Principles of RF and Microwave Design 1630816493, 9781630816490

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Principles of RF and Microwave Design

Morgan FM.indd i

10/22/2019 10:33:30 AM

For a complete listing of titles in the Artech House Microwave Library, turn to the back of this book.

Morgan FM.indd ii

10/22/2019 10:33:49 AM

Principles of RF and Microwave Design Matthew A. Morgan

Morgan FM.indd iii

10/22/2019 10:33:49 AM

Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the U.S. Library of Congress. British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library. Cover design by John Gomes

ISBN 13: 978-1-63081-649-0

© 2020 ARTECH HOUSE 685 Canton Street Norwood, MA 02062

All rights reserved. Printed and bound in the United States of America. No part of this book may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording, or by any information storage and retrieval system, without permission in writing from the publisher. All terms mentioned in this book that are known to be trademarks or service marks have been appropriately capitalized. Artech House cannot attest to the accuracy of this information. Use of a term in this book should not be regarded as affecting the validity of any trademark or service mark.

10 9 8 7 6 5 4 3 2 1

Morgan FM.indd iv

10/22/2019 10:33:49 AM

To my family, whom I treasure

Morgan FM.indd v

10/22/2019 10:33:49 AM

Morgan FM.indd vi

10/22/2019 10:33:49 AM

Contents Preface

xix

Acknowledgments

xxi

Chapter 1

1 1 2 3 4 6 8 9 10 13 13 16 18 20 23 24 26 28 28 30 35

Fields and Waves 1.1 Maxwell’s Equations 1.1.1 Large-Scale Form and Constitutive Relations 1.1.2 Physical Interpretation 1.1.3 The Continuity Equation 1.1.4 Boundary Conditions on a Perfect Conductor 1.1.5 Boundary Conditions at a Dielectric Interface 1.1.6 Differential Form 1.1.7 Differential Boundary Conditions 1.2 Static Fields 1.2.1 Electrostatics 1.2.2 Magnetostatics 1.2.3 Static Fields in a Coaxial Geometry 1.3 Wave Solutions 1.4 Spectral Analysis 1.4.1 Maxwell’s Equations in the Frequency Domain 1.4.2 Phasor Notation 1.5 Wave-Boundary Interactions 1.5.1 Reflection from a Conducting Boundary 1.5.2 Reflection and Refraction from a Dielectric Boundary 1.6 Lossy Materials

vii

viii

Chapter 2

Principles of RF and Microwave Design

1.6.1 Dielectric Polarization 1.6.2 Magnetic Polarization 1.6.3 Conduction 1.6.4 The Skin Effect 1.7 Lorentz Reciprocity 1.8 From Fields to Circuits Problems References

36 37 37 39 41 42 43 45

Lumped Elements 2.1 Voltage, Current, and Kirchhoff’s Laws 2.1.1 Kirchhoff’s Voltage Law 2.1.2 Kirchhoff’s Current Law 2.2 Lumped-Element Devices 2.2.1 Resistors 2.2.2 Reactive Elements 2.2.3 Capacitors 2.2.4 Inductors 2.2.5 Transformers 2.3 Energy and Power 2.3.1 Power Dissipation in Resistors 2.3.2 Energy Storage in a Capacitor 2.3.3 Energy Storage in an Inductor 2.3.4 Energy Transfer in a Transformer 2.4 Terminal Parameters in the Frequency Domain 2.4.1 Impedance and Admittance 2.4.2 Complex Power 2.5 Resonators 2.5.1 Series RLC Resonator 2.5.2 Parallel RLC Resonator 2.5.3 Loaded Q 2.6 Sources 2.7 Tellegen’s Theorem and Its Implications 2.7.1 Network Reciprocity 2.8 Parasitics 2.8.1 Realistic Capacitors 2.8.2 Realistic Inductors 2.8.3 Realistic Transformers

47 48 48 50 51 51 52 53 55 56 59 59 60 63 64 65 66 68 70 70 73 74 75 78 80 83 84 85 86

Contents

Problems References

ix

88 90

Chapter 3

Transmission Lines 3.1 Construction from Lumped Elements 3.1.1 Distributed Inductance and Resistance 3.1.2 Distributed Capacitance and Conductance 3.1.3 The Telegrapher’s Equations 3.2 Construction from Field Equations 3.2.1 Separation of Variables 3.2.2 Relating Terminal Parameters to Field Quantities 3.3 Transmission Lines in Circuits 3.3.1 The Terminated Transmission Line 3.3.2 Stubs and Quarter-Wave Transformers 3.3.3 Infinite T-Lines and the Characteristic Impedance 3.3.4 Signal Transmission with Transmission Lines 3.3.5 Loss in Transmission Lines 3.3.6 The 50Ω Standard 3.4 Transmission-Line Resonators 3.4.1 Half-Wave Resonators 3.4.2 Quarter-Wave Resonators 3.4.3 Coupling to a Transmission-Line Resonator 3.5 Printed Circuit Technologies 3.5.1 Microstrip Design Equations 3.5.2 Planar Junctions and Discontinuities 3.5.3 Radial Stubs 3.5.4 Coupled Lines Problems References

91 91 92 92 93 97 98 101 105 105 108 109 111 112 114 117 118 120 121 122 122 125 128 129 130 132

Chapter 4

Network Parameters 4.1 Immittance Network Parameters 4.1.1 Impedance Parameters 4.1.2 Admittance Parameters 4.1.3 ABCD-Parameters 4.1.4 Hybrid and Inverse-Hybrid Parameters 4.2 Wave Network Parameters 4.2.1 Scattering Parameters 4.2.2 Scattering Transfer Parameters

133 133 134 139 142 144 147 149 154

x

Chapter 5

Principles of RF and Microwave Design

4.3

Properties of Networks 4.3.1 Reciprocity 4.3.2 Symmetry 4.3.3 Antimetry 4.3.4 Losslessness 4.3.5 Passivity 4.3.6 A Constraint on Three-Port Networks 4.4 Signal-Flow Graphs 4.4.1 Principles of Signal-Flow Graphs 4.4.2 Decomposition Rules 4.4.3 Mason’s Rule 4.4.4 Equivalent Matrix Form 4.5 Even and Odd-Mode Analysis 4.5.1 Two-Port Example 4.5.2 Coupled Lines 4.6 Normalization of the Scattering Parameters 4.6.1 Generalized Scattering Parameters 4.6.2 Renormalization 4.7 Parameter-Defined Networks 4.7.1 Gyrators 4.7.2 Impedance Inverters Problems References

155 156 157 159 160 161 161 163 163 165 167 168 170 172 174 177 177 179 181 181 184 187 189

Transformations and Identities 5.1 Transformations 5.1.1 Impedance Scaling 5.1.2 Frequency Scaling 5.1.3 Frequency Inversion (High-Pass Transformation) 5.1.4 Band-Pass Transformation 5.1.5 Band-Stop Transformation 5.1.6 Richard’s Transformation 5.1.7 Immittance Inversion (Duality Transformation) 5.2 Identities 5.2.1 Delta-Wye Identity 5.2.2 Star-Mesh Identity 5.2.3 Inverter-Dual Identity 5.2.4 Equivalent Impedance Groups

191 191 192 193 194 197 199 200 203 208 208 211 212 213

Contents

5.2.5 5.2.6 5.2.7 5.2.8 Problems References

Transformer Identities Kuroda’s Identities Coupled-Line Identities A Three-Port Transmission-Line Identity

xi

214 216 220 221 223 224

Chapter 6

Impedance Matching 6.1 Single Frequency Matching 6.1.1 Voltage Standing-Wave Ratio (VSWR) 6.1.2 The Smith Chart 6.1.3 Movements on the Smith Chart 6.1.4 Two-Parameter Matching Networks 6.1.5 Two Transmission-Line Matching Networks 6.2 Broadband Impedance Matching 6.2.1 Comparison of Two-Parameter Matching Networks 6.2.2 Carter Chart 6.2.3 Immittance Folding 6.2.4 The Bode-Fano Criteria 6.3 Resistance Matching 6.3.1 Multisection Transformers 6.3.2 Maximally Flat (Binomial) Matching Transformers 6.3.3 Equiripple (Chebyshev) Matching Transformers 6.3.4 Tapered Transformers 6.3.5 Exponential Taper 6.3.6 Klopfenstein Taper Problems References

225 225 228 230 231 236 241 244 244 247 249 255 257 258 261 264 267 269 270 273 275

Chapter 7

Waveguides 7.1 Waveguide Modes 7.1.1 Cutoff Frequencies 7.1.2 Transverse Fields 7.1.3 Backward-Traveling Waves 7.1.4 Power Flow in Waveguide Modes 7.1.5 TE and TM Modes 7.2 Rectangular Waveguide 7.2.1 TM Modes 7.2.2 TE Modes

277 277 279 280 283 284 284 285 287 290

xii

Chapter 8

Principles of RF and Microwave Design

7.2.3 Charge and Current Distributions 7.2.4 Mode Plots 7.2.5 Dispersion Diagrams 7.2.6 Phase Velocity and Group Velocity 7.2.7 Losses 7.2.8 Modal Discontinuity Analysis 7.2.9 Waveguide Discontinuity Equivalent Circuits 7.3 Circular Waveguide 7.3.1 TM Modes 7.3.2 TE Modes 7.3.3 Modes, Dispersion, and Shielding 7.3.4 Losses 7.4 Waveguide Cavity Resonators 7.4.1 Rectangular Cavity Resonators 7.4.2 Cylindrical Cavity Resonators 7.4.3 Stored Energy in a Cavity 7.4.4 Quality Factors of Cavity Resonators 7.4.5 Other Cavity and Resonator Types 7.5 Ridged Waveguide 7.5.1 Double-Ridged Waveguide 7.5.2 Quad-Ridged and Triple-Ridged Waveguides 7.6 Coaxial Waveguide 7.7 Periodic Waveguide and Floquet Modes 7.7.1 Surface Impedance Approximation 7.7.2 Hybrid Modes 7.7.3 The Balanced Hybrid Condition 7.7.4 Fast, Slow, Forward, Backward, and Complex Waves 7.8 Dielectric Waveguides 7.8.1 Confinement by Total Internal Reflection 7.8.2 Fiber Cladding 7.8.3 LP Modes in Optical Fiber Problems References

292 293 296 299 301 304 311 313 313 316 318 320 324 324 327 330 332 333 333 333 336 337 340 341 344 346 347 349 350 351 352 356 357

Launchers and Transitions 8.1 Microstrip-to-Coax and CPW-to-Coax Transitions 8.1.1 End-Launch Transition 8.1.2 Right-Angle Transition

359 360 360 361

Contents

Chapter 9

xiii

8.2

All-Planar Transitions 8.2.1 Microstrip to CPW and GCPW 8.2.2 Microstrip to Slotline 8.2.3 CPW to Slotline 8.3 Balanced to Unbalanced 8.4 Coax to Waveguide 8.4.1 Coax to Rectangular Waveguide 8.4.2 Coax to Double-Ridged Waveguide 8.5 Microstrip to Waveguide 8.5.1 Longitudinal Probes 8.5.2 Vertical Probes 8.6 Waveguide Tapers 8.6.1 Trapped Mode Resonances 8.7 Hybrid-Mode Launcher 8.8 Beam Coupling to Optical Fibers 8.8.1 Numerical Aperture 8.9 Beam Generation Problems References

363 363 365 365 366 368 368 369 371 371 373 375 375 380 382 382 383 383 385

Antennas and Quasioptics 9.1 Parallel-Plate Waveguide 9.1.1 TEM Modes 9.1.2 TE Modes 9.1.3 TM Modes 9.2 Fourier Optics 9.2.1 Planar Fourier Optics 9.2.2 Rotman Lens 9.2.3 Fourier Optics in Free Space 9.3 Gaussian Beams 9.3.1 The Paraxial Helmholtz Equation 9.3.2 The Gaussian Mode Solution 9.3.3 Terms of the Gaussian Beam Equation 9.3.4 Complex Beam Parameter 9.4 Ray Transfer Matrices 9.4.1 Rays in Gaussian Beams 9.4.2 Matrices for Common Optical Elements 9.4.3 Application to Gaussian Beams

387 387 388 390 391 394 394 398 399 399 400 402 405 408 410 411 411 415

xiv

Chapter 10

Principles of RF and Microwave Design

9.5

Field Equations for Antennas 9.5.1 Potential Formulation of Maxwell’s Equations 9.5.2 Retarded Potentials 9.6 Wire Antennas 9.6.1 Short Dipoles 9.6.2 Half-Wave Dipoles 9.6.3 Gain and Directivity 9.6.4 Radiation Resistance 9.6.5 Other Wire Antennas 9.7 Complementary Antennas 9.7.1 Half-Wave Slot 9.8 Planar Antennas 9.8.1 Microstrip Patch Antenna 9.8.2 Planar Inverted-F Antenna (PIFA) 9.9 Horn Antennas 9.9.1 Pyramidal Horn 9.9.2 Conical Horn 9.9.3 Potter Horn 9.9.4 Conical Corrugated Feedhorn 9.10 Metrics for Directive Beams 9.10.1 Beam Area 9.10.2 Beam Efficiency and Stray Factor 9.10.3 Effective Area 9.11 Friis Transmission Equation Problems References

418 418 421 424 424 426 429 430 431 433 433 437 438 439 440 441 443 444 445 447 447 448 449 451 453 454

Flat-Frequency Components 10.1 Terminations 10.1.1 Printed-Circuit Terminations 10.1.2 Waveguide Terminations 10.1.3 Absorber 10.2 Attenuators 10.3 Splitters 10.3.1 Tee Junctions 10.3.2 Resistive Splitters 10.3.3 Wilkinson Power Dividers 10.3.4 Gysel Power Divider

457 457 458 460 461 464 465 466 467 468 471

Contents

Chapter 11

xv

10.3.5 Rectangular Waveguide Combiners 10.4 Phase Shifters 10.4.1 Lumped-Element High-Pass and Low-Pass 10.4.2 Schiffman Phase Shifters 10.4.3 Corrugated Phase Shifters 10.5 Directional Couplers 10.5.1 Coupled-Line Coupler 10.5.2 Multisection Couplers 10.5.3 Codirectional Couplers 10.5.4 Multi-Aperture Waveguide Couplers 10.5.5 Beam Splitters 10.6 Quadrature Hybrids 10.6.1 Branchline Hybrids 10.6.2 Lange Couplers and Tandem Couplers 10.7 180◦ Hybrids 10.7.1 Rat-Race Hybrids 10.7.2 Waveguide Magic Tee 10.7.3 Lumped-Element 180◦ Hybrids 10.8 Nonreciprocal Components 10.8.1 Physical Mechanism of Nonreciprocity 10.8.2 Faraday Rotation Devices 10.8.3 Resonant Absorption Isolators 10.8.4 Stripline Circulators and Isolators Problems References

472 474 475 476 477 478 478 480 482 484 487 490 491 494 495 496 498 499 499 500 501 503 504 505 506

Frequency-Selective Components 11.1 Equalizers 11.1.1 Lumped-Element Equalizers 11.1.2 Transmission-Line Equalizers 11.2 Foundations of Electronic Filters 11.2.1 Periodic Networks 11.2.2 Canonical Filter Responses 11.2.3 Lumped-Element Ladders 11.3 Reflectionless Filters 11.3.1 Topological Basis of Reflectionless Filters 11.3.2 Mitigation of Negative Elements 11.3.3 All-Pole Reflectionless Filters

509 509 510 513 515 515 519 521 527 527 530 533

xvi

Chapter 12

Principles of RF and Microwave Design

11.4 Distributed Filters 11.4.1 Stepped-Impedance Filters 11.4.2 Coupled Resonators 11.4.3 Susceptance Slope and Band-Pass Scaling 11.4.4 Edge-Coupled Band-Pass Filters 11.4.5 General Coupling-Routing Topologies 11.4.6 Empirical Extraction of Couplings 11.5 Waveguide Filters 11.5.1 Half-Wave Resonator Waveguide Filters 11.5.2 Stepped-Height Waveguide Filter 11.5.3 Waffle-Iron Low-Pass Filter 11.6 Frequency-Selective Surfaces 11.7 Time-Domain or Pulse-Shaping Filters 11.7.1 Gaussian Filters 11.7.2 Bessel-Thomson Filters 11.8 Multiplexers Problems References

534 535 536 541 543 546 547 549 550 551 552 553 554 556 556 557 560 562

Amplifiers 12.1 Gain and Stability 12.1.1 Power Gain Factors 12.1.2 Oscillations 12.1.3 Stability Circles 12.1.4 Rollet Stability Factor 12.1.5 Geometric Stability Factors 12.1.6 Maximum Gains 12.1.7 Balanced Amplifiers 12.2 Low-Noise Amplifiers 12.2.1 Noise Temperature and Power 12.2.2 Noise Models of Two-Port Networks 12.2.3 Noise Parameters and Matching 12.3 Power Amplifiers 12.3.1 Saturated Power 12.3.2 Linearity 12.3.3 Efficiency 12.4 Multistage Amplifiers Problems

565 565 566 570 572 574 575 576 578 579 580 583 584 592 593 596 600 602 603

Contents

xvii

References

605

Frequency Conversion 13.1 General Nonlinearity 13.2 Multipliers 13.2.1 Single-Element Multipliers 13.2.2 Selective Harmonic Suppression 13.3 Mixers 13.3.1 Single-Diode Mixers 13.3.2 Single-Balanced 13.3.3 Double-Balanced 13.3.4 Triple-Balanced 13.3.5 Sideband-Separating 13.3.6 Subharmonic 13.4 Fundamental Limits Upon Conversion Efficiency Problems References

607 607 611 611 612 614 615 616 619 620 622 627 628 631 632

Appendix A Mathematical Identities A.1 Trigonometric Identities A.1.1 Pythagorean Identities A.1.2 Sum-Angle Formulas A.1.3 Double-Angle and Triple-Angle Formulas A.1.4 Half-Angle Formulas A.1.5 Power Reduction A.1.6 Product-to-Sum Formulas (Prosthaphaeresis) A.1.7 Sum-to-Product Formulas A.1.8 Composite-Angle Formulas A.1.9 Outphase Summation A.1.10 Trigonometric Products A.1.11 Exponential Forms (Euler’s Formula) A.2 Hyperbolic Functions and Identities A.2.1 Relationships to Standard Trigonometric Functions A.2.2 Pythagorean Identities A.2.3 Sum-Argument Formulas A.2.4 Double-Argument Formulas A.2.5 Half-Argument Formulas A.2.6 Sum-to-Product Formulas A.2.7 Exponential Forms

633 633 633 633 634 634 634 635 635 635 636 636 637 637 637 637 637 638 638 638 638

Chapter 13

xviii

Principles of RF and Microwave Design

A.3 Other Theorems A.3.1 Binomial Theorem A.3.2 Law of Sines and Cosines A.4 Integrals A.4.1 Trigonometric Integrals A.4.2 Other Useful Integrals

638 638 639 639 639 640

Appendix B Integral Theorems B.1 The Divergence Theorem B.2 Stokes’ Theorem

641 641 642

Appendix C Vector Identities

643

Appendix D Vector Operator Forms D.1 Cartesian Coordinates (x, y, z) D.2 Cylindrical Coordinates (r, θ, z) D.3 Spherical Coordinates (r, θ, φ)

645 645 646 647

Appendix E Delta-Wye Identities

649

Appendix F Transmission-Line Identities References

651 657

Appendix G Special Functions G.1 Chebyshev Polynomials G.2 Bessel Functions G.2.1 Recurrence Formulas G.2.2 Zeros and Extrema G.2.3 Asymptotic Approximations G.2.4 Modified Bessel Functions References

659 659 661 663 664 665 666 667

About the Author

669

Index

671

Preface I suppose the question I need to answer is, “Why another microwave engineering book?” There are already so many — Pozar, Rizzi, Collin; these were good enough for me, why not for everyone else? The answer is threefold. First — and this is perhaps one of the best reasons for writing a book — I rather like the subject. I find that writing is one of the best ways to firm up one’s own conceptual understanding of things. I recently discovered an interview of a fellow author, and a personal inspiration of mine, Dr. Stephen Maas, who admitted that he frequently goes back to his own books to find information. As he put it, there is really no substitute for a cleanly written reference that describes things in exactly the way the reader thinks about them. I could not agree more. My publications have become the go-to sources for me to refresh my memory of things that I once knew but have not used in a while. That is reason enough for me to write this book. I know not how many others may find it worth keeping on their shelves, but for myself, at least, it will always be close at hand. The second reason for writing this book is that in the course of my career I have encountered a surprising number of young students and engineers — and even a few not so young — who have learned what steps are required to solve a particular microwave engineering problem, but lack the deeper insight to understand why. They know how to run a simulator and to optimize a given design, but then fail to notice the small, simple change in topology that could make the design work better. They can calculate the characteristic impedance of a transmission line, but then wonder how it can be real-valued without the line being lossy. They know how to draw a waveguide splitter, but fail to anticipate the resonance that appears due to a trapped higher-order mode. In short, they know what formulas to use, but not where those formulas came from, nor what implicit assumptions were made

xix

xx

Principles of RF and Microwave Design

to derive them. To be clear, mathematics is an indispensable tool for us to shine a light on the truths hidden beneath the surface of physical laws, and it shall not be neglected in these pages, but neither shall we ignore the imperative to connect what the math teaches us with the clarity of an accurate mental picture. Math without intuition is clumsy. Intuition without math is impotent. We must have both in order to realize our full creative potential as engineers, and it is my hope to offer a way of looking at things that helps the next generation of electronics innovators do exactly that. Finally, I have come to realize that most of the microwave engineers I have met fall into one of two groups: those who think primarily in terms of lumped elements, and those who think primarily in terms of transmission lines, as if one were a truer and more pure description of reality than the other. The former group is forever recasting problems — even those that pertain to waveguides and antennas — into their lumped-element equivalents. The latter group sees all as wave propagation, knowing in their hearts that all inductors are merely tightly coupled transmission lines in disguise, and believing (erroneously) that waveguides are transmission lines also, but perhaps less ideal in their characteristics than coax. The truth, as I see it, is that all such models are merely useful idealizations, and that none has any greater claim to truthfulness or sophistication than another. Preferring one model over another is sort of like being right or left-handed — a simple preference that, in this case, may arise in part from our particular way of thinking about the world, or as a consequence of which teachers we had for which courses. As I have said in other writings, the most successful microwave engineer is the one that is adept at adjusting his or her viewpoint from one physical model to another as is best suited to a particular problem. It is for this reason that I reintroduce lumped-element concepts in Chapter 2 as a reduction of Maxwell’s equations — even though most readers will surely have learned the former before mastering the latter. I then develop transmission lines, waveguides, and quasioptical components in the same way, building up our repertoire of available circuit elements all on equal footing, each the master of their own particular corner of parameter space. Only then do I talk about engineered components such as couplers, baluns, filters, and active devices and describe their construction using all types of elements (lumped, distributed, and quasi-optical) side by side so that the reader may become accustomed to choosing the right approach for each individual application.

Acknowledgments I am once again indebted to my friend and colleague, Dr. Shing Kuo Pan, for his meticulous review of the derivations found throughout this book. Thanks to him, a host of my embarrassing mistakes will never see the light of day!

xxi

Chapter 1 Fields and Waves Although it is assumed that the readers of this book will already have a working familiarity with electromagnetic concepts, it is deemed useful to review those concepts here so that the derivations that follow in later chapters are well supported by sound, physical principles. It is further the intent of this chapter to acclimate the reader to my mathematical and conceptual style in a familiar setting, so that newer concepts introduced in later chapters are more easily grasped. We therefore begin with an overview of the foundational laws of electromagnetics and the mathematical principles upon which all other technologies in this book will be built. Some notational conventions will be established, and the physical processes that lead to the macroscopic properties of the materials used to construct our circuits will be described.

1.1

MAXWELL’S EQUATIONS

The foundation of all electronic circuits — and arguably, by extension, of modern civilization1 — is Maxwell’s equations. Known by many equivalent formulations (integral, differential, time harmonic, relativistic), these classical equations govern the relationship and behavior of the electric and magnetic fields, which in turn describe the forces that charges, either static or in motion, exert upon one another. 1

The great physicist and teacher, Richard Feynman, famously stated, “From a long view of the history of mankind, seen from, say, ten thousand years from now, there can be little doubt that the most significant event of the 19th century will be judged as Maxwell’s discovery of the laws of electrodynamics. The American Civil War will pale into provincial insignificance in comparison with this important scientific event of the same decade.”

1

2

Principles of RF and Microwave Design

1.1.1

Large-Scale Form and Constitutive Relations

The integral (large-scale) form of Maxwell’s equations is given below2 , ‹ ˚ D · dS = ρdV S

(1.1a)

V

‹ B · dS = 0 S

˛

E · dl = − ˛

l

¨

H · dl = l

∂ ∂t

(1.1b)

¨

J · dS + S

B · dS S

∂ ∂t

(1.1c)

¨ D · dS

(1.1d)

S

where E and H are the electric and magnetic field vectors, respectively. D is the electric flux density or electric displacement, which is related to the electric field via the dielectric permittivity, ε, of the constituent medium, D = εE

(1.2)

Similarly, B is the magnetic flux density, defined by the constitutive relation B = µH

(1.3)

where µ is the magnetic permeability of the medium. These fields, in principle, are generated by charges and their motions, such that the charge density, ρ, and current density, J, are referred to in these equations as sources. The integrals are taken on the surface S (with outward directed vector S) enclosing an arbitrary volume V , or on the closed loop l (with directed arc length vector l) enclosing an open surface S as selected to suit the boundary conditions of the problem under consideration. These subscripts may be omitted from the integrals in question when the region of integration is clear from context. The electric and magnetic fields in turn exert a force, F, on charged particles according to the Lorentz law, F = q (E + v × B) 2

(1.4)

For the purposes of this book, we will work almost exclusively with the SI unit system, which is most prevalent in engineering fields, as opposed to the Gaussian-CGS units, which in the past were popular in certain scientific disciplines such as theoretical physics and astronomy [1].

3

Fields and Waves

εE

μH

ρ

(a)

(b)

E μH

H J, εE

(c)

(d)

Figure 1.1 Visual representation of Maxwell’s equations. (a) Gauss’s law, eq (1.1a). (b) Nonexistence of magnetic charge, (1.1b). (c) Faraday’s law, (1.1c). (d) Generalized Amp`ere’s law, (1.1d).

where q is the electric charge and v is the velocity of motion of that particle, if any. In general, the permittivity and permeability are anisotropic (directiondependent) functions of space, time, and even field strength in nonlinear media; however, it is sufficient in most cases to consider them scalar constants on the temporal and spatial domain as well as the energy scale of a particular solution, as written above. Similarly, as with any classical field theory, it is now recognized that a correction (quantum electrodynamics) is needed to account for certain phenomena observed at the smallest scales, such as photon-photon interactions. However, for the purposes of all cases discussed in this book, the above formulation of Maxwell’s equations, the constitutive relations, and the associated forces may be considered as exact and fundamentally true [2]. 1.1.2

Physical Interpretation

A visual representation of Maxwell’s equations is given in Figure 1.1. The first, (1.1a), illustrated in Figure 1.1(a), is known as Gauss’s law and states that the total electric field strength emanating from a finite volume of space is equal to the net electric charge within that region. The second3 , (1.1b), illustrated in Figure 1.1(b), has a similar form, describing the total magnetic field strength emanating from a volume of space. As there is no magnetic charge, the magnetic field lines must be continuous throughout the region, and the number of lines entering it must be the same as the number leaving it. The third, (1.1c), illustrated in Figure 1.1(c), is Faraday’s law of induction, which states that the total electric field around any closed loop is associated with 3

There is no universally accepted name for the second of Maxwell’s equations. Some call it “Gauss’s law for magnetism,” in recognition of its similarity to the original Gauss’s law, others call it the “Absence/Nonexistence of Magnetic Poles/Charge” or the “Magnetic Nondivergence,” and still others simply concede that it has no name.

4

Principles of RF and Microwave Design

an opposing increase of the magnetic field threading through that loop (opposing in this case because the time derivative of the magnetic field is negative when oriented according to the right-hand rule [3]). Once again, there is a similar relation for magnetic field loops in (1.1d), illustrated in Figure 1.1(d), known as Amp`ere’s law. In this case the center of the loop under consideration may be threaded by either a changing electric field, or an electric current, or both. To summarize, • The electric field emanates from electrical charge (1.1a) and circulates around changing magnetic fields (1.1c). • The magnetic field, in turn, emanates from nowhere (1.1b) but forms closed loops threaded by electrical currents and changing electric fields (1.1d). Note that in a source-free region, where ρ = 0 and J = 0, the set of equations becomes symmetric in E and H, in that one may be exchanged for the other without altering their relationships (with exception of a sign change in (1.1d) and some scaling factors associated with the impedance of the medium). This is a condition known as duality which plays an elegant role in the solution of many electromagnetic and circuit problems [4]. It shows that a field solution for given stimuli and boundary conditions applies equally well to the dual problem involving magnetic boundaries in place of electric boundaries, E in place of H, and vice versa, with a reversal of polarity in the coupled fields resulting from the sign change. Duality will be discussed in the context of electrical networks in Section 5.1.7. In the original form of Amp`ere’s law, discovered experimentally, the second term involving the time derivative of the electric field was missed. This term (or its spatially localized form) is known as the displacement current and was introduced by James Clerk Maxwell to resolve certain inconsistencies in the remaining set of equations. One such inconsistency is the continuity of electric charge. 1.1.3

The Continuity Equation

Consider a surface, S, enclosing a finite volume of space with a small cutout bounded by a loop, l, as shown in Figure 1.2(a). We may apply (1.1d) to this geometry by integrating the magnetic field, H, around the closed loop on the left side of the equation, and the physical current density, J, plus the displacement current density, dD/dt, through the surface on the right side of the equation. Let us further take the limit as the cutout defined by the loop, l, becomes vanishingly small. Since the magnetic field must be finite, the path integral on the left side

5

Fields and Waves

J

J

H

l

H

H

S

S2

S1

(a)

(b)

Figure 1.2 Derivation of the continuity equation for electric charge. (a) Evaluation of the path integral around a vanishingly small closed loop. (b) Summing the contributions from two coincident path integrals in opposite directions.

of (1.1d) approaches zero as the loop circumference diminishes. In this limit, the surface S also becomes closed, allowing us to substitute (1.1d) into (1.1a) such that ‹ 0=

J · dS +

∂ ∂t

‹

¨ D · dS =

J · dS +

‹ J · dS = −

∴

∂ ∂t

∂ ∂t

˚ ρdV

(1.5a)

˚ ρdV

(1.5b)

This result, known as the continuity equation, makes explicit the conservation of electric charge. A net current entering a certain region of space must be associated with an accumulation of charge there, whereas a current exiting must lead to charge depletion. If the reader is troubled by the concept of a path integral around a vanishingly small closed loop, an alternate derivation may proceed as follows. Instead of allowing the loop l to shrink in size, we evaluate the path integral in both directions, counter clockwise for surface S1 and clockwise for surface S2 , as shown in Figure 1.2(b). By summing the contributions from both integrals, we obtain ¨

¨ J · dS +

S1

∂ J · dS + ∂t

S2

¨ S1

¨ ∂ D · dS + D · dS ∂t S2 ˛ ˛ = H · dl − H · dl = 0 (1.6a) l

‹ ∴ S1 +S2

∂ J · dS + ∂t

l

‹ D · dS = 0 S1 +S2

(1.6b)

6

Principles of RF and Microwave Design

n

Js

S

(a)

l

(b)

Figure 1.3 (a) Closed surface for determination of normal (perpendicular) fields at a conducting boundary. (b) Closed loop for determination of tangential fields at a conducting boundary.

‹ S1 +S2

∂ J · dS = − ∂t

˚ ρdV

(1.6c)

V

To some, the truth of the continuity equation may seem rather obvious. Of significance, however, is that this result would not have been obtained without the displacement current term added by Maxwell, for in that case we would have found that the net current entering or leaving a finite region of space must always be zero. Put another way, if magnetic fields did not respond to changing electric fields as Maxwell suggested, it would be physically impossible for a non-neutral charge to accumulate anywhere in space (nor would it be possible to discharge any accumulated electrostatic potential leftover somehow by the initial conditions of the universe). 1.1.4

Boundary Conditions on a Perfect Conductor

To simplify the derivation of the behaviors of many kinds of circuit elements, it is useful to enumerate first the solutions of Maxwell’s equations at the interface between conductive and nonconductive media (e.g., between metal and a dielectric insulator). For many metals, it is a good approximation to consider their conductivity as infinite for the initial field solution and then account for their losses as a small perturbation. Such a medium will be a called a perfect electric conductor (PEC) or electric wall. It can be shown that both the electric and magnetic fields must vanish inside a perfect conductor, so we will be concerned primarily with the normal (perpendicular) and tangential components of the electromagnetic fields just outside the interface to the conductor. Consider the closed surface drawn in Figure 1.3(a). Sometimes called a pillbox, we assume that it is vanishingly small in height and small enough (but finite) in diameter that the fields may be considered constant throughout. By applying

Fields and Waves

(1.1a), we find

7

‹ D · dS = (n · D) A = ρs A

(1.7a)

ρs (1.7b) ε where A is the area of the top of the pillbox, n is the unit surface normal, and ρs is the surface (as opposed to volumetric) charge density. Similarly, by applying (1.1b), we find for the magnetic field n·B=0 (1.8a) n·E=

∴n·H=0

(1.8b)

To probe the tangential fields at the surface, we use a small closed loop that runs parallel to the interface lengthwise above and below the conducting boundary, as shown in Figure 1.3(b). In this case we assume that the length of loop parallel to the surface is small enough that the fields are constant along it, while the height of the loop is vanishingly small. Consider (1.1c) on this path, from which we can derive the tangential component of the electric field. The surface integral of B vanishes because the field is finite and the area enclosed by the loop is infinitesimal. The derivative of this surface integral also vanishes because we assume that the fields cannot change value infinitely fast. Therefore, we have ˛ E · dl = (n × E) L = 0 (1.9a) ∴n×E=0

(1.9b)

where L is the small but finite length of the loop. Similarly, applying (1.1d) for the magnetic field, ˛ H · dl = (n × H) L = Js L ∴ n × H = Js

(1.10a) (1.10b)

where Js is the current density at the surface, and we have used the fact that ¨ D · dS = 0 (1.11) owing to the vanishingly small size of the area enclosed by the loop. In conclusion, we find that the electric field must always be perpendicular to the surface of a perfect electric conductor with strength proportional to the charge concentrated at the surface, while the magnetic field must be entirely parallel to it, but perpendicular to the direction of current flow, with strength equal to the surface current density [5].

8

1.1.5

Principles of RF and Microwave Design

Boundary Conditions at a Dielectric Interface

Similar loop and pillbox test structures may be used to derive the boundary conditions from Maxwell’s equations at an interface between two dielectric media instead of between a dielectric and a conductor. Let us consider the fields, E1 and H1 , in the first dielectric medium having constituent parameters ε1 and µ1 , and the fields in the second medium, E2 and H2 , where the second medium has constituent parameters ε2 and µ2 . Since dielectric media are generally nonconducting, there is normally no current at the interface, nor any mechanism for a buildup of unbound electric charge. For this derivation, then, we assume that the interface is source free. The pillbox gives us the normal component of the electric field via (1.1a) ‹ D · dS = (ε1 n · E1 − ε2 n · E2 ) A = 0 ∴ n · (ε1 E1 − ε2 E2 ) = 0

(1.12a) (1.12b)

Similarly, the normal component of the magnetic field is found by (1.1b) ‹ B · dS = (µ1 n · H1 − µ2 n · H2 ) A = 0 ∴ n · (µ1 H1 − µ2 H2 ) = 0

(1.13a) (1.13b)

The tangential fields are found by evaluating (1.1c) and (1.1d) around a small loop at the surface ˛ E · dl = (n × E1 − n × E2 ) L = 0

˛

∴ n × (E1 − E2 ) = 0 H · dl = (n × H1 − n × H2 ) L = 0 ∴ n × (H1 − H2 ) = 0

(1.14a) (1.14b) (1.14c) (1.14d)

In conclusion, the tangential electric and magnetic field components are continuous across a dielectric boundary, while the normal field components change discontinuously at the boundary in proportion to the relevant constituent parameters (ε and µ) [5].

9

Fields and Waves

1.1.6

Differential Form

It is often convenient to work with Maxwell’s equations in differential form instead of the integral form. The differential form of Gauss’s law may be found by application of the divergence theorem (also called Gauss’s theorem [6] or Ostrogradsky’s theorem) to the surface integral of the electric flux density, ‹

˚ D · dS =

˚ (∇ · D) dV =

ρdV

(1.15)

The symbol ∇, called the del operator, is given in Cartesian coordinates by ∇=

∂ ∂ ∂ x+ y+ z ∂x ∂y ∂z

(1.16)

where x, y, and z, are the unit vectors in the direction of the spatial coordinates x, y, and z, respectively. The del operator applied directly to a scalar function is referred to as the gradient, while the dot product of the del operator with a vector field is the divergence of that field, and the cross product is its curl. Since (1.16) must hold for any arbitrary choice of the enclosing surface, S, the arguments inside the volume integrals must be equivalent at all points. We may therefore remove the integrals, and are left with ∇·D=ρ

(1.17)

Similarly, for the magnetic flux density, we have ∇·B=0

(1.18)

We may also apply Stokes’ theorem [6] to the path integrals in (1.1c) and (1.1d), such that ˛ ¨ ¨ ∂ E · dl = (∇ × E) · dS = − B · dS (1.19a) ∂t ∴∇×E=−

∂B ∂t

(1.19b)

and ˛

¨ H · dl =

¨ (∇ × H) · dS =

∂ J · dS + ∂t

¨ D · ds

(1.20a)

10

Principles of RF and Microwave Design

∂D (1.20b) ∂t The differential form of the continuity equation then follows directly by taking the divergence of (1.20b), noting that the divergence of the curl of any vector field must be zero [6], and then substituting from (1.17), ∴∇×H=J+

∇ · (∇ × H) = ∇ · J +

∂ (∇ · D) ∂t

(1.21a)

∂ρ (1.21b) ∂t ∂ρ ∴∇·J=− (1.21c) ∂t It is worth pointing out that Maxwell’s equations are not mutually independent. Take, for example, the divergence of (1.19b), 0=∇·J+

∇ · (∇ × E) = −

∂ (∇ · B) = 0 ∂t

∴ ∇ · B = ρm

(1.22a) (1.22b)

where ρm is a placeholder for a fictitious magnetic charge, a dual counterpart to electric charge. Mathematically, this is consistent, but since no such charge is known to exist, we traditionally set this term equal to zero, and (1.22b) becomes equivalent to (1.18). It should be noted, however, that if such a charge is ever proven to exist, ρm may be reintroduced without any further disruption to the accuracy or selfconsistency of Maxwell’s equations. 1.1.7

Differential Boundary Conditions

In Section 1.1.4, boundary conditions were derived for fields adjacent to a PEC. A useful special case occurs wh