a–c). In Fig (18.31a), a unit length line is broken into 3 equal similar segments. The size of each smaller segment, called the copy, is 1/3. For n number of copies, the scale-factor is r = 1/n. If the self-similar copy is repeated along a line, without changing the direction, the 1D space is filled in and a long line is created.

3 similar sections of line

(a) Creation of 3 similar scaled (1/3) copies of the line section.

Original square

4 similar scaled squares

(b) Creation of 4 similar 1/2 scaled squares.

Original cube

Figure 18.31

8 similar scaled cubes (c) Creation of 8 similar 1/2 scaled cube.

Creation of scaled copies of objects in Euclidean space.

675

676

18 Planar Resonating Structures

Figure (18.31b) shows four similar segments (copies) of the original unit square. If it is broken into n number of copies, the scale-factor of a copy is r = 1 n. Thus, for n = 4, r = 1/2, i.e. sides of the original unit square are reduced by a factor 1/2. Finally, Figure (18.31c) shows eight similar cubic segments (copies) of a unit cube. The scale-factor of a cube copy is r = 1 3 n . So, for n = 8, r = 1/2, i.e. each side of the original unit cube is reduced by a factor 1/2. Using the induction method, the following expression is written for the scale-factor (r), relating it to the number of self-similar segments (copies, n), and space dimension d: 1 1 a , No of copies, n = d b r n ln n Dimension, d = c − ln r 18 7 1 Scale − factor, r =

d

In the above equation, the scale-factor r is a fraction giving a positive number for the dimension. The above equation provides the similarity dimension of an object, if its scale-factor and the number of copies involved in its creation are known. In generating a copy, the rule of self-similarity has to be followed. So, the dimension obtained is known as the similarity dimension. The self-similar, i.e. the similarity, dimension d can be defined using the following more general expression, involving more number of variables that occur in some fractal curves [J.64]: n1

1 r1

d

+ n2

1 r2

d

+

+ nm

1 rm

d

= 1, 18 7 2

where m is the number of scales, used in the initiator of the fractal curve. The above expression shows that a fractal curve can have more than one scale-factor. The scale-factor rm is an integer, and a dimension is a positive number. For only one kind of scale-factor m = 1, the above equation is reduced to equation (18.7.1). It is noted that the scale-factor r in equation (18.7.1) is a fractional number, whereas the scale-factor rm of equation (18.7.2) is an integer. Both the scale-factors are reciprocal to each other. Both forms are used in further discussion. The above concept is applied to a regular closed geometrical area to get a finite area, with infinite broken irregular boundary, i.e. the fractal boundary. The irregular fractal curve can also be generated using the same process. Likewise, the 2D-space is filled with the broken curve of an infinite length to get 2D-like surface. Alternatively, the filled 2D geometrical space can be depleted by the self-similar copies. These are all irregular broken fractal shapes used for the development of the compact planar resonators, filters, antennas, etc.

The following four popular fractal constructions are considered in further discussion:

•• ••

Koch curves and Koch islands. Minkowski curves and Minkowski islands. Hilbert curves. Sierpinski triangles and Sierpinski carpet.

The above fractal geometrical shapes can be generated using MATLAB codes. The coordinate details of the fractal curve generated in MATLAB are transferred to the EM-Simulator for the development of the fractal components [J.65]. The self-similarity and space-filling, or space consumption, and scaling are the important basic characteristics for the creation of the fractal curves and fractal areas. The fractal curve is generated by an infinite repetition of the self-similar copy, with changing scale-factor. In practice, very fine structures are ignored, and the recursive process of fractal curve generation is terminated with 2–6 iterations. This method of fractal curve generation is known as the iterated function system (IFS). It is based on a series of affine transformation, where the fractal elements undergo the rotation, shearing, translation, and scaling [J.64, J.66, J.67]. The basic characteristics of the fractal shapes have the following implication in the microwave fractal technology:

• • •

The self-similarity helps to design the multiband antenna. In the case of filters, it provides better stopband rejection and generates the transmission zeros to improve the selectivity. The space-filling ability helps to develop an electrically small antenna and resonators. The area fractals and boundary fractals can provide high directivity antenna and antenna-array with low-side levels. The space-filling curves have been used to develop the high-impedance EBG surfaces and metasurfaces.

Koch Curves and Koch Islands

The equilateral triangular shape is a generator that acts on an initial line segment of unit length to create the Koch curve. In this case, the initial line is the initiator of the Koch curve. Figure (18.32) explains the process. A unit line length is divided into three segments, creating self-similar three copies of a line, with the reduced scale-factor (r = 1/3). During the first iteration, the generator replaces the middle part of the line segment with an equilateral triangle without a base. So, the original line segment containing 3 self-similar copies is replaced by the four self-similar copies. The segment numbers are marked on the figure. During the second and subsequent iterations, each line section is replaced by four self-similar

18.7 2D Fractal Resonators

(1) 0

(2)

(3)

1/3

Intiator

2/3

(2)

A

Unit length Line segment

A

1

1

3

(3)

B

(4)

(1)

C

B First iteration

Generator

C

(a) Initiator: Equilateral Triangle.

2 (b) The first iteration.

Third iteration

(c) The second iteration. Higher iteration

Figure 18.32

Creation of a Koch curve from a line segment.

copies by applying the generator shown in Figure (18.32). It results in the Koch curve, created using the broken lines. Figure (18.32) shows the third and higher-order Koch curve. It is a non-differentiable curve. It is interesting to note that within the 1D dimension of the original line, the Koch curve has an infinitely large number of line sections, although its 1D Euclidian dimension is fixed. The Koch curve has a peculiar property – with an increasing number of iterations, the curve length increases to infinity, while its enclosed area is finite. After nth iteration, the perimeter ℓ and enclosed area (A) of the Koch curve are given by the following expressions: ℓ=

Figure 18.33

4 3

n

a,

A=1+

4 9

n−1

b 18 7 3

The limiting value of the enclosed area is 1.8. However, after a few iterations, say 3, the Koch structure becomes so fine that the iteration can be terminated for any practical application. The terminated fractal is known as the pre-fractal. The pre-fractal ignores the fine structure of the arbitrarily small scale. The Koch curve has the number of copies n = 4 and the scale-factor r = 1/3. So, its dimension d, using equation (18.7.1), is 1.2619.

(d) The third iteration.

Creation of Koch island or Koch Snowflake from an equilateral triangle.

In the above discussion, the Koch curve generation process is applied to a straight line. It can also be applied to an initiator equilateral triangle and other polygons. It generates the Koch island or Koch Snowflake. Figure (18.33a–d) demonstrates it through three iterations for an equilateral triangle, where the middle part of each arm is replaced by a triangle without a base. The equilateral triangle without a base acts as a generator. The standard Koch curve is obtained using a triangular generator. However, the triangle can be replaced by a square of size w. The parameter w is called the indentation length. In Fig (18.34), the modified version of Koch generator is applied to all four sides of a square. The resulted loop, or island, is also known as the Minkowski island. Its perimeter, after nth iterations, is given by the following expression: ℓn =

1+

2 w 3

n

ℓn − 1

18 7 4

Minkowski Curves

Figure (18.35) shows the recursive process for the generation of the 1st, 2nd, and 3rd iteration of the Minkowski curve, from a line initiator. The eight-sided double square acts as a generator. The generator itself is the first iteration, that is obtained by getting eight copies of a line, with the reduced scale-factor 4.

677

678

18 Planar Resonating Structures

W

Generator

Initiator

Figure 18.34

First iteration

Second iteration

Higher iteration

Creation of square Koch island or square Koch Snowflake from a square.

Initiator line 1

3

2

3

2

4

4

1 Generator:

5

8

6

1st iteration Figure 18.35

7

2nd iteration

3rd iteration

Generation of Minkowski curve from the 8-sided pulse generator.

The initiator line is divided into four sections, and sections 2 and 3 are replaced by the squares without a base to get the first iteration of the Minkowski curve. Subsequently, each arm of the Minkowski curve is replaced by the eight-sided Minkowski generator of the previous iteration step. At each iteration step, the line segment length is reduced by a factor 4, and to get total perimeter, it is multiplied by 8. In each step, the total curve length is increased by a factor of 2. So, the length of the Minkowski curve increases to infinite with increasing order of iterations. Its dimension d, with n = 8 and r = 4, is 1.5. It has better space-filling ability, as compared to the Koch curve that has dimension 1.2619. Likewise, one can use the square Sierpinski fractal curves to fill a given area. It looks like Minkowski fractal with a 2 : 3 ratio between the inner and outer squares [J.68]. In place of a line section, a planar area such as a triangle, or square, can be taken as the initiator in the Minkowski process. It results in the Minkowski islands [J.69]. The single-sided square [J.65], or a rectangular, generator is also used to generate the Minkowski island. Its square format, shown in Figure (18.34), is also known as the Koch island. In Figure (18.36), the five-sided Minkowski square or rectangular self-similar generator is applied to each of the four sides of a square patch resonator [B.25]. The rectangular pulse has height ℓ, known as the indentation width. The pulse width is g.

Figure (18.36) shows the results of the first two iterations. The initiator rectangle can also be replaced by a square ring initiator [J.64]. In the case of a rectangular generator, two scale-factors are involved due to three horizontal segments (n1 = 3) with size g (r1 = 3), and two vertical segments (n2 = 2) with size ℓ (r2 = 3/ℓ). The ℓ-dependent dimension of Minkowski island is computed by taking two terms of equation (18.7.2). In the case of a square generating pulse, the dimension is d = ln 5/ln 3 = 1.465. For ℓ in between 0 and 0.9, the dimension is in between 1.0 and 1.413. The perimeter of the Minkowski island increases with each iteration, and for the mth iteration, it is given by the following expression: pm =

1+

2 ℓ 3

m

pm − 1 ,

p0 = 4 ,

18 7 5

where m is the number of iteration. Normally, the prefractal Minkowski islands, up to 3–4 iterations, are used in the microwave circuits and antenna applications. Hilbert Curves

Hilbert curve is a space-filling curve. It is a continuous 1D line, formed by joining the broken U-shaped line structure. It fills in the prescribed 2D area with several iterations. In the process, the perimeter of the Hilbert curve increases infinitely. Figure (18.37a–d) shows a few iterations of the Hilbert curve, within a S × S square.

g g

L

Square generator

18.7 2D Fractal Resonators

Initiator 1

2

1st iteration 2 3

4 5 3 g L Rectangular generator 1

L

Figure 18.36

2nd iteration

ℓ

Generation of Minkowski island from the 5-sided square/rectangular generator.

4

1

Connecting lines

S

S Initiator

2

3

S (a) Initiator and 1st iteration.

(c) 3rd iteration. Figure 18.37

S (b) 2nd iteration.

(d) 4th iteration.

Generation of Hilbert curves using U shape generator.

679

680

18 Planar Resonating Structures

The bent U- shape line is an initiator of the Hilbert curve, and the generator is also a reduced scaled U-shape with 1/2 of vertical and horizontal lengths. The total length of a scaled U-shape generator is 1/2 of the length of the initiator. So, the scale-factor (r) of the Hilbert curve is 2. The first iteration, sometimes taken as the zeroth-order iteration, is just the enlarged U-shape of the size S × S. Figure (18.37b) shows that during the second iteration at each corner of the S × S size U-shape, four copies of the scaled U-shapes (thick dark lines) are placed, with proper orientation. Four U-shaped copies are connected using additional three lines. For n = 4, the dimension of the Hilbert curve is d = log 4/log 2 = 2. It shows that the Hilbert curve has the ability to fill in completely the 2D Euclidean space with a higher order of iterations. Hilbert curve is created by joining centers of the background grids at each stage of iteration. The grid size G(m) reduces with m number of iterations. During the iteration process, the length L(m) of the curve also increases. These are obtained using the following expressions: Gm =

S 2 −1 m

a,

L m = 2m + 1 S

b

18 7 6 The similarity ratio can also be changed, like a Minkowski rectangular generator, by changing the width

(a) Initiator, (n = 0).

(b) 1st iteration, (n = 1).

(d) 3rd iteration, (n = 3). Figure 18.38

Generation of Sierpinski triangles.

of the U-shaped generator. Sometimes Peano curves are also used as the space-filling curve to realize compact microstrip filters [J.70]. The Peano-curve algorithm has a relatively higher compression rate than the Hilbertcurve algorithm to fill-in a 2-D region. So, for the same area of a fractal resonator, it has a lower resonance frequency. Sierpinski Triangles and Sierpinski Carpet Sierpinski Triangles

Sierpinski triangles are also self-similar geometrical structures. These structures have been used for the development of compact multiband planar monopoles. However, it is not a space-filling structure. Normally, it is used to create sieve-type structures with an iterative process, shown in Fig (18.38a–e). There is some modification of the standard Sierpinski triangles. In the present case, the initiator is an equilateral triangle of side length a. During the first iteration, three halfsized black triangles are created, and the central, white triangular area is removed. During the second iteration, every three black triangles create another half-sized three numbers of black triangles, while dropping central white triangles. At this stage, nine black triangles are created. Likewise, during the 3rd and 4th iterations, 27 and 81 triangles are created. During each step of

(c) 2nd iteration, (n = 2).

(e) 4th iteration, (n = 4).

18.7 2D Fractal Resonators

the iteration, 1, 3, 9….3(n − 1) central triangles are dropped. It forms a geometric series, and the total number of triangles removed at the nth iteration is (3n − 1)/2. This formation is known as the Sierpinski triangle or Sierpinski gasket. All removed triangles are inverse (top-down) with respect to the initial triangle. At the nth iteration, the sum total of removed triangular area, and the total perimeter of retained triangles are given by the following expressions: 3 4 3 Total perimeter = 9 2 Area removed =

1−

a2

3 4

n

a

n

−1

b 18 7 7

It is seen from the above that for n ∞ the removed area is a2 3 4 that is the same as the area of the initial triangle. Thus, after the infinite number of iterations, the area of the Sierpinski triangle is reduced to zero, whereas its perimeter increases to infinite. However, in practice, 2–3 iterations are used to develop a compact multiband monopole antenna. The multiband operation is due to

(a) Initiator-blank square.

(b) 1st iteration.

(c) 2nd iteration.

(d) 3rd iteration.

Figure 18.39

Generation of Sierpinski carpet.

the self-similarity of the triangle at the half-scale, i.e. scale-factor (r = 2) with n = 3 copies at each stage of iteration. It gives us the dimension of the Sierpinski triangle as, d = ln(3)/ln(2) = 1.585. Sierpinski Carpet

Sierpinski carpet is another popular fractal structure. It is used to design the microstrip array with a low sidelobe level [J.67]. Figure (18.39a–d) shows the generation of a Sierpinski carpet from a blank square. During the first iteration, the blank square is divided into nine squares through three horizontal and three vertical divisions. The eight numbers of the white square of 1/3 size are retained, giving the dimension of Sierpinski square as ln 8/ln 3 = 1.8928. The 1/9 black square is removed. During the second iteration, each of 8 blank squares is further divided into 8 white squares giving 64 white squares. The process continues and generates the square Sierpinski carpet. Likewise, the pentagonal and hexagonal Sierpinski carpets are also obtained. Similar to the previous case, with infinite iterations, the area of Sierpinski square is reduced to zero, while its perimeter becomes infinitely large.

681

18 Planar Resonating Structures

18.7.2

overall height, Koch monopole shows lower resonance frequency, as compared to the identical height of a simple wire monopole, i.e. the Euclidean monopole. It means to maintain a fixed resonance frequency, obtained from the Euclidean monopole, the height of Koch monopole should be reduced. Both the resonance frequency and 2 : 1 VSWR bandwidth decrease with an increase in the order of iterations of Koch monopole. However, this characteristic is not unique to Koch geometry. Better compactness, i.e. more decrease in the resonant frequency, could be obtained on using the meander line and normal mode helix monopoles. Of course, their 2 : 1 VSWR BW is less than that of Koch monopole. In all the cases of folding, the wire length increases with an increase in the number of iterations. It results in a decrease in resonance frequency. Figure (18.40) shows five iterations of three kinds of monopoles. K0 is the Euclidian wire monopole. The resonance frequency for them decreases with an increase in the number of iterations [J.74]. It is noted that up to the second iteration much difference is not seen for a reduction in the resonance frequency for all three kinds of monopoles.

Fractal Resonator Antenna

The above-discussed fractal geometrical structures have been used to develop compact multiband antenna, antenna array, resonators, filters, and multiband PBG structures. Space-filling ability provides compactness to the structure, while increasing the resonating length. The self-similarity provides multiband behavior. The monopole, dipole, and loop antennas are primarily resonating structures to which fractal geometrical arrangements have been applied. Their use provides a reduction in the resonant frequency. Thus, we can get more compact structures, as compared to their Euclidian counterparts. However, such a reduction is not unique to fractal geometry. Other kinds of folded structures also provide identical compactness, or sometimes even better compactness. We summarize below the results of some of the important fractal geometries in respect of a reduction in the resonance frequency only. Details could be followed through the published literature [J.64, J.66, J.67, J.71–J.73]. Koch fractal geometry has been used for the development of compact resonant monopoles. For the same

K0

K1 K2 K3 K4 (a) Koch fractal monopoles.

M1

K5

M2

M3

M4

M5

(b) Meander line monopoles. 1250 Resonant frequency (MHz)

682

1050 850 Koch fractal 650

Meander

450

Helix Euclidean

250 H1

H2

H3

H4

(c) Normal-mode helix monopoles. Figure 18.40

H5

0

1

2 3 Antenna iteration

4

5

(d) Resonance frequency variation of 4-types of monopoles with several iterations.

Resonance frequency behavior of Koch and other folded monopoles. Source: From Best [J.71]. © 2003, IEEE.

18.7 2D Fractal Resonators

However, above the 4th iteration, frequency reduction gets saturated for Koch fractals, whereas in the case of the meander, and normal helix monopole further frequency reduction can be achieved. The Euclidian monopole provides more decreases. Koch geometry has also been used to get the compact dipole printed antenna. The Q-factor decreases with an increase in the number of iterations [J.67]. Even Hilbert fractals have been used to get the compact dipole antenna [J.73]. However, a meander dipole provides more reduction in the frequency as compared to that of Hilbert fractals. In the case of Hilbert fractals geometry, the currents in adjacent wires are in opposite directions, reducing the effective inductance of the Hilbert fractals loops. Minkowski fractal, shown in Fig (18.36), has been used to get a compact loop antenna, as compared to that of a circular or rectangular loop antenna [J.64, J.66, J.72]. The decrease in the resonance frequency is more for a higher number of iterations. However, it saturates after 2–3 iterations. The indentation width also controls the lowering of the resonance frequency. However, such behavior is not unique to Minkowski fractal or any other fractal. Other kinds of non-fractal folded geometry can provide even more or identical resonance frequency reduction. The self-similar Sierpinski gasket has also been used to get the multiband antenna [J.71]. The multiband response increases with an increase in the self-similar iterations. However, it has been noted that a significant portion of the standard self-similar Sierpinski gasket can be either removed or altered, without changing the multiband nature. It shows that the standard Sierpinski gasket is not an optimum or unique geometrical arrangement to get the multiband antenna.

18.7.3

Fractal Resonators

The planar fractal resonators have been used to develop high-performance compact filters. The case of a microstrip fractal resonator is more complicated than that of the fractal monopole in the air medium. The fractal iterations increase the microstrip line length in a fixed area. It results in a lowering of the resonance frequency, with increasing iterations. So, the length and area of a resonator can be reduced significantly to maintain the same prescribed frequency. Due to the longer microstrip line, packed in a smaller area, the energy storage ability of a fractal microstrip resonator is more. It increases the Q-factor of a fractal λg/2-resonator 10 times, as compared to a conventional resonator. However, with increasing iterations, to pack longer microstrip lengths, the conductor width is reduced. It increases the conductor loss. On reducing the spacing between the conductor tracks, the conductor loss is further increased due to the proximity effect. Normally, after 3 iterations, the Qfactor of a microstrip fractal resonator is reduced. Also, due to the increase in the discontinuities with iterations, both the radiation and surface-wave losses increase, leading to a reduction in the Q-factor [J.65, J.75]. The resonating structures are characterized using four parameters – compactness, resonance frequency, Q-factor, and appearance of next resonating mode. The fractal geometry has been used both for the patch and line resonators. We summarize below results on a few planar fractal resonators with the main focus on the Minkowski and Hilbert resonators. The details can be followed from the available literature [B.25, J.65, J.68–J.70, J.75–J.80]. Figure (18.36) shows the 5-sided Minkowski prefractal generator as applied to a square patch resonator. Figure (18.41a) shows the change in the resonance

00 Patch

–10 –20 8.5 mm

–30

6.7 mm

–40

Patch

1

5.82 mm 5.82 mm

0 –5 –10 –15 –20 –25 –30 –35

6.7 mm

1

8.5 mm

2

|S21| (dB)

S11 (dB)

3

2

–50 8

9

10 11 12 13 Frequency (GHz)

14

15

(a) Effect of increasing iterations on the resonance frequency of a patch resonator[B.25]. Figure 18.41

5

10

15

20 25 Frequency (GHz)

30

35

40

(b) Effect of increasing iterations on 2nd passband [J.76].

Behavior of microstrip Minkowski fractal patch resonator. Source: From Jarry et al. [B.25]. © 2009, John Wiley & Sons. From Hanna et al. [J.76]. © 2006, IEEE.

683

684

18 Planar Resonating Structures

Table 18.1 Resonance frequency compression of fractal planar resonators.

f2 (GHz)

f3 (GHz)

Δf1/f0 (%)

Δf2/f0 (%)

Δf3/f0 (%)

Dimension

10.18

9.60

9.46

26.2

4.2

1.02

1.893

11.00

10.40

9.80

19.1

4.4

4.4

1.465

4.3

2.90

1.85

–

32.6

24

–

2.000

3.7

2.85

2.58

–

23.0

7.3

–

1.262

Fractal/freq.

f0 (GHz)

f1 (GHz)

Sierpinski

13.8

Minkowski

13.6

Hilbert Koch

frequencies of Minkowski pre-fractal square microstrip patch resonator for 3 iterations. During each of the iteration processes, a portion of the patch is removed, while maintaining the overall patch area. The initial patch resonance frequency 13.6 GHz is reduced at each stage of iteration. The results are shown in Table (18.1). The first iteration reduces the patch frequency to 11 GHz (19.1%), whereas the 2nd and 3rd iterations only add another 4.4% reduction in the resonance frequency. The saturation in the frequency reduction is reached at the 2nd iteration. If it is desired to maintain a constant resonance frequency, with Minkowski fractal geometry, the patch area is reduced, and a compact resonator is obtained. Figure (18.41b) demonstrates this aspect of Minkowski fractal patch resonator, using the shielded suspended microstrip medium. The resonator is designed on an RT-Duroid 5880 substrate with εr = 2.2, tan δ = 0.0009, h = 0.254 mm [J.76]. The unperturbed patch resonance frequency 14 GHz is maintained during both steps of iterations, by reducing the patch area from 8.54 × 8.54 mm2 to 6.76 × 6.76 mm2, and to 5.82 × 5.82 mm2. However, the perimeter increases from 34.16 mm to 40.48 mm, and next to 48.56 mm. In the process, the unloaded Q-factor increases from 353.84 to 424.3, and next to 425.8. At the first iteration, the patch area is reduced by 37.34%, while the perimeter is increased by 18.5%. At the second iteration, the patch area is reduced by an additional 16.21%, and the perimeter is further increased by 23.65%. However, the Qfactor increases by 70.46 at the first iteration, whereas it further increases only by 1.5 at the second iteration. Thus, it is noted that the rate of area saving gets reduced at the second iterations. Despite a long perimeter, the Q-factor comes to almost saturation. Any further iteration reduces the Q-factor. For instance, the 3rd and 4th iterations reduce the area to 5.7 × 5.7 mm2 and 4.74 × 4.74 mm2, and the corresponding Q-factors reduce to 403.4 and 344.1 [B.25]. So, Minkowski fractal patch resonator is useful only up to the second iteration.

Figure (18.41b) also shows suppression of the 2nd harmonics for both iterations, and further appearance of a transmission zero at the second iteration. These features are useful to the design of the bandpass filters. In the WLAN band, the patch area reduction up to 64% is also reported for the Minkowski fractal patch resonator [J.81]. Sierpinski square fractal has also been used to reduce the patch area. Table (18.1) shows the results. Sierpinski square fractal has better frequency compression, as compared to that of Minkowski fractal patch resonator. Koch fractal has been applied to a circular patch resonator to achieve a 60% reduction in the area, while its unloaded Q-factor has increased by 30% [J.75]. It is due to the localization of energy in Koch fractal. Hilbert and Koch’s fractals have also been used to miniaturize the microstrip line resonators. Figure (18.42a) shows the decreasing resonance frequency of the U-type line resonator on applications of Hilbert fractal iterations. The frequency lowering does not saturate for two iterations. Table (18.1) summarizes the results for both Hilbert and Koch fractals. It also shows the dimensions of four kinds of fractal geometries. Let us recall that the dimension is a space-filling ability. The frequency reduction, during the 1st iteration of three cases, almost follows the increasing order of the dimension. The 2nd iteration has more exceptions. However, the increasing dimensional order works for the patch and line resonators differently. Figure (18.42b and c) shows a decrease in the resonance frequency and an increase in the unloaded Q-factor of the superconducting Hilbert resonators for increasing side length. The resonator, shown in Figure (18.37), is developed using the YBCO HTS at 770 K used on a MgO substrate εr = 9.6, h = 0.508 mm, tan δ = 10−6. The external side length (S) is related to microstrip width w, and separation g between the strip conductors, and the number of iterations n as follows [J.69]: S = 2n w + g − g

18 7 8

0

1st

Res. frequency (GHz)

2nd

U-Resonator

18.7 2D Fractal Resonators

S11 (dB)

–10 –20 –30 –40 –50

4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.5

1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 Frequency (GHz) (a) Effect of increasing order of iterations on the resonance frequency of a line resonator[B.25].

n=3

n=4

n=5

3

4

5 6 7 8 External side S mm

9

10

(b) Decrease in resonance freq. of Hilbert resonator with an increased side length. n is the number of iterations[J.69].

106

Unloaded Q-factor (Qo)

n=3 n=4

105

104

n=5

3

5 6 7 8 9 10 External side S mm (c) Increase in Q0 of Hilbert resonator with an increased side length. n is the number of iterations[J.69]. Figure 18.42

4

(i) Microstrip and Hilbert resonator on the same substrate.

(ii) Microstrip and Hilbert resonator at two different substrate. (d) Hilbert resonator and broadside-coupled microstrip and Hilbert resonators[J.78].

Behavior of microstrip Hilbert fractal line resonators – simple and superconducting. Source: From Jarry etal. [B.25]. © 2009, John Wiley & Sons. From Barra et al. [J.69]. © 2005, IEEE. From Crnojevic-Bengin and Budimir [J.78]. © 2005, IEEE.

Like the previous case, for a fixed side length (S), the resonance frequency decreases with an increase in the number of iterations from n = 3 to 5. However, the Qfactor decreases significantly with an increase in the number of iterations, from n = 3 to 5. It is due to the increase in current density at the edges, causing significant conductor loss. The Q-factor also decreases with a decrease in the w/g ratio. Thus, at 2 GHz for n = 3, and w/g = 1.5, 1.0, 0.5, the corresponding unloaded Qfactors are 67 700, 64 200, 56 100. At n = 5, these are reduced to 10 400, 8 700, 6 800 [J.69]. Figure (18.42b) also shows that the frequency compression, with higher iteration, is more effective at the lower value of S, i.e. at the higher resonating frequency, as compared to its effectiveness at the lower resonating frequency.

Finally, Figure (18.42d) shows two configurations of the capacitively coupled 3rd iteration λg/2-Hilbert line resonator on an alumina substrate εr = 9.6, h = 0.625 mm. The resonator width w = 1.96 mm, g = 0.1 mm, and length L consist of several sections (N) of square Hilbert geometry [J.78]. The feeding capacitive gap is 0.2 mm. In the first case, Hilbert line resonator is on the same substrate as that of the feed microstrip. In the second case, Hilbert line resonator is located at another lower substrate, while the upper substrate supports another microstrip resonator and feed lines. Both substrates are alumina, and the separation between the broadside coupled resonators is 0.05 mm. The simulation results for both cases are summarized in Table (18.2). Hilbert resonator has about 84% line length

685

686

18 Planar Resonating Structures

Table 18.2 Simulation results of two configurations of microstrip Hilbert line resonator. Case-1: Hilbert resonator N

3

4

5

6

L (mm)

4.7

6.3

7.9

9.45

fr (GHz)

3.403

2.646

2.162

1.827

S21 (dB)

−1.65

−2.4

−3.32

−4.41

QL

315

441

569

716

Q0

997

1038

1065

1123

Case-II: Coupled Hilbert and microstrip resonators L (mm)

4.7

6.3

7.9

9.45

fr (GHz)

1.884

1.428

1.148

0.960

S21 (dB)

−8.0

−11.0

−13.5

−16.6

QL

1047

1231

1400

1433

Q0

1244

1337

1465

1467

reductions, as compared to a conventional microstrip line resonator. Of course, there is some reduction in Q-factor also for Hilbert resonators. The above discussions on the fractal resonators demonstrate that the fractal geometry can provide significant compactness to the resonators, even with higher Qfactors for a few numbers of iterations. However, such performances can also be obtained from other nonfractal folded geometries. Moreover, the suitability of particular fractal geometry has to be decided by the designer. There is no unique answer to these important questions. It is difficult to arrive at any definite correlation between the geometrical parameters, such as the dimensionality and self-similarity, of the fractal shapes to the electro-dynamical property of the line resonators, i.e. to the resonance frequency, Q-factor, and bandwidth. However, a large volume of literature is available for the design of compact microwave components using the fractal structures.

18.8

Dual-Mode Resonators

The planar symmetrical resonating resonators, a square and circular patch, and their ring versions, always support two orthogonal degenerate modes. The degenerate modes have identical resonance frequencies. However, their field patterns are orthogonal to each other. Two orthogonal resonating modes act as two independent resonators, without any mutual coupling, although both resonators are packed in the same physical body of a patch. Normally, a physical resonating structure is used as a single-mode device for the development of the

bandpass filters. Several such coupled single-mode resonators are required for the development of a narrowband high selectivity BPF. If we can control the coupling between two degenerate modes, then the number of physical resonators can be reduced to half for the design of a BPF. This arrangement provides a compact highperformance BPF, with low insertion loss. A single cavity with five modes has been also used to design a BPF [J.79]. This section presents the basic functioning of the dualmode resonators – both in form of the patch and ring. The concept of the dual mode also accommodates the SIR and the fractal resonators discussed previously. 18.8.1

Dual-Mode Patch Resonators

Figure (18.43a and b) shows the square microstrip capacitively coupled patch resonators, excited from the horizontal side and vertical side, respectively. The square patch supports the TM10 and TM01 dual degenerate modes at the same resonant frequency; their fields are mutually orthogonal. The arrow shows the current on the patch. Figure (18.43c) shows the excitation of the dual orthogonal resonating modes. Figure (18.43d) shows that each mode could be modeled as an LC resonant circuit, placed orthogonal to each other. The resistance R accounts for the losses incurred in a resonator. The magnetic flux lines from the inductor #1 of one resonator do not couple to the inductor #2 of another resonator. So, the dual orthogonal modes are uncoupled modes. However, to use the dual degenerate modes supporting patch resonator as a two-pole BPF, there must be a coupling between both the modes. Such coupling can be achieved by breaking the orthogonality of the dual modes, using any of the following methods:

•• •

Deforming the symmetrical shape of resonators. Introducing a perturbation in a resonator. Changing the location of the port from its orthogonal position.

All three methods rotate the equivalent resonator circuit from its orthogonal position. It permits coupling between two resonators, i.e. the coupling between the dual degenerate modes. In the process of the coupling, the single frequency splits into two frequencies, corresponding to the even and odd modes. The even- and odd-mode frequencies have larger separation for strong coupling. The coupling coefficient can be computed from two frequency peaks obtained on an EM-Simulator. The process is discussed in section (18.5). We discuss below the dual-mode patch resonators, in the square, circular, and triangular forms.

18.8 Dual-Mode Resonators

C#1

Orthogonal mode#2

R#1 Mode#1 (d) Uncoupled two orthogonal modes.

(g) Dual-mode fractal square [J.84]. Figure 18.43

(b) Excitation of orthogonal mode #2.

(c) Excitation of orthogonal dual modes.

(e) Inductive perturbation.

(f) Capacitive perturbation.

(h) Dual-mode meander line resonator [J.88].

(i) Dual-mode Sierpinski carpet resonator [J.89].

R#2

L#1

L#2

C#2

(a) Excitation of mode #1.

Excitation of orthogonal dual unperturbed and perturbed modes in a square patch resonator. Source: Based Cassinese et al. [J.84]. © 2001, IOP Publishing. Based on Roy [J.88]. Based on Yeh et al. [J.89].

Dual-mode Square Patch Resonator

A slightly deformed square is a rectangle that supports the spitting of dual modes, i.e. the TM10 and TM01 modes. The impedance locus of a corner-fed square microstrip patch resonator, with both side arms = 11.30 cm on a 1/16 Rexolite 2200 substrate (εr = 2.62, tan δ = 0.001) is a circle [J.45]. However, the same square patch deformed into a rectangular patch, with arms 11.30 cm and 11.20 cm, provides splitting of the single resonant frequency of the dual modes. Figure (18.44a) shows a cusp in the impedance locus of the deformed square. It gets evolved into a larger loop, and finally into two circular loci. The unloaded Q-factor of the dual-mode resonator is computed from the impedance loop [J.80]. The splitting of the resonance

frequency can be also achieved by cutting a corner of the square patch [J.38]. The size of the perturbation, i.e. the cut of the corner, influences the frequency of the dual modes. Figure (18.43e and f ) shows perturbations, at 45 offset from its port, in the square patch by removing a portion of the square, and by adding a small patch at the corner. The first one is the inductive perturbation, while the second one is the capacitive perturbation. They produce the inductive and capacitive effects due to the enhanced magnetic and electric field, respectively. So, they determine the inductive and capacitive coupling between the dual degenerate modes [J.74, J.82, J.83]. A patch resonator can handle larger power and has moderate Q-factor (100) without a shield, and 266 with

687

18 Planar Resonating Structures

0

790

–10 805 810 815 820

Amplitude |S21| (dB)

688

Star-type patch 900 MHz

Conventional square patch 2.2 GHz

–20 –30 –40 –50 –60

830

(a) Coupled degenerate modes of rectangular patch (arms: 11.3cm, 11.2cm) [J.45]. Figure 18.44

–70 Transmission zeros 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 Frequency (GHz) (b) Frequency response of conventional and star-type dual-mode resonators [J.82].

Impedance locus of corner-fed deformed square patch and frequency response of star-type dual-mode resonator. Source: From Lo et al. [J.45]. © 1979, IEEE. From Banciu et al. [J.82]. © 2004, IEEE.

a shield. However, its size is large [B.6]. The resonating patch can be made compact by using the meander line or fractal geometry. Figure (18.43g) shows a star-type arrangement of the slots with a square patch resonator [J.82, J.84, J.85]. The patch with central orthogonal slots reduces the resonant frequency by more than 30%. If the slot lengths are unequal, it creates a necessary field perturbation to split the resonance frequency of the dualmode slotted square. An additional 15% frequency reduction is achieved by etching four transverse slots at the center of each arm of the 1/3rd of arm-length. It is a Koch, or a Minkowski, type fractal square patch. The center slotted Koch square has unloaded Q-factor between 180 and 310, depending on the crossed slot length – longer the slot length, higher is the unloaded Q-factor. The direct coupling of the external circuits provides 10% fractional BW, whereas the capacitive coupling provides 1% fractional BW. Several other interesting variations are available for the slotted square patch with side slot-cuts. These structures provide the 2pole BPF having the symmetrical response with transmission zeros on both sides of the passband [J.86, J.87]. Figure (18.44b) compares the typical frequency of a conventional square patch of side arm 20.5 cm on a substrate with εr = 10.8, h = 0.635 mm and the startype dual-mode square patch of the same size. There is a decrease in the resonant frequency from 2.2 GHz to 900 MHz. Two transmission zeros also appear in improving the BPF characteristics [J.82]. Figure (18.43h) further shows a capacitively coupled dual-mode compact 2D square resonator created

through the microstrip meandered line. Its size is 35 × 35 mm. It has two field perturbation patches at the corners. The resonator acts like a 2-pole BPF at 605 MHz, with 27 MHz bandwidth, and 1.6 dB insertion loss. Its return-loss is better than 25 dB, and the out of band rejection is also better than 35 dB [J.88]. Finally, Fig (18.43i) shows a compact dual-mode Sierpinski carpet resonator. The Minkowski island dualmode resonators have also been used to get the wideband BPF [J.89]. Dual-mode Circular Patch Resonator

A circular microstrip patch resonator can also support the orthogonal dual modes – mode #1 and mode #2, traveling in the opposite direction. Figure (18.45a) explains the dual-mode operation [J.81]. A circular patch supports two orthogonal modes #1 and 2 with orthogonal E-field vectors. Again, the circular patch can be loaded with either the inductive or a capacitive perturbation, as shown in Fig (18.45b). The diametrically oppositely placed two perturbations can also be used. An equivalent circuit of the dual resonator is also obtained, where the inductive perturbations are in series with the branch line, and the capacitive perturbations are shunt connected. Four kinds of dual-mode structures are formed out of their combinations [J.81]. Figure (18.45c) shows that in the place of an inductive slit, a circular slot can be used to create the perturbed field [J.90]. Figure (18.45d) shows the frequency response of such a dual-mode resonator at 4.55 GHz designed on a substrate with εr = 9.6, h = 0.8 mm.

18.8 Dual-Mode Resonators Circular patch supporting orthonal modes + + +

E-field

– – – E-field + + +

– – –

Mode #1 Substrate

Peturbation

Peturbation

Orthogonal mode #2

Port#2

Port#2

Port#1

Opposite traveling two modes

(a) Opposite rotating dual modes.

Port#1

(b) Inductive and capacitive perturbations.

ro

45°

50 Ω feed line

Amplitude |S21| (dB)

0 –10 –20 –30 –40 Transmission zeros 50 Ω feed line (c) Circular perturbation [J.90]. Figure 18.45

–50

2

3

4 5 6 Frequency (GHz)

7

8

(d) Frequency response of Fig (c) [J.90].

Dual-mode circular resonators. Source: From Li et al. [J.90]. © 2005, John Wiley & Sons.

The radii of the patch and slot are 22.6 mm and 5.75 mm, respectively. Separation of the circular slot gap, from the edge of the patch, influences the upper transmission zero frequency, and the passband bandwidth of the 2pole BPF. The circular slot can be replaced by several other shapes to get even better responses.

mode is almost unchanged with changing the size of the perturbation, whereas the resonance frequency of another mode increases with a larger size of the cut, i.e. for a smaller value of b.

18.8.2 Dual-mode Triangular Patch Resonator

The equilateral triangular patch resonator, shown in Figure (18.46a), is also a symmetrical structure. However, in the XY-coordinate system, it provides only one mode, and its orthogonal dual degenerate mode cannot be obtained [J.91, J.92]. The orthogonal dual mode can be obtained in the rotated X Y coordinate system. Figure (18.46b) shows the current distribution on the patch for both the mode #1 and mode #2. It also shows the orthogonality of the modal currents. Figure (18.46c) shows an equilateral triangular patch of side arm a = 15 cm, with a triangular cut perturbation, on a substrate with εr = 10.8, h = 1.27 mm. Figure (18.46d) demonstrates the asymmetrical mode spitting, one resonant

Dual-Mode Ring Resonators

The square ring and circular ring resonators have higher unperturbed Q-factors (161 and 167) as compared to the patch resonators. Figure (18.47a) shows a square ring dual-mode resonator. Like a square patch, the unperturbed square ring also supports the uncoupled degenerate orthogonal dual modes due to its symmetry. Any kind of perturbation, either in form of a cut or in form of a patch, at its diagonal corner located at 135 from the port #1, disturbs the modal field. It results in the mode splitting. The following two aspects of a perturbation are examined: (i) effect of the size of a perturbation on the modal frequency splitting and (ii) effect of nature of coupling due to corner cut, and the addition of a small patch at the corner.

689

18 Planar Resonating Structures

Y

X′

X A

Y′

(b) Current distributions for mode #1 and orthogonal mode #2 [J.91].

(a) Equilateral triangular patch and rotated coordinate.

10 b = 11.75 cm

0 Pertubation for dual modes coupling

b

Amplitude (dB)

690

a = 15 cm

11.25 cm

10.75 cm

–10 –20 –30 –40

a

–50 3.8 (c) Triangular perturbation cut for dual modes coupling. Figure 18.46

3.9

4.0

4.1 4.2 4.3 Frequency (GHz)

4.4

4.5

4.6

(d) Control of mode splitting by controlling the height of cut [J.91].

Dual-mode equilateral triangular resonator. Source: From Hong and Li [J.91]. © 2003, IEEE.

Figure (18.47a)) considers a square ring on an RTDuriod substrate with εr = 10.8, h = 1.27 mm. An inner triangular patch of variable size d = 2 mm, 3 mm, 4mm is attached at the corner of the ring. Figure (18.47b) shows its effect on the response at the port #2 [J.93]. Even at d = 2 mm, the degenerate mode frequency breaks into the even- and odd-mode frequencies. Their mutual coupling produces the 2-pole BPF response with a small ripple in the center. Once perturbation size is increased to 3mm, the coupling between dual modes is stronger that gives a larger size ripple, and further separation of two modal frequencies. For 4 mm size perturbation, this effect is enhanced. The data can be generated with the help of an EM-Simulator and the coupling factor is computed. It is needed for the design of a BPF. This has already been discussed in section (18.5).

Figure (18.47c) shows three cases of perturbations: (i) square shape corner-cut, (ii) square shape inner corner attached with the patch, and (iii) square shape outer corner attached with a patch [J.83]. Again, the microstrip square ring is on a substrate with εr = 10.8, h = 1.27 mm. The length and width of conducting track of square arm are 20 mm and 2 mm. The perturbation size is d = 2 mm, and the coupling gap is 0.25 mm. The corner cut increases the perimeter forming a longer current path, enhancing the magnetic field. It results in the inductive coupling between the dual modes. Likewise, the addition of a small corner patch, inner or outer, enhances the electric field. It results in the capacitive coupling, i.e. the electric coupling, between the dual modes. Figure (18.47d) of the case-1 shows that magnetic coupling produces no transmission zero and the BPF response is like a Chebyshev response. However, the

18.8 Dual-Mode Resonators

0 Insertion loss (dB)

20 mm

16 mm

Port#1

d

–5 d = 4 mm –10 d = 3 mm

–15 d = 2 mm –20 1.50

1.65 1.60 Frequency (GHz)

1.55

Port#2 (a) Square ring resonator with triangular patch perturbation.

1.70

1.75

(b) Mode splitting with the size of perturbation [J.93].

Port#2

(1) Inductive perturbation.

Insertion loss (dB)

Port#1

Port#1

Port#1

0

Port#2

(2) Capacitive perturbation.

Port#2

(3) Capacitive perturbation.

(1): Inductive S21

–20 –30 –40 –50

(c) Square ring resonator with square perturbation. Figure 18.47

–10

–60 1.4

S11 (2): Internal capacitive (3): External capacitive 1.5

1.6 1.7 Frequency (GHz)

1.8

(d) Responses of square ring resonators for a different type of perturbations [J.83].

Dual-mode square ring resonators. Source: From Görür [J.83]. © 2004, IEEE. From Hong and Lancaster [J.93]. © 1995, IET.

electric coupling produces two symmetrically located transmission zeros. The inner attached patch of the case-2 moves the response to the right, whereas the outer attached patch of the case-3 shifts it to the left. Thus, the perturbation’s shape and size determine the nature, and also the strength of the coupling. The dual-mode resonator can be analyzed using the transmission line model. An equivalent circuit model of the lumped coupled resonator can also be developed. The dual-mode square ring has also been used to design the CPW-based BPF [J.94]. Figure (18.48a) shows the dual-mode circular ring resonator of diameter 11.6 cm. It is like a square ring resonator on a Polyguide substrate with εr = 2.33 [J.95]. However, in this case, the mode splitting takes place due to the non-orthogonal asymmetrical placement of

ports. Figure (18.48a) also shows the coupled mode’s response at the port #2. It can be controlled by changing the angle of ports placement. Figure (18.48b) further shows that the same ring with a 3 mm wide cut of 4 mm depth. It provides a strong coupling between the dual modes, as two resonance peaks are almost completely separated. The response is controlled by tuning the size of the cut. Finally, Fig (18.48c) shows that the perturbation can also be achieved using the step impedance. The MW and EW can be placed at the plane of symmetry to get even- and odd-mode equivalent circuits, shown in Fig (18.48d). The analysis can be completed, as it has been done for the SIR, discussed in subsection (18.1.4) to get the resonance frequencies [J.25]. Several other variations of the dual mode ring resonator are also available [J.96].

691

18 Planar Resonating Structures

40 Ω

40 Ω

11°

3 mm Port#2

Port#2

Port#1

Port#1

(a) Dual-mode ring resonator using feed asymmetry [J.95].

(b) Dual-mode ring resonator using inductive perturbation [J.95].

Z02 θS θS θL

C

D

Plane of symmetry

Z01

Port#1

692

Z01, θL = π – θS

Z02,θS

Even mode

π/2 D

Port#2

(c) Dual-mode ring resonator using step impedance. Figure 18.48

(d) Equivalent circuits for split modes.

B.7 Simon, R.N.: Coplanar Waveguide Circuits

Books B.1 Chang, K.; Hsieh, L.H.: Microwave Ring Circuits and

B.3 B.4 B.5

B.6

C

Dual-mode ring resonators. Source: From Wolff [J.95]. © 1972, IET. By permission of IET.

References

B.2

Odd mode

Related Structures, 2nd Edition, John Wiley & Sons, New York, 2004. Garg, R.; I. Bahl; P. Bhatia; Ittipiboon, A.: Microstrip Antenna Design Handbook, Artech House, Boston, 2000. Hoffmann, R.: Microwave Integrated Circuit Handbook, Artech House, Boston, 1985. Chang, K. (Editor), Handbook of Microwave and Optical, Vol. 1, John Wiley & Sons, New York, 1989. Lee, K.F.; Dahele, J.S. (Ed. James, J.R.; Hall, P.S.): Handbook of Microstrip Antennas, Vol. 1 IEE Publication, 1989. Hong, J.S.; Lancaster, M.J.: Microstrip Filters for RF/Microwave Applications, John Wiley & Sons, New York, 2001.

B.8 B.9

B.10 B.11

B.12 B.13 B.14

Components and Systems, John Wiley & Sons, New York 2001. Munk, B.A, Frequency selective surfaces: Theory & Design, John Wiley & Sons, New York, 2000. Rizzi, P.A.: Microwave Engineering – Passive Circuits, Prentice-Hall International Edition, Englewood Cliff, NJ, 1988. Pozar, D.M.: Microwave Engineering, 2nd Edition, John Wiley & Sons, Singapore, 1999. Ramo, S.; Whinnery, J.R.; Van Duzer, T.: Fields, and Waves in Communication Electronics, 3rd Edition, John Wiley & Sons, Singapore, 1994. Collin, R.E.: Foundations for Microwave Engineering, 2nd Edition, McGraw-Hill, Inc., New York 1992. Ludwig, R.; Bretchko, P.: RF Circuit Design, Pearson Education Asia, New Delhi, 2000. Rao, N.N.: Elements of Engineering Electromagnetics, 3rd Edition, Prentice-Hall, Englewood Cliff, NJ, 1991.

References

B.15 Staelin, D.H.; Morgenthaler A.W.; Kong J.A.:

B.16

B.17

B.18 B.19

B.20 B.21 B.22

B.23 B.24

B.25

Electromagnetic Waves, Prentice-Hall, Englewood Cliff, NJ, 1994. Elliott, R.S.: An Introduction to Guided-waves and Microwave Circuits, Prentice-Hall, Englewood Cliff, NJ, 1993. Jordan, E.C.; Balmain Keith G. E.: Electromagnetic Wave and Radiating System, Prentice-Hall India, New Dehi, 1989. Edward, T.C.: Foundations for Microstrip Circuit Design, John Wiley & Sons, New York, 1987. Sharma, A.; Khanna, A.P.S.; (Ed. Bahl, I.; Bhatia, P.): Microwave Solid State Circuit Design, John Wiley & Sons, New York, 1989. Schelkunoff, S.A.: Electromagnetic waves, D. Van Nostrand Company, 1943. Balanis, C.A.: Antenna Theory Analysis and Design, 3rd Edition, John Wiley & Sons, 2005. Falconer, K, Fractal Geometry: Mathematical Foundations and Applications, 2nd Edition, John Wiley & Sons Ltd, 2003. Mandelbrot, B.B.: The Fractal Geometry of Nature, Freeman, San Francisco, 1982. Jaggard, D.L.: Fractal Electrodynamics: From Super Antenna to Superlattices, Fractal in Engineering, pp. 204–221, Springer-Verlog, New York, 1997. Jarry, P.; Bennett, P.: Design, and Realizations of Miniaturized Fractal RF and Microwave Filters, John Wiley & Sons Ltd, 2009.

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J.9

J.10

J.11

J.12

J.13

J.14

J.15

J.16

J.17

J.1 Troughton, P.: Measurement techniques in microstrip,

Electron. Lett., Vol. 5, No. 2, pp. 25–26, 1969. J.2 Gopinath, A.: Maximum Q-factor of microstrip

J.3 J.4 J.5

J.6

J.7

resonators, IEEE Trans. Microwave Theory Tech., Vol. MTT- 29, No. 2, pp. 128–131, 1995. Troughton, P.: High- Q-factor resonators in microstrip, Electron. Lett., Vol. 4, No. 24, pp. 520–5216, 1968. Hammerstad, E.O.: Equations for microstrip circuit design, Eu. Microwave Conf. Proc., pp. 268–272, 1975. Krischning, M.; Jansen, R.H.; Koster, N.H.L.: Accurate model for open-end circuit terminations, Electron. Lett., Vol. 17, No. 3, pp. 123–124, 1981. Sengupta, D.L.: Transmission line model analysis of rectangular patch antennas,” Electromagnetics, vol. 4, No. 4, pp. 355–376, 1984. Verma, A.K.; Nasimuddin Accurate determination of input parameters of rectangular microstrip patch antenna on thick substrate and comparison of several

J.18

J.19

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697

19 Planar Periodic Transmission Lines Introduction The unloaded planar lines support the continuous spectrum of wave motion, whereas the reactively loaded 1D periodic lines, also reactively loaded 2D periodic surfaces, exhibit a discrete spectrum of wave motion through alternate passband – stopband phenomena. The stopband frequency region inhibits the wave propagation, thus, creating the electromagnetic frequency bandgap, so the periodic structures are called the electromagnetic bandgap (EBG) structures. These are also sometimes called the photonic bandgap (PBG) structures, using the terminology from optics. These nomenclatures are due to their parallelism with the electronic bandgap in materials discussed using the quantum mechanics-based Brillouin Zone (BZ) [J.1–J.3]. BZ’s description and the language of crystallography are useful for the analysis of the EBG lines and surfaces. The EBG surfaces are discussed in chapter 20. The concept of the periodic lines and periodic surfaces are also associated with the metamaterials lines, i.e. the metalines and metasurfaces discussed in chapter 22. This chapter is concerned with periodic lines, also known as the artificial lines. The periodic lines have the following interesting features:

• • •

Creation of alternate passband and stopband. It is useful for the development of filters and other components. Supporting the slow-waves propagation. It is useful to develop a compact, i.e. miniaturized, microwave devices. Supporting backward wave propagation. It helps to develop a transmission line-based metamaterials.

Objectives

•

To understand the wave propagation on the 1D and 2D lattice structures.

••

To understand the Floquet theorem and space harmonics To discuss the 1D EBG circuit models and realization of the 1D planar EBG in microstrip and CPW.

19.1

1D and 2D Lattice Structures

The crystal structure of the natural materials has a periodic arrangement of atoms and molecules. It provides a periodic medium for the EM-waves propagation. Below the X-rays wavelengths, the medium is homogeneous and reflection and transmission of waves take place. However, at the X-rays wavelengths, the phenomenon of diffraction takes place. The reactively loaded lines and surfaces are viewed as the scaled-up periodic structures that mimic the one-dimensional (1D) crystal structures. A brief account of the X-rays propagation in the lattice structure is helpful to appreciate the EM-wave propagation on the artificial electromagnetic bandgap (EBG)-based lines and surfaces. 19.1.1

Bragg’s Law of Diffraction

The periodic array of atoms is located in the stack of parallel planes. The EM-wave propagating through such a periodic structure gets partially reflected from each level, i.e. the plane, of the three-dimensional (3D) crystal. Figure (19.1a) shows the constructive interference of the reflected waves, from three layers of atoms/molecules, resulting in the formation of a diffraction pattern. It is described by the famous Bragg’s law [B.1–B.4]. Figure (19.1a) shows the rays of incident wave normal forming an angle θ at the location O of the first reflecting plane. The path length A–B–C of the second beam is 2d sin θ. The path length A –B –C of the third beam, with respect to the first beam, is 2 × 2d sin θ. The needed path length is n λ to get constructive interference, i.e. the

Introduction to Modern Planar Transmission Lines: Physical, Analytical, and Circuit Models Approach, First Edition. Anand K. Verma. © 2021 John Wiley & Sons, Inc. Published 2021 by John Wiley & Sons, Inc.

19 Planar Periodic Transmission Lines

the EM-wave can propagate for a range of frequency, with the corresponding wavenumber, within the BZ. Only at the edge of the BZ, i.e. at the Bragg’s plane (BP), reflection occurs. These concepts are useful to the modern 1D and 2D EBG structures, described in the language of the crystal lattice. Therefore, it is appropriate to discuss briefly the crystal lattice structure.

in-phase interference of beams. So the Bragg’s law of diffraction can be written as follows: 2d sin θ = n λ, n an integer wavenumber, k = 2π λ

a b, 19 1 1

where d is the separation between two reflecting planes. Figure (19.1b) shows an arrangement of the scatterers, i.e. molecules, at equal distance d to create the 1D periodic structure. Figure (19.1b) also shows the reflecting planes, known as the Bragg’s planes (BP), satisfying the condition given by equation (19.1.1a). For the normal wave incidence, the angle of incidence is θ = 90 . The following expression is obtained for the distance between the reflecting planes: d=n

λ 2

a,

nπ , d

k=

n = 1, 2, …

19.1.2

Crystal Lattice Structures

An artificial crystal structure is a periodic arrangement of scatterers (i.e. inclusions) in one, two, or threedimensional host medium. The crystal structure is described using the geometrical arrangements called the lattice. The identical inclusions, also called basis, are attached at each lattice point. Thus, a physical crystal is formed by a combination of the lattice and basis. This arrangement is called the direct lattice in the real space. The EM-wave propagating in periodic real space is described in the reciprocal space, where the wavevector, i.e. the k-vector, forms its own lattice. Both the direct and reciprocal lattices are correlated [B.1].

b, 19 1 2

where k is the propagation constant, i.e. the wavenumber of the incident wave. The reflected wave is discretized by the periodic medium. Thus, the incoming continuous incident wave acquires periodicity due to its propagation through a periodic medium. Figure (19.1b) shows a periodic arrangement of the atoms/molecules in the real space, i.e. the direct space, with a periodicity d. It is also called the lattice constant d. A pair of the Bragg’s plane defines the unit cell of the periodic structure. The wave phenomenon is described in the wave-vector space, also called the k-space. It is shown in Fig. (19.1c). The wave acquires the periodicity π/d in the k-space. The k-space is further known as the reciprocal space, as its periodicity is reciprocal to that of the lattice constant d of the direct space. The region between the Bragg’s planes in the k-space (−π/d < k < π/d) is known as the Brillouin zone (BZ). So

Direct – Lattice

Figure (19.2a and c) shows the geometrical arrangements of the 2D and 1D direct lattice. The 2D direct lattice is described by two direct primitive vectors

1

1

2

2

θ

3

O

d

They form the following translational vector T to generate the 2D lattice: T = u1 a 1 + u2 a 2 ,

d

d

O

d

d X

(b) 1D lattice in real space.

C

A B Aʹ

3

Reflected wave

Normal

Cʹ

d Vertical Bragg’s planes (BP)

Bʹ AB = BC = d sin(θ) and AʹBʹ = BʹCʹ = 2d sin(θ) (a) Cross-section of the periodic arrangement of reflecting layers.

O –2π d

a 1 , a 2 . The

vectors could be mutually orthogonal or nonorthogonal.

Vertical Bragg’s planes (BP) Incident wave

698

–π Brillouin π d zone (BZ) d

k ( β) 2π d

(c) 1D lattice in k-space (reciprocal space).

Figure 19.1

19 1 3

Wave propagation in the 1D periodic medium.

19.1 1D and 2D Lattice Structures

Unit cell a⃗2

b⃗ 2 a⃗1 b⃗ 1

(b) 2D lattice in reciprocal (k) space.

(a) 2D lattice in direct (real) space. y

Unit cell d

x

d

d

a⃗1 = d

z

x

(c) 1D lattice in direct (real) space.

k b⃗ 1 = 2 π d (d) 1D lattice in reciprocal (k) space.

Figure 19.2 The lattice structures in real and reciprocal space.

where u 1 and u2 are integers. The translation vector T spans the 2D physical space to form the 2D periodic surface. The continuous EM-wave, propagating through the 2D crystal lattice, acquires the periodicity. For instance, the electric field E 0 located at the distance r is expressed as a function of the position vector r , i.e. E 0 r . The position vector r is given by r =x a1 +y a2

19 1 4

Reciprocal Lattice

The EM-wave propagation in a periodic structure has discrete values of the propagation constant. It is described in the 2D and 1D reciprocal lattice space, i.e. in the k-space, as shown in Fig. (19.2b and d), respectively. Figure (19.2b) shows the primitive vectors b 1 , b 2 of the 2D lattice, defining the reciprocal lattice. Each point of the reciprocal lattice represents a set of planes, identified as the Bragg’s planes for the wave reflection. The reciprocal space is mapped by another translation vector G defined as

In a 2D lattice, E 0 r is a periodic function of r with periods a 1 and a 2 along the direction of two crystal axes. Thus, the electric field E 0 r

acquires its perio-

= E0 r

where v1 and v2 are integers. Figure (19.2d) shows the 1D reciprocal space with the 1D reciprocal lattice. In the

19 1 5

primitive vectors, and tive

The electric field at a lattice point of an infinite periodic structure is invariant under the translation T . The 1D direct lattice, Fig. (19.2c), is described by the 1D direct primitive vector a 1 . The direct primitive vectors a 1, a 2

19 1 6

case of a 3D crystal lattice, we have

dicity from the periodicity of the direct lattice: E0 r + T

G = v1 b 1 + v2 b 2 ,

and a 1 , of the 2D and 1D periodic space

respectively also define the 2D and 1D unit cells. Their infinite repetition creates the infinite extent 2D and 1D periodic lattice. The distances a1 and a2 are the lattice constants.

vectors.

b 1, b 2, b 3

The

a 1 , a 2 , a 3 direct

b 1, b 2, b 3 reciprocal

reciprocal primiprimitive

vectors

correspond to the vector components

kx x, ky y, kz z of the wavenumber

k

in the k-space.

However, the real wave propagates in the physical 3D periodic direct space. Both sets of primitive vectors, describing the direct space and reciprocal space (i.e. k-space) are related through the following orthogonality expression: b i a j = 2 π δij =

2 π, 0,

i=j i

j

19 1 7

699

700

19 Planar Periodic Transmission Lines

1D Reciprocal Lattice

Expression (19.1.7) shows that the vector b 1 is normal b 1 a 1 = 2 π,

Figure (19.2c) shows the 1D direct space with the lattice constant (i.e. periodicity) d, giving the direct primitive

b 1 a 2 = 0, b 1 a 3 = 0. Similarly, the vector b 2 is nor-

vector, a 1 = d x. The reciprocal primitive vector b 1 is obtained from equation (19.1.8a) as follows:

to

both

a 2, a 3

mal to both a 1, a 2

vectors,

i.e.

a 1 , a 3 vectors, and b 3 is normal to both

b1 = 2π

vectors. Following this rule, Fig. (19.2b) b 1 , b 2 in the 2D reciprocal space.

shows the vectors

The reciprocal primitive vectors

b1 = 2π b2 = 2π b3 = 2π

a1

a2 × a3 b

a2 × a3 a1 × a2

a1

k-space, so the primitive vector b 1 could be identified with the wavenumber kx or propagation constant βx in the k-space. The x-component of the propagation con-

a

a3 × a1 a1

vector b 1 is also directed along the x-axis, and it is normal to the plane (y–z). It also meets the condition given by equation (19.1.7). As the reciprocal space is also the

a 1 , a 2 , a 3 as follows:

a2 × a3

stant β is βx for the wave motion in the x-direction.

19 1 8

2D Reciprocal Lattice

c

a2 × a3

The 2D direct periodic lattice could be constructed in five different ways, forming five distinct Bravais lattices [B.1, B.2, B.4]. However, the square, rectangular, and hexagonal lattices, shown in Fig. (19.3a and c), are more common formations of the planar EBG structures for the

Expressions (19.1.8a–c) are also applicable to the 1D and 2D lattice structures. This chapter is concerned with the 1D and 2D cases only.

Figure 19.3 Some 2D direct and reciprocal lattices in the k-space.

Unit cell a⃗2

z

y a⃗1x

y

b⃗ 2(ky) b⃗ 1(kx)

x

(a) Direct rectangular lattice for square lattice, a⃗1 = a⃗2.

y o

y z

x

19 1 9

As there is no periodicity along the y- and z-directions, so we have taken a 2 = y and a 3 = z. It is seen that the

b 1 , b 2 , b 3 of the 3D

reciprocal lattice are constructed from the 3D direct primitive vectors

2π y×z x = d dx y × z

B a2 Ax a1 c

150°

(b) Reciprocal rectangular lattice for square lattice, b⃗ 1= b⃗ 2. ky

90° b2 b1

a2 kx

Unit cell

(c) Direct lattice hexogonal.

(d) Reciprocal lattice hexgonal.

19.1 1D and 2D Lattice Structures

microwave applications [B.5]. Figure (19.3b and d) also show the corresponding 2D reciprocal lattices. The 2D crystal structures are located in the x–y plane, and the z-axis, with unit vector, z, and kz = 1 is normal to it.

rectangular unit cell in the reciprocal space. The primitive vectors

b 1 , b 2 are identified with the components

(kx, ky) of the wavenumber vector k in the k-space for the 2D wave motion in the x–y plane.

2D Rectangular Lattice

Figure (19.3a) shows the 2D direct rectangular lattice in the (x–y) plane with the primitive vectors

a 1, a 2 ,

directed along the x-axis and y-axis. The primitive vectors scan the space to form the 2D periodic structure a 1, a 2

of an infinite extent. The primitive vectors

form the rectangular unit cell. The square unit cell is formed for the case a 1 = a 2. The corresponding primitive vectors

b 1, b 2

in the reciprocal space are

obtained from the equation (19.1.8) as follows: b1 = 2π b1 =

a2 ×z a1 2π x a1

= 2π

a2 ×z a,

2D Hexagonal Lattice

Figure (19.3c) shows the direct hexagonal unit cell of the 2D oblique crystal structure. It is formed by the equilateral triangular lattice OAB with a1 = a2 = a. The reciprocal hexagonal unit cell, shown in Fig. (19.3d), is formed by the b 1 and b 2 reciprocal primitive vectors. The reciprocal hexagonal unit cell is rotated by 30 with respect to the direct hexagonal unit cell. The primitive direct and reciprocal translation vectors for the 2D case are given by the following expressions:

a 2 sin 90 x a1 x

a 2 sin 90 x

likewise, b 2 =

2π y a2

b

T = u1 a 1 + u2 a 2

a

G = v1 b 1 + v2 b 2

b

a 3 = z unit vector

c

The primitive vectors b 1 and b 2 are computed in the reciprocal space, using equation (19.1.8), as follows:

19 1 10 The a 3 = z, kz = 1(unit vector), a1, and a2 are the lat-

b1 = 2π

tice constants. The reciprocal primitive vector b 1 is normal to the direct primitive vector a 2 and parallel to a 1. Likewise, the reciprocal primitive vector b 2 is normal to a 1 and parallel to a 2 . Figure (19.3b) shows the 2π

b1 =

2π c, a cos 150

c=

a cos a 1 , c Likewise, b 2 = 2 π b2 = 2π

z× a1

a2 ×z a1

a2 ×z

= 2π

a sin 90 c , ax a sin 90 c

where the unit vector c is normal to the vectors a 2 and z, making an angle 150 with respect to the x-axis. Thus,

= − b1

4π c 3a

a

a1 = ax

19 1 12

a2 z× a1 ay a 2 ay

= 2π

ay

=

a cos a 2 , y

So the vector b 2 is directed along the y-axis. The reciprocal hexagonal unit cell has sides of length 4 π 3a . It is rotated by 30 from the vector a 2 . It is shown in Fig. (19.3d). The same results are obtained by using equation (19.1.7). The primitive vectors

b 1, b 2

are

again identified with the components (kx, ky) of the wavenumber vector k in the k-space.

19 1 11

2πy 4π y = a cos 30 3a

19.1.3

b

Concept of Brillouin Zone

BZ is constructed in the reciprocal space, i.e. the k-space. Bragg’s diffraction occurs at the boundary of BZ. Therefore, the boundaries of the BZ are a pair of parallel Bragg’s planes (BP), acting as the reflecting surfaces. The reciprocal translation vector G, with primitive vectors b 1 , b 2 , b 3 , is identified as the wave vector k with the

701

702

19 Planar Periodic Transmission Lines

b⃗ 1

b⃗ 1 d

d

d O a⃗1= d

d

Figure 19.4 Formation of 1D first BZ.

Г

x

kx( G⃗ ) π O + BP d Bragg plane Г X BZ: Shaded area IBZ: Г to X

π BP – Bragg plane d

(a) 1D direct lattice with lattice constant d.

(b) 1D reciprocal lattice showing the first Brillouin zone.

components (kx, ky, kz) in the 3D space. We discuss below the 1D and 2D BZ, as applied to the 1D and 2D periodic structures, supporting the EM-wave propagation [B.1, B.4].

2D Brillouin Zone

1D Brillouin Zone

BZ for the 2D periodic structure, in the reciprocal space, i.e. the k-space, is constructed through the geometrical process in four steps. Figure (19.5a) shows the direct rectangular lattice ABCD and Fig. (19.5b) demonstrates the formation of the first BZ in the reciprocal space.

Figure (19.4a) shows the 1D lattice with period d along the x-axis. The wave propagates in the x-direction with propagation constant kx. The propagation is described in the 1D reciprocal space, i.e. in the k-space with a prim-

Step #I: A lattice point in the reciprocal space is taken as the origin of the BZ. Step #II: The reciprocal primitive lattice vectors b 1 , b 2 , belonging to the nearest neighboring

itive vector b 1 = k 1 = kx x. Using equation (19.1.7), the wave vector is kx = 2 π/d. It is the period of the 1D reciprocal lattice, shown in Fig. (19.4b). Bragg’s condition, equation (19.1.2), is satisfied at the Bragg’s reflecting planes (BP). Bragg’s planes around any lattice point in the k-space are normal to the axis of the lattice and are located at the bisectors of the reciprocal primitive

points A , B , C , D to the origin, are connected to the origin Γ(0, 0). Step #III: The perpendicular bisectors for the vectors ± b 1 , ± b 2 are drawn. The smallest closed polygon, in the present case, a rectangle, formed is the first BZ. The first BZ, a lightly shaded area, corresponds to the Wigner– Seitz cell of the reciprocal lattice [B.1]. Step #IV: Further, by taking into account the rotational symmetry, BZ is reduced to the smallest primitive unit cell called the IBZ. For the rectangular lattice, shown in Fig. (19.5b), the IBZ is a darker shaded triangular area.

vectors ± b 1 = ± k 1 = ± kx x = 2π d x , so Bragg’s planes are located at b 1 2 = ± π d x . The region between two Bragg’s planes, −π/d ≤ kx ≤ π/d, is the first BZ, and the region Γ ≤ kx ≤ X, with Γ = 0 and X = π/d is called the irreducible Brillouin zone (IBZ) with a minimum distinct dimension that follows the Bragg’s law.

Figure 19.5

ky Dʹ

Y

M D Aʹ a⃗2 A

a⃗1

O a⃗2

a⃗1

B

C

X

–X

Г

X

kx

b⃗ 2 b⃗ 1 Bʹ Ist BZ: Shaded area π π π Г = (0,0); X = ( ,0):M =( , ) a1

(a) Direct lattice.

Cʹ

IBZ

a1 a2

(b) Reciprocal lattice.

Formation of the BZ and IBZ for a rectangular unit cell.

19.2 Space Harmonics of Periodic Structures

Figure 19.6 Formation of BZ and IBZ for a hexagonal unit cell.

ky A 3ʹ

2 M

1

B

K

1

IRZ

3

F 2ʹ

Г 2 C

d

BZ 1ʹ

1ʹ 3

kx

3ʹ E

2ʹ

D Г = (0,0); K = (2π/3d, 2π/√3d); M = (0, 2π/√3d) (b) Reciprocal lattice with BZ and IBZ.

(a) Direct lattice.

Figure (19.6a and b) further explains the formation of BZ and triangular IBZ of the hexagonal direct lattice. The coordinates of the IBZ, obtained from equations (19.1.10) and (19.1.12), are shown in Figs. (19.5b and 19.6b). Figure (19.6b) shows the origin Γ(0, 0) of the lattice in the reciprocal space. The locations A, …, F are the neighboring points of the lattice to the origin Γ(0, 0) with AΓ, …, FΓ connecting lines and 1–1, 1 –1 , etc. their perpendicular bisectors for creating the BZ shown as a light gray hexagon. The triangle ΓKM of darker gray shade is the IBZ. The fundamental wave propagation property, i.e. the (ω − β) or (k − β) dispersion diagram, is obtained from the IBZ unit cell that is periodically extended to the whole of the k-space. In a case of the EBG structure, the eigenmode solver of EM-simulator is used for this purpose.

19.2 Space Harmonics of Periodic Structures Floquet–Bloch theorem provides the periodic boundary condition to solve Maxwell’s wave equation and also the transmission line wave equation, in an infinite extent periodic medium. The solution of the wave equation, in the form of the eigenfunction, provides a discrete solution of a source free periodic medium. The Floquet–Bloch theorem is applied to the 1D, 2D, and 3D unit cell of the direct lattice in the real space. The plane EM-wave propagating through the unit cell acquires its periodicity on the application of the periodic boundary condition, using the Floquet–Bloch theorem. The homogeneous uniform host medium may not have any dispersion. However, a periodic medium, periodically loaded with inclusions, is a

dispersive medium and its propagation behavior is described in the reciprocal, i.e. the k-space. It gives the Brillouin dispersion diagram for the discrete mode of propagation, known as Bloch mode or the Bloch wave [B.3– B.12]. In this section, the Floquet–Bloch theorem is formulated to get the general nature of the Bloch wave in terms of space harmonics. In the next section, it is applied to a cascaded chain of unit cells of the 1D artificial periodic transmission lines. In chapter 20, it is applied to the 2D artificial periodic surface. The EM-simulators also solve the eigenmode problem of the periodic medium under the periodic boundary condition, using the Floquet–Bloch theorem.

19.2.1 Floquet–Bloch Theorem and Space Harmonics Figure (19.7) shows the periodically arranged basis (inclusions) in a 1D host medium. It is a cascading of the unit cells of lattice constant (period) d. The unit cell, between two Bragg’s planes (BP), contains one inclusion. The Bloch wave propagates along the z-axis with the complex propagation constant γ = α + j β0. The fields inside each unit cell are identical and at the boundary of the unit cell, i.e. at z = 0, d, 2d, …, they have the same amplitude. They differ only in the phase at the boundary, due to the propagation factor e−γd. The periodic physical situation has the following two consequences:

•

The field F(x, y, z) is a periodic function of d, i.e. F x, y, z + nd = Fp x, y, z ,

19 2 1

where n is an integer. The subscript p is used to emphasize that the field F(x, y, z) is a periodic one.

703

704

19 Planar Periodic Transmission Lines

Figure 19.7

Unit cell

Cascaded unit cells of infinitely long 1D periodic line.

Inclusion

Host medium

Z=0

•

d

d d

d 2d

3d

nd

The propagation factor e−γd, indicating the phase shift, is associated with the field function at each right-hand side plane of the unit cell, shown in Figure (19.7), with respect to the previous plane.

The above mentioned two properties of the propagating Bloch wave, in the 1D periodic medium, are combined to form the statement of Floquet–Bloch theorem: F x, y, z + d = Fp x, y, z e − γd , where, γ = α + j β0 19 2 2 As the field Fp(x, y, z) = F(x, y, z + d) is a periodic function of d, so it can be expanded in form of Fourier series: + ∞

Fn x, y e − j

Fp x, y, z =

Z

d

2 πn z d ,

19 2 3

medium can support the TEM, TE, TM, or HE mode. If the same medium is periodically loaded, then TEM, TE, TM, or HE modes have their space harmonics. The space harmonics constitute one group or packet with single group velocity to transport energy from the source to the load, say in the positive z-direction. On the other hand, the phase velocity of each space harmonics is different. They are in the ±n pair, i.e. both in the positive z and negative z directions. A periodic structure supports both the forward wave (both group and phase velocities in the same direction) and the backward wave (the group and phase velocities in the opposite direction). These waves are discussed in section (3.3) of chapter 3. The phase velocity vpn and group velocity vg of the nth harmonics are given below:

n= −∞

where Fourier coefficient Fn(x,y) is the field amplitude in the x–y plane. It is a solution of the wave equation in the unit cell [B.6–B.8]. The following 1D periodic function is obtained using the above equations:

vpn =

vg =

ω = βn

ω 2π n β0 + d

1 ∂βn ∂ω

=

1 ∂ω = ∂ β0 + 2π n d ∂ω ∂β0

+ ∞

F x, y, z = e − αz

n= −∞ + ∞

Fn x, y e − j β0 + Fn x, y e − j βn z

2π n d

z

a

n= −∞

where, βn = β0 +

2π n d

b

d

1 F x, y, z ej and, Fn x, y = d

2π n d

z dz

b

19 2 5

F x, y, z + d = F x, y, z = Fp x, y, z e − α + j β0 z F x, y, z = e − αz

a

c

For a slow-wave structure, the +n gives the phase velocity less than the phase velocity in the host medium, say in the free space, i.e. vpn < c (=ω/β0). However, the group velocity is independent of n. For the fast-wave, –n gives us vpn > c (ω/β0) causing Cherenkov-type radiation forming the leaky wave. Cherenkov-type radiation is discussed in subsection (9.7.3) of chapter 9. These are obtained from the Brillouin dispersion diagram [J.4–J.7].

0

19 2 4 If the periodic wave function F(x, y, z) is multiplied by e , then the propagating Bloch wave, in the 1D periodic medium, is obtained. Bloch wave is a superposition of the ±n space harmonics, also called the spatial harmonics. Each space harmonics, with positive n, has its phase velocity in the positive z-direction, whereas, with negative n, each space harmonics has separate phase velocity in negative z-direction. The number of space harmonics with the positive and negative phase velocity is the same. However, space harmonics are not distinct individual modes, like the waveguide modes [B.12]. A uniform host jωt

19.3 Circuit Models of 1D Periodic Transmission Line The transmission line is a homogeneous 1D EM-wave supporting host medium. It can be periodically loaded with the inclusions – the series reactance or shunt susceptance. It can also be loaded with more complex inclusions, such as the L, T, and π-network of lumped reactive elements. Section (3.4) of chapter 3 presents a discussion on the circuit models of the reactance loaded transmission lines, without any consideration of periodicity.

19.3 Circuit Models of 1D Periodic Transmission Line

wave, in the form of voltage/current wave, is supported by an artificial periodic transmission line, with the complex propagation constant γ = α + jβ and the characteristic impedance, called the Bloch impedance ZB. Due to the periodic loading, the propagation constant β of the Bloch wave has to accommodate the propagation constants of the space harmonics βn also. As discussed previously, the propagation constant β of a periodic line is located in the reciprocal space, i.e. in the k-space. The nth unit cell section is described by the [ABCD] matrix of three numbers of the cascaded sections: two sections of length d/2 and one section of the normalized susceptance/reactance. The [A, B, C, D] matrix is discussed in subsection (3.1.3) of chapter 3. The input and output port voltages and currents, shown in Fig. (19.8c), are related as follows:

Several 1D periodic media are also modeled, without solving Maxwell’s wave equation, as the reactance/susceptance loaded artificial transmission lines [B.6, B.9, B.11]. In some cases, even the 2D periodic medium is modeled as the 1D periodic medium [J.8, J.9]. This section is concerned with the propagation characteristics and the circuit models of the 1D periodic transmission lines, loaded with several kinds of reactive/susceptive inclusions. 19.3.1

Periodically Loaded Artificial Lines

Figure (19.8a and b) shows the shunt susceptance and series reactance loaded periodic transmission lines of an infinite extent. The figures also show the unit cells of these loaded lines with lattice constant (periodicity) d. A unit cell is composed of the lumped susceptance (B)/reactance (X) in between two host line sections of length d/2. The characteristic impedance and propagation constant of the unloaded lossless host medium are Z0 and k = ω με, respectively. The normalized susceptance, b = B/Y0 = BZ0, and normalized reactance, x = X/Z0 = XY0 are used as the inclusions. Bloch

d

d Inclusion jB

Z0, k d 2

jX

jB Host medium

In + 1

B D

Vn + 1 In + 1

19 3 1

Unit cell jX

jX = Z

Inclusion Host medium μ, ε

Z0, k μ, ε

d 2

d 2

In

A C

=

The unit cell of a periodically loaded line is modeled as the uniform artificial transmission line of length d,

Unit cell Y= jB

jB

Vn In

d 2

In + 1

In Z

jB = Y Vn

Z0, k

Vn + 1

μ, ε

Vn

Vn

In + 1 A B C D

μ, ε

d 2

d d Unit cell 2 2 (a) Shunt loaded line. In

Vn + 1

Z0, k

Vn + 1

d Unit cell 2 (b) Series loaded line.

In

In + 1

Vn

ZB, γ ϕ = γd

(c) Modeling of unit cell by an artificial transmission line. Figure 19.8 Susceptance/reactance loaded infinite periodic lines, showing the unit cells.

Vn + 1

705

706

19 Planar Periodic Transmission Lines

(ϕ = γ d). It is shown in Fig. (19.8c). It supports the Bloch waves with propagation constant γ and Bloch impedance ZB. Following Floquet–Bloch theorem, the voltage, and current of the Bloch waves, propagating in the +tive zdirection, at the nth and (n+1)th ports are related as follows: Vn + 1 = Vn e − γd Vn In

=

a,

In + 1 = In e − γd

eγd

0

Vn + 1

0

eγd

In + 1

Vn + 1 In + 1

Passband

Stopband

c

In case of a periodic line, the voltage/current V(z)/I(z) is a periodic function F(z). The Floquet–Bloch theorem is applicable in this case. The following expression, giving the eigenvalue, is obtained from equations (19.3.1) and (19.3.2c): B D − eγd

=0

19 3 3

The nontrivial solution of equation (19.3.3) is obtained for the determinant Δ = 0: AD − A + D eγd + e2γd − BC = 0

Stopband

eγd + e − γd A+D = 2 2 A+D a cosh γd = 2 cosh γd = A b

Equation (19.3.4a) applies to an asymmetrical network, and equation (19.3.4b) is for a symmetrical network. These expressions help to compute the complex propagation γ = α + jβ of the Bloch-wave modes provided the elements of the [ABCD]-matrix of the unit cell is known. Once the operating frequency is increased from zero onward, a periodic structure exhibits the passband over a certain frequency range, followed by the stopband over another frequency range. The second passband is followed by the second stopband, and so on. For a circuit arrangement (series capacitor/shunt inductor), the sequence can start with the stopband followed by the passband. For the passband, α = 0, cosh(γd) = cosh(jβd) = cos(βd). In this case, the Bloch wave travels through the periodic line. In the case of the stopband, β = 0, cosh(γd) = cosh (αd). Also at βd = π, i.e. at the Bragg’s plane, we get the stopband. In this case, the Bloch wave is reflected and

b

c

Therefore, the frequency-dependent propagation constant, i.e. the dispersion and loss, of a periodic line is characterized by the unit cell. The dispersion behavior β (ω) is periodic. Next, the task is to evaluate the characteristic impedance, i.e. Bloch impedance, of an equivalent uniform transmission line, i.e. the artificial transmission line. The normalized Bloch impedance (ZB,n) at the nth port of the unit cell is given as follows: ZB Vn + 1 = , Z0 In + 1

ZB = Z0

Vn + 1 In + 1 19 3 6

Equation (19.3.3) provides the following expression: A − eγd Vn + 1 + B In + 1 = 0

e2γd − A + D eγd + 1 = 0,

19 3 4

a

19 3 5

ZB,n =

The reciprocal network meets the condition, AD − BC = 1 [B.13]. Under this condition, the above equation is reduced to

also, for A = D,

A+D , A+D ≤2 2 cos βd = α=0 A+D , 2 cosh αd = A+D >2 β=0 A+D , 2 A + D < −2 cosh αd = βd = π

b

19 3 2

A − eγd C

the periodic line supports only the attenuated evanescent mode. However, the factor (A + D)/2 must meet the following conditions to get the passband/stopband:

19 3 7

Bloch impedance is obtained from equations (19.3.6) and (19.3.7), and for In + 1 0: ZB =

− B Z0 A − eγd

19 3 8

Equation (19.3.4) is solved for eγd: e

γd

=

A + D 2−4

A+D ± 2

19 3 9

Bloch impedance for both the asymmetrical (A D) and symmetrical (A = D) unit cells are obtained from equations (19.3.8) and (19.3.9): case A

D

case A = D

Zasym = B Zsym = B

− 2B Z0 A−D ± B Z0 A2 − 1

a

A + D 2 −4 b 19 3 10

For a lossless periodic structure, the parameter B is an imaginary quantity and in the passband A < 1, leading

19.3 Circuit Models of 1D Periodic Transmission Line

passband and stopband, even if the host line is lossless. Similarly, the characteristic impedance of a normal line is either frequency-independent or mildly frequency-dependent. However, Bloch impedance, i.e. the characteristic impedance of a periodic line, is highly dispersive giving real value in the passband and an imaginary value in the stopband. Despite these differences, both lines have some similarities. For the matching condition in the passband, ZL = ZB, and full power is transferred from the source to the load. If the periodic line length is ℓ = mλg/2, i. e. β ℓ = mπ, m = 1, 2, …, the input impedance is ZL, similar to the usual transmission line. The line transformation and design of components in the passband work as usual, giving the compact circuits, as a periodic line can support the slow-wave.

to a real value of the Bloch impedance ZB. The positive ZB is for Bloch wave propagating in the positive z-direction, whereas the negative ZB shows that it is propagating in the negative z-direction. In the stopband |A| > 1, the Bloch impedance is an imaginary quantity. It shows no propagation of the wave in the stopband, due to the reflection. If a finite extent periodic line is terminated in a load ZL, and the unit cell is symmetric, the reflection coefficient at the input port is computed as Γ=

ZL − ZB ZL + ZB

19 3 11

To avoid reflection on a periodic line, ZL = ZB. Further, normally the [ABCD] – parameter of a unit cell is frequency-dependent, giving highly dispersive Bloch impedance. The increase or decrease of ZB with frequency depends on the inductive or capacitive loading of a line. However, at the band edge, ZB = 0, or ∞, leading to high reflection and creation of the stopband. In the above discussion, the periodic line is viewed as a normal homogeneous transmission line with the complex propagation constant γ = α + j β and Bloch impedance ZB, as its characteristic impedance. However, both lines are not similar in their behavior. For instance, the normal lossless line has α = 0 at all operating frequency, without showing any stopband. A periodic line has alternating Shunt Y A B C D Series Z A B C

D

= Y – shunt

= Z – series

cos θ 2

j sin θ 2

j sin θ 2

cos θ 2

cos θ 2

j sin θ 2

j sin θ 2

cos θ 2

×

×

The normalized characteristic impedance and admittance are Z0 = 1 and Y0 = 1. Also, the normalized lumped impedance and admittance are z = Z Z0 and y = Y Y0 = YZ0, respectively. The propagation constant

19.3.2

[ABCD] Parameters of Unit Cell

The [ABCD] – parameters of a unit cell is required, to compute the propagation constant and Bloch impedance using equations (19.3.5) and (19.3.10). Figure (19.8a) shows a unit cell comprised of two host line sections of length d/2 (i. e. θ/2 = kd/2) and a shunt admittance Y as an inclusion. Figure (19.8b) shows the unit cell with a series impedance Z as an inclusion. Using equations (3.1.23)–(3.1.25) of chapter 3, the [ABCD] – matrices of both the cases, using the normalized components, are obtained as follows:

1

0

y

1

1

z

0 1

×

×

cos θ 2

j sin θ 2

j sin θ 2

cos θ 2

cos θ 2

j sin θ 2

j sin θ 2

cos θ 2

a 19 3 12 b

in the host medium is k = ω με = k0 εr. In the case of microstrip, or CPW, as the host medium εr εreff is taken. The electrical length of the unit cell is θ = kd.

Shunt Y A

B

C

D

y y cos θ − 2 2 y cos θ + j sin θ 2

a

z z z sin θ j sin θ + cos θ + 2 2 2 z z z cos θ + j sin θ j sin θ + cos θ − 2 2 2

b

cos θ + j = Y – shunt

y sin θ 2

y y j sin θ + cos θ + 2 2

j sin θ +

19 3 13

Series Z A

B

C

D

cos θ + j

= Z – series

707

708

19 Planar Periodic Transmission Lines

In order to return to Z0 of the host line, the B element of the above [ABCD] – matrix is multiplied by Z0 and C element by (1/Z0). Equating matrix element A from equations (19.3.4) and (19.3.13), the following dispersion relations of the periodic transmissions are obtained: Shunt Y Series Z

y sin θ 2 z cosh γd = cos θ + j sin θ 2 cosh γd = cos θ + j

a b 19 3 14

For the capacitive susceptance and inductive reactance as the loading elements, the normalized values are used as, y = jb = jωCZ0 and z = jx = jωL Z0 . The equations (19.3.14a, b) are reduced to the following dispersion expressions: Shunt Y Series Z

b sin θ 2 x cosh γd = cos θ − sin θ 2 cosh γd = cos θ −

a b 19 3 15

The dispersion and loss of a shunt capacitor loaded line are computed as follows:

sinh αd sin βd = 0

19 3 17

Expression (19.3.17) is the matrix element A of the equation (19.3.13). At any frequency θ = kd is known, as k = ω με = ω c εr , where c is the velocity of EM-wave in the free space. The shunt loading element is b = ωCZ0. Therefore, the above expression gives an infinite number of solutions for βd, when cos (θ) − (b/2) sin θ ≤ 1, i.e. for −1 ≤ [cos (θ) − (b/2) sin (θ)] ≤ + 1. These βn solutions correspond to the propagation constants of the space harmonics. The fundamental propagation constant is computed for the range −π ≤ βd ≤ + π, i.e. within the first BZ, shown in Figure (19.4b). Other solutions are periodic extensions. Stopband: The stopband frequency range has sinβd = 0 βnd = nπ (n = 0, 1, 2, …) and sinh(αd) 0. Therefore, the propagation constants at the lower and upper edge of the first passband are β = 0, and β = π/d. The range 0 ≤ β ≤ π/d defines the IBZ, shown in Figure (19.4b). The β = π/d also determines the lower edge of the first stopband with α 0. The attenuation constant in the stopband is computed from the following expression:

19 3 18

b sin θ 2

On separating the real and imaginary parts, the following expressions are obtained: cosh αd cos βd = cos θ −

b sin θ 2

cosh αd = cos θ − b 2 sin θ ≥ 1

cosh αd + jβd = cosh αd cos βd + j sinh αd sin βd = cos θ −

cos βd = cos θ −

b sin θ 2

a b 19 3 16

Expressions (19.3.16) provide the passband and stopband characteristic impedance of the periodic line. Passband: For sin βd 0, sinh(αd) = 0, therefore, in the passband β 0 and α = 0. The following dispersion expression for the propagating wave in the passband is obtained for α = 0:

Expression (19.3.18) is the matrix element A of equation (19.3.13). The attenuation constant α has only one solution, as cosh (αd) is not a periodic function. For the +z direction going wave, the attenuation constant is α > 0, and for the negative z-direction going wave, the attenuation constant is α < 0. It is noted that at the band edge, βd = π, d = λg/2. It shows that the loading elements are λg/2 apart and the load appears to be connected directly to the source with β = 0. The above analysis is valid for the series impedance loaded line also. The b is replaced by x = ωL/Z0. The dispersion relation (19.3.17) can be recast in the following compact form, applicable to both cases of the period loading:

cos βd = cos kd − p kd sin kd = A vph CZ0 c for shunt capacitor p = , vph = 2d εr vph L , for microstrip CPW, εr εreff for series inductor p = 2Z0 d Z0 εr = 1 Z0 ε r = εreff , for microstrip CPW, εr εr

a b c d

19 3 19

19.3 Circuit Models of 1D Periodic Transmission Line

In equations (19.3.19a–d), Z0 is the characteristic impedance of the host transmission line. The matrix element A belongs to the transmission matrix [ABCD]. The parameter p could be treated as a loading factor of the reactively loaded line. The loading factor p depends on the type of loading element and its series/parallel connection. For p = 0, there is no periodic loading and (19.3.19a) for the loaded line is reduced to the unloaded line with β = k. The β is propagation constant of a periodic line and k is the wavenumber (propagation constant) of the host medium. Bloch Impedance: Once the [ABCD] – parameters of a unit cell are known, Bloch impedance of both cases of the periodic line can be evaluated. For the symmetrical and reciprocal periodic line, (A = D, AD − BC = 1) and equation (19.3.10b) is reduced to B C

Zsym = ± Z0 B

19 3 20

Using the parameters B and C from equation (19.3.13) with the above equation, Bloch impedance for both cases of the periodic line are obtained as sym

= ± Z0

j sin θ + y 2 cos θ − y 2 j sin θ + y 2 cos θ + y 2

Shunt Y

ZB

Series Z

Zsym = ± Z0 B

1 2

a

j sin θ + z 2 cos θ + z 2 j sin θ + z 2 cos θ − z 2

1 2

b

19 3 21 For the lumped shunt connected capacitor (C) and the series-connected lumped inductor (L), the above equations are reduced to Shunt Y

Zsym = ± Z0 B

Series Z

ZB

sym

= ± Z0

j sin θ + p kd cos θ − p kd j sin θ + p kd cos θ + p kd

1 2

j sin θ + p kd cos θ + p kd j sin θ + p kd cos θ − p kd

1 2

a b

19 3 22 The parameter p for both the cases is different and is given in equation (19.3.19).

19.3.3

Dispersion in Periodic Lines

The dispersion behavior of a periodic transmission line, such as a reactively loaded quasi-TEM mode microstrip or a reactively loaded non-TEM waveguide, is obtained through the (ω − β) or (k − β) dispersion diagram. On a periodic line, apart from the fundamental mode, the space harmonics are also present. However, the dispersion diagram of the space harmonics, i.e. the (ω − βn)

diagram, is periodically related to the fundamental mode in the k-space. The range −π ≤ βd ≤ π is the domain of the first BZ. Figure (19.9a) considers a hypothetical 1D lattice, without any inclusion, of the lattice constant (period) d along the x-axis. The period of the propagation vector in the k-space, from the equation (19.1.9), is b1 = β = 2π/d. Figure (19.9a) further shows a pair of reflecting Bragg’s planes (BP) located at βd = ± π, forming the first BZ. The domains of βd, i.e. (0, π) and (0, − π), are symmetrically located, so they contain identical information on the dispersion of a periodic medium. In this way, only the domain (0, π) of βd is important for the dispersion analysis. This smallest domain is called the IBZ. Figure (19.9a) shows the dispersion diagram of a host homogeneous medium, both air and dielectric medium with relative permittivity εr. The vertical axis of the dispersion diagram is kd, where k is the propagation constant in the nonperiodic homogenous host medium. The parameter kd is also treated as the normalized frequency. Thus, in the air medium k0d = π, i.e. ω = πc/d and in the homogeneous dielectric medium, kd = π, i e ω = πc d εr, where c is the velocity of the EM-wave in free space. The horizontal axis is βd. The propagation constant β is the propagation constant of wave in the periodically loaded medium. The frequency can be treated as a function of β, i.e. the dispersion diagram is a plot of the ω = ω(β) function over the IBZ, 0 ≤ βd ≤ π. In the case of an air medium, the dispersion diagram is a k = ± β0, (ω = cβ0) line, known as the light lines. It is shown as a pair of thick black lines in Fig. (19.9a–c). The light lines divide the (k − β) dispersion diagram into four regions. The first and fourth regions support the forward and backward traveling slow-waves, respectively. The second and third regions support the forward and backward traveling fast-waves, respectively. Figure (19.9a) shows the linear dispersion behavior of the hypothetical unit cell, without any inclusion or loading. Its gradient, i.e. the gradient of the light line, at the origin gives the phase velocity vp(air) = c = (ω/β) in the air medium (thick black lines) and vp die = c εr in the dielectric medium (thin grey lines). The condition vp(die) < c provides the slow-waves propagation in the first region. The slow-waves can also propagate in the negative z-direction, i.e. along the −βd direction in the k-space. In the negative z-direction, the phase velocity is negative. Figure (19.9b) is the dispersion diagrams of the microstrip type host medium loaded with the periodic inclusions. It supports the forward traveling slow-wave in

709

19 Planar Periodic Transmission Lines

Light line in air medium BP

3

2

π 3

1

4

4

(a) Hypothetical periodic medium without inclusion.

π βd

BP

2

1

0 (b) Loaded microstrip periodic medium.

–βd –π

kd

–βd –π

BP

Light line in dielectric medium

π

Light line in air medium BP

kd

Light line in air medium

kd

BP Dielectric medium Light line in

π βd

BP

π 3

4 –βd –π Figure 19.9

2

ωc

1

Light line in air medium

710

π βd 0 (c) Loaded waveguide periodic medium.

Dispersion diagram of the fundamental wave of the periodic media.

the region #1 and backward traveling slow-wave in the region #4. Figure (19.9c) is the dispersion diagrams of the waveguide type host medium loaded with the periodic inclusions. It has a cut-off frequency ωc. The dispersion curve is in the region #2 and region #3, supporting the forward and backward traveling fast-wave, respectively, as vp > c. In case, the dispersion diagram of the open periodic transmission line is in the region #2 and region #3, the fast wave may result in the leaky wave, i.e. Cherenkov-type radiation, radiating the power to the free space. In the case of a periodic microstrip, the leaky wave can also radiate in the substrate exciting the surface waves [J.4–J.6]. Figure (19.10) demonstrates the space harmonics dispersion diagram of a hypothetical 1D periodic structure, without any inclusion, in the air medium. The horizontal axis shows the period of the propagation constant of the periodic medium, i.e.βd = βnd = β0 + 2πn/d, due to the Floquet–Bloch theorem. Its first period is 0 ≤ βd ≤ 2π. The Floquet–Bloch boundary condition is satisfied at βd = 0, ± 2π, ± 4π, …. A pair of Bragg’s planes at βd = ± π, defining the first BZ is shown as the shaded area. It contains the dispersion lines of a pair of the forward and backward moving waves. Figure (19.10) also shows the IBZ.

Figure (19.10) shows the forward-moving Bloch waves ( ) in the full lines: A…E, from the left to right. It assumed that the source is located on the left-hand side, so the energy is always transported from the source to load, i.e. the group velocity Vfg is always positive, irrespective of the direction of the phase velocity Vp. It is also assumed that the source is located at the origin 0. The phase – velocity of all space harmonics for (n = 0, +1, +2, …) are positive. The phase velocities of all space harmonics for (n = 0, –1, –2, …) are negative. The waves A and B are the forward traveling forward waves with positive vfg and positive Vp. However, the waves D and E are the forward traveling backward waves with positive vfg and negative Vp. The periodic line also supports the backward traveling Bloch wave ( ), dashed lines: A …D , from the right to left. These are the reflected waves from the inclusions located at the 1D direct lattice points. They carry energy with the group velocity vbg and treated as a negative quantity; so group velocities vfg and vbg are opposite to each other. In the case of a matched periodic line, there is no backward traveling wave on the line. However, in Fig. (19.10), the waves B and C are reflected backward traveling backward waves with negative vbg and positive Vp. Finally, the

19.3 Circuit Models of 1D Periodic Transmission Line

the energy always travels from the source to load, i.e. from left to right giving positive group velocity vg for both the forward wave and backward wave. However, their phase velocities are opposite to each other, positive for the forward wave and negative for the backward wave. Figure (19.9a–c) show that if these waves are outside the light line cone, they are bounded slow-waves. However, if these are inside the light lines cone, they are the fast-waves and may turn into the leaky modes. The backward wave supporting medium is also treated as the metamaterials with both εr and μr as negative quantities. It has an interesting consequence. Suppose a λg/4 line, supporting the forward wave, is cascaded with the same line length supporting the backward wave, i.e. the metamaterials based line. The total line length forms a λg/2 resonator with the zero phase shift at the output, unlike the π phase shift for a normal line supporting only the forward wave. The first section gives the lagging, i.e. −π/2, phase shift, while the next section gives the leading, i.e. +π/2 phase shift, providing overall zero phase-shift at the end of the resonator. Figure (19.12a) shows, over one period 0 ≤ βd ≤ 2π (−π ≤ βd ≤ π), the wave splitting of the shunt capacitors/ series inductors loaded TEM mode line. The dispersion curve starts at kd = 0, i.e. at the normalized first lower cut-off frequency ωLc1 = 0 of the first passband. In the

n=1

Light line in air medium

Vfg

Vbg

4π

Vbg

M1 n=2

n=–2 Vfg

L2 Vfg

Vfg

BZ

Vbg

3π

L1

Vfg

Vbg

Q3

Bragg’s plane

n=0

Bragg’s plane

kd

waves A and D are the reflected backward traveling forward waves with both vbg and Vp negative. An infinitely long periodic line supports an equal number of the forward and backward waves. Even over one period in the k-space, i.e. βd = 2π, one forward traveling forward wave is obtained, and another wave is the backward traveling backward wave. Both these waves interact at locations P1, P2, P3, P4, … for kd = π. They also interact at locations Q1, Q2, Q3, … for kd = 2π, at locations L1, L2, … for kd = 3π, at locations M1, … for kd = 4π. At these locations, both waves get coupled and produce the standing waves on a periodic line. In the case of a real periodic line with inclusions, the waves split into two traveling Bloch waves separated by the frequency bandgap, i.e. by a stopband. In summary, a periodic line supports four kinds of waves: the forward traveling forward wave, backward traveling forward wave, forward traveling backward wave, and backward traveling backward wave. These waves are discussed in chapter 3, section (3.4). Figure (19.11) summarizes the condition of their existence on a line. In the case of the forward wave, the factor (vg × vp) is positive, whereas for the backward wave the factor (vg × vp) is negative. The positive direction of the movement of waves is always from the source to load. Thus, for a real finitelength line, with a source located at the left-hand side,

n=–1

Vfg

n=0

Vbg Light line in Vfg air medium

Vbg

Vfg

Q1

2π

Vbg

Q2

n=1

n=–1 Vfg

Vbg

P4 D′

E Vfg –4π –βd

–2π

P2 B′

A

0

π IBZ One period

Vfg 2π

(-) Forward moving wave vfg; (--) Backward moving wave vbg .

Figure 19.10

Dispersion diagram of the 1D periodic structure.

C′

B

Vfg

Vbg –π

Vbg

P1

A′

Vfg

Vfg

Vbg π

D

Vbg –3π

Vfg

P3

Vbg 3π

4π βd

711

712

19 Planar Periodic Transmission Lines

IBZ region, the dispersion curve increases with frequency and bends to reach the maximum value of βd = π, corresponding to the first upper cut-off frequency ωU c1 . , supports the forward The first passband, ωLc1 to ωU c1 slow-wave propagation in the forward direction. It is Bloch mode #1. Figure (19.12a) also shows that the second passband, supporting Bloch mode #2, starts at the βd = π, corresponding to kd = π. Bloch mode #2 is the backward wave that is partly fast-wave within the light-line cone and partly slow-wave outside the light-line cone. The normalized lower cut-off frequency at βd = π is ωLc2 = ω d c = π. The upper cut-off frequency of the second passband occurs at ωU c2, corresponding to βd = 0, or 2π. There is a stopband for Bloch wave over the frequency L band from ωU c1 to ωc2. The stopband supports the evanescent mode with the attenuation factor α, and β = 0 or π/d. It is noted that the propagation constant is symmetrical over the IBZ range 0 ≤ βd ≤ π and −π ≤ βd ≤ 0. Over −π ≤ βd ≤ 0, the replica supports the backward moving slow-waves in the first passband. For a matched terminated line reflected backward moving wave is absent. The dispersion is computed only over the IBZ. Figure (19.12b) shows the dispersion behavior of the series capacitors/shunt inductors loaded TEM type host transmission line. In this case, Bloch mode #1 inside the passband #1 is a backward wave, and mostly the fast-wave with the lower cut-off frequency at ωLc1 and the upper cut-off frequency at ωU c1 , corresponding to kd = π, i.e. for βd = 2π, i. e. 0 at Bragg’s plane. However, the Bloch mode #2 inside the passband #2 is the forward U fast-wave in the band ωLc2 to ωU c2 . Frequency ωc2 corresponds to kd = 2π, for βd = π at Bragg’s plane. Again, two Bloch modes are separated by a bandgap. The series

Vg Forward moving backward waves

Forward moving forward waves

Vp Backward moving forward waves

Figure 19.11

Backward moving backward waves

Conditions for the existence of four kinds of waves on a periodic structure.

capacitor/shunt inductor loaded line, supporting the backward wave propagation, is metamaterials-based. Finally, Fig. (19.9c) shows the dispersion behavior of the waveguide medium with capacitive loading. The host waveguide has a cut-off frequency ωc at ωLc1 for the fundamental TE10 mode. The dispersion behavior is obtained for the waveguide periodically loaded with the capacitive irises. In the IBZ, Bloch mode #1 inside the passband #1 is the forward fast-wave between ωLc1 and ωU c1 corresponding to βd = π. Bloch mode #2 inside the passband #2, between ωLc2 to ωU c2, is the backward fast-mode. Other higher-order Bloch modes can also be obtained. The Bloch modes are always separated by a bandgap. In summary, the propagation constant of the Bloch wave is scanned over the IBZ, 0 ≤ βd ≤ π, the corresponding frequencies are obtained for the propagating Bloch modes #1, 2, 3, …, separated by the nonpropagating decaying evanescent modes in the stopband. It also provides the cut-off frequencies, defining the passband and stopband. Using the symmetry about the kd-axis, the dispersion is obtained in the range,−π ≤ βd ≤ 0. It is a replica of the dispersion in the IBZ. Combined together, the dispersion in the first BZ, −π ≤ βd ≤ + π, for the fundamental Bloch wave is obtained. Using this process, EM-simulator in the eigenmode format provides the dispersion behavior of an infinitely long periodic structure. However, the cut-off frequency, propagation constant, attenuation constant, and also Bloch impedance of a periodically loaded planar transmission line are easily computed with the help of the transmission line model, discussed in the next section. The above discussion is centered around the fundamental mode of the space harmonics of the Bloch mode in the IBZ. In the passbands, all the space harmonics of Bloch modes travel at the same group velocity; however, at different phase velocity. At any frequency, the propagation constants βn of all space harmonics are obtained, if the β of the fundamental mode in the IBZ is known. Figure (19.12c) shows Bloch modes #1, 2, with the space harmonics of the periodically loaded rectangular waveguide with capacitive irises.

19.3.4

Characteristics of 1D Periodic Lines

This subsection discuses both the dispersion and Bloch impedance of the periodic lines. The dispersion characteristics of the fundamental Bloch mode in the passband are computed using equation (19.3.19) for 0 ≤ βd ≤ π over the IBZ. Beyond IBZ, for other periods, the dispersion is obtained as the periodic extension. The attenuation factor αd in the stopband is computed using

19.3 Circuit Models of 1D Periodic Transmission Line

BP

BP

kd Stopband #2

ωU c2 β=k

ωU c1

vfg

vbg

Passband #1

ωLc1 =0 –π

0

c

ωnor =

Stopband #1

ωU c1

π

Vp > c

vbg Passband #1

vfg

ωLc1 Stopband #1

0

π βd

IBZ

vfg

Stopband #2 ωd

Passband #2

ωLc2 π

Light line in air medium

c

vbg

ωnor =

ωd

vbg Passband #2

ωLc2 vfg

BP

kd

2π ωU c2

–π

0

IBZ

Light line in β=k air medium

BP 2π

π βd

One period

One period (a) Series inductor/shunt capacitor loading.

(b) Series capacitor/shunt inductor loading. IBZ One period

kd (rad)

Light line/air line

4π ωU c2

Passband

ωLc2 2π ωU c1

Stopband

Passband ωLc1 0.0

Stopband –4π

–3π

–2π

–π

0.0

π

2π

3π

4π

βd (rad) (c) Capacitive iris-loaded waveguide. Figure 19.12

Slow-wave/fast-wave supporting periodic TEM-type/waveguide structures.

equation (19.3.18). It is a nonperiodic function. A few loading cases are considered below. Dispersion Characteristics Loading Elements: Series Inductor/Shunt Capacitor

Figure (19.8a and b) shows the periodic loading of lines using the shunt and series-connected inclusions. Equations (19.3.19a–c)provides their dispersion relations. The parameter p determined by the reactive loading of the host line is the loading-factor. Figure (19.12a) shows

similar dispersion behavior for both the loading elements. Figure (19.13a–c) demonstrates computation of the dispersion and attenuation of Bloch modes in the passband and stopband, respectively, using the matrix element A given in equations (19.3.18) and (19.3.19). The computation of the matrix element A is done with respect to kd, i.e. the normalized frequency variable kd = ωnor , ω = ωnor vp d , where, vp = c εr . In a case of the unloaded line, i.e. for the uniform host medium, loading-factor, p = 0. The passband is obtained

713

19 Planar Periodic Transmission Lines

for A = cos (βd), − 1 ≤ A ≤ + 1 and the stopband for A = cos (βd), |A| > 1. The shaded region of Fig. (19.13a) shows the passband. Figure (19.13b) shows the corresponding dispersion diagrams of the propagating Bloch modes #1, 2, 3 in three passbands, for p = 0.0, 0.25 and 0.5, i.e. for no loading, the light and strong loading, respectively. The passband is reduced for the increased loading, i.e. for a higher value of the loading-factor. Further, there is a decreasing bandwidth for the higherorder Bloch modes. Figure (19.13c) shows the computed attenuation factor in the stopband. The higher loading (p = 0.5) provides higher attenuation and wider stopband. However, the upper edge of the stopband, i.e. the lower edge of the passband, remains fixed, as it is decided by the lattice constant d. The stopband bandwidth increases for the higher Bloch modes. The dispersion diagram shows the alternating passband/stopband with the lower and upper band-edge cut-off frequencies, ωLc,n , ωU c,n ; 1, 2, … of the passband. These edge frequencies also provide the band-edge of the stopband.

passband. Therefore, the lattice constant, i.e. the period, d controls the lower edge of the passband, i.e. the upper edge of the stopband. The Upper Edge of the Passband The upper cut-off frequencies of the n number of passbands ωU c,n are obtained at the edge of the IBZ, i.e. for β d = π, 0, etc. The first upper cut-off frequency of the first passband is obtained for A = –1 in equation (19.3.19a). It is rewritten as follows:

cos ωnor − p ωnor sin ωnor + 1 = 0

3π

a, 19 3 23

where ω = kd. For a given value of the parameter p, n number of upper cut-off frequencies are obtained corresponding to the odd-numbered passband, i.e. n = 1, 3, …. Let us illustrate the computation of band structures for p = 2. In this case, by numerically solving equation (19.3.23a), the principal value, i.e. the first normalized cut-off frequency of the passband, is computed U nor ωnor c,1 = 0 9602, giving ωc,1 = ωc,1 c d εr , where c is the velocity of EM-wave in free space. For a line in the air medium, εr = 1. However, for microstrip or CPW relative permittivity is replaced by the effective relative permittivity, εr εreff. Normally, the static value of the effective relative permittivity εreff is used. However, the dispersion in the host microstrip or CPW medium can nor

The Lower Edge of the Passband It occurs for A = ± 1, i.e. at the normalized frequency ωnor = kd = 0, π, 2π, 3π, … corresponding to the IBZ edge, βd = 0, π, 0, π, …. It is noted that βd = 2π is identical to βd = 0 and βd = 3π is identical to βd = π. The lower band-edges of the n numbers of the passband are ωLc,n = 0, πvp d, 2πvp d, etc , where n = 1, 2, … is the order of the

3π

3π

Stopband

p=0 p = 0.25

ωU c3 ωLc3 2π

ωU c2 π

Mode #2

ωU c1 Bloch Mode #1

ωLc1=0

0.0

Passband

ωLc2 π

p = 0.5

2π

Passband Stopband

kd (rad)

Mode #3

kd (rad)

2π

kd (rad)

714

π

Stopband

Passband

Stopband Passband Stopband

Passband Stopband Passband

0

0.0 0.5 1.0 1.5 2.0 2.5 0 1 2 3 2π 3π 0.0 π αd (Np) βd (rad) Value of A (a) Computation of A (b) Computation of dispersion. (c) Computation of attenuation. (.....): No loading (p = 0),(---): Light loading (p = 0.25),(–): Strong loading (p=0.5) (→) part of dispersion shows forward traveling Bloch wave.

–5 –4 –3 –2 –1

Figure 19.13

Computation of dispersion and attenuation of shunt capacitors/series inductors loaded transmission line.

19.3 Circuit Models of 1D Periodic Transmission Line

be easily accounted for. If the period of inclusion is d = 1 cm and εr = 1, then f U c,1 is 4.5846 GHz. The first lower cutoff frequency of the first passband ωLc,1 is 0. Thus, the bandwidth of the first passband is

L fU c,1 − f c,1 =

4.5846 GHz. Likewise, the lower frequency of the second passband occurs at kd = ωnor = π, i.e. for ωLc,2 = π c d εr. For the present case, f Lc,2 is 15 GHz. The upper cut-off frequency ωU c,2 of the second passband is computed for A = 1, i.e. βd = 0 from the following equation: cos ωnor − p ωnor sin ωnor − 1 = 0

b

corresponding to βd = 0. For p = 1, using equation (19.3.24a) ωLc,1 is 1.3065. The upper cut-off frequency nor ωU c,1 of the first passband occurs at ωc,1 = π. Likewise, L the lower cut-off frequency ωc,2 of the second passband occurs at βd = π. Using the above equation, it is determined as ωLc,2 = 3 6773 . Its upper cut-off frequency occurs at ωU c,2 = 2π = 6 2831 . For the assumed loading factor p, the value of the series capacitor/shunt inductor is computed, if the characteristic impedance Z0(εr) of the unloaded host line is known. The cut-off frequency in GHz is easily computed, provided the period d, and εr or εreff are also known.

19 3 23 The numerical solution of equation (19.3.23) provides the upper cut-off frequencies for the even-numbered passband, i.e. for n = 2, 4, …. For the present U case, ωnor GHz. The 2 = 3 431 , giving f c,2 = 16 382 bandwidth of the second passband is U L f c,2 − f c,2 = 16 382 − 15 000 = 1 832 GHz. In a case of the loading factor p = 0.5, the first lower cut-off frequency is ωLc,1 = 0. The normalized first upper cut-off frequency ωnor c,1 obtained from equation (19.3.23a)

Loading Elements: Shunt Resonant Circuit/Series Resonant Circuit

is 8.216 GHz, that is 1.7207. For d = 1 cm and εr = 1, is the bandwidth of the first passband also. As computed, the f Lc,2 = 15 GHz and ωnor c,2 = 4 0457 for the upper band

Figure (19.14a) shows the side view of the 2D EBG structure, between a two-dielectric layered parallel plate waveguide, supporting the quasi-TEM mode. It is used to suppress the simultaneous switching noise (SSN) in the high-speed PCB technology [J.9]. The structure is modeled through the 1D unit cell, shown in Fig. (19.14a), using a shunt connected series L1C1 resonant circuit. The C1 capacitance is realized by the parallel plate capacitor of thickness h2 and relative permittivity εr2 between the top metal conductor and the patch. The grounded vias contribute to the series inductance L1. The parallel plate line has characteristic impedance Z0(εr) and wavenumber k. The shunt susceptance of the series L1C1 resonant circuit is

fU c,1

edge of the second passband. It gives f U c,2 = 19 373 GHz and 4.373 GHz wide second passband.

Loading Elements: Series Capacitor/Shunt Inductor

In this case, the host line is periodically embedded with the series capacitors or the shunt inductors. For the series capacitor, the normalized reactance is z = jx = − j ωCZ0, and for the shunt inductor, the normalized susceptance is y = j b = − jZ0 ωL. The dispersion relation, using equation (19.3.19), of these loaded lines is written as follows: cos βd = cos ωnor + p ωnor sin ωnor for the series capacitor for the shunt inductor

p = d 2vp CZ0 εr p = d Z0 εr 2vp L

a b c, 19 3 24

εr where ωnor = kd, vp = c εr , Z0 εr = Z0 εr = 1 and in a case of the microstrip/CPW εr εreff. Figure (19.12b) shows the dispersion diagram, i.e. Brillouin diagram. It starts with the first stopband, at zero lower cut-off frequency. Its upper cut-off frequency is the lower cut-off frequency of the first passband, i.e. ωLc,1

Sometimes, the reactance/susceptance loading elements are replaced by the LC series resonant circuit or by the LC parallel resonant circuit. Such loadings are involved in the modeling of the 2D planar EBG structures, and even in the modeling of the stub loaded 1D periodic line. Two cases are discussed below: i) Shunt-connected series resonant circuit loaded line

1 + j ωL1 , j ωC1 j ωC1 , where ω0 = Y= 1 − ω ω0 2 Z=

1 L1 C 1

19 3 25

In equation (19.3.25), ω0 is the resonance frequency. At ω < ω0, Y is capacitive, at ω = ω0, Y = ∞, and at ω > ω0, Y is inductive. At frequency ω < ω0, βd follows the shunt capacitance loading. However, at a frequency ω > ω0, it follows the shunt inductor loading in the IBZ range of a unit cell. The dispersion relation, from the equation (19.3.19a), is obtained:

715

19 Planar Periodic Transmission Lines

cos βd = cos kd − p kd sin kd ≤ 1 vph C1 Z0 εr c where, p = 2 , vph = εreff 2d 1 − ω ω0 cosh αd = cos kd − p kd sin kd > 1

shows the attenuation in stopband for better clarity. The dispersion is shown over IBZ, i.e. in the range 0 ≤ βxd ≤ π. For the period d = 5.59 mm, the range is [0, 562] rad/m. The resonance frequency of the series L1C1 resonant circuit is 3.65 GHz. In the first passband, from f Lc1 = 0 to f U c1 = 2 GHz at βd = π, the loading is capacitive. The lower edge of the first stopband is at 2 GHz and its upper edge is at the lower edge of the second passband at f Lc2 = 13 1 GHz , giving 11.1 GHz wide stopband. However, within the stopband itself, high attenuation occurs due to the resonance at 3.65 GHz. Above the frequency 3.65 GHz, shunt loading is inductive. The second

a b c 19 3 26

In expression (19.3.26), the frequency-dependent shunt loading capacitance C = C1/(1 − (ω/ω0)2) is obtained from equation (19.3.25). Figure (19.14b) shows the dispersion diagram (black line) in the passband for Bloch mode, and also the attenuation (gray line) in the stopband for a periodic line with such loading. Figure (19.14c) again

g

C1

h2 z

εr1

E H

d

s

εr2

h1

kx 2a

L1 C1

x

y

h

Z0,β

Z0,β d 2

d 2

(a) 1D structure and unit cell. BP

40

αx

80

IBZ

30

Passband

βx

27 25

in htl Lig

αx 20 18 15 13.1

Attenuation per unit cell (dB)

35

Frequency (GHz)

716

e

βx Passband

10

αx

5 3.65 2

3.65GHz

70 60 50 40 30 20 10

0

100

200

300

Passband βx 400 500

600

700

Propagation constant (βx) rad/m and attenuation constant (αx) Np/m (b) Dispersion diagram over the IBZ [J.9] (Reproduce from [J.9],©2005 IEEE). Figure 19.14

800

0

0

5

10

15 20 25 30 Frequency (GHz)

35

40

(c) Attenuation diagram [J.9].

Dispersion and attenuation diagrams of the periodic line with shunt-connected series resonant circuit. Source: From Rogers [J.9]. © 2005, IEEE.

19.3 Circuit Models of 1D Periodic Transmission Line

passband ends at its upper band edge at f U c2 = 18 GHz corresponding to βd = π, giving 4.9 GHz bandwidth. The second stopband starts at 18 GHz and ends at 27 GHz, i.e. the third passband starts at 27 GHz and ends with an upper cut-off at f U c3 = 35 GHz. The edge frequencies are marked in Fig. (19.14). ii) Series connected short-circuited stub loaded line Figure (19.15a) shows the series-connected shortcircuited stubs of length ℓ. The stubs periodically load the transmission line with a lattice constant d. The characteristic impedance and propagation constant of the host line are Z0(εr) and k, respectively. The characteristic impedance and propagation constant of the stub are Zd(εr) and k1. The short-circuited series stub could be modeled as a series-connected parallel resonant circuit. It is discussed in section (17.3.1) of chapter 17. The equivalent lumped circuit element could be obtained. It offers series load Z1. At a frequency below the

resonance frequency, i.e. ℓ < λg/4, the stub offers a series inductive loading to the line, where λg is the guided wavelength of the stub. At resonance, ℓ = λg/4, there is a sudden change in the type of loading, from inductive to capacitive, offering a change in βd by π. The attenuation is also very high due to the open-circuited loading. Above the resonance frequency, i.e. λg/4 < ℓ < λg/2, the line is loaded with a series capacitor. At ℓ = λg/2, series resonance occurs. The cycle is repeated at high frequency. In the case of a planar line, normally shunt connected open-circuited stubs are used for the periodic loading [B.13–B.15]. An open-circuited stub is modeled as the shunt connected series resonant circuit offering a capacitive loading below the resonance. It is again discussed in section (17.3.1) of chapter 17. Using equation (19.3.14b), the following dispersion and attenuation expression are obtained for the short-circuited seriesconnected periodic line shown in Fig. (19.15a):

Z1 = j Z01 tan k1 ℓ , where, k1 = 2π λg

Series connected load

a

cos βd = cos kd + j Z1 2Z0 sin kd cosh αd = cos kd + j Z1 2Z0 sin kd > 1

In case, the line is in the air medium, we have εr = 1, k = k1 = k0. Figure (19.15b) shows the dispersion behavior of the periodic line, in the air medium, with the characteristic impedance of stub Z01 = 2Z0 and ℓ/d = 1.2 [J.7]. The first passband is governed by the inductive loading between ωLc1 = 0, and ωU c1 at βd = π. The stopband U L starts at ωc1 and ends at ωc2 for βd = 2π. At βd = π, the series-connected parallel resonance occurs at a

frequency ωr1 and βd is changed by π. The electrical length βd is jumped to 2π. The high attenuation at frequency ωr1 occurs within the stopband itself. The stopband continues up to ωLc2 , i.e. at the edge of the second passband at βd = 2π. In the second passband, ωLc2 − ωU c2 , the short-circuited stub offers the seriesconnected capacitor that decides the nature of dispersion. The series resonance occurs at ωr2.

Light line

SC Stub

Loading

Zd (εr)

π

19 3 27

b c

ωU c2

ℓ

d L

kd

Z0 (εr) Host line

Passband Stub length λg/2

No Vp = c loading

ωLc2 ωr ωU c1

Jump in β by π

Stopband

Stub length λg/4

Cut-off C

(a) Series connected stub. Figure 19.15

Passband ωLc1 = 0

0

π

2π

βd (b) Dispersion diagram.

Nature of dispersion in the series-connected short-circuited stub-loaded periodic line.

3π

717

19 Planar Periodic Transmission Lines

Figure (19.16a) also suggests that in the case of the odd-numbered passbands, the Bloch impedance is zero at the band edge and its mid-band value decreases with increasing order of the odd Bloch modes. In the case of the even Bloch mode, Zsym B is infinitely large at the band edge. The middle frequency band value increases with the increasing order of the even Bloch modes [J.10]. For an asymmetrical unit cell, even in the passband, the Bloch impedance is not a real quantity. It is a complex quantity. However, the propagation constant of the periodic line is not influenced by the asymmetry in the unit cell [B.11, J.11].

Characteristics of Bloch Impedance

Bloch impedance ZB, i.e. the characteristic impedance of a periodic line, is highly dispersive. Its nature of dispersion is decided by the capacitive and inductive loading. Equations (19.3.10a) and (19.3.10b) are used to compute ZB for the asymmetrical and symmetrical unit cells. Using equation (19.3.13), the elements A and B of [ABCD] matrix, of the symmetrical unit cell, are written below: A = cos βd = cos kd − p kd sin kd B = j sin kd + p kd cos kd − 1

a b, 19 3 28

19.3.5 Some Loading Elements of 1D Periodic Lines

where p is the loading-factor is given by equation (19.3.19). Bloch impedance Zsym B of the symmetrical unit cell is a real quantity in the passband, for |A| ≤ 1. For the shunt capacitive loading factor p = 0.1, Fig. (19.16a) shows the frequency-dependent Bloch impedance diagram with four passbands (P), corresponding to four propagating Bloch modes [J.10]. Two modes are separated by a stopband (S), supporting the nonpropagating evanescent mode. At the lower frequency, Zsym starts with 50 Ω, B i.e. the characteristic impedance of the host line. In decreases with frethe first passband, the real Zsym B quency, as the loading element is a shunt capacitor and reaches zero at the upper band edge of the passband is an imaginary quantity #1. In the stopband, Zsym B (shown in dotted line). In the next passband #2, Zsym B is a negative real quantity, showing that the second Bloch mode is the backward wave, as suggested in the dispersion diagram Fig. (19.16b) for p = 0.1.

In the above discussion, the series impedance Z and shunt admittance Y have been taken as the loading elements, i.e. inclusions embedded in the host line. The Z and Y could be a single element reactance/susceptance, or these could be the series/parallel resonant circuit. The L, T, and π networks of the reactive elements, shown in Fig. (19.17a–c), can also be used as inclusions [J.11, J.12]. These networks are also used to model several kinds of loading elements. Figure (19.17d–i) shows a few more loading elements. Some of these elements are useful to realize the metamaterials also, discussed in chapters 21 and 22. We summarize below the [ABCD] parameters of the unit cells of the periodic lines loaded with these inclusions. The L, T, and π networks of the reactive elements are basic structures. By using the capacitors (C) and inductors (L), in the series/shunt, different kinds of inclusions are created. These have different propagation characteristics. The

5

80 S

P 40

S

P

S

P

P

4

0 Loading factor: p = 0,1 P; Pass band S; Stop band

–40 –80

3

Stopband

2

Passband

0 1

2

3

4 kd

5

6

7

8

(a) Frequency-dependent Bloch impedance. Real ZB(-), Img ZB(...). Figure 19.16

Passband

1

–120 0

Loading-factor (p) = 0.1

kd

Bloch impedance (Ω)

718

9

10

0

0.6

1.2

1.8

2.4

3

βd (b) Dispersion for p = 0.1.

Dispersion diagram and Bloch impedance of shunt capacitance-loaded line. Source: From Takagi [J.10]. © 2001, IEEE.

19.3 Circuit Models of 1D Periodic Transmission Line

periodic line can also be formed by cascading these loading elements. Figure (19.17d) shows the T-network type inclusion with series L and shunt C between a pair of host line sections of lengths d/2 (θ/2 = kd/2). It supports the forward traveling slow-wave, whereas a line with L-network type inclusion with series C and shunt L, shown in Fig. (19.17e), supports the backward wave. The transformer loaded line, shown in Fig. (19.17f ), is also used to model the periodic inclusions [J.13]. The step impedance structure, shown in Fig. (19.17g), is also used to get the 1D periodic transmission medium. The L and C of the L, T, and π networks could further be separated by the line sections, shown in Fig. (19.17h and i), supporting the forward and backward waves, respectively. Using the equations from subsection (3.1.3) of chapter 3, the [ABCD] parameters of the T-network, Fig. (19.17a), are obtained as follows: A

B

C

D

=

1 Z 2

1

0

1

Z 2

0

Y 1

0

1

1

For the L-network shown in Fig. (19.17b), these are A

B

C

D

=

1

Z

1

0

0

1

Y 1

=

1 + ZY

Z

Y

1 19 3 30

The [ABCD] parameters of the T-network based unit cell of period d, shown in Fig. (19.17d), are

A

B

C

D

=

cos kd 2

j Z0 sin kd 2

j Y0 sin kd 2

cos kd 2

×

1

j ωL 2

1

0

1 j ωL 2

0

1

j ωC

1

0 1

cos kd 2

j Z0 sin kd 2

j Y0 sin kd 2

cos kd 2

×

19 3 31 1 + ZY 2 = Y

Z/2

Z 1 + ZY 4 1 + ZY 2

19 3 29

Z/2

Z

Z

Y

d (a) Symmetrical T-network. (θ/2) d/2

L/2

Z0 k

L/2

(θ/2) d/2

(θ/2) d/2

C

Z0 k

Z0 k

Z0

Zd

(θ/2) d/2

θ1

Z0 k

Z01

L

n:1

a Z0

(g) Line section loaded line.

Z0 C k

Z0 k

Z0 k

d (θ = kd) (h) Composite shunt C loaded and series L loaded line.

Unit cells of some loading elements (inclusions).

1:n

Z0

θ1 Z01

(f) Transformer loaded line.

Cap loaded Ind loaded line section line section d1/2 d1/2 d2/2 L d2/2 Z0 k

θ

d

(e) L-network (C-L type) loaded line.

d

Figure 19.17

(c) Symmetrical π-network.

d (θ = kd)

(d) T-network (L-C type) loaded line. b

C

Y

d

d (b) L-network.

d (θ = kd)

a

Y

Y

(θ/2) 2L

C

Z0

(θ/2) Z0

d (θ = kd) (i) Composite series C loaded and shunt L loaded line.

2L

719

720

19 Planar Periodic Transmission Lines

The evaluated [ABCD] parameters are summarized below: A = D = 1 − ω2 LC 2 cos kd − ωCZ0 + 2 − ω2 LC 2 B=j

ωCZ20 +j

2 + 2 − ω LC 2

ωL 2Z0

1 − ω LC 2 Z0 sin kd − ωCZ20 2 + 2 − ω2 LC 2 ωL

C = j ωC 2 + 2 − ω LC 2

4Z20

ωL 4

19 3 32

cos kd

sin kd Z0 + ωC 2 − 2 − ω2 LC 2

1 − ω LC 2 2

sin kd 2

cos kd

2

2

+j

ωL 4

2

ωL 4Z20

The [ABCD] parameters of the transformer loaded line, shown in Fig. (19.17f ), are A

B

C

D

=

j Z01 sin θ1

cos θ1 j Y01 sin θ1 ×

cos θ1

1 n 0

cos θ1

0

j Y01 sin θ1

n

n 0

cos θ

0

j Y0 sin θ

1 n

j Z0 sin θ cos θ

j Z01 sin θ1

19 3 33

cos θ1

The [ABCD] parameters of the lossless line section (Zd, kd) loaded line, shown in Fig. (19.17g), are summarized below: A = D = cos kd b cos 2kd −

Z2d + Z20

B = j Z0 cos kd b sin 2kd + j sin kd b

2Zd Z0 sin kd b sin 2kd Z2d Z0 cos 2 kd − Z20 Zd sin 2 kd

C = j 1 Z0 cos kd b sin 2kd + j sin kd b

Z20

Zd cos kd − 2

Zd Z20

19 3 34

2

sin kd

The dispersion relation and Bloch impedance of the unit cell of Fig. (19.17i) are summarized below: A = cos βd = 1 + ZY 4 cos θ + j 1 4 ZY0 + Z0 Y sin θ + ZY 4

a

where, Z = 1 j ωC and Y 2 = 1 jω 2L

b

ZB = ±

2

Y + ZY 4 +

ZY20

Z 2 cos θ + j Z0 sin θ + Z 2 2 cos θ + j Y0 + ZYY0 2 + Z0 Y2 4 sin θ + ZY2 8 − ZY20 2

c 19 3 35

The elements of the [ABCD] parameters are used to get the dispersion relation and Bloch impedance of a periodic line.

19.3.6

Realization of Planar Loading Elements

Figures (19.18)–(19.20) consider some of the structures to realize the capacitor, inductor, and resonating inclusions, respectively, in the microstrip technology.

19.3 Circuit Models of 1D Periodic Transmission Line

The physical size of these elements is within λg/10 to treat them as the lumped elements. The approximate closed-form models are summarized, although more accurate results could be obtained using the EM-simulator and the parameter extraction [B.16]. More elaborate closed-form models of some of these lumped elements are also available [B.17].

Realization of Capacitors in Microstrip

•

Patch capacitor: It is a usual parallel plate capacitor, shown in Fig. (19.18a). It also appears in the process of modeling more complex inclusions, forming the unit cells. It can acquire different shapes, such as square, rectangle, circle, triangle. An accurate computation of capacitance of a patch requires the computation of its main central capacitance, between the patch and ground conductor, and also the fringe capacitance outside the perimeter of the patch [J.14, J.15]. The total capacitance of a rectangular patch capacitor is computed using the following expressions:

T1

S

T2

w

T1

CS

1 Z0 W, εr = 1, h L ε0 εr A − 2 h c Z20 W, εr , h 1 Z0 L, εr = 1, h W ε0 εr A − , Cf2 = 2 h c Z20 L, εr , h

where c is the velocity of the EM-wave in free space. The central capacitance C0 and the fringe capacitances Cf1 and Cf2 p.u.l. of the rectangular patch are computed from the W and L side of the patch. The characteristic impedance of the patch on the air and dielectric substrates are Z0(εr = 1) and Z0(εr), respectively.

•

Gap capacitor: Figure (19.18b) shows a gap S between the microstrip sections. It is modeled as the π-network between the planes of de-embedding T1 and T2. The series capacitance Cs is the main capacitance, whereas the shunt capacitance, Cp is the fringe capacitance between the strip ends and the ground conductor. These capacitances are computed using the following set of equations [B.18, B.19]:

T1

T2 s w

CP

T2

T1

T2 Rs

Cs

T1 Width: w ℓ

T1

T2

Zc, β

Z0

εr

d

(b) Gap capacitor. ℓ T1 T2

Z0 T2

T1 T1

T2 Rs

T1

jX/2

Cs

jX/2

T2

Cp

jB T2

(d) MIM capacitor. Figure 19.18

T1

T2

(c) Interdigital capacitor.

ℓ 0 05 Rs = Rsur ℓ 2 w + t

1 4 + 0 217 ln w 5t , for 5 < w t < 100

In equation (19.3.42), Rsur is the surface resistance, h is the substrate thickness, and the factor Kg accounts for the decrease in the inductance due to the ground plane of a microstrip. The inductance of a microstrip section increases by cutting an aperture in the ground plane, behind the strip conductor.

•

or L = 2 0 × 10

ℓ ln

ℓ+

r2 + ℓ

2

a

r + 1 5 ℓ−

•

Meander line inductor: To increase the line inductance, a longer microstrip is bent to form the meandered microstrip line inductor, shown in Fig. (19.19c). The strip width (w) is normally smaller than the substrate thickness (h). The mutual inductance between the adjacent arms, carrying the oppositely flowing currents, is ignored, if the spacing (s) between the adjacent arms is more than three times of the strip width. However, for a compact meandering, the inductance cannot be computed from the total length of an unfolded line, as the mutual inductance cannot be ignored. The corners of the bends also offer additional discontinuities. A more complex model is available for the computation of the inductance of a meander line [J.17, J.20]. High-impedance line section as an inductor: A short length high-impedance (90 Ω – 120 Ω) microstrip section, between two low impedance line sections, acts as a series inductor. The line section, shown in Fig. (19.19d), is modeled as the π-network

r2 + ℓ

2

19 3 43

b,

[B.16, B.19]. The main inductive reactance, giving the series inductor, and the associated fringe capacitive susceptance are obtained from the following expressions

where r is the radius of via. The substrate thickness is the length of the via. Another expression is also available to compute the wire inductance [B.20].

•

c

The inductance of via (pin): Figure (19.19b) shows a via, i.e. a pin, of the length ℓ and diameter d, connecting the microstrip patch or line to the ground plane. Its inductance in Henry is estimated using the following expression [B.20, J.8, J.19]:

L = 2 0 × 10 − 7 ℓ ln 4 ℓ d + 0 5 d ℓ − 0 75 −7

19 3 42

b

X = ZL sin

2π 2π ℓ L ℓ ≈ ZL λg λLg

where, phase velocity vp = c B 1 π 1 = tan L ℓ ≈ 2 ZL Z λg L

π ℓ λLg

= ωL,

L=

εreff ,

λLg = λ0

ZL ℓ vp εreff

a b c

19 3 44

In equations (19.3.4a–c), ZL is the characteristic impedance of the inductive microstrip section and λLg is its propagation constant. The above approximation is valid for ℓ < λLg 8.

•

Short-circuited stub as an inductor: Figure (19.19e) shows a short-circuited short length stub. It is used as a shunt inductor Lp. For the short-circuited line,

723

19 Planar Periodic Transmission Lines

High-impedance microstrip

50 Ω microstrip

w

s

ℓ

h =ℓ

Aperture in ground plane Ls Rs

εr Via

L w

d (b) Shunt inductor.

(a) Line inductor.

(c) Meander line inductor.

High-impedance microstrip

50 Ω microstrip

w ℓ

di

s ℓ f0, the loading is a series-connected capacitor, and the line supports the fast-waves. The unit cell is simulated on an EM-simulator to get the de-embedded |S21|-response. It provides both the 3 dB cut-off frequency fc and pole frequency f0 in GHz. The plane of de-embedding is at the center of the spur slot, similar to the case of a DGS [J.24]. The inductance

4.0 Passband slow-wave

3.5

Stopband

3.0 2.5

Uniform line

2.0 β α

1.5 1.0

εr = 10.2 h = 0.635 mm

0.5

Bandstop i ii iii

Passband fast-wave

0.0 2

4

Complex characteristics impedance Z*0 (Ω)

However, the center of the stopband is maintained at 10.2 GHz, while the width of the stopband widens. The period, i.e. the lattice constant, d determines the central stopband frequency. The attenuation in the stopband follows the same order. The second passband supports Bloch mode #2 that is a fast-wave, as its SWF is below the value of SWF of the host microstrip line. It is the leaky wave in the form of the surface wave of the microstrip line. Figure (19.27b) shows that Re{ZB(f )} is highly dispersive. In the first passband, its value, from 52Ω at 2 GHz, increases to a very high value, 350Ω at the band-edge. The initial 52Ω is due to the inductive loading. Re {ZB(f )} in the second passband is always below 50Ω. In the stopband Re{ZB(f )} = 0, while Img{ZB(f )} starts from a very high value and decreases to zero. Figure (19.27c) shows the increasing order of attenuation in the stopband for the structures of Fig. (19.26a-i

Complex propagation constant γ = α+jβ

8 10 12 14 16 Frequency (GHz) (a) Complex propagation constant of three type EBG. period d = 5.6 mm.

350 300 Im (Z*0) Re (Z*0)

250 200 150

Structure-iii ii i

100 50 0.0

6

Uniform line 2

4

6

8 10 12 Frequency (GHz)

14

16

(b) Complex Bloch impedance of three type EBG. period d = 5.6 mm.

00 Structure

–05 5-cell MS circuits

–10 Insertion loss (dB)

736

i MoM-SOC Momentum

–15 –20

ii

–25 –30 –35

iii

6 8 10 12 14 16 Frequency (GHz) (c) S-parameters of the complex propagation constant of thee type EBG period d = 5.6 mm. Figure 19.27

2

4

Propagation characteristics of three periodic lines shown in Fig. (15.26a). Source: Reproduced from Zhu [J.52]. © 2004, John Wiley & Sons.

19.4 1D Planar EBG Structures

θs f = Δθ f + θm f 2π Δθ f π 2π + d d= λg 180 λ λ0 Δθ f λ0 λ0 + = λg 360 d λ Δθ f λ0 + εreff SWF f = 360 d

(L) and capacitance (C) of the RLC circuit model are obtained using equation (19.3.49). At the pole frequency f0, R of the model is evaluated as follows: 2 Z0 , where at resonance, Z = R 2 Z0 + Z 1 − S21 f 0 R = 2 Z0 S21 f 0 19 4 4 S21 =

θm f = kd =

and

2π d, λg λ0 λ0 , SWF = εreff λg

where θs f = βd =

2π d, λ

λ=

19 4 5 The λg, λ, and λ0 are the guided wavelength of microstrip with a spur, without spur and in the free space. The periodicity is d. The following expressions are obtained from the above equations:

2.6

Slow-wave factor

Slow-wave zone 2.0 DMS DGS

1.6

Uniform line 1.2

Fast-wave zone

0.8

19 4 6

c

0

bʹ gʹ

–10 |S11| & | S21| (dB)

aʹ DGS

2.4

b

The slowing-factor demonstrates the nature of the Bloch wave propagation, and the S-parameters response helps to get the passband/ stopband characteristics. Let us consider the unit cell for the host 50Ω microstrip on a substrate with εr = 2.33, h = 0.787mm. Figure (19.28a) compares the performance of the spur loaded microstrip, i.e. the DMS, against the DGS loaded microstrip. The dimensions of the spur resonator are a = 5.4 mm, b = 0.5728 mm, g = 0.2 mm, and the DGS has a = b = 1.4405 mm and g = 0.2 mm. Figure (19.28a) compares the slowing-factor of both cases at identical pole frequency f0 = 17 GHz. Both the structures have almost identical behavior. At frequency f < 17 GHz, due to series-connected inductive loading, the propagating Bloch mode is in the slow-wave zone, as the SWF of the periodic line is above the SWF of the host microstrip. However, at frequency f > 17 GHz, the SWF of the loaded line is below that of the host microstrip, giving the fast-wave propagation. Figure (19.28b) shows the S-parameters response of the three spur based unit cells periodic line. It creates a wide stopband, 11.5 GHz–31.75 GHz, and also ripples at the passband edge due to mismatching of Bloch impedance at the band-edge.

The series impedance Z of the spur resonator, forming the unit cell, is located between two microstrip sections of length θ/2 = kd/2. The alternative analysis method is also available [J.53]. The unit cell is used to obtain the (kd − βd) dispersion diagram. However, the slowingwave factor (SWF) of the unit cell can also be obtained from its ∠S21-phase response with spur and without spur, using an EM-simulator as follows: Let θs(f ) and θm(f ) are the phase angles in degree for a unit cell with spur and without spur, i.e. only microstrip line. Due to the loading, θs(f ) > θm(f ), and the differential phase is obtained: Δθ f = θs f − θm f ;

a

Stopband

–20 –30 –40

S11 S21

–50

0

15 17 20 25 30 35 Frequency (GHz) (a) Slow-wave/fast-wave factor of the single unit cell DMS and DGS. Figure 19.28

5

10

0

5

10

15 20 25 30 35 Frequency (GHz) (b) S-parameter response of three unit cells DMS.

Propagation characteristics of the periodic line of Fig. (19.26b). Source: Reproduced from Kazerooni et al. [J.53]. © 2010, John Wiley & Sons.

737

738

19 Planar Periodic Transmission Lines

Figure (19.26c and d) show two more DMS, giving a high value of the loading series inductor to a 50Ω microstrip line. Figure (19.26d) provides a high value of SWF and wider stopband bandwidth [J.54, J.55]. Several other geometrical shapes, including the fractal shapes, have been reported in the literature [J.56].

Bc = ω C =

Figure (19.29a–d) illustrate four cases of the shunt connected capacitively-loaded host microstrip line. The shunt connected capacitors are realized by the straight-line stub, T-stub, or even by the radial stub. A unit cell is comprised of a stub located between two identical length 50Ω microstrip sections [B.13–B.15 J.4, J.6, J.57, J.58]. Bloch wave on these periodic lines, with capacitive inclusions, is a modification of the quasi-TEM mode of the host microstrip line. The propagation behavior is discussed below: Case #I: It uses the single-sided open stub loaded 50Ω microstrip. Figure (19.29a) shows the loaded line, with a unit cell. The stub characteristic impedance Zoc could be 50Ω or less, such as 20Ω. The loading shunt capacitor, for the stub length ℓc < λgc/4, is computed as follows:

• w1

kd/2 kd/2

Case #II: In this case, the single-sided capacitive stubs are replaced by the double-sided stubs, shown in

s ℓ1

– w ℓ

w2 Unit cell

(a) Single-sided open stub loaded microstrip. 3.2 mm

(b) Double-sided open stub loaded microstrip.

1.1 mm 0.42 mm (i)

0.42 mm

0.42 mm 0.42mm 1.6 mm

1.1 mm 0.42 mm

0.42 mm

(ii)

Aperture in ground

0.42 mm

(c) Compact T-type open stub loaded microstrip. Figure 19.29

19 4 7

However, discontinuity can alter the component value to some extent. It can be accounted for by using a better circuit model [B.19]. The periodic structure supports the slow-wave propagation in the first passband. The SWF increases with frequency up to the edge of the stopband, while the Bloch impedance ZB is reduced below 50Ω, and it further decreases with frequency up to the lower edge of the stopband. This frequency-dependent behavior of ZB for a shunt capacitor-loaded microstrip is opposite to that of the series inductor loaded microstrip. It is discussed earlier that the series inductor line also supports the slow-waves; however, its Bloch impedance is higher than Z0 (50Ω) that increases with frequency up to the lower band-edge of the stopband. The second passband of the shunt capacitor-loaded microstrip supports the fast-wave, like a series inductor loaded line, that may cause the leaky wave. It is further discussed for the case-III, shown in Fig. (19.29c). An interesting application of the doubled-sided stub-loaded microstrip is discussed below:

ii) Periodically loaded microstrip (shunt capacitor)

•

1 2 π ℓc tan λgc Zoc

(d) Compact T-type stub and series inductor loaded microstrip.

Some shunt capacitor-loaded microstrip periodic line.

19.4 1D Planar EBG Structures

Fig. (19.29b) [J.58]. The low-frequency behavior of the stub length ℓc < λgc/4 is the same as that of the case-I; however, with increased capacitive loading. At higher frequency, leading to ℓc = λgc/4, the closely placed stubs present the short-circuits, i.e. the electric walls at both sides of the host microstrip. Therefore, over a certain frequency band, around ℓc = λgc/4, the double-sided stubloaded periodic microstrip acts as a substrate integrated waveguide (SIW), described in section (7.8) of chapter 7. In this version of the SIW, the realization of the electric walls does not require any shorting pins or vias, used in the standard SIW. Even a brittle alumina substrate can be easily used to realize the SIW in its newer version. The structure is called the corrugated SIW (CSIW), and the standard one is the pin SIW (PSIW). In the case of the CSIW, we use low impedance, i.e. wide width, microstrip. Its width w2 is computed at the desired center frequency of the CSIW [J.58]. The periodicity of the CSIW, s ≤ λg/8 and the stub width w1, s/w1 2. The stub length is ℓ1 = λg/4 − Δℓ, where Δℓ is the open-end discontinuity of the stub. It is computed using the following expression [B.18]: Δℓ εreff + 0 3 = 0 412 h εreff − 0 258

w 1 h + 0 262 w 1 h + 0 813

19 4 8

λg is taken at the center frequency of the CSIW. Figure (19.30a) shows both the EM-simulated and experimentally obtained S-parameters of a CSIW designed on Taconic TXL substrate – εr = 2.55, tan δ = 0.0019, h = 1.0mm. The w2 = 10.5 mm wide CSIW is 62 mm long. The w1 = 0.8 mm wide stub is ℓ1 = 4.5 mm long and its period is s = 2.0 mm. The 12.2 GHz–17.2 GHz wide 3 dB operating BW is observed for the CSIW. Its insertion loss is better than 1.6 dB, while |S11| is better than 14.0 dB. The lower cut-off region is observed.

•

Case #III: Figure (19.29c) shows periodic microstrip using the double-sided T-type compact stubs. The w2 = 0.42 mm wide microstrip, on a substrate εr = 2.95, h = 0.508 mm, is the host medium. For the light capacitive loading, the period is d = 3.2 mm. Other dimensions are shown in Fig. (19.29c-i). For a compact microstrip periodic structure of four unit cells, the period is reduced to d = 1.6 mm, shown in Fig. (19.29c-ii). In a case of the light loading, the loading capacitors are isolated. However, for the compact case, there is a mutual coupling between the loading

stubs that strongly influences the propagation characteristic, discussed below. Figure (19.30b–d) shows the new Brillouin diagrams of lightly and strongly loaded microstrip with T-stubs [J.4– J.6]. In place of one linear light line, it has three pairs of such lines. The line-1 and its replica are dispersion lines of the unloaded host microstrip medium, supporting the quasiTEM mode with propagation constant, k = k0 εreff . The dispersion line-2 and its replica correspond to the TM0 – surface wave mode, supported by the conductor backed dielectric substrate. The dispersion line-3 and its replica shows free space propagation, i.e. the radiation. The dispersion lines-2 and -3, with β = ± kTM0 and β = k0, respectively, are above the k = k0 εreff line. The modal representation of waves on the new dispersion diagram is different from the standard (ω − β) Brillouin diagram. It shows the passband/ stopband regions with the slow-wave and fast-wave presence. Besides, it also explains the origin of the leaky modes, i.e. conversion of any space-harmonics of the Bloch mode to the surface wave modes-TM0, and other higher-order surface wave modes. It further explains their conversion to the radiation mode in the free space. It is the basis of the leaky-wave antenna that is due to Cherenkov-type radiation discussed in subsection (5.5.7) of chapter 5 and subsection (9.7.3) of chapter 9. The dispersion line-1 divides the Brillouin dispersion diagram into two regions – the bound region, below the line-1, supporting the propagating mode, and the unbound region above it, supporting the leaky mode wave. The bound region has the first passband, supporting the slow-wave Bloch mode #1 and the closed (nonradiating) stopband. In the passband, all components of the space-harmonics have a real propagation constant, kz,n = βn, for no dielectric and conductor losses. It is the slow-wave region, as |βn| > k0. In the case of the light loading, shown in Fig. (19.29c-i), for the period d = 3.2 mm, Bragg’s reflection, corresponding to βd = − π, occurs at f = 30 GHz as shown in Fig. (19.30b-i). It is the upper edge of the closed-stopband. Due to the capacitive loading, the slow-wave supporting passband bandwidth is 0 GHz–18 GHz. The EM-simulated stopband bandwidth is 18 GHz–31.5 GHz. The frequency deviates from an estimated 30 GHz, as the T-stub is not purely capacitive. It requires better modeling. Figure (19.30c) shows the S-parameters of the four unit cells (d = 3.2 mm) periodic line of total length

739

19 Planar Periodic Transmission Lines

Light lines 1

–10

1

–20 –30

|S11| Measured Simulated BW of CSIW

–40

10

8

12 14 16 Frequency (GHz)

Stopband

40

Stopband

20 10

Passband

0 –2 18

–1.5

–1 –0.5 0 0 0.5 βd/π α/ko (i) (ii) (b) Brillouin and attenuation diagrams of Fig. (19.29c-i) [J.6]

20

(a) S-parameters response of CSIW. [J.58].

60

–10

50 Frequency (GHz)

0

Radiation zone

–20 –30 Stopband

Passband

–40

c

Passband 30

–50

|S11| & |S21| (dB)

23 3 2

50 |S21| Frequency (GHz)

Scattering paramters (dB)

0

32

Lines

40

1

Light lines 1 1 2 3

Passband Stopband

30 20 10 Passband

–50 10

0 –2

20 25 30 35 40 Frequency (GHz) (c) S-parameter of four cell filter of Fig. (19.29c-i) [J.6].

15

6

0

5

–10

3

1

Stopband Passband

2

|S11| & |S21| (dB)

4 α/k0

740

Passband

0 20

25

30

35 40 45 Frequency (GHz)

(e) Attenuation of Fig. (19.29c-ii) [J.6]. Figure 19.30

–1.5

–1 –0.5 βd/π (d) Brillouin diagram of Fig. (19.29c-ii) [J.6].

–20 –30 –40 Passband

50

55

0

–50 10

15

20

Stopband

25 30 35 40 Frequency (GHz)

45

50

55

(f) S-parameter of four cell filter of Fig. (19.29c-ii) [J.6].

Propagation characteristics of the periodic line of Fig. (19.29). Source: Reproduced from Liu et al. [J.58]. © 2016,Microwave Optical Tech. Letters. Reproduced from Baccarelli et al. [J.6]. © 2007 ISMOT.

19.4 1D Planar EBG Structures

ℓ = 12.8 mm. It confirms the first passband between 0 and 18 GHz. It shows the stopband, corresponding to |S21| = − 10dB, from 20 GHz to 32 GHz. The transition region is from 18 GHz to 20 GHz. The (ω − β) dispersion diagram is obtained for an infinitely long periodic line, whereas the S-parameter is for the finite length periodic line. The bandwidth of passband/stopband obtained from both methods have some difference, as ideally the (ω − β)-diagram does not show any transition region between the passband and the stopband. Figure (19.30bii) shows the (ω − α) diagram, giving the attenuation in the stopband. In the closed-stopband, space-harmonics propagation constant kz,n is complex. However, the mode of propagation is still the slow-wave, as |βn| > k0. For a lossless microstrip, α = 0 in the first passband. The second passband is from 31.5 GHz to 39.5 GHz. It has two regions. The first one provides the narrow nonradiating passband, shown in Fig. (19.30b), from 31.5 GHz to 33 GHz. It supports the backward fast-wave because |βn| < k0. The second one provides the leaky wave region. Above the crossing-point C, one or more space-harmonics meet the condition βn < k0 < kTM0 and leak to the free space in the form of radiation, or leak to the substrate in the form of the surface-wave mode TM0, etc. The peaks in the response of the S-parameter, shown in Fig. (19.30c), show a presence of the leaky modes. If βnα > 0 for any space-harmonics, the leakage increases exponentially in the vertical direction. If βnα < 0, space-harmonics leakage decreases. The new Brillouin diagram considers a number of the spaceharmonics also [J.4]. Figure (19.29c-ii) shows a very compact three-unit cells periodic line with period d = 1.6 mm. It provides stronger loading. Other dimensions are shown in Fig. (19.29c-ii). Figure (19.30d) shows the new (ω − β)-dispersion diagram with a widening of both the first and second passband, 0 GHz–26.8 GHz, and 39.5 GHz–55 GHz. The stopband starts around 26.8 GHz, (βd = π). However, stopband, supporting the complex propagation constant, is the involved one, as shown in the (ω − α) diagram of Fig. (19.30e). In both stopbands, we have evanescent modes with two different values of attenuation constant α and same βn. Above 40 GHz, i.e. in the second passband, α is very high, extending the stopband, as shown in Fig. (19.30f ). The extended stopband is in the radiating region [J.6]. iii) Periodically modulated strip conductor of microstrip The centers of the multiple stopbands, of a periodic line, are harmonically related. However, on several occasions, for the filter’s applications, only one stopband is

needed, without disturbing the high-frequency band. The sinusoidally modulated slot in the ground plane provides such a response [J.42]. However, it has a packaging problem and its realization in MMIC is difficult. Normally, the modulated strip conductor of a microstrip also provides multiple passband-stopband characteristics [J.59]. However, if the characteristic impedance of a microstrip follows the sinusoidal variation, say 50Ω to Zmin back to 50Ω, and to Zmax and back to 50Ω creating 0 0 a unit cell, we get only one stopband, suppressing other harmonically related stopbands [J.60–J.62]. The modulated strip profile is obtained from the variation in the characteristic impedance. The modulated microstrips also create the EBG high impedance surface, controlling the surface wave propagation [J.63]. Figure (19.31a) shows characteristic impedance variation of a microstrip over one unit cell of period d. Over the first half period, 0 ≤ p ≤ d/2, the characteristic impedance is below 50Ω, and over the next half period, max d/2 ≤ p ≤ d, it is above 50Ω. The Zmin pair is 0 , Z0 shown in Fig. (19.31a). The sinusoidal variation in characteristic impedance is governed by the following expression: Z0 ≤ 50Ω Z0 = 50 − 50 − Zmin sin 2πp d , 0 for 0 ≤ p ≤ d 2 Z0 ≥ 50Ω Z0 = 50 + also,

Z0 =

Zmax 0

max Zmin 0 Z0

a

− 50 sin 2πp d , for d 2 ≤ p ≤ d

b

= 50Ω

c 19 4 9

The variation in the strip width (w/h) at the discrete locations is obtained using the synthesis expression (8.3.38) of a microstrip line given in chapter 8. Figure (19.31b) shows one such structure over one period. The microstrip width W50, WZmin, and WZmax corresponding to the characteristic impedance 50Ω, minimum characteristic impedance Zmin, and maximum characteristic impedance Zmax are shown in Fig. (19.31a and b). Figure (19.31c) shows six numbers of cascaded unit cells on a substrate with εr = 2.2, h = 0.508 mm, t = 0.11 mm. Figure (19.31d–e) shows the S-parameter response for the characteristic impedance pairs (Zmin, Zmax): (25Ω, 100Ω) and (15.5Ω, 161Ω), period d = 20 mm, giving the bandstop at 5.5 GHz. For both cases, the stopband does not occur at the harmonics 11 GHz and 16.5 GHz. Also, the stopband attenuation and bandwidth are more for the second case, due to the large Zmax Zmin ratio. 0 0

741

19 Planar Periodic Transmission Lines

150

WZmin

(15.5 Ω, 161 Ω )

WZmax

Z0 (Ω)

W50

W50

(25 Ω, 100 Ω )

100

(b) Shape of one cell.

(31 Ω, 80 Ω )

50

0 0.0

0.5 Position along one cell (p/d) (a) Variation of characteristic impedance.

1.0

Dielectric substrate Microstrip line (c) Six cells EBG.

0.0

0.0 –10 Stopband

S-parameters (dB)

–10 S-parameters (dB)

742

–20 –30 –40

–20

Stopband

–30 –40 –50 –60 –70

–50

–80 1

3

5

7 9 11 13 Frequency (GHz)

15

17

(d) S-parameter of six cells EBG(25 Ω, 100 Ω). Thick line measured and thin line simulated results on (ε = 2.2, h = 0.508 mm). Figure 19.31

19.4.2

1

3

5

7 9 11 13 Frequency (GHz)

15

17

(e) S-parameter of six cells EBG (Z0 = 15.5 Ω, 161 Ω).

Periodic microstrip of modulated strip conductor. Source: Reproduced from Nesic and Nesic [J.60]. © 2001 John Wiley & Sons.

1D CPW EBG Line

The CPW, like a microstrip, could be also periodically loaded with series inductor/shunt capacitor or shunt inductor/series capacitor to get the periodic CPW structures. Such loading is also done for the CPS structures. Furthermore, these loaded lines are used to get a high Q-factor resonator [J.64]. The loading is carried out both for the central strip conductor and also for the ground conductors. Figure (19.32a–j) illustrates several reactive structures to load the central strip conductor of a host CPW, without disturbing the ground plane conductors. The geometrical structures of reactive loading elements, i.e. the inclusion, have a strong influence on the characteristics of a periodic CPW. The loading elements, in some cases, can be modeled as the lumped elements, using the closed-form expressions [B.21]. Alternatively, the loading elements can be modeled using the T or π-networks.

Their component’s values are extracted from the Z or Yparameters obtained using an EM-simulator [B.16]. Figure (19.33a–f ) show several structures for the shunt-connected stubs. These stubs are accommodated by creating slots in the ground conductors. Only slots in the ground conductors, without any stub, also provide series inductors loading of a CPW. Similarly, the DGS in the ground plane also loads the CPW with the parallel resonant circuit. The open-circuited shunt stub acts as a shunt capacitor, or even as a shunt connected series resonant circuit. The stubs could be connected to either one side or both sides of the strip conductor. The radial stubs are also used to get a compact periodic CPW. Both the central strip and ground conductors can also be reactively loaded. There are several variations in the shape of the loading elements forming the unit cells. This section considers one case each for the series/shunt loading of the

19.4 1D Planar EBG Structures

L

C

C

(a) Shunt capacitor. C

(c) Series capacitor.

(b) Series inductor.

C

L

(d) Series inter digital capacitor.

(e) Series spiral inductor.

L

L

(f) Series MIM capacitor. L

C

(g) Series short-circuited stub (series inductor) loading.

(h) Series combination of L and C using s/c and o/c stubs.

(i) Parallel combination of L and C.

L

(j) Shunt inductor using short-circuited central strip conductor. Figure 19.32

Periodic loading of the central strip of the CPW.

inductor/capacitor of the CPW periodic structures, shown in Fig. (19.34a–d). Series Inductor Loading

Figure (19.34a-i-iii) show three types of series inductor loading of a 50Ω CPW on the substrate with εr = 10.2, h = 0.635 mm – (i) double-sided inductive T-slots in

the ground plane, (ii) double-sided inductive T-slots in the strip conductor, and (iii) double-sided inductive Tslots in both the ground plane and strip conductor. Figure (19.34a) shows three cells of each type of loading with a period d = 5.08 mm, slot dimension Δ = 2.54 mm, and slot gap, g = 0.508 mm [J.65]. The (ω − β) dispersion diagram and Bloch impedance ZB are determined by the

743

744

19 Planar Periodic Transmission Lines

(a) DGS series inductor loading.

(b) Shunt inductor loading using s/c stub.

(c) Shunt capacitor loading using o/c stub.

(d) Series inductor loading using slots in the ground plane.

(e) Shunt radial stub loading.

(f) Double-sided radial stub loading.

Figure 19.33

Patterned ground planes and CPW stubs to form the unit cells.

Type A

Type B s w s

(ii)

g

d

d = 5.0 8 mm Δ = 2.54mm

d = 5.08 mm Δ = 2.54 mm (i)

(i) Z0 β0

Type C

Cg Cp

Cp

(ii) d = 5.08 mm Δ = 2.54 mm (iii)

(b) Series capacitor loading [J.66].

(a) Series inductor loading [J.65].

Unit cell C Z+0

(i) Geometry L

Z–0

ZB βB (ii) Equivalent circuit of unit cell (c) Seven-cells shunt capacitor loading [J.67]. Figure 19.34

(d) Series capacitor and shunt inductor loading [J.68].

Series/shunt-loaded periodic CPW. Source: Reproduced from Zhu [J.65]. © 2003, IEEE. Reproduced from Zhu and Shi [J.66]. © 2005 John Wiley & Sons. Reproduced from Martın et al. [J.67]. © 2003, John Wiley & Sons. Reproduced from Gao and Zhu [J.68]. © 2003, IEEE.

19.4 1D Planar EBG Structures

de-embedded [ABCD] parameters of the cascaded N unit cells: AN

BN

CN

DN

=

cosh γNd =

cosh γNd

ZB sinh γNd

YB sinh γNd

cosh γNd

AN + B N 2

b,

ZB =

BN CN

a c 19 4 10

The [ABCD] parameters are obtained from the EMsimulated de-embedded S-parameters. The passband– stopband characteristics are obtained by plotting the extracted value of AN over a frequency range, 2 GHz– 16 GHz, shown in Fig. (19.35a). In the stopband range 8.2 GHz–13.8 GHz, |AN| 1. It is shown in the shaded area. The upper edge of the stopband is nearly fixed at about 13.8 GHz. It is decided only by the lattice constant d. However, the lower edge of the stopband decreases from the type-i to type-iii, creating increasing wider stopband bandwidth. It is due to the increased inductance of the slot in the ground plane (case-i), slot in the strip conductor (case-ii), and slots in both the strip and ground conductors (case-iii). The slot in the strip conductor offers higher inductance, due to the higher current density in the strip, as compared to that of in the ground conductor. Figure (19.35b) shows the slow-wave region in the passband, as the normalized propagation constant, β/k0 is above to that of the uniform CPW host medium. The β is propagation constant of the Bloch wave of the periodic line, whereas k0 is the propagation constant of the host medium. In the stopband range, the propagation constant γ is complex and Bragg’s condition is met at βd = π. The first passband of Fig. (19.35c) shows an increase in Bloch impedance with frequency, up to the edge of the stopband. It is always above the 50 Ω impedance of the host CPW. In the second passband, it is much below the 50 Ω impedance. It is the characteristics of a series inductor loaded periodic line. Finally, Fig. (19.35d) compares |S21| of three structures consisting of N = 6 cells. The structure-iii provides wide stopband bandwidth and high attenuation level in the stopband due to stronger inductive loading. The ripples at the band-edges are due to the mismatch of Bloch impedance to the 50 Ω impedance of the CPW. The ripples are controlled by tapering the slots [J.69].

The series inductor loading of a CPW is also obtained by using several kinds of the DGS slots and stubs, including the radial stub in the ground plane [J.70, J.71]. More complex loading of both the strip and ground conductors have also been done [J.72]. Series Capacitor Loading

Figure (19.34b) shows four unit cells of the series loaded capacitors to the central strip conductor of a CPW. The loading capacitor is realized by a gap in the strip conductor, modeled by the π-network. Figure (19.36a) shows the extracted value of shunt (Cp) and series (Cg) capacitances of the π-network, using the MOM and deembedding process [J.66]. These capacitances can also be evaluated, through the de-embedded Y-parameters of the gap discontinuity, using an EM-simulator [B.16]. The value of the main series capacitance Cg is much higher than that of the Cp. The gap capacitance is in the middle of the unit cell of a period d. Two CPW sections have a length (d − g)/2, where g is the slot gap. The [ABCD] parameters of the four unit cells cascaded CPW is obtained using equation (19.4.10). Figure (19.36b) shows the results of the propagation constant and attenuation of the Bloch wave. The periodic structure acts as a bandpass filter between two stopbands. In the first stopband, we get β = 0, α 0. However, in the second stopband, the complex mode exists with β 0, α 0, and the passband, occurs between 5.8 GHz and 9.4 GHz. As β/k0 of the Bloch wave in frequency range 6.5 GHz–9.4 GHz is below that of the host CPW, the passband supports the fast-wave. In the frequency 5.8 GHz–6.5 GHz, β/k0 < 1, i.e. the normalized propagation constant is below that of the air-medium. In this band, the Bloch wave is a leaky radiation mode. Figure (19.36c) shows that Bloch impedance in the first stopband is capacitive. In the passband, the Re (ZB) maximum occurs at 20Ω, i.e. much below the 50Ω characteristic impedance of an unloaded host CPW. The value of Re (ZB) can be controlled by changing the slot-gap g and line gap s of a CPW to some extent. However, it is difficult to obtain 50Ω for a periodically loaded CPW to get proper matching at input and output terminations. Both ends are terminated in 32Ω, λg/4 CPW sections for matching. The |S21|-parameter is computed from [ABCD] parameters of the periodic line, using the following expression [J.66]:

745

19 Planar Periodic Transmission Lines

1

Passband γ/k0 = jβ/k0 + α/k0

AN

0 (i) (ii)

–0.5

(iii) Propagation: |A|1

–1.5

Stopband

3 Uniform CPW 2 Wc

β/k0 α/k0

1

Stopband 0

–2 2

4

6

8 10 12 Frequency (GHz)

14

16

2

4

8

6

10

12

14

16

Frequency (GHz)

(a) Calculated A-parameter for three types of the unit cells.

(b) Complex propagation constant of type iii-unit cell. 0

200

Insertion loss (dB)

–10

Re (ZB) Im (ZB)

150

Stopband

100

Wc

50 0

Passband

4

0.5

Bloch impedance (ZB) Ω

746

(i)

–20 Stopband –30

MoM-SOC Momentum Experiment

–40

Uniform CPW

(iii) Stopband

–50 2

4

6

8 10 12 Frequency (GHz)

14

16

4

(c) Bloch impedance of type iii unit cell. Figure 19.35

(ii)

6

8 10 12 Frequency (GHz)

14

16

(d) Insertion loss of six-cells EBG.

Characteristics of series inductor-loaded periodic CPW of Fig. (19.34a). Source: Reproduced from Zhu [J.65]. © 2003 IEEE.

S21 = S21 2 = and S21 2 =

2 2 cosh γNd + ZB Z0 + Z0 ZB sinh γNd 4+ 4+

Im ZB Z0 Re ZB Z0

2

2

4 + Z0 Im ZB

2

4 Re ZB

2

+ Z0

sinh 2 αNd

, in both stopbands

− 2 sinh 2 βNd

19 4 11

, in passband

Shunt Capacitor Loading

Figure (19.36d) shows the |S21|-passband response of the matched terminated periodic line. In place of the simple gap capacitors, the inter-digital capacitors or the MIM capacitors can be used to form the series capacitors-loaded periodic CPW. This arrangement provides stronger loading.

Figure (19.34c) shows the seven-sections shunt capacitorsloaded CPW on a substrate with εr = 10.2, h = 1.27 mm. The 95Ω CPW has a width w = 11.5 mm of strip conductor and g = 3.15 mm gap. After the shunt capacitor loading, Bloch impedance is about 50Ω. It avoids the λg/4 matching sections at both ends of the periodic CPW.

19.4 1D Planar EBG Structures

0.20 0.15

Cg

0.10

Cp Cp

0.05 5

10

15

20

25

30

35

40

45

β0/k0 Uniform CPW

Air medium 2

3

4

5

Passband fast-wave

6.5

6 7 8 9 Frequency (GHz)

10 11 12

(b) Complex propagation constant of unit-cell.

00

100 Re ZB Im ZB

50

Passband

–20

Passband

Z0u(uniform)

|S21| (dB)

Bloch impedance ZB (Ω)

2

0

CPW slot width (mil) (a) Extracted value of equivalent π-network.

0 W = 70mil S = 20 mil t = 5mil T = 305 mil

–50

–100

2

3

4

5 6 7 8 9 Frequency (GHz)

Figure 19.36

–40 –60 MoM-SOC Momentum Measurement

–80

10 11 12

–100

(c) Bloch impedance of unit cell.

2

3

4

5

6 7 8 9 Frequency (GHz)

10 11 12

(d) Insertion loss of six cells EBG.

Characteristics of series capacitor-loaded periodic CPW shown in Fig. (19.34b). Source: Reproduced from Zhu and Shi [J.66]. © 2005 John Wiley & Sons.

0

0

–10

–10

Stopband

–20

|S11| dB

|S21| dB

3

1

Cp

0.00

β/k0 α/k0

4 γ/k0 = α/k0+jβ/k0

Equivalent capacitance Cp, Cg pF

εr = 10.2, h = 25 mil t = 5mil, W+2S = 110 mil Cg

Leaky radiation mode

5

0.25

–30 –40 –50

Stopband

–20 –30

–40 –50 –60

–60 0

1 2 3 4 5 Frequency (GHz) (a) S21-parameter of seven-cell EBG. Figure 19.37

6

0

1 2 3 4 5 Frequency (GHz) (b) S11-parameter of seven-cell EBG.

6

Characteristics of seven-cell shunt capacitor-loaded periodic CPW of Fig. (19.34c). Source: Reproduced from Martın et al. [J.67]. © 2003 John Wiley & Sons.

The shunt capacitances are realized by the T-type open-ended stubs, separated by a distance d = 13.3 mm. The total length of the periodic line is 9.2 mm [J.67]. The value of shunt capacitor Csh = 2.3 pF is

extracted using an EM-simulator. The (ω − β) diagram of the structure is obtained as discussed previously. In the passband, the periodic line supports the slow-wave with SWF = 2.3. Figure (19.37a,b) shows |S21| and |S11|

747

748

19 Planar Periodic Transmission Lines

responses giving the passband and deep attenuation stopband. The bandwidth of the stopband can be further extended by using the dual periodicity line [J.73]. The use of the shunt connected varactor diode provides a tunable CPW resonator [J.74].

B.11 Martín, F.: Artificial Transmission Lines for RF and

B.12

B.13

Shunt Inductor Loading

The strip conductor of a CPW can be connected to both the ground conductors, using the short-circuited stubs. These short-circuited stubs provide the shunt inductor loading to a CPW. This structure shows bandpass filter behavior. The series-connected capacitors and the shunt-connected inductors are used to get the 1D metamaterials line structures. Figure (19.34d) shows one such structure [J.68]. The metamaterials lines, i.e. the metalines, are discussed in subsection (22.1.5) of chapter 22.

B.14

B.15

B.16

B.17 B.18

References

B.19

Books B.1 Kittel, C.: Introduction to Solid State Physics, John

Wiley & Sons, New York, 1986. B.2 Ashcroft, N.W.; Mermin, N.D.: Solid State Physics, B.3

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Holt. Rinehart and Winston, USA, 1976. Yariv, A.; Yeh, P.: Optical Waves in Crystals: Propagation and Control of Laser Radiation, WileyInterscience, John Wiley & Sons, 1984. Brillouin, L.: Wave Propagation in Periodic Structures – Electric Filters and Crystal Lattice, 2nd Edition, Dover Pub, USA, 1953. Fang, F.; Rahmat-Samii, Y.: Electromagnetic Band Gap Structures in Antenna Engineering, Cambridge University Press, UK, 2009. Collin, R.E.: Foundations for Microwave Engineering, 2nd Edition, McGraw-Hill, Inc., New York, 1992. Collin, R.E.: Field Theory of Guided Waves, IEEE Press, New York, 1991. Hutter, R.G.E.: Beam and Wave Electronics in Microwave Tubes, D. Van Nostrand Co., Princeton, NJ, 1960. Russer, P.: Electromagnetics, Microwave Circuit and Antenna Design for Communication Engineering, 2nd Edition, Artech House, USA, 2006. Hwang, R.-B.: Periodic Structures: Mode-Matching Approach and Applications in Electromagnetic Engineering, John Wiley & Sons, Hoboken, NJ, 2013.

B.20 B.21

Microwave Applications, John Wiley & Sons, Hoboken, NJ, 2015. Ramo, S.; Whinnery, J.R.; Van Duzer, T.: Fields and Waves in Communication Electronics, 3rd Edition, John Wiley & Sons, Singapore, 1994. Pozar, D.M.: Microwave Engineering, 2nd Edition, John Wiley & Sons, Singapore, 1999. Elliott, R.S.: An Introduction to Guided-Waves and Microwave Circuits, Prentice-Hall, Englewood Cliff, NJ, 1993. Rizzi, P.A.: Microwave Engineering – Passive Circuits, Prentice-Hall International Edition, Englewood Cliff, NJ, 1988. Swanson, D.G.; Hoefer, W.J.R.: Microwave Circuit Modeling Using Electromagnetic Field Simulation, Artech House, USA, 2003. Bahl, I.: Lumped Elements for RF and Microwave Circuits, Artech House, USA, 2003. Gupta, K.C.; Garg, R.; Bahl, I.; Bhartia, P.: Microstrip Lines, and Slot Lines, 2nd Edition, Artech House, Boston, 1996. Hong, J.S.: Microstrip Filters for RF/Microwave Applications, 2nd Edition, John Wiley & Sons, USA, 2011. Bahl, I.J. Bhatia, P.: Microwave Solid State Circuit Design, John Wiley & Sons, New York, 1988. Simon, R.N.: Coplanar Waveguide Circuits Components and Systems, John Wiley & Sons, New York, 2001.

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gap terminology: historical perspectives, 29th European Microwave Conf., pp. 295–298, Munich 1999. Maagt, P. de; Gonzalo, R.; Vardaxoglou, J.: Review of electromagnetic bandgap technology and applications, Electromagnetics Division, European Space Organization. Rahmat-Samii, Y.: The marvels of electromagnetic bandgap (EBG) structures, ACES J., Vol. 18, No. 4, pp. 1–10, Nov. 2003 Baccarelli, P.; Pallotto, S.; Jackson, D.R.; Oliner, A.A.: A new Brillouin dispersion diagram for 1-D periodic printed structures, IEEE Trans. Microwave Theory Tech., Vol. 55, No. 7, pp. 1484–1495, July 2007. Baccarelli, P.; Pallotto, S.; Jackson, D.R.; Oliner, A.A.: Analysis of printed periodic structures on a grounded substrate: a new Brillouin dispersion diagram, Digest MTT-S Int. Microwave Symp., pp. 12–17, June 2005.

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technique for innovative design of ultra-broadband microstrip bandpass filter, MTT-S Int. Microwave Symp., pp. 315–318, TU4C-2, 2000. Karmakar, N.C.; Mollah, M.N.: Investigation into nonuniform photonic bandgap microstrip line low-pass filters, IEEE Trans. Microwave Theory Tech., Vol. 51, No.2, pp. 564–572, Feb.2003. Huang, S.Y.; Lee, Y.H.: Tapered dual-plane compact electromagnetic bandgap microstrip filter structures, IEEE Trans. Microwave Theory Tech., Vol. 53, No.9, pp. 2656–2664, Sept. 2005. Siakavara, K.: Novel microwave microstrip filters using photonic bandgap ground plane with fractal periodic pattern, Optical Tech. Letters, Vol. 43, No. 4, pp. 273– 276, Nov. 2004. Liu, H.-W.; Sun, X.-W.; Li, Z.-F.; Yoshimasu, T.: Propagation characteristics of microstrip lines with 2D defected ground structures, Int. J. RF Microwave Computer-Aided Engineering, pp. 280–286, Feb. 2006. Qian, Y.; Itoh, T.: Planar periodic structures for microwave and millimeter wave circuit applications, MTT-S Int. Microwave Symp., pp. 1533–1536, TH2C3, 1999. Zhu, L.: Guided-wave characteristics of periodic microstrip lines with inductive loading: slow-wave and bandstop behaviors, Microwave Optical Tech. Letters, Vol. 41, No. 2, pp. 77–79, April 2004. Kazerooni, M.; Cheldavi, A.; Kamarei, M.: Analysis, modeling, and design of cascaded defected microstrip structure for planar circuits, Int. J. RF Microwave Computer-Aided Engineering, pp. 170–182, Jan. 2010. Xue, Q.; Shum, K.M.; Chan, C.H.: Novel 1-D microstrip PBG cells, IEEE Microwave Guided Waves Lett., Vol. 10, No. 10, pp. 403–405, Oct. 2000. Xue, Q.; Shum, K.M.; Chan, C.H.: Novel perforated microstrip PBG cell, Microwave Optical Tech. Letters, Vol. 26, No. 5, pp. 325–327, Sept. 2000. Liu, H.W.; Sun, X.W.; Li, Z.F.: A low-pass filter of wide stopband with a novel multilayer fractal photonic bandgap structure, Optical Tech. Letters, Vol. 40, No. 5, pp.431–432, Mar. 2004. Lee, C.O.; Shin, C.-C.: Enhanced periodic structures with compact size, Electronics Letters, Vol. 40, No. 8, pp. 479–481, April 2004. Liu, Z.; Xiao, G.; Zhu, L.: Numerical de-embedding and experimental validation of propagation properties of corrugated substrate integrated waveguide, Microwave Optical Tech. Letters, Vol. 58, No.10, pp. 2456–2060, Oct. 2016. Glandorf, F.J.; Wolff, I.: A spectral-domain analysis of periodically nonuniform microstrip lines, IEEE Trans.

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Microwave Theory Tech., Vol. 35, No. 3, pp.336–343, Mar.1987. Nesic, D.; Nesic, A.: Bandstop microstrip PBG filter with sinusoidal variation of the characteristic impedance and without etching in the ground plane, Microwave Optical Tech. Letters, Vol. 29, No.6, pp. 418–420, June 2001. Laso, M.A.G.; Lopetegi, T.; Erro, M.J.; Benito, D.; Garde, M.J.; Sorolla Ayza, M.: Multiple frequency tuned photonic bandgap microstrip structures, IEEE Microwave Guided Wave Lett., Vol. 10, pp. 220– 222, 2000. Lopetegi, T.; Laso, M.A.G.; Erro, M.J.; Benito, D.; Garde, M.J.; Falcone, F.; Sorolla, M.: Novel photonic bandgap microstrip structures using network topology, Microwave Optical Tech. Letters, Vol. 25, pp. 33–36, 2000. Matekovits, L.; Colemè, G.V.; Orefice, M.: Controlling the band limits of TE-surface wave propagation along a modulated microstripline-based high impedance surface, IEEE Trans. Antennas Propag., Vol. 56, No. 8, pp. 2555–2562, Aug. 2008. Yun, T.Y.; Chang, K.: One-dimensional photonic bandgap resonators and varactor tuned resonators, IEEE Int. Microwave Sym., MTT(s) Digest, TH3B-4, pp. 1629–1632. Zhu, L.: Guided-wave characteristics of periodic coplanar waveguides with inductive loading – unit-length transmission parameters, IEEE Trans. Microwave Theory Tech., Vol. 51, No.10, pp. 2133–2138, Oct. 2003. Zhu, L; Shi, H.: Frequency-dependent guided-wave characteristics of periodically series-capacitive loaded coplanar waveguides, Optical Tech. Letters, Vol. 46, No. 1, pp. 54–58, Jul. 2005. Martın, F.; Falcone, F.; Bonache, J.; Lopetegi, T.; Laso, M.A. G.; Sorolla, M.: New CPW low-pass filter based on

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20 Planar Periodic Surfaces Introduction The planar periodic surface is the 2D-extension of the 1Dperiodic line discussed in chapter 19. These are the reactively loaded 2D periodic surfaces, exhibiting the discrete spectrum of wave motion in the 2D K-space through alternate passband–stopband phenomena. In this case, Brillouin Zone is two-dimensional that provides complete suppression of the surface waves on a substrate. The metasurfaces are also the engineered 2D periodic surface. However, the metasurfaces have no bandgap. It is discussed in chapter 22. So, the planar periodic surface is in general called the electronic bandgap (EBG) surface. It has several forms depending upon the applications. The EBG surface has the following interesting features:

• • • •

Creation of alternate 2D passband–stopband over the engineered surface. A natural surface supports the continuous RF spectrum. Suppression of surface waves on the planar surface, and creation of the high impedance; or artificial magnetic conductor, i.e. AMC surface, to improve the performances of the planar antennas. Supporting the fast-wave propagation leading to Cherenkov-type radiation. It is useful for the design of the leaky-wave antenna. Creation of the frequency selective surfaces (FSSs) used in the antenna technology.

Objectives

• •

To discuss characteristics of some engineered planar 2D EBG surface. To discuss the 2D EBG circuit models and realization of the EBG planar surfaces.

20.1

2D Planar EBG Surfaces

The basic characteristics of the EBG surfaces are obtained through simulations on the EM-Simulators.

However, it is a time-consuming process. The initial design and optimization of the EBG surfaces are carried out with the help of the circuit models. The propagating 2D Bloch modes and complete surface waves bandgap are treated through the 2D dispersion diagram in the k-space. The frequency-dependent phase response, revealing the surface property of the EBG, is explored through the reflection phase diagram. This section discusses the basic characteristics of two generic EBG surfaces – the mushroom-type EBG and the uniplanar compact EBG, i.e. the UC-EBG. The reflection and dispersion diagrams are presented to understand the characteristics of the EBG surface. The models of the EBG surface are further discussed in the next section (20.2). 20.1.1

General Introduction of EBG Surfaces

The 2D-EBG surface is a natural extension of the 1Dperiodic transmission lines. The microstrip patch antenna and planar circuits are designed on the conductor-backed dielectric substrates, supporting the undesired surface waves of both TE and TM polarizations. The presence of the surface waves degrades the performance of both the microstrip circuits and antennas. Likewise, horizontally placed dipole near reflecting conducting planes, i.e. near a perfect electric conductor (PEC), suffers a reduction in gain and impedance matching. In general, a special type of artificial surface is needed to suppress the surface waves of both polarizations, in all azimuthal directions. In case such an artificial surface is used as a reflector, it should provide inphase reflection to improve the gain of a wire antenna or other antennas. Such in-phase reflection is possible with the help of a non-existing perfect magnetic conductor (PMC). The 2D-planar EBG creates such artificial surfaces, with both characteristics – suppression of the surface waves and in-phase reflection, over a certain frequency band. The EBG surface with unique property is a man-made surface, beyond (meta) the natural material

Introduction to Modern Planar Transmission Lines: Physical, Analytical, and Circuit Models Approach, First Edition. Anand K. Verma. © 2021 John Wiley & Sons, Inc. Published 2021 by John Wiley & Sons, Inc.

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20 Planar Periodic Surfaces

surface. So, it is also called the metasurface. However, the periodic surfaces without any bandgap property are treated as the metasurfaces. These are discussed in chapter 22. As a reflecting artificial surface, the EBG or metasurface nearly behaves as the PMC. The EBG surface over a certain frequency band is designated as the artificial magnetic surface (AMC) [B.1, J.1–J.4]. The EBG surfaces are the well-known frequency selective surfaces (FSSs), placed on the top of the conductor-backed substrate. The FSS has several applications in antenna technology. It is also used as an ultra-wideband shield [B.2, J.5]. The EBG surface shows alternate 2D passband– stopband in all azimuthal directions on its surface. Each passband corresponds to one Bloch mode, propagating in the form of the TM/TE surface waves. The 2Ddispersion characteristics of the Bloch modes are obtained with the help of the 2D-irreducible Brillouin zone (IBZ) in the k-space, discussed in section (19.1.3) of chapter 19. The unloaded conductor-backed substrate is the host medium. The reactive loading elements, i.e. conducting patches and strips, are inclusions arranged two-dimensionally over the host medium. The square and rectangular lattices are commonly used to get the isotropic and anisotropic EBG surfaces. However, sometimes hexagonal lattice is also used. The isotropic EBG surface has identical reflection coefficients, for the normally incident plane waves, in both the x-direction and y-direction on an EBG plane. The reflection coefficient of the anisotropic EBG surface is direction-dependent. The anisotropic surface is used to convert the linearly polarized wire antenna to a circularly polarized antenna [B.1, B.3, J.6–J.10]. Conceptually, the EBG surface is described by the LC parallel resonant circuit distributed over the surface. It works as the planar bandpass and bandstop filters to create alternate passband–stopband for the 2D Bloch mode surface waves. At resonance, the EBG surface has an almost infinite surface impedance, with in-phase reflection. So, at resonance, it acts as a PMC surface. However, within its reflection bandwidth, it acts as an AMC surface with high surface impedance. The AMC surface is also called the high impedance surface (HIS). The HIS has a surface impedance equal to, or more than, the intrinsic impedance of free space. The reflection phase bandwidth for the AMC surface is defined as the frequency range corresponding to the ±90∘ reflection phase variation [B.1, B.2, J.4, J.11, J.12]. Outside the ±90∘ reflection phase bandwidth, the EBG surface is still reactive and within another BW; over the (90∘ ± 45∘) reflection phase, the surface is defined as the reactive impedance surface (RIS) [J.13]. A compact

antenna has better impedance matching over the RIS bandwidth. So, 90∘ ± 45∘ reflection phase bandwidth is also called the matching BW, whereas the ±90∘ reflection phase bandwidth is called the AMC bandwidth. The AMC bandwidth may or may not coincide with the stopband bandwidth, i.e. the bandgap, obtained from the complete suppression of the surface waves. The complete suppression of the surface waves shows their suppression in all azimuthal directions on the EBG surface. The partial suppression of surface waves occurs in a specific direction only. The dispersion diagram provides the stopband bandwidth. The |S21|-response also provides the information on its stopband, i.e. bandgap [J.6, J.7]. The periodic surface, i.e. the metasurface, in this capacity is called the electromagnetic bandgap (EBG) structures. The EBG surface is used as a holographic surface [J.14]. The meaning of various electromagnetic surfaces, derived from the EBG surface or metasurface, and their bandwidths are further explained and computed using the closed-form models. Figure (20.1a–f ) shows two generic EBG surfaces – the mushroom-type EBG [J.6, J.7] and the uniplanar compact EBG, i.e. the UC-EBG [J.15] and some of their variations. Figure (20.1a–i) shows the structure of a mushroom-type EBG. It is created by connecting an array of square or rectangular patches, located at the substrate, to the ground conductor, using the metallic vias/pins. The hexagonal patches are also used [J.10]. Figure (20.1a–ii) shows the development of a simple parallel LC resonant circuit model. It is distributed over the surface and is used to model the surface impedance of a mushroom EBG. The capacitance of the LC-model is due to the gap-capacitance between adjacent patches. The inductance is caused by two vias and the ground conductor. Figure (20.1b) shows another capacitive patch-loaded EBG surface. In this arrangement, vias are absent. This structure does not provide any stopband characteristics for the surface waves. However, it has a frequencydependent resonant reflection phase response, as the conductor-backed substrate provides an inductive load at the surface [B.3]. The EBG surface is a RIS, useful for designing the compact antenna with good impedance matching [J.13]. Figure (20.1c) is a further modification of the open mushroom structure. In this case, a top shield is added. The mushroom EBG structure is located within a parallel-plate waveguide (PPW). In place of a simple ground conductor, this structure is used as a composite ground plane for the high-speed digital circuits. It suppresses the simultaneous switching noise (SSN) or the ground board noise (GBN) of the high-speed digital circuits [J.16, J.17].

20.1 2D Planar EBG Surfaces

x

C L

a

b

c

d

C L z

Top view y

εr2 = 1

Square patch

w

εr1 = εr Via

Top view

g d

C h

Ground conductor

Front view (Reactive impedance surface, i.e. RIS)

z

Front view (i) Structure

L (ii) Circuit model

(a) Mushroom EBG.

(b) Capacitive EBG.

Cg/2 L Cg/2

Unit cell Cg/2 L

Top view of inside EBG

Top view

Cg/2 L

Cg

Front view (i) Structure (c) Mushroom EBG inside PPW.

(ii) Circuit model (d) UC-EBG.

g w w L

Top view of EBG at the interface

C εr

t Front view (e) Sandwiched EBG. Figure 20.1 Some 2D-planar EBG surfaces.

2r (f) Lumped element-loaded mushroom EBG.

755

20 Planar Periodic Surfaces

Figure (20.1d-i) shows the structure of a uniplanar compact EBG (UC-EBG), different in concept from that of the mushroom type. There is no via connecting the patch to the ground. A narrow microstrip line connects two unit cells, providing an inductance across the gap capacitance. Figure (20.1d-ii) shows the development of a parallel LC-resonant circuit model for the surface impedance of the structure. Unlike the mushroom-type EBG, the UC-EBG can be easily fabricated even on alumina or on a semiconductor substrate. Figure (20.1e) shows that the capacitive EBG can also be embedded inside the dielectric media. It is located at the interface of two dielectric layers, forming the sandwiched EBG. Over the top of the second layer, an antenna is mounted to improve its performance, say to improve its circular polarization [J.18, J.19]. Figure (20.1f ) shows the mounting of the lumped chip capacitors across the gaps between two patches to increase the value of the gap capacitance. It brings down the resonance frequency, so that an EBG surface can operate even in the UHF range. However, this arrangement reduces the BW of the EBG surface. Occasionally, the lumped inductances are also added to the vias, before connecting them to a ground plane. The resonance frequency is further reduced and the BW of the EBG is improved [J.20]. In place of capacitors, varactor diodes are used to get a tunable EBG surface. It is useful for the beam steering of an antenna array [J.21, J.22]. 20.1.2

Characteristics of EBG Surface

The artificial periodic surfaces, i.e. the metasurfaces, are characterized with the help of the dispersion diagram

and reflection phase diagram. The EM-simulators, such as the CST Microwave Studio, Ansoft HFSS, etc., generate both characteristics diagrams, using their eigenmode solver. Figure (20.2a and b) shows the geometrical setups of a rectangular/square unit cell of the EBG surface for simulating its dispersion and reflection phase characteristics, respectively. The periodic boundary conditions (PBCs) are applied to all four faces in the x- and y-directions. An absorbing airbox is located at the top of the setup in the far-field region. Likewise, Floquet port illuminates the EBG surface with polarization-dependent and incident angle-dependent plane waves. In the case of a rectangular lattice, the lattice constant is different in both directions. However, for a square one, it is identical in both the directions. The user’s manual of an EMsimulator provides necessary information on such a simulation setup and process. A periodic metasurface does not need to act simultaneously both as the AMC and EBG, over an identical frequency band. Both frequency bands, i.e. the surface wave bandgap and in-phase reflection band, may not even overlap. It happens only for certain geometrical parameters of a periodic surface. The resonance frequency fr, giving zero reflection phase, ϕEBG(fr) = 0∘, could be within the stopband, or it may be outside the frequency band [J.22, J.23]. The reflection phase of the EBG surface can be polarization-dependent, i.e. an EBG surface can provide different phase variations in the mutually orthogonal directions on its surface. Its resonance frequency can also be polarization-dependent. Such EBG surfaces are anisotropic. This kind of surface is used to get the circularly polarized waves, from a

Floquet port

PML

E

De-embedding

Z

H

θ

X

Plane waves

PBC

Γ

PBC

M PBC

756

X TM - polarization

H

H

E EBG surface

z y

x

y d

(a) Dispersion simulation setup. Figure 20.2

EBG surface

z

h

x d

Z θ

X TE - polarization

(b) Reflection-phase simulation setup.

EBG surface setup for EM-simulation.

20.1 2D Planar EBG Surfaces

linearly polarized wire antenna. These are also used to get wideband circular polarization from a narrow band circularly polarized patch antenna [J.8, J.9, J.18, J.19, J.24]. The EBG resonator antenna can provide high gain [J.25]. The EBG surface with the polarizationindependent reflection phase is known as the isotropic EBG surface. Likewise, the EBG surface can support several kinds of Bloch mode propagations, such as the slowwave, fast-wave, in both the forward mode and backward mode. For an open structure, it also supports the leaky waves, just like the 1D artificial periodic lines. The ω − β Dispersion Diagram

The simulation setup shown in Fig (20.2a) is simulated in .the eigenmode solver of the EM-simulators to get the (ω − β) diagram of a unit cell of an infinite extent

periodic structure. The simulation is carried out over the perimeter of an IBZ, described in the k-space with Γ, X, M as the symmetric points. The x-directed propagation phase βxd of Bloch modes is varied in steps, from Γ (0, 0) to X (π, 0), while taking βyd = 0. For each value of βxd, the eigenmode solver numerically solves the eigenfrequencies of several selected number of Bloch modes of the TM/TE types. It generates the ω − β dispersion diagram for the component of the surface waves propagating in the x-direction. In a case of no real solution, the surface waves do not propagate and the stopband is created in the x-direction. Next, the edge of IBZ, in the XM direction, is scanned by varying the propagation phase βyd from X (π, 0) to M (π, π), while taking βxd = π. It generates the ω − β dispersion diagram for the component of the surface wave propagating in the y-direction. Finally, –20

18

ht Lig

Frequency (GHz)

14 12

5.80 GHz

Surface wave bandgap

8.40 GHz

–30

S21 (dB)

lin

e

16

TM0 TE1 TE2

10 Q 8 Surface wave bandgap (5.771–8.45 GHz)

–40

–50

6 P

4 2 0

Γ

EBG TM EBG TE Metal TM

–60

M X

–70 Γ

X

3

Γ

M

4

6 7 8 9 10 Frequency (GHz) (b) S-parameter response of EBG surface.

(a) ω - β diagram of mushroom EBG. 20

14

Reflection phase (Deg.)

lin ht Lig

Frequency (GHz)

16

180

TM0 TE1 TE2

e

18

12 10 No surface wave bandgap

8

M

6 4

Γ

2 0

X

r= 0.04 Via Radius (in λ6 GHz)

TM

135

11

band edge

90

0.03

45

fr

0

0.02 0.01

In-phase reflection band

TE 0.0 band edge

–45 –90 –135

X

–180

Without Vias Γ

5

M

(c) ω- β diagram of capacitive EBG surface.

Γ

0

2

4

6

8 10 12 14 16 18 20 Frequency (GHz) (d) Reflection phase of EBG surface with via radius(r = 0.0 to 0.04) as a parameter.

Figure 20.3 Dispersion and reflection phase diagram of the EBG surface. Source: Reproduced from Li et al. [J.26]. © 2008 with permission of AIP Publishing.

757

20 Planar Periodic Surfaces

the edge MΓ is scanned by varying both βxd and βyd simultaneously, from M (π, π) to Γ (0, 0), to get the dispersion in the diagonal direction. The common stopband BW provides the complete stopband BW for the surface wave. It is taken as the bandgap of the 2D EBG surface [B.1, J.26, J.27]. Figure (20.3a–c) illustrates the ω − β dispersion diagrams of the mushroom-type EBG and capacitive surface (mushroom EBG without vias). These are obtained using the EM-simulator [J.23, J.26]. The parametric study has been done for five structural parameters: patch width (w), patch gap (g), a substrate thickness (h), via radius (r), and relative permittivity (εr). Figure (20.1a-i) shows the isotropic EBG surface with a period of d = w + g. The following initial parameters, at 6 GHz, are taken for the mushroom EBG unit cell: w = 0.14 λ6GHz, g = 0.006 λ6GHz, h = 0.03 λ6GHz, r = 0.01λ6GHz, and εr = 2.65, where λ6GHz is the free space wavelength at 6 GHz. Figure (20.3a) shows that the first propagating Bloch mode is TM0 surface wave, with no cutoff frequency. It is below the light line, and closely follows it up to point P at the stopband edge, showing the forward slow-wave propagation, and then it shows a backward slow-wave propagation with the negative group velocity. The second Bloch mode (TE1 surface wave mode) commences at the point Q, i.e. at the upper edge of the stopband. The Bloch modes, outside the light cone, are bound modes. The stopband of width 5.771 GHz– 8.45 GHz is visible in the dispersion diagram. The third Bloch mode (TE2 surface wave mode) also appears, without any bandgap. Inside the light cone, TE1 and TE2 modes are the fast-waves. They are the leaky modes of the open EBG surface. Figure (20.3b) shows the measured S21 response of the 20 × 20 unit cells; size 146 mm ×

90 45

–50

0 –45

–60

–90 –135

–70

6.3 GHz 4

5

6

9.65 GHz

–180

7 8 9 10 11 12 13 Frequency (GHz)

(a) TM band edge outside +90° reflection phase.

Figure 20.4

180 135 90

│S21│ (dB)

–40

Bandgap

–30

180 135

146 mm. The TM and TE surface modes are excited and received by a pair of very short vertical dipoles and a pair of loops, respectively [J.3, J.28, J.29]. It shows 5.80 GHz– 8.4 GHz wide stopband, nearly the same as obtained from the ω − β diagram. The nature of the Bloch modes, as the TM/TE mode, could be decided from their field structures on the EM-simulator [J.23]. Figure (20.3c) shows the dispersion diagram of the capacitive EBG surface of Fig (20.1b). It does not exhibit any bandgap. Therefore, the vias are necessary for the creation of a bandgap. However, Fig (20.3d) shows the reflection phase diagram with an (±90∘) in-phase reflection band created for capacitive EBG structure (via radius = 0). Figure (20.3d) also shows the reflection phase of the mushroom-type EBG surface with finite-size via radius. The small cross, marked on the reflection diagram, shows the TM and TE bandedges of the ω − β diagram, corresponding to the lower and upper edge of the bandgap of the surface waves. An increase in the radius of via brings the TM band-edge near to (+90 ) reflection phase. Otherwise, the bandgap and in-phase reflection band do not normally overlap. However, in the case of d/h ≤ 2, there is a simultaneous occurrence of the surface wave stopband and in-phase reflection bandwidth. Figure (20.4a and b) demonstrates it. For Fig (20.4a), the EBG structure has d/h = 4.0, and for Fig (20.4b) d/h = 2.0. For the former case, the TM band edge occurs at reflection phase +135 , and for the latter case, it occurs at +90 . The lower and upper band edges of the bandgap, also the reflection-phase BW, are influenced by the structural parameters of the EBG surface [J.23, J.26]. The simulation process of the reflection diagram is discussed below.

–40

11.6 GHz 45 0

9.3 GHz

–50

–45 –90

WG PBG

–60

–135

Reflection phase (deg)

Bandgap

Reflection phase (deg)

–30

│S21│ (dB)

758

–180 –70 5

6

7

8

9

10 11 12 13 14

Frequency (GHz) (b) TM band edge within +90° reflection phase.

Comparison of surface wave bandgap and reflection bandwidth of the mushroom EBG. Source: Reproduced from Li et al. [J.26]. © 2008 with permission of AIP Publishing.

20.1 2D Planar EBG Surfaces

The Reflection Phase Diagram

In a case of reflection phase simulation, the Floquet port, located at the height z = H, shown in Fig (20.2b), illuminates the EBG surface with a normal incidence of the TM/TE polarizations. Both the normal incidence and oblique incidence of the plane waves on the EBG surface are used to know the stability of reflection-phase behavior with respect to the angle of incidence. In the case of the TE-polarization, Ey component is normal to the vias. The vias are not excited, and the presence or absence of vias does not influence the reflection-phase response of the TE-polarization. For the normal incidence, the Ex component of the TM-polarization is normal to the vias. Again, the vias are not excited, and both the normal incident TE- and TM-polarized waves have identical reflection phase response.

0

0

180

120

120 90 60

180

0

240

–60 –90 –120

d = 180 mil d = 120 mil

300 360 0

5

10 Frequency (GHz)

PEC –5 ϕEBG(Z = 0) (Degree) S11 (dB)

–60 Δϕ (Degree)

The reflection phase is needed at the surface of the EBG, as its characteristics. So, the de-embedded phase information at the surface is to be obtained. The EMsimulator provides it. However, understanding of the de-embedding process is useful for the measurement of a reflection phase of an EBG surface [J.3, J.28–J.30]. First, the reflection phase of a conducting surface (PEC) is obtained at height H in the far-field region. Next, at the same height, the reflection phase response of the EBG surface is recorded. At the certain resonance frequency, the reflection phase difference of the PMC is π radian, i.e. 180 , with respect to the reflection phase of the reference PEC. We compute the reflection-phase variation of the EBG surface with frequency from the phase difference of the EBG and PEC surfaces as follows:

–180 20

15

PMC

–10 –15 EBG

–20 –25 –30 10

12

14

16

18

Freq (GHz) (b) Return-loss of dipole over PEC, PMC, EBG (RIS) [J.31].

(a) Reflection phase of isotropic UC-EBG [J.30]. 200

0

–10 100

S11 (dB)

Reflection phase (Degree)

–5 150

50

–15 0.60 –20 –25 –30

0

0.26 0.54 0.32 0.48

0.42

–35 –50 10

12

14 Freq (GHz)

16

(c) Reflection phase BW corresponding to 90°± 45° for mushrooms EBG(RIS) [J.31].

Figure 20.5

18

–40 10

12

0.36

14

16

18

Freq (GHz) (d) RL of dipoles with resonant lengths, shown in the unit of, λ12GHz within the reflection-phase BW of RIS [J.31].

Reflection phase of the EBG surface. Source: Reproduced from Yang et al. [J.30]. © 1999 IEEE. Reproduced from Yang and Rahmat-Sammi [J.31]. © 2003 IEEE.

759

20 Planar Periodic Surfaces

ϕPEC z = H = 2kH + π

a

20.1.3

ϕEBG z = H = 2kH + ϕEBG z = 0 ϕEBG z = 0 = Δϕ + π = Δϕ + 180∘

b c

A horizontally polarized wire antenna placed near the perfectly conducting surface (PEC) does not radiate properly due to out of phase reflection. A PMC surface provides constructive interference, and such an antenna radiates in a better way. However, Fig (20.5b) shows that the return-loss is not good for an antenna placed over the PMC. The EBG surface, defined by a bandwidth corresponding to the reflection phase ϕEBG(z = 0) = 90∘ ± 45∘, improves the return-loss [J.13, J.31]. Figure (20.5c) shows the reflection-phase diagram with a gray strip defining the reflection-phase bandwidth over ϕEBG(z = 0) = 90∘ ± 45∘. The reflection-phase response is obtained for the square mushroom-type EBG shown in Fig (20.1a). The structure has the following dimensions: w = 0.12 λ12GHz, g = 0.02 λ12GHz, h = 0.04 λ12GHz, r = 0.005 λ12GHz, εr = 2.20. Over the frequency band, corresponding to ϕEBG(z = 0) = 90∘ ± 45∘, the

where, Δϕ = ϕEBG z = H − ϕPEC z = H

d 20 1 1

Figure (20.5a) illustrates the simulated frequency dependence of the reflection phase difference Δϕ. The EBG reflection phase ϕEBG(z = 0) at the surface of the UC-EBG is also shown. The isotropic UC-EBG is designed on a substrate with εr = 10.2, h = 25 mil and period d = 180mil and 120 mils [J.30, J.33]. The EBG surface acts as a PMC at resonance frequencies 13.4 GHz and 9.5 GHz, corresponding to d = 120 mil and 180 mils, respectively. Over the reflection bandwidths, shown in gray strips, the EBG surface acts as an AMC surface [J.4].

Y X

Reflection phase (Deg.)

200

Top view

Horizontal Wire Dipole Near EBG Surface

Square EBG Rect. EBG, Y pol Rect. EBG, X pol

150 100 50 0 –50 –100 –150 –200

2

2.5

3

3.5

4

4.5

Cross-sectional view Frequency (GHz) (a) Rectangular patches.

(b) Response of rectangular patch EBG. 200

W

4

150

Y X

Reflection phase (Deg.)

760

3 100 3

50

2

2 1

4 0 –50 1 2 3 4

–100 –150

Top view Cross-sectional view (c) Square patches with displaced via. Figure 20.6

–200

2

2.5

3

Via at 1/2 W Via at 3/8 W Via at 1/4 W Via at 0

3.5

4

4.5

Frequency (GHz) (d) Response of displaced via square EBG.

Reflection phase of the anisotropic mushroom EBG surface. Source: Reproduced from Yang and Rahmat-Sammi [J.32]. © 2004 John Wiley & Sons.

20.1 2D Planar EBG Surfaces

EBG surface is inductive. It is known as the reactive impedance surface (RIS). Usually, the RIS is an inductive surface. Figure (20.5d) shows that any horizontally placed dipole, with a resonant length within the BW of the RIS, has proper return-loss. The frequency band corresponding to ϕEBG(z = 0) = 90∘ ± 45∘ is known as the matching bandwidth or the RIS bandwidth. Figure (20.5b) compares results of a dipole of length 0.40 λ12GHz placed over the PEC, PMC, and EBG (RIS) [J.31].

also acquire linear polarization. The PEC, PMC, and isotropic EBG surfaces change the polarization of the reflected waves to the LHCP. Likewise, a rectangular patch anisotropic mushroom EBG also acts as the ground plane to an antenna located near to it. In this case, it is used to get the circularly polarized radiated waves from a linearly polarized wire antenna placed at the 45 orientation on the EBG surface, shown in Fig (20.7c) [J.9, J.34]. Both aspects of the anisotropic EBG surface are examined below.

Anisotropic EBG Surface

EBG as a Reflecting Surface

The anisotropic EBG surface has polarization-dependent, and also incidence angle-dependent, reflectionphase response. It provides different resonance frequencies and different frequency-dependent phase behavior even for the orthogonal polarizations in the plane of the EBG. The anisotropy is realized by having different lattice constants in the x- and y-directions, or by using the asymmetric loading elements. For instance, a mushroom-type EBG can have anisotropic surface by using a rectangular patch, shown in Fig (20.6a). It is also realized by using rectangular slots-loaded square patch, or by using asymmetrically placed via connected to square patches, shown in Fig (20.6c) [J.9, J.32, J.34]. The rectangular patch has a dimension 0.24λ3GHz × 0.16λ3GHz on a substrate with εr = 2.20, h = 0.04λ3GHz. The gap between adjacent patches is 0.02 λ3GHz and the radius of vias is 0.0025 λ3GHz. The λ3GHz is the free-space wavelength at 3 GHz. Figure (20.6b) shows the reflection-phase responses of both the rectangular anisotropic and square isotropic EBG surfaces. The isotropic surface has a square patch of dimension 0.16λ3GHz × 0.16λ3GHz. The x-polarized normal incident wave has lower resonance frequency, as compared to that of y-polarized incident waves. It is due to the longer length of the rectangular patch along the x-direction. The identical phase response is obtained for y-polarized waves incident on both the rectangular and square patches. At nearly 2.9 GHz, the 90∘ reflection phase occurs for the x- and y-polarized waves. Figure (20.6d) shows that the displaced via, along the width w of the patch in the x-direction, has dual resonances with dual reflection BW. The maximum frequency difference between dual resonances is for vias at the edge of the patches. The anisotropic EBG surface acts both as a reflecting surface and as a ground plane. The rectangular patch anisotropic mushroom EBG, as a reflecting surface, controls the polarization of the incident right-hand circularly polarized (RHCP) waves [J.32]. The reflected waves can have either RHCP or elliptical polarization. It can

Figure (20.7a) shows a reflecting EBG surface with normally incident RHCP waves and reflected waves. The reflecting EBG surface is in the (x-y)-plane. The incident and reflected waves are in the ( ) z-direction. The electric fields of the normally incident RHCP and reflected waves are in

E

= xejkz + yjejkz

a

= xe − jkz + jθx + yje − jkz + jθy

b,

ref

E

20 1 2 where θx and θy are the reflection phases of the EBG surface, for the x-polarized and y-polarized waves, respectively. The reflected waves travel away from the observer, i.e. away from the EBG surface. The timedependent factor ej ωt is suppressed in the above equations. The reflected waves can be rewritten as a linear combination of two circularly polarized waves [B.4]: ref

E

+ ER

where, EL =

1 + ej

= e − j kz ej θx EL

x+jy 2

1 − ej

θy − θx

2 θy − θx

a

2 b,

ER =

x−j y 2

c 20 1 3

In the above equations, EL and ER are the unit vectors for the LHCP and RHCP waves, respectively. For the PEC reflecting surface, θx = θy = π, and for the PMC reflecting surface, θx = θy = 0. In both cases, the coefficient of the unit vector ER is zero. Thus, in both cases, the reflected waves are the LHCP waves, because the RHCP is zero. To retain the RHCP, even after reflection, we must have (θy − θx) = 180∘. It results in zero value for the coefficient of EL , resulting in the RHCP for the reflected waves. At 2.9 GHz, Fig (20.6b), the

761

20 Planar Periodic Surfaces

4 3

Ein

Reflected waves

2 Axial ratio

Incident waves

ERef

Z Y

1 0 –1 –2

LHCP,AR = 1

RHCP AR = –1

3dB RHCP

–3

X

–4

2

(a) Reflecting EBG surface.

4

2.5 3 3.5 Frequency (GHz)

4.5

(b) Polarization control [J.32]. 10 8

100 mm Axial ratio

762

34 mm y

6 4 2 0 1

x (c) EBG ground plane supporting 45° inclined wire dipole [J.9]. Figure 20.7

2

= −

4

5

(d) Circularly polarized waves from the wire antenna [J.9].

EBG reflection surface and ground plane for polarization control. Source: Reproduced from Yang and Rahmat-Samii [J.9]. © 2005, IEEE. Reproduced from Yang and Rahmat-Samii [J.32]. © 2004, John Wiley & Sons.

phase difference 180 is achieved after the reflection. The axial ratio (AR) of the reflected waves is obtained as follows: AR = −

3 Frequency (GHz)

ER + EL ER − EL 1 − ej

θy − θx

1 − ej

θy − θx

EBG as a Ground Plane Surface

Figure (20.7c) shows a 45 inclined wire antenna on a rectangular patch mushroom EBG surface that acts as a virtual anisotropic ground plane [J.9]. The dipole is located at a small distance h = 0.02 λ above the ground rad

plane. The radiated wave E 2 +

1 + ej

θy − θx

2 −

1 + ej

θy − θx

is a summation of the

dir

from the dipole and the reflected wave

2

direct wave E

2

E from the conducting ground plane, located at distance h below the EBG surface:

ref

20 1 4

rad

E Figure (20.7b) shows the variation in the AR with frequency. At 2.90 GHz and 3.30 GHz, AR = −1, and a pure RHCP wave is obtained after reflection because (θy − θx) = 180∘. It has 3 dB ARBW from 2.81 GHz to 3.43 GHz (19.9%). At 2.58 GHz, the phase difference is 90 . It gives the linear polarization with AR = ∞. Figure (20.7b) also shows the range of LHCP.

dir

= E +

ref

+ E

=

E0 xe − jkz + ye − jkz 2

E0 xe − jkz − 2jkd h + jθx + ye − jkz − 2jkd h + jθy 2 20 1 5

In the present case, kdh ≈ 0. If EBG acts as a PEC, then θx = θy = π. In this case, the above expression provides

20.2 Circuit Models of Mushroom-Type EBG

zero radiated field. If EBG acts as a PMC, then θx = θy = 0∘. rad

− jkz

x + y , i.e. the radiated wave In this case, E = E0 e is linearly polarized. However, if at certain frequency θx = + 90∘, θy = − 90∘ simultaneously, then the above expression for the radiated field is rad

E

=

E0 − jkz e x + y + j x−y 2

20 1 6

In this case, the radiated field is RHCP waves. The mushroom EBG is designed on the RT-Duriod 6002 substrate, εr = 2.94 ± 0.04, h = 6.1 mm, with rectangular patches of size 8 mm × 13 mm. The gap between patches is 2mm in the x-direction, and 1mm along the y-direction. The size of the surface is 100 mm × 100 mm, with 9 × 6 rectangular patches. The length of the dipole is 34 mm at the height 3 mm and the radius of the wire is 0.34 mm. It is fed by a 50Ω coaxial cable. Figure (20.7d) shows a 2 dB AR in the broadside direction. It occurs at 3.56 GHz with 3 dB AR BW 5.6% (3.45 GHz–3.65GHz). The performance of the antenna as a CP antenna is not satisfactory. However, it demonstrates the concept of the EBG surface as the polarization converter.

20.2 Circuit Models of Mushroom-Type EBG

C=

This section describes lumped circuit models also called the effective surface models. They provide the conceptual understanding of the EBG responses, both the reflection and dispersion responses, presented in the previous subsections. More realistic and accurate 2D transmission line-based models are discussed in section (20.4). 20.2.1

w εr1 + εr2 d cosh − 1 π g

Figure (20.1a-i) shows a mushroom EBG of (w × w) square patch on a substrate with relative permittivity εr, and thickness h. Its lattice constant, i.e. period, is d. Its basic circuit model, shown in Fig (20.1a-ii), is the

a,

L = μr μ0 h

b

20 2 1 The present structure has εr1 = εr, εr2 = 1, μr = 1. The LC-circuit model provides the following expression for evaluating the surface impedance Zs Ω/sq of the lossless EBG surface: Zs ω =

Basic Circuit Model

Figure 20.8 Circuit model of the radiating antenna located over the EBG surface.

LC parallel resonant circuit distributed over the surface. The gap, between adjacent patches, contributes toward capacitance C, and via provides the inductance. The series resistance R can be further added with an inductor and conductance G across the capacitor to account for the conductor and dielectric losses, respectively. The EBG surface acts as the 2D bandstop filter to suppress the TM surface waves, around the resonance frequency ω0. The EBG structure creates multiple passband/stopband that suppresses the TE surface waves also. However, the basic LC-circuit model explains the creation of the first bandgap only, and does not capture the higher-order bandgap. The surface impedance (Zs), without the EBG, is very low, and it supports the propagation of the fundamental TM surface waves. However, the EBG surface shown in Fig (20.8a) provides a HIS suppressing the fundamental TM surface wave mode. At the resonance frequency ω0 and over a bandwidth Δω = ωH − ωL around ω0, the reactive surface impedance is very high. At the upper and lower band frequencies, (ωH, ωL), the surface impedance is equal to the intrinsic impedance of free space, i.e. |Zs| = |±jXs| = η0. The surface capacitance C F/Sq. and surface inductor L H/Sq. of the basic circuit model are approximately evaluated using the following expressions [J.6, J.7]:

jωL 1 − ω ω0

2

a,

where,

ω0 =

1 LC

b

20 2 2 At the resonance frequency, ω = ω0, the surface impedance becomes infinite Zs ∞ in the absence of losses. For

Radiation in space Normal incident plane waves Zs g εr1= εr d (--): EBG surface with Zs (a) Normal incidence on EBG.

η0 L C

Antenna as a current source

Zs: Surface impedaance (b) Antenna over AMC.

763

764

20 Planar Periodic Surfaces

ω < ω0, Zs = j XL, i.e. the EBG surface is inductive, whereas for ω > ω0, Zs = j XC, i.e. the EBG surface is capacitive. In the case of the normal incidence of the plane waves, shown in Fig (20.8a), both kinds of reactive surfaces totally reflect the incident waves. However, reflection phases from them are different. The EBG surface contributes to the reflection coefficient. It is computed using the equivalent transmission line concept as follows: Γω =

Zs ω − η0 Zs ω + η0

at ω = ω0 , Γ ω0 = + 1 0∘ for ω < ω0 , ΓL ω =

b

j X s ω − η0 = 1 ϕL j X s ω + η0

where, ϕL = π − 2 tan − 1 Xs ω η0 for ω > ω0 , ΓL ω =

a

c

− j X s ω − η0 = 1 ϕH − j X s ω + η0

where, ϕH = − π + 2 tan − 1 Xs ω η0

d 20 2 3

In the above equation, free space impedance is η0 = 120 π . The frequency-dependent surface impedance Zs(ω) of equation (20.2.2a) provides the inductive surface reactance j Xs(ω). The reflection-phase is frequencydependent. At the resonance frequency, ω0, the EBG surface acts as a PMC with the reflection phase ϕ = 0∘ and the surface impedance Zs(ω = 0) ∞. The reflection phase (ϕL) is positive forming an inductive surface at the lower frequency below ω0, i.e. for ω 0, ϕL + π. Whereas, at the higher frequency above ω0, the phase (ϕH) is negative forming a capacitive surface, i.e. for ω ∞ , ϕH − π. Therefore, at very low and at very high frequencies, the EBG surface acts as a PEC as the surface impedance Zs(ω 0) = Zs(ω ∞) = 0. Thus, the EBG surface can create an artificial electric conductor (AEC) with out-of-phase reflection. At certain lower and higher frequencies, ωL and ωH, the relation |jXL| = |jXH| = η0 is obtained. At ωL and ωH, the reflection phases are ϕL = π/2 and ϕH = − π/2. Thus, the reflection bandwidth Δω is defined over the reflection phase ϕ = ± π/2. In the frequency range (ωH, ωL), the EBG surface acts as a HIS, as over this frequency band η0 ≤ |Zs| < ∞. Almost in-phase reflection is maintained within Δω. The EBG surface over (ωH, ωL) frequency band acts as an artificial magnetic conductor (AMC) [J.4]. The circuit model explains the bandgap for the suppression of surface waves. The bandgap occurs within, or in the neighborhood, of the reflection bandwidth. However, the results of EM-simulations show that the surface wave bandgap and reflection bandwidth may not be identical. In both lower frequency range 0 < ω ≤ ωL and upper frequency ωH ≤ ω < ∞ range, |Zs| < ηo; the EBG surface is a

low impedance surface (LIS) with an out-of-phase reflection. The impedance matching of an antenna located over the PMC, or even over AMC, is not satisfactory, as surface impedance is high compared to free space impedance. Better impedance matching occurs over the frequency band defined by the reflection phase, ϕ(ω) = (90∘ ± 45∘) [J.13, J.31]. It is the range of the matched inductive RIS surface. The phase angle ϕ(ω) is measured with respect to the positive Re(Γ) axis. The polar reflection diagram, shown in Fig (20.9), demonstrates the frequency-dependent behavior of the reflection phase, and corresponding changes in the nature of the EBG surface. The outer circle shows the clockwise increasing frequency variation from ω = 0 to ω = ∞. The inner circle shows the corresponding variation in the reflection phase from ϕ = + π to ϕ = − π. The inner area of the circle shows the corresponding nature of the periodic surface. It is noted that usual matched inductive RIS is created around ωL; however, around ωH another matched capacitive RIS could be developed. Figure (20.8b) models the radiating antenna as a Norton current source, in the presence of the AMC surface. The current source feeds radiated power to the load η0, i.e. to free space. The radiated power drops by 3 dB at the reflection bandwidth edge, ωL and ωH, for |Zs(ω)| = η0. It helps to obtain an expression for the radiation bandwidth (BW) in terms of circuit elements as follows: jωL 1 − ω ω0

2

= η0 ,

ω2 =

1 1 + 2 2 ± LC 2η0 C

1 1 1 + 2 2 η0 C LC 4η0 C

1 2

20 2 4

Normally L is about 1nH/Sq., and C is in the range of 0.05–10pF/sq. The term 1 η20 C2 can be ignored, giving the following expression: ω2 ≈

1 1 , ± LC η0 C LC

ω = ω0

1±

Z0 Z0 , ω ≈ ω0 1 ± η0 2η0

20 2 5 In the above expression, Z0 = L C is a kind of characteristic impedance Z0 < < η0; and ω0 = 1 LC. The EBG bandwidth is given as BW =

ωH − ωL 1 Z0 1 Z0 Z0 − 1− = = 1+ ω0 η0 2 η0 2 η0

BW =

1 η0

L C 20 2 6

To lower the resonance frequency of an AMC surface, the C and L can be increased together, or individually.

20.2 Circuit Models of Mushroom-Type EBG

ω = ωL

Figure 20.9 Polar reflection diagram of EBG surface showing the nature of the surface impedance and reflection BW. AMC is from ϕL = + π/2 to ϕH = − π/2.

ϕL = + π/2 η = jXs

ϕ=+π ϕ=–π

Capacitive LIS

PEC (Zs = 0) – 3π/4

In-phase reflection

ω=0 ω=∞

ω Inductive LIS

Out of phase reflection

Impedance matching Inductive RIS

3π/4

ϕ

π/4

AMC Inductive HIS Capacitive HIS

Impedance matching Capacitive RIS

Re(Γ)

ω = ω0

Im(Γ)

PMC (Zs = ∞) – π/4

η = –jXs ϕH = – π/2 ω = ωH

Lower mushroom Upper mushroom

Patch of mushroom Via

Capacitor

Via Half -loop spiral inductor Ground conductor (a) Two-level mushroom EBG. Figure 20.10

(b) Half-loop spiral inductor with via.

Schemes to increase C and L of a mushroom EBG surface.

However, increasing C reduces the BW, whereas increasing L improves it. Figure (20.10a) shows a multilayer mushroom scheme to increase the capacitance [J.35]. Figure (20.10b) shows another scheme to increase the inductance of a via adding one-turn spiral inductor. The multi-turn inductor is also used to get a higher value of inductance [J.36, J.37]. These arrangements provide compact EBG at a lower frequency. A simple EBG operates around 10 GHz for a few mm-thick substrates with relative permittivity εr in the range 2–10. However, the structure of Fig (20.10a) has a bandgap between 2.2

GHz and 2.5 GHz. The scheme of Fig (20.10b) operates in the frequency range of 1.1 GHz–2.7 GHz. The inductance can also be increased by using the magnetodielectric substrate with the permeability of more than one [J.38]. However, it is difficult to get such substrates. The simple circuit model also provides the dispersion relation for the first two Bloch modes propagating on the EBG surface. However, it is valid only for 1D wave propagation, without any Bragg’s reflection. The inductive surface supports the TM-type Bloch mode, whereas the capacitive surface supports the TE-type Bloch mode.

765

20 Planar Periodic Surfaces

k2o = μo ε0 ω2 = β2x + β2y + β2z

20 2 7

The EBG surface of Fig (20.1a) is in the (x-z)–plane, where the z-axis is in the direction of propagation of surface waves, and βx = 0. In the y-direction, the field is confined in the evanescent mode, i.e. βy = j α, where α is the attenuation constant. The above general dispersion relation is reduced to β2z = μo ε0 ω2 + α2

20 2 8

The surface impedance of the TM and TE-polarized surface waves, given by equation (7.6.1) of chapter 7, is summarized below: ZTM = s

jαTM ωε0

ZTE s =

a,

− jωμ0 αTE

b

20 2 9

Using equation (20.2.8) with the above equations, the dispersion relations, for both surface waves modes, are obtained below: ZTM = s ZTE s =

βz,TM =

j β2 − μo ε0 ω2 ωε0 z,TM

1 2

a

− jωμ0 β2z,TE

− μo ε0 ω2

ω ZTM 1− s η0 c

b

1 2 2 1 2

=

ω 1+ c

XTM s η0

2 1 2

c

π

βz,TE

–π/2

Resonance Frequency

Capacitive surface

0

5

10

15

20

Frequency (GHz) (a) Refection phase diagram Figure 20.11

20

25

30

ω 1+ = c

η0 XTE s

2 1 2

d

High impedance region

Slow-wave TE waves Capacitive surface resonance frequency

15 TM waves

10

Inductive surface slow-wave region

5

–π 0

Fast-wave

Frequency (GHz)

Inductitive surface

π/2

2 1 2

= ZTE The surface impedance, ZTM s s = Zs, is taken from equation (20.2.2). The basic circuit model does not provide polarization-sensitive surface impedance. For ω < ω0, surface impedance is inductive and an EBG supports the TM-mode with propagation constant βz,TM. While for ω > ω0, surface impedance is capacitive and the EBG supports the TE-mode with propagation constant βz,TE. Figure (20.11a and b) shows the reflection phase diagram and dispersion diagram of an EBG surface for the sheet capacitance C = 0.05pF/sq and sheet inductance L = 2nH/sq [J.6, J.7]. In the frequency range,0 < ω < ω0, the surface impedance is inductive, i.e. ZTM = j XTM s s . It increases with frequency, resulting in the decreasing reflection phase, as shown in Figs (20.9 and 20.11a), from π to π/2 (at ωL), then to 0 at ω0. Further increase in the frequency results into the capacitive EBG surface and its reflection phase is ϕ(ω) = − π/2 at ωH and moves to −π at ω ∞. Figure (20.11b) shows the nature of the dispersion of the first two propagating modes supported by the EBG surface. Initially the propagation constant βZ,TM of the TM surface wave increases and follows the light line, as βZ,TM = ω c, for XTM < < η0. At ωL, XTM = η0, cors s responding to ϕL = π/2, and the propagation constant is βZ,TM = ω c 2 For a further increase in frequency, the βZ,TM bends horizontally, showing a presence of the TM slow-waves. It is a bounded non-radiating first

30 25

ω η0 1 − TE = c Zs

20 2 10

light line

The AMC does not support any surface wave propagation, creating the bandgap. The wave vector k0 follows the following general dispersion expression:

Reflection phase (Radians)

766

0

0

200 400 600 800 1000 1200 Propagation constant, βz (1/cm) (b) Dispersion diagram

Reflection and dispersion diagrams of mushroom EBG for C = 0.05pF/Sq and L = 2nH/Sq. Source: Reproduced from Sievenpiper et al. [J.7]. © 1999, IEEE.

20.2 Circuit Models of Mushroom-Type EBG

Bloch mode. At ω0, XTM ∞ ; giving ϕ(ω) = 0∘ and s βZ,TM ∞. The TM mode stops propagation. At ω0, the propagation constant is βZ,TE = ω/c for the TE surface waves. It starts its propagation at ω = ω0, and passes through XTE = η0 , corresponding to ϕH = − π/2 at s ωH. The reflection phase decreases monotonically with frequency, from 0 to −π/2, and next to −π. The TE surface wave mode is the next Bloch wave. It is also a slow-wave. The basic circuit model, i.e. the effective surface impedance model, does not show any bandgap, as the TMmode ends at ω0, and the TE-mode starts from there. The basic circuit model does not use the Floquet–Bloch PBC creating the Brillouin zone that is responsible for the bandgap. So, the bandgap is not developed in the basic circuit model. The basic circuit model also does not account for βZ,TE above the light line, showing an absence of the fast-wave as the leaky-wave radiation mode. It also does not provide any information on higher-order Bloch modes. However, the reflection phase diagram, Fig (20.11a), shows approximate bandwidth, in the gray strip, of the AMC surface. The reflection phase diagram is valid only for the normal

incidence of the plane waves on the EBG surface. It does not provide information on the polarization-sensitive reflection for the oblique incidence. It is taken into account by the improved dynamic circuit model. However, complete 2D characterization of the EBG surface is possible only by using the full-wave methods, such as the FEM or the FDTD method [B.1]. The approximate characterization of the EBG surface could be done using the 2D-circuit model discussed in section (20.4). 20.2.2

Dynamic Circuit Model

The basic circuit model of the EBG surface does not consider the interaction between unit cells. The dynamic circuit model is an analytical full-wave model. It is valid for both the TE and TM polarizations of the obliquely and also normal, incident plane waves. The dynamic model is applicable to both the mushroom-type EBG and the capacitive RIS, as shown in Fig (20.1b). The capacitive RIS, without vias, is a special case of the mushroom EBG with vias [B.3, J.39–J.44]. Figure (20.12a) shows a mushroom EBG structure on a substrate with relative permittivity εr1 = εr and thickness

x g1 z

E d1 y

d2

εr2

H

(a) Top view of the anisotropic EBG surface.

εr2 εr1

H

k η1

θ1

h

z

z

(b) TM-polarization.

Free space εr =1 Zs η Zg 0

Free space εr =1 η Zg 0

θ2

h kd

TE θ1

θ1 θ2

εr1 η1

y

θ1

z

x

E

k

TM

εr

h

Zd

kd

Surface of air-dielectric interface

g2

y

i. Transmission line model ii. Surface impedance model (c) TE-polarization. Figure 20.12

Oblique incidence of plane waves on mushroom EBG.

(d) Circuit model.

767

768

20 Planar Periodic Surfaces

h. The capacitive patch is located at the interface of the dielectric and air media, εr2 = 1. The structure is in the (x-z)-plane and the EBG has lattice constants d1 and d2 in the z- and x-directions. The gaps between adjacent patches are g1 and g2 in the z- and x-directions. Thus, the EBG surface is anisotropic. In case of the isotropic surface, d1 = d2 = d and g1 = g2 = g. The isotropic EBG surface has an identical reflection phase response for both the z- and x-polarized incident plane waves. Whereas, for an anisotropic surface, the reflection phases for both polarizations are different [J.9, J.31, J.34]. Figure (20.12b and c) shows the TM- and TEpolarizations of the incident plane waves. The plane (y-z) is the plane of incidence of the plane waves, striking obliquely to the EBG surface. The angle θ1 = θ is the angle of incidence. Inside the substrate, following Snell’s law, waves undergo refraction [B.5]. The plane waves have wavenumber k0 and kd in the air and dielectric media, respectively. The conductor-backed substrate, h < λd/4, appears as an inductive surface, with Zd dielectric surface impedance. The matrix of patches, located at the interface, is modeled as the capacitive grid impedance Zg. The surface impedance Zs of the patch matrixloaded conductor-backed substrate is a parallel combination of Zd and Zg. The surface impedance is again a parallel resonant circuit, like the simple LC-model, distributed over the surface. Figure (20.12d) demonstrates the modeling process of the surface impedance. First, the transmission line model is obtained; next, it is used to get the surface impedance model. The substrate relative permittivity is εr without a matrix of vias. The presence of vias modifies the substrate relative permittivity. The modified permittivity of the wired dielectric medium is modeled as an anisotropic medium [B.3, J.43]. The vias interact differently with the TM and TE polarized obliquely incident plane waves. For the TMpolarized waves, Fig (20.12b), both Ey and Ez components are present. The Ez component, normal to the via, does not excite it. However, the parallel Ey component induces the current in vias. The induced charges are accumulated at the ends of the vias, creating the induced dipoles. These dipoles are responsible for the creation of anisotropic relative permittivity of a wire medium, that is treated as the plasma medium. The wire medium is further discussed in section (21.1) of chapter 21. However, the densely arranged wire medium, with the period much less than a wavelength, could be treated as the PPW model. Two rows of vias arranged in the x-direction form a PPW for the Ez-component of the TM-polarized waves. The PPW is like a substrate integrated waveguide (SIW), discussed in subsection (7.8) of chapter 7. The TE-polarized waves, shown in Fig (20.12c), have only an Ex-field component that is orthogonal to the vias. So, the vias are not

excited. The TE-polarized waves have an identical response to the EBG with and without vias. For a normal incidence, the reflection-phase responses of both the TEand TM-polarized waves are identical; and are not influenced by the presence of vias. Grid Impedance of a Matrix of Patches

Figure (20.12a) shows a matrix of the rectangular patches on a dielectric substrate, forming an anisotropic EBG surface. The tangential Ez- and Ex-field components create the polarization-dependent gap capacitance, Cg. The tangential field components also induce current density Jz and Jx on the EBG surface. The polarization-dependent grid impedances, ZTM g = Ez Jz and TE Zg = Ex Jx , are obtained using the average boundary condition and also the modified Babinet principle. The TM/TE grid impedances for a matrix of the rectangular patches are summarized below [B.3, J.40, J.44]: ZTM g =

− jηeff 2αTM

a − jηeff

ZTE g = 2αTE

1 ko 1− 2 keff

2

b,

sin 2 θ d2 1 + d2 d1 d1

20 2 11 where θ is an angle of incidence. The grid parameters, αTM and αTE, are computed from the following expressions: αTM =

keff d1 1 ln π sin πg1 2d1

a

αTE =

keff d2 1 ln π sin πg2 2d2

b 20 2 12

For d1 >> g1 and d2 >> g2, the above expressions, in the simplified forms, are used for the TM- and TEpolarizations: αTM =

keff d1 2d1 ln π πg1

a,

αTE =

keff d2 2d2 ln π πg2

b

20 2 13 An anisotropic matrix of rectangular patches provides the anisotropic grid impedance, for d1 d2 and g1 g2. However, for the isotropic EBG surface, we have d1 = d2 = d, and g1 = g2 = g. Also for the normal inciTE dence, θ = 0∘ , ZTM g = Zg = Zg . The effective intrinsic impedance of the inhomogeneous medium and the effective wavenumber keff are given as ηeff = η0

εreff ,

where, εreff = εr + 1 2

keff = k0 εreff

a b

20 2 14

20.2 Circuit Models of Mushroom-Type EBG

The characteristic impedance η0 and wavenumber k0 = ω μ0 ε0 = ω c are for free space. Dielectric Surface Impedance

Figure (20.12d) shows that the conductor-backed substrate appears as an inductive surface, Zd = j ηd tan (kdh), at the EBG plane. In the case of the oblique incidence, the Zd is polarization-dependent. The waves entering the dielectric region undergo refraction and follow Snell’s law. The TE- and TM-polarized obliquely incident plane waves view the modified intrinsic impedances of the dielectric medium [B.5]. Following the discussion of subsection (5.2.4) of chapter 5, these modified intrinsic impedances are written as follows: ηd cos θ2

ηTE d =

ηd = η0 ε r ,

where,

ηTM d = ηd cos θ2

a,

sin θ2 =

c,

sin θ εr

b

Using the above equations and Fig (20.12b and c), the TE-and TM-polarized dielectric surface impedances are obtained as follows:

=j

ωμ0 tan kyd h , cos θ2 kd

ZTE d = jωμ0

ZTE d =j

tan kyd h ωμ0 cos θ2 kyd cos θ2

tan kyd h kyd

a

ωμ0 cos θ2 tan kyd h kd ωμ0 =j cos θ2 tan kyd h kyd cos θ2

ωμ = j 0 cos 2 θ2 tan kyd h kyd

b

The TE-polarized dielectric surface impedance is independent of the angle of incidence θ, whereas for the TM polarization, it is dependent on θ. The y-directed normal component of the wave vector, kyd, is obtained from the following dispersion relation: +

k2yd

+

k2z

=

TE TM TE TM Zg TE TM TM Zd + ZTE g

Zd

= ω μ0 ε0 εr 2

ZTE s =

jωμ0 tan kyd h kyd tan kyd h F 1 − 2keff αTE kyd

where, F = 1 − = ZTM s

1 ko 2 keff

2

a

sin 2 θ d2 1 + d2 d1 d1

jωμ0 tan kyd h cos 2 θ2 kyd tan kyd h cos 2 θ2 1 − 2keff αTM kyd

where, cos 2 θ2 = 1 −

sin 2 θ εr

b

Similarly, the surface impedance of an anisotropic EBG surface can also be obtained. Another set of approximate expressions for the polarization-dependent isotropic surface impedance of the mushroom EBG is also available. In this case, the polarizationTM are treated as dependent grid impedances ZTE g and Zg the impedances of a FSS on an air-substrate, parallel connected to the dielectric surface impedance [B.3]: − jη0 2α cos 2 θ

a,

ZTM g =

− jη0 2α

1 2

20 2 18

b

20 2 21

The TE -polarized incident waves are not influenced by vias, and for the TM-polarized waves, it is modeled as a PPW. On using the above expression, with the expressions for Zd and approximation ωμ0 tan(kydh)/kyd ≈ ωη0h, the following alternate expressions are obtained for the surface impedance of an isotropic EBG surface [B.3]: jk0 η0 h a ZTE s ω, θ = 1 − 2k0 hα cos 2 θ ω, θ = ZTM s

jk0 η0 h 1 − 2k0 hα

b

20 2 17

For the plane waves incident in the (y-z)-plane, the surface waves propagate in the z-direction, i.e. kx = 0 and kz = k0 sin θ. The above equation provides kyd: k2yd = ω μ0 ε0 εr − sin 2 θ

20 2 19

Using equations (20.2.11) and (20.2.16), with the above equation, the following expressions are obtained for the isotropic surface impedance of the EBG surface for both the polarizations:

ZTE g =

20 2 16

k2xd

TM

20 2 20

=j

ZTM d

ZTE s

ηd tan kyd h cos θ2

TM ZTM d = jηd tan kyd h = jηd cos θ2 tan kyd h

ZTM d

Figure (20.12d) shows that the polarization-dependent surface impedance is a parallel combination of Zg and Zd:

d 20 2 15

TE ZTE d = jηd tan kyd h = j

Surface Impedance of EBG Surface

20 2 22 ∘

For the normal incidence (θ = 0 ), the surface impedance is polarization-independent. At the resonance frequency of the EBG surface, the surface impedance has a pole, i.e. the denominator of the surface impedance is zero. The resonance frequency for the TE-polarization

769

20 Planar Periodic Surfaces

is θ dependent, whereas for the TM-polarization, it almost θ independent. Reflection Coefficient and Dispersion Relation of EBG Surface

The plane waves, striking the isotropic EBG surface, view the intrinsic impedance of the free space as η0/ cos θ and η0 cos θ for the TE- and TM-polarized waves, respectively. The polarization-dependent reflection coefficients of both polarizations, following subsection (5.2.4) of chapter 5, are obtained as follows: ΓTE ω, θ =

ZTE s ω, θ − η0 cos θ ZTE s ω, θ + η0 cos θ

a

ΓTM ω, θ =

ZTM ω, θ − η0 cos θ s ZTM ω, θ + η0 cos θ s

b 20 2 23

3 2

100

θ = 0° θ = 60° Model θ = 85° Fourier (TE) all MOM (TE) θ

1 2 3

1 50

In the above equations, surface impedance TM ZTE ω, θ is taken from either equation (20.2.20) s or (20.2.22). For the normal incidence (θ = 0∘) on the isotropic surface, the reflection phase is polarization-independent. However, for the oblique incidence, the reflection phase ∠ΓTE(ω, θ) depends on the angle θ. Such dependence is weak for the reflection phase ∠ΓTM(ω, θ). The surface impedance, equation (20.2.22), also shows unstable θ dependent resonance frequency of the mushroom EBG surface [B.3]. However, more accurate expression (20.2.20) provides more stable resonance frequency. Figure (20.13a and b) shows polarization-dependent reflection phases for the obliquely incident plane waves on the isotropic EBG of the patch grid [J.40]. The isotropic EBG has lattice constant d = 2 mm, gap g = 0.2 mm on a substrate with εr = 10.2, h = 1 mm. The analytical model closely follows two full-wave results. The

150 Reflection phase (Deg.)

150 Reflection phase (Deg.)

0 –50

–100

1 2 3

1 2

100 50

3

θ = 0° θ = 60° Model θ = 85° Fourier (TM) all MOM (TM) θ

0 –50 –100 –150

–150 8

10

12 14 Frequency (GHz)

16

18

8

10

(a) Reflection of TE-polarization.

12 14 Frequency (GHz)

16

18

(b) Reflection of TM-polarization.

25 TM mode (Analytical) 1 TE mode (Analytical) 2 TM mode (HFSS) TE mode (HFSS)

20 Frequency (GHz)

770

Light line

2

15

1

10

5 0

100

200

300 400 β (1/m)

500

600

700

(c) Dispersion in EBG. Figure 20.13

Polarization-dependent reflection coefficient and dispersion of the isotropic EBG surface. Source: Reproduced from Luukkonen et al. [J.40]. © 2008, IEEE.

20.3 Uniplanar EBG Structures

resonance frequency of the TE- polarization is almost stable with respect to the angle of incidence θ, whereas for the TM-polarization, it is more unstable.

capacitances can be used with the basic LC-circuit model, to compute the resonance frequencies, and also the surface impedances for both the anisotropic and isotropic EBG surfaces.

Dispersion Model of EBG Surface

The dynamic analytical circuit model can also compute the dispersion of the first two TM and TE Bloch modes, with better accuracy as compared to the basic model. Using the transverse resonance method, discussed in section (7.7) of chapter 7, the dispersion relation for the propagation constant kz is obtained from the following expression: 1 TE TM η0

+

1 TM ZTE s

=0

20 2 24

For the oblique incidence, frequency-independent TE TM η0

is taken as η0/ cos θ, and η0 cos θ for the TEand TM-polarized waves, respectively. The surface TM impedance ZTE is taken from equation (20.2.20), or s also from equation (20.2.22). However, like the basic model, it also provides only the 1D dispersion, without any Bragg’s reflection. Again, it is only applicable to the slow-waves outside the light-line cone. Figure (20.13c) compares accurately the results of the dynamic circuit model against the results of HFSS. Again, the dynamic circuit model is not able to account for the fast-waves and leaky waves existing within the light-line cone [J.40]. Polarization-Dependent Patch Capacitance

The dynamic analytical circuit model also provides the polarization-dependent patch capacitances. The polarization-dependent capacitive grid impedances can be written using the polarization-dependent patch capacitances: ZTE g

TM

=

1 TM jωCTE g

CTE g

,

TM

=

1 TM jωZTE g

20 2 25 Using equations (20.2.11) to (20.2.13) with equaTM are obtained as tion (20.2.22), expressions for CTE g CTE g = ×

d2 ε0 εr + 1 π

2d2 πg

d2 d1 sin 2 θ ε r + 1 1 + d2 d1

a

d1 ε0 εr + 1 2d1 ln πg π

b

1−

CTM g =

ln

20 2 26 The isotropic surface has d1 = d2 = d (square patch), TM and for the normal incidence, CTE g = Cg . The patch

20.3

Uniplanar EBG Structures

The uniplanar EBG structure shown in Fig (20.1d-i) is popularly called the UC-EBG. It was introduced to create a patterned ground plane, showing EBG behavior, in place of a normal solid ground conductor for the microstrip-based circuits and antennas. By suppressing the surface waves, it improves the stopband characteristics of microstrip-based components and circuits, provided the stopband of the circuits comes within the bandgap of the EBG. It also creates the AMC/RIS surface to improve the radiation performances of an antenna [J.4, J.12]. Figure (20.1d-ii) shows the LC-parallel resonant-type circuit model of the surface impedance of a UC-EBG. It is like the case of a mushroom-type EBG. Figure (20.14a-i) shows that two adjacent unit cells are connected by a narrow microstrip line, providing the inductance L of the model. The value of L is estimated by equation (19.3.54b) of chapter 19, or simply by equation (20.2.1b). The capacitance C is the gap capacitance between two adjacent patches, given by equation (20.2.1a). Alternatively, equation (20.2.26) can also be used. The surface impedance resonant frequency and associated bandwidths are estimated using equations (20.2.2) and (20.2.6), respectively. Due to the presence of the ground plane, additional shunt capacitances Cp are added to get the π-type equivalent circuit model, shown in Fig (20.14a-ii) of a UC-EBG. The LC resonant circuit is in the series arm of the π-model. The Cp is a parallel plate capacitor between the UC-EBG and the ground conductor. The AMC is created around the resonance frequency, between ±90∘ reflection phases. The UC-EBG constructed on an RT-Duriod 6010 substrate, with εr = 10.2, h = 25 mil has the stopband at resonance frequency 12 GHz. Substrates of very high relative permittivity εr = 140 or 200 are used to get lower stopband resonance frequency up to 5 GHz [J.45]. Figure (20.14b and c) suggests a few schemes to inductively load the UC-EBG that help to bring down the resonance frequency, and also to enhance the bandwidth of the EBG surface [J.44–J.47]. Figure (20.14d) demonstrates an application of UC-EBG. The UC-EBG surrounds a patch antenna to improve its directive gain, and also its radiation characteristics, by suppressing the surface waves on the substrate [J.33]. Figure (20.14e) presents an UC-EBG with the fractal unit cell to get the multiband AMC. The smallest patch of the

771

772

20 Planar Periodic Surfaces Unit cell

w

L

Cg

CP

L

Cg

Thin microstrip line b s

a CP

g a a = 620, b = 200, s = 20, g = 20, ii. π - circuit model of a w = 420,d = 400 mils. Width unit cell 50 Ω microstrip = 48 mils. ii. Very compact EBG i. Two adjacent cells of EBG [J.46]. [J.47]

i. Unit cells connected by narrow microstrip

(a) Standard UC-EBG.

Conductive patch

(b) Meandered line loaded EBG.

Conductive meander line

Substrate

εr, δ

h

Ground plane

(c) Meander line connected array of patches [J.45].

(d) UC-EBG surrounded microstrip patch antenna [J.33]. g

Wl

2 Wf

Top layer

Li

2 Wi

W

h1 h2

Arlon AD1000 Caclad 6250

h3

Floating layer

Arlon AD1000

Ground plane

(a) EBG surface.

(b) Unit cell.

(c) Top surface.

(d) Cross-section showing 3 layers. Cgap

2CMIM

Conventional uni-planar EBG

Line

2CMIM

Circuit model 3 patches comprising the unit cell (f) Multilayer EBG structure. (e) Fractal EBG for multiband AMC. Figure 20.14

Some UC-EBG structures. Source: Reproduced from Coccioli et al. [J.33]. © 1999, IEEE. Reproduced from Iravani and Ramahi [J.45]. © 2010, IEEE. Reproduced from Abedin and Ali [J.46]. © 2005, IEEE. Reproduced from Waterhouse and Novak [J.47]. © 2006, IEEE.

unit cell contributes to a higher band of the AMC. The next lower band is due to the larger patch. The complete structure of a fractal patch provides the lowest AMC band [J.48].

Figure (20.14f ) shows the multilayer uniplanar EBG structure and its equivalent circuit model. The structure has used the MIM (metal insulator metal)-type capacitor loading, to get the EBG resonance in the range of a few

20.4 2D Circuit Models of EBG Structures

MHz [J.48, J.49]. It is a three-layered UC-EBG structure. The first layer h1 is Arlon AD 1000 with εr = 9.1, h = 0.127 mm. The second layer h2 is a floating metallic layer of Cuclad 250 with εr = 2.5, h = 38 μm. It rests on the third layer h3 that is again Arlon AD 1000 with εr = 10.9, h = 3.2 mm. The UC-EBG is placed at the top of the first layer. Overlap of UC-EBG with the floating conductor forms two MIM capacitors (CMIM), and are in series connected through the floating conductor. The combined CMIM appears across the LC-resonant circuit of the UC-EBG. The equivalent circuit, also given in Fig (20.14f ), computes the resonance frequency of the UC-EBG using the expression ω0 = 1

L Cg + CMIM .

The multilayer EBG has low resonance frequency and its AMC bandwidth is from 170 MHz to 320 MHz.

20.4 2D Circuit Models of EBG Structures The lumped circuit models of the EBG surface, presented in section (20.2), do not show the existence of the bandgap. However, the 1D-periodic line model of the EBG surface shows the partial bandgap. It is discussed in subsection (19.3.4) of chapter 19. The more realistic and accurate 2D circuit models of the EBG surfaces provide the complete bandgap and correct nature of the first two Bloch modes [J.50–J.54]. The 2D-unit cell, using the 2D Floquet–Bloch theorem is analyzed to get the Bloch modes propagation characteristics of the infinitely large 2D-periodic surface. During the 2D circuit-model-based simulation process, the 2D-IBZ is used to get the dispersion characteristics of Bloch modes. The simulation also computes the attenuation diagram. The 2D models are developed in two forms: i) Shunt-connected 2D planar EBG circuit model. ii) Series-connected 2D planar EBG circuit model.

20.4.1 Shunt-Connected 2D Planar EBG Circuit Model Figure (20.15a) shows the 2D circuit model of a unit cell of the mushroom-type EBG structure with and without vias. In case the EBG is made of capacitively coupled large patches over the conductor-backed ground plane, the shunt inductor, due to via, is absent. Figure (20.1ai) shows the boundaries of the 2D unit cell, a-b-c-d marked on the top view of the EBG surface. The magnetic wall-type boundary is located at the center of the coupling gap g. So, the series gap capacitance is taken as 2C in Fig (20.15a). The line section length is taken

as kw 2, k = k0 εr . The line section is formed from the PPW between the patch and the ground plane. The isotropic EBG surface is realized for identical periodicity, i.e. lattice constant d in both x- and y-directions. For an anisotropic surface, the lattice constants are d1 and d2 in the x-direction and the y-direction, respectively. The period of the isotropic EBG surface is d = w + g, where w is the width of the square patch, and g is the gap between adjacent patches. Figure (20.15b) shows the 2D [ABCD] transmission matrix representation, i.e. the pair of input (T1, T2) and pair of output (T3, T4) matrices of a unit cell. Figure (20.15c) shows the 4-port matrix network with the periodic boundaries (PB) satisfying Floquet–Bloch periodic conditions at the ports. At the node #O, the outputs of transmission matrix pair (T1, T2), inputs of the transmission matrix pair # (T3, T4), and the fifth matrix network (T5), a model of the grounded via, are connected in parallel. The transmission matrices, like the 1D-periodic line case, are obtained from the cascaded matrices of the series-connected capacitor (2C) and the transmission line section kd/2 of characteristic impedance Z0. The transmission matrices are reciprocal matrices resulting in the following expression for their determinants: Δ1 = Δ2 = Δ3 = Δ4 = 1. The positions of the diagonal elements of output matrices # T3 and T4 are interchanged, i.e. A3 D3 and A4 D4, due to the order of physical placements of the capacitors and line sections with respect to the node # O. In general, the 2D circuit applies to the anisotropic and nonreciprocal surfaces also. However, the further discussion considers only the isotropic and reciprocal EBG surface, due to the simplicity and popularity. The input transmission matrix pair (T1, T2) and the output transmission matrix pair (T3, T4), in the x- and y-directions, are defined as follows:

Input matrix along x − diection

Input matrix along y − direction

Output matrix along x − diection

Output matrix along y − direction

T1 x =

T2 y =

T3 x =

T4 y =

A1

B1

C1

D1

A2

B2

C2

D2

D3

B3

C3

A3

D4

B4

C4

A4

a

b

c

d

20 4 1 Figure (20.15c) shows the port voltages and port currents of the 2D unit cell. The voltage and current at the ports #1, #2, #3, and #4; taking care of the 2D

773

20 Planar Periodic Surfaces

Y

2C TL

Z0

Z0

Z0 PB

TL

X

TL

PB

kd/2

B1

C1

D1

––

(i) Input transmission matrix #T1. –jβxd I″x + + Ix e D3 B3 V″x Vx e–jβxd C3 A3 – –

2C

L

A1

V′x

–

TL

Z0

2C

Vx

PB

kd/2 O

+ I′x

+

Ix

2C

PB

(ii) Output transmission matrix #T3.

PB

+

+ Vx –

+ –βyd Vy e –

I″y + I′x

T1

I′y

+

+O

V′x –

Vo

V′y

–

+

PB Iv

T2

–

T4

V″y –

+ I″x

V″x

–

PB Ix e–jβxd

T3

–

+

Vy

Iv e–βyd

PB

Output port #4

Input port #1

Ix

(b) x-directed input and output transmission matrices.

Output port #3

(a) 2D - circuit model of mushroom EBG unit cell.

Input port #2

774

+ I′z

Vx e–jβxd

+

y

– T5

Iz

–

z

(c) Analytical model of a 2D unit cell. Figure 20.15

4-Port network of 2D-circuit model of mushroom-type EBG surface in (x-y) plane.

Bloch–Floquet periodic conditions, at the periodic port (PB), are defined as follows: V1 = Vx , V3 = Vx e − jβx d ; I1 = Ix , I3 = Ix e − jβx d V2 = Vy , V4 = Vy e

− jβy d

; I2 = Iy , I4 = Iy e

− jβy d

Vx Ix Vx Ix

a b 20 4 2

Figure (20.15b-i) shows x-directed voltage and current pair Vx , Ix at the output of the input transmission matrix # T1. Similar relation holds for y-directed voltage and current pair Vy , Iy at the output of the input transmission matrix # T2. The voltage and current are computed as follows:

Vx =

=

=

A1

B1

Vx

C1

D1

Ix

1 Δx

D1 − C1

D1 B1 Vx − Ix Δx Δx

Likewise, Vy = Iy = − where,

c,

a

− B1 A1

Vx Ix

b

Ix = −

C1 A1 Vx + Ix Δx Δx

d

D2 B2 Vy − Iy Δy Δy

e

C2 A2 Vy + Iy Δy Δy Δx = A1 D1 − B1 C1

f g,

Δy = A2 D2 − B2 C2

h

20 4 3

20.4 2D Circuit Models of EBG Structures

Figure (20.15b-ii) shows the output transmission matrix T3 with the input voltage and current pair Vx , Ix and the output voltage and current pair − jβx d , Ix e− jβx d . The voltage and current pairs are corVx e related as follows: Vx

C1 D5 D3 e − jβx d Vx − C3 e − jβx d − Δx B5 A1 D5 − A3 e − jβx d − B3 e − jβx d Ix Δx B5

B3

Vx e − jβx d

+

C3

A3

Ix e − jβx d

+ −

Vx = D3 e − jβx d Vx + B3 e − jβx d Ix Ix = C3 e

−

D3 =

Ix

Use of the KCL at node #O, i.e. Ix + Iy − Ix − Iy − Iz = 0, results in the following equation:

− jβx d

Vx + A3 e

− jβx d

C2 A2 − C4 e − jβy d Vy + − A4 e − jβy d Iy = 0 Δy Δy

a

Ix

b

Vy = D4 e − jβy d Vy + B4 e − jβy d Iy

c

Likewise, in the y − direction,

20 4 8b By combining equations (20.4.5) and (20.4.8), the following matrix equation is obtained: Vx Vy

M

Iy = C4 e − jβy d Vy + A4 e − jβy d Iy

Ix Iy

d 20 4 4

The voltages at node # O are continuous, i.e. Vx = Vx , and Vy = Vy. Under the continuity condition, the following equations are obtained from equations (20.4.3) and (20.4.4): D1 B1 Vx − Ix = D3 e − jβx d Vx + B3 e − jβx d Ix Δx Δx D1 B1 − D3 e − jβx d Vx − + B3 e − jβx d Ix = 0 a Δx Δx D2 B2 Likewise, − D4 e − jβy d Vy − + B4 e − jβy d Iy = 0 b Δy Δy

= 0,

V0 Iz Iz =

=

A5 C5

B5 D5

0 , Iz

b 20 4 6

At node #O, V0 = Vx , and use of equation (20.4.4a) with the above equation provides current in the shunt arm, Iz =

D5 B5

D3 e − jβx d Vx + B3 e − jβx d Ix

20 4 7

At node#O, again Vx = Vy = V0 , and use of equation (20.4.4a) provides D3 e − jβx d Vx + B3 e − jβx d Ix − D4 e − jβy d Vy − B4 e − jβy d Iy = 0

Vx

0 f 31

0 f 32

f 23 f 33

f 24 f 34

Ix

f 41

f 42

f 43

f 44

Vy Iy

D1 − D3 e − jβx d , Δx

f 12 = −

B1 + B3 e − jβx d Δx

f 23 =

D2 − D4 e − jβy d , Δy

f 24 = −

B2 + B4 e − jβy d Δy

f 31 = D3 e − jβx d ,

= 0,

f 32 = B3 e − jβx d , f 33 = − D4 e − jβy d

f 34 = − B4 e − jβy d , f 41 = − f 42 =

C1 D5 D3 e − jβx d − C3 e − jβx d − Δx B5

A1 D5 − A3 e − jβx d − B3 e − jβx d Δx B5 C2 A2 − C4 e − jβy d , f 44 = − A4 e − jβy d Δy Δy

20 4 9b

By eliminating Ix and Iy from equation (20.4.9a), equations are obtained involving only Vx and Vy. From the first and second rows of the above equation, the following expressions are obtained: f 11 Vx + f 12 Ix = 0, D1 − D3 Δx e − jβx d Ix = Vx B1 + B3 Δx e − jβx d Likewise, D2 − D4 Δx e − jβx d Vy Iy = B2 + B4 Δx e − jβx d

a

b 20 4 10

However, the reciprocal network has Δx = A1D1 − B1C1 = 1. The above equation is reduced to Ix =

20 4 8a

0

f 11 =

a

D5 V0 B5

0

where

f 43 = − V0 = B5 Iz

f 12

20 4 9a

20 4 5 Across the node #O, the following relation holds for the matrix [T5]:

f 11

D1 − D3 e − jβx d B1 + B3 e − jβx d

a,

Iy =

D2 − D4 e − jβx d b B2 + B4 e − jβx d 20 4 11

775

776

20 Planar Periodic Surfaces

On substituting Ix and Iy in the third and fourth rows of equation (20.4.9a), the expressions are obtained, involving Vx and Vy. These equations are arranged in the following form:

g11 =

B1 D3 + B3 D1 e − jβx d B1 + B3 e − jβx d

a,

g12 =

Vx

G

Vy

=

g11 g21

− g12

Vx

g22

Vy

= 0,

20 4 12

where

B2 D4 + B4 D2 e − jβy d B2 + B4 e − jβy d

b

g21 =

1 + e − j2βx d − e − jβx d B1 C3 + C1 B3 + A1 D3 + D1 A3 + B1 D3 D5 + D1 B3 D5 B5 B1 + B3 e − jβx d

c

g22 =

1 + e − j2βy d − e − jβy d B2 C4 + C2 B4 + A4 D2 + D4 A2 B2 + B4 e − jβy d

d 20 4 13

For the nontrivial solution of equation (20.4.12), ΔG = 0. It leads to the following characteristic equation: g11 g22 + g12 g21 = 0 Me − jβx d 1 + e − j2βy d − Pe − jβy d

cos βx d + cos βy d + 2 − 4AD − BDD5 B5 = 0

M 2 cos βy d − P + N 2 cos βx d − Q = 0,

M = B1 D3 + B3 D1 , N = B2 D4 + B4 D2 , P = B2 C4 + B4 C2 + A4 D2 + A2 D4 Q = B1 C3 + B3 C1 + A1 D3 + A3 D1

cosh γx d + cosh γy d − 2 BC + AD − BDD5 B5 = 0

+ B 1 D3 D5 + D1 B 3 D5 B 5 For an isotropic 2D-EBG surface, [T1] = [T2] = [T3] = [T4], i.e. A = A1 = …A4; B = B1 = … B4; C = C1 = … C4; and D = D1 = … D4. However, for an anisotropic surface, the x- and y-directed matrices are different. For isotropic EBG surface, the variables M, N, P, and Q are reduced to M = 2BD, N = 2BD, P = 2 BC + AD

cos

kd 2

20 4 15

+

j kd sin 2 Z0 2

b

The above expressions show the dispersion relation of the 2D lossless, isotropic, and reciprocal EBG medium. In the case of a lossy EBG surface, the above dispersion relation is modified as

where

B = D

a

20 4 16

20 4 14

A C

cos βx d + cos βy d − 2 BC + AD − BDD5 B5 = 0 or using AD – BC = 1 for a reciprocal network,

+ Ne − jβy d 1 + e − j2βx d − Qe − jβx d = 0,

Q = 2 BC + AD + BDD5 B5

Equation (20.4.14) is reduced to the following dispersion relations [J.35]:

1 kd sin 2 Z0 2ω C 2

20 4 17 For the 2D circuit of an isotropic EBG surface, shown in Fig (20.15a), the [ABCD] matrix is obtained from the cascading of the [ABCD] matrices of a series capacitor 2C, followed by the line section kd/2 of characteristic impedance Z0. For the mushroom EBG, kd/2 = (w + g)k/2d ≈ wk/2d. The physical line length is d and k is the propagation constant, k = ω μ0 ε0 εr , of the host line. The results are given below: kd 1 kd − cos 2 2ω C 2

j Z0 sin cos

The [ABCD] matrix of the shunt branch of Fig (20.15c) is A5 B 5 1 jω L = 20 4 19 C5 D5 0 1 On substituting the above equations in the dispersion relation (20.4.16), the following dispersion relation is obtained for the 2D circuit shown in Fig (20.15a):

20 4 18

kd 2

sin 2 −

βx d 2

+ sin 2

1 kd cos Z0 ωC 2

βy d 2 2 sin

=

1 kd 2 sin 2 2

kd Z0 kd cos − 2ωL 2 2 20 4 20

The dispersion diagram, shown in Fig (20.16a), is plotted for the variation in the x-directed propagation

20.4 2D Circuit Models of EBG Structures

ky d

c b a

d2

Gray strips: Stopbands

y

Bloch mode #2 M Γ

0Γ (0,0)

X (π,0)

M (π,π)

Γ w1 d1

x

X

kx

Brillouin zone

10

X

Backward wave band Bloch mode #1

M

Y

w2

kd = π/9 at 1 GHz Z0 = 100 Ω L = 11.278 nH C = 3.009 pF

a Γ (0,0)

Propagation constant (βd)

Frequency (GHz)

kd = ωd/Vp (Radians)

π

8 6 4 2 0

M Y (π,π) (0,π)

Γ (0,0)

X (π,0)

M (π,π)

Γ (0,0)

Propagation constant (βd) (a) 2D-dispersion diagram of backward wave supporting EBG [J.50]. Figure 20.16

(b) 2D- dispersion diagram of forward waves supporting EBG. [J.36].

Dispersion diagram of the 2D EBG surface. The gray strips show the complete stopband in the azimuthal plane. Source: Reproduced from Kamgaing and Ramahi [J.36]. © 2005, IEEE. Reproduced from Gribic and Eleftheriades, [J.50]. © 2003, IEEE.

(π/d, 0); the variation in the βy from the location X(π/d, 0) to M(π/d, π/d); and finally the variation in the (βx, βy) from M(π/d, π/d) to Γ(0, 0). This process scans the edges of the IBZ of the square lattice, shown in the inset of Fig (20.16a). The assumed component values are also shown there [J.50]. In the region (Γ − X), βx is decreasing with increasing frequency, causing propagation of Bloch mode #1 as the backward wave (BW). It is due to the series capacitor and shunt inductor loading. Its passband #1 starts at the frequency location (a), corresponding to the location M, on Fig (20.16a). The bandgap #1 exists from frequency 0 to a. The partial bandgap #1, in the range Γ to X, is wider. Bloch mode #1 is followed by a narrow complete (i.e. the global) bandgap #2 between frequencies b and c. The Bloch mode #2, from frequency c to π, supports the forward-mode propagation in the passband #2. The dispersion expression (20.4.17) is applicable to a lossy mushroom EBG also, inside the PPW, as shown in Fig (20.1c). This EBG structure is used to suppress the switching noises of the high-speed digital circuits [J.53]. The dispersion expression can be rewritten for the series-connected impedance Z1 and the shuntconnected impedance Z2, shown in the inset of Fig (20.17a), as follows:

cosh γx d + cosh γy d = +

2Znor 1 +

2+

Znor 1 cosh γd 2Znor 2

1 Znor 1 nor sinh γd + 2Znor 2Z2 2

a

nor where, Znor 1 = Z1 Z0 , Z2 = Z2 Z0 , Z0 = η0 h d εr ,

γ = α + jk = j ω c

εr

b 20 4 21

In the above equation, η0 is the intrinsic impedance of free space, c is the velocity of EM-wave in free space, and the substrate is taken as lossless with α = 0. The structure without the top shield is considered first. The structure has a period of d=10mm. A square patch of width w = 9.6 mm and gap g = 0.2 mm from the adjacent patch is taken on a substrate with εr = 2.33, h = 3.08. Figure (20.17a and b) show the dispersion diagram and attenuation diagram, respectively [J.53]. Figure (20.17a) also shows the side-view of the structure and its equivalent circuit. The impedance Z1 is due to the coupling capacitance Cc, and impedance Z2 models centrally located via as an inductor. Subsection (19.3.6) of chapter 19 summarizes the model for a via. The computed values of components are shown in Fig (20.17b). Bragg’s boundary points, “a” to “e,” showing edges of

777

20 Planar Periodic Surfaces

12

12

3rd bandgap d

w

d

w

εr

8

M–Γ X–M Γ–X

Forward waves

b

6

d

10

h

Z1

Z1 Z0, γ

Z2

4

f

Frequency (GHz)

10

Frequency (GHz)

Cgap = 782 fF Lvia = 1.72 nH Z0 = 80 Ω

8

b

6 4

f e

2nd bandgap

Backward waves

2

a 0

Γ

e

2 0

Γ

(a) Dispersion diagram without the top shield.

bandgap (2D model & FEM)

bandgap (1D model) d

w

h1=16 μm h2 = 0.1mm

f0

Frequency (GHz)

16

d = 2.2 mm W=2 mm

εr1 = 30 εr2 = 2.33

M–Γ Γ–X X–M

12

Z0, γ C2

8

C1 L

4

f02

f01 X M Propagation constant

Γ

(c) Dispersion diagram with a top shield. Figure 20.17

10

C1 = 72.9 pF C2 = 0.91 pF L = 49.8 pH

1D 2D FEM

12

0 Γ

2 4 6 8 Attenuation constant Np

20

16

4

0

(b) Attenuation diagram without the top shield.

20

8

a c

c

1st bandgap

X M Propagation constant

Frequency (GHz)

778

0

0

1

2 3 4 5 6 7 Attenuation constant Np

8

9

(d) Attenuation diagram with a top shield.

The dispersion and attenuation diagrams of the mushroom-type EBG outside/inside a PPW. Source: Reproduced from Tavallaee and Abhari [J.53]. © 2007, IET.

frequency bands, are shown in the diagram. The dispersion diagram follows the previous discussion. In this case, at several frequencies, between 0 and 10 GHz, both the propagation constant and the attenuation constant are computed, using the expression (20.4.21a). Both the complete and partial bandgap are obtained. The global complete bandgap is shown in gray. In bandgap region, attenuation αx, αy, αxy are nonzeros, showing the presence of the evanescent mode in all directions.

In the range c-a, attenuation is not present and the partial bandgap is enhanced in some directions. Figure (20.17c and d) shows the dispersion and attenuation diagrams of the EBG inside a PPW. The PPW structure is shown in the inset of Fig (20.17c). The circuit model of the unit cell is modified. It is shown in the inset of Fig (20.17d), along with the values of components. In this case, Z1 = 0, and the Z2 is a series combination of capacitance C1 and C2-L parallel circuit. This model is

20.4 2D Circuit Models of EBG Structures

a little different from that given in subsection (19.3.4) of chapter 19. In the present model, C1 is the parallel-plate capacitor between the patch and top shield, C2 is the coupling capacitance Cc between adjacent patches, and L is due to the inductance of via. The PPW is having an inhomogeneous dielectric medium that is replaced with an equivalent homogeneous medium of relative permittivity εreq, and thickness h = h1 + h2. The PPW is modeled as a transmission line section of length γd/ 2 and characteristic impedance Z0. The expressions used in the modeling are summarized below: Z2 =

1 1 + jωC1 j ωC2 − 1 ωL

εreq =

a

jB D tan jB = − D tan

ZBloch = − x Likweise, ZBloch y

1 βx d 2 1

a b

βy d 2

20 4 25 Bloch impedance provides a relation between the voltage and current at the same port. However, a relation between the cross-ports, i.e. between Vx and Vy, is calculated by using equations (20.4.12), (20.4.13a), and (20.4.13b): Vx =

g12 B2 D4 + B4 D2 Vy = g11 B4 + B2 e + j βy d

B3 + B1 e + j βx d Vy B1 D3 + B3 D

a

For an isotropic EBG surface,

h1 + h2 h1 εr1 + h2 εr2

b 20 4 22

Vx =

1 + e + j βx d Vy 1 + e + j βy d

b 20 4 26

The propagation relation for this case, for Z1 = 0, is obtained from equation (20.4.21). Figure (20.17d) shows two resonances giving no attenuation (at the parallel resonance, f01) and high attenuation (at the series resonance, f02) in the attenuation diagram. Figure (20.17c) shows that, unlike the previous case, this structure supports the forward Bloch mode, and also a wide bandgap. The dispersions computed from the 1D model and the FEM also are shown in Fig (20.17c) [J.55, J.56]. The 2D circuit model is closer to the results of the FEM method, whereas the 1D circuit model is not so accurate.

Therefore, for the known Vy, the Vx is obtained. Likewise, for the known Ix, the Iy is obtained. The voltages and currents of 4-ports are related through βxd and βyd. Special Case – Uniplanar 2D EBG Structure

The uniplanar EBG surface, shown in Fig (20.16b), is a special case of the mushroom-type EBG. The square capacitive patches are interconnected by the narrow inductive microstrip sections of the patch that form the UC-EBG shown in Fig (20.1d). The shunt arm transmission [T5] matrix, shown in Fig (20.15c), is absent, i.e. the shunt arm is an open-circuited terminal, Zshunt ∞,

Bloch Impedance

Bloch impedances of Bloch waves, both in the x- and ydirections are computed below [J.50]. Bloch impedances are defined as ZBloch = x

Vx Ix

a,

ZBloch = y

Vy Iy

b 20 4 23

= − ZBloch x

f 23 Vy + f 24 Iy = 0

b

− j βx d

Vx f 12 B1 + B3 e = = for Δx = 1 Ix f 11 D1 − D3 e − j βx d

For an isotropic 2D EBG surface, B1 = B3 = B and D1 = D3 = D. The above expression of Bloch impedance is reduced to

D5

f 41 = −

1

Zshunt

0

1

,

B5 = ∞

C1 − C3 e − jβx d , Δx

f 42 =

A1 − A3 e − jβx d Δx 20 4 28

Likewise, g11, g12, and g22 of equation (20.4.13) remain unchanged. However, g21 is modified to

c

20 4 24

C5

=

The parameters of equation (20.4.9) remain unchanged, except f41 and f42. These are modified using B5 ∞ for the uniplanar EBG, as follows:

The following expressions are obtained from the 1 two rows of equation (20.4.9a): a,

B5

20 4 27

st

f 11 Vx + f 12 Ix = 0

A5

g21 =

1 + e − j2βx d − e − jβx d B1 C3 + C1 B3 + A1 D3 + D1 A3 B1 + B3 e − jβx d 20 4 29

The reciprocal and isotropic EBG surface has d1 = d2 = d; A = A1 = … A4; B = B1 = … B4; C = C1 = … C4;

779

780

20 Planar Periodic Surfaces

and D = D1 = … D4, and Δx = Δy = 1. The expressions for g11, g12, g21, and g22, given in equations (20.4.13) and (20.4.29), are reduced to g11 = g21 =

2De − jβx d 1 + e − jβx d

a,

g12 =

2D e − jβy d 1 + e − jβy d

1 + e − j2βx d − 2 BC + AD e − jβx d B 1 + e − jβx d

g22 =

1 + e − j2βy d − 2 BC + AD e − jβy d B 1 + e − jβy d

b c d 20 4 30

The dispersion relations for both the lossless and lossy uniplanar EBG surfaces are obtained by substituting the above parameters in the following expression from equation (20.4.12):

center, the position of matrix elements A and D are interchanged for the output matrices in the x- and ydirections. The present analysis is applied to the isotropic and reciprocal EBG surface. However, the method, in general, applies to the anisotropic and non-reciprocal EBG surface also. Figure (20.18a) shows the transmission matrices network of the unit cell of the EBG surface of Fig (20.18b). Figure (20.18c) further shows the equivalent 2D circuit diagram of the EBG surface. Applying the KCL and KVL to the network of Fig (20.18a), the following relations are obtained at the junctions of outputs of the input matrices and inputs of the output matrices: KCL Ix = Ix , Iy = Iy , Ix = − Iy ,

a

KVL Vx + Vy − Vx − Vy = 0

b 20 4 32

g11 g22 + g12 g21 = 0 cos βx d + cos βy d = 2 AD + BC

a

Or accounting for loss cosh γx d + cos γy d = 2 AD + BC

b 20 4 31

The expressions of Bloch impedances, equation (20.4.25), remain unchanged. Moreover, this model is also applicable to UC-EBG, as is shown in Fig (20.1d). The inset of Fig (20.1d) shows another form of the UC-EBG, large metallic patches are connected by the narrow metallic strips. The circuit model is also applicable to the planar lumped element EBG, as shown in Fig (20.1f ). In all cases, the matrix elements are evaluated using appropriate circuit element models. Figure (20.16b) shows that the dispersion in the uniplanar EBG starts with a passband, supporting the forward Bloch mode #1 [J.33, J.36, J.51, J.53]. This characteristic is different from that of the mushroom-type structure, where Bloch mode #1 is a BW. 20.4.2 Series-Connected 2D Planar EBG Circuit Model Figure (20.18a) is a series-connected 4-port network, corresponding to the 2D EBG surface with the seriesconnected loading elements, shown in Fig (20.18b). The EBG constitutes of reactively loaded coplanar strip lines to develop the isotropic surface. The interdigital capacitors and meandered inductors are connected in series with the coplanar strip lines. Figure (20.18c) further shows its equivalent circuit. The equivalent circuit is analyzed below, using the [ABCD] network model [J.54]. Due to the isotropy, the matrices are identical. They further satisfy the reciprocity relation Δ = AD − BC = 1 for the reciprocal EBG surface. Due to the physical placement of components, with respect to the

The set of equations (20.4.3) provide relations between the input and output voltages/currents of an individual matrix. Equations (20.4.4a) to (20.4.4d) relate the input and output of the x- and y-directed voltages/currents, using Floquet–Bloch PBCs. On using the set of equations (20.4.3) and (20.4.4), with equations (20.4.32a) and (20.4.32b), the following expressions are obtained: C 1 + e − jβx d Vx + A e − jβx d − 1 Ix = 0 C 1+e Ce

− jβx d

− jβy d

Vy + A e

Vx + Ae

+ Ae − jβy d Iy = 0

− jβx d

− jβy d

− 1 Iy = 0

Ix + Ce

− jβy d

b Vy

c , D 1 − e − jβx d Vx − B 1 + e − jβx d Ix

+ D e − jβy d − 1 Vy + B 1 + e − jβy d Iy where,

a

d

A1 , …A4 = A; B1 , …B4 = B; C1 , …C4 = C; D1 , …

D4 = D; Δ = 1 20 4 33 The above equations are rearranged in the matrix form, given by equation (20.4.9a), where the following expressions compute the matrix elements: f 11 = C 1 + e − jβx d , f 12 = A e − jβx d − 1 , f 23 = C 1 + e − jβy d f 24 = A e − jβy d − 1 , f 31 = Ce − jβx d , f 32 = Ae − jβx d , f 33 = Ce − jβy d f 34 = Ae − jβy d , f 41 = D 1 − e − jβx d , f 42 = − B, e − jβx d + 1 f 43 = D e − jβy d − 1 , f 44 = B e − jβy d + 1

20 4 34 Using the above relations with the first two rows of equation (20.4.9a), the following expressions are obtained for the Ix and Iy:

20.4 2D Circuit Models of EBG Structures

Vye–jβy PB#4

CPS strips 1.2 mm

Iye–jβy

10 mm

D B C A

I′x

Ix

+ Vx – PB#1

A B C D

Y

Interdigital figure length: 0.5 mm figure width:100 μm figure gap:100 μm

I″y – + I″ + V″y + x V′x V″x – – – + V′y I′y

Ix e–jβx + D B Vx e–jβx C A – PB#3

A B C D

X

Meandered line width: 100 μm

Substrate

Iy PB#2 – + Vy (a) 2D unit cell of series-connected 4 ports network.

(b) Layout of stripline-based EBG loaded with the series capacitor shunt inductor.

5.0 FW band

4.0

2L z

2C

2C 2C

y

2C

2C

βd/2, Z0

2C

Frequency (GHz)

2L

100 μm

2L

2C 2C

100 μm 1.2 mm

βd = –0.3954 at 1.2 GHz Z0= 100 Ω d = 10 mm L = 15 nH C = 4 pF

3.0 μeff and εeff defined

M Γ

2.0

X

BW band 1.0

100 μm

x

2L

0

Γ

X

Wave vector (d) Dispersion diagram of structure - (d).

(c) 2D equivalent circuit of the unit cell. Figure 20.18

Γ

M

Series-connected 2D uniplanar EBG. Source: Reproduced from Elek and Eleftheriades [J.54]. © 2007, IET.

f 11 Vx + f 12 Ix = 0,

f 23 Vy + f 24 Iy = 0

− jβx d

C 1+e Vx Ix = A 1 − e − jβx d

a,

C 1 + e − jβyd Iy = Vy A 1 − e − jβy d

b

20 4 35 The further relation between Vx and Vy is obtained by substituting Ix and Iy in the expressions from the third and fourth rows of equation (20.4.9a): f 31 Vx + f 32 Ix + f 33 Vy + f 34 Iy = 0 f 41 Vx + f 42 Ix + f 43 Vy + f 44 Iy = 0 g11 Vx + g12 Vy = 0, g21 Vx + g22 Vy = 0

g11

g12

Vx

g21

g22

Vy

=

0 , 0

20 4 36

where g-parameters are given as follows: g11 = e − jβx d 1 − e − jβy d

a , g12 = e − jβy d 1 − e − jβx d

b

g21 =

1 + e − j2βx d − AD + BC e − jβx d 1 − e − jβx d

c

g22 =

− 1 + e − j2βy d − AD + BC e − jβy d 1 − e − jβy d

d 20 4 37

781

782

20 Planar Periodic Surfaces

The nontrivial solution of equation (20.4.36) is obtained from Δg = 0,

John Wiley & Sons, New York, 1989.

for lossless EBG cos βx d + cos βy d

B.5 Simon, R.; Whinnery John R.; Theodore, V.D.: Fields

= 2 AD + BC

and Waves in Communication Electronics, 3rd Edition, John Wiley & Sons, Singapore, 1994.

a

cosh γx d + cos γy d

for lossy EBG

= 2 AD + BC

b, 20 4 38

where γx/y = αx/y + j βx/y. The above dispersion relation is the same as that of given in equation (20.4.31), for the uniplanar EBG. However, the Bloch impedances, from equations (20.4.23) and (20.4.35), given below, are different: A 1 − e − jβx d C 1 + e − jβx d

ZBloch =j x

Electromagnetics, Artech House, Boston, USA, 2000. B.4 Balanis, C.: Advanced Engineering Electromagnetics,

g11 g22 − g12 g21 = 0

ZBloch = x

B.3 Tretyakov, S.: Analytical Modeling on Applied

A βd tan x C 2

A 1 − e − jβyd C 1 + e − jβy d βy d A = j tan 2 C

Journals J.1 Lim, J.-S.; Kim, H.-S.; Park, J.-S.; Ahn, D.;

J.2

a , ZBloch = y

b

c , ZBloch y

d

J.3

20 4 39 A relation between the port voltages Vx and Vy are also obtained from equations (20.4.36) and (20.4.37): g11 Vx + g12 Vy = 0,

g Vx = − 12 Vy g11 e − j2βx d − 1 Vx = Vy 1 − e − j2βy d

J.4

J.5

20 4 40 In case the wave propagates along the x-direction, βxd 0, βyd = 0, the above equation provides Vy = 0. The above equation is not needed in this, and the 2D EBG surface is treated as a 1D EBG line with transversely loaded short-circuited two stubs, each of length d/2. The layout, shown in Fig (20.18b), provides Bloch mode #1, supporting the BW, in its first passband. It is shown in the dispersion diagram of Fig (20.18d). Next, the Bloch mode #2 is a forward wave (FW). Two numbers of the bandgap are shown as the gray strips.

References

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Books B.1 Fang, F.; Rahmat-Samii, Y.: Electromagnetic Band Gap

Structures in Antenna Engineering, Cambridge University Press, UK, 2009. B.2 Munk, B.A.: Frequency Selective Surfaces -Theory and Design, John Wiley & Sons, New York,2000.

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Nam, S.: A power amplifier with efficiency improved using defected ground structure, IEEE Microw. Wirel. Compon. Lett., Vol. 11, No. 4, pp. 170–172, Apr. 2001. Rahmat-Samii, Y.: The marvels of electromagnetic band gap (EBG) structures. Appl. Comput. Electromagn. Soc. J., Vol. 18, No. 4, pp. 1–10, Nov. 2003. Sievenpiper, D.; Zhang, L.; Broas, R.; Yablonovich, E.: High-impedance electromagnetic surfaces with forbidden bands at radio and microwave frequencies, SPIE Conf. on THz and GHz Photonics, Vol. 3795, pp. 154–161, July 1999. Ma, K.P.; Hirose, K.; Yang, F. R., Qian, Y.; Itoh, T.: Realization of magnetic conducting surface using novel photonic bandgap structure, Electron. Lett., Vol. 34, No. 21, pp. 2041–2042, Oct. 1998. Syed, I.S.; Ranga, Y.; Matekovits, L.; Esselle, K.P.; Hay, S.G.: A single-layer frequency-selective surface for ultrawideband electromagnetic shielding, IEEE Trans. Electromag. Compat., Vol. 56, No. 6, pp. 1404–1409, Dec. 2014. Sievenpiper, D.F.: High-Impedance Electromagnetic Surfaces, UCLA, Ph.D. dissertation, Jan. 1999. Sievenpiper, V.; Zhang, L.; Broas, R. J.; Alexópolous, N.; Yablonovitch, E.: High-Impedance electromagnetic surfaces with a forbidden frequency band, IEEE Trans. Microw. Theory Tech., Vol. 47, pp. 2059–2074, Nov. 1999. Yang, F.; Rahmat-Samii, Y.: Polarization dependent electromagnetic bandgap characteristics, design, and applications, IEEE AP – S., Int. Sym. Digest, Vol. 3, pp. 339–342, June 2003. Yang, F.; Rahmat-Samii, Y.: A low profile single dipole antenna radiating circularly polarized waves, IEEE Trans. Antennas Propagat., Vol. AP-53, No. 9, pp. 3083–3086, Sept. 2005. Lee, R.; Vardaxoglou, Y.C.; Budimir, D.: Microwave bandgap and bandpass structures using planar metallodielectric periodic arrays, www.armms.org/ media/uploads/1271348629.pdf.

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phase of the scattered and/or radiated field from a high impedance surface of quasi-periodic sequences, IEEE Antennas Wirel. Propag. Lett., Vol. 12, pp. 321– 323, 2013. Matekovits, L; Colemè, G.V.; Orefice, M.: Controlling the band limits of TE-surface wave propagation along a modulated microstrip line- based high impedance surface, IEEE Trans. Antennas Propag., Vol. 56, No. 8, pp. 2555–2562, Aug. 2008. Mosallaei, H.; Sarabandi, K.: Antenna miniaturization and bandwidth enhancement using a reactive impedance surface, IEEE Trans. Antennas Propagat., Vol. AP-52, No. 9, pp. 2403–2414, Sept. 2004. Matekovits, L.: Analytically expressed dispersion diagram of unit cells for a novel type of holographic surface, IEEE Antennas Wirel. Propag. Lett., Vol. 9, pp. 1251–1254, 2010. Yang, F.R.; Ma, K.P.; Qian, Y.; Itoh, T.: A uniplanar compact photonic-bandgap (UC-PBG) structure and its applications for microwave circuits, IEEE Trans. Microw. Theory Tech., Vol. 47, No. 8, pp. 1509–1514, Aug. 1999. Scogna, A.C.; Suppression of simultaneous switching noise in power and ground plane pairs, Conformity, Vol.37, pp. 37–43, June 2007. Ki Hyuk Kim, K.H.; Schutt-Ainé, J.E.: Analysis and modeling of hybrid planar-type electromagneticbandgap structures and feasibility study on power distribution network applications, IEEE Trans. Microw. Theory Tech., Vol. 56, No. 1, pp. 178–186, Aug. 1999. Agarwal, K.; Nasimuddin, Alphones, A.: RIS-based compact circularly polarized microstrip antennas, IEEE Trans. Antennas Propagat., Vol. AP-61, No. 2, pp.547– 554, Feb. 2013. Xu, H.-X.; Wang, G.-M., Liang, J.-G.; Qi, M.Q.; Gao, X.: Compact circularly polarized antennas combining meta-surfaces and strong space-filling meta-resonators, IEEE Trans. Antennas Propag., Vol. AP-61, No. 7, pp.3442–3450, July 2013. Yanagi, T.; Takeshi Oshima, T.; Ohashi, H.; Konishi, Y.; Murakami, S.; Itoh, K.; Sanada, A.: Lumpedelement loaded EBG structure with an enhanced bandgap and homogeneity, Proc. iWAT2008, P 311, pp. 458–461, Chiba. Mias, C.; Yap, J.H.: A varactor-tunable high impedance surface with a resistive-lumped-element biasing grid, IEEE Trans. Antennas Propagat., Vol. AP-55, No. 7, pp. 1955–1962, July 2007. Sievenpiper, D.F.; Schaffner, J.H.; Song, H.J.; Lo, R.Y.; Tangonan, G.: Two-dimensional beam steering using an electrically tunable impedance surface, IEEE Trans.

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Antennas Propagat., Vol. AP-51, No. 10, pp. 2713– 2722, Oct.2003. Raza, S.: Characterization of the reflection and dispersion properties of `mushroom’-related structures and their application to antennas, M.Sc. Thesis, Department of Electrical And Computer Engineering, University of Toronto, 2012. Verma, A.K.; Kumar, A.: Design of low pass filters using some defected ground structures, Int. J. Electron. Comm. (AEU), Vol. 65, No. 10, pp. 864–872, 2011. Raheel, M.; Hashmi, R.M.; Zeb, B.A.; Esselle, K.P.: Wideband high-gain EBG resonator antennas with small footprints and all-dielectric superstructures, IEEE Trans. Antennas Propag., Vol. 62, No. 6, pp. 2970–2976, June 2014. Li, L.; Chen, Q.; Yuan, Q.; Liang, C.; Sawaya, K.: Surface wave suppression bandgap and plane-wave reflection phase band of mushroom-like photonic bandgap structures, J. Appl. Phys., Vol. 103, No. 023513, pp.103–110, 2008. Remski, R.: Analysis of photonic bandgap surfaces using Ansoft HFSS, Microw. J., Vol. 43, pp.190–198, Sept. 2003. Sievenpiper, D.F.; Zhang, L.J.; Broas, R.F.J.; Alexopolous, N.G.; Yablonovitch, E.: High-impedance electromagnetic surfaces with a forbidden frequency band, IEEE Trans. Microw. Theory Tech., Vol. 47, pp. 2059–2074, 1999. Sievenpiper, D.F.: High-impedance surfaces, Ph.D. thesis, University of California, LA, USA, 1999. Yang, F.R.; Ma, K.P.; Yongxi Qian, Y.; Itoh, T.: A novel TEM waveguide using Uniplanar Compact PhotonicBandgap (UC-PBG) structure, IEEE Trans. Microw. Theory Tech., Vol. 47, No. 11, pp. 2092–2098, Nov. 1999. Yang, F.; Rahmat-Sammi, Y.: Reflection phase characterizations of the EBG ground plane for low profile wire antenna application, IEEE Trans. Antennas Propag., Vol. 51, No. 10, pp. 2691–2703, Oct. 2003. Yang, F.; Rahmat-Sam, Y.: Polarization-dependent electromagnetic bandgap (PDEBG) structures: Designs and applications, Microw. Opt. Technol. Lett., Vol. 41, No. 6, pp. 439–444, June 2004. Coccioli, R.; Yang, F.R.; Ma, K.P.; Itoh, T.: Aperturecoupled patch antenna on UC-PBG substrate, IEEE Trans. Microw. Theory Tech., Vol. 47, pp. 2123–2130, Nov. 1999. Yang, F.; Rahmat-Samii, Y.: Polarization dependent electromagnetic bang-gap surfaces: characterization, design, and applications, IEEE Trans. Antennas Propag. Int. Sym. Digest, Vol. 3, pp. 339–342, 2003. Sievenpiper, V.; Zhang, L.; Broas, R.J.; Yablonovitch, E.: High-Impedance electromagnetic surfaces with forbidden frequency bands at radio and microwave

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frequencies, SPIE Conf. on THz and GHz Photonics, Vol. 33795, pp. 154–165, July 1999. Kamgaing, T.; Ramahi, O.M.: Design and modeling of High-impedance electromagnetic surfaces for switching noise suppression in power planes, IEEE Trans. Electromagn. Compat., Vol. 47, No. 3, pp. 478–488, Aug. 2005. Kamgaing, T.; Ramahi, O. M.: High-impedance electromagnetic surfaces for parallel-plate mode suppression in high-speed digital systems, IEEE 11th Topical Meeting Electr. Perform. Electron. Packag., pp. 279–282, 2002. Yousefi, L.; Mohajer-Iravani, B.; Ramahi, O.M.: Enhanced bandwidth artificial magnetic ground plane for low-profile antennas, IEEE Antennas Wirel. Propag. Lett., Vol. 6, pp. 289–292, 2007. Tretyakov, S.A.; Simovski, C.R.: Dynamic model of artificial reactive impedance surfaces, J. Electromagn. Waves Appl., Vol. 17, No. 1, pp. 131–145, 2003 Luukkonen, O.; Constantin Simovski, C.; Granet, G.; Goussetis, G.; Dmitri Lioubtchenko, D.; Antti V. Räisänen, A.V.; Sergei A. Tretyakov, S.A.: Simple and accurate analytical model of planar grids and highimpedance surfaces comprising metal strips or patches, IEEE Trans. Antennas Propag., Vol. 56, No. 6, pp. 1624–1632, June 2008. Luukkonen, O.; Constantin Simovski, C.; Granet, G.; Goussetis, G.; Dmitri Lioubtchenko, D.; Räisänen, A. V.; Sergei A. Tretyakov, S.A: Corrections to “Simple and accurate analytical model of planar grids and highimpedance surfaces comprising metal strips or patches”, IEEE Trans. Antennas Propag., Vol. 58, No. 6, pp. 2162–2162, June 2010. Luukkonen, O.; Alitalo, P.; Simovski, C.R.; Tretyakov, S.A.: Experimental verification of analytical model for high impedance surfaces, Electron. Lett., Vol. 45, No. 14, July 2009. Luukkonen, O.; Simovski, C.R.; Räisänen, A.V.; Tretyakov, S.A.: An efficient and simple analytical model for analysis of propagation properties in impedance waveguides, IEEE Trans. Microw. Theory Tech., Vol. 56, No. 7, pp.1624–1632, July 2008. Luukkonen, O.: Artificial impedance surfaces, Ph.D. thesis, Helsinki University of Technology, Dec. 2009. Iravani, B.M.; Ramahi, O.M.: Wideband circuit model for planar EBG structures, IEEE Trans. Adv. Packaging, Vol. 33, No. 1, pp. 169–179, Feb. 2010.

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785

21 Metamaterials Realization and Circuit Models – I (Basic Structural Elements and Bulk Metamaterials)

Introduction The engineered metamaterials [J.1–J.5] are the outcome of the artificial dielectrics discussed in subsection (6.2.2) of chapter 6. Their significant properties are presented in section (5.5) of chapter 5. The materials of the EBG group are also artificially engineered materials with unique properties. These are discussed in chapters 19 and 20. However, metamaterials are different from them. In the case of the EBG/PBG bandgap materials, the lattice constant is a half-wavelength long, so these are not treated as homogeneous materials. In the case of the metamaterials, the size of inclusion and lattice constant are less than one-tenth of a wavelength, and Bragg diffraction is avoided. Therefore, the metamaterials are treated as a homogeneous medium, like a natural material. The homogenized metamaterials are characterized by the effective relative permittivity and effective relative permeability, with unique properties normally not found in nature. The metamaterials are wideband materials grouped into two basic categories – resonant type, and nonresonant type. The resonant type metamaterials are narrow-band materials, whereas the nonresonant types are broadband. This chapter presents the basic resonant metallic inclusion structures for the realization of the 3D bulk epsilon negative (ENG), the mu negative (MNG), and double-negative (DNG) metamaterials. The resonant type inclusions, both metallic and dielectric, embedded in the host dielectric medium is treated as artificial molecules. The nonresonant metamaterials are more commonly used for the 1D metalines, also for the 2D metasurfaces. The metalines and metasurfaces are discussed in chapter 22. Our primary concern in this chapter is to discuss the circuit models and the realization of metamaterials using the planar structures. The homogenization and extraction of the electrical parameters of the metamaterials are also discussed. The detailed theory and applications of the metamaterials are presented in the textbooks [B.1–B.8].

Objectives

• • • •

To discuss the realization of the ENG, MNG, and DNG metamaterials using the wire medium, loop medium, and wire-loop combined medium. To present a summary of some metallic inclusion structures in the planar technology for the realization of metamaterials. To discuss the parameter retrieval process of metamaterials. To discuss the homogenization and the dynamic Maxwell Garnett model of the metamaterials.

21.1

Artificial Electric Medium

The artificial electric media, i.e. the epsilon negative (ENG) materials, the epsilon near zero (ENZ) materials, and the high permittivity dielectrics, are realized using the wire medium. It is also called the rodded medium. Figure (21.1a–c) shows the 1D, 2D, and 3D bulk wire medium-based artificial electric medium. The wires (rods) are the inclusions embedded in the host air medium. The arrangement is important for the realization of the bulk metamaterials [J.6, J.7]. Physically, all three arrangements appear to be three-dimensional. However, electrically they are different. The 1D wire medium of Fig (21.1a), also known as the single-wire medium (SWM), responds to the z-polarized Ez field component, i.e. the parallel polarized incident waves, propagating in the y-direction with wavenumber k. The wave can equally propagate in the x-direction with the same wavenumber. The wires of radius r are arranged in the 2D-square lattice with the lattice constant d. The Ez field component induces the emf on the conducting wire that causes the z-directed electric polarization in the wire. Thus, the permittivity of a wire medium is different along the length of the wires. However, in the (x–y)-plane, i.e. in the directions normal to

Introduction to Modern Planar Transmission Lines: Physical, Analytical, and Circuit Models Approach, First Edition. Anand K. Verma. © 2021 John Wiley & Sons, Inc. Published 2021 by John Wiley & Sons, Inc.

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21 Metamaterials Realization and Circuit Models – I

the thin wires, no polarization takes place leaving permittivity of the host air medium unchanged. Therefore, the 1D wire medium is a uniaxial anisotropic medium. The infinitely long, thin wire medium acts as an electric plasma medium with a cut-off frequency, i.e. the medium acts as a high-pass filter. The plasma medium, described by the Drude model, is discussed in section (6.5.2) of chapter 6. Below the plasma frequency fp, the relative permittivity of the wire medium is negative, forming the ENG medium. Above the plasma frequency, the relative permittivity is a positive quantity with the value less than unity. In this frequency range, the wire medium acts as the ENZ medium. The ENZ dielectric medium supports the fast-waves, unlike the natural dielectrics that always support the slowwaves due to its refractive index more than unity. The ENG and ENZ media are discussed in section (5.5) of chapter 5. The wire medium also suffers from the spatial dispersion due to a component of the EM-wave, TEM/TM type, propagating along the wire. Thus, the z-component of the effective permittivity of the wire medium becomes wavenumber-dependent, and the spatial dispersion occurs in the wire medium [B.7, B.9]. In the case of the temporal dispersion, the permittivity is frequencydependent, whereas for spatial dispersion the permittivity is wavenumber-dependent. Physically, the finite length wires are supported on dielectric foam. The rectangular waveguide also supports the wire medium with some restrictions [B.10, B.11]. In a planar technology, the wires are replaced by the printed conducting strips of the rectangular cross-section. However, a high permittivity substrate influences the characteristics of the wire medium. The modeling of the 1D wire medium, using in the circuit model, is discussed in this section. Figure (21.1b) shows the 2D-double wire medium (DWM). It responds to the Ex and Ez field components of the incident waves with polarization in the (x–z)plane. Likewise, Fig (21.1c) shows the 3D-triple wire medium (TWM). Both the DWM and TWM could be the connected and disconnected wires, i.e. at the crossing points, the wires could be electrically joined together or remain electrically isolated. Both arrangements have different nature of the wave propagation. They form the 2D and 3D plasma media. The 2D wire medium can act as the lightweight reflecting metallic-plate. The TWM supports any arbitrary polarization and could form the isotropic medium [J.7, J.8, B.4]. However, to get a homogeneous medium, the radius r of the wires and also the lattice constant d must be very small, about one-tenth or less than a wavelength.

21.1.1

Polarization in the Wire Medium

Figure (21.2a) shows the cross-section of a part of the arrangement of wires forming the wire medium. The wires, shown as black circles, are arranged periodically using the cubic unit cell of the lattice constant d. The infinitely long thin wire element A of radius r, parallel to the z-axis, is located at the center of the arrangement of five wires in the (x–y)-plane. The considered wire length ℓ is within the cubic unit cell (d × d × d). The four-wire elements around the element A are taken from the neighboring unit cells (UC). The voltage V is induced in the wire due to the time-harmonic external field, Ez = Eozejωt. It is assumed that the magnetic field Hϕ, created by the current I flowing in the wire, extends up to the wires of the neighboring UC without interacting with them. Thus, the wires of the wire medium are isolated and noninteracting. The wire has the self-inductance L per unit length and resistance R p.u.l. The resistance R accounts for the losses in the wire medium, giving its relative permittivity by equation ε∗r = 1 − j σ ε0 ω . Figure (21.2b) shows the RL circuit model of the wire in the unit cell. The current I(t) = I0ejωt flowing in a wire of length ℓ in the cubic unit cell is obtained as follows: Lℓ

dI dI + Rℓ I = V, L + RI = Ez dt dt Ez = R + jωL I

a b 21 1 1

The displaced charges on the wire create the dipole due to the electric polarization Pz. The local field causing the polarization is assumed to be identical to the external field Ez. This simplifying assumption is not valid for the densely packed wire medium. The displacement vector Dz is obtained from equation (6.1.6) of chapter 6. It is also related to the external field through the complex relative permittivity of the wire medium. Both relations are mentioned below, giving an expression for the polarization vector Pz: Dz = ε0 Ez + Pz Dz = ε0 ε∗r Ez = ε0

a σ Ez 1−j ωε0

from the above equations Pz =

b σEz Jz I = = jω jω jωd2

c

21 1 2 The current density Jz is averaged over the area of the unit cell, i.e. Jz = I/d2. On substituting current from equation (21.1.1b) in equation (21.1.2c), the polarization Pz is obtained: Pz =

1 Ez 2 R + jωL jωd

21 1 3

21.1 Artificial Electric Medium

Ez z Ez

z Ez x Hx

k y

d

x

y Hx k

d

d d

d

(a) 1D-wire medium, 2D-lattice, Ez-polarization.

(b) 2D-double wire medium (DWM), 3D-lattice, electric polarization in the (z–y) plane.

PMC

z x

y z E z

Hx

PEC y k d PMC x PEC d

(c) 3D-triple wire medium (TWM), 3D-lattice, arbitrary electric polarization.

(d) 1D-wire medium in the hypothetical parallel plate waveguide medium. The top and bottom gray walls are PEC, and the sidewalls (white) are the PMC.

Figure 21.1 Geometry of the wire-medium.

The complex relative permittivity of the wire medium is obtained by using equation (21.1.3) with the expression (21.1.2a): 1 Ez = ε0 Ez + 2 jωd R + jωL 1 ε∗r = 1 − 2 2 ε0 d L ω − j R L ω

ε0 ε∗r Ez

1 ε0 d2 L 1 where, R = σπr2 ω2ep =

a,

γe =

R L

b

c 21 1 5

21 1 4

The wire medium could be treated as the electric plasma medium with electric plasma frequency ωep. Normally, the plasma frequency is expressed as ωp. However, electric plasma frequency is expressed by the symbol ωep to distinguish it from magnetic plasma frequency ωmp discussed subsection (21.2.2). On comparing equation (21.1.4) against equation (6.5.16c) of chapter 6, the following expressions are obtained for the electric plasma frequency ωep and the electric damping coefficient γe:

The expression for R is obtained from equation (4.2.19) of chapter 4. The self-inductance L of the wire is approximately computed using the model shown in Fig (21.2a). The model could be viewed as a coaxial cable of unit length with inner radius r and outer radius d. The coaxial cable model computes the inductance L p.u.l. of the wire as follows: d

d

r

r

ψ μ μ I L= = 0 Hϕ dρ = 0 dρ I I 2πρ I L=

μ0 d ln 2π r

21 1 6

787

788

21 Metamaterials Realization and Circuit Models – I

d

ρ

r d

I

R

V

L

d A d

Hϕ

x y (a) Arrangement of a section of the wire medium in the (x–y)-plane. Figure 21.2

(b) Circuit model of an element of wire medium.

Circuit model of wire-medium.

We can also compute the inductance of the wire A in presence of neighboring conductors. At distance ρ = d/2, the magnetic fields due to the current in the wire A and adjacent wires cancel each other, leading to the creation of the square-shaped magnetic walls. So the self-inductance is computed as follows after getting the effective magnetic field Hϕ in the region r < ρ < d/2 due to two current sources separated by a distance d [J.9]: Hϕ =

L=

I 1 1 − 2π ρ d − ρ μ0 I

a

d 2

r

L≈

I 1 1 − dρ, 2π ρ d − ρ

L=

μ0 d − 1 387 , ln r 2π

for

μ0 d2 ln 2π 4r d − r

b

r < ωep, equations (3.4.10a) and (3.4.10b) of chapter 3 show that the wire medium supports the fast-wave with the group velocity below the velocity of light c. The predicted Drude model type permittivity response of the wire medium has been experimentally confirmed in the microwave range 7–14 GHz [J.12]. The arrays of microwires of radius r = 10, 30 μm and lattice constant d = 4, 5, 6 mm have been arranged on a stack comprising of 20 numbers of Rohacell HF 51 foam plates with low permittivity εr = 1.07 material. The dimension of stack Rohacell HF 51 plates is 200 × 200 × 100 mm3. Each plate contains 39 wires. The wire medium is mounted in a window frame surrounded by the absorbing material. Figure (21.3a) shows the |S21| response of the wire medium for r = 10 μm and d = 5 mm. The dashed response is the reference response at −15 dB without the sample of the wire medium. The response of the normal polarized (E⊥) incident waves is identical to the reference response, as there is no polarization in the wire medium. However, below the plasma frequency fep, the response of the parallel polarized (E ) EM-waves has an attenuation level of less than −60 dB. The plasma frequency fep is about 9.8 GHz at −35 dB below the

789

21 Metamaterials Realization and Circuit Models – I

0

0

–20

E⊥

–20 d =5 mm r = 10 μm

E∥

–60 –80

7

8

9 10 Frequency [GHz]

d =5 mm

d = 4 mm

–80

reference value wire medium

ENG medium

d = 6 mm

–60

ENZ medium

–100 –120

–40

S21 [dB]

–40 S21 [dB]

11

–100 –120

12

r = 10 μm

7

(a) Response of two different polarization of the incident wave. 0

8

9

10 11 12 Frequency [GHz]

13

14

(b) Response of wire medium for three lattice constants.

–20 P=5

–40 S21 [dB]

790

P = 10

–60 P = 20

–80 –100 –120

ENG 7

8

ENZ

d = 6 mm r = 30 μm

10 11 9 Frequency [GHz]

12

(c) Response of wire medium for an increasing number of wire medium plates. Figure 21.3 Experimental response of wire-medium. Source: Reproduced Gay-Balmaz et al. [J.12], with the permission of AIP Publishing. © 2002, AIP Publishing.

reference level. The nonpropagating ENG region is shown in gray. The propagating wave in the ENZ region is above fep. Figure (21.3b) shows the wire medium response for the lattice constant d = 6, 5, and 4 mm. It is noted that with a decrease in the size of the unit cell, the plasma frequency increases from 8.2 to 9.0 to 12.4 GHz. The plasma frequency also shows an increase in fep from 8.2 to 9.1 GHz for an increase in the radius of wire from r = 10 to 30 μm. Finally, Fig (21.3c) shows an increase in attenuation level in the ENG medium with an increase in the number of wire carrying plates P, i.e. an increase in the thickness of the wire medium. The accuracy of the computed plasma frequency, using available expressions for the wire inductance, could be examined against the experimental results. Table 21.1 compares the results of these expressions.

Pendry [J.2] provides a 1.5–4.8% deviation from the experimental results. There is an increase in the deviation with an increasing radius of wire, and increasing lattice constant. The expression for inductance given in reference [J.13] provides a little-improved result, 0.8–4.2%. The expression (21.1.8b) results in a high deviation for a small radius of wire, 7.8–11.5%, whereas, for the thicker wire, it is only 4.8%. The expression of Pendry could be used for the most of applications using the wire medium in the air host medium. However, the computed plasma frequency may have more deviation for the printed wire medium on a substrate. The present discussion illustrates that the engineered artificial ENG and ENZ materials could be realized using the circuit models. The RL circuit model is extended below to the reactively loaded wire medium.

21.1 Artificial Electric Medium

Table 21.1

Computed and exp. fp.

r in μm

d in mm

Exp. fp GHz [J.12]

Com. fp GHz [J.2]

Com. fp GHz [J.9]

Com. fp GHz [J.13]

10

4

12.4

12.22

13.82

12.30

10

5

9.80

9.60

10.80

9.65

10

6

8.20

7.88

8.84

7.92

30

6

9.10

8.66

9.09

8.71

21.1.3

Reactance Loaded Wire Medium

Inductively Loaded Wire Medium

Figure (21.4a) shows the impedance loaded wire medium. The loading element could be a lumped inductor, loop or spiral inductor, lumped capacitor, metallic structures of several geometrical forms. It could be even the active elements [J.13–J.17]. Figure (21.4b) shows the inductive loading element LL in the form of a square, or a circular loop. Figure (21.4c) shows the printed circuit version of the wire medium with the rectangular cross-section. It also shows its RL circuit model. Figure (21.4d) shows the gap capacitor (CL) loaded printed wire with the RLCL circuit model. Following the process discussed above, the circuit model is used below to obtain the effective permittivity of the impedance loaded wire medium material. The equation (21.1.4) is modified below to compute the effective relative permittivity of the loaded wire medium with impedance ZL: ε∗r = 1 +

1 jε0 ωA R + jωL + ZL

21 1 14

In expression (21.1.14), A = d2 is the area of the unit cell in the (x–y)-plane. The ZL is on a per unit length basis. Expression (21.1.14) does not account for the interaction between the wires of the medium [J.17, J.18]. There are five cases for the load impedance ZL: 1) ZL is short-circuited, reducing the loaded wire medium to an unloaded wire medium described by the Drude model as a plasma medium; 2) Inductor loaded wire medium with ZL = jωLL, giving again a plasma medium with lower plasma frequency; 3) Capacitively loaded wire medium with ZL = 1/jωCL, giving the Lorentz type effective permittivity medium; 4) Series resonant circuit loaded wire medium with ZL = j(ωLL − 1/ωCL), again giving the Lorentz type medium; and 5) Parallel resonant circuit loaded wire medium, ZL = j(ωCL − 1/ωLL)−1.

Figure (21.4b) shows the inductively loaded wire medium, using either a square or circular loop. The loops are constructed with the wire of circular crosssection of radius r. The total length of loaded line is ℓ. The loading inductance LL p.u.l. for both cases can be estimated from the following approximate expressions [J.13, J.19, B.12]: μo R 8R ln − 1 75 ℓ r μ 4a 4a Square loop LL = o sinh − 1 πℓ r

Circular loop J 13

a

LL =

b

21 1 15 In expressions (21.1.15), LL is the inductance of the loop. The square loop has side length “a”, and circular loop radius is R. The expression for the square loop of wire of circular cross-section is obtained from the approximate expression of the square loop made of wire of square crosssection [J.20]. The total inductance in the unit cell is Ltotal = LL + LW p.u.l., where wire inductance L = Lw is given equation (21.1.6), or equation (21.1.8a, b). Using total inductance, the plasma frequency of the loop loaded wire medium, from equation (21.1.5), is written as follows: Circular loop J 13 ω2ep =

2π c d 2 2πR ℓ ln 8R r − 1 75 + ln d r

a

Square loop ω2ep =

2π c d 2 8a ℓ sinh − 1 4a r + ln d r

b 21 1 16

The inductively loaded wire medium also behaves like a plasma medium, at the reduced frequency. The loading inductor reduces the effective radius of the wire that increases the wire inductance and lowers the plasma frequency [J.13, J.18]. Figure (21.5a) shows the transmission response of the empty test chamber of parallel conducting plates separated by ℓ = 10 mm. The radius

791

21 Metamaterials Realization and Circuit Models – I

ZL ZL Ez

z Ez

ZL

ZL

R

ZL ZL

ZL

a ZL

LL

d d

Ez

(b) Inductive loop-loaded wire medium.

Ez

R z

z Ez k

L

R

Ez

g k y

Hx

y

x

L

x d

(c) 1D printed wire medium, Ez-polarization. Figure 21.4

L

ZL

ZL

(a) 1D-impedance-loaded wire medium, 2D-lattice, Ez-polarization.

Hx

ℓ

R

a

ZL

ZL

y ZL k Hx x

ZL ZL

ZL

a

b

CL

(d) 1D capacitor-loaded printed wire medium, Ez-polarization.

Impedance loaded wire medium.

The unloaded wire medium has an electric plasma frequency fep above 12 GHz. Therefore, attenuation is more than −50 dB in the lower frequency range, due to the nonpropagating waves in the negative permittivity wire-based plasma medium. However, the

r of unloaded straight wire medium is 0.2 mm, and lattice constant d of the square lattice is 5 mm. The loaded circular loops have radius R = 1.3 mm, r = 0.2 mm. The array has 5 rows normal to the incident waves, incident on the chamber, and 30 rows in the transverse direction.

–20

Effective rel. permittivity (εr)

–15 Transmitted power (dBm)

792

Empty chamber

–25 –30 Loop-loaded wire medium

–35 –40

Straight wire medium

–45 –50 –65 9

9.5

10 10.5 11 Frequency (GHz)

11.5

12

(a) Response of straight wire, and loop-loaded wire medium[J.13]. Figure 21.5

1000

0.2 0.1

500

Lorentz model for cut-wire medium g = 0.3 mm

0

ω0e

–500 –1000

Drude model for wire medium

–1500

g=0

–2000 0.0

0.2

0.4 0.6 0.8 Frequency (GHz)

1.0

(b) Permittivity response of straight wire, and capacitor-loaded wire medium [J.15].

Response of inductor and capacitor loaded wire-medium. Source: Reproduced from Smith et al. [J.13], with the permission of AIP Publishing. © 1999, AIP Publishing. Reproduced from Takano et al. [J.15], with the permission of AIP Publishing. © 2010.

21.2 Artificial Magnetic Medium

computed value of the electric plasma frequency fep for the loop loaded wire medium is 10.7 GHz. Therefore, above about 10.7 GHz, the response shows the transmission of the waves. For the square loop of side length a = 2 mm, and other parameters remaining unchanged, the computed plasma frequency, using equation (21.1.16b), is 6.24 GHz, whereas the simulated one is 6.8 GHz. The deviation is not high, even if expression (21.1.16b) is approximate.

Table 21.2

Extracted RLCL parameters. L (10−7 H/m)

CL (10−18 F/m)

R (104 Ω/m)

0.0

0.78

>1000

3.66

0.1

1.13

7.67

8.20

0.2

1.92

2.55

6.05

0.3

2.44

1.70

6.38

g (mm)

Source: Reproduced from Takano et al. [J.15]. © 2002, AIP Publishing.

Capacitively Loaded Wire Medium

The capacitive loading of a wire medium is achieved by periodically cutting the wire and inserting a lumped capacitance. Even a cut in the wire provides the loading through the gap capacitance. The capacitor loading creates series resonance in the wire medium below its plasma frequency, and the Drude type wire medium is converted into the Lorentz type wire medium [J.15]. Such a medium, as shown in Fig (21.4d), is known as the cut-wire medium [J.15]. The Lorentz type permittivity response is discussed in section (6.5) of chapter 6. The external LC-resonant circuit loading also provides the same type of response with lower resonant and lower plasma frequencies. For the capacitor-loaded wire medium, the load impedance is ZL = − j/ωCL. On substituting it in equation (21.1.14), the complex effective relative permittivity of the capacitor-loaded wire medium is expressed as follows: ε∗r = 1 + where ω2ep

ω2ep

ω20e − ω2 + jωγe 1 1 , ω20e = = , ε0 AL LCL

a γe =

R L

substrate paper, using conductive ink, is 2.5 μm. The width of the printed line, the lattice constant, and the capacitive gap are a = 0.2 mm, d = 0.4 mm, and g = 0.1, 0.2, 0.3 mm respectively. Figure (21.5b) shows the experimental results of the Drude type response of the unloaded wire medium. It also shows the Lorentz type response of the gap-capacitor-loaded wire medium. The resonance frequency is less for a smaller gap, i.e. for a larger value of the loading capacitance. By fitting the parameters in the Lorentz model, the RLCL parameters are extracted. On using these parameters, Lorentz model responses are shown in Fig (21.5b). Table 21.2 shows the extracted RLCL parameters of the Lorentz model. Below the resonance frequency, the positive-valued permittivity is dispersive. The Lorentz type artificial dielectric has higher valued permittivity just below the resonance. The permittivity response is similar to the response shown in Fig (6.9b) of chapter 6. In the frequency range ω0e < ω < ωep, the permittivity is negative, i.e. the medium is ENG, and above ωep, it is the ENZ medium.

b 21 1 17

In expression (21.1.17a, b), ω0e is the characteristic electric resonance frequency of the Lorentz medium. The positive permittivity below the resonance frequency ω0e is obtained, giving the limiting value as follows: ω 0, ε∗r 1 + CL ε0 A . Figure (6.9b) of chapter 6 shows the response of the Lorentz medium, so the capacitor-loaded wire medium acts as the Lorentz artificial dielectric medium. The wire medium, and also the capacitor-loaded wire medium, can be realized even in the THz frequency range in the planar technology on a substrate paper. Figure (21.4c) shows the printed wire medium [J.15]. Figure (21.4d) shows the cut-wire grid of cut-width g, creating the loading capacitor. The substrate paper is 0.07 mm thick. It has a refractive index of 1.5 + j0.05. The conductor thickness b of the printed line on

21.2

Artificial Magnetic Medium

The artificial magnetic medium with effective relative permeability is engineered by using a periodic array of the current-carrying loops. Such a medium could be called the loop medium. A small current-carrying loop, forming the magnetic dipole moment (m) given by equation (6.1.24a) of chapter 6, is the source of diamagnetism. Figure (21.6a) shows the induced current I in a loop excited by the external z-directed axial magnetic field Hext z . The array of the current-carrying loops creates the mu near-zero (MNZ) material with the effective relative permeability less than unity, i.e. 0 < μr < 1. In such an arrangement, the paramagnetism showing μr > 1 is not achieved. Similarly, the negative permeability is also not realized using the current-carrying continuous loop. However, the capacitor-loaded resonating loop, shown in Fig (21.6b), can provide Lorentz type magnetic

793

794

21 Metamaterials Realization and Circuit Models – I

equivalent circuit, shown in Fig (21.6d), models the split ring resonator (SRR) [J.21, B.14].

behavior, similar to the behavior of the ferromagnetic materials [B.13]. The Lorentz model is discussed in section (6.5) of chapter 6. The capacitor-loaded resonating loop can act as an artificial magnetic molecule, or magnetic element, to create a material with high permeability. It also creates the negative permeability needed by the MNG medium. Figure (21.6b) shows the capacitor-loaded circular loop that exhibits series resonance. The time-varying external magnetic field excites the resonant current I in the loop. The loop provides the inductance, and the lumped capacitor is externally added to get the series resonance [B.10]. The loop has conductor loss, also a little radiation loss [B.8]. However, the capacitor-loaded loop is usually not a suitable choice for the realization of the MNG and DNG materials, as the lumped capacitors are bulky and lossy, and are not suitable for forming a large array. Figure (21.6c) solves the problem by the use of the concentric split rings resonator (SRR) structures. The ring has usually either circular or square shape. The splits in the loops break the flow of the induced conduction currents in the loops. However, the displacement current flows from the outer ring to the inner ring due to a higher valued induced potential at the outer ring. In this arrangement, the narrow gap between two rings acts as the loading capacitance, and the inductance is provided by current flowing in the strip conductors. The RLC

21.2.1

Characteristics of the SRR

Figure (21.7a) shows the double rings circular SRR excited by the external time-varying axial magnetic field Hext z . The induced voltage and current in both rings are also shown in Fig (21.7a). The SRR is divided into the symmetrical upper and lower parts due to the splits, i.e. due to gap in the strip rings. The splits are placed at the diametrically opposite ends on the outer and inner rings. The induced voltages, Vout, and Vin at the outer ring and the inner ring, respectively, cause the displacement current Id to flow from the outer ring to the inner ring, as Vout > Vin. It helps the flow of continuous currents Iout and Iin in the outer and inner rings. An electric wall (EW) P1P2 Q2Q1, passing through the middle of the splits, could be placed at the diameter of the SRR. The formation of capacitors shows the accumulation of charges on the rings. Further, the currents of rings have the positional variations along the circumference of the rings. However, the total current at any location on the rings remains unchanged. The voltage and current variations across the rings are shown in Fig (21.7b). The locations of the voltage and current maxima and minima are also indicated in both Fig (21.7a, b). The maxima of

Y I

Hzext Z

Hzext C

X

(b) Capacitor-loaded circular loop

(a) Circular loop excited by Hzext-field.

f

f I

d

r

g

g

d

Reff

g

g

b

b

b

b 2r d

d

(c) Circular and square split-rings resonators (SRR) Figure 21.6

I

V

Leff Ceff

(d) Equivalent circuit of SRR

Artificial magnetic molecules or elements excited by external axial magnetic fields.

21.2 Artificial Magnetic Medium

Upper half of SRR Ioutϕ +

+ b +

+

b –

–

– Id p2

max

Iout

+ Vin –

P1 min

Y

rm r ϕ

P2

Vout

EW ro +

k

p1 Q1 max Vout

X

Hz Z

+

+ + Iout(ϕ) f

–

– Vout +

Q2

rext

Hzext

Ey

min

Iout

–

–

–

Lower half of SRR (a) SRR showing induced voltage, current, and electric dipole moments of the outer and inner rings (p1, p2).

Vout

0 Q2

π P1

–2π Q2 ϕ

Q2

Iin

Current

Voltage

Q1

P1

Q2

P2

Iout 0 Q1

–π P2

–2π Q1 ϕ

Current distributions

Vin P2 Voltage distributions

(b) Voltage and current distributions along the circumference of both rings. Figure 21.7 The split rings resonator (SRR).

voltages occur at the gaps in the rings and currents are zero at the gaps. The maxima of currents are at the positions opposite to gaps. At these locations, voltages are zero. The detailed circuit model of the SRR is available [B.5, B.15]. The present section focuses attention on a simplified model to understand its working. The double-ring SRR could be viewed as a single current-carrying loop of mean radius rm = r + b + f/2, where b is the width of both the inner and outer strips, and f is the circumferential gap between outer and inner rings [J.22]. The current-carrying loop of the mean radius rm acts as a

magnetic dipole. The arrays of these magnetic dipoles create artificial magnetic materials. The SRR-based magnetic material is primarily a bianisotropic magnetic medium. The bianisotropic medium is discussed in section (4.2) of chapter 4. However, ignoring the bianisotropy, we consider below it as an isotropic medium modeled through the RLC – circuit. The section (21.2.4) examines the bianisotropy [J.23]. 21.2.2

Circuit Model of the SRR

In our discussion, we ignore capacitance due to the split gaps. Figure (21.6d) shows the series resonance RLC

795

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21 Metamaterials Realization and Circuit Models – I

circuit model of the SRR. The circumferential gap f between the strip rings forms the upper and lower capacitances CU and CL of equal value, i.e. CU = CL = C. Both capacitances are connected in series giving the effective gap capacitance as Ceff = C/2. The current-carrying conducting strips offer the inductance Leff, and the resistance Reff accounts for the losses in the SRR. The axial timeext jωt harmonic magnetic field Hext induces the z = Hzo e emf in the equivalent single loop, for a narrow gap f b, of mean radius rm. The half capacitance is C = πrmCpul, where Cpul is the slot gap capacitance p. u. l.

Figure (21.6d) shows the circuit model to obtain the magnetic polarizability αm of a loop medium due to the axial magnetic field Hext z . It is needed to compute the effective relative permeability μr of the artificial magnetic medium. The loop medium is treated as a homogeneous medium, as the lattice constant d of the periodic loop medium, as shown in Fig (21.8a), the sizes of the SRR 2rext, and separation ℓ are much less than the operating wavelength at the resonance frequency of the SRR. The induced emf Vind, and corresponding loop current I in the SRR, due to the external time-varying magnetic flux, is obtained below with the help of the equivalent circuit shown in Fig (21.6d): dϕ = − jωμ0 SHext z dt Vind I= Reff + jωLeff + 1 jωCeff μ ω2 ω2 SCeff Hext I = 2 0 2 0m ω0m − ω + jωReff Leff z μ ω2 S Leff I= 2 0 2 Hext ω0m − ω + jωγm z Reff damping factor γm = Leff

a

b c 21 2 1

where S is the area of the single equivalent loop and ω0m = 1 Leff Ceff = 2 Leff C is the characteristic magnetic resonance frequency of the SRR. Using equation (6.1.24a) of chapter 6, the z-oriented magnetic dipole moment mz, and corresponding magnetic polarizability αm are obtained as follows: mz = IS = αm =

μ0 ω2 S2 Leff Hext ω20m − ω2 + jωγm z

mz μ0 ω2 S2 Leff = ω20m − ω2 + jωγm Hext z

μ0 ω2 S2 d3 Leff ω20m − ω2 + jωγm Fω2 μ∗r = 1 − 2 ω − ω20m − jωγm

μ∗r = 1 +

μ∗r = 1 −

Computation of Effective Relative Permeability of Loop Medium

Vind = −

The volume of the cubic unit cell, containing one SRR inclusion, is d3. Therefore, the magnetic dipole density of the artificial magnetic material is N = 1/d3. Using the modified equation (6.1.27c), i.e. μ∗r = 1 + Nαm , the following expression is obtained for the complex effective relative permeability of the artificial magnetic materials:

a b 21 2 2

+j

a

Fω2 ω2 − ω20m 2

ω2 − ω20m + ωγm Fω3 γm 2

2

ω2 − ω20m + ωγm 2 where, filling – factor : μ π2 r4 μ π2 r4 ω2 Ceff F = 03 m = 0 m 3 0m , S = πr2m d Leff d Fω2 Also, μ∗r = 1 + χ∗r = 1 + 2 ω0m − ω2 + jω ω0m Q ωom where, Q = γm

b

c

d, 21 2 3

where rm is the mean radius of the equivalent single ring loop of SRR. Expression (21.2.3) of the complex relative permeability μ∗r applies to a low-density SRR based medium. It does not account for the effect of the local field. Using γm = 1/3 the Clausius–Mossotti expression (6.1.28c) of chapter 6, i.e. the expression μ∗r = 1 + Nαm 1 − Nαm 3 approximately accounts for the effect of the local field [J.24]. In equations (21.2.3a–d), both μ∗r and F are dimensionless quantities. The filling-factor F is less than unity. On comparing equation (21.2.3a) against equation (6.5.5b) of chapter 6, it is noted that the SRR-based artificial magnetic materials follow the Lorentz type resonance, except the presence of ω2 in the numerator. Also, the filling-factor F is not the magnetic plasma frequency ωmp, although it has some relation to it. The permeability response of the SRR element is determined by the parameters F, ω0m and γm. To obtain the permeability response analytically, these parameters are determined using the closed-form expressions. It is difficult to measure the damping factor γm. In equation (21.2.3d), it is replaced by the Q-factor. The Q-factor is a measurable quantity for the SRR element. Moreover, the measured Q value of the SRR element can also be used for creating an array to develop the artificial magnetic material. So for a fixed value of the

21.2 Artificial Magnetic Medium

Hz

2rext

e

X

d

Ex Z Hz

d k Y

μr

Passband paramagnetism

Stopband MNG

Passband diamagnetism MNZ

Frequency (ω)

(a) Cubic lattice (ℓ= d) of SRR-based artificial magnetic medium.

1

ωmp

(1-F)

ω0m

ωom

ωmp

ω

Fastwave Light line Bandgap

Slow-wave Propagation const (β) (b) Permeability response the SRRbased artificial magnetic medium.

(c) Dispersion diagram of the SRRbased artificial magnetic medium.

Figure 21.8 The SRR lattice and permeability response.

filling-factor F, the measured Q can be used to get the permeability response of the artificial magnetic material accurately [J.25]. However, the maximum value of Re χ∗r ≈ FQ 2 is associated with a high-loss. Therefore, for a fixed value of F and high Q-factor of the SRR, a high value of μr is realized over a narrow bandwidth. The value of F must be above 0.2 to reduce losses in the artificial magnetic materials [J.26, J.27]. The nature of permeability response of a lossless SRR medium, with damping factor γm = 0, is explained from the real part of relative permeability:

μr = 1 −

Fω2 , 2 ω − ω20m

μr =

ω2 − ω2mp ω2 − ω20m

where, magnetic plasma frequency ω2mp =

1−F

a

ω20m b 1−F 21 2 4

Figure (21.8b) shows the frequency response of the permeability function with a pole at the characteristic magnetic resonance frequency ω0m and zero at the magnetic plasma frequency ωmp. The SRR-based medium is the Lorentz type magnetic plasma medium. In the case of a lossy magnetic

797

798

21 Metamaterials Realization and Circuit Models – I

medium, the infinitely large value of the relative permeability μr is replaced by the resonance peak. Figure (21.8b) demonstrates the limiting cases: for ω 0, μr 1, for ω ωmp, μr 0, and for ω ∞ , μr (1 − F). In the present case, the SRRs are in the host air medium. It is noted that in the frequency range 0 < ω < ω0m, the medium behaves as a dispersive paramagnetic material with a positive permeability. Near the characteristic resonance frequency ω0m, the permeability has a high value. In a narrow frequency band ω0m < ω < ωmp, the permeability is negative for creating the MNG material. In the highfrequency range above the plasma frequency, i.e. for ω > ωmp the permeability is again positive. However, its value is less than unity, creating the MNZ material medium. The strength of the permeability is controlled by the filling-factor F, i.e. by the size of the SRR inclusion within the unit cell of size d × d. The square-shaped SRR inclusion has the maximum value of F. The above analysis is approximate, ignoring the effect of depolarization due to the neighboring SRR elements of the array. It could be accounted for through the interaction constant [J.28, B.16]. The damping factor γm could be treated as the interaction constant used in the Clausius–Mossotti model discussed in subsection (6.1.2) of the chapter 6. The above analysis also ignores the spatial dispersion [J.29]. However, it demonstrates the nature of the low-density resonant artificial magnetic medium. The propagation constant of the EM-waves in the artificial magnetic medium is obtained as ω β= μr , εr = 1, c

ω β= c

1−F

ω2 − ω2mp

1 2

ω2 − ω20m

where rm = r + b +

21.2.3 Computation of Equivalent Circuit Parameters of SRR The five sets of expressions to compute the circuit elements of the equivalent circuit of the SRR element are summarized below: Set #1: The effective inductance used in the equivalent circuit of a circular SRR, shown in Fig (21.6d), could be approximately computed by using the following expression:

32rm −2 , b

f 2

21 2 6 Sometimes, the effective inductance is approximated by Leff ≈ μ0rm. Alternate approximate expressions for the computation of the inductance can also be used [B.5]. The variational method is also available to compute more accurately the needed inductance [J.22]. The capacitance Ceff of the circuit model is computed by treating the concentric strip rings as the parallel conducting strips, ignoring the effect of the curvature, printed on a dielectric sheet. The capacitance of the parallel strips is computed by using equation (9.4.13), or equation (9.4.20) of chapter 9. Other expressions are also available [J.22]. Finally, the resistance Reff of the circuit model is estimated by the following expression:

Reff =

2πr , bhσ πr , bδσ

for

h ωmp. So the artificial magnetic medium behaves as a filter medium with passband–stopband– passband characteristics.

Leff = μ0 rm ln

Effective inductance J 30

Ceff =

2πrm K k ε0 , 3 Kk

where k =

1 − k2 μ πr2 = 0 m ℓ

a

k = Leff

f 2b + f

b

for cubic lattice ℓ = d F =

πr2m d2

c

2πrm Zs ω b

Reff = Re Z ω

= Re

where, Zs ω =

1−j σωδω

d, 21 2 8

where Zs(ω) is the surface impedance of the strip conductor. Usually, the skin depth δ(ω) is less than the ring conductor thickness, and conductivity σ(ω) is frequencyindependent. However, in the optical frequency range, it

21.2 Artificial Magnetic Medium

structure [J.32]. The asymmetrical coupled lines-based model is also available to account for the unequal arbitrary strip widths of two SRR [J.33]. The following set of curve-fitted expressions can be used to determine ω0m, between 8 and 12 GHz, with accuracy within 10% [J.34]:

is frequency-dependent and computed by using the expression σ ω = ω2mp ε0 γ − jω [J.31]. Set #3: More accurate expressions, with accuracy in the range 6–12%, are available to compute the characteristic resonance frequency ω0m of the circular SRR Leff = 2 57e − w3 Ceff = 0 217 + 3 3367e

2

πrext − 2 2f 1 − π 2

0 059 2rext + εr − 5 − 3 2d1

− 0 1955e

+ 0 05εr − 0 218 + ω0m =

a 0 432w 1 − 0

+0

07w32

− 0 47h

0 599h 0 0248 + h − 0 599

1 2π Leff Ceff , 1 2π Leff Ceff +

317w 22

b

for rext > 5 2mm 5 2 − rext 2 ,

c

for rext < 5 2 mm

21 2 9

Figure (21.7a) shows dimensions of an SRR within the unit cell of size d × d × ℓ, ρ is the resistance per unit length of conducting strip, and Ceff is the effective capacitance p.u.l. of gap f between strip rings, treated as the parallel conducting strips. For the cubic unit cell, shown in Fig (21.8a), the height in the z-direction is ℓ = d. Figure (21.9a, b) shows the permeability response for the following parameters:

In expression (21.2.9a–c), the parameters are w1 = b, w2 = 2b, w3 = b, f1 = f for f < 1 mm, and w1 = b/3, w2 = 1.414b, w3 = b/1.414, f1 = 1.414f for f > 1 mm. Set #4: Pendry Expression for Relative Permeability of Loop Medium Pendry has obtained another set of analytical expressions to compute the relative permeability of the circular SRR-based artificial magnetic medium by averaging the magnetic field within the unit cell. The expressions are summarized below with an illustrative computation [J.3]: F μ∗r = 1 − 2 a ω − ω20m − jωγm πr2 2ℓρ b , γm = c F = 2m μ0 r m d 3ℓ d ω20m = μ0 π2 Ceff r3m 1 2b e Ceff = ln πμ0 c2light f

2

μreal

2 μimg

1

1

0

0

–1

0

5

10 15 Frequency (GHz)

–1 20

(a) Copper ring resistance ρ = 200 Ω p.u.l. Figure 21.9

Real and img. parts of relative permeability

Real and img. parts of relative permeability

3

b = 1 0 × 10 − 3 m

f = 1 0 × 10 − 4 m,

ℓ = 2 0 × 10 − 3 m

rm = 2 0 × 10 − 3 m, for copper − strip ring, ρ = 200 for resistive strip ring, ρ = 2000 The computed characteristic resonance frequency f0m is 13.41 GHz. Pendry expression computes the resonance frequency f0m within 10% accuracy. Figure (21.9a, b) demonstrate the paramagnetic, MNG, and diamagnetic behavior of the SRR based artificial magnetic materials.

21 2 10 3

d = 1 0 × 10 − 2 m,

3

3 2

μreal

μimg

2

1

1

0

0

–1 0

5

10 15 Frequency (GHz)

–1 20

(b) High resistance ring ρ = 2000 Ω p.u.l.

The permeability response of the cubic lattice of the SRR based magnetic materials. Source: Reproduced from Pendry et al. [J.3]. © 1999, IEEE.

799

800

21 Metamaterials Realization and Circuit Models – I

The response of Fig (21.9b) also shows high losses associated with the highly dispersive lossy medium. The magnitude of (±μr) is reduced with an increase in loss and finally vanishes for higher losses. Therefore, lowloss is an important condition for the realization of the high-value permeability and also MNG materials. Set #5: Circuit Parameters of Square Split-ring Resonators Figure (21.6c) also shows the square split-ring resonator of arm’s length 2r within the square unit cell of size “d”. The rings of width b are separated by f, and end gap g is the split gap. It is printed on the PCB board. In the vertical direction, the unit cell height is ℓ. For the cubic lattice, it is ℓ = d. This structure provides the largest value of the filling-factor F. The complex effective relative permeability is computed by using equation (21.2.3a). The parameters are computed using the following expressions [J.35]: ω20m

=

3ℓc2light

π ln 2b f r3 2ℓρ γm = μ0 r 21.2.4

a,

4r2 F= 2 d

induced charges on the outer and inner split-rings. The inner ring is smaller than the outer one. Therefore, a complete cancellation of the effect of electric dipoles is not possible for the concentric coplanar rings SRR, as the induced voltages Vout and Vin at Q1 and P2 are not equal. The cancellation of the effect of electric dipoles is achieved with the use of the broadside coupled SRR, mentioned in subsection (21.2.5). Following the discussion in subsection (6.1.4) of chapter 6, the electric and magnetic dipole moments due to the polarization and cross-polarization could be written as follows: ext em ext py = αee yy Ey + αyz Hz

mz =

The Onsager symmetry condition provides = − αem yz , and equation (21.2.12) is written as [B.5], =

mz

21 2 11

The external time-varying axial field, Hext z applied to the SRR as shown in Fig (21.7a), induces a current in the loop creating the z-oriented magnetic polarizability αmm It also induces the charges on the split rings, creatzz ing the y-directed electric polarizability αem yz . Similarly, the SRR can also respond to the external time-varying electric field Eext y , creating the y-directed electric polarizabilee ity αyy , and also the z-directed magnetic polarizability αme zy . Thus, the SRR has magneto-electric coupling due to the cross-polarization [J.36–J.38]. The crosspolarization is discussed in subsection (6.1.4) of chapter 6. It shows that the external time-varying magnetic field induces both the magnetic and electric polarization. Similarly, the external time-varying electric field generates both of these polarizations. Usually, bianisotropy is disadvantageous. It is stronger in a single split-ring. However, it is reduced in the concentric coplanar double SRR with diametrically opposite splits that partially cancels the electric polarization. However, the z-directed induced magnetic dipoles of both the split-rings are additive, as the direction of the circulating currents in them are in the same direction. It is shown in Fig (21.7a). It further shows oppositely directed induced electric dipole moments, p1, and p2 at locations Q1 and P2 due to the

b

αme zy

b

Bi-anisotropy in the SRR Medium

+

a

ext αmm zz Hz

21 2 12

py

c

ext αme zy Ey

αee yy

αem yz

Eext y

− αem yz

αmm zz

Hext z

21 2 13

The elements of polarization matrix equation (21.2.13) are dyadic as shown in equation (6.1.31) of chapter 6. The edge-coupled SRR creates a bianisotropic medium. The circuit model has been used to obtain the polarization elements [J.30]. Further, the Eext x field applied to the SRR also creates the x-directed non-resonant electric 3 polarizability αee xx = ε0 16 3 rext . The polarizabilities of the SRR, creating the lossy bi-isotropic magnetic medium are summarized below [J.22, J.23, B.5]: αmm zz =

π2 r4m ω2 2 Leff ω0m − ω2 + jωγm

me 3 αem yz = − αzy = − j2πrm f eff Cpul

a, ω20m ω

ε0 16 3 r 3 ext ω2 ω20m − ω2 + jωγm αee xx =

b c

where f eff ≈ b + f αee yy =

ε0 16 3 ω2 rext + 4f 2eff r2m C2pul Leff 0m 3 ω

2

1 ω20m ω2 − 1

d

21 2 14 For the lossless SRR-based medium, the damping factor em γm is ignored, i.e. γm = 0. The polarizabilities αmm zz , αyz , ee and αyy have resonant nature. The presence of the polarizabilities in the SRR-based medium is observed through the stopband in the transmission response, i.e. in the S21 response. The nature of stopband in the S21-response is decided by the polarization of the incident waves, and the orientation of the SRR with respect to the direction of the polarizing incident field. The transmission measurement has been performed in the free space on the SRR based magnetic medium block of the 25 × 25 × 25 UC.

21.2 Artificial Magnetic Medium

is seen in the transmission response. Finally, case #iv generates the electric dipole, creating a stopband in the transmission response. There could be the other two cases for the SRR orientation in the (H–k), i.e. (z–x) plane. These are also the cases of nonpolarization. A similar study has been conducted by mounting the SRR in the TE10 mode rectangular waveguide [J.3, B.17]. The stopbands show the presence of the negative permeability in the Lorentz magnetic materials. The responses of all four orientations have been further experimentally confirmed [J.38].

It is also obtained using the numerical method. Figure (21.10a) shows four orientation of the square SRR with respect to the Ey and Hz field components. The wave propagates in the X-direction with the wavevector k. Figure (21.10b, c) show the S21 response of four orientations of the SRR due to different polarization and cross-polarization. Figure (21.10d) shows the current and charge distributions, for both the noncirculating and circulating currents, on a single-ring SRR [J.36]. The case #i of Figure (21.10a) creates the magnetic dipole, giving the resonant negative permeability, due to the Hz-field component. It appears as the narrow stopband in the simulated and experimental transmission response shown in Fig (21.10b, c). However, for the case #ii both the magnetic and electric resonant polarizations take place at nearby resonance frequencies. In this case, the Ey-field component induces the charges across the split gap shown in Fig (21.10d) creating the electric dipole. Figure (21.10d-ii) also shows the induced circulating current in the ring creating the magnetic dipole. Thus, the case #ii creates a wider stopband. The case #iii does not generate any polarization, so no stopband

21.2.5

Variations in SRR Structure

The size of the SRR is about λ0/10. It is just the limit of homogenization. To reduce its size further, some variations in the SRR are suggested. The SRR shown in Fig (21.7a) is an edge-coupled ring giving a smaller value of the effective capacitance. It also suffers the crosscoupling of the external electric and magnetic fields responsible for the bianisotropy. The effective capacitance has been significantly increased, and the crosscoupling is controlled by the use of the broadside coupled 0

–30 f

(ii) k X

HZ E

E

(iii) k

H

k

H

H

(iv) k

a b a a = 3 mm f = g b = 0.33 mm

i

ii

–60 –90 iv

–120 0.06

0.065

0.07

(g: Gap in split rings, f: Gap between two split rings.)

Transmission |S21| (dB)

(a) Spatial orientation of the SRR.

+ ΔQ i

+ ΔQ

0.085

0.09

+ ΔQ′

+ ΔQ

iii – ΔQ′

ii –40

0.08

(b) Simulated transmission through the SRR slab [J.36].

0 –20

0.075 ωα/c

+ ΔQ′

iv

Current

(i)

iii

g

E

T [dB]

E Y

ΔQ′ ωmp, the medium is a positive MNZ medium with 0 < μr < 1. In the case of a composite permittivity–permeability medium, the frequency band ω0m < ω < ωmp creates the DNG medium, and the transmission response shows a passband due to the negative value of both the permittivity and permeability functions. The DNG passband supports the backward wave propagation. For ω > ωep, the composite medium is a DPS medium giving another passband, supporting propagation of the standard forward wave. Figure (21.13a, b) demonstrate the envisaged the material and transmission response of Table 21.3. Figure (21.13a) shows the sketched Drude type permittivity response of the strip-wire medium. It also shows the permeability response of the Lorentz type SRR medium. The medium has negative permittivity and permeability, in the range ω0m < ω < ωmp creating the DNG type medium. In the range, ωmp < ω < ωep stopband is created due to the negative permittivity. Figure (21.13b) shows the transmission response in the complete frequency range, with a bandgap between two pass-bands. The first passband is due to the DNG supporting the backward wave propagation and the negative refraction, while the second passband supports the forward wave propagation and the positive refraction. In the case of the identical value of the electric and magnetic plasma frequency (ωmp = ωep), the stopband disappears and a continuous DNG–DPS medium is obtained. Thus, within a combined wideband passband, the wave propagation is seamlessly varied from the backward wave to forward wave with an increase in frequency. Such a medium is realized by the 1D CRLH metamaterials. It finds application in designing the leaky-wave antenna, discussed in subsection (22.4.1) of chapter 22.

Frequency response of the composite permittivity–permeability function (Drude–Lorentz function). Sign of permeability function

Frequency range

ω2 − ω20m

Sign of permittivity function

ω2 − ω2mp

μr sign

ω2 − ω2ep

Nature of band

0 < ω < ω0m

(−)

(−)

(+)

(−)

Stop

ω0m < ω < ωmp

(+)

(−)

(−)

(−)

DNG pass

ωmp < ω < ωep

(+)

(+)

(+)

(−)

Stop

ω > ωep

(+)

(+)

(+)

(+)

DPS pass

Bandgap ENG

DNG

ENG

DPS

ωep ωmp Frequency (b) Transmission response of composite SRR-SW medium.

(a) Material response of composite SRR-SW medium.

Material and transmission responses of strip wire (SW) and SRR metamaterial.

The above discussion shows that the generation of a passband in the stopband region of the magnetic medium does not ensure the creation of the DNG medium. The DPS medium also supports a passband with the forward wave propagation. The passband of the DNG medium supports the backward wave. The nature of the unknown medium – DPS or DNG, is decided by the differential phase Δφ of the wave passing through two distances L1 and L2 (L2 > L1) in the medium. It is obtained as follows: ω ϕi = − βg Li = − n ω Li , i = 1, 2 a c ω Δϕ = − n ω L2 − L1 b, c 21 3 3 where n(ω) is a dispersive refractive index of the composite medium. For the DPS medium, n(ω) is a positive quantity giving negative differential phase Δϕ, whereas for the DNG medium, n(ω) is negative, giving the positive value of Δϕ. Using the EM-simulator, the differential phase Δϕ is easily obtained from the transmission phase of the wave across the metamaterial medium of two unequal lengths. Lorentz–Lorentz Function

Following the above discussion, the Lorentz–Lorentz composite function and the propagation constant of the EM-wave in the Lorentz–Lorentz composite medium could be written as follows: ε r μr =

ω2 − ω2ep

ω2 − ω2mp

ω2

ω2 − ω20m

ω βg = c

DPS

ωom

Frequency

Figure 21.13

DNG

Bandgap

ωep Bandgap

ωom

Bandgap

0

Transmission

μr εr ωmp

Real εr and real μr

21.3 Double Negative Metamaterials

− ω2oe

ω2 − ω2mp

ω2 − ω2ep

ω2 − ω20m ω2 − ω2oe

a 1 2

b

21 3 4

where 0 < ω0e < ω0m < ωmp < ωep < ∞. The characteristic electric and magnetic resonance frequencies could be identical also. Another band performance table, similar to Table (21.3), can be prepared for the Lorentz– Lorentz arrangement. The realization of both the Drude–Lorentz and Lorentz–Lorentz DNG media is discussed in the next subsection. Further examples also are given in subsection (21.3.3).

21.3.2 Realization of Composite DNG Metamaterials The previous sections have discussed the realization of the negative permittivity and negative permeability material with the help of the Drude electric and Lorentz magnetic type inclusions. The DNG can also be realized by using the Lorentz electric and Lorentz magnetic inclusions. This section discusses a few examples for the realization of the composite DNG (C-DNG) metamaterials by using a unit cell of both arrangements – Drude–Lorentz inclusions, and Lorentz–Lorentz inclusions, with negative permittivity and negative permeability over common frequency band. The DNG metamaterial was first demonstrated by Smith and coworkers, using the rodded (wire) medium with SRR [J.5]. Next, it was realized in the planar technology replacing the rod-SRR composite by the composite structure of strip wire-SRR (SW-SRR). It is difficult to solder the strip-wire at the top and bottom conductors in a parallel plate waveguide, so the SW has been replaced by the capacitively loaded strip (CLS). Thus, the DNG medium has also been realized by using the CLS–SRR composite. The complex SRR has been replaced by a simpler capacitively loaded loop (CLL). The CLS–CLL composite provides another structure to realize the DNG medium. The

805

21 Metamaterials Realization and Circuit Models – I

follows: the size of inner square ring r = 0.506 mm, strip width of SRR b = 0.124 mm, the gap between rings f = 0.15 mm, and split gap g = 0.114 mm. The structure is constructed on a substrate with εr = 3.02, h = 0.5 mm. The metallic strip of width w1 = 0.5 mm is also printed on the backside of the substrate. It is located in the middle of the SRR. The Z-polarized incident EM-wave on the metamaterial propagates along the y-axis. In the y-direction, 4 UC are considered to get the transmission parameter S21. The incident wave has Ez-field component parallel to the SW and its Hx-field component is normal to the plane of the SRR [J.41]. Figure (21.14b) shows the magnitude of the transmission (|S21|) response. The transmission peak in the DNG region, shown in gray, is at 22.6 GHz. The frequency range of the DNG medium is

three C-DNG structures are briefly discussed below. The details could be obtained from the literature [J.41–J.44]. Strip Wire and SRR (SW-SRR) Composite Element

The unit cell of the SW-SRR composite element is shown in Fig (21.14a). It is the Drude–Lorentz arrangement. The size of unit cell is 2.5 mm × 2.5 mm × 2.5 mm. The electric field component Ez is responsible to create the electric dipole in the strip wire (SW) and the magnetic field component Hx creates the magnetic dipole in the SRR. The z-polarized incident wave propagates in the ydirection with wavevector ky. The perfect conductor surfaces (PEC) are placed at the top and bottom of the cubic unit cell in the (x–y)-plane. The perfect magnetic surfaces (PMC) are placed at the sidewalls of the cubic unit cell in the (y–z)-planes. The dimensions of SRR are as d w1 h εr g

d

f

–10

w

r

Ez z

ky

–20

–50

d Cubic unit cell

–60 21

20

Hx

250

24.5

150

1 Unit cell 2 Unit cells 3 Unit cells 4 Unit cells

UC 4

100 50

DPS DNG UC 1

0 –50

20

21 22 Frequency (GHz)

DPS

23

(c) Transmission phase response [J.41]. Figure 21.14

Frequency (GHz)

200

22 23 Frequency (GHz)

24

(b) Transmission magnitude response [J.41].

(a) Unit cell of composite SW-SRR metamaterials.

22.6 GHz

DPS

DNG

–40

y

x

DPS

–30

22.6 GHz

b f

00

∣S21∣ (dB)

b

S21 phase in deg.

806

24

24

I Light line

II Light line P

23.5

HFSS Light line TLM model

23 22.5 –180 –120

–60 0 60 Phase in deg.

120

180

(d) Dispersion diagram [J.41].

Transmission and dispersion responses of strip wire and SRR metamaterial. Source: Reprinted (figure) with permission from Woodley et al. [J.41]. © 2005 APS.

21.3 Double Negative Metamaterials

obtained from the phase response shown in Fig (21.14c). Outside the DNG, both in the lower and upperfrequency bands, the composite SW-SRR medium behaves like a DPS medium. High attenuation level on the left side of the DNG shows that the DPS of the lower frequency band is the nonperfect ENG due to the negative permittivity of the strip wire medium. Therefore, the electromagnetic nature of the composite cannot be determined from the magnitude of the transmission response alone. Additional confirmation is needed to accept the gray region of Fig (21.14b) as the DNG medium created by the composite SW-SRR. It could be achieved by any of the following processes:

• • • •

By showing that the EM-wave passing through the probable DNG medium has the leading phase, unlike the lagging phase in a normal DPS medium. The dispersion diagram should exhibit the backward wave propagation in the probable DNG medium. The probable DNG medium should offer negative refraction for the wave passing through it. Finally, the real part of the extracted permittivity and permeability parameters must be negative in the probable DNG medium.

Figure (21.14c) shows the phase responses for four lengths of the composite metamaterial medium. The length is expressed in terms of the number of UC. We note that during the frequency interval shown in gray, the differential phase Δϕ is positive and the refractive index n(ω) is negative according to equation (21.3.3b). Thus, the gray interval shows the creation of the DNG material. It is in between two DPS regions. The value of the differential phase Δϕ is frequency-dependent, i.e. the refractive index is dispersive. Figure (21.14c) also shows the lowering of the transition frequency in the lower frequency band of DNG with an increase in the number of UC. However, the upper transition frequency of the DNG bandwidth is fixed, i.e. independent of the number of UC. Figure (21.14d) shows the dispersion diagram obtained from the eigensolver of the HFSS EM-Simulator and also using the TLM model. The periodic boundary conditions are applied at all faces of the unit-cell and phase is varied only along the y-direction. The gray area is above the light line. The direction of the local slope at the location P in the quadrant-II, and the slope at the origin O shows the anti-parallel phase and group velocities. It indicates the propagation of the forward traveling backward wave in the DNG medium. The wave in the quadrant-I is the backward traveling backward wave. The HFSS-simulated dispersion diagram, Fig (21.14d), shows a bandgap due

to the intersection of the Bloch mode with the light line at the location P. However, the TLM model does not show any bandgap. The DNG frequency range obtained from the phase diagram is different from that of the dispersion diagram. It is noted that the phase diagram of Fig (21.14c) is obtained for a finite structure, whereas the dispersion diagram of Fig (21.14d) is generated for the infinite structure. CLS–SRR Composite Element

The strip-wire is replaced by the capacitively loaded strip (CLS) to create the DNG medium using the CLS–SRR composite designed on a dielectric substrate [J.42]. It is the Lorentz–Lorentz arrangement of inclusions. The composite structure mounted in the parallel plate waveguide (PPW), shown in Fig (21.15a), does not require soldering of the CLS at the top and bottom of the structure. The CLS–SRR composite is mounted in the (y–z)-plane of the PPW along the length (y-axis) of the waveguide. The PPW supports the TEM mode wave propagating in the y-direction with Ez and Hx-field components. The Ez-field component interacts with the CLS structure. The CLS creates the Lorentz type electric plasma medium with resonance at the frequency ω0e. It is like a strip cut-wire medium. It could be modeled by the LC resonant circuit. The vertical strip wire of CLS provides the inductance L, and its two horizontal strip wires provide the capacitance C of the equivalent LC resonant circuit. Both the CLS and SRR structure are fabricated on the same substrate. Figure (21.15a) shows the simple DNG medium with an SRR between two CLS. The Hxfield component interacts with the SRR loop. The SRR creates the Lorentz type magnetic plasma medium. The incident wave propagating in the y-direction of the parallel-plate waveguide has Ez and Hx-field components, exciting the electric and magnetic dipoles in the CLS and SRR, respectively. The dimensions of the EM-simulated CLS–SRR structure are available in the reference [J.42]. The complex refractive index response, shown in Fig (21.15b), is extracted from the S-parameter response. It clearly shows the negative refraction over a narrow frequency band around 9.5 GHz, and the medium is DNG type. At this frequency, the extracted value of the permittivity and permeability also exhibits negative value due to the Lorentz type response. The more elaborate arrangement of the structure has also been examined in the reference [J.42]. CLS–CLL Composite Element

Figure (21.15c) shows that the capacitively loaded loop (CLL) that has replaced the SRR of the CLS–SRR

807

21 Metamaterials Realization and Circuit Models – I

1.0 0.5

CLS

Refractive index

0.0

CLS

Ez z Substrate

k y x Hx

–0.5 –1.0 –1.5 –2.0

SRR

z

nRe nImg

–2.5

y

–3.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0 13.0 14.0 15.0 Frequency (GHz) (b) Refractive index response of CLS–SRR metamaterial slab[J.43].

(a) CLS–SRR metamaterial elements in the (y–z)-plane.

2500 nRe nImg

2000 1500

y

z

Ez Substrate #A

k

CLS

z

y x Hx

y

Refractive index

808

1000 500 0 –500 –1000 –1500 –2000

Substrate #B

CLL

z

–2500 3.520

(c) CLS–CLL metamaterial elements the (y–z) plane. The CLL is arranged behind the CLS on another substrate. Figure 21.15

3.526

3.530 3.534 Frequency (GHz)

3.540

(d) Permittivity–permeability response of CLS– CLL metamaterial block [J.42].

CLS–SRR and CLS–CLL metamaterials responses of the material parameters. Source: Reproduced from Cheng and Ziolkowski [J.42], © 2003, IEEE). Reproduced from Ziolkowski [J.43]. © 2003, IEEE.

composite discussed above. The CLS and CLL are constructed on separate substrates #A and B. A block consisting of the alternate layers of the CLS on substrate #A and CLL on substrate #B synthesizes the DNG material. It is the Lorentz–Lorentz arrangement. The CLS– CLL composite structure, located in the (y–z)-plane, could be embedded in the substrate region of a microstrip line supporting the quasi-TEM mode wave propagating in the y-direction. The quasi-TEM mode wave has a normal Ez-field component and horizontal Hx-field component. The Ez-field component of the microstrip

excites the CLS creating the Lorentz type permittivity response. The Hx-field component of the microstrip excites the CLL creating the Lorentz type permeability response [J.43]. The resonant refractive index, with its both real and imaginary parts, is shown in Fig (21.15d). The narrow bandwidth negative refractive index material is created at 3.5297 GHz. The details of the CLS– CLL structure are available in the reference [J.43]. The CLL has also been used to create the artificial magnetic conductor (AMC) as a reflector to enhance the gain of an antenna [J.44].

21.3 Double Negative Metamaterials

equation (21.2.12), like the SRR. The closed-form expressions for the bianisotropic polarizabilities have been obtained using the antenna model [J.45, J.46]. However, the electric and magnetic resonances are coupled for the Ω-particles-based metamaterials. The presence of the electric response can significantly influence the strength of the magnetic response in a way that can destroy its negative permeability [J.47]. Even the composite Ω-SW medium may not create the DNG medium. Sometimes, it creates the passband of a DPS medium [J.48]. The unit cell of the broadside coupled (BC) Ω-particles, shown in Fig (21.16a-ii), is a composite structure of reversely printed two Ω-particles on both sides of a substrate. This arrangement cancels the inherent bianisotropy of single Ω-particle. In the case of the BC Ω-particles, the electric and magnetic responses are fairly decoupled. Thus, the broadside coupled Ω-particle is a suitable structure for the synthesis of a low-loss and broadband metamaterials [J.49–J.51]. More numbers of BC Ω-particles, connected in series with lattice constant d in the y-direction, form one column of the metamaterial. The periodic repetition of the column in the x-direction, with a lattice constant d, forms the square lattice in the (y–x)-plane. The planar lattice is formed on a PCB substrate with (εr, tan δ), and thickness h. The periodic arrangement of the PCB cards in the z-direction forms the volumetric metamaterials, as shown in Fig (21.16b). However, one blank substrate is placed between two printed PCB to form the volumetric metamaterials. The whole assembly is hot-pressed to get the 3D solid metamaterials [J.49]. The Ey-polarized incident waves propagate in the x-direction. The Ey-field component

21.3.3 Realization of Single-Structure DNG Metamaterials The single physical inclusions can also produce both the electric and magnetic response, creating the DNG medium. The composite double array of wire-SRR, over the frequency range ω0m < ω < ωmp, forms the narrow bandwidth C-DNG metamaterials. It also suffers from high losses. The DNG and DPS media are separated by a bandgap. However, sometimes a seamless transition, from the DNG state to DPS state with a change in frequency, is needed for practical applications. It is normally obtained in the microstrip line structure using the composite right and left-handed (CRLH) structure, discussed in subsection (22.1.5) of chapter 22. The present subsection discusses the magneto-electric particles, Ω-particle, and S-ring particles with both the electric and magnetic responses excited by the incident EMwave of suitable polarization. Both these particles are used to get the low-loss and wideband S-DNG type metamaterials.

Ω-Particle Metamaterials

Figure (21.16a-i) shows one element of the Ω-particle that is printed on a PCB card without backside metallization. It consists of two parts: the C-structure, responding to the magnetic field normal to its plane, and upper and lower connected strip wires (SW), responding to the electric field parallel to the SW. It is the Drude–Lorentz arrangement for the electric and magnetic responses. However, the metamaterial based on the Ω-particle has bi-anisotropic property, expressed through

PMC

SW

W Ey y

S SW

R

k

C

(i)

z Input port

g

d

x Hz

R

Output port

d

PEC

w

y

Substrate PMC

h

Ey x

(ii)

(a) Ω and broadside-coupled Ω-particles. Figure 21.16

PEC

Hz

z

k

(b) 2D-array of broadside coupled Ω-particles.

Broadside coupled Ω-particles-based metamaterial.

809

810

21 Metamaterials Realization and Circuit Models – I

excites the Drude type electric polarization, and the Hzfield components excite the Lorentz type magnetic polarization in the metamaterial of Fig (21.16b). To get the seamless DNG and DPS medium, without any bandgap, the structure must operate in the balanced mode, with identical electric and magnetic plasma frequencies, i.e. ωep = ωmp. In the case ωep ωmp, the medium operates in the unbalanced mode and a bandgap appears between the DNG and DPS [J.51]. The magnetic resonance and magnetic plasma frequencies, ω0m and ωmp, can be estimated by the following expressions: 1 ω0m a , ωmp = 1−F LC πR2 F= 2 d Cp ε0 εr w 2R + w g = π − C= 2 2h h 2 2 μ πR L= 0 h

ω0m =

b

insertion loss (IL) of about 7.5 dB. However, the insertion loss of the DNG is high. The refractive index, permittivity, and permeability of the metamaterial medium have been extracted by treating the unit cell as a two-port network. The extracted material parameters, from the de-embedded S-parameters, help to compute the propagation constant and the characteristic impedance of the medium. Figure (21.17c–f ) shows the frequency-dependent extracted parameters. The negative refractive index, permittivity, and permeability are shown in light gray. The hatched area shows the lower cut-off frequency. The process of extraction of the material parameters is discussed in section (21.4).

c S-ring Particle Metamaterials

d e, 21 3 5

where Cp is the upper and lower C-shape capacitance, treating the structure as the parallel plate capacitor. The dimensions are shown in Fig (21.16a-ii). The electric plasma frequency of the SW medium can be estimated from Pendry formula, or any other expression discussed in subsection (21.1.1). However, dimensions have to be fine-tuned on EM-simulator to equalize both plasma frequencies. It is illustrated below. The BC Ω-structure-based metamaterial has been engineered on FR-4 substrate of thickness h = 0.8 mm and εr = 4.0, tan δ = 0.02. The strip width, slitgap, radius R of C-section of the Ω-particle, and lattice constant d are as follows: w = g = 0.5 mm, R = 1.1 mm, d = 3.9 mm. The unit cell with front and back PMC, and top and bottom PEC are simulated in the EM-simulator to get the dispersion diagram shown in Fig (21.17a) [J.50]. The metamaterial supports the wideband, approximately from 8.0 to 12.8 GHz, backward wave. At the location P on the dispersion diagram, the local slope and the slope at origin O are oppositely directed supporting the backward wave. The medium also supports the forward wave approximately from 12.8 to 16 GHz, as the slopes at Q and O are in the same direction. So the seamless passband is obtained, between 8.0 and 16.0 GHz, from the DNG to the DPS medium. There is a cut-off frequency of around 8.0 GHz. It is confirmed by the experimental results shown in Fig (21.17b). The experiment has been conducted in the parallel plate waveguide environment [J.49, J.50, B.17, B.19]. It shows the transmission band from 8.3 to 16.0 GHz with minimum

The light gray card of Fig (21.18a-i) shows a column of oppositely printed pairs of S-rings on both sides of a substrate of relative permittivity εr and thickness h. It forms the broadside coupled (BC) S-ring particle. The cascading of the PCB cards with a blank separating substrate, or air-gap, forms a column. Other PCB cards and blank PCB cards are not shown in Fig (21.18a-i). The BC-S-ring is a self-resonant metallic structure. It is the Lorentz–Lorentz type arrangement exhibiting negative permittivity, and also negative permeability in the overlapping frequency band, so the negative refractive index is created over a wideband with small insertion loss. It is unlike the narrowband C-DNG, with higher insertion loss, obtained from the composite SW-SRR structure. The S-ring does not require electrical contact when mounted inside a parallel plate waveguide (PPW). It is a significant advantage for the fabrication and measurement of the metamaterials [J.52–J.55, B.11]. Figure (21.18a-i) shows, in gray, one unit cell of the BCS inclusion on a substrate. The unit cell has dimensions a, b, and d. The dimensions of the S-rings are as follows: conducting strip width c, height p, and S-ring width w. The front and back S-rings are separated by the substrate thickness h. The relative permeability of the structure has been obtained using the circuit model. The results for the identical upper and lower rings are summarized below in terms of the magnetic resonance frequency ωom and the magnetic plasma frequency ωmp [J.52, J.54]: 2F + jA 1 − ωom ω 2 − B + jG 1 = μ0 abF 3d Cs 1 = ω2om 1 − 2F

μreff = 1 − where, ω2om ω2mp

a b c

21.3 Double Negative Metamaterials

18

0

Backward-wave Q

βd αd

–10 S12 (dB)

Frequency (GHz)

15 12

–20

P 9

–40 –3 –2 –1 0 1 2 3 4 Propagation and attenuation constants (a) Dispersion diagram for DNG and DPS medium without bandgap.

6

9 12 15 Frequency (GHz)

18

(b) Transmission response. 6

1.0 0.5

Refractive index (n)

Re (z) Im (z)

0.0 –0.5 –1.0 6

9 12 15 Frequency (GHz)

2 0 –2 –6

permeability (μ)

15 0

18

Negative permittivity

Positive permittivity

Re (μ) Re (μ)

3 0 Negative permeability

–3

Positive permeability

–6 9 12 15 Frequency (GHz)

18

6

(e) Permittivity response. Figure 21.17

12 15 Frequency (GHz)

6 Re (ε) Im (ε)

30

–30 6

9

(d) Refractive index response.

45

–15

Positive refractive index

Negative refractive index

–4 –8 6

18

Re (n) Im (n)

4

(c) Impedance response.

Permittivity (ε)

DPS

DNG

–30

Forward-wave

6 –4

Impedance (Z)

Sim Exp

9

12 15 Frequency (GHz)

18

(f) Permeability response.

Responses of broadside coupled Ω-particles-based metamaterial. Source: Reproduced from Zhang et al. [J.50], with the permission of AIP Publishing. © 2008, AIP Publishing.

pw Filling – factor F = ab εr ε0 wc εr ε0 wc + Cs = h p−h

d

A=

2ωμ0 RF2 pab ωμ0 Fab 2 1 − 1 ωμ0 FabCs

a

B=

Rd 2 1 − 1 ωμ0 FabCs

b

e 21 3 6 G=

The remaining parameters, used in equation (21.3.6a), are given in the following expressions:

ωμ0 Fab

2

Rd 2ωμ0 Fab − 2d2 Cs ω ωμ0 Fab

2

1 − 1 ωμ0 FabCs

c 21 3 7

811

21 Metamaterials Realization and Circuit Models – I

details are as follows: a = 5.2 mm, b = 4.0 mm, p = 5.0 mm, w = 2.8 mm, c = 0.4 mm, h = 0.508 mm, d = 2.0 mm. The size of the UC and S-ring are less than λ/5. The 10 × 50 UC are fabricated on each PCB card and 120 equally spaced cards are stacked to fabricate the 3D-metamaterial. It is used to construct a prism for the refractive index measurement of the engineered metamaterial. Figure (21.18d) shows the far-field free space transmission measurement in the X-Ku band. It shows wideband, 11.5–13.5 GHz, transmission due to both the DPS–DNG nature without any bandgap. The nature of the transmission band has been examined through the refractive index measurement by using the metamaterial in the prism form. Figure (21.18b) shows the dispersive nature of the positive and negative refractive indices without any bandgap. The resonance frequency is at 12.5 GHz. The negative refraction occurs in the frequency range 12.5–15 GHz, i.e. over the 2.5 GHz bandwidth. It also shows high-valued positive refractive index. The insertion loss of the metamaterial is 1.69 dB/cm (0.422 dB/cell).

In equation (21.2.7), R is the resistance of the strip conductor. The magnetic resonance frequency of the S-loop is estimated by minimizing the imaginary part of the permeability function. The accuracy of the model has been tested in the X (8–12.5 GHz), Ku (12.5–18 GHz), and K (18—27 GHz) bands [J.52]. The permittivity of the S-ring rodded medium of Fig (21.18a-ii) follows the Lorentz model: εreff = 1 −

ω2ep − ω2oe

21 3 8

ω2 − ω2oe + jγe ω

In the case of the single S-ring structure, the electric resonance frequency ωoe is zero and the medium follows the Drude model. However, ωoe occurs in the S-type cut-wire structure, shown in Fig (21.18a-ii). The S-type cut-wire structure shows the Lorentz electric response, and also the Lorentz magnetic response. Figure (21.18c) shows the fabricated BC-S ring metamaterial on a low-loss thin R04003C substrate. It also shows the substrate details. For the X-Ku band the structural

h

Refractive index (n)

w

p

p

c

c

g

a

z Ez w b

d εr

y Hy x

k

(i)

5 4 3 2 1 0 –1 –2 –3 –4 –5 8

(ii)

(a) One column of BC-S-ring metamaterial and gap coupled S-ring particles.

R04003C

εr = 3.55 tan δ = 0.0021 –0.0027

h = 0.508 mm

Figure 21.18

Positive (n > 0)

–5

DNG 9

10 11 12 13 Frequency (GHz)

Negative (n < 0)

14

15

S21 HFSS S21 Experimental

–10 –15 –20 –25 –30 –35 –40

(c) Fabricated S-ring metamaterial[J.52].

DPS

(b) Refractive index of the S-ring metamaterials [J.52].

0 Transmission (S21) dB

812

8

9

10

11

12 13 14 15 Frequency (GHz)

16

17

18

(d) Simulation and experimental transmission response of the Sring metamaterials [J.52].

Performance of the broadside coupled S-ring-based metamaterials. Source: Reproduced from Lee et al. [J.52], with the permission of AIP Publishing.© 2015, AIP Publishing.

21.3 Double Negative Metamaterials

The BC S-rings are also used on the artificially engineered magnetic substrate to enhance the DNG bandwidth. Figure (21.19a) shows the complete metamaterial on the artificial magnetic substrate, i.e. the meta-substrate, that is created by the metallic closed-rings on an FR-4 substrate with εr = 4.0, h = 1.0 mm, μ1 = μ0. The periodic unit cell in the (x–z)-plane is 4.0 mm × 4.0 mm. It has periodicity 2.5 mm in the y-direction. The ring size is rin = 3.2 mm and rout = 3.6 mm. The fractional filling-factor Fc = 0.57 of the closed-rings creates the artificial magnetic substrate. The artificially acquired permeability of the engineered magnetic substrate is μ2 = (1 − Fc) μ0 = 0.43μ0.

The BC S-ring creates the metamaterial on the magnetic meta-substrate. The relative permeability of the metamaterial is computed by using the following expression [J.55]: μreff =

1 − Fc 1 − F −

ωmp = ωom 1 + ωom =

1 LC

1 − ωom

1 F 1 − Fc 1 − F

F ω 2 + jR ωL

a b c 21 3 9

In equation (21.2.9), F is the filling factor of the BC S-ring of size, p = 10.8 mm, w = 5.6 mm, c = 0.8 mm,

w Ez z

rout

xk p

y Hy

rin

c

dx

h

(a) The series-connected BC S-ring particles metamaterials on an artificial magnetic substrate. Without meta-substrate With meta-substrate Receiver

0

Source

10

–10 Sample

–30

–50 –60

10 Real (μ) Img (μ)

5 0 –5

–10 4

–70

Real (ε) Img (ε) 4.5

–80 2

Effective parameters

–40

Effective parameters

∣S21∣ (dB)

–20

3

5 5.5 Frequency

4

6

10 Real (μ) Img (μ)

5 0 –5 –10 4

5 Frequency

Real (ε) Img (ε) 5 5.5 4.5 Frequency

6

6

7

8

(b) Transmission response of the BC S-ring type metamaterials on the conventional and artificial magnetic substrates. Figure 21.19

Performance of the broadside coupled S-ring particle-based metamaterials on an artificial magnetic substrate. Source: Chen et al. [J.55]. © 2014, Springer Nature.

813

814

21 Metamaterials Realization and Circuit Models – I

dx = 10.0 mm. The structure has 10 UC along the direction of propagation. Figure (21.19b) shows the experimental results of the transmission measurement in free space. It also shows the extracted permeability and permittivity responses in insets. The parameters are extracted from the measured S-parameters response of the engineered slab. Both the parameters are negative forming the DNG over the frequency band 4.29–5.61 GHz (30.8%) for the BC S-rings on the conventional FR-4 substrate, whereas it is 44.9% (4.08–5.91 GHz) for the BC S-rings on the meta-substrate. The theoretical bandwidths for the conventional and meta-substrate, using the above equations, are 28 and 44%, respectively. Figure (21.19b) also exhibits a small insertion loss of the S-ring metamaterial.

21.4 Homogenization and Parameters Extraction The above discussion has treated the periodically arranged discrete metallic structure-based metamaterial medium as a homogeneous medium under the condition kd 1, where k is wavenumber inside the metamaterial and d is the lattice constant of a unit cell. Thus, the hypothetical electromagnetically continuous medium, forming an effective medium, can replace the discrete arrangement of the periodic inclusions in the host medium. The effective medium is described by the primary material constants: complex effective relative permittivity and complex effective relative permeability, or complex refractive index. The medium is also described by the equivalent secondary material parameters: the complex intrinsic impedance and complex propagation constant, i.e. the wavenumber. At the molecular level, the natural materials are also discrete. Similarly, the metamaterials are based on the discrete metallic and nonmetallic inclusions acting as the artificial atoms or artificial molecules. The permittivity and permeability of the effective homogeneous medium created by these discrete inclusions could be numerically computed from the computation of the average field over the unit cell [J.3, J.56, J.57]. However, the numerical computation process is demanding and inefficient. In practice, these parameters are extracted, i.e. retrieved, from the measured S-parameters. The experimental S-parameters of a slab of the metamaterial is obtained from the free space measurement [J.43, J.58], by mounting the slab in a parallel plate waveguide (PPW) [J.59, B.11], or mounting the slab in the TE10 mode supporting rectangular waveguide [J.60–J.62]. Such

characterization of the metamaterials is useful for their practical applications. The S-parameters of the inhomogeneous engineered metamaterials is also obtained using the EM-simulators [J.43, J.63–J.67]. Therefore, the homogenization of the metamaterials is realized through the S-parameter method. The homogenization achieved using the circuit models, discussed above, is approximate, whereas the homogenization obtained from the S-parameter method is more accurate and dynamic. The metamaterial could also be treated as a mixture of inclusions in a host medium. The Maxwell Garnett mixture model of homogenization, discussed in sections (6.3.3) and (6.5.3) of chapter 6 has been extended to metamaterials based on the Mie scattering-based inclusions. This section presents a brief discussion on the parameter extraction, i.e. the parameter retrieval from EM-simulated, or experimentally obtained, S-parameters. This section further presents the extension of the Maxwell Garnett mixture model to the metamaterials to get the dynamic frequency-dependent parameters of the homogenized medium. The interested reader should consult the original literature for the detailed discussions [J.60–J.65, J.68–J.74]. 21.4.1

Nicolson–Ross–Weir (NRW) Method

The classical NRW method has been used to obtain the complex relative permittivity and complex relative permeability of a homogeneous isotropic material slab [J.60, J.75]. The method has been further extended to get these parameters of the homogenized metamaterials [J.43, J.62–J.66]. Figure (21.20a) shows that the inhomogeneous metamaterial slab of thickness d could be replaced by the homogenized equivalent metamaterial slab of identical thickness d such that both have almost identical S-parameters response, both in the magnitude and phase. We can characterize the equivalent homogeneous slab by the effective complex relative permittivity and permeability. These parameters are extracted from the EM-simulated/measured de-embedded complex S-parameters, i.e. magnitude and phase response of the S-parameters. Inside the material slab of finite thickness d, multiple reflections take place. However, these multiple reflections finally provide the reflection and transmission at the input–output interfaces of the slab. The NRW is the three stages process discussed below: At the first stage of the parameter retrieval, the reflection and transmission coefficients are obtained at the interfaces of the slab-air from the measured/simulated de-embedded S-parameters. At the second stage, the normalized intrinsic impedance and the refractive index of

21.4 Homogenization and Parameters Extraction

S22

S12

Metamaterial inhomogeneous slab.

Inc. wave

Metamaterial equivalent homogeneous slab.

Inc. wave S21

S11

S22

S12

d

S21

S11 d

(a) Equivalent homogeneous slab.

De-embedded S-paramters at interface of slab

Reflection and transmssion coeff. at interface of slab

Extraction of intrinsic impedance and refractive index

Extraction of complex εr, μr

Compare amplitude and phase of S-parameters of homogeneous slab against source data

(b) Schematic diagram of the parameter extraction process. Figure 21.20

Homogenization and extraction process of material parameters.

the slab are obtained. Finally, at the third stage, the complex relative permittivity and complex relative permeability of the slab are extracted. Such parameters extraction, at the discrete frequencies, is carried out over a frequency band. The schematic diagram of Fig (21.20b) shows the extraction process and verification of its accuracy. To verify the correctness of the parameter extraction, the extracted metamaterial parameters are used to obtain the S-parameters of the metamaterial slab. The computed S-parameters are compared, both magnitude and phase, against the original S-parameters available at the first stage. The useful expressions are obtained below for the extraction process as applied to both the homogeneous and inhomogeneous slabs in the free space or mounted in the TEM type PPW.

Computation of Relative Permittivity and Permeability of the Slab

Figure (21.21a–c) shows three arrangements of UC of the homogeneous and inhomogeneous material slabs of thickness d. Under the condition kd 1, a metamaterial slab of thickness d could be replaced by the equivalent homogeneous isotropic material slab of the same thickness. The wavenumber within the unit cell is k. Figure (21.21a) shows a slab of the homogeneous, symmetric, and reciprocal structure with S11 = S22, S21 = S12. The relative permittivity and permeability of the equivalent medium can be extracted from the deembedded S-parameters at two faces of the slab. To obtain these material parameters, the intrinsic impedance/Bloch characteristic impedance and refractive

index/propagation constant of the equivalent homogeneous slab is computed from the S-parameters. Figure (21.21b) shows an inhomogeneous, isotropic, and asymmetric structure with S11 S22. In this, the Bloch characteristic impedance of the equivalent medium is not uniquely defined, so the relative permittivity and permeability cannot be extracted uniquely. However, Fig (21.21c) shows an inhomogeneous, isotropic, and symmetric slab with S11 = S22, S21 = S12. In this case, the Bloch characteristic impedance is uniquely defined and the relative permittivity and permeability of the equivalent homogeneous medium can be extracted uniquely. The slab can also consist of several layers of the material sheets and effective material parameters could be extracted for such a structure also [J.76]. The periodic arrangement of UC can be treated either through the transfer matrix method (TMM) [J.64] or by using the [ABCD] formulation discussed in subsection (19.3.2) of chapter 19. The needed expressions are presented below for extracting the material parameters of the slab, shown in Fig (21.21a). The expressions correlate the material parameters with the reflection coefficient Γ and the transmission coefficient τ at the interfaces A–A and B–B of the slab of thickness d. The slab material has an intrinsic impedance ηm, refractive index n, relative permittivity εr, and relative permeability μr. Normally, these are complex quantities due to the losses. The normalized intrinsic impedance of the slab is ηnor μr εr. In the case of a semiinfinite slab m = ηm η 0 = (d ∞), the reflection coefficient at the interface A–A and the intrinsic impedance are related as follows:

815

816

21 Metamaterials Realization and Circuit Models – I

X A S12

X

(Air) η0 Inc. wave

E k

H

Z

Y

B

η0 (Air)

η m, β εr, μr A

S22

A S11

S21 B

S11

η0, β0

Equivalent two-port network

η0, β0

A

Z

d

B

ηm, β

S21

B

(a) S-parameter description of a homogeneous slab. X A

S22

B

Air

X

S22

B

Air

Air

Inc. wave

Air

Inc. wave A

S11

d1

d2

S21 B

A Z

d

d1

d2

S11

(b) S-parameter description of an asymmetric inhomogeneous slab. Figure 21.21

A

S21

d1 B

Z

d

(c) S-parameter description of symmetric inhomogeneous slab.

S-parameters descriptions of a unit cell of a slab.

μr ε r − 1

ηm − η0 ηnor m −1 = = nor ηm + η0 ηm + 1 Γ+1 ηm = η0 Γ−1 Γ=

μr εr + 1

a b 21 4 1

In equations (21.4.1a, b), η0 is the intrinsic impedance of free space. The transmission coefficient at the interface B–B of the slab of thickness d is τ = e − jβd = e − jβ0

is obtained from the transmission coefficient τ using an equation (21.4.2). However, there is difficulty in extracting a unique value of the refractive index, as the exponential function with j sign is a periodic function giving multiple solutions. On taking the logarithmic value of the complex transmission coefficient: ln τ = Re ln τ

= − j β0 n d − 2πm − n β0 d

μr εr d

= e − jβ0 nd = e − jβ0 n d e − n τ = e − j β0 n d − 2πm e − n m = 0, ± 1, ± 2, …,

β0 d

β0 d

21 4 2

where Z-directed propagation constant of the slab of the complex refractive index n = n − jn is β = k = β0n, and β0 = k0 is the wavenumber in free space. The passive medium has n > 0; so that the wave must decay in the direction of propagation. The DPS medium has |n | > 0, giving the lagging phase in the direction of propagation. However, the DNG medium has |n | < 0, giving the leading phase in the direction of propagation. NRW Method

Once the reflection and transmission coefficients are known, the Nicolson–Ross–Weir (NRW) method [J.60, J.64] helps to extract the complex relative permeability and permittivity of the slab, through the extracted intrinsic impedance and refractive index. The complex intrinsic impedance is obtained from the reflection coefficient Γ using (21.4.1b), whereas the complex refractive index

− jImg ln τ

On separating the real and imaginary parts of the transmission coefficient, the following expressions are obtained: n = −

1 Re ln τ β0 d

1 Img ln τ β0 d where, m = 0, ± 1, ± 2, … n =

a + 2πm b 21 4 3

It is obvious from equations (21.4.3a, b) that the imaginary part of the refractive index n is uniquely determined. However, the value of the real part of the refractive index n depends on the correct choice of integer m. This is known as the branching problem [J.65]. It is not a trivial issue. The choice of m depends on the slab thickness d. However, for the slab thickness, d < λ/2 the principal value m = 0 is taken. Even in this case, the wave length λ inside the unit cell is unknown, as the material parameters of the slab are unknown. The branching problem is discussed separately. Assuming that the complex intrinsic impedance

21.4 Homogenization and Parameters Extraction

μr = n − jn

and refractive index have been computed correctly, the material parameters are determined as follows: η m = η0

μr , εr

η μr = n m η0

μr = εr

ηm η0

nor

= n η m −n η

2

= nηnor m

μr εr

a,

n=

c,

η εr = n 0 ηm

b =

n

Re μr = μr =

d

ηnor m

The real and imaginary parts of the complex intrinsic impedance and the refractive index of a passive medium should satisfy the following passivity condition for the complex relative permeability μ∗r = μr − jμr and complex relative permittivity ε∗r = εr − jεr : n − jn η nor m − jη

=

nor m

1 Re εr = εr =

Img εr = εr =

ηnor m

1

nor

2 ηnor m

nor m

nor m

< 0,

for DNG medium

ηnor m 1

nor

n η m −n η

2 ηnor m

nor m

> 0,

nor

≤n ηm

21 4 7

The S-parameters of the isotropic symmetric slab, S11 = S22, S21 = S12 at the interfaces A–A and B–B are obtained from equation (5.4.15) of chapter 5, equations (21.4.1), and (21.4.2):

=

1 − τ2 Γ 1 − Γ2 τ2

S21

4η0 ηm = ηm + η0 =

>0 a

for DNG medium :

nor m

>0

b

nor m

21 4 6

for both DPS and DNG media

b

To use the NRW method for extracting the material parameters, expressions (21.4.8a–c) have to be inverted, i.e. the reflection Γ and transmission τ coefficients have to be expressed in terms of the known (measured/EMsimulated) de-embedded S-parameters. The following variables are introduced for the ease of inversion: V1 = S21 + S11

a

V2 = S21 − S11

b

1 − V1 V2 Define another variable, X = V1 − V2 2 2 1 − S21 + S11 X= 2S11

c d 21 4 9

1 − e − jβd 1−

By using equations (21.4.8a, b), expressions (21.4.9a– d) for the variables V1, V2, and X are rewritten as follows:

ηm + η0 2 e − j2βd

2

η m − η0

a

2

1−

η m − η0

η − η0 where, 1 − Γ = 1 − m ηm + η0

2

ηm + η0 2 e − j2βd b

2

4η0 ηm = ηm + η0

τ+Γ τ−Γ a , V2 = 1 + Γτ 1 − Γτ 1 − τ + Γ 1 + Γτ τ−Γ X= τ + Γ 1 + Γτ − τ − Γ 1 + Γ2 X= 2Γ

b

V1 =

e − jβd

1 − Γ2 τ 1 − Γ2 τ2 2

nor

−j n η m −n η

nor

The Relation between S-parameters and Reflection– Transmission Coefficients

ηm − η0 ηm + η0

nor m

a

1

In equation (21.4.7), n and η nor m are positive quantities for the wave propagation in a lossy medium.

S11 =

nor m

for DPS medium

Therefore, the imaginary parts of the extracted μr and εr of the DNG should meet the following passivity condition: nη

n η nor m −n η

Similarly,

> 0,

2

nor m

21 4 5

nor m

nηm +n η

nor

−j n η m + n η for DPS medium :

nor

nor

nηm +n η

nor m

Img μr = μr = n η m + n η

nηm +n η

2

nor m

nor n η nor λm/2, has nonzero branching index m. It has to be determined correctly for the correct extraction of the material parameters. Several methods have been

819

820

21 Metamaterials Realization and Circuit Models – I

suggested for this purpose. However, every method has a limitation. A few methods are summarized below. Using Two Thicknesses of a Slab

Even for a thin slab, several thicknesses should be considered for the parameter extraction and consistent results should be obtained [J.63]. If we consider two thickness d1 and d2 such that (d2 − d1) < λm/2, then the branching index m could be estimated from the following expression, using the transmission phases ∠τ1 and ∠τ2 of slab, corresponding to the thickness d1 and d2 [J.66]: m=

d2 τ 1 − d1 τ 2 2π d1 − d2

21 4 21

However, the condition (d2 − d1) < λm/2 is uncertain as the λmis unknown inside the slab. Iterative Method

This method evaluates the real part of the refractive index at two adjacent frequency f0 and f1 (f1 > f0) using expression (21.4.12) for the transmission coefficient [J.65]. The branching index m is estimated by approximating the transmission coefficient at f1 using the Taylor expansion of the expression (21.4.12):

e − jβ0

e − jβ0

f1 n f1 d

f1 n f1 d

= e − jβ0

f0 n f0 d

= e − jβ0

f0 n f0 d Δ

≈ e − jβ0

ej β0

f 0 n f 0 d − β0 f 1 n f 1 d

e

f0 n f0 d

1+Δ+

Δ2 2

where, Δ = j β0 f 0 n f 0 d − β1 f 1 n f 1 d

a b 21 4 22

To evaluate the parameter, Δ the complex refractive index n(f0) at the first frequency f0 has to be computed that also has a branching problem. Using equation (21.4.3a), the Img{n(f0)}, i.e. n (f0), is determined uniquely. The computation of Re{n(f0)}, i.e. n (f0) involves the determination of the branching index m. The value of n (f0) is determined for a few branching indexes, say m = 0, 1, 2. The normalized intrinsic impedance is computed using equation (21.4.13d). Each computed value of a complex refractive index corresponding to a distinct value of m must satisfy the passivity condition (21.4.7). If only one value of m satisfies the passivity condition, it is accepted as a correct branching index “m.” In the case of multiple values for branching index m satisfying the passivity condition, the refractive index is computed for each value of m at several other frequencies, and passivity condition is tested. The correctly selected value of m should satisfy the

passivity condition at several frequencies. The value of n (f0) at frequency f0 is accepted for the correct branching index. It is used in equation (21.4.22a). The transmission coefficient e − jβ0 f 1 n f 1 d at frequency f1 appearing in the quadratic equation (21.4.22a), for the variable Δ, is computed from equation (21.4.12). The approximate quadratic equation (21.4.22) provides two complex roots – Δ1(f1), Δ2(f1) at frequency f1. Using two roots, two approximate solutions are obtained for two complex refractive indexes n1appx f 1 and n2appx f 1 . Both of complex refractive indexes have their imaginary parts. However, the correct imaginary part n (f1) of the refractive index n(f1) is already known, so the correct choice of the root should provide the corresponding approximate value of the imaginary part, closer to the known value of n (f1). The correct choice of root helps to compute the approximate refractive index, i.e. approximate real part nappx(f1) also. At this stage, a few values of real n (f1) are computed, using equation (21.4.3b), for a few values of m = 0, 1, 2… One particular value of a real refractive index n (f1), corresponding to the correct value of m, is closer to nappx(f1) obtained by using the correct root of equation (21.4.22a). Once correct branching index m is determined, the correct value of the refractive index is computed using equation (21.4.3b). As both the intrinsic impedance of the material slab and its refractive index are known, the complex relative permittivity and permeability are extracted using equations (21.4.4c, d). The extracted parameters should have μr ≥ 0, εr ≥ 0, as the medium is passive. The process has to be repeated at each of the frequency points in the considered range of frequencies. The computer code could be written for this repetitive parameter extraction process. The extracted complex relative permittivity and permeability of the equivalent slab are used to simulate the S-parameters, and it should closely follow the S-parameters of the original inhomogeneous slab. It decides the correctness of the extracted parameters. Using Kramers–Kronig Relations

This method approximately recovers the real part of the refractive index at a frequency ω from the known imaginary part of the refractive index at frequency ω [J.68, J.77]: n

appx

2 ω =1+ P π

∞

0

ωn ω dω, ω2 − ω 2

21 4 23

where P denotes the principal value of improper integral. The imaginary part of the refractive index n (ω) is

21.4 Homogenization and Parameters Extraction

uniquely extracted using equation (21.4.3a). The limits of the integral are replaced by the finite limits of frequency over which the S-parameters are available either experimentally or using EM-simulator. So the process involves the truncation error, and only the approximate value of the refractive index n appx(ω ) is recovered. However, once it is substituted in equation (21.4.3b), the branching index m is obtained from the following approximation: m = Round

n

appx

ω − n ω, m = 0

β0 d 2π 21 4 24

In equation (21.4.24), n (ω, m = 0) is a real part of the refractive index for the principal value m = 0 obtained from equation (21.4.3b). The Round function provides the least integer for the branching index m. The MATLAB code is available for this process [J.68, J.78]. Another code is also available for the extractions process, including the detailed process of the EM-simulation [J.67]. Using the above-computed branching index m, equation (21.4.3b) extracts the correct value of the real part n (ω) of the refractive index n(ω). Again, the complex permittivity and permeability are extracted as discussed above for other methods. The extracted parameters may show a nonphysical discontinuity in the response [J.68]. It should be removed. Any parameter extraction process can have such discontinuity in the response. However, using the Drude–Lorentz model for the permittivity and permeability the discontinuity in the response can be removed [J.69]. A few references could be consulted to examine the extracted parameters using various methods [J.63–J.66, J.68, J.69]. The above discussion is limited to the isotropic slabs with normal incidence. However, the retrieval process has been developed for the oblique incidence also. The parameters of the isotropic homogeneous materials should be independent of the angle of incidence and polarization states of the incident EM-waves. It is difficult to develop such metamaterials. Likewise, the methods of parameter retrieval of the anisotropic and bianisotropic slabs have also been investigated by the researchers [J.71, J.79]. 21.4.2

Dynamic Maxwell Garnett Model

The metallic inclusions, i.e. the strip conductors-based structures have been primarily used to develop both the SNG and DNG engineered metamaterials in the microwave and mm-wave frequency ranges. However, such metamaterials are lossy in the THz to optical frequency ranges. The low-loss SNG and DNG, useful for

the microwave to the optical frequency, could be engineered by the all nonmetallic inclusions in the dielectric host medium. The spherical dielectric inclusion particles are commonly considered. However, other shapes, such as cylindrical and cubic shaped dielectric resonators, have also been examined. These structures are attractive for the realization of superlens and clocking devices [J.80, J.81]. The dielectric resonators-based metamaterials can be homogenized by using the modified version of the Maxwell Garnett (MG) model discussed subsection (6.3.3) of chapter 6. The modified MG model is frequencydependent that computes both the effective relative permittivity and effective relative permeability of a homogenized medium. The spherical dielectric inclusions are modeled using the Mie scattering that creates both the magnetic and electric type Mie resonances induced by the incident high-frequency waves. The resonating inclusions behavior is also responsible for generating negative permeability and negative permittivity. They follow the Lorentz model response. The negative permittivity is also obtained by using the conducting inclusion, i.e. the plasmonic inclusion modeled by the Drude model, discussed in subsection (6.5.2) of chapter 6. The MG model can accommodate these negative parameters, although the classical MG model has been used only for the positive material parameters [J.82–J.84]. Mie Scattering

The composite material could be developed using the spherical dielectric inclusions of radius “a” with relative permittivity εri and relative permeability μri embedded in the host medium with relative permittivity εrh and relative permeability μrh. The incident EM-wave on the composite material induces both the magnetic and electric dipoles, and also multipoles on the spherical dielectric inclusions. These dipoles and multipoles create the scattered field known as the Mie scattering. The scattered field of the dielectric spherical inclusions is treated as the superposition of the TM and TE normal modes with following Mie coefficients [J.85, B.19]: mψn mx ψn x − ψn x ψn mx mψn mx ξn x − ξn x ψn mx ψn mx ψn x − mψn x ψn mx bn = ψn mx ξn x − mξn x ψn mx εri μri ni where, m = = nh εrh μrh ω a εrh μrh a = 2πnh x= c λ an =

a b c d 21 4 25

821

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21 Metamaterials Realization and Circuit Models – I

In equations (21.4.25a–d) m is the contrast ratio of the refractive indices, and x is the size parameter of the inclusion. The functions ψn and ξn are the nth order Riccati–Bessel function. The derivatives of ψn and ξn, with respect to the arguments, are ψn and ξn respectively. The complete solution of the scattered field is presented by Bohren and Huffman in their beautiful book [B.19]. The coefficients an and bn determine magnitudes of fields of the scattered normal modes. The coefficient an provides a large value of the E-radial field component of the TM-mode under the resonance condition. The Hradial field is zero. Likewise, the coefficient bn corresponds to the excited TE-mode with a high value of the H-radial field component. In this case, the E-radial field is zero. To excite these modes by the incident EM-field, the denominators of equation (21.4.25) are nearly zero, giving the following resonance conditions: For TM – mode For TE – mode

ψn mx 1 ψn mx = ξn x m ξn x ψn mx ψ mx =m n ξn x ξn x

a b 21 4 26

The resonance frequency of the TM and TE modes are computed from equations (21.4.26a, b). The TM011 mode is the fundamental resonant mode with the lowest frequency, and the TE011 mode is the next resonant mode at a higher frequency. Both equations given above are identical except the location of the contrast-ratio (m) in the denominator in one equation, and in the numerator for the other. Figure (21.22a, b) show the sketched electric field lines of the TM011 and TE011 modes on the surface of the dielectric spherical inclusion. The E-field

circular loops of the TM011 mode suggest the presence of the circular loops of displacement current on the dielectric sphere. The magnetic field is normal to the loops. Thus, the magnetic field excited sphere in the TM011 mode behaves as a magnetic dipole. The host medium could be treated as loaded with the magnetic inclusions, with the magnetic polarizability αm. Likewise, the E-field lines of the TE011 mode of the dielectric sphere behave like an electric dipole. In this case, the host medium could be treated as loaded with the electric inclusions, with the electric polarizability αe. The above description suggests that the composite medium behaves as a magnetic medium at a lower resonant frequency with a narrow bandwidth. The composite medium also behaves like an electric medium at the higher resonance frequency. Using the Lorentz model for the magnetic and electric medium, the negative permeability and negative permittivity can be obtained above their respective resonance frequencies. Thus, at a single frequency, the SNG medium is obtained with inclusions of a uniform fixed value of radius. The DNG medium is not obtained in this way. However, the magnetic and electric polarizabilities of the spherical inclusion could be used to obtain the dynamic Maxwell Garnett model #I. The dynamic Maxwell Garnett model #II could also be obtained using the modified relative permittivity and permeability of the spherical inclusions. A summary of both models is given below.

Dynamic Maxwell Garnett Model # I

The Mie coefficients a1 and b1 of the spherical inclusions could be replaced by the following electric and magnetic polarizabilities, corresponding to the electric and

(+)

H

(–) E-field lines (a) Closed-loop of E-field lines of TM011 mode around the sphere. H-field normal to the plane. Figure 21.22

E-field lines (b) E-field lines of TE011 mode on the sphere.

The electric field lines of the first two normal modes of Mie resonance.

21.4 Homogenization and Parameters Extraction

magnetic fields excited electric and magnetic dipoles [J.86, J.87]: αe = j4πε0

3a3 a1 2x3

a,

αm = j4πμ0

3a3 b1 2x3

b 21 4 27

In equation (21.4.27), the radius of the spherical inclusion is a, and x is the size-parameter given by equation (21.4.25). For n = 1, the Mie coefficients a1 and b1 are computed from equations (21.4.25a, b). The FORTRAN code is available to compute the Mie scattering coefficients [B.19]. The Ricatti–Bessel functions and their derivatives are expressed in terms of the trigonometric functions [J.88, J.89]: sin ρ 1 cos ρ − cos ρ, ψ1 = sin ρ 1 − 2 + ρ ρ ρ cos ρ 1 + sin ρ , ξ1 = ψ1 + j ψ1 + 2 cos ρ ξ1 = ψ1 − j ρ ρ ψ1 =

21 4 28 The spherical inclusions, as the electric dipoles, are arranged in the cubic lattice of volume Δv. The fillingfactor f of inclusion is obtained as follows: f=

vi 4π 3 = a N, Δv 3

N=

3f , 4πa3

21 4 29

where N is the inclusion density. The effective relative permittivity εreff, i.e. the equivalent relative permittivity, of the composite medium, i.e. the mixture medium of host and inclusion, in terms of the electric polarizability of the inclusion, is obtained from the Clausius–Mossotti equation (6.1.13) of chapter 6, and equation (21.4.29): εreff − εrh Nαe = = 3ε0 εreff + 2εrh 2fαe εrh +1 4πε0 a3 εreff = e fα 1− 4πε0 a3

f αe 4πε0 a3

Similarly, the magnetic composite medium is created by the magnetic dipoles of spherical dielectric inclusions, resonating in the TM011 mode. The effective relative permeability of the magnetic composite medium is obtained by recasting equation (6.1.28b) of chapter 6 as follows: μreff − μrh Nαm = = 3μ0 μreff + 2μrh εrh 2fαm μrh +1 4πμ0 a3 μreff = fαm 1− 4πμ0 a3

21 4 32 By substituting αm from equation (21.4.27b) in equation (21.4.32), the dynamic MG model #1 for the effective permeability is obtained:

μreff =

3f b1 + 1 μ x3 + j3fb1 x3 = rh3 3f x − j 3 2 fb1 1 − j 3 b1 2x

μrh j

21 4 33

Dynamic Maxwell Garnett Model # II

Lewin has suggested a different expression for the dynamic MG model # II by modifying the relative permittivity and permeability of the spherical dielectric inclusion. Thus, relative permittivity εri and relative permeability μri of the spherical inclusion are replaced by the frequency-dependent modified relative permittivity and permeability μmod in the static MG model given εmod ri ri by equations (6.3.13a) and (6.3.17a) of chapter 6 to get the dynamic Maxwell Garnett model #II [J.90]: εreff = εrh 1 + εreff = εrh 1 +

εmod ri

3f a1 + 1 x3 3 1 − j 3 a1 2x

εrh j εreff =

21 4 31

Expression (21.4.31) is the high-frequency extension of the classical MG model obtained from the Clausius– Mossotti model.

μreff = μrh 1 + μreff = μrh 1 +

3f εmod − εrh ri + 2εrh − f εmod − εrh ri

εmod + 2εrh ri

21 4 30 On substituting the electric polarizability of the spherical inclusion, resonating in the TE011 mode, from equation (21.4.27a) in equation (21.4.30), the following expression of the effective relative permittivity εreff is obtained for the dynamic MG model #1:

f αm 4πμ0 a3

μmod ri

1 − εrh − 1 3 3f εmod ri

a

3f μmod − μrh ri + 2μrh − f μmod − μrh ri

μmod + 2μrh ri

1 − μrh − 1 3 3f μmod ri

= F θ εri , μmod = F θ μri , F θ where, εmod ri ri 2 sin θ − θ cos θ = 2 θ − 1 sin θ + θ cos θ

b

c

21 4 34 In equations (21.4.34a–c), the frequency-dependent parameter is θ = k0 a εri μri , and the frequencydependent function F(θ) is approximately a periodic resonant function that is responsible for the resonant

823

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21 Metamaterials Realization and Circuit Models – I

nature of the inclusion. For a spherical dielectric inclusion in the cubic lattice of lattice constant d, the filling-factor f is 4/3π(a/d)3. The wavenumber of the incident field in free space is k0 = 2π/λ0. For a spherical inclusion, the maximum filling-factor in the cubic lattice is f = vi/Δv = ((4/3)πa3)/8a3 = π/6 ≈ 0.5. The MG model #II works satisfactorily around the filling-factor f ≈ 0.3. The TE-mode Mie resonance creates the effective permittivity, while the TM-mode resonance is responsible for the effective permeability of the composite medium with Mie resonance frequency f TM < f TE r r for an inclusion particle. Around the Mie resonance, the effective permittivity and permeability of the composite medium are very large and the εreff and μreff approach to εrh and μrh of the host medium. Thus, around the modal Mie resonance frequencies, the εmod and f = f TE ri r TM mod μri f = f r of a spherical inclusion are obtained from equation (21.4.34) follows: εmod + 2εrh 1 ri − =0 mod 3 3f εri − εrh = − εrh f = f TE εmod ri r

2+f 1−f

a

μmod + 2μrh 1 ri − =0 3 3f μmod − μ rh ri = − μrh f = f TM μmod ri r

2+f 1−f

b 21 4 35

The above negative values are required for the modified electrical parameters of the inclusion to achieve the resonance for the TM and TE modes of the scattered field. It helps to get an effective homogeneous medium. Using equation (21.4.34), at the given resonance frequency of TE/TM mode, the radius “a” of the spherical inclusion is determined from the relation θ = k0 a

mod . εmod ri μri

Once the electrical parameters of the inclusion and host medium are known, the parameter θ is obtained from equations (21.4.34c) and (21.4.35). Lewin has considered two important limiting cases for the modified relative permittivity and permeability [J.90]: Case # a: At low frequency of the incident wave, the parameter θ is small with F(θ) = 1. It leads to εmod εri , and μmod μri . Under this condition, ri ri the dynamic MG model #II of equation (21.4.34) is reduced to the static MG model, discussed in chapter 6. Case # b: The parameter θ could be large and complex. It applies to a metallic, i.e. plasmonic, inclusion with F(θ) ≈ 2/jθ. For this case, the modified parameters of inclusion are given below:

εmod f = ri

2 j

εri μri

λ0 2πa

a

μmod f = ri

2 j

μri εri

λ0 2πa

b 21 4 36

Normally, the frequency-dependent modified material parameters of the inclusion εmod f and μmod f are much ri ri smaller than the real material parameters εri and μri of an inclusion particle. However, around the Mie resonance frequency f TE r of TE-mode, the modified relative permitmod tivity εri f of the inclusion is much higher than the relative permittivity εrh of the host medium, i.e. εmod f εrh . The resonance frequency f TE is the offri r resonance frequency for the TM-mode responsible for the creation of the modified relative permeability μmod f of the inclusion. Thus, around resonance freri mod f is much less than the relative permequency f TE r , μri ability of the host medium, i.e. μmod f < < μrh . Under ri these conditions around the resonance frequency f TE r , the modified MG model #II of equation (21.4.34) provides the following approximate expression for the effective permittivity and permeability of the mixture: 3f 1−f 3f 1− 1−f

εmod = εrh 1 + ri

a

= μrh μmod ri

b 21 4 37

The accuracy of both forms of the dynamic MG has been tested against the extracted parameters of the effective homogeneous medium from the simulated S-parameter of the cubic unit cell using the NRW method. The deep sub-wavelength composite of spherical Ag inclusions, radius a = 3 nm, arranged in a cubic lattice with lattice constant d = 15 nm, in the SiO2 host medium has been simulated on CST to get the S-parameter of unit-cell. Figure (21.23a) compares the results of both MG models against the extracted results from the S-parameters. All three results show very good agreement for the effective relative permittivity. It shows that the mixture follows the Lorentz model behavior. The effective relative permeability is unity [J.88]. The conditions discussed also prevail around the Mie resonance frequency f TM for the resonating TM mode, c creating the dominant relative permeability in the inclusion and a much smaller value for the relative permittivity, i.e. μmod > > μrh and εmod < < εrh . Again, the ri ri approximate expressions for effective permittivity and permeability of the mixture could be obtained. Several schemes have been suggested and demonstrated to develop the Mie resonance-based DNG

21.4 Homogenization and Parameters Extraction

4

Re(εr) MG-II Im(εr) MG-II

Relative permittivity (εr)

Re(εr) MG-I 3

Im(εr) MG-I

2 Re(εr) Sim Im(εr) Sim

1 0 0.25

0.5

1 0.75 Frequency (GHz)

1.25

(a) Comparison of two MG models against the extracted results of S-parameters. Host medium – SiO2 and Ag spherical inclusion. 6 a. Re(εr) Sim b. Im(εr) Sim c. Re(μr) Sim d. Im(μr) Sim

4 E k H

Relative permittivity (εr)

Relative permittivity (εr)

6

a

2

c 0

b

d

4

5

7.5

10

12.5

15

a

2 c

5

Frequency (GHz) (b) Extracted parameters of the SRR inclusion. Figure 21.23

b

d

0 –2

–2

a. Re(εr) MG-I b. Im(εr) MG-I c. Re(εr) MG-II d. Im(εr) MG-II

7.5

10

12.5

15

Frequency (GHz) (c) Parameters of the equivalent spherical dielectric inclusion.

Effective medium parameters using the dynamic MG models. Source: Reproduced from Szabo and Fuzi [J.88]. © 2014, IEEE.

materials. The Mie resonators could be in several shapes – spherical, cylindrical, cubic, etc. Some of these schemes are nicely summarized in a review paper [J.80]. It is noted above that the magnetic and electric Mie resonances of a dielectric sphere occur at two different frequencies. To obtain the electric resonance and magnetic resonance at the same frequency, two spherical inclusions of the same material but of different sizes could be used within a cubic unit cell [J.91]. The larger size inclusions are needed to bring down the permittivity resonance frequency near to the magnetic resonance of a smaller inclusion. A pair of inclusions of the same size but made of different materials can also achieve identical electric and magnetic resonance frequencies. Normally, the dielectric inclusion resonates at the fundamental frequency giving negative permeability. It is the MNG type of material. However, a metallic pin mounted near or inside the dielectric inclusion provides the negative permittivity [J.92]. The composite inclusion

provides the DNG response. The dielectric inclusions can be arranged in a waveguide section also [J.93]. The negative permeability is generated by the Mie resonance and the TE cut-off mode adds the negative permittivity. The combined response is DNG. Similarly, the metal-coated dielectric inclusion provides the DNG response [J.94]. The inner dielectric core responds to the magnetic excitation, while negative permittivity is generated by the outer plasmonic shell. The dynamic Maxwell Garnett model could be helpful to get the homogenized effective medium. Tailoring the effective medium property could be more convenient with the dynamic MG model. Equivalent Mie Magnetic Resonator of SRR Inclusion

Normally the dynamic MG model, based on the Mie scattering, is not suitable for the metallic SRR type inclusion. However, once the effective permeability of the SRR-based material has been extracted, using the

825

21 Metamaterials Realization and Circuit Models – I

the original SRR response, shown in Fig (21.23b). However, as expected, its permittivity response occurs at a higher frequency, so it is a magnetic medium only. By adding additional wire-medium, the DNG response can be obtained.

20 Re(μreff) model Img(μreff) model Re(μreff) EMG and mie

15 10

Img(μreff) EMG and mie

05 00

–5.0 –10 7.0

7.2

7.4 7.6 Frequency (GHz)

7.8

Effective rel.permeability (μreff)

Effective rel.permeability (μreff)

simulated S-parameters, the radius and permittivity of the equivalent spherical inclusions, filling-factor, and permittivity of the host medium can be optimized using the dynamic MG model to meet the permeability response of the SRR [J.88]. Figure (21.23b) shows the extracted permeability, and also permittivity of the copper SRR in a cubic unit cell of lattice constant d = 5 mm. The dimensions of the SRR printed on a substrate with εr = 3.84, h = 0.25 mm are given: the external size of ring 3 mm, the width of the strip 0.25 mm, the size of the split gap and separation between rings 0.5 mm, and strip thickness 0.02 mm. The optimized equivalent radius of spherical inclusion is a = 2.31 mm and filling-factor f = 0.13. The permittivity of the inclusion is 37.67 and the host medium is air. The equivalent inclusion can be implemented practically. Figure (21.23c) shows the response of the equivalent Mie resonance-based medium. Its permeability response is almost identical to that of

Equivalent RLC Circuit Model of Mie Magnetic Resonator

The Mie resonance-based dynamic MG model computes the effective permeability of the composite medium with dielectric spherical inclusions. However, computation of the resonance frequency, the bandwidth of negative permeability response, and computation of magnetic response by varying the geometrical parameters are not done directly. The equivalent RLC circuit model is available for the direct computation of these parameters. The RLC equivalent circuit, as shown in Fig (21.6d), for the SRR resonator is presented in section (21.2.2). The circuit elements of the equivalent RLC, around the first

20 Re(μreff) model Img(μreff) model Re(μreff) simulation

15 10

Img(μreff) simulation

05 00 –5.0 –10

8.0

7.0

(a) Complex permeability as computed by the circuit model and dynamic MG model#II.

7.2

7.4 7.6 7.8 8.0 Frequency (GHz) (b) Complex permeability as computed by the circuit model and EM-simulator.

0.6

7.5

Simulation EMG Model Approximate model

7.4 7.3

0.5

Simulation EMG Model

0.4 Δ

ωr /2π (GHz)

826

7.2

0.3 0.2

7.1

0.1 7.0 0.0 6.9 0.0

0.1

0.2

0.3

0.4

0.5

f (c) Resonance frequency computed by four methods. Figure 21.24

0.0

0.1

0.2

0.3

0.4

0.5

f (d) Fractional BW computed by four methods.

Computation magnetic response by three methods. Source: From Liu et al. [J.95], reproduced courtesy of The Electromagnetics Academy. © 2011, The Electromagnetics Academy.

References

resonance, could be computed from the following expressions [J.95]: R=

μ0 π4 cni 6ni

a,

C=

12an 2i μ0 π5 c2

c

L=

μ0 π3 a 12

Books b

B.1 Engheta, N.; Ziolkowski, R.W. (Editors):

21 4 38 In equation (21.4.38), spherical inclusion has a radius a and its low-loss refractive index is n∗i = ni − jni ni < < ni . The velocity of EM-wave in free space is c. The resonance frequency of the inclusion is ω0 = 1 LC . The effective permeability could be obtained from equation (21.2.3). For the cubic lattice, parameters F, γm and frequency ω0m used in equation (21.2.3) are computed from the following expressions [J.95]: 9f F= 3f + π2 ωom =

a,

2π3 cni γm = 2 n i a 3f + π2

π2 c ni a

3f +

a,

ωr ≈

πc 3 1− 2f ni a 2π

B.4

B.6

In expression (21.4.39), f is the volume fraction of inclusion. The relative bandwidth Δ of the negative permeability, and also approximate resonance frequency, can be estimated from the following expressions [J.95]: 9f π2 − 6f

B.3

b

21 4 39

Δ=

B.2

B.5

c

π2

References

B.7 B.8 B.9

b 21 4 40

The accuracy of the circuit model is tested for the spherical Barium Strontium Titanate (BST) material with complex relative permittivity (εri = 1600 − j4.8). The sphere of BST with radius a = 0.5 mm is located in the cubic lattice of the lattice constant d = 2 mm. The relative permittivity of the host medium is εrh = 2. Figure (21.24a) compares the circuit model-based computed results of complex effective permeability against the results of the dynamic MG model #II. Likewise, Fig (21.24b) compares the results of the circuit model against the parameter extracted from the simulated Sparameters. Only a minor variation is seen among the results from three sources. Figure (21.24c, d) further compares the circuit model-based results of resonance frequency and fractional BW of negative permeability of a composite medium against the results obtained from two other sources – dynamic MG model, and EM-simulator. The resonance frequency decreases and fractional BW increases with an increase in the volume fraction.

B.10

B.11

B.12

B.13 B.14 B.15

B.16

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metallic powders in the microwave range, Modell. Simul. Mater. Sci. Eng. IOP Pub., Vol. 18, No. 025015, pp. 13, 2010. Cohen, D.; Shavit, R.: Bi-anisotropic metamaterials effective constitutive parameters extraction using oblique incidence S-parameters method, IEEE Trans. Antennas Propag., Vol. 63, No. 5, pp. 2071–2078, May. 2005. Nicolson, A.M.; Ross, G.F.: Measurement of the intrinsic properties of materials by time-domain techniques, IEEE Trans. Instrum. Meas., Vol. IM-19, pp. 377–382, Nov. 1970. Shi, Y.; Li, Z.Y.; Li, L.; Liang, C.H.: A retrieval method of effective electromagnetic parameters for inhomogeneous metamaterials, IEEE Trans. Microwave Theory Tech., Vol. 65, No. 4, pp. 1160–1178, Sept. 2009. Peiponen, K.E.; Lucarini, V.; Vartiainen, E.M.; Saarinen, J.J.: Kramers–Kronig relations and sum rules of negative refractive index media, Eur. Phys. J. B Condens. Matter Complex Syst., Vol. 41, No. 1, pp. 61– 65, Sept. 2004. http://effmetamatparam.sourceforge.net/net Jin, J.; Liu, S.; Lin, Z.; Chui, S.T.: Effective – medium theory for anisotropic magnetic materials, Phys. Rev. B, Vol. 80, No. 115101, pp. 1–6, 2009. Zhao, Q.; Zhoul, J.; Zhang, F.; Lippens, D.: Mie resonance-based dielectric metamaterials, Mater. Today, Vol. 12, pp. 60–69, Dec. 2009. Holloway, C.L.; Kuester, E.F.; Baker-Jarvis, J.; Kabos, P.: A double negative (DNG) composite medium composed of magnetodielectric spherical particles embedded in a matrix, IEEE Trans. Antennas Propag., Vol. 51, No. 10, pp. 2596–2603, Oct. 2003. Vinogradov, A.P.; Dorofeenko, A.V.; Zouhdi, S.: On the problem of the effective parameters of metamaterials, Phys. Uspechi, Vol. 51, No. 5, pp. 485–492, 2008. Wallén, H.; Kettunen, H.; Qi, J.; Sihvola, A.: Some Interesting Effects when Homogenizing Plasmonic Composites, XXXII Finnish URSI Convention on Radio Science and Electromagnetics 2010, pp. 47–50, Oulu, Finland, 2010. http://www.ursi.fi/2010/papers/ A33_Wallen. Sihvola, A.: Six-dimensional view of dielectric mixtures as metamaterials, Eur. Phys. J. Appl. Phys. Vol. 46, No. 32602, pp. 1–5, 2009.

J.85 Mie, G.: Beitrage zur optic truber medien speziell

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kolloidaler metallosungen, Ann. Phys., Vol. 25, pp. 377– 445, 1908. Doyle, W.T.: Optical properties of a suspension of metal particles, Phys. Rev. B, Vol. 39, No. 14, pp. 9852–9858, 1989. Ruppin, R.: Evaluation of extended Maxwell Garnett theories, Opt. Commun., Vol. 182, pp. 273–279, Aug. 2000. Szabo, Z.; Fuzi, J.: Equivalence of magnetic metamaterials and composites in the view of the effective medium theories, IEEE Trans. Magn., Vol. 50, No. 4, pp. 2501104-1–2501104-4, Apr. 2014. Aspnes, D.E.: Local field effects and effective – medium theory: a microscopic perspective, Am. J. Phys., Vol. 50, No. 8, pp. 704–709, 1982. Lewin, L.: The electrical constants of a material loaded with spherical particles, Proc. Inst. Elec. Eng., Vol. 94, pp. 65–68, 1947. Vendik, I.; Vendik, O.; Kolmakov, I.; Odit, M.: Modelling of isotropic double negative media for microwave applications, Opto-Electron. Rev., Vol. 14, No. 3, pp. 179–186, 2006. Zhao, O.; Kang, L.; Du, B.; Zhao, H.; Xie, Q.; Huang, X.; Li, B.; Zhou, J.; Li, L.: Experimental demonstration of isotropic negative permeability in a three-dimensional dielectric composite Phys. Rev. Lett., Vol. 101, pp. 027402, July 2008. Ueda, T.; Lai, A.; Itoh, T.: Demonstration of negative refraction in a cut-off parallel-plate waveguide loaded with 2-D square lattice of dielectric resonators, IEEE Trans. Microwave Theory Tech., Vol. 55, No. 6, pp. 1280–1287, July 2007. Andrea Alùa, A.; Enghetab, N.: Polarizabilities and effective parameters for collections of spherical nanoparticles formed by pairs of concentric doublenegative, single-negative, and/or double-positive metamaterial layers, J. Appl. Phys., Vol. 97, pp. 0943101–094310-12, 2005. Liu, L.Y.; Sun, J.B.; Fu, X.J.; Zhou, J.; Zhao, Q.: Artificial magnetic properties of dielectric metamaterials in terms of effective circuit model, Progr. Electromagn. Res. PIERS, Vol. 116, pp. 159–170, 2011.

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22 Metamaterials Realization and Circuit Models – II (Metalines and Metasurfaces)

Introduction The electromagnetic properties of the metamaterials are discussed in the section (5.5) of chapter 5. Chapter 21 presents their realization and characterization as the 3D-bulk engineered materials. However, the 1D and 2D forms of the metamaterials, also known as the metalines and metasurfaces, have their specific properties and applications, from the microwave to optical frequency ranges. In practice, the metalines and metasurfaces are realized by the periodic arrangement of the inclusions in the host medium. However, they are different from the artificially engineered EBG/ PBG lines and surfaces, discussed in chapters 19 and 20. The EBG/PBG materials are the bandgap materials as their lattice constant is half-wavelength, offering the Bragg diffraction. However, for the metalines and metasurfaces, the size of inclusion and lattice constant both are preferably less than one-tenth of a wavelength, or certainly less than a fourth of a wavelength. Therefore, the Bragg diffraction is avoided in the metalines and metasurfaces. It helps to get the homogeneous effective medium characterized by the effective (equivalent) permittivity and permeability. The metalines and metasurfaces are modeled using the circuit models. The metalines are realized by both the resonant and nonresonant inclusions providing wide operational bandwidth. The 1D metalines are realized in the microstrip, CPW, and waveguide technology. These are useful for their ability to miniaturize microwave components with controllable dispersion characteristics. The 2Dmetasurfaces are also realized in planar technology. This chapter presents the characteristics and realization of the metalines and metasurfaces in planar technology. Several textbooks are available to cover various aspects and applications of the 1D and 2D metamaterials in detail [B.1–B.8].

Objectives

• • • • • •

To present circuit models to synthesize 1D DNG and other metamaterials. To discuss the realization of nonresonant metalines in microstrip technology. To discuss the realization of resonant metalines in microstrip and CPW technologies. To present a few illustrative application examples of metalines. To present the characterization and control of 2D-metasurfaces. To discuss a few applications of 2D-metasurfaces.

22.1

Circuit Models of 1D-Metamaterials

The 1D-metamaterials, also called the metalines [B.7, B.8], are normally designed in the microstrip, CPW, and waveguide technology. The standard 1D transmission line host medium supports the forward wave propagation. These DPS media are loaded with the periodic nonresonant and resonant type inclusions to realize the metalines. Chapter 19 presents the modeling of the 1D-periodic lines using the 1D transmission matrix, i.e. the [ABCD] matrix. However, the periodic medium should be homogenized to get the effective medium described by the effective permeability and effective permittivity. The terminology – “effective” appears to be confusing in the context of the microstrip-based metalines, as the effective permittivity/permeability is normally used to describe the propagation characteristics of an inhomogeneous microstrip line. To avoid confusion, we prefer to use the terminology equivalent permeability and equivalent permittivity to characterize the effective medium.

Introduction to Modern Planar Transmission Lines: Physical, Analytical, and Circuit Models Approach, First Edition. Anand K. Verma. © 2021 John Wiley & Sons, Inc. Published 2021 by John Wiley & Sons, Inc.

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22 Metamaterials Realization and Circuit Models – II

following equation for the propagation constant of the effective medium:

This section considers the basic 1D-circuit models to get the artificial DPS, DNG, ENG, and MNG material media. The electromagnetic properties of these media are discussed in section (5.5) of chapter 5.

22.1.1

Homogenization of the 1D-medium

Zpul Ypul A+D = 1 + se sh 2 2 pul Y , 4 sin 2 βd 2 = − Zpul se sh

εr Medium μr

H

Z

Figure 22.1

X β

Y

β=

pul − jωLpul =ω se jωCsh

22 1 1

22.1.2

Zse

Circuit Equivalence of Material Medium

Figure (22.1) shows the electromagnetic modeling of a continuous medium in terms of the continuous equivalent transmission line with characteristic impedance Z0 and propagation constant γ, and also in terms of the discrete circuit elements: series impedance Zpul se and pul shunt admittance Ysh of the equivalent asymmetrical L-network to model a material medium. However, the symmetrical T-network, and occasionally π-network, is also used for this purpose [B.2, B.3, B.5, B.7]. For the engineered effective medium, the process is in the reverse order, i.e. the discrete circuit networks are cascaded to synthesize the effective (equivalent) homogeneous

pul

Z0, γ

pul

Ysh

pul

Lse

pul

Csh

X Eq. Tr. line

Circuit equivalence of a material medium.

pul Lpul se Csh

Expression (22.1.2) is a standard expression of the propagation constant of a continuous transmission line discussed in chapter 2. Thus, under the homogeneity condition, the discrete periodic line structure behaves as a continuous effective medium. Of course, under Bragg’s condition, the concept of the effective medium is not applicable and the medium exhibits the stopband characteristics [B.2, B.6, B.7]. Equation (22.1.1) is a more general expression. The series impedance and shunt admittance of a unit cell could be formed by any combination of inductance and capacitance to create several kinds of lossless effective media. The effective medium is characterized by the equivalent (i.e. effective) permeability μeq and equivalent (i.e. effective) permittivity εeq. These material medium parameters could be computed from series impedance and shunt admittance of the equivalent transmission line model of a DPS medium. However, the equivalence is also applicable to other engineered media.

where β is the propagation constant of the loaded transmission line medium, d is the period of the unit cell formed by the L-network with series impedance Zpul se and shunt admittance Ypul sh on the per unit length (p.u. l.) basis. Expression (22.1.1) is valid for the symmetrical T-network also. It is obvious that to get the real solution of β, the series and shunt arm impedances must be imaginary quantities with opposite signs. If the impedance (Zse) and admittance (Ysh) are lumped elements, then pul Zpul se = Zse/d, and Ysh = Ysh/d. Using the homogeneity condition d/λ < < 1, equation (22.1.1) is reduced to the

Y E

− Zse d Ysh d =

22 1 2

The usual transmission line and open space are continuous media, with or without losses, supporting the forward wave propagation. However, the 1D medium, both the DPS and DNG types, are engineered by the periodic arrangement of the series/shunt lumped inductors and shunt/series capacitor network. Even the series and parallel resonating structures, connected in series and parallel configurations, could be used to get the periodic lines. Thus, an artificial loaded line constitutes the discrete medium. To get the homogeneous effective medium, the size of the inclusions and the period must be very small as compared to the operating wavelength. The periodic lines could be modeled by the cascaded symmetric unit cells of the T-network, π-network, or asymmetric L-network. Using equations (19.3.4a) and (19.3.30) of chapter 19, the following dispersion equation is obtained for the lossless 1D medium, formed by the loaded periodic lines, expressed as the L-network [J.1, B.3]: cos βd =

pul − Zpul se Ysh

β=

Eq. L-network

Eq. LC-network

22.1 Circuit Models of 1D-Metamaterials

medium with μeq, εeq under the homogeneity condition. The correspondence between the material parameters and the primary line constants is discussed in section (4.3) of chapter 4, and also in subsection (5.5.3) of chapter 5. However, it could also be obtained by using the correspondence between the electric field wave equation of the effective medium and the voltage wave equation of the equivalent transmission line: d Ey = − β2 Ey = − ω2 μeq εeq Ey = jωμeq jωεeq Ey dx2 d2 V pul pul pul = Zpul jBpul jωCpul se Ysh V = jXse sh V = jωLse sh V dx2

a,

μeq =

Xpul se = Lpul se ω

b

εeq =

Ypul sh jω

c,

εeq =

Bpul sh = Cpul se ω

d

In the above equations, the series inductance and shunt capacitance model the transmission line. Therefore, the equivalent line, shown in Fig (22.1), is a usual LC-line that models the DPS medium. However, the above relations (22.1.4a, c) are also valid for other engineered lines, such as the ENG, DNG, and MNG. These engineered media are discussed in chapter 5 using Fig (5.10). The propagation constant β and characteristic impedance Zc of the equivalent line and effective medium are computed from the following expressions:

Effective Medium β = ω

pul a , Z0 = Zpul se Ysh

μeq εeq c , Z0 =

Zpul se Ypul sh μeq εeq

pul Lpul se Csh ,

a,

εeq =

c,

Z0 =

jωCpul sh = Cpul sh jω

b

Lpul se

d

Cpul sh

22 1 6

22 1 4

Transmission line γ =

jωLpul se = Lpul se jω

μeq =

b,

where Ey is the electric field vector in the medium and V is the voltage across the equivalent line. Figure (22.1) shows the series impedance/reactance p.u.l and shunt admittance/susceptance p.u.l. of the equivalent L-network, corresponding to electrical parameters of a medium. On comparing the above equations, the equivalent permeability μeq and equivalent permittivity εeq of the effective medium are obtained in terms of the line parameters: Zpul se jω

DPS-Medium (L–C) Line (L-Series, C-Shunt)

a

22 1 3

μeq =

following expressions of the effective media are obtained by using the equations (22.1.4a,c) and (22.1.5):

β=ω

2

835

ENG-Medium (L–L) Line (L-Series, L-Shunt) pul

μeq =

1 jωLsh jωLpul 1 se = Lpul = − a , εeq = se pul jω jω ω2 Lsh Lpul se

β=j

pul Lsh

c , Z0 = jω

,

pul Lpul se Lsh

d

22 1 5 The effective medium model is also valid for a lossy medium [J.2, B.2]. However, only the lossless medium is considered below. Figure (5.10) of chapter 5 shows that the four kinds of artificial media could be developed by cascading of an infinite number of the unit cells. The

d

22 1 7

DNG-Medium (C–L) Line (C-Series, L-Shunt)

μeq =

1 jωCpul 1 se = − jω ω2 Cpul se

a

εeq =

1 jωLpul 1 sh = − 2 jω ω Lpul sh

b

β= −

1 ω

1 pul Lpul sh Cse

,

c,

Z0 =

Lpul sh Cpul se

d 22 1 8

MNG-Medium (C–C) Line (C-Series, C-Shunt)

μeq =

1 jωCpul 1 se =− jω ω2 Cpul se

pul

a , εeq =

jωCsh = Cpul b sh jω

c , Z0 =

j ω

pul

b

b

β=j

Csh , Cpul se

1 pul Cpul se Csh

22 1 9 pul In the above equations Lpul se and Lsh are the series inductance and shunt inductance of the equivalent L-network pul on p.u.l. basis. Likewise, Cpul se and Csh are the series capacitance and shunt capacitance. It is noted from the above equations that for the DPS medium both the equivalent permeability μeq and equivalent permittivity εeq are positive, giving positive real values for the propagation constant and characteristic impedance

d

836

22 Metamaterials Realization and Circuit Models – II

supporting the forward waves with a positive phase and group velocities. The lossless DPS medium is nondispersive. For the ENG medium εeq is negative, leading to imaginary values for both β and Z0. Thus, the medium supports only the nonpropagating evanescent mode. Even during the evanescent mode, the ENG medium is dispersive. For the DNG medium, both equivalent permeability μeq and equivalent permittivity εeq are negative quantities, giving the negative value of the frequency-dependent β, and positive value of Z0. The DNG medium supports the backward wave propagation with phase velocity in the negative direction, and group velocity, indicating power flow, in a positive direction. The direction of group velocity is in the direction of power flow that is from source to load. The lossless DNG medium is dispersive. Finally, for the MNG medium, μeq is negative, leading to imaginary values for both β and Zc. Thus, the MNG medium also supports only the nonpropagating evanescent mode. It is a dispersive medium.

networks forming the 1D DPS medium. Figure (22.2a, c) shows the unit cells of the reactively loaded host line with series arm loading capacitance Cpul se in series or parallel to host line inductance Lpul . Figure (22.2b, d) shows h the shunt arm loading inductance Lpul sh in parallel or in series to host line capacitance Cpul . The loading elements h are shown in gray boxes. In practice, the host line could be a microstrip, CPW, or waveguide. The reactive loading is achieved by the planar structures, or by the lumped SMT components. Depending upon arrangement and the type of reactive loading, the loaded line creates the MNG, MNZ, ENG, and ENZ type engineered effective medium with either the Drude type or Lorentz type response. Section (3.4) of chapter 3 presents the propagation characteristics of some of the reactively loaded lines [B.9]. However, the 1D-effective medium under the homogeneity condition is not discussed at that stage. The creations of some effective media are discussed below.

Series Capacitance Cpul se in the Series Arm

22.1.3

Single Reactive Loading of Host Medium

The host transmission line could be treated as a cascaded pul network of the unit cells of the L-type Lpul h Ch – circuit

Figure (22.2a) shows the circuit configuration forming an effective medium. The series arm loading modifies the equivalent permeability of the host medium, while its equivalent permittivity remains unchanged. Using

pul

Lh

pul

pul

Cse

Lh

pul

Ch

(a) Series capacitance loading in the series arm.

pul

Ch

pul

Lsh

(b) Parallel inductance loading in the shunt arm. pul

pul

Lh

Cse

pul

Ch pul

Lh

pul

pul

Ch

(c) Parallel capacitance loading in the series arm. Figure 22.2

Lsh

(d) Series inductance loading in the shunt arm.

Single-arm reactive loading of the host LC-line. Loading components are in gray boxes.

22.1 Circuit Models of 1D-Metamaterials

the equations (22.1.4a, c), these are computed as follows: Series impedance Zpul se =

jωLpul h +

= jωLpul 1− h

1 jωCpul se 1

a

pul ω2 Lpul h Cse

1− Equivalent Permeability μeq = μ0 μr,eq = Lpul h

ω2mp

b

ω2

where, Lpul h = μh = μo μr,h 1 ωmp = pul pul Lh Cse

c d

Shunt admittance Ysh = jωCpul h εeq = ε0 εr,h = Cpul h

e

22 1 10

1 0

MNZ

MNG

Eq. Relative permittivity εr,eq

Eq. Relative permeability μr,eq

In the case of the air host medium, μh μ0 and εh ε0. The line loaded with series capacitance Cpul se behaves as an artificial magnetic plasma medium with magnetic plasma frequency ωmp. Therefore, the line loaded with

Figure (22.2b) shows the loading of a host line with parallel inductance Lpul sh inclusion in the shunt arm of the Lnetwork. The unloaded L-network corresponds to the

ENZ

ENG ωep Frequency (ω)

0

(b) Drude type permittivity response. Eq. Relative permittivity εr,eq

Eq. Relative permeability μr,eq

(a) Drude type permeability response.

DPS

Parallel Inductance Lpul sh in the Shunt Arm

1 0

ωmp Frequency (ω)

0

a series capacitance follows the Drude model discussed in subsection (6.5.2) of chapter 6. The propagation constant and characteristic impedance are computed using equations (22.1.5a, b). Next, phase velocity and group velocity can also be computed using equations (3.3.10) and (3.3.11) of chapter 3. Figure (22.3a) shows the Drude model type permeability response of the effective medium. It has negative permeability below the magnetic plasma frequency ωmp. Thus, for frequency range ω < ωmp, the effective medium behaves as an MNG medium created by the C–C line. For the host air medium (μr = 1) above ωmp, the effective medium acts as an MNZ medium with 0 < μr, eq < 1. The permittivity of the host air medium (εr = 1) remains unchanged by the series capacitor loading. The Drude type engineered medium is a wideband nonresonant negative permeability material below ωmp. It is an MNZ medium above ωmp.

DPS

1

ωom MNG

(ω) Frequency

(c) Lorentz type permeability response.

0 0

ωoe ENG

(ω) Frequency

(d) Lorentz type permittivity response.

Figure 22.3 The permeability and permittivity response of the effective medium.

837

838

22 Metamaterials Realization and Circuit Models – II

DPS host medium. The permeability of the host medium remains unchanged. However, the equivalent permittivity of the effective medium is modified. The equivalent permittivity is obtained as follows by using equation (22.1.4c): pul Shunt admittance Ypul 1− sh = jωCh

= jωCpul 1− h

1 pul ω2 Lpul sh Ch

ω2ep

a

ω2

Equivalent permittivity εeq = ε0 εr,eq = Cpul 1− h

where, Cpul h = εh = εo εr,h ,

ω2ep ω2

1

ωep =

pul Lpul sh Ch

b

c

22 1 11 Again, Fig (22.3b) shows the Drude type equivalent permittivity of the effective medium with electric plasma frequency ωep. For the frequency range ω < ωep, the ENG medium with negative permittivity is created by the L–L line. The air host medium loaded with shunt inductor Lpul sh acts as an ENZ medium (0 < εr,eq < 1) above the electric plasma frequency ωep. The propagation constant, characteristic impedance, phase velocity, and the group velocity can be easily computed. Thus, Fig (22.3b) describes the electric plasma-type wire medium discussed in subsection (21.1.2) of chapter 21. Parallel Capacitance Cpul se in the Series Arm

Figure (22.2c) shows the circuit configuration of an effective medium with modified equivalent permeability. It is achieved by loading the host medium line with pul a capacitance Cpul se parallel to Lh connected in the series arm. The permittivity of the host medium remains unchanged. Following the previous process, the equivalent permeability is obtained: Series impedance Zpul se =

2 jωLpul h ω0m ω20m − ω2

a

Equivalent permeability μeq = μ0 μr,eq = where, Lpul h = μh = μo μr,h ,

ω0m =

2 Lpul h ω0m 2 ω0m − ω2

1 pul Lpul h Cse

b c

22 1 12 Figure (22.3c) shows the Lorentz type permeability response of the effective medium. It has no plasma frequency. However, the medium has a magnetic resonant frequency ωom. In the low-pass region ω < ω0m, the medium has a high value of positive permeability near

the resonance. In practice, μr, eq is finite due to the losses. In the high-pass frequency range ω > ω0m, the medium acts as the MNG medium created by the C–C line. Thus, for the frequency ω < ω0m, medium supports the forward wave propagation, whereas for ω > ω0m, it is in the evanescent mode. Just below ω0m, the high value of permeability can be realized in the DPS medium. Series Inductance Lpul sh in the Shunt Arm

Figure (22.2d) shows the reactively loaded host line with the series-connected inductor Lpul sh inclusion in the shunt arm of the L-network corresponding to the host medium. Again, it provides the Lorentz type of equivalent permittivity. In the process of loading, the permeability of the medium remains unchanged. The equivalent permittivity of the effective medium is obtained: Shunt admittance Ypul sh =

1 jωLpul 1− sh

Equivalent permittivity εeq = ε0 εr,eq = where, Cpul h = εh = εo εr,h ,

ω0e =

a

ω20e ω2 2 Cpul h ω0e ω20e − ω2

1

b c

pul Lpul sh Ch

22 1 13 Figure (22.3d) shows the Lorentz type equivalent permittivity response, with the electric resonance frequency ω0e, of the effective medium without any plasma frequency. For ω < ω0e, the medium is in the low-pass mode and exhibits high positive permittivity near the lower side of the resonance. However, for the frequency ω > ω0e, the ENG effective medium is created by the L–L line. 22.1.4 Single Reactive Loading of Host Medium with Coupling The effective media discussed above do not account for the coupling between the host line and reactive inclusions. The reactively loaded coupled medium creates the Lorentz type effective medium with the resonance and plasma frequencies. The coupling could be either inductive or capacitive to get the MNG or ENG response respectively below the plasma frequency and above the resonance frequency. The MNG/ENG is created in a narrow frequency band. Above the plasma frequency, the MNZ or ENZ medium is created. Figure (22.4a) shows a loading capacitor Cpul se , in the series arm, parallel to the host medium inductor Lpul h . It also shows that the loaded host line is coupled through

22.1 Circuit Models of 1D-Metamaterials pul

Lh

pul

Cse pul

pul

Lcoup

pul

Ch

Lsh

pul

Lh

pul

pul

Ccoup

Ch

(a) Series inductive coupling.

(b) Shunt capacitive coupling.

Figure 22.4 Coupling between host line and reactive loadings. Loading is shown in gray boxes.

the series inductor Lpul coup to an external circuit. The coupling element is shown in a dotted box. The circuit configuration provides the modified equivalent permeability of the effective medium. It could be computed by the series impedance, following the process discussed above. Below the magnetic resonance frequency pul Lpul h Cse , the medium is in the low-pass

DPS

⊥

Circuit Models of 1D Metalines

The series or shunt reactive loading of the host transmission line can produce only the SNG, i.e. the ENG/MNG, response. Depending upon the nature of the reactive

DPS

⊥

MNZ ωom

ωmp

(ω) Frequency

MNG

O

22.1.5

Eq. Relative permittivity εr,eq

Eq. Relative permeability μr,eq

region with higher permeability. However, at the frequency ω > ω0m, the parallel resonant circuit behaves as a capacitive reactance. It further produces series resonance with the coupling inductance Lpul coup to create the magnetic plasma frequency ωmp > ω0m; so in a narrow frequency band, ω0m < ω < ωmp, the MNG medium is created. Above the plasma frequency ω > ωmp, the MNZ medium is created. It is noted that the presence of the inductive coupling Lpul coup modifies the permeability response of Fig (22.3c) in the MNG region. Figure (22.5a) shows the modified Lorentz type permeability response of the effective medium. In Fig (22.4a), the series capacitive coupling could also be used, in place to inductive coupling, to realize the bandpass filter type response in the DGS loaded microstrip. The series-connected parallel resonant circuit is realized by a DGS and the

(a) Lorentz type permeability response.

O

ENZ ωoe

ωep

(ω) Frequency

ENG

ω0m = 1

coupling capacitor is realized by the gap discontinuity in the microstrip line located at the neck of the DGS [J.3]. Figure (22.4b) shows the parallel inductor loading, i.e. inductive inclusion, in the shunt arm of the host medium. The coupling between the host line and inclusion occurs through the capacitor Cpul coup in the shunt arm. Again, the shunt admittance is used to get the Lorentz type equivalent permittivity response shown in Fig (22.5b) with both the electric resonance frequency ω0e and the electric plasma frequency ωep. In the lowpass frequency range ω < ω0e, the medium is a DPS medium with high permittivity. In a narrow frequency range ω0e < ω < ωep, the medium is an ENG medium. Finally, above the plasma ω > ωep, the effective medium is the ENZ medium. It shows that the presence of the coupling capacitance Cpul coup modifies the permittivity response of Fig (22.3b) in the ENG region.

(b) Lorentz type permittivity response.

Figure 22.5 Complete Lorentz type response and bandpass response of the effective medium.

839

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22 Metamaterials Realization and Circuit Models – II

combination of Fig (22.2b, c) involving both the Drude and Lorentz responses, so the circuit configuration of Fig. (22.6c) provides the Lorentz–Drude type response. It has a narrow band DNG response. Lastly, the series–series configuration is a combination of Fig. (22.2a, d), involving again both the Drude and Lorentz responses. Figure (22.6d) created the Drude–Lorentz type response. This configuration also has a narrowband DNG response. It is concluded that only the CRLH, i.e. the series– parallel circuit configuration can be effectively used to engineer the 1D-DNG medium supporting the backward wave propagation.

loading, the response of the medium could be either the Drude model type broadband response or it could be the resonance type Lorentz narrow band response. However, the cascaded MNG and ENG, with an overlapped frequency band, provide DNG response. The 1Dmetalines with the DNG property could be realized by the reactive inclusion loading of both the series and shunt arms of the host L-network unit cell. The doubly loaded host lines can have four configurations: (i) Series– Parallel configuration, (ii) Parallel–Series configuration, (iii) Parallel–Parallel configuration, and (iv) Series–Series configuration [J.4]. Figure (22.6a–d) show four circuit configurations for the synthesis of four kinds of metalines. The four circuit configurations create four kinds of material responses. The series–parallel configuration is developed by combining the circuits of Fig (22.2a,b). It provides the Drude–Drude type response. This configuration is also known as the Composite Right/Left Hand (CRLH) configuration. It has a wideband DNG response [J.5–J.9, B.2, B.3]. Similarly, the dual-CRLH (D-CRLH) medium, i.e. the parallel–series configuration, is obtained by combining Fig (22.2c,d). It involves the Lorentz–Lorentz type response. It has a limited bandwidth DNG response [J.10]. The parallel–parallel configuration is a

Series–Parallel Configuration

Figure (22.6a) shows the unit cell of the series–parallel configuration of the CRLH metalines. The unit cell of pul the host medium Lpul is loaded with a series h , Ch capacitance Cpul in the series arm, and also with a parse allel inductance Lpul in the shunt arm. The loading sh inclusions are shown in gray boxes. The period d is much smaller than the operating wavelength λ. The condition d < < λ is called the homogeneity condition. The series impedance Zpul model and shunt admittance Ypul se sh

pul

Lh

pul

pul

Cse

Lh

pul

pul

Ch

pul

Cse

pul

Ch

Lsh

pul

Lsh

d

d (a) Series parallel configuration.

(b) Parallel series configuration. pul

Lh

pul

Lh

pul

Cse

pul

Ch

pul

Cse

pul

Ch

pul

Lsh

d (c) Parallel parallel configuration.

pul

Lsh

d (d) Series series configuration.

Figure 22.6 Double arms reactive inclusion loading of the host LC-line forming the unit cell of metalines. Loading elements are shown in gray boxes.

22.1 Circuit Models of 1D-Metamaterials

the equivalent permeability and equivalent permittivity of the 1D effective medium. These are given below: pul Zpul 1− se = jωLh

ω2mp

pul 1− a , Ypul sh = jωCh

ω2 1

where, ωmp =

c , ωep =

pul Lpul h Cse

ω2ep

b

ω2

1

d

pul Lpul sh Ch

22 1 14 In equation (22.1.14), ωmp and ωep are the magnetic and electric plasma frequencies, responsible for creating the MNG and ENG part of the DNG medium. The equivalent permeability and permittivity of the effective medium are obtained by using equation (22.1.14) with the equations (22.1.4a, c): μeq = μ0 μr,eq = Lpul 1− h μr,eq = μr,h 1 − εeq = ε0 εr,eq =

where, μr,h εr,h

Lpul h Cpul h

a

ω2mp

b

ω2

Cpul h

εr,eq = εr,h 1 −

ω2mp ω2

1−

ω2ep

c

ω2

ω2ep μ0

e

ε0

f

Equations (22.1.15b, d) also compute the equivalent relative permeability μr, eq and equivalent relative permittivity εr,eq of the homogenized effective medium. The relative permeability and relative permittivity of the host medium are μr,h and εr,h respectively. In the case of the air host medium, these quantities are unity. However, if either microstrip or CPW is the host medium, the geometric parameters-dependent effective permittivity and permeability of these structures could be treated as the permittivity and permeability of the host medium. The propagation constant βsp and the characteristic impedance Zsp of the series–parallel (SP) effective medium are obtained by using the above equations with equations (22.1.5c, d): μeq εeq = ω 1−

Zsp =

ω2mp ω2

μeq = εeq

propagation constant ω impedance

pul Lpul h Ch ×

1 2

1−

ω2ep

1 2

a

ω2

ω2 − ω2mp Lpul h pul ω2 − ω2 Ch ep

1 2

b 22 1 16

pul Lpul and characteristic h Ch

Lpul Cpul h h of a host medium are modified

by the plasmonic effect created by double reactive loading. The modifying factors are enclosed in the brackets [] in the above equations. The phase velocity vp,sp, and the group velocity vg,sp of the EM-wave on the metaline, i.e. 1D SP-effective medium could also be obtained from βsp [J.4]:

vp,sp =

ω = vp,h βsp

vg,sp = vg,h

1−

ω2mp ω2

1−

1− ω2mp

1−

ω2mp ω2

1 2

ω2 ω2ep ω2

1−

1−

1− ω2ep ω2 ω2ep ω2

ω2ep

a

ω2

1 2

×

1−

ω2mp

−1

b

ω2

where phase and group velocities in host medium 1 vp,h = vg,h = pul pul Lh C h

c

22 1 17

d

ω2

22 1 15

βsp = ω

It is noted from equation (22.1.16) that both the

The host medium is nondispersion with equal phase and group velocities. However, the reactively loaded SP-metaline medium is highly dispersive. In equation (22.1.17b), (−) is used for ω < ωmp, ω < ωep and (+) is used for ω > ωmp, ω > ωep. In equations (22.1.17), the magnetic and electric plasma frequencies are unequal, i.e. ωmp ωep giving the unbalanced metaline. In case both plasma frequencies are equal, i.e. ωmp = ωep, the balanced metaline, also called matched frequency metaline, is obtained. The balanced metaline is a more useful configuration [J.5–J.8, J.11, B.2, B.3]. Unbalanced CRLH Metaline

Figure (22.7a) shows the (ω − β) dispersion diagram of the unbalance series–parallel, i.e. the CRLH metaline. It has three distinct frequency regions ω < ωmp < ωep; ωmp < ω < ωep, or ωep < ω < ωmp; ω > ωep > ωmp. These frequency regions have their own characteristics discussed below. Region ω < ωmp < ωep

Under the above condition, Fig (22.6a) is reduced to the C–L line supporting the backward wave propagation. Equations (22.1.16a) and (22.1.17a) provide negative values for both the propagation constant βsp and the phase velocity vp,sp, whereas from equations (22.1.16b) and (22.1.17b) both the characteristic impedance Zsp

841

22 Metamaterials Realization and Circuit Models – II

ω DPS bandgap

Band -gap ωmp

Negative β

ωcl

ωp DNG Slow-wave

Negative β

Light line

ωep Backward wave

Bandgap

DPS Slow-wave Negative β

0

ω

MNG

ωom

Positive β

DNG

Slow-wave

ωom

ENG π

(c) Dispersion diagram of the parallel series (D-CRLH) metaline. Figure 22.7

Positive β

0

ωoe ω0

–π

ωcl

(b) Dispersion diagram of the balanced series parallel (CRLH) metaline.

Fast-wave

DNG

Fast-wave

(a) Dispersion diagram of the unbalanced series parallel (CRLH) metaline. ω

DPS Slow-wave

DNG Bandgap Positive β

0

cone

Fast-wave

DNG DNG Bandgap

Slow-wave DPS

Light

Fast-wave

Forward wave

Fast-wave ωep

Fast-wave

Backward wave

Lightcone

Slow-wave

DPS bandgap

ωc2 Backward wave

ωc2

Forward wave

ω

Slow-wave

842

–π

Negative β

0

β

(d) Dispersion diagram of the parallel parallel metaline.

The (ω − β) dispersion diagrams of the metalines.

and the group velocity vg,sp are positive. The upper sign of the factor ( ) is taken in this case. It demonstrates the presence of the DNG effective medium in this frequency region, as both the permeability and permittivity are negative from equations (22.1.15a) and (22.1.15c). The DNG medium exists on the left-hand side in Fig (22.7a). It is noted that at ω = ωmp and ω = ωep, leading to βsp = 0, showing no propagation. Fig (22.7a) also shows the existence of the first stopband region with a cut-off frequency ωc1. It limits the lower frequency range of the DNG medium. The above-discussed expressions do not show such a cut-off phenomenon. Region (ωep < ω < ωmp), or (ωmp < ω < ωep)

For the first case (ωep < ω < ωmp), the operating frequency ω is above the shunt resonance and below the series resonance. In this frequency range, the circuit of Fig (22.6a) is

reduced to the C–C line. The C–C line corresponds to the MNG medium. However, for the second case (ωmp < ω < ωep), the operating frequency ω is above the series resonance and below the shunt resonance. The circuit of Fig (22.6a) is reduced to the L–L line, creating the ENG medium. Thus, Fig (22.7a) shows a bandgap in the frequency band |ωep − ωmp| for the SNG type medium. It supports only the nonpropagating evanescent mode.

Region ω > ωep > ωmp

In this region, the circuit of Fig (22.6a) is reduced to the standard L–C line. The propagation constant βsp and characteristic impedance Zsp are positive quantities. Figure (22.7a) and equations (22.1.17a, b) show that both the phase velocity vp,sp, and the group velocity vg,sp are positive. Thus, in this frequency range, the effective

22.1 Circuit Models of 1D-Metamaterials

medium is DPS type supporting the forward wave propagation, as shown in the right-hand side of Fig (22.7a). Balanced CRLH metalines

Under the balanced condition, the magnetic and electric plasma frequencies are equal, i.e. ωmp = ωep = ωp.The effective medium is a plasma medium with the plasmon frequency ωp. Using equation (22.1.15), the equivalent permeability and permittivity are zero, μeq = εeq = 0 at the plasma frequency ωp. Likewise, from equation (22.1.17), the phase velocity is infinite at ωp, whereas the group velocity is finite: vg,sp =

1 2

1 pul Lpul h Ch

=

1 2

1 pul Lpul sh Cse

22 1 18

The characteristics impedance of the unbalanced CRLH metaline is frequency-dependent. However, it is frequency-independent for the balanced case. It remains the same for both the DNG and DPS regions. The propagation constant and characteristics impedance of the balanced CRLH metaline are obtained from equations (22.1.16a, b): βbal sp = ω

pul Lpul 1− h Ch

ω2p ω2

regions are within the light-cone supporting the fastwave. It is the leaky radiation mode, useful for the design of the frequency scanning leaky-wave antenna. It scans the upper half of the space from backfire radiation to endfire radiation of the leaky-wave antenna. It is illustrated in Fig (22.20). Outside the light-cone, the medium supports the bounded backward and forward slow-wave (SW) propagation. It is useful for the development of several unique and compact microwave devices [J.5, J.7, J.8, J.11, B.2, B.4]. The analysis is valid for the effective medium under the homogeneity condition. However, a more accurate analysis has been done using the [ABCD] transmission matrix by treating the metaline as a periodically cascaded structure. The [ABCD] matrix analysis provides the bandgap in the DNG medium called the DNG bandgap at the lower cut-off frequency ωc1. It also computes DPS bandgap at the upper cut-off frequency ωc2. The DPS bandgap limits the DPS medium upper-frequency range. Figure (22.7a, b) shows both bandgaps with cut-off frequencies and also tapering of the dispersion diagram, shown by the dashed line, at the band-edge. The expressions are available to compute these cut-off frequencies and bandwidth of the DNG and DPS regions [B.2, B.4].

a Parallel–Series Configuration

Zbal sp DPS =

Zbal sp DNG =

Lpul h Cpul h Lpul sh Cpul se

bal Zbal sp DPS = Zsp DNG

b

c d 22 1 19

Figure (22.7a,b) shows the (ω − β)-dispersion diagram of the balanced CRLH-metaline with a light-cone. In Figure (22.7b), the second bandgap of the unbalanced case disappears and a continuous change in the nature of the medium, from the DNG to DPS, is obtained with an increase in the operating frequency. For ω < ωp, the effective medium is DNG, and for the frequency ω > ωp, the medium is DPS. It is also evident from expression (22.1.19a). The frequency-independent and equal characteristics impedance, in both the DNG and DPS regions, results in the broadband impedance matching. Figure (22.7b) further shows that a part of the backward wave and forward wave of the DNG and DPS

Figure (22.6b) shows a unit-cell of the parallel–series configuration. It is a dual network of the series–parallel network discussed above, so it is also called the dualCRLH, i.e. the D-CRLH metaline [J.10, J.12]. The circuit configuration of Fig. (22.6b) corresponds to the LorentzLorentz type material with the permeability resonance frequency ω0m and the permittivity resonance frequency ω0e. Even the balanced D-CRLH line, under identical resonance frequency in the series and shunt arms, i.e. ω0m = ω0e = ω0, creates a bandgap due to the opencircuit condition in the series arm and the short-circuit condition with the shunt arm. Figure (22.7c) shows that at ω0, the balanced D-CRLH metaline has a narrow bandgap shown within the dashed-lines. It widens for the unbalanced case. In the low-pass region, the circuit of Fig (22.6b) is reduced to the L–C line showing the DPS effective medium. For the high-pass region, it is reduced to C–L metaline showing the DNG effective medium. Following the previous process, the equivalent permeability and equivalent permittivity of the effective DCRLH medium are obtained from the series impedance and shunt admittance p.u.l. of the Fig (22.6b):

843

844

22 Metamaterials Realization and Circuit Models – II

pul Zpul 1− se = jωLh

ω2 ω20m

−1

pul 1− Ypul sh = jωCh

ω2 ω20e

−1

where, ω0m = ω0e =

a b

1

c

pul Lpul h Cse

1

d

pul Lpul sh Ch

22 1 20

1− μeq = Lpul h εeq = Cpul 1− h

−1

ω2 ω20m

a −1

ω2 ω20e

b 22 1 21

Equations (22.1.20) and (22.1.21) show that the parallel– series (ps) medium, i.e. the D-CRLH medium, is a Lorentz–Lorentz type effective medium, without plasma frequency. Its propagation constant βps and characteristic impedance Zps, phase velocity vp,ps, and group velocity vg,ps are obtained as [J.4], βps = ω

Zps =

pul Lpul h Ch

Lpul h Cpul h

1−

ω ω20m 2

−1 2

1−

ω ω20e 2

−1 2

a

vg,ps = vg,h

1−

ω2 −1 ω20m

ω2 1 − 0m ω2

ω2 ω20m

b

1 2

−1

+

ω2 ω20e

1−

ω2 −1 ω20e ω2 1 − 0e ω2

Lt∞ vp,sp = −

Lt vg,sp = ∞

ω2 ω0m ω0e

ω2 ω0m ω0e

1 pul Lpul h Ch

1 pul Lpul h Ch

a

b 22 1 24

1 2 1 − ω ω20e 1 − ω2 ω20m 2

ω = vp,h βsp

ω

ω

22 1 22

vp,ps =

outside the light-cone supports the SW. In the frequency region ω < ω0m < ω0e, the propagation constant, characteristic impedance, phase velocity, and group velocity are positive. Thus, the effective medium is DPS. At ω = ω0m, βps ∞ , Zps = ∞ , vp,ps = vg,ps = 0 and the series arm is open-circuited. Likewise, at ω0e, the same nature is seen as the shunt arm is short-circuited and Zps = 0. For the frequency band ω0m < ω < ω0e, the effective medium is MNG, and stopband is created without any wave propagation. For the case ω0e < ω < ω0m, stopband is an effective ENG medium. However, for the case ω > ω02 > ω01, Fig (22.6b) is reduced to the C–L metaline, forming the DNG effective medium. In this frequency range, the propagation constant and phase velocity are negative, whereas the characteristic impedance and group velocity are positive. Unlike the CRLH, the D-CRLH has a narrow stopband even for the balanced case ω0m = ω0e = ω0. It is shown within the dashed-line in Fig (22.7c). In the high-frequency limiting case with the DNG region, the following results are obtained for the phase velocity and group velocity:

a

Expression (22.1.24) shows that both the phase velocity and group velocity approach to infinity with ω ∞. This is unphysical condition. In practice, they have finite values due to the parasitic effect of the components. However, the very high value of group velocity can be realized for the D-CRLH metaline. The D-CRLH metaline has better impedance matching after the steep change in the characteristic impedance [J.10].

1 2

× −1

−1

where in host medium vp,ps = vg,h =

Parallel–Parallel Configuration

−1

b 1

pul Lpul h Ch

c

22 1 23 Figure (22.7c) shows the dispersion diagram of the DCRLH metaline with a light-cone. The region within the light-cone supports the fast-wave, and the region

Figure (22.6c) shows the unit cell of the parallel–parallel (pp) type of artificial metaline. It has the Lorentz–Drude type response. The permeability exhibits Lorentz response with a magnetic resonant frequency ω0m, whereas the permittivity exhibits Drude response with electric plasma frequency ωep. By following the process applied to the CRLH metaline, the equivalent permeability and permittivity parameters of the parallel– parallel (pp) effective medium could also be computed by obtaining the series impedance and shunt admittance p.u.l. of the loaded host transmission lines:

22.1 Circuit Models of 1D-Metamaterials

pul 1− Ypul sh = jωCh

where, ω0m =

ωep =

−1

ω2 ω20m

pul Zpul 1− se = jωLh

a

ω2ep

b

ω2

1

c

pul Lpul h Cse

1

d

pul Lpul sh Ch

22 1 25

1− μeq = Lpul h

ω2 ω20m

1− εeq = Cpul h

ω2ep ω2

−1

a Series–Series Configuration

b 22 1 26

The propagation constant βpp and characteristic impedance Zpp of the EM-wave in the parallel–parallel effective medium are obtained as,

βpp = ω

Zpp =

pul Lpul h Ch

1 − ω2ep ω2

Cpul h

1 2

a

1 − ω2 ω20m 1 2

Lpul h

1 1 − ω2 ω20m

to either the DNG or DPS medium, depending on the relative value of the resonance frequencies in two arms of the circuit. The DNG bandwidth can be controlled by controlling both resonance frequencies. The dispersion diagram of Fig (22.7d) shows the real negative value of the propagation constant of the DNG medium in the frequency range ω0m < ω < ωep. Above the light-line, the medium supports the radiative fast-wave. However, below the light-line, the medium supports the backward wave propagating in the slow-wave mode. Outside the bandpass frequency range, the propagation constant is an imaginary quantity shown in dashed-lines. It shows that the wave is in the nonpropagating evanescent mode.

1 − ω2ep ω2

b 22 1 27

The phase and group velocities can also be obtained for this medium. However, the characteristic of the medium can be estimated from the circuit shown in Fig (22.6c). The dispersion diagram is shown in Fig (22.7d). In the frequency range ω < ω0m < ωep, the circuit Fig (22.6c) is reduced to the L–L circuit and the effective medium is ENG. Likewise, in the frequency range ω > ωep > ω0m, the parallel–parallel circuit is reduced to the C–C circuit and the effective medium is MNG. Equation (22.1.27) provide the propagation constant and characteristic impedance as the imaginary quantities in both these frequency ranges, creating the stopband. However, in the middle frequency band ω0m < ω < ωep, the parallel–parallel circuit is reduced to the C–L line creating the DNG medium. For the case ωep < ω < ω0m, the L–C line is obtained, and the DPS medium is created. Thus, the passband could belong

Figure (22.6d) shows the series–series (SS) configuration of the artificial line. It has the Drude–Lorentz response. The permeability exhibits the Drude type response with magnetic plasma frequency ωmp, whereas the permittivity exhibits the Lorentz type response with a resonant frequency ω0e. Like the previous cases, equivalent permeability and equivalent permittivity of the effective medium can be obtained from the circuit components. Its propagation characteristics are identical to the propagation characteristics of the parallel–parallel configuration. It has stopband in the frequency range ω < ωmp < ω0e and ω > ω0e > ωmp, and a passband in the middle-frequency range ωmp < ω < ω0e or ω0e < ω < ωmp. However, the nature of the frequency bands is different. In the frequency range ω < ωmp < ω0e, the circuit shows the C–C line, i.e. the MNG effective medium, whereas in the frequency range ω > ω0e > ωmp, the circuit is reduced to L–L, creating the ENG effective medium. Likewise, in the passband, ωmp < ω < ω0e the DPS medium is created, and in the range, ω0e < ω < ωmp the DNG medium is created. However, the dispersion characteristics, as shown in Fig (22.7d), are also applicable to the series–series configuration. The bandwidth of the DNG could also be controlled by controlling the resonance frequencies. The present section has considered only the lossless nonresonant metalines structures by embedding the C and L inclusions in the host transmission lines. The losses could be accounted for by associating resistance and conductance with the inductor and capacitor used in the circuit configuration. They have a noticeable influence on the performance of the engineered media [J.2, J.7]. The practical realizations of the engineered nonresonant media are discussed in section (22.2). The resonant metalines are also developed by using the resonant inclusions in the host line [J.13, B.3, B.7]. It is discussed in section (22.3).

845

846

22 Metamaterials Realization and Circuit Models – II

22.2 Nonresonant Microstrip Metalines The capacitor and inductor type inclusion structures are loaded to the host microstrip and CPW to realize the metalines. However, the present discussion is limited to the microstrip. The extended discussion to the CPW-based metaline is available in reference [J.14– J.16]. The microstrip realization of three kinds of the metalines: CRLH, a combination of the ENG-MNG structures, and D-CRLH are discussed in this section. 22.2.1

Series–Parallel (CRLH) Metalines

Figure (22.8a-i) suggests the realization of the CRLH, i.e. the series–parallel configuration, metaline by loading the host microstrip with the series-connected semilumped gap capacitor and via. The grounded via forms the shunt connected inductor. Figure (22.6a) shows the loading inclusions Cse and Lsh. The inclusions could be distributed over the period d of the unit cell to get pul Cpul se = Cse d and Lsh = Lsh d. The host microstrip proand Cpul vides inductance and capacitance Lpul h h .

However, in this arrangement limited values of loading elements are realized. Figure (22.8b) shows that the series-connected gap capacitors could be replaced by the chip capacitors, or by the semi-lumped interdigital capacitor as shown in Fig (22.8a)-ii, to get a higher value of series capacitance. However, the interdigital capacitor has self-resonance that limits its operational bandwidth with higher losses. Its physical size is also large to realize the high value of capacitance. Moreover, the interdigital capacitor-based CRLH metalines is an asymmetrical structure with different values of the Bloch impedance at the input and output ports. Thus, the design of the interdigital capacitors at both ends of the line, shown in Fig (22.8c), is different. The end capacitors are different from the other interdigital capacitors. The low-loss compact chip capacitors solve some of the problems associated with interdigital capacitors. However, these are available only for limited values and normally used up to 3–6 GHz. The metal-insulator-metal (MIM) capacitors are also used to realize higher values for the loading capacitors. The MIM capacitors are helpful to realize the broadband CRLH and

i. series gap capacitance and shunt via inductance

ii. Series interdigital capacitance and grounded shunt stub. (a) Series capacitance and shunt inductance loaded microstrip.

(b) Six series chip capacitors and five grounded shunt stub inductances loaded microstrip.

(c) Seven unit cells CL loaded microstrip [J.2]. Figure 22.8

Series capacitance and shunt inductance loaded microstrip metalines. Source: Caloz and Itoh [J.2]. © 2002, IEEE.

22.2 Nonresonant Microstrip Metalines

parameters magnitude and the phase responses from almost DC to 5.0 GHz. Both the simulated and experimental results are shown. It is the case of the symmetrical CRLH without any bandgap. Its transition frequency is 2.97 GHz. The medium is DNG between 1.0 and 2.97 GHz, and DPS from 2.97 to 5.0 GHz. In the DNG range, the propagation constant β is negative, whereas in the DPS range, it is a positive quantity confirming the backward and forward wave propagation in the DNG and DPS ranges. In the DPS region, almost smooth linear increase of propagation constant β is noted with an increase in frequency. However, the DNG region is dispersive. Figure (22.8c) shows the 7-cell CRLH metaline developed using eight numbers of interdigital capacitors and seven numbers of the grounded stubs [J.2]. The host microstrip is designed on the RT/Duroid 5880 substrate εr = 2.2, h = 1.57 mm. At 1.5 GHz, the interdigital capacitors have a value of 2.06 pF, and the stub has an inductance of value 4.62 nH. The cell length is 12.2 mm. Figure (22.9c,d) shows the experimental magnitude

also D-CRLH metalines. However, the MIM-based metalines require two-level of the substrate [J.10, J.14, J.17]. Figure (22.8a-ii) shows that simple via can be replaced by the high impedance grounded stub to realize higher values for the loading shunt inductors. The microstrip section of the metaline is reduced to a minimum length to support the grounded stub, as shown in Fig (22.8c) [J.2]. In place of the stubs, the meandered-line inductors can also be used to get larger values for the loading inductors. The grounding vias can be replaced by a large patch capacitor to provide the virtual ground to the shunt inductor [J.5]. Figure (22.8b) shows the layout of the chip capacitorsbased symmetrical CRLH-metalines. It has six loading series-connected ATC 600S chip capacitors. The end capacitors have a value of 5.6 pF, and other capacitors have a value of 1.8 pF. The host microstrip line sections are 5.14 mm long and 2.78 mm wide and designed on an F4B-1/2 substrate with εr = 2.65, h = 1 mm [B.10]. The length and width of the grounded stubs are 9.47 mm and 1 mm, respectively. Figure (22.9a, b) shows its S-

00

5 Frequency (GHz)

–10 ∣S11∣/∣S21∣ (dB)

DPS

DNG

–20 –30 –40 Exp. S11 Sim. S11

4 3 2

DNG

S12 S12

1 –400

–200 0 200 β (1/m) (b) Phase response of S21-parameters [B.10].

3 4 5 Frequency (GHz) (a) Magnitude response of S-parameters [B.10]. 1

2

10

500

microstrip theory fmin

0 –10

0

≈ fc

–20

fmax

β (1/m)

∣S11∣/∣S21∣ (dB)

DPS

Measured Simulated

–30 –40

LH range

–50 –60

–1000

LH-Range

–1500 S11 S21

–70 –80

–500

0

0.5

1

1.5

2

2.5

3

Frequency (GHz) (c) Magnitude response of S-parameters [J.2].

3.5

–2000

β(ω) = – ω 0

1

2

1 L' C'

3

4

5

Frequency (GHz) (d) Phase response of S21-parameters [J.2].

Figure 22.9 Magnitude and phase response of microstrip metalines. Source: From Liu and Huang [B.10]. Published in [short citation] under CC BY 3.0 license. Public Domain. Reproduced from Caloz and Itoh [J.2], © 2002, IEEE.

847

22 Metamaterials Realization and Circuit Models – II

operating in the MNG and ENG modes, the composite MNG–ENG metalines are designed and tunneling is obtained. Its application for the design of a higher-order bandpass filter has also been demonstrated. The CRLHbased circuit model and its realization in microstrip technology are discussed below [J.18–J.21]. Figure (22.10a) considers the cascaded slabs of MNG– ENG [J.20]. Each slab is designed using four unit cells. The unit cells are modeled using the CRLH circuit configuration shown in Fig (22.6a). It reduces the C–C and L–L networks to get the MNG and ENG mediums, respectively. At the interface resonance ω0, the maximum potential occurs at the interface, and tunneling takes place. Under the tunneling condition, the input signal is transferred to the output. The tunneling

and phase responses of the S-parameters. The highly dispersive DNG has a bandwidth from 1.0 to 3.0 GHz with negative β. Figure (22.9d) also shows the lumped circuit phase response. In the DPS region, the response is not satisfactory. However, the improved response has been obtained for the 24 unit-cell CRLH structure [J.5, B.2]. 22.2.2

Cascaded MNG–ENG (CRLH) Metalines

Individually, the MNG and ENG slabs do not support wave propagation. However, the cascaded MNG–ENG slab or the ENZ–MNZ slab over the common overlapped frequency band supports the wave propagation. This is known as the wave tunneling that occurs when tunneling conditions are satisfied. Using the CRLH unit Interface ENG

MNG (MNG + ENG) Unit cells (U.C.)

UC UC UC UC UC UC UC UC

eng

mng

mng

Cse

Cse

Lh

eng

Lh

eng

eng

Lsh

mng

Ch

Ch

Lsh

mng

Lumped CRLH composite circuit

ℓc1

CRLH #2

ℓc2 ωc2 ℓs1 Shorted (via) stub

ωc1 (MNG + ENG) Composite medium

Interdigital capacitor C–C

5 0 Magnitude (dB)

–5 –10 –15 –20

Fullwave sim S11 Fullwave sim S21 Measured S11 Measured S21

–25 –30

2.5

3

3.5

Frequency (Hz)

–5 –10 –15 Fullwave sim S11 Fullwave sim S21 Measured S11 Measured S21

–20 –25 –30

4 ×109

(c) Exp. and Sim. S-parameters response [J.20]. Figure 22.10

ωs2

(b) Realization of cascaded MNG–ENG metaline [J.20].

To adjacent figure

0

ℓs2

ωs1

L–L

(a) Cascaded MNG–ENG metaline. 5

ENG

MNG

CRLH #1

Magnitude (dB)

848

3

3.05

3.1

3.15

3.2

3.25

Frequency (Hz)

3.3

3.35 ×109

(d) Exp. and Sim. S-parameters response of overlapped MNG–ENG region [J.20].

(MNG–ENG) Cascaded metalines and tunneling response. Source: Fujishiqe et al. [J.20]. © 2005, John Wiley & Sons.

22.2 Nonresonant Microstrip Metalines

bandwidth is determined by the overlapped frequency band of the MNG and ENG. The homogenized equivalent medium is expressed through the medium paramng mng eng eng meters μeq , εeq , μeq , and εeq . Using equations (22.1.14) and (22.1.15), these are expressed in terms of the circuit elements of Fig (22.10a): mng

mng

For the MNG medium: μeq < 0, εeq > 0 mng

μmng eq = Lh

1−

mng

εmng eq = Ch

1−

a

ω2 ω2ep1

b

ω2

c

mng Lh Cmng se

mng

= Cmng se

Lsh Ch mng Lh

eng

d

eng

mng

ωep1 < ω < ωmp1

Lh Ceng se eng Ch

e eng

mng

ω0 =

mng

Lh ω2mp2 + Lh ω2mp1 mng

Lh

ω0 =

ωmp2 = ωep2 =

1−

ω2

1

pul a , εeng 1− eq = Ch

ω2ep2

b

ω2

c

eng Lh Ceng se

1

mng

ωmp2 < ω < ωep2

d,

eng eng Lsh Ch

e

22 1 29 The host lines of the unloaded MNG and ENG lines sections are the normal DPS lines with identical propagation constant: mng

β0

eng

= β0 ,

mng

mng

Lh Ch

eng

eng

= Lh Ch

22 1 30

The wave tunneling occurs through the composite MNG–ENG metaline under the tunneling conditions that result in the identical operating bandwidth for both the MNG and ENG lines [J.18]: Tunnelling condition eng μmng eq = − μeq

a,

mng

Lh

1 2

a

eng

+ Lh

1 eng + Lh

1 1 + mng Ceng C se se

1 2

eng εmng eq = − εeq

b

Upper frequency ωmp1 = ωep2

c

Lower frequency ωep1 = ωmp2

d 22 1 31

Satisfying the tunneling conditions, the following relations between the circuit elements of the MNG line are obtained from equations (22.1.29)–(22.1.31) in terms of the known circuit elements of the ENG line:

b 22 1 33

eng

For the ENG medium: μeq > 0, εeq < 0. eng

c

The inductance Lh of the host MNG line is arbitrarily chosen. The complete transparency, i.e. full tunneling occurs at the frequency ω0. It is the interface resonance frequency. It is obtained by substituting equations (22.1.28a) and (22.1.29a) in equation (22.1.31a):

22 1 28

μeng eq = Lh

b

mng

mng

Lsh =

Lsh Ch

ω2mp2

a

eng

1

ωep1 =

eng

Lh Ch mng Lh

22 1 32

ω2mp1

1

ωmp1 =

eng

=

mng

Ch

The tunneling bandwidth is (ωep1 − ωmp1) or (ωmp2 − ωep2) around the tunneling frequency. Figure (22.10b) shows the microstrip implementation of the composite MNG and metaline using four unit cells of each MNG and ENG slabs. The series capacitor inclusions are implemented in the interdigital configuration and the shunt inductor inclusions are implemented through the grounded stubs. The component values and dimensions of the unit cell, on the RT/Duroid substrate-5880, εr = 2.2, h = 1.575 mm, are given the reference [J.20]. Figure (22.10c) shows the simulated and experimental results of the S-parameters. Figure (22.10d) shows the expanded version of the transparency region. The experimental and simulated values for the tunneling frequency f0 are 3.18 and 3.13 GHz, respectively. At f0, the insertion loss is −1.9 with −27 dB return loss. 22.2.3

Parallel–Series (D-CRLH) Metalines

Figure (22.11) presents the realization of the D-CRLH metaline in microstrip technology. The scheme-I, shown in Fig (22.11a), is based on the MIM capacitor Cse in the series arm and Ch in the shunt arm. The inductances Lh in the series arm and Lsh in the shunt arm are implemented by the high impedance narrow width lines [J.10]. The scheme-II, shown in Fig (22.11b), is based on the series-connected interdigital capacitor Cse. The shunt capacitor Ch is obtained from the patch capacitor. The inductances Lh and Lsh in the series and shunt are again obtained from the high-impedance microstrip sections [B.10]. The components are tuned to get a wideband

849

22 Metamaterials Realization and Circuit Models – II

Cse Ch Lsh

Lh

Lsh

Lh

Cse Ch

Sub#2 Sub#1 (a) Two-level implementation of D-CRLH using MIM capacitors [J.10].

meas

Frequency (GHz)

10 0 –10 –20 –30 –40 –50 –60 –70 –80

S11

LC

DPS

meas

S21 0

1

2

LC S21

S11

DNG

3 4 5 6 7 Frequency (GHz)

8

9 10

(a) S-parameters response of the D-CRLH. Figure 22.12

(b) Single level implementation of D-CRLH using a series interdigital capacitor with parallel inductive strip wire.

Microstrip implementation of the D-CRLH metalines. Source: Caloz and Nguyen [J.10]. © 2007, Springer Nature.

Bandgap

Figure 22.11

S-parameters (dB)

850

10 9 CL βLH 8 7 DNG 6 5 4 3 2 1 0 –π –π/2

λg = ∞ mea

βLH

Bandgap

meas

βRH

LC

DPS βRH 0

π/2

π

βd (b) Phase response of the D-CRLH.

S-parameters and phase response of the D-CRLH metalines. Source: Caloz and Nguyen [J.10]. © 2007, Springer Nature.

impedance match at both the input and output ports. The complete structure is embedded in the microstrip line. Figure (22.12a, b) shows the experimental and simulated results based on the LC lumped circuit model for the five-unit cells D-CRLH metaline of the scheme-I [J.10]. A bandgap is present for the D-CRLH metaline. The lower passband is DPS type supporting the forward wave propagation and the upper passband is DNG type supporting the backward wave propagation. There is an insertion loss in the DNG region due to the leaky-wave radiation caused by the fast wave. Figure (22.12b) shows that the experimentally obtained propagation constant βmeas LH of the DNG has crossed into the DPS region. It is due to the parasitic effect in a practical circuit. However, for the propagation constant βCL LH , computed from the simplified circuit model, the crossing of regions does not occur. The region crossing of propagation constant is a useful phenomenon to design the broadside leakywave antenna [J.5, B.2].

22.3

Resonant Metalines

In the previous section, the CRLH-metalines are realized by the series capacitors and shunt inductors embedded in the host lines as inclusions. These are nonresonating inclusions forming wideband metalines. Of course, combined with the host line series inductance and shunt capacitance, the loading elements create series and shunt resonances defining the magnetic and electric plasma frequencies, respectively. The resonance in-line structure is important for the generation of negative permittivity and negative permeability. The CRLH-type metalines are also realized in the microstrip and CPW host lines by embedding the resonant inclusions. Figure (22.13a–d) shows four kinds of the resonating inclusions – (i) SRR (split-ring resonator) type metallic strips resonator printed on the dielectric substrate, (ii) Complementary SRR (CSRR) type resonating apertures cut in the metallic conductor of a metalized

22.3 Resonant Metalines

y

Ls

Ey

Φm

Cc

Φe

L0/2 z

Hz

x

C0/2

C0/2 Conducting plane

Dielectric substrate

Cs = C0/4

(a) SRR strip on the substrate and its equivalent circuit.

L0/2 Lc = L0/4

(b) CSRR aperture in the metal conductor and its equivalent circuit. Lc = L0/2. Conducting plane

Ls

Co

Lo

Cc Dielectric substrate (c) The O-SRR strip on the substrate and its equivalent circuit. Figure 22.13

(d) O-CSRR aperture in the metal conductor and its equivalent circuit.

Four-kinds of resonating inclusions and their equivalent circuits.

substrate, (iii) Open SRR (O-SRR) type metallic strips resonator printed on the dielectric substrate, and (iv) Open complementary SRR (O-CSRR) type resonating apertures cut in the metallic conductor of the metalized substrate [J.22–J.27, B.7, B.11, B.12]. The SRR is a magnetic particle described by the magnetic dipole and the CSRR is an electric particle described by the electric dipole. Other resonating inclusions, such as the spiral or even the DGS, have also been used to realize the planar metalines [J.28–J31]. This section summarizes the basic property of these four resonating inclusions embedded in the host microstrip and CPW to realize the metalines. Normally, the resonant metalines are narrow bandwidth lines. However, by the suitable design process, the wideband components have been developed [J.23]. 22.3.1

Resonant Inclusions

Figure (22.13a) shows the sub-wavelength size resonant SRR inclusion etched on the dielectric substrate. It also shows the equivalent resonant circuit. It is excited by the time-varying Hz axial magnetic field creating the magnetic dipole moment. The MNG type effective medium is created due to negative permeability between the

magnetic resonance frequency ω0m and magnetic plasma frequency ωmp. It is shown in Fig (22.5a). The inductance Ls is due to the time-varying magnetic flux Φm and capacitance Cs is a series combination of the slot capacitance C0/2 in the upper and lower halves of the SRR. The detailed operation and modeling of the SRR are discussed in subsection 21.2.2 of chapter 21. In the case of a microstrip line, the SRR is located by the side of a strip conductor so that it is magnetically coupled to the microstrip. However, for a CPW structure, the SRRs are located at its backside, underneath the CPW slots to achieve the magnetic coupling. Figure (22.13c) shows a variation of the SRR by opening the gaps of the SRR and connecting them to the strip conductor. It called the open SRR (O-SRR) inclusion [J.32]. Figure (22.13c) also shows its equivalent series resonant circuit. Its resonance frequency is half of the resonance frequency of an SRR. It is a more compact resonator. Likewise, Fig (22.13b) shows the aperture type SRR etched in the conducting ground plane of a microstrip line. It is a complementary structure of the SRR, so it is called the complementary SRR (CSRR). Figure (22.13b) also shows its equivalent circuit. It is excited by the axial time-varying Ez electric field, i.e. electric flux Φe and

851

22 Metamaterials Realization and Circuit Models – II

behaves like an electric dipole, creating the ENG type medium with negative permittivity above its resonance frequency ω0e and below the electric plasma frequency ωep [J.26]. The electric flux is responsible for the capacitance Cc, and inductance Lc is a parallel combination of the upper and lower halves of the CSRR. The CSRR is electrically coupled to microstrip or CPW. Figure (22.13d) shows a variation in the CSRR structure, again by opening the ring aperture. It is called the open CSRR (O-CSRR) [J.33]. Its equivalent parallel resonant circuit is also shown in Fig (22.13d). It is also a compact resonator.

22.3.2

CSRR-Based Microstrip Metaline

Figure (22.14a) shows the loaded microstrip line with the unit cell of CSRR. The CSRR aperture is etched in the ground plane of a microstrip. As noted above, a periodic arrangement of the CSRR loaded microstrip line creates the narrowband ENG medium. However, the broadband MNG with negative permeability is created by a gap discontinuity of width Wg in the microstrip conductor. To get a larger value of gap capacitance, the gap discontinuity has been replaced by the interdigital capacitor shown in Fig (22.14b). Figure (22.14c) shows the T-type equivalent circuit model with Lh − Ch elements corresponding to the host microstrip. It also shows the π-capacitive network, i.e. the (Cg − Cf) network that models the gap discontinuity or interdigital capacitor. In the π− capacitive network, the gap capacitance is Cg and the fringe capacitance is Cf. Figure (22.14c) also shows the parallel resonator tank circuit, i.e. the (Lc − Cc) circuit, coupled to the line. The (Lc − Cc) circuit models the CSRR inclusion.

Microstrip Resonant Metalines

The microstrip resonant metalines are developed using both the CSRR resonant aperture in the ground plane of microstrip and placing the SRR by the side of the strip conductor of a microstrip.

d

Microstrip

Wg

Wm

(a) CSRR loaded microstrip with a series gap capacitor. CSRR in the ground plane. Gap discontinuity Cg

Lh/2 Ch

Cf

CSRR

Lc

Cf

Lh/2

Capacitor

(b) CSRR loaded microstrip with a series gap interdigital capacitor. CSRR in the ground plane. Lh/2

2 Cg

Ch

Cc

(c) Equivalent circuit of CSRR and gap embedded in microstrip. Figure 22.14

Interdigital

Microstrip

Ground plane

852

CSRR and gap capacitor loaded CRLH-metaline.

2 Cg

Lh/2

C

Lc

Cc

(d) Equivalent circuit of CSRR and gap loaded microstrip forming the CRLH–type line.

22.3 Resonant Metalines

Fig (22.15b) shows that it is converted into a passband on the addition of the gap capacitor. The gap capacitor is responsible for the wideband plasma-type MNG medium with negative permeability. In a narrow band, the medium acts as a DNG medium. However, using the interdigital capacitor, in place of a gap capacitor, the wideband S21 response is obtained. It is shown in Fig (22.15d). Figure (22.15c) shows the dispersion diagram of the CRLH circuit working under a nearly symmetrical condition. A small stopband region appears due to small asymmetry. The wideband passband is due to the combined DNG–DPS mode operation [J.23].

Figure (22.14d) shows that the circuit of Fig (22.14c) can be transformed into the symmetrical T-type CRLH line, except the presence of a capacitor (C). The circuit model explains the presence of the transmission zero frequency (fz) at displaced locations in the simulated results of the CRLH line. The frequency fz is shown both in the absence and also in the presence of gap-capacitor. Figure (22.15a, b) shows the displaced locations of fz. The values of the five components of the circuit of Fig (22.14d) are extracted from the simulated results [J.23, J.26, J.34, B.11]. The reference [J.23] provides the extracted values of components of the circuit model for the CSRR loaded microstrip, as shown in Fig (22.14a), on Rogers RO3010 substrate with εr = 10.2, h = 1.27 mm. Figure (22.15a) shows the EM-simulated and circuit model-based S-parameters response of the CSRR loaded microstrip unit cell without a capacitive gap. It shows the presence of a narrow stopband due to the formation of ENG medium with negative permittivity. However,

The microstrip can also be loaded with the SRR inclusions as shown in Fig (22.16a). It forms a bandstop filter due to the negative permeability of the MNG medium. The SRRs placed by the sides of a microstrip are magnetically coupled to it. The equivalent circuit is discussed in

0

0

–10

–10

–20

S21

∣S11∣/∣S21∣ dB

–30 fz

–40 S11

–50

EM-simulation Circuit simulation

–60 –70 1.0

1.5

2.0

2.5

–20 –30

–50

–70 0.5

3.5

fz 1.0

Frequency (GHz)

0

2.0 1.5 Circuit simulation EM-simulation Measurement

–0.5 0.0 Phase ϕ (rad/π)

0.5

1.0

(c) Dispersion diagram of CSRR and interdigital capacitor loaded microstrip [J.23]. Figure 22.15

2.5

3.0

–20

2.5

S21 (dB)

Frequency (GHz)

3.0

–1.0

1.5 2.0 Frequency (GHz)

S21

3.5

0.5

EM-simulation Circuit simulation

(b) S-parameter response of CSRR loaded microstrip with a gap capacitor [J.23].

(a) S-parameter response of CSRR loaded microstrip without a gap capacitor [J.23].

1.0

S11

–40

–60

3.0

S21

–40

0

–60

–20 –40 S11

–80 0

1

Circuit simulation –60 EM-simulation Measurement

2 3 Frequency (GHz)

S11 (dB)

∣S11∣/∣S21∣ dB

SRR-Based Microstrip Metaline

4

(d) S-parameter response of CSRR and interdigital capacitor loaded microstrip [J.23].

Response of CSRR and gap capacitor loaded CRLH-metaline. Source: Reproduced from Aznar et al. [J.23]. © 2008, Springer Nature.

853

22 Metamaterials Realization and Circuit Models – II

λ/4

λ/4 Reference line

854

Lm Lr

W2

Reference plane

C1

Z

L′2 C′2

L1

Substrate#1 W1 Z0

Lg Unit cell S Substrate#2

(b) BC-SRR-loaded and via loaded microstrip [J.35].

Topology of SRR loaded microstrip and SRR-via inductor loaded CRLH-metaline. Source: Mao et al. [J.35]. © 2005, IEEE. Lee and Lee [J.36]. © 2006, John Wiley & Sons.

[J.36]. The single-turn SRR can also be used to get the bandstop response [J.37]. The SRR based on tapered width strip conductor improves the rejection bandwidth of a bandstop filter [J.38]. However, by adding via between microstrip and ground plane the DNG passband response could be also obtained. The via creates the plasma medium with negative permittivity. Figure (22.16b) shows such a structure using the broadside-coupled SRR (BC-SRR) that avoids the bianisotropy of the SRR. The four unit cells of this structure have been used to design a leaky-wave antenna [J.35]. The SRR loaded microstrip lowpass filter has been used to realize the MNZ medium [J.39]. 22.3.3

Reference plane Ground plane

(a) SRR-loaded microstrip bandstop filter [J.36]. Figure 22.16

h1, εr1 h2, εr2

W3

CPW-Resonant Metalines

Following the development of microstrip metalines, the resonant CPW metalines are also designed using the SRR at the back of the unmetallized face of a CPW, and also by embedding the CSRR aperture in the conducting strips of a CPW. SRR-Based CPW Metaline

Figure (22.17a, b) show two schemes of periodic loading of the host CPW structure [J.26, J.40, J.41, B.12]. Scheme #1 is a uniplanar structure, i.e. SRRs and CPW strip conductors are on the same side of the substrate. However, the slot regions of the CPW have been enlarged to accommodate the SRRs. The wide slot-gap CPW structure has very high characteristics impedance, in the range of 190 Ω. Therefore, impedance matching with

50 Ω input/output ports is difficult. Scheme #2 is a biplanar structure. In this structure, the SRRs are located on the backside of the substrate, just behind the slots of the CPW. The loaded CPW can easily achieve a characteristic impedance of 50 Ω. The SRRs inclusions are magnetically coupled to the CPW. Figure (22.17c-i) shows the basic equivalent circuit of the unit cell of the magnetically coupled SRR. The SRR is modeled by the parallel LC resonant circuit. The π-network models the host DPS line structure. Figure (22.17c-ii) shows the simplified equivalent circuit. The SRR inclusion appears in the series arm that is responsible for the creation of an MNG medium with negative permeability over a narrow frequency band. It is similar to Fig (22.4a). The SRRs loaded structure can provide only the stopband characteristics. The topology of Fig (22.17b) shows the strip conductor loading of both slot-gaps of a CPW. The shunt inductor, due to the strip conductors, loaded line acts as an electric plasma medium with negative permittivity. The combined circuit model, shown in Fig (22.17c-iii), is the model of the CRLH configuration, acting as a DNG medium with a passband. The electric plasma frequency should be higher than the resonance frequency of the SRR. Figure (22.18a) shows the S21 response of the SRRloaded CPW in absence of the grounded strip conductors. It has a stopband response near 8.0 GHz due to negative permeability. However, Fig (22.18b) shows the passband response in the same frequency region

22.3 Resonant Metalines

(a) Uniplanar SRRs loaded CPW. SRRs located in wide slot regions [J.41].

(b) Biplanar SRRs loading at the backside of CPW and strip loading in slot regions [ J.26].

Cs Ls

SRR M

2Lh CPW

Ch/4

2Lh

C's

2Lh

L's

L's Ch/4

(i) Unit cell of SRR loaded CPW

C's

Ch/4

Ch/4

Lsh

Ch/4

(ii) Simplified equivalent unit cell

Lsh

Ch/4

(iii) Equivalent unit cell of SRR and shunt inductor loaded CPW

(c) Equivalent circuit of SRR loaded, and also inductor loaded CPW. Topology of SRR loaded CPW, SRR-strip inductor loaded CPW-metaline and circuit model. Source: Baena et al. [J.26]. © 2005, IEEE. Falcone et al. [J.41] © 2004, John Wiley & Sons.

0

0

–10

–10 ∣S21∣ (dB)

∣S21∣ (dB)

Figure 22.17

–20 –30 Sim Exp.

–40

2

4 6 8 Frequency (GHz)

10

12

(a) ∣S21∣ bandstop response of the SRR loaded CPW.

Figure 22.18

–20

Sim Exp.

7.5

8.0

8.5

9.0

–30 –40

–50 0

0 –5 –10 –15 –20 –25 –30 7.0

–50

0

2

4 6 8 Frequency (GHz)

10

12

(b) ∣S21∣ bandpass response of the SRR and strip loaded CPW.

The |S21| response of SRR-strip loaded CPW. Source: Baena et al. [J.26]. © 2005, IEEE.

once the metallic strips are added to create negative permittivity. For the case of the SRR and metallic strip-loaded CPW, the medium is DNG. The details of the structures are available in reference [J.26].

CSRR-Based CPW Metalines

The SRR-loaded CPW metaline is not a matched line with uniplanar metallization. In the case of the MMIC and LTCC process, it causes additional complexity.

855

22 Metamaterials Realization and Circuit Models – II

CSRR

CSRR

(a) CSRR loaded CPW.

0

∣S21∣ (dB)

–20 Simulated

–20 –30 –40

3 4 5 Frequency (GHz)

(c) Stopband response of CSRR loaded CPW. Figure 22.19

Sim Exp.

–10

–10

–30

SRR

(b) CSRR and backside SRR loaded CPW.

0

∣S21∣ (dB)

856

SC 2

3

4

5

6

Frequency (GHz) (d) Passband response of CSRR and SRR loaded CPW.

Topology of CSRR and CSSR-SRR loaded CPW and S21 response. Source: Al-Naib et al. [B.12].© 2012, In Tech.

A pair of CSRR is partly embedded in the ground conductor and partly in the central strip conductor of a CPW. It forms the unit cell of the CSRR loaded CPW, shown in Fig (22.19a). The CSRR inclusion provides a deeper rejection band shown in Fig (22.19c) [J.40, J.41]. It is due to the negative permittivity in the stopband region. The detail of the structure on the FR4 substrate is available in reference [B.12]. Figure (22.19c) shows good impedance matching below the rejection band. However, this structure does not provide a clean passband above the stopband. It has a spurious stopband at about 9.0 GHz, not shown in Fig (22.19c). It can be controlled by using the complementary U-shaped split resonator (CUSR) [B.12]. The structure can be further compacted by using complementary spirals in place of the CSRR. By cascading several unit cells, the rejection bandwidth can be improved. Figure (22.19b) shows the loading of CPW by the combined CSRR–SRR resonant structures. The SRRs are located at the back of the unmetallized face of the substrate. The CSRR provides negative permittivity and SRR provides negative permeability. The DNG medium is

created within the passband, around 4.0 GHz, shown in Fig (22.19d).

22.4

Some Applications of Metalines

The metalines, both nonresonant and resonant types, have been widely used to improve the performance of classical microstrip and CPW-based components and antenna. The metalines have been used to get compact microwave components operating over enhanced bandwidth. These are also used to get the dual-band and quad-band components. The design details of these components are available in numerous open literature and published books [B.1– B.8]. The present section provides just brief accounts of four examples to illustrate novel applications of metalines. The reader can follow the design details from the available resources. 22.4.1

Backfire to Endfire Leaky Wave Antenna

The microstrip based leaky-wave antenna is a traveling wave antenna. It radiates when the antenna operates in

22.4 Some Applications of Metalines

Backward at 3.4 GHz 0 120 –10 150 –20 –30 180 –40 –50 210 240 –60

90

30 0 330 270

1

Figure 22.20

300

270

270 H-plane

Interdigital capacitor

X

Forward at 4.3 GHz 90 60

Broadside at 3.9 GHz 90

E-plane

24

Shorted stub

24 unit cells CRLH based leaky-wave antenna scanning upper half of sky. Source: Liu et al. [J.43]. © 2002, IET.

the higher-order mode. Its propagation constant β is positive, causing radiation in the forward direction with the increase in frequency. Thus, scanning the complete upper half of the sky is not possible. However, the balanced CRLH-based microstrip line, realized by loading the host microstrip line with a series interdigital capacitor and shunt grounded stub, supports both the

backward wave propagation with negative β in the DNG region for f < f0 and the forward-wave propagation with positive β in the DPS region for f > f0. For β = 0, transition in the nature of wave-supporting medium occurs, without any bandgap, at the transition frequency f0. Thus, the metaline can act as a backfire to endfire leaky-wave antenna scanning complete upper half of the sky [J.42]. Figure (22.20) illustrates the CRLH 24 unit cells-based leaky-wave antenna at f0 = 3.9 GHz. At a frequency below 3.9 GHz, the backward radiation is obtained, at 3.9 GHz the radiation is in the broadside direction, and at a frequency above 3.9 GHz forward radiation occurs. The details of the antenna structure on substrate RT/Duroid 5880 with εr = 2.2, h = 1.57 mm are available in reference [J.43, B.2]. 22.4.2 Metaline-Based Microstrip Directional Coupler The usual edge coupled microstrip coupler has a broad bandwidth (>25%). However, it has a weak coupling level of −10 dB or less. The metalines-based couplers can provide an arbitrary level of coupling, even nearly up to 0 dB, while retaining the broadband characteristics. There are three kinds of metalines-based directional coupler: (i) CRLH–CRLH coupler, (ii) CRLH–microstrip coupler and (iii) (D-CRLH)–CRLH coupler. Figure (22.21a–c) show the fabricated images of these couplers. Figure (22.21d) shows a schematic diagram of the third case. These Coupled

Coupled 3

1

Isolated 4

3

4

2

1

2

Through

Input

(a) 9-cells CRLH-CRLH 0 dB coupler [J.44].

Coupled

Input

Isolated

Through

Input

(b) 3-cells CRLH-CRLH 3 dB coupler [J.44]. 2

Isolated Port#3

Port#1

k

s

Port#2

Through

CRLH

1 CRLH

(c) 9-cells CRLH-microstrip 0 dB coupler [J.45]. Figure 22.21

4

Port#4

3 D-CRLH

D-CRLH

(d) Dual-band 9-cells(D-CRLH)-CRLH coupler.

Configurations of metalines directional couplers. Source: Reproduced from Caloz et al. [J.44], © 2004 IEEE. reproduced from Caloz and Itoh [J.45], © 2004 IEEE.

857

22 Metamaterials Realization and Circuit Models – II

even–odd mode gaps. The coupler operates within these gaps. The couplers are designed such that the gaps overlap each other providing the coupling. The line length, i.e. the number of a unit cell is determined to achieve the desired level of coupling. Figure (22.21a) shows the nine unit cells long coupler with 0 dB coupling, whereas 3 dB coupler using the three unit cells is shown in Fig (22.21b). Both these couplers are developed on Rogers RT/Duroid 5880 substrate with εr = 2.2, h = 1.57 mm. The coupling gap between the two lines is 0.3 mm which is a comfortable gap-width from the fabrication point of view. The coupling can be controlled by changing the coupling gap and the number of unit cells. Figure (22.22a) shows the S-parameters response of the 0 dB coupler. The 0 dB coupling, in practice about 0.5 dB coupling, is achieved over a bandwidth of 36% from 3.2 to 4.6 GHz. Its directivity is about 25 dB. Figure (22.22b) shows the S-parameters response of

couplers are analyzed using the coupled-mode theory. The design methods, fabrication details, and performance results of these couplers are available in the open literature [J.5, J.10, J.44–J.46, B.2–B.4]. We consider these couplers briefly to illustrate the novelty of design provided by the metalines. Figure (22.21a) shows the nine unit-cells-long fabricated CRLH–CRLH 0 dB coupler. The mainline with ports 1–2 and the coupled line with port 3–4 are microstrip-based CRLH metalines. Port #1 is the input port and port #3 is the coupled port. The ports #2 and 4 are the through and isolated ports, respectively. The uncoupled CRLH metaline is described by the equivalent circuit shown in Fig (22.6a). However, the magnetic and electric couplings contribute toward the additional coupling inductor and the coupling capacitor. It is the symmetrical coupler analyzed by the even–odd mode concepts [B.1, J.44]. The even and odd mode CRLH equivalent circuits are not identical. It results in the

10 0 dB Coupling

10

band

–10

S-parameters (dB)

S-parameters (dB)

0

–20 –30

S11 S21 S31 S41

–40 –50

0 –3 –10

S11 S21 S31 S41

–20 –30

3 dB Coupling band

–40 –50 –60

–60 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

0

1

Frequency (GHz) (a) Response of CRLH – CRLH 0 dB coupler [J.44].

f1 = 1.98 Ghz

0 –10 –15 –20 –25 –35

0 dB Coupling

S11 S21 S31 S41

–40 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 Frequency (GHz) (c) Response of CRLH – microstrip 0 dB coupler [J.45]. Figure 22.22

S-parameters (dB)

–5

–30

3 4 2 Frequency (GHz)

5

6

(b) Response of CRLH – CRLH 3 dB coupler [J.44].

5

S-parameters (dB)

858

0 –1.5 –5 –10 –15 –20 –25 –30 –35 –40 –45 –50 0 1

f2 = 5.04 Ghz S31

S21

S41 2

3 4 5 6 Frequency (GHz)

7

8

(d) Dual band response of CRLH – microstrip 0 dB coupler [J.10].

Frequency response of metalines based directional couplers of Figure (22.21). Source: Caloz and Nguyen [J.10]. © 2007, Springer-Verlag. Caloz et al. [J.44]. © 2004, IEEE. Modified from Caloz and Itoh [J.45].

22.5 Modeling and Characterization of Metsurfaces

the 3 dB coupler. It has a 50% bandwidth from 3.0–5.8 GHz and 20 dB directivity. Figure (22.22c) shows the S-parameters response of the nine unit cells long (62 mm) CRLH–microstrip 0dB coupler. The CRLH line supports the backward wave, whereas the microstrip supports the forward wave. Thus, it is an asymmetrical coupler that is analyzed using the c-mode and π-mode concept [J.45]. This is also fabricated on Rogers RT/Duroid 5880 substrate with a 0.3 mm coupling gap. The fabricated coupler has a 0.7 dB coupling with 53% bandwidth from 2.2 to 3.8 GHz. It has excellent 30 dB directivity. Again, by reducing the number of unit cells, a compact 3 dB coupler can be designed. The above-mentioned couplers are single band couplers. However, the 15 unit cells (D-CRLH)–(CRLH) coupler, shown in Fig (22.21d), is a dual-band coupler [J.10]. It also shows the equivalent circuits of both the CRLH and D-CRLH isolated lines. These circuits are modified by the coupling. Figure (22.21d) shows dual coupling bands of (−1.5 dB) coupling, around f1 = 1.98 GHz and f2 = 5.04 GHz bands. Its S-parameters are, to some extent, influenced by the choice of the line as the mainline or a coupled line. 22.4.3

Multiband Metaline-Based Components

Usually, microstrip components work in a single specified band. However, for some applications, the microwave components are required to operate in the dual and more frequency bands. These bands are not harmonically related. The dual-band microstrip-based components are designed using the CRLH-metalines [B.2, B.4, B.7]. The unit cell of balanced CRLH-metaline exhibits dual-band response as its dispersion diagram continuously moves, with an increase in frequency, from the DNG region to the DPS region. The dispersion diagram is nonlinear and can be tailored to meet the dual-band requirement. The concept of dispersion diagram tailoring has helped to conceive the concept of dispersion engineering [J.7, J.8, J.13, J.47]. Dual-band Components

The microstrip sections are used to realize microwave components. The standard DPS microstrip section has needed 90 electrical length at a single frequency. However, the CRLH-metaline can operate at two harmonically unrelated frequencies, as shown in Fig (22.23a), for the dispersion diagrams of a CRLHmetaline. It also shows the working of a DPS microstrip line at harmonically related frequencies. The −90 line section is obtained at frequency f1 and also −270

(+90 ) at frequency f2 using the metaline, where f2/f1 is either integer or noninteger. It is not possible with a DPS line. Figure (22.23b) shows a realization of a metaline section, using SMD (surface-mounted devices) discrete elements, embedded in the microstrip host line [J.48]. It also shows the equivalent circuit of the CRLH line section. Such a line section can be used as an opencircuited λ/4 stub to a microstrip line. Figure (22.23c) shows that the stub creates dual stopband at 890 and 1670 MHz. This concept has been used to design dualband branch-line coupler and rat-race coupler [J.48]. The resonant inclusions-based metalines are also used to design other dual-band components [J.13, B.7]. Quad-band Components

Figure (22.24a) shows the passband type quad-band metaline. It is realized by embedding the D-CRLH configuration, shown within gray square boxes, in the CRLH type of metaline, shown within light gray elliptical boxes. Let us consider the case when ω0 is the resonance frequency of all four resonators in the series and shunt arms of the quad-band metaline. In this case, at the operating frequency, ω < ω0 the composite network is reduced to the CRLH line #1. At frequency, ω > ω0, it is reduced to another CRLH line #2. Each CRLH line supports the dual-band, giving the quad-band for the unit cell of metaline shown in Fig. (22.24a). The dispersion diagram of Fig (22.24b) exhibits an equal phase response at four frequencies. Both CRLH#1 and #2 lines are balanced metalines without bandgap and crossing from the DNG to DPS medium occurs at 1.5 and 2.5 GHz, respectively [J.47]. Figure (22.24c) shows the realization of quad-band metaline in the passband mode. The series resonator in the shunt arm is formed by a combination of the patch and meandered inductor. The parallel resonator in the series arm is formed by a combination of meandered inductor and chip capacitor. Figure (22.24d) shows a wide bandwidth dual passband response, as four passbands have merged into two pass-bands. The dual of the configuration shown in Fig (22.24a) can also be obtained giving quad stopband response [J.47].

22.5 Modeling and Characterization of Metsurfaces The metasurfaces are considered the 2D equivalents of volumetric 3D metamaterials discussed in chapter 21. Fabrication of the 3D metamaterials is a difficult and expensive process. The fabrication process of the

859

22 Metamaterials Realization and Circuit Models – II

Viahole

Viahole

Phase response

CRLH (DNG)– Metaline DPS line

0°

LL

ℓR

–90°

2CL

RH-TL 2CL CL –270°

LL

ℓR RH-TL

LH-TL f1

0

f2

3f1 (b) Lumped elements metaline.

Frequency (a) Dispersion diagram of lines. 0 –5 S-parameters (dB)

860

–10 –15 –20 –25 –30

S21 Sim, Exp.

S11 Sim, Exp.

–35 –40 0.6

0.8

1

1.2

1.4

1.6

1.8

2

Frequency (GHz) (c) Response of the open stub. Figure 22.23

Metaline for the dual-band components. Source: Hsiang Lin et al. [J.48]. © 2004, IEEE.

monolayer or a few layered metasurfaces is much simpler. Like the metamaterials, the metasurfaces are engineered by periodically arranging the subwavelength metallic/dielectric inclusions on the surface of the host dielectric substrates. These metasurfaces are also engineered using the apertures as inclusions in the metallic screen. Recently, the metasurfaces have been widely explored for their unique applications in the microwave to optical frequency ranges. Reviews of their characteristics, modeling, and applications are available in recent publications [J.49–J.52, B.13]. The periods of inclusions on metasurfaces are at the subwavelength scale. It avoids the Bragg type diffraction.

In this respect, the metasurfaces are different from the EBG surfaces, discussed in chapter 20. Under the excitation of the EM-waves, from microwave to optical frequencies, the inclusions get polarized, electrically or magnetically, creating the electric and magnetic dipoles distributed over the metasurfaces. Thus, the electric metasurfaces are realized with the electric dipoles and the magnetic metasurfaces are engineered using the magnetic dipoles, whereas the electromagnetic metasurfaces are engineered using both the electric and magnetic dipoles. The array of polarized inclusions, also called the array of scatterers or meta-atoms, are treated as the array of antenna causing the reflection and refraction

22.5 Modeling and Characterization of Metsurfaces

D-CRLH

D-CRLH

CRLH

Frequency (GHz)

CRLH

5 4.5 4 3.5 3 2.5 2 1.5 1 0.5 0

D-CRLH

CRLH

f4 f3 f2 f1 f5

0

0.5

1

2

1.5

2.5

3

3.5

βd

(b) Dispersion diagram of dual CRLH-unit cell [J.47].

(a) Passband type dual CRLH-metaline unit cell.

0

–40

Passband

Passband

│S21│ dB

–20

–60 –80 0

1

2

3

4

5

6

7

Frequency (GHz) (c) Realization of dual CRLH-unit cell [J.47]. Figure 22.24

(d) S21- response of dual CRLH-unit cell [J.47].

Quad-band metaline. Source: Eleftheriades and Islam [J.47]. © 2007, Elsevier.

of the EM-waves. The inclusions have variable geometric parameters, i.e. variation in their shape, size, and orientation on the metasurfaces. They control the amplitude, phase, and polarization state of the incident waves due to the spatially varying arrangement of inclusions. Thus, the metasurfaces support the anomalous reflections and refractions governed by the generalized Snell’s law resulting in their wavefront controlling ability in the 3D space. The metasurfaces also support the surface-waves propagation. Space-waves can be coupled to the metasurfaces as the surface-waves and further radiated in the desired manner as the leaky-waves. In this way, the metasurfaces are different from similarlooking frequency selective surfaces (FSS). In principle, the thickness of a metasurface should be almost zero, i.e. very small compared to the operating wavelength. In this case, engineering of the electric dipoles of the surface is simple. However, creating magnetic dipoles on a monolayer metasurface is difficult, as it requires circulatory electric current in a plane normal to the surface of inclusions. It requires at least two-

layered structure or presence of a conductor backing to the electric dipoles to generate magnetic dipoles. The electric dipole and its oppositely polarized image dipole create such a current loop acting as the magnetic dipole. The natures of the metasurfaces are determined by the nature of the polarizable inclusions and their periodic arrangements, so the metasurfaces could be isotropic, anisotropic, bi-isotropic, and bianisotropic. The materials based on such inclusions are discussed in chapter 6. Under the time-harmonic incident fields, the electric and magnetic dipoles create discrete electric and magnetic polarization currents. However, subwavelength size inclusions and periods of their 2D lattices help the homogenization of the metasurfaces. On the homogenized metasurface, both polarization currents are substituted by the equivalent continuous electric and magnetic surface currents. The surface currents act as the secondary radiators, like the secondary sources on the Huygens surfaces. The metasurfaces supporting the surface currents could be treated by Schelkunoff’s equivalent surface principle [B.14].

861

862

22 Metamaterials Realization and Circuit Models – II

continuous electric J s and magnetic Ms currents. Thus, the equivalent surface is obtained through the process of homogenization of an artificially engineered discrete metasurface. The metasurface is located at the interface of medium #1 and medium #2. The surface currents present discontinuity to the EM-fields E 1 , H 1 and

The 3D bulk metamaterials are characterized by the equivalent (effective) permittivity and permeability. However, usually, it is not a proper description of the 2D metasurface, as it is based on the surface polarization phenomenon. As a practical measure, in some cases, such characterization is used. The metasurfaces are characterized by surface polarizabilities, surface susceptibility, and surface admittances/impedances. The surface admittances/impedances are useful to appreciate reflection, refraction, and impedance matching with free space [J.50]. The present selection discusses some of these characterization methods of metasurfaces. The next section (22.6) presents a few illustrative examples for the applications of metasurfaces in the microwave frequency ranges. 22.5.1

E 2 , H 2 existing in two media: E 2 − E 1 × n = Ms

a

n × H2 − H1 = J s

22 5 1 where n is the normal unit vector. The field quantities E 1, H 1

account for the total incident and reflected

field in the medium #1, and field quantities E 2 , H 2 account for the transmitted (refracted) field in the medium #2.

Characterization of Metasurface

The wave phenomena of reflection, refraction (transmission), and diffraction from the metasurface are caused by the induced time-harmonic electric and magnetic dipoles periodically arranged on a dielectric sheet. Thus, the incident time-harmonic EM-waves induce the discrete polarization currents and charges distributed over the metasurfaces. The scattering of the fields by metasurfaces is modeled as Schelkunoff’s equivalent surface to understand the scattering of EM-waves. Further, the metasurface is modeled by several different parameters. These aspects are discussed below.

Modeling of Metasurface

Figure (22.26) shows stages of modeling of the engineered metasurface. Each stage of modeling is discussed below: Stage – 1

It is the physical step. The incident EM-waves induces the time-harmonic electric dipoles in periodically arranged inclusions located at the interface. In a volumetric material, the polarization and magnetization (magnetic polarization) are expressed as the dipoles per unit volume. It is discussed in chapter 6. However, in the case of a metasurface, the surface polarization is considered as the dipoles per unit surface area [J.50]. The surface electric and magnetic polarizations are further related to the surface electric and magnetic susceptibilities. Following the equations (6.1.3), (6.1.5), and

Schelkunoff’s Equivalent Surface

Figure (22.25) shows that a metasurface, supporting the physically realizable discrete electric and magnetic polarization currents could be replaced with Schelkunoff’s equivalent surface. It supports the equivalent Transmitted (refracted) field (E2, H2)

Medium #2

Js = nˆ × (H2 – H1) nˆ

Ms = –nˆ × (E2 – E1)

Electric and magnetic dipoles Metasurface as equivalent Schekunoff surface

Figure 22.25

(E1, H1) Reflected field

b,

Medium #1 Incident field

Metasurface supporting time-harmonic electric and magnetic dipoles. The corresponding electric and magnetic polarization currents are treated as continuous electric and magnetic currents on Schelkunoff’s equivalent surface.

22.5 Modeling and Characterization of Metsurfaces

Surface electric and magnetic polarizabilities

Discrete surface electric and magnetic polarization currents

[αe] [αm] Figure 22.26

sur/dt dPsesur/dt, dPsm

Eq. continuous surface currents

Js

Surface electric and magnetic susceptibilities

[χes] [χms]

Ms

Surface electric and magnetic admittance

[Yes]

[Yms]

Reflection and transmission coefficients

[Γ] [τ]

Stages of metasurface characterization.

(6.1.24) of chapter 6, the following expressions for the time-harmonic induced dipole moments and susceptibilities could be written: Electric/magnetic dipole moments: loc

loc

p = αe E 0 ejωt

m = αm H 0 ejωt

a,

where αe and αm are electric and magnetic polarizability respectively. Electric/magnetic surface polarization (Pse, Psm) and surface susceptibility χes , χm are obtained as follows: s loc αe E 0 ejωt S

p = S

P sm =

m = S

loc αm H 0 ejωt S

a b

loc

also, P se = χes E 0 ejωt

c

loc

jωt P sm = χm s H0 e αe and, χes = S m α χm s = S

inc

of the incident and reflected fields inc

H

d e f ,

E

ref

; and the field in the upper-medium #2 is tra

the transmitted field ( E

tra

), (H ).

Stage – 2

The time-harmonic electric and magnetic dipoles constitute the discrete electric and magnetic polarization currents. Each inclusion is treated as a miniature radiator or antenna radiating in both the lower and upper media resulting in the reflection and transmission, so the metasurfaces behave as reflectarrays and transmit-arrays. The surface polarization currents, due to the local tangential fields, are obtained as follows from the equations (22.5.3a, b): Electric/magnetic discrete polarization currents: d P se jωαe loc = E t = jωαes Eloc t dt S loc d P sm jωαm loc = H t = jωαm s Ht dt S

loc

av

Et = Et = loc

H

av

= Ht =

E

+ E− 2

a

H + + H− 2

b

+

tra U

U

where, E

= E2

+

sur

sur

=

E

sur

H − = H1

sur

ref

+ E

d

tra U

=H

e sur inc

L

=

H

Stage – 3

At this stage, the discrete metasurface is treated as Schelkunoff’s equivalent surface supporting the continuous electric and magnetic surface currents. These are equivalent to their corresponding physical polarizations currents giving the following relations: loc

J s = jωε0 χes E t

ref

a,

loc Ms = jωμ0 χm s Ht

b 22 5 6

L sur

U

H + = H2

c

b,

where the electric and magnetic surface polarizabilities are αes and αm s .

sur

inc

L

E − = E1

= E

a

22 5 5

22 5 3 where S is an area of a unit cell, χes and χm s are surface electric and magnetic susceptibilities, respectively. The tangential local fields, causing the polarization, at the interface are treated as the average fields of the fields from the upper (+) and lower (−) sides of the interface:

,

,E

ref

,H

b, 22 5 2

P se =

The total field in the lower-medium #1 is a combination

L

+H

f sur

22 5 4

Stage – 4

At this stage, the metasurfaces are characterized by the electric and magnetic surface susceptibilities. These are related to their surface polarizabilities, as the polarization currents are also obtained in terms of the susceptibilities. Therefore, from equations (22.5.5) and (22.5.6), the following relations are obtained:

863

864

22 Metamaterials Realization and Circuit Models – II loc dPse = jωε0 χes E t a , also jωαes = jωε0 χes b dt dPsm loc m Ms = = jωμ0 χm c , also jωαm d s Ht s = jωμ0 χs dt

Js=

22 5 7

Stage – 5

The metasurfaces support both the surface currents and the tangential field components. Thus, the metasurfaces are also characterized by the electric and magnetic surface admittances Yes , Ym or surface impedances s Zes , Zm . The surface impedance/admittances have their s origin in the polarizabilities of dipoles and corresponding susceptibilities. Using equation (22.5.7), the following relations are obtained among surface electric and magnetic impedance/admittances, surface electric and magnetic susceptibilities, and surface electric and magnetic polarizabilities of inclusions creating electric and magnetic dipoles: loc

Anisotropic Metasurface

In subsection (4.2.3) of chapter 4, it is discussed that the permittivity and permeability of anisotropic materials are tensor quantities. Likewise, using equations (22.5.2) and (22.5.8) in tensor form, the anisotropic metasurfaces are characterized by the electric and magnetic dipole moments p and m described by the 2 × 2 tensors (dyadic) of the surface electric and magnetic polarizabilities, and the corresponding surface electric and magnetic admittances as follow: p

Zes = 1 Yes

c , M s = Zm s Ht

0

0

αm s

Et a

m

Ht

where electric polarizability tensor =

a , also E t = Zes J s

αes =

loc

J s = Yes E t loc

chapter 4, the results related to such metasurfaces are summarized below:

αes

=

αes

=

b

loc

d f h

=

αm s

=

αm s

=

22 5 8

Stage – 6

The surface impedances/admittances are responsible for the reflection, transmission, and also impedance matching of the metasurfaces with free space. The impedancematched metasurfaces avoid the reflection and fully transmit the incident waves. Such metasurfaces are known as the Huygens metasurfaces. Thus, the metasurfaces are also characterized by the reflection and transmission coefficients. In the case of the isotropic metasurface, the reflection coefficient (Γ) and transmission coefficient (τ) are scalar quantities. Their computation for the normally incident plane waves is discussed in section (22.5.2). The above discussion is limited to isotropic metasurface. In this case, the surface polarizability, surface susceptibility, and surface admittance/impedance are scalar quantities. However, the metasurfaces could also be anisotropic, bi-isotropic, and bi-anisotropic in nature. In these cases, the metasurface characterizing parameters are described by 2 × 2 matrices and tensors, shown in Figure (2.26). Following the above discussion, and also section (4.2) of

αexy,s

αeyx,s ,

αeyy,s

b

magnetic polarizability tensor

m

m e , Zm also H t = Ym s Ms s = 1 Ys e e e m m Ys = jωε0 χs = jωαs g , Zs = jωμ0 χm s = jωαs

αexx,s ,

αm xx,s ,

αm xy,s

αm yx,s ,

αm yy,s

c 22 5 9

Yes

Js

0

Et

=

a Zm s

0

Ms = Yes

= Zm s

=

Yes

=

= = Zm s

Ht

Yexx,s ,

Yexy,s

Yeyx,s ,

Yeyy,s

Zm xx,s ,

Zm xy,s

Zm yx,s ,

Zm yy,s

b

c 22 5 10

In equations (22.5.9) and (22.5.10), the surface admittance and impedance tensors (dyadic) [Yes] and [Zm s ] correspond to the electric and magnetic surface polarizabilities. The electric and magnetic polarizabilities are discussed in section (6.1) of chapter 6. For properly aligned metasurfaces, the cross-coupling terms, i.e. the (xy) and (yx) terms, are zero. The reflection and transmission coefficients of anisotropic surfaces are also tensor quantities:

22.5 Modeling and Characterization of Metsurfaces =

inc

= Γ E Γxx , = Γ = Γyx ,

tra

a, Γxy

c,

Γyy

E

=

inc

= τ E τxx , = τ = τyx ,

b τxy τyy

d 22 5 11

The anisotropic metasurface is a useful device for polarization conversion. Bianisotropic Metasurface

The general linear tensor metasurfaces are characterized with the help of general linear relations of induced surface electric and magnetic dipole moments created by the average local tangential fields:

p m

αee s

αem s

αme s

αmm s

=

αem = − αme s s

Et

a

Ht

T

b, 22 5 12

22.5.2 Reflection and Transmission Coefficients of Isotropic Metasurfaces The reflection and transmission of normally incident plane waves at the interface of two different media are discussed in section (5.1.1) of chapter 5. However, in that discussion, polarizing inclusions are not present at the interface of media. The metasurface as the interface has the specific feature of accommodating polarizable inclusions at the interface. The metasurface could be present even in a homogeneous medium such as air medium. In the present analysis, the metasurface, i.e. Schelkunoff’s equivalent surface is characterized by the electric and magnetic surface admittances supporting the electric and magnetic surface currents. The boundary condition (22.5.1) is used in terms of admittances given by the equation (22.5.10a). Figure (22.27) shows the interface of the metasurface located in the (y–z)-plane. The y-polarized EM-wave propagating in the x-direction is normally incident at the metasurface. The total field in the left-hand halfspace (medium #1) is a summation of the incident and reflected waves, whereas the transmitted wave is in the right-hand space (medium #2): Etot = E0 e − jkx + Γejkx y E0 − jkx e − Γejkx z, Htot = η0

mm me where [αee ], [αem s ], [αs s ], and [αs ] are the electric, magnetic, electromagnetic, and magnetoelectric surface polarizabilities tensors, respectively. In matrix equation (22.5.12), T denotes the transpose matrix. The presence of the magnetoelectric/electromagnetic polarizability is another unique characteristic of bianisotropic metasurfaces as against the anisotropic metasurfaces. It provides additional facility for manipulation of the wavefront by such metasurfaces [J.52, J.53]. Likewise, the general tensor metasurfaces are also characterized by the electric, magnetic, and magnetoelectric/electromagnetic tensor admittances:

Ms where

Et Ht

Yee s

Zem s

Et

Yme s

Zmm s

Ht

Medium#1

a

inc

=

n× E inc

b

n×H 22 5 13

In equation (22.5.13), the surface admittance tensor mm [Yee s ], and impedance tensor [Zs ] correspond to the surface electric and magnetic susceptibility or polarizability tensor, respectively. The surface impedance tensor [Zem s ] and surface admittance tensor [Yme s ] correspond to the magnetoelectric and electromagnetic polarizability or susceptibility tensor, respectively.

x0

d

Medium#2 η0

η0 Reflected waves

=

a

22 5 14

Incident waves

Js

Etra = τE0 e − jkx y τE0 − jkx z, e Htra = η0

E

ref

H ref k

E

inc

y k

z

Metasurface

ref

E

(y–z)plane

Transmitted waves E tra

H

tra

k

x

H inc

Figure 22.27

Reflection and transmission at the interface of the metasurface.

865

866

22 Metamaterials Realization and Circuit Models – II

The time-harmonic variation ejωt is suppressed in expressions (22.5.14). The boundary conditions at the interface are given below: tra

x×

H

x −H

tot

x

= Js

The transmission and reflections coefficients are obtained from equations (22.5.17b) and (22.5.19b): τ=

e 4 − Zm s Ys 2 + η0 Yes 2 + Zm s η0

a

Γ=

e 2 Zm s η0 − η0 Y s 2 + η0 Yes 2 + Zm s η0

b

a

x=0 tra

x×

E

x −E

tot

= − Ms

x

b

x=0

22 5 15 On expanding equation (22.5.15a), we get the following expression: y

Htot z

x

− Htra z

x

x=0

+z

Htra y

x

− Htot y

x

22 5 16 tot As Htra y x = Hy x = 0, so Jsz = 0, and using equation (22.5.14) the following relation is obtained:

b

Likewise, on expanding equation (22.5.15b), we get the following expression: x=0

tot + z Etra y x − Ey x

x=0

= − yMsy + zMsz tot Etra y x − Ey x

x=0

c

Zm s =

2η0 1 − τ + Γ 1 + τ−Γ

d 22 5 20

The metasurface constituted of both electric and magnetic surface polarizabilities can provide the perfect reflection and perfect transmission. The conditions for perfect reflection and perfect transmission are obtained from equation (22.5.20). Perfect Reflection

a

22 5 17

tra y Etot z x − Ez x

2 1−τ−Γ η0 1 + τ + Γ

x=0

= yJsy + zJsz

tra e av Htot z x − Hz x x = 0 = Jsy = jωαs E jωαes tra E x + Etot x x = 0 = 2 E0 Ye 1 − Γ − τ = s τ + 1 + Γ E0 η0 2 e 1 − η0 Y s 2 τ+Γ= 1 + η0 Yes 2

Yes =

The perfect reflection from the metasurface requires zero transmission, i.e. τ = 0 in equation (22.5.20a). Thus, using equation (22.5.20a), the condition for the total reflection of normally incident plane wave at the metasurface is e Zm s Ys = 4

a,

tra as Etot z x = Ez x = Msy = 0

22 5 18 On using expressions (22.5.7), (22.5.14), and other related expressions the following relation is obtained: jωαm tot s Htra E0 τ − 1 + Γ = = − z + Hz 2 Zm E0 a = − s τ + 1−Γ 2 η0 − Zm s τ−Γ + 1 τ−Γ −1 = 2η0 1 − Zm s 2η0 τ−Γ = b 1 + Zm s 2η0 av − jωαm s H

22 5 19

b 22 5 21

Equations (22.5.8g, h) are used to get equation (22.5.21b). The complex reflection coefficient of the metasurface under the above condition is obtained from equation (22.5.20d): Γ = Γ ϕref =

= − Msz

2 αes αm s = −4 ω

Γ= − Γ = 1,

Zm s − 2η0 Zm s + 2η0

1 − jωαm s 2η0 1 + jωαm s 2η0 ϕref = 2 tan − 1 ωαm s 2η0

a b c 22 5 22

Likewise, the complete reflection can be also achieved using electric polarizability. The reflection coefficient of the metasurface under the perfect reflection acts like the PEC/PMC surface. However, unlike these surfaces, the metasurface facilitates control of the reflection phase from −π to π, i.e. full phase control of 2π. Thus, the magnetic and electric polarizabilities of each inclusion can be tailored at a needed frequency to achieve perfect reflection and the desired reflection phase. The oblique incidence at

22.5 Modeling and Characterization of Metsurfaces

the interface of two natural media also provides perfect reflection at a critical angle. However, the reflection phase from the interface of natural media cannot be tailored. It is discussed in section (5.3) of chapter 5. Perfect Transmission

The perfect transmission from the metasurface requires zero reflection, i.e. Γ = 0 at metasurface. In this case, the metasurface is matched with free space. The metasurface without reflection is known as the Huygens metasurface [J.54]. The concept of Huygens metasurface is similar to the concept of Huygens load, discussed in subsection (2.1.4) of chapter 2. Equation (22.5.20b), under the requirement Γ = 0, provides the following condition for the matching: η0 =

Zm s Yes

a,

η0 =

αm s αes

b 22 5 23

The complex transmission coefficient under no reflection condition is obtained from equation (22.5.20c) using the electric polarizability: τ = τ φtra = τ=

2 − η0 Yes 2 + η0 Yes

1 − j ωη0 αes 2 1 + j ωη0 αes 2

τ = 1, ϕtra = − 2 tan − 1 ωη0 αes 2

a

b c 22 5 24

A similar result could be obtained for the magnetic polarizability. The perfectly transmitted waves from the Huygens metasurface can acquire any phase between −π to +π by tailoring the polarizabilities of each inclusion forming the metasurface. The perfect transmission also occurs at the Brewster angle for the obliquely incident plane waves at the interface of two natural media. However, the phase of the transmitted waves cannot be tailored. The Brewster angle is discussed in section (5.3.1) of chapter 5. Metasurface with only Electric Polarization

It is difficult to design an inclusion with the magnetic response, as it requires a current loop normal to the plane of a metasurface. Therefore, only electric type polarization is commonly used to design metasurfaces. In this case, the magnetic polarizability (αm s ) is zero, lead= 0. The transmission and reflection ing to Zm s

coefficients of the electric type metasurface are obtained from equation (22.5.20): τ=

2 2 + η0 Yes

a,

Γ=

− η0 Yes 2 + η0 Yes

b

τ=

2 2 + jωη0 αes

c,

Γ=

− jωη0 αes 2 + jωη0 αes

d 22 5 25

For the electric type metasurface, both transmission and reflection occur simultaneously. The phase control also has a limited range, i.e. −π/2 to π/2. Similar behavior is obtained for the magnetic metasurface designed with inclusions of only magnetic polarizability. However, the electric type metasurfaces backed by conducting sheets act as the reflectarrays and provide 0 to 2π reflection phase control [J.51]. 22.5.3

Phase Control of Metasurface

The above discussion shows that the reflection and transmission phases can be tailored by controlling the phase of individual inclusions. It helps to control the direction of the reflected and transmitted waves from the metasurface. The realization of such phase control is discussed in the present subsection. In the case of a material slab, the phase of the refracted wave is acquired by the incident wave while traversing the slab. In the case of a metasurface, the inclusions constituting the metasurface act like dipoles receiving the incident waves and reradiating in the forward direction as the refracted waves, and in the backward direction as the reflected waves. However, control of shape, size, and orientation of the inclusions at each location on the metasurface in a periodic manner provides the mechanism of the phase control of the scattered incident waves resulting in the anomalous reflection and refraction. The anomalous reflection and refraction are governed by generalized Snell’s law that is discussed in next subsection (22.5.4). The working of the metasurface is analogous to the working of the reflect/transmit antenna arrays [J.55]. In the case of the metasurfaces, full 2π i.e. 360∘ wavefront control of the reflected and refracted waves have been realized. Moreover, metasurfaces are homogenized surfaces with subwavelength size periodicity. The metasurfaces can have a uniform phase distribution of inclusions. The uniform metasurface is shown in Fig (22.38a). In this case, there is no wavefront control of the reflected and refracted waves. However, as discussed above, the metasurfaces can have nonuniform phase distribution of inclusions, i.e. the 1D and 2D phases can be the functions of positions along two orthogonal axes

867

868

22 Metamaterials Realization and Circuit Models – II

over the metasurfaces. Such nonuniform phase metasurfaces are known as the phase-gradient metasurfaces (GM). The phase variations of the metasurfaces are obtained by two mechanisms (i) Controlling the size of inclusions, (ii) Controlling rotation (orientation) of identical inclusions [J.49, J.51]. These mechanisms are applied to electric/magnetic polarization-based inclusions and also to the Huygens metasurfaces using both electrically and magnetically polarizable inclusions. Controlling Size of Inclusions over a Spatial Period

Figure (22.30a) shows the 1D array of variable-sized resonating metallic inclusions of the identical shapes without any rotation. The in-phase current is induced in one inclusion resonating as a half-wavelength electric dipole at the frequency of the incident waves. In other inclusions, with a size smaller or larger than halfwavelength dipole inclusion, the currents are induced with leading/lagging phases. Thus, the total phase of backward radiated (reflected) and forward radiated (transmitted/refracted) fields are controlled by varying the phase of the reradiating induced current. As each inclusion has a different resonating frequency, so it is called the metallic antenna dispersion method to control the phase of the metasurface [J.56]. Figure (22.30a) shows that n numbers of inclusions are periodically arranged in square unit cells of period d. A supercell of length Λ = n d is formed with a spatial period Λ. The supercell is repeated both horizontally and vertically to form the phase gradient metasurface. In the present arrangement, the induced currents on inclusions acquire the ±π/2 lead/lag phase. It results in π, i.e. 180∘ phase control of reflected/transmitted waves from the metasurface. The full 360∘ phase control of the wavefront is not achieved by using only electric type inclusions. It is achieved in the presence of a conductor backing. Over a spatial period Λ = n d, 2π phase control is obtained. z'

Controlling Rotation of Inclusions over a Spatial Period

Figure (22.33c) shows the 1D array of variable rotation of the identical size of resonating metallic inclusions forming one spatial period, i.e. one supercell. The horizontal and vertical repetition of the array forms another type of the phase-gradient metasurface. It is called the Pancharatnam–Berry (PB) phase-gradient metasurface. The Pancharatnam–Berry phase is associated with polarization conversion. It is realized by the anisotropic subwavelength inclusions of identical geometry (size and shape) parameter; however, with spatially varying orientation (rotation) of inclusions. The full 2π, i.e. 360∘ phase control of the reradiated wavefront has been achieved by the PB phase-gradient metasurface. The response of a rotated anisotropic inclusion, in terms of the Jones matrix of the anisotropic inclusion without rotation, is presented below. The Jones matrix [J] of the inclusion or metasurface signifies its transmission/ reflection coefficient matrix, i.e. [τ]/[Γ] matrix. It is discussed in subsection (4.6.4) of chapter 4.

The Response of Rotated Anisotropic Inclusion

Figure (22.28a) considers normally incident plane EMwaves incident on an inclusion (scatterer) located in the (y–z) plane. The inclusion is realized by a narrow aperture cut in the metallic sheet. The normally incident wave on the aperture inclusion is moving away from an observer in the negative x-direction. The Jones matrix of inclusion, without rotation, is given by equation (4.6.14) of chapter 4. For simple anisotropic inclusion, the crosscoupling terms are zero, i.e. Jyz = Jzy = 0. Figure (22.28b) considers the incident plane wave of any polarization incident on a rotated inclusion with an angle θ. On using the identities cos2θ = (1 + cos 2θ)/2 and sin2θ = (1 − cos 2θ)/2 with equation (4.6.24) of chapter 4, the rotated Jones matrix [J(θ)] of the rotated anisotropic inclusion is given by the following equation:

z θ y

y Metallic screen (a) Aperture inclusion (scatterer) in metallic screen. Figure 22.28

Metallic screen (b) Rotated aperture inclusion at θ.

An inclusion scattering the incident waves.

22.5 Modeling and Characterization of Metsurfaces

Janiso θ

=

1 2

Jyy + Jzz + Jyy − Jzz cos 2θ

Jyy − Jzz sin 2θ

Jyy − Jzz sin 2θ

For inclusion without rotation,

Janiso θ = 0

=

For the RHCP waves’ incident at rotated anisotropic inclusion, the above expression of Jones matrix is used to compute the transmitted waves. In the process, the

tra

= τaniso θ

E Etra y Etra z Etra y Etra z tra

E

0

Jzz

b

Jones matrix is treated as the transmission matrix [τaniso(θ)] of the metasurface.

1

inc

, For incident RHCP wave E

τyy + τzz + τyy − τzz cos 2θ

=

j

inc

E0

τyy − τzz sin 2θ

1

τyy + τzz + τyy − τzz cos 2θ

j

=

1 2

=

inc 1 τyy + τzz + τyy − τzz e E0 2 j τyy + τzz − j τyy − τzz ej2θ

τyy − τzz sin 2θ

22 5 26

inc

E0

j2θ

inc τyy + τzz τyy − τzz ej 2θ E0 + + 2 2 inc τyy + τzz τyy − τzz ej 2θ − E0 zj 2 2

= tra

E

0

tra = yEtra y + zEz = y

tra

E

Jyy

in

E

a

Jyy + Jzz + Jzz − Jyy cos 2θ

22 5 27

inc inc τyy + τzz τyy − τzz j2θ y + jz E 0 + e y − jz E0 2 2 inc inc τyy + τzz 1 τyy − τzz j2θ 1 = E0 E0 e + 2 2 −j j

In − phase RHCP For inclusion without rotation

τaniso

a

LHCP with leading phase 2θ τyy 0 θ=0 = 0 τzz

In the present case, the normally incident wave on the aperture inclusion is moving away from the observer in the (−x)-direction. Therefore, in expression (22.5.27), the column matrices have (+j) for the RHCP and (−j) for the LHCP waves. The present description deviates from the discussion given in subsection (4.6.2) of chapter 6. Equation (22.5.27) is rearranged using Jones vectors. Expression (22.5.27) shows that the transmitted waves constitute the combined circular polarizations, i.e. it has in-phase co-polarized RHCP component and the cross-polarized LHCP component with a leading 2θ Pancharatnam–Berry phase, i.e. the twice the rotation angle of the inclusion. The transmitted wave is traveling away

b

from an observer. Thus, a supercell of n members of rotated inclusions, with variable rotation angles in the range of 0∘ ≤ θ ≤ 180∘, can control the wavefront of transmitted waves from 0∘ to 360∘. However, the copolarized component has to be suppressed and power has to be transferred to the cross-polarized wave to enhance the transmission efficiency in the desired direction. Expression (22.5.27) can be rearranged and the information of the LHCP incident waves also added in the following illustrative form: tra

E

+ tra

E−

inc

τ+ = τ−

+ +

τ+ − τ− −

E

+ inc

E−

22 5 28

869

870

22 Metamaterials Realization and Circuit Models – II

In expression (22.5.28), the signs (+/−) with the field vectors show the RHCP/LHCP incident waves, respectively. The transmission coefficient components with signs (++/−−) show the copolarized RHCP/LHCP components, whereas the cross-polarized τ with signs (+−/−+) shows the polarization conversion from the LHCP to RHCP, and from the RHCP to LHCP wave after the transmission. The off-diagonal matrix elements are maximized, i.e. the cross-polarized transmitted wave is maximized to achieve the wavefront control of the transmitted wave with high efficiency. The above discussion is also applicable to the anisotropic reflective inclusion or an anisotropic metasurface. The reflected waves from the rotated inclusion could be written as follows: ref

E

+ ref

inc

=

E− also

Γ y−z =

Γ+

+

Γ+

Γ−

+

Γ− −

Γyy

0

0

Γzz

−

E

+ inc

a

E− b 22 5 29

Equation (22.5.29b) provides the reflection coefficient matrix of the anisotropic inclusion/metasurface, without rotation, located in the (y–z)-plane. The reflected waves again have both the copolarized and crosspolarized wave components. The polarization conversion ratio (PCR) shows the efficiency of the metasurface as a polarization converter, for the transmission and reflection mode operation. It is defined as follows [J.57]: Transmission mode PCR + = PCR − = Reflection mode PCR + = PCR − =

τ2+

τ2+ + +

+ τ2− −

τ2− − + τ2− −

τ2+ + Γ2+ Γ2+

a

explained with the help of generalized Snell’s laws discussed in next section (22.5.4). 22.5.4

The generalized Snell’s laws for the metasurface are an extension of the classical Snell’s laws applicable to the material surface without any phase discontinuity. The classical Snell’s laws explain only in-plane reflection and refraction. It is discussed in section (5.2) of chapter 5. However, the metasurfaces are phase discontinuous interface and follow the generalized Snell’s laws. These laws are obtained by conserving the wavevectors across the boundary of the metasurfaces. The laws are obtained more rigorously by using Fermat’s principle of least time, formulated as the principle of stationary phase [J.56, J.56, J.60]. These laws explain the anomalous reflection and refraction occurring at the metasurfaces. The 2D metasurfaces can provide negative refraction, as well as the negative reflection, whereas the 3D bulk metamaterial slabs provide only negative refraction. It is discussed in subsection (5.5.2) of chapter 5. Figure (22.29a) shows the 3D reflection and refraction (transmission) of obliquely incident plane wave lying in the (x–z)-plane. The incident wave strikes at the phasegradient metasurface located in the (x–y)-plane. The wavevector components in three mutually perpendicular planes are as follows:

•

+

+ Γ2− −

(x–z)-plane: It is the plane containing the incident inc

wave with the incident wavevector k = n1 k 0 , where n1 is the refractive index of the upper halfinc

•

Γ2+ + 2 + + Γ− − Γ2− −

Generalized Snell’s Laws of Metasurfaces

space, i.e. the medium #1. The wavevector k subtends an angle θinc with respect to the z-axis. (x–y)-plane: It accommodates the phase gradient metasurface with the phase gradients dϕ/dx along the x-axis, and dϕ/dy along the y-axis. These phase gradients are added to the x- and y-components of inc

the incident wavevector k at the surface of the metasurface. Thus, the modified components of the incident wavevector are written as

b 22 5 30

The above analysis has been carried out for the anisotropic metasurface without rotation and without crosscoupling terms. However, it can be applied to the metasurfaces with cross-coupling terms also. Such metasurfaces provide asymmetric transmission, i.e. the wave transmission from the left-hand side to right-hand could be different from the wave transmission from the righthand side to the left-hand side [J.58]. The unique reflection and transmission (refraction) of the incident waves from the metasurfaces are beautifully

dϕ , dx dϕ dϕ = + dy dy

inc kinc mod,x = kx +

a

inc kinc mod,y = ky

b 22 5 31

In equation (22.5.31b), kinc y =0

as the incident

inc

wavevector k

•

is in the (x–z) plane.

(y–z)-plane: It is the orthogonal plane containing the out-of-plane reflected and refracted (transmitted)

22.5 Modeling and Characterization of Metsurfaces

z

kref

z

Medium #1 n1

θinc ϕref

P

y

θref

(x–z) plane

(x–y) plane

Reflected wave

θtra

(x–y) plane

Refracted wave ktra

ϕtra

x

Medium #2 n2

O

y

P1 Plane of metasurface x

ky

kr

(x–y) plane

z y

(b) Wavevector diagram.

(a) Anomalous reflection and refraction in 3D-space.

Out-of-plane anomalous reflection and refraction at the phase-gradient metasurface.

waves in the upper-half and lower-half of the space. The lower half-space is medium #2 with refractive index n2. The reflected and transmitted wavevectors in media #1 ref

and #2 are k

ϕ1

x

P2

ϕ2

kx

Metasurface

Figure 22.29

k θ

(y–z) plane

Incident wave

Orthogonal plane

kinc

Plane of incidence (x–z) plane

x − y − plane kix = ki sin θi cos ϕi1

a

kiy = ki sin θi sin ϕi1

b

tra

= n1 k o and k

= n2 k o , respectively.

22 5 32

ref

The reflected wavevector k subtends an angle of reflections θref with the z-axis. Its projection in the (x–y)-plane has an angle ϕref = ϕ1 with respect to the x-axis. The reflection angle ϕref = ϕ2 with respect to the z-axis in the (z–y)-plane is also considered. The

y − z − plane kiy = ki cos θi sin ϕi2

a

kiz = ki cos θi cos ϕi2

b 22 5 33

tra

refracted (transmitted) wavevector k = n2 k o subtends an angle of refraction (transmission) θtra with the z-axis and ϕtra = ϕ1 with the x-axis in the (x–y)-plane. Its projection in the (y–z)-plane subtends ϕtra = ϕ2. The angles of wavevectors could be seen with better clarity on the wavevector diagram. Figure (22.29b) shows the wavevector diagram of arbitrary wavevector k with its components in the (x–y)-plane, i.e. in the i

plane of the metasurface. The wavevector k = k (i = inc, ref, tra) could be used for the incident, reflected, and transmitted waves. The angle of the wavevector OP = k subtends the angle θ = θi (i = inc, ref, tra) with the z-axis. In the planes – (x–y), and (y–z), its projections O P 1 and O P 2 subtend angles ϕ1 = ϕi1 , ϕ2 = ϕi2 with the

i

x-axis and z-axis, respectively. The components of the k in the (x–y) and (y–z) planes could be written from Fig (22.29b):

i

In this discussion, the incident wavevector k is in the (x–z) plane. The reflected and transmitted (refracted) plane waves are in the upper-half medium #1 and lower-half medium #2 respectively, as shown in Fig (22.29a). The components of the wavevectors on the metasurface located in the (x–y) plane are obtained as follows:

• •

Incident wavevector in the (x–z)-plane, ϕinc 1 = 0: inc kinc x = n1 k0 sin θ

a,

kinc y =0

b 22 5 34

Reflected wavevector in the (x–y)-plane: ref kref x = n1 ko sin θ

kref y

= n1 ko cos θ

ref

a sin ϕ

ref

b 22 5 35

871

872

22 Metamaterials Realization and Circuit Models – II

Equation (22.5.35) for kref is obtained from y ref equation (22.5.33a) and the angle ϕref is the 2 =ϕ out-of-plane angle of reflection in the (y–z)-plane.

•

Refracted wavevector in the (x–y)-plane: tra ktra x = n2 ko sin θ

ktra y

= n2 ko cos θ

tra

a sin ϕ

tra

b 22 5 36

tra is the out-of-plane angle of refracThe angle ϕtra 2 =ϕ tion in the (y–z)-plane At the plane of the phase-gradient metasurface, the xand y-components of the modified incident wavevector inc

k mod are obtained and (22.5.34):

from

dϕ dx

inc kinc + mod,x = n1 k0 sin θ

a,

equations

kinc mod,y =

(22.5.31) dϕ b dy 22 5 37

The generalized Snell’s laws of anomalous reflection and refraction are obtained from the phase matching of both x-and y-components of the modified incident wavevector, in the (x–y)-plane of the metasurface, with the components of the wavevectors of the reflected and refracted waves. Anomalous Reflection Law

Using equations (22.5.35) and (22.5.37), the phase matching of reflected and incident wavevectors provides the anomalous reflection laws: kref x

=

kinc mod,x ,

kref y

a,

=

kinc mod,y

1 dϕ sin θref − sin θinc = n1 ko dx 1 dϕ cos θref sin ϕref = n1 ko dy

b c d 22 5 38

Anomalous Refraction Law

Again, using equations (22.5.36) and (22.5.37), the phase matching of the refracted and incident wavevectors at the plane of the metasurface, the anomalous refraction laws are obtained: inc ktra x = kmod,x

a,

inc ktra y = kmod,y

1 dϕ n2 sin θtra − n1 sin θinc = ko dx 1 dϕ tra tra cos θ sin ϕ = n2 ko dy

b c d 22 5 39

Equations (22.5.38c, d) are the generalized Snell’s laws of reflection. Equations (22.5.39c, d) are the generalized Snell’s laws of refraction. The phase-gradient dϕ/dx is responsible for the in-plane, i.e. in the plane of incidence, anomalous reflections, and refraction over complete upper and lower halves space respectively. The phase-gradient dϕ/dy is responsible for the out-of-plane, i.e. the 3D anomalous reflections and refraction. In case dϕ/dy = 0, the anomalous reflection/refraction laws are limited to the in-plane anomalous reflection/refraction laws. The phase-gradient could be both positive and negative. The negative phase gradients are responsible for the negative reflection and negative refraction. For both dϕ/dx = dϕ/dy = 0, the generalized laws are reduced to the classical Snell’s laws. The generalized Snell’s laws have been confirmed experimentally [J.49, J.50, J.56, J.59, J.60]. These are illustrated in section (22.6.1). The linear phase variations, i.e. ϕ(x) = ξx x + ϕ0 wand ϕ(y) = ξy y + φ0 provide the mechanism to deflect the incident plane waves into the desired direction. The plane waves could be linearly or circularly polarized. Likewise, complete transmission of the incident waves, without reflection, in any direction is possible. It is achieved with Huygens surfaces [J.54]. By using the quadratic phase profile ϕ(x, y)= 2π λ x2 + y2 + f 2 − f , the flat lens has been realized with focal length f. Several other wavefront controls have been obtained using the metasurfaces [J.49, J.51, J.56]. 22.5.5

Surface Waves on Metasurface

The above discussion has explored the ability of the metasurface to control the wavefront of free space waves (FSW). The metasurface also supports the SW propagation that is also controlled by tailoring the inclusions of metasurfaces. The surface-waves are nonradiating bounded modes. It is discussed in section (7.5) of chapter 7. However, under a certain condition, they can become radiating leaky-waves. Thus, there is a coupling mechanism between the FSW to the surface waves, i.e. the FSW–SW coupling, and also SW–FWS coupling. This subsection discusses the surface waves on both the uniform metasurface and nonuniform metasurface [J.50, J.61].

Surface Waves on Uniform Metasurface

In this case, there is no phase gradient on the uniform metasurface. It is just like a homogenized EBG surface without any bandgap. However, the DNG type metasurface, supporting the backward wave can have a bandgap from zero to cut-off frequency commencing the surface wave. Normally, in the microwave range, the uniform

22.6 Applications of Metasurfaces

metasurface is a 2D periodic arrangement of the metallic inclusions at the surface of a dielectric sheet. Its 2D circuit model, shown in Fig (20.15c), is discussed in section (20.4) of chapter 20. Equation (20.4.20) of chapter 20 presents the 2D dispersion relation that gives both the propagating surface waves and nonpropagating Bloch mode. Under the homogeneity condition, i.e. for electrically short line section kd < < 1 and also for a subwavelength period of the unit cell βxd < < 1, βyd < < 1, equation (20.4.20) is reduced to β2x + β2y =

k−

1 Z0 ωCd

2k −

Z0 ωLd

The metasurfaces with unidirectional phase gradient and matching of the wavevectors can couple free-space waves to surface-waves without re-radiation. The propagation constant of the surface-waves is in the bound mode for the condition β > k0. However, for β < k0, the metasurfaces are in the radiation mode, supporting the Cerenkov type leaky-wave radiation. The Cerenkov radiation is explained in subsection (5.5.7) of chapter 5. These conditions could be realized by the proper design of inclusions. Moreover, the inclusions also control the surface impedance of the metasurfaces. By creating the holographic impedance surface, the holographic leaky-wave antenna has been designed [J.63–J.65].

22 5 40 The homogenized metasurface is described by the equivalent permeability and equivalent permittivity obtained below. Let us take βx = β cos θ, and βy = β sin θ. The wavevector β of the surface wave, in the (x–y)-plane, subtends an angle θ with the x-axis. Using characteristic impedance Z0 = μ ε, and phase velocity v = 1 με of the host medium, the above dispersion relation of surface-wave can be expressed in the 2D effective medium: β = ω μeq εeq where μeq =

μ−

1 ω2 Cd

a b , εeq =

2ε −

1 ω2 Ld

c

22 5 41 Relation (22.5.41) shows that the homogenized metasurface is isotropic and homogeneous. The loading series capacitor can provide negative magnetic susceptibility and loading shunt inductor can provide negative electric susceptibility. In this case, the metasurface supports the bound backward wave propagation. At a certain higher frequency, both quantities could be positive numbers supporting the usual forward wave propagation. Of course, in some frequency bands, one of the quantities could be a negative number, and a nonpropagating band is created [J.62].

Surface Waves on Nonuniform Metasurfaces

The effective medium concept is only conceptual. The metasurfaces can have spatial variation. The metasurfaces are useful to realize the mode conversion between two bound surface modes, or for the conversion between the bound SW and radiating FSW. Such characteristics are obtained by the nonuniform metasurfaces, tailoring their phase gradient [J.61].

22.6

Applications of Metasurfaces

The metasurfaces have been used to develop a wide variety of components and devices from microwave to optical frequency ranges. This section covers a few examples of metasurfaces to illustrate some of their unique characteristics, such as the anomalous reflection and refraction exhibiting bending of the EM-waves in the desired direction, complete transmission of waves using the concept of Huygens’ surfaces, designing anisotropic metasurfaces, and showing their ability to convert the polarization states. 22.6.1 Demonstration of Anomalous Reflection and Refraction of Metasurfaces The 1D and 2D linear phase-gradients of gradient metasurface (GM) are controlled to get the anomalous reflection and refraction. Both the reflection type and transmission type GM are discussed below. Reflection Type Metasurfaces

The equal amplitude reflective gradient metasurface has been designed by controlling the phase-gradient of the conductor backed periodically arranged microstrip square inclusions. The metallic patch shows electric polarization. However, due to the conductor backing, it has a magnetic response also and the reflection phase can be controlled over 0–2π in the plane of incidence for the 1D GM. However, 2D phase-gradients provide the out-of-plane reflection also. 1D Phase-gradient Metasurface

The gradient metasurface is realized by n numbers of unit cells with different geometrical parameters of the metallic inclusions without any rotation.

873

22 Metamaterials Realization and Circuit Models – II

θref = sin − 1 ξx k0 ,

Figure (22.30a) shows one supercell of n = 6 numbers of inclusions of the square microstrip patch resonators of varying sizes. Each inclusion is within a unit cell of period dx = dy = d. The periodicity, i.e. length of the supercell is Λ, i.e. Λ = n × d. Figure (22.30b) shows that isotropic metasurface is formed by the repetition of the supercells, both horizontally and vertically, on a substrate with conductor backing. The FR-4 substrate (εr = 4.3, tan δ = 0.025) of thickness h = 1.0 mm is used to design the metasurface. It is located in the (x–y)-plane and the EM-wave with its wave vector k0 is normally incident on it, i.e. θinc = 0. The incident wave is reflected at an angle θref. The xdirected phase gradient of the metasurface is ξx = dϕ/dx. Following equation (22.5.38c), the angle of reflection in the air medium (n1 = 1) is

dy y z

ξx < k0

a

ξx = k0 sin θref

b 22 6 1

The n numbers of unit cells of period d are used to achieve linear discrete 2π phase variation over a supercell of the space periodicity Λ. The phase gradient is obtained below: Phase variation over one unit cell Δϕx = Δϕ = 2π n dϕ Δϕ 2π ≈ = Phase gradient ξx = ξ = dx dx nd

a 5.5 mm

7.3 mm

7.6 mm

7.8 mm

8.1 mm

Incident waves

9.0 mm

Super cell (Λ = 60 mm)

Reflected waves

(a) Supercell of 6 unit cells with square patches.

at 8.0 GHz

θref = 39°

Gradient metasurface 5×5 Super cells (300 mm × 300 mm)

Z y X

x

Gradient metasurface

(c) Gradient metasurface of 5 × 5 supercells [J.66].

(b) Gradient metasurface of 5 × 5 supercells.

8.4 GHz

0

–5 –10 Reflection (dB)

–5 –10 –15 –20 –25 7

8 8.4 9 10 Frequency (GHz)

12

(d) Measured reflectivity of the grad. metasurface [J.66]. Figure 22.30

–20 –30

–40 11

b

The angle of reflection, controlled by the phasegradient of the metasurface, is obtained from equations (22.6.1) and (22.6.2):

a x

a

22 6 2

Unit cell

dx = 10 mm

Refletivity (dB)

874

7

8

9 10 Frequency (GHz)

11

12

(e) Measured reflected power at θ tra = 40° [J.66].

Anomalous reflection of the 1D gradient metasurface. Source: Wang et al. [J.66]. © 2014, IEEE.

22.6 Applications of Metasurfaces

2π k0 nd

θref = sin − 1

λ0 nd

= sin − 1

= sin − 1

λ0 , Λ 22 6 3

where λ0 is the wavelength of the incident wave. A gradient metasurface using the variable size of n = 6 numbers of square microstrip patches is examined at 8.0 GHz for the unit cell period of 10 mm [J.66]. The variable patch sizes are marked below patches of Figure (22.30a). The 6 numbers of unit cells constitute one supercell of length (period) Λ = n d = 60 mm. The length of the metasurface in the x-direction is 10Λ = 600 mm. The width of the metasurface in the y-direction is 2Λ = 120 mm. At 8.0 GHz, i.e. λ0 = 37.5 mm, the discrete phase change by each patch is 2π/n = 360∘/6 = 60∘. The theoretical angle of reflection from equation (22.6.3) is θref = sin−1(37.5 mm/60 mm) ≈ 38.7∘. The EM-simulation, shown in Fig (22.30c), confirms the theoretical result of the generalized Snell’s reflection law with the angle of reflection 39 . Finally, the metasurface is fabricated on the FR-4 substrate. Its size is 5Λ × 5Λ = 300 mm × 300 mm. Figure (22.30d) shows that at 8.4 GHz its reflectivity is −20 dB in the direction of normal. Figure (22.30e) experimentally validates the angle of reflection θref = 40∘. 2D Phase Gradient Metasurface

The anisotropic gradient metasurface with phase gradients ξx = dϕ/dx and ξy = dϕ/dy in the x- and y-directions, respectively, facilitates an arbitrary control of reflection direction in the 3D upper space. It has both the in-plane and out-of-plane reflections. Figure (22.29a) shows the in-plane angle of the reflection angle θref in the (x–z)plane. It also shows the out-of-plane reflection angle ϕref in the (y–z)-plane. Following the wavevector diagram of Fig (22.29b) and using equation (22.5.37), both angles of reflections are written as kr = k0 sin θref = ξ2x + ξ2y tan ϕref =

kinc mod,y kinc mod,x

=

1 2

a

ξy ξx

b 22 6 4

Following equation (22.6.2), both phase-gradients are computed. The above expressions are rewritten as follows: θ

ref

= sin

−1

ϕref = tan − 1 where, ξx =

2π nx d x

λ0

1 nx dx

2

+

nx d x n y dy c,

1 2

1 ny d y

2

a b

ξy =

2π n y dy

d 22 6 5

In equation (22.6.5), nx and ny are the numbers of unit cells in the x- and y-directions, respectively. Likewise, dx and dy are the periods of unit cells in the x- and ydirections, respectively. In the case of anisotropic metasurfaces, these are unequal quantities. However, for the isotropic metasurface has nxdx = nydy = nd. The above equations are reduced to θref = sin − 1

2λ0 nd

a,

ϕref = 45∘

b 22 6 6

For the 10Λ × 10Λ = 600 mm × 600 mm gradient metasurface at 8.0 GHz, the computed reflection angles are θref = 62.12∘ and ϕref = 45∘. The experimentally obtained angles θref = 61∘ and ϕref = 45∘ validate the general Snell’s laws for the out-of-plane reflection [J.66].

Transmission Type Metasurfaces

Figure (22.31a) shows the multilayered 1D isotropic gradient metasurface to deflect the normally incident plane waves as refracted (transmitted) waves in a predetermined direction. The metasurface structure is constructed with the metallic perforated layer separated by the dielectric layers. It is designed using the PCB of thickness h = 1.575 mm, εr = 2.65, and the metal layers have thickness t = 0.035 mm. The perforations are electrically polarizable coaxial annular apertures in all three layers and aligned properly [J.70]. The annular apertures are periodically arranged in the (x–y)-plane with nine columns as shown in Fig (22.31b) with 12 rows. Only three rows are shown in Fig (22.31b). A thin metallic film perforated with a subwavelength array of apertures exhibits extraordinary transmission due to the localized resonances. The multilayer arrangement can enhance the transmission over a wider bandwidth [J.67–J.69]. In this case, each aperture is located within a square lattice unit cell of periodicity dx = dy = d = 12 mm. The n = 9 numbers of unit cells constitute one supercell of the period Λx = 9 × 12 = 108 mm along the x-axis. Likewise, in the direction of the y-axis, the supercell period is Λy = 12 × 12 = 144 mm. To achieve, the in-plane full control of the transmitted waves over 0 − 2π phase along the x-axis, n = 9 apertures should have a phase difference Δϕ = 2π/9 ≈ π/4. The x-directed phase gradient obtained from equation (22.6.2) is ξx = dϕ/dx = Δϕ/dx = π/4d. There is no phase gradient along the y-axis, i.e. ξy = dϕ/dy = 0. The gradient metasurface is located in the air medium, i.e. n1 = n2 = 1. For the normally incident plane waves with θinc = 0, equation (22.5.39c) provides the angle of in-plane refraction:

875

22 Metamaterials Realization and Circuit Models – II

Normal

Refracted waves

Perforated metal sheet H y

E x

H

y z

x p (b) 9 columns of annular ring apertures [J.70].

×π

(a) 3 metallic perforated layered metasurface. 0.0

0.8

–0.5

0.6 –1.0 0.4 –1.5

1

2

3 4 5 6 7 8 Inclusion index number (i)

θtra = sin − 1

140

100

80

60 40

Measurement

160

20

2

180

0

1 200 220

–2.0 9

(c) Transmission amplitudes and phases of each inclusion. Phase diff. between inclusions π/4 [J.70]. Figure 22.31

120

Transmission phase (rad)

1.0

0.0

Rn

r

Incident waves

0.2

E

E

Dielectric spacer sheet

k

H

k

ri

z

k

Ctra

R

Transmission amplitude

876

1. Without metasurface 2. With metasurface 240

260

280

340 320

300

(d) Radiation pattern of incident waves in the air medium without metasurface and refracted (transmitted) waves at θ tra = 18° [J.70].

Anomalous refraction (transmission) of the 1D multilayer gradient metasurface. Source: For the use of figure 1b from Wei et al. [J.70]. © 2013, The Optical Society.

λ0 ξ 2π x

a,

ϕtra = 0, as ξy = 0

b 22 6 7

The operating frequency of the incident wave is at 10 GHz, i.e. λ0 = 30 mm. Therefore, in the present case, the theoretically calculated angle of refraction is θtra = sin−1[(30/2π) (π/4 × 12)=18.2 . To design the phase-gradient metasurface, 9 numbers of annular apertures are tuned to achieve the phase difference π/4 between the adjacent elements. The outer radius of the aperture is fixed at R = 5.8 mm and the inner radii of nine apertures are tuned at the EM-

simulator to achieve the phase difference π/4. The tuned radii in mm are as follows: 1.98, 2.76, 3.66, 4.26, 4.56, 4.76, 4.86, 4.94, and 5.08 [J.70]. In the process, the transmission amplitude of each aperture is maintained above 0.8. Figure (22.31c) shows the amplitude response and the x-directed linear phase response of a supercell. The FDTD simulation of the gradient metasurface with one period along the y-axis, using the periodic boundary condition, provides the refraction at 18 with respect to the z-axis. It has a high transmission efficiency. The metasurface of size 108 mm × 144 mm has been illuminated with normally incident plane waves. Figure (22.31d) experimentally validates the

22.6 Applications of Metasurfaces

angle of refraction at 18 of the normally incident waves working at 10 GHz. The experimentally obtained transmission efficiency is 65%. The polarization-sensitive multilayer metasurfaces have also been developed to get the full control of both the reflected and transmitted waves simultaneously [J.71]. The strategy to improve the transmission efficiency of the metasurface is discussed in section (22.6.2).

22.6.2 Reflectionless Transmission of Metasurfaces The concepts of Huygens metasurfaces and Fabry–Perot resonance are used to enhance the transmission efficiencies of metasurfaces.

Huygens Metasurfaces

The electric type metasurfaces, exclusively based on the electrically polarizable inclusions, do not provide impedance matching with free space. Therefore, reflection occurs at the surface planes of the metasurfaces and their transmission efficiency degrades. However, the Huygens’ metasurfaces, based on both the electric and magnetic type polarizable inclusions, provide such impedance matching by controlling the ratio of the imaginary parts of the surface impedance Zms and surface admittance Yes. Equation (22.5.23) states the matching condition. The polarizabilities of both electric and magnetic inclusions are located at the same location on the metasurfaces. These inclusions are simultaneously controlled to achieve the matching by tuning their dimensions. Moreover, to transmit (refract) the waves in the desired direction, an appropriate phase gradient is also created by varying the dimensions of the inclusion. Figure (22.32a) shows a unit cell consisting of electrically and magnetically polarizable inclusions. The upper face of the substrate, Rogers RO4003 (εr = 3.55, tan δ = 0.0027), supports the capacitively and inductively loaded conducting strips. It is electrically polarized by a y-directed electric field, creating Yes. The lower face of the substrate supports the SRR (capacitively-loaded loop). It is magnetically polarized by the z-directed magnetic field, creating Zms [J.54]. Figure (22.32b-i) shows a fabricated (226 mm × 226 mm) metasurface in the (y-z)- plane at 10 GHz. It is composed of a vertical stack of 58 identical PCB strips, with 2.35 mm air gaps between PCB strips. The Huygens’ metasurface is developed by the periodic repetition of supercells. One period length Λy, in the direction of the y-axis, is shown in Fig (22.32b-ii) for both electric and magnetic inclusions. Using equations (22.5.39c) and (22.6.2), the period length Λy of the metasurface

in the air medium is computed from the specified angles of incidence and transmission: sin θtra − sin θinc =

1 dϕ λ0 λ0 = ≈ Λy ko dy nd

22 6 8

The TM polarized EM-wave, at 10 GHz (λ0 = 30 mm), is normally incident at the metasurface from its bottom. The intended transmission is required at the refraction angle θtra = 45∘. Equation suggests a period of the supercell Λy = 42.36 mm. The supercell is composed of n = 12 numbers of unit cells of d = 3.53 mm size. The supercell covers 0–2π phase variation over its length. Each inclusion provides the phase change Δϕ = π/6. The inclusions of Fig (22.32a) are tuned on the EM-simulator to get around unity transmission and needed phase change. The transmission coefficient τ and reflection coefficient (Γ) of each inclusion pair are also determined from the EM-simulator. Equations (22.5.20c, d) are used to compute complex Yes and complex Zms of each of 12 numbers of unit cells forming a supercell of the Huygens’ metasurface. Figure (22.32c) shows the computed real and imaginary parts of the complex Yes and Zms. For a typical unit cell, shown in Fig (22.32a), the values are Yes = (0.02 + j3.14)/λ0) and Zms = (0.07 + j2.3)λ0). Figure (22.32d) shows the EM-simulated refraction at 45 for the normally incident plane waves at the metasurface. It confirms the design. Finally, Fig (22.32e) shows the transmission efficiency of the fabricated (226 mm × 226 mm) metasurface. Some refraction takes place in the undesired directions also. The peak transmission efficiency is 86% at 10.5 GHz, and the 3 dB bandwidth is 24.2%. Metasurfaces Using Fabry–Perot Resonance-Based Inclusions

The inclusion, i.e. the scatterer for the design of a metasurface has to be carefully selected to achieve nearly complete transmission of the incident waves. Next, the Panchratanum–Berry rotational arrangement of inclusions, using an identical number of inclusions, is adapted to refract the beam in the desired direction. Both aspects are discussed below. Choice of Multilayered Inclusion

Nearly total transmission is achieved by a multilayered C-shaped electric type inclusion, i.e. the split ring resonator (SRR) with a gap g, shown in Fig (22.33a-i). The Cshaped inclusions and the metasurface constituted by their periodic arrangement, shown in Fig (22.33c), are located in the (y–z)-plane. The multilayered inclusion is formed by coaxially arranged 4 numbers of C-shaped metallic rings separated by the dielectric spacers of

877

22 Metamaterials Realization and Circuit Models – II

Λy = 4.3 cm

6.00 mm y z 3.90 mm

(i) Huygens’ metasurface. 1.54 mm

1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6 7 8 9 10 11 12 x

3.53 mm

y

z

(ii) Supercell of electric and magnetic inclusions.

(a) Unit cell of electric and magnetic inclusions.

Zms/η0 and Yes η0

(b) Fabricated metasurface structure at (λ0 = 30 mm) showing one supercell.

Hz

Re{Yes} Im{Yes} Re{Zms} Im{Zms}

Normal

1.2

ϕ Refracted waves

0.6 0

x Incident waves

–0.6 –0.4 –0.2 0.0 0.2 0.4 0.6 y/λ0 i.e. (n d/λ0) (c) Complex surface impedance/admittance of each inclusion of the supercell. Transmission efficiency (%)

878

–0.6 y –1.2

(d) Simulated refraction of normally incident plane waves.

100 Measured Simulated

80 Refracted Direction

60 40

Undesired Directions

20 0 9

9.5

10

10.5

11

11.5

12

Frequency (GHz) (e) Transmission efficiency of Huygens’ metasurface. Figure 22.32

Anomalous refraction of Huygens’ gradients metasurface. Source: Pfeiffer et al. [J.54]. © 2013, American Physical Society.

22.6 Applications of Metasurfaces

1.0 p

y y' d

y Ey Hz

g θ r R d

k0 iii x z

z

sim

τLHCP

mea

τLHCP

0.6 mea

τRHCP 0.4

sim

τRHCP

0.2 ii 0.0 8

(a) C-shaped SRR and its multilayer inclusion.

9

10 11 Frequency (GHz)

(b) Measured and simulated transmission of LHCP and RHCP waves for normally incident RHCP waves [J.72]. 1.0

–180

LHCP

d d

Transmission

0.9 Super cell Λ

–120

0.8

–60 LHCP

0.7 0.6

0.4 z (c) Multilayered gradient metasurface [J.72]. Figure 22.33

0

RHCP

60

0.5

y

12

120 1

2

3 4 5 Unit cell (n)

6

Phase shift (deg.)

p

0.8 z'

Transmission

p

180

(d) Amplitude and phase shift of 6 unit cells (n = 1 to 6) under normally incident RHCP and LHCP waves [J.72].

Transmission of normally incident RHCP waves on the multilayered gradient metasurface. Source: From Liu et al. [J.72] Springer Nature. Licensed under CC- BY- 4.0. Public Domain.

thickness p. The multilayered inclusion is located within a square lattice of period d. The y-polarized EM-wave, propagating in the x-direction, electrically excites the inclusion. The electric current on the metallic strips and displacement current in the dielectric spacers creates a circulating current loop in the (x–y)-plane. The loop is excited by the Hz-magnetic field component and acts as a magnetically polarized inclusion. At local Fabry–Perot type resonance, the impedance of the multilayered inclusion is matched to free space. It provides nearly total transmission of the normally incident waves. Thus, the present arrangement also acts as a Huygens’ metasurface. Figure (22.33a-ii) shows the inner and outer radii r and R of an inclusion. Further, the inclusion could be rotated by an angle θ with respect to the z-axis. It provides the Pancharatnam–Berry phase angle 2θ for its transmitted crossed-polarized (CP)-wave component. The copolarized component of the transmitted wave has no phase change. It is discussed in section (22.5.3).

The transmission characteristics of a metasurface developed at 11.0 GHz with a (5 × 5)-array of inclusions, without rotation, have been examined both experimentally and through EM-Simulation [J.72]. The fabrication details of the metasurface on the FR4 substrate are as follows: r = 1.5 mm, R = 2.5 mm, g = 0.75 mm, p = 2.5, and d = 6.5. The simulated τsim and experimental transmission τmea responses, between 8.0 and 12.0 GHz for the right-hand CP (RHCP) wave, are shown in Fig (22.33b). At the resonance frequency, the effective length of the strip forming C-shaped inclusion is Leff ≈ λ0/2. The maximum transmission of the cross-polarized LHCP transmitted waves at 11.0 GHz is 90% and over broadband (10.3–11.7 GHz) is above 80%. The transmitted level of the copolarized RHCP wave is below 20%. Further improvement in transmission efficiency and bandwidth has been obtained with low-loss Rogers 5880 substrate. The high transmission efficiency is maintained for the oblique angle of incidence between 0∘ and 60∘.

879

22 Metamaterials Realization and Circuit Models – II

Puncharatnum–Berry Phase Control

Figure (22.33c) shows the gradient metasurface on the FR-5 substrate using the supercell of n = 6 numbers of unit cells. The period of the supercell is Λz = n d. Using equation (22.6.7), the angle of refraction of the normally incident RHCP wave at 11.0 GHz is θtra = sin−1(27.273 mm/6 × 6.5 mm) = 44.37∘. In the case of the Puncharatnum–Berry phasing arrangement by the rotation of inclusion, θ = 0∘ − 180∘ rotation covers full 360 phase control of the cross-polarized LHCP refracted waves. Therefore, in Fig (22.33c) each of six scatterers is rotated by π/d = 180∘/6 = 30∘ with respect to its neighbor. The transmission characteristics of the metasurface of Fig (22.33c) are experimentally examined for the normally incident RHCP wave. The results are shown in Fig (22.34a–d). The measured magnetic field Hz in the (x–z)-plane of Fig (22.34b) shows that the LHCP wave is refracted anti-clockwise at (−45 ) validating the theoretically calculated angle of refraction. Figure (22.34a) shows that the transmission efficiency is about 80%. Figure (22.34c) shows that the normally incident LHCP is refracted clockwise at +45 . In Fig (22.34d), it is

Transmission

880

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0

interesting to note that the normally incident linearly polarized (LP) wave is split into two beams transmitted as the LHCP and RHCP waves at angles θtra = 45∘. The LP wave could be treated as a linear combination of the LHCP and RHCP waves.

22.6.3 Polarization Conversion of Incident Plane Wave The polarization conversions from linear to circular [J.73–J.76], LHCP to RHCP, [J.77] and rotation of linear polarization [J.78] are accomplished with subwavelength thick metasurfaces. The polarization conversions have been achieved with anomalous reflection/refraction in the desired direction. Both the reflection-type and transmission-type metasurfaces are used to get polarization conversions. Furthermore, the metasurfaces are also used as the substrate and superstrate to convert the linearly polarized microstrip antennas and slot antenna into the circularly polarized antenna [J.79, J.80]. The present subsection discusses briefly illustrative examples

Normal mea

τ LHCP

–45° Metasurface LHCP X

–90 –75 –60 –45 –30 –15 0

RHCP –2

15 30 45 60 75 90

–1

0 Z/λ0

Angle of refraction (deg.) (a) In-plane angle of refraction θ tra of LHCP transmitted waves.

2

(b) Incident wave RHCP. Refracted waves LHCP. Normal

Normal –45°

+ 45° Metasurface

+ 45° RHCP

LHCP

RHCP X

X

LP

LHCP –2

–1

0 Z/λ0

1

Metasurface 2

(c) Incident wave LHCP. Refracted waves RHCP.

Figure 22.34

1

–2

–1

0 Z/λ0

1

2

(d) Incident wave LP. Refracted waves divided in LPCH and RHCP.

Experimental refracted Hz-fields of the normally incident waves on a gradient metasurface. Source: From Liu et al. [J.72] Springer Nature. Licensed under CC- BY- 4.0. Public Domain.

22.6 Applications of Metasurfaces

of the reflection-type and transmission-type metasurfaces and their applications to the microstrip patch and slot antenna.

Reflection-type Metasurfaces

There are several possible shapes of the microstrip patch type inclusions (scatterers) to develop high-efficiency reflection-type anisotropic metasurface that converts linearly polarized incident waves to the circularly polarized waves. The behavior of the reflection phase of a metasurface is similar to that of the EBG surface, discussed in subsection (20.1.3) of chapter 20. The phasegradient of metasurface further directs to the CP waves in the desired direction. Figure (22.35a) shows a unit cell of the orthogonal Ishaped metallic patch inclusions – horizontal Iy and vertical Iz inclusion, printed on a conductor backed F4B substrate with εr = 2.65, tan δ = 0.001, h = 2 mm. The structural details of the unit cell are as follows: a = 2 mm, w = 0.4 mm, d = 6 mm. The arm length pz of the Iz-inclusion is taken at 3 mm and length py is varied from 2 to 5 mm [J.74]. Figure (22.35b) shows at 10 GHz phase response ϕz(y) and ϕy(y) of the Ez and Ey polarized normally incident plane waves, propagating in the x-direction. Due to conductor backing, the magnitudes of both the horizontal and vertical polarization reflection coefficients are |Γyy| = |Γzz| ≈ 1. It is noted that the phase response of the reflection phase ϕz(y) is not influenced by the variation in pz. However, the reflection phase ϕy(y) of the Iy-inclusion decreases from 130 to −200 with increasing pz. At a certain length, the inclusion resonates showing zero reflection phase, like an artificial magnetic conductor (AMC). Thus, the reflection phases of the orthogonal I-shaped inclusion can be controlled independently of each other by varying their lengths. The anisotropic reflection-type phase-gradient metasurface is designed with 3 × 3 supercells. Each supercell is composed of n = 12 unit cells of period 6 mm. The top of Fig (22.35c) shows one supercell Λy = n d with varying size of the inclusions to cover full reflection phase 0 – 360 at 10 GHz. The inclusion lengths py and pz are adjusted to achieve the reflection phase difference, i.e. the differential phase Δϕ = |ϕy − ϕz| for each of 12 inclusions at 10 GHz. The adjusted dimensions are as follows: py (mm) = 4.95, 5.1, 5.15, 2.0, 3.7, 4.1, 4.3, 4.45, 4.5, 4.6, 4.75, 4.8; pz (mm) = 2.0, 3.7, 4.1, 4.3, 4.45, 4.5, 4.6, 4.75, 4.8, 4.95, 5.1, 5.15. Figure (22.35c) shows the EMsimulation of the phase responses of y and z-polarized normally incident EM-waves, for three periods of the supercells at 10 GHz. The differential reflection phase

Δϕ = |ϕy − ϕx| is maintained at 90 to achieve circular polarization. For a linearly polarized wave incident at 45 , the circularly polarized wave is obtained after reflection, as the magnitudes of the reflected Ey and Ez field components are nearly unity with 90 phase difference. It is the anomalous reflection with an identical angle of reflection for both polarizations. The angle of reflection, for both horizontal (H) and vertical (V) polarization, is computed from equation (22.6.3) θref = sin−1(λ0/ nd) = sin−1(30/12 × 6) = 24.620. The radiation pattern of Fig. (22.35d-i) experimentally confirms the angle of reflection at 24 for both polarizations. Figure (22.35dii) further demonstrates the experimentally obtained circular polarization. The bandwidth of the reflection-type linear to the circular converter is about 15% (9.5–11.0 GHz) within 1.0 dB axial ratio [J.73]. This metasurface has also been used to rotate the linearly polarized wave, and further also to split the incident waves after reflection [J.73, J.74]. An ultra-wideband LP to CP converter, (4.7–21.7 GHz) within 3 dB axial ratio, has also been developed using the via connected L-type inclusions [J.75]. Transmission-type Metasurfaces

Figure (22.36a) shows a three-layered unit cell, also called the transmit-arrays, of an anisotropic transmission-type metasurface. It is the symmetrical subwavelength thick structure with identical front and back layers, and different middle layer [J.76]. Each layer (i = 1, 2, 3) is characterized by the diagonalized tensor = sheet or surface admittance Y s,i : =

Y s,i =

Yzz s,i

0

0

Yyy s,i

,

i = 1, 2, 3

22 6 9

The surface admittances are imaginary quantities, as the low-loss structure is ideally treated as lossless. The transmission-type metasurface is a Huygens’ surface. There= fore, the final surface admittance Y s at the top level of the structure is a combination of the electric and magnetic type surface admittances arising out of the electric and magnetic polarizabilities discussed in subsection (22.5.2). The condition of complete transmission is given by equation (22.5.23). Both diagonal elements of the admittance tensor can be tuned independently to achieve the matching condition of each unit cells. The tuning also provides the phase difference Δϕ = |ϕy − ϕz| = 90 to get the refracted CP-waves from the incident LP-waves. Moreover, n number of unit cells can form a supercell of Λ periodicity. The unit cells of the supercell can be tuned individually to refract

881

22 Metamaterials Realization and Circuit Models – II

100 d

Ref. phase response ϕz(y) / ϕy(y) (deg)

t

a h pz

py

d z x

w y

ϕz(y)

f0 = 10 GHz 0 ϕy(y) 100

ϕz(y) ϕy(y)

–200 3

2

(a) The unit cell of orthogonal I-inclusion.

4

5

py (mm) (b) Variation in reflection phase response with the length of Iy-inclusion.

2

1

z

3

4

5

6

7

8

9 10 11 12

0 y 100 ϕy(y) / ϕz(y) (deg)

882

0 90° –100 ϕy(y)

–200 0

50

ϕz(y) 100

150

216

y/z (mm) (c) Phase variation along 3 super cells. 120

90 1 0.8

60

60

1

0.6

150

90 1.5

120 30

0.4

150

30

0.5

0.2

0

180

0

180 f =10.0 GHz

210

330

10.0 GHz-H 10.0 GHz-V

240 270

300

330

210 240 270

i. Reflected radiation at 24°

300

ii. Circular polarization

(d) Experimental radiation patterns of reflected waves. Figure 22.35

Anisotropic reflection – type gradient metasurface to convert incident linearly polarized waves into reflected circularly polarized waves. Source: From Ma et al. [J.74], Springer Nature. Licensed under CC- BY- 4.0. Public Domain.

(transmit) the CP-waves in the desired direction following the generalized Snell’s Law. Figure (22.36a) also shows that each layer is a patterned substrate of metallic structures, and separated

from other layers by a substrate and bonding layers. Rogers Ultralam 3850 (εr = 3.19, tan δ = 0.0045, h = 0.05 mm and Rogers 2929 are used as the substrate and bonding film. Figure (22.36b) shows the metallic

Ys = Ys1

= Ys2

= Ys1

βp, ηd

z

y

2929 Bonding Film

0yy

3850 Core 3850 Core

3850 Core 3850 Core

0.018 0.100 0.050 0.050 0.050 0.050 0.050 0.050 0.018

0.2

= Yszz Ys = 0

0.25

22.6 Applications of Metasurfaces

YS1 Hy

Js –

z

z

5.5mm

y x (a)

(c) Induced electric currrent. Figure 22.36

P2

C 0.1

z

P1 L 0.15 y

(i) Outer sheet

P2

L P2

(ii) Inner sheet

(d) Fabricated metasurface.

(b) Outer and inner inclusion sheet. All dimensions in mm. Efficiency, axial ratio (dB)

Hy

P2

0.79

Total: 0.436mm (λ0/9.7)

x

YS2

0.35 0.1

P1

(a) The unit cell of 3 cascaded sheet admittances.

YS1

P1 0.1 P1

6 Measured 5 Simulated 4 Axial ratio 3 2 1 0 –1 Efficiency –2 –3 –4 60 65 70 75 80 85 90 95 100 105 110 Frequency (GHz) (e) Axial ratio and efficiency of the metasurface.

Multilayered anisotropic transmission – type metasurface to convert incident linearly polarized waves into refracted circularly polarized waves. Source: Pfeiffer and Grbic [J.76]. © 2013, IEEE.

patterns (inclusions/scatterers) of the outer and inner layers. For the outer layer centrally located, thin vertical strip L provides an inductive response that is enhanced by 4 numbers of corner located P1-patches. The horizontal strip C is a capacitive stub. Likewise, at the inner layer thin horizontal strip is inductive and the capacitive response is obtained by 4 numbers of P2 patches. The dimensions, shown in Figure (22.36b), provide Δϕ = 90 differential phase to convert the incident linearly polar− k0 x ized wave Einc x = Einc to get the refracted o y ± z e CP wave. Figure (22.36c) shows that the incident time-varying electric fields induce the surface current density Js on the outer layer inclusions. The induced field creates electric polarizability and generates an electric response. The induced surface current also forms a circulating loop current through the displacement current between the inner and outer layers. Figure (22.36c) further shows that the effect of surface currents of inner layers is canceled, and a larger current loop is formed between two outer layers. This current loop is responsible for the

magnetic polarizabilities and magnetic response at the surface of the three-layered metasurface. The [ABCD] transmission matrix has been used to extract the surface admittances from the simulated S-parameters of the structures on the EM-simulator [J.76].

Quarter – Wave Plate

The above-discussed metasurface is used to develop a quarter-wave plate to get the circular polarization from the incoming linearly polarized wave. It is designed at 77 GHz. Figure (22.36d) shows 81.4 mm × 81.4 mm fabricated metasurface with n = 5 supercells of size (Λ = 5.5 mm). Figure (22.36e) shows the simulated and experimentally obtained axial ratio and efficiency performances of the transmission-type metasurface in the W-band. Its peak axial ratio is 3.7 dB at 85 GHz. The cross-polarization discrimination level is 13.6 dB and the transmission efficiency is better than −1 dB.

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22 Metamaterials Realization and Circuit Models – II

at an angle θtra after conversion to circular polarization. It follows that the generalized Snell’s Laws of refraction given by equation (22.5.39c). In the present case, a normally incident 77 GHz LP wave is refracted at an angle θtra = sin−1(3.896/5.5) = 45.1∘. For the incident LP wave, − k0 x , the refracted wave is LHCP/ Einc x = Einc o y ±z e RHCP. Figure (22.37d) shows the variations in the axial ratio and high transmission efficiency between 71 and 84.5 GHz. It shows the wideband high-efficiency transmission of the circularly polarized waves. Further, the metasurface can scan the upper space in the range 51 –42 by changing the frequency between 70 and 80 GHz. Further, the multilayer parabolic phase distributed metasurface has been used to design the flat lens in the infrared range [J.81].

Beam Refracting Gradient Metasurface

Figure (22.37a) shows a three-layered unit cell of different geometry to realize anisotropic transmission-type gradient metasurface to convert the incident LP-waves into the CP-wave. It refracts the waves in a predetermined direction. The working of the unit cell is similar to the previous one. The supercell of length Λ = 5.5 mm is composed of n = 5 numbers of unit cells. It spans the complete 0–2π transmission phase. Figure (22.37b) shows the outer and inner layer of a supercell of variable sizes to design the gradient metasurface. The y-directed transmission phases ϕy, n (n = 1.5) of each of five inclusions, located at the outer and inner layers, are tuned to get a π/2.5 phase angle with respect to each other. The tuning also achieves the matching condition for the complete transmission of incident waves. Similar phase tuning is done for the unit cells along the z-axis. The dimensions of tuned unit cells of a supercell are available in reference [J.76]. Figure (22.37c) shows a 81.4 mm × 81.4 mm fabricated anisotropic gradient metasurface. It also shows that the linearly polarized wave incident, from the bottom side, on the metasurface at an angle θinc is refracted

Z Y

The reflection-type metasurface can be used as a substrate to design a circularly polarized patch antenna using an LP microstrip patch. Figure (22.38a) shows a sandwiched anisotropic metasurface created by the Outer sheets (Ys1)

Wz

Y

Sz

Gy

Circularly Polarized Microstrip Patch Antenna on Metasurface Substrate

Gz

Z

1.1 mm

X

Wy

Sy

z

Middle sheets (Ys2)

0.1 mm

1.1 mm (λ0/3.5)

0.4 mm (λ0/3.5)

y

1.1 mm

(b) Supercell of 5 unit cells.

θtra Circular

z

5.5 mm y

x y θinc

TM

TE

(c) Fabricated metasurface and conversion of incident LP waves to the refracted CP waves. Figure 22.37

Efficiency, Axial ratio (dB)

(a) 3D view of a unit cell and dimensions of inner and outer layers.

Grad. metasurface

884

4 3 2 1 0 –1 –2 –3 –4 –5 –6 70

Measured Simulated

Axial ratio

Efficiency

80 75 Frequency (GHz)

85

(d) Axial ratio and efficiency of a grad. metasurface.

Multilayered anisotropic transmission – type gradient metasurface to convert incident linearly polarized waves into circularly polarized refracted waves in the desired direction. Source: Pfeiffer and Grbic [J.76]. © 2013, IEEE.

22.6 Applications of Metasurfaces Substrate-3RT duroid. h3 = 0.762 mm

7 mm Height (h)

Metasurface

10 mm

Patch Substrate-1 FR-4, h1=1.6 mm Z

Substrate-2 FR-4, h1= 3.2 mm X

Partially reflecting surface (PRS) Patch

Substrate-2 FR-4, h2 = 1.6 mm

Metasurface

Substrate-1 FR-4, h1 = 3.2 mm

Z

GND

GND

X

Y X (b) Elliptical patch on sandwiched metasurface substrate and PRS superstrate. 10

0

8

–10

6

–20 –30 –40

4

Simulated S11 Measured S11 Simulated Gain Measured Gain

3

3.5

4

4.5

2 5

0

4 Axial ratio ∣dB∣

10

Gam (dBi)

S11 (dB)

(a) Elliptical patch on sandwiched metasurface substrate.

3 2 1 0

3

Simulated 3.5 4

Measured 4.5

Frequency (GHz)

Frequency (GHz)

(c) Return-loss and gain response.

(d) Axial-ratio response.

Figure 22.38

5

Circularly polarized elliptical patch antenna on sandwiched metasurface substrate and PRS superstrate. Source: Srivastava et al. [J.79]. © 2019, IET.

electric type elliptical rings arranged in the rectangular unit cells. FR-4 substrates (εr = 4.4, tan δ = 0.02), with two thicknesses h1 = 3.2 mm and h2 = 1.6 mm, form a composite layer substrate. The linearly polarized patch is the primary radiating element. The dimensions of the elliptical ring inclusion are tuned on the EMsimulator to achieve the differential reflection phase Δϕ = |ϕy − ϕz| = 90∘ for the reflected waves to realize the circularly polarized radiation. The dimensional and other details are available in the reference [J.79]. To enhance the directive gain of the radiator a transmission-type metasurface, designed with a 2D array of square metallic patches, is used as a partially reflecting surface (PRS). The PRS is tuned to achieve both high gain and wider axial ratio BW (ARBW). Figure (22.38b) shows the designed and fabricated

structure. Figure (22.38c) shows the |S11|, and gain responses, and Figure (22.38d) shows the axial ratio response. The final circularly polarized radiator has 35.3% impedance matching BW, 24.7% 3 dB ARBW with gain 9.3 dBi. Circularly Polarized Slot Antenna Using Metasurface Superstrate

The transmission-type anisotropic metasurface can be used as a superstrate with a linearly polarized microstrip-fed slot antenna to get the circularly polarized radiation of enhanced gain [J.80]. Figure (22.39a) shows the 3D perspective view of a three-layered structure. The antenna is designed at 5.8 GHz. The ground plane of the first substrate with εr = 3.6, tan δ = 0.003, h1 = 0.762 mm supports the LP-slot antenna. It is fed

885

22 Metamaterials Realization and Circuit Models – II

Substrate #2

Metasurface

w mw

mg

Sw Sl

l

lc Ground plane with slot Feeding line y

z

h2

fw x

Substrate #1

fl

gl

h1

gw g

(a) Perspective view of slot radiator with metasurface as a polarization converter superstrate.

(b) Superstrate, ground conductor with a slot radiator, microstrip feed to a slot, two-layered substrate.

10.0

12

7.5

9

Simulation Measurement Axial ratio (dB)

Gain total (dBi)

886

5.0 2.5 0.0 4.5

Figure 22.39

6 3

Simulation Measurement 0 5.0

5.5

6.0

6.5

7.0

4.5

5.0

5.5

6.0

6.5

Frequency (GHz)

Frequency (GHz)

(c) Gain response.

(d) Axial ratio response.

7.0

Circularly polarized slot radiator using metasurface as a superstrate. Source: Huang et al. [J.80]. © 2016, AIP Publishing.

by a microstrip line on the second substrate. The cornercut square patches are used to design the metasurfacebased superstrate on the second substrate. It has a thickness h2 = 4.0 mm. Figure (22.39b) shows the dimensional details with following optimized values: sw = 28, sℓ = 2.5, fw = 1.6, fℓ = 27, gw = 6, gℓ = 14, g = 0.6, w = 39.2, ℓ = 40, mw = 8.8, mg = 1 (all dimensions in mm). Figure (22.39c) shows the gain response. Figure (22.39d) shows the axial ratio response of the circularly polarized antenna. The experimentally obtained impedance matching BW of the CP antenna is 32% (4.7–6.5 GHz) and its ARBW is 20.6% (5.14–6.32 GHz). The antenna has a directive gain of about 8.0 dBi. The present subsection has discussed only a few illustrative application examples of metasurfaces. The

metasurfaces have been used for many other purposes, for instance, as the polarization-sensitive beam splitters [J.74, J.82], RCS reduction devices [J.83], microwave absorbers [J.84, J.85], holographic antenna [J.63, J.64], and surface-waves-based devices and antenna [J.49, J.50, J.65].

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891

893

Index a [ABCD] matrix coupled transmission lines 444–447 vs. [S] parameters 60–61 abnormal/anomalous dispersive medium 64 admittance matrix excitation 48 response 48 short-circuited parameters 48 admittance type Green’s function 579–580 Aharonov–Bohm effect 6 air-filled waveguide sections 357 aluminum oxide ceramic 457 Ampere’s circuital law 3 Ampere’s law 94 amplitude matching 127 angle of incidence Brewster angle 132 critical angle 133–134 TE polarization 134–135 TM-polarization 135–136 anisotropy substrate 276–278 anomalous dispersive medium 65 anomalous reflection law 872 anomalous refraction law 872 anti-symmetric anisotropic materials 83 aperture coupling 391 approximate TEM mode 265 artificial dielectric materials 159, 161, 168 artificial dielectrics 79 artificial electric conductor (AEC) 136 artificial electric medium equivalent parallel plate waveguide model 789–791 polarization 786–789 reactance loaded wire medium 791–793 artificial linear dispersive transmission lines 63 artificial magnetic conductor (AMC) 124, 135 artificial magnetic materials 165

artificial magnetic medium. see split ring resonator (SRR) artificial transmission line 22 asymmetrical coplanar stripline (ACPS) 326 asymmetrical coplanar waveguide (ACPW) 301 asymmetrical coupled lines 392 asymmetrical coupled lines network parameters 4-port [ABCD] matrix parameters C-mode Voltage 449 determination 448–452 open circuit condition 448 short-circuit condition 450 π-mode voltage 449 asymmetrical/symmetrical structure 412 asymmetric coupled microstrips frequency-dependent line parameters 426–430 static parameters 424–426 attenuation constant 25

b backward wave (BW) medium 140 supporting line 72–74 bandgap materials and metamaterials 159 bandpass filter 486–489 beam refracting gradient metasurface 884 Bessel equation 356 bianisotropic electromagnetic materials 84 bi-anisotropic materials 167 bilateral slot line 354 binder material 461 Birchak rule 178 BMSL. see buried microstrip line (BMSL) boundary surface 219 boundary value problem 25 boxed coplanar waveguide N-terms basis function 534–537 one term basis function 532–534 boxed microstrip line, 503–507

Introduction to Modern Planar Transmission Lines: Physical, Analytical, and Circuit Models Approach, First Edition. Anand K. Verma. © 2021 John Wiley & Sons, Inc. Published 2021 by John Wiley & Sons, Inc.

894

Index

boxed microstrip line (cont’d) bottom and top ground planes 507–508 central conductor 507 discrete Fourier domain 528–529 line capacitance 529–531 Bragg’s law of diffraction 697–698 branch-line coupler 486 Brewster angle 132 Brillouin zone 1D Brillouin zone 702 2D Brillouin zone 702–703 broadside coupled lines 391 broadside microstrip coupler 486 Bruggeman formula 178 bulk micromachining 471 buried microstrip line (BMSL) 468–470

c capacitive coupling capacitance matrix 399–400 coefficient 400–401 direct capacitance 400 even-mode excitation magnetic wall 397 open-circuited structure 397 odd-mode excitation electric wall 397 Maxwell’s capacitance matrix 399 nomenclature C-mode 399 short circuited structure 399 zero (cipher) phase difference 399 capacitive coupling coefficient 400–401 capacitive input admittance 362 capacitive input susceptance 366 capacitively loaded loop (CLL) 807–808 capacitively loaded strip (CLS) CLL 807–808 SRR 807 capacitor-loaded hairpin resonator 650 Cauchy–Riemann conditions 303 cavity resonators 489 ceramic substrate 456 Cerenkov radiation 152, 342 characteristic admittance 362 characteristic impedance 25 charge distribution functions 500–503 chemical etching 262, 457 Cherenkov-type leaky mode radiation 343, 344 circuit model lossy dielectric medium 88–90

lossy magnetic medium 90–91 circuit models, CPW and CPS 345–346 circuit models, dielectric material parallel RC 194–195 parallel series combined 195–196 series combination of two parallel RC equivalent relative permittivity 198–200 limiting values 197–198 upper layer lossless 196–197 upper layer lossy 198 series RC 193–194 series RLC resonant 200–202 circular patch 670–671 circular polarization 357 classical transmission line EM-theory 1, 2–6 Heaviside transmission line equation 7–8 Kelvin’s cable theory 6–7 propagation medium 8 telegraph 1–2 CLL. see capacitively loaded loop (CLL) closed-form dispersion model 336–337 codirectional coupling 395 coefficients of capacitance matrix 399–400 Cohn’s equivalent waveguide model 377 common equivalent single-layer substrate 554 complementary line 301 complementary SRR (CSRR) resonant metalines 852–853 complementary structure 353 complete elliptic integrals 379 complex propagation constant 25 composite RH-LH (CRLH)-transmission line 73 composite substrate microstrip line 547–548 composite substrate slot line 354 computer- aided design (CAD) 266 concept of reflectance (reflectivity) 123 concept of transmittance (transmissivity) 123 conductor- backed CPW (CB-CPW) 302 conductor-backed dielectric surface 240 conductor-backed slot line (CBSL) 354 conductor loss 22, 26, 234, 337–341 multilayer coupled microstrips 558–559 multilayer microstrip lines 552 conductor loss, microstrip line conformal mapping method 291 diffusion equation 288 internal inductance 289 lineal current density per unit width 289 perturbation method 291–292

Index

series of impedance 288 surface impedance 287 surface impedance of the conductor 292–293 surface resistance 289 waveguide model 289–290 Wheeler’s incremental inductance rule 293–295 conductors 80 conductor thickness 422 conformal mapping method 13, 291 analytic function Cauchy–Riemann conditions 303 conformal mapping 304 conjugate functions 304 electrostatic potential 304 Laplace’s equations 303 transformation (mapping) 304 complex variable dependent 302 independent 302 mapping function 303 mapping/image of curve 303 single- valued 302 elliptic sine function. see elliptic sine function properties electrostatic potential 305 EW 305 geometrical transformation process 305 MW 305 orthogonal one 304 physical transformation process 305 SC transformation 306–307 constant charge distribution 500 constitutive relations, flux fields 78–79 continuous coupling 391 conventional thick-film technology 461–462 coplanar ground planes 326–327 coplanar multistrip microstrip lines 598–599 coplanar multistrip structure 596–598 coplanar stripline (CPS) asymmetric CPS infinitely wide ground plane 326 characteristics impedance 327 coplanar ground planes 326–327 CPS–CPG line 328 dispersion closed-form dispersion model 336–337 effective relative permittivity 327 effect of conductor thickness 330 on finite, ACPW and ACPS ACPS line 326

ACPW line 324–326 thickness substrate 324 on finite thick substrate alternate conformal mapping 322–324 symmetrical 321–322 gradual approach 329 on infinitely thick substrate asymmetrical 320–321 symmetrical 319–320 modal field structure 332–334 coplanar waveguide (CPW) 261, 485–486 admittance type Green’s function 579–580 boundary condition 576–577 closed-form dispersion model on a finitely thick substrate 334 frequency-dependent characteristic impedance improved dispersion model 334–336 conductor-backed 317–319 continuity condition 577 effect of conductor thickness 329 finite ground planes 313–315 Fourier transform even mode 595–596 odd mode 595–596 Galerkin’s method 589–590 impedance type Green’s function 580–584 infinite extent 310–311 infinite ground planes 311–313 longitudinal current density 578–579 modal field structure conductor-backed 332 dispersion 330–332 on a finite substrate 330–332 slot-field 577 static characteristics finite dielectric thickness 316 infinite dielectric thickness 315 strip current densities 577 substrate with finite thickness 311–313 top shield 316–317 transverse current density 579 coplanar waveguides (CPW) 9, 13 1D planar EBG structures series capacitor loading 745–746 series inductor loading 743–745 shunt capacitor loading 746–748 resonant metalines CSRR 855–856 SRR 854–855 Coulomb’s law 2, 3

336

895

896

Index

counter-directional (reverse) coupler 395 coupled coplanar waveguide (CPW) shielded broadside 432–433 symmetric edge 430–432 coupled microstrip line 486 multilayer dielectric medium space domain analysis 519–523 spectral domain analysis 523–525 coupled microstrip line resonator 659–661 circuit model electric (capacitive) coupling 659–662 hybrid coupling 663–664 magnetic (inductive) coupling 662–663 structures 664–666 coupled-mode analysis 392–394 coupled transmission line circuit models capacitive coupling. see capacitive coupling inductive coupling. see inductive coupling coupling coefficient 408–409 even–odd mode, analysis method coupled strip lines 407–408 input impedance matching 406 isolation between ports 407 S-parameters 405–406 structure. see structure, coupled transmission line wave equation admittance 414–416 characteristic impedances 414–416 Kelvin–Heaviside transmission line equations 409–411 solution of coupled 411–414 coupled transmission line equations 392 CPW. see coplanar waveguide (CPW) CPW resonator 651–653 critical angle 133–134 cross-coupling factor 84 cross-coupling gyrotropic factor 113 cross-talk 391 crystal lattice structures direct–lattice 698–699 1D reciprocal lattice 700 reciprocal lattice 699–700 2D hexagonal lattice 701 2D reciprocal lattice 700–701 2D rectangular lattice 701 cubic charge distribution 501 current node equation 23 cut-off property 64

d Debye dispersion law in time-domain 180 Debye relaxation model 179, 183 de-embedding process 59, 61–62 deep reactive ion etching (DRIE) 472 defected microstrip structure (DMS) 726 delayed oscillation 19 deltagap voltage source 62 depolarization process 180 DFT. see discrete Fourier transform (DFT) diamagnetic materials 166 dielectric constant of medium 79 dielectric loss 23, 26, 341–342 multilayer coupled microstrips 557–558 multilayer microstrip lines 551–552 dielectric loss, microstrip line lossy dielectric-medium capacitor 286 effective loss-tangent 287 EM-wave 285–287 dielectric medium/material 79 artificial dielectric computation 172–173 electromagnetic band-gap 169 equivalent permeability 169 homogenization 168 lattice constant 169 1D artificial dielectrics 169, 170 phase advance dielectrics 171 phase delay dielectrics 169 photonic bandgap 169 static relative permittivity 172 TEM wave 172 2D artificial dielectrics 169, 170 circuit models. see circuit models, dielectric material frequency response. see frequency response, dielectric material interfacial polarization. see interfacial polarization macroscopic concept 159 mixture. see mixture, dielectric medium modeling. see modeling, dielectric medium natural nonpolar 168 polar 168 resonance response. see resonance response, dielectric material dielectric polarization bound charges 159 external field 160

Index

ferroelectric materials 160 free charges 160 local field 160 dielectric slab, EM - waves normal incidence 138–139 oblique incidence 136, 136–138 dipole moment 161 Dirac’s delta function 3 direct capacitance 400 direct-coupled resonator 655–656 direct inductances 401–402 directivity of a coupler 396 discrete coupling 391 discrete Fourier transform 502–503 derivatives 528 even function 527 odd function 526–527 sine and cosine discrete Fourier transforms 526 discrete Fourier transform (DFT) 498 dispersion, microstrip line dispersion law correction factor, LDM 283–284 Schneider’s conditions 283 frequency-dependent characteristic impedance 284–285 inflection frequency 279 Kirschning–Jansen 284 Schneider’s condition 279 synthesis of 285 waveguide model 279 frequency-dependent characteristic impedance 281–282 frequency-dependent equivalent width 281 higher-order modes 282 dispersion model 544–545 multilayer ACPW/CPW 564–565 dispersive lines 261 dispersive transmission lines 25 backward wave supporting line 72–74 series capacitor loaded LC line 74–75 series-connected capacitor 74 shunt inductor loaded line 72 displacement current 77 DMS. see defected microstrip structure (DMS) DNG. see double negative (DNG) Doppler effect 151–152 double negative (DNG) 73 composite metamaterials CLS–CLL 807–809 CLS–SRR 807 strip wire and SRR 806–807

composite permittivity–permeability functions Drude–Lorentz function 803–805 Lorentz–Lorentz function 805 single-structure S-ring particle metamaterials 810–814 Ω-particle metamaterials 809–810 double negative medium (DNG) 140 double-positive (DPS) 73 medium 139 double refraction/birefringence 112 DRIE. see deep reactive ion etching (DRIE) Drude–Lorentz function 803–805 Drude–Lorentz model 6 Drude model 139 dry etching 472 dual-band metaline 859 dual-mode circular patch resonator 688–689 dual-mode patch resonators 686–689 dual-mode resonators dual-mode patch resonators 686–689 dual-mode ring resonators 689–692 dual-mode ring resonators 689–692 dual-mode square patch resonator 687–688 dual-mode triangular patch resonator 689 dual step impedance resonator 640 dyadic Green’s function 563–564 dynamic closed-form models 266 dynamic electromagnetic theory 4 dynamic Maxwell Garnett model Mie magnetic resonator RLC circuit model 826–827 SRR inclusion 825–826 Mie scattering 821–825

e edge coupled lines 391 edge-singularity 332, 337 effective relative permittivity 268 effluvium 2 eigenfunctions 225 eigenvalues 225 electrets 160 electrically gyrotropic medium 83 electrical parameters 159 electrical properties, dielectric medium. see dielectric medium electric and magnetic scalar potentials 213 electric circuit theory 3 electric displacement vector 78 electric field intensity (E) 77

897

898

Index

electric field intensity (E) (cont’d) Ohm’s law 79 electric flux density 78 electric plasma medium 171 electric polarization 6, 79, 162 electric vector potential 6 hypothetical magnetic current 215 Lorentz gauge condition 216 scalar magnetic function 215 TEz mode 216–217 electric walls (EW) 219, 280, 305, 354, 431 electromagnetic bandgap (EBG) 133 electromagnetic (EM) fields boundary surface, close waveguide interface of two media 221 perfect electric conductor 219–220 perfect magnetic conductor 220–221 classifications HE and EH mode 213 TEM mode 213 TE mode 213 TM mode 213 electric vector potential 215–216 hybrid mode for TEy mode 218–219 for TMy mode 219 magnetic vector potential 214–215 Maxwell’s equations 214 vector potentials 214 waveguide structures open waveguide 219 electromagnetic wave equation 4 electromagnetic waves Hertzian spark-gap 5 loaded dipole 4 electromagnetic (EM) theory 11 Drude–Lorentz model 6 electrostatics and scalar potential 2–3 generation and transmission of electromagnetic waves 4–5 Hertzian potentials 5–6 magnetic effect of current 3 magnetic vector potential 4 Maxwell’s dynamic electromagnetic theory 4 Ohm’s law 3 time-varying magnetic field 3 electronic band-gap (EBG) 9, 14, 140 electronic/photonic bandgap (EBG/PBG) structures electronic polarization 161, 162, 164–165 electroscope 2

electrostatics and scalar potential effluvium 2 gravitational field 2 Green’s theorem 3 Laplace’s equation 2 one-fluid model 2 Poisson’s equation 2 potential 2 static electricity 2 elliptical polarization 357 elliptic sine function 302 elliptic integral 1st kind, forms 307–310 approximation 309 complete 308–309 embedded microstrip line 468 embedded (buried) slot line 354 energy method 495–497 epitaxial growth method 464–465 epsilon-near-zero (ENZ) materials 79, 167 medium 139 epsilon negative (ENG) medium 79, 139, 188 equilateral triangular patch 672–674 equivalent parallel plate waveguide model 789–791 equivalent permeability 169 equivalent single-layer substrate 553 equivalent transmission line 244 equivalent waveguide model 254, 257–258 evanescent mode 43 evanescent wave 225 even-mode characteristics impedance 421 frequency-dependent models 423–424 top shield 421 even-mode coplanar waveguide (CPW) capacitance 430–431 even-mode effective relative permittivity 419–420 frequency-dependent models 423 even-mode excitation 397 even-mode filling-factor 420 even–odd mode analysis 392 extraction process. see homogenization

f

237

fabrication process epitaxial growth method 464–465 ion implantation method 465 isolation implantation method 465–466 mesa etching method 465 Faraday rotation 114

Index

fast-wave normal dispersive medium 68 fast-waves 112 fictitious magnetic current 355 filling-factor 268 finite difference time-domain (FDTD) 175, 265 finite element method (FEM) 265 fin line 353 flexible polyethersulfone (PES) substrate 468 Floquet–Bloch theorem 703–704 folded hairpin resonator 651 force fields 77 forward/counter directional coupling 394–395 forward traveling wave 225 forward wave (FW) medium 140 Fourier coefficients 362 Fourier domain line capacitance 499–500 Poisson equation 498–499 Fourier transform 500–503 fractal geometry dimension and characteristics 675–676 Hilbert curves 678–680 Koch curves 676–677 Koch islands 676–677 Minkowski curves 677–678 Sierpinski carpet 681 Sierpinski triangles 680–681 fractal resonator antenna 682–683 fractal resonators 683–686 free charge carriers 79 frequency-dependent characteristic impedance 281 frequency-dependent equivalent width 281 frequency-dependent line parameters 426–430 frequency-dependent models even-mode characteristics impedance 423–424 even-mode effective relative permittivity 423 odd-mode characteristics impedance 424 odd-mode effective relative permittivity 424 frequency-dependent process 161 frequency response, dielectric material Debye dispersion relation dipolar polarization 182 electronic polarization 182 frequency-dependent 181 ionic polarization 182 polarization and depolarization process 180 polarization law polarization decay law 181 polarization rise law 181 relaxation 179

Fresnel’s equations fringing field 263

127, 129

g Galerkin’s method CPW structure 589–590 line capacitance 529–531 microstrip 587–589 galvanometer 3 gap capacitor 721–722 Gauss divergence theorem 94 Gauss’s law electric flux 78 magnetic flux 78 general wave equation 21 geometrical transformation process 305 Getsinger’s dispersion models C-mode excitations 428–430 π-mode excitations 428–430 Getsinger’s model 281 Getsinger’s parameter 281 Green’s function 3, 497–498 discrete Fourier domain 528–529 single-layer planar line CPW 576–584 formulation 574–576 microstrip structures 576–584 Green’s Theorem 3 group velocity 66 guided-wavelength 227 gyroelectric medium 83–84 gyromagnetic medium 83–84 gyromagnetic ratio 84

h half-mode SIW (HM-SIW) 256–258 half-waveplate 112 Hammerstad–Jensen (H-J) closed-form expressions 271, 278 Hankel functions 356 Harmonic oscillator model 184 Heaviside’s condition 27 Heaviside transmission line equation harmonic excitation 7 Huygens’s load 8 Laplace transform 7 line conductance 7 matrix method 7 RC model 7 Hertzian potentials 5

899

900

Index

heteroepitaxy process 464 higher-order guided modes 266 higher-order modes 282 high-pass filter (HPF) 43, 72 Hilbert curves 678–680 HMIC technology. see hybrid microwave integrated circuit (HMIC) technology Holloway and Kuester (HK) 292 homogeneous effective medium 207 homogeneous medium 261 symmetrical coupled line even-mode excitation 435–436 4-port coupled lines 434–435 odd-mode excitation 436–437 2-port coupled lines 437–438 symmetrical coupled microstrip line current on conductor 442–444 current wave 441–442 even–odd mode excitation 438–439, 442 voltage on conductor 439–441 voltage wave 439 Y parameters 441 Z parameters 438 homogenization 168, 193, 254 dynamic Maxwell Garnett model 821–827 NRW method 814–821 homogenization of a material 159 host material 461 host medium and inclusion 159 Huygens’ load 25 Hybrid microwave integrated circuit (HMIC) 13 hybrid microwave integrated circuit (HMIC) technology hybrid fabrication process 457–459 MIC fabrication process 457–459 substrates 456–457 thick-film process 460–462 thin-film process 459–460 hybrid mode 21, 218, 218–219, 263 hybrid (HE and EH) mode 213 hyperbolic form 28 hyperlens 150–151 hypermedium 118 hypothetical spherical cavity 163

i immittance approach 605–610 multilevel strip conductors 605–610 single-level strip conductors 600–605 impedance type Green’s function 580–583 incident power variable 52

inclusions 168 induced displacement current 409 inductance matrix 401 induction effect 3 inductive coupling coefficient 402 direct inductances 401–402 inductance matrix 401 inductive (magnetic) coupling 391 inductive coupling coefficient 402 inhomogeneous medium 261 insertion loss (IL) 57, 396 integral equation method 265 integrated model 295 interaction constant 163–164 interdigital capacitor 722 interface, EM-waves dielectric and perfect conductor 124 normal incidence 121–124 transmission line model, composite medium 124–125 interfacial/Maxwell–Wagner polarization 162 interfacial polarization transient polarization current 190 two-layered capacitor boundary conditions 190–191 steady-state condition 191 transient condition 192 internal surface inductance 293 intrinsic impedance 363 inverse Cerenkov radiation 152–153 inversr Doppler effect 152 inverted conductor-backed slot line (ICBSL) 354 inward moving wave 356 ionic polarization 161, 162, 165 ion implantation method 465 iris-loaded rectangular waveguide 357 isofrequency spherical surfaces 117 isofrequency surface concept 115 isolated (decoupled) port 396 isolation factor 396 isolation implantation method 465–466 isotropic anisotropic medium 129 isotropic medium 81 isotropic medium, EM waves 1D wave equation 97–98 EM-wave equation 96–97 Maxwell’s equations 101–102 plane waves linear lossless 98–101 lossy conductor 102–104

Index

j Jones matrix, polarization Cartesian coordinate system;, 108–109 half-waveplate 110 Jones vector 106–107 linear polarizer 107–108 quarter-wave retarder 110 transforms the vector components 108 wave retarder 109–110 Jones vector 106–107

k Kelvin-Heaviside transmission line equations 19, 22, 23, 223, 409–411 Koch curves 676–677 Koch islands 676–677 Kramers–Krönig relations 159, 179, 820–821 Krowne method 380–381

l laminated waveguide 253 Laplace’s equations 303 leaky modes 266 leaky-wave antenna 856–857 left-hand circularly polarized (LHCP) waves 113 Legendre polynomials 591 LIGA technology 478 linear charge distribution 500–501 linear graph 21 line capacitance Fourier domain 498–503 Galerkin’s method 529–531 line parameters asymmetric coupled microstrips 424–430 coupled CPW 430–433 symmetric edge coupled microstrips 419–424 line resonator direct-coupled resonator 655–656 planar transmission line resonator 659 reactively coupled line resonator 656–658 tapped line resonator 658–659 location of sources finite length line and infinite length line 38–39 series voltage 39–40 longitudinal section electric mode (LSE) 214 equivalent circuits 600–601 equivalent transverse transmission line 600–601 longitudinal section magnetic mode (LSM) 214 equivalent circuits 600–601 equivalent transverse transmission line 602–603

Looyenga rule 178 Lorentz gauge condition 5, 215 Lorentz–Lorentz function 805 Lorentz type harmonic oscillator model 179 losses, CPW and CPS structures conductor loss CPS structure 341 CPW structure 339–341 perturbation method 339 Wheeler’s incremental inductance method 338–339 dielectric loss 341–342 substrate radiation loss 342–345 lossless exponential transmission line 42–43 lossless nonuniform line 42 lossy multilayer microstrip lines 542–544 lossy transmission line characteristic impedance 26–27 propagation constant 27 low-pass filter (LPF) 40, 72 low temperature cofired ceramic (LTCC) circuit fabrication 482–484 components 484–489 materials and process 480–482 planar transmission line 484–489 waveguide and cavity resonators 489 low-temperature co-fired ceramics (LTCC) 9 LSE. see longitudinal section electric mode (LSE) LSM. see longitudinal section magnetic mode (LSM) LTCC. see low temperature cofired ceramic (LTCC) lumped circuit elements model 19 lumped element equivalent circuit model 23 lumped parallel resonant circuit 617–619 lumped series resonant circuit 615–617

m macroscopic concept 159 magnetically gyrotropic medium 83 magnetic charge density 78 magnetic current model Bessel equation 356 circular polarization 357 elliptical polarization 357 equivalent isotropic medium 356 fictitious magnetic current 355 Hankel functions 356 inward moving wave 356 outward moving wave 356 quasi-static solution 356 zeroth-order solution 357 magnetic dipole moment 166

901

902

Index

magnetic displacement vector 78 magnetic effect of current 3 magnetic field intensity (H) 77 magnetic permeability 6 magnetic vector potential 4, 214 Lorentz gauge condition 215 TMz mode 217–218 vector wave equation 214 wavenumber 214 magnetic walls (MW) 221, 280, 305, 354, 430 magnetization 162 process 79 vector 166 magneto-dielectric (MD) materials 84 magneto-dielectric-medium 207 magnetoelectric materials 84–85 material medium, waves anti-symmetric anisotropic materials gyroelectric medium 83–84 gyromagnetic medium 83–84 magneto-dielectric (MD) materials 84 magnetoelectric materials 84–85 dispersive medium 85–86 isotropic medium 81 nondispersive medium 85–86 non-lossy and lossy medium 86–87 symmetric anisotropic materials 82–83 materials, waves conductors 80 dielectric materials 79 magnetic materials 79–80 Maxwell equations, 214 Ampere’s law 94 conduction current density 91 constitutive relations 92 continuity equation 93 electric displacement current density 91 flux field quantities 92 force field quantities 92 Gauss divergence theorem 94 Lorentz’s force 92 magnetic charge 92 magnetic displacement current density 91 phasor form 93 power and energy relation 94–96 Stokes theorem 94 Maxwell Garnett–Debye model 189 Maxwell Garnett–Drude model 190 Maxwell Garnett–Lorentz model 189–190 Maxwell Garnett (MG) mixing rule 164

Maxwell–Heaviside–Hertz equations 4 Maxwell model 176 Maxwell’s charge distribution function 501–502 Maxwell’s distribution function 500 Maxwell’s dynamic electromagnetic theory 4 Maxwell’s equations, algebraic form 101–102 Maxwell’s induction law 94 MCM. see multichip module (MCM) mechanical etching 457 mechanical milling 262 medium coupling 391 mediumwave (MW) 1 mega molecules 168 membrane technology 473–475 MEMS. see microelectro-mechanical systems (MEMS) mesa etching method 465 metalines. see 1D-metamaterials metal insulator semiconductor (MIS) 65, 161, 467 metamaterial perfect absorber (MPA) 153 metamaterials 161, 169 metamaterials medium, EM - waves DNG medium amplitude-compensation 147–149 Cerenkov radiation 152 Doppler effect 151–152 hyperlens 150–151 inverse Cerenkov radiation 152–153 inversr Doppler effect 152 lossless 141 negative refraction 143–144 Pendry superlens lens 150 refractive index 142–143 transmission line model 144–145 Veselago flat lens 149–150 wave impedance 143 wave propagation 146–147 lossy DNG slab, with Conductor Backing 155–156 lossy DNG slab, without Conductor Backing 156 MPA DNG slab absorber 155 metasurface absorber 154–155 salisbury absorber 154 metasubstrates 207–208 metasurface absorber 154–155 method of moment (MoM) 266, 378, 498 method of partial capacitance 277 method of the line (MOL) 272 metsurfaces anomalous reflection and refraction 873–877 characterization

Index

anisotropic materials 864–865 bianisotropic materials 865 modeling 862–864 Schelkunoff’s equivalent surface 862 generalized Snell’s laws 870–872 isotropic metasurfaces electric polarization 867 perfect reflection 866–867 perfect transmission 867 phase control response 868–870 rotation 868 size 868 quarter 883 reflection 881 reflectionless transmission Fabry–Perot resonance 877–880 Huygens metasurfaces 877 surface waves nonuniform metasurfaces 873 uniform metasurface 872–873 transmission 881–883 micro coplanar line (MCL) 326 micro-coplanar strip (MCS) 262 microelectro-mechanical systems (MEMS) 9 fabrication process 471–473 transmission line structures 473–478 micromachined transmission line technology MEMS fabrication process 471–473 transmission line structures 473–478 microshield technology 475–477 microstrip 485 Galerkin’s method 587–589 impedance type Green’s function 580–583 legendre polynomials 591 realization capacitors 721–722 inductors 723–726 resonant circuits 726–728 singular Chebyshev polynomial basis functions 592–593 singular sinusoidal basis functions 591–592 transition structure 485–486 microstrip edge-coupled resonator 640 microstrip hairpin resonator 649–651 microstrip integrated circuit (MIC) technology 9 microstrip line 24 anisotropy substrate anisotropic-ratio 277 equivalent substrate 278

misalignment angle 276 partial capacitance method 277 w/hratio 277 circuit model 295–296 dispersion. see dispersion, microstrip line losses 285 conductor loss. see conductor loss, microstrip line dielectric loss. see dielectric loss, microstrip line non-TEM 264 1D planar EBG structures periodic loading of ground conductor 731–742 periodic loading of substrate 728–731 parameters 266–268 pure TEM mode 264 quasi-TEM mode closed-form models 266 full-wave methods 265 static methods 266 static characteristic impedance 270–271, 272 static parameters effect of conductor thickness 272–274 effect of shield 274–276 results 271–272 variational method 503–512 Wheeler’s transformation concept of partial capacitance 268 equivalent homogeneous medium 269 filling-factor 269–270 inverse 270 Laplace’s equation 269 microstrip line resonator microstrip hairpin resonator 649–651 microstrip ring resonator 643–645 microstrip step impedance resonator 645–649 open-end microstrip resonator 640–643 short-circuited ends microstrip resonator 643 microstrip line with superstrate 549 microstrip patch resonators circular patch 670–671 equilateral triangular patch 672–674 MWM 667–670 rectangular patch 667 ring patch 671–672 microstrip ring resonator 643–645 microstrip step impedance resonator 645–649 microstrip structures 262 Mie magnetic resonator RLC circuit model 826–827 SRR inclusion 825–826 Mie scattering 821–825

903

904

Index

Minkowski curves 677–678 MIS. see metal insulator semiconductor (MIS) mixture, dielectric medium Bruggeman formula 178 complex equivalent relative permittivity 178 mass basis and volume basis 173 Maxwell Garnett formula 175–177 parallel capacitor model 174 series capacitor model 175 Sihvola formula 177–178 variational method 175 Wiener model 174 MMWIC. see monolithic millimeter-wave integrated circuits (MMWIC) modeling, dielectric medium Clausius–Mossotti model 163 dielectric polarization electronic 161 ionic 161 orientational/dipolar 161 space charge (interfacial) 161–162 Lorentz local electric field model 163–164 material magnetization 165–167 polarizability electronic 164–165 ionic 165 orientational/dipolar 165 relative permittivity 162–163 susceptibility 162–163 mode-matching method 265 modified Maxwell’s distribution function 502 modified Wolff model (MWM) 667–670 monochromatic wave 20 monolithic microwave integrated circuit (MMIC) 9, 462 fabrication process 464–466 lines 468 planar transmission lines 466–471 Mossotti local electric field 164 multichip module (MCM) 463 multilayer asymmetric coplanar waveguide/coplanar waveguide SLR process dispersion models 564–565 loss models 565–566 SLR 560–561 static SDA 561–564 multilayer broadside coupled microstrip line 522–523 multilayer coupled microstrip lines SLR process characteristic impedance and synthesis 556–557

dispersion model 555–556 equivalent single-layer substrate 553–555 loss models 557–559 multilayer dielectric medium coupled microstrip line space domain analysis 519–523 spectral domain analysis 523–525 multilayer edge coupled microstrip line 520–522 multilayer microstrip line anisotropic dielectric layers 518 space domain analysis 512–515 TTL method 515–518 multilayer microstrip lines SLR process characteristic impedance and synthesis 549–550 circuit model 553 dispersion model 544–549 losses 550–552 lossy structure 542–544 multilayer planar transmission lines immittance approach multilevel strip conductors 605–610 single-level strip conductors 600–605 multilevel multiconductor microstrip 599 multilevel strip conductors 605–610 multiphase mixtures 168 multisection transmission lines 36–38 multiterm Debye models 205 mu near-zero (MNZ) materials 80 mu near-zero (MNZ) medium 139 mu negative (MNG) materials 80 mu-negative (MNG) medium 139 mushroom-type EBG circuit model 763–767 dynamic circuit model dielectric surface impedance 769 dispersion model 771 dispersion relation 770–771 grid impedance 768–769 reflection coefficient 770–771 surface impedance 769–770 mushroom type resonators 726 MWM. see modified Wolff model (MWM)

n near-zero (NZ) medium 139 negative refractive index materials 143 negative uniaxial medium 118 network parameters asymmetrical coupled lines 447–452

Index

coupled line section 433–447 Newtonian mechanics 159 Nicolson–Ross–Weir (NRW) method branching problem 819–820 equivalent homogeneous slab 818 iterative method 820 Kramers–Kronig relations 820–821 parameters retrieval 818–819 relative permittivity and permeability 815–816 s-parameters vs. reflection transmission coefficients 817–818 thin slab 820 nihility 139 nondispersive 21 nondispersive lines 261 non-Euclidean dimension 674 nonmagnetic plasma medium 64 nonpolar solids 168 nonpropagation higher-order modes 366 nonresonant microstrip metalines cascaded MNG–ENG 848–849 parallel–series 849–850 series–parallel 846–848 nonuniform coupling 392 nonuniform transmission line 40–41 nonuniform transmission lines lossless exponential 42–43 wave equation 40–42 normal dispersive medium 64 normal incidence 138–139 normal incidence, plane waves 121–124 normalized current wave 52 normalized dimension 613 normalized voltage wave 52 Norton current 34 NRW method. see Nicolson–Ross–Weir (NRW) method

o oblique incidence 125, 136–138 oblique incidence, plane waves angle of incidence. see angle of incidence dispersion, refracted waves isotropic medium 129–130, 130 uniaxial anisotropic medium 129–130 perfect electric conductor 127–128 refection/transmission coefficients 131–132 TE (perpendicular) polarization 125–127 TM (parallel) polarization 128, 128–129 wave vs. characteristic impedance 130–131

odd-mode characteristic impedance 421 top shield 421–422 odd-mode characteristics impedance frequency-dependent models 424 odd-mode coplanar waveguide (CPW) capacitance 431–432 odd-mode effective relative permittivity 420–421 frequency-dependent models 424 odd-mode excitation 397–399 odd-mode filling-factor 420 one-dimensional resonator 614 1D artificial dielectrics 169, 170 1D lattice structures 697–703 1D-metamaterials circuit models balanced CRLH parallel–parallel configuration 844–845 parallel–series configuration 843–844 series–series configuration 845 series–parallel configuration 840–841 unbalanced CRLH 841–843 endfire leaky wave antenna 856–857 homogenization 834 material medium 834–836 microstrip directional coupler 857–859 single reactive loading host medium 836–838 host medium with coupling 838–839 1D periodic transmission line [ABCD] parameters of unit cell 707–709 characteristics bloch impedance 718 series capacitor/shunt inductor 715 series inductor/shunt capacitor 713–715 shunt resonant circuit/series resonant circuit 715–718 dispersion 709–712 elements 718–720 periodically loaded artificial lines 705–707 planar loading elements 720–728 1D planar EBG structures CPW series capacitor loading 745–746 series inductor loading 743–745 shunt capacitor loading 746–748 microstrip line periodic loading of ground conductor 731–742 periodic loading of substrate 728–731 one-port reflection-type resonator 620–623 open-circuit parameters 46 open-end microstrip resonator 640–643

905

906

Index

open microstrip line 261, 508–510 conductor thickness 510 energy method 510–512 Green’s function 508–510 optimum resonant length 649 orientational/dipolar polarizability 161, 165 oscillation type circuits 45 oscillator model 159 outward moving wave 356 overhead single-wire 1

p paired split-ring resonators (PRR) 802 parallel capacitor model 174 parallel conductor transmission line 266 parallel plate waveguide (PPW) 223, 789–791 paramagnetic materials 166 parasitic microstrip mode 334 patch capacitor 721 Pendry superlens lens 150 perfect electric conductor (PEC) scalar electric potential 219–220 scalar magnetic potential 220 surface boundary conditions 219–220 perfect electric conductors, electric walls (EWs) modes characteristics attenuation 232 conductor loss 232–234 dielectric loss 234–235 modal field 229 power density of TEz mn mode 232 power density of TMz mn mode 232 power transfer 231–232 propagation constants 227 surface current 229–231 wave impedance 228–229 wave velocities 227–228 TEz modes cut-off wavenumber 224 eigenvalue and eigenfunction 224–226 evanescent wave 225 forward traveling wave 225 modal scalar magnetic wave functions 226 TMz modes eigenvalue and eigenfunction 226 field components 227 perfect magnetic conductor (PMC) MW 221 scalar electric potential 221 scalar magnetic potential 221

surface boundary conditions 220–221 perfect magnetic conductors, magnetic walls (MWs) TEz mode 235–236 TMz mode 236–237 periodically loaded artificial lines 705–707 perturbation method 291 PES substrate. see flexible polyethersulfone (PES) substrate phase advance dielectrics 171 phase constant β, 20, 25 phase delay dielectrics 169 phase shift constant 20, 25 phase shift properties 55–59 phase velocity vp 20, 63–66 phosphor silicate glass (PSG) 473 photoimageable technology 461 photo-imageable thick-film technology 462 photonic bandgap (PBG)/electromagnetic band-gap (EBG) materials 169 photosensitive emulsion 461 physical transformation process 305 physical vapor deposition (PVD) process 460 planar line structure 12 planar transmission line resonator 659 planar transmission lines analytical methods conformal mapping method 10 edge-coupled strip lines 10 BMSL 468–470 coplanar stripline 8 electrodes of active devices 470–471 EM-simulators 10 MIC technology 9 MIS/MOS structure 467 multilayer MMIC lines 468 passive components and interconnect 466 PES substrate 468 Schottky structure 467 Si/GaAs-substrates 467 thin-film structure 467–468 waveguide 8 plane of incidence 125 plastic substrate 456 Poisson equation 2, 499–500 polarization, EM-wave circular 105–106 elliptical 106 Jones matrix. see Jones matrix, polarization linear 104–105 polarization current density 188 polarization decay law 181

Index

polarization process 159, 180 polarization rise law 181 polarized microstrip patch antenna 884–885 polarized slot antenna 885–886 polar solid 168 positive photolithography process 457 positive uniaxial medium 118 power–current (pi) 378 power–voltage (pv) 378 PPW. see parallel plate waveguide (PPW) primary constants 24 process of deposition 262 process of etching 262 propagation characteristics 62–63 propagation constant 25 PRR. see paired split-ring resonators (PRR) PSG. see phosphor silicate glass (PSG) PVD process. see physical vapor deposition (PVD) process

q quad-band metaline 859 quadrature coupler 396 quarter-waveplate 112 quasi-static Wheeler’s incremental inductance method 338 quasi-TE10 mode 353 quasi-TEM mode 264

r radiation loss 26 model 257 modes 266 reactance loaded wire medium capacitance 793 inductance 791–793 reactive ion etching (RIE) 472 reactively coupled line resonator capacitive coupling 657–658 inductive coupling 656–657 reciprocal properties 54 rectangular patch 667 rectangular waveguide composite electric and MWs 237–240 perfect electric conductors, EWs 223–235 perfect magnetic conductors, MWs 235–237 refection/transmission coefficients 131–132 reference plane 55 reflected power variable 53 reflection coefficient 32, 54

reflection factor 396 reflection-type parallel resonator 622–623 reflection-type series resonator 620–622 relative permittivity of a low-density medium 163 relative permittivity of medium 79 relaxation response 179 resistance factor (RF) 293 resonance response 179 resonance response, dielectric material Drude model 188 Lorentz oscillator model 183 c-kind harmonic oscillator 183 Harmonic oscillator model 184 high-frequency relative permittivity 186–188 local electric field 183 low-frequency relative permittivity 185–186 Maxwell–Wagner interfacial polarization 187 relative permittivity near resonance 186 wire medium 185 Maxwell Garnett–Debye model 189 Maxwell Garnett–Drude model 190 Maxwell Garnett–Lorentz model 189–190 plasma 188 resonant metalines CPW 854–856 CSRR 852–853 inclusions 851–852 microstrips 852 SRR 853–854 resonant/nonresonant metamaterials 140 resonating structures 613–615 resonator with external circuit 619–620 retardation 4 return-loss 32 reverse directional coupling 394–395 RIE. see reactive ion etching (RIE) right-hand circularly polarized (RHCP) waves 113 ring patch 671–672 ring resonator 640 rodded medium 785

s salisbury absorber 154 sandwich slot line 354, 370–371 scalar electric potential 214, 217 scalar magnetic potential 214, 218 scalar network analyzer 46 scattering [S] parameters incident power variable 52 normalized current wave 52

907

908

Index

scattering [S] parameters (cont’d) normalized voltage wave 52 properties, [S] matrix phase shift 55–59 reciprocal 54 unitary 55 reflected power variable 53 reflection coefficient 54 [S] matrix 53–54 transmission coefficient 54 Schneider’s physical conditions 281 Schottky structure 65, 467 Schwarz–Christoffel (SC) transformation 302, 306–307 screen printing 460 SDA. see spectral domain analysis (SDA) secondary line parameter 266 secondary parameter γ, 25, 26 separate equivalent even-/odd-mode single-layer substrate 554–555 series capacitor loaded LC line 74–75 series capacitor model 175 series-connected 2D planar EBG circuit model 780–782 series-connected capacitor 74 series-connected parallel resonant circuit 626–627 series-connected series resonant circuit 624–626 shielded broadside coupled coplanar waveguide (CPW) even mode 433 odd mode 432–433 shielded composite substrate microstrip lin 548–549 shielded microstrip line 545–546 shielded slot line input admittance higher-order modes 373–376 TE10 mode 372–373 shielded suspended slot line 354 short-circuited ends microstrip resonator 643 short wave (SW) 1 s/h-ratio 377 shunt-connected 2D planar EBG circuit model bloch impedance 779 uniplanar structure 779–780 shunt-connected parallel resonator 627–628 shunt-connected series resonance 628–629 shunt current source 27–28 shunt inductor loaded line 72 shunt inductor loaded line structure 69, 71 Sierpinski carpet 681 Sierpinski triangles 680–681 Si/GaAs-substrates 467 sine and cosine discrete Fourier transforms 526

single isolated microstrip line 420 single-layer planar line Green’s function CPW 576–584 formulation 574–576 microstrip structures 576–584 single-layer reduction (SLR) 466, 560–561 formulation 13, 341 single-level strip conductors 600–, 600–605 single slot slot-line 598 single strip microstrip 598 single-term Debye model 202–204 single-term Lorentz model 204–205 single-wire medium (SWM) 785 single-wire transmission line 1 singular Chebyshev polynomial basis functions 592–593 singular sinusoidal basis functions 591–592 SIR. see step impedance resonator (SIR); step-impedance resonator (SIR) slab waveguide TEz even mode 251 TEz odd mode 251 TMz even mode 252 TMz odd mode 252 slotline 593 slot line characteristic 376–378 conformal mapping method composite substrate 380 sandwich 379–380 standard 379 integrated model 379, 381–384 Krowne method characteristic impedance 381 slowing-factor 380 magnetic current model. see magnetic current model structures open slot line 353–354 shielded slot line 354–355 waveguide model. see waveguide model slot line resonator 653–654 slowing-factor 368–369 slow-wave type surface wave propagation 135 SLR. see single-layer reduction (SLR) SLR process formulation 566–567 multilayer ACPW/CPW dispersion models 564–565 loss models 565–566

Index

SLR 560–561 static SDA 561–564 multilayer coupled microstrip lines characteristic impedance and synthesis 556–557 dispersion model 555–556 equivalent single-layer substrate 553–555 loss models 557–559 multilayer microstrip lines characteristic impedance and synthesis 549–550 circuit model 553 dispersion model 544–549 losses 550–552 lossy structure 542–544 Snell’s laws 14, 126 SOC. see system-on-chip (SOC) solenoid 3 solid conducting sphere 178 space-charge polarization 161–162 space domain analysis multilayer broadside coupled microstrip line 522–523 multilayer edge coupled microstrip line 520–522 space harmonics 703–704 spatial dispersion 21 spatial period 20 spectral domain analysis 523–525 spectral domain analysis (SDA) 11, 213, 265, 355, 571–574 spectral domain method 302 split ring resonator (SRR) 167 Bi-anisotropy 800–801 characteristics 794–795 circuit model 796–798 equivalent circuit parameters 798–800 resonant metalines 853–854 variations 801–803 spurious harmonic frequency 648–649 spur microstrip line resonator 639 sputtering process 460 square split-ring resonators 800 SRR. see split ring resonator (SRR) standard slot line characteristic impedance 369 equivalent waveguide formation of height 358–359 formation of width 358 expressions in waveguide 359–361 input admittance higher-order modes 363–364 TE10 mode 363 modes in waveguide 359 slowing-factor 368–369

susceptances 361–362 TE mode 364 TM mode 364–368 standing wave 32 static closed-form models 266 static electricity 1 static/low-frequency electric polarizability 172 static parameters 424–426 static permittivity 178 static relative permittivity 172 static spectral domain (SDA) 498, 518–519, 561–564 step impedance resonator (SIR) 639, 646, 726 Stokes theorem 94 stopping distance 291 strip wire-SRR (SW-SRR) 806–807 strong coupling 391 structures, coplanar stripline structures (CPS) on effect of conductor thickness 330 modal field structure 332–334 structures, coupled transmission line aperture coupling 391 asymmetrical coupled lines 392 broadside coupled lines 391 capacitive (electric) coupling 391 continuous coupling 391 coupled-mode analysis 392 coupled transmission line equations 392 discrete coupling 391 edge coupled lines 391 even–odd mode analysis 392 inductive (magnetic) coupling 391 medium coupling 391 nonuniform coupling 392 strong coupling 391 symmetrical coupled lines 392 uniform coupling 392 weak coupling 391 substrate integrated waveguide (SIW) 213, 253 substrate leaky-mode radiation loss 344 substrate radiation loss 342–345 superstrate 261 surface current density field 590 hybrid modes 590–596 surface-micromachining 472–473 surface wave waveguide cut-off wavenumber 243 frequency-dependent effective relative permittivity 243 TEz mode 242–244 TMz mode 240–242

909

910

Index

susceptibility 6 suspended composite substrate 364 slot line 371 suspended microstrip line 546–547 SWM. see single-wire medium (SWM) symmetrical coupled lines 392 homogeneous medium 434–438 symmetrical coupled microstrip line 597–598 homogeneous medium 438–444 symmetrical coupled transmission lines 444–447 symmetrical directional coupler 395–396 symmetrically coupled microstrips 426 symmetric anisotropic materials 82–83 symmetric edge coupled coplanar waveguide (CPW) even-mode capacitance 430–431 odd-mode capacitance 431–432 symmetric edge coupled microstrips characteristic impedances 419–423 even-and odd-mode relative permittivity 419–423 frequency-dependent models 423–424 synthesis, coplanar stripline structures (CPS) 347–348 synthesis, coupled coplanar waveguide (CPW) 346–347 system-on-chip (SOC) 463

t tapped line resonator 658–659 telegraph 1–2 telegrapher’s equations 22 telephonic transmission 1 temporal dispersion 21 temporal period 20 TFMS. see thin-film microstrip (TFMS) thermal evaporation process 459–460 Thevenin’s theorem 33 thin film microstrip (TFMS) 262, 464, 467–468 thin film microstrip line (TFML) 292 thin-film process sputtering process 460 thermal evaporation process 459–460 thin-film structure 467–468 three-dimensional resonator 615 3D symmetrical Bruggeman formula 177 time-dependent electric polarization (P) 93 time-dependent magnetic polarization (M) 93 time-varying magnetic field 3 TMz and TEz modes 217 topshielded microstrip line 261 trans-admittance 49 transatlantic telegraphy 1 transmission [ABCD] parameters

current ratio 49 trans-admittance 49 trans-impedance 49 voltage ratio 49 transmission coefficient 54 transmission line classical. see Classical transmission line communication technology system 1 monopole and dipole antenna 1 objectives 1 planar 1 telegraph 1–2 transmission line, composite medium 124–125 transmission line model 641 transmission line resonator lumped resonator modeling 630–634 transmission matrix 45 transmission [T] matrix 56 transverse electric (TE) mode 213 perpendicular polarization 125–127 polarization 125, 134–135 waves 111 transverse electromagnetic (TEM) mode 213 transverse magnetic (TM) mode 213 parallel polarization 128–129 polarization 128, 135–136 waves 111 transverse resonance condition 368 transverse resonance method (TRM) 244, 357, 427, 771 transverse transmission line (TTL) multilayer microstrip line 515–518 Tripathi waveguides dispersion model 426–428 two-dimensional resonator 614 2D artificial dielectrics 169, 170 2D fractal resonators antenna 682–683 fractal resonators 683–686 geometry 674–681 2D lattice structures 697–703 2D planar EBG circuit model series circuit model 780–782 shunt circuit model 773–779 bloch impedance 779 uniplanar structure 779–780 2D planar EBG surfaces 753–756 characteristics reflection phase diagram 759–760 ω−β dispersion diagram 757–758

Index

horizontal wire dipole anisotropic 761 ground plane surface 762–763 reflecting surface 761–762 two-frequency wave-packet 66–69 two-port reaction-type resonator 629 two-port ring resonator 644 two-port transmission-type resonator 623–629

u UIR. see uniform impedance resonator (UIR) ultra-high frequency (UHF) 1 uniaxial anisotropic medium 129 uniform coupling 392 uniform impedance resonator (UIR) 639 uniform transmission lines circuit model 21–23 Kelvin–Heaviside transmission line equations frequency-domain 24–26 time domain 23–24 lossy transmission line, characteristic of 26–27 lumped circuit elements model 19 power relation 34–36 Thevenin’s theorem, application of 33–34 voltage and current-wave equation 28–33 wave equation with source 27–28 wave motion 19–21 uniplanar EBG structures 771–773 uniplanar transmission lines 301, 353 unique waveguide model 13 unitary properties 55

v values for coefficient 348 variation 493–495 variational expressions Fourier domain 498–503 microstrip line 503–512 transmission line energy method 495–497 Green’s function method 497–498 variational method 302 vector network analyzer (VNA) 46 vector potentials 214 vehicle material 461 very high-frequency (VHF) 1 Veselago flat lens 149–150 V-groove CMOS technology 477–478 virtual heights 378 voltage–current (vi) 378

voltage loop equation 23 voltage reflection coefficient 32 voltage standing wave ratio 32 voltage wave equation 24 voltaic pile 1

w wave admittance 362 wave equation 20 wave function 21 waveguide dispersion model 426 waveguide model sandwich slot line 370–371 shielded slot line. see shielded slot line standard slot line. see standard slot line waveguides equivalent circuit model power-current 245 power-voltage 245 transmission line model 245–247 wave vs. characteristic impedance 244–245 rectangular. see rectangular waveguide SIW complete mode 253–256 half mode 256–258 surface wave. see surface wave waveguide TEM-mode circuit relations 223 Kelvin-Heaviside transmission line equations parallel-plate waveguide 222 transverse resonance method dielectric-loaded 248–249 double-layer dielectric sheet backed,conductor 252–253 slab. see slab waveguide standard rectangular 247–248 waveguides as propagation medium 8 waveguide technology 478, 489 wave impedance 130, 228 wave motion 19–21 wave number of EM waves 214 wave propagation dispersion biaxial medium 114–115 uniaxial medium 116–118 isofrequency contours, concept 115 isofrequency surface, concept 115 uniaxial gyroelectric medium Faraday rotation 114 LHCP and RHCP wave 113

223

911

912

Index

wave propagation (cont’d) uniaxial medium mode of propagation 111 optical axis 111 phase shifters 112 TE waves 111 TM waves 111 waveplates 112 wave type circuits 45 wave velocity group velocity two-frequency wave-packet 66–69 phase velocity 63–66 wave vs. characteristic impedance 130–131 weak coupling 391 wet etching 471–472 Wheeler’s (Wh.) closed-form expressions 271 Wheeler’s incremental inductance method 339 Wheeler’s incremental inductance rule 293–295 Wheeler’s transformation 268, 542 w/h-ratio 377 wideband Debye models 205–207 Wiener model 174

wire medium equivalent parallel plate waveguide model polarization 786–789 reactance loaded wire medium 791–793

789–791

z zero-dimensional lumped resonator lumped parallel resonant circuit 617–619 lumped series resonant circuit 615–617 one-port reflection-type resonator 620–623 resonator with external circuit 619–620 two-port reaction-type resonator 629 two-port transmission-type resonator 623–629 zero-dimensional resonator 613 zeroth-order solution 357 [Z] parameters open-circuit parameters 46 port current (IN) 46 port voltage (VN) 46 self-impedance 46 symmetrical network 47 [Z] vs. [ABCD] parameters 59–60

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