Handbook of RF-Microwave Components and Engineering [1 ed.] 0471390569, 2003016089

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HANDB □□ K CF I

RF/MI. CR □ W:AVE · .

Ca M

~

•

.

P □ -N ENTS AND

.

ENGINEERING

I

HANDBOOK OF RF/MICROWAVE COMPONE TS AND E GINEERING

Handbook of RF/Microwave Components and Engineering Editor-in-Chief

Kai Chang _

Department of Electrical Engineering Texas A&M University College Station, Texas

Members of Editorial Board T.T. Fong G.I. Haddad N.C. Luhmann R.D. Nevels F.K. Schwering H .F. Taylor S.K. Yao

HANDBOOK OF RF/MICROWAVE COMPONENTS AND ENGINEERING

Edited by

KAI CHANG

~WILEY-INTERSCIENCE •

A JOHN WILEY & SONS, INC., PUBLICATION

Copyright © 2003 by Jobn Wiley & Sons, Inc. All rights reserved. Published by John Wiley & Sons, Inc., Hoboken, New Jersey. Published simultaneously in Canada. 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, scanning, or otherwise. except as permitted under Section 107 or 108 of the 1976 United States Copyright Act. without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 R osewood Drive, Danvers, MA 01923, 978-750-8400, fax 978-750-4470, or on the web at www.copyrigbt.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-60 11. fax (201) 748-6008. e-mail: permreq @wiley.com . Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. Yo u should consult with a professional where appropriate. Neither the publisher nor author shall be liable for any loss of profit or any o ther commercial damages, including but not limited to s pecial, incidental. consequential. or o ther damages. For general information on our other products and service please contact o ur Customer Care Department within the U.S . at 877-762-2974, outside the U.S. at 317-572-3993 or fax 317-572-4002. Wiley also publishes its books in a variety of electronic formats. Some content that appears in print. however, may not be available in electronic format.

Library of Congress Cataloging-in-Publicatwn Data: Handbook of RF/microwave compo nen ts and engineering / Kai Chang~ Editor. p. cm . Includes index. ISBN 0-471-39056-9 (Cloth) 1. Radio circuits. 2. Microwave circuits. 3. R adio frequency integrated circuits. 4. Antennas (Electronics) 5. Passive components. L Chang, Kai, 1948-

TK 6560.H343 2003 621.384'12-dc22 Printed in the United States of America.

10 9 8 7 6 5 4 3 2 1

2003016089

HANDBOOK OF RF/MICROWAVE COMPONENTS AND ENGINEERING 1. Transmi ion Lines 2. Tran mi ion-Line Di continuities

3. Filter , Hybrid and Coupler , Power Combiners, and Matching Networks

4. Cavities and Re onators 5. Ferrite Control Component

6. Surface Acou tic Wave Devices 7. Quasi-Optical Technique 8. Components for Surveillance and Electronic Warfare Receiver 9. Antennas I : Fundamentals and

umerical Methods

10. Antennas II: Reflector, Lens, Hom, and Other Microwave Antennas of Conventional Configuration 11. Antennas ID: Array, Mi11imeter Wave, and Integrated Antennas 12. Antennas IV: Micro trip Antennas

13. Antennas V: Active Integrated Antennas 14. Mixers and Detectors 15. Multipliers and Parametric Device 16. Semiconductor Control Devices: PIN Diodes 17. Semiconductor Control Devices: Phase Shifters and Switches 18. Transferred Electron Devices

19. IMPA'IT and Related Transit-Tune Devices 20. Microwave Silicon Bipolar Transistors and Monolithic Integrated Circuits 21. FETs: P ower Applications

22. FETs: Low-Noise Applications 23. High-Electron-Mobility Transistors: Principles and Applications 24. Heterojunction Bipolar Transistors and Applications

25. Oscillators and Frequency Synthesizers 26. RF Components 27. Microwave Superconductors 28. Microwave MEMS and Micromachining

HANDBOOK OF OPTICAL COMPONENTS AND ENGINEERING 1. Optical Wave Propagation 2. Infrared Techniques 3. Optical Lenses

4. Optical Resonators 5. Spatial Filters and Fourier Optics 6. Semiconductor Lasers 7. Solid-State Lasers 8. Liquid Lasers 9. Gas Lasers 10. Optical Fiber Transmission Technology 11 . Optical Channel Waveguides and Waveguide Couplers 12. Planar Optical Waveguides and Applications 13. Optical Attenuators, Isolators, Circulators, and Polarizers 14. Optical Filters for Telecommunication Applications 15. Wavelength Division Multiplexers and Demultiplexers 16. Wide-Bandwidth Optical Intensity Modulators 17. Optical Modulation: Acousto-Optical Devices 18. Optical Modulation: Magneto-Optical Devices 19. Optical Detectors 20. Acousto-Optic Modulators and Switches 21. Optical Amplifiers

CONTENTS

Preface Contributors

xv xvii

1. Transmission Lines

1

lnder J. Bahl

1.1

1.2 1.3 1.4 1.5

Basics of Transmission Lines 1 Characteristics of Conventional Transmission Structures Characteristics of Planar Transmission Lines 19 Quasi-Planar Transmission Lines 42 Coupled Lines 54 References 61

2. Transmission-Line Discontinuities

2

67

KC. Gupta

2.1

2.2 2.3 2.4 2.5

2.6 2. 7

Introduction 67 Coaxial-Line Discontinuities 68 Rectangular Waveguide Discontinuities 74 Stripline Discontinuities 97 Microstrip Discontinuities 103 Finline Discontinuities 116 Discontinuities in Other Planar Transmission Structures 120 References 121

vii

viii

CONTENTS

127

3. Filters, Hybrids and Couplers, Power Combiners, and Matching Networks Jnder J. Bahl

3.1 3.2 3.3 3.4

Filters 127 Hybrids and Couplers 157 Power Combiners 186 Impedance,.Matching Networks References 201

190

209

4. Cavities and Resonators Michael Dydyk

4.1 4.2 4.3 4.4 4.5 4.6

Introduction 209 Basic Definitions 210 Design of and Coupling to Waveguide-Cavity Resonators 214 Design of Planar Resonators 226 Microstrip Dielectric Resonators 238 Effects of Surface Finish on Quality Factors 252 References 253 Appendix A 255

259

5. Ferrite Control Components William E. Hord

5.1 5.2 5.3 5.4 5.5

Introduction 259 Isolators 270 Circulators 273 Phase Shifters 279 Subassemblies Using Ferrite Phase Shifters 289 References 292

6. Surface Acoustic Wave Devices

295

David P. Morgan

6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9

Introduction 295 Basic SAW Devices 299 Propagation Effects and Materials 316 Nonreflective SAW Transducers 327 P-Matrix 336 Internal Reflectivity - Gratings and Transducers Coupled-Mode Equations 349 Low-Loss Filters -Nonresonant 353 Resonators and Resonator Filters 364

341

CONTENTS

6.10

ix

Concluding Remarks 376 Acknowledgments 378 Reference 3 7 8

7. Quasi-Optical Techniques

387

Paul F. Goldsmith. Tatsuo Iroh, Karl D. Stephan, and Amir Mortazawi

7 .1 7 .2 7.3 7.4 7 .5

Gau ian Beam Propagation 387 Quasi-Optical Interface to Planar Circuits 397 Active Quasi-Optical Devices 400 Quasi-Optical/Spatial Power Combining 405 Quasi-Optical/Spatial Amplifier Arrays 408 References 422

8. Components for Surveillance and Electronic Warfare Receivers

427

James B. Y. Tsui and Charles H. Krueger

8.1 Introduction 427 8.2 Basics of EW Receivers (1, 2] 427 8.3 Superbeterodyne Receivers 428 8.4 Instantaneous Frequency Measurement Receivers 434 8.5 Introduction to Channelized Receivers [l , 2] 443 8.6 Compressive Receivers (60-69] 458 8.7 Bragg Cell Receivers [82-86] 465 8.8 Cueing Receivers 470 8.9 Frequency-Selective Limiters [109-112] 473 8.10 Signal-To-Noise Enhancers 475 8. 11 Digital Electronic Warfare (EW) Receivers [113-118] 476 8.12 Summary 481 References 482

9. Antennas I: Fundamentals and Numerical Methods

489

F. K. Schwering, A. W. Glisson, and M. A. Morgan

9.1 9.2

Fundamentals 489 References 507 Numerical Techniques References 552

508

1O. Antennas II: Reflector, Lens, Horn, and Other Microwave Antennas of Conventional Configuration Donald G. Bodnar, J. J. Lee, G. L James, F. K. Schwering, and J. W. Mink

10.1

Reflector Antennas References 589

562 ·

561

x

CONTENTS

10.2 Lens Antennas 59 1 References 620 10.3 Hom Antennas 622 References 641 10.4 Dipole and Monopole Antennas 643 References 671 10.5 Other Microwave Antennas 673 References 699 701

11. Antennas III: Array, Millimeter Wave, and Integrated Antennas Robert J. Mailloux, F. K. Schwering, and A. A. O/iner

11.1

Array Antennas 701 References 729 11.2 Millimeter Wave Antennas References 7 69

730

775

12. Antennas IV: Microstrip Antennas Y. T. Lo, S. M. Wright, J. A. Navarro, and M. Davidovitz

12.1 General Introduction 775 12.2 Various Theories for Microstrip Antenna Elements 12.3 Analysis of Infinite Arrays of Microstrip Antenna Elements 806 12.4 Applications and Designs 850 References 899 13. Antennas V: Active Integrated Antennas

780

905

Jonathan D. Fredrick and Tatsuo Itoh

13.1 13.2 13.3 13.4 13.5

Introduction and History 905 Integrated Antenna Oscillator 906 Coupled Oscillators and Phase Control 909 Amplifying Antennas 911 Signal Processing Arrays 915 References 920

14. Mixers and Detectors Erik L Kol/berg

14. l 14.2 14.3 14.4

Introduction 923 Some Common Detector and Mixer Devices 923 Optimization of Diode Detectors 937 Mixers: Simple Theory and Basic Definitions 941

923

CONTENTS

14.5 14.6 14.7 14.8 14.9 14.10 14.11 14.12 14.13

xi

Theoretical Modeling of Schottky Barrier Mixer 949 Single- and Multiple-Diode Mixer 956 Intermodulation in Microwave Mixer 964 Practical Implementation of Microwave Mixer 967 Practical Implementation of Millimeter Wave Schottky Barrier Mi er 973 Submillimeter Wave Schottky Diode Mixer 979 Superconducting Millimeter Wave Mixer 983 Fet Mixer 987 Further A pect on Microwave and Millimeter Wave Mixers 995 References 999

15. Multipliers and Parametric Devices

1007

J. \-V. A rcher and R. A . Batchelor

15 .1 15.2 15 .3 15.4 15.5 15.6

Introduction 1007 Manley-Rowe Relationships 1008 Practical onlinear Reactance Devices 1009 Multiplier Design Using Varactor Diodes 1018 Multipliers Using Step-Recovery Diodes 1035 Parametric Amplifiers and Varactor Frequency Converters 1044 References I 054

1059

16. Semiconductor Control Devices: PIN Diodes Joseph F. White

16. 1 The PIN Diode- an Extension of the Pn Junction 16.2 Microwave Eqmvalent Circuit 1067 16.3 High Rf Power Limits 1090 References 1095

1059

17. Semiconductor Control Devices: Phase Shifters and Switches Ajay I. Sreenivas

17.1 17 .2 17.3 17.4 17.5 17.6 17.7

Introduction 1099 Loaded-Line Phase Shifters 1099 Reflection Phase Shifter 1109 Switched-Line Phase Shifters 1116 High-Pass/Low-Pass Phase Shifter 1120 Analog Phase Shifters [50- 56] 1125 Miscellaneous Considerations 1126 References 1130

1099

xii

CONTENTS

18. Transferred Electron Devices

1135

Cheng Sun

18.1 18.2 18.3 18.4 18.5 l !b.6 18.7

Introduction 1135 Negative Differential Mobility 1136 Modes of Operation for Oscillators 1139 Modes of Operation for Amplifiers 1145 EquaJ-Are-a- Rule 1146 Equivalent-CircuitModels 1149 Oscillator/ Amplifier Circuits and Monolithic Device Development 1153 18.8 Power Combining 1163 18.9 Epitaxial Material Growth and Device Fabrication 1166 18. 10 Conclusions and Recent Developments 1171 References 1173

19. IMPATT and Related Transit-Time Devices

1179

Kai Chang and H. John Kuno

19.l 19.2 19.3 19.4 19.5 19.6 19.7

Introduction 1179 Device Physics and Modeling 1180 Device Design and Fabrication 1191 Oscillators 1207 Power Amplifiers 1220 Power Combiners 1227 IMPATT Diode for Frequency Multiplication and Conversion 1240 19.8 Noise Characteristics and Spurious Oscillations 1241 19.9 Related Transit-Time Devices (BARITT, TRAPATI, TUNNETT, and MITTAT) 1247 19.10 Other Devices 1253 19 .11 Recent Development 125 8 19. 12 Summary and Future Trends 1261 Reference 1261 20. Microwave Silicon Bipolar Transistors and Monolithic Integrated Circuits Craig P. Snapp

20.1 20.2 20.3 20.4 20.5

Introduction 1271 Bipolar Transistor Structure and Modeling 1275 Discrete Transistor Design and Performance 1283 Monolithic Microwave Integrated Circuits 1293 Projections 1308 References 1311

1271

CO TENTS

21. FETs: Power Applications

xiii

1315

H. A . H ung

21. l 21.2 21.3 21.4 21.5 21.6 21.7 2 1.8 21 .9 21.10 21.1 1

Introduction 1315 Statu of PowerFET and Amplifier. 13 16 Power FET Modeling 13 19 Power FET De ign 1336 Matching Circuit De ign 134 1 Fabrication Proce e 1355 Mea uremeotTechniquc 136 1 Thermal Con ideration 1366 Packaging De ign 1369 Reliability and R adiation Effects 137 1 Conclu ion 1372 Acknowledgment 1373 Appendix A 1373 Appendix B 1375 Appendix C 1379 Reference 13 81

1389

22. FETS: Low-Noise Applications Thomas A. Midford

22. 1 22.2 22.3 22.4 22.5 22.6

Introduction 1389 Low- oi e MESFET Device Physics 1392 oise Performance and Models 1399 Low- o ise MESFET Design and Fabrication Low- o ise Amplifier Design 143 1 Conclusions and Outlook 1443 References 1449

141 1

23. High-Electron-Mobility Transistors: Principles and Applications Jacques Zimmermann and Georges Sa/mer

23.1 23.2 23.3 23.4 23.5 23.6 23.7 23.8 23.9 23.10

Introduction 1455 Principle and Model of the HEMT 1457 Small-Signal Equivalent Circuit 1463 Technology of the HEMT 1465 Physical Analysis of the HEMT 1468 Low-Temperature Behavior 1473 Low-Noise Amplification 1476 Power Amplification 1482 New HEMT Structures 1486 Conclusion 1493 References 1493

1455

xiv

CONTENTS

1497

24. Heterojunction Bipolar Transistors and Applications J. S. Yuan

24.1 24.2 24.3 24.4

RF Device Parameters 1500 Transistor Reliability 1504 HBT Models 1517 Circuit Applications 1521 References.. 1541

1547

25. Oscillators and Frequency Synthesizers Ulrich L. Rohde

25.1 25.2 25.3 25.4 25.5 25.6

Introduction 1547 Frequency Synthesizer Fundamentals 1548 Important Characteristics of Synthesizers 1556 Building Blocks of Synthesizers 1560 Phase Locked Loop Designs 1575 The Fractional-N Principle 1596 References 1602

1609

26. RF Components Vijay Nair

26.1 Active Devices 1609 26.2 Two-Port Description of a Three-Terminal Device 26.3 Passive Components 1665 References 1669

1658

27. Microwave Superconductors

1673

Carles Sans and Guo-Chun Liang

27 .1 Superconductor Technology 1673 27 .2 Superconducting Devices 1685 27.3 Superconducting Systems and Applications References 1707

1694

28. Microwave MEMS and Micromachining H. J. De Los Santos, S. J. Cunningham, A. S. Morris, Ill, S. F. Bart,

1711

and W. S. Best

28.1 28.2 28.3 28.4 28.5 28.6

Index

Introduction 1711 MEMS Fabrication Techniques 1712 MEMS-Based Devices 1720 MEMS-Based Circuits and Systems 1734 MEMS Packaging 1743 Summary 1752 References 1752

1759

PREFACE

The four-volume Handbooks of Microwave and Optical Components were published in 1989 and 1990. Since then. we have witnessed a rapid deve]opment in the microwave and optical areas. Radio frequency and microwave wireless personal communications has become one of the hotte t growth areas. Optical device have been used in a wide variety of applications from compact disk players to communication systems. With the e expanding applications, it is time to update the handbooks with new material in response to the new developments. In the second edition, Volume 1 and Volume 2 of the first edition have been combined into one book entitled Handbook of RF/Microwave Components and Engineering. Volume 3 and Volume 4 have been combined into one book entitled Handbook of Optical Components and Engineering. New chapters have been added, and many old chapters have been revi ed. This handbook i intended to erve as a compendium of principles and de ign data for practicing microwave and optical engineers. Although it is expected to be most useful to engineers actively engaged in designing microwave and optical systems, it should also be of considerable value to engineers in other disciplines who have a desire to understand the capabilities and limitations of microwave and optical y terns. To achieve these goals, this handbook covers almost all important components. Theoretical discussion s and mathematical formulations are given only where essential . Whenever possible, design results are presented in graphic and tabular form ; references are given for further study. The book provides, in practical fashion, a wealth of essential principles, methods, de ign information, and references to help solve problems in highfrequency spectra. Each chapter is written as a self-contained unit with its own list of references. Some overlap is inevitable among chapters, but it has been kept to a mjnimum. It is hoped that this comprehensive handbook will offer the type of detailed infonnation necessary for use in today's complex and rapidly changing high-frequency engineering. The authors who have contributed chapters to this handbook have done an excellent job of condensing mountains of material into readable accounts of their respective areas.

xv

xvi

PREFACE

The emphasis throughout has been to provide an overview and practical information of each subject. I would like to thank all members of the editorial board for their advice and suggestions, Dr. Felix Schwering for organizing the antenna chapters, Ms. Cassie Craig of Wiley for managing this project, and Mr. George Telecki, our Wiley editor, for his constant encouragement. I wish especially to thank my wife, Suh-jan, for her patience and support.

Kai Chang College Station, Texas

CONTRIBUTORS

J. W. Archer Di ision of Radiophysics CSIRO, Sydney. Australia lnder J. Bahl M/A-COM Roanoke. Virginia

S.F.Bart Draper Lab Cambridge, Massachusetts R. A. Batchelor Division of Radiophysics CSIRO, Sydney, Australia W. S. Best Triangle ElectroMagnetics, LLC Morri ville, North Carolina Donald G. Bodnar Georgia Tech Research Institute Georgia Institute of Technology Atlanta, Georgia Kai Chang Texas A&M University College Station, Texas

S. J. Cunningham Coventor, Inc. Cary, North Carolina

M. Davidovitz Department of Electrical Engineering University of Minnesota Minneapolis, Minnesota (formerly Electromagnetics Laboratory, Univ. of IL at Urbana-Champaign) H. J. De Los Santos NanoMEMS Research, LLC Irvine, California Michael Dydyk Government Electronics Group Motorola Inc. Scottsdale, Arizona Jonathan D. Fredrick Department of Electrical Engineering University of California Los Angeles, California A. W. Glisson Department of Electrical Engineering University of Mississippi University, Mississippi Paul F. Goldsmith Department of Astronomy Cornell University Ithaca, NY

xvii

xviii

CONTRIBUTORS

K. C. Gupta Department of Electrical and Computer Engineering University of Colorado Boulder, Colorado

Robert J. Mailloux Air Force Research Laboratory Sensors Directorate Hanscom Air Force Base, Massachusetts

William E. Hord Microwave Applications Group Santa Maria, California

Thomas A. Mid:ford Formerly GaAs Product Line Raytheon Corporation

H. A. Hung Army Research Laboratory Adelphi, Maryland

J.

Tatsuo Itoh Electrical Engineering Department University of California Los Angeles, California G. L. James Division of Radiophysics CSIRO Epping, New South Wales Australia Erik L. Kollberg Chalmers University of Technology Gothenburg, Sweden Charles H. Krueger Air Force Research Laboratory Wright Patterson Air Force Base Dayton, Ohio

H. John Kono QuinStar Technology Inc. Torrance, California

J. J. Lee Hughes Aircraft Company Fullerton, California Guo-Chun Liang Allrizon Communications Corporation Shanghai, China Y.T.Lo Electromagnetics Laboratory Department of Electrical and Computer Engineering University of Illinois at Urbana-Champaign Urbana, Illinois

W. Mink Electronics Division U. S. Army Research Office Research Triangle Park, North Carolina David P. Morgan Impulse Consulting Northampton, United Kingdom M.A. Morgan Naval Postgraduate School Monterey, California A. S. Morris Coventor, Inc. Cary, North Carolina

Amir Mortazawi Electrical Engineering and Computer Science Department University of Michigan AnnArbor, Michigan Vijay Nair Motorola Labs Tempe, Arizona

J.

A. Navarro Boeing-Phantom Works RF Technology & Phased Array Group Seattle, Washington A. A. Oliner Polytechnic Institute of New York Brooklyn, New York Ulrich L. Rohde Synergy Microwave Corporation Paterson, New Jersey

CONTRIB UTORS

Georges Salmer Centre Hyperfrequence et Semiconducteur Univer ite de Science et Technique de Lille-Flandre -Artoi Villeneu\'e d' cq. France Carle an School of Telecommunication Engineering UniYersitat Pompeu Fabra Barcelona. pain F. K. Sch\l ering U.S. Ann y CECOM Fort Monmouth. ew Jer ey 1

Craig P. Snapp Avantek. Inc. Microwave Semiconductor Group Advanced Bipolar Products ewark. California

xix

Cheng Sun California Polytechnic State Univer ity an Lui. Obi po. California Jame B. Y. Tsui ir Force Re earch Laboratory Wright Patter on Air Force Ba e Dayton. Ohio Joseph F. White JFW Technology. Inc. Orlean . Ma achu ett S. l\1. Wright Electromagnetic and Microwave Laboratory Department of Electrical Engineering Texa A&M Univer ity College Station, Texas (formerly Electromagnetic Laboratory. Uni . of IL at Urbana-Champaign)

J. S. Yuan Aja. I. Sreenivas Ball Aero pace & Technologie Corp. Boulder. Colorado Karl D. Stephan Department of Technology Soutbwe t Texa State University San Marcos, Texas

Univer ity of Central Florida Orlando, Florida Jacques Zimmermann Centre Hyperfrequence et Semiconducteur Univer ite des Sciences et Techniques de Lille-Flandres-Artois Villeneuve d'A cq, France

1 TRANSMISSION LINES lNDER

J. B AHL

M/A-COM Roanoke, Virginia

Transmi ion line in microwave and millimeter wave circuits are normally used to carry information or energy from one ppint to the other and as circuit elements for pasive circuits such as filters , impedance transformer , couplers, delay lines, and baluns. Passive elements in conventional microwave circuits are mostly distributed and employ sections of transmi ion lines and waveguides. This is because the sizes of discrete lumped elements (re i tor , inductor , and capacitor ) used in electronic circuits at lower frequencie become comp arable to the wavelength at microwave frequencies. However. when the sizes of lumped elements are reduced to dimensions much smaller than the wavelength. they are also used at microwave frequencies. In this chapter variou types of microwave and millimeter wave transmi ion media and their basic characteristics are described.

1.1 BASICS OF TRANSMISSION LINES Multiconductor structures that support transverse electromagnetic (TEM) or non-TEM mode of propagation are commonly referred as transmission lines. Waveguide (single conductor) or dielectric rod (nonconductor) or their derivatives support non-TEM mode of propagation. The TEM transmission lines are characterized by four basic parameters: the characteristic impedance Zo, the phase velocity vp (or guide wavelength >..8 = vp / f), the attenuation constant a and peak power-handling capability Prov. in terms of physical parameters (e.g., geometrical cross section), and properties of the dielectric and the conductor materials used. In a TEM line (perfectly terminated), the ratio of the voltage to the current at any point along the line is constant, having the units of resistance for a lossless medium. This ratio is defined as the characteristic impedance. The propagation constant for Handbook of RF/Microwave Componenls and Engineering, ISBN 0-471-39056-9 © 2003 John Wuey & Sons, Inc.

Edited by Kai Chang

1

2

TRANSMISSION LINES

Figure 1.1

Lumped-circuit representation of a transmission line.

a lossy transmission structure is a complex quantity, comprising a real part known as the attenuation constant (contains information about dissipation due to conductor and dielectric losses) and an imaginary part known as the phase constant (contains information about phase velocity). The attenuation constant is defined as

a=

average power lost per unit length 2(power transmitted)

( 1.1)

Consider a uniform transmission line with series resistance (R), series inductance (L ), shunt conductance ( G), and shunt capacitance (C), all defined per unit length of the line as shown in Fig. 1.1 Important transmission-line expressions are summarized in Table 1. 1. The voltage standing-wave ratio (VSWR) is the ratio of maximum voltage to the minimum voltage amplitude. Figure 1.2 shows nomograph of VSWR. The VSWR is directly measured by using a slotted line. Other quantities shown in this figure are directly measured using network analyzers. An extensive variety of transmission and waveguide structures are used at microwave frequencies. Figure 1.3 shows cross-sectional views of commonly used structures. Half-wavelength, quarter-wavelength, or smaller sections of these lines form the basic building blocks in most microwave and milHmeter wave circuits. The power-handling capability of a transmission line is limited by dielectric breakdown and by heating due to attenuation. The electrical breakdown limits the peak power, while the increase in temperature due to conductor and dielectric losses limits the average power. At normal temperature and pressure, the breakdown electric field of dry air is 2. 9 x 106 V/m. Using this and by calculating the maximum field strength, the peak power-handling capability of a transmission line is readily determined. The average power-handling capability of a transmission line is determined by the temperature rise of the line in an air environment The parameters that play major roles in the calculation of average power capacity are (a) attenuation constant, (b) surface area of the line, (c) maximum tolerable temperature rise, and (d) ambient temperature (i.e., the temperature of the medium surrounding the transmission structure). Thus the average power rating can be raised, if desired, by choosing a higher temperature limit and using forced cooling or cooling fins.

1.2 CHARACTERISTICS OF CONVENTIONAL TRANSMISSION STRUCTURES Coaxial lines are frequently used in microwave circuits. Open-wire lines have higher radiation losses and therefore not used for the transmission of microwave energy.

TABLE 1.1

General Formulas for Transmission Lines"

Quantity

General Linc

Propagation constant. y Phase constant,

= a + j f3

)

/J

0

Impedance of shorted line lrn~dance of open line

Impedance of quarter-wave line Impedance of half-wave line Voltage along line, V (z)

Current along line, 1 (z)

Reflection coefficient, p Standing-wave ratio 0

(.JJ

II

-

ff

o tanh yl

j Zo tan fJI

Zo colh yl

I; cosh Y Z -

z~ sinh YZ

>..

RG OSSible to obtain as much as a 4: I ratio between cutoff frequencies for the TE20 and TE 10 modes [4]. Figure 1.13 shows cutoff wavelength of single- and double-ridged waveguides. Tables 1.12' and 1.13 give the essential characteristics of these guides.

....oc

(a)

(b)

TE01

™ 01

Ac= 1.640 a

Ac= 2.613 a

.-t1.U. t~-~....t1tf,r ' .. 1I I I III 7

--.,dH, .... a.. . . . :_.. .. ..., _._ '--t--"--1 -

1.

......

;Iii~i

...

...... ...... :,:::: ... . .

:,•Pt

t

--t-·1-r· 1-I- • · - TT t 1-· I I I I · -1-~~t 1 · • - 1-rtt+1-· 3. I I I. I I

2.

--t ·r~H+-

- -l°T T~ r1 -

..

.-r-t111-I-- • I-H1 •I- ~ I I I, I I II I I I I

~

LO')

s

Ac= 3.412 a

1• / , ,

1. •t• \.--L~~

11111 ; \

r1.r ...1·., :'\ \, -H·H·', I \ I I I I '. /

.,~1 ,,J \t«_t- ~: "%'1711 1 ·'fu

0·1y-'\

1. Cross-sectional view 2. Longitudinal view through plane 1-1 3. Surface view from s-s

1.

--- Current E-Field ·-· ·- H-Field

Ac = 1.224 a

......+ ~-

__

- :....; ..... ~......;-,. +-- ..... ~;"_._ ..... , ~

,":..'-'

~,-r·\' ·1-Jf.11· ~-1-, ~\ r ·,- +.i 1J ··.· ·-'· · 1 ; ··· · ·••·• 3 , 1✓.
..o < 0.025, full-wave rigorous results are very close to the results obtained by the planar waveguide model approach.

2.5.4 Bends A right-angled bend is normally used for introducing flexibility in the layout of the circuit design. The equivalent circuit of Fig. 2.15c i used for a right-angled bend (without any change in line impedance at the bend). In thi case we denote the hunt capacitance by Cb and the series inductances by Lb/2 each. The do ed-form expre ion are given below [3] C

_!!_ (pF/m)

w

Lb h(nH/m)

=

{

(14Er + 12.5) W/ h -(1.83Er - 2.25) 0.02Er -------;::;:;:::;::;::;::------+--

for W / h < 1

(2.233)

(9.5Er+ l.25 )W / h + 5.2Er+1.0

for W/ h ~ 1

(2.234)

✓ W/ h

-= 100(4JW/ h-

4.21)

W/ h

(2.235)

Equations (2.233) and (2.234) are accurate to within 5% for 2.5 ~Er~ 15 and 0.1 ~ W/ h ~ 5. The accuracy of (2.235) is within about 3% for 0.5 ~ W/ h ~ 2.0 when compared with the numerical results of Thom on and Gopinath [52] . For E= 9.9 and at f = 5.0GHz, it i een that Cb/ W increases from 15.0 to 200 pF/m as W/h is increased from 0.1 to 2.0. For the same set of parameters, Lb/ hLw increases from -0.37 to +0.5.

2.5

MJCROSTR LP DISCONTINUITIES

11 I

d

'

Figure 2.18 Geometry of a chamfered bend.

Several other inve tigarions on characterization of microstrip bends have been reported [53-56]. Discontinuity reactances caused by a right-angle bend can be minimized by chamfering the bend (i.e., by chopping off the outer comer). Data on the optimum amount of chamfering have been given in Ref. 54 as M

= 52 + 65 exp( -1.35

W / h)

(2.236)

for W / h ~ 0.25 and E r~ 25, where M is the percentage chamfer given by (X / d) x 100% with x and d shown in Fig. 2.18. For a chamfered bend, the reflection coefficient Sll ~ 1, but the discontinuity reactances cause a reduction llb in length compared to that measured along the centerlines of the microstrip lines. A closed-form expression for this reduction in length may be written as [40] (2.237) where D is the equivalent parallel-plate line width given by

D = (TJo/ ,/E;;)h/ Zo

(2.238)

and fp is the first higher-mode cutoff frequency given by fp(GHz)

= 0.4Zo/ h (mm)

(2.239)

Microstrip bends have also been analyzed by using the planar waveguide approach [25, 26, 51], as have 120° bends (57), arbitrary angled bends [58], and twofold truncated bends [59] . A typical frequency variation of bend reactances based on this approach is shown in Fig. 2.19. Various values of IS11 I and l/h for a chamfered bend are also included in this figure.

2.5.S T-Junctions Quasi-static analyses of capacitances and inductances associated with microstrip Tjunctions have been reported [54, 60-62). The circuits used for modeling microstrip T -junctions differ from (but are essentially equivalent to) that discussed for stripline. An equivalent circuit is shown in Fig. 2.15d and consists of series inductances L1 in the main line and L 2 in the stub arm, and junction capacitance Cr. In the design of

TRANSMlSSION-LINE DISCONTINUITIES

112

1.0

W= 1.882 mm h =0.65mm

0.9

€,

I

I

= 6.781

I

w 0.8

son

0.7

~

/!

'fi ::::f / \/

- - 1S,,I - - - - 1/h

I

I

3.6

I

I

I

3.2

I

.(:

::::::

I

(3.7b)

0

'\

~

-0 "&

-%

i

,m

CD "0

:1 G',

' ~ 0

', , -

~

bt

~

,:,

' ,

(/) {/)

5

5 0 0.30

~

5 0.35 0 0.40

t:' C:

.Q , C:

'0

I

,

I

;

'\

25 20 0.60 15 0.70 10 0.80

I

I

I

I

I

,''·t:

Q)

(/)

C:

0.10 0.08 0.06 0.04 0.03 0.02 0 .01

Figure 3.4 Nomograph for selecting number of sections of maximally flat filter for given insertion loss in stopband. This chart is separated into stopband (left-hand side) and passband regions (right-hand side). (From T. Milligan, "Nomographs Aid the Filter Designer," Microwaves & RF, Vol. 24, October 1985.)

133

3.1 FILTERS

Figure 3.5

TABLE 3.2

Prototype low-pass filter.

Element Values for a Maximally Flat Low-Pass Prototype Filter'1

Value

of n 1 2 3 4 5

6 7 8 9 10 0

gg

81

2 .000

1.000

l.414 1.000 0.7654 0.6180 0.5176 0.4450 0.3902 0.3473 0.3129

1.414 2.000 1.848 1.618 1.414 1.247 1.111 1.000 0.908

go = 1, w~

1.000 1.000 1.848 2.000 1.932 1.802 1.663 1.532 1.414

1.000 0.7654 1.618 1.932 2.000 1.962 1.879 1.782

1.000 0.6180 1.414 1.802 1.962 2.000 1.975

1.000 0.51 76 1.247 1.663 1.879 1.975

1.000 0.4450 1.111 1.532 1.782

810

811

1.000 0.3902 1.000 1.000 0.3473 1.000 1.414 0.9080 0.3129 1.000

= 1.

The maximally fiat filter design is optimum in the sense that it provides an insertionloss response with a maximum number of frequency derivatives equal to zero at zero frequency with the rejection monotonically increasing as rapidly as possible with frequency. At microwave frequencies, maximally flat design is not as popular as the Chebyshev design. Chebyshev Response. For Chebyshev response, the attenuation loss (in decibels) for w~ = l and w1 ~ l , may be expressed as A

= 10 log[l + (l0A,,, /lO -

1) cos2 (n cos- • w')]

(3.8)

where n is the order of the filter, Am the ripple magnitude in decibels, and w~ the bandwidth over which the insertion loss has maximum ripple Am. The Chebyshev filter nomograph (8] is shown in Fig. 3.6, illustrating Oeft to right) four variables: normalized frequency w' / number of sections n, stopband insertion loss, and passband ripple. For example, for passband ripple of 0.5 dB and ~, and u>~ are the passband edge, stopband edge, transmission zeros, and minimum insertion-loss frequencies, respectively. Element values for the doubly terminated low-pass prototype network shown in Fig. 3.8b are given in Table 3.5 [9] for n = 5, 7, 9, and 11 and various values of return loss in the passband and insertion loss in the stopband.

w;,

3.1.2 Filter Transfonnations In principle, one could build low-pass prototype designs exactly, having the source and load resistance at l n and the upper edge of the passband at I rad/s. However, the

TABLE 3.4

Low-Pass Filter Element Values for Elliptic Function Response0

n=5

n =3

VSWR

w'l

Lm(dB)

1.02

2.79 4.1 3 2.00 2.92 4. 13 2.00 2.46 3.63 l.47 2.00 2 .92 4. 13 1.24 l.62 2.28 3.24

10.4 20.7 10 .3 20 .8 30.2 14.5 20.4 3 1. l 10 .7 20.4 31.3 40.7 LO.O 20.0 30.4 40.2

1.06 I

1.10

1.22

... 1.50

L1

= L3

0.2490 0 .3050 0.3646 0.45 18 0 .489 1 0.49 15 0.5438 0.5968 0 .5776 0.7160 0.7920 0 .8233 0.7539 0.9689 1.0855 1.1 395

L2

C2

VSWR

0 .2227 0.0803 0.3456 0. 1222 0.0552 0.2760 0. 1585 0.0640 0.5885 0.2230 0.0892 0.0420 0.9958 0.3549 0.1466 0.0669

0.4399 0 .5509 0.56 15 0.7290 0.8007 0.703 1 0 .8004 0.8991 0.6362 0.870 1 0 .9992 1.0525 0.55 12 0.8466 1.0090 1.0844

1.02

1.06

I. LO

1.22

J.50

I (,t)I

1.56 2.0 2.92 1.3 1 1.56 2.0 2.92 1.22 1.41 2.0 2.92 1.15 1.31 1.74 2.00 2.92 1.15 1.31 1.74 2.00 2.92

L,,. (dB)

/ ., I

L2

Ci

l \

/.,4

C4

L5

21.9 35.3 53.4 20.5 31.4 44.8 63.0 20.0 30.2 49.2 67.4 21.4 3 l.0 48.2 55.3 73.5 27.5 37.1 54.3 61.4 79.6

0.3566 0.4159 0 .4559 0.4709 0 .5428 0.5942 0.6290 0.5460 0 .6265 0.7072 0.7406 0.7083 0.8003 0.8932 0.9 151 0.9479 l.0247 1. 1203 1.2 176 1.2407 1.275 1

0. I 910 0.0985 0.0416 0.2716 0.1560 0.0822 0.0351 0.3242 0. 1920 0.0772 0.0330 0.3837 0.2332 0. 1022 0 .0733 0.0315 0.3789 0.2342 0.1036 0.07446 0.0321

0 .8809 0 .9605 1.0106 0.9699 1.0782 1.1502 1.1985 0.9858 1.1099 1.2257 1.2732 0.9948 1.130 I 1.2603 1.2909 1.3366 1.0076 1.1251 1.2429 1.2709 1.3129

1.0426 1.0979 1. 1629 1. 1365 1.2211 1.3 155 1.3934 1.1656 1.2706 1.4348 1.51 82 1.2204 1.3675 1.5768 J .6348 1.7270 1.41 25 1.6134 1.8677 1.9355 2.0418

0.8135 0 .320 1 0. 1182 1.1820 0.5257 0.2422 0.0962 1.4187 0.6535 0 .2206 0 .0895 1.5864 0.7740 0.2914 0.2034 0.0843 1.3297 0.7148 0.2867 0.2024 0.0852

0.4701 0.7 156 0.8999 0.4654 0.7274 0 .9457 1.1061 0.4467 0.7 125 1.0383 1.1884 0.450 1 0.7108 1.0395 1.1264 1.2621 0.5370 0.7697 1.0565 1.1320 1.2498

0.0396 0.2555 0.3886 0 .0715 0.3055 0.4659 0.5731 0. 1073 0.3494 0.5884 0.6883 0.2483 0.4888 0.7418 0.8039 0.8983 0.5944 0.8 177 1.0661 1. 1287 1.2249

(Continued )

~

....:a

; TABLE 3.4

(Continued ) n= 7

VSWR

w'l

Lm(dB)

Li

L2

C2

L3

L4

C4

Ls

L6

c6

L1

1.02

1.31 1.56

1.15 1.31 1.56

36.1 51.4 70.1 38.8 6 1.0 79.7 36.6 50.l 65.4

2.00

84.1

1.15 1.31 1.56 2.00 1. 15 1.31 1.56

42.7 56.2 71.5 90.2

0.4249 0 .4691 0.4996 0.5742 0.6394 0.6669 0.6556 0.7107 0.7488 0 .7755 0.8602 0.9143 0.9520 0.9785 1. 1768 1.2336 J .2735 1.3017

0.1489 0.0876 0.0466 0.1642 0.0764 0.0409 0.1972 0.1226 0.0729 0.039 1 0.1 892 0 .11 83 0.0706 0.0379 0.1939 0.1218 0.0728 0.039 1

1.0331 1.0914 1.1317 1.1570 1.2510 1.2907 1.1747 1.2552 1.311 0 1.3502 1.224 1. .3006 1.3542 1.3920 1.1 947 l.2638 1.3127 1.3473

1.0440 1.1938 1.3112 1.1405 l.3792 1.4979 1. 1371 1.3239 1.4780 1.5992 1.2869 1.4840 1.6465 1.7744 1.5 113 1.7309 1.9 118 2.0541

0.7284 0.3849 0.1939 0.8265 0.3419 0.1767 1.0475 0.5856 0.3310 0.1725 0.9889 0.57 l 1 0.3278 0.1724 1.0110 0.5966 0.3462 0.1833

0.7801 1.0325 1.2336 0,789 1 1.1625 1.3538 0.6988 0.9699 1.2009 1.3868 0.7402 0.9949 1.2126 1.3879 0.7240

0.9720 1.1235 1.2633 1.0193 1.2934 1.445 1 0.9854 1.1951 1.3860 1.5440 1.0966 1.3357 1.5449 1.7148 1.2778 1.5567 1.7952 1.9867

0.6776 0.3404 0. 1654 0.6938 0.2762 0.1400 0.8389 0.4628 0.2572 0.1322 0.7384 0.4259 0.2426 0. 1266 0.712 1 0.4235 0 .2456 0.1295

0.6 157 0.8167 0.9750 0.7103 1.0067 1. 15 18 0.6805 0.90 15 1.08 10 1.2196 0.773 1 0.9796 1. 1458 1.2737 0.8017 0.9850 1. 1319 1.2448

0.1016 0.2773 0 .3966 0.2470 0.4776 0.5780 0.2864 0.4657 0.5962 0.6910 0 .5 188 0.6829 0.8060 0.8971 0.8360 0.9992 1.243 1.2179

2.00

1.06

1.22 1.56 2.00

1.10

1.22

1.50

2.00

48.8

62.6 77.6 96.3

I

I

0.9525 1. 1480 1.3055

II

VSWR

w'l

Lm(dB)

L1

L2

Ci.

L,

l.,4

C4

Ls

L 50

-

--

---

-·----

11 RL. ~ 26 dB

L'" ) 50

L,,.) 40

(dB)

(dB)

(dB)

(dB)

1.04297

1.04428

0 843579

084511 5

0.846644

1.01792 0.935065 0.729441 0.9843 13 0.958626 0.787121

0.840267 1.39582 0.6044 1

0.994819 0.710293 0.918352

0.907422 0.959304 0.760814

0.804735 1.36035 0.590439

Lo(8) L2(8) C2(8)

0.90863 1 0.786767 0.6175 1 0.794605 1.13239 0.975041

0.661279 1.04755 0.80535

0.995387 0.877259 0.755266 0.57681 1 0.736593 0.959 128 1.13087 0.990848 0.837434

Lo(6) L2(6) C2 (6)

0.93471 0.61751 1.13239

0.672323 1.04755 0.80535

1.03347 0.908381 0.57681 1 0.736593 1.13087 0.990848

0.77411 0.959 128 0.837434

Lo(4) L2(4) C2(4)

0.908631 0.786767 0.661279 0.72944 1 0.984313 1.39582 0.958626 0.787 121 0.6044 1

0.995387 0.710293 0.9 18352

0.877259 0.959304 0.7608 14

0.755266 1.36035 0.590439

Lo(2)

1.01792

0.935065

0.840267

0.994819

0.907422

0.804735

Ca ( I)

1.0416

1.04297

1.04428

0.843579

0.845115

0.846644

Ca ( 11)

1.0416

l o( IO) £.2( 10) C2(IO)

l,,.

40

I 206H 0.227543 1.37499

(dB)

L,,,

~

0.778435

(dB)

R.L. ~ 20 dB Element Values

I.,..

0.773927

n

L,,. ) 40

I ,.. ' SO

0 77 1053

(dB)

-

26 (dB)

--

60

I.'"

(dB)

(dH)

I J0282

Rt. • 26 dB CdB)

__cc

n=9 Element Values

SO

-

RI.. , 20 dB

l'" '.> 40 (dB)

Element Values

5

ti

0.807267 0.794605 0.975041

L,,.

~ 40

l,,.

~

60

142

ALTERS, HYBRIDS AND COUPLERS, POWER COMBINERS, AND MATCHING NETWORKS

TABLE 3.5(b)

Generalized Chebysbev Values RL.

~

20(dB)

R.L.

Lm ~ 50 (dB)

Lm ~ 40 (dB)

Lm ~ 60

~

26(dB)

(dB)

Lm ~ 50 (dB)

Lm ~ 40 (dB)

Degree

Frequency

Lm;;:;: 60 (dB)

5

w'0 w'L

2.68541 2.52739

2.17845 2.04812

1.7878 1.69345

3.05766 2.88554

2.46817 2.3175

2 .01033 1.90012

7

Wo

1.44559 1.33773

1.29516 1.21229

1.77125 1.61703

1.55497 1.43071

1.38122

{J)~

1.63565 1.4999

9

w'0 w'L

1.32599 1.21737

1.22745 1.14178

1.14922 1.0848

1.3958 1.27249

1.28424 1.18496

1.19397 1.11696

11

w'0

CUJ-u>

= low-pass bandwidth, (J)' /w;

2

Wo

I Ck = - - - -

g121r BW' Zo

Series-tuned shunt elements

Lt= 2JrB~Zo g1cwo C _

21rBW' Zo

Zo

Lt=

gk21rBW'

Kk 2JrBWZo

C1 = gk

2JrBW' Wo22 0

= (J) j wt.,p.

= high-pass band edge, (J) / (J)~ = - 15,000 at 10 GHz) cavity bandpass filters. The TEo11 low-loss cavities, in which the circumferentially flowing currents in the cavity walls are minimized, lead to rather bulky filters compared with dual-mode structures. In a circular waveguide, the dominant TE 11 mode has the property of orthogonal symmetry; that is, a cylindrical cavity can support two modes with 90° orientation to each other. li both modes are excited in a cavity, the filter is known as a dual-mode filter. The work on ellliptic function and linear-phase waveguide filters using dual-mode cavities [20-22] with cross couplings was particularly noteworthy and significant. The combination of high-Q factor, elliptic function response, delay equalization (when desired), and small size (due to the dual-mode cavities) has lead to the virtual standardization of these designs for satellite transponders. A schematic of a four-resonator filter consisting of two TEn 1 dual-mode cavities is shown in Fig. 3.13c. It is seen that

3.1 ALTERS

149

modes 1 and 4 may be coupled to each other either in-phase or out-of-phase using a suitable coupling screw arrangement [16) . Other dual-mode filters include the quare cavity TE 103 and TErn5 mode , and the cylindrical TE113 mode [23]. It has been reported that the TE 113 mode gives an unloaded Q of about 16 000 at 12 GHz, and the TE 113 mode fi lter has the highest performance by a wide margin, as shown in Table 3.9. The major advantage of waveguide filter are low-loss performance and high powerhandling capability. The commonly used material are brass, aluminum, and lnvar. A]] three material are u ually ilver-plated to en ure lowe t possible passband attenuation. Invar provides the be t temperature stability, due to its low-thermal-expansion coefficient. while aluminun1 provide a low-weight solution.

Finline and Metal Insert Filters. Finline and metal insert filters consist of ladder-type inserts centered in the £-plane of rectangular waveguides. A finline filter configuration is hown in Fig. 3.14. Meta1 insert filters have a similar geometry (Fig. 3.14) except that the substrate thickness is zero. Finline filters are easily fabricated using metallized substrate by photoetching techniques and thus are compatible with integrated circuit technology. Metal in ert filters (also known as £-plane filters) are fabricated on a metal TABLE 3.9 J\lfidband Loss, Unloaded Q, and Resonator Volume Comparison for 12-GH.z Filters Waveguide Mode

Midband Filter Type

Loss

Canonical dual mode Canonical dual mode

0.80

0.60

Folded waveguide Canonical dual mode

0.50 0.36

Calculated Unloaded Q 0

0

Total Res onator Volume (in.3)

Square TE 103

TE1os

7,300 9,700

7.3

11 ,600

9.2

16,300

4.8

4.4

Cylindrical

TF.ou TE113 0

Accuracy is ±10%. Source: E. L. Griffin and F. A. Young, "A Comparison of F our Overmoded Canonical Narrowband Bandpass Filters at 12 GHz," IEEE MTT-S Im. Microwave Symp. Dig., 1978.

b

d=b Dielectric

~++ , + + + ++~ 7 4½ la

Figure 3.14

l4

l.s . la

l1

la lg

Finline filter configuration.

'

150

FTLTERS. HYBRIDS AND COUPLERS, POWER COMBINERS , AND MATCHING NETWORKS

sheet by photoetching, pressing, or stamping. Because dielectric losses are absent in £-plane filters, they are suitable for narrow-band high-Q applications. The design parameters of finline filters having three resonators (i.e., four metal inserts) for midband frequencies of about 15, 34, and 66 GHz are given in Table 3.10 (24]. The corresponding wavegujde housings are WR62 (Ku-band, a= 15.799 mm, b = 7.899 mm), WR28 (Ka-band, a= 7.112 m , b = 3.556 mm), and WR15 (V-band, a = 3.76 mm,) b = 1.88 mm). The substrate is RT/Duroid 5880 (Er = 2.22). The design of E-plane filters has been described in Refs. 25-33.

Coaxial Filters. Coaxial low-pass and bandpass filters are very important components in low-power microwave systems [l]. The bandpass filters have interdigital and combine configurations as shown in Fig. 3.15. The interdigital filters use quarter-wave-long resonators, while a combline filter uses about one-eighth-wavelength resonators loaded with lumped capacitors at the open circuit end. The bandwidth of the filter depends on the coupling between the resonators and the separation between the resonators and the ground plane. The combline filter has the advantage of being physically smaller for the same specification. The coaxial filters are best suited for narrow-band ( 1 to 30%) applications in the frequency range 100 MHz through 20 GHz. However, coaxial filters having larger band-widths and also working beyond 40 GHz have been demonstrated. Printed Circuit Filters. Satellite, airborne communications, and EW systems have requirements for small size, lightweight, low-cost filters . Microstrip and stripline filters are very suitable for wide-band applications where the demand on selectivity is not severe. Various kinds of filters, shown in Figs. 3 .16 and 3 .17, can be realized using microstrip-type structures. TABLE 3.10

Computer Optimized Design Data for Finline Filters0

Finline Filter

Substrate thickness l1 = l9 Ii= ls l3 = l1 J4 = 16 (in.) (mm) (mm) (mm) (mm)

Bandwidth

ls

Jo

~lb

(mm)

(GHz)

(MHz)

Ku-band a= 15 .799 mm b 7.899 mm

0.005 0.01 0.02 1/32 1/16

27.079 27.565 28.127 13.706 15.660

2.178 2.075 1.976 1.860 l.500

8.390 8.400 8.350 8.300 8.000

8.148 7.760 7.372 6.984 5.840

8.400 15.19 8.400 15.285 8.350 15.252 8.300 15.207 8.000 15.190

560 601 560 557 524

Ka-band a=7.112mm b = 3.556 mm

0.005 0.01 0.02 1/32

19.5575 19.928 20.580 21.260

0.715 0.660 0.550 0.480

3.740 3.740 3.630 3.525

4.115 3.800 3.420 2.970

3.745 3.745 3.640 3.530

34.02 33.989 33.982 33.99

620 698 675 635

V-baod a= 3.76 mm b = 1.88 mm

0.005 0.01

14.770 14.824

0 .347 0.353

1.895 l.855

2.038 2.038

1.900 66.09 l.860 65.940

1055 1080

=

= 2.22, conductor thickness = 17 .5 µm. b3-dB limits. Source: F. Arndt et al., "Theory and Design of Low Insertion-Loss Finline Filters," IEEE Trans. Microwave Theory Tech., Vol. MTT-30, February 1982.

u fr

3. 1 FILTERS

1S1

(b)

I Figure 3.15

I

I

Coaxial-line filters: (a) intercligital; (b) combline.

Filter configuration

Properties

Stub loaded

~

1. fspe =3 fc

---~

2. Pass band loss

=1dB

3. Low pass

Stepped impedance

1. fspe =3 fc 2. Pass band loss

=1dB

3. Low pass Direct-coupled resonator c::::J C==:J c::J c:.::: c::J c:::J

=

1. fspe 2 fo 2. Pass band loss

=1dB

3. Band pass Parallel-coupled

,-------.:::::,

1. fspe =3 f0

c:=···

2. Pass band loss

=1dB

3. Band pass

c:;___J Combline

lnterdigital

]~ ~ ~--.~~ D~-- - ~~ ~~

1. fspe = 3 f0 2. Pass band loss

=1dB

3. Band pass

Grounded stub direct-coupled

Js'Jt;2C'~SJ1

1. fspe = 3 fc 2. Pass band loss

=1dB

3. High pass

Figure 3.16 Microstrip-type filter configurations. (From L J. Bahl and P. Bhartia, ,Microwave Solid State Circuit Design, Wtley, New York, 1988.)

152

Fil..TERS, HYBRIDS AND COUPLERS, POWER COMBINERS, AND MATCHING NEfWORKS .:.

(a)

1

2

• • •

T

N

L = Al4

--.....J'\Outputl -::.-

(b)

I

I

1

2

-=

--::.-

I

• • •

T

L:zA/4

N

Outp1

Input --::.-

(c)

Input

1

2

• ••

N

T

l

\

•I

Figure 3.17 Tapped-line filter configurations: (a) interdigital; (b) combline; (c) hairpin-line; (d) parallel coupled.

The suspended microstrip provides a higher Q than microstrip or tripline, as most of the energy is propagating in the air. This results in lower-lo s filters with sharper band edges. The wide range of impedance values achievable makes this medium particularly suitable for low-pass and broadband bandpass filters. An account of the development of many different types of filters in this medium was given in Ref. 34. Filters can be realized using lumped elements or by employing microstrip sections. It is difficult to simulate accurately a series inductor in microstrip line. Kuroda's identities [35, 36] allow one to realize low-pass structures using shunt elements with the identical response. Richards transformation [36, 37], which establishes a imple relationship between lumped and distributed circuit elements, enables one to design filters using distributed circuits. In many microwave filter designs, a length of transmission line terminated in either an open circuit or a short circuit is often used as a resonator. Figure 3.18 illustrates four such resonators with their equivalent LRC networks, which were determined by equating slope parameters for both of these configurations at resonance w WQ. Low-pass filters in stripline form are very important components in microwave systems. A very good account of the early development is given in the classic book of

=

3.1 FILTERS Stub configuration

Equivalent LAC network

2o = Ti

Zo =

0

J.

iZ

0

:-Jr

R = Z 0a. l

O="'oL = L.

R Wo = _1_ .JLG

').. /4 short circuited

2o: ~~Io

Element values w0 L =

')../4 open circuited

153

4a. l

=l Yo Zo= l ,Jf

Woe L

C

G = Y0a. l 0 = ~=-1L G 4at

')../2 open circuited

WoC = ~Yo

T

Zo= f,Jf

t

G = Y0a. t

2at WoL = ~Zo 0=

>../2 short circuited

~ IC

Zo= ~-J[ R = Zoa,,!. 0 - .JL - 2at

Figure 3.18 Equivalent circuits for TEM strip line resonators. (From I. J. Bahl and P. Bhartia, Microwave Solid State Circuit Design, Second Editio~ Wtley, New Yor~ 2003.)

Matthaei et al [1]. TEM structures such as stripline and microstrip are ideal for lowpass filters, and the design is approximated to an idealized lumped-elements circuit. The low-pass filter can be realized using the microstrip or stripline sections given in Table 3. 11 [7]. Popular configurations for printed circuit filters (Fig. 3.16 and 3.17) are direct coupled, parallel coup}~ interdigital, combline, and hairpin-line. Direct-coupled resonator filters [1] have excessive length. The dimensions can be reduced by a factor of 2 with the introduction of parallel-coupled lines. Parallel coupling is much stronger than end coupling, so that realizable bandwidths could be much greater (35, 38]. In this configuration the first spurious response occurs at three times the center frequency, and a much larger gap is permitted between parallel adjacent strips. The gap tolerance is also reduced, permitting a broader bandwidth for a given tolerance. Filters can be designed with reasonable accuracy using design infonnation in Refs. 1, 38-43. To fabricate compact filters, resonators are placed side by side. Interdigital, combline, and hairpin-line filters are realized using this concept. Accurate design analysis of these filters, interdigital [1, 44, 45], combline [1, 46-48] and hairpin-line [49, 50] are also available. Advanced RF/microwave bandpass filters such as microstrip cascaded quadruplet and trisection filters, as shown in Fig. 3.19, to meet stringent requirements for wireless

'

154

FILTERS, HYBRIDS AND COUPLERS, POWER COMBINERS, AND MATCHING NETWORKS

TABLE 3.11

Distributed Inductors and Capacitors Expressions

Equivalent Circuit

Distributed Element

Approximate Expressions

Mainly Inductive L

ZoL . L= sm w

• 0

Low Zoe

High ZoL - - Low Zoe

CL=

(2rr/L)

1 tan wZoL

-Agl

(7 ) -r/L >..8 L

Mainly Capaciti ve

r-fc, - ~hZ H19 oL

/

Low Zoe

Fr

C= _1 sin (21rle) 0c tan (rrle -Z - ) wZ0c

High ZoL 0

0

Le =

w

A ge

Age

Source: I. J. Bahl and P. Bhartia, Microwave Solid State Circuit Design, Second Edition, WLley, 2003, Chap. 6.

ew York,

communications systems have also been described in the literature [51]. These filters use microstrip open-loop resonators and are arranged in such a way that each resonator has one cross-coupling. The cascaded quadruplet filter contains cascaded sections of four resonators, whereas the trisection filter consists of three resonators. In such filters, the cross-couplings are adjusted to realize a pair of attenuation poles at finite frequencies to improve the insertion loss and the selectivity for a given number of resonators. Figure 3.20 shows the measured perfonnance of a narrowband trisection filter shown in Fig. 3.19(b) and fabricated on a 1.27-mm alumina substrate using copper conductors. The measured insertion loss is about 1.2 dB over the 860-910-MHz frequency range.

Dielectric Resonator FilJers. MIC structures have suffered from a lack of high-Q miniature elements, which are required to construct high-performance, highly stable, narrow-band filters . Filters such as bandpass and bandstop are frequently realized using high-quality dielectric resonators. The dielectric constant is around 40, and resonator Q (~ 1/ tan8) lies between 5,000 and 10,000 in the frequency range 2 to 7 GHz [52, 53). These days, the commercially available dielectric resonators have temperature stability as good as that of Invar. Table 3.12 [7] lists properties of commonly used materials in dielectric resonators. Filters may be constructed in all the common transmission media, ranging from waveguide to microstrip. Such filters are small in size, lightweigh~ and low in cost. Most of the early work was carried on in the early 1960s as summarized in Ref. 54. The basic bandpass filter topology consists of an evanescent-mode waveguide section (waveguide below cutoff) in which the dielectric resonators are housed. The most commonly used mode in the resonators is the TEo 1 6 • The dual-mode or hybridmode HEo1 ~ resonators find applications in sophisticated elliptic-function dielectric filters. Various kinds of dielectric resonator filters have been treated in the literature [55-59].

3. 1 FILTERS

(a)

155

20 Tapped fee line

✓

1.0

1.75 (b)

23.0 Unit: mm

1.4

?""

Figure 3.19 Physical layouts of the cross-coupled microstrip filters. (a) cascaded quadruplet and (b) trisection. All dimensions are calculated for an alumina substrate (Er = 10.8 and

thickness = 1.27mm).

156

FILTERS, HYBRIDS AND COUPLERS, POWER COMBINERS, AND MATCHING NETWORKS

\

\ \ \

- 10

I I I

~,,."'

\

-40

-50 700

750

800

850

900

950

1000 1050 1100

Frequency (MHz)

Figure 3.20 Measured performance of a microstrip trisection filter.

TABLE 3.12

Properties of Dielectric Resonators

Q

Material (Zr, Sn)TiO4 type C Ba(Zr, Zn, Ta)O3 type C Zr/Sn titanate D-8515 type Barium tetratitante D-8512 type (Zr, Zn)TiO4 E-2036 type

f,

ct

CLE

(ppmfC)

(ppmfC)

Company

15,000 (4 GHz)

37.3 ±0.5

6.5

-13±2

Murata

10,000 (10 GHz)

28.6 ± 0.5

10.2

- 20.4 ± 4

Murata

10,000 (4 GHz)

36.0 ± 0.5

5.6

-6.9

Trans Tech

10,000 (4 GHz)

38.6 ± 0.6

9.5

-10.4

Trans Tech

4,000 ( 10 GHz)

37.0 ± 0.4

5

- 10± 1

Thomson-CSF

Source : I. J. Bahl and P. Bhartia, Microwave Solid State Circuit Design, Second Edition, Wiley, 2003, Chap. 6.

cw York,

3.1.7 Compact Filters Several techniques to design miniature planar bandpass filters at L-band have been used. These incJude lumped elements, ceramic block [60-64], meander or folded resonators [65, 66], u e of high-K dielectric materials [67, 68], multilayer dielectric and metallization circuit configuration [69], and stepped impedance resonator circuit [70- 73]. Pseudo-interdigital [74], hairpin resonator, and open-loop resonator [75] filters do not require any grounding of the resonators. Therefore, these filters are more suitable for low-cost and high-volume production, as a large number of filters can be simultaneously printed on a single substrate board. Their high-temperature superconducting versions are suitable for satellite applications [76]. Pramanick [68] provides a step-by-step design of hairpin resonator filters on high-K materials.

3.2

HYBRIDS AND COUPLERS

157

At RF frequencies , printed resonators u ing thin- and thick-film hybrid technologies have been succe sfully u ed to develop miniature and low-cost filters. The Q of microstrip resonator can be increased by 20% to 50% by employing a muJ tilayer filter structure. To achieve lower in ertion lo with narrower bandwidths and low cost, low-temperature cofired ceramic (LTCC) technology is preferred. LTCC technology using high-K material such as BiCaNbO with a dielectric con tant value of about 60 is capable of producing low-lo ( < 2 dB), miniature, and narrowband fil ters for mobile applications.

Lumped Element Filters. At RF frequencies and at the lower end of the microwave frequency band, filter have been realized using lumped elements (chip/coil inductors and chip capacitors), printed inductor , and discrete chip capacitors, and employing a multilayer printed circuit technique such as LTCC or printed circuit boards. Lumped element filters can be implemented easily, and using currently available surface-mounted components one can meet size and cost targets in high-volume production. Ceramic Block Filters. Coaxial interdigital and combline filters using ceramic blocks are commonly used for 200-3000 MHz applications. These filters are based on high dielectric constant ceramic technology, in which filter size reduction is accomplished by a reduction in guide wavelength. A number of very high dielectric constant (er rv 35-100) ceramic materials (e .g., barium titanates and zirconates) with very low loss (Q factor - 5000) are currently available. The filters using these materials have typically 2-dB insertion loss and have bandwidth from l % to 20%. Ceramic block filters are temperature stable (3 ppml°C), and their temperature range of operation is normally from -30°C to +85°C. They are surface mountable, and in high volume, their cost is $2-$5. Their specific applications include cellular radio, mobile radio, wireless LAN, PCN, GPS, CATV, and ISM band. Figure 3.21 shows a three-resonator combline ceramic block filter. The coupling between pairs of adjacent resonators jg realized by circular/rectangular air holes. The inhomogeneous interface between the high dielectric constant ceramic and the air hole gives rise to different phase velocities for the even and odd modes of the coupled lines. This difference provides the required coupling between the resonators to realize a filter and effectively reduces its size and cost. The design of such filters, although straightforward, requires numerical methods such as EM simulators to determine the coupling between the resonators. Normally, filters are designed empirically and tuned after fabrication using ceramic grinders and metal scrapers. Analysis, design, and test results for various ceramic block filters have been discussed by many authors [60- 64] .

3.2 HYBRIDS AND COUPLERS Hybrids and couplers are indispensable components in rapidly growing applications of microwaves in electronic warfare, radars, and communications systems. The circuits are used frequently in frequency discriminators, balanced amplifiers, balanced mixers, automatic level control, and many other applications. A hybrid or a directional coupler can in principle be represented as a multiport network, as shown in Fig. 3.22. The structure has four ports: input, direct, coupled, and isolated. If P1 is the power fed into port I (which is matched to the generator impedance) and P2, P3, and P4 are the powers

....

UI 00

High-K ceramic block Coupling pad

Input

I

"VfA

I

A

@

I I I

I I I

I

I

I I

I I I --'

L _, _

l

I

B

C

II

®1 I I I

I I'

I I

I I

I I

I

I

I

I

1 1

I I

I

I

I I

I

I

.J. _, 1 _ 1_ \

I1 \ I I ,, 1 \I I \... J

I I

l

~

.J.

@. I I

..J _ 1-1_

I\ I I \ I I ,, --' \I

1 1. , I

W/\\.

I I I

I I I

I

Ii

Output

I

I

I _1 _J I I\ I I I \ _ _, \

.L

l

\..,-a

I

L

\

\ Un metallized coupling holes (circular or slot)

Figure 3.21 High-K ceramic block combline band-pass filter. All the surfaces are metallized except the top surface. A, 8 , and C are metallized coaxial resonators. MetalUzed side waJls of the ceramic block act as outer conductors.

3.2 Input

HYBRIDS AND COUPLERS

159

Diract ~

1

2 ;

-

4-

3 Cou p led

Isolated

Figure 3.22

Four-port network.

available at the ports 2, 3, and 4. respectively (while each of the ports is terminated by its image impedance), the two most important parameters that describe the performance of this network are its coupling factor and directivity, defined as follows:

~ . f actor (dB) = C = l0logcoup1mg P3

(3. 19)

= l0log -P3

(3.20)

/ = 10 log Pi = D + C

(3.21 )

. . (dB) direcuv1ty

=

D

P4

The isolation and transmitted power are given by isolation (dB) =

P4

2

transmitted power (dB) = T = 10 log p P1

(3.22)

As a general rule, the performance of these circuit elements is specified in terms

of coupling, directivity, and the terminating impedance at the center frequency of operating frequency band. Usually, the isolated port is terminated in a matched load. For many applications, the single-section coupler proves to be of inadequate bandwidth. A multisection design which is a cascaded combination of more than one single-section coupler, each being a quarter wavelength long at the center frequency of the band results in a larger bandwidths. The number of sections to be used depends on the tolerable insertion loss, bandwidth, and the available physical space.

3.2.1 Hybrids Many types of hybrids are used in microwave and millimeter wave subsystems. Some of the more commonly used are 90° hybrids (branch-line), hybrid-ring (rat-race), and matched hybrid tee (often called a magic tee). Design aspects of these circuits are discussed in this subsection.

90° Hybrids. The branch-line type of hybrid shown in Fig. 3.23 is one of the simplest structures for a 90° hybrid. The geometry is readily realizable in any transmission medium. Branch-line hybrids have narrow bandwidths, on the order of 10%. As shown in Fig. 3.23, the two quarter-wavelength-long sections spaced one-quarter wavelength apart divide the input signal from port 1 so that no signal appears at port 4. The coupled signals at ports 2 and 3 are equal in magnitude, but out of phase by 90°. The coupling

160

FILTERS, HYBRJDS AND COUPLERS, POWER COMBINERS, AND MATCHING NETWORKS

z,

1,__

I

_,,

r l---.._____

_..,.-2

I

.I

Series arm

)./4

Zo

i....-Shuntarm

-----.3

-----------1

4----

1 ...1 - - -)J-4--

Figure 3.23

Single-section branch-line hybrid.

factor is determined by the ratio of the impedance of the shunt and series arms and is optimized to maintain proper match over the required bandwidth. For 90° lossless hybrids the following conditions hold good:

(3.23)

!: =

2 [

] 1/ 2

(3.24)

(:: ) + I

For 3-d.B coupling, the characteristic impedances of the shunt and series arms are Zo and Z 0 / ,J2., respectively, for optimum performance of the coupler. Z0 is the characteristic impedance of the input and output ports. When matched hybrids are cascaded in series as shown in Fig. 3.24, by adjusting immittance (shunt admittance and series impedance) values, the transfer function of the complete hybrid can be realized either Chebyshev or maximally flat. Each branch in the hybrids is A/ 4 at the center frequency. The hybrid also has end-to-end symmetry (i.e., Hn+l H1, etc., and Kn= K 1, etc.). Normalizing all the immittances with respect to the terminating immittances Kn+l =Ko= 1, various midband formulas for branch-line hybrids up to four branches or three sections are given in Table 3.13 [77, 78]. In the derivation of these expressions, junction effects, losses, and so on. have been neglected. Such hybrids have also been analyzed by Reed and Wheeler [79] and Reed [80]. The values of H' s and K's for maximally flat response for two and three sections are given in Table 3.14. Values of immittances for multisection hybrids and different bandwidths

=

+r;'41H2 Ko

Ko Figure 3.24

K1

K,

K2

~

,_1

--- K· ----

Ki

Ki+ 1

I f-· H,

------K,_1

K,

K1+ 1

=K1

Ko

-~~~:=E~. ---

K

K

-

--- Kn =K1 Kn+l = Ko

N sections cascaded branch-line hybrids with n + 1 branches.

3.2

TABLE 3.13

161

HYBRIDS AND COUPLERS

Midband Formulas for Branch-Line Couplers up to Four Branches Two Branches

Condition for perfect match and directivity

2

K1

Three Branches

Four Branches

= 1 + H .,j 2H, K f + H2(1 - Hf)

1- H l1

+ K I2

2(K r - H 1H2)

Hi

= ..fi.-1 Kf = H2..fi. H1

Conditions for 3-d.B directional coupler

Conditions for 0-dB directional coupler

=

K f (l - H 1) 1 + 2H1 - H~

H2.J2. K2 = - l - H1 if H 1 < 1

ot possible ( H 1, K 1 -+ oo)

Source : L. Young, "Branch Guide Directional Couplers," Proc. Nat. Electron Conf., Vol. 12, 1956.

TABLE 3.14 Sections R

Tmmitances of Maximally Flat Branch-Line Couplers for n K 'I

K'2

=2

and 3

H1

H2

0.1010 0.1715 0.2251 0.2679 0.3333 0.3819 0.4202 0.4775 0.5194

0.2062 0.3639 0.4983 0.6188 0.8333 1.0249 1.2008 1.5196 1.8070

0.0501 0.0840 0.1086 0.1274 0.1542 0.1732 0.1854 0.2029 0.2138

0.1539 0.2694 0.3656 0.4498 0.5957 0.7220 0.8351 1.0344 1.2091

n =2 1.50 2.00 2.50 3.00 4.00 5.00 6.00 8.00 10.00

1.0153 1.0449 1.0783 1.1124 1.1785 1.2399 1.2965 1.3978 1.4861 n=3

1.50 2.00 2.50 3.00 4.00 5.00 6.00 8.00 10.00

1.0089 1.0258 1.0446 1.0634 1.0988 1.1307 1.1594 1.2087 1.2501

1.0206 1.0606 1.1067 1.1546 1.2499 1.3416 1.4288 1.5909 1.7392

162

FILTERS, HYBRIDS AND COUPLERS, POWER COMBINERS, AND MATCHING NETWORKS

TABLE 3.15 Parameters 3-dB Hybrids Parameter

Two-Section

for

Cbebyshev

Maximally Aat

0.4141 0.7071 1.0 120.1 n 10.1 n 50.0 n

5.8486 0.4141 l.1742 0.7765° 120.1 n 42.6 n 38.8 n

R

H1,3 H2 K 1.2 Z shuntl ,3 Z sbunt2 Z seriesl.2

°For maximally Oat K 1.2

= 1/ Ki_2 .

are also given in Ref. 77. Here coupling is expressed in terms of R as follows: P 2. 1

R+1 = 20 log R _ l

(3.25)

As an example, the line impedances of a 3-dB two-section hybrid having 50-n source and load impedances are given in Table 3.15. A computer-roded design technique that is suitable for an optimum design of multisection hybrids is also described in the literature (82]. Using matching techniques, the bandwidth of a single-section hybrid has been extended from 10% to an octave (83]. In MMICs, lumped capacitors can be easily realized and have become attractive in reducing the size of passive components. Reduced-size branch-line hybrids that use only lumped capacitors and small sections of transmission lines (smaller than )...8 / 4) have also been reported [84]. The size of these hybrids is about 80% smaller than those for conventional hybrids and is therefore suitable for MMICs. The reduced-size branch-line hybrid is shown in Fig. 3.25. The value of capacitance Cb is given by (3.26) where Zo is the impedance of the various ports of a branch-line hybrid and Wo denotes the radian frequency corresponding to the center frequency of the branch-line coupler. The size of the branch-line hybrid can further be reduced by using the lumpedelement approach [85-87]. The lumped-element 90° hybrid can be realized either in a "pi" or a "tee" equivalent network. In MMICs, a "pi" network is preferred to a "tee" because it uses fewer inductive elements, which have lower Q and occupy more space. Figure 3.26 shows a lumped-element 90° hybrid. The values of lumped inductive and capacitive elements of a 3-dB hybrid are given by (3.27a) and (3.27b)

3.2

HYBRIDS AND COUPLERS

,_

b

).g/12

0

I

_.._

-.

0

1 I

_..,

>.gt8

0

163

70.70

--

70.70 I

I

-. I

Figure 3~5

--

0

Reduced-size branch-line hybrid.

Hybrid Rings (Rat-Race Hybrids). A hybrid ring or rat-race circuit consists of an annular line 1.5A in circumference to sustain standing waves, with four arms connected at appropriate points, as shown in Fig. 3.27. The circuit may be realized using waveguide, microstrip, or dielectric guides. However, the performance of the dielectric guide rat race is poor due to radiation losses from the curved structures. The characteristics of the hybrid ring are similar to those of the 90° hybrid except that the output ports 2 and 3 have 180° phase difference instead of 90°. For 3-dB coupling, Z1 = Z2 = -v'2Zo and has a broader bandwidth (>20%) than the branch-line hybrid. The bandwidth of this device can be increased to approximately an octave by using a shorted quarterwavelength parallel coupled line for the 3A/ 4 section in Fig. 3.27. This circuit would then make a ring of one wavelength circumference, as shown in Fig. 3.28. To design this hybrid, only additional information required is for the even-mode characteristic impedance Z0e = 2.41221 . For a multisection and power split of more than 6 dB, most planar transmission lines require too wide a range of impedances. These may sometimes be difficult to realize

164

FILTERS, HYBRIDS AND COUPLERS, POWER COMBINERS, AND MATCHING NETWORKS

z,

--/------I I

1

L1

C1

I I I I I I

C1

------

----

I I I

Z

Ci

)

P

4

2

~

c;

I

I I I

I I

----

------

I I

I I I I

~Z I I I I

P

3

__L ---7------

z, Figure 3.26

Lumped-element equivalent circuit model for the 90° hybrid shown in Fig. 3.23.

I

I

4

Figure 3.27 Hybrid-ring configuration.

physically. On the other hand, very wide line widths may require unreasonable aspect ratios at higher frequencies, due to shorter wavelengths. The physically unrealizable high-impedance line can be avoided by using a modified hybrid ring (88], as shown in Fig. 3.29. The characteristic impedances of the lines for both conventional and modified

3.2

HYBRIDS AND COUPLERS

165

Zo Figure 3.28

Broadband hybrid ring using a sh orted parallel coupled ).../ 4 section.

2

\

Figure 3.29 Modified hybrid-ring configuration.

hybrid rings are given in Table 3.16 for Zo = 50 Q . The 3-dB hybrid-ring configuration has about 27% bandwidth when the amplitude unbalance is 0.4 dB and isolation and return Joss are better than 20 dB. An improved bandwidth hybrid-ring design bas been reported by Kim and Naito [89]. The configuration is shown in Fig. 3.30, and design parameters are listed in Table 3.17. Similar to branch-line hybrid, compact size, reduced-size [84], and lumped-element [85-87] rat-race hybrids have been developed. Figure 3.31 shows a lumped-element 180° hybrid. Following the same procedure as discussed in Ref. 85, the lumped elements for the low-pass "pi" section and the high-pass "tee" section can be expressed as

(3.28a)

166

FILTERS, HYBRIDS AND COUPLERS, POWER COMBINERS, AND MATCIDNG NETWORKS

TABLE 3.16 Characteristic Impedances of the Lines for Two Hybrid-Ring Directional Couplers Conven ti onal H ybrid Ring

Modified Hybrid Ring

Power-Split Ratio (dB)

Z1

Z2

Z1

Z2

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0 13.0 14.0 15.0

70.7 75.1 80.4 86.5 93.7 102.0 111.6 122.6 135.2 149.5 165.8 184.3 205.2 228.9 255.5 285.6

70.7 67.0 63.9 61.3 59.l 57.4 55.9 54.8 53.8 53.1 52.4 51.9 51.6 51.2 51.0 50.8

70.7 72.3 74.5 77.3 80.8 84.9 89.8 95.6 102.3 109.9 118.6 128.4 139.4 151.9 165.9 181.6

70.7 66.9 63.8 61.1 59.0 57.1 55.6 54.4 53.5 52.7 52.1 51.6 51.2 50.9 50.7 50.5

Source: A. K. Agrawal and G. F. Mikucki, "A Printed Circujt Hybrid rung Directional Coupler for Arbitrary Power Divisions," IEEE Trans. Microwave Theory Tech., Vol. MTT-34, December 1986.

Figure 3.30 Improved bandwidth hybrid-ring configuration.

1

Ci= - - -

v'2Zow

L

I - cos 01 I + cos01

= _ v'2Zo

(3.28b)

(3.28c)

w sin 02

1 + cos02 1 - cos02

(3.28d)

When 01 = 90° and 02 = 270° or -90°, element values for a 500 system become L1 =Li= 11.25// nH and C1 = C2 = 2.25 / f pF, where f is in GHz.

3.2

HYBRIDS AND COUPLERS

167

TABLE 3.17 Optimum Values of the Parameters and Bandwidth of the Improved 3-dB Hybrid-Ring Directional Couplersa

Y2

Y3

Y4

Y 1c

Y2c

Bandwidth

0.88034

1.34850

3.57850

7 .7350

1.05570

1.72860

50.67%

Optimized

0.88023

1.07190

1.82070

3.0

1.07160

1.34580

45.33%

0.85517 0 .86087 0.67781

1.07060 0.98403 0.82789

Optimized by specifying Y4

1.66050 1.36840 1.15230

2 .5 2.0 1.5

1.05500 1.05890 0.93429

1.37250 1.24130 1.18500

44.44% 43 .55% 39.11 %

0.77460

1.01140

1.12800

0.99623

1.37580

36.44%

Y1

Remarks

Optimized under Y3 = Y4

0

Yi through Y2c are normalized admittances. Source : D . I . Kim and Y. aito, "Broad-band Design of Improved Hybrid-Ring 3-dB Directional Coupler," IEEE Trans. Microwave Theory Tech., Vol. MTT-30, November 1982.

-2C1 T

--

T

2C1

2

4

3

C1.I_

.I. -

C1

-

Figure 3.31

Lumped-element 180° hy brid.

Matched Hybrid Tees. A matched hybrid tee (often called a magic tee) is an interesting example of a 3-dB waveguide directional coupler. The circuit is used in frequency discriminator circuits, balanced mixers, and other applications. The circuit configuration is shown in Fig. 3.32 and can be seen to be a combination of E- and H-plane tees. When a TE 10 mode is incident at port 4, the electric field has even symmetry about the midplane and hence no power is coupled to port 3. However, the waves coupled to ports 1 and 2 are of equal magnitude anct are in phase. Similarly, when a TE 1o mode is incident at port 3, the electric field has odd symmetry about the midplane and hence

168

FILTERS, HYBRIDS AND COUPLERS, POWER COMBINERS, AND MATCHING NETWORKS Port .3

Port 2

£-plane arm

H-plane arm

Port 1

Figure 3.32

Hybrid T configuration.

no power is coupled at port 4. The waves coupled from port 3 into ports 1 and 2 are equal in magnitude and in phase opposition. The scattering matrix of a magic tee is given by 0 0 1 I 1 0 0 -1 1 [S]=(3.29) v'2 1 -1 0 0 1

1

0

0

(a)

(b)

~)

Primary waveguide

A

f+-(2n - 1) :

•I

(--:::------Port 1 - -

-

7 - __._ - 7 - - - -

Port 2

r - ---.~- - - ~- -- -~dd ~ - Port 4

Port3 Canc!L _ - - ~ __,,._ -

zo, and the junction effects between the connecting guides have been neglected. In many designs, the latter is considered small and the former may be taken into account by replacing l with Zeff, which is given by

Leff = l

2L1z'

+-

"Jr

The integration limit cally negligible.

3.2.S

z'

[f3e(z) - f3o(z)]dz

(3.51)

Zo

is chosen to be the point at which the coupling is practi-

Wilkinson Couplers

A Wilkinson coupler [116, 117], also known as a two-way power divider, offers broad bandwidth and equal phase characteristics at each of its output ports. Figure 3.45 (inset) shows the schematic diagram of a Wilkinson coupler. The output port isolation is obtained by series terminating the output ports. Each of the quarter-wave lines has the characteristic impedance of .J2Z0 and the output ports are terminated by a resistor of 2Zo ohms, Zo being the system impedance. A Wilkinson power divider offers a bandwidth of about one octave as shown in Fig. 3.45. The performance of a Wilkinson coupler can be further improved, depending on the availability of space, by the addition of a A/ 4 transformer in front of the powerdivision step.

182

FILTERS. HYBRIDS AND COUPLERS, POWER COMBINERS, AND MATCHING NETWORKS

Zo

Z=O

(a)

z' 2

1

0

Plane of - - - symmetry

L...___,~.I

4

3

y (b)

y

If

Even mode

♦ Two-orthogonal modes X

Waveguide

M

y

Fin-structure

Magnetic wall

_ __._......._,. Ey ,Hx Odd mode y

(c)

rb ► X

''

L--------4► Ey,Hx

2a

Even mode

Figure 3.44 (a) Schematic of distributed coupler; (b) coupled finlines and the field distributions; (c) coupled image lines and the field distributions.

Multisection Wilkinson Couplers. The octave bandwidth of a single-section coupler proves to be inadequate in many applications. Therefore, Cohn [117] proposed the use of multisections for bandwidth expansion. The use of multisections makes it possible to obtain a decade bandwidth. A multisection Wilkinson power-divider-coupler consists of a number of quarter-wave sections with resistive terminations at the end

3.2

rt4

~

Input

Cl)

>

183

Output 1

'124

Zo

a:

HYBRIDS AND COUPLERS

Input

..r22a

1.5

Uncompensated

1.2 1.0 0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

1.4

1.5

Normalized frequency

Figure 3.45

Frequency r esponse of Wtlkinson coupler.

of every section, as hown schematically in Fig. 3.46. Larger bandwidth and greater isolation are obtained when a larger number of sections are used. Design of such couplers is given in Refs. 118 and 119. Figure 3.47 shows maximum input/output VSWR versus bandwidth for multisection Wtlkinson couplers. Parameters of multisection stripline Wilkinson couplers on a low-dielectric-constant substrate are given in Table 3.22 (119].

Unequal-Power-Split Wilkinson Couplers. Figure 3.48 shows the schematic of the unequal-power-split Wtlkinson couplers (120]. As can be seen from the figure, output impedance transformers are also required, in contrast to the equal-power-split case. The design equations for the uncompensated and compensated cases are as follows:

l + K2 R =Zo K Zo Zs =,JK

')J4 TYP.

7

Figure 3.46 Multisection Wtlkinson coupler.

184

FILTERS. HYBRIDS AND COUPLERS, POWER COMBINERS, AND MATCHlNG NETWORKS 1.4 - -- - . - - - - - , - - - - -, r - - - - r - - - - r - r - - - - - .

1.3

3 section (input)

2 section (input}

a: ~

4 section (input)

Cl)

>

E 1.2

::,

E

·x

ctS ~

2 section (output}

1.1

4 section (output)

1.0

2.0

3.0

5.0

4.0

6.0

Bandwidth - f2 /f1

Figure 3.47 Maximum input/output VSWR versus bandwidth for a multi ection Wtlkinson coupler. (From H. Howe, Jr.• Stripline Circuit Design. Artech Hou e, Dedham, MA, 1974 .)

Uncompensated

Compensated

r/ 4

Z 1 = Zo [ l : K 2 22

= Zo[K3f4(1 + K 2)1 4]

Z3 = Zo

(1

+ K 2) 1/ 4 K 5f4

The e coupler provide two in-phase i olated output with a constant arbitrary power division over a wide bandwidth. Design equation for arbitrary power divi ion and termination impedances for a three-port power divider are also available [121]. The size of the power divider can be reduced by capacitive loading similar to miniaturizing hybrids. Schematic of a miniaturized Wilkinson power divider is shown

3.2

HYBRIDS AND COUPLERS

185

TABLE 3.22 Parameters of Power Divider and Common Design Parameters for Both Wilkinson and Improved Version Power Dividers

Number

z

R £rpected Section

1

400.000 211.-+60 107.180

2

3

482.160 291.630 172.620 103.165

2

3 4

4-+2.480 616.145 446.230 319.900 217.580 129.620 248.260

2

3 4 5

6

7

2.506 1.729 1.096

=4 55.785 64.785 77.175 89.630

Expecred Section l

=3 57A85 70.710 86.980

Expected Section 1

Width 0 (mm)

2.547 1.973 1.401 0.985

=1 56.370 60.255 65.085 70.710 76.820 82.985 88.700

2.415 2.16] 1.887 1.615 1.365 1. 150 0.978

Epsilon = 2.150, thickness of sub= 1.600 mm. Source: C. Q. Li. S. H. Li, and R. G. Basisio, "CAD/CAE Design of an Improved Wideband Wilkinson Power Divider," Microwave J., Vol. 27, ovember 1984. 0

(a)

r

)J4T/J4__, ~

Zo

Z4

Zo

(b)

Figure 3.48 Schematic of unequal-power-split Wilkinson coupler: (a) uncompensated; (b) compensated. (From L. I. Parad and R. L. Moynihan, "Split-Tee Power Divider," IEEE Trans. Microwave Theory Tech., Vol. MTT-13, January 1965.)

186

FILTERS, HYBRIDS AND COUPLERS, POWER COMBINERS, AND MATCHING NE1WORKS

Figure 3.49

Configuration of a reduced-size two-way divider.

in Fig. 3.49. The design equations for this configuration are given below [122) :

.J"izo

= sin(flol)

(3.52a)

cos(f3ol ) C1= - --

(3.52b)

Zo1

Wo.J"i

where f3o (= 2rr / )...) is the propagation constan4 2-0 is the characteristic impedance of the system, and wo(= 2rr/ 0) is the angular frequency at the design frequency Jo. The termination resistance R = 22-0 and C2 = 2 C 1• For example, for f3ol = )../ 4, ).../ 8, and ).../ 12, the values of C 1 are 0, 0.16, and 0.195 pF, and the values of 2-01 are 70.7, 100, and 141.4 Q respectively.

3.3

POWER COMBINERS

Solid-state devices are low power, and with increasing frequency, the power output from a single solid-state device decreases rapidly. In many applications radio-frequency (RF) power levels are required that far exceed the capability of any single device or amplifier. It is therefore desirable to extend the power level by combining techniques to take advantage of the many desirable features of solid-state devices, such as small size and weight, reliability, and performance in a broader range of applications. Although there are fundamental limitations to the power that can be generated from a single device, the achievable power levels can be significantly increased by combining a number of devices operating coherently or by accumulating the power from a number of discrete devices. This may be done in one of two ways: either by combining power at the device level or at the circuit level. Most combining techniques can provide "graceful degradation" in the case of failure of one or more devices in the combiner.

3.3

POWER COMB INERS

187

Microwave [123] and millimeter wave [124] power-combining techniques have been reviewed in the literature. In thi section only circuit-level power-combining techniques as classified in Fig. 3.50 are discussed. Each combining scheme consists of dividers and combiners, and in general, the combining and iliviiling circuits are identical. The de irable characteristic of a combining tructure are minimum loss in the matching element , minimum los in the combining networks, minimum amplitude and phase imbalance, a good combiner input VSWR, even distribution of dissipated heat in the device . and efficient removal of ilissipated heat.

3.3.1

Resonant Cavity Combiners

In re onant cavity tructures cylindrical or rectangular cavities are used to combine the power output from a number of devices. These ilividers/combiners have low loss (~0.2 dB) and a combining efficiency of 85 to 90%. These combiners have proven to be the mo t successful for narrow-band applications up to 220 GHz [124]. An 80-W FET amplifier using an eight-way divider-combiner working from 5.9 to 6.4 GHz has been developed [125].

3.3.2

onresonant Combiners

There are two categories of nonresonant combiners: (1) those that combine the output of N devices in a single step (known as N-way combiners) and (2) tree or chain combining structures (known as corporate combiners). Nonresonant combiners have been developed for wide-band systems up to 60 GHz.

N -Way Combiners. Many nonresonant N-way combining techniques are available. Essentially, three types of N-way combiners are used for combining large numbers of amplifiers: Wtlkinson, radial, and planar. The N-way Wilkinson divider (116] shown in Fig. 3.51a has the advantage of low loss, moderate bandwidth, and good amplitude

Combining techniques

Resonant cavity combiners

Rectangular

waveguide resonant cavity combiners

Nonresonant combiners

orporate combiners

Cylindrical resonant cavity combiners

Combiners

Radial line

Wilkinson combiners

Hybrid coupled

Figure 3.50 Different circuit combining techniques.

188

FILTERS, HYBRIDS AND COUPLERS, POWER COMBINERS, AND MATCHING NETWORKS

'4-)../4--1 (a)

---

z Z =Z0\!N

z

z

Zo

- - --1,..___...,.____.

Input/Output I

2

3

4

I

I I

z

N

(b)

Zo 1

Input

,..__........_1~

Output

I I I I I I

Planar 1 1 L__~mbin~ __ j

Figure 3.51

(a) Wilkinson N -way divider-combiner; (b) modified N-way divider-combiner.

and phase balance. However, its major disadvantage for power applications is the "floating starpoint" isolation resistors. These resistors require a nonplanar crossover configuration, which limits the power-handling capability of the combiner. Fortunately, a simplified version as shown in Fig. 3.51b can also be used [126, 127]. This particular arrangement has a combining efficiency on the order of 90% and shows much promise for chip combining as well as for MMIC applications. Radial line and planar N -way combiners are shown in Fig. 3.52. The radial-line [128] combiner has low loss, inherent phase symmetry, and good isolation. Its main disadvantage is that it requires a threedimensional structure. On the other hand, the planar N -way combiner divider [129] requires (N - I) x N quarter-wave sections for maximum isolation and thus is very large. A compact structure using a tapered microstrip line has also been described [130]. The inherent redundancy in the N -way combiner makes it possible to obtain a graceful degradation characteristic. In an N -amplifier combiner with F failed amplifiers, the power output, relative to maximum output power, is given by Po/ Pmax = (N - F)/ N 2 •

Corporate Combiners. A corporate structure for combining power from two-way adders or combiners as shown in Fig. 3.53a is known as tree combiner. The loss

3.3

(a)

POWER COMBINERS

l--.l./4-l z

189

1

R

z

2

R

z

Zo

3

R

Input/output

z

R

I

R

4

I I

R

I

z (b)

~.lJ~

Zo1

, I

Zo1 Figure 3.52

l-.lJ4-,

H~ R, R, R,

2'oM 1

R2

Zo2 I

I

I

I

N

I

Zo2

----

R2 R2

----~

RM

ZoM

2 I I

RM

2'oM

I

1

I

I

iRM

M= N- 1

I

oN

N-way divider-combiner: (a) radial line; (b) planar.

in adders limit the combining efficiency. Figure 3.53b (123] illustrates the combining efficiency of the corporate structure versus the number of devices for various loss per adder values. The number of devices combined in this way is binary. Examples of two-way adders are directional couplers, hybrids, and the two-way Wilkinson combiner and are described in the preceding section. Among the two-way adders, the Lange coupler is usually preferred because of its good isolation and wide-band properties. However, cascading these to obtain high-order combining becomes impractical beyond a four-way combiner due to the relatively high coupler loss. A serial or chain combiner is shown in Fig. 3.54a. Here each successive stage of an N -way combiner adds 1/ N of the power delivered to the output. The number of the stage determines the required coupling coefficients for that stage~ as indicated in the figure. One advantage of the chain structure is that another stage can be added simply by connecting the new source to the line after the Nth stage through a coupler with l0log(N + 1) coupling coefficients. The roles of input and output ports are changed for the divider structure. Losses in the couplers reduce the combining efficiency and bandwidth attainable with this approach. Combining efficiency for the chain structure for each path is shown in Fig. 3.54b [123]. The four-way structure on microstrip can be used to realize combining efficiency on the order of 90% over an octave or greater bandwidth. A comparison of various circuit-level power-combining techniques appears in Table 3~23 [131].

190

FILTERS, HYBRIDS AND COUPLERS, POWER COMBINERS , AND MATCHING NETWORKS

(a)

+

T

I

I

I I

I I

I

I

+

I

Pout

I .L

I

•2N I 2 N- 1 Devices A~ders

2N -2

22

Adders

2 Adders

Adders

1

Adders

'

y

N stages of power combining

100

(b)

------------------...a-..--------, ..__

a,,-,. .---._

~ .... 90

-

""' "" ' ' ' ''u. ' ~, '

/\ '- '

'

~ 80 ~

Cl)

'

·5

'n '

'

◊... '

C

·c

u0

.... . . ,~

~

'

70

C>

:0 E

0 .1

'°' ',,' ', ...

C

~

,..

------

---e..;;

'

, .... ,

0.2

.... ~

.. _

03 . ' , 'A..

'"O

....

--. __

....

'

' , 0.4 .... ,

'- '

60

- .... _-- .

- ---- -

Loss per adder (dB)= 0.5

'o.. ,

'

-.,

'-6,

', ',

'6

□, ....

'~ ', 50

'o.

'-a '

.......

'"

40 ~_.__ _ _....__ __.__ _ _....__ __._________.____ 2

4

8 16 32 Number of devices combined

64

128

Figure 3.53 (a) Corporate combining structure; (b) combining efficiency for a corporate combining structure. (From K. J. Russel, "Microwave Power Combining Techniques," IEEE Trans. Microwave Theory Tech., Vol. MIT-27, May 1979.)

3.4

IMPEDANCE-MATCHING NETWORKS

An impedance-matching network is perhaps the most extensively used circuit element in passive and active circuits. When two sections or components of different

impedances are connected together, an impedance-matching network is invariably

3.4

In

(a)

In

Coupling ___ coefficient

In

_3 dB

IMPEDANCE-MATCHING NETWOR KS

In

-4.78 dB

-6dB

191

In

-~--out

- 10 log N dB

100 r - - - - - - - - - - - - - - - - - - - - - -

(b)

90

---

#- 80 >-

0 C:

(l)

c3

i

70

O>

.s C :0

§

(_)

Loss (dB)

60

=

50

2

4

8

16 32 Number of devices

64

128

Figure 3.54 (a) Serial combining structure; (b) combinmg efficiency for the chain combining structure. Loss in decibels refers to the Joss in each power path in each stage's coupler. (From K. J. Russel, "Microwave Power Combining Techniques," IEEE Trans, Microwave Theory Tech. , Vol. MIT-27, May 1979.)

required to ensure maximum power transfer from one port to another. In solid-state circuits such as amplifiers and mixers, matching networks play a very important role when low noise and broadband characteristics are desired. Design of matching networks at microwave frequencies employ both lumped elements and distributed circuit elements while at millimeter wavelengths they use small lengths of transmission lines or waveguides. Single or multiple stubs and one or more quarter-wave sections are used as impedance matching circuit elements. An excellent treatment of impedance-matching networks is given in Ref. 132.

192

ALTERS, HYBRIDS AND COUPLERS, POWER COMBINERS, AND MATCHING NETWORKS

TABLE 3.23

Comparison of Circuit-Level Power-Combining Techniques Disadvantages

Advantages

Combining Technique

N-way waveguide cavity

Low loss High efficiency

Nonplanar Complex assembly Narrow band

N-way Wilkinson

Low loss Moderate bandwidth Good isolation High efficiency

Nonplanar Low power

N-way radial line

Low loss Good isolation

Nonplanar Complex assembly

N -way planar

Large bandwidth Good isolation Moderate loss

Large size Low efficiency

Corporate structure

Good isolation Large bandwidth

Impractical beyond four-way due to low efficiency

Chain structure

More flexible Octave or greater bandwidth Good efficiency Good isolation

High-resolution fabrication required Complex design

Source: I. J. Bahl and P. Bbartia, Microwave Solid Stare Circuit Design, Second Edition. Wtley, York, 2003.

3.4.1

ew

Transmission-Line Matching Schemes

In this section the commonly used impedance-transforming properties of transmission lines for designing narrow- as well as broadband matching networks are reviewed. A standard transmission line, such as a microstrip, can be used as a series transmission line, as an open-circuited or a short-circuited stub, and as a quarter-wavelength transformer section for impedance matching. Series Single Section. Simplest of all matching networks is a eries transmission line of electrical length 0 and characteristic impedance Zm used to match a complex load Z L ( = RL + jXL) to a real resistance Ras shown in Fig. 3.55a. The required transmissionline parameters are given as [133]

Zm and

✓RRL - (R;_ + Xi) = -------1 - RL / R

✓(1 - RL / R)[RRL - (R f + Xl)] tan0 = - - - - - - - - - - - - XL

(3.53a)

(3.53b)

It is a narrow-band matching technique with limited use, since only those impedances can be matched that result in a real value of Zm in (3.53a).

3.4 (a) R _

lMPEDANCE-MATCHING NETWORKS

193

_,

,

__

(}

(b) R . _ __

,..

___;::JJ.L,__

90°

Z'm

• 1• 45~

Figure 3.55 Two-port impedance transformers: (a) single section; (b) two sections. (From H. A. Atwater, ''Reflection Coefficient Transformations for Phase-Shift Circuits," IEEE Trans. Microwave 77ieory Tech. . Vol. MIT-28, June 1980.)

In a practical design, the characteristic impedance of the transformer is limited by the type of transmi ion line used. For example, the impedance values lie between 20 and 100 n when microstrip or triplines are used. Using (3.53), one can show that with these restrictions on the impedance values tuning can be done over approximately 22% of the Smith chart [134]. This range can be enhanced by using two transmission lines in series [ 135]. One section brings the impedance into the matchable area and the other one matches it to the source impedance. The two-line section transformer with characteristic impedances limited within 20 and 100 n can match loads over 75% of the Smith chart. Expressions for the design of transmission-line transformers between complex loads and complex source impedances, for maximum power transfer or a conjugate match, have also been given [136]. These are

Zm

= (Rs 1ZLl 2 -

Ri lZs1 Ri -Rs

112

2 )

(3.54)

and (3.55) where Zs = Rs + j Xs is the complex source impedance. Figure 3.55b shows a matching scheme using two transmission-line sections [137]. The initial section (45°) of this network has a characteristic impedance equal to the magnitude of the terminating impedance (3.56) and the final section (90°) has the characteristic impedance

(3.57)

194

Fil..TERS, HYBRIDS AND COUPLERS, POWER COMBINERS, AND MATCHING NETWORKS

Stub Matching. Alternatively, single-stub or multiple-stub tuners are used to match complex impedances, such as in a feeder connection between a transmitter and an antenna or between the generator and the input of an active device or output of the device and the load. Double-stub matching is needed in cases where it is impractical to place a single stub physically in the ideal location. Location and length of opencircuited or short-circuited stubs is easily calculated using graphical techniques on a Smith chart. The stub-tuned matching circuits can, of course, be designed into a microstrip or stripline circuit. Once again, the desired impedance matching of a complex load is achieved over a narrow bandwidth. When multiple stubs are used for impedance matching over a large frequency range, CADs may be used. Single-Stub Matching. Since the stub-matching problems involve parallel connections on the transmission line, it is easier to design the circuit using admittance values rather than impedances. In a single-stub-matching networ~ the total admittance of the terminated line and the stub is matched with the admittance of the source Y = 1/ R for maximum power transfer. For example, in Fig. 3.56a, the stub is located at AA in such a way that YAA

= Y = Ys + Yv

(3.58)

where Ys is the admittance of the stub (snort circuited or open circuited) of length l , and YD is the admittance of the load transformed at AA location. Double-Stub Matching. A double-stub-matching network consists of two stubs (shortcircuited sections preferred, because they are easier to obtain than a good open circuit) connected in parallel with a fixed length between them (Fig. 3.56b). Usually, the length of the transmission line between the stubs is ½, or of a wavelength. Lengths and impedances of both stubs and the location of the stubs closest to the load can be adjusted to get a perfect match over a narrow bandwidth. In Fig. 3.56b,

i, i

(3.59) where YB B is the total admittance at the left of the second stub, Ys2 the admittance of the second stub, Yv2 the admittance YAA(= Ys 1 + Yv 1) at location AA transformed to location BB by the line section between the two stubs, Ys 1 the admittance of stub one, and Yv1 the load admittance transformed to the location of the first stub. Quarter-Wavelength Double-Stub Transformers. Figure 3.57 shows a configuration for a quarter-wavelength double-stub transformer (137] . The electrical lengths of the stubs are

= tan- 1[-bt =f J gi (1 02 = =f tan- I[(l - gL)/ gL] 01

gi)]

(3.60)

(3.61)

Quarter-Wavelength Transformers. Another important class of impedance-matching network is the quarter-wave impedance transformer, which is used to match a real

3.4

(a)

IMPEDANCE-MATCHING NETWORK S

195

A

I I I

y

I

I

'

(b)

B

~

1/' ~~

/-

A

I

Stub2

Stub 1

I

Y51

Ys2

'

l

--

1h01

Yss

y

,.

B

Figure 3.56

ZL

YAAI

.,A

L

(a) Single-stub matching configuration; (b) double-stub matching configuration.

goo

I• R

•I

Zm Zm

1

Zm

i

ZL

-

Figure 3.57 Double-stub impedance transfonner with quarter-wavelength spacing. (From H . A. Atwater, "Reflection Coefficient Transformations for Phase-Shift Circuits," IEEE Trans. Microwave Theory Tech., Vol. MTT-28, June 1980.)

196

Fil..TERS, HYBRIDS AND COUPLERS, POWER COMBINERS, AND MATCHING NETWORKS

impedance to another real impedance. For narrow-band applications, a single section is adequate, while for wider bandwidths, multisections are required.

Single Section. The input impedance of a lossless network shown in Fig. 3.55a is given by (3.62) When Zin= R = Zo at 0 = 1r/ 2, (3.63)

If Zm is chosen equal to JZoZL, the load impedance ZL is matched to Zo at the center frequency, where the transformer length corresponds to (A/ 4 + nA/ 2). For TEM transmission lines, 0 = 1rl/(2lo), where lo is the frequency for which 0 = 1r/ 2. In this case, the fractional bandwidth can be expressed as

Lil = lo

2(1 - L) = 2(1 - :!:em) lo 1r

(3.64)

where (3.65) and

rm is the maximum value of reflection coefficient in the required passband.

Multisection. Ann-section impedance transformer is given in Fig. 3.58. The reflection coefficient at the input of the matching network is given by a sum of the series of multiple partial reflections arising at each of the impedance discontinuities. A minimum passband VSWR for a given input/output impedance ratio and bandwidth is pos ible when the partial reflection coefficients are chosen to have ratios corresponding to the like terms in a Nth-order Chebyshev polynomial [138]. The value for the (m + l)th-section impedance in the transformer is given in terms of mth-section impedance to be Gm

ln(ZN+2/Z1)

(3.66)

N+I

La; i=1

Z1

~

~

;G?: ~~ ~~ 1-- 0

.,.

Figure 3.58

8

., ..

z..

ZN+1

~

------

r) -----0~

N +1

t--0~

N-sectioo impedance transformer.

ZN +2=ZL

3.4

TABLE 3.24 N

a1

1 2 3 4 5 6

XQ

oXO

.,

xx~ 0 4

x~ Xo .x 0

=

IMPEDANCE-MATCH ING NETWORKS

197

Proportionality Constants a; for the Cbebyshev Transformer° a2

a3

04

2\·2 -2

3x~ 4x-t 5x~ 0 6xg -

3x0 4x:?

6x04 - 8x02 + 2 lOxJ - 15x5 + 5xo 15x06 - 24x04 + 9x02

5x~ 0 6xri

20x06

-

36x-t0

+

l 8x02

-

2

ec 91 ; 81 = :r / 2 - wrr /4: w = fractionaJ bandwidth.

where the a; are the proportionality constants related to the Cbebysbev polynomial, ZN+2 the load impedance, and 2 1 the source impedance as shown in Fig. 3.58. Table 3.24 lists the value of these constants up to N = 6 from a simple recursion formula given by Cohn [138].

Tapered Transmission Lines. An alternative approach to realizing a broadband transformer is to u e a tapered-line section with a characteristic impedance that is variable with longitudinal di tance. Some of the tapers that have been used for impedance matching are of the linear, the exponential, and the Chebyshev designs [139, 140]. The synthesis of a tapered-line transformer is done at the lowest frequency of interest because the derivative of the reflection coefficient decreases rapidly with frequency. The length of the exponential taper is given as [140] (3.67) where r m is the maximum reflection coefficient, Z L is the normalized (with respect to source impedance) value of the load impedance to be matched, and Ag is the guide wavelength at the lowest frequency of interest The normalized impedance profile as a function of length parameter l is given as Z (l)

= exp{ln[Zi(l/ L)]}

(3.68)

The design of linear and Chebyshev tapered-line transformers has been described in Ref. 139.

Coupled-Line Transformers. Quarter-wavelength coupled lines have been used as filter elements and directional couplers, and their other important applications are in broad-band impedance-matching transformers and de blocks. If Z s and ZL are the source and load impedances to be matched by a A/ 4 section of a coupled line shown in Fig. 3.59a, at center frequency the relation between the terminating impedances and the even- and odd-mode impedances of the coupled section are given by (3.69)

198

FILTERS, HYBRIDS AND COUPLERS, POWER COMBINERS, AND MATCHING NEfWORKS

(a)

I Zs

1

1----o z,

w

-i.__. .t. . .w_ _ 1--t :::)J4

-I

(b}

Figure 3.59

Quarter-wave-coupled line section: (a) impedance transformer; (b) de block.

The quarter wavelength of the coupled section at the center frequency may be calculated from the following relation [141]:

l=------n_/_ 2_ _ _ __ K

+ [(Zoe -

Z0o)/(Z0e

+ Z0o]~K

(3.70)

where

K ~K

= (/Je + /Jo) / 2 = (/Je -

/Jo) / 2

/Je = 2rr/JcKoe

/3 = 2rr/ / cKoo 0

where / is the center frequency, c the velocity of light, and Koe and K0o the velocity factors for the even and odd modes, respectively. Table 3.25 summarizes electrical and physical parameters for typical microstrip coupled-line transformers. These transformers also act as de blocks in microwave circuits. When Z s = ZL, Zo = (Z0e - Z0o)/2. Such a configuration (Fig. 3.59b) can easily provide about an octave bandwidth with insertion loss within 0.2 dB up to 20 GHz. 3.4.2 Lumped-Element Matching Circuits. Inductors and capacitors connected in L-shaped configuration are widely used as impedance-matching circuit elements.[87] There are four possible arrangements of inductors and capacitors which can be utilized in T-Configuration. The range of impedance transformation depends on the value of original inductors or capacitors in the T-configuration. The addition of one more element to the simple T-section matching circuit gives the designer much greater control over bandwidth [142] and also pennits the use of more practical circuit elements. The circuit configurations and circuit element values

TABLE 3.2S Parameters for Different Transformation Ratios° Transformation

Serial

Number

1 2 3 4

...

5 6 7 8

9 10 1_1 0

~

i

fr

'

From:

To:

Q

Q

50.0 50.0 50.0 50.0 50.0 50.0 50.0 50.0 50.0 50.0 50.0

= 9.8, t = 3 to 4 µm , h

75.0 75.0 53.0 53.0 35.0 35.0 25.0 25.0 10.0 10.0 10.0

= 635 µm .

Desired Impedances (

Zo" Q

162.5 152.5 142.96 132.96 123.67 113.67 110.7 1 100.71 84.72 74.72 64.72

~

',

Actual Impedances

Z0o

Zo~

Zoo

Q

Q

Q

40.0 30.0 40.0 30.0 40.0 30.0 40.0 30.0 40.0 30.0 20.0

162.40 153.8 1 143.06 132.50 123.24 113.75 110.28 100.69 84.66 74.57 65.11

40.59 30.24 40.47 30.72 39.83 29.96 40.03 30.07 39.81 30.17 20. 14

:.

Velocity Factors I

Physical Dimensions

Koe

Kao

W/ h

S/ h

0.404 0.403 0.401 0.399 0.397 0.394 0.394 0.391 0.385 0.381 0.376

0.4300 0.4298 0.4298 0.4296 0.4293 0.4291 0.4287 0.4284 0.4261 0.4253 0.4246

0.120 0. 159 0.175 0.240 0.260 0.350 0.335 0.455 0.565 0.790 1.05

0.048 0 .018 0.070 0.030 0.135 0.040 0.135 0.055 0.250 0.115 0.01

N

8

(1)

(2)

c;

C1

R1

(3)

L1

~

R2

z

- 1) -

1

L2 = R

2

NR 1

L 3=

~~

L1

Ri

L1= A 1 ✓N -1 u.lo

C 1 = (woA 1 ✓N -1 r 1

C2= ( tooR2J

L2

o

(4.3)

An important parameter of a resonant circuit is the quality factor Q, which specifies the frequency selectivity and performance in general and is defined as

Q

energy stored

W

energy dissipated

P

= w- - - -- - = w-

(4.4)

where W is the maximum stored energy and P is the average power loss. The peak value of electric energy stored in the capacitor is 2 We and occurs when the energy stored in the inductor is zero, and vice versa, and is given by (4.5)

The dissipated power, in turn is (4 .6)

Hence, for the circuit of Fig. 4.2, Q

= a>oCoRo

(4 .7)

If the resistor Ro of the transformer coupled resonator represent the losses in the resonant circuit only, the Q given by Eq. (4.7) is called the unloaded quality factor, Q 0 • If the resonant circuit is coupled to an external load (as shown in Fig. 4.2) that absorbs a certain amount of power, this loading effect will change the net resistance and consequently the quality factor. The Q, called the loaded quality factor and denoted QL, for the circuit of Fig. 4.2 is given by

QL

=

euoRoCo 1 + Ro/N 2Zo

(4.8)

The external quality factor, denoted Qexr, is defined to be the quality factor that would result if the resonant circuit were loss-free and the external loading was present. Thus (4.9) The use of the definitions above shows that

1/ Qi

= 1/ Qext + 1/ Qo

(4.10a)

4.2

BASIC DEFINITTO S

213

The loaded quality factor, Q L for the system of Fig. 4.2 can be al o written as follows: Ql

where

/3 i

= Qo/ (1 + /3)

(4. 10b)

defined as a coupling coefficient and is given by (4.1 1)

There are three types of coupling coefficient :

• Critical Coupling. tor; then

This occurs when the resonator is matched to the genera-

/3 =

(4.12)

l

• Overcoupling. Thi occur when the resonator's terminals are at a voltage maximum. The normalized impedance at the voltage maximum is equal to the reciprocal of the standing-wave ratio (VSWR). That is,

f3 = 1/ VSWR

(4.13)

• Undercoupling. This occurs when the resonator's terminals are at the voltage minimum and the input terminal impedance is equal to the standing-wave ratio (VSWR). That is, (4.14) f3 = VSWR,

In the distributed circuit area, the total energy stored in the resonator is obtained by integrating the energy density over the volume of the resonator: E 2 -IEI d v =Wm =

We = /.V

2

!.µ

-I HI2 d v

V

2

(4.15)

where E and H are the peak values of the field intensities and E and µ are the permittivity and _permeability, respectively. The average power less in the resonator is evaluated by integrating the power density over the inner surface of the resonator. Hence P

1

= -Rs IH, I2 da 2

(4 .16)

s

where Hr is the peak value of the tangential magnetic intensity and Rs is the swface resistance of the resonator. Substitution of Eqs. (4.15) and (4.16) into (4.4) yields

wµJ.1Hl 2 dv Q=

V

R, 11H,l 2 da

(4.17)

214

CAVITIES AND RESONATORS

4.2.3 General Formulas for Quality Factors The Q 's of simple series- and shunt-resonant circuits have been treated quite extensively in literature and consequently well known. Frequently, it is necessary to find the Q of a more complex circujt, such as a distributed resonator and a lumped coupling mechanisms. A general formula for this is easily derived from the fundamental definition of Q and is written as [l] Wo

Q

aB

= -2G- x -aw

(J)~

wo

ax

2R

aw

(4.18)

=- X -

w~

where B and G are the shunt susceptances and conductances while X and R are the series reactances and resistances at the desired tenninal. Equation (4.18) gives the general Q for a system whose impedance or admittance function are known at a given terminal and their derivatives evaluated. The various Q' s follow by appropriate choices of G or R . To find Q 0 , the loss (and hence the value of G or R ) is chosen to correspond to the internally dissipated energy. To determine Q ext or QL , the loss is chosen to correspond, respectively, to the externally dissipated energy or to the total dissipation. This will be further clarified by examples in Section 4.3.3.

4.3 DESIGN OF AND COUPLING TO WAVEGUIDE-CAVITY RESONATORS 4.3.1

Rectangular-Cavity Resonators

Figure 4.3 illustrates a rectangular cavity of height b, width a, and length, d . This structure may be considered to be a section of rectangular waveguide terminated in a short circuit at z = d. If the length of thi rectangular waveguide equal a multiple of a half guide wavelength at the frequency, f, the resultant standing-wave pattern is such that the x and y components of electric field are zero at : = 0. This means that a short circuit can be placed at z = 0 as well. The resultant structure is known as rectangular cavity. The resonant frequency of a cavity resonator is obtruned by solutions of Maxwell's equations [2] or transverse resonance technique [3] which satisfy the boundary conditions impo ed by the resonator. It i always assumed that the cavity is made of a

Y ♦

----.b

I

#_/, .r---

_t_:--------~--- --►x

z

Figure 4.3 Rectangular cavity resonator.

4.3

DESIGN OF AND COUPLING TO WAVEGUIDE-CAVITY RESONATORS

215

perfect conductor. The boundary conditions which must be met are that no tangential electric field and no normal magnetic field exist at the surface of the cavity walls. Application of either approach provides the ame result for the resonant frequency of a rectangular waveguide resonator for either TE or TM modes and is given by

(4.19)

where d

A = _!_ P

(4 .20)

2

A _ g -

AO ✓-;::::l====(A==o== / A==c== .nm==):::;::2

(4.21)

2

Ac nm

'

= -;::= =2 = == =2 j(n/ a ) + (m/ b)

(4.22)

and Ao is the free-space wavelength. The mode subscripts nmp pertain to the number of half-sinusoid variations in the standing-wave pattern along the x , y , and z axes, respective!y. Scrutiny of Eq. (4.19) reveals that there is an infinite number of resonant frequencies corresponding to different field distributions. If b < a < d is satisfied, the cavity will support the TE101, dominant mode, which is the lowest resonant frequency and corresponds to the TE 10 mode in rectangular waveguide. The greatest separation between the dominant and the next higher-order mode is obtained for a square-base cavity (a = d) with height one-half or less of the base length (a / b ~ 2). In this case, the second resonance is 1.58 times the first (lowest) resonance. Additional information regarding higher-order excitation and mode separation can be found in Harrington [4]. To determine the cavity unloaded quality factor, Q, the losses caused by the finite conductivity must be evaluated. This is done with the aid of Eq. (4.17) of Section 4.2.2. Introduction of the appropriate fields and integrating [5] gives us

Q0n:nlllP

=

2 m-p)2 + (m)2 sad [( - + (n)22] + r,bd [(n- -p)2+ ((m)2 - + (n)2 - )] ad

a

+ab(~)2

b

[er +Gr]

bd

s=

17 = {

b

(4.23)

where

1 ifn#O { ½ ifn=O

a

1 ifm;,fO ½ ifm=O

216

CAVITIES AND RESONATORS

and Zw =

.JiiTi =

Q°™mn;,

377. 7 Q = the free-space impedance. Also,

=

with

(:)2 b(ya + d ) + Gf a(yb+d)

(4.24)

1 if p,':0 y = { ½ ifp=0

If the cavity is filled with a dielectric material of permittivity

=€

€

I

-

•

]€

II

(4.25)

the quality factor becomes (4.26) where (4 .27)

Exampk 1. Design a cavity resonator to operate at 10 GHz using the TE 1o1 , dominant mode, and standard rectangular waveguide. Determine the first higher-order mode resonant frequency and the unloaded quality factor, Q 0 , assuming the waveguide is made from silver.

Solution a. Standard rectangular waveguide for 10-GHz operation is designated as WR-90 with the following dimensions [6] :

= 0.9 in. b = 0.4 in.

a

(4.28) (4.29)

Application of Eq. (4.21) for the case of thls example where n d =Ag/2

(4.30)

Ago= 2d d

= p = 1 and m = 0: (4.31)

= 0.781 in.

(4.32)

Solution b. Based on previous discussion, the first higher-order mode is the TEo 11 and its resonant frequency is given by

(bt) + (dt) 2

/~11

= c2

2

(4.33)

4.3

DESIO

OF AND COUPLING TO WAVEGUIDE-CAVITY RESO ATORS

217

TABLE 4.1 Surface Resistivity of Frequently Useable Metals° 2.52 X 10- 7 ./7 26 1 X lQ- 7 ./7 3.26 X 10- 7 ./7 3.0 X 10- 7 .Jf

Silver Copper Gold Aluminum 0

fin hertz.

Source: S. Adam, Microwave Theory and Applications. Prentice-Hall. Englewood Cliffs. J. 1969.

Substituting the appropriate value yield

= 16.57 GHz

Jr-rr=_ · •'-iJI I

(4.34)

Solution c. The unloaded quality-factor equation for the dominant mode, TE101 , is deduced from Eq. (4.23) and is given by

Q 0n;;101

=

bZw(k101ad) 3 21'(2 R s(2a3b + 2d3b + a 3d

(4.35)

+ d 3a)

where k 101

= ]'( J(l/a) 2 + (l/ d) 2

(4.36)

Combining the results of Eqs. (4.28)- (4.36) and Table 4.1 for surface resistivity of silver, the unloaded quality factor, Q o is calculated to be Q

(5.326

_ rEtot -

X

0.9

m2 X 2.52 X +9

3

X

0.781) 3

10- 7 .Jioio(2 X 0.93 0.781 + 0.781 3 X 0.9)

= 8119 4.3.2

X

X

0.4

X

0.4 + 2

X

377.7 X

0.781 3

X

0.4

(4.37)

Cylindrical Cavity Resonators

The cylindrical cavity resonator is formed from a section of circular waveguide of length d and radius a0 with short-circuiting plates at each end, as shown in Fig. 4.4. This type of cavity resonator has seen a great deal of use in power-combining techniques (7, 8], where N active devices (such as IMPATT or GUNN diodes) are inserted into the cavity resonator to generate N times the amount of power of a single device. The TE- and TM-mode resonant frequencies of a cylindrical cavity resonator filled with a lossless medium are determined from the relation [9] :

(4.38)

218

CAVITIES AND RESONATORS

T d

l_ Figure 4.4

Cylindrical cavity resonator.

where the values of p correspond to the zeros and extrema of Bessel functions that is, (4.39) and (4.40)

where the prime denote the derivative of the function with respect to its argument. The first subscript refers to the order of the Bessel function and the second to the order of the zero. The useful order of these zeros are tabulated in Tables 4.2 and 4.3. In his treatment of cylindrical resonators, Harrington [4] shows that ford / a < 2 the dominant mode is the TMo 1o, while for d / a > 2 the TE 11 1 mode is dominant, which corresponds to the dominant TE11 mode in the circular guide. Also, if d / a < 1, the

TABLE 4.2 Ze~ of B~l Functions of the First Kind N M

l

2

0 l 2

2.40483 3.03171 5.13562

5.52008 7.01559

TABLE 4.3 Extrema of Bes.wl Functions of the First Kind N M

l

2

0 1 2

3.8317 1.8412 3.0542

7.0156 5.3314

4.3

DESIG

OF AND COUPLING TO WAVEGUIDE-CAVITY RESO ATORS

219

fir t higher resonance is 1.59 times the fir t re onant frequency . This is very similar to the square-ba e rectangular cavity, where the mode eparation i 1.58. Of all the mode , the TEo 11 i perhap the most intere ting ince it ha onl y circumferential current in both the cylindrical wall and the end plate . This means that the end plate of the cavity re onator can be free to move to adjust the length, d , for frequency tuning purposes without introducing deletariou effects to the unloaded quality factor. It i neces ary to point out that the TEo 11 mode i not the dominant mode; con equently, care mu t be exerci ed to choose the coupling cheme such that no higher-order mode are excited. The mode of operation in cylindrical cavity resonator of greatest intere t to the power-combining technology is the TM010 mode, the dominant mode for a short, fat cylindrical cavity. The field configurations in this mode have no variation in the axial or aximuthal direction . The unloaded quality factor for the cylindrical cavity resonator i obtained in the u ual power lo and energy stored with the following result:

(4.41)

(4.42) with

if q # 0 if q = 0

Example 2. (a) Design a cylindrical cavity resonator to operate at 10 GHz in the dominant TM010 mode. (b) Determine the expected unloaded quality factor, Qo, for this cavity resonator, which is made from silver. Solution a. To assure operation in the dominant TMo10 mode, it is necessary to make the len~ d , of the cylindrical cavity resonator smaller than a quarter-wavelength by at least a factor of 2. This gives

= d

Ao 8

= _.!_ = 8 Jo

11.8 X 109 8 x 1010

The resonant frequency from Eq. (4.38) for the

= 0 .1475

in.

(4.43)

TMo10 is given by

TM

froio

c P01

= 2-;--

(4.44)

Inverting this equation and introducing the first Bessel function zero yields C

2.405

21T

fro10

ao=- X

(4.45)

220

CAVITIES AND RESONATORS

By substitution the radius of the cylindrical resonator becomes

ao

=

11.8

109

X

2rr

2.405 __ 0

x 101o

Solution b. The unloaded quality factor for the

. 451 . m.

TMo10

mode from Eq. (4.42) is

Zw(2.405)

2Rs(l

(4.46)

+ ao / d)

(4.47)

Upon substitution, we obtain

= Qo

4.3.3

2.405 2

X

2.52

X

10-7

X

X

377.7

-Jioio(l + 0.451/ 0.1475)

= 1404_6

(4.48)

Coupling to Cavity Resonators

In the previous sections we have discussed briefly the type of electromagnetic waves that may exist in the cavity resonators without addressing the excitation mechanisms. Obviously, the cavity resonators must be excited by coupling electromagnetic energy, which is introduced from the outside. Some of these coupling mechanisms are: (a) Insertion of a conducting probe inside the cavity resonator, which is driven by an external transmission line (b) Insertion of a conducting loop

(c) Insertion of a slit or iris between the cavity and a driving waveguide The coupling coefficient enables the designer to control the coupling mechanism . In this section we introduce a technique for determining the dependency of coupling mechanisms on its scenario. This is done by first determining the external quality factor with the aid of the general formulas for Q-factors, di cussed in Section 4.2.3, followed by the use of Eq. (4.11). The most important aspect of thi technique is the modeling of the cavity resonator and the coupling mechanism. Several example follow : slit- and iris-coupled rectangular waveguide, inductive- and probe-coupled cylindrical cavity resonators.

Slit-Coupled Cavity Resonators. Con ider a rectangular cavity re onator coupled to a rectangular waveguide by means of a slit in the transver e plane, as hown in Fig. 4.5. Such a slit behaves as a hunt capacitive usceptance with a normalized value given by [10] b

= 2k101b In (c crrt) 1r

2b

where b

= waveguide height

t

= slit opening

k 101

ko

= Jk5-

= w/c

(rr/a)2

= propagation constant for the TE mode

(4.49)

4.3

DESIGN OF AND COUPLING TO WAVEGUlDE-CAVITY RESONATORS

221

(a)

(b)

Figure 4.5 circuit.

(a) Slit-coupled rectangular waveguide cavity resonator realization; (b) equivalent

The equivalent circuit of the slit-coupled cavity resonator may be modeled as shown in Fig. 4.5b - as a hart-circuited transmission line of length d shunted by a normalized u ceptance b. By neglecting the losses in a short-circuiting plate in comparison to tho e in the waveguide of length d , the total input admittance becomes yin - = jb+coth yd Yo

(4 .50)

where

y=a+jk101 k 101

2rr = Ag

(4.51) (4.52)

and a is the attenuation constant Assuming that the waveguide is made of a good conductor, so that the attenuation constant, a, is small, then

tanb a d

~

ad

(4.53)

and (4.54) The Q of this circuit and consequently the coupling coefficient can be obtained from Eq. (4.18), where B is the total input susceptance and G the conductance. For the case of this coupling mechanism B

= Yo -

Yo cotk101d

>.. - )2 ( 2 a\ Jw = ( ; ) bYo(l + trb) "" tr ; b Yo

aB

>..

(4.55) (4.56)

222

CAVITIES AND RESONATORS

The external quality factor Q ext the shunt conductance is Yo, so (4 .57) For the unloaded quality factor, Q 0 , the shunt conductance is given by

= Yo(ad)b-2

(4.58)

= rr (Ag)2_1

(4.59)

G

and

Qo

2

)..

ad

The coupling coefficient is then

1

Qo

fJ =

Q ext

=

(ad)b2

-

1 (ad) {(4b/ )..g) ln[csc(rrt / 2b)]} 2

(4.60)

Inverting (4.60) and solving for the slit opening, t , gives t

2b . _ 1

=-

rr

SID

1

(4 .61)

1n- 1 [()..g/4b) ✓l /(ad)/J]

This type of coupling mechanism is excellent for tight coupling (i.e., /J > 100).

Example 3. Determine the required slit opening, t , to achieve a coupling coefficient /J, of 300 for the cavity resonator designed in Example 1. Solution. To solve this problem, it is necessary to determine the attenuation constant, a, of rectangular waveguide, which is given by [4]

O'.TE10

=

RsAO rr2 Zw

[(a) + (AO)2] N/cm rr

2

2rr

(4.62)

Substituting the appropriate values in Eq. (4.62), the attenuation con tant for this example becomes

O'.TE10

=

2.52 X 10-

rr2

X

7

.Jwio X 1.18 [(

377.7

X

2.54

T[

)2 (

0.9

2rr )

2

1.18

]

= 4.025

X

10

_4

N/cm

(4.63) The number 2.54 appearing in Eq. (4.63) is a conversion from English to metric units. Using this result with the rest of the information given, the slit opening is calculated to be t

2 X 0.4 . - 1 =-- sin rr

l l n

1 (

1.562 ✓ 4 X 0.4 4.025

X 10- 4 X

= 34.7 mils This slit opening is quite practical and easy to realize.

1 0.781

) X

2.54 X 30()

(4.64)

4.3

DESIGN OF AND COUPLING TO WAVEGUIDE-CAVITY RESONATORS

223

Iris-Coupled Cavity Resonators. The iris-coupling mechanism is very similar to the slit-coupling mechanism except that the iris i repre ented as a hunt inductive susceptance with a normalized value given by [l 0] -

3ab

b= - - 8k101 rJ

(4.65)

where ro is the iris radius. Following the procedure as for the slit, the coupling coefficient is derived to be /3 = 16JrrJ (4.66) (ad)ab).. 8

Inverting this equation and olving for the radius of the iris yields

ro =

1

-2

/3 (ad)ab)..8

3

(4 .67)

2Jr

This coupling mechanism is excellent for realizing undercoupled and critically coupled coupling coefficients.

Example 4. Determine the radius of the iris necessary to realize critical coupling for Example 1. Solution. Application of Eq. (4.67) for the case of critical coupling gives 1

ro = -

3

4.025

X

10- 4

X

0.781

X

2

2.54

X

0.4

X

0.9

X

1.562

2Jr

= 20.75 mils

(4.68)

Inductive Coupling to the Cylindrical Cavity Resonator. Consider extending the inner conductor of the coaxial transmission line into a cylindrical resonator as shown in Fig. 4.6. At frequencies for which d is substantially smaller than a quarter-wavelength, the current and the fields in the inner conductor are practically uniform and in the direction parallel to d. Hence, in the analysis leading to the coupling coefficient, it is only necessary to consjder the mode of excitation for which the electric field intensity is parallel to d . The dominant TMo10 is one such mode. In a completely general treatment of the input impedance of the cavity resonator, contributions from modes other than TMo10 should be included. A current filament having a spatially uniform magnitude parallel to the cylindrical cavity resonator axis d will produce nonzero excitation of any TMmno mode. These modes will contribute small resistive and reactive terms to the input impedance. Assuming a shunt-transformer-coupled representation with dissipation, as shown in Fig. 4 .2, the input impedance is Zm =

.

jCtJ

- - - - - 2- - - - - N2Co[wi - w

+ j (w:--- 1

(4.103)

There are three types of coupling mechanisms to the half-wave resonator: capacitive, coupled line, and tap point, as shown in Figs. 4.1, 4.12, and 4.13. The capacitive coupling mechanism is characterized using previously established techniques, with the following result: Q ext

rrYo)

Yo (

1 + wCs

(4. 104)

2QowCs Yo(I + rrYo/ wCs)

(4.105)

= 2liJEs

and

fJ=----

In the above, equivalent lengthening was used to take into account the fringing capacitance. To determine the dependency and limitations of the coupled-line coupling mechanism, the circuit is modeled as shown in Fig. 4.12. This circuit consists of a coupled line of 0 electrical degrees with ports 2 and 4 open circuited, port 3 tenninated with an open-circuited line of 0 electrical degrees, and the input port terminated in the characteristic impedance, 2 0 • It is possible to determine the input impedance of this circuit, impose a resonance condition, and determine the normalized real component of the input, impedance that by definition is the desired coupling coefficient. The analysis commences with the Z -matrix representation of the four-port network given by [ 17]

rr -

V1 V2 V3 V4

Zll Z12 Z13 Z14

Z12 Z11 Z14 Zt3

Zt3 Zt4 Z11

Z14 Zn

lI

Z12

13

Z12

Zt 1

0

~~~~~~~~~

~""""~~~~~

i

0

f

~. ~ 2

Figure 4.12 Coupled-line coupling mechanism.

Figure 4.13 Direct-coupled (tap point) coupling mechanism.

(4.106)

4.4

DES IG

OF PLANAR RESO ATORS

231

With the specific loads of Fig. 4.12, the input impedance is determined to be 2

zl3

Zin= Z11 Zll

(4.107)

+ Z3

where Z11

=

= Z3 =

Z13

Zoe+ Z0o coth yl0 2 Z0e + Z0o cschy l0 2 Zocoth yl

(4. 108) (4.109) (4.110)

y =cx+Jk

(4.111)

Zoe+ Zoo 2 2J'C 0= - lo Ag

(4.112)

Zo =

(4.1 13)

Ago l=--fo 2

(4.114)

By making appropriate substitutions and performing the necessary operations, the coupling coefficient is deduced to be

k

2

28

/3=---sinh(J'C / 2Qo)

(4.115)

Jr

By giving due consideration to the maximum k that can be readily reproduced, it is concluded that this coupling mechanism is good for low values of coupling coefficient (/3 < 10). To minimize the effects of the T-junction, it is appropriate to taper the input line as shown in Fig. 4.14. The length of the tapered transmission line is not arbitrary and

-b:-e-.i---7-------i•.-tl

T Ag

2

l Figure 4.14 Tapered transmission line in the tap point coupling mechanism.

232

CAVITIES AND RESO ATORS

l

g

4

+

Figure 4.15

Figure 4.16

Tap point coupling mechanism with transformer.

Equivalent circuit of the tap point coupling mechani m.

has to be at least one half-wavelength. An alternative approach i to u e a quarterwavelength transformer a shown in Fig. 4.15 . Thi ha the advantage of having a smaller overall ize. Modeling this coupling mechanism as two hunt open-circuited transmission lines (Fig. 4.16) and following already e tabli bed procedures, the coupling coefficient becomes (4.116) where Z1i is the input impedance of the tapered tran mi ion line. The coupling coefficient to the half-wave re onator with a tran former (Fig. 4.15) is given by 2

2Qo fJ= ( -Zo ) - co 2 0 Zr

rr

(4.117)

where Zr i the di tributed tran former cbaracteri tic impedance. The tap point coupling mechani m can be u ed to realize very large coupling coefficients (/J > 10).

4.4.4 Planar Radial Resonators Planar re onanl tructure discu sed in the previous ections lose energy by radiation and consequently have poor unloaded quality factors. In this section we discuss the condition of resonance in circular re onant structures that are expected to have a better

DES IG

-t.4

OF PLANAR RESONATORS

233

Substrate permittivity =E, Eo permittivity =µ, 0

Circular disk defining upper resonant structure

Ground-plate conductivity =u

Planar radial re onator.

Figure 4.17

unloaded quality factor than tho e obtained with transmission-line section [18] . That tructure is the circular di k resonator and i shown in Fig. 4.17. There are e eral publications [18, 19, 20] that present a method to calculate the re onant frequency of thi re onator using a magnetic wall model of radius a0 filled with a medium of dielectric con tant Er. Becau e of the small height, h , of the substrate material currently in use, all fields in the resonator are of TM mode with respect to the Z-axi of a cylindrical coordinate system. Watkins (18] has shown that the resonant frequency of uch a re onator with magnetic wall is calculated from

kao

2nfr ao

= ---Fr = Gnm C

(4.118)

where anm is the mth zero of the Bessel function of order n . The lowest-order mode for this type of resonator is the TM11 o mode, on which discussion focuses in this section. Comparison of theoretical and experimental resonant frequency results reveal. a deviation of several percent. The primary reason for this deviation is the edge field effect. which was neglected in this model. Obviously, a more exact theory is required to describe the resonance phenomena properly. To accomplish this, a new resonator model bas been advanced and shown in Fig. 4.18. This model has an effective radius, a eff, and a dynamic dielectric constant edyn · The effective radius takes into account the electric and magnetic stray fields in a homogeneous medium (Er 1), whereas the dynamic dielectric constant takes into account the influence of the stray fields as well as the inhomogeneous field distribution of the different modes in the resonator. The effective raruus is determined via a disk capacitor filled with air and given by [21] 2 h aeff = ao [1 + nao + 1.7726)] L/ Z (4.119) nao 2h

=

(1n

The dynamic dielectric constant is calculated from the quotient Cc!yn(E tdyo-=

= EoEr)

Cdyn(E

= €0)

(4.120)

234

CAVITIES AND RESONATORS

T h

€0€c1yn

Figure 4.18 New planar radial resonator model. (From L Wolff and N. Knoppik, "Rectangular and Circular Microstrip Disk Capacitors and Resonators," IEEE M1T-S Int. Microwave Symp. Dig., 1974; copyright IEEE.)

where the total dynamic capacitance is Cdyn

= C o,dyn + C e,dyn

(4.121)

= 0.3525Co,stat

(4.122)

with

Co,dyn

(4.123) C

_ Ce.stat 2

e,dyn -

(4.124)

To find tbe fringing capacitance, Ce.stat, the radial planar resonator is considered as a degenerate microstrip line having a line width of 2a0 • As is well known, the characteristic impedance of the microstrip line can be calculated by a function Z (2ao, h, t , Er) deduced by Wheeler (22] or Schneider (23]. This function renders a static edge capacitance of the planar radial resonator: (4.125) The phase velocity of a quasi-TEM mode in microstrip line is Z(2ao,h,t,Er)

Vph

= C Z(2ao, h, t, Er = l)

(4.126)

This results in the dynamic dielectric constant being dependent on the dimensions a0 and h of the resonator, the dielectric constant Er, and the field distributions of the mode of interest. The resonant frequency of the new resonator model can then be computed from

f "™,o

= l.84lc/ 2rraeff~

and the accuracy is better than 1%.

(4.127)

4.4

DESIGN OF PLANAR RESONATORS

235

The unloaded quality factor of the planar radial resonator operating in the dominant TM11 0 mode can be deduced in the ame manner as has been described in previous sections, with the following result [18, 24]: Qo

1 = - - - ---()..oRs/n Z wh ) + tan 8

(4.127a)

Equation (4.127a) include the loss mechanism (tan 8) of the substrate material. However, the imple analy is u ed to arrive at Eq. (4. 127) does not include radiation from the edge of the radial re onator. There are two coupling mechani m to the planar radial resonator, that is, coaxial and microstrip transmission line overlap as shown in Figs. 4.19 and 4.20, respectively. To determine the coupling coefficient for the coaxial transmission-line coupling mechan.i m, let the inner conductor of the coaxial transmission media extend into the resonator as shown in Fig. 4.19. At frequencie for which the substrate height (h) is substantially smaller than a quarter-wavelength, the current and the fields in the inner conductor are practically uniform and in the direction parallel to h . Under these conditions, the re onator will operate in the dominant mode. Again, taking advantage of established procedures, the coupling coefficient for the inductive coupling mechanism is given by [24] (4.128) This equation states that the coupling coefficient (/3) will be minimum when (4. 129)

r=O (i.e. at the center of the resonator).

Coax

Figure 4.19

Coaxial transmission line-to-radial resonator coupling mechanism.

Figure 4.20

Microstrip-to-radial resonator overlap coupling mechanism.

236

CAVITIES AND RESONATORS

To arrive at the coupling coefficient for the microstrip overlap coupling mechanism shown in Fig. 4.20, a somewhat different approach (not yet discussed) has to be undertaken. First, it is necessary to assume the existence of a cosinusoidal electric field distribution with one period around the periphery of the radial resonator [24], given by (4. 130) Further, it is necessary to assume that the tangential component of the magnetic field at radius r = aeff is a constant over the width of the microstrip inner conductor, W, and zero elsewhere. This permits representation of the tangential component of the magnetic field with a Fourier series,

H(aeff, lfr)

H11fr ~ (sinn,ft) = -+ 2H1 ~ - - cos Jr n1r

(4.13 1)

n= 1

where

sinlfr

=-

w

(4.132)

2aeff

Another solution for Hy, is deduced from Maxwell's equation and is given by (4. 133) where (4. 134) Comparing Eq. (4.131) with (4.133), coefficient A 1 becomes

. 2H1 sinv, A 1 =-1 - - - - Jr Ye1{(kaeff)

(4. 135)

At this point, it is possible to determine the input admittance

(4.136) This result can then be used to detennine the external quality factor (Qext) and ultimately, the coupling coefficient, given by

/J = (4QoYo sin 1/1) Jr Ye

(

kaeff ) 2 (kaeff) - 1

(4.137)

4.4.S Microstrip Ring Resonators The ring resonator, shown in Fig. 4.21 has been widely used as a tool in the characterization of microstrip transmission line. The measurement accuracy in dispersion,

4.4

Figure 4.21

DESIGN OF PLANAR RESONATORS

237

Micro trip ring resonator.

phase velocity, and effective dielectric constant are enhanced with the ring resonator as compared to the linear re onator approach. The primary reason for this improvement is that the ring resonator is free from open-end effects. Many current and future system applications require a stable, electronically tunable o cillator. Mo t frequency tabilized circuits use dielectric resonators or waveguide cavity circuit which are generally difficult to tune electronically (25-27]. Recently (28], circuits have been proposed using the ring resonator which can be easily fabricated and electronically tuned over a wide frequency range using varactor diodes. Unlike the dielectric resonators the ring resonator/varactor circuits are amenable to monolithic implementation. Ring resonators can also be used, as filter elements. The microstrip ring resonators have been studied extensively in the open literature [28-34]. Most studies use a field theory approach to investigate the effects of line width, curvature, and dispersion on the resonant frequency. An open-ring resonator with a gap inside the ring was also investigated using a magnetic model and perturbation analysi [33, 34]. However, the effects of the coupling gap on the resonant frequency have been neglected except for Ref. 28. The design of the ring dimensions is quite straightforward. A microstrip ring structure resonates if its electrical length is an integral multiple of the guide wavelength. This may be expressed as nJ..g = 2nr

for n = 1, 2, 3, ...

where r is the mean radius of the ring and Ag is the guide wavelength. To use the ring, the ring has to be coupled to an external circuit. This coupling loads the ring and thus changes slightly the resonant frequency.

Figure 4.22 Equivalent circuit of ring resonator. (From K. Chang, T. S. Martin, F. Wang, and J. L. Klein, "On the Study of Microstrip Ring and Varactor-Tuned Ring Circuits," IEEE MTI'-S Int. Microwave Symp. Dig., December 1987; copyright IEEE.)

238

CAVITIES AND RESONATORS

To access the effects of the coupling gap on the resonant frequency, an equivalent circuit for the ring resonator and the coupling gap has been developed, as shown in Fig. 4.22. It was found that the resonant frequency decreases slightly as the coupling gap becomes smaller. For most ranges of gap size, the effects on resonant frequency are small and negligible. The equivalent circuit of the microstrip ring resonator, as shown in Fig. 4.22, can also be used to develop the external quality factor ( Qex1) and consequently coupling coefficient (/3). This information is not available at this time.

4.5 MICROSTRIP DIELECTRIC RESONATORS A dielectric resonator is a piece of unmetallized ceramic with a high dielectric constant in which the electromagnetic fields are confined to the dielectric region and its immediate vicinity by reflections at the dielectric-air interface. The dielectric properties required for practical microwave/millimeter applications are high unloaded quality factors, Q0 ; low temperature coefficient of dielectric constant; and high relative dielectric constant The size of a dielectric resonator is considerably smaller than the size of an empty resonant cavity operating at the same frequency, provided that the relative dielectric constant of the material is substantially larger than unity. Only within the last 10 years, materials having a dielectric constant between 30 and 40 with good temperature stability and low dielectric losses have become available. The shape of a dielectric resonator is usually a short cylinder, but other shapes have been used. A commonly used resonant mode in cylindrical dielectric resonators is the dominant mode, denoted as TEo icS • When the relative dielectric constant i about 40, more than 95% of the stored electric energy of the dominant mode, as well as a great part of the stored magnetic energy (typically over 60%), are located within the cylinder [35]. The remaining energy is distributed in the air around the resonator, decaying rapidly with distance away from the resonator surface.

4.5.1

Geometry Determination of the Microstrip Dielectric Resonator

The design of a dielectric resonator, like the design of a metal cavity, depends on its natural resonant frequencies. Since exact solutions of dielectric re onators with shapes other than spheres or doughnuts cannot be rigorously computed approximate techniques are necessary to solve the problem. Both H and E resonant modes can be excited in a dielectric resonator. The H mode has a large normal magnetic field component at the boundary surfaces, while the E mode contains no predominant normal magnetic field component at the surfaces. Dielectric resonator resonant frequencies are usually computed by assuming that the dielectric resonator is placed in unbounded space. In real conditions, however, dielectric resonators are placed in microwave/millimeter wave structures, such as waveguides, striplines, or microstrip transmission lines. Because these microwave/millimeter wave structures are close to the resonators, they disturb the resonator's external fields and alter their resonant frequency and reduce the unloaded quality factor. A dielectric resonator structure commonly used in practical microwave integrated circuits (MICs) is shown in Fig. 4.23. This structure is composed of a cylindrical

4.5

MJCROSTRlP DIELECTRIC RESO ATORS

239

Upper ground plane

d

Lower ground plane

Figure 4.23

Basic micro trip dielectric resonator structure.

dielectric sample placed on a dielectric substrate, one side of which is metallized as a ground plane, and an upper ground plane placed above the dielectric resonator sample. The first-order resonant frequency of this structure is determined by assuming that the resonator is housed in a contiguous magnetic-wall, cylindrical waveguide below cutoff. This waveguide is terminated by two conducting surfaces, representing the ground plane and the top cover of a microstrip transmission line, with a dielectric between the resonator and the ground plane. In the dielectric region, the guide is above cutoff, and a standing wave exi ts at resonance. In the air, and in the substrate region, the field decay since this area appears as a waveguide below cutoff. A complete solution for the resonant frequency of the microstrip dielectric resonator can be obtained either through the solution of the boundary value problem or the transverse re onance procedure. Because of its simplicity, the transverse resonance procedure will be introduced in this chapter [36]. Carrying out the details of the boundary value problem, it becomes clear that the procedures, even in the transverse direction, are similar to those in any transmissionline problem. The solutions in the transverse directions consist of standing waves, with the transverse transmission line at resonance [37], or expressed mathematically

-

~

Zin= Z+Z=O

(4.138)

where the Z + Z are the impedances looking to the right and left of an arbitrary reference plane. When this procedure is applied to the microstrip dielectric resonator, with an equivalent circuit as shown in Fig. 4.24, the following transcendental equation results: tan y;t

=

y;(Y0a tanh Yosh + Yos tanh Y0ad)

(4.139)

y/ (tanh Yosh ) (tanh Y0a d ) - Yos Y0a

Definitions of the symbols are listed in Appendix A. Equation (4.139) determines the resonant frequency when Es, Er, R, h, and d are known, or the thickness of the resonator when the frequency is known. In this technique

/,

240

CAVITIES AND RESONATORS

Figure 4.24

Equivalent-circuit representation of the microstrip dielectric resonator.

the radius, R , of the microstrip dielectric resonator must be selected a priori. The best way to do this is to find the radii that cause the following conditions to occur: Yi= Yos

=0

(4.140)

and R

= 1.2025 -C

(

-1

(J)

Fr

+-1

Fs

)

(4. 141)

These theoretical results were compared with an experimental microstrip dielectric resonator detailed in Table 4.4. The measured resonant frequency of this resonator was 11.5 GHz -15% higher than the desired result. To improve this, it is necessary to include the decaying fields that exist around the cylindrical surface. This analysis starts by assuming the existence of an external magnetic field, which leads to the following result [38]: (4. 142) The objective is to find the effect of the decaying field on the roots of the Bessel function, specifically /3c; R. For a given frequency and resonator radius, this is accomplished with Eq. (4.142) and the definitions of /3ci and ctco · The solution i completed when the ratio (2.405/R), appearing in all the definitions dealing with Eq. (4.139), is replaced with the newly found f3c;. With this modification, the theoretical and experimental comparison is within I %, as shown in Fig. 4.25 . TABLE 4.4 Microstrip Dielectric Resonator Design Specifications Resonant frequency Resonator material Relative dielectric constant of resonator material Substrate material Relative dielectric constant of substrate material Thickness of substrate Thickness of resonator [from Eq. (4.2)] Thickness of air above resonator Radius of dielectric resonator

10.0 GHz Barium tetratitanate 38.0 Teflon fiberglass 2.54 31 mils 70 mils 100 mils 107.5 mils

4.5

MICROSTRIP DlELECTRIC RESONATORS

12

Tl [5Z]

j_

b = 0.202" T

! T, = 0.06959"

-

..L

241

I

T

Tm= 0.070"

h = 0.031"

N

:I:

-

CJ 11 >u

T, = Theoretical Tm= Experimental

C: (1)

:J CT

-.... .... (1)

C:

m C:

0

(/'J

Q) a: 10

9 .050

{ T1 = 0.1007" Tm= 0. 101 "

150

100

Height of resonator (mil)

Figure 4.25 Theoretical and experimental resonant frequency comparison of microstrip dielectric resonator.

Instead of selecting the radius (R ) of the microstrip dielectric resonator a priori, it is possible to develop a general continuous solution by rearranging Eq. (4.139) to [39] tan y;t

(Yoa / Y;) cotb Y0ad + (Yos/Y;) cotb Yosh =-------------

(4.143)

1 - (YosYoa!Y?) coth Yos h cotb Y0ad

The right side of Eq. (4 .143) is actually the sum of two angles: -t 2R

=

1 [ tan- 1 ( Y0a R coth Y0a R -d ) 2y;R y; R R

YosR cotb YosR -h ) ] + tan- 1 ( YiR

R

(4 .144)

Based on these expressions, and the appropriate propagation constant, it is possible to generate a family of normalized curves. A typical example using barium tetratitanate on Cu-Flon (Polyflon Corp., New Rochelle, NY) with the upper ground plane removed (d / R = oo) and variable h/R is shown in Fig. 4.26. Any radius-to-wavelength ratio selected from Fig. 4.26 inherently provides the necessary height-to-radius ratio of the dielectric resonator. It is a simple matter, then, to solve for the parameters of the dielectric resonator, since the desired resonant frequency is known. The information can also be used to evaluate secondary parameters, such as mechanical tuning range. Dielectric resonatois have no basic frequency limitations and can even be applied to microstrip circuits operating in V and W bands of the millimeter wave frequency range.

J

242

CAVITIES AND RESONATORS

1.0 0.9 0.8 0.7 0.6

l(t ~ C\I

I

0

.-


-40

Q.

co

::,

~

-0.1

e

(!)

-60

30

35

40

45

-0.2

Frequency [MHz]

Figure 6.8 Response of VSB filter, from [24]. Insertion loss is 30 dB. Copyright © 1993 IEEE.

308

SURFACE ACOUSTIC WAVE DEVICES

rejection etc.), but it has high insertion loss. For low losses, other types are used; the performance is not so good, and to an extent, it depends on the filter type. Hence, in considering a particular requirement, it is important to bear in mind the variety of filter types and their respective strengths and weaknesses. Temperature effects always need to be considered, and they often lead to the conclusion that the device bandwidth needs to be somewhat larger than the specified bandwidth. SAW devices come in a variety of shapes and sizes and, hence, a variety of package types. Many packages are the metal dual-in-line or TO types, but ceramic surface-mount types are also common, particularly for high frequencies. At gigahertz frequencies, packages as small as 2 x 2 mm2 are used. The frequency response of the device depends on the impedances with which the two ports are terminated, and the required impedances are specified by the device supplier. Some devices can be connected directly to a 50 Q source and load, but often it is necessary to add inductors to tune out the transducer capacitances. It is also common to use a two-component circuit at each end, to transform the impedance in addition to tuning the capacitances. Stray capacitance or inductance is often significant, especially at high frequencies, and the terminating components may need adjustment to allow for these. Another vital issue is electromagnetic feedthrough between the input and output, which limits the stopband rejection. The feedthrough within the package has been given extensive theoretical studies. In this chapter, the devices are presented in an ''unbalanced" form; that is. at each port, there is one live terminal and one ground terminal. However, many devices can have a balanced port, in which there are two live terminals designed to receive voltages with equal magnitude but opposite sign. Balanced drives are common in some communication systems, and many SAW devices are designed for this purpose. Some devices can be optionally operated either balanced or unbalanced. In some cases, the device can be balanced at one port and unbalanced at the other, o that it has the function of a balun transformer in addition to the main filtering function.

6.2.3 Correlation In the context of SAW devices, correlation refers to the process of applying a complex waveform to a linear device whose impulse response is the time-rever e of the input waveform. The device is known as a " matched filter." The output waveform has a prominent peak called the "correlation peak." If the input waveform is accompanied by noise, as is always the case for a radio receiver, the correlation process increases the signal-to-noise ratio (SNR); taking the signal to be measured at the peak, the correlation process gives the best SNR obtainable by linear proces ing, assuming the noise to be white. Several types of SAW device have been developed for this purpo e. The techniques involved are mostly well established and well covered in the literature.

Pulse Compression. This technique, used in radar systems, increases the range at which the radar can detect a target without needing an increase in the peak transmitted power. In principle, any type of complex waveform is applicable, provided the timebandwidth product (TB) is large, where T is the waveform duration and B is its bandwidth. The matched filter increases the signal-to-noise power ratio by a factor TB,

6.2 (a)

BASIC SAW DEVICES

309

(b)

IillJ

11111111111

Figure 6.9

(a) Pulse compression filter. (b) Slanted version.

equivalent to increasing the tran mitted peak power by thi factor. For radar systems, TB is typically 50 to 500. In practice the waveform i usually a chirp, that is a waveform whose frequency varie monotonically with time. The matched filter needs to have its impulse response as the time-rever e of thi , another chirp, and this can be regarded as a dispersive delay line. Figure 6.9a how a typical arrangement, where the right transducer has graded periodicity. For a single-frequency SAW, the transducer responds most strongly where its pitch correspond to the SAW wavelength so that different delays are obtained for different frequencie . The left transducer can be uniform, as shown, and short enough to give a bandwidth covering the band of the right transducer. Alternatively, it can be another graded-periodicity transducer with pitch varying in the opposite direction, so that the device is approximately symmetrical about the center. The device shown i described as "down-chirp" because the frequency of its impulse response decreases with time. Usually the frequency is a linear function of time. A pulse compression filter can also be used to generate a chirp pulse by applying a short wideband pulse to it, a process called "passive generation." Thus, the radar transmitter can use an up-chirp (down-chirp) SAW device, while the receiver uses a down-chirp (up-chirp) device for correlation. However, the generation process has the disadvantage of producing a rather weak signal, and digital generation techniques have become more attractive in recent years. For pulse generation, the system requires the impulse response of the filter to have an almost flat envelope. If the transducers are unapodized, as indicated in Fig. 6.9a, the impulse response is found to have a slope. This is because the response at a given frequency turns out to be proportional to f 312 , as can be shown from the analysis in Section 6.4. It is usually necessary to apodize one of the transducers (a chirp transducer) to compensate for this effect and for second-order effects such as the circuit effect and diffraction. In the correlator, in the receiver, there is another reason for apodization. If the receiver filter response has a flat envelope, the output waveform has a (sin x)/x form, with sidelobes 13 dB below the main peak. These are usually unacceptable because they could be falsely interpreted as extra targets. To suppress them, additional weighting is applied in the form of apodization, using typically a Taylor function. In this way, the sidelobes may be reduced to 35 dB or more below the main peak. Typically, bandwjdths are in the region 10 to 100 MHz, durations (T) are 5 to 20 µ s, and TB products are 50 to 500. For the longer devices, quartz is the preferred material because temperature stability is needed, while shorter devices use strong piezoelectrics (lithium niobate or tantalate) to obtain acceptable losses.

310

SURFACE ACOUSTIC WAVE DEVICES

---

In

W] -- W] Out

~\\\\\\\\ HF' 'LF //////////II \

\

Grooves

Figure 6.10 RAC principle.

As for bandpass filters, considered earlier, pulse compression filters usually avoid the use of single-electrode transducers because of the problem of electrode reflections. However, an alternative solution is to use slanted transducers, as indicated in Fig. 6.9b. Here, signals at a given frequency travel through relatively few electrodes. reducing the severity of the effect. The use of single-electrode transducers enables higher operating frequencies to be obtained, as illustrated by some 1.5-GHz device [25]. Interdigital devices generally give larger insertion losses as the TB product is increased. This is because at any one frequency, only part of the chirp transducer is active; the rest acts as a capacitive shunt. This limitation is overcome in the reflective array compressor (RAC). As shown in Fig. 6.10, the RAC has two arrays of inclined grooves, each reflecting the waves through 90°. The groove arrays reflect strongly when the pitch corresponds to the wavelength, and the RAC has graded pitch so that different frequencies are subject to different path lengths and hence delays. Apart from minor second-order effects (propagation loss, diffraction etc.), the groove arrays can be extended indefinitely without increasing the insertion loss. Practical RACs have generally used the strongly piezoelectric substrate lithium niobate, in order to obtain acceptable transducer losses with wide bandwidths. TB products up to 16,000 have been produced [26]. Weighting, needed to minimize the sidelobes, is obtained by varying the depths of the groove , which are fabricated by ion etching through a movable mask. The fabrication is a slow, and therefore costly, process, and the physical size of substrate is another cost factor. RACs often need to be enclosed in an oven to regulate the temperature. For narrowband devices, it is sometimes possible to obviate these problems by using a quartz sub trate with metal strips as reflectors.

PSK Fillers. Although chirp waveforms are common in radar systems, communication systems often use another type of complex waveform. This is the phase-shift-keyed (PSK) waveform and other related waveforms. A PSK waveform consists of a sequence of contiguous pulses of carrier, with relative phase 0° or 180°. In spread-spectrum communications, each bit of data can be represented by a equence of this type, and correlation is needed to optimize the SNR. This can be achieved using a SAW tapped delay line, as in Fig. 6.11 . This is called a PSK filter, or sometimes an analogue matched filter (AMF). The device has a sequence of short identical transducers, the taps, with regular spacing, and an input transducer with length equal to the tap spacing. When the input transducer is impulsed, a burst of SAW energy travels along the surface exciting the taps in turn, so that a continuous output waveform is produced Inverting a tap changes

6.2

BASJC SAW DEVICES

311

=nm Figure 6.11

SAW PSK fi lter (AMF).

the phase of its output by 180°, o elected taps can be reversed to corre pond to the code required. Thi device i easily analyzed by methods given later, in Section 6.5. Good results are obtainable with everal hundred taps. However, flexibility in the coding is an important factor. and devices have been made with the taps connected via wire bonds so that the coding can be chosen when the device is bonded. Extending this logic, it is attractive to be able to adjust the coding electronically, and this can be done by connecting each tap to an electronic switch.

Convolvers. Another type of correlator is the nonlinear convolver, shown in Fig. 6.12. In this device, RF signals are introduced at both ends of a lithium niobate substrate, so that they propagate toward each other. Where they overlap, a weak nonlinear mechanism causes mixing and a sum-frequency signal is produced, with amplitude proportional to the product of the input wave amplitudes. The nonlinearity is an intrinic property of the material. The product signal appears as an electric field that can be sensed by a uniform metal electrode extending over most of the area between the input transducers. If the input waveforms are / 1 (t ) and / 2 (t), the wave amplitudes within the device have the forms / 1 (t - x / v) and / 2 (t + x / v) (for simplicity, this ignores delays associated with the transducer positions). The output waveform g(t ) is proportional to the integrated product of these, so that g(t )

3400 33000

90

180

Propagation direction (deg)

Figure 6.16

SAW velocities on Y-cut lithium niobate, as functions of propagation direction.

6.3.2 Diffraction Diffraction has been di cussed in detail by, for instance, Slobodnik [42]. In SAW devices, it is usually analyzed by methods analogous to familiar optical diffraction. In particular the wave amplitude is represented by a scalar 1/1, which does not need to be physjcally identified as, for example, surface displacement or potential. This approximation is usually adequate even though SAW sources can have very small apertures. in the region of one wavelength or less. The scalar approximation enables the required results to be obtained from velocity data, which are expressed in terms of the slowness curve, Fig. 6.17. To allow for anisotropy, the disturbance is represented as an angular spectrum of plane waves (ASPW). Thus, we consider plane waves at all angles, with complex amplitude \JI () dependent on the propagation direction¢. Here is the angle relative to the principal propagation direction, which makes an angle 0 with a reference direction such as a crystal axis. With a proper choice of 'll(), any amplitude distribution v,(x 1 , y) at a particular x = x 1 can be represented as an infinite sum of plane waves. The amplitude elsewhere is obtained by using the known propagation velocity of the plane-wave constituents. It is necessary to assume that, for a principal propagation direction along x( = 0), the ampUtudes are negligible for directions in the left half-plane (l I ~ 90°). This method was first used by Kharusi and Farnell [43]. The diffraction analysis [4] is not given here, but we describe the main results. These follow from the ASPW analysis, and they are valid for any type of wave in an anisotropic medium. The waves are assumed to be propagating on a free surface, without reflections from components such as transducers. The wave amplitude is written as the integral (6.12)

where w (ky) is the distribution of the constituent plane waves and kx is regarded as a function of ky- It is assumed that kx (k y) is a single-valued function. From this equation, it can be seen that 1/1(0, y) is the -F ourier transform of 'V(ky). Hence, \ll(ky) can be

318

SURFACE ACOUSTIC WAVE DEVICES

Slowness curve

Beam direction

Figure 6.17

The slowness curve and the energy flow direction.

obtained as the inverse Fourier transform of l/,(0, y), which we regard as the source distribution. Toe form of this source is arbitrary.

Beam Steering. The direction of energy flow in an anisotropic material is not generally in the same direction as the wave vector, in contrast to the isotropic ca e. As shown in Fig. 6.17, the energy flow direction is perpendicular to the slowness curve, which is a polar plot of the wavenumber. For a point on the curve, the vector from the origin represents the direction of the wavenumber k and its length represents lkl = (J)/ v, where v is the phase velocity of plane waves. Toe angle y between the energy flow direction and the x-axis is the "beam steering angle," also called the '·power flow angle," and it is given by 1dv (6. 13) tan y = - v d0 A SAW beam generated by a uniform transducer will shift ideway by this angle, o the receiving transducer must be displaced sideways to receive it properly. However, the substrate material is usually chosen such that y = 0, o that the effect doe not appear.

Minimal Diffraction Orientations. For some cases, the ani otropy is such a to reduce the amount of diffraction spreading, almost as if a collimating Jen had been used. Considering a wave whose principal propagation direction is x, minimal diffraction occurs when the slowness curve is normal to the x-axis for directions along and near to x. That is, the x-component of wavenumber, kx, i independent of propagation direction for a range of angles, including the x-axis. Minimal diffraction spreading occurs because the energy flow directions, normal to the slowness curve, are all parallel to the x-axis. For this to be true, the beam steering angle must sati fy the condition

d y / d0

= -1

for minimal diffraction

(6.14)

This condition is feasible only for a finite range of angles. It occurs approximately in a number of practical SAW materials, notably, Y-Z lithium niobate in which dy / d0 -1.08. For many devices, this is good enough to avoid the need for diffraction compensation during the design process. Its significance is not just that some time is saved;

=

6.3

PROPAGATIO

EFFECTS AND MATERIALS

319

diffraction compen ation i not an exact proce , and avoiding the need for it can lead to better device performance . For a minimal diffraction condition. the pha e velocity u is proportional to cos 0.

Parabolic Approximation - Beam Steering and Scaling. In many cases, it is adequate to approximate the slowoe curve by a quadratic function of the form (6. 15) where ko i the alue of k'C for plane wave along x . This is an adequate approximation for selected material , as pecified later. With y defined a above, the constants a and b are given by a

= tan y;

b

= 1 + (1 / v)(d 2 u/ d0 2 ) = (1 + dy / d 0) sec2 0

(6.16)

An isotropic material gi e a = 0 and b = I, and a minimal diffraction orientation has b = 0. For a material ati fying Eq. (6.15), there are two important conclusions. Firstly, ubstitution in Eq. (6.12) show that the constant a simply shifts the diffraction pattern ideways. Mathematically, if v,0 (x y) is the pattern for the case a = 0, then for a =I= 0. the pattern is v, (x , y) = 1/to(x y - ax) , for all x. Again, this demonstrates beam steering. The econd conclusion is that the diffraction pattern is a scaled version of the pattern given by anisotropicmaterial (b = 1). For simplicity, assume a= 0. Define 1/t; (x, y) as the diffraction pattern for the isotropic case (a= 0, b = 1). Then substituting Eq. (6. 15) into Eq. (6.12) gives v, (x , y)

= V,;(bx , y ) · exp[j(b -

l)xko]

(6.17)

Hence, the diffraction pattern is a scaled version of the isotropic pattern, except for a simple phase change.

Diffraction Pattern for an Aperture with Uniform lllumination. This case corresponds to SAW excitation by a source within a transducer. For this purpose, it is usually adequate to regard the overlap region between two electrodes as a localized source. We take v,(0, y) = 1 for IYI < W / 2, and zero elsewhere. Using the parabolic approximation and taking a = 0, Eq. (6.12) gives ,ft(x , y)

= ~ exp(-jkox + jn/ 4) {A+exp(-j1ru2 / 2) du

lA-

v2

(6.18)

where A±= (y ± W/ 2)[k0 / (nxb)] 1l 2 . It is assumed here that b ~ 0, which is nearly always true for SAW materials. In the near-field (Fresnel) region, Eq. (6.18) gives a distribution similar to the source, but with some ripples. In the far-field (Fraunhofer) region, it gives a sin(X) / X fonn, which is the Fourier transform of the source distribution. The integral is related to the Fresnel integrals C (x) and S (x), which are defined by the function F(x)

= C(x)- jS(x) =

f

exp(- jrru 2 / 2) du

(6.19)

320

SURFACE ACOUSTIC WAVE DEVICES

In a two-transducer device, the receiving transducer is regarded as a sequence of receivers defined by electrode overlaps. Consider a receiver at location x, with aperture Wr and displaced a distanced from the x-axis. The output signal is found by integrating v,(x. y) of Eq. (6.18) from d - Wr / 2 to d + Wr/2. Denoting the integral by R , this is

found to give (6.20)

where 77

= 1/ [rrxkob] 1l2 and the function X(x ) is X (x) =

1'

2

(6.21 )

F(t) dt = x • F(x) - (j / 1r ) [exp(- j,rx / 2) - 1]

The parameters Bn are defined as Bn = (wn / 2 + d)17ko , with Wt = W + Wr , w2 = W - Wr, w3 = Wr - W , and W4 = -W - Wr. If there is no diffraction (b-+- 0), Eq. (6.20) can be simplified to give R = Wr exp(- jxko), provided the receiver is fully overlapped by the launching source. Diffraction calculations are generally very time consuming because the number of calculations is the product of the number of sources in the two transducers. However, the speed can be improved by precalculating a table for the function X (x), and then using linear interpolation. The result in Eq. (6.20) is derived using the parabolic approximation. When this approximation is not valid, it is necessary to revert to the ASPW, Eq. (6.12). to analyze diffraction. In SAW bandpass filters, diffraction tends to produce some di tortion of the passband and some unwanted signals in the stopband above the pa band. Design technique to compensate for this have been studied extensively, and they have been reviewed by Seifert et al. [44]. In narrowband filters, diffraction can be minimized by using withdrawal weighting (Section 6.2.2). Data for some common materials are given in Table 6.1.

6.3.3 Temperature and Other Propagation Effects For well-poUshed crystals, the free-surface attenuation and di persion (due to imperfections) are generally negligible except at frequencies above say 500 MHz [42]. Propagation in metal electrode (usually aluminum) is more relevant for practical devices.

TABLE 6.1

Diffraction Data (see Table 6.2 for other data)

Material

dy / d9

Parabolic theory valid? Minimal diffraction?

Lithium niobate, Y-Z Ditto, 128° Y-X Quartz, ST-X Lithium tantalate, X-l12°Y Bi12Ge020, E = (45, 40.04, 90°) Langasite, E = (0, 138.5, 26.6°)

- 1.08 - 0.35 0.38 - 0.29 - 1.00 - 1.00

no yes ye no approximately

E

= Euler angles

yes no no no yes yes

6.3

PROPAGATIO

EFFECTS AND MATERIALS

321

This depend on metal thickne , but again the effects are not usually ignificant except at high frequencies. Temperature change cause SAW velocity changes, but for SAW device , a more relevant factor i the delay change in a device, which also involve the thermal expansion. The temperature coefficient of delay (TCD) can be predicted theoretically, using temperature coefficient of the bulk piezoelastic constant to derive the temperature variation of velocity. In most cases, the delay varie approximately linearly with temperature, and the temperature coefficient is not much affected by the SAW transducers. However, for the special case of ST-X quartz, the delay T is a quadratic function of temperature 0. with a "turnover temperature" (at which dT / d B = 0) near room temperature. This gives excellent tability. Toe orientation is a rotated Y-cut as shown in Fig. 6.18 where X, Y, and Z are the cry tal axe , z is the urface normal, and x is the SAW propagation direction. For ST-X q ~ the rotation angle t/1 is 42.75°, giving a turnover temperature 00 of 21 °C. For temperature 0 , the delay T has the form T (0 )

= T (Bo)[I + c (0 -

00)2]

(6.22)

with c = 32 x 10- 9 (°C)-2 • Toe turnover temperature depends on the crystal rotation angle [45], decreasing at a rate of around l0°C per degree of rotation (although the relation is not very linear). Smaller rotation angles, typically 34° to 38°, are often used because the metallization and the crystal mounting reduce 00 , and because the midpoint of the operating range is often higher than 21 °C. All of these orientations are referred to as ST-X quartz. The effect of a temperature change is to scale the impulse response of the device. If the delay between two particular points changes from T to (1 + E) T, where s is small, the impulse response changes from h(t) to h' (t) = h[t / (1 + s)]. From Fourier transform theory, the frequency response changes from H (w) to H ' (w) = H[(l + s)w] (neglecting an inconsequential amplitude change). Hence, the frequency response is also scaled. The center frequency Jo of a bandpass filter will change according to l1Jo/ Jo = -1:1 T / T, and its bandwidth is also scaled. Writing the frequency response as H (w) = A(w) expU(w)], the phase is changed from¢(w) to '(w) = ¢[(1 + t:)w]. This relation is useful when considering the effect of temperature on pulse compression filters. The above equations for temperature change also apply for a velocity change. They can be used to establish the velocity accuracy needed The velocity can vary slightly

z

z

y

X,x

Figure 6.18 Rotated Y-cut orientation. The X and x axes are toward the observer, and x is the SAW propagation direction.

322

SURFACE ACOUSTIC WAVE DEVICES

because of small changes in substrate composition, inaccuracy of orientation, and variation in the electrode geometry (width, thickness) and composition.

6.3.4 Specification of Orientation - Euler Angles As een earlier, simple orientations are specified by quoting the surface normal followed by the propagation direction. Thus, "Y-Z lithium niobate" indicates that the substrate is Y-cut and the wave propagates along Z, where Y and Z are crystal axes. Rotated Y-cut such a the- sT-X case of Fig. 6.18 can be written in the form 42.75° Y-X, when 1/t = 42.75°. For more general cases, Euler's convention can be used, as defined by Slobodnik (42] and Goldstein [46]. Take X, Y, Z a the crystal axes, and x, y, z as the device orientation, where z is the outward-directed normal to the surface, and x is the propagation direction. Initially, x, y, and z are the same as X, Y, and Z, and the three Euler angles are zero. The crystal axes are imagined to have fixed directions. We define up to three rotations: (1) Rotate the

x, y, z axes anticlockwise about z through an angle

>.. (i.e.,

x and y

rotate anticlockwise as seen by an observer looking down the z-axis toward the origin). The new axes are called x 1 , Y1, and z1 (where z1 = Z). y 1, and z1 axes anticlockwise about the new axes x2, Y2, and z2.

(2) Rotate the

x 1,

x1

through an angleµ,. Call

(3) Rotate the x 2 , Y2, and z2 axes anticlockwise about z2 through an angle 0. Call the new axes x3, y3, and z;3.

In the final orientation, z3 is the surface normal and x 3 is the propagation direction. If there are less than three rotations, we use x j , etc., where j < 3. The angles >.. and µ, define the substrate orientation, wherea 0 define the propagation direction in the surface. Rotated orientation may be shown diagrammatically using the stereographic projection familiar to crystallographers. Thi shows three-dimensional directions unambiguously on a two-dimensional diagram. The method i shown in Fig. 6.19 (left). Imagine a sphere with a line drawn from the origin in the direction to be represented. The line meets the surface at a point P . From P , draw a line to the outh pole of the phere, and the point where this cro es the equatorial plane i denoted by p. The tereogram is a drawing of the equatorial plane with all the point p marked. If the point P is in the lower hemisphere, the line is drawn to the north pole and the point p is marked by an open circle in tead of a clo ed one. A plane in three-dimen ional space becomes a straight or curved line on the tereogram. The right diagram in Fig. 6.19 show the tereogram for the Euler angle , after three rotation . For a rotated Y-cut, uch as Fig. 6.18, the Euler angles are ).. = 0 = O and µ, =

1/t - 90°. 6.3.5 Saw Materials Established Materials. Data for SAW materials are summarized in Table 6.2. The parameter e govern the tran ducer capacitance, as described in Section 6.4. Among the common materials, lithium niobate has strong coupling but poor temperature

6.3

PROPAGATION EFFECTS AND MATERIALS

323

X

Figure 6.19 Left stereographic projection. Right: Euler angle rotations.

performance; it is best suited for wideband devices. The Y-Z case has minimal diffraction but generates unwanted bulk waves rather badly, so early TV filters often used this with an msc to reject the bulk waves. The 128° case gives much less bulk wave generation and enabled the size to be reduced because an msc was not needed, and this was preferred although the diffraction is not minimized. ST-X quartz is at the other extreme, giving weak coupling but good temperature stability. For cut angles 34° to 43°, the SAW properties do not vary much, apart from the change of turnover temperature. Lithium tantalate has an X-cut case with propagation 112° from the Y-axis_, giving intermediate properties [47, 48]. Zinc oxide films on glass substrates have been developed for economical filters, although they are not very reproducible. The piezoelectric ceramic PZT is also usable, but above about 50 MHz, its loss becomes prohibitive. These last three cases have also been used for TV filters. Lithium tetraborate has intermediate coupling and an orientation giving a zero TCD [49- 51]. This material is slightly soluble in water and therefore needs special processing methods, but it is becoming widely used.

Transverse Waves and Leaky Waves. All cases considered so far have used piezoelectric Rayleigh waves, that is, waves similar to the Rayleigh wave in an isotropic material but with an associated electric field. The particle motion is usually in or nearly in the sagittal plane. In contrast, it has been shown that some piezoelectrics can support a wave with the motion in the surface plane, normal to the propagation direction. This solution is called the Bleustein- Gulyaev- Shimizu wave after its discoverers. It occurs when there is a two-fold axis of symmetry normal to the sagittal plane. However, it has not had much practical usage in devices. In quartz, Lewis [58] showed that, for some orientations, bulk shear waves could be used in devices similar to SAW devices, and named them surface skimming bulk waves (SSBW). In general, a bulk wave excited at the surface does not satisfy the boundary conditions and, therefore, leaks energy into the bulk. But at particular orientations, this leakage can be very small, giving a wave whose attenuation is for practical purposes

SURFACE ACOUSTIC WAVE DEVlCES

324

TABLE 6.2

Data for SAW Materials Euler angles A, µ, 0°

Material

VJ mis

6 v/ v £00/£0

Teoppml°C

Ref.

A. Materials in Common Use Lithium niobate, LiNbO3, Y-Z Ditto, 128°Y -X Quartz, SiO2, ST-X (42.75°Y-X) Lithium tantalate, LiTaO3, X-l 12°Y PZT ceramic, Z-X Ditto, X-Y ZnO/glass, fundamental mode Lithium tetraborate, Lh B4O7 45°X-Z LiNbO3, 64°Y-X, (LSAW) Ditto, 41 °Y-X, (LSAW) LiTaO3, 36°Y-X (LSAW) Quartz, 36°Y-X+90° (STW)

94 75

[40] [52]

0, -90, - 90 0, 38, 0

3488 2.4% 3979 2.7%

46 56

0,-47.25, 0 90, 90, 112

3159 0.06% 3300 0 .35%

5.6 48

0,0,0

135, 90, -90

2360 2400 2576 3350

1% 11 % 0.7% 0.45%

11

0, 0, 0, 0,

- 26, 0 -49, 0 -54, 0 -54, 90

4742 4792 4212 5100

5.5% 8.5% 2.4% n/a

52 63 50 5.6

80 80 32 0(60)

[57]

0, 0,45 45, 40.04, 90

1681 1827 2741 2342 3950

0.7% 0.3% 0.25% 0.15% 0.05%

46

120

[42]

0 0 0(-6)

[61] [63] [64]

0(32) 18

[53] [54, 55]

40 9 11

0(270)

[56] [49]

[58, 59. 60] [58]

B. Other Established Materials Bismuth Germanium oxide, Bi12 GeO20, (001 , 110) Ditto Berlinite, AlPO4, 144.5°Y-X Gallium phosphate, GaPO4 Quartz, LST cut, - 75°Y-X

o, 54.5, 0 0, - 165, 0

46

5.5

(LSAW) Langasite, La3GasSiO14 ++ Potassium niobate, KNbO3,

0, 138.5, 26.6 2730 0.16% 0, -30, 0 4000 17%

[62] 0 0(1830) [65]

60°Y-X Lithium tetraborate (HVPSAW) 0, 47.3, 90 ZnOJsapphire AlN/sapphlre ZnO/diamond 0, 38,0 SiO2/LiNbO3 128°Y-X

6790 5500 5910 6500 3990

0.6% 2% 0.5% 1.5% 3%

-3 43 0 0

[67] [68) [70] [71] [72]

Note : For films, parameters depend on film thickness; repre entative values are shown. All ca es have

=

=

PFA 0. LSAW leaky surface acoustic wave. + numbers in brackets are c x 109 {°C)- 2 , c defined in Eq. (6.22). ++this orientation of langasite shows the natural SPUDT effect.

negligible; theoretically, it might be 10- 4 dB per wavelength or less. The wave then behaves rather like a SAW. The SSBWs often have high velocities. A case in point is 36° Y-cut quartz, with propagation nonnal to the X-axis. This gives an exceptionally high velocity of 5100 mis, attractive for high-frequency devices, and a TCD of zero. The delay is quadratic, as in Eq. (6.22), and the constant c is about 60 x 10- 9 (°C)- 2 . This case also gives zero coupling to a Rayleigh-type SAW. In another development, Auld [73, 74] showed that, in an isotropic material, a bulk shear wave can be guided

6.3

PROPAGATIO

EFFECTS AND MATERIALS

325

along a surface by an array of groove parallel to the wavefront. In ome modem device , the e two concepts are combined in the Surface Transverse Wave (STW), in which the abo e cut of quartz i u ed and urface guidance is obtained by the metal electrodes of tran ducer and grating . High-frequency re onators with excellent performance have been made by u ing this wave (Section 6.2.4). Another cut of quartz gives an SSBW with exceptional temperature stability [58]. Thi i -50.5° Y-X + 90° orientation. The delay i approximately a cubic function of 0. with a ariation of 30 ppm from Oto 140°C. Thi i much better than ST-X quartz, which would gi e 150-ppm variation over this 0 range. Another case in quartz, the LST cut, supports a leaky wave with exceptional temperature stability, although device in ertion lo e are found to increa e with temperature [64, 75]. A related case occur in 36° Y-X lithium tantalate (58, 59, 57]. This is usually called a ' leaky wave:' or ·p eudo- urface wave" (PSAW). For rotation angles near 36°. wave olution with complex wavenumbers are found, with small attenuation. The attenuation vani he near 36°, for both free and metallized surfaces. At this point, the velodty i high and the l1v/ v value is 2.4%. Compared with SAWs on the same material the coupling i tronger while the temperature coefficient is similar (and therefore les than that for SAWs on lithium niobate). Velocity and attenuation curves are bown on Fig. 6.20. The Rayleigh wave, at 3100 to 3400 mis, has negligible

~OO---r----.-----.-.---r--.---.-----.-.--,-~

Bulk shear wave

en 4000

Leaky wave, free Leaky wave, metallized

-g E

~

~ 3500

Bulk shear wave

30000

10

20

30

40

50

60

50

60

Rotation angle (deg}

1.0

-

0.8

~

-

0.6

"'::::,C"

0.4

CD -0

C 0

'

.:

Free surface

"

0.2 0

y

' /"" Metallized

Q)

i

'

0

10

20

30

40

Rotation angle (deg}

Figure 6.20

Velocities and attenuation for leaky waves on rotated Y-cut lithium tantalate.

326

SURFACE ACOUSTIC WAVE DEVICES

piezoelectric coupling. The leaky wave gives higher power handling capability than a SAW because it has a larger penetration depth. It has been used extensively for high-frequency resonator filters. These devices use thick metallization, which affects the wave somewhat, and for these, it has been found that lower insertion losses are obtained using 42° rotation instead of 36° [76]. Similar leaky waves occur in rotated Y-cut lithium niobate, again giving elevated f).v / v values (Table 6.2). In this case, there are two cuts of interest, because zero attenuation occurs for the free surface at 41 ° and for the metallized surface at 64° [57] . Despite this rather confusing picture, both cases have been used successfully for devices.

Recent Developments. Growth techniques for the crystal langasite (La3GasSiO14) have recently made it a practical possibility for SAW devices, and it offers properties rather like ST-X quartz but with stronger coupling. With Euler angles (0°, 138.5°, 26.6°), it gives good temperature stability and minimal diffraction, although it does show the natural SPUDT effect [62]. Da Cunha [63] quotes several other cuts. Similar results are found for langanite (La3Gas.sNbo.sO 14) and gallium phosphate (GaPO4) [63] ; for the latter, the orientation in the table has temperature stability better than ST-X quartz. There is also a family of new silicates with similar properties [77] . Another approach for improving temperature stability is to add a SiO 2 film. On 128° Y-X lithium niobate, this gives TCD ~ 0 when h / A = 0.3, with little effect on the velocity or f).v / v. Used in 1-GHz filters, the film caused only 1-dB increase of insertion loss [72] . Exceptionally strong coupling occurs in potassium niobate, KNbO3 , which has orientations with !:l.v/v up to 25% and has a zero TCD orientation [65]. Recent years have also seen renewed interest in leaky wave [66], and one has been found in lithium tetraborate [67]. This is a longitudinal leaky wave; that is, its particle motion is almost parallel to the wave vector, in the plane of the surface. Being trongly related to the bulk longitudinal wave, it has a very high velocity and is known as a high-velocity PSAW, or HVPSAW. The orientation is -43°Y-X + 90°, and it gives v1 = 1000 mis and !:l.v /v = 0.7%. The high velocity i attractive for high-frequency devices, and a 1.5-GHz IIDT filter was demonstrate~ with 2-dB in ertion loss. Other HVPSAWs are quoted by, for example, da Cunha [78], who gives several examples on lithium tetraborate and quartz. Piezoelectric films have a long history in SAW, enabling a nonpiezoelectric substrate to be used. For analysis, a matrix transformation can be u ed to allow for each layer, leading to SAW and leaky-wave solutions [79]. Zinc oxide on glass has been used for TV IF filters, and recently, the technology has advanced to enable these films to be used in high-frequency devices, using sapphire ub trates that give high velocities (Table 6.2). A 1.5-GHz IIDT filter gave 1.3-dB insertion loss [68, 69]. Also on sapphire, aluminum nitride films have been used [70]. Another high-velocity nonpiezoelectric substrate is diamond, which can be grown on a silicon ubstrate giving a SAW velocity of 11,000 mis. With a ZnO film, the velocity is reduced, and there i a tradeoff between velocity reduction and !:l.v / v, depending on thickness; typically, v1 values are 7000 to 10,000 mis and !:l. v /v values are 1% to 2% [71]. The TCD can be reduced to zero by adding a SiO2 film [71].

6.4

NONREFLECTIVE SAW TRANSDUCERS

327

6.4 NONREFLECTIVE SAW TRANSDUCERS The delta-function analy i introduced in Section 6.2. 1 is an excellent approach for illu tration of basic concept . However. for practical purpose , it ha erious limitations, being unable to give the absolute re pon e of a transducer or its impedance or its reflection coefficient. To remedy these deficiencie , we need to probe more deeply. This section is mainly ba ed on the effective permittivity approach, which has yielded a great deal of useful algebraic re ult . The transducers here are assumed to be nonreflective o that they do not reflect SAW when horted, for example, the double-electrode transducer (Fig. 6.2b). Reflective transducers are con idered in later sections. We consider unapodized transducers with aperture large enough for transverse variations (in the y-direction) to be ignored, thu excluding diffraction, for example. As before, the surface of the substrate is at z = 0, and the waves propagate along ±x. The electrodes are taken to be of negligible thickness and zero resistivity. Practical transducers typically have electrode thickne ses of I % to 2 % of the wavelength. The analysis here remains valid for such cases, although not for leaky-wave substrates considered later (Section 6.9.4).

6.4.1

Electrostatic Solution

We first take the substrate to be nonpiezoelectric, so that the theory is concerned only with electrostatic effects. This problem has an important role in the piezoelectric case considered later. Becau e the fields are electrostatic, the solutions will be valid for all frequencies and we can take Cu= 0 without losing generality. In order to analyze transducers and other structures on a surface, we need to consider the electric field or potential at the surface and the currents flowing in the electrodes. The currents are determined by the charge density a (x) on the electrodes, which is taken to include charges on both sides. These quantities are functions of x . As in many problems concerning linear phenomena, much simplification is obtained by first considering one Fourier component; the analysis can later be extended to more general cas es by using Fourier synthesis. Therefore, considering a particular surface potential - 1.0 ns

N = 250

Q)

0

0.5 N = 40

I

------------

. 104

is

99

100

101

102

Frequency (MHz)

Figure 6.29 Reflection coefficient and delay of periodic grating, center frequency 100 MHz.

6.6

INTERNAL REFLECTMTY - GRATINGS AND TRANSDUCERS

345

An important result concerns the group delay r of the reflected wave, given by r = - d / d w, where is the phase of P 11 • It can be shown from Eq . (6.67) that~for a semi-infinite grating at the center frequency, r = ~t / lrs1- Here ~t = p / ve is the strip spacing in time units. Thi result is needed to evaluate the mode spacing in resonators. The above analysis as ume power conservation, but a more general formulation is given in [168].

6.6.3 Evaluation of Strip Reflection Coefficient rs In practical gratings. the reflecting strip are usually either deposited metal or grooves, and we consider particularly the former. A metal strip on the surface gives reflectivity becau e it perturbs the wave, and two types of perturbation are involved: (1) electrical and (2) mechanical. Both effects are generally weak, and to a good approximation, they may be evaluated independently and then added. Hence, the strip reflection coefficient can be written as (6.71) where re and rm are the electrical and mechanical terms, each evaluated as if the other was absent~ In a weakly piezoelectric substrate such as quartz, the mechanical effects dominate and re can often be ignored. In a strongly piezoelectric substrate such as lithium niobate, the electrical term is often dominant, but rm can also be significant if the film is thick. Electrical effects arise because the electrode shorts out the electric field at the surface. For practical purposes, this process is independent of the electrode thickness. However, it does depend on electrical connections made to the strips, and we consider two cases: (1) shorted strips (all strips connected electrically by a bus bar), and (2) open-circuit strips (all strips isolated electrically). The solution can be found approximately using the equation (x) = G (x) * a(x), Eq. (6.26), where the Green's function G (x) has electrostatic and SAW tenns as in Section 6.4.2. At the center frequency of the first stopband, where the strip spacing p is )../2, Datta shows that the result is [87] (6.72) where the upper sign is taken for open-circuit electrodes and the lower sign for shorted electrodes. As before, ll.. = 1ra/ p. These functions vary slowly with frequency, but the variation is usually insignificant The derivation accounts for neighboring electrodes -the reflection coefficient of one isolated electrode is quite different. Figure 6.30 shows re for the two cases, as functions of the metallization ratio alp. Note that, apart from the coupling parameter ~v/v, these functions are independent of the choice of material. For a/ p = 1/ 2, both cases give lrel = 0.718~v/ v. The mechanical term rm depends on film thickness and on the material and orientation. Unlike the electrical term, it is independent of the electrical connections of the electrodes. Datta (88, 89] gives the reflection coefficient as (6.73)

346

SURFACE ACOUSTIC WAVE DEVICES -s;' ~

0.6

,::, CD

-~ 0.4 m

E

z0

a lp = 0.5

0.2

0.2

0.4

0.6

0.8

1.0

NonnaJized frequency, flf5

Figure 6.31 (VJ - Vm),

Velocity variation due to electrical loading. Normalized velocity Vn

= (Ve -

Vm) /

348

SURFACE ACOUSTIC WAVE DEVlCES

The mechanical effect depends on film thickness h. In many cases, the velocity is known to have the form (6.76) For small h , the first term dominates. The constant Dm is given by (88, 89, 100, 101] 2

2

Dm = [21r(~v/v)/t:00Hlu 1/ 1 2 (a 1 - pv}) + lu2/ l (a 2 - pv}) - ju3/¢ 1 p v}} (6.77) For aluminium electrodes on ST-X quartz, this gives Dm = -0.13. The K 11 term in Eq. (6.76) is often known as the "stored energy" term because it was attributed to storage of energy at the electrode edges [ I 02]. For a Ip = 1/ 2, experimental results for ST-X quartz [103, 95] gave K 11 values around 8.0. Theoretical result were similar [ 104, l 05], although Datta found that K II varies a little with h / A. Early studies of gratings often modeled them as transmission lines, with repetitively mismatched sections to generate the reflections. To repre ent stored energy, susceptances were connected in parallel at locations corresponding to electrode edges [102]. This model should not be taken too rigidly. However, it did correspond quite well with experiment For example, in grooves the velocity shift was found to be proportional to (h/ )..) 2 , in agreement with Eq. (6.76) because the electrical term should be omitted in thi case.

6.6.5

Single-EJectrode Transducers (Regular Electrodes)

We have already considered nonreflective transducers in the quasi-static analysis of Section 6.4. In single-electrode transducers, there is the added complication of reflections from the electrodes. For gratings, electrode reflection have been considered in Section 6.6.2. Hence, the analysis of single-electrode transducer i basically a question of combining the quasi-static analy is with the reflecting grating analysis. As een above, grating properties (reflection coefficient and velocity shift) depend on the electrical connection of the trips (open- or hort-circuit). Fortran ducer analysis, the grating propertie need to be evaluated as if the trip were all honed. This is easily seen by considering a shorted tran ducer, which evidently upports the wave in the same way as a shorted grating. The behavior is closely related to that of a launching transducer by the reciprocity relation, Eq. (6.34). The transducer behavior i expre ed in term of the P-matrix. Parts of thi are as calculated for a grating, because the transducer behave as a grating when it is shorted. This applies to P11, P12 , P21 (= P12), and P22. which are given by Eqs. (6.67) and (6.68). When shorted, the transducer is ymmetric, so P22 = P11 • The remaining terms involve tran duction. To analyze thi , we consider a shorted transducer. A sume first that the electrode do not reflect. so that the quasi-static analysis is valid. Consider a transducer with only one electrode connected to the live bus bar. When this transducer is shorted and a wave with amplitude ( urface potential) ; is incident, the current entering this electrode is [106] (6.78) This can be shown from the analysis in Section 6.4, with p 1 (k~) given by Eq. (6.41). The phase of the potential ; is assumed to be referenced to the center of the electrode.

6.7

COUPLED-MODE EQUATIONS

349

Equation (6.78) i accurate if the wave propagation i not disturbed by the electrodes, a valid assumption in the quasi-static case. Here the electrodes reflect the waves. It will be assumed that the electrode reflection coefficient is not too large, so that the wave amplitude varie slowly and Eq. (6.78) remains approximately true. Referring back to the grating analy i , illustrated in Fig. 6.27, the waves incident on electrode 11 have amplitude Cn - 1 and bn, and both contribute to the current. Because they are defined at the port po ition , they need to be multiplied by exp( - j kep / 2) to refer to the electrode center. Also, becau e the amplitude of each wave changes as it passes the electrode, it i better to take the average of the amplitude on both sides. Finally, Eq. (6.78) ex pre es the current in terms of the wave potential ¢; , and for the P-matrix analysis, the amplitude Cn and bn mu t be defined as for the amplitude A of Eq. (6.48), 2 o that lcn 1 / 2 and lbn12 / 2 are the SAW powers. With these modifications, the current for electrode n i In

= - j P1(ke) [cuWI's/2] 112 [(cn-1 + bn) exp(-jkep / 2) +(en+ bn-1) exp(jkep / 2)]

(6.79) The transducer can be analyzed by tarting at the left end (port 1) and assuming no output wave (so that bo = 0). Then Eq. (6.63) can be used repeatedly to obtain the wave throughout the structure, and Eq. (6.79) can be used to give the current for each electrode connected to the live bu bar. We then have P32 = I /cN , where / is the total current. Also. P 22 = CN/ bN and P12 = c0 / bN. Similar cascading in the other direction gives P 31 Pu . and P21. In addition, reciprocity gives P13 = - P 3 1/ 2 and Pn = - P32/ 2. This completes the P-matrix except for the admittance P33 , which equals G 0 + j Ba + j cuC, . The capacitance C, is obtained from the electrostatics, as explained earlier. For G 0 • we can use the power conservation, Eq. (6.53), so that Ga(w) = IPnl 2 + IP231 2 • The susceptance Ba can then be found by Hilbert transformation of Ga (Sections 6.2 .1 and 6.4.5). This method has the advantage that electrostatic effects are modeled quite accurately, so that the results show correctly the effects of changing the electrode width, and the harmonic responses are also simulated correctly. However, the assumption of power con ervation is somewhat restricting, because propagation loss can be significant at high frequencies. The method is valid for arbitrary polarity sequences Pn. In single-electrode transducers, the electrode reflections have a prominent effect on the transducer conductance. Figure 6.32 compares the Ga curves for a transducer with and without electrode reflections, using data for ST-X quartz. The reflections cause the curve to peak up at a frequency below the nominal center frequency, an effect that is very significant in many high-frequency devices. This also provides a clear experimental test for the sign of rs. If r,/j is positive instead of negative, the peak of G a moves to higher frequencies instead of to lower frequencies. The case without reflections could be obtained by recessing the electrodes, as in some resonators. A

6.7 COUPLED-MODE EQUATIONS These equations, · also called the coupling-of-modes (COM) equations, are of great value in analyzing reflective transducers and gratings. They apply to a system in which freely propagating waves are coupled together weakly, causing the wave amplitudes

350

SURFACE ACOUSTIC WAVE DEVICES

0.20

-.....--~--.-----.--r----r---r-----r-ir--r---r--i

0.15 0 .r; E

-

E 0.10

I I

I

I

I

-

I

0.05 0

I

L-==::.,J....CJ......:::::...i.L...-.L--...l.-....J-~~~~==t:::.-.J

96

100

104

Frequency (MHz)

Figure 6.32

Effect of electrode reflections on conductance of single-electrode IDT.

to vary slowly. They have been used for microwave devices since the 1950s. SAW gratings can be analyzed using COM equations similar to those established earlier, but in transducers, the presence of transduction as well as reflection causes complications, and COM equations for this case were first considered by Koyamada and Yoshikawa (107]. Justification of the equations is given by [93, 108] and, from the physics, by Akcakaya [ 109]. The coupled-mode equations, as given by Chen and Haus [93], have the form

= -j8C(x) + c12B(x) + a 1V dB(x)/d x = j8B(x) + cT2 C(x) + af V

dC (x)/d x

di (x)/d x

= 2af C (x) -

2a1 B(x) + jwC1 V

(6.80a) (6.80b) (6.80c)

The parameter 8 = ke - ko is a frequency variable; ko is the wavenumber at the Bragg frequency where strong reflections occur. In a grating or ingle-electrode transducer at the first stopband, we have ko =JC/ p , because the spacing p equal )J2 at this frequency. C(x) and B(x) are wave amplitude , defined as C(x) = c(x) exp(jkox);

B(x) = b(x) exp(-Jkox)

(6.81)

where c(x) and b(x) are the "actual" wave amplitudes, with c(x) being the wave traveling to the right (+x direction) and b(x) to the left. The interaction occurs for frequencies close to the Bragg frequency (ke ~ k0 ), o C (x ) and B(x) are slowly varying functions. The amplitudes are defined such that ICl 2 / 2 is the power of one wave, and similarly for B. Also, c 12 is a reflection coefficient per unit length in a differential sense, so that C12- d x is the reflection in a distance d x . Similarly, a 1 is a transduction coefficient per unit length. The presence of these parameters at various places in the equations is a consequence of reciprocity. In the third equation, C 1 is the capacitance per unit length. / is the current in the bus bar, and it can be interpreted such that d I is the current flowing into a section of the transducer extending from x to x + dx. The total current is thus the integral of di / dx along the transducer length, with 1 set to zero at one end of the transducer.

6.7

COUPLED-MODE EQUATIONS

351

For clarity, suppose there is no reflection (c 12 = 0) and no applied voltage ( V = 0). Then the fir t equation has a solution C (x) = exp(-j8x), giving c(x) = exp(-jkex). So c(x) is just the con tant-amplitude wave expected in this situation, with wavenumber ke. A velocity shift due to the electrode can be included by choosing a suitable value for ke. The equations probably repre ent the simplest possible way of describing a system with both transduction and reflection; yet, they can give solutions with good accuracy, adequate for many practical purposes. Useful algebraic re ults can be obtained if we assume a uniform transducer, so that a 1 and C12 are independent of x . To olve the equations, we first set V = 0, which i applicable for a grating or a shorted tran ducer. Consider a solution of the form C ex exp(- j sx) and B ex exp(-jsx). Substituting into Eqs. (6.80) gives (6.82) and B/ C = j(o - s)/ c 12. This is a grating-mode solution, similar to that found using the RAM (Section 6.6.2). For real 8, s becomes imaginary when 181 < !c12 1, giving a stop band. Unlike the RAM analysis the COM gives only one stopband, and this is a consequence of the approximations made in deriving Eqs. (6.80). The stopband edges are at 8 = ±Jc12 1. For a single-electrode transducer, we show below that c12 = p and this gives the fractional width of the stopband as !::,,.f/ Jo = 21rs 1/rr, in agreement with the RAM [Eq. (6.66)]. When V is finite, Eqs. (6.80a, 6.80b) have a solution independent of x. This is easily found to be C (x) = K 1 V and B (x) = K 2 V, with

±r:/

(6.83) This solution is the "particular integral,'' which, although valid, does not have enough unknown constants. For a general solution, we can add two solutions with the previous form exp(±jsx) (the "complementary function"), so that the general solution is

= h 1 exp(- jsx) + h2expUsx) + Ki V B (x) = h 1p 1exp(-jsx) +h2p2expUsx) + K2 V

C(x)

(6.84)

where p 1 =j (8-s)/c 12 and P2=j(8+s)/c12; scan be taken as either solution of Eq. (6.82). The parameters h 1 and h2 are constants, dependent on the boundary conditions. From this solution, we need to find the transducer P-matrix, defined by Eq. (6.49). The transducer is taken to have length L, extending from x = 0 to x = L. Thus, Art = b(O), A 12 = c(L), and so on. Defining a denominator D = s cos(sL) + j8 sin(sL), the P-matrix is found to be

P11

= -cr2 sin(sL)/D

= s exp(-jkoL)/ D P22 = c12 sin(sL) exp(-2j/coL)/ D D • P31 = 2ai sin(sL) - 2 sK2[cos(sL) -1] P12

352

SURFACE ACOUSTIC WAVE DEVICES

= - 2a 1 sin(sL) + 2sK1[l - cos(sL)] P 33 = - Ki P 31 - K 2 P 32 exp(JkoL) + 2(a; K 1 -

D • P32 exp(jk0 L)

a 1K 2) L

+ jwC,

(6.85) where C, = L • C I is the transducer capacitance. These equations satisfy the reciprocity relations, Eqs. (6.50). They also satisfy power conservation, Eqs. (6.51) and (6.52), if 8 is real . However, propagation loss can be included by allowing ke, and therefore 8, to become complex. Equations (6.85) assume a uniform transducer (c12 and a1 independent of x ) ._A nonuniform transducer can be analyzed by using Eqs. (6.85) for small uniform sections and then cascading using Eqs. (6.59). Some equations valid at the center frequency (8 = 0) are given in Section 6.8.2. Compared with the RAM method, the COM has the advantage of giving closedform formulas for the Pii , it can allow for propagation loss, and it does not require the rather clumsy Hilbert transform to obtain B0 • Disadvantages are that the equations are less accurate, and parameter values (for c 12 and a 1) need to be supplied for each transducer type. In contrast, the RAM formulation gives transduction for a wide variety of transducer types by including electrostatic effects in the function p 1 (/3).

6.7.1

Parameters for Single-Electrode Transducer

The COM parameters for a single-electrode transducer, in its fundamental passband, follow from the RAM theory. It is common to take the ports to be at the centers of the end electrodes, but they could alternatively be just outside the end electrodes as in the RAM theory above. To make the correspondence, we consider a short transducer operating at the center frequency, for which the COM equations give (6.86) where the expression for P13 ignores the reflectivity. Assume that port 1 is outside the transducer, at a position p / 2 from the center of the first electrode (as in the RAM analysis of Section 6.6.2). In the RAM method, a "transducer" of length p, with only one electrode, gives Pll = -rs, where the sign follows because the port is at a distance p/ 2 from the electrode center. Using Eq. (6.86), we find c 12 ~ p. H port 1 is taken to be at the center of the first electrode, this becomes c 12 ~ p. For the transduction, consider a "transducer" of length 2p, with one live electrode located centrally. This will have ports at the centers of the end electrodes. In the quasistatic theory, the transduction is given by ~ (6.32) or (6.78). Deducing p 13 from this, and equating with the COM formula, gives

r;/ -r; /

(6.87) This applies if the first electrode is grounded and port 1 is at the center of this electrode. If port 1 is outside, at a distance p / 2 from the electrode center, the above a 1 should be multiplied by - j. If the first electrode is live instead of grounded, the sign of a 1 should be changed. For both types of port position, the factor exp(- JkoL) in Eqs. (6.85) is ±1 for all w. It is usually adequate to treat c12 and a 1 as constants, evaluated at the center frequency.

6.8

LOW-LOSS FTLTERS -

ONRESONANT

353

The above analysis al o applies for a grating, in which case the Pii with i or j equal to 3 are not required.

6.8

LOW-LOSS FILTERS-NONRESONANT

A explained in Section 6.2.2, early SAW bandpas filters were subject to unwanted multiple-tran it ignal . and hence. high insertion losses had to be accepted, although the performance could otherwi e be excellent. However, the growth of wireless communications from about 1985 onward gave ri e to demands for low-loss filters, in order to improve y tern noise figures and to reduce the required amplifier gains and, hence, power con umption. Many low-loss techniques have been developed, and most of them fall in one of three categories: (1) Arrange for a transducer to have waves incident from both directions; this can

cancel reflectivity (ring filters ; IIDTs). (2) Arrange for the transducer to have directivity; this also can cancel the reflectivity (SPUDT filter ). (3) Use resonators as the basis for filter realization, thus exploiting the multiple

reflections (transverse coupled resonators; longitudinally coupled resonators). For IF applications, there is also the objective of size reduction, because conventional filters would be much too large for portable systems such as mobile phones. For RF devices, there is a particular emphasis on low insertion loss, which often needs to be less than 2 dB. In this section, we consider nonresonant filters using methods (1) and (2). Further low-Joss filters using resonators, are described in Section 6.9.

6.8.1

Three-Transducer Filters and IIDTs

It is well known that an unapodized symmetrical transducer, with identical waves incident from both sides, does not reflect the waves if it is electrically matched [110]. This follows from the P-matrix equations in Section 6.5. From Eq. (6.54a) with P23 = P13 , we find ( P11 + P 12) = -P13/ Pt,y Using this and the first of Eqs. (6.49), with A;i = An, gives A,1 = Pn(V -An / Pt3). Setting V = - /Yi, where Yi is the load admittance, and using the third of Eqs. (6.49) to find V, we obtain A, 1 An

=

Pn [ Pt 3

2G

a

P 33

+ YL

_

l]

(6.88)

which also makes use of Ga = 21P13 j2 . When the transducer is matched, we have Yi = P3*3 and, hence, P33 + Yi = 2Ga, so that Att is zero. This result does not depend on the transducer being uniform or nonreflective. One way of using this result is a three-transducer device of Fig. 6.33a [110], in which the central transducer, giving the output, is symmetrical and matched. With all transducers matched, the loss is 3 dB (due to bi directionality of the input transducers) and the multiple-transit signals are eliminated, at least at one frequency . The loss can be reduced further by using SPUDTs, described below, for the outer transducers. A

>

354

SURFACE ACOUSTIC WAVE DEVlCES (b )

(a)

Figure 6.33 Low-loss devices: (a) three-transducer filter; (b) extra track added using self-resonant transducers.

variant of this is a two-track device in which the central transducers are connected, as in Fig. 6.33b . A long uniform IDT can have an admittance that is real at one frequency, because there is a region of w where Ba is negative and approximately equal to -wC,. When this is the case, two such identical transducers are electrically matched to each other, providing a coupling mechanism with little loss and with some frequency selectivity. Ring filters provide another implementation. As shown in Fig. 6.34a, two transducers in different tracks are coupled by "reflecting trackchangers." The latter device transfers a SAW from one track to an adjacent one, and it changes its direction. There are several implementations for this, one being an m.s.c. with strip spacing )../3. In the upper track, the wave phases at the electrode locations are 0°, 120°, 240°, and so on, and the connections are such that the phases in the lower track are 0°, 240°, 120°, and so on, equivalent to 0°, -120°, -240°, and o on. This gives an output wave traveling in the opposite direction. Filters on 128°Y-X lithium niobate had 170-MHz center frequency and 1- to 2-dB loss [111]. lnterdigiJaJed lnterdigital Transducer (IIDT). This device [112] can be seen as an extension of the above principle. As shown in Fig. 6.34b , it has a sequence of IDTs

(a)

(b)

In

j,,,

!

Out

Figure 6.34

Low-loss filters: (a) ring filter using multistrip trackcbangers; (b) DDT.

6.8

TABLE 6.5

LO\V-LOSS FILTERS -

ONRESONANT

355

Examples of Bandpas Filter Performance

Type

Material

Tran versal, with msc DART filter 2-track DART filter Z-path filter TCRF IIDT DDT LCR LCR

Y-Z LiNbO3

IEF IEF 2-track reflector filter Ring filter

ST-X quartz ST-X quartz

Center 3 dB freq uency bandwidth (MHz) (MHz) 37 197 220.38

6 1.4 1.5

Insertion loss Rejection (dB) (dB ) 30

Ref.

55

(24]

8.6 8.0

60 42

[136) [125]

ST-X quartz ST-X quartz 36°Y-X LiTaO3 36°Y-X LiTaO3 36°Y-X LiTaO3 45°X-Z Lh B4O7 64°Y-X LiNbO3 36°Y -X LiTaO3 128°Y-X LiNbO3

210.38 471 935 878 914 71 850 2450 200

1.4 0.35 41 35 4 0.3 40 100 24

7.0 3.4 3.5 1.8 3 4.3 2 3 4

43 45 30 30 40 45 50 45 35

[127] [24] (137] [138] [139] (142) (140) [141] [128]

128°Y-X LiNbO3

172.5

15.5

1.7

55

(1 11]

connected altemate]y to the input and output ports. For simplicity, assume the transducer are all identical. The principle is seen if we imagine the three output transducers to be connected to a load with admittance three times that needed to match one transducer alone. In this condition, each transducer is electrically matched. It receives identical input waves from the two sides and, therefore, it absorbs the incident SAW energy, converting it all into electrical output energy. The same applies to the input transducers, except for tho e at the ends, where there is some loss because of waves radiated outward. The insertion loss, in decibels, is ideally l0log[(N, - 1)/(N, + I)], where N, is the number of transducers, typically 5 to 9. The response is very complex because, when the transducers are not matched, there is a large number of SAW paths from the input to the output. However, the good performance obtainable is illustrated by low-loss RF filters using leaky waves, as shown in Table 6.5. These devices can be modified in a variety of ways. For example, the IIDT can have reflecting gratings added at the ends to reduce the loss. Devices with two tracks can use self-resonant transducers, similar to Fig. 6.33b, to couple the tracks. This gives extra stopband rejection with little extra insertion loss.

6.8.2 Unidirectional Transducers -Principles General Discussion. The objective of unidirectional transducers is to eliminate the triple transit signal by designing transducers to give an acoustic reflection coefficient of zero. For a nonreflective transducer ( P11 = 0) connected to a load with admittance YL, Eq. (6.56) shows that the reflection coefficient has magnitude R = Ga / lP33 + YL IHowever, this is of no practical interest because zero reflection requires YL = oo, so that no power is transferred to the load. Hence, we conclude that internal reflections are necessary, so that Pu :/= 0. This problem is discussed by Hartmann and others (113,

356

SURFACE ACOUSTIC WAVE DEVICES

114, 115, 116]. The solution takes the fonn of various types of Single-Phase Unidirectional Transducers (SPUDT ). These are widely used in IF applications (50- 500 MHz). They are not well suited for RF applications because they have relati vely narrow electrode widths. Before discussing this, we briefly mention multiphase unidirectional transducers, in which each period has three or more electrodes and three or more voltages are applied, with different phases. A simple network can provide the phases needed. The first of these was Hartmann's three-electrode type [117). Although effective, these transducers had the disadvantage that crossovers were required, leading to complex fabrication. This was the motivation behind the search for single-phase types, i.e., types with only one relative voltage. However, Yamanouchi's group-type multiphase transducer [118) used a meander line instead of crossovers, and this seems more practical. Regarding SPUDTs, we make the following statements without proof [115, 116): (1 ) To usefully obtain zero acoustic reflection, the transducer must have internal reflections, as seen above. (2) For a transducer to usefully give zero acoustic reflection, it must be directional. That is, with directivity D defined by D = IP13/ P23I, we need D > 1. For D = 1, the loading required is purely reactive, so that there is no power transfer.

(3) For any transducer with D > 1, there is a load admittance such that the acoustic reflection coefficient is zero. The real part of the load admittance is positive. (4) The loading required in (3) does not match the transducer electrically. However, when Dis large, the loading approaches the matched condition; i.e., YL ~ P33 . We then have low conversion loss and low acoustic reflections simultaneously. Hence, we need large D values for low-loss devices. Typically, 20 log(D ) will be 6 dB or more. (5) The directivity depends on the internal reflectivity and the relative phase of the transduction and reflection processes, as clarified later. For a particular reflectivity IPH I, the largest possible directivity is

(6.89) Hence, for large D, we need strong reflectivity, with I P11 1 close to unity. When D is maximized in this way, the load admittance for zero reflection is YL = Ga / lPll I - j · Im(P33}. To make IP11 I close to unity, the SPUDT needs to be quite long, typically 100 ).. or more. Consequently, it is best suited to narrowband applications, and hence, the substrate is usually quartz in order to obtain sufficient temperature stability. The directivity of a transducer can be affected by the symmetry of the substrate. This is explained in Section 6.8.5, which discus es the Natural-SPUDT effect. For the present, this effect is ignored, so that we assume that the substrate behaves symmetrically. A symmetrical transducer structure will therefore be bidirectional.

SPUDT Principles -the Group-Type SPUDT. It is convenient to start with a very simple SPUDT consisting of a transducer with a reflecting strip on the right, as in Fig. 6.35a. The transducer is assumed to have no internal reflections and to have symmetrical geometry. The strip also has symmetrical geometry. As shown Section 6.6.2,

6.8 (a)

Port 1 I

'I

A1 I ._ I I I I Forward

~ I

Port 2

-

I I I I

d

LOW-LOSS ALTERS - NONRESO ANT

357

(b)

,_A

rn ~

I I I I

I

2

11 1111

111111

111111

I

NAo

MAo

Simple SPUDT

Group-type SPUDT

Figure 6.35

SPUDT principle .

its reflection coefficient rs, referenced to its center, is imagjnary. The velocity is taken to be the same everywhere. We consider the response only at the center frequency, where the wavelength is Ao, and the di tances between the transducer center and the two ports (broken line ) are taken as multiples of Ao. The strip reflection coefficient rs is taken to be negati e imaginary. this being the most common case. The di tance d between the center of the transducer and the strip is set so as to reinforce the wave to the left and reduce the wave to the right. For unit-applied voltage. we define A as the amplitude of wave generated by the transducer, in either direction, with phase referenced to the transducer center. Then the total wave leaving at the left port is A 1 = A+ A · rs • exp(-2jkd), and the wave leaving at the right is A2 = A · ts where ts is the (real) transmission coefficient of the strip. The transducer has unity transmission coefficient (P12 ), because Pu = 0. To maximize the wave to the left, we take d = (n/ 2 - 1/ 8)Ao , and this wave is then A(l + lrsl) . The ratio of wave amplitude is thus (6.90) in agreement with Eq. (6.89). If rs/} was positive, we would take d = (n / 2 + 1/ 8)). 0 to get the directivity in the same direction. For practical strips, lrsI is small, typically 1 % or 2%, so there is little directivity here. However, the principle is the same if we add extra strips to make a grating with pitch )...0/2, giving stronger reflectivity. This form of SPUDT is quite practicable, although it is limited to rather small bandwidths. A better arrangement is to repeat the structure, as shown in Fig. 6.35b, which is Lewis' group-type SPUDT [I 19]. Because the port separation is a multiple of Ao in Fig. 6.35a, the transducer separation is a multiple of Ao in Fig. 6.35b. The gratings are typically shorted metal electrodes. With the appropriate spacing between transducer and grating centers, it is easy to show that for any one transducer and any one grating, the wave amplitudes obey Eq. (6.90) (with rs, t3 replaced by the grating coefficients). What is not so clear is the behavior of the SPUDT as a whole. To clarify this, we can use the COM equations of Section 6.7, assuming no loss so that 8 is real. Considering the center frequency, we set 8 = 0 so that s= Jlc12I- Define 0 and as the phases of the transduction parameter a1 and the reflection parameter c 12 , so that cx 1 = lcx1 I exp(j0), and c12 = Icul exp(j) . Then Eqs. (6.85) are found to give

Pn P23

=e

_2 j 0

[

coth(lc121L/ 2) + ei> ] coth(lc121L/ 2) - e-i(W-¢,)

(6.91)

358

SURFACE ACOUSTIC WAVE DEVICES

in agreement with Hartmann (11 3]. Maximum directivity occurs when 20 - so that exp[j (20 - ¢ )] = 1. We then find

= 2nn , (6.92)

Thus, if D 1 i the directivity for one cell in the transducer, the directivity for N cells is DN = Df. For the same conditions, Eqs. (6.85) give (6.93) and from these, we find (1 +I P 11 1)/1P12I = exp(lc12IL), agreeing with Eq. (6.89). These COM results can be applied to any type of transducer, of course. In particular, they apply to a reflective single-electrode transducer, for which Section 6. 7 showed that 0 and¢ are both ±rr / 2 (taking port 1 to be at the center of the first electrode). In this case, we find exp[j (20 - ¢ )] = ±j and Eq. (6.91) gives IP13/ P 23 I = 1, so that there is no directivity. The COM equations are not very suitable for the group-type SPUDT because the cells are many wavelengths long. However, the equations are applicable at the center frequency. Given a reasonably high directi vity, the transducer can be electrically matched to give low acoustic reflections and low conversion loss, as noted earlier. Hence, with two such transducers, a device can simultaneously give good suppression of multiple-transit signals and low insertion loss.

Reflection and Transduction Centers. The group-type SPUDT has characteristic points at the centers of the transducers and gratings. The transduction process - wave generation and reception - can be regarded as occurring at a "transduction center," at the center of each transducer. The reflections can be regarded a occurring at the center of each grati ng. In other types of SPUDT, the locations of these centers are less obvious, but the concepts are of much value in analyzing the transducer. To find a reflection center, we first imagine the SPUDT to be shorted, o that no transduction occurs. Considering some particular structure inside the tran ducer, the reflection coefficients for waves incident from the right and the left must have the same magnitude. This follows from power conservation, Eq. (6.51), u ing P 21 P 12 . The waves can be regarded as being reflected from a localized poin 4 and by choosing the location of this point, the two reflection coefficients can be made the same in phase as well as amplitude. The point is then the reflection center. Relative to this, the reflection coefficient must be imaginary (Section 6.6.2). To find a transduction center, we have to imagine a transducer with no reflection. This is actually nonphysical, but for one cell, the reflection is usually quite weak, so the transduction can be defined approximately. Considering a localized structure, the waves generated in the two directions have equal amplitude, as hown by Eq. (6.55). This means that it is possible to identify a point such that, relative to that point, the waves have the same phase. This is the transduction center. Relative to the transduction center, the amplitudes of the generated waves are in quadrature with the applied voltage. This follows from Eq. (6.54a), which gives P 23 = - P 13 when P11 = 0 and P 12 1 (and P23 = P 13 when the observation points are equidistant from the source). Transduction can be analyzed by a quasi-static analysis as in Section 6.4 (120, 121]. Sometimes these points can be identified from the symmetry. If the geometry shows symmetry about a point when the electrode voltages are ignored (i.e., for a shorted

=

=

6.8 I I I

RC

LOW-LOSS FILTERS-NONRESONANT

359

TC

I I I

I

I I I I

X=O port

Figure 6.36

One cell of a general SPUDT. in terms of transduction and reflection centers.

tran ducer), then that point must be a reflection center. If the geometry is symmetric about a point when the voltages are taken into account, the point is a transduction center. In a periodic SPUDT, each period has two choices for the position of the trans duction center, and four for the reflection center. To give directivity, the two centers mu t be at different places, and optimally, they are spaced by (n / 2 ± l / 8)Ao. In a group-type SPUDT, the two centers are clear, as stated above. In a single-electrode transducer both centers are at the centers of the electrodes, so that their spacing is nAo/2 and there is no directivity. To locate these centers, it is sufficient to find points where the reflection and transduction coefficients appear imaginary. Assuming that each period (of length Ao) can be represented as having one "transducer" and one "reflector," it can be modeled as in Fig. 6.36. The reflector and transducer are at positions XR and XT, and port 1 is at x = 0. With transduction switched off, the transducer has P11 = j AR exp(- j2kxR), where j AR is the reflection coefficient of the reflector. For transduction, assume the "transducer" generates waves of amplitude j AT when unit voltage is applied. This gives P13 = j A r exp( - j kxr ). The center-frequency COM results are in Eq. (6.86), and equating with these gives (6.94) where Ar and AR are real. In the notation of Eq. (6.91), we have 0 = kxr + n / 2 and ¢ = 2kxR + rr /2. If xr - XR = A/ 8, this gives 20 -

0.5

for A ➔ 0, waist ➔ aperture .1 ➔ large, waist ➔ apex P .s 0.2 'A. in mode launcher; .s 0.5 "- in

output flare d = 'A.14

Figure 7.S Feed horns for coupling quasi-optical beams to waveguide and their equi vale nt Gaussian beam waist radii , waist locati on, and other design data.

396

QUASI-OPTICAL TECHNIQUES

to approximately 10% for horns a few wavelengths in diameter using a stepped-mode launcher. A dual-mode horn with a cylindrical output section is limited to an aperture diameter between ....., 1.3). and 1.9)., so that the TM 11 but not too many other modes will be excited. Larger-aperture horns can be made by adding a flared output section after the desired modes have been launched in the cylindrical section, but the phasing criterion results in a useful bandwidth restricted to a few percent. For dual-mode horns operating within the band where the pha ing condition is satisfied, the Gaussian waist radius is equal to 0.64a, where a is the aperture radius. Dual-mode horns work well only with small aperture phase error so that the phase center is near the horn aperture.

Corrugated Feed Horns. Corrugated, or scalar feed horns are based on propagation of the HE 11 hybrid mode which has an electric field distribution quite similar to that of the fu ndamental-mode Gaussian distribution [Eq. (7. 1)]. Consequently, the radiation pattern is highly Gaussian, with very low sidelobes. Other advantages include high polarization purity and low cross-polarized radiation. This type of feed horn requires a launcher section to launch the desired hybrid mode. The most widely used design consists of a series of slots tapering gradually in depth from ~)./ 2 to )./ 4. The flared section of the feed horn requires the presence of "-'). / 4 deep slots to present the proper reactive impedance to support the propagating hybrid mode. In all cases there must be at least two slots per wavelength along the axis of the horn. Scalar feed horns can be divided into two categories, depending on the aperture phase error [Eq. (7.9)]. Horns with small (6 ;S 0.3) aperture phase error have a radiation pattern essentially determined by diffraction from the aperture. The effective Gaussian beam waist radius is given in terms of the aperture radiu a and the aperture phase error 6 by the expression wo

= 0.64a / (1 + 6.776 2) 112

(7. 10)

Defining S as the location of the beam waist behind the horn aperture, expres ed as a fraction of the horn slant length Ls, we find that

S=

1 1 + (0.38/ 6 ) 2

(7 .11)

These two expressions allow the efficient u e of small-phase-error scalar feeds in Gaussian beam systems. Large-aperture phase-error scalar feed horns have 6 > 0.5 ; the large curvature of the wavefront in the horn aperture results in a radiation pattern which i essentially independent of frequency ; the angle of the - 17 dB relative power i equal to 00 , the half flare angle of the horn. The equivalent Gaus ian beam waist radiu is thus given by Wo

= 0.45 ()./00)

(7.12)

The location of the beam waist is located slightly in front of the feed-horn apex, about 12 % of the distance between the apex and aperture. Since the wai t radius of a low-phase-error scalar feedhorn is essentially independent of A, and that for a large-pha e-error horn proportional to )., it can be seen that the Gaussian beam properties of one can be transformed to those of the other using an optical system with input and output distances equal to lens focal length [Eq. (7 .4c)].

7.2

QUASl-OPTICAL 1NTERFACES TO PLANAR CIRCUITS

397

The required lens focal length i f = 0.91 a/ Bo, where a is the aperture radius of the small-pha e-error feed horn and Bo the half flare angle of the large-phase-error feed horn.

7.2

QUASI-OPTICAL INTERFACES TO PLANAR CIRCUITS

Mo t microwave circuit designs today u e planar media in preference to waveguide. The lower co t and often higher precision afforded by photolithograpby can make planar circuitry attractive even at millimeter wavelengths where waveguide has traditionally held way. Although a transition from a planar circuit to a quasi-optical system can always be made via waveguide and a horn antenna, a simpler solution is to fabricate an antenna directly onto the planar substrate, eliminating the loss and expen e a sociated with waveguide. In this section we treat a variety of planar antennas suitable for serving as interfaces between planar circuits and quasi-optical systems.

7.2.1 Resonant Antennas Resonant antennas depend for their operation on a more-or-less fixed relationship between the wavelength of the radiated energy and the physical dimensions of the structure. This means that their usable bandwidth is relatively small, on the order of I 0% or less of the designed center frequency. This is nevertheless sufficient for many applications. Figure 7 .6 summarizes the most common planar resonant antennas which have been used in quasi-optical sy stems to date. Under "type," an illustration showing the physical configuration also indicates the direction of radiated electric field polarization and critical dimensions in terms of guide wavelength Ag. This differs from the free-space wavelength AO by a factor that depends on the thickness and dielectric constant of the substrate. A first-order approximation for a semi-infinite substrate of relative dielectric constant Er is A

g

~

Ao ✓;:;:(E==r==+=;1::;:::::)/:;;:2

(7 .13)

A precaution to be observed with all planar antennas having main lobes perpendicular to the substrate is the avoidance of surface-wave excitation. In contrast to the desired wave radiating perpendicular to the substrate, surface waves propagate parallel to the substrate, rob power from the desired radiative mode, and cause high sidelobes via scattering and diffraction effects. Although the lowest-order TMo surface-wave mode has a cutoff frequency of zero, the next-highest surface-wave mode has a cutoff frequency of (7.14)

where c is the velocity of light in vacuum and t is the thickness of the grounded substrate. Operation below this cutoff frequency will tend to minimize, although not necessarily eUminate, surface-wave effects. The slot antennas described in Fig. 7 ,6 all radiate bidirectionally, with a slightly greater intensity on the side away from the ground plane. This property is usually

398

QUASI-OPTICAL TECHNlQUES Resonant planar antennas tor quasi-optical systems Feed method

Type

Comments

Microstrip patch

E

I,

_ J.. g

• Microstrip line ■ Probe from beneath ground plane

■

■

Balanced microstrip

• Medium bandwidth ■ Unequal E- and H-plane beamwidths

■

Balanced microstrip

■

Relatively narrow bandwidth for

lowVSWR ■

Unequal E- and H-plane beamwidths

2

-

...

Half-wave dipole

~ Full-wave dipole array

,

■

f,- J.. g

--=::::::::-

J.. g

-2

Medium to narrow bandwidth Nearly equal E- and H-plane beam widths

'::"

.

Half-wave slot • Microstrip probe • Slot line

• Bidirectional radiation • Unequal E- and H-plane beamwidths

-~ 2 No

7

Full-wave slot array • Coplanar waveguide • Bidirectional radiation ■

Nearly equal E- and H-plane beamwidths

Slot ring • Coplanar waveguide • Fairly wide bandwidth • Microstrip probe • Similar E- and H-plane beamwidths • Bidirectional radiation • Dual-mode feed permits two orthogonal polarizations from single structure

Figure 7.6

Resonant planar antennas for quasi-optical ystems.

undesirable, and most of the slot antennas shown can be made unidirectional by means of a cavity or reflector to suppress radiation on one side. A useful technique due to Rutledge is to place a planar antenna on the surface of a large dielectric lens. For slot antennas having the slot width much less than >..8 , Rutledge finds that the ratio of power Pa radiated into the air to power Pd radiated into a semi-infinite dielectric

7 .2

QUAST-OPTICAL INTERFACES TO PLA AR CIRCUITS

medium having a relative dielectric con tant of Pa I pd

Er

399

i given by

_- Er- 3/ 2

(7. 15)

For dipole , the relation i more comple ·, but mo t of the radiated power still travels into the dielectric medium, where a uitably curved surface will transfer it to the quasi-optical y tern with good efficiency.

7.2.2

onresonant Antennas

onresonant antennas are usually based on principles fir t outlined by Carrel for the design of frequency-independent antennas. A truly frequency-independent antenna would be infinite in size, but practical de ign can achieve usable bandwidths of several octave . The three type hown in Fig. 7.7 are suitable for planar fabrication using photolithography. The bow-tie and log-periodic antennas share with the slot antennas the property of bidirectionality and mu t be backed by a cavity or reflector if unidirectional operation is desired. The end-fire tapered slot antenna is somewhat different in that it i basically a traveling-wave device, showing a fair level of gain over a broad bandwidth. The dielectric plays a pas ive role in the first two designs, but is an e ential part of the wave-slowing proce s in the end-fire tapered slot.

Nonresonant planar antennas for quasi-optical systems Type

Feed method

Comments

Bowtie • Balanced open line

• Broad bandwidth ■ Modified dipole radiation pattern • Impedance adjustable with angle 6

• Balanced open line

• Broad bandwidth

• Slot line

• Yery broad bandwidth

• Microstrip probe

• Edge oriented endfire radiation • Relatively high gain, narrow beamwidth over 1-2 octaves

Log periodic

Tapered slot

Figure 7.7 Nonresonant planar antennas for quasi-optical systems.

400

7.3

7.3.1

QUASI-OPTICAL TECHNIQUES

ACTIVE QUASI-OPTICAL DEVICES

Detectors

Once energy has been transferred from a quasi-optical system to a conventional guidedwave medium, the signal processing operations of detecting or mixing can be performed with conventionally designed circuits. Rather than discuss such traditional approaches, in this section we describe quasi-optical detectors and mixers whose basic form is determined by their use in a quasi-optical system. As the examples will show, considerable simplification can result from taJ0 C

Q)

::l CT

Q)

Lt

Ideal case

Delay time

Figure 8.33

t

Linearity of a delay line.

the delay line. The delay time is referred to the differential delay time. For example, if a delay line operates from 1000 to 1500 MHz, the delay time at 1000 MHz is 1100 ns and at 1500 MHz is 1000 ns, the bandwidth is 500 MHz, and the differential delay time is 100 ns. Therefore, the time bandwidth is 500 MHz x 100 ns = 50. Note that the time-bandwidth product is a dimensionless number. In general, a higher time- bandwidth product is desired in a dispersive delay line. If a dispersive line is used in a compressive receiver , the bandwidth of the delay line is directly related to the input bandwidth of the receiver and the differential delay time is related to the frequency resolution of the receiver. The linearity of a dispersive delay line is measured by 8// 8t or the derivative of frequency with respect to time. Figure 8.33 shows the linearity of a delay line. If the 8// 8t is a constant, it is easier to match the oscillator to the delay line. The insertion loss of the delay line should be kept at a minimum. To keep the sensitivity of the receiver constant, the insertion loss of the delay line should also be constant across the operating band. The spurious output in the time domain (i.e., the triple transit) should be minimized in order to have a higher dynamic range. Electromagnetic (EM) dispersive delay lines, also referred to as meander lines because of their structure, were popularly used in compressive receivers before the development of SAW dispersive delay .lines. Meander lines are bulky and difficult to fabricate and probably are not suitable for mass production. Thus they have generally been replaced by SAW dispersive delay lines. Due to the recent development in superconductivity, a superconducting dispersive delay line is potentially applicable to compressive receivers. In the next two sections we discuss SAW and superconducting dispersive delay lines.

8.6.2 SAW Dispersive Delay Lines [70-78] The operating principle of a SAW diseersive delay line is the same as that of a SAW filter. The input electric signal is changed into an acoustic wave at the input transducer

462

COMPONENTS FOR SURVEILLANCE AND ELECTRONIC WARFARE RECEIVERS

and changed back into electric signal at the output transducer. In the acoustic-wave mode, the wave travels dispersively; that is, the low-frequency wave travels a longer distance than the high-frequency wave. In other words, the lower frequency has a longer delay than the high frequency. This arrangement can minimize the variation of insertion loss across the band because the high insertion loss at high frequency can be offset by the short path, which causes less loss. Simple frequency-dependent transducers can provide the dispersive delay as shown in Fig. 8.34a and b because the high frequency will be excited at the tips of the transducers which are closer. Thus the high frequency travels a shorter distance than the low frequency which is excited at the far ends of the transducers. Frequency-dependent transducers with frequencydependent reflecting gratings as shown in Fig. 8.34c can also produce a frequency

(a)

Dispersive grating

(b)

(c)

High-frequency path Dispersive grating

Input )

Output)

n.-.&---u

o-,.f~Llu

Figure 8.34 Dis~rsiv~ delay lines: (a) with tilted chirp transducers; (b) with straight chirp transducers; (c) with chirp transducers and reflecting gratings. (From Ref. 1.)

8.6

COMPRESSIVE RECETVERS (60- 69]

463

versu time delay. The paths of the high and low frequencies are shown in Fig. 8.34c . The grating can be either metal trip depo ited on the ubstrate or grooves etched in the ubstrate. Although the arrangement in Fig. 8.34c i more complicated to fabricate, thi i a more desirable approach becau e the added control makes it easier to attain the performance desired. The SAW de ices u ually operate under 2 GHz becau e insertion loss increases with increa ing frequency. Another limitation on the operating frequency of SAW devices is that at high frequency the di tance between fingers of the transducer is very close. It is difficult to fabricate ubmicron- ized transducers with high yield. As a result, the SAW di persi e delay lines often have a narrow bandwidth, usually under 1 GHz. The

Output

•

Amplifier

SIN Delay line

enhancer

r Amplifier

Figure 8.50 Microwave memory loop.

At a low input signal level, most of the power is coupled into the magnetostatic wave, which travels to the edge of the film and is converted into heat When the input is above a certain level (in the range 0.1 to 10 mW), the magnetostatic wave starts to saturate and less power will be absorbed by the YIG film . As a result, less power is taken away from the microstrip line and there is a relatively low percentage of power loss. The only known potential use of a signal-to-noise enhancer is in a microwave memory loop, as shown in Fig. 8.50. In this circuit the input microwave signal is amplified, delayed, and again fed into the amplifier. Therefore, the microwave signal, usually a pulsed signal, will be stored in the memory loop. In this arrangement the noise in the circuit will be amplified and finally reaches a very high level to mask the signal. The signal-to-noise enhancer in the circuit will keep the signal above the noise level by suppressing the low-level signals.

8.11

DIGITAL ELECTRONIC WARFARE (EW) RECEIVERS (113-118]

With the continuing performance improvements in analog-to-digital converters (ADCs) and signal processing, it appears that all receivers in the future will be implemented in

8. 11

DIGITAL ELECTRO IC WARFARE (EW) RECEIVERS [1 13- 118]

477

digital technology. The major ad antage of a digital receiver are better performance, versatility of de ign, and ease of maintenance. Once the input analog signal i converted into a digital ignal, there are no ignificant performance variation due to temperature drift or aging problems. Al o, digital recei er normally have ignificant advantage in co t, ize. and weight. It i predicted that future narrowband receiver will be implemented through oftware algorithm on digital hardware. Thi wiJJ enable changing the function of a receiver by imply changing the receiver oftware program . For example, to change from an amplitude modulation (.At\ 1) receiver to a frequency modulation (FM) receiver. one can u e the .. ame hardware and imply change the oftware. It has been demon crated in the laboratory that it i po ible to build a wideband receiver with digital cheme . The major difference between analog and digital receivers are hown in Figure .51 . In an analog receiver, the input -ignal pa ses through the RF down converter and RF ection and i converted into a video signal through crystal ,rideo detectors. The video ignal i digitized and further processed to obtain the pul e de criptor words for measured signal characteri tics such as frequency, pulse width, amplitude, and o on. Becau e the video ignal i narrow band, the performance requirements on the ADC are not tringent In a digital receiver, the input ignal i converted in the RF down converter to an intermediate frequency (IF) and an ADC is u ed to digitize the IF signal As the IF normally has very wide bandwidth, the requirements on the ADC are very stringent. The ADC will be further di cus ed in the following sections. Toe spectrum e timator i used to find the frequency information on the input signal, which is one of the most important signal orting parameters generated by an EW receiver. There are many different approaches to obtain frequency information. The mo t common approach is through use of the Fast Fourier Transform (FFI'). Although many modem e timation method can provide better results than the FFf, they are usually computationally inten ive. Also, because EW receivers must operate in real

\/

Video signal

IF

Digital \ words 1

~

-RF section

RF converter

! :

Parameter encoder

!

Digital processor

Analog EW receiver

\I

IF

I

I

Digitized data

I

• RF converter

ADCs

--

o-1g1t1 ·· zed data

-I

Spectrum estimator !,...___ _ _ _ __

D'191'tal l words i

Parameter encoder

il •

l

iI

!

iI

i _ _ _ _ _ _ _ _ _.....;i !

Digital EW receiver

Figure 8.Sl

Comparison of analog and digital receivers.

..

Digital processor

478

COMPONENTS FOR SURVEILLANCE AND ELECTRONIC WARFARE RECEIVERS

time and the ADC sampling frequency is in the gigahertz range, the performance of modem spectrum estimators is normally inadequate except for special applications. In most EW receivers, the frequency estimation still uses FFf or FFf-related operations. In the design of both analog and digital EW receivers, the parameter encoder is one of the most demanding components in a receiver design. It must determine the correct number of signals without reporting false signals, and it must generate the correct pulse descriptor words on simultaneous signals from the output of the spectrum estimator. The design of the encoder is very unique for each receiver and, therefore, will not be discussed.

8.11.1

ADC

The performance of a receiver depends on the receiver design and the performance of the ADC. The receiver performance cannot surpass the performance of the ADC used in the receiver. The ADC sampling frequency is closely related to the bandwidth of the receiver. If the IF bandwidth is ~B, the Nyquist sampling frequency is fs

= 2~B

(8.66)

In general, the sampling frequency used in a receiver is about 2 .5~B. If the bandwidth of the receiver is 1 GHz, the minimum desired sampling frequency is 2.5 GHz. This requirement constrains the EW receiver applications to only high-frequency ADCs. Not only must the ADC operate at a high sampling frequency, but the input signal frequency of the ADC must also be high. In some receiver designs, it is desirable to have the IF bandwidth be less than an octave in width. For example, with a sampling frequency of 2.5 GHz, the maximum input bandwidth is 1.25 GHz, but the IF maybe placed in the 1.25 to 2.5 GHz range (rather than in the O to 1.25 GHz range). This requirement that the input frequency be higher than Is/2 places difficult demands on the ADC performance. The dynamic range of the receiver is directly related to the amplitude resolution provided by the ADC. The amplitude resolution of an ADC is the number of steps into which the input signal range is divided. The resolution is usually expressed as bits (b), and the number of steps is 2 to the power of b. The dynamic range (DR) of the receiver can be written as DR ~ 6.02b dB. (8.67) Using this equation, one can see that an 8-bit ADC can provide a dynamic range of approximately 48 dB. Another important ADC performance characteristic is the maximum signal-to-noise ratio that can be achieved, which can be expressed as (SN R)max

= 1.76 + 6.02b dB

(8.68)

The effective number of bits (beff) is usually less or equal to the number of bits of the ADC. Equation 8.68 can be used to find the number of effective bits as b

_ (SN R)dB - 1.76 ~ff 6.02

where (SN R)dB is the maximum signal-to-noise ratio and a measured quantity.

(8.69)

8. 11

DIGJTAL ELECTRONlC WARFARE ( EW) RECEIVERS l 113- 11 8]

479

The output ignal frequency domain of a n ADC contain many puriou. ignal (or pur~) that mu t be con idered in the de ign of the receiver. It i rea, onable to include all the puriou ignal in obtajning the number of effective bit . Thus, the above equation can be modified a b eff

_ (SI AD )dB - 1.76 6.02

(8 .70)

where (SJ 'AD )d 8 i the ignal-to-noi e and di tortion ratio at the output of the ADC. Figure 8.52 how the re ul ts of ome mea urement of the FFf output of an 8-bit ADC. The ADC ampling ignal frequency i at 2801.664 MHz with a power level of 8.5 dBm. The input ignal (FJ) has a frequency of 2553.543 MHz with a power level of -3.6 dBm. The magnitude of the output i plotted a a function of the baseband frequency in MHz. The cale of the ordinate i labeled "'dBFS ," where dBFS i ..dB below full cale .. , ote that the "0'' alue of the ordinate is et to the peak of the input ignal, Fl . The FFf length i 16384 (2 14 ) points. The FFf length for thi measurement i much longer than the FFf length normal1y e mployed in an actual receiver, approximately 256 points, in order to be compatible with the minimum pul e width anticipated. As the input ignal i a real value (as opposed to a complex value), there are only 8192 (16384/2) frequency bin . With 8 bits, the calculated (SNR )d B from Equation (8.6 ) i 49.9 dB ( 1.76 + 6.02 x 8). If the calculated (SNR)dB is referenced to a single frequency bin (49 .9 + 10log8192), the resulting (SNR)dB is 89.1 dB. ote that the -49.9 dB and -89.2 dB value are indicated on the Magnitude scale in Figure 8.52. In Figure 8.52, the nine largest signals are labeled 1- 9 and are arbitrarily

Single-tone power spectrum

TRW 8-bit ADC (MCM CCA), F1 = 2553.543 MHz @ - 3.6 dBm Clk = 2801.664 [email protected] dBm, 08-05-1997, 13:24:48, Avg code -= 127.3, FFT size = 16384

0

SFDR = 57.5 dB, SNR = 43.2 dB, SINAD = 42.3 dB

- 10 - 20 •

en

u..

- 30 -

CD

-40

a> 'O ::,

- 50

cC>

-60 -

'O

...

SNRq = -49.9 dB ---~------:----------------, 1 2 4 3 8 5 7 6 9

I is the loss factor of the antenna* [see Eq. (9.8)); TJ P < l is a polarization mismatch factor; and TJ N < I is an impedance mismatch factor. An explicit expression for TJp can be found in Ref. 3; TJN is given by

TJN

=

4Re(ZR)Re(ZA) IZR + ZAl 2

(9.14)

where ZA and ZR are the impedances of antenna and receiver, respectively. If an impedance-transforming network is used as indicated in Fig. 9.2, then Z A in Eq. (9.14) must be replaced by the antenna impedance after transformation (i.e., as it is seen at the input port of the receiver).

Point-to-Point Transmission Equation and Radar Range Equation. The effective area is useful for calculating the power transfered between the two antennas of a free-space radio link. Usually, a radio link will operate under far-field conditions, the antennas will be pointed at each other, and requirements (a)-(c) for optimum reception will be satisfied in reasonable approximation. Under these conditions, the received power is with Eqs. (9.11) and (9.12) given by (9.1 5a) where SP is the absolute value of the Poynting vector of the transmitted field at the location of the receive antenna. Evidently,

.

(9~15b)

* In Eqs. (9.12) and (9.13), GA and

L ,.. are assumed to include all conduction and dielectric losses of the

antenna and the dissipation losses of the impedance-matching network; but they do not include mismatch losses (which are accounted for by T/N ).

498

ANTENNAS I: FUNDAMENTALS AND NUMERICAL METHODS

where Ptr is the transmitted power and d is the distance between the antennas; G, and Gr are the gains of the transmit and receive antennas, respectively. Combining Eqs. (9.15a) and (9.15b) yields the ratio of the received and transmitted power: (9.16) By a similar calculation one obtains for the radar return of a target with the radar cross section O"rct P ree

Ptr

(9. 17)

which is the well-known radar range equation. In this equation, d is the distance of the target from the radar antenna, and the radar cross section crrd is defined by the equation S~ec = O"rctS;/ 4rrd 2 (9.18) where ~ is the Poynting vector of the transmitted radar beam at the location of the target, and s~ec is the Poynting vector of the field, scattered by the target, at the location of the receive antenna. Equation (9 .17) holds for both monostatic and bistatic radar operation, the difference between the two modes of operation taken into account through O"rct and its dependence on the directions, as seen from the target, in which the transmit and receive antennas are located. Bandwidth. The bandwidth of antennas is difficult to define in general terms. Most antenna characteristics, including gain, beamwidth, sidelobe level, polarization, and impedance, are functions of frequency, and the variation of each of these parameters may limit the useful frequency band of an antenna designed for a specific purpose. Frequently used measures of antenna bandwidth are the impedance bandwidth, which indicates the frequency band over which the SWR at the circuit port of the antenna remains below a given value, and the pattern bandwidth, which defines the frequency range in which the pattern properties (beamwidth, sidelobe level, polarization) remain within given tolerances. But it is not uncommon that the lower limit of the frequency band is determined by one antenna parameter and the upper limit by a different one. The only appropriate definition of antenna bandwidth seems to be the one given by Jasik as "that frequency range within which the antenna meets a given set of specifications" [1]. Hertz Dipole. The most elementary radiating structure is the Hertz dipole, which can be described as an oscillating dipole of charge Q and length l, where l is infinitesimally small and Q is large, so that the dipole moment p = Ql is a finite quantity. Alternatively, a Hertz dipole may be regarded as a current element of length land current / = j w Q. Dipo]e antennas are not widely used at microwave frequencies, but the Hertz dipole is discussed here since its field has basic properties which are common to all antennas and since the field radiated by any given antenna-as soon as its current distribution is known - can be obtained as a superposition of the fields radiated by its current elements, that is, it can be obtained as an integral over a distribution of local Hertz dipole sources. This is reflected, for example, in Eq. (9.3).

9.1

FUNDAMENTALS

499

z

I I

r

I I I I I

~--------:-'-----,---►Y I I

' ''

/

/

II I

', ',

',

//

/

/

I / I//

------ - - - - - - - - - __ .::-.a¥ X

Figure 9 .4

Hertz dipole of moment p

= pez: coordinate systems and field strength components.

Assume that a Hertz dipole of moment p = pe, is located at the origin of a spherical coordinate system r , B, > 1, the first terms in Eqs. (9 .19) dominate and we have

Ee =

H¢,

w2µo

e~kr.

= - -4n- p -r - smB

for

kr

>>

1

(9.20)

while E,, which~is proportional to (kr)- 2 , is negligible. Thus, in the far-field region, both E and H have the r-dependence e-:ikr /r and form an outward-traveling spherical wave. Furthermore, E and H are directed normal to each other and to the direction

500

ANTENNAS I: FUNDAMENTALS AND NUMERICAL METHODS

of propagation (so that the Poynting vector has the radial direction away from the antenna). Finally, E and H are in phase and their amplitudes are related by the freespace wave impedance Zo = (µo / Eo) 112 = 120rrn. These properties apply to the far field of any antenna, regardless of its configuration. In the specific case of a Hertz dipole, the far field is linearly polarized, and the radiation pattern is proportional to sin 0 and uniform in phase. The radiated power is given by

and for the power radiation pattern one obtains, with Eq. (9.5), G(0)

= ~ sin2 0

The pattern is omnidirectional, with maximum radiation occurring in the azimuth plane and zero radiation in the upward and downward directions. The half-power beamwidth in the elevation plan is 90° and the directivity gain is 1.5.

Aperture Antennas; Near Field and Far Field. Microwave antennas are usually designed to provide high directivity gain and narrow beamwidth. Most of these antennas belong to the class of aperture antennas which includes well-known radiating structures such as reflector, lens, horn, and planar array antennas. For these antennas an ap~rture plane can be defined in which the tangential electric or magnetic field strength distribution is known or can be determined in good approximation. Typically, the aperture plane coincides with, or is very close to, the physical aperture of the antenna, and the field strength distribution in this plane is significantly different from zero over a finite area only, while outside this area the field strength is small and often can be neglected. In the following, we assume that the aperture plane is the plane z = 0 (see Fig. 9.5) and that the antenna structure is to the left of this plane (i.e., in the space range z < 0). The field distribution in the space range z > 0 can then be determined from the relations

E,(x, y, z)

=

ff Ex(x' ,

y',O)g(x -x' , y-y', z)dx' dy'

(9.21a)

SA

Ey(x , y, z)

=ff E (x', y' , O)g(x - x' , y - y', z) dx' dy' 1

for

z> 0

(9.21b)

SA

where Ex(x' , y' , 0) , Ey(x' , y', 0) is the known tangential electric field strength distribution in the plane z = 0 and

g(x - x' , y -y' , z)

= - 2~ :z

e:ltR)

R

= [(x _

x')2

+ (y

_ y')2

+ z2]1/2

(9.22) is the normal derivative of the scalar Green's function of this plane. The integration is performed over the area SA of the aperture plane where Ex and Ey are significantly

9.1

501

FUNDAMENTALS

y'

--- -- -o

0

I

I

- -- - - - - ~I!- - -- -- -

-

_

cf>

___,► - z

I

I I

~0~

0~,s I

I

I : ,. z =0

--

r, 0

I

I l~'l> O can be found directly from Maxwell 's equations. If the tangential magnetic (rather than electric) field strength is known in the aperture plane, Ex and E y on both sides of Eqs. (9.21) are replaced by Hx and Hy, respectively. For highly directive antennas, two regions can be distinguished in the space range z > 0, the near-field or Fresnel region and the far-field or Fraunhofer region; see Fig. 9.5.* In the Fresnel region, the cross-sectional amplitude and phase distributions of the radiated beam do not vary strongly as the beam travels in the z-direction, t and the beam diameter increases slowly. In the far-field region, on the other hand, the antenna beam shows the typical diffractional beam spread and the usual far-field behavior as discussed in connection with the field of a Hertz dipole. The transition between the two regions is not precisely defined but occurs at a distance from the antenna of about z = 2D2 / Ao, where D is the diameter of the antenna aperture or., * The Fresnel region is sometimes referred to as the radiating near-field regfon to distinguish it from the reactwe near-field region, where the stored electric and magnetic fieJd energies differ significantly. A strong _reactive near-field region is observed. for example, for dipole and loop antennas. In the case of an aperture antenna it may exist about the edge of the aperture but is limited in range and usually not very pronounced. We assume here that the field distribution in the aperture does not vary rapidly, with the possible exception of a linear phase progression in the case of a s~rable beam antenna. t We assume here that the main-beam axis of the antenna is the z-axis.

502

ANTENNAS I: FUNDAMENTALS AND NUMERlCAL METHODS

more general, an effective linear dimension. Evidently, for antennas of large diameter, the Fresnel region can extend over many wavelengths. In the near-field region, Eqs. (9.21) and (9.22) can be simplified by introducing the Fresnel approximation

R

~ { zz + -1 (x---- -+-(y - y')2 -x ' )2

z

2

Thus

g (x - x , , y - y , , z)

~

z) =

in phase terms

_ -j/c . -Jk -e ~ exp { - -J -k [ (x - x ,)2 + (y - y ')2]} 2rr z 2z

~r

and Ex,y (x, y,

in amplitude terms

j'k

2

rr

e -jkz

-z-

I

(9.23)

(9 .24)

I

Ex,y(x , y, 0)

SA

~

• exp {-; [(x - x')2

+ (y - y'}2]}

dx' d y'

(9.25)

The Fresnel approximation (9.23) was originally introduced in optics to describe the field near the beam axis (z-axis) at large distances from the radiating aperture (i.e., for z >> D ). The validity of Eq. (9.23) for this region is evident. That this approximation also leads to a reasonably accurate description of the field in the region 0 ~ z < 2D2 /'Ao (for arbitrary x and y ) is not immediately obvious. But it can be shown [4] that a valid approximation is indeed obtained under the condition that the plane-wave representation of the radiated field consists of a narrow directional spectrum of propagating waves centered about the main-beam direction (i.e., provided that the main beam has a small beamwidth and that sidelobe amplitudes decrease rapidly away from the axis). In the far field region, the Fraunhofer approximation can be used. that is

R

~ { rr -xx'- +--yy' r

in amplitude terms

in phase terms

(9.26)

where

is the radial distance from the center of the antenna aperture. Since in the far field r > > lx'I, IY'I, the approximation is obviously valid. Equations (9.21) and (9.22) then reduce to

jk e -jkr ( xx' + yy' ) z g(x-x' ,y-y' ,z) ~ ---exp j k - - - 21r r r r

(9.27)

and (9.28a)

9.1 FUNDAMENTALS

503

(9.28b)

with

\119(0 , )

= + 2jkrr 11{{ [Ex(x ' , y' , 0) co

+ Ey(x' , y' , 0)

in]

s~

x e pLJk(x ' co + y' in) in 0] dx' d y' "1¢(0. ¢ )

= - ~: co

0

ff

(9.29a)

[E, (x' , y' , 0) sin¢ - E, (x ' , y' , 0) co ¢]

s" x exp Uk (x ' cos + y' in) in 0] dx' d y'

(9 .29b)

We ha e introduced here the pherical coordinates r, 0 , indicated in Fig. 9 .5. The functions \P8 and '119 are the 0- and -components of the field strength radiation pattern of the antenna. ote that the radiation pattern is related to the aperture illumination E x(x ' , y' , 0) , E> (x' . y' , 0) by a spatial Fourier transform: Eq. (9.29) represent the pattern functions '118 and Wq, as the k-space spectrum of the aperture distribution functions Ex and Ey- It is evident from this relationship that a large antenna is required to produce a pattern of narrow beamwidth, while a small antenna will radiate a broad beam. With Eqs. (9 .5), the power radiation pattern of the antenna is obtained as G ( , ) 0

=

21r

2

rr

4rr[l'lle(0, ) 1

!. !. [

Iwe(0,

q,=0

2

) 1

+ l'11q, (0, )1 2]

(9.30)

+ I'II4> ( 0 , ) 12 ] sin 0 d 0 d

B=O

and the directivity gain may be written in the form* 2

ff + ff k Vfii_ ;;,----------~ Re[!! dy'] E,(x', y', 0)dx' dy'

s,.

2

E 1 (x', y' , 0) dx' d y'

2

GD = G (0) =

/.Lo

:tr

s,.

(9.3 la)

(ExH; - E 1 H;),=-0 dx'

• The approximate expression (9.31b) for G O is obtained by assuming that the aperture SA has dimension in the order of a few wavelengths or more and that the tangential electric and magnetic _fields in the aperture are related as in a plane TEM wave. In the integral in the denominator of expression (9.3 la) one may then approximate (ExH; - EyH;),=0 ~ .Jfo/µo(ExE; + EyE;)z=O, which leads to expression (9.31b).

504

ANTENNAS I: FUNDAMENTALS AND NUMERICAL METHODS

ff

2

Ex dx' d y'

4rr s,..

+

ff

2

E 1 dx' d y'

sA

ff (1 Ex1 + IE, 1 2

2

)

(9.31b) dx' d y'

SA

where the expression in the denominator of Eq. (9.31a) is the total real power transmitted through the ~tenna aperture SA. As before, we are assuming that the mainbeam axis has the direction normal to the aperture plane. Table 9. l lists pattern data for several different aperture illuminations in normalized form [1]. It is assumed that the antenna has a rectangular aperture of area 2D1 x 2D2 and that only one field component, say Ex, is different from zero, so that the aperture field is linearly polarized. Furthermore, it is assumed that the aperture field is separable in x and y. Thus

.!::!

Qi~

I

(9.32)

D1' D2 "'

elsewhere The radiation pattern \Jl8 , \Jlcp then can be written in the form jk

\Jle(0, )

= +-\Jl1(u1)\Jl2(u2) cos¢ 2rr

\Jlcp(0, ¢ )

= _jk'l11 (u 1)\Jl2(u2)sin¢cos0 2rr

with u1

= kD1 cos¢sin0

(9.33a)

u2

= k D2 sin sin 0

(9 .33b)

where with i = 1, 2,

ti= x/ D1 ,

~2

= y/ D2

(9.34) In the table the pattern function \Jl;(u;) is listed for a number of illumination functions E i(~;) (the subscript i is suppressed). In addition, the beamwidth, sidelobe level, and directivity gain associated with these pattern functions are shown. The gain is expressed in terms of a gain factor a : If a 1 is associated with the illumination function E 1 and a2 with E2, the directivity gain of an aperture antenna with the illumination (9.32) is (9.35) where A= 4D 1 D2 is the aperture area. The a-values of Table 9.1 are approximations in accordance with Eq. (9.3 lb), which hold for sufficiently large apertures with D1•2 in the order of a few wavelength or more. For an antenna with a square aperture, the E- and H-plane beamwidths are typically 50 to 8OAo/ ../A (in degrees), depending on the aperture illumination. For a circular aperture of the same area and of similar illumination, the beamwidth is moderately smaller and the directivity gain somewhat higher.

Normal Gain and Supergain Antennas. The question may be asked: What is the maximum directivity gain that can be obtained with an aperture antenna of given size?

9.1 FUNDAMENTALS TABLE 9.1

Pattern Data for Rectangular Aperture Antennas

Aperture Di tribution, E (~)(- 1 ~; ~ + l )

Radiation Pattern Function. \ll (u )

E (~)

0

+1

= 1-

1/ ~

0

E (t)

Directivity Gain Factor, a

25.4~ D

13.2

1.0

27.8>-o D

17.1

0.97

33.0Ao

20.6

0.83

23

0.8 1

32

0.67

Half-Power Beamwidth in deg

2

2D (1

-1

Intensi ty of Fir t Sidelobe, (dB below Main Beam)

A

2D mu u

-1

+

d i-, ) sin u du u

D

+1

= l - t2 1'C

D

cosu ,

(n:/ 2) - - u

2

34.4 .l.o

D

41.6),.0

D -1 E ($)

505

0

+1

= cos2 -1'2C$

Source : After Jasik [1]. © 1961 McGraw-Hill. Reprinted with permission of McGraw-Hill.

The answer is that, in the mathematical sense, the problem doe_s not have a solution. In theory, an arbitrarily high gain can be achieved with an arbitrarily small antenna. Such antennas are called supergain antennas and, while in principle they are feasible, they are not useful as practical antennas. Supergain antennas may be defined simply as antennas whose directivity gain exceeds the maximum normal gain. In the case of a normal gain antenna the amplitude and phase of the aperture illumination do not vary rapidly across the aperture and the radiated field maintains a degree of uniformity throughout the Fresnel region

506

ANTENNAS 1: FUNDAMENTALS AND NUMERICAL METHODS

of the antenna. Maximum normal gain occurs when both the amplitude and phase of the aperture field are constants that do not vary across the aperture. The mainbeam direction in this case is the forward direction normal to the aperture plane (z-axis) and, because of the uniform phase distribution, all surface elements of the aperture contribute constructively to the field in this direction. The maximum normal gain can be calculated from Eq. (9.31b). One obtains (9.36) where A is the aperture area. Comparison with Eq. (9.12) shows that the effective receiving aperture Ae of an antenna of maximum normal gain equals its geometrical aperture area. In the case of a supergain antenna, both the amplitude and phase of the aperture illumination vary significantly across the aperture and destructive interference occurs in all directions, including the mainbeam direction. But if the amplitude and phase distribution are chosen appropriately, this destructive interference will have a much smaller effect on the radiation in the forward direction than on that in other directions, with the net result that the directivity gain of the antenna is increased. Supergain antennas, however, are not practical, for several interrelated reasons: (a) The field distribution in the aperture is very complicated and is very difficult to realize. But it is critical; small deviations from the exact aperture illumination will destroy the supergain. (b) The energy stored in the near field is very high Harrington [5] has estimated that achieving a supergain of 3 dB above maximum normal gain will require the antenna Q to be as high* as 1()5 to 106 . Very large field strength will occur in the reactive near-field region.

(c) The bandwidth is very small

(!:J.///o = Q- 1).

On the other hand, one should not overlook that certain receiving array antennas, where the signals received by the individual elements are processed to a degree before they are combined to the output signal, operate in effect as supergain antennas. But as transmit antennas, such arrays would not be useful. Antennas of maximum normal gain have one disadvantage; their sidelobe level is high. Table 9.1 shows that an aperture illumination of uniform amplitude and phase leads to a radiation pattern with first sidelobes that are down from the main-beam level by only 13.2 dB, which would not be advantageous for many applications. But this problem can be corrected easily by changing to a tapered amplitude distribution which decreases smoothly from the center toward the rim of the aperture. (The phase distribution is maintained uniform across the aperture to ensure constructive interference in the * In Ref. 5 the antenna Q is defined as for We> Wm for Wm> Wl'

where We and Wm are the stored electric and magnetic near-field energies, respectively, and prod is the radiated power.

REFERE1 CES

507

forward direction.) A hown in Table 9.1, a quadratic amplitude distribution reduce the idelobe level to 20.6 dB and a co ine- quared taper re ult in fir t idelobe that are 32 dB below the main-beam maximum. The e re ults appl y to rectanguJar aperture ~ for circular aperture v ith imilar illumination taper , the idelobe level would be reduced omewhat funher. The price to pay i a moderate decrea e in directivity gain in the order of 2 to 3 dB or, if the gain i to be maintained, the aperture area mu t be somewhat increa ed. The directivity gain of uch antennas can be written as

where 77 i the so-called aperture efficiency, which for well-de igned antennas i typically 60 to 90%. depending on the de ired idelobe level.

Antenna literature. For further information on microwave antennas, the reader i referred to the e cellent new antenna books published in recent year by Elliott [2], Stutzman and Thiele [6], Balani [7]. and Blake [8]. An in-depth pre entation of antenna theory can be found in the two-volume text edited by Collin and Zucker [9], and an exten ive treatment of phased-array antenna in the three-volume text by Hansen [10]. In addition, three comprehensive and very well written handbooks on antenna theory and design have recently appeared edited by Rudge et al. [11], Johnson and Ja ik (12] , and Lo and Lee [13]. The e books al o provide an exten ive bibliography on antennas and related ubjects.

REFERE CES 1. H . Jasik,Ed.AntennaEngineering Handbook , McGraw-Hill,NewYork, 1961 , Chaps, 1, 2, 31.

2. R S . Elliott, Antenna Theory and Design , Prentice-Hall, Englewood Cliffs, J. 1981. 3. Reference Data f or Radio Engineers, 6th ed, Howard W. Sams, Indianapolis, IN, 1975, Chap. 27. 4. F. Schwering, "On the Range of Validity of Fresnel-Kirchhoffs Approximation Formula;• IRE Trans. Antennas Propag., AP-10, January 1962. 5. R. F. Harrington, A. T. Villeneuve, and M. K Hu, Antenna Research , Tech. Rep. No. EE6 l 9593P2, Syracuse University Research Institute, Syracuse, NY, March 1959. 6. W. L. Stutzman and G. A. Thiele, Antenna Theory and Design, Wiley, New York, 1981. 7. C. A. Balanis. Antenna Theory, Analysis and Design, Harper & Row, New York, 1982. 8. L. V. Blake., Antennas. Artecb House, Norwood, MA, 1984. 9. R. E. Collin and F. J. Zucker, Antenna Theory , Parts 1 and 2, McGraw-Hill, New York. 1969. 10. R. C. Hansen, Microwave Scanning Antennas, Vol. 1, Academic Press, New York, 1964; Vols. 2 and 3, 1966. 11. A. W. Rudge, K . Milne, A. D . Olver, andP. Knight, Ed., The Handbook ofAntenna Design, Peter Peregrinus, London, 1982. 12. R. C. Johnson and H. Jas~ Ed, Antenna Engineering Handbook, 2nd ed., McGraw-Hill, New York, 1984. 13. Y . T. Lo and S. W. Lee, Ed., Handbook of Antenna Theory and Design , Van Nostrand Reinhold, New York, 1988.

508

9.2

ANTENNAS I: FUNDAMENTALS AND NUMERICAL METHODS

NUMERICAL TECHNIQUES

A. W. GussoN AND M . A. M ORGAN

9.2.1

Introduction

A wide variety of numerical techniques are available for the analysis of antenna problems. These methods may generally be classified as either time-domain or frequencydomain methods, and these may be further grouped under the broad headings of integral equation methods, differential equation methods, and ray tracing methods. The choice of a particular analysis technique is often dictated by the form of the antenna, the antenna environment, and the frequency range of interest. In this section, we consider the fundamental aspects of frequency-domain numerical solution techniques. To obtain solutions for all frequency ranges and antenna configurations of interes~ it may be necessary to use a combination of techniques. For antennas whose dimensions are on the order of the wavelength, the method of moments (MM) [ 1] is a well-known frequency domain solution procedure. The MM is usually considered to be an integral equation technique that can be used to model electromagnetic radiation and scattering problems involving conducting wires and surfaces as well as homogeneous and inhomogeneous material bodies. Application of the method involves the solution of a matrix equation comprising a "full" matrix in integral equation formulations. It is therefore a computationally intensive and storageintensive method when large numbers of unknowns are required in the numerical model. Differential equation-based methods are usually employed for electromagnetic computations involving penetrable inhomogeneous material structures, where numerical solutions are obtained through the use of the "finite methodsu (i.e., finite elements and finite differences) [2]. Finite methods usually produce very sparse system matrice , resulting in significant reductions of needed memory and computation time vis-a-vi the full matrix inversion embodied in volume integral equation formulations. For given computational constraints of time and memory, the finite methods thus have the potential to solve relatively larger volume electromagnetic problems, having characteristic dimensions up to several wavelengths. Matrix solution techniques developed from integral equation and differential equation formulations become impractical when the antenna is large in terms of wavelengths, although their useful region continues to expand with improvements in computer hardware and the advent of the so-called "fast methods" (3, 4]. An often used method for the analysis of large antenna structures is the geometrical theory of diffraction (GTD) [5]. GTD is a ray tracing technique which includes the direct and reflected ray contributions of the geometrical optics field as well as diffracted ray contributions generated by scatterer surface di continuities and curved surface . Time-domain solution technique provide an alternative to the frequency-domain methods. The finite-difference time-domain method (FDTD) is a widely used marchingin-time solution to the differential form of Maxwell's equations [6, 7]. The transmissionline modeling method (TLM) (8, 9) and finite element time domain methods (FETD) are also available [10]. Time domain integral equation solution methods have also been studied extensively [ 11], and new approaches to increasing the efficiency of time domain integral equation solutions have been developed [12, 13]. The basic concepts nece sary for the application of the different frequency-domain methods are presented in this section. Many details, as well as the relevant theoretical

9.2

NUMERICAL TECHNIQUES

509

discussion . are omitted in the interest of providing a wide range of application information for different type of antenna configurations. For additional details the reader i referred to Ref . I, 2, and 5, the reference located therein, and to the other specific reference provided in thi ection. For the application of time-domain solution techniques. the reader i referred to [6-13].

9.2.2

Integral Equation Approach

Method of i\1oments. The method of moments is a widely u ed technique for modeling variou types of antenna and catterer whose maximum dimensions are on the order of the wavelength at the frequency of interest. It is usually applied to integral equation formulations for radiation and scattering problems, but it may also be applied to differential equation formulations. The application of the MM in integral equation formulations can generally be summarized by the procedural steps indicated below. 1. Obtain an equivalent problem in which the physical structures are replaced by equi alent source (who e values are to be determined) radiating in an infinite homogeneous region or a region for which the Green's function is analytically known. 2. Apply boundary conditions appropriate to the problem (e.g., total electric field tangential to a perfectly conducting surface is zero) to obtain an equation which the equivalent sources must satisfy. 3. Di cretize the domain over which the equivalent sources exist (e.g., the antenna surface) and expand the equivalent sources in terms of basis functions defined on the discretized domain. The coefficients of the basis functions are the unknown quantities to be determined. 4. Enforce the boundary condition equation from step 2 in some sense over the various regions of the discretized structure to obtain a system of simultaneous equations. This is often referred to as the '~testing" procedure. 5. Solve the system of equations for the unknown coefficients of the basis set representing the equivalent sources. 6. Compute other quantities of interest that depend on the equivalent source distribution, such as antenna impedance and radiation pattern.

Wire Antennas. The steps above are described briefly in the following as they might be applied to the analysis of a perfectly conducting wire antenna, which is driven by a time-harmonic voltage source V and which is of arbitrary configuration, such as that illustrated in Fig. 9 .6a. Simple wire antennas such as the one shown in the figure are not widely used at microwave frequencies, but more complex wire radiators may be of interest1 and the wire geometry provides for a simple example. The first step is to obtain an equivalent problem in which all physical structures have been removed. The equivalence principle (14] is applied to replace the wire with an electric surface current density J that resides in infinite homogeneous space and on the surface originally defined by the physical wire surface (Fig. 9.6b). We assume here that the wire is a solid metal rod. If the wire is a tube with vanishingly thin walls, J represents the vector sum of the current on the interior and exterior surfaces. For the solid metal rod, the current J which produces the correct "scattered" field is

510

ANTENNAS I: FUNDAMENTALS AND NUMERICAL METHODS

Figure 9.6 (a) Arbitrary thin-wire antenna in homogeneous region; (b) repre entation of wire by equivalent current distribution; (c) segmentation of wire for application of method of moments.

known, from the equivalence principle, to be given by J = fi x H, where H is the total magnetic field at the wire surface and n is the outward-directed unit normal to the surface. The magnetic field is not known initially, so we seek to determine J numerically. The total field anywhere external to the wire can be represented as the sum of the scattered field (Es, Hs) radiated by J and the excitation field (Ee, He), which is defined to be the field produced by the excitation sources in the absence of the wire structure.

9.2

NUMERICAL TECHNIQUES

511

An integral equation for J can be obtained by applying the boundary condition that the total electric field tangential to the surface S of the wire is zero: (9.36) where Es i the field radiated by J and Ee i the excitation field. The field Es may be represented, for example, either through the dyadic Green's function for the electric field or through the magnetic vector and electric calar potential functions to obtain two forms of the boundary condition equation (9.36):

1rrry/ix f,J (r') · (k2 I + VV)G (r , r' ) dS' = fixE'(r)

rES

(9.37)

and nx UwA(r)

+ v' (r )] = fi X E e(r )

r

E

s

(9.38)

where I is the identity dyad and G (r , r') is the homogeneous space Green's function: e-iklr- r'I

G(r r')

= - --

(9.39)

Ir - r'I

A (r) and (r) are the magnetic vector and electric scalar potentials defined by A(r)

= .!:!:_ f J(r')G(r, r') dS' 4rr ls

(9.40)

ct> (r)

= _I_

(9.41)

4rrE

{ q(r')G(r, r') dS'

ls

where q is the equivalent surface charge density related to J through the continuity equation. Equation (9.37) involves an integral that can be evaluated only in the principal value sense, as is denoted in the equation by the bar through the integral operator. In general, it is necessary to use basis functions having continuous first derivatives for the current representation to obtain good results with (9.37). Excellent results can then be obtained with this equation if appropriate care is taken in the evaluation of the expression. The well-known Numerical Electromagnetic Code (NEC) [15] employs this approach to model wires. For this discussion, however, we employ the potential fonn of (9.38). For wires having a radius much less than the wavelength, the equations are somewhat simplified by applying the so-called thin-wire approximations. Under these conditions the current transverse to the axis of the wire is neglected and the boundary condition is enforced only in the axial direction. The axially directed current is also assumed to be circumferentially invariant and it is assumed that it can be represented as a filamentary current flowing along the wire axis (or, alternatively, one may assume that the boundary condition is enforced along the wire axis). Equation (9.38) may then be written as

jwA1(I)

+ :I

[4>(1)]

= E{(l)

(9.42)

512

ANTENNAS I: FUNDAMENTALS AND NUMERICAL METHODS

where l denotes the arc length along the wire axis or, when used as a subscript, the axial component of the potential or the field. The excitation Ee indicated in the equation varies in form according to the source model used. The most commonly used source in wire problems is probably the deltagap source Ee = V 8(l -18 ), where 8 is the Dirac delta function and 18 is the location of the feed point along the wire length. This model is easy to use and usually produces results of sufficient accuracy when used appropriately. Its primary disadvantage is that it is fairly sensitive to the number of unknowns used to model the current distribution in the feed region. As the number of unknowns approaches infinity, the imaginary component of the feed current will approach infinity also. Nevertheless, when a reasonable number of unknowns is used, quite good results are generally obtained. The delta-gap source produces an excitation field which is localized to the feed point. It is assumed not to radiate, so its contribution at other points along the wire length is zero, as is its contribution to the antenna fields. Alternative source models may provide more accurate results, particularly with regard to input impedance, in some situations [16] . The magnetic frill feed [17-19] is one such source in which an annular disk of circumferentially directed magnetic current is placed around the wire feed point. This model is well suited for antennas which are fed at a ground plane from a coaxial transmission line, where it closely models the physical situation. The magnetic frill source, however, produces a nonzero excitation field Ee which must be computed as radiation from the frill source. Another model that has been used with some success is the current-slope-discontinuity source [15, 20], although there are data to indicate that it may not necessarily be a better choice [21]. This source is based on a biconical transmission-line model of the feed region. A relatively simple source model that has been shown to compare very well to the magnetic frill feed has also been described in [22] . The method of moments may be applied to the equivalent problem of Fig. 9.6b, for which the boundary condition Eq. (9.42) is to be used, by first subdividing the domain over which the equivalent current exists into subregions. A wire may be divided into segments as shown in Fig. 9.6c. Basis functions fn(l) which exist over one or more subregions are then chosen for use in representing the current distribution / (/) = 2n a J (l) in the form N

I (l)i ~

L

lnfn (l)

(9.43)

n=l

where the In's are the unknown coefficients of the current representation and i is the unit vector in the direction of the wire axis. The fn 's may be of either the subdomain type or the entire-domain type. As the name implies, entire-domain basis functions generally exist (are nonzero) over the entire structure, or at least over some very large portion of the structure. A subdomain basis function, on the other band, is generally nonzero over only a very small portion of the structure and usually has a maximum dimension of one-quarter of the excitation wavelength or less. Entire-domain basis functions are most useful for very regular structures, such as long straight wires, loops, bodies of revolution, bodies of translation, and so on [23-27]. For these types of structures the use of entire-domain basis functions may yield satisfactory results for relatively few unknown terms in the current expansion (9.43). Even for such regular geometries, however, a significant amount of care may be required in the testing procedure to obtain a well-conditioned matrix.

9.2 NUMERICAL TECHNIQUES

513

Subdomain basi function provide for more flexibility in modeling complex geometrical tructure and generally produce well-conditioned matrix formulation . The form of the basi function hould be cho en to reasonably represent the actual current di tribution. The u e of mooth ba i function , uch a pline functions, will generally provide more accurate elution for fewer unknown if the actual current distribution has imilar '' moothne "propertie . The inclu ion of the appropriate ingular behavior of the current, when pre ent. al o enhance convergence properties of the numerical elution [28]. The use of more ophi ticated basi function uch a these, however, often increase programming complexity and reduce program generaljty_ For example, the current on a tep-radiu wire appear to exhibit a lope discontinuity at the radiu change (unJe it i viewed on a greatly expanded cale). Smooth basis functions cannot model thi behavior well unle the amount of the slope change i known and thi lope change is included in the basi function at the radiu tep [15, 29]. Triangle basi functions can model uch behavior without modification, however. A combination of ubdomain and entire-domain-type ba is function can also be employed to improve solution efficiency and accuracy in some cases. An example of this for wire antennas and catterers i the u e of a polynomial approximation for the current basi function on wire egments [30, 31 ], where a wire egment may repreent an entire wire antenna or only a portion of a wire antenna, and the degree of the polynomial approximation may be specified independently on each wire segment. The polynomial basis functions in thi approach provide for a smooth current representation over each wire segment, and by increasing the degree of the polynomial approximation on a egment, a good repre entation of the current can be obtained using longer wire egments. However, increasing the polynomial approximation order too far again results in poor matrix conditioning. In [30], pulse testing functions are used to generate the system of simultaneous equations and use of the lowest order polynomial approximation yields a triangle expansion function set and a pulse testing function set as described in the following. For further discussion we assume here that the basis functions fn(L) are triangle functions such as those illustrated in Fig. 9.7. This leads to a piecewise linear representation of the current distribution along the wire, as is also indicated in the figure. Note that 0 at the wire ends is the basis set is chosen so that the boundary condition that I automatically satisfied. This choice also leads to a piecewise constant representation for the charge q, which is the source for the scalar potential term in (9.42). Generation of a system of simultaneous equations from which one may olve for the unknown coefficients of the wire current representation is accomplished by " testing" (9.42) with a set of testing functions tn (L). The testing procedure involves forming the inner product of each testing function with the boundary condition equation (9.42), where the inner product of two vector functions f(l) and t(l) is defined as

=

(f(/), t (l ))

=

l

f(I) . t(I) di

(9.44)

The testing path C in the testing procedure is chosen to be along the wire surface in the direction of the wue axis. The testing procedure has the effect of enforcing (9 .42) over the domain of the testing functions in a weighted-average sense. The testing functions should be chosen such that they can reasonably represent the range of the vector operators appearing in (9.42). In practice, one common choice is to use the basis function set as the testing function also, but other choices are possible and may

514

ANTENNAS 1: FUNDAMENTALS AND NUMERJCAL METHODS

Figure 9.7 Representation of wire current distribution by triangle basis functions and placement of pulse testing functions along wire.

even be preferable [32, 33]. When the set of testing functions is chosen to be the same as the set of current basis functions, the resulting solution procedure is referred to as Galerkin' s method. Frequently employed basis testing sets for Galerkin approaches are the triangle [24] and the piecewise sinusoidal [34] sets. The MM matrix obtained from a Galerkin procedure can be shown to be symmetric via reciprocity if the integrations over the testing region and the equivalent sources are performed with the same degree of accuracy. Another common choice for the set of testing functions i the pulse testing set tn (l), which is shown in Fig. 9.7 (the pulses are shown beneath the wire for illustration purposes only). The MININEC computer code for wire analysis [35] employs the triangle basis set/pulse testing set (with appropriate approximations) shown in Fig. 9.7 and is based on the work of Glisson and Wilton [33, 36]. This approach is simple, it converges well, and it has other desirable features [29, 33]. It represents a "minimal" approach to the solution of an equation of the form (9.42) in the sense that one cannot reduce the smoothness of the basis set (or the testing set) without degrading the olution convergence properties significantly unless one compen ate by increasing the smoothness of the testing set (or the basis set). Figure 9.8 illu trates this by a convergence comparison of this approach with one in which a pulse basis set is used with point matching (17] (delta-function testing functions; also known as collocation) for a resonant-length wire antenna problem. In the figure the magnitude of the feed current is plotted as a function of 1/ N, where N i the number of unknowns used to model the wire current. The triangle basjs/pulse testing procedure clearly provides a better result for a reasonable number of unknowns. The NEC code, however, produces excellent results by employing delta-function te ting functions (less smooth than pulses) and spline basis function (smoother than triangles) [15]. Convergence properties for some of the possible choices for basis and testing functions bave been investigated in Ref. 37. To avoid excessive time spent in the numerical evaluation of the testing or source region integrals, further approximations are often made. For the triangle basis/pulse testing approach, for example, the integration over the magnetic vector potential A introduced by the testing procedure may be approximated as described in Ref. 33. Integration of the pulse testing functions against the gradient of the scalar potential is

9.2

UMERICAL TECHNIQUES

SIS

151~-= - ==~========7 ~ 12.5 Triangle basis set, pulse testing set

-0

:::J

.-:: C

O>

Pulse basis set, point-matching

(0

-........ E

10

C

(1)

:::J (.)

~

-0 Q)

7.5

if.

L = 0.47 1

!1 a~_l

a = 0.005 1 5 ,____ _ _ _,____ _ __ ,____-,---_ ___.1.....-_ _ _ 0 0.05 0.1

0.15

0.2

__,J

(1/N)

Figure 9.8 Con vergence of numerical solution for current on resonant-length dipole as function of the inverse of the number of unknowns in model for two different numerical procedures.

evaluated analytically. The integration of a triangle basis function against the Green's function in the calculation of A is also often approximated as integration of a pulse of equal moment against the Green's function . In the triangle basis/triangle testing approach of Mautz and Harrington each source triangle is approximated as a series of four pulses [24] and further approximations are made in the integration over these pulses. The testing function triangles are approximated by a series of four impulse functions, so that the testing integrations can be performed analytically. The application of the triangle basis/pulse testing procedure described above to (9.38) or (9.42), using a delta-gap voltage source forcing function and with the appropriate approximations, yields the following expressions for the impedance matrix elements and the forcing function vector (for details, see Ref. 33, 35, or 36):

Zmn

=

JW~ 1?'

4 _

~

~

(rm~l/ 2 - rm- J/2) •(1n- l/2V'm,n- l/2.n + ln+I/2V'm,n.n+ l/2)

j

( V'm+ l/ 2,n,n+I _

4,r(IJ€

l!:.ln+l/2

V'm+ l / 2,n- 1,n _ V'm - 1/ 2.n,n+ l

l!:.ln - 1/2

l!:.ln +l/2

+ V'm - 1/ 2,n- l ,n) l!:.ln- 1/2

(9.45)

if lg E (ln- l / 2, ln +l / 2) otherwise

(9.46)

where 1/1 is an integral function defined by (9.47)

516

ANTENNAS I: FUNDAMENTALS AND NUMERICAL METHODS

with 1

k(r m, r')

= -2

JT

Rm

I rr e- j kRm R d 0 may then be computed as radiation by the magnetic current sheet and the equivalent electric currents. The calculations would only be valid for z > O, of course, since the model in Fig. 9.14b is valid only in that region. The aperture chosen in the preceding discussion, however, does not represent the only possible choice. If the field distribution at the waveguide end (z = d) were known more accurately, the short could have been placed at the end of the waveguide. The

9.2

NUMERlCAL TECHNIQUES

523

(a)

Waveguidl ~

--00 PEC ground plane at z = 0

(b)

Z=O Ground plane

/ dPEC short -

(c)

00

-

-

2d--

PEC waveguide) section (d)

J

--~;-

---------J

Figure 9.14 (a) Geometry for waveguide extending through perfectly conducting ground plane; (b) mode] equivalent to original problem for z > 0; (c) application of image theory to remove ground plane; (d) application of equivalence principle to obtain equivalent sources in homogeneous medium.

solution procedure would follow in the same manner with the application of image theory, use of the equivalence principle to represent the waveguide section, and solution via the method of moments. An internal resonance problem might arise with this choice, however, unless the CFIE is used because the shorted waveguide with its image would form a closed body. If the current distribution Min Fig. 9.14b is not known, one must also develop a situation that is equivalent to the original situation of Fig. 9.14a for z < 0. This may be done by shorting the "aperture" in the waveguide at z = 0 and placing a magnetic current -M over the short as shown in Fig. 9.15a. The use of -M ensures continuity of tangential E across the aperture. Image theory may then be applied to obtain an

524

ANTENNAS I: FUNDAMENTALS AND NUMERICAL METHODS

Z=O

(a)

Waveguide

~

--00

'1...-PEC short

Model valid in waveguide for z < O

Infinite waveguide

(b)

1

--00 Figure 9.15

00-

(a) Model equivalent to the original problem of Fig. 9.14a for z < O; (b) app-

lication of image theory to obtain magnetic source in infinite waveguide.

infinite waveguide with a magnetic current sheet at z = 0 as in Fig. 9.15b. If M were known, the fields in the waveguide could be determined for z < 0. In particular, the tangential aperture magnetic field HA at z = o- can be determined in terms of the unknown current distribution -Mand the waveguide excitation. Similarly, HA can be determined at z = o+ in terms of the equivalent currents M and J of Fig. 9. l 4d. The additional equation necessary to determine M is therefore obtained by requiring HA as computed in the two different equivalent situations to be equal at z = 0. For additional information, see, for example, Refs. 61-64.

Material Bodies. The method of moments can also be applied to antennas that include material regions with constitutive parameters other than those of the surrounding space. For a single homogeneous material body the equivalence principle may be applied to develop two equivalent situations, one valid for the body interior and one valid for the region external to the body, in terms of equivalent surface current densities J =fix Hand M =Ex ft. In the exterior equivalence the body may be removed, its space filled with material of the same type as the exterior region, and the equivalent surface current densities placed on the surface originally defined by the body. In the interior equivalence, everything exterior to the body may be discarded and replaced by homogeneous material of the same type as the body, and the equivalent currents again are placed on the surface originally defined by the body. As in the preceding aperture problem, the interior and exterior equivalent currents can be related by a minus sign because the tangential electric and magnetic fields in the original situation must be continuous across the material interface. The fields in each region can be computed, using the appropriate equivalent situation, as radiation by the equivalent surface currents J and M once these currents are known. Since the fields within each region can be expressed in terms of these currents (and any other sources within the region), one can obtain equations from which J and M may be determined by enforcing

9.2

NUMERICAL TECHNIQUES

525

the boundary condition that tangential E and H , as computed using the different expre ion valid in each region, mu t be continuou aero the dielectric interface [33, 65]. Variou form of equation (EFIE, CFIE, etc.) have been derived and implemented for homogeneou bodie [65, 66]. The olution procedure for homogeneous bodies may also ea ily be e tended to treat bodie involving multiple homogeneous material region [67, 68] . Approximate boundary condition may be used under certain conditions, such a for highly conducting material to reduce the number of unknown and simplify the solution procedure for material bodie [69, 70]. They may also be used to model other type of urface uch a corrugated urface and rough urface [71, 72]. The triangular urface patch modeling method [44] has been applied to homogeneous material bodie [73] and to bodie with different type of boundary conditions [49, 74]. Compie bodie compri ing both conducting and material structures such as a microstrip patch antenna situated on a dielectric ubstrate are al o of significant interest. Systematic procedure for the formulation and application of triangular patch models to such compo ite tructure are available [75, 76]. Depending on the type of structure and frequency range of intere t, different method may be employed for the analysis. For example, if the dielectric ubstrate can be approximated as being infinite in extent, a Green· s function approach may be u ed so that the only unknowns are the currents induced on the patch surface [77] . For a finite substrate, a full model of the dielectric urface may be required [78], or an approximate volume model of the dielectric might be applied [79]. To model complex multilayer structures of finite extent, such as printed circuit boards, model based on aperture theory may be applicable [80] . Several diverse modeling methods are available for treating problems involving inhomogeneous dielectric regions. One possibility is to model the dielectric region as layer of homogeneous material [67] . This option is most useful when the inhomogeneity of the region is relatively smooth and slowly varying, except possibly for abrupt changes in the medium parameters, and when the regions that are approximated by constant parameters form reasonable contours. For more general inhomogeneous regions the approache employed more often involve the solution of a differential equation formulation, which is di cus ed sub equently, or of a volume integral equation (VIE) formulation [81). In the VIE formulation a dielectric region is replaced by a volume polarization current (9.60) J(r) = jw[i(r) - Eo]E (r) where

i is the complex dielectric constant defined by i(r)

= E(r) -

(9.61)

j[a(r)/ w]

and where € and a are the medium permittivity and conductivity. Separating the electric field into a scattered field and an excitation field leads to the VIE

J(r) jw[i(r) - Eo]

+ jwA(r) + V(r)

= E e(r)

(9.62)

where A and are defined as in (9.40) and (9.41) with E replaced by E and with the surface integration replaced by integration over the volume of the dielectric region. The VIE (9.62) is valid throughout the dielectric region. The method of moments may

526

ANTENNAS I: FUNDAMENTALS AND NUMERICAL METHODS

be applied to this equation by subdividing the region into volume elements, expanding the unknown volume polarization current J in basis functions defined on these elements, and "testing" (9 .62) to obtain a set of simultaneous equations as described previously. The VIE approach has been appljed to problems involving scattering by arbitrarily shaped inhomogeneous dielectric bodies by several investigators [82-85]. In Refs. 82-84 cubical and rectangular volume element cells have been used with a pulse expansion set for the current and a point-matching testing procedure. A tetrahedral cell volume element approach is presentyd in Ref. 85. This approach employs basis functions and testing functions which are extensions of those previously described for use on triangular patches for perfectly conducting surfaces. Because of the large number of unknowns required to model a volume, the application of the VIE has generally been limited to the modeling of fairly small regions or to the use of very crude models. Hybrid differential equation-integral equation formulations and volumesurface integral equation formulations have also been developed to model problems involving inhomogeneous regions [86, 87].

9.2.3 Differential Equation Approach Differential equation-based numerical methods are, by necessity, formulated as boundary value problems. As such, the solution of scattering and radiation problems in unbounded spatial regions requires a mechanism for applying proper conditions on the boundary that encloses the solution region. Several methods have been developed to provide needed boundary conditions. These include the following general procedures: (1) coupling the closed region solution to field representations in the unbounded region (unimoment and infinite elements); (2) supplying the needed boundary conditions via surface integral equations (hybrid boundary elements, etc.); (3) employing field relationships on multiple surfaces via the equivalence theorem (field-feedback formulation) ; (4) deducing a governing modal field relationship on the boundary (measured equation of invariance); (5) inserting phantom absorbing media to enclose the region (artificial absorbing boundary); and (6) implementing special nonreflecting differential equations at the boundary (radiation boundary conditions). The unimoment method, as developed by Mei [88], provides a self-consistent approach to coupling interior and exterior field problems through a separable surface interface. This method was employed by Stovall and Mei [89] and Morgan and Mei [90] to both antenna radiation and scattering problems involving inhomogeneous axisymmetric dielectrics. Further applications of the unimoment method, using finite elements, were made by Morgan [91, 92] to problems involving raindrop scattering and microwave energy deposition in the human head. For many applications the use of a separable surface, as is needed in the unimoment method, results in numerical inefficiencies and other difficulties. To allow the use of more generalized surfaces, a hybrid method was developed by Morgan et al. [93]. This approach, termed the finite element boundary integral (FEBI) method, combines a finite element interior solution with a surface integral equation on the boundary. A more recent innovation is that of the field feedback formulation (F3), which replaces the integral equation on the boundary in the FEBI approach with a discrete feedback relationship between adjacent boundaries [94, 95]. An early technique for employing finite method solutions in open-region problems was to simply impose an artificial zero boundary condition at a sufficiently large distance from the antenna or scatterer [90]. Another method employs "infinite elements,"

9.2

NUMER1CAL TECHNIQUES

527

where an a umption i made a to the behavior of the exterior field [97]. 1n both of the e method the finite element me h region mu t be extended sufficiently far from the material object to aJidate the far-field type of a umptions that are being made. Thi e ·tended me b ize can re ult in low numerical efficiency. Several additional techniques have been developed in recent year to minimize the numerical elution region by placing the boundary clo e to the object. The most popular clas of method employ radiation boundary condition [98, 99] . These schemes are ba ed on nomefiecting differential o perator derived for a urned wave behavior at a pecified boundary geometry. Although imple and efficient to employ, they can be inexact when applied to wave behavior or boundaries that violate their defining condition . Another recent innovation for boundary termination i the measured equation of invariance (MEI) [100 101], which imbeds the specific boundary geometry in a derived differential operator. The MEI method ha been used with finite elements to compute field from an electrically large reflector [102] and with the fi nite difference technique to olve for fulJ-wave propagation over irregular or random terrain [103]. Due to the par e matrice invol ed, finite methods are highly efficient for computing fields in\'Olving inhomogeneou and even anisotropic materials [104, 105] as well as problem involving complex geometries [106]. Iterative numerical methods both for the finite method olution within the interior region [ 107] and for the associate boundary condition operator [108] increa e the electrical size and complexity of problems that can be handled. To further extend the realm of computation, significant research bas been devoted [86, 109-112] to hybrid computational methods that partition complex real-world problems into several regions, each with specialized geometry or material properties. Fields or currents in each region are then solved by an optimum method such as finite elements an integral equation, or an asymptotic approximation. The e solutions are then systematically interfaced and recombined into a global solution. The discussion to follow will introduce the fundamental concepts employed in finite difference and finite element solutions (termed finite methods) and will consider various applications using the unimoment method, hybrid boundary elements and the field feedback formulation. Textbooks are available that provide extensive overviews of finite method formulations, boundary termination schemes, and applications [113-115]. Several commercial software packages are also available that use 2-D and 3-D finite methods in both the frequency and time-domains. Some notable examples are OPERA and the CONCERTO suite, both marketed by Vector Fields (www.vectorfields.com), FEKO (www.feko.co.za), and QuickField (www.quickfield. com).

Finite Method Approximations. The finite methods may be classified as the various techniques that provide discrete-wise approximate solutions to continuous boundary value problems [2]. As such, the finite methods offer a means to approximate the solution of specified differential equations in one or more dimensions, including time. The most common method has been that of finite differences, which obtains discrete approximations to partial derivatives by differentiating a piecewise polynomial , or other approximating function, which has been point-matched to the actual unknown function at the "nodes" of the problem. As a simple example, consider the finite difference approximation of the various derivatives, up to the nth order, of an unknown function, f (x ), in one dimension. The domain of~ is partitioned into general) y unequal segments, l , 2 , .. . In the region of x that contains the separated by ordered nodes, xk, for k

=

528

ANTENNAS I: FUNDAMENTALS AND NUMERICAL METHODS

ith node, x; , let f (x ) be approximated by an nth-order polynomial, (9.63) The coefficients can be found as linear functions of n + 1 of the unknown nodal values off (x). Usually, these particular nodes will be taken to bracket the ith node. The linear relationship between the coefficients and the nodal values of / (x) is developed by point matching the polynomial in Eq. (9 .63) at the n + 1 nodes, resulting in the linear system defined by n

L xfaP = f (xk)

for n

+ 1 distinct values of k

(9.64)

p=O

After inverting this syste~ the resultant linear functional form for each ap can be substituted into Eq. (9.63). The finite difference formulas for each derivative up to the nth order can then be obtained by analytically differentiating Eq. (9.63), followed by an evaluation at x = x; . These derivative approximations are of the form of a linear weighted function of the n + 1 nodal values of f (x ). For example, if n = 2, with equal segments, b..x, then a quadratic expression in (9.63) yields the well-known formula for the second derivative, d 2 f(x;) dx 2

f(x; +1) - 2/(xJ

+ f(x; -; )

(b..x) 2

(9.65)

The finite difference solution of a boundary value problem is set up by replacing the analytical derivatives contained in the differential equation by finite difference formulas at each nodal point where the solution function is to be found. Thus there results a system of linear equations relating the unknown nodal values of the solution function to both the known excitations (drivers) of the differential equation and the known boundary values of the solution function. The finite difference technique can be thought of as a special case of the finite element method. The finite element method (FEM) may be considered as the numerical implementation of the "weighted residual" technique [116], as applied to boundary value problems using basis and testing functions having compact support. Aside from the use of differential operators, the weighted residual approach is conceptually identical to that of the method of moments, originating in conjunction with finite element solutions to structural and fluid dynamics problems. The FEM provides the numerical solution of a differential operator equation in an M-climensional "volume," Vo, which bounds r = (r 1 , r 2 , ..• , rM) where M is usually no more than 4, D(r)• f (r ) = g(r) for r inside Vo (9.66) D(r) is the differential operator, / is the unknown vector function, and g is the known driving vector. Es ential boundary data concerning f is known on a surface, So, which encloses V0 . An example is the scalar Helmhotz equation in two dimensions, (V

2

+ k5)f (x, y) = g(x, y)

(9.67)

9.2

NUMERJCAL TECHNIQUES

529

To find the approximate numerical elution of thi equation, we u e a basis function expansion to repre ent /, N

f (r )

= L C,, U

11

(r )

(9.68)

n= l

where the et of ba i functions {U,,} should, idealJy, have the same order of differentiability a doe the e act olution. A N i increa ed, the approximate expansion hould converge in a pointwi e en e to /(r ). This last condition depends on how completely the et of basi function pan the ubspace of function s occupied by the various elution to (9.66). Thi quality is reflected in the requisite differentiability and the linear independence of the basi et. Upon ub tituting Eq. (9.68) into (9.66). there results N

D· /(r )

= L C,,{D(r ) -U,,(r )} = g(r )

(9.69)

n=I

To olve for the N coefficients, we enforce this equation with respect to a succession of N weighted integration over V0 • N

{Wt(r ) . D· f (r )}

=

L C,, (Wk(r ). D (r ) •U,, (r)} = (Wk(r ), g(r )}

(9.70)

n=l

for k = 1 to N. The inner product, (W(r) . u(r )), indicates an integration of the dot product of the two indicated vector functions over V0 . The set {Wk(r)} is termed the weighting (or testing) functions. In setting up the N x N linear system indicated by (9.70) there are some additional considerations. One of these concerns the support region of the basis functions: either full range (over all of Vo) or compact (each being over only a portion of V0 ). An example of full-range basis functions is the set of complex exponentials employed in Fourier series, where the C,, 's are termed the spectra of the expansion. Compact support basis functions are more common to finite element applications. Usually, these basis functions are selected so that at each node of the discretized problem, all basis functions, except one, are zero. At its associated node the basis function will usually be set to unity. In such a case the compact support basis set conveniently yields Cn 's, which represent the solution values of /(r) at the N nodes. Another consideration involve the set of weighting functions. There is obviously an unlimited selection available. Three of the more common types are:

Point Collocation. which uses a delta function at each ordered node coordinate, rk, for the associated Wk. The effect of this is to reduce the integration "moments" in Eq. (9.70) to simple point matching at the respective nodes, resulting in nothing more than the finite difference method. An advantage of this method is its relative simplicity in generating the matrix elements from (9.70), since integrations are reduced to enforcing the approximation at the node points. On the other hand, there is no control on the behavior of the solution in between the nodes. Subdomain Collocation. which employs a set of mutually exclusive constant functions, Wk(r) = I, in a defined region around the kth node. These regions around each

530

ANTENNAS I: FUNDAMENTALS AND NUMERICAL METHODS

node are nonoverlapping and usually are directly adjacent to one another, without unfilled space. This approximation is usually more accurate than point collocatio~ but not as accurate as Galerkin's method, when self-adjoint operators are involved. Galerkin's Method. which uses the same set of functions for both basis and testing, Wk(r) = Uk(r). For the case of a self-adjoint operator, D(r), it can be shown that the functional defined by Q(f) = {f , D • f ) - 2{f, g } is stationary about the solution to the original operator equation, in (9.66) (117]. This variational principle, when enforced on the basis function expansion in Eq. (9.68), is termed the "Rayleigh- Ritz method" and yields the Galerkin's result for the weighted residual approach. This procedure usually provides the most accurate solution and forms the foundation for most of the FEM work that has been done. Another means of achieving this same result (the Galerkin equations, with W.t = Uk) is by way of the classical Euler-Lagrange variational formulation. This has the advantage of reducing the order of differentiation on the basis functions, vis-a-vis a direct Galerkin approach, and will be demonstrated in the following section. An excellent discussion of the error bounds and rates of convergence for these three methods is given in [117] .

Finite Element Example. Consider the simple problem of the undriven (g = 0) Helmholtz equation in Eq. (9.67) in a rectangular region with mixed boundary conditions (BCs), as shown in Fig. 9.16. The rectangular problem space is discretized into a grid of doubly ordered nodes, with triangular finite elements, as in Fig. 9.17. The unknowns are thus the values of f (x;, y i) at the nonboundary nodes. This is a simple example of scalar-valued 2-D finite elements. It is also possible to employ vector-valued elements in 2-D and 3-D [118, 119] whose edge segments form vector "basis functions" akin to the scalar functions in (9.71) below. We will employ piecewise linear "pyramid" basis functions, U;,j (x, y), to represent the solution, 6

f(x , y)

7

=LL li.jui.j(X , y)

(9.71 )

i = l j=1

The support region for U; ,i is all elements that share the (i, j) node, as illustrated in Fig. 9.18 for an interior node. In the lth element, we will locally number the associated three nodes, k = 1, 2, 3, as shown in Fig. 9.19. Within this lth element, the linear basis function associated with the kth node is given by the matrix product Uk(x, y)

= [x, y, 1] · [Lh

(9.72)

where [ L ]k is the kth column of the element coordinate matrix,

(9.73)

where D1

= det[ L] is twice the area of the triangle.

9.2 NUMERICAL TECHNIQUES

531

y

l!.. = 0

ay

f(0, y) = f2 (y)

f (a, y) =0

Inside of V0

OL__ _ _ _ _ _ _ _ _ _ __jaL_---x f (x, O) = f1(x)

Figure 9.16 Helmholtz equation with mixed boundary conditions .

. . . - - - - - - Node(2, 5)

7 6 5 4

3

2 j == 1 i= 1

2

3

4

5

6

Figure 9.17 Rectangular region finite element mesh.

Employing the Euler -Lagrange formulation, we seek the nodal values in (9.71) for which the quadratic functional below is stationary, Q(f)

= {V /, V /

)-

J?o(/, / }

(9.74)

Note that this functional has only first-order derivatives inside the inner product {•, •} integrals. This result follows directly from the classical variational formulation and

532

ANTENNAS I: FUNDAMENTALS AND NUMERICAL METHODS Uii has linear

variation in each element

where

Uii = o on the boundary ...._____ of the element group

Figure 9.18

Node with surrounding element group.

t -th element

Figure 9.19

Local coordinates in an element

can be obtained through applying Green's theorem (or multiple integration by parts) to the Galerkin equations. The stationary solution is found by substituting (9.71) into (9.74) and then differentiating Q with respect to each of the unknowns. Setting these derivatives to zero results in the linear system of equations, 6

7

LL f;,j {(VU;,j, VUm,n) -

k5(Ui.j• Um ,nn

=0

(9.75)

i= l j = l

form= 2 to 5 and n = 2 to 7. Note that for a given (i, j) node, only (m, n) nodes sharing at least one common element will provide a nonzero contribution to the moment integrations. Thus the matrix defined by the terms in braces in (9.75) will usually be quite sparse. This feature is produced by all finite methods when using compact support basis and testing functions.

9.2

Denoting the four unknowns at the node aero in Fig 9 .1 7 by the vector

NUMERICAL TECHNIQUES

533

the jth horizontal row of the me b

(9.76) where [·] 7 indicate matrix tran po e. the matrix equation implied by Eq. (9.75) can be written as a linear matrix-ve tor relation hip between adjacent row vector , (9.77a) where the block matrice are banded in tructure,

[A] :

X

X

x

x X

X

X

O O [B] :

X

X

X

0

X

0

X

X

X

X

X

X

X

[CJ:

X

X

0

X

0 X

X

(9.77b)

X

The nonzero matrix elements, indicated by x, as well as the elements of the boundary condition vector, Pi , are obtained in terms of the element integrals within the braces in (9.75). Within the lth element, having L-matrix defined by (9.73), the integrands can be obtained directly from Eq. (9.72). Denoting the relationship between local (in element l ) and global node coordinate by k = (m, n) and q = (i, j), there results

(9.78)

(9.79)

The matrix contributions are thus assembled from element integrations of the form

P,.,

=ff x'y' dxdy

(9.80)

t,.

These are available in tabular form in a number of references on finite elements (e.g., Ref. 116). Having loaded the block matrices relating adjacent row vectors of unknowns, the global matrix structure will be of the tri-block form

[Bh [Ah

[CJ2 [Bh

0

[Ch

(9.81)

0

[Ah

[Bl,

This sparse matrix allows highly economical inversio~ for even very large matrix order, by any of a number of different algorithms [120]. In addition, by ordering the

534

ANTENNAS I: FUNDAMENTALS AND NUMERICAL METHODS

nodes properly, the matrix can often be made to have a block structure, as it does in this case. A block-structured matrix can be very readily inverted by way of the Riccati transform algorithm [90]. This example was presented in the spirit of demonstrating some of the elementary numerical procedures that accompany the application of the finite element method. We will now tum our attention to some of the techniques for employing finite method solutions to field problems in open, unbounded regions. Unimoment Method. As developed in the unimoment method, the exterior region fields are represented by a functional expansion in one of the separable coordinate systems for the vector Helmholtz equation [121]. The spatial interface for coupling the interior numerical solution to the unbounded exterior region is thus a constant coordinate surface of the separable system employed in the outside expansion. Spherical interfaces were used in Refs. 89-92 and 122- 124 due to the relative ease of generation of exterior region spherical harmonic field expansions. To understand the conceptual basis of the unimoment method, consider the solution of an unbounded field problem involving a cylindrical two-dimensional inhomogeneous penetrable scatterer having arbitrary cross section. The known impressed field E; and Hi, which may be due to a localized source or be represented as an incident plane wave, is decomposable into TE and TM components (z-directed H; or E;, respectively). In either case, the homogeneous scalar Helmholtz equation [Eq. (9.67) with g = O] is applicable, but with a variable wave number within the object,

{V2 + k2 (r , 0)}/(r, 0)

=0

(9.82)

where we are using circular coordinates (r , 0). The unimoment solution proceeds by enclosing the scattering body within a circularcylindrical geometric surface, as illustrated in Fig. 9.20. Notice that there are two concentric cylinders, having radii r1 and r2, each of which enclose the scattering object.

... :.: . . .. ·\ . . . .. . . . .

. Inhomogeneous: : : ••dielectric cylinder:. •• • • • • E (r. 9) •.: •. : • •• • _J _ •• • •• • •••

. ·. .. .. . . .

Figure 9.20

Unimoment matching contours for cylindrical problem.

9.2

NUMER1CAL TECHNIQUES

535

With pecified Dirichlet boundary conditions (BCs) on the outer boundary, r = r,, a finite method i u ed to solve for the nodal values of f (r, 0) for r < r 2 • A necessary attribute of the interior me h construction is that a et of the solution nodes lie on the inner cylinder r = r 2 . The total field outside the mallest cylinder that enclose the catterer may be represented by the sum of the known incident field and a cylindrical harmonic expansion for the unknown canered fields, N

f(r, 0)

= f ;(r, 0) + Co(r) + L

anCn(r 0)

+ bnSn(r, 0)

(9.83a)

n= l

where

= H~2>(kor ) cos n0 Sn (r. 0) = H; 2>(kor ) sin n0

Cn (r , 0)

(9.83b) (9.83c)

2

with H~ > the Hankel function of the second kind. In practice, this truncated series will converge rapidly to the scattered field if N > kor2. The unknown scattered field coefficients {an} and {bb}, are found by first solving the interior region problem for BCs on r = r 1 which are composed of the incident field and each of the scattered field modes in the expansion. The numerical solutions for r < r 1, corresponding to each of these applied BCs, is indicated by a circumflex overbar. For example, a BC of Ji (r 1 , 0) produces an interior solution of / i (r , 0), while a BC of Sn(r1 , 0) produces Sn(r, 0). Using the principle of superposition, the numerical solution for the total field inside the outer boundary will be given by Eq. (9.83), but with numerical solutions replacing Ji C-n, and Sn . To solve for the coefficients, we simply equate the numerical solution to the analytical solution along the circular contour, r = r 2 , resulting in N

Lan{Cn(r2, 0) - Cn(r2, 0)}

+ bn{Sn(r2 , 0) -

Sn(r2, 0)}

n=l

(9.84) The unknown scattered field coefficients may be obtained by a weighted residual approach~ where (•) indicates, in this case, an integration on 0 from O to 2rr and the ~-functions below indicate the respective function differences in (9.84), N

Lan {Wm(0), ~Cn (0)) +bn{Wm(0), ~Sn(0 )) n=l

(9.85) with 2N + 1 linearly independent weighting functions, {Wm}. By selecting deltafunction weights at 2N + I nodes along the matching contour, r = r2, Eq. (9.85) will provide the point-matching form of the solution. A more accurate method is to enforce (9.84) in the least-squares sense over the circle. This results in the weighting functions

536

ANTENNAS T: FUNDAMENTALS AND NUMERICAL METHODS

being proportional to the complex conjugates of the function differences within the braces. The integrations to evaluate the matrix elements in (9.85) are performed either numerically or semianalytically by using the basis function expansions employed in the interior solution to represent both the difference functions and the weighting functions . In either case, the resultant 2N + l square matrix can be inverted to obtain the scattered field coefficients. The scattered field may then be obtained from the expansion in (9.83). The interior field can also be found by using the weighted superposition of the stored interior field solutions. The unimoment-method has been employed in several computational efforts. One of the earliest of these involved the finite difference solution for radiation and input impedance of a finite-length biconical antenna, loaded by inhomogeneous dielectric [89]. This structure is depicted in Fig. 9.21. Since both the fields and material structure are axisymmetric in form (invariant to the phi-coordinate) the solution can be reduced to a single meridian plane. (r , 0) in spherical coordinates. A section of the finite difference mesh is shown in Fig. 9.22. The interior region solution for this antenna problem was formulated using a special case of the coupled azimuthal potential (CAP) formulation, where the fields are generated from the azimuthal components of E and H [125]. A sample result is shown in Fig. 9.23, wruch compares the computed and

Metal cone

/

Ground plane

t

Coaxial feed

Figure 9.21

Experimental dielectrically loaded biconical antenna.

\

Increasing/

\ - -Increasing i- --

Figore 9.22 Finite difference mesh for the biconical antenna.

9.2

NUMERICAL TECHNIQUES

537

75

eal

50

-

- - Computed - - - Measured

( /J

E

.c 0

25

C

Kr°

- 25 .___ __ __ ___.____ _ _..,,__ _

__,JL..,.__ _.....J

0

Figure 9.23

1

2

3

4

5

Comparison of measured and computed input impedance.

measured input impedance of a plexiglass loaded biconical antenna. The computation was performed at discrete frequencies over a 10: 1 range, wherein the bicone height ranged from 0.16 to 1.6 wavelengths. The CAP formulation has also been combined with the MEI boundary termination approach to compute fields from an electrically large parabolic reflector [102]. A second application of the uni.moment method is that of scattering by inhomogeneous bodies of revolution (90]. This effort employed a triregional finite element mesh in the (r , 0) meridional plane, as shown in Fig. 9.24. The CAP formulation was employed to represent the nonaxisymmetric fields using a Fourier series in the phicoordinate. Spherical harmonic expansions were used to represent both the scattered field outside the mesh and the total fields within the spherical "core" region. The sets of coefficients used in these field expansions were found by applying the various expansion modes for the potentials as BCs along the contours r = a and r = b. In addition, the various incident fields being considered were applied along r = b. A finite element solution for each BC was then evaluated along the inner contours, r = r 1 and r = r 2 • The total fields were then assembled from these numerical solutions and equated in the least-squares sense to the original analytical expansions, resulting in a matrix equation for the coefficients. Numerous comparisons to experiments were made for solid and hollow dielectric bodies of various shapes. A typical result is shown in Fig. 9.25, where the bistatic scattering is from a plexiglass body having cylindrical, conical, and hemispherical portions.

538

ANTENNAS I: FUNDAMENTALS AND NUMERICAL METHODS

Finite element region

Exterior region

r=b - ..,1..-:::.l---

Figure 9.24

Semiannular conformal finite element mesh.

25

-...

R ine

~h,

Q)

Q)

E

-

uj

0

>

TE incidence a =135°

20

15

H-plane pattern - - - -

I

"O

:ea. E cu

C>

- - - Computed

I

Q)

Measured

I

I

10

C

·c

Q) ~

~

CJ)

5

0 .___ _..___ ___.__ _.....__ _ ___.___ ___,____...,___ __.__ ___._ _....______

- 180

- 120

-60

0

60

120

~

180

Theta (degrees)

Figure 9.25 Scattering amplitude comparison for a plexiglass object computed using the unimoment method.

9.2

UMERICAL TECHNIQUES

539

Additional re earch efforts invol ing the uni.moment method include scattering by dielectric cylinder [122] and axi ymmetric raindrop [91 ], a well a cattering by multiple bodie of revolution [123] and even buried object (12➔]. In another ca e, the cattering problem wa o lved for a complex Jo y dje]ectric model of the hu man head, in order to evaluate the interior field di tribution a a function of frequency, incident aspect, and polarization (92]. It become numerically inefficient to u e a spherical me h to enclo e hape that occupy only a mall ponion of the pherical volume (e.g., thjn cylinders and flat disk ). Although it i po ible to utilize a eparable nonspherical urface to increa e the numerical efficiency of the interior region olution, this will be off et by additional requirements in generating the pecial function that are needed in the exte1ior expansion.

Field Feedback and Boundary Integral Methods. A technique to circumvent the need for a eparable boundary interface wa developed by Morgan et al. [93]. Thi hybrid finite element boundary integral (FEBn method combine a finite element solution of the interior region with the urface integral equation found in the extended boundary condition method. Toe FEBI allow the u e of a surface interface that conforms to the outer boundary of the cattering object, a shown in Fig. 9.26. The finite element solution proceeds in a imilar manner to that of the unimoment method with the incident field and cattered field spherical harmonic expansion modes being applied as BC at the outer boundary, SG. Numerical solutions are then found at the surface of the cattering body, S 8 , for each of the e applied BCs. To evaluate the expansion

Z-axis

t

SG ~-#--#--Jtl--11-__.~ ~

/

Direction of evolution

Figure 9.26 Surface conforming finite element mesh.

540

ANTENNAS I: FUNDAMENTALS AND NUMERJCAL METHODS

coefficients for the boundary field, a combined field integral equation may be used, as employed by Waterman in the "extended boundary condition formulation" (126]. This integral equation relates the tangential field just inside the boundary, Ss, to that just outside and does not make use of a knowledge of the material structure inside Ss. The FEBI method works well for diffraction calculations involving moderately elongated lossy dielectric and magnetic objects. An example computation, compared to that performed using the extended boundary condition method, is shown in Fig. 9.27. Solution convergenre becomes elusive, however, if the surface interface is sufficiently elongated or flattened (i.e., length-to-diameter ratios outside the range of about 0.1 to 10). This failure results because of the incomplete nature of the exterior region spherical harmonic expansion which is employed to represent the field over the nonspberical surface of the scattering body (127]. The field feedback formulation (F3) has been developed [94] to mitigate the restrictions inherent in coupling interior and exterior region field solutions. Using F3, the interior boundary value problem is initially decoupled from the outside region. The interior problem may then be formulated and computed using the most expedient approach that can accurately accommodate the level of material complexity that is present. The exterior region field is represented in terms of modes generated from surface integrations involving equivalent currents obtained from the interior region solution. These modal fields, which satisfy the radiation conditions, do not rely on the use of separable coordinate surfaces for their completeness. A primary advantage of this method is its modular nature, where the forward and feedback transfer matrices can be computed independently.

Horizontal pol., E-plane

Vertical pol., H-plane

1.0 ....,.....,...........,_..,..~-T-""1--.---.........--.---...................-..--..................-

..................- - - - - - 1

0

0• ~

E,+ /H

___v

I

I

2a = 0.5 Ao

- - - - - - FEBI

- ',,,

alb = 2

E,= 4 -

- - - - - FBCM

j/

......

' ,,....,

.... _--.,__ _______ _

0 ....._...._........._~.&.....1.....a&.i.....i.-L-..L....L.....&......L...J--'-...&....1.....i.....i.....L....L.-L-.L....L.....L..J.....J.-L..J....J-1....J...JU

180

90

0

90

8 (degrees)

Figure 9.27

Scattering amplitude comparison for FEBI computation.

180

9.2

0

NUMERICAL TECHNIQUES

lj

Rs

Forward A

+

seal ¢>(,.)

o

N

541

Far-field

r

~

-

=n=1 1: Cn 4>n(ro) -

.

N

K(r) = K'nc+ l: C K Feedback

n= 1 n

n

/J Figure 9.28

Field feedback formulation equivalent system.

3

The F scattering elution method may be conceptualized by considering the simple feedback system that is depicted in Fig. 9.28. The input to the forward transfer function, • , i an array composed of boundary nodal values of the field on the outer surface, SG as was shown in Fig. 9.26. The A-operator represents the finite method solution to attain the tangential field values on the inner surface, S 8 , in terms of any specified boundary data on Sc. These numerically derived tangential fields on S8 can then be u ed to form equivalent electric and magnetic currents (combined to form an array K) that generate the scattered field from the object. Green' s function integrations of these equivalent currents on SB , which are represented by the feedback operator, yield the scattered fields on the outer surface, SG. The overall transfer matrix of the F3 system in Fig. 9.28 is readily obtained and is similar in form to that from the analysis of a simple recursive control syste~ (9.86) where [I] is an identity matrix. The unknown coefficients may be found from this equation through the method of weighted residuals, where both sides are integrated over SG with respect to a set of N linearly independent weighting functions. Once the coefficients are obtained, the scattered field is found through far-field Green's function integrations of the assembled equivalent currents on SB . These integrations are denoted by the f-operator in Fig. 9.28. A simple demonstration of the F3 is scattering by a metallic thin wire [94]. As shown in Fig. 9.29, there is only a single column of finite elements in the mesh. This produces a banded global system matrix whose diagonal is only three matrix elements wide. The Riccati transform provides ultrafast inversion of this narrow-banded matrix. Comparisons of the magnitude and phase of current on a one-wavelength-long wire, as computed from the F3 and Hallen's integral equation, are shown in Fig. 9.30. In this particular F3 computation, some error resulted from the use of piecewise linear basis functions to represent the magnetic field. In the immediate vicinity of the wire, the actual magnetic field is characterized by a rapidly decaying evanescence which is not accurately "trackable" using linear functions . More specialized basis functions would allow a better natural convergence of the numerical solution. In a later effort [95], application of F 3 to scattering by resonant-sized arbitrarily shaped lossy inhomogeneous dielectric objects produced RCS calculations with typical errors of 1 % .

542

ANTENNAS I: FUNDAMENTALS AND NUMERICAL METHODS

z

L -b--'

2

I M+ N

M

;11.. =

-

---

p~~i~- [1!......---.

Triangulars elements

i\l!·------4 ---- R

0

\l:------4 :::::: :❖:•

~ \t -----L

-2

Figure 9.29

lit

n= 1

r

n=M + 1

Ordered

a = .02.l.

nodes

Finite element mesh for thin-wire scattering using F3 •

9.2.4 Geometrical Theory of Diffraction The method of moments and the differential equation-based method are most applicable when antenna sizes are on the order of no more than a few wavelength . For much larger antennas such as reflector antennas, a high-frequency method such as the geometrical theory of diffraction (GTD) or the uniform theory of diffraction (UTD) is generally more efficient to use because of the large number of unknowns that would be required for a subsectional modeling solution method and the consequent large amount of computer time that would be required to obtain the solution. GTD and UTD are ray tracing techniques which include the direct and reflected ray contributions of the geometrical optics field, but which add contributions to the field from diffracted rays. The diffracted rays are launched by surface discontinuities such as edges, comers, and tips, as well as by curved surfaces. These discontinuities may essentially be viewed as new "sources" that contribute to the field, but the strength and form of the sources are dependent on the type of discontinuity, on the type of illumination incident on the discontinuity, and on the incident angle of the illumination. Curved surfaces produce diffraction terms that appear to be distributed sources. GTD and UTD are approximate techniques that can be made increasingly accurate, within certain restrictions, as additional terms such as curved surface diffraction

9.2

NUMERICAL TECHNIQUES

543

4

Magnitu/

180

3

90

~

-

-....

E

a, "O

( /)

Q) Q)

.2

cO> as E

2

-........

0

O>

Q)

"O Q)

(/)

«s

C

.c

a,

a.

::,

X

l)

- - - F3

-90

xxxxxxx IE

1

..

-180

L =). N=70

0 _ _ _.;;..____..___ _ _----1_ _ _ _____..._ _ _ ___.

-0.5

-0.25

0

0.25

0.5

Z-axis (wavelengths)

Figure 9.30

Induced current comparison for integral equation and F3 computations.

terms and multiple diffraction terms are included. However, they are based on the assumption that diffraction is a localized phenomenon dependent only on the surface characteristics and the incident illumination in the immediate vicinity of the diffracting structure. The GTD description of various diffraction phenomena is obtained from the asymptotic solution of canonical scattering problems. Thus one should not use these techniques to compute fields very near the diffracting structures of an object, although in many situations reasonably accurate results can be obtained when the observation point is relatively close to the diffraction point (on the order of a wavelength, for example). In its simplest form, GTD adds a diffraction term to the geometrical optics (GO) field [17] due to a surface discontinuity such as the tip of the perfectly conducting wedge shown in Fig. 9.31. For a given source location the observed field due to GO contributions will be discontinuous as one crosses the shadow boundaries indicated in the figure, because as one crosses these boundaries, terms in the GO field representation are abruptly dropped. This is clear when one notes that in region II, the GO field is equal to the incident field, while in region ill, the GO field is zero. GTD adds to the GO field a diffraction term, which essential} y appears to be a dependent source emanating from the wedge tip. The line source radiates into all regions shown in Fig. 9.31. The contribution of the source, however, is dependent not only on the incident source

544

ANTENNAS I: FUNDAMENTALS AND NUMERICAL METHODS

\

Observation point

Region I

\ \ Diffracted ray

\

\

\ Reflection - shadow boundary

~\ \

\

I

Region II

I

I

Conducting wedge

=0

I I

//\ /

I I

Figure 9.31

Incident shadow boundary

Region Ill

Geometrical parameters for wedge diffraction problem.

type, polarization, and locatio~ but also on the observation distance and angle relative to the diffraction point, and on the type of structure causing the diffraction. The contribution of this source is known as the diffraction coefficient. It is determined as a correction to the GO field for an asymptotic solution of a canonical problem, such as the wedge geometry shown, and then it is applied to other more complex geometries. The diffraction coefficient must be discontinuous at shadow boundaries to offset the GO field discontinuity so that a continuous variation of the field is observed as one crosses the shadow boundaries. The diffraction coefficient expressions originally developed by Keller [128], however, were not accurate in the vicinity of the shadow boundaries. The term GTD is often used today to refer also to an improved theory known as UTD [129, 130], which provides for diffraction coefficients that produce accurate results even along shadow boundaries. For a given source and observation location, there is a point of diffraction along the wedge tip from which the diffracted rays emanate. The diffraction point is determined by minimizing the distance between the source and observation points along a path with a point on the wedge tip. If the source and observation points do not lie in a plane perpendicular to the diffracting edge, the diffraction rays emanate in a cone surrounding the wedge tip, where the vertex of the cone is located at the diffraction point and where the half-angle of the cone is equal to the angle between the incident ray and the line formed by the wedge tip. The diffracted field Ed can be written in

9.2

545

NUMERJCAL TECHNlQUES

general form as (9.87) where E;(r d) i the incident illumination at the point of diffraction r d, D i the dyadic diffraction coefficient, and A i a patial attenuation factor along the diffracted ray. The arguments s and s' are the di tance from the observation and source points, re pectively. to the diffraction point r d. The diffraction field is conveniently eparated into two vector components which are parallel and perpendicular to the plane of diffraction, which j a plane containing the diffraction ray and the diffracting edge. The incident illumination i treated in the ame way for the plane of incidence, which i a plane containing the diffracting edge and the incident ray. When this is done, the dyadic D 1 a diagonal dyadic who e coefficients are gi en by (9.88) where 0' i the incident angle with respect to the diffracting edge and ' is the azimuthal angle of the ource relative to the diffracting edge; together with the distances' of the ource from the point of diffraction, these quantities uniquely define the location of the ource in a spherical coordinate sy tern (s' . 0', ') centered at the diffraction point. ote that the diffraction point r d must be determined before (9.88) can be applied and the equation is therefore alid in what is referred to as a ray-fixed coordinate system. The u 8 functions in (9 .88) are generalized versions of the canonical solutions for the diffracted field in the wedge problem of Fig. 9.31 subject to plane-wave illumination. For a wedge formed by the plane surfaces= 0 and= nrr as shown in Fig. 9.3 1, these functions may be written as e-j (rr/ 4Je -jkL {

1C

+a

uB(L , a)=------;:==- cot--F[kLg+(a)] 2n ✓2rrkL 2n

+ cot

Jr -

2n

a

F[kLg-(a) ]

}

(9.89)

with (9.90) and (9.91) In (9 .91 ), N ± are integers which most nearly satisfy the equations

2n1r N + - ( ± ') 2nrr N - - (

= rr

± ' ) = - 1r

(9.92) (9.93)

The quantity L appearing in the expressions is a generalized distance parameter resulting from UTD work (129]. It is determined by requiring the total field to be continuous at shadow boundaries. It may be defined succinctly in terms of the radii of curvature of the incident wavefront, or for simplicity, in tenns of different types of illuminations

546

ANTENNAS I: FUNDAMENTALS AND NUMERICAL METHODS

as given below:

s sin2 0' pp' L=

p

for plane waves for cylindrical waves

+ p'

ss' sin2 0' s + s'

(9.94)

for conical and spherical waves

In (9.94) p and p' represent the usual cylindrical coordinate distances from the diffracting edge to the observation and source points, respectively. The spatial attenuation factor A used in (9.94) is defined as 1 A(s , s')

=

for plane, cylindrical, and conical wave illumination

Js I [

s(s: s')

]

(9.95)

J/ 2

for spherical wave illumination

The application of the basic methods of GTD as described here is a relatively straightforward task which can be applied to a wide range of problems with excellent results. Increasingly sophisticated methods and models may be developed with GTD, however. Additional information concerning advanced procedures can be found in Refs. 5, 130 and 131. For diffraction from curved edges, for example, slightly modified forms of the diffraction coefficients given here provide more accuracy [130]. Diffraction coefficients have also been obtained for higher-order edges [132]. Diffraction from tips, such as a cone vertex, may be included in some special cases [130]. The inclusion of multiple diffraction terms, in which a diffracted ray is itself diffracted by another surface discontinuity (or the same one, after reflection, for example), increases the accuracy of the GTD model, but the "bookkeeping" task in the computer program also becomes correspondingly more difficult. GTD has also been applied to problems in which the scattering surfaces are not perfect conductors [133, 134]. Smoothly curved surfaces also "diffract" energy in the sense that if an incident ray grazes a smooth curve, a portion of the energy of the ray is carried around the surface into the shadow region [129, 130]. This phenomenon, referred to as a creeping wave, may be viewed as a wave that attaches itself to the surface and which sheds energy as its propagation direction follows the curve of the surface. Implementation of the creeping wave phenomenon is generally much more complex than that of the simple edge diffraction form described here. The creeping wave follows a geodesic path along the surface of the body as it sheds energy. Except for very specialized geometries, the computation of the geodesic path followed by a creeping wave is a difficult task that hinders the implementation of this feature in GTD solutions. The GTD approach as presented here still suffers from one serious drawback-it cannot be used to compute the field at a caustic. A caustic is a point or line along which the rays converge to create a singular (and nonphysical) solution. The caustic region arises when curved surfaces or curved edges focus the rays. Alternative formulations must be used to avoid this difficulty. One possible approach to overcome this problem for diffracted fields is to use the concept of equivalent currents [135, 136]. The equivalent current method is employed essentially by eliminating the offending diffracting edge and replacing it with equivalent electric and/or magnetic line sources.

9.2

NUMERICAL TECHNIQUES

547

The strength of the equivalent ource are determined by again olving a canonical problem, and the equivalent current are found to depend on parameters in a manner imilar to that of the diffraction coefficient . Once the form for the equivalent currents are determined, the diffracted field quantitie due to the edge are computed in the usual manner through radiation integral . Another approach to remove thi difficulty is the use of a cau tic-corrected olution [I 37]. Alternative formulations such as the Incremental Theory of Diffraction (ITD) that are valid at caustics and shadow boundaries have al o been developed [138] .

9.2.S

Antenna Environment

Antennas are rarely located in the homogeneous free- pace environment that one would prefer for numerical analy i . There are almost always other cattering bodies or surface in the vicinity that hould be included in the antenna model to obtain the most accurate re ults. These may include for example, the perfectly conducting ground plane or po sibly the imperfectly conducting earth ground above which an antenna is located, the satelli~e body on which an antenna i mounted, the mounting platform or feed struts for a reflector antenn~ or imply other nearby structures, such as bulkheads or other antennas. In ome situations it may be assumed that such scattering structures do not significantly influence the antenna radiation pattern or input impedance, and they are simply ignored. In other cases approximations may be made to obtain a more manageable "'scatterer·' geometry. For example, an antenna situated above a fairly large finite ground plane may be modeled as an antenna above an infinite ground plane. Image theory may then be used, of course, and the analysis problem is greatly simplified. When all else fails. one actually attempts to model the environment more accurately. Various approaches may be taken to include the presence of nearby scattering structures in the model, depending on the geometry of the scattering surfaces. For a very few specialized geometries, the Green's function for an elementary dipole source radiating in the presence of the body may be known such as is the case for a perfectly conducting or dielectric sphere illuminated by an elementary dipole or a plane wave. This information may be used in a method of moments solution procedure. For example, Fig. 9.32 illu trates a situation in which a scatterer is assumed to be located near the surface of an antenna to be modeled by the method of moments. In cases such as this, the total electric field Er at the antenna surface may be written as (cf. Figure 9.32) (9.96) where Ee is the field produced at r by the excitation source J e in the absence of all scattering surfaces, Ee8 is the field produced at r due to the scattering of the excitation field by the scatterer with the antenna removed, Es is the field produced at the antenna surface due to radiation by the current elements on the antenna surface with the scattering body removed, and Es8 is the field produced at the antenna surface due to indirect radiation by the antenna current elements (i.e., radiation by the current elements which is scattered back to the antenna surface by the scattering body). When the Green's function for the scattering body is known, Ee8 and E sB can be represented analytically if the excitation and antenna current sources can be modeled by plane waves or elementary sources. When the "scatterer" is an infinite ground plane, for example, Ee8 represents the portion of the incident field reflected by the ground plane

548

ANTENNAS I: FUNDAMENTALS AND NUMERICAL METHODS

~ Antenna

~catterer

Source J8

Figure 9.31. Direct and scattered contribution to antenna surface field for scatterer located near antenna.

to· the antenna surface, while E sB represents the field due to the images of the currents induced on the antenna surface. If the analytical form of the Green's function for the scatterer is not known, the terms E eB and E sB may instead be calculated numerically for various source and observation point locations via solution of a scattering problem and the results then stored in a matrix [41] . This approach yields a matrix representation for a portion of the Green's function for the body and it is sometimes referred to as a "numerical Green's function." The numerical Green's function approach may be useful in some situations to reduce computation time, storage requirements, or to provide flexibility in a solution procedure. Equation (9 .96) is also relevant to a hybrid :rviM/GTD model for scattering problems [17, 139). In the hybrid approach GTD provides an approximate Green's function for the elementary MM sources radiating in the presence of the scattering body. Figure 9.33 schematically illustrates the various contributions to the MM forcing function vector (Fig. 9.33a) and to the moment matrix (Fig. 9.33b) for a sleeve dipole antenna fed by a magnetic current ring source, for example, radiating in the presence of a wedge-shaped ground plane. For the hybrid approach, the terms with superscript B appearing in (9.96) are conveniently broken down further into diffraction and reflection terms as indicated in the figure. The MM solution obtained in this case would comprise only the values of the wire current coefficient basis set Equation (9.96) must be used again to determine the radiation field of the antenna mounted on the wedge-shaped ground plane. In many antenna calculations it is also necessary to include the effect of the imperfect ground (earth or seawater, for example) above which the antenna is located. If one assumes that the imperfect ground is planar and that the illumination is an incident plane wave, the E eB term in (9 .96) is easily calculated for this case as reflection from a dielectric interface. For elementary excitation sources and for the elementary equivalent sources modeling the antenna, precise computations involve the evaluation of Sommerfeld integrals, a task that has been studied widely (140-143]. Approximate methods involving the use of asymptotic expansions and reflection coefficient models,

9.2 NUMERICAL TECHNIQUES

549

(a) Antenna with magnetic frill feed

E eB dif EeB ref

Wedge ground plane

(b)

ESB dif

ESB ref

Wedge ground plane

Figure 9.33 Schematic representation of direct, reflected, and diffracted contributions from (a) frill source and (b) equivalent current source for antenna on wedge-shaped ground plane.

for example, for sources located above conducting media, are useful alternatives in some cases [144-146]. The reflection coefficient approach is particularly simple and is very efficient to use, but it may not provide sufficient accuracy for some parameter ranges, such as when the observer and source are close to the ground. In this approach image sources are used to model the effect of the ground, with the strength of the

550

ANTENNAS I: FUNDAMENTALS AND NUMERICAL METHODS

image sources being related to the reflection coefficient for an incident plane wave with the appropriate polarization. Other hybrid methodologies are also available for the analysis of antennas in a complex environment. A review of some of these methods is provided in [147] . Advances also continue to be made in improving and implementing various hybrid techniques using Physical Optics (PO) [148], Shooting and Bouncing Rays (SBR) [149], iterative methods [150] , and GTD/UTD [151, 152].

9.2.6 Antenna Synthesis The methods that have been described briefly in this section are antenna analysis methods. They are generally applied to an antenna of known configuration to determine the current distribution on the antenna and/or the radiation characteristics of the antenna. The antenna synthesis problem usually involves the design of an antenna configuration that will possess a desired set of radiation characteristics. The problem may consist not only of choosing the general antenna configuration, but also of choosing, for example, appropriate dimensions for the antenna elements and their spacings as well as choosing the appropriate excitation distribution to provide the desired antenna characteristics. In addition, there may be constraints placed on the overall dimensions of the antenna, the dimensions of some antenna elements, or on the excitation distribution to permit physical realizability. An antenna design is generally begun by choosing a particular class of antenna configurations that the designer believes has the potential to satisfy the physical constraints required and to produce the desired radiation characteristics. One might begin, for example, by choosing a linear array antenna as the basic configuration. The elements of the linear array might be either aperture, surface, or wire radiators, depending on the physical requirements of the problem. Various synthesis approaches are then available to obtain array factors that satisfy some design criterion. For example, the Schelkunoff polynomial method is useful for designing arrays whose patterns have nulls in specified directions [153] . The Fourier transform and the Woodward [154, 155] methods are useful for synthesizing a radiation pattern over the entire visible range. The Dolph-Chebyshev [156] and Taylor [157] methods are useful for obtaining patterns having narrow main beams as well as low sidelobe levels. Extensions of some of these methods are also available for treating, for example, particular classes of rectangular arrays [158, 159]. Each of the preceding procedures, however, involves the u e of array elements for which the current distribution is known or can be computed via numerical techniques. When it must be computed, the current distribution on an array element is usually obtained either for the element radiating alone or as an element of an infinite array. If the distribution for an element radiating alone is used, the interactions between array elements are ignored and results may or may not be accurate, depending on the particular configuration of array elements and the particular type of element. For very large arrays the use of an infinite array model does provide information on the mutual interactions, although the mutual interaction information may not be accurate for elements near the edges of the array. Furthermore, these methods assume that the array elements are all of the same type. For more general types of arrays, numerical techniques may provide a useful alternative. For a single antenna or an array antenna that can be modeled using numerical techniques such as those described in this section, a self-consistent approach to the

9.2

NUMERICAL TECHNIQUES

551

synthesis problem can be developed using nonlinear optimization methods. In thi s type of procedure one actually olves analysis problems for various values of antenna parameters. A multidimen ional earch procedure is employed to adjust the antenna parameters until a "cost" or objective function involving the desired radiation characteristics is minimized. The antenna model in thi procedure can be made complete in the sense that all mutual interactions between antenna elements may be included. The problems, of course, are that the antenna or array must be modeled in its entirety, that a numerical solution for the radiated fields (for example) must be computed many time using different parameter in the antenna configuration, and that a suitable objective function must be developed. Since there may be a very large number of parameters that might be varied for any given antenna configuration the optimization i generally performed with respect to one parameter or a few specific parameters. For example, a linear array consisting of wire elements might be optimized only with respect to wire placemen~ wire radii, or wire lengths; or one may choose to perform the optimization with respect to all three of these parameters, but not to optimize with re pect to more complex parameters such as wire orientation. One should also note that the objective function may possess local minima, so results of the first optimization may not represent the best solution. Starting the search process from a different set of initial parameter values may produce a better solution. Another means of obtaining a better solution from the optimization process is to perform the optimization several times, each time using different objective functions involving modifications of the acceptable radiation characteristics or of the importance of each characteristic. Because one or more antenna analysis problems must be solved at each step of the process, optimization can be a rather costly procedure in terms of computer time. The method provides a great deal of flexibility in the synthesis process, however, in terms of the many different antenna parameters which can be modified in the search procedure. It also provides the designer with the ability to optimize the antenna with respect to more than one radiation characteristic by using an objective function that includes the desired radiation characteristics weighted in an appropriate fashion. Optimization of a Yagi antenna using the method of moments and the conjugate direction optimization method has been presented in Refs. 160 and 161. In Ref. 160 the Yagi antenna is optimized with respect to a composite cost function involving forward directivity and mean-square sidelobe level. In Ref. 161 the optimization procedure is employed to increase the bandwidth of the Yagi antenna. Significant advances have also been made in antenna synthesis techniques through the application of global optimization procedures such as genetic algorithms (162, 163] and simulated annealing [164]. Genetic algorithms are based on evolution of a solution through a natural selection process, where the most fit solutions are selected or given preference at each step of the evolution. Simulated annealing is based on the analogy of the slow cooling of a liquid into its lowest energy state along with the Boltzmann probability that a particular energy state will exist at the next lower temperature level. Both of these methods are "global" in the sense that they permit random variations in the solution that enable the solution search process to escape from local minima. They may require more computation time than direct optimization methods, however, because a sufficient sampling of the solution space must be carried out to ensure that a global minimum is found. Genetic algorithms and simulated annealing have been applied to a wide variety of antenna optimization problems [165-172] .

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REFERENCES 1. R. F. Harrington, Field Computation by Moment Methods , Macmillan, New York, 1968. 2. K. K . Mei, M. A. Morgan, and S. K. Chang, "Finite Methods in Electromagnetic Scattering," in P. L. E. Ushlenghi, Ed., Electromagnetic Scattering, Academic Press, New York, 1978, Chap. 10. 3. N. Engheta, W. D. Murphy, V. Rokhlin, and M . S. Vassiliou, ''The Fast Multipole Method (FMM) for Electromagnetic Scattering Problems," IEEE Trans. Antennas Propagat., 40, pp. 634-641, Jl!!}e 1992. 4. J. Song, C. Lu, and W . C. Chew, "Multilevel Fast Multipole Algorithm for Electromagnetic Scattering by Large Complex Objects," IEEE Trans. Antennas Propagat., 45, pp. 1488- 1493, October 1997. 5. G. L. James, Geometrical Theory of Diffraction for Electromagnetic Waves , Peter Peregrinus, London, 1976. 6. A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-/Jomain Method, 2nd Ed, Artech House, Norwood, MA, 2000. 7. K. S. Kunz and R. J. Luebbers, The Finite Difference Time Domain Method for Electromagnetics , CRC Press, Boca Raton, FL, 1993. 8. W. J. R Hoefer, "The Transmission Line Matrix (TLM) Method" in T. Itoh, Ed., Numerical Techniques for Microwave and Millimeter-Wave Passive Structures, Wiley, New York, 1989, pp. 496-591. 9. C. Christopoulos, The Transmission-Line Modeling Method: TIM , IEEE Press, Piscataway, NJ, 1995. 10. J. Lee, R. Lee, and A. Cangellaris, "Time-Domain Finite-Element Methods," IEEE Trans. Antennas Propagat., 45, pp. 430-442, March 1997. 11. S. M. Rao, Ed., Time Domain Electromagnetics, Academic Press, San Diego, CA, 1999. 12. A. A Ergin, B . Shanker, and E . Michielssen, "Plane-Wave Time-Domain Algorithms," in W. C. Chew, J.-M. Jin, E . Michielssen, and J. Song, Eds., Fast and Efficient Algorithms in ComputationalElectromagnetics, Artecb House, Norwood, MA, 2001 , pp. 815-858. 13. B. Shanker, A. A. Ergin, K. Aygiln, and E. Michielssen, "Plane-Wave Trme-Domain Algorithm Enhanced Time-Domain Integral Equation Solvers," in W . C. Chew, J.-M. Jin, E. Michielssen, and J. Song, Eds., Fast and Efficient Algorithms in Computational Electromagnetics, Artecb House, Norwood, MA, 2001, pp. 859-892. 14. R . F. Harrington, 1ime Harmonic Electromagnetic Fields, McGraw-Hill, New York, 1961. 15. G. J . Burke and A. J. Poggio, Numerical Electromagnetic Code (NEC)-Method of Moments , Parts I, II, and Ill, Tech. Doc. NOSC TD 116, Naval Ocean Systems Center, San Diego, CA, July 1977 (revised: January 1980). 16. G. A. Thiele, "Wire Antennas," in R . Mittra, Ed., Computer Techniques for Electromagnetics, Pergamon Press, Oxford, 1973, Chap. 2. 17. W . L. Stutzman and G. A Thiele, Antenna Theory and Design, Wiley, New York~ 1981. 18. C. M. Butler and L. L. Tsai, "An Alternate Frill Feed Formulation," IEEE Trans. Antennas Propag., AP-21, pp. 115-116, January 1973. 19. L. L . Tsai, "A Numerical Solution for the Near and Far Fields of an Annular Ring of Magnetic CUrrent," IEEE Trans. Antennas Propag., AP-20, pp. 569-576, September 1972. 20. R. W . Adams, A. J. Poggio, and E. K. Miller, Study of a New Antenna Source Model, Tech. Rep. UCRL-51693, Lawrence Livermore Lab., Lawrence, CA, October 1974. 21. J. K . Breakall and A. P. Ludwigsen, "A Comparison of NEC and MININEC Results with Experimental Measurements," Conj. Proc. 2nd Ann. Rev. Prog. Appl. Comput. Electromagn., Monterey, CA, March 1986.

REFERENCES

553

22. G. P. Junker, A. W . Glisson. and A. A. Kishk, "A Modified Extended Delta Source Model,'' Microwave and Optical Tech. Lett., 12 pp. 140- 144, June 1996. 23. P . G. Rogers and M. W. Gunn, "An Entire-Domain Galerkin AnaJysis of the Moderately Thick Dipole," IEEE Trans. Antennas Propag. , AP-28, pp. 11 7- 121 , January 1980. 24. J. R. M autz and R. F. Harrington, ''Radiation and Scatteri ng from Bodies of Revolution," Appl. Sci. Res., 20, pp. 405-435, June 1969. 25. L. . Medgye i-Mitschang and C. Eftimiu, "Scattering from Wires and Open Circular Cylinder of Finite Le ngth U ing Entire Domain Galerkin Expansions," IEEE Trans. Amennas Propag., AP-30, pp. 628-636, July 1982.

26. L . . Medgye i-Mitschang and J. M. Putnam, "Scattering from Finite Bodies of Translation: Plate , Curved Surface , and NoncircuJar Cylinders," IEEE Trans. Antennas Propag.. AP-31 . pp. 847-852, · ovember 1983. 27. L. . M edgyesi-Mitschang and J. M . Putnam, "Electromagnetic Scattering from Extended Wire and 1\vo- and Three-Dimensional Surfaces," IEEE Trans. Antennas Propag., AP-33, pp. 1090-1100, October 1985. 28. D . R. Wilton and S . Govind, ·Tocorporation of Edge Conditions in Moment Method Solutions." IEEE Trans. Antennas Propag., AP-25, pp. 845-850, November 1977. 29. A . W . Glis on and D. R. Wuton, Numerical Procedures for Handling Stepped-Radius Wire Junctions, Final Rep .. Contract N66001-77-C-0156, Naval Ocean Systems Center, San Diego, CA, March 1979. 30. A . R. Djordjevic, MB. B azdar, G. M . Vitosevic, T. K. Sarkar, and R. F. Harrington, Alralysis of Wire Antennas and Scatterers, Software and User 's Manual , Artech House, orwood. MA 1990. 31. A . R . Djordjevic, M . B . Bazdar, T . K. Sarkar, and R. F. Harrington, "Solution of TwoPotential Equation for Wire Structures Using Polynomial Expansion and Pulse Testing Functions," Proceedings of International Co,iference on Antennas and Propagation, pp. 540- 543, Yor~ 1991.

32. E. K. Miller and F. J. Deadrick. "Some Computational Aspects of Thin-Wire Modeling," in R. Mittra. Ed., Numerical and Asymptotic Techniques in Electromagnetics, SpringerVerlag, New York, 1975, Chapter 4 . 33. A . W. Glisson and D . R. Wilton, "Simple and Efficient Numerical Methods for Problems of Electromagnetic Radiation and Scattering from Surfaces," IEEE Trans. Antennas Propag.• AP-28, pp. 593-603, September 1980. 34. J. H . Richmond. "Coupled Li near Antennas with Skew Orientation," IEEE Trans. Antennas Propag., AP-18, pp. 694-696, September 1970. 35. J. C. Logan and J. W . Rockway, The New MININEC (Version 3): A Mini-numerical Electromagnetic Code, Tech. Doc. NOSC TD 938, Naval Ocean Systems Center. San Diego, CA, September 1986. 36. D. R. Wilton, Wire Problems, Leet. Notes for "Short Course on Computational Methods in Electromagnetics," SCEEE, St. CJoud, FL, November 1980. 37. C. M. Butler and D . R. Wilton, "Analysis of Various Numerical Techniques Applied to Tb.in-Wrre Scatterers," IEEE Trans. Antennas Propag., AP-23, pp. 534- 540, July 1975. 38. C. M . Butler, D . R. Wilton, and A . W. Glisson, Fundamentals ofNumerical Solution Techniques in Electromagnetics, Short Course Notes, University of Mississippi University, October 1984.

39. A. W. Glisson and D . R. Wilton, Simple and Efficient Numerical Techniques for Treating Bodies of Revolution, Tech. Rep. RADC-TR-79-22, Rome Air Development Center, Griffiss AFB, Rome, NY, March 1979.

554

ANTENNAS I: FUNDAMENTALS AND NUMERICAL METHODS

40. J. R. Mautz and R. F. Harrington, An Improved E-Field Solution for a Conducting Body of Revolution , Tech. Rep. TR-80-1, Syracuse University, Syracuse, NY, 1980. 41. A. W. Glisson and C. M. Butler, "Analysis of a Wire Antenna in the Presence of a Body of Revolution," IEEE Trans. Antennas Propag., AP-28, pp. 604-609, September 1980. 42. J. F. Shaeffer and L. N. Medgyesi-Mitschang, ''Radiation from Wire Antennas Attached to Bodies of Revolution: The Junction Problem," IEEE Trans. Antennas Propag., AP-29, pp. 479-487, May 1981. 43. G. P. Junker, A. A. Kishk, and A. W. Glisson, "Input Impedance of Dielectric Resonator Antennas Excited by a Coaxial Probe," IEEE Trans. Antennas Propagation, 42, pp. 960-966, July 1994. 44. S. M. Rao, D . R. Wilton, and A. W. Glisson, "Electromagnetic Scattering by Surfaces of Arbitrary Shape," IEEE Trans. Antennas Propag., AP-30, pp. 409-418, May 1982. 45. E. H. Newman, P. Alexandropoulos, and E . K Walton, "Polygonal Plate Modeling of Realistic Structures," IEEE Trans. Antennas Propag., AP-32, pp. 742-747, July 1984. 46. S. M. Rao, "Electromagnetic Scattering and Radiation of Arbitrarily-Shaped Surfaces by Triangular Patch Modeling," Ph. D . Dissertation, University of Mississippi, University, August 1980. 47. A. W . Glisson, D. R. Wilton, and D . H . Schaubert, "Integral Equation Methods for Electromagnetic Scattering by Arbitrarily Shaped Bodies," Proc. Workshop Electromagn. Field Comput., Schenectady, NY, pp. F-25-F-30, October 1986. 48. J. R. Mautz and R. F. Harrington, " H -Field, £-Field, and Combined-Field Solutions for Conducting Bodies of Revolution/' Arch. Elektr. Ubertr. , 32, pp. 157-164, April 1978. 49. A. W . Glisson, M . Orman, F. Falco, and D. Koppel, "Electromagnetic Scattering by an Arbitrarily Shaped Surface with an Anisotropic Impedance Boundary Condition," Appl. Comp. Electromag. Soc. J., 10(3), pp. 93- 106, 1995. 50. W. A. Johnson, D. R. Wilton, and R. M . Sharpe, "Modeling Scattering from and Radiation by Arbitrary Shaped Objects with the Electric Field Integral Equation Triangular Surface PATCH Code," Electromagnetics, 10(1- 2), pp. 41 - 63. Jan.-June 1990. 51. S. U. Hwu, D . R Wilton, and S. M. Rao, "Electromagnetic Scattering and Radiation by Arbitrary Conducting Wire/Surface Configurations," 1988 JEEE Int. Antennas Propagat. Symp. Dig., 26, pp. 890-893, June 1988. 52. I. Tekin and E. H. Newman, ''Method of Moments Solution for a Wue Attached to an Arbitrary Faceted Surface," IEEE Trans. Antennas Propagat. , 46. pp. 559- 562 April 1998. 53. S. Bhattacharya, S. A. Long, and D . R. Wilton, "The Input Impedance of a Monopole Antenna Mou~ted on a Cubical Conducting Box," IEEE Trans. Antennas Propagat., 3S, pp. 756- 762, July 1987. 54. M. I. Sancer, R. L. McClary, and K. J. Glover, "Electromagnetic Computation using Parametric Geometry," Electromagnetics , 10(1 - 2), pp. 85- 103, Jan.-June 1990. 55. D. L. Wilkes and C. Cha, ''Method of Moments Solution with Parametric Curved Triangular Patches," 1991 IEEE Int. Antennas Propagat. Symp. Dig., 29, pp. 1512- 1515, June 1991. 56. S. Wandzura, "Electric Current Basis Functions for Curved Surfaces," Electromagnetics, 12( 1), pp. 77- 91, 1992. 57. B. D. Popovic and B. M. Kolundzija, Analysis of Metallic Antennas and Scatterers, IEE, London, 1994. 58. R. D. Graglia, D. R. Wilton, and A. F. Peterson, ''Higher Order Interpolatory Vector Bases for Computational Electromagnetics," IEEE Trans. Antennas Propagat., 45, pp. 329-342, March 1997.

REFERE CES

555

59. W . J. Brown and D . R. Wtlton, •'Singular Basi Function and Curvilinear Triangle in the Solution of the Electric Field Integral Equation," IEEE Trans. Antennas Propagat., 47. pp. 347-353, February 1999. 60. B. M. Kolundfija and A. Djordje ic. Analysis and Synthesis of Composite Metallic and Dielectric Stn,cwres, Artech Hou e. Norwood, MA, 2002. 61. C. M . Butler and K. R. Umashankar. "Electromagnetic Excitation of a Wire Through an Aperture-Perforated Conducting Screen." IEEE Tran . Ante1111as Propag., AP-24, pp. 456- 462. July 1976. 62. S . . Sinha, D. K. Mehra, and R. P. Agarwal. "Radiation from a Wavegu ide-B acked Aperture in an Infinite Ground Plane in the Pre ence of a Thin Conducting Plate." IEEE Trans. Antennas Propag., AP-34, pp. 539- 545, April 1986. 63. T . Wang, R. F. Harrington. and J. R. Mautz, "Electromagnetic Scattering from and Transmission through Arbitrary Aperture in Conducting Bodie ," IEEE Trans. Antennas Propagat.. 38, pp. 1805-1814. ovember 1990.

64. A. W. Gli on. 'Tu}uivalent Current Excitation for an Aperture Antenna Embedded in an Arbitrarily Shaped impedance Surface." IEEE Trans. Antennas Propagat., SO, pp. 966-970, July 2002. 65. J. R. Mautz and R. F. Harrington. ·'Electromagnetic Scattering from a Homogeneou Material Body of Revolution;' Arch. E/ektr. Ubertr., 33, pp. 71-80, February 1979. 66. A. A. Kishk and L. Sbafai, "Different Formulations for Numerical Solution of Single or Multibodie of Revolution with Mixed Boundary Condition s," IEEE Trans. Antennas Propag.. AP-34, pp. 666-673, May 1986. 67. S . Govind, D . R. WLlton, and A. W. Glisson, "Scattering from Inhomogeneous Penetrable Bodies of Revolution,', lEEE Trans. Antennas Propag.. AP-32, pp. 1163-1173, November 1984. 68. L. . Medgyesi-Mitschang and J. M. Putnam, "Electromagnetic Scattering from Axially Inhomogeneous Bodies of Revolution," IEEE Trans. Antennas Propag., AP-32, pp. 797-806, August 1984. 69. M . A. Leontovich, Investigations on Radiowave Propagation, Part LI , Printing House of the Academy of Sciences, Mo cow, Russia, 1948. 70. T . B. A. Senior, "Impedance Boundary Conditions for Imperfectly Conducting Surfaces," Appl. Sci. Res., Sect. B(8), pp. 418-436, 1960. 71. K. A. Iskander, L . Sbafai, A. Frandsen, and J. E. Hansen, "Application of Impedance Boundary Conditions to Numerical Solution of Conugated Circular Homs," IEEE Trans. Antennas Propagat., 30, pp. 366-372, May 1982. 72. T. B. A. Senior, "Impedance Boundary Conditions for Statistically Rough Surfaces," Appl. Sci. Res. , Sect. B(8), pp. 437-462, 1960. 73. K. Umashankar, A. Taflove, and S. M . Rao, "Electromagnetic Scattering by Arbitrary Shaped Three-dimensional Homogeneous Lossy Dielectric Objects," IEEE Trans. Antennas Propagat., 34, pp. 758-766, June 1986. 74 . A . W. Glisson, "Electromagnetic Scattering by Arbitrarily Shaped Surfaces with Impedance Boundary Conditions," Radio Science , 27(6), pp. 935-943, 1992. 75. J.M. Putnam and L. N. Medgyesi-Mitschang, "Generalized Method of Moments for Three-dimensional Penetrable Scatterers," J. Opt. Soc. Am. A , 11, pp. 1384- 1398, April 1994. 76. J. Shin, A. W. Glisson, and A A. Kishk, "Modeling of General Surface Junctions of Composite Objects in an SIFJMoM Formulation," Proc. of the 16th Annual Review of Progress in Applied Computational Electromagnetics, pp. 683 - 690, Monterey, CA, March 2000.

556

ANTENNAS I: FUNDAMENTALS AND NUMERICAL METHODS

77. K. A. Michalski and J. R. Mosig, "Multilayered Media Green's Functions in Integral Equation Formulations," IEEE Trans. Antennas Propagat., 45, pp. 508-519, March 1997. 78. J. Shin, A. A. K.ish.k, and A. W . Glisson, "Analysis of Rectangular Dielectric Resonator Antennas Excited through a Slot over a Finite Ground Plane," 2000 IEEE Int. Antennas Propagat. Symp. Dig., 38, pp. 2076- 2079, July 2000. 79. W. L. Wu, "Analysis of Electrically Small Antennas and Scatterers in the Presence of Thin Material Layers," Ph.D. Dissertation, University of Mississippi, University, MS 38677, May 1995. 80. M R Abdul-Qaffoor, H. K . Smith, A . A . Kishk, and A. W . Glisson, "Simple and Efficient Full-Wave Modeling of Electromagnetic Coupling in Realistic RF Multilayer PCB Layouts," IEEE Trans. Microwave Theory Tech., 50, pp. 1445-1457, June 2002. 81. J. H . Richmond, "Scattering by a Dielectric Cylinder of Arbitrary Cross Section Shape," IEEE Trans. Antennas Propag_, AP-13, pp. 334-341, May 1965. 82. D. E. Liversay and K. M . Chen, "Electromagnetic Fields Induced Inside Arbitrarily Shaped Biological Bodies," IEEE Trans. Microwave Theory Tech., MTT-22, pp. 1273- 1280, December 1974. 83. M . J. Hagmann, 0. P. Ghandhi, and C.H. Durney, "Numerical Calculation of Electromagnetic Energy Deposition for a Realistic Model of Man," IEEE Trans. Microwave Theory Tech. , MTT-27, pp. 804-809, September 1979. 84. S. Rukspollmuang and K. M. Chen, "Heating of Spherical versu s Realistic M odels of Human and Infrahuman Heads by Electromagnetic Waves," Radio Sci., 14(6S), pp. 51 - 62, November-December 1979. 85. D . H. Schaubert, D. R Wilton, and A. W . Glisson, "A Tetrahedral Modeling Method for Electromagnetic Scattering by Arbitrarily Shaped Inhomogeneous Dielectric Bodies," IEEE Trans. Antennas Propag., AP-32, pp. 77-85, January 1984. 86. X. Sheng, J. Jin, J. Song, C. Lu, and W. C. Chew, "On the Formulation of Hybrid FiniteElement and Boundary-Integral Methods for 3-D Scattering." IEEE Trans. Antennas Propagat. , 46, pp. 303-311, March 1998. 87. C. C. Lu and W. C. Chew, "A Coupled Surface-Volume Integral Equation Approach for the Calculation of Electromagnetic Scattering from Compo ite Metallic and Material Targets/' IEEE Trans. Antennas Propagat., 48, pp. 1866- 1868, December 2000. 88. K . K. Mei, '1.Jnimoment Method of Solving Antenna and Scattering Problems, ' IEEE Trans. Antennas Propag., AP-22, pp. 760-766, November 1974. 89. R. E. Stovall and K . K. Mei, "Application of a Unimoment Technique to a Biconical Antenna with Inhomogeneous Dielectric Loading,'' IEEE Trans. Antennas Propag., AP-23, pp. 335-341, May 1975. 90. M. A. Morgan and K. K. Mei, "Finite Element Computation of Scattering by Inhomogeneous Penetrable Bodies of Revolution,'· IEEE Trans. Antennas Propag., AP-27, pp. 202- 214, March 1979. 91. M . A. Morgan "Finite Element Computation of Microwave Scattering by Raindrops," Radio Sci. 15, pp. 1109- 1119, November 1980. 92. M. A. Morgan, "Finite Element Calculation of Microwave Absorption by the Cranial Structure," IEEE Trans. Biomed. Eng., BME-28, pp. 687- 695, October 1981. 93. M. A. Morgan, C. H . Chen, S. C. Hill, and P. W. Barber, "Finite Element-Boundary Integral Formulation for Electromagnetic Scattering," J. Wave Motion, 6, pp. 91-103, January 1984. 94. M. A. Morgan and B. E. Welch, "The Field Feedback Formulation for Electromagnetic Scattering Problems," IEEE Trans. Antennas Propag., AP-34, pp. 1377-1382, December 1986.

REFERENCES

557

95. M . A. M organ and T. B. Welch, "Field Feedback Computation of Scattering by 2-D Penetrable Objects," IEEE Trans. Antermas Propagat., 40, pp. 445- 450, April 1992. 96. N . Mabaya. P. E. Lagas e, and P. Vandenbulcke, "Finite Element Analysis of Optical Waveguides:• IEEE Trans. Microwave Theory Tech.. MTT-29, pp. 600- 605, 1981. 97. P . Bette . "Infinite Elements,·• Int. J. Numer. Methods Eng., 11, pp. 53- 64, 1977. 98. K. S. Komi arek, . N. Wang, A. K. Dominek and R. Hann, " An Investigation of New FETD/ ABC Method of Computation of Scattering from Three-Dimensional Material Objects." IEEE Trans. Antennas Propagat., 47, pp. 1572- 1585, October 1999. 99. 0 . M. Ramabi. "Absorbing Boundary Conditions for the Three-Dimensional Vector Wave Equation." IEEE Trans. Antennas Propagat., 47, pp. 1735-1736, November 1999. 100. K . K. Mei, R . Pou , Z. Chen. Y .-W. Liu, and M . D. Prouty, "Measured Equation of Invariance: A New Concept in Field Computations," IEEE Trans. A11tennas Propagat., 42, pp. 320- 328, March 1994. 101. Y.-S. Xu and H .-M . Chen, "Validity of the Measured Equation of Invariance," IEEE Trans. A11tennas Propagai.. 47, pp. 1814-1819, December 1999. 102. Y.-L. Luo, K.-M . Luk, K. K. Mei, and E. K.-N. Yung, "Finite Difference Analysis of Electrically Large Parabolic Reflector Antennas," IEEE Trans. Antennas Propagat. , 50, pp. 266-276, M arch 2002. 103. R. Janaswamy, '·A Fast Finite Difference Method for Propagation Predictions over Irregular. Inhomogeneous Terrain," IEEE Trans. Antennas Propagat., 42, pp. 1257-1267, September 1994. 104. C. M . Krowne. ''Vector VariationaJ and Weighted Residual Finite Element Procedures for Highly Anisotropic Media," IEEE Trans. Antennas Propagat., 42, pp. 642-650, May 1994. 105. W. Sun and C. A. Balanis ''Edge-Based FEM Solution of Scattering from Inhomogeneous and Anisotropic Objects," IEEE Trans. Antennas Propagat., 42, pp. 627-632, May 1994. 106. R . P. Jedlicka. S. P. Castillo, and L . K. Warne, "Coupling Through Tortuous Path Narrow Slot Apertures into Complex Cavities," IEEE Trans. Antennas Propagat., 48, pp. 456- 466, March 2000. 107. P. Soudais, "Iterative Solution of a 3-D Scattering Problem from Arbitrary Shaped Multidielectric and Multiconducting Bodiest IEEE Trans. Antennas Propagat., 42, pp. 954-959, July 1994. 108. S . Alfonzetti and G. Borzi, ''Finite-Element Solution to Electromagnetic Scattering Problems by Means of the Robin Boundary Condition Iteration Method," IEEE Trans. Antennas Propagat., SO, pp. 132- 140, February 2002. 109. R. Lee and V. Chupongstimun, "A Partitioning Technique for the Finite Element Solution of Electromagnetic Scattering from Electrically Large Dielectric Cylinders, IEEE Trans. Antennas Propagat., 42, pp. 737-741, May 1994. 110. T. Cwik, "Coupling Finite Element and Integral Equation Solutions Using Decoupled Boundary Meshes," IEEE Trans. Antennas Propagat., 40, pp. 1496- 1504, December 1992. 111. J. Liu and J.-M. Jin, " A Novel Hybridization of Higher-Order Finite Element and Boundary Integral Methods for Electromagnetic Scattering and Radiation Problems," IEEE Trans. Antennas Propagat., 49, pp. 1794- 1806, December 2001. 112. T. Rylander and A. Bonderson, "Application of Stable FEM-FDTD Hybrid to Scattering Problems/' IEEE Trans. Antennas Propagat., SO, pp. 141 - 144, February 2002. 113. M. A. Morgan, Ed., Finite Element and Finite Difference Methods in Electromagnetic Scattering , Elsevier, New York, 1990. 114. J. L . Volakis. A Chatterjee, and L. C. Kempel, Finite Element Method/or Electromagnetics , IEEE Press, Piscataway, NJ, 1998. 115. J. J~ The Finite Element Method in Electromagnetics , 2nd Ed, Wtley, New York, 2002.

558

ANTENNAS I: FUNDAMENTALS AND NUMERICAL METHODS

116. L. Lapidus and G. F. Pinder, Numerical Solution of Partial Differential Equations in Science and Engineering , Wiley, New York, 1982, pp. 49- 79. 117. G. Strang and G. Fix, An Analysis of the Finite Element Method , Prentice-Hall, Englewood Cliffs, NJ, 1973. 11 8. A. F. Peterson, ''Vector Finite Element Formulation for Scattering from Two-Dimensional Heterogeneous Bodies," IEEE Trans. Antennas Propagat., 42, pp. 357-365, March 1994. 119. L. S. Andersen and J. L. Volakis, "Adaptive Multiresolution Antenna Modeling Using Hierarchical Mixed-Order Tangential Vector Finite Elements," IEEE Trans. Antennas Propagat., 49, pp. 2ll - 222, February 2001. 120. R. P. Tewarson, Sparse Matrices, Academic Press, New York, 1973. 121. P. M . Morse and H. Feshbach, Methods a/ Theoretical Physics, 1, Sect. 5.1, McGraw-Hill, New York, 1953. 122. S. K. Chang and K. K. Mei, "Application of the Unimoment Method to Electromagnetic Scattering of Dielectric Cylinders," IEEE Trans. Antennas Propag. , AP-24, pp. 35- 42, January 1976. 123. J. F. Hunka and K. K. Mei, ''Electromagnetic Scattering by Two Bodies of Revolution," Electromagnetics , 1(3), pp. 329-347, July-September 1981. 124. S. K. Chang and K. K. Mei, ''Multipole Expansion Technique for Electromagnetic Scattering by Buried Objects," Electromagnetics , 1(1), pp. 73-89, January-March 1981. 125. M . A. Morgan, K. K. Mei, and S. K . Chang, "Coupled Azimuthal Potentials for Electromagnetic Field Problems in Inhomogeneous Axially-Symmetric Media." IEEE Trans. Antennas Propag., AP-25, pp. 413 - 417, May 1977. 126. P. C. Waterman, "Scattering by Dielectric Obstacles," Alta Freq., 38 (Spec.), pp. 348-352. 1969. 127. R. F. Miller, "Rayleigh Hypothesis in Scattering Problems," Electron. Lett.. 5(17), pp. 416-418, 1969. 128. J. B. Keller, "Geometrical Theory of Diffraction," J. Opt. Soc. Am.• 52, pp. 116- 130, February 1962. 129. R. G. Kouyoumjian and P. H . Pathak. "A Uniform Geometrical Theory of Diffraction for an Edge in a Perfectly Conducting Surface," Proc. IEEE , 62. pp. 1448-1461, November 1974. 130. R G. Kouyoumjian, ''The Geometrical Theory of Diffraction and Its Application," in R. Mittra, Ed., Numerical and Asymptotic Techniques in Electromagnetics , Springer-Verlag, New York, 1975, Chap. 6. 131. R. C. Hansen, Ed., Geometric Theory of Diffraction, IEEE Press, New York, 1981. 132. T. B. A. Senior, ''The Diffraction Matrix for a Discontinuity in Curvature." IEEE Trans. Antennas Propag., AP-20, pp. 326-333, May 1972. 133. R . Tiberio, G. Pelosi, and G . Manara, "A Uniform GTD Formulation for the Diffraction by a Wedge with Impedance Faces," IEEE Trans. Antennas Propag., AP-33, pp. 867- 873, August 1985. 134. J. A. Volakis, "A Uniform Geometrical Theory of Diffraction for an Imperfectly Conducting Half-Plane," IEEE Trans. Antennas Propag., AP-34, pp. 172- 180, February 1986. 135. C. E. Ryan, Jr., and L. Peters, Jr., "Evaluation of Edge-Diffracted Fields Including Equivalent Current for the Caustic Regions," IEEE Trans. Antennas Propag., AP-17, pp. 292- 299, May 1969 (correction: AP-18, p. 275, March 1970). 136. G. L. James and V. Kerdemelidis, "Reflector Antenna Radiation Pattern Analysis by Equivalent Edge Currents," IEEE Trans. Antennas Propag., AP-21, pp. 19-24, January 1973 (correction: AP-21, p. 756, September 1973).

REFERENCES

559

137. J. H. Meloling and R. J. Marhefka, "A Caustic Corrected UTD Solution for the Fields Radiated by a Source on a Flat Plate with a Curved Edge," IEEE Trans. Antennas Propagar., 45, pp. 1839- 1849, December 1997. 138. R. Tiberio. S. Maci, and A. Toccafondi, 'An Incremental Theory of Diffraction: Electromagnetic Formulation," IEEE Trans. Antennas Propagat., 43, pp. 87-96, January 1995. 139. G. A. Thiele and T . H. Newhouse, "A Hybrid Technique for Combining Moment Methods with the Geometrical Theory of Diffraction," IEEE Trans. Antennas Propag., AP-23, pp. 62-69. January 1975. 140. P. Parhami, Y. Rahmat-Samii. and R. Mittra, "An Efficient Approach for Evaluating Sommerfeld Integrals Encountered in the Problem of a Current Element Radiating over Lossy Groun~·• IEEE Trans. Antennas Propag., AP-28, pp. 100-104, January 1980. 141. A. Moh en, "On the Evaluation of Sommerfeld Integrals," Proc. Inst. Electr. Eng., 129B, pp. 177-182, August 1982.

142. I. V. Lindell and E. Alanen, ·'Exact Image Theory for the Sommerfeld Half-Space Proble~ Part III: General Formulation," IEEE Trans. Antennas Propag., AP-32, pp. 1027-1032, October 1984. 143.

A. Michalski. "On the Efficient Evaluation of Integrals Arising in the Sommerfeld Half-Space Problem:' Proc. Inst. Electr. Eng., 132H, pp. 312-318, August 1985. K._

144. A. Banos. Jr.. Dipole Radiation in the Presence of a Conducting Half-Space, Pergamon Press, Oxford, 1966. 145. E. K. Miller, A . J. Poggio, G. J. Burke, and E. S. Selden, "Analysis of Wire Antennas in the Presence of a Conducting Half-Space. Part II. The Horizontal Antenna in Free Space," Can. J. Pltys., 50, pp. 2614-2627. 1972.

146. R. Mittra, P. Parhami, and Y. Rahmat-Sami4 "Solving the Current Element Problem over Lossy Half-Space without Sommerfeld Integrals," IEEE Trans. Antennas Propag., AP-27, pp. 778-792, November 1979. 147. G. A Thiele, "Overview of Selected Hybrid Methods in Radiating Systems Analysis," Proc. IEEE. 80, pp. 66-78, Jan. 1992. 148. U. Jakobus and F. M . Landstorfer, "Improvement of the PO-MoM Hybrid Method by Accounting for Effects of Perfectly Conducting Wedges," IEEE Trans. Antennas Propagat., 43, pp. 1123-1129, October 1995. 149. 1. J~ F . Ling, S. T . Carolan, J.M. Song, W. C. Gibson, W. C. Chew, C. Lu, and R. Kipp, "A Hybrid SBR/MoM Technique for Analysis of Scattering from Small Protrusions on a Large Conducting Body,'' IEEE Trans. Antennas Propagat., 46, pp. 1349- 1357, September 1998. 150. R. E. Hodges and Y. Rahmat-Samii, "An Iterative Current-Based Hybrid Method for Complex Structures," IEEE Trans. Antennas Propagat., 45, pp. 265-276, February 1997. 151. C. J. Reddy, M. D . Deshpande, C. R. Cockrell, and F. B. Beck, "Radiation Characteristics of Cavity Backed Aperture Antennas in Finite Ground Plane using the Hybrid FEM/MoM Technique and Geometrical Theory of Diffraction," IEEE Trans. Antennas Propagat., 44, pp. 1327-1333, October 1996. 152. I. P. Theron, D . B. Davidson, and U. Jakobus, ''Extensions to the Hybrid Method of Moments/Uniform GTD Formulation for Sources Located Close to a Smooth Convex Surface," IEEE Trans. Antennas Propagat., 48, pp. 940-945, June 2000. 153. S. A. Schelkunoff, "A Mathematical Theory of Linear Arrays," Bell Syst. Tech. J., 22, pp. 80-107,_ 1943. 154. P . M . Woodward, "A Method for Calculating the Field over a Plane Aperture Required to Produce a Given Polar Diagram," J. Inst. Electr. Eng., Part 3A , 93, pp. 1554-1558, 1946.

560

ANTENNAS I: FUNDAMENTALS AND NUMERICAL METHODS

155. P. M. Woodward and J. D. Lawson, 'The Theoretical Precision with which an Arbitrary Radiation-Pattern may be Obtained from a Source of Finite Extent/' J. Inst. Electr. Eng., Part 2 , pp. 363-370, September 1948. 156. C. L. Dolph, "A Current Distribution for Broadside Arrays which Optimizes the Relationship Between Beam Width and Side-Lobe Level," Proc. IRE , 34, pp. 335-348, June 1946. 157. T. T . Taylor, "Design of Line-Source Antennas for Narrow Beamwidth and Low Sidelobes," IRE Trans. Antennas Propag., AP-3, pp. 16- 28, January 1955. 158. T. T. Taylor, "D~sign of Circular Apertures for Narrow Beamwidth and Low Sidelobes/' IRE Trans. Antennas Propag., AP-8, pp. 17- 22, January 1960. 159. F. I. Tseng and D. K. Cheng, "Optimum Scannable Planar Arrays with an Invariant Sidelobe Level," Proc. IEEE, 56, pp. 1771 - 1778, 1968. 160. D . Kajfez, ''Nonlinear Optimization Reduces the Sidelobes of Yagi Antenna," IEEE Trans. Antennas Propag., AP-21, pp. 714-715, September 1973. 161. D. Kajfez, "Nonlinear Optimization Extends the Bandwidth of Yagi Antenna," IEEE Trans. Antennas Propag., AP-23, pp. 287-289, March 1975. 162. R. L. Haupt and S. E. Haupt, Practical Genetic Algorithms , Wtley, New York, 1998. 163. Y. Rahmat-Samii and E. Michielssen, Eds., Electromagnetic Optimization by Genetic A lgorithms , Wiley, New York, 1999. 164. L. Davis (Ed.), Genetic Algorithms and Simulated Annealing, Morgan Kaufmann, Los Altos, CA, 1987. 165. C. S. Ruf, "Numerical Annealing of Low-Redundancy Linear Arrays," IEEE Trans. Antennas Propagat., 41, pp. 85-90, January 1993. 166. F. Hsu, P. Chang, and K. Chan, "Optimization of Two-Dimensional Radome Boresight Error Performance using Simulated Annealing Technique," IEEE Trans. Antennas Propagat., 41, pp. 1195-1203, September 1993. 167. R. L. Haupt, ''Thinned Arrays using Genetic Algorithms," IEEE Trans. Antennas Propagat., 42. pp. 993-999, July 1994. 168. Z. Altman, R. Mittra, and A. Boag, "New Designs of Ultra Wide-Band Communication Antennas using a Genetic Algorithm," IEEE Trans. Antennas Propagat.. 45, pp. 1494- 1501, October 1997. 169. F. J. Ares-Pena, J. A. Rodriguez-Gonzalez, E. Villanueva-Lopez, and S. R . Rengarajan, "Genetic Algorithms in the Design and Optimization of Antenna Array Patterns,·· IEEE Trans. Antennas Propagat., 47, pp. 506- 510, March 1999. 170. H. Mosallaei and Y. Rabmat-Samii, "Nonuniform Luneburg and Two-Shell Lens Antennas: Radiation Characteristics and Design Optimization," IEEE Trans. Antennas Propagat.. 49, pp. 60- 69, January 2001 . 171. Z. Li, Y. E . Erdemli, J. L. Volakis, and P. Y . Papalambro ...De ign Optimization of Conformal Antennas by Integrating Stochastic Algorithms with the Hybrid Finite-Element Method," IEEE Trans. Antennas Propagat., 50, pp. 676- 684, May 2002. 172. G. Washington, H.-S. Yoon, M. Angelino, and W. H. Theuni en, "Design, Modeling, and Optimization of Mechanically Reconfigurable Aperture Antennas," IEEE Trans. Antennas Propagat. , 50, pp. 628- 637, May 2002.

10 ANTENNAS II: REFLECTOR, LENS, HORN, AND OTHER MICROWAVE ANTENNAS OF CONVENTIONAL CONFIGURATION D ONALD G. Boo AR Georgia Tech Research lnsrinae Georgia Institute of Technology Atlanta, Georgia

J. J. L EE Hughes Aircraft Company Fullenon, California

G. L. JAMES Division of Radiophysics CS/RO Epping, New South Wales Australia

F. K . S CHWERING V. S. Army CECOM Fon Monmouth, New Jersey

J. W. MINK Electronics Division V. S. Army Research Office Research Triangle Park, North Carolina

Handbook of RF/Microwave Components and Engineering, ISBN 0-471 -39056-9 © 2003 John Wtley & Sons, Inc.

Edited by Kai Chang

561

562

ANTENNAS II: REFLECTOR, LENS, HORN, AND OTHER M1CROWAVE ANTENNAS

10.1

REFLECTOR ANTENNAS

D ONALD

G. B ODNAR

10.1.1 Introduction Reflector antennas have a long and colorful history dating back to the early experiments of Hertz in 1888. Hertz used parabolic reflectors both to send energy from a spark transmitter and to collect incoming waves and concentrate them on a detector [1]. He used this equipmen to demonstrate "action at a distance." Microwaves and reflector antennas took a back seat to submegahertz frequencies and wire antennas, which became very popular for radio-telegraph communication activities early in the twentieth century. Utilization of microwaves and of reflector antennas began again in earnest during World War II, during which time great theoretical and experimental advances were made for these antennas [2]. Interest in reflector antennas waned after World War II but began again in the late 1950s motivated by a number of interests, including (1 ) the need for large collecting areas for radio astronomy antennas, (2) the need for high-gain ground-based antennas for deep-space communications, and (3) the need for shaped beams from satellites used to communicate with the earth. In the 1970s the reflector antenna captured the majority of the business for high-efficiency ground-based antennas used for satellite communications. Although phased array antennas have received considerable attention during recent years, their main use is in applications that require rapid changes (on the order of milliseconds) in beam-pointing direction. Since the cost of a phased-array antenna is often one to two orders of magnitude more than that of a reflector antenn~ reflector antennas are often chosen over phased arrays on the basis of cost when the sophisticated beam-pointing performance of the phased array is not absolutely required. Reflector antennas will be classified in this chapter first by the number of reflecting surfaces used and then subdivided according to the shape of the urface. Single reflector antennas are discussed first and are usually preferred over double reflector antennas due to (1) the decreased cost in manufacturing only one reflector and (2) the greater simplicity in aligning the feed and one reflector compared to that required for two reflectors and a feed. Performance advantage , however, of a double reflector may outweigh its cost disadvantages in many cases. Feeding at the focal point of the reflector will be covered first followed by a discussion of beam scanning produced by feed motion away from the focal point. Techniques for evaluating reflector antenna performance are given in Ref. 2 and will not be covered in this section. Only the results of such analy es will be presented. The areas of application of commonly used analysis techniques can be summarized as follows. Geometrical optics is usually employed in the design of the shape of a reflector antenna surface. In determining the pattern of the reflector antenn~ physical optics and similar techniques are used first to find either the current distribution on the reflector surface or its aperture field distribution. Wave theory is then utilized to transform either of these distributions into the radiation pattern of the antenna. The relationship between the aperture distributions and the far-field pattern is a Fourier transform relationship if the aperture is planar. The e relationships are shown more explicitly in Eq. (10.3) and (10.29).

10.1 REFLECTOR ANTENNAS

563

10.1.2 Single-Reflector Antennas Single-reflector antenna (often called di b antennas) are comprised of one reflecting urface and a feed. The feed may be a single radiator or may consist of multiple radiator , a di cussed in Section 10.1.8. The surface of the reflector may be described by a imple mathematical function or can be pecially shaped and described by a table of numerical value . The edge of the reflector (which determines the aperture shape) may be cut into any hape, including the comrnonJy occurring circular and elliptical edge . The most common form of single reflector is the paraboloid, which is di cus ed next.

Center-Fed Paraboloid. A paraboloid i formed by spinning a parabola about its axis as hown in Fig. IO.l a . The equation of the surface of a paraboloid with its vertex at the origin in rectangular and spherical coordinates is

x2 + y2

= 4/z p = f sec2 (1/l/ 2)

(10. la) (IO.lb)

where f is the focal length of the paraboloid, which is equal to the distance OF. It is a geometrical property of a paraboloid that all rays which emanate from its focal point and are reflected off the paraboloidal surface will leave the reflector parallel to the reflector axis, as illustrated in Fig. IO.lb. In addition, all path lengths from the focus to the aperture plane are equal (see Fig. IO.lb). Thus the paraboloid produces a constantphase field distribution over the aperture plane if the phase center of the feed is placed at the focal point of the reflector and if this feed radiates a spherical wavefront. Because of this constant-path-length property, the paraboloid is inherently a broadband device. Its performance is limited at low frequencies by the size of the reflector becoming small in terms of wavelengths. Usually, a reflector diameter of two to three wavelengths is required for satisfactory performance. The high-frequency performance is limited by surface roughness, as discussed in Section 10.1.7. At all frequencies the phase center of the feed must remain at the focal point for proper performance. Practical feeds do this only approximately since their phase center moves with frequency, so perfect collimation is not obtained at all frequencies from such feeds. From aperture theory, one recalls that a higher-gain and narrower-beamwidth antenna is obtained as its aperture size is increased. Thus the size of the reflector aperture is determined by the desired gain and beamwidth which for a circular aperture can be determined from the following two equations:

where

D

= (A./1r)v'G/rJ

(10.2)

D

= k>..jHPBW

(10.3)

= diameter of the antenna aperture G = gain of the antenna relative to that of an isotropic radiator

D

HPBW TJ

= half-power beamwidth of the main lobe of the antenna in degrees = efficiency of the antenna

ANTENNAS II: REFLECTOR, LENS , HORN, AND OTHER MICROWAVE ANTENNAS

564

X

(a)

Far field point (R,¢,0)

·-\ - - -'- - -- I

R

~

\ (p \ \ \ \

\ \

I

I I I I I I I

ItsCP=±~~\'L_j__

D

\I

z

I

I

I

l __ ...J ______=-~

y

I I

--- - - - - - f -- - - -- (b)

Parabolic reflector

A------------.v

, r

Plane wavefront

I I

Aperature plane

Figure 10.1 (a) Geometry of paraboloidal reflector; (b) typical energy paths from a feed to the aperture plane in a paraboloid.

k

= beamwidth constant of the antenna in the same units as HPBW

).. = wavelength of operation in the same units as D

Equation (10.2) specifies the diameter of the antenna in terms of the desired antenna gain while Eq. (10.3) specifies the diameter in terms of the desired beamwidth. Usually, a minimum gain and a maximum beamwidth are specified for an antenna. Consequently, the larger value of D from Eqs. (10.2) and (10.3) is chosen to satisfy both conditions simultaneously. Values of 71 and k for various aperture distribution are given in Ref. 3. Values of 1J = 0.5 and k = 70° are common estimates used for radar antennas when little is known about the details of the particular antenna of interest. The gain of an antenna is often stated in terms of decibels relative to isotropic, which is related to G by GdBi = 10 log 10 (G).

10. l

565

REFLECTOR ANTENNAS

The feed selected for use with a paraboloid must radiate energy to all parts of the reflector in order to achieve the gain and beamwidth specified by (10.2) and (10.3). Thus the angle Bs over which the feed must radiate energy is given by (10.4) where 2v,o is the angle subtended by the reflector as seen from the focal point and f is the focal length of the reflector= distance OF. Equation (10.4) is plotted in Fig. 10.2, which shows that use of a larger f ID ratio antenna (e.g., a longer focal length for a fixed diameter) requires a feed with a narrower beamwidth. This, in turn, requires a larger feed, which increa es aperture blockage as discussed in Section 10.1.7 and results in lower gain and increased sidelobe levels. Thus there is a trade-off between performance and f ID ratio. The angle Bs in (10.4) is the beamwidth required from the feed, and the feed pattern should be chosen such that the illumination at the edge of the reflector is 10 dB below the maximum illumination at the center. It has been found in practice that this illumination produces near-maximum gain from the paraboloid. Larger edge tapers (on the order of - 20 dB) produce lower sidelobes at the expense of slightly lower gain. The edge illumination of the reflector can be determined in the following fashion. First note that it is the product (sum in decibels) of the feed taper and space attenuation. Feed taper is the ratio of the feed power pattern in the direction of the reflector edge to its value in the direction of the vertex. The spreading of the feed energy as it travels from the feed to the reflector (1/ R 2 change in power level) is referred to as space attenuation even though there is no ohmic loss of power. The ratio squared of the focal length relative to the distance from the focal poin.t to the reflector edge is called space attenuation. The space attenuation factor is plotted in Fig. 10.3 versus f/D ratio.

220

\

200

-.... f/'J

(I) (I)

\

180

C,

-

(I) -c 160

~

N

\

140

(I)

O> C

ca 120 -c (I) C

sf/'J

\ \

~

~

~

100

.0 ~

en

80

"'

"

~

60 0.2

0.4

'

0.6

~

0.8

---.....

~

r----

1.0

----

-

1.2

1.4

f/D ratio

Figure 10.2 Subtended angle of a paraboloid as seen from the focal point versus f ID ratio.

566

ANTENNAS 11: REFLECTOR LENS , HORN, AND OTHER MICROWAVE ANTENNAS

-6

-

------.----,,---~--r-----,---,-----.---:;i

-5 p

co

-0

C: 0

-4

Paraboloid

('(S

C: Q)

.L= 20 log sec2 ; p

f

:;::i

::,

A = 20 log

-3

i:::: ('(S Q)

0

('(S

-2

Q. (f)

-1 0 J..--=::::::I::._

0

10

__J__ _L __

20

_L__

__J__

_ _ _ _ , J ~ - . . . . 1 . . - - _ _ J __

30 40 50 60 Angle off axis 1/, (degrees)

70

80

__J

90

Figure 10.3 Space attenuation for a paraboloidal reflector.

It can be seen, for example, from this figure that 6 dB of extra taper is produced by

space attenuation for a 180° subtended angle (f/ D = 0 .25). The feed taper at the reflector edge can be determined from Fig. 10.2 and the known pattern of the feed. This value added (in decibels) to the space attenuation obtained from Fig. 10.3 gives the total aperture taper. If a certain aperture taper (e.g., -10 dB) is desired, the space attenuation obtained from Fig. 10.3 is subtracted from this value to find the required feed taper. Next one uses the accurate approximation that the main beam of the feed can be approximated by the parabolic function P(v,)

=-

10(1/1/1/110/

(10.5)

where P(1/I) is the power pattern of the feed in decibels and v, 1o is the 10-dB beamwidth of the feed which is plotted in Fig. 10.4. From these curves the IO-dB beamwidth of the feed can be determined so as to produce the required feed taper at the reflector edge. The feed can then be designed using the references in Section 10.1.4. A different type of feed is required for high-efficiency atellite communication antennas. These antennas try to achieve a uniformly illuminated aperture distribution rather than a tapered one since such a distribution produces the maximum possible gain. A uniform aperture distribution can be obtained from a paraboloid if a feed pattern of the form E(v,) = sec2 (v, / 2) (10.6) can be produced since this feed pattern exactly compensates for space attenuation. This pattern has a minimum value on axis and increases toward the edge of the reflector. Higher-order modes can be added to a feed horn [4] to approximate this behavior. Alternatively, shaping of the reflector surfaces can be used to obtain a nearly uniform distribution, as discussed in Section 10.1.8. One of the undesirable features of a center-fed reflector is the blockage of aperture energy by the feed and feed support. This blockage increases the level of the

10.1

REFLECTOR ANTENNAS

567

Or---=:::::::::r----.----,-----,---.-------.------.--~

-4 ,---r---+--~~ --+---+------!---LJ co ~ ... CD

-8 r - - - - t - - - - - t - - - - - t - - - +-----.:str--~----l----j__J

~

0 C.

~ ~ ~

;

- 12 ,-------r---+-----+----+----1----~-

, ✓ Measured ,

Calculated pattern

- 16 r-------t-----t------f.-----+Eds

-20~--_.__ _ _......__ _---L_ _ _ 0 0.2 0.4 0.6 0.8

=-1o(J!..) 1/110

pattern

' 2

-..1...__ _____,1L..,__ ___J__

1.0

1.2

_ _l..L..J

1.4

Relative angle J!_

1/110 Figure 10.4 Standard feed-horn pattern.

sidelobes and decreases the gain of the antenna [3] . Blockage effects are discussed in Section 10.1.7. The deleterious effects of blockage can be reduced by decreasing the size of the blockage relative to the aperture. Alternatively, the blockage can be eliminated by removing the blockage from the aperture region through the use of an offset reflector.

Offset-Fed Paraboloid The offset paraboloidal reflector shown in Fig. 10.5 not only eliminates blockage effects of the feed and its supports but also reduces reflection of energy from the reflector back into the feed to a very low level. In addition, it lends itself to the use of longer focal lengths, which is advantageous in certain applications. The major disadvantages of the offset paraboloid are the increased cost of the backup structure required to hold the reflector in its proper shape and the additional cross polarization generated from it for a linearly polarized feed (see Section 10.1.6). The co-polarized characteristics of the offset parabolic reflector are very similar to those of the center-fed paraboloid just discussed. The main difference is that the feed is pointed near the center of the reflector in the offset direction of the reflector instead of at its vertex in order to keep the feed energy on the reflector. The feed is actually pointed slightly beyond the center of the reflector in order to balance the larger space attenuation at the outer edge of the reflector compared to that at the inner edge. In addition, as mentioned above, blockage is not present to degrade sidelobe performance. An excellent review of offset paraboloidal reflectors is given in Ref. 5.

Ellipse. Elliptical reflectors have two ~ocal points at finite distances from the reflector (see Fig. 10.6), in contrast to the paraboloid, which has one finite distance focal point

568

ANTENNAS II: REFLECTOR, LENS, HORN, AND OTHER MICROWAVE ANTENNAS (a)

Reflector support Feed support

Feed waveguide

Center fed reflector Reflector

(b) Reflector support

Feed waveguide

Feed support Offset fed reflector

Figure 10.S

F2 Second focal point ~

(a) Center- and (b) offset-fed reflector geometries.

F1

L

First focal point

Figure 10.6 Typical energy path from a feed in an ellipsojdal reflector.

I 0. 1

REFLECTOR A TENN AS

569

and one focal point at infinity. Thu the ellip e can take energy from a feed located at one focal point and reconcentrate it at the other focal point. Thi property can be used to ad antage in a number of application . Hjgh-power energy, for example, can be inputted into a feed at one focal point and then concentrated at the . econd focal point. where a ample of material can be located. In thi manner, the field around the ample are not influenced by the metal part of the feed . Another application of the ernp e i in the low-lo tran mi ion of energy from one part of a microwave y tern to another [6], often called a beam-waveguide feed. Thi i very important in millimeter wa e antenna y tern , where lo e in waveguide can be prohibitively high. Energy i tran mitted between the foci without wall lo e a in conventional waveguide. Elliptical reflector are al o u ed a ubreflectors in Gregorian reflector antennas ( ee Section 10.1.3). For such antenna the feed i placed at one focaJ point of the ellip e, while the econd focal point of the ellip e i made to coincide with the focal point of the paraboloidal main reflector.

Hyperbola. The hyperbola al o has two foci . However, one focal point is real and the other i imaginary. The field radiated by a feed placed at the real focal point will be reflected off the hyperbola in uch a manner that it appear to be coming from the virtual focal point of the hyperbola ( ee Fig. 10.7). This feature is used in a Cas egrain antenna. where the feed i placed at the real focal point of the hyperbolic subreflector and the virtual focal point i made to coincide with the focal point of the paraboloidal main reflector. Sphere. The sphere is not a perfect collimator, o there is no point at which a simple feed can be located to produce a collimated beam. However, a Taylor series expansion of the surface of a phere shows that the phere can be approximated locally by a paraboloid with a focal length equal to half the phere radius. Thus the feed for a sphere is placed approximately halfway between the center and the surface of the sphere. Only the central portion of the spherical surface is illuminated by the feed since the higher-order surface terms produce phase distortion as shown in Fig. 10.8 which prevents the e outer areas from providing collimated energy. Notice from the figure that the phase error increases as more of the reflector is illuminated and as the

F2 Virtual focal point

~

F,

L

Real focal point

Figure 10.7 Typical paths for energy reflected off a hyperboloid.

570

ANTENNAS II: REFLECTOR, LENS, HORN, AND OTHER MICROWAVE ANTENNAS

0.020

I

I

I I

i/ I II I

I

0.015

I

0.010

~ -~ -

f = 0.5 R

0.005

V

J

)

I

I

:::::..

_[__y'

~ "'~,

0

""

-0.005

~

~

0

0.1

0.2

/ /0.48 R /

-~

""' ',

-0.010

I

II

/

V I Vo.4665y ~

..........

0.3

I

V

I

J

V ,..__0,-45'3.,/ I

0.4

0.5

0.6

rtR

Figure 10.8 Phase error in a spherical reflector. (After Ref. 7.)

feed is moved away from the halfway point. But a certain compensation of errors is possible by appropriate choice of a and f . The optimum focal length for an aperture of radius a from a sphere of radius R is [7, p. 340]

(10.7) One of the attractive features of a spherical reflector is that its beam can be scanned over wide angular sectors without distortion [7] . This occurs since the feed always sees the same geometry as long as it moves along a circular arc centered on the center of the sphere and as long as it points along a radial. Line source feeds have also been u sed with spherical reflectors in order to utilize a larger portion of the reflector surface.

Parabolic Cylinder. The preceding reflectors have utilized a point source feed . In contrast, the parabolic cylinder reflector can use either a point source or a line source feed. This reflector is parabolic in one plane and cylindrical in the other plane. The parabolic cylinder is often used to provide beam collimation in the plane orthogonal to a collimating parallel plate structure such as a geodesic lens, pillbox antenna, or Lewis scanner [8]. When a point source feed is used with a parabolic cylinder, a collimated beam is produced in the parabolic dimension and the feed beamwidth is mirrored in the cylinder direction. Other Single-Reflector Types. A number of novel reflector shapes have been developed for wide-angle scanning applications. Scanning properties of conventional paraboloidal

10.1

REFLECTOR ANTENNAS

571

and Ca egrain reflectors are covered in Section 10. l.5 . The hourglas antenna [8] consists of an hourgla - haped reflector with a circular array of element wrapped around the out ide of the reflector and radiating toward it. The concave reflecting surface i formed by rotating a parabolic arc about a vertical axi which is parallel to the latu rectum of the arc. A mall number of the feeds are energized with the proper phase to produce a bean1. The beam i canned by switching in successive groups of element . Scanner of this type have been made to operate over a 2: 1 frequency band. A parabolic-toru antenna [8] i formed in the arne manner as the hourglass antenna except that the parabolic generating arc is concave toward the axis of rotation, in contrast to the hourglas reflector. where it is convex toward the axis. The feed i located in ide the reflector and only a portion of the reflector arc is present. The parabolic-torn antenna combines the focusing properties of a parabola with the widecan propertie of the phere. The antenna i scanned by moving a feed along a circular arc and can sector of 120° have been obtained. The barrel reflector [9, Sec. 1.7] is formed by rotating a ector of a parabola about a vertical axis and constructing the reflecting surface from wires oriented at 45° to the vertical. A feed on the inside oriented at 45° i collimated by the side of the reflector which it illuminates (and which has its wires parallel to the feed polarization). Energy reflected from this side of the reflector passes through the other side of the reflector since these wires are perpendicular to the fir t ones.

l0.1.3

Double-Reflector Antennas

Two-reflector antenna systems provide technical advantages over their single-reflector counterparts, but these advantages are obtained through increased cost. One of the main advantages of a double-reflector antenna is that it allows a rear-mounted feed to be used, which can result in less blockage, easier access to the feed electronic equipment, and less weight in front of the reflector (thus permitting more rapid motion of the antenna). The extra cost is associated with fabricating the second reflector and the extra cost of supporting and aligning this second reflector. Cassegrain. The Cassegrain antenna consists of a paraboloidal main reflector and a hyperbolic subrefiector, as discussed in Section 10.1.2. The classical Cassegrain utilizes a concave paraboloid and a convex hyperboloid as shown in the top drawing of Fig. 10.9. However, other forms of a Cassegrain are possible, as shown in this figure [10]. In particular, a concave hyperboloid is also possible with a concave paraboloid. The latter geometry requires either a larger subreflector or a longer focal length for the main reflector, both of which are undesirable from a blockage and structural point of view. Hence the convex hyperboloid is norma11y seen in practice. The first series of drawings in Fig. I 0.9 have the diameter of the main reflector held fixed as the feed beamwidth increases. The convex hyperboloid degenerates into a plane and finally into a concave hyperboloid as the feed beamwidth increases. Further increase in feed beamwidth causes the main reflector to become flat and eventually convex. Evaluation of the performance of the Cassegrain antenna can be performed by using the equivalent paraboloid concept (10]. According to this concept, the feed and Cassegrain reflector system are replaced by the same feed and an equivalent paraboloid. The focal length of the equivalent paraboloid, fe, is related to the focal length, f, of

572

ANTENNAS Il: REFLECTOR, LENS, HORN, AND OTHER MICROWAVE ANTENNAS

Illustration

-

.,,. .......

.......

.......

0vf0, and feffm

fm and fc

e

>1

>0

>1

\

I

I I I

.......

Convex subdish (classical form)

PAR

PAR

1

\

I

>0

00

Flat subdish

(/J

E ... .g

...

...... .......

0

~

....... .......

>O

< -1

>0 Concave subdish

O> Q)

(/J (/J

as

u

0

1

00

Flat main dish

-1

-1

Convex main dish

... ...

.......

0

.......

~

\ I I I

(/J

E

.g

PAR

........

........

I

>1

>O

>0

.:;

as

ci>

a: - 30

-40 t...UI.U.....-..L--ll-.---'-----'-__..........._.__..____.,_~....._.....__.....,

0

2

4

6

8

10

12

ka sin 0

Figure 10.57

Maximum cross-polar radiation patterns in the 45° plane for

-

0.8

~o

0.6

a conical born.

ea>

-co u,

a>

,
C (.) ....

co s~

---~E Co

0.2 0.0 0.0

0.2

0.6

0.4

0.8

1.0

/J.

Figure 10.58 Phase center position for the principal planes of a conical horn.

628

ANTENNAS U: REFLECTOR, LENS , HORN, AND OTHER MICROWAVE ANTENNAS

increasing values of 6., the phase center position moves toward the throat region of the horn at differing rates for the two principal planes, with the greatest discrepancy occurring where 6. ~ 0.3. For large values of 6. the phase center is essentially at the apex of the horn. The beam efficiency of the total radiation pattern is plotted in Fig. 10.59 for various values of 6.. As expected from observing the radiation patterns in Figs. 10.55 and I 0.56, the beam efficiency for a given radiation angle 0 decreases as 6. increases from zero. The efficiency at the -12-dB half-beamwidth points on the main beam in the principal planes is indicated on Fig. 10.59. The gain of a conical horn has been studied by several authors, the most convenient design data being given by Jakes [4]. From his data we can express the gain by the approximate empirical formula gain

= 20 log[0.915ka exp(-1.636. 2 ) ]

dBi,

6. ;S 0.5

(10.91)

From the earlier work of King [5] we can deduce from his curves that the optimum gain horn occurs when 6. ~ 0.35, giving a gain of (20 log(ka) - 2.5] dBi. The accuracy of Eq. (10.91) has not been verified for all cases, but compared with the limited data given by King, the agreement is within the range -0.1 to +0.2 dB of the measured gain, provided that 6. < 0.4. Larger values of 6. gave increasingly poorer agreement. All of the results presented so far have applied where the far-field is on or near bore-sight. For an indication of the radiation performance of conical horns into the far-out sidelobes and in the near-field, see Refs. 6-8.

~

C

Q)

0.8

c3 !E Q)

~

Q)

al

A = 1.0

0.6

2

4

6

8

10

12

ka sin 0

Figure 10.59 Beam efficiency for a conical hom. The beam efficiency at the - 12-dB level on the main beam in the principal planes is shown: asterisk, H -plane; filled circle, E -plane.

10.3

1.8

TE 11

1.6 \

a:

>

1.2

629

c,:--,r--J :---,,----,----..----.. \

3: 1.4 Cl)

HORN ANTENNAS

-

7

----

Theory

•

Measured points

\~ \

.

'\

' '...

',, .....

.......... . . . ._____

--

fl

1.0 ,.___ __.__ _' - - ' ---L-- - L --2.0 2.2 2.4 2. 6 2.8

-- -- -~•II

3.0

ka1

Figure 10.60

VSWR of a circular waveguide with an infinite plane flange.

There is not a great deal of information on the impedance match of conical horns. For values of 0 ;S 30°, and provided that the frequency is somewhat above cutoff of the TE 11 mode in the exciting waveguide, the mismatch to the horn is usually negligibly small for most application s. With increasing 0 the mismatch becomes more evident, particularly for small values of ka;. For the extreme angle of e = 90° we can refer to the mismatch of a circular waveguide terminating in a large flange. Figure 10.60 shows both measured and theoretical VSWR results for the 90° horn, where for ka; > 2.5 the VSWR remains below 1.1.

Higher-Order Modes. As the frequency increases, higher-order modes may propagate. Of particular interest in horn design is the TM 11 mode. This mode has a radiation pattern [9] Eco = f (u ) sin2

~ 0

a. (I)

;>

- 20

(1'

a> a:

-30

-40 ___.__--L_....,__.__.__.._.__.__.__L...L..1.....___.._____. 0

2

4

6

8

10

12

ka 8 sin 0

Figure 10.63 Normalized E -plane radiation patterns for pyramidal and sectoral horns.

Fig. 10.65 for pyramidal horns, where, with ae fixed, ah is chosen to ensure that the principal planes have the same beamwidth at the 12-dB level below boresight. As seen in the figure, the beam efficiency of pyramidal horns is not as high as that for the other principal horn types discussed in this chapter. Pyramidal horns are often used as primary gain standards. The gain in the far-field as calculated by Schelkunoff [19] can be expressed as (10.94)

632

ANTENNAS II: REFLECTOR, LE S, HORN, AND OTHER MICROWAVE ANTENNAS

Figure 10.64 Phase center position for the principal planes of pyramidal and sectoral horn .

>, 0

i

·o

0.8

!E Cl)

E «s a, m

0.6

2

4

6

8

10

12

Figure 10.65 Beam efficiency for a pyramidal horn with aeU ( A fixed and ah(A cho en for equal -12-dB bearnwidths in the principal planes. The filled circles indicate the - 12-dB level.

where Le and Lh are gain reduction factors given in Table 10.1. These factors account for the reduction in gain as a result of the phase error introduced aero the aperture when flaring the horn. Equation (10.94) can also be used at a finite-range distance r / )... simply by redefining t:,.e,h in Table 10. l . Instead of u ing Eq. (10.93), which holds for r ~ oo, the expression for t:,.e,h is generalized to take the form (20]

The accuracy of Eq. (10.94) would appear to be within ±o. l dB, provided that the horn gain is ""20 dBi or better (21, 22]. For horns having gains less than 20 dBi, the

103

HORN ANTENNAS

633

TABLE 10.1 Gain Reduction Factors for Sectoral and Pyramidal Horns

Le

Lh

6.~.h

(dB)

(dB)

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95

0.00 0.04 0.15 0.35 0.62 0.97 1.40 l.92 2.54 3.24

0.00 0.02 0.07 0.17 0.29 0.45 0.65 0.88

1.00

4.04

4.93 5.91 6.96 8.04 9.08 9.98 10.60 10.87 10.81 10.50

1.14

1.43 1.75 2.09 2.45 2.82 3.20 3.58 3.95 4.31

4.65 4.97 5.25

error becomes progressively worse, as discussed in these references. Further, since Eq. (10.94) is calculated on the basis of a quadratic phase term in the aperture, it cannot be expected to be accurate for large values of 1:1 . For sectoral horns either Le or Lh is zero. However, since one of the aperture dimensions is smaIL Eq. (10.94) can be substantially in error. This is discussed in Ref. 23, where a gain correction for E-plane sectoral horns is proposed. Finally, an optimum-gain horn occurs when l:ie ~ 0.25 and l:ih ~ 0.375, giving a gain of [2.09 + 10 log(aeah/A 2)] dBi. As with smooth-walled conical horns, there is not a great deal of information on the impedance match of pyramidal horns. However, provided that the flare angles are moderate and the aperture dimensions ae,h/A are ;::l , the resulting impedance match of the horn should be sufficiently low for most applications. For some measured data, see Ref. 22. With sectoral horns, reflection from the horn aperture provides a significant contribution to the horn mismatch. A number of authors have investigated the match of sectoral horns, particularly E-plane sectoral horns [16, 24, 25]. For H-plane sectoral horns the throat region mismatch is much less than that at the aperture, while in £ plane sectoral horns both the throat and aperture mismatch are significant, resulting in large variations in VSWR-with values up to 1.5-as the length of the horn or the frequency changes.

634

ANTENNAS II: REFLECTOR, LENS, HORN, AND OTHER MICROWAVE ANTENNAS

10.3.4 Corrugated Horns The conical corrugated horn supporting hybrid modes has received intensive investigation over the past 20 years in response to the demands of satellite communication and radio astronomy. They are extensively used as feeds in reflector antenna installations because of their desirable radiation properties of good pattern symmetry, low cross-polar sidelobe levels, and a high beam efficiency with low sidelobes. The design of corrugated horns is more complicated than that of smooth-walled horns, and in the space available here we can only give broad design principles. Detailed information on corrugated horns is to be found in Ref. 26, and a number of useful design papers have appeared in Refs. 27-31. When a noncircular beam with low sidelobes is required, it is possible to use corrugated rectangular or elliptical feeds to take advantage of the high-performance capabilities of hybrid modes. These horns, however, have received considerably less attention than circular aperture horns, since they are more difficult to analyze and design and, in the case of elliptical horns, considerably more difficult to manufacture. We shall only consider conical corrugated horns here, but for a survey of noncircular corrugated horns, see Chapter 7 in Ref. 26. An interesting alternative to the corrugated horn is the dielectric-loaded hybrid mode first proposed in [32] and developed further in (33]. The basic principle of operation is similar to the corrugated horn in that a waveguide partially filled with dielectric supports hybrid modes (described in the following section) as occurs in a corrugated waveguide. Thus, the results given later in this section for radiation from a corrugated horn as a function of aperture diameter and t:,,. when the mode is "balanced" (see below) also apply to a dielectric-loaded hybrid mode horn of similar dimensions. The dielectric-loaded hybrid mode horn is much simpler in construction and potentially considerably cheaper to manufacture. By its nature, it is capable of very wideband operation compared to the corrugated horn. An example is given in [34], where, with one structure, a bandwidth ratio of over 30: 1 was achieved with respect to return loss, cross polarization, pattern symmetry, and phase center stability. Toe main variation with frequency was in the general reduction in beamwidth with frequency. However, a difficulty with the dielectric-loaded horn is obtaining suitable dielectric materials with a permittivity of "'1.2 with an acceptable loss tangent "' I 0- 4 or less. In practice, it is often necessary to simulate such a low-loss dielectric by artificial means such as using alternate layers of low-loss Teflon and low-loss, low-density expanded polystyrene foam in a sandwich construction. By adjusting the relative thickness of the two materials, a simulated dielectric of the required permittivity with low loss can be achieved in the horn axial direction. This approach and other means of creating artificial dielectrics need to be developed before use of the dielectric-loaded hybrid mode horn becomes more widespread.

Hybrid Modes. Hybrid modes are basically a combination of TE and TM modes, taking the form TMmn + yTEmn, where m is > 1 and y is a dimensionless real parameter known as the mode-content factor. For this mode combination to propagate as a single entity with a common propagating velocity, the horn or waveguide must have anisotropic surface reactance properties [9] . One of the most practical means of achieving this is to provide metal corrugations at the surface. Figure 10.66 shows a cross-sectional view of conical corrugated horns where there are typically 5 to IO corrugations per wavelength, the slot-to-pitch ratio w/p is ~0.75, and the slot depth dis

10.3

HORN ANTENNAS

635

(a) \

\ W '

a/)..

I I

-

-

--

_,___ - -- - -

. -

I·-

Ii

-

//)._ .._,__ _ __j___ ;I

di>..

{b) '

'\

'

\

\

\

\

\

a/).. \

\

I

\I

/i-j

I

~

'

I

' ~~"-

I

I

I

I

' ;' I

/

I

/

Figure 10.66 Cross-section view of corrugated horn showing usual slot configurations for (a) E> small; (b) E> large.

given by the empirical formula (10.95) where AO is the .wavelength of the frequency where the hybrid modes are required to be "balanced" (see below), ko = 2rr /"Ao, and an is the radius in wavelengths of the corrugated surface at the nth slot.

636

ANTENNAS II: REFLECTOR, LENS, HORN, AND OTHER MICROWAVE ANTENNAS

There are two distinct classes of hybrid modes: HEmn is the nomenclature used when y is positive, and EHmn when y is negative. Of particular interest is the behavior of hyb1id modes within the horn when Iy I ,...., 1. The modes are said to be balanced when Iy I = I, and it is for this condition that many of the following results are given. Of particular interest for horns is the HE 11 mode, while the EH1n modes are particularly undesir able because of their high cross-polarization characteristics. Doniinant Mode. The hybrid mode normally required in circular or conical corrugated waveguide is the HE11 mode, and as for the smooth-walled conical horn, we assume here that a/)..~ l. When the mode is balanced (y = 1) the fields within the horn have a radial dependence 10 (2.4050 / 8) independent of the azimuthal angle. The resultant radiated field from the horn is totally co-polarized (E x = 0) with complete pattern symmetry. As the frequency varies from the value when y = 1, pattern symmetry is lost and the cross-polar level, which is a maximum in the 45° plane, gradually increases. The radiation patterns for the balanced HE 11 mode in any plane is shown with D.. as parameter in Fig. 10.67. These patterns are very similar to the H -plane patterns of the smooth-walled horns (see Fig. 10.55). Note that for D.. ~ 0.4, the pattern, while becoming broader, does not change significantly in shape. If we change the ordinate and plot the radiation pattern against 0 / e for large Ii, as in Fig. 10.68 [27], we find that the radiation pattern is largely independent of /1 and hence of aperture size. This is sometimes referred to as the gain-saturated condition, in which the horn gain and beamwidth remain essentially the same irrespective of increase in aperture size. Consequently, when /1 > 0.4 we refer to the horn as wideband. When D.. < 0.4 the radiation pattern becomes sensitive to frequency, and the horn is referred to as narrow band. In the balanced hybrid condition at the frequency Jo, the HE11 mode does not produce any cross-polar radiated field. As the frequency departs from Jo the crosspolar level, which is a function of k0 a, increases, as shown in Fig. 10.69. These values are somewhat idealized, since in practice a certain amount of higher-order

-

CI)

--10

-0 ~

Q)

~

0

a. - 20 Q)

-

.::: ctl

cii

a:

- 30

Figure 10.67 Normalized co-polar radiation patterns for the conical corrugated horn.

10.3

0

.. ~ '·, ~ -,. ' ~ ' .' '

--~-~ '

-

,

~

HORN ANTENNAS

637

= 1.25

1.5

CD

"O

.Cl)... - 10 ~ 0

,~,

0.75

~

Cl)

=a,> a5

a:

0.75

-20

.

1.0

.. -30

1.5 0

0.5

1.0

'

'' 1.5

0/0

Figure 10.68 fore< 70°.

Normalized co-polar radiation patterns for wide-band corrugated conical horns

-CD

"O

-20

10

Cl)

>

Cl)

....a, 0

~ I

~

20

::,

E E

30

xa,

40

.... - 30 (.) 0

~

20

50 -40

0.6

0.8

1.0

1.2

1.4

30 40 1.6

f/f0

Figure 10.69 Theoretical cross-polar sidelobe level in the 45° plane of the corrugated conical horn due to departure from HEu mode balance at fo. Slot width-to-pitch ratio is 0.75.

EH1n modes are generated, particularly in the throat region of the horn. For a carefully constructed horn, cross-polar levels as low as -50 dB are possible over limited bandwidths [35], increasing to around -30 to -40 dB for horns used over wider bandwidths [31].

638

ANTENNAS U: REFLECTOR, LENS, HORN, AND OTHER MICROWAVE ANTENNAS

-©

en

0

()

1.0

c> ).., H

(10.99a)

where \11 (9)

= - k sine

le

+H

- H

for 0 ~

J (z) . - -e+1b.cose dz IN

~ 1r

(10.99b)

is the field strength radiation pattern of the dipole antenna. When written in vector form, \II has the O direction. IN is an (arbitrary) current normalization factor that typically is chosen to be the input current of the antenna. The spherical coordinate system r , O, has its origin at the center point of the antenna; see Fig. 10.75.

z

! y

X

Figure 10.75

Spherical coordinate system used in Equation (10.99).

10.4

DCPOLE AND MO OPOLE ANTENNAS

647

The exact method as outlined here i traightforward. With the use of modem computer and the commercial availability of ophisticated, user-friendly modeling codes, it is also highly accurate, efficient, and easy to implement; it should be very usefuJ for design purpo e . A econd method for the characterization of dipole antennas is approximate. It is based on the induced EMF method (7-9] and a surne a inu oidal current distribution:

= IN sink(H

I (z)

-

1:1)

for

1:1

~ H

(10. 100)

The method i useful in that it re ults in a clo ed-form analytical expre sion for the input impedance of cylindrical dipole antennas:

ZA where

F (x)

Zo

l

= -4rr m . 2 (kH ) [4F(k H ) cos(k H ) -

1

2x = tJx /.u=O

-:

- ju

F (2kH )]

[

du - 2} ka cosx

+ 1n

(10. l0la)

(2 ) ] k:

sin x

or equivalently ZA

= _'lo .

l J2rr sm (kH) 2

[(I+

2cos 2 (kH ))ka

+ e1 2lcH)(Si (2kH) -

- (1

+

sin (2kH) ln H

a

jCin (2kH))

(10.l0lb) where

Si (x)

=

!.

sinu

x

-- du ,

u=O

Cin (x)

=

U

!.

x

u=O

1-

COSU

- - - du U

Equations (10.101) are valid up to frequencies somewhat above the first resonance of the dipole antenna, i.e. , for O < kH < 2, which is the practically useful range. To avoid unnecessary computations, Tai [10] has reduced Equation (10.101) by a series expansion, and Elliott [8] has simplified the formula further by a curve fitting procedure using second-degree polynomials. The resulting expression

Z A = [122.65 - 204. lkH 2

- j [ 120 (in :

+ l IO(k H )2 ]

- 1) cotg (kH) - 162.5

+ 140kH -

40(kH )2] (10.102a)

is valid in the region

1.3

~

kH

~

1.7;

0.001588

~

a

-

)..

~

0.009525

(10.102b)

and is very easy to use. The field strength radiation pattern associated with the assumed sinusoidal current distribution (10.100) takes the form -

'11(0)

cos 0) - cos(kH) = - 2cos(kH - - - ' - - --Slil-._E) _ _ __

(10.103)

648

ANTENNAS Il: REFLECTOR, LENS, HORN, AND OTHER MICROWAVE ANTENNAS

Near and below the first resonance at kH = re, the pattern has the typical toroidal 2 shape with a zero in the axial directions e = 0 and 180°, and a maximum in the plane normal to the antenna axis (E> = 90°). At larger antenna heights or frequencies, the radiation pattern will develop a lobing structure. But such patterns usually are not desirable for communications. A third method uses Hallen' s integral equation formulation of the cylindrical antenna problem, which utilizes a representation of the antenna field in terms of the retarded scalar and vector potentials (11 ]. This technique results in graphs that show the resistive and reactive parts of the antenna impedance ZA, as a function of frequency (kH), with the length-to-radius ration H la as a parameter. These diagrams are very instructive. They show the sequence of series and parallel resonances occurring for dipole antennas as frequency increases, and allow one to estimate the bandwidth of these antennas as a function of their slenderness ratio H la. As they are well known, these graphs are not reproduced here; they may be found, for example, in [7, 8]. A critique of the three methods discussed here (and some additional techniques) for the calculation of the characteristics of dipole antennas has been given by Elliott [8]. This review also includes a comparison with King' s very accurate results. Individual dipole and monopole antennas radiate at linear polarization. Circular polarization can be obtained by the use of crossed dipoles as indicated in Fig. 10.76. The two dipoles will be fed such that their input currents are equal in amplitude but differ in phase by 90°. If the crossed dipole antenna is placed over a large metallic ground plane, it will radiate a unidirectional pattern. Using the sinusoidal current distribution (10.100) and assuming an infinite ground plane the field strength radiation

I

-j/ (y)

I

I

X

Figure 10.76 Crossed dipole antenna over metal ground plane. The two dipoles are assumed to be fed in phase quadrature, resulting in circular polarized radiation in the upward direction.

10.4 DIPOLE AND MO OPOLE ANTENNAS

649

pattern take the form \11 (8. )= 4sin(kz 0 co 8) X

[cee inco e + ,.

+j (ee co

co

for O ~ 8

~

e-

"")cos(k H

e

co ....,

e

. "" ) lil ....,

in sin 0) - cos(k H ) l - sin 2 sin 2 8

co (k H co in 8) - cos(kH ) ] 1 - cos 2 sin 2 8

j(

2

(10.104a)

where 2H i the dipole length, ~o i the height above the ground plane, e is the elevation angle counted from the z-a.xi , and is the azimuth angle counted from the x -axi · and e are the unit vector in the 0 and directions, respectfully. Near E> = 0. Equation (10. 104a) can be implified

ee

(10.104b) Evidently, the radjation pattern is circularly polarized in this region, but it approaches linear polarization for 0 ➔ 90°. The angular region over which the pattern is in effect circularly polarized can be extended by using drooped dipoles such as inverted-V or inverted-U dipoles. For increased impedance bandwidth, the linear dipoles may be replaced by a pair of crossed (and drooped) bow-tie antennas. Equation (10.104a) holds under the assumption of an infinite ground plane. For a finite ground plane, pattern distortions will occur in particular in the 0-region near 90° (the radiation field now extends into the lower half space). Most dipole and monopole antennas are of circular cross section, and in the theory of tubular antennas, this is usually assumed. If the cross section deviates from circular, the radiation pattern of the antenna is not noticeably affected; it is determined in effect by the antenna's height only and so is the radiation resistance, as long as the cross-section dimensions are small in comparison to H and A. The reactive part of the antenna impedance, on the other hand, is related to the near-field stored energy of the antenna, and thus, it will show a moderate dependence on the cross-section shape. This dependence may be accounted for by replacing an antenna of noncircular cro s section by a circular cross-section antenna of eqwvalent radius. Balanis has determined the equivalent radii, ae, for a number of cross sections. Results are shown in Table 10.2 [12].

10.4.3

Vector Effective Height and Effective Area of Dipole Antennas

A quantity very useful for the characterization of linear antennas operating in the receive mode is the vector effective height [10, 13). Although especially useful for linear antennas, the concept applies to any passive, reciprocal antenna; it is closely related to the effective area of these antennas, a quantity discussed in the chapter on Antenna Fundamentals.

650

ANTENNAS II: REFLECTOR, LENS, HORN, AND OTHER MICROWAVE ANTENNAS

TABLE 10.2 Equivalent Radii for Conductors of Noncircular Cross Section (From Balanis [12), © 1982 Harper & Row) Geometrical Shape

Electrical Equivalent Radius

ae = 0.25a

ae ~ 0.2(a + b)

ae = 0.59a

0.6 ca ~ 0.4

.,.

_.,... 02

~-

-0.4

I.,

""""

-- -I

0.6

0.8

1.0

b/a

I V \

V

\

?- - '

1 lo ae~ - - - -2 (Si+ S2) X [Sf ln a,+ lo a 2 + 2S1S2 In s] S1, S2 = peripheries of conductors C 1, C2 a 1, a2 = equivalent radii of conductors C 1, C2

sf

10.4

DIPOLE AND MO OPOLE ANTE

AS

651

The vector effective height i defined in term of the open circuit voltage at the antenna terminal when the antenna i operated in the receive mode. A ume that a plane wave i incident upon the antenna from the direction 0 0 , 0 , a ' indicated in Fig. 10.77 . Pa 1 e antenna are linear device , and the open circuit vo]tage, V0 c , can be written as (1 0. 105) where E inc i the (unperturbed) incident electric field at the location of the terminal of the antenna and the factor h~ defines the vector effective height of the antenna. Both E inc and h~ are treated here a two-dimen ional vector becau e E inc i orthogonal to the (known) direction of incidence. By u e of the reciprocity theorem h e, can be expre ed in term of the field trength radiation pattern of the antenna in the tran mit mode. If a tran mit voltage yrr i applied to the antenna terminal , the input current at the terminal will be (1 0.106a) where Z A is the input impedance of the antenna, and the antenna will radiate a field that in the far- field region take the form 1

-jkr

E " = - ZoroxH rr = --. Z0 Irr e 4TCJ

0

\II ( 0 , )

for kr

r

~ oo

---------'------,----,.---- y I

I

I I

I /

I

I/

X

Figure 10.77 Plane wave incident from direction 0o, tf,o.

( 10.106b)

652

ANTENNAS Il: REFLECTOR, LENS, HORN, AND OTHER MICROWAVE ANTENNAS

where \fl (E> , ) is the field strength radiation pattern of the antenna, r is the radial distance from the antenna terminals, r0 is the unit vector in the radial direction, and Zo = 120n Q. We have r0 • \fl = 0, and power conservation requires that Zo (4n) 2

[[

11

1'11 (0 , )1 2 sin0d0d

= Rrad

(10.107)

41l'

where Rrad is the radiation resistance of the antenna* . By a straightforward application of the reciprocity theorem to the transmit vs. the receive case, one finds (10.108) which defines the vector effective height in terms of the field strength radiation pattern of the antenna. Furthermore, with the effective area of the antenna given by (see Equation (9.12) of chapter on Antenna Fundamentals):

(10.109) where 4n l'11 (0 , )1 2

D(E> , ) = - - - - - - - -

1f

(10.110)

2

1"1(0, )1 sin 0d0d

41l'

is the directivity function of the antenna and TJA combining Equation (10.107)-(10.110)

= Rrad / RA its efficiency, one finds by (10.111)

which shows the relationship between he and A e. Both A e and lhel 2 have the same directional dependence, i.e., that of the directivity D(E>, ) of the antenna But note that the vector effective height, a complex vector quantity, provides more information on the receiving characteristics of the antenna than does the effective are~ which is a real scalar. The power delivered by the antenna to a receiver of impedance ZR= RR+ j XR is

(10.112)

• The resistive pan of the input impedance of the antenna consists of two parts Re (ZA) where R rat.1 accounts for the radiated power and Rioss for any antenna losses.

= RA = Rrad + R 1oss

10.4

DIPOLE AND MONOPOLE ANTENNAS

653

which my be rewritten as

Pru =

12 2 Cz:R:::12)c~~,2·.~E:~~2)G!:lh,1)cE;: )

= T1fT] pAes:c Here

TJr

TJp

(10. 113)

4RARR

+ ZR l2

=

IZA

=

lh . E incl2 lh el 2 IEincli : polarization mismatch factor between antenna

:

impedance mismatch factor between antenna and receiver

and incident field

e

The antenna efficiency TJA = Rrad/ RA does not explicitly appear in Equation (10.113) because it is included already in the effective area Ae; see Equation (10.109). In the case of perfect impedance and polarization match, one has T/ 1, TJ p = l and Equation ( 10.113) reduces to the well-known relation Pree = AeS~C, where Ae is the effective area (10.111) of the antenna and ~ c is the magnitude of the Poynting vector of the incident wave. The discussion so far has been concerned with the general class of passive, reciprocal antennas. For cylindrical dipole antennas, the vector effective height is with Equations (10.108) and (10.99) (10.114) where / (z) is the antenna current in the transmit mode and / N = I ( 0 ) = Jtr. Approximating I (z) by a sinusoidal current distribution, i.e., by I (z)

= 1(0) sin~k; lzl)

for k H


= 21r / A at frequency f.

(11.8)

11 . l

ARRAY ANTENNAS

705

The In are complex weights a igned to each element, and fn (0, ¢) is the radiation pattern (or element pattern) of the nth element in the presence of the other array elements. If all element patterns are a sumed to be the same (an assumption that array edge effects and hence interelement coupling i negligible), one can factor a unique element pattern /(0, ) out of the expression (11.8) and the pattern is seen as the product of an array factor F (0, ¢) and the element pattern. A (0, )

= f (0 , ) F (0, )

where F (0, )

= L In exp(j kndxu)

(1 1.9)

In this case one can create a maximum of A (0, :.= «s Q)

cc - 40

0.6

0.4

0.2

0.8

uor v

Figure 11.14

Cbebyshev array pattern.

where NT= number of elements = 2N + 1 for NT odd = 2N for NT even m

= 0, I , 2, 3, . ..

8 = l for NT even and 8 form= 0 est= 1 for all other m =2

= 0 for NT odd and the constant

Stegen's formulas are obtained by expanding the Chebyshev radiation pattern in a Fourier series, and are far more convenient and stable to compute than the original equation of Dolph or those derived prior to Stegen' s work. The Chebyshev pattern synthesis procedure has received much attention in the literature. Stegen and others (21-23] give equations for beamwidth, and there are several convenient expressions for array gain valid for large arrays. For spacings greater than )../ 2, Drane [22] gives the following equation for the directivity of a large array:

2r 2 D= - - - - 2- - - - 1 + ('A/ L)r [1n(2r )/ ,r]I /2

(11.31)

In this expression L is the array length L = (N - I )dx. This result is in close agreement with the equation of Elliott (23], and shows that the directivity does not increase indefinitely with L but reaches a maximum value 2r2 , or 3 dB greater than the specified sidelobe level. Although the Chebyshev pattern is a classic synthesis procedure and is well documented and conveniently tabulated, it is not useful for large arrays because of the gain limitation mentioned earlier. The stipulation that the sidelobes remain constant for large angles leads to a maximum in the directivity and then reduced directivity with further increases in array length. In addition, for increasingly large arrays, this requires a nonmonotonic aperture illumination with peaks at the array edges and cannot be excited efficiently.

11.l

ARRAY ANTENNAS

721

Taylor Line Source Synthesis. The most commonly u ed method of ynthesizing lowidelobe pattern was developed by Taylor [24] and assumes a continues line source instead of a set of discrete radiator . Taylor showed that equal sidelobe patterns could not be synthesized exactly with a continuou di tribution, but could be approximated by a function of two parameters, A and llbar (often denoted as n with an overbar). The family of patterns derived by Taylor have the first n bar sidelobes at the desired level, and all idelobes beyond n = nbar fall off like (sinrr)/rrz for z = Lu/A for a line ource of length L. The synthe ized pattern, normalized to unity, is .

F (z, A , llbar)

=

TI

nbar- 1

SlilJr Z

rr z

n=I

2

2

1 - Z / Zn 2 n2 1- z/

for z

= uL /A

(11.32)

The number z11 are the zero of the function and are given by Zn=

±aJ A 2 + (n - 1/ 2) 2 for 1 ~ n ~ nbar

where

(11.33)

nbar

a=-------[A2 + (nbar - 1/ 2)2] 1/ 2

The parameter A is defined in terms of the voltage sidelobe ratio r as (11.34) The "dilation factor,, a is applied to move the first few (nbar) zeros of the sin(rrz)/nz pattern to correspond closely to those of the Chebyshev pattern, and so to a first approximation the beam.width is increased by that same factor. An approximation for the beamwidth is therefore given by (11.35) where

The aperture distribution required to produce Taylor patterns is nblr--]

g(x)

= F(O, A, nbar) + 2

L

F(m, A, n bar) cos(2mrrx / L )

for - L /2 ~ x ~ L/ 2

m=l

(11.36) The coefficients F(m, A,nbar) are the pattern values of Eq. (11.32) at the integers

I 2

[(nbar - 1) .] F(m,A,nbar)=( ) I( l )' nbar - 1 + m . nbar - m .

TI (l -m2/ 2)

n1,ar- l n=l

Zn

(11.37)

Figure 11.15 shows the pattern of a ~ontinuous source computed using F.q. (11.32). The pattern has a -40-dB Taylor taper, length L = 8>.., and nbar = 4. The pattern of

ANTENNAS 111: ARRAY, MILLIMETER WAVE, AND INTEGRATED ANTENNAS

722

co

-0 ~

Q)

-20

~

0

a. Q)

> .:

10 dB) at fundamental mode excitation is 9.3%, 8.8%, and 7 .6%, re pectively. Thus, to allow for a bandwidth of 10%. Er hould not exceed 10; but for a bandwidth of 5% it may be as high as 30. The tudies indicate furthermore that by careful selection of the dimensions and location of the feed probe, good impedance match to 50 Q can be achieved at low VSWR o er the bandwidth indicated above. In addition, careful selection of the probe position will help in uppre ing higher-order modes which otherwise may be excited together with the fundamental mode resulting in pattern distortions. Since dielectric resonator antenna are electrically small, their radiation patterns in good approximation will be tho e of a short electric dipole or short magnetic dipole (or a combination of the e patterns) depending on the resonant mode that is used. For the three antennas of Fig. 11.30, when operated in their fundamental modes, the co-polarized radiation pattern is close to that of a magnetic dipole pointed in the xdirection parallel to the ground plane as indicated by the arrows in Fig. 11.30. This holds for all three antennas. The cross-polarized patterns would be more difficult to predict. The co-polarized (magnetic dipole) pattern is omnidirectional in the £-plane (y-z plane); in the H-plane (x-z plane) it has a broad maximum at zenith and decreases with 0 reaching a zero at 0 = ±90°, where 0 is the elevation angle counted from the z-axis. Figs. 11 .31 and 11.32, taken from a paper by Shum and Luk (36], show the computed input impedance and the calculated and measured radiation patterns for a cylindrical dielectric resonator antenna of the type sketched in Fig. 11.30b. The antenna, designed for the 6 GHz region, is excited in the fundamental HEM11s mode and its parameters are Er= 9.2, d = 2a = 12.2 mm, b = 4.15 mm, l = 5.1 mm, and r0 = 0.51mm; see Fig. 11.31a. For the measurements the antenna was placed on a 6Ao x 6Ao ground plane where Ao is the free-space wavelength. Shum and Luk make several interesting comments on these figures including the following [36] . Fabrication imperfections may introduce an air gap of width tP between the feed probe and the dielectric medium of the resonator, and an air gap of height tg between the resonator and the ground plane; see Fig. 11.31a. Since Eaorcn is discontinuous at the air-dielectric interface, a high electric field strength will exist in these gaps, which- as is evident from Fig. 11.31 b- can result in a significant shift in resonance frequency and input impedance. Thus, in the theory and design of dielectric resonator antennas such air gaps have to be accounted for accurately (36]. Furthermore, the co-polarized radiation patterns in Fig. 11.32- as expected- are close to those of a horizontal magnetic dipole, but the £-plane pattern shows a noticeable asymmetry which is attributed to the presence of higher-order modes. Also, in the H -plane, the cross-polarized pattern is only about 10 dB below the maximum of the co-polarized pattern, which may not be satisfactory. The study has shown, however, that as the distance b of the feed probe from the antenna axis is increased the asymmetry of the £-plane pattern is reduced, and so is the magnitude of the cross-polarized pattern in both the E- and H-planes. Hence the feed probe should be located as close to the edge of the resonator as possible. The resonance frequency will be little affected; it varies only slightly with probe position (36].

750

ANTENNAS III: ARRAY, MILLIMETER WAVE, AND INTEGRATED ANTENNAS

z

(a)

t

Dielectric ,,..- ~ resonator ~

8

7 .T

d

I

100

(b)

-a

.._,

tg =

p=

80 60

Q)

0 C

40

"C

20

n,

Q)

a.

E

....:,

0

a. -20 C

-40 -60

5

5.5

6.5

6

7

7.5

Frequency (GHz)

Figure 11.31 Input impedance of probe-fed cylindrical dielectric resonator antenna: (a) antenna geometry and (b) impedance vs. frequency (from Shum and Luk (36], © 1998 IEEE).

(a)

.,-- 0--._

(b)

0

180

180

- - Copol (computed); - - - - - Copol (measured); - ·-· -·- Xpol (computed);

•··········· Xpol (measured)

Figure 11.32 Radiation pattern of probe-fed cylindrical dielectric resonator antenna: (a) Eplane pattern and (b) H -plaoe pattern (from Shum and Luk (36], © 1998 IEEE).

l J .2

MILLIMETER WAVE ANTENNAS

751

z

(a)

Metallized surface

(b)

Metal

X X

Figure 11.33 Alternative feed arrangements for dielectric resonator antennas: (a) aperturecoupled micro trip feed (after Leung et aJ. (38]. © 1995 IEEE) and (b) conformal metal strip feed (after Leung [39], © 2000 IEEE).

In Fig. 11.30 the antennas are fed by a coaxial probe. Other useful feed arrangements are hown in Fig. 11.33. The e include excitation by a slot-coupled microstrip line [38] or by an external metallic strip placed on the surface of the (hemispherical) resonator [39] . At the bottom end the strip is connected to the inner conductor of the coaxial feed line. The microstrip feed arrangement, Fig. 11.33a, provides a simple and effective way for interfacing dielectric resonator antennas with integrated circuits. At the same time this arrangement shields the integrated circuits from the antenna radiation [38]. The main advantage of the external metal strip feed metho~ Fig. 11.33b, is its simplicity [39]. The effort of drilling a hole for the coaxial probe into a small resonator and the error in impedance and resonance frequency that an oversized hole produces (see above) are avoided. In addition, this feed technique naturally satisfies the condition that the feed probe be placed as far off-axis as possible, and thus is likely to result in a largely distortion-free co-polarization pattern and a minimum cross polarization level [36, 39] . Dielectric resonator antennas should be useful in particular for communication in the microwave and mm-wave bands when omnidirectional coverage is required and small antenna size in combination with high efficiency is important. They should also be well suited as radiating elements for integrated phased arrays when a wide scan range is specified and high efficiency is a major design goal.

Traveling-Wave Antennas Derived from Open Millimeter Waveguides. Most mi11imeter waveguides are open guiding structures; to minimize structural complexity and conduction losses, they do not enclose the guided fields on all sides by metal walls. But this means that if the uniformity or the symmetry of the guiding structure is perturbed, or the guide is not excited in the appropriate mode, radiation will occur and part of the guided energy will be leaked away from the waveguide. This leakage effect may be used to advantage for the design of antennas, by intentionally introducing perturbations in an open waveguide such that radiation occurs in a controlled manner. Antennas of this kind are typically leaky-wave antennas or modified surface-wave antennas. Similar to the guides from which they are derived, they are structurally siinple and easy to fabricate; they are directly compatible with these guides and suitable for integrated designs.

752

ANTENNAS ill: ARRAY, MlLLIMETER WAVE, AND INTEGRATED ANTENNAS

In this section we first present a number of typical antenna types in this class that are basically line sources. Next, we discuss several examples of arrays that are based on these line sources and permit two-dimensional scanning. Finally, we consider how pattern shaping and sidelobe control may be achieved with some of these antennas. Part of the work on this class of antennas is very recent. A presentation of the theories underlying the performance of these antennas would go beyond the scope of a handbook. We therefore restrict ourselves here to a brief description of the operating principles of the antennas and refer to the literature for a discussion of their theory and design guidelines. In particular, Ref. I provides a systematic review of many of these antennas. This reference may not be up-to-date with respect to all of the antenna structures, but it contains all of the principles and includes an extensive bibliography on this class of antennas. All of the antennas discussed here show significant differences in their operating principles, and each may be regarded as typical for a subclass of open-waveguide traveling-wave antennas. Some millimeter waveguides, in particular metal waveguide and finline, are shielded rather than open guiding structures, and the design of antennas that are directly compatible with these guides takes a somewhat different approach. At the end of this section we will briefly discuss some finline antennas that may be viewed as rudimentary horns. The first three antenna types that belong to the class of open-waveguide travelingwave antennas discussed above are derived from dielectric waveguide, and they are sketched in Fig. 11 .34. The first antenna is a tapered dielectric rod antenna, and the other two are periodically perturbed dielectric image guides. These antennas have been known for some time - tapered rod antennas for many years -but they have found renewed interest as millimeter wave antennas. The analytical treatment of the antennas in Fig. 11.34 is difficult, but due to recent systematic work in this area, the theory of the periodic dielectric antennas can be regarded as well understood by now and detailed design information on these antennas has become available. This is due primarily to the work of Peng and Schwering on the antenna of Fig. 11.34b and to the work of Oliner and Guglielmi on the antenna of Fig. 11 .34c. Tapered dielectric rod antennas are well known and have been used, on a limited scale, in the microwave region (e.g., as "poly rod" antennas). These antennas consist of a dielectric waveguide whose cross section decreases monotonically in the forward direction. The cross section may have a rectangular or circular shape leading to an antenna of pyramidal or conical shape, respectively. The antenna may be fed by a dielectric waveguid~ of the same permittivity. A surface wave incident from the guide will gradually spread out as it travels along the tapered section and thus be transformed from a strongly bound mode to a free-space wave. The antenna is a typical surfacewave antenna whose phase velocity is smaller than the free-space wave velocity c, so that it radiates in the forward (end-fire) direction. The directivity gain that can be achieved with these antennas is limited to 18 to 20 dB, corresponding to a beamwidth of 17 to 20° [44-47] . But good bandwidth can be obtained, and by careful design, sidelobes may be kept at a level more than 25 dB below the main peak [46]. The theory of these antennas is not very well developed, but empirical design guidelines have been available for some time [44, 45]. The taper of the antenna does not have to be linear, as assumed in Fig. 11.34a, but can take any shape. By choosing this shape carefully, one can minimize the power reflected back to the feed and maximize the power radiated. It is important to produce a gradual transition from the feed to the tapered section, and then to terminate the

11.2

MILLIMETER WAVE ANTENNAS

753

(a) X

(b)

(c)

Metal strips

Figure 11.34 Traveling-wave antennas derived from dielectric waveguides: (a) tapered dielectric rod antenna; (b) periodic leaky-wave dielectric antenna with rectangular grooves; (c) periodic leaky-wave dielectric antenna with a metal-strip grating.

taper in a sharp tip. For a long, gradual taper, the linear profile is found to be very effective [47]. On the other hand, the antenna profile may be shaped to achieve a phase velocity of the guided wave that approaches-over a substantial portion of the antenna length-the value postulated by the Hansen- Woodyard condition for end-fire arrays of enhanced directivity, which will result in an antenna of maximum gain. A problem with maximum-gain antennas is their pattern quality, which is distorted by high side-lobes. For a discussion of these antennas, we refer to Zucker [45]. Alternatively, the directivity of tapered dielectric rod antennas can be increased by using linear or planar arrays of these antennas with the (parallel oriented) element antennas excited in equal phase. [45]. The directivity of a planar array is determined by the aperture area of the array and would not appreciably exceed that of a dipole array of the same aperture size. But since tapered rod antennas already have a substantial directivity, they can be spaced much farther apart than dipole elements. Hence fewer

754

ANTENNAS ill: ARRAY, MILLIMETER WAVE, AND INTEGRATED ANTENNAS

elements are needed and mutual coupling is much reduced. The antenna spacing is limited, on the other hand, by the need to keep grating lobes of the array pattern at a sufficiently low level. Ideally, the first grating lobe should be collocated with the first zero of the element pattern, which in effect would suppress this lobe [45] . Periodic dielectric antennas [48-52] consist of a uniform dielectric waveguide with a periodic surface perturbation. The waveguide is excited in the fundamental mode [l] and its width (in the y direction of Fig. 11.34) is usually chosen comparatively small to guard against the excitation of higher-order modes. If a narrow beamwidth in both principal planes of the antenna is desired, the width of the dielectric waveguide may be chosen to be large; single-mode excitation must then be ensured by the use of an appropriate feed a1Tangement. The surface perturbation of the waveguide may take the form of a dielectric grating, as indicated in Fig. 11.34b, or one may use a grating of metal strips, as in Fig. 11.34c, or metal disks. Diffraction at this grating will transform the basic guided mode into a leaky wave and the waveguide becomes an antenna The grating periodicity excites an infinite number of space harmonics of the basic slow wave type on the dielectric waveguide, and the period of the grating is chosen relative to the wavelength such that then = - l space harmonic becomes fast but all the others remain slow. As the frequency is increased, this fast space harmonic produces a radiating beam that scans from backfire through broadside and into the forward quadrant, even reaching end fire if the parameters are chosen appropriately. The theory underlying the operation of the antenna in Fig. 11.34b is presented in Ref. 51. A summary is also given in Ref. 1, which contains many details, both with respect to the phase and leakage constant behavior and the radiation pattern performance, together with design information. In practice, there is an upper limit to the value of leakage constant obtainable with grooved dielectric antennas, so that the radiated beams from uch antennas are generally narrow. In most cases, narrow beams are desired, however. Metal-strip gratings of the type shown in Fig. 11.34c permit both wide and narrow beams, so that they are more versatile. Early experimental studies were made at the U.S. Army Electronics Laboratories at Fort Monmouth [48), and the first complete theory for the metal-grating antenna in Fig. 11.34c appeared in Ref. 52. The details of the underlying theory are treated thoroughly in [90, 91, 92]. Another accurate method of analysis for these antennas was presented by Encinar [53). Many of the general design consideration appropriate to grooved dielectric antennas apply as well to those with metal-strip gratings, although the expressions for the phase and leakage constants are different. of course. A scan rate of 20 to 40° for a l 0% frequency variation is typical [48, 51]. Note, however, that these grating antennas, as all traveling wave periodic antennas, do not radiate in the exact broadside direction. As broadside conditions are approached, an internal resonance develops which inhibits radiation. The associated "open stopband" of the antenna, however, is narrow and, since practical antennas have a finite beamwidth, the only effect usually noticed when scanning through broadside is an increase in SWR, and therefore a dip in the radiated power level. Leaky-wave antennas based on uniform open waveguides form the next category, where the three most common of the open waveguides are the groove guide, the nonradiative dielectric (NRD) guide (which is an improved variant of H-guide but was conceived independently), and microstrip line operated in its first higher-order mode. The antennas shown in Fig. 11.35 fall into this category. Some of the recent improvements on these basic structures are described shortly.

11.2

MILLIMETER WAVE ANTENNAS

(a)

755

z

/ E _..

(b)

e.,

(c)

e.,

4-T-m~~""'"""""""'~i

>

~

~

Q)

a:

I

11 H•plane

- 30.00

0 0

g

0 0

ci (0

0 0

g

~

- 20 00 .

ci

8

.. . :,- ::

. ) l/

E-plane

0 0

. = : :

1

\:

8

8

g g g

Observation angle (deg.)

-30.00

0 0

g

0 0

E-plane

H-pfane

g

8 ~

8

ci

8

ci t"')

8

8

g g

Observation angle (deg.)

Figure 11.45 Theoretical and measured radiation pattern of tapered slot antennas: (a) antenna ~eometry, (b) linear-tapered slot antenna on 20-mil Duroid substrate, and (c) Vivaldi (exponenllally tapered) slot antenna on 0.5" styrofoam substrate (from Janaswamy and Schaubert [88] © 1987 IEEE). '

.REFERENCES

769

and Schaubert [88]. For thin as well as truck substrates of low Er the main beam is nearly circular symmetric de pite the planar geometry of the antenna. Fig. 11.45 shows theoretical E-plane and H-plane pattern for a linear-tapered lot antenna on a 20-mil Duroid ub trate and a Vivaldi antenna on a 0.5" styrofoam ubstrate. The measured patterns, taken at 10 GHz. confinn the circular symmetry of the main beam.

REFERENCES 1. F. Schwering and A. A. Oliner. ''Millimeter Wave Antennas,''

in Y. T. Lo and S. W . Lee,

Ed ., Handbook on Antenna Theory and Design , Van Nostrand Rei nhold, New York, 1988, Chap. 17. 2. P. Bhartia and I. J. Bahl. Millimeter Wa ve Engineering and Applications , Wiley, New York, 1984, Chap. 9.

3. R. B. Dybdal, ·%llimeter Wave Antenna Development," Proc. Antenna Appl. Symp., University of Illinois, Urbana. September 22-24, 1982. 4. K. Solbach, "Various Millimeter Wave Antennas for Radar and Communication," Int. Work-

shop MM-Waves, University of Rome "Tor Vergata," Rome, Italy, April 2-4, 1986. 5. J. J. Lee, "Dielectric Lens Shaping and Coma-Correction Zoning. Part I. Analysis," IEEE Trans. Antennas Propag., AP-31 , pp. 211-216, January 1983. 6. J. J. Lee and R. L. Carlise, "A Coma-Corrected Multibeam Shaped Lens Antenna. Part II. Experiments,' IEEE Trans. Antennas Propag., AP-31 , pp. 216-220, January 1983. 7. W . Rotman. '"EHF Dielectric Lens Antenna for Multiple Beam MILSATCOM Applications," Dig. Int. IEEEIAPS Symp., Albuquerque, NM, June 1982, pp. 132-135. 8. T . L. ap Rhys, ''The Design of Radially Symmetric Lenses," IEEE Trans. Antennas Propag., AP-18, pp. 497- 506, July 1970. 9. T. Ohtera and H. Ujiie. ''Radiation Performance of a Modified Rhombic Dielectric Plate Antenna," IEEE Trans. Antennas Propag., AP-29, pp. 660-662, July 1981. 10. A. Rombach, "Dielectric Feeds with Low Cross Polarization," Proc. Int. URSI Symp. , 1980, Munich, Germany, August 26-29, 1980. 11. F. G. Farrar, ''Millimeter Wave W-Band Slotted Waveguide Antennas," Dig. IEEFIAPS Int. Sym., Los Angeles, June 16-19, 1981, pp. 436-439. 12. J. L. Hilburn and F. H. Prestwood, "K-Band Frequency Scanned Waveguide Array," IEEE Trans. Antennas Propag., AP-22, pp. 340-342, March 1974. 13. K. Solbacb, "Some Millimeter-Wave Slotted Array Antennas," Proc. 14th Eur. Microwave Conf, Luttich, Belgium, 1984. 14. K Solbach, ''Hybrid Design Proves Effective for Flat Millimeter Wave Antennas," Microwave Systems News of Communication Technology, 15, pp. 123 -138, June 1985. 15. B. R. Rao, "94 Gigahertz Slotted Waveguide Array Fabricated by Photolithographic Techniques," Dig. Int. IEEE-APS Symp. , Houston, May 23-26, 1983, pp. 688-689. 16. M.A. Weiss, "Microstrip Antennas for Millimeter Waves,." IEEE Trans. Antennas Propag., AP-29, pp. 171-174, January 1981. 17. F. Lalezari and T. Pett, Millimeter MicrostripAntennasfor Use in MIL-SPEC Environment, Final Report to Battelle Columbus Labs./U.S. Anny CECOM, Fort Monmouth, NJ, November 21, 1983. 18. M. A. Weiss and R. B . Cassell, Microstrip Millimeter Wave Antenna Study. R&D Tech. Rep., CORAPCOM-77-0158-F, U.S. Army CORADCOM, Fort Monmouth, NJ, April 1979. 19. A. Henderson, A E. England, and J. R. James, "New Low-Loss Millimeter Wave Hybrid Microstrip Antenna Array," 11th Eur. Microwave Conf., Amsterdam, September 1981.

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ANTENNAS ill: ARRAY, MILLIMETER WAVE, AND INTEGRATED ANTENNAS

20. J. R. James and A. Henderson, "A Critical Review of Millimeter Planar Arrays for Military Applications," Mil. Microwave Conj., London, October 20-22, 1982. 21. J. R. James, G. John, and C. M. Hall, "Millimeter-Wave Hybrid Dielectric-Microstrip Antenna Array." IEE Proceedings, 131, Part H, No. 6, pp. 341-350, D ecember 1984. 22. J. R. James and C. M. Hal1, Investigation of New Concepts for Designing Millimeter-Wave Antennas, Final Tech. Rep. on Contract DAJA37-80-C-0183, U.S. Army European Research Office, London, September 1983. 23. N. G. Alexopoulos, P. B. Katehi, and D. B. Rutledge, "Substrate Optimization for Integrated Circuit ~tennas," IEEE Trans, Microwave Theory Tech., MTT-31, pp. 550-557, July 1983. 24. Z. Rav-Noy, C. Zah, U. Schreter, D. B. Rutledge, T. C. Wang, S. E. Schwarz, and T . F. Kuech, "Monolithic Schottky Diode Imaging Arrays at 94 GHz," Dig. Infrared Millimeter Wave Conj., Miami Beach, FL, December 1983. 25. H . R. Fetterman, T . C. L. G. Sollner, P. T . Parrish, C. D. Parker, RH. Mathews and P. E. Tannenwald, "Printed Dipo]e Millimeter Wave Antenna for Imaging Array Applications," Electromagnetics, 3, pp. 209-215, 1983. 26. P. B. Katehi and N. G. Alexopoulos, "On the Modeling of Electromagnetically Coupled Microstrip Antennas-The Printed Strip Dipole," IEEE Trans. Antennas Propag., AP-32, pp. 1179-1186, November 1984. 27. N. G. Alexopoulos and D. R. Jackson, "Fundamental Superstrate (Cover) Effects on Printed Circuit Antennas," IEEE Trans. Antennas Propag., AP-32, pp. 809-816, August 1984. 28. N. G. Alexopoulos and D . R. Jackson, "Gain Enhancement Methods for Printed Circuit Antennas," IEEE Trans. Antennas Propag., AP-33, pp. 976-987, September 1985. 29. H. Y. Yang and N . G. Alexopoulos, "Gain Enhancement Methods for Printed Circuit Antennas through Multiple Superstrates," IEEE Trans. Antennas Propag., AP-35 , pp. 860-863, July 1987. 30. D. R. Jackson and A. A. Oliner, "A Leaky-Wave Analysis of the High-Gain Printed Antenna Configuration," IEEE Trans. Antennas Propag., AP-36, pp. 905-910, July 1988. 31. G. P. Gauthier, J. P. Raskin, L. P. B. Katehi, and G. M. Rebeiz, "A 94 GHz ApertureCoupled Micromachined Microstrip Antenna," IEEE Trans. A11tennas and Propag., AP-47, pp. 1761 - 1766, 1999. 32. J. C. Yook and L. P. B. Katehi, "Micromachined Microstrip Patch Antenna with Controlled Mutual Coupling and Surface Waves," IEEE Trans. Antennas and Propag., AP-49, pp. 1282-1289, 2001. 33. S. A. Long, M. W. McAllister, and L. C. Shen, "The Resonant Cylindrical Dielectric Cavity Antenna," IEEE Trans. Antennas Propag.• AP-31, pp. 406-412, 1983.

34. S. J. Fiedziuszko and S. Holme. "Dielectric Resonators," IEEE Microwave Magazine, 2, pp. 51-60, September 2001 .

35. K. W. Leung, K. M. Luk, K. Y. A. Lai, and D. Lin, ''Theory and Experiment of a Coaxial Probe-Fed Hemispherical Dielectric Resonator Antenna," IEEE Trans. Antennas Propag., AP-41, pp. 1390-1398, 1993. 36. S. M. Shum and K. M . Luk, "FDTD Analysis of a Probe-Fed Cylindrical Dielectric Resonator Antenna," IEEE Trans. Antennas Propag., AP-46, pp. 325-333, 1998. 37. R. K. Mongia and A. Ittipiboon, "Theoretical and Experimental Investigation on Rectangular Dielectric Resonator Antennas," IEEE Trans. Antennas Propag., AP-4S, pp. 1348-1356, 1997. 38. K. W. Leung, K. M . Luk, K . Y . A. Lai, and D. Lin, "Theory and Experiment of an Aperture Coupled Hemispherical Dielectric Resonator Antenna," IEEE Trans. Antennas Propag., AP43, pp. 1192- 1198, 1995.

REFERENCES

771

39. K. W . Leung, "Conformal Strip E citation of Dielectri c Resonator Antenna," IEEE Trans. Antennas Propag .. AP-48 , pp. 96 L-967, 2000. 40. K . W . Leung. K . K. Tse, K . M . Luk, and E. K. N. Yung, "Cross-Polarizalion Characteristic of a Probe-Fed Hemi pherical Dielectric Resonator Antenna," IEEE Trans. Antennas Propag.. AP-47, pp. 1128- 1230. 1999. 41. D. R. Jack on, A. A. Oliner, and A. Ip. ''Leaky-Wave Propaga ti on and Radfation for a arrow-Beam Multiple-Layer Dielectric Structure." IEEE Trans. Antennas Propag., AP-41, pp. 344-348, March 1993. 42. H. 0 tner, J. Detlefsen, and D. R. Jackso n, " Radiation from One-Dimensional Dielectric Leaky-Wave Antenna ," IEEE Trans. Antennas Propag., AP-43, pp. 331 - 338, April 1995. 43. M. Thevenot, C. Cheype, A. Rei neix. and B. Jecko, "Directive Photonic-Bandgap Antenna ," IEEE Trans. Microwave Theory Tech.. MTT-47, pp. 2115 - 2122, November 1999. 44. D . G. Kiely, Dielectric Aerials, Methuen, London, 1952. 45. P. J. Zucker. ·'Surface Wave Antenna ," in R. C. Johnson, Ed., Antennas Engineering Handbook, 3rd ed. McGraw-Hill, New York, 1993, Chap. 12.

46. S. Kobayashi, R. Mittra. and R. Lampe, "Dielectric Tapered Rod Antennas for Millimeter Wave Applications," IEEE Trans. Antennas Propag., AP-30, pp. 54-58, January 1982. 47. S . T . Peng and P. Schwering, "Effect of Taper Profile on Performance of Dielectric Taper Antennas," Dig. URS! Natl. Radio Sci. Meet., Seattle, WA, June 18-22, 1979, p . 96. 48. K . L. K1obn. R . E. Hom, H. Jacobs, and E. Freibergs, "Silicon Waveguide Frequency Scanning Linear Array Antenna," IEEE Trans. Microwave Theory Tech., MTT-26, pp. 764-773, October 1978. 49. T . Itoh and B. Adelseck, ''Trapped Image Guide Leaky-Wave Antennas for Millimeter Wave Applications," IEEE Trans. Antennas Propag., AP-30, pp. 505-509, May 1982. 50. S. T. Peng andF. Schwering, Dielectric Grating Antennas, R&D Tech. Rep. CORADCOM78-3, Forth Monmouth, NJ, July 1978. 51. F. Schwering and S. T. Peng, "Design of Dielectric Grating Antennas for Millimeter Wave Applications,'' IEEE Trans. Microwave Theory Tech., MTT-31, pp. 199-209, February 1983. 52. M. Guglielmi and A. A. Oliner, "A Practical Theory for Image Guide Leaky-Wave Antennas Loaded by Periodic Metal Strips," Proc. 17th Eur. Microwave Conf, Rome, September 7-11 , 1987, pp. 549- 554. 53. J. A. Encinar, ·'Mode-Matching and Point-Matching Techniques Applied to the Analysis of Metal-Strip-Loaded Dielectric Antennas," IEEE Trans. Antennas Propag., AP-38, pp. 1405 - 1412, September 1990. 54. A. Sanchez and A. A. Oliner, "A New Leaky Waveguide for Millimeter Waves Using Nonradiative Dielectric (NRD) Waveguide. Part I. Accurate Theory," IEEE Trans. Microwave Theory Tech., MTT-35, pp. 737-747, August 1987. 55. A . Sanchez and A. A. Oliner, "Accurate Theory for a New Leaky-Wave Antenna for Millimeter Waves Using Nonradiative Dielectric Waveguide," Radio Science, 19, 1225-1228, Sept-Oct. 1984. 56. Q. Han, A. A. Oliner, and A. Sanchez, "A New Leaky Waveguide for Millimeter Waves Using Nonracliative Dielectric (NRD) Waveguide. Part Il. Comparison with Experiments," IEEE Trans. Microwave Theory Tech., MTT-35, pp. 748-752, August 1987. 57. P. Lampariello and A. A. Oliner, "A New Leaky Wave Antenna for Millimeter Waves Using an Asymmetric Strip in Groove Guide. Part I. Theory," IEEE Trans. Antennas Propag., AP-33, pp. 1285-1294, December 1985.

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ANTENNAS ill: ARRAY, MILLIMETER WAVE, AND INTEGRATED ANTENNAS

58. P. Lampariello and A. A. Oliner, "A New Leaky-Wave Antenna for Millimeter Waves Using an Asymmetric Strip in Groove Guide. Part II. Design Considerations," IEEE Trans. Antennas Propag., AP-33, pp. 1295-1303, December 1985. 59. A. A. Oliner, S. T. Peng, and K. M. Sheng, "Leakage from a Gap i n NRD Guide," Dig. IEEE Int. Microwave Symp. St. Louis, June 3-7, 1985, pp. 619-622. 60. H. Shigesawa, M . Tsuji, and A. A. Oliner, "Coupling Effects in an NRD Guide Leaky Wave Antenna," Dig. Natl. Radio Sci. Meet, Philadelphia, June 9-13, 1986, p. 27. 61. P. LamparieUo, F. Frezza, H. Shigesawa, M. Tsuji, and A. A. Oliner, "Guidance and Leakage Properties of Offset Groove Guide," Digest. IEEE Int. Microwave Symp., Las Vegas, NV, June 9-11, 1987, pp. 731-734.

62. H. Shigesawa, M. Tsuji, and A. A. Oliner, "Theoretical and Experimental Study of an Offset Groove Guide Leaky Wave Antenna," Dig. IEEF.IAPS Int. Symp. Blacksburg, VA, June 15-19, 1987, pp. 628-632. 63. A. A. Oliner, "Leaky-Wave Antennas," in Antenna Engineering Handbook, Third Edition, R. C. Johnson, Ed., McGraw-Hill, New York, 1993, Chap. 10. 64. A . A. Oliner, "Recent Developments in Millimeter-Wave Antennas," Alta Frequenza, LVill, pp. 491-505, September-December 1989.

65. A. A. Oliner, "Scannable Millimeter Wave Arrays," Final Report on RADC Contract No. Fl 9628-84-K-0025, Polytechnic University, 528 pages in two volumes, September 30, 1988. 66. P. Lampariello, F. Frezza, H. Shigesawa, M . Tsuji, and A. A. Oliner, "A Versatile LeakyWave Antenna Based on Stub-Loaded Rectangular Waveguide: Part I-Theory," IEEE Trans. Antennas Propag., AP-46, pp. 1032-1041, July 1998. 67. F. Frezza, P. Lampariello, H . Shigesawa, M . Tsuji, and A . A. Oliner, "A Versatile LeakyWave Antenna Based on Stub-Loaded Rectangular Waveguide: Part II-Effects of Flanges and Finite Stub Length," IEEE Trans. Antennas Propag., AP-46, pp. 1042-1046, July 1998. 68. M. Tsuji, H . Shigesawa, F. Frezza, P. Lampariello, and A . A. Oliner, "A Versatile LeakyWave Antenna Based on Stub-Loaded Rectangular Waveguide: Part ill-Comparisons with Measurements," IEEE Trans. Antennas Propag., AP-46, pp. 1047-1055, July 1998. 69. J. Green, H. Sbnitkin, and P. Bertalan, "Asymmetric Ridge Waveguide Radiating Element for a Scanned Planar Array," IEEE Trans. Antennas Propag., AP-38, pp. 1161 - 1165, August 1990.

70. F. Frezza, M. Guglielmi, and P. Lampariello, "Millimetre-Wave Leaky-Wave Antennas Based on Slitted Asymmetric Ridge Waveguides," TEE Proc. - Mierow. Antennas Propag., 141, pp. 175-180, June 1994. 71. C. Di Nallo, F. Frezza, A. Galli, and P. Lampariello, "Complete Characterization of LeakyWave Antennas Based on Stepped Rectangular Waveguides," Proc. European Microwave Conj., pp. 1062- 1067, Bologna, Italy, 1995. 72. C. Di Nallo, F. Frezza, A. Galli, G. Gerosa, and P. Lampariello, "Stepped Leaky-Wave Antennas for Microwave and Millimeter Wave Applications," Annales des Telecommunications, 52, pp. 202- 208, March-April 1997. 73. C. Di Nallo, F. Frezza, A. Galli, and P. Lampariello, "Theoretical and Experimental Investigations on the 'Stepped' Leaky-Wave Antennas," Digest IEEE AP-S lntemat. Symp., 2, pp. 1446-1449, Montreal, Canada, 1997. 74. A. A. Oliner and K. S. Lee, "The Nature of the Leakage from Higher Modes on Microstrip Line," Dig. IEEE Int. Microwave Symp. Baltimore, June 2-4, 1986, pp. 57-60. 75. A. A. Oliner and K . S. Lee, "Microstrip Leaky Wave Strip Antennas," Dig. TEEF./APS Int. Symp., Philadelphia, June 8-13, 1986, pp. 443-446.

76. W. Menzel, "A New Traveling Wave Antenna in Microstrip," Arch. Elektr. Ubertr., 33, pp. 137- 140, April 1979.

REFERENCES

773

77. W . Menzel, "A New Travelling Wave Antenna in Microstri p," Proc. 8th European Microwave Conference, pp. 302- 306, Paris. France, 4 - 8 Sept. 1978. 78. Y.-D . Lin, J.-W. Sheen, and C.-K. C. Tzuang, ·'Anal ysis and Design of Feeding Structures for Microstrip Leaky Wave Antenna," IEEE Trans. Microwave Theory Tech. , MTT-44, pp. 1540-1547, September 1996. 79. P. Lampariello and A. A. Oliner, "Bound and Leaky Modes in Symmetrical Open Groove Guide.' Alta Freq., 52(3), pp. 164- 166, 1983. 80. P . Lampariello and A. A. Oliner, "Leal,' Modes of Symmetrical Groove Guide," Dig. IEEE Int. Microwave Symp. Boston, M ay 30- June 3, 1983, pp. 390-392. 81. A. A. Oliner and S. J. Xu, •'A ovel Phased Array of Leaky-Wave NRD Guides," Dig. Natl. Radio Sci. Meet., Blacksburg, VA, June 15-19, 1987, p. 139.

82. P. Lampariello and A. A . Oliner, "A Novel Phased Array of Printed-Circllit Leaky-Wave Line Sources,'' Proc. 17th Eur. Microwave Conf. , Rome, September 7- 11, 1987, pp. 555-560. 83. M . Guglielmi and A. A. Oliner, "A Linear Phased Array of Periodic Printed-Circuit LeakyWave Line Sources That Permits Wide 1\vo-Dimensional Scan Coverage," Dig. Natl. Radio Sci. Meet., Syracuse, NY, June 6 - 10. 1988. p. 420. 84. K. Solbach, "Hybrid Design Proves Effective for Flat Millimeter Wave Antennas," MSN & CT, 15, pp. 123-138, June 1985. 85. S. Kobayas~ R. Lampe, R. Mitt:ra. and S. Ray, "Dielectric Rod Leaky-Wave Antennas for Millimeter Wave Applications," IEEE Trans. Antennas Propag., AP-29, pp. 822-824, September 1981. 86. J. A. Encinar, M . Guglielmi, and A A. Oliner, "Taper Optimization for Sidelobe Control in Mi11imeter-Wave Metal-Strip-Loaded Dielectric Antennas," Dig. Natl. Radio Sci. Meet., Syracuse, NY, June 6-10, 1988, p. 379. 87. K. S. Yngvesson, D. H. Schaubert, T. L. Korzeniows~ E . L. Kollberg, T. Thungren, and J. F. Johansso~ "Endfire Tapered Slot Antennas on Dielectric Substrates," IEEE Trans. Antennas Propag., AP-33, pp. 1392-1400, December 1985. 88. R. Janaswamy and D. H . Schaubert, "Analysis of Tapered Slot Antennas," IEEE Trans. Antennas Propag., AP-35, pp. 1058- 1065, 1987. 89. R. J. Mailloux, Phased Array Antenna Handbook, Artech House, Boston, 1994, Chap. 5. 90. M. Guglielmi and A. A. Oliner, ''Multi.mode Network Description of a Planar Periodic Metal-Strip Orating at a Dielectric Interface- Part I: Network Formulation," IEEE Trans. Microwave Theory Techn. , MTT-37, pp. 534-541 , March 1989.

91. M. Gugliehni :and A. A Oliner, "Multi.mode Network Description of a Planar Periodic Metal-Strip Grating at a Dielectric Interface-Part II: Small Aperture and Small-Obstacle Solnti.ons.n IEEE Trans. Microwave Theory Techn., MTT-37. pp. 542- 552, March 1989. 92 M.. Guglielmi and R Hochstadt, 'Multimode Network Description of a Planar Periodic Metal-Strip Grating at a Dielectric Interface- Part ill: Rigorous Solution," IEEE Trans. Microwav~ Techn.., MTI-37, pp. 902-909, May 1989.

12 ANTENNAS IV: MICROSTRIP ANTENNAS Y.T.Lo Electromagnetics Laboratory Depanmeni of Electrical and Computer Engineering University of lllinois at Urbana-Champaign Urbana, Illinois

S. M.

WRIGHT Electromagnetics and Microwave Laboratory Depanment of Elecrrical Engineering Texas A&M University College Station, Texas (formerly Electromagnetics Laboratory, Univ. of IL at Urbana-Champaign)

J. A.

NAVARRO Boeing-Phantom Works RF Technology & Phased Array Group Seattle, Washington

M.

DAVIDOVITZ Department of Elecrrical Engineering University of Minnesota Minneapolis, Minnesota (formerly Electromagnerics Laboratory, Univ. of IL at Urbana-Champaign)

12.1

GENERAL INTRODUCTION

Microstrip antennas were introduced some 50 years ago [l, 2]. They have received little attention until the early 1970s [3-5], when there came a new demand for compact and Handboolc of RF/Microwave CompoMnts and EngiMering, Edited by Kai Chang ISBN 0-471-39056-9 © 2003 John Wiley & Sons, Inc.

775

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ANTENNAS TV: MICROSTRJP ANTENNAS

lightweight antennas in many applications. Prompted largely by the effort of the Rome Air Development Center, Hanscom AFB, Massachusetts, vigorous research activities have taken place in the last decade, and the development of this family of antennas has finally evolved into a new era with products vastly different from their ancestor. It is believed that, today, the radiation mechanism and physical behavior of these antennas are well understood and their analytic theory is well established (see the References). Microstrip antenna (also called patch antenna) is a form of small antenna, printed over a grounded dielectric substrate with a thickness commonly very small in wavelength, as shown in Fig. 12.1. It can also be considered as a form of slot antennas. Despite the severe limitation in bandwidth, these antennas have many unique attractive features: compact in structure, light in weight, low in profile, conformable with host vehicle geometry, easy for reproduction, and integrable with feeding networks and micro-wave devices, particularly compatible with the modem MMIC technology. Lately, much effort has been made to search for new designs, such as the so-called multifunctional microstrip antennas. For example, a single patch can be designed to function for any polarization state without power-dividing networks and phase shifters, even possibly at two different frequencies (see Section 12.4.4). However, these antennas are inherently inefficient and narrow-band radiators, which could have significant dielectric loss, copper loss, and surface-wave loss. Thus for these antennas, a low Q-factor, or SWR at the input port, does not necessarily imply a good antenna. It is therefore important to emphasize that their bandwidth can be meaningfully defined only by examining over the band the stability of all antenna performances, including gain, pattern, and polarization, in addition to the commonly used criterion of the input impedance. Attempts have also been made to broaden the bandwidth, and some partial success was reported, but mainly at the expense of compactness (which, in fact, has made these antennas unique) with an overall antenna thickness as much as almost two-tenths of a wavelength. In such a case, a printed circuit horizontal dipole over the ground plane will probably perform just as well, if not better. This development appears to have returned to the age-old well-known technology. One of the most attractive applications is the large MMIC phased microstrip antenna arrays, particularly for space-borne systems where compactness and deployability are of

Figure 12.1 Rectangular microstrip antenna. (After Richards et al., [6], © 1981 IEEE.)

12. l

GENERAL INTRODUCTION

777

major concern. In this application, the capability of these antennas to perform multiple functions will make them even more attractive from the cost-saving point of view. For such large arrays, probe feed i clearly undesirable. More recently, much attention has been turned to the electromagnetic coupling, including the use of a microstrip line either andwiched between the patch and the ground plane, or below the ground plane and coupled through an aperture to the patch (see later for detail). Both of the e techniques require multiple dielectric layers. The patch can also be excited with slotlines or coplanar lines etched on the ground plane. In this case only a single dielectric layer is sufficient. However for most of the e designs, there is a certain amount of radiation leakage under the ground plane.

12.1.1

Overview of the Theories and Applications of Microstrip Antennas

The analytic theories for microstrip antennas can generally be divided into three groups according to the models assumed. These models have different levels of complexity and require vastly different computation efforts. Although there is no simple clearcut rule as to which one of the three is best to use, the first guideline would be the antenna thicknes . Others are the particular antenna performance to be evaluated, the antenna geometry. the feeding system, and so on. For example, the radiation pattern can generally be predicted very well simply by using the equivalent magnetic current in the cavity model theory. On the other hand, the antenna input impedance cannot, in general, be predicted accurately even with the so-called rigorous full-wave theory. The difficulty can be traced to the notorious problem in defining an impedance in a nonconservative field. The difficulty manifests itself more seriously at higher frequencies. (We return to this issue later.) The first theory, called the transmission-line model (TLM) theory, is based on the observation that a rectangular patch antenna is simply a wide open-circuit microstrip line, in which the radiation is leaked through the open end edges. However, even for a commonly used microstrip line there is no rigorous analytic solution available, and one must therefore resort to some approximate and empirical solutions. This approach is not only limited to the rectangular geometry, but is also inadequate to predict many antenna characteristics [6] . Nevertheless, it is the simplest theory to apply and requires little computational effort. The second is the cavity model (CM) theory, which is based on the assumption that for thin microstrip antennas the field under the patch should differ little from that of a cavity with appropriate boundary conditions. This theory requires more computation effort than the TLM theory but much less than the third approach to be discussed later. This theory can predict all antenna properties quite accurately, provided that the antenna thickness is no more than a few hundredths of a substrate dielectric wavelength. One advantage of this theory is that it can provide much physical insight into the antenna which have led to many new designs and applications. The third theory is based on a rigorous full-wave (FW) formulation, which is applicable, in principle, to microstrip antennas of any thickness, even with superstrate, and of any geometry. But it requires a considerable amount of computation. Even with this so-called most rigorous theory, accurate results are not always assured. The basic difficulty lies in the dubious definition of antenna impedance as stated previously and in the modeling of the practical complex feed structure accurately. In many published works, the computations were made only for some idealized excitations. However,

778

ANTENNAS TV: MICROSTRIP ANTENNAS

for many electromagnetic feeding systems and very large arrays, this theory seems to be the only feasible method for possible solutions. (To illustrate this, the problem of a disk patch fed electromagnetically with a microstrip line is considered in detail in Section 12.2.3.) Efforts have also been made to improve and to extend the aforementioned theories. For example, in the CM theory there are two major assumptions: a magnetic wall around the patch and an idealized current ribbon in place of the actual probe feed. Suggestion has been made to replace the magnetic wall with an impedance wall [7] . Unfortunately, in thjs regard there are two major questions needed to be answered. First, what is the correct wall impedance? Since the ratio of tangential electric and magnetic field components at the wall is a very complicated function, one is confronted with the difficulty to define the wall impedance. Second, even with the knowledge of the impedance, the modes cannot be made to satisfy the boundary condition mode-wise due to the nonseparability of the problem. Some constant capacitive wall reactance has been proposed, but this seems tantamount to a small correction for the patch dimensions as used by others [see Eq. (12.3)] to account for the fringing field effect. Perhaps it should be noted that the so-called "wall impedance" can, in principle, be rigorously formulated by resorting to Huygens' principle in the following manner. Let the cavity be Huygens' surface, which divides the space into two regions: the cavity itself and the semi-infinite region above the ground plane and external to the cavity. The effect of one region on the other can be accounted for by placing appropriate Huygens' electric and magnetic current sources on Huygens' surface. Their ratio is the required true wall impedance. Unfortunately, these sources are unknown. But from Huygen's principle this pair of sources can be replaced by only one of them in the presence of a magnetic or an electrical wall. The field in the cavity can then be regarded as the sum of that due to the feed and that due to one of the surface sources at the wall, while the field in the exterior region is that due to the latter only. Imposing the continuity condition at the wall should result, in principle, in a "rigorously" correct solution as far as the wall impedance concept is concerned. But it is doubtful that this approach is simpler than the FW analysis. One may, in fact, expect that for thin microstrip antennas, the contribution in the cavity due to Huygens' source at the magnetic wall is not significant. This point of view can be used to justify the validity of CM theory. The modeling of feed structure and the definition of antenna impedance have been the most puzzling problems in antenna analysis, due to the nonconservative nature of the electric field. Since the so-called impedance depends strongly on the local field at the feed, it is not surprising to find disagreement between the measured and the theoretically computed impedance based on an idealized excitation. To alleviate this difficulty, a circular patch fed off-center with a thin cable was considered [8, 9). This structure is of particular academic interest since in the small cable the only propagating mode is the TEM wave. Only for this wave the input impedance can be meaningfully defined as well a measured. To make the computation more manageable, in Ref. 9 the magnetic wall is again assumed as in the CM theory. This simplification is made on the belief that the satisfaction of boundary condition at the feed is probably much more important than that of the wall, since the impedance depends strongly on the local field of the feed. (In fact, this speculation was verified later, by using the multiple scattering approach and thence made it possible to e~tend the analysis to a rectangular patch with a cable feed [8, 14].) For this model, the modes that satisfy all the boundary conditions in the cavity can be found. These modes are then matched to the well-known modes

12. l

GENERAL lNTRODUCTlON

779

in the cable at the feed-cavity junction. It is found that with thi modification, the CM theory become applicable to patche as thick as one-tenth of a sub trate dielectric wavelength. This tudy how the imple fact that the impedance must be defined and evaluated in the feed line, where there i clearly a dominant TEM wave. This concept has been extended successfully to electromagnetically excited patches. A brief review of these theories with basic design formulas, when available, is given in Section 12.2. In particular, Section 12.2. 1 deals with the TLM theory; Section 12.2.2, the CM theory including the e tended CM theory; and Section 12.2.3, the FW analysis which is based on the Green' function (GF) formulation and uses the moment method for numerical olutions. Although ome of the theorie are generall y applicable to patche of any geometry, for concrete numerical results only a few commonly used hape are considered in detail. The analysis of a micro trip antenna array of N elements can be pursued, as for any array, in four different way . First, the most rigorous approach is to apply the FW formulation to all elements in much the same manner as for a single element, except that the number of unknowns is increased by a factor of N and the size of the Z-matrix, which re ult from the interactions between basis functions in the moment methmi is increased by a factor of N 2 . Even for a moderately large array the required computational effort could become prohibitively expensive. The second approach is based on the so-called two-element mode4 in which only the interaction between two elements of various spacings is considered, usually under the assumption that all other elements are ab ent. The third approach is a single-element model in which interactions among all the elements are ignored. Clearly, this is the simplest approach and, in so doing, the design of an array is reduced essentially to that of a circuit. This method could provide a useful design, particularly when the element spacings are large and the microstrip antenna elements are thin. However, the anomalies that originate from mutual couplings could not be predicted with this approximate theory. The fourth approach is to approximate a large array as a truncated portion of an infinite array. Invoking the Floquet theorem, the analysis can be reduced to that of a single element. This approach will automatically take all element interactions into account and can therefore predict the coupling effect. However, to apply the Floquet theorem, the array must have a periodic structure and uniform excitations. (In principle, the techniques could be extended to nonuniform excitation, but the amount of computation would be prohibitively large.) In Section 12.3, the FW theory is applied to infinite microstrip antenna arrays. The major difficulty in all FW analyses is the computation of very slowly convergent, sometimes even nonconvergent, series or integrals. The method for resolving this difficulty is discussed in some detail. As an example, numerical results for printed dipole arrays are presented together with experimentally measured results for comparison. The reason for choosing printed dipoles is simply that experiments can easily be conducted in a waveguide simulator. Section 12.4 is devoted to designs and applications. Most design concepts and formulas presented in this section are derived from the CM theory. Although the theory is not rigorous and the formulas not exact, they can give results often accurate wjthin a very few percent, particularly for thin patches. In contrast, as far as design is conceme~ the use of the FW theory is somewhat limited. For example, if it is desired to find the geometrical configuration and dimensions of a patch antenna for a given frequency, bandwidth, polarization, and some radiation characteristics, the FW approach

780

ANTENNAS IV: MICROSTRIP ANTENNAS

would not be able to provide a solution readily, not to mention the time-consuming costly computations. In this respect, the CM theory can play a very important role in the initial design stage. Section 12.4. 1 discusses the design process for a few commonly used LP patch antennas, while Section 12.4.2 deals with the design of dual-frequency elements. In Section 12.4.3, the theory, design formulas, and design procedures of CP microstrip antennas are presented in detail with several examples, including those for dual polarization, dual frequency, and multiplexing operations. Finally, several four-element array modules with novelfeeding systems are discussed in Section 12.4.4.

12.2 VARIOUS THEORIES FOR MICROSTRIP ANTENNA ELEMENTS

In this section three theories at different levels of rigor and complexity will be presented in some detail along with design formulas whenever available. The TLM theory is the simplest and can provide some first-order design information with practically no computational effort. However, analytical simplicity is obtained at the expense of accuracy and restriction of the range of applicability. Furthermore, it is restricted to rectangular geometry. The CM theory overcomes many shortcomings of TLM and is capable of predicting the antenna performance accurately if the patch is not more than a few hundredths of a wavelength thick. One of the primary advantages of this theory is the physical insight it allows into the field distribution under the patch. This understanding is useful in many new designs. However, the CM analysis requires an empirical edge extension factor to account for the fringing field effect and also an idealized excitation at the feed. The FW theory is the most rigorous. It utilizes the dyadic Green's function pertaining to the grounded dielectric slab for the derivation of a set of integral equations for the unknown current on the patch. The technique can easily be extended to multiple dielectric layers. The moment method is then used to solve the integrations numerically, commonly in the Fourier transform domain. The accuracy of this approach is, in principle, limited by the availability of computer resources. However, even for the simplest antenna configuration, the numerical treatment becomes prohibitively inefficient and expensive. To take advantage of the relatively simple CM theory, an improvement is made to extend its validity to thicker antennas by imposing the feed boundary condition. This extended CM theory is also discussed in this section. However, this technique is restricted to the probe feed. An electromagnetically excited patch is considered in the last part of this section. For this case, unfortunately, only the FW analysis is applicable. The major difficulty in this approach lies in numerical computation, which is in essence the same as that in the array analysis. Details of the basic technique for improving the computational efficiency are discussed in Section 12.3.

12.2.1 Transmission-Line Model The rectangular patch microstrip antenna can be viewed as a resonant section of a relatively wide transmission line, radiating primarily from the two open-circuited ends. Figure 12.2 shows the physical rectangular patch antenna, along with its equivalentcircuit model. It is assumed that the fields vary sinusoidally along the side of the patch

12.2

VARIOUS THEORIES FOR MICROSTRIP ANTENNA ELEMENTS

781

l J , 1/;·Jvi~ ~/

-,

Radiating edges

~

Feed / point

Patch

b_J

l-c

L

Slot 1

Patch

Slot 2

~ Substrate

~a+pHJ~S_lde r- ...-------.- - view t

I

r

~

< f

- G + jB Yc 1

Ground plane

G,,+ jB-,

1

Transmissionhne model

z;n

L LJ n G-,+iS,

After transformation

G1+ j8 1

Figure 12.2 Trans mission-line model of the rectangular patch antenna. (After Carver and Mink, [7], © 1981 IEEE.)

possessing the resonant dimension b. Along the width a, of the patch, the fields are assumed to have no variation. It can be shown that the most significant contribution to broadside radiation comes from the two nonresonant ends, appropriately termed the radiating edges. The edges of resonant length are correspondingly known as the non-radiating edges or ends. The radiating edges are treated as slots with dimensions a, t, having constant electric field aperture distributions. The slots are connected by a microstrip line. The characteristic impedance of the line, denoted by Zc, can be expressed as follows:

Zo t

Zc = -

-,

t/a

F,a

Er -

1(

(12.1)

2

12!)-l/ &.e • l . . . . = euective re ative peroutt1v1ty where Ee = - - + - - 1 + 2 2 a Zo = 1201r = free-space impedance Er = relative permittivity of the substrate t = substrate thickness Er+

1

AO where Ao is the free-space wavelength.

(12.2)

782

ANTENNAS IV: MICROSTRIP ANTENNAS

The reactive part of the aperture admittance is capacitive and can be calculated using the edge extension formula for the microstrip transmission-line open-circuit truncation. The susceptance is given by (12.3) where fl.l

ko

0.3 (a/t) + 0.264 . = 0A 12t-Ee+ - - - - - - - - = edge extension Ee - 0.258 (a/ t ) + 0.8

(12.3a)

= 2rr/Ao

For the case in which the feed is located along the radiating edge, the antenna input impedance is computed simply by transforming one of the slot admittances through the length of line b and adding it to the other slot admittance. The resulting equation for Yin is given by Ya+ jYc tan {Jb (12.4) Yin = Ya + Yc-Y.c _+_J__Y._a_tan--{3-b where Ya f3

= Ga + j Ba = aperture admittance = (2n /Ao),./Ee = transmission-line propagation constant

At resonance, the imaginary part of Yin goes to zero. This condition yields the following formula for calculating the resonant frequency, given the antenna dimensions and the substrate permittivity:

2BaYc

2Ba

tan/3 b -- B 2 + G2 - y 2 ~ - Ye a a c

(12.5a)

where the fact that in most cases Ba , Ga c

Measured locus

x Computed locus

Increment: 0. 1 GHz (increasing frequency is clockwise)

Figure 12.13 Measuredandcomputedimpedancelocifor c = 13.0 mm, t = 4.1 mm, E1 = 2.62, (1) e = 7.5 mm and (2) e = 10.5 mm. (After M . Davidovitz and Y. T. Lo, IEEE Trans. Antennas Propag., AP-34(7), July 1986; © IEEE, reproduced by permission.)

796

ANTENNAS IV: MlCROSTRIP ANTENNAS

where k1

= transverse vector wave number = kxx +kyy = kr(cosax+sinay) = kr k.,;k, = lk,I

and apply (12.25a) to transfonn the Maxwell equations. Now, let the transforms E and H of the electric and the magnetic fields respectively, be decomposed in the following manner:

E = E,z+ E1 fl = Hzz + fl, E, = V'k.1 + V"(k., xz) fl, = I'(zxk,) + I"kr

(12.26a) (12.26b) (12.26c)

This decomposition is completely analogous to the E (or TMz) and H (or TEz) modal representations of the field in waveguides. Each plane wave in the Fourier spectrum of the field can be considered as a mode with the corresponding wave number k1 • V ' and I ' play the role of the E (TMz) model coefficients and V" and J" of the H {TEz) coefficients. Substitution of (12.26) into the transformed Maxwell equations can be shown (22) to reduce the latter to two sets of independent scalar equations having the following form : d I . I -V(z, z) + JkzZl(z , z) dz d d z l (z, z') + jk, YV(z, z' )

=0

(12.27a)

=i

(12.27b)

Equations (12.27) are valid for both E- and H-modal coefficients. and the quantities appearing therein are defined as follows:

z = observation point

= source pomt i = {8 (z - z') z, z' are in the same layer I

•

z

z, z' are not in the same layer

0

= Dirac delta function Z = I / Y = { Z ' = k,jwE Z" = wµ, / k,

8 ( ·)

k,

= j k2 -

2

k;; k

(12.27c) E -modes H-modes

= w2 Eµ, (the branch of the square root is chosen such

as to satisfy the radiation condition) The original vector problem is thus reduced to a scalar one. governed by standard transmission-line equations (12.27). In this context, the individual dielectric layers are viewed as sections of transmission lines having characteristic impedance z and propagation constant k, . Continuity conditions at interfaces and conditions at boundary and source can be applied to the scalar quantities V and I.

12.2

VARIOUS THEORIES FOR MlCROSTRlP ANTENNA ELEMENTS

797

With the definitions above. the FT of the electric-field dyadk Green's function for a transver e electric current heet take on the following form : ::::::

G(k, ; Z, z')

=-

V '(z,

z')lc,k, -

V" (z , z' )(k, xz)(k, xz) + (ktfws) l '(z,

z')zk,

(12.28) Derivation of the Green· function for a z-directed magnetic current i completely analogou to that hown above [22]. In phy ical pace (x, y, : ). the Green' function can be used to cast the microstrip antenna problem in the form of a vector integral equation. Figure 12.14 depicts a micro trip tructure con isting of a patch SP and a microstrip feed S1 , which can be located on the ame plane as the patch or at a different level. Enforcement of the boundary condition, requiring the total electric field to vanish on the microstrip, yields the de ired integral equation,

zx [E'(r ) + E(r)] = 0,

r E [Sp+ S1 ]

(12.29)

where E1 (r) is the electric field at r due to the impressed source, and

Eer)

=

ff

Ger, r') · Jper')dr' +

s,

ff

Ger, r') · J1er') dr'

e12.30a)

s,

where JPand JI are the currents on the patch and the feed, respectively. The microstrip is assumed to be made of infinitesimally thin conductors; hence JPand JI are actually sums of currents flowing on its top and bottom surfaces. Equation (12.30b) is obtained from (12.30a) by application of the convolution theorem.

Ground plane

Figure 12.14 Patch antenna electromagnetically excited by a microstrip transmission line.

798

ANTENNAS IV: MICROSTRIP ANTENNAS

Analytical solution of Eqs. (12.29) and (12.30) is precluded even for simple geometries. The method of moments [23] can be applied to obtain a numerical solution. The accuracy and efficiency of the solution largely depend on the choice of the expansion and testing functions. In the method of moments this choice can be guided by previous investigations of similar geometries and dictated by the inherent geometrical features, such as symmetries and edges and also numerical convenience. The method of moments reduces the integral equation to a set of linear algebraic equations of the form [Z][/] = [V] (12.31) where [/] [ V]

[Z]

= vector of undetermined expansion function coefficients = excitation vector = MM matrix

The elements of [Z], denoted by Zmn, are given by

ff

dr

(12.32a)

G(k,; z, z') .J. (k,) dk,

(12.32b)

G(r, r') • J.(r')dt'

(Jn)

4!2ff t,(k,). 00

=

-00

where Jn (r') Jm (r')

D (•) *

= expansion basis function = testing function = domain of the function

= complex conjugation

The elements of [V] can be written as

Vm

=-

If

E1.m(r) •Jm (r) dr

(12.33a)

D (Jm)

ff

00

= 41/2 E~ (k,) .j~ (k,) dk,

(12.33b)

-00

where E~ (r) is the component of the impressed field transverse to the z-axis. Equations (12.32b) and (12.33b) are obtained from their respective predecessors (12.32a) and (12.33a) by application of convolution and Parseval theorems. The former are more advantageous from the computational standpoint, as they require fewer numerical integrations to be performed. Often, the expansion and testing functions are drawn from the same set. The MM solution in that case is known as the Galerkin technique. As an example illustrating the solution of Eqs. ( 12.27), consider the case of a twolayered slab backed by a ground plane. Figure 12.15 shows the physical configuration alongside its equivalent transmission-line model. The dielectric layers are characterized

12.2

VARIOUS THEORIES FOR MICROSTRIP ANTENNA ELEMENTS

799

00

t Yo kzo Z = Z2

1A

Z =Z

Z=

z1

Z =O - - - - - < i

Ground plane

Short circuit

Figure 12.15 Two-layered grounded dielectric slab and its equivalent transmission-line circuit representation.

by their permittivities E 1 and E2 and respective thicknesses t 1 and t2 . The source point z' is located in the second layer (i.e., z 1 ~ z' ~ z2 ). The modal Green's function for this example is given by

V(z , z' )

=

V2(z2, z' )e-j~(1.-,i)

(12.34a)

V2(z, z') , sin k;.1z V2(Z1, Z) . k sm z1Z1

(12.34b)

,

1 (z,

1

d

(12.34c)

,

z ) = - j kz Z dz V (z, z )

where V2(z, z')

=~

1

(12.34d)

[cosk,2(z< - z1)

~,

+ }Y (z1) sinkz2(z< -

z1)]

Y (z1)

x [cosk,2(z> -z1)- }Y'(z1)sink,2(z> -z1)]

(12.35) ..,

Y1

Y (z 1)

= -}-. cotk,1t1

#

= Y (z1) + Y (z 1)

Y (z1)

Y2

kzq = and

..

(Eq~ -

.. Y (z1)

-

kf) 112

..,

= Y2Y (z1)

q = l or 2

z> = max(z, z') z< · min(z, z')

800

ANTENNAS IV: MICROSTRIP ANTENNAS

Note that Eqs. (12.34) and (12.35) are equally valid for both E- and H-model coefficients. V' (z, z' ) and V" (z, z' ) are obtained, therefore, by substitution of the respective modal admittances Yq, as defined in Eqs. (12.27c), for the region with Erq, q = 0, I , 2. The basis functions employed in the Green' s function - moment method solutions of the microstrip antenna problem, as indeed in all MM solutions, are of two varieties: entire domain or subsectional. As implied by the nomenclature, in the first case the unknowns are expanded in a complete set of basis functions, having the entire structure (e.g., entire patch or feed line) as the domain. In the second case, the currents are approximated piecewise by simple functions whose domains are considerably smaller than the structure itself and the wavelength. Examples of the entire-domain basis functions are

Rectangular patch (Fig. 12.16a) . (1) l x

Jy (ii)

=

sinr1r (x / a ) sin(s - ½)1r (y / b) - ;::::===- -----;:~====--- , ✓a2-x2 J b2 - y2

r, s

=

cos(r - l)1r(x / a ) cos(s - ½)1r (y / b ) --;::=====-- - ---;::::====-- , ✓a 2 _ x 2 J b2 _ y2

r, s

= l , 2, . . .

=-

J.y" = J Jy

l , 2, . ..

a 2 - x2 Um (x) --;::::== Tn(Y / b) ' a J b2 _ y2

m,n

= 0, 1, 2, ...

J b2 - y 2un (y / b) ,

m, n

= 0, l , 2, . ..

= ✓Tm(x / a ) a2

-x 2

( 12.36)

where Tm and Un are the Chebyshev polynomials of the first and second kind, respectively.

Circular patch (Fig. 12.16b) I p = J 1 - (p / c) 2 Uu(p / c) cos m°

=

~

1 2 3 4 5

~

1 2 3 4 5

~

1 2 3 4 5 0

1.5

2.0

2.5

3.0

0.80509 6.37652 12.61287 18.88053 25.15599

0.67734 3.28247 6.35324 9.47133 12.60124

0.58471 2.26364 4.27330 6.33924 8.41951

0.51362 1.75777 3.23611 4.77494 6.32991

4.0

5.0

6.0

7.0

0.41112 1.25112 2.20208 3.21274 4.24176

0.34102 0.99217 1.68661 3.19865 3.19865

0.29042 0.83062 1.37743 1.96588 2.57334

0.25240 0.71780 1.17060 1.65479 2.15782

8.0

10.0

12.0

15.0

0.22291 0.63335 1.02177 1.43264 1.35955

0.18035 0.51371 0.81992 1.13588 1.32858

0.15123 0.43223 0.68749 1.10714

0.12101 0.34893 0.55478 0.96701

m = 1, A = a/ b.

a circled cross, is chosen on the radial line = 0° for some desired impedance at J12 = 1620 MHz as shown in Fig. 12.51. Second, a shorting pin is placed some angle o away on the nodal circle of the (1, 2) mode so that it will have some effect on the ( 1, 1) mode impedance as shown in Fig. 12.52 for three different values of () C Cl) ::::,

4~

CT

()

-~ ~

5 Cl)

()

,.

1250

.,j

0

CD

-

0

a:

1200

- 1.0

-0.8

--0.6

--0.4

-0.2

0

0.2

0.4

0.6

0.8

1.0

Normalized shorting pin location

Figure 12.65 location.

Resonant frequency of the antenna shown in Fig. 12.64 versus shorting pin

From the two results above, one finds

1)

k0 1 - k 1o= p + q= lk 11 I ( A +A

k'

= k io + p = k 10 ( 1 -

and finally (recall that k10 =

= k' - (A + I/ A ) 2Q

1 2Q A

--

)-l

(12.127)

(12.128)

1r/a, ko1 = 1r/ b), (12.129)

12.4

APPLICATIONS AND DESIGNS

873

. - - - Patch "C" Shaped slot Substrate y

Ground plane

Figure 12.66

Rectangular antenna tuned with a wrapped around microstrip line.

RHCP

□

LHCP

Figure 12.67

Square CP microstrip antenna fed with a printed circuit hybrid.

jk "

k-plane p

k'

q

ko1

-+-~----.:"---,-------:;:,r---k'

0

k"/ k' = -1/20

k=k' +jk"

Figure 12.68 Geometrical relations of phasors k10, koi, and k ink-plane for a nearly square CP microstrip antenna.

874

ANTENNAS IV: MICROSTRIP ANTENNAS

Equations (12.123), (12.128), and (12.129) constitute the basic design formulas. For a given material and patch dimension a x b x t , Q can be either computed or measured. Then the feed must be located on the locus:

y' =

!cos-

1

[

A

cos{: x')] or x' = ; cos- [!cos(: l )] 1

(12.130)

where A is one of the two solutions of (12.129), given by (12.131) For a nearly square patch, c

..o) I

Ix

3

-~ 0 ~

"O ~ "O

2

I

C

ca

I

I

I

I

I

I

I

I

I

X

.0

a..

0

1 I

I

I

I

I

I

I

x(/ #

0

0.02

0.04

0.06

0.08

0.1

t Figure 12.71

CP bandwidth versus thickness of a CP microstrip antenna.

878

ANTENNAS IV: MICROSTRIP ANTENNAS

1.05

-0

1.04 0 = 40

---

~

.._

"O

~

1.02

cij

E .... 0

z 1.01

Aspect ratio (a l b)

Figure 12.72 CP frequencies (normalized with respect to the (1, 0) mode resonant frequency) versus the aspect ratio (a / b) of a nearly square microstrip antenna for its Q 40, 60, 80, and 100.

=

Other CP Microstrip Antennas. From the theory developed in the preceding subsection, it is not difficult to create many new designs. First the patch must be able to carry two degenerate modes with linearly polarized radiation fields in orthogonal directions. Let the eigenvalues of the two modes be k_ and kt . Then, to achieve CP, a proper perturbation must be introduced to the patch such that either k_ or k changes by an amount satisfying the following equation: k_ -kt _ _!_ k' Q

where

k'

(12.140)

2rrfo = (J),./j:u = ko,.Jµ;E; = --,.Jµ;E; C

Jo = the operating frequency µ.,,,

E,

Q

= relative permeability and permittivity of the sub trate material = quality factor of the path

Some examples are shown in Fig. 12.73. In particular, an exact square with two opposite comers cut off as shown in Fig. 12.73a is of great intere t becau e of the ease in fine tuning of the dimensions for the CP operation. In this case, the two orthogonal modal fields are the sum VI+ = (V101 + V110) and difference VI- = (V101 - V1 1o) of the (0, 1) and (1, 0) modes pertaining to the quare patch. The e modes will produce an LP field in the two diagonal directions. By simply sketching the E:. distribution of modes VI+ and VI- , one find that IE: 1reaches the maximum at two opposite comers and zero at the other two comers for, say, the VI+ mode. The exact opposite situation is obtained for the difference mode VI - . From the perturbation theory one sees that the physical truncations of two opposite comers will have insignificant effect on the wave number of the mode where IE, I = 0 there, but a strong effect on the mode whose I£;: I is maximum there. Thus if the two opposite comers are trimmed off by an amount

12.4 (a)

I
., :-{: "j. J ,- ... I =·~ ~..£ for a square patch over a substrate of Rexolite 2200 with Er = 2.62. Measurement was made in the S-band by Seung Choi.

Industrial synergy

Figure 12.87

Industrial synergy.

12.4

APPLICATIONS AND DESIGNS

893

• Commercial vehicle sensor • Back-up proximity sensors • Blind-side proximity sensors • True ground-speed indicator (K-band) • Radar • Police radar (X-, K-, Ka-band) • Commercial vehicle radar (K-, V-, W-band) • Collision warning and/or avoidance • Automatic cruise control • Automatic master/ lave convoys The next few paragraphs will give a brief overview of some applications of microstrip antennas in each of these areas.

Communications. Microstrip antennas have been used for wireless communications in cellular ystems as well as point-to-point data, video, and voice links. Although not as efficient as slotted waveguide arrays or horns, microstrip patch arrays have been produced for UHF cellular communication base stations and point-to-point links at L-, S-, C-. and X-band. The lower profile, conformal nature, and aesthetics along with lower costs in volume have gradually led to an increase in its market share. Figure 12.88a shows several Ka-band patch arrays with increasing gain fed by the same connector (i.e., no change in cross section or depth). ~Although more efficient, a horn antenna with an equivalent field of view would require significantly more depth as the field of view is reduced. Figure 12.88b shows two Ka-band transceivers with nearly identical half-power beam widths used in police radar applications. As shown, the microstrip array is several inches shorter in depth for a lower profile front-end. Typical systems tend to use the area behind the microstrip patch array to accommodate electronics. Simple electronic integration is another advantage of microstrip that will continue its use in the future [93]. RFID applications [94] at S- and C-bands are instrumental in wireless tagging and monitoring applications. The cost requirements of RFID and tagging systems tend to drive these systems toward complete integration, making the use of microstrip ideal for this application. ISM bands are typically narrow enough to accommodate the bandwidth limits of thin microstrip antennas and arrays. Bluetooth applications for computers and peripherals [95] is another related area that will continue to proliferate. Already, wireless local area networks and mouseto-computer interfaces can be accommodated using integrated patch technology. As wiring is often heavy, tedious, and costly for large platforms, sensors (i.e., temperature, pressure, light, chemical, etc.) will continue to be distributed and linked through wireless local area networks. Single patch antennas and arrays of patches are currently used for several dual-use Global Positioning System applications. Clever uses of microstrip patch modes and multilayer fabrication allow the use of multiple frequencies for enhanced system performance. Several OPS products are now being used for car-rental vehicle location, anti-theft systems, and several pedestrian uses. The use of adaptive antennas and phased

894

ANTENNAS JV : MICROSTRIP ANTENNAS (a)

Sample microstrip patch antenna arrays. (b)

Ka-band transceiver comparisons.

Figure 12.88

Microstrip array samples and waveguide horn comparison.

arrays in communication systems requires higher packing den ities for improved scanning performance and wider instantaneous bandwidth for increased data rates. The lower cost potential of integrated microstrip array continues to make thi an attractive option [96]. Commercial Microwave Sensors [97]. The area of inexpen ive ensors ha benefited greatly from the advancement of material and component technology. Originally applied to automatic door openers, Doppler ensor have naturally moved into home/office security, surveillance, and automotive applications. The volumes required in the automatic door opener application helped to drive co t lower and expand its use in other markets. To reduce component count and cost , the e sen or are typically homodyne transceiver . A single o cillator erves as the transmitter and pumps a mixer device that serves as the receiver. The return ignal is directly converted to ba eband and processed to detect motion. The first production tran ceiver used packaged Gunn and Schottky mixer diodes in waveguide. The packaged diode within the e tran ceivers are gradually being replaced by surface-mount devices on planar media. This allows for lower profile ensors, lower component co ts, and reduced a sembly labor. The gradual evolution of the packaging of the Doppler sensor is shown in Fig. 12.89. The original package include a waveguide tran ceiver, packaged diodes. filter, and shaped flange to control radiation pattern. An FR4 substrate was used to provide DC power to the Gunn diode, while the received signal was amplified and processed

12.4

APPLICATIONS AND DESIGNS

895

Heterogeneous integration

\

' ' 'L;;....;;.;;,;;;;.;;.L~-

----FR4 circuitry

(DC power, IF circuitry)

'lator

FR4 circuitry

'

&

,,i..;.;;;=:..a:

Mlcrostrlp an &

(DC power, IF ciroultry)

Waveguide

FR4 circuitry

oscillator & mixer

(DC power, IF circuitry)

Waveguide oscillator & mixer

Figure 12.89 Heterogeneous integration of a bomodyne sensor.

using leaded components and devices. The leaded components were replaced with surface mount devices and the number of devices required was minimized. The metallic flange and filters were replaced by a planar antenna array. Furthermore, the packaged mixer diode can be replaced with a surface mount device on a microstrip circuit. Similarly, the packaged Gunn diode can be replaced with a surface mount diode or transistor on a microstrip transceiver circuit. As a result, all electronic components, RF distribution networks/antennas, DC power, and logic can reside on a planar multilayer printed wiring board. Automatic pick-and-place machines, solder reflow, and other mass-production techniques can be readily applied to accommodate larger volumes. These changes provide a more versatile, cost-effective, conformal sensor that can be applied to even more markets. The automotive market is now ready to use types of transceivers as backup proximity sensors, as well as blind-spot sensors to alert drivers of nearby obstacles. Wrreless interconnection of multiple sensors, as described earlier, reduces overall system costs and makes them more adaptable to various platforms or structures. Figs. 12.90 and 12.91 show several X- and K-band examples of highly integrated sensors. Radar. The automotive market naturally sought the use of Doppler sensors for safer driving. Originally intended for collision avoidance, then collision warning, and ultimately automatic cruise control, the use of these sensors for a true radar has provided a significant challenge to industry. Many mechanical reflector variations, switched beams, beamformers [98] and even phased arrays were built for proof-of-concept systems [99]. Unlike military radars, this commercial radar must operate in cluttered environments

896

ANTENNAS IV: MICROSTRIP ANTENNAS X-band

K-band Rx

Tx

Figure 12.90

Integrated homodyne ensors for commercial applications.

Back side (pre-assembly)

Waveguide transceiver and cover not shown

Ba completed

Figure 12.91

Integrated homodyne true-ground peed ensor for automotive application .

(for instance, overpasses, trees, pedestrians, bicycles, and trucks all meshed together). The radar must also discern ignals that vary over a wide dynamic range. At the same time, the automakers require that the entire system cost be in the few hundred dollar range! There are many examples and radar prototypes that have sought to fill this need as shown in Fig. 12.92. Pattern predictions and measurements are shown in Fig. 12.93.

12.4

~ ...... ~ Circulator

897

APPLICATIONS AND DESIGNS

I

I I I I Back side

Front side

Mechanically steered Single-plane monopulse receiver

Figure 12.92

Sample W-band radars for automotive applications.

II:az (theory vs. measured)l 0

-r·-=-~ ~:::,::::::::;r=::,::=:;r==::::;:::::::;=~i'--.~ffl----Jlllaz (theory vs. measured)

I ·nV

I

·' 1

I

· 1

'

J

r

I

L

.

," 0

30

:

60

90

Angle (degrees)

Figure 12.93 Theory vs. measured for monopulse receiver in Fig. 12.92.

Each of the applications shown above has benefited from a reduction in piece parts count and integration. The use of photolithographic processes and surface mount attachment has been shown in high-scale production at K-band and in prototypes up through W-band.

898

A TE

AS IV: MJCROSTRIP ANTEN AS

The next challenge for industry i in the area of "low-cost" electrically scanned antennas. Many complex pha ed-array military radars have been built and are in production as de cribed by Brookner (100]. Prototypes have shown complete wafer-scale integration [101] or digital beamforming [102]. Other have been used in military communication on-the-move exercises [103, 104]. Electrically scanned apertures are being sought for on-the-move communications and network centric capabilities [105]. To date, a typical phased array has been prohibitively expensive with a large number of piece parts, many a sembly teps, and 100% testing required throughout the build. Although acceptabje for most criticaJ military radar application , commercial application have sought to minimize cost while maintaining performance (106]. On-the-move operations with full hemispherical coverage operating in harsh mechanical and electrical environments have moved toward mechanical augmentation [107]. Many phased array developers require volume and automation (i.e., as in automatic door openers) to lower the overall costs. Others have proposed solutions that combine mechan.icaJ/electrical canning, variable mechanical lenses, variable electrical loading [108] and o on. Each of these has strengths and weaknes e that may make it applicable to some specific platform or application. Difficult electrical canning requirements (i.e., wide instantaneou bandwidth, >±65° beam canning, low axial ratio over scan volume, etc.) often exclude the use of rnicrostrip radiators. For example, Figure 12.94 shows Connexion One, a Boeing 737-400 aircraft equipped with Ku-band tran mit and receive phased array antennas (109]. Each array can maintain atellite communication with Ku-band satellites up to a "-'70° scanning angle. The transmit antenna is capable of a ingle arbitrary linearly polarized beam while the receive array has imultaneou left-hand and right-hand circularly polarized beams. These phased-arrays are part of the Connexion by Boeing system, which is a mobile information services provider that i bringing the

Figure 12.94

Commercial mobile SATCOM.

REFERENCES

899

fastest available high-speed Internet, data and entertainment connectivity to aircraft in flight. The service is available in the executive ervices market in the U.S., which includes operators of private and government aircraft. Service demonstrations with Lufthansa German Airline and British Airways have now been completed. In addition, Japan Airlines and Scandinavian Air System have announced plans to equip long-range aircraft in their fleets with Connexion by Boeing beginning in 2004. The antenna element used by Boeing to meet the stringent scanning requirements are dielectric-loaded circular-waveguide radiators in a tightly packed triangular lattice. The elements are integrated with low-noise amplifiers and/or power amplifiers along with phase-shifters to achieve beam-scanning. Although microstrip patch antennas were not used in the application above, the low-profile, low-cost, and conformal nature of microstrip antenna keeps this medium in the trade matrices. Continued advancements in design, analysis, manufacturing, and testing should bring about a day when an integrated microstrip phased array aperture has sufficient bandwidth and scanning range for mo t commercial and military applications (110]. Similar to the personal computer market, it will then be a readily available commodity which will create the next boom in communications connectivity.

Acknowledgments A major portion of the materials presented in this chapter is derived from the work sponsored by the RADC/EEA, Hanscom AFB, Massachusetts, and NASA Lewis Research Center, Clevelan~ Ohio. The facilities and support provided by the Alexander von Humboldt-Stiftung, DFVLR West Germany, to one of the authors (MD) for completing his contribution to the chapter are also gratefully acknowledged.

REFERENCES 1. G. A Deschamps. "Microstrip Microwave Antennas," 3rd USAF Symp. Antennas, 1953.

2 . H . Gutton and G . Baissinot, '~at Aerial for Ultra High Frequencies," French Patent 703,113, 1955. 3. J. Q. Howell, "Microstrip Antennas," IEEE AP-S Int. Symp. Dig. 1972, Williamsburg, VA, December 1972, pp. 177 -180. 4. R. E. Munson, "Conformal Microstrip Antennas and Microstrip Phased Arrays," IEEE Trans. Antennas Propag., AP-22(1), pp. 74-78, January 1974. 5. RE. Munson, "Single Slot Cavity Antennas Assembly," U.S. Patent 3,713,162, January 23, 1973. 6. W. F. Richards, Y. T. Lo, and D. D. Harrison , "An Improved Theory ofMicrostrip Antennas and Applications/' IEEE Trans. Antennas Propag., AP-29(1), pp. 38-46, January 1981 . 7 . K. R. Carver and J. W. Mink, ''Microstrip Antenna Technology," IEEE Trans. Antennas Propag., AP..29(1), pp. 2-24, January 1981. 8. M. Davidovitz, "Feed Analysis for Microstrip Antennas," Ph.D. Dissertation, Department of Electrical and Computer Engineering, University of Illinois at Urbana-Champaign, Urbana, August 1986. 9. M. Davidovitzand Y. T . Lo, "InputlmpedanceofaProbe-FedCircularMicrostripAntenna with Thick Substrate,U IEEE Trans. Ante nnas Propag., 34(7), pp. 905-911, July 1986. 10. M. D. Deshpande and M. C. Baily, "Input Impedance of Microstrip Antennas," IEEE Trans. Antennas Propag., AP-30(4), pp. 645-650, July 1982.

900

ANTENNAS IV: MICROSTRJP ANTENNAS

11 . I. J. Bahl and P. Bhartia, Microstrip Antennas, Artech House, Dedham, MA, 1980. 12. Y. T. Lo, D . Solomon, and W. F. Richards, ''Theory and Experiment on Microstrip Antennas," IEEE Trans. Antennas Propag., AP-27(2), pp. 137-145, March 1979. 13. Y. T. Lo, W. F. Richards, and P. Simon, "Design and Theory of Circularly Polarized Microstrip Antennas," IEEE AP-S Int. Symp. Dig. June 1979, pp. 117-120. 14. M. Davidovitz and Y. T. Lo, "Cutoff Wavenumber and Modes for Annular Cross-Section Waveguide with Eccentric Inner Conductor of Small Radius," IEEE Trans. Microwave Theory and Techniques , MTT-35, pp. 510-515, May 1987. 15. W. F. Richards. and Y. T. Lo, 'Theoretical and Experimental Investigation of a Microstrip Radiator with Multiple Lumped Linear Loads," Electromagnetics, 3(3-4), pp. 371-385, July-December 1983. 16. W. F. Richards, "Microstrip Antennas," in Y. T. Lo and S. W. Lee, Eds., Antenna Handbook: Theory, Applications, and Design, Van Nostrand Reinhold, New York. 1988, Chap. 10. 17. E. 0 . Hammerstad, "Equations for Microstrip Circuit Design." Proc. 5th Eur. Microwave Conf , September 1975, pp. 268-272. 18. S. A. Long and L. C. Shen, 'The Circular Disc, Printed Circuit Antenna," IEEE AP-S Int. Symp. Dig., June 1977, pp. 100-103. 19. W . F. Richards and Y . T. Lo, "A FORTRAN Program for Rectangular Microstrip Anten nas," Interim Rep. RADC-TR-82-78, April 1982. 20. N. Marcuvitz, Ed., Waveguide Handbook, MIT Radiat. Lab. Ser. Vol 10, McGraw-Hill. New York, 1951. 21. L . C. Shen, S. A. Long, M. R. Allerding, and M. D . Walto~ ''Resonant Frequency of a Circ ular-Disc, Printed-Circuit Antenna.'' IEEE Trans. Antennas Propag.. AP-25(4). pp. 595-596, July 1977. 22. L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves. Prentice Hall, Englewood Cliffs, NJ, 1973. 23. R F. Harrington, Fiel.d Computation by Moment Methods. Macmillan. New York, 1968 (Reprinted: R. E. Krieger, Melbourne, FL, 1982). 24. P. L. Sullivan and D. H. Schaubert, "Analy is of an Aperture Coupled Microstrip Antenna," IEEE Trans. Antennas Propag., AP-34, pp. 977-984, August 1986. 25. P. J. Davies and P. Rabinowitz, Methods of Numerical Integration , Academic Press, New York, 1975. 26. P. M. Morse and H. Feshbach, Methods of Theoretical Physics . M cGraw-Hill, New York,

1953. 27. W . F. Richards, "Anisotropy, Birefringence. and Dispersion in Artificial Dielectrics," Ph.D. Dissertation, Department of Electrical and Computer Engineering, University of Illinoi at Urbana-Champaign, Urbana, 1977. 28. S. M. Wright, ''Efficient Analy is of Infinite Microstrip Arrays on Electrically Thick Substrates,'' Ph.D. Dissertation, Department of Electrical Engineering. University of Illinois, Urbana- Champaign, Urbana, 1984. 29. D. M . Pozar, "Improved Computational Efficiency for the Moment Method Solution of printed Dipoles and Patches," Electromagnetics. 3(3-4), pp. 299-309, JulyDecember 1983. 30. D. R. Jackson and N. G. Alexopoulos, "An Asymptotic Extraction Technique for Evaluating Sommerfeld-Type Integrals," IEEE Trans. Antennas Propag., AP-34(12), pp. 1467-1470, December 1986. 31. R. C. Hanson, Ed .. Microwave Scanning Antennas, Vol. 2, Array Theory and Practice , Academic Press, New York, 1966.

REFERENCES

901

32. E. H . Newman and J. E . Tehan . ..Analysi of a Microstrip Array and Feed Network.,, IEEE Trans. Antennas Propag. , AP-33(4), pp. 397-403, April 1985. 33. D . M. Pozar, ··Analysis of Finite Phased Array of Printed Dipoles," IEEE Trans. Antennas Propag., AP-33( 10), pp. 1045- 1053, October 1985. 34. D . M. Pozar, " Finite Pha ed Array of Rectangular Microstrip Patches," IEEE Trans. Antennas Propag., AP-34(5), pp. 658 - 665, May 1986. 35. H. A. Wheeler, "The Radiation Re istance of an Antenna in an Infinite Array or Waveguide," Proc. JR£, pp. 478- 487, April 1948. 36. Amitay, N., Galindo, V., and Wu, C. P., Theory and Analysis of Phased Array Antennas. New York: Wtley-lnterscience. 1972. 37. Diamond. B. L., ·'Phased Array Radar Studies, 1 January 1963 to 1 July 1964." MIT Lincoln Laboratories, Tech. Rep. No. TR381 , by Group 44. 38. E . C. Dufon , "Finite Scattering Matrix for an Infinite Ante nna Array," Radio Sci., 2, pp. 19-27. January 1967. 39. P. C. Clemmow. The Plane Wave Spectn,m Representation of Electromagnetic Fields, Pergamon Press, Oxford, 1966.

40. R. F. Harrington, Time-Harmonic Electromagnetic Fields , McGraw-Hill, New York, 1961 , p. 186.

41. T. Itoh and W. Menzel. ·•A Full-Wave Analysis Method for Open Microstrip Structures," IEEE Trans. Antennas Propag .. AP-29(1), pp. 63-68, January 1981. 42. S. M Wright and Y. T . Lo, ''Efficient Analysis for Infinite Microstrip Dipole Array," Electron. Lett., 19(24). pp. 1043-1045, November 24, 1983. 43. D. M . Pozar and D . H. Schaubert, uscan Blindness in Infinite Phased Arrays of Printed Dipoles," Trans. Antennas Propag., AP-32, pp. 602-610, June 1984. 44. B. J. Rubin and R L. Bertoni. "Reflection from a Periodically Perforated Plane Using a Subsectional Current Approximation," IEEE Trans. Antennas Propag., AP-31, pp. 829-836, November 1986. 44 a. W. L . Stutzman, and G. A. Thiele.Antenna Theory and Design , Wtley, New York; 1981. 45. P. W. Hannan and M . A. Balfour, "Simulation of a Phased-Array Antenna in a Waveguide," IEEE Trans. Antennas Propag., AP-13, pp. 342- 353, May 1965. 46. G. H. Knittel, A H essel, and A. A. Oliner, "Element Pattern Nulls in Phased Arrays and their Relation to Guided Waves.'' Proc. IEEE, S6(11), pp. 1822- 1836, November 1968. 47. RF. Frazi.ta, "Surface-Wave Behavior of a Phased Array Analyzed by a Grating-Lobe Series,'' IEEE Trans. Antennas Propag., AP-1S, pp. 823- 824, November 1967. 48. D. M . Pozar, "General Relations for a Phased of Printed Antennas Derived from Infinite Current Sheets," IEEE Trans. Antennas Propag. , AP-33(5), pp. 498- 504, May 1985. 49. L. Stark, "Radiation Impedance of a Dipole in an infinite Planar Phased Array," Radio Sci., 1(3), pp. 361-377, March 1966. 50. D. M. Pozar and D. R Schaubert, "Analysis of an Infinite Array of Rectangular Microstrip Patches with Idealized Probe Feeds," IEEE Trans. Antennas Propag., AP-32(10), pp. 1101 - 1107, October 1984. 51 . E. C. Jordan andK. G. Ba]maio, Electromagnetic Waves and Radiating Systems, PrenticeHaJL Englewood Cliffs, NJ, 1968. 52. W. C. Chew and J. A. Kong, "Analysis of a Circular Microstrip Disk Antenna with a Thick Dielectric Substrate," IEEE Trans. Antennas Propag., AP-29( 1), pp. 68- 76, January 1981. 53. M. D. Deshpande and P. D. R. Prabhakar, "Analysis of Dielectric Covered Infinite Array of Rectangular Microstrip Antennas," IEEE Trans. Antennas Propag., AP-35(6), pp. 732-736, June 1987.

902

ANTENNAS IV: MICROSTRIP ANTENNAS

54. C. C. Liu, J. Shmoys, A. He el, J. D. Hanfling, and J. M. Usoff, "Plane Wave Reflection from Microstrip-Patch Arrays -Theory and Experiment," IEEE Trans. Antennas Propag., AP-33(4), pp. 426-435, April 1985. 55. V . W. H. Chang, "Infinite Phased Dipole Array," Proc. IEEE , 56( 11 ), pp. 1892-1900, November 1968. 56. C. M. Butler and T . L. Keshavamurthy, "Inve tigation of a radial, Parallel-Plate Waveguide with an Annular Slot," Radio Sci., 16(2), pp. 159- 169, March-April 1981. 57. N. G. Alexopoulos and I. E. Rana, "Mutual Impedance Computation between Printed Dipoles," IEEE Trans. Antennas Propag., AP-29(1), pp. 106-111, January 1981. 58. J. N. Brittingham, E. K. Miller. and J. T. Okada, SOMINT: An Improved Model for Study ing Conducting Objects Near Lossy Half-Spaces, Rep. No. UCRL-52423, Lawrence Livermore Lab., University of California, Livermore, 1978. 59. W. F. Richards, J. R. Zinecker, D. R. Wilton, S. Singh, Y. T . Lo. and S. M. Wright, "Acceleration of Periodic Green 's Functions in Free Space," URSJ Symp. Proc., Houston, May 23-26, 1983, p. 83. 60. S. M. Wright, Y. T . Lo, W . F. Richards, and J. R. Zinecker, "Efficient Evaluation of the Periodic Green's Function for a Grounded Dielectric Substrate," URSI Symp. Proc. , Houston, May 23-26, 1983, p. 82. 61. Y. T. Lo, S. M. Wright and M. Davidovitz, "Antenna IV: Microstrip Antennas," Chapter 13 in Handbook of Microwave and Optical Components." K. Chang, Ed., 1st Ed. Wtley, New York, 1989. 62. R. C. Hansen, "Finite array scan impedance at the grating lobe angle.'· JEE. Tenth International Conference 0 11 Antennas and Propagation (Conf. Puhl. No. 436). Edinburgh~ UK, pp. 196- 199, 1997. 63. H. Holter and H. SteyskaJ, "On the Size Requirement for Finite Phased-Array Models," IEEE Transactions on Antennas and Propagation , 50(6), pp. 836-840, 2002. 64. R. Mittra, W. L. Ko, and Y. Rahmat-Samii, ..Transform Approach to Electromagnetic Scattering," Proceedings of the IEEE, 67(11 ), pp. 1486- 1503. 1979. 65. R. Mittra, W. L. Ko, and Y. Rahmat-Samii. "Solution of Electromagnetic Scattering and Radiation Problems Using a Spectral Domain Approach - Review,'' Wave Motion, 1(2), pp. 95- 106, 1979. 66. J. L. Volaki and K. Barke hli, "Application of the conjugate gradient FFr method to radiation and scattering," in T. K. Sarkar (ed.), PIER 5, Applications of Conjugate Gradient Method to Electromagnetics and Signal Analysis. New York: Elsevier, 1991. 67. C. F. Wang, F. Ling, and J. M. Jin, " A Fast Full-Wave Analy i of Scattering and Radiation from Large Finite Array of Micro trip Antennas," IEEE Transactions on Antennas and Propagation , 46(10), pp. 1467- 1474, Oct. 1998. 68. S. W. Lee, "Radiation from an Infinite Periodic Array of Parallel-Plate Waveguide ," IEEE Transactions on Antennas and Propagation, 1S, pp. 598- 606. 1967. 69. B. A. Munk and G. A. Burrell, "Plane wave expansion for arrays of arbitrarily oriented piecewise linear elements and its application in determining the impedance of a single antenna in a lossy half- pace,'' IEEE Transactions on Antennas and Propagation, 27(3), pp. 331 -343, 1979. 70. A. I himaru, "Finite periodic structure approach to large canning array problems," IEEE Transactions on Antennas and Propagation, 33(0096- 1973), pp. 1213- 1220, 1985. 71. A. Ishimaru, G. E. Miller, and W. P. Geren. "Application of the finite periodic structure approach to free excitation of large pha ed array systems." Proc.. IEEE Antennas and Propagation Society International Symposium, Vancouver, BC, Canada, Vol. 1, pp. 217-220, 1985.

REFERENCES

903

72. A. J. Roscoe and R. A. Perrott, "Large fini te array analysis u ing infinite array data," IEEE Transactions on Antennas and Propagation, 42(7), pp. 983 - 992, July 1994. 73. A. D. Khzmalyan, "Finite Array Antenna. Scattering, and Transmission-Line Analysis Using the FFT," IEEE Antennas and Propagatioll Magazine, 42(1045-9243), pp. 41 - 53. 2000. 74. V. V. S. Prakash~ R. Mittra, and J. Yeo, "Radiation from finite microstrip patch arrays using infinite array approach.'' Proc. lEEE Amennas and Propagation Society International Symposium, San Antonio. TX. Vol. 2, pp. 192- 195, 2002. 75. A. K Skrivervic and J. R. Mo ig, ''Finite Phased Array of Microstrip Patch Antennas: the Infini te Array Approach." IEEE Transactions on Antennas and Propagation, 40(5), pp. 579- 582, May 1992. 76. S. M Wright and D. G. Twine, "Improved FFT analysis of finite periodic antenna arrays,'' Proc. 1990 IEEE APSIURSI Symposium, p. 2, 1990. 77. S. M . Wright. ''Analysis of edge effects in finite phased array antennas," Proc., 1990 Allenon Antenna Applications Symposium Rome Lab Technical Report 91-156 (ADA 23705 6), Vol. 1. pp. 217- 243, April 1991. 78. S. M. Wright "A DFT synthesis method for finite arrays of dipoles on layered media," Proc., 1992Allenon Antenna Applications Symposium, Rome Lab Technical Report 93- 119 (ADA 268167), Vol. 1. pp. 307- 336, June 1993. 79. J. Huang and D. M . Pozar, "Microstrip arrays: analysis, design, and applications," in K . F. Lee and W. Chen (eds.)~ Advances in M icrostrip and Printed Antennas, New York: WileyInterscieace. 1997. pp. 123-162. 80. 0 . A. Civi, V. B. Erturk. P.H. Pathak, P. Janpugdee, and H.-T. Chou, "A hybrid UTDMoM approach for the efficient analysis of radiation/scattering from large, printed finite phased arrays," Proc. Antennas and Prop agation Society 2001 IEEE International Symp., Vol. 2, pp. 806-809J 2001. 81. S. M . Wright and R. L. M agin, "Analysis of planar microstrip arrays for microwave byperthermia," P roc., Progress in Electromagnetics Research Symposium (PIERS), Cambridge, MA 1991. 82. S . S. Zhong and Y. T. Lo, "Single-Element Rectangular Microstrip Antenna for DualFrequency Operation," Electron. Lett., 19(8), pp. 298-300. April 14, 1983. 83. B . F. Wang and Y. T. Lo, ''Microstrip Antennas for Dual-Frequency Operation," IEEE Trans. Antennas Propag., AP-32(9), pp. 938-943, September 1984. 84. J. L. Kerr, "Microstrip Polarization Techniques," Proc. Antenna Appl. Symp., Allerton Park, University of Illinois at Urbana- Champaig~ Urbana, September 1978. 85. Y. T. Lo, B. Engst, and R. Q. H. Lee, "Circularly Polarized Microstrip Antennas," Proc. Antenna Appl. Symp., Allerton Parle, University of Illinois, at Urbana- Champaign, Urbana, September 22- 24, 1984. 86. Y. T. Lo and W. F. Richards, "Perturbation Approach to Design of Circularly Po1arized Microstrip Antennas," Electron. Lett., 17(11 ), pp. 383 - 385, May 28, 1981.

87. B. F. Wang and Y. T. Lo, Multiplex Microstrip Antennas for Dual Polarizations, Interim Tech. Rep. to Magnovox Electronic Systems Company, Fort Wayne, IN, March 1985. 88. J. Kita and Y. T. Lo, Multiplex Circularly Polarized Microstrip Antennas, Final Rep. to Magnovox Electronic Systems Company, Fort Wayne, IN, December 1985. 89. M . L. Oberhart, Y. T. Lo, and R. Q. H. Lee, "New Simple Feed Network for an Array Module of four Microstrip Elements," Electron. Lett., 23(9), pp. 436- 437, April 23, 1987. 90. M. L. Obe~ Y. T. Lo, and R. Q. H. Lee, "Study of Microstripline Fed Microstrip Antennas and Microstrip Antenna Arrays/' Proc. Antenna Appl. Symp., Allerton Park, University of Illinois at Urbana-Cbampai~ Urbana, September 1986.

904

ANTE

AS JV : MICROSTRIP ANTENNAS

91. P. LaMarche. Micrometrics Corporation. (http:/Avww.micrometrics.com). 92. J. F. Zurcher and F. E. Gardiol, Broadband Patch Antennas, New York: Artech House, J995. 93. R. Flynt. L. Fan. J. A. avarro. and K. Chang. "Low Cost and Compact Active Integrated Antenna Tran ceiver for System Application .'' IEEE Transactions 011 Microwave Theory and Techniques, pp. 1642- 1649. Oct. 1996. 94. M. Kosse!. H. Benedickter. W . Bachtold. R. Kung, and J. Han en. "Circularly-Polarized Aperture-Coupled Patch Antennas for a 2.4 GHz RFID System." Microwave Journal, pp. 20- 44. ov. 1999. 95. C. Sora . M . Karaboikis. G . Tsachtsiris, and V. Mak.io , "Analy is and De ign of an Inverted-F Antenna Printed on a PCMCIA Card for the 2.4 GHz ISM Band," IEEE Antennas and Propagation Magazine, 44(1), pp. 37-44, Feb. 2002. 96. A. W., Ma t, J. W .. Shipley, D. E .. Heckaman, and W . Whybrew. ''Multi-Tile Configured Phased Array Antenna Archi tecture" United States Patent #6.166,705, December 26, 2000, Harris Corp.-Melboume. FL. 97. P. Heide. "Commercial Microwave Sen or Technology: An Emerging Bu iness;' Microwave Journal , pp. 348-352, May 1999. 98. M. E. Rus el, A. Crain, A. Curran, R. A. Campbell, C. A. Drubin, and W. F. Miccioli. "Millimeter-Wave Radar Sensor for Automotive Intelligent Crui e Control." IEEE Transactions on Microwave Theory and Techniques , 45(12), pp. 2444-2453. December 1997. 99. D. D. Li, S. C. Luo, and R. M. Knox, "Millimeter-Wave FMCW Radar Tran ceiver/ Antenna for Automotive Applications," Applied Microwave and Wireless. pp. 58-68. June 1999. 100. " Phased Arrays for the New Millennium:· 2001 IEEE Antennas and Propagation/URS/ Symposium, July 8-13, 2001. Bo ton. MA. 101. J. M. Colin, "Pha ed Array Radars in France: Present and Future:· IEEE Inteniationa/ Symposium on Phased Array Systems and Technology. October 15-18, 1996. pp. 458-462, Bo ton, MA. 102. "Multi-Channel Receiver and Optical Data Link for Radar Sy terns with Digital Beamfonning," IEEE 1995 International Radar Conference. May 8-11. 1995, pp. 201-206. Alexandria, VA. 103. G. J. Erickson, G. W. Fitz immon . S. H. Goodman. D. T. Harvey, G. E. Miller, D. Rasmu en. and D. E. Riemer, "Integrated circuit Active Phased Array Antenna for Millimeter wa\-e Communication Application :· Antennas & Propagation Society lmemational Symposium, pp. 848-851. June 19-24. 1994. 104. D. E. Riemer, D. T. Harvey, J. H. McCandle. , and G. F. Fitz immon , "Low-Co t Communication Phased-Array Antenna," United States Patent#S.886,671, Boeing CompanySeattlc, WA. 105. "The Cooperative Engagement Capabilitie :• Johns Hopkins Applied Physics Laboratory Technical Digest. 16(4), pp. 377-396. 1995. 106. D. E. Riemer, "Packaging De ign of Wide-Angle Pha ed-Array Antenna for Frequencie Above 20 GHz,'' IEEE Transactions 011 Antennas and Propagarior,. 43(9). Sept. 1995. A SYSTEM FOR O 107. U.S. Army-Electronic Command. ' Ka-BAND SATCOM ANTE THE MOVE (OTM) OPERATION," FBO#0334, ovember 01, 2002.

108. "Development of a Voltage-Variable Dielectric (VVD). Electronic Scan Antenna," IEE 1997 Imernational Radar Conference, Edinburgh. Scotland. October 14- 16, 1997, pp. 383-385. 109. "Company unveils Satellite TV for Cars'' (http:/Avww.cm1.com/2003ffECH/ptech/OJ/08/tv. car.reut/). 110. Connexion By Boeing (http://l-vlvw.boeing.com/con11exion).

13 ANTENNAS V: ACTIVE INTEGRATED ANTENNAS Jo ATHAN D. FREDRICK AND TATSUO !TOH

13.1 INTRODUCTIO 13.1.1

AND HISTORY

Introduction

A microwave design reaches maturity levels not yet encountered in other technologie , the microwave engineer is more often forced to explore highly unconventional approache toward solving problems. Since the earlie t day of microwave engineering, the designer has been eeking ways to improve on existing design goals, such as physical dimensions, power consumption, RF output power, noise performance, and linearity. to name just a handful Thi chapter, on the topic of microwave active integrated antennas (AIAs), investigates recent steps taken to advance the state of the art in AIA design. The dual-facetted viewpoint of antennas and active circuitry is explored through functions such as radiators, diplexers, filters, resonators, amplifiers, and mixers, and individual devices, uch as diodes and transistors. The dual-purpose viewpoint of antennas and traditional circuit elements has Jed to variations of microwave engineering. Traditional microwave circuit engineers would progress through their design routine with the knowledge that the output of their work would be connected to nothing more than a 50 Q load. It does not matter to them if the load is an antenna or a dummy termination. Recently, as microwave circuit engineers consider the possibility of AIAs, new avenues of improvement open. The output of the original circuit does not necessarily end in a single-ended 50 ~1 termination. The output of a circuit may be connected to another block that provides signal processing-beyond radiation. Similarly, antenna engineers have long designed their antennas to work with 50 Q systems. It did not matter whether the connecting device was a filter, amplifier, Handbook of RF/Microwave Components and E flKineering. ISBN 0-471-39056-9 © 2003 John Wtley & Sons. .Inc.

Edited by Kai Chang

905

906

ANTENNAS V: ACTIVE INTEGRATED ANTENNAS

or mixer. Now the antenna engineer may view the circuit beyond as a signal processing ystem, possibly providing beamforming or signal processing. Additionally, the antenna de igner may decide to alter the antenna's feed to provide a signal processing function of its own.

13.1.2 Historical Background A far back as 1928 transmitter designs were considered with antenna and active device in mind [l]. In this aged design, a 1-MHz resonant circuit was coupled to an antenna through an electron tube to form a transmitter. Although this is an elementary design, it captures the essence of AIA design, where the antenna and circuitry are considered together. However elegant we view this design now, it was nearly a half of a century later before interest in AIA was renewed. During the 1960s and 1970s, with the maturity of solid state devices, numerous studies of the affects of interaction between antenna and active circuitry were performed (2- 10]. These works all address performance-related issues such as antenna or system bandwidth, lengthening of small antennas, coupling effects, noise performance, and efficiency.

13.1.3 Recent Works In recent years, most of the AlA research has gone in the direction of quasi-optical power combining. Quasi-optical power combining utilizes carefully arrayed active devices and antenna elements to add output power in free space instead of in a guided wave structure, with the primary benefit being reduction in combining losses (11, 12]. These benefits become particularly attractive at millimeter wave frequencies [13, 14, 15]. As this topic has been dealt with in several other ources, it will not be addressed here. In the past decade, new AlA designs have emerged to address performance needs in microwave and millimeter wave systems. Topics of particular interest are beamforming techniques, signal processing and smart antennas, high-power- high-efficiency transmitters, low noise receivers, and coupled or phased o cillator arrays. As illustrated herein, the AIA design concept, as applied to these goals, provides benefits such as reduction in loss, smaller circuit size, increased efficiency, and the elimination of expensive RF components. Moreover, the benefits of the AIA concept become pronounced at millimeter wave frequencies. This chapter gives an overview of recent advances through research into the AIA concept. Although the AIA field is very extensive, we address the most commonly researched aspects of the design process, namely, integrated oscillators and arrays, amplifying antennas, and signal processing arrays. Eac~ major section discusses several state-of-the-art designs.

13.2 INTEGRATED ANTENNA OSCILLATORS The first design demonstrating characteristics of an integrated antenna oscillator (IAO) was during the 1920 [1]. The IAO is one of the breeds of AIA researched thoroughly because it offers a practical solution to common transmitter problems. In many systems, oscillators are designed to function independently of their system-wide role. Circuitry is

13.2

907

INTEGRATED ANTENNA OSCILLATORS

designed to provide the negative re i lance needed for o cillation while being matched to a tandard impedance at the o cillator output. Such de ign method may be was teful . It i po ible to achieve teady tate o cillation while coupling directly to an antenna, thereby eliminating a dedicated matching network. To achieve the oscillation condition, the antenna may be modified to meet certain feed re trictions or impedance criteria. The mo t commonly u ed active device for IAO are olid tate devices such as Gunn or IMPATT diode . or more recently, transi tor such as MESFET, HEMT, and o on. Conventional de ign approach ugge t antenna and o cillator be considered separately only to be connected later by a tandard tran mis ion Une. IAO design anticipates the complex input impedance of the radiating element and attempt to create the nece ary condition for o cillation with it. Circuit design and simulation must take place in an integrated fa hion: i.e. antenna input characteristic must be part of the oscillator imulation . The IAO ha the previou ly mentioned advantages of AIAs, namely, smaller ize, lower cost, and lower loss as compared to the conventional approach. Toe general integrated o cillator circuit is shown in Fig. 13. 1. For an active device,

Zo (f) =Ro(/)+ j X D(f)

(13.1)

i the input impedance looking into the device with one port terminated. The load impedance. including the antenna' feed network and radiating aperture, is expressed as

(13.2) Oscillation occurs when the well-known Kurokawa conditions are satisfied [16]:

X L(Jo)

+ X D(fo) = 0

(13.3)

RL (fo) ~ IRv(fo)I

(13.4)

where Jo is the oscillation frequency determined by the resonance of the circuit. The first conilition simply states that the circuit must be in resonance. The second conilition requires that the negative device resistance is greater than the load resistance, thereby providing gain. The location of the active device in an antenna needs to be considered carefully during design in order to satisfy the above conditions.

Active device

Figure 13.1 General integrated antenna oscillator circuit

908

ANTENNAS V: ACTIVE INTEGRATED ANTENNAS

Active patch antenna

Ground plane

Slot/

Cavity

Sliding plunger

Figure 13.2 Tunable cavity-backed microstrip patch IAO. Source: (22].

Early integrated active antenna concepts surfaced during the 1960s. Antennas integrated with a parametric amplifier, tunnel diode, or transistor were reported (17-19]. The idea found very little practical use until the mid-1980s when integrated circuit antennas became popular for compact mobile systems and spatial power combining. Recently, the integrated antenna oscillator concept has been extended to achieve compactness (20]. During the 1990s, the IAO matured further. Numerous designs were proposed to address the need to generate millimeter wave oscillations for radiation, while minimizing the losses associated with transmitting power from oscillator to antenna via a guided wave structure. Chou et al. (21] designed an IAO utilizing a leaky mode microstrip antenna. In this work, the microstrip antenna is designed with leaky wave radiation in mind. The first higher mode of the microstrip line (TE 01 ) is a radiative mode and is used to design a leaky wave antenna. In this design, a slotline underneath the center of the leaky microstrip is used as the antenna feed, thereby allowing active circuitry to be uniplanar. The active devices are integrated on the back of the antenna on CPW, and a CPW to slotline transition is implemented. The advantages of this design are reduction in cross coupling, no via holes required, ease of device mounting, and low losses. The authors use an NE42484 low-noise HEMT to produce negative resistance when embedded in a series feedback circuit. An injection signal is used to stabilize the oscillation frequency, which is set by end-coupled resonators at the HEMT gate. A phase noise measurement of - 100 dBc/Hz at a 100 KHz offset is obtained at 9.17 GHz with 8.6-dBm output power. Antenna cross polarization is - 15 dB. Work has been conducted by Zheng et al. (22] to investigate IAO stability and phase noise reduction. In this work, the authors stabilized microstrip patch IAOs using cavity-backed techniques. In the cavity-backed experiment, the microstrip patch oscillator is placed above a tunable cavity that couples to the patch through a slot in the substrate ground plane, as illustrated in Fig. 13.2. Measurements indicated that a very slight change in oscillation frequency occurred after putting the patch onto the cavity. However, when the cavity was tuned such that its TE 101 resonance was at the oscillation frequency of the antenna a significant reduction in phase noise was observed. At an offset of 8 KHz, a 25-d.B reduction in phase noise was achieved.

13.3

13.3

COUPLED OSCILLATORS AND PHASE CONTROL

909

COUPLED OSCILLATORS AND PHASE CONTROL

Injection-locking technique may be u ed to achieve ynchronou operation of a number of IAO element . The igruficant point of contrast between a coupled o cillator array and a conventional phase array i that Lhe pha ed array derives all its power and phase coherence from a single ource divided between the element , wberea the coupled array generally has a ource for each element [23]. Each of the individual oscillator in a coupled o cillator array need to be phase coupled. In addition to achieving phase coherence for power combining purpo e , it has been found that such techniques also allow for beamforming function without additional pha e- hifting circuitry, suggesting a potential for low-co t radar systems. A few of the coupling techniques are described herein. Figure 13.3 how three po ibilitie for ynchronization of IAOs by injectionlocking. Each array element has a elf-contained voltage-controlled oscillator that include an antenna as the re onator and load. In Fig. 13.3a , the oscillators are all coupled to a common ignal (the desired output signal) that is distributed using a corporate feed network. In Fig. 13.3b , the o cillator are coupled through a serial feed network. This type of feed allows a progressive phase change between each element to can the radiation pattern. The third type of synchronization scheme is illustrated in Fig. 13.3c. Thi type of feed network relie on oscillator coupling through a dielectric resonator at the fundamental oscillation frequency to generate a stable second harmonic for 60-GHz applications. The architecture of Fig. 13.3a has been used to implement a simple Doppler radar ystem with tracking functions (24]. In this work, the coupled oscillators serve in quasi-optical power combining. frequency conversion (as a self-oscillating mixer, and directional tracking. The in-phase injection locking of the oscillators is crucial for effective quasi-optical power combining at broadside. The injection locking of each oscillator to uJo ensures that the oscillators will be in-phase at eu0 as per Kurokawa's theory (25]. For the system to perform Doppler tracking, a se1f-oscil1ating mixer (SOM) is implemented at the coupled oscillators as well. The Doppler-shifted return signal is directly mixed down with the coupled oscillator signals. As the oscillators are always in-phase with each other, the two Doppler shift signals are obtained by passing the IFs through a hybrid to produce sum and difference channels. The sum channel retains information about the relative velocity from the Doppler shift. The difference channel provides angular target information if the signal is analyzed in a vectorial fashion. The architecture illustrated in Fig. 13.3b may be utilized in applications similar to the parallel feed architecture; however,_ progressive phase shifts between elements must be accounted for. The DR-coupled SOMs illustrated in Fig. 13.3c are part of a system that suggests coupled oscillators, and antennas may be designed to serve more than one integration goal [26]. Toe DR stabilizes a 30-GHz push-pull oscillator, which serves as a balanced second harmonic mixer as well as generating the LO. The push-pull oscillator is antiphase at the fundamental and in-phase at the second harmonic. The RF signal is received in an antiphase manner due to the antenna's balanced feed nature. As this antenna is broadband, the SOM is excited in an antiphase manner over a broadband, without the use of a balun. The IF channels are then combined with an off-chip hybrid and bandstop filter. •

910

ANTENNAS V: ACTIVE INTEGRATED ANTENNAS

(a)

In-phase power dMder

Injection locking signal Corporate feed (b)

Injection signal Series feed

(c)

Quasi-Yagi

antenna

Dielectric resonator coupled feed

Figure 13.J Various coupled oscillator feed networks.

J_.4

13.4

MP IFYI

911

T

In re nt

ar fm n r ·1th d mand r 1 me lin · h ari n. On of th m m1n n t h · in am · i hann ni tu · fr q p prim pp rtu r harnioni tuning. D d irn 'th harm . . ·1 bl in th . · In addition t appli d t n · · link; it an ith qual ti e n pr th utiliz · u i larizati hara t ri ti and ant nna fe d ' fi Id on.figuration t bt in 1 p and hio-hl impl t fl ti m thod pr du in m nopuls radi tion p tt rn .

13..1

cti e Transmittin

ntennas

driv n b b tt m-lin figur n wada than an oth r , r consumption b tion an mak or br aka budget at oth r time it m d · u to impl i n in hands a .re ult of r arch and , · , r- ha\i been p rform d t d plifi r n i hplifi are ntial ompon n f · nt ·o well t and li h · tran mitt r in wir communication per en imp m n ·o po r-added · c an if it an b de igned ithout the major d gradation in lin arity. Ohir t al. [27] h n that impro in the PAE of an on-board -KW o1id- tat . pow r amplifi r in a · · ellit from % o 0% will r du wa t hea ob tantiall from .7 n de ign ar hitectur for higb-effici n and good tin arity power amplifi r h b n in tiaat d. Th AJA on p i a promi jn t hniqu for achie ing high e n and minimum ir uit iz f, r tran mitter . In thi b me th antenna i u ed a barmoni runina I ado a pow r amplifier in addition to it original role as the radiating el m nt. mentioned in [28] antennae u ed in the AIA approach mu t radiat ffi ientl with a c ptabl radiation pattern . In additi n, in a high] ffici n po er amplifi r d ign the load im edance in tb.1 e th antenna hould pro id a ptm I r a ti e termination at the higher harmonic . An ear] demonstration of a econd ingle-end d AIA amplifi d ign mp1oy a modified circular egm nt micro trip antenna a .55 GHz [29]. bl k di gr m f th A1A i hown in Fig. 1 .4a. rom the input impedance plotted in Fig. 1 .4b it i clear tbi antenna rea ti ely temrinate both econd and third harmonic . Additionall th real part of the input impedan is zero at the cond and third harm nic . A r lati ely high PAE of 6 % wa a hi v d a 2.55 GHz with an outpu p wer of 24.4 dBm with no major degradati n in the antenna radiation patterns. More recently the concept ha been extended to pu h-pull pow r amplifi r de ign where the power of two antiph e dri en cl -B p wer amplifier · dire tl y combined through a dual-feed planar antenna [30 l]. In a traditional mi rowa e frequency pu h-pull power amplifier two FET de jce combine output pow ~ through a broadband 180° hybrid or a ba1un. Howe er the lo associated with the output hybrid limits the practical efficiency of this type of power amplifier at microwa e and e pecially rniJUmeter wa e frequencie . In the AIA approach, active device are directly d ign

1

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6.5

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igure 13.4

ir ular

gm nt mi ro trip p

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all wing th ant nn to rv a po r ombin and a hannonicall tuned load. Th function are in a diti n t i original 1 a radi tin element thu minimi ing ir uit ize and in rti n 1 In th m t r nt pu h-pull P d ign [ ] th amplifi r i int grat d with a p bl f re ti I mrinatin th ond modified qua i- a · ant nna, whi h i hannoni . A 1 k diagram of thi de ign i bown in Fi . The ation in th tnmcat d ground plan to uppr in-ph f th antenna and the a ciat d ond harmoni radi tion. ak P at an utput po r of .2 dBm h b n a hi d at .15 GHz. dditionall nd harm ni radiation dB b l w th fundamental in both E and H plan . ound to be

in gnu d

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nteno

be utili d to impro rlonnan of r e1 er as ell Th f rec i th t ar ften of on noi e figure compactnes and th m t m t ei er de ign whil maintaining fun tionality. In rd r to a hi low noi figure eral n w de ign techniqu ha e n d eloped and d monstrated. In th hniqu the I w noi d vice are integra d closely with the antenna el ment l! a hi e hi h rforman e goal . Th paramet r

nc pt ma

13.4

AMPLIFYl G ANTE

AS

913

Corrugations

Rat-race hybrid

Truncated ground plane

Figure 13.5

- -- i

Dual-feed quasi-Yagi antenna with integrated push-pull power amplifiers.

13.4.2.1 Integrated Low Noise Receiver. Unlike high-power amplifiers used in the transmitter side. as discussed in 13.4.1, the receiver front end discussed here is concerned with noise performance. A low noise receiver does not generally require a harmonic tuning network; however, it does require a special impedance match in order to achieve a desired performance. In [33], a novel circularly polarized low noise receiver is introduced. The circularly polarized receiver provides an ideal example of the benefits of the integration of an antenna into the low noise receiver front end. Toe integration of the low noise amplifier (LNA) with the antenna leads to a further reduction in system noise figure because the LNAs are before the combining network. Thus, an improvement in performance is obtained by setting the noise figure of the system at an earlier stage, thereby rendering losses in the combining network negligible. The design presented in [33] utilizes a square microstrip patch antenna at 5.75 GHz to receive circularly polarized waves. The low noise devices are integrated with the antenna's non-50 Q impedance in order to obtain the minimum noise figure match without the use of any tuning stubs, impedance steps, or inset feeds. The author reports a minimum measured noise figure of 0.4 dB, and a gain of 11 dB is obtained at 5.75 GHz with good agreement with simulations. Measured results are shown in Fig. 13.6. A block diagram of the receiver front end is shown in Fig. 13.7. 13.4.2.2 Single Element Active Antenna Monopulse Receiver In addition to utilizing the antenna as the low noise figure match point for the active device, the antenna may serve other functions in receiver front ends. In [34], a revolutionary step is made in using a single antenna element to form monopulse sum and difference patterns for automotive radar applications. A high degr~ of interest remains in developing economical and compact radar systems for automotive applications, as such systems are still only available in high-end luxury cars and experimental systems. The cost and complexity of such systems still

914

ANTENNAS V: ACTIVE INTEGRATED ANTE

AS

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(a)

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a:i'

3.5

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5.9

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25

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,

~

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1........1..--L.......L-&..-..L__,__...l...-l.___.,____._....__..___.,_....._...___,__._....._....__.__,_..............__.__,_...................____

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Figure 13.6

Measured performance of the CP low noi e active receiver.

must be reduced. Thus, the concept of AIA, in which the circuit is matched directly to the antenna, rather than a 50 common interface, can be used advantageously to enhance the design and perfonnance of the radar front end. In this work, Lin demonstrates that by properly combining the dual-feed nature of the quasi-Yagi antenna (35] with a hybrid circuit; excellent monopulse operation can be achieved with the simplicity of using a single antenna element.

n

A conventional quasi-Yagi antenna, as reported by Qian [35), has a single microstrip feed with a microstrip-to-coplanar strips (CPS) transition that acts as a balun. Thus,

13.5

S IGNAL PROCESS ING ARRAYS

915

Microstrip patch

- --

LNAs -

-

Quadrature hybrid

RHCP

Figure 13.7

LHCP

Block diagram of the low noise circularly polarized active antenna receiver.

in single-feed operation, each dipole driver element is driven out-of-phase, and the antenna operates similar to a printed dipole antenna. Additionally, when operated with two feeds, it is possible for each driver element to operate independently, similar to two monopoles. This characteristic is exp]ojted in this work to form sum and difference radiation pattern by combining the outputs from each feed out-of-phase and in-phase, respectively. Figure 13.8a shows the simulated radiation pattern of each feed of the antenna. Note that the peak of each pattern is not centered at 0°, but rather shifted by ±21 °. Therefore, a sum and difference pattern can be obtained by combining the two patterns with appropriate excitation phases. In addition to using the dual-feed nature of the quasi-Yagi antenna, an LNA is integrated to each of the feed lines to provide gain and low noise figure. The measured results of the sum and difference channels is given in Fig. 13.8b . A noise figure of 3.6 dB and gain of 7.7 dB were measured in this C-band design.

13.5

SIGNAL PROCESSING ARRAYS

In recent years, the area of phased array antennas has undergone radical advances in functionality. In the past, phased array antennas were set to a fixed beam scan or were scanned utilizing expensive RF components such as phase shifters and variable attenuators. The expense of the RF components used in old phased array systems comes in the form of high losses, design and manufacturing expenses, and limited functional range. The recent revolution in signal processing arrays, such as retrodirective arrays and smart antenna arrays, has been motivated as a solution to these problems. These recent advances wed modem digital signal processing techniques to formal antenna

916

ANTENNAS V: ACTIVE INTEGRATED ANTENNAS

(a)

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Figure 13.8 Measured perfonnance of the dual-feed quasi-Yagi antenna in an active monopulse radar system.

array synthesis. The result of this union is a powerful new breed of signal processing array that offers advanced functions with very modest design costs. The first type of signal processing array discussed in this chapter is that of the retrodirective array. Retrodirective arrays have seen a new interest in recent years as

13.5

SIGNAL PROCESSfNG ARRAYS

917

well. As a type of analog or RF signal processing array, the retroclirective functionality comes about from proper manipulation of signal phases within the array. The second type of signal processing array di cu sed i that of the smart antenna type. In the smart antenna design process, one or more RF antenna array function is carefully moved to a lower frequency for digital signal processing (DSP). The use of DSP allows function s such as beamforming, peak finding, nulling, and so on to be performed s.imply and quickly, with high level of adaptability.

13.5.1

Retrodirective Arrays

Retrodirective arrays represent a type of special antenna array that reflects any incident signal back toward the source without prior knowledge of the source's location. They do not rely on the sophisticated digital signal processing algorithms as utilized by "smart antennas.'' A retrodirective array can provide omnidirectional coverage, while maintaining a high level of antenna gain. This unique property makes retrodirective arrays important in a wide range of applications, such as self-steering antennas, radar transponders, search and rescue and mobile communication systems (36- 38]. Retrodirectivity may be realized when each element in the array radiates an outgoing wave whose phase is conjugate to that of the incoming signal relative to a common reference (39]. The classic example of retrodirective array is the Van Atta array, where the conjugated elements of a symmetric array are connected by transmission lines of equal length [40] . However, this classic example has its limitations on symmetry of the array and uniformity of the phase-front. To overcome these limitations, a more general approach of phase conjugation based on heterodyne mixing was proposed [41, 42]. Phase conjugation with heterodyne mixing is a simple and effective technique to achieve retrodirectivity using an LO that is twice the RF frequency. In this scheme, the lower sideband product has the same frequency as the RF, but the phase is conjugated. When combined with an antenna and placed in an array, the phase-conjugated signal from each antenna element will be radiated toward the source direction. However, because the RF and IF share the same frequency in this scheme, good RF/IF isolation cannot be achieved using a filter. Altemative approaches must be used [42, 43]. More recently, an active retrodirective array circuit topology was demonstrated. The use of MESFETs in phase-conjugated circuitry is attractive because these active devices can provide conversion gain in addition to the mixing operation. This allows an array to send amplified signals toward the source location without dedicated amplifiers, resulting in compact circuit size and lower cost (44]. Figure 13.9 shows a photo of a prototype 4-element retrodirective antenna array using the circuit topology proposed in [45). The experimental results have shown excellent retrodirective performance and may be seen Fig. 13.10. This type of self-tracking system can be used in advanced wireless applications such as RF ID tags and remote information retrieval.

13.5.2 Smart Antenna Arrays Wireless local area networks have received increased popularity due to their flexibility and convenience. However, to meet multimedia bandwidth requirements, higher frequencies, higher data rate, and higher user density become a reality. In these situations, multipath fading and cross-interference become a serious issue, resulting in degradation

918

ANTE

AS V: ACTIVE INTEGRATED ANTE

Figure 13.9 (a)

C~D

cc -0

-en

(.)

AS

Photo of the prototype 4-element retrodirective array.

0

-

-5

-I

I

~

- 10

L

-

Source at broadside -Measured -··- ·· Theory .__ ___,

______

L

- 15

r

a: .Q co - 20

-"' al

-

- ,!•I • • I !, :

I

ii! : ;

-

- 25

•

\

L

- t L. -

~!

-60

-

\ ~

. I

.L\ _

- 'J..j~ :

I

• I

~!

-30 - 90

,.

i

·n'-

\• I:

- 30

0

I

30

.

'\

60

90

Scattering angle (deg.) Source at 0° (b)

Q) ~

cc -0

-en

(.)

a:

0

'

Source at 45 deg. Measured ····· ·,··· ·····:·· , ! -··- ·· Theory ;•.

- 5 ---- - 10

t

.

.

' . ··-•• ··••·····•r····•·····~

..... ! . ·.

:c

s -20 Cl)

/

: •\..

:

!

;

- 15

0

cc

,, . ,. ...

. .. •

:

•

f

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•

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.: i

.

•

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•

.:

:-~ •·-i-·'·----:. ! .

~

.! ,. ii

.

.. i;

•

.

.. ' •

-- --~· . -·····---- .•·-· -···----

.

:; !:

·t : : : : ! -25 ·····--- \.i• -· t• ······:·--· 1,:•····~-··-tf- ··-·· i · ···-·· · ··· t ··--· · ·---• II • •• • -i ,

:.

- 30 - 90

' \

-60

I

! - 30

:=i'

0

:♦

30

60

90

Scatter angle (deg.) Source at 45°

Figure 13.10

Bi latic RCS performance of the retrodirective array with various source angles.

13.5

SIGNAL PROCESSING ARRAYS

919

of the bit error rate. To deal with these problems and to achieve higher communication capacity, smart antenna y tern with adaptive beamforming capability have become a popular topic for research and development [46). Two types of beamfonning in smart antenna ystems exist. The anaJog architecture is usually based on expen ive RF phase shifters, whereas digital beamforming systems commonly consi t of a digital ignal processor that caJculates and applies weighting vectors to each sampled data. The latter approach ha the advantages of simplicity, flexibility, and lower power dissipation. However, current DSP technology has a speed bottleneck in data 1/0 with contrast to the high CPU speed it offers. For a typicaJ DSP system the input-proce -output procedure takes about 20 clock cycles. Therefore, the maximum data throughput is around 50 MBytes/second even if the state-of-the-art DSP chip with CPU clocks up to a few gigahertz is used. In smart antenna systems, the data throughput i the product of the number of channels and the data rate. Thus, congestion in the DSP 1/0 happens for an 8-channel system with a data transfer rate around 5 Mbps [47] . Jeon has proposed a smart antenna system with a novel hybrid analog- digital beamforming network [47]. The authors idea is to separate throughput intensive and computation-intensive tasks. Based on RF quadrature down-converters, throughput intensive tasks such as real-time beamforming, are carried out using analog multipliers at the IF. whereas computation-intensive tasks like the calculation of the complex weighting coefficients can be performed using the DSP by sampling the signaJ periodically. This way, the DSP I/O congestion is relieved. In this work, a smart antenna system is built using planar circuits. Direction of arrival (DOA) estimation is performe~ and the beamform.ing result is presented. A block diagram for the system is shown in Fig. 13.11 .

RF receiver

Analog beamformer

VQ downconv.

• • 8 elements • •

• •

IFout

1/Q downconv.

Digital signal processor DigitaJ signal processing

Figure 13.11 Block diagram of the adaptive beam forming smart antenna array. Source: [47].

920

ANTENNAS V: ACTIVE INTEGRATED ANTENNAS

SNOI

-+- Measured

-

-60

-40

-20

0

Simulation

20

40

60

Angle, (degree]

Figure 13.12

Beamforming results from adaptive beam forming smart antenna array.

DOA estimation was conducted at various angles by using two algorithms, ESPRIT and MUSIC, in an environment that simulated a cluttered office. The DOA in - 60° ~ + 50° range was estimated approximately within 5° error. When the experiment was conducted, a signal-of-interest (SOI) was placed at + 20° and a signal-not-of-interest (SNOI) was placed at -20°. After running the algorithms and measuring the IF signals from + 60° to - 60° the results, the SOI was found to be at + 20.2° and the SNOI at - 22.6°. Using the results from this experiment, complex weighting coefficients were applied to the digital beamformer to point the main beam at +20° and a null at - 22°. Measured results are shown in Fig. 13 .12.

REFERENCES 1. H. A. Wheeler, "Small Antennas," IEEE Trans. Antennas Prop., AP-23, pp. 462-469, July

(1975).

2. H. H. Meinke, "Active Antennas," Nachrichtentech Z. , 19, pp. 697- 705, (1966). 3. A. P. Anderson, W. S. Davies, M . M . Dawoud. and D. E. Galanaki , " otes on Transistor- Fed Active Array Antennas,'' IEEE Trans. Antennas Prop., AP-19, pp. 537- 539, July (1971). 4. J. R. Copeland, W . J. Robertson, and R. G. Verstraete, "Antennafier Array ," IEEE Trans. Antennas Prop., AP-12, pp. 227- 233, March (1964). 5. M . I. Kontorovich, "Active Antennas," Radio Eng. Electron. Phys., 19, pp. 126- 127, (1974).

6. A. P. Anderson and M. M. Dawoud, ''The Performance of Transistor Fed Monopoles in Active Antennas," IEEE Trans. Antennas Prop., AP-21, pp. 371 - 374, July (1973). 7. T. S. M. Maclean and P. A. Ramsdale, "Short Active Aerials for Transmission," Int. J. Electronics, 36, pp. 261- 269, February (1974). 8. J. P. Daniel and C. Terrel, ''Mutual Coupling between Antennas - Optimization of Transistor Parameters in Active Antenna Design," IEEE Trans. Antennas Prop., AP-23, pp. 513-516, July (1975).

REFERENCES

921

9. M . M. Dawoud and A. P . Anderson, 'Experimental Verification of the Reduced Frequency Dependence of Active Receiving Arrays," IEEE Trans. Antennas Prop., AP-22, pp. 362-344, M arch (I 974). 10. P . K . Rangole and S. S. Midha, "Short Antenna with Active Inductance," Electron. Lett., 10, pp. 462-463, October (1974). 11. J. Lin and T . Itoh, " Active Integrated Antennas," IEEE Trans. Microwave Theory Tech., MTT-42, pp. 2 186- 2194, December (1994). 12. R. A. York and Z. B. Popovic, Active and quasi-optical arrays fo r solid state power combining . Wiley. ew York, (1997). 13. L. Wandinger and V. albandian, '·Millimeter-Wave Power Combining Using Quasi-Optical Techniques." IEEE Trans. Microwave Theory Tech., MTT-31, pp. 189- 193, February ( 1983).

14. J. W. ~ ''Quasi-Optical Power Combining of Solid State Millimeter Wave Sources," IEEE Trans . ,\ 1icrow ave Theory Tech. , MIT-34, pp. 273- 279, February (1986).

15. Y. Qian and T . Itoh, "Progress in Active Integrated Antennas and Their Applications," IEEE Trans. Microwave Theory Tech. , MIT-46, pp. 1891- 1900, November ( 1998). 16. K . Kurokawa. "Some Basic Characteristics of Broadband Negative Resistance Oscillator Circuits;' B ell System Tech. Journal , 48, pp. 1937- 1955, July-August (1969).

17. A. D . Forst. "Parametric Amplifier Antenna/' P roc. IRE , 48, p . 1163, (1960). 18. J. R. Copeland and W. J. Robertson, ·'Design of Antennaverters and Antennafiers," Electronics, pp. 68- 71, October (1961).

19. K. Fujimoto, "Actives Antennas: Tunnel-Diode-Loaded Dipole Antennas," IEEE Proc., 53, p. 174, (1965). 20. D . Bonefacic and J. Bartolic, "Compact Active Integrated Antenna with Transistor Oscillator and Line Impedance Transformer," Electron.. Lett., 36, pp. 1519 -1521 , (2000).

21. G. Chou and C. C. Tzuang, "Oscillator-Type Active-Integrated Antenna: The LeakyMode Approach/' IEEE Trans. M icrowave Theory Tech. , MTT-44, pp. 2265- 2272, December (1996).

22. M. Zheng, J. W. Andrews, P. S. Hall, P. Gardner, Q. Chen, and V. F. Fusco~ "Active Integrated Antenna Oscillator Stability and Phase Noise Reduction,'' ICMMT Proc., pp. 285-288, (1998). 23. K. D . Stephan, "Inter-Injection-Locked Oscillators for Power Combining and Phased Arrays," IEEE Traru. Microwave Theory Tech. , MTT-34, pp. 1017- 1025, October (1986).

24. S. T. Chew and T. Itoh, "An Activ e Antenna Phased Array Doppler Radar with Tracking Capability," IEEE Antennas and Prop. Soc. Symp. 3, pp. 1368- 1371, June (1995). 25. K. Kurokawa, "Injection-Locking of Solid-State Microwave Oscillators," IEEE Proc. , 61, pp. 1386- 1409, October ( 1973). 26. M . Sironen, Y. Qian, and T . ltoh, " A Sub- Harmonic Self-Oscillating Mixer with Integrated Antenna for 60-GHz Wrreless Applications," IEEE Trans. Mic rowave Theory Tech., MTT49, pp. 442-450, March (2001). 27. T. Ohira, K. Ueno, K. Horikawa, and H . Ogawa, "Onboard active phase array techniques for high-performance communication satellites," MWE '97 Microwave Wkshp. Dig., Yokohama, Japan, pp. 339-345, Dec (1997). 28. V . Radisic, Y. Qian, and T. ltoh, ''Novel Architectures for High-Efficiency Amplifiers for Wrreless Applications,'' IEEE Trans. Microwave Theory Tech., MTT-46, pp. 1901 - 1909, November ( 1928). 29. V. Radisic, Y. Qian, and T. Ito~ "Class-F Power Amplifier Integrated with Circular Sector Microstrip Antenna," IEEE M1T-S Intl. Microwave Symp. Dig., 2, pp. 687-690, June (1997).

922

ANTE

AS V: ACTIVE CNTEGRATED ANTENNAS

30. C. Y. Hang, W. R. Deal, Y. Qian, and T . Itoh, "Push-Pull Power Amplifier Integrated with Microstrip Leaky-Wave Antenna," Electron. Lett., 35, pp. 1891 - 1892, October (1999). 31. C. Y. Hang, W. R. Deal, Y. Qian, and T . ltoh, "Push-PulJ Power Amplifier Integrated with Quasi-Yagi Antenna For Power Combining and Harmonic Tuning," IEEE MIT-S Intl. Microwave Symp. Dig., 1, pp. 533-536, June (2000). 32. C. Y. Hang, W. R. Deal, Y. Qian, and T. Itoh, "High- Efficiency Push-Pull Power Amplifier Integrated with Quasi- Yagi Antenna," IEEE Trans. Microwave Theory Tech., MTT-49, pp. 1155-1 161, June (2001 ). 33. J. D. Fredrick, Y. Qian, and T. Itoh, "Novel Design Technique for a Low Noise Receiver Front End with Integrated Circularly Polarized Patch Antenna," 30th European Microwave Con/. Symp. Dig., 2, pp. 333-336, October (2000). 34. S. Lin, Y. Qian, and T. Itoh, "A Low Noise Active Integrated Antenna Receiver for Monopulse Radar Applications," IEEE MIT-S Intl. Microwave Symp. Dig., 2, pp. 1395- 1398, May (2000). 35. Y. Qian, W. R. Deal, N. Kaneda, and T . Itoh, ..Microstrip-Fed Quasi-Yagi Antenna with Broadband Characteristics," Electronics Lett., 34, pp. 2194-2196, November (1998). 36. D . L . Margerum, "Self- Phased Arrays", Microwave Scanning Antennas, R C. Hansen, Ed., Los Altos, CA 1985, Vol. 3, Ch. 5. 37. B. S. Hewitt, 'The Evolution of Radar Technology into Commercial Systems", IEEE MIT-S Intl. Microwave Symp. Dig., 2, pp. 1271 - 1274, June (1994). 38. S. L. Karode and V. F. Fusco, "Self-Tracking Duplex Communication Link Using Planar Retrodirective Antennas", IEEE Trans. Antennas Prop. , AP-47, pp. 993- 1000, June (1999). 39. M . I. Skolnik and D. D. King, "Self-Phasing Array Antennas", IEEE Trans. Antennas Prop., AP-12, pp. 142-149, March (1964).

40. E. D. Sharp and M . A. Diab, 'Van Atta Reflector Array", IRE Trans. Antennas Prop., AP-8~ pp. 436- 438, July (1960). 41. C. Y. Pon, "Retrodirective Array Using the Heterodyne Technique", IEEE Trans. Antennas Prop., AP-12, pp. 176-180, March (1964). 42. C. W. Pobanz and T. Itob, "A Conformal Retrodirective Array for Radar Applications Using a Heterodyne Phased Scattering Element", IEEE MTI-S Intl. Microwave Symp. Dig., 2, pp. 905- 908, May (] 995). 43. S. L. Karode and V. F. Fusco, "Novel Retrodirective Beam Forming Techniques'', 27th European Microwave Conj. Symp. Dig. , 1, pp. 81-86, September (1997). 44. Y. Chang, H. R Fetterman, I. Newberg, and S . K . Panaretos, ·•Microwave Phase Conjugation Using Antenna Arrays", IEEE Trans. Microwave Theory Tech., MTT-46, pp. 1910- 1919, November (1998). 45. R. Y. Miyamoto, Y. Qian, and T. ltoh, '"Active Retrodirective Array for Remote Tagging and Wirele s Sen or Application ", IEEE MIT-S Intl. Microwave Symp. Dig., 3, pp. 1431 - 1434, June (2000).

46. W. L. Stutzman, J. H. Reed, C. B. Dietrich, K. B. Kim, and D. G. Sweeney, "Recent Results from Smart Antenna Experiments-Ba e Station and Handheld Terminals", IEEE Radio and Wireless Conj. Dig., 1, pp. 139-142, September (2000). 47. S. Jeon, Y. Wang, Y. Qian. and T. ltoh, "A Novel Planar Array Smart Antenna System with Hybrid Analog-Digital Beamforming", IEEE MIT-S Intl. Microwave Symp. Dig., 1, pp. 121-124, May (200 1).

14 MIXERS AND DETECTORS ERIK L.

KOLLBERG Chalmers University of Technology Gothenburg, Sweden

14.1 INTRODUCTION In any application of microwaves, millimeter waves, or submillirneter waves, detection is a must! Detection can be performed by detector elements based on a variety of physical principles, some of which are described below. Some of these principles work over very large frequency ranges: that is why detectors that one usually associates with infrared frequencies are also mentioned in this chapter. In some applications, straight detection is not good enough, and converting the signal to a lower frequency, using a mixer, is required. This is the case, for example, in applications when the phase of the signal must be preserved for further information processing, or when the sensitivity of the system requires extremely narrowband filters that are impossible to realize at the signal frequency. The major part of this chapter is devoted to mixers. Definitions, analysis, various topologies, and so on, are topics dealt with in some detail below. For general references concerning mixers and detectors, see Refs. 1-4.

14.2 SOME COMMON DETECTOR AND MIXER DEVICES 14.2.1 Introductory Remarks In this section we discuss different types of devices that can be used as detectors and/or as nonlinear elements in mixers. The frequency range considered is from a few hundred megahertz to terahertz frequencies. Important detector and mixer devices are described briefly, with particular emphasis on sensitivity and frequency response. Important also is the ability of the detector to respond to fast modulation. Generally, two types of Handbook of RF'/Microwave ComponenJs and Engineering, Edited by Kai Chang ISBN 0-471-39056-9 © 2003 John Wtley & Sons, Inc.

923

924

MIXERS AND DETECTORS

detector are used for microwave and millimeter waves: rectifying diode detectors and thermal detectors. In particular, in mixer applications, the response time of the devices (nonlinear element) must be much less than the inverse of the intermediate frequency. Since thermal detectors have a slow response, they cannot be used in mixer applications. 14.2.2 pn-Type Diodes There are quite a number of different ways of making diodes for detection and mixer application out of erniconductor materials, and some of the more important methods are briefly discusses in this section. The Schottky diode, which is the most important for microwave and millimeter wave applications, is discussed separately in Section 14.2.3. pn-Diodes. A pn-diode is essentially made from a semiconductor (e.g., Si) with a pdoped region and an n-doped region, each with an ohmic contact (see Fig. 14.1). The doping profile can be realized using different techniques [3], and the diode properties can be tailored by proper choice of the exact doping profile. In Fig. 14.2 is shown the doping profile of the ordinary pn-diode, the backward diode, and the tunnel diode. The basic physics of pn-diodes can be studied in, for example, Sze' s book on semiconductor devices [3]. The ordinary pn-diode is essentially a "low-frequency" diode u eful up to about 1 GHz, while the tunnel and backward diodes can in principle be used up to mi11imeter wave frequencies. The equivalent circuit of a pn-junction is a nonlinear resistance in parallel with a nonlinear capacitance (compare with Section 14.2.3 and Fig. 14.4). For the ordinary pn-diode, the nonlinear resistance is directly related to the de characteristics, that is,

i where q k Vi T

= i0 [exp(q Vi/ kT) -

l]

( 14.1)

= charge of the electron = Boltzmann con tant = voltage applied over the pn-junction

= physical temperature of the junction Metal

Ohmic contact

p-doped area by diffusion or ion-implantation

n

-----------------------------------------~~!lffll11li Metal

. Ohmic contact

Figure 14.1 Schematic outline of a pn-junction diode. The semiconductor is usually silicon.

14.2 (a) pn-diode

925

SOME COMMON DETECTOR AND MIXER DEVICES

(b) Backward diode

(c) Tunnel diode

Doping profile

NA

Band diagram, zero bias

E Electrons

-,L_ Current - voltage characteristic

I

I

Figure 14.2 Doping profiles, band diagrams, and N characteristics of (a) pn, (b) backward, and (c) tunnel diodes. Notice that the doping in the tunnel diode is high enough to make the semiconductor degenerate.

The nonlinear capacitance for an "abrupt junction" (the doping changes abruptly from p - to n-type) is expressed as C=

Co

Ji-

(14.2)

(VBi -~kT / q )

where V8 ; is the built-in potential, which like Co is dependent on the type of semiconductor and the amount of p- and n-type doping (3). For linear graded junctions or other types of doping profiles, the capacitance voltage dependence will become slightly different from Eq. (14.2). These formulas are a consequence of the depletion of charges, which causes the potential barrier V8 i (Fig. 14.2) to hinder an excess flow of majority charges. The capacitance is essentially the capacitance between the p- and then-doped region assuming the depleted region to be equivalent to an insulator, with a dielectric constant equal to that of the semiconductor (Er = 11.9 for Si and Er = 13.1 for GaAs). However, the charge transport mechanism over the barrier is related to a flow of majority carriers, electrons and holes, respectively, that have to disappear when they have reached the p- and the n-region, respectively. The required mechanism is a recombination of charges, which is not instant: the time constants involved are of

926

MIXERS AND DETECTORS

the order of nanoseconds. This phenomenon gives rise to a frequency dependence of the nonlinear resistance, and an excess capacitance in parallel with the depletion capacitance, usually referred to as the diffusion capacitance [3]. This means that at high enough frequencies, the switching capabilities of the diode are washed out. In practice, the ordinary pn-diode is not used above frequencies of the order of 1 GHz.

Tunnel Diodes. The tunnel diode is a pn-diode which is made up of such heavily doped p- and n-type material that tunneling will dominate over thermal emission and diffusion. This will cause the IV characteristic to show a negative resistance region (Fig. 14.2c) [3] . Tunneling is a majority-carrier phenomenon and is not governed by the conventional transit-time concept. Also, recombination is not a problem since the electrons can go straight into empty energy states either in the conduction band or the valence band (see Fig. 14.2). Hence tunneling devices should, in principle, work well into the millimeter wave frequency region. However, there are other limitations at high frequencies, partly related to the comparatively large depletion capacitance, which will be described in Section 14.13.2. The tunnel diode has been used as a ensitive and low-noise detector, as well as a mixer element, for microwave frequencies. The tunnel diode mixer is described in Section 14.13.2. Backward Diodes. The backward diode is a special type of tunnel diode. The doping concentrations on the p and n sides are barely large enough to make the semiconductor degenerate (which is also the case for the tunnel diode). Therefore, a tunneling current will occur in the back-biased diode which is significantly larger than the current for the corresponding forward bias, Fig. 14.2b. The backward diode, which has very low 1/ f noise ( ee Section 14.2.3}, can be used as a low-noise detector and as a mixer diode. Since there i no minority carrier torage effect (diffusion capacitance), it al o has a good frequency response. Furthermore, since tunneling is dominant, its N properties are insensitive to temperature change and radiation effects [3]. The magnitude of the nonlinearity, defined as

y

=

d 2 i / dv 2 di / dv

for the backward diode can be made to exceed that of the pn-junction or the Schottky diode (y = q / r,kT) (see Section 14.2.3).

Planar Doped Barrier Diodes. The planar doped barrier diode [5] is an interesting device, since it may have a symmetrical IV characteri tic similar to what is obtained when two ordinary diodes are coupled in antiparallel (see Fig. 14.27). Such a diode would be ideal for subharmonically pumped mixers (Section 14.6.7). Both contacts of the diode are ohmic, and an n+-i-p+-i-n + doping profile with an extremely thin, fully depleted accepter layer (p+) fonns a triangular barrier. By tailoring the widths of the different doping layers, the IV characteristic may be haped, and the symmetrical N characteristic is obtained when the doping profile is symmetric (see Section 14.6.7 for further comments).

14.2

14.2.3

SOME COMMON DETECTOR AND MIXER DEVICES

927

Schottky Barrier Diode

By far the most common mixer and detector element for microwave, millimeter waves, and submillimeter wave is the Schottky barrier diode (sometimes referred to as the hot carrier diode).

Characteristics of the Schottky Barrier Diode. The nonlinear characteristics of the Schottky barrier diode are the re ult of the properties of metal-semiconductor interface [3]. In fac~ the detectors often used at the beginning of this century, consisting of a metal cat whi ker contacting a silicon or a germanium crystal, was nothing but a (low-quality) Schottky diode. The difference between that diode and those we frequently use today is, of course, that modern diodes are of a much superior quality, ince the properties of the metal-semiconductor interface are infinitely better controlled. However, even today, the remaining problems in manufacturing high-quality diodes can be traced back to the problem of producing perfect metal-semiconductor interfaces [6-9]. The diode properties can be understood by using a band diagram model, as shown in Fig. 14.3, for a metal contact on an n-type semiconductor. At the metal-semiconductor surface charge will form a dipole layer such that a potential drop ¢, on the order of ½ to 1 V is created between the semiconductor and the metal. This voltage drop is called the barrier height The exact value of the barrier height is determined by the detailed physics near (within a few tens of Angstroms) the interface, and varies depending on a number of things, such as the particular metal, the semiconductor material, the doping, the processing technique, and so on. Hence it is possible to get diodes with different barrier heights. which means that the N as well as the CV characteristics will depend on the choice of diode. In order to have charge neutrality, the semiconductor will be depleted of electrons (of holes in p-type material) up to a certain distance W from the interface. This is the reason for the parabolic form of the potential. Electrons with enough energy can pass from the semiconductor to the metal, and vice versa. For no bias, there is of course a balance (i.e., there are as many electrons passing either way, yielding a zero current). When the diode is biased, the balanced situation is upset. Referring to Fig. 14.3, it is obvious that for forward bias, the barrier is lower, as seen from the semiconductor side,

Conduction band edge

T ct>

w n-type semiconductor

Fanni level

Metal

V;> 0

V;=O

Forward bias

Zero bias

Backward bias

Figure 14.3 Band diagram of the Schottky diode for three bias conditions. Io this case the semiconductor is n-type, which means that forward-biased electrons will go from the semiconductor to the metal.

928

MIXERS AND DETECTORS

letting more electrons go from the semiconductor to the metal. The current-voltage characteristic can be calculated theoretically using models that are appropriate for slightly different situations (see, e.g., Sze's book, Ref. 3). However, for all cases the same general form is obtained, that is, 1

= I0 [exp (q Vj / rJkT) -

J]

(14.3)

where rJ is the ideality factor, which for a good diode at room temperature is close to 1. The very existence of the depleted region implies that there will be a capacitance inversely proportional to the width W of the depletion region. This capacitance depends ma imilar way on the bias voltage (Vj) as for the pn-diode [3]: (14.4)

where Co is the zero-bias capacitance, ¢ the barrier height, and Vn the Fermi voltage. Equivalent Circuit of the Schottky Diode. The junction properties of a Schottky diode (and a pn-diode) can be modeled with a nonlinear resistance in parallel with the junction capacitance. In low-level detectors and mixers, the junction small-signal differential resistance is simply 8Vj

kTJT

r1- -- -81 -- -qi-

(14.5)

Io practice, the diode must be modeled with an equivalent circuit comprising not only the junction itself, but also the spreading resistance from the junction area to the ohmic contact(s), and a contact resistance due to the ohmic contact itself. Figure 14.4 hows a schematic diagram of a GaAs Schottky diode chip and the corresponding equivalent circuit. Notice that the RF current is flowing around the urface of the diode. Hence the RF serie resistance is slightly larger than the de re istance. Al o notice that the

Si02

Depleted region

W'( \'})

Schottky metal (anode contact)

Cathode contact (ohmic)

Figure 14.4 Schematic diagram of a GaAs Schottky barrier diode and its equivalent circuit. Notice that the Schottky metal is in the fonn of a bathtub, which will improve its performance (7, 8].

142

SOME COMMO

DETECTOR AND MIXER DEVICES

929

contribution to the series re i tance from the undepleted part of the epitaxial layer is actually bia dependent, a fact that is u ually neglected in device models used for analyzing detector and mixer properties. At ubmillimeter wave . further modifications in the device model will become necessary ( ee Section 14.10). The epitaxial layer doping i cho en for optimum performance, that is, the doping mu t neither be too low, ince then the erie re i tance will become high due to low conductivity in the epitaxial layer nor must it be too high, which will cau e tunneling to dominate over thermionic emi ion [6], making the diode noisy (high 77) and the capacitance large. The doping of the ubstrate hould be as high as possible in order to maximize the conductivity and con equently minimize the substrate contribution to the erie re i tance. The Mott diode ha an epitaxial layer thickness such that at zero bias the depleted region will just reach ub trate. In thi way, the contribution to the series resistance from the le s conductive epilayer is minimized, and the capacitance of the diode will not vary as much for the backward-biased diode. The smaller capacitance variation will cause le parametric effect in mixer applications, a fact that may be favorable in designing low-noise mixers. A common measure of the hlgh-frequency response of a particular diode is the cutoff frequency. which i defined for the zero-biased diode and determined from the constraint that the applied RF voltage i divided equally over the series resistance and the junction capacitance. Hence the diode cutoff frequency is simply (14.6)

where R s usually is defined at de. It is evident that the Mott diode has an improved cutoff frequency. The cutoff frequency of GaAs diodes is in general higher than for Si diodes, due to the higher mobility of GaAs, yielding a lower R s.

Noise in Schottky Diodes. There are several types of noise sources identified in semiconductor diodes. The most important ones are men tioned below [8-13]. Shot Noise. The shot noise is caused by the fact that the current flowing through the diode is due to the transport of individual electrons with a finite charge. The fact that their arrival at the anode can be described statistically by the Poisson distribution means that the root-mean-square (rms) fluctuation in the current o/ 2 is proportional to the current: (14.7) 012 = 21 q t,.f where t,.J is an infinitesimally small frequency interval. The noise power from the diode can be calculated using Eqs. (14.5) and ( 14.7) as

!

- o12 k T t,.f Pn - 4(81 /ov) - 2 11

(14.8)

By identifying this expression with the ordinary expression for noise power (i.e., Pn = kT t,.f), it is~seen that the equivalent noise temperature due to shot noise is (14.9)

930

MlXERS AND DETECTORS

1/f Noise. There usually is a noise component proportional to 1/ f, which is significant at quite low frequencies, and therefore is an important factor when detector nojse is considered. For mixer applications this noise is usually less important, since the intermediate frequency is high enough that the 1/ f noise is negligible. There are several physical phenomena that can cause the 1/ f noise. It is most often related to something which is not perfect in the junction area (8-10] . For instance, traps at the metal-semiconductor interface, possibly located in a thin oxide, will cause 1/f noise. At high forward-biased currents, traps in the undepleted part of the epilayer may also cause 1/ f noise. In this case, however, the time constants involved are sufficiently short that this noise may cause problems at frequencies of several gigahertz. Thermal Noise. Since the series resistance is due substantially to the finite resistivity of the semiconductor substrate, it will essentially contribute thermal noise. However, the ohmic contacts may also contribute some noise related to shot and 1/ f noise. Hot Electron Noise and lntervalley Scattering Noise. With strong forward bias, the electrons may gain enough energy to become significantly more energetic than at thermal equilibrium. Hence the electron temperature, which of course is intimately related to the experimentally measured noise temperature, will exceed the physical temperature of the device [10-12]. This noise is referred to as hot electron noise. In GaAs devices, energetic electrons may enter the satellite valley [3]. Since the mobility of electrons in the satellite valley is much lower than the mobility of electrons in the main valley, the transfer of electrons from the main valley into the satellite valley will cause fluctuations in the current. This noise is called intervalley scattering noise [12]. Total Diode Noise. Using the equivalent circuit of Fig. 14.4, the noise temperature of the diode (at low frequencies, where the influence of the diode capacitance can be neglected) can be expressed as follows: r j Tsh + R epi Tepi + R sub Tsub ~mooe~_;_ ___;_~--- -

,.,.

rj

+ R epi + R sub

(14.10)

At very high frequencies, the influence of the diode capacitance has to be taken into account [13] [i.e., Eq. (14.10) will be modified].

Cooling Schottky Barrier Diodes for Low-Noise Operation. According to the preceding section, both shot noise and thermal noise should be directly proportional to temperature. However, shot noise will, in practice, decrease with temperature, as indicated in Fig. 14.5. The reason the shot noise will not be reduced below a certain value is that at low temperatures, the tunneling of electrons through the barrier will become more important than thermionic emission. In simple words the reason is that the tunneling current can be expressed using a formula which is essentially the same as for thermionic emission, except that temperature is exchanged for an equivalent temperature e, proportional to the square root of the doping concentration (see Fig. 14.5). Hence, operating the diode at 15 K, whlch is a typical temperature obtained in commercial closed-cycle cooling machines, will substantially decrease the diode noise. Another effect of the cooling is that the log[i] - v characteristic will become steeper, since

14.2

SOME COMMO

DETECTOR AND MIXER DEVICES

931

Shot noise K

100 According to tunneling - qfl ( No 1 112 theory, Tiunnel - 2 k 4Em*)

50 Tiunnel

100

200

300

Temperature K

Figure 14.5

Shot noise temperature versu physical temperature for a typical GaAs diode with n -type doping of about 3 x 10 16 cm- 3 •

the exponent in Eq. (14.3) includes r,T = e. This will improve the device properties when used in detector application , but will not improve as much in mixer applications. Notice that for cooling only GaAs will do, since carrier freeze-out will become a seriou problem in low-doped Si at low temperatures. For further information, see Section 14.9.3, and Refs. 6. 7, and 9.

Types of Diode Construction and Packaging. Many methods of constructing diodes and diode packages exist. The reader should consult the various manufacturers' catalogs for details. Here will be presented a very brief description of a few designs related to Schottky barrier detector and mixer diodes. In Fig. 14.6 are shown schematics of different constructions. Naked diode chips are available for bonding into hybrid circuits, or whisker contacting in waveguide circuits (Fig. 14.6a and d). Figure 14.6b shows a typical outline of a beam lead diode. Also, pair and quads of beam lead diodes are available (Fig. 14.6c) for use in balanced, double balanced, and double-double balanced mixers (see Section 14.6). Each type of construction bas advantages and disadvantages, depending on the application. Practical considerations are, of course, important. The performance is affected by the parasitics (parasitic capacitance, lead inductance, series resistance), which together with zero-bias capacitance, cutoff frequency, and other relevant data, are usually well described and quantified in the manufacturer's catalog. Notice that the higher mobility of GaAs makes GaAs diodes superior to Si diodes, in particular for miUimeter wave applications. In Table 14.1 are shown different diode configurations available as monolithic beam lead devices and their respective applications. Bonding of beam lead diodes is an important art, and an overview of recommendations is given in Table 14.2. For further information, consult Ref. 14 and application notes from the various manufacturers (e.g., M/A-COM, Hewlett-Packard). One should always follow the manufacturer's recommendations. Various encapsulations of diodes are available, both for single- and multiple-diode configurations (again see manufacturers' catalogs). An example of an encapsulated diode is shown in Fig. 14.7. Important parasitics are the capsule capacitance and the lead inductances, quantified by the manufacturer. Characterization of Diodes. By "characterization" we mean the determination of the equivalent circuit of the diode (see Fig. 14.4). As was mentioned in Section 14.2.3, the

932

MIXERS AND DETECTORS Gold leads

(b)

(a)

Cathode c::::::::\

Silicon

7.____,_._~

Glass

0.1 mm 0.5mm

{c)

_ _,/ _ _,(

\._

\._

0.5mm

Figure 14.6 Some diode constructions: (a) diode chip with one bonding pad for hybrid integrated circuits; (b) IC beam lead detector/mixer diode; (c) IC beam lead diode quad for balanced and double balanced mixers; (d) electron micrograph of a diode chip (0.2 x 0 .2 x 0 .1 mm) with 2-µm-diameter diodes to be whisker contacted for millimeter wave detector/mixer applications (see Fig. 14.37). (Courtesy of Farran Technology Ltd., Cork, Ireland.)

series resistance varies with the bias, and is also frequency dependent. Assuming these factors can be neglected, one assumes the series resistance to be a constant circuit parameter, and the series resistance can be determined from the IV characteristic. Plotting log[/] versus voltage yields a curve as shown in Fig. 14.8. The series resistance Rs can now be calculated from the excess voltage drop at high forward currents. There are alternative techniques for determining Rs, particularly for small-area millimeter wave diodes, described in Ref. 13. The capacitance can be determined by a suitable commercial capacitance meter, capable of accurately measuring the capacitance down to a fraction of a femtofarad if

14.2

TABLE 14.1

SOME COMMO

DETECTOR AND MDCER DEVICES

933

Commercially Available Monolithic Beam Lead Diode Devices

Device

Electrical Circuit

Single

~

Features Ideal for u e a mixers or detector on MICs.

Series pair

Ideal for u e anywhere that a clo ely matched pair of diodes is required.

Reverse erie~ pair

Same as above except for polarity.

Common-cathode pair

Ideal for signal comparison detectors.

Antiparallel pair

Ideal for ubharmonically pumped mixers.

Split pair

Ideal for temperature-compensated detector use.

Four-junction pair

Ideal for high-level up-converters.

Star quad

Ideal for use in star mixer circuits that do not require an IF balun.

Quad bridge

Ideal for use in termination-insen itive mixers or biased mixers.

Quad ring

Ideal for use in double balanced mixers.

Eight-junction ring

Ideal for use in double balanced mixers requiring higher compression point and/or better IM performance.

Twelve-junction ring

Ideal for use in double balanced mixers wh ere highest compression point and/or best IM performance is required.

Source: Courtesy of Alpha Industries, Inc.

necessary. H Fig. 14.9 is shown the capacitance versus bias voltage for an ordinary Schottky diode and a Mon diode.

14.2.4 Thermal Detectors Calorimeters. It is possible to measure power by letting the radiation be absorbed in any type of matched load and just measuring the temperature rise (e.g., with a thermal

934

MIXERS AND DETECTORS

TABLE 14.2 Some Methods for Bonding Diode Chips Beam lead diodes Conducting epoxy or polyimid Parallel gap welding Thennocompression bonding Ultrasonic bonding Wobbe] bonding Chip diodes For chip die-down bonding techniques use, e.g., Conducting epoxy or polyimid Eutectic or soft soldering For lead bonding use, e.g., Ball (wire) or wedge (wire and ribbon) bonding of either thermocompression or ultrasonic type Always follow the recommendations from the manufacturer!

Metal cap Metal ring Ribbon inductance

Ceramic ring

Package capacitance

Connecting ribbon

1 mm

Figure 14.7

Typical diode encapsulation, ceramic pill type.

10mA

1 mA

100µA

10µA

11 =

i1lv1 1 µA L..._,j~_..___ _.....___ _ 500

600

700

~

800

_ __.___ _

900

mV

Figure 14.8 Determination of the ideality factor '1 and the series resistance Rs from the measured log[ I] - V characteristic.

14.2

SOME COMMON DETECTOR AND MIXER DEVICES

935

fF

Capacitance

20 15

Mottdiode

- o - o -0 Ordinary sch.diode _o S"

-

-o--o---

Bias

-3

-2

-1

1

Volt

Figure 14.9 Experimental capacitance versu bias voltage for an ordinary Schottky diode [compare. e.g .. Eq. (14.4)] and a Mott diode.

pile) of the load. Since the power absorption can be achieved quite independently of the type of temperature-measuring device, a calorimeter can be made extremely broadband. A major problem in constructing calorimeter detectors is to avoid spurious response due to temperature drift. The temperature rise may, for example, be measured with a thermocouple, in which case the temperature reference point should be located so that the influence of ambient-temperature fluctuations will become minimal. The problem can also be tackled by using a balanced configuration, that is, measuring the response of the power-ab orbing detector with respect to a calibrating twin element [15]. The amount of temperature rise, as in the bolometer case, will depend on the thermal resistance from the load to the surroundings, and the time constant will be determined by the product of the thermal mass of the load and the thermal resistance (16]. Calibration of the power meter can be accomplished by including a separate de heating element as close to the power-absorbing load as possible [15] .

Bolometers. A bolometer is a type of detector that is constructed from a material whose resistance varies rapidly with temperature (16]. When radiation is absorbed~ the temperature of the detector rises, causing a measurable change in the resistance. For example, a thin wire of platinum in a waveguide can be mounted to absorb the microwave power, and the wire resistance will then increase due to the temperature rise. Since the temperature rise will become larger if the thermal resistance to the surroundings is made large, it will help if the bolometer is mounted in an evacuated chamber. Doing so will cause the time constant of the bolometer to increase. A feature of practical importance is that it is not very difficult to match the bolometer detector over a broad band. Io fact, power meters based on bolometer detectors may cover a frequency band much broader than a waveguide band. Certain types of detectors will become extremely sensitive when cooled to very low temperatures. In particular, bolometer-type cryogenically cooled detectors are common and quite important for submi11imeter and infrared waves. The sensitivity is an effect not only of less temperature fluctuations and lower thermal noise, but also of access to materials with very large temperature coefficients. In particular, the resistivities of semiconductors_have an extremely large temperature dependence. Semiconducting materials such as germanium or silicon with various dopants have been used as temperature-sensitive absorbers. The sensitivity of this type of detector is expressed in

936

MIXERS AND DETECTORS

terms of noise equivalent power (NEP) W / .Jttz, which is a common way of indicating the sensitivity of infrared detectors. The sensitivity expressed in NEP of the semiconductor bolometers can reach 6 x 10- 16 W /.Jilz (17] when cooled to 0.3 K.

Golay Cell: A Pneumatic Detector. The Golay cell is a thermal type of detector (16, 17], which senses the pressure change in a gas cell heated by the incident radiation. The detector is used in particular at submillimeter wavelengths. However, the lower cutoff frequency of the detector, determined by the window opening, is typically as low as 60 GHz. The achievable sensitivity of the Golay cell at room temperature is within a factor of 10 of the theoretical limit, which is about 10- 10 W / .JHz [17]. Despite a slow response and severe microphony, it is a common and commercially available device for laboratory use. The problem with the microphonics is to some extent overcome by chopping the signal at the input of the detector and using a narrowband amplifier followed by a phase detector after the detector. Pyroelectric Detector. The pyroelectric detector [ 17, 18] is essentially an IR thermal detector, but can be used at millimeter and suhmillimeter frequencies as well. The pyroelectric effect exists in crystals exhibiting ferroelectricity (i.e., crystals showing a spontaneous electric polarization). Since this polarization is temperature dependent, the surface charge will change when the crystal is heated by absorbed radiation. The spectral range is limited in practice by the choice of material, absorber, and window size. The maximum sensitivity is typically 5 x 10- 10 W /.Jilz (18].

14.2.S Further Detector and Mixer Devices In this section we mention briefly other, less common mixer and detector devices. Hot Electron Detectors. Ordinary photoconductive materials can be used only for detecting radiation in the infrared; they cannot be used beyond a wavelength of about 250 µm. For submillimeter and millimeter wavelengths, a pecial type of photoconductivity can be used, based on the energy-dependent mobility of conduction band electrons. This "hot electron" bolometer detector also has to be cooled to liquid helium temperatures [17, 19]. The hot electron detector can reach a sensitivity of 6 x 10- 13 W /.Jilz [17]. The hot electron detector can be used in mixer applications, as discussed in Section 14.13.4. Superconducting Detectors. The most sensitive mixer systems that have been built up to now are based on superconducting elements in which two superconductors are separated with a very thin oxide (typically, 20 A). Two types of tunneling phenomena can occur in these elements: superconducting electrons forming Cooper pairs can tunnel through the junction, a phenomenon called the Joseph on effect; and Cooper pairs can break up and tunnel as single electrons (quasi-particles), a phenomenon called quasiparticle tunneling. Both phenomena have been demonstrated to be useful in detecting devices as well as in mixer devices. Superconducting tunnel elements are discussed further in Section 14.11.

14.3

14.3 14.3.1

OPTIMIZATIO

OF DIODE DETECTORS

937

OPTIMIZATION OF DIODE DETECTORS Introduction

Diode detector are rectifiers used to convert RF ignal , u ually modulated and of low amplitude, co (modulated) de. Diode detector are the mo t common type of detector in microwa e and millimeter wave y terns. Although both pn and Schottky diodes can be u ed, the latter type i much more common than the former. We discuss below how to optimize diode detectors for maximum en itivity. For further information it is recommended that the reader con ult application note from detector diode manufacturers (e.g .. Alpha, Hewlett-Packard. M/A-COM).

14.3.2

Theory of Low-Level Detection

At low levels, diode detectors act a square-law detectors, that is, the output voltage (current) is proportional to the RF input power (square of the input RF voltage). At higher ignal level . the detector will become linear, and at still higher levels the detector aturate ( ee Fig. 14.10). In Fig. 14.11 a simple detector circuit i hown. Since the bias circuit is designed to et the current through the diode equal to / bias for no input power, the signal will cau e the bias voltage to decrease. Let us assume that the junction voltage is V00 when no signal is pre ent and decreases by 8vo when an RF vo]tage 8vs cos(wst) is applied Equation 14.3 then yields (Vo = k11T / q ) :

.

l

V00-+8vo t) ] = l bjas + St. = ((10· {exp [ -+-8vs -cos(w,r --Vo

-

i}))

1

10-1 Linear

j1

0 > CJ)

0)

Square law

10- 2

a,

=: 0 >

...0

-- 1o-4 ( .)

a, a,

0

10-6

1o-8

I/Ill/

Noise

L--f----+---+----t---t-- 20 0 +20 -60 -40 Power dBm

Figure 14.10 Typical diode detector output characteristic.

938

MIXERS AND DETECTORS

Current regulated bias supply

Low noise video amplifier

Impedance transformer

RF source

DC return

Figure 14.11

Typical detector circuit.

8vo

8vs

Vo

Vo

= ({ (/ bias + io) [ I + - + -

'.::::'. (/bias + io)

OVo 1 1 + Vo + 4

[

RF bypass

cos(wst )

( )2] OVs

Vo

(14.11)

- io

Hence, since the diode differential resistance is r i '.::::'. Vo/ / bias (the series resistance is neglected), we have (14.12) Consulting Fig. 14.12a and considering Eq. (14.12), it is obvious that the diode can be considered a voltage source with an internal resistance of ri and a voltage amplitude of ov;/ 4Vo. Assuming that the video amplifier input resi tance Rv is infinite, the de current remains unaltered (oi = 0) and we have 1 ovs = ___ 2

8vo

(14.13)

4 Vo

Since 8v; is proportional to the available input power times the real part of the input impedance (Re{Z8 )), 8vs is maximized by choosing Re{Zg} as high as possible, and

(a)

-- --- -----.-.I

(b)

----------

I G

B~v•

Detector diode

I

I I I I I

L-----------

M

~Vo

+ Rs Detector diode

~--------~

Figure 14.12 Detector circuit representation: (a) RF; (b) de.

Rv

14.3

OPTIMIZATIO

OF DIODE DETECTORS

939

the diode impedance (determined by the bia oltage) to be of the same order as the input impedance. A more detailed analy is. talcing into account the diode capacitance and eries resi tance, can be done referring to Fig. 14.12. The generator is a sumed to be matched to the diode, that i , the generator impedance Z 8 i a complex conjugate of the diode impedance Zd. The available igna1 power then can be expres ed as (14.14) that is. (14.15) where, according to Eq. (2.4). B

= wCd c

~

wCo

---:::===== ✓l - Yoo / ¢

(14.16)

v;

8 0 i actually a function of the matching circuit and increases with Re{ Za}, and is made maximum by the proper choice of G [Eq. (14.14)]. From Fig. 14. 12 we may calculate the video response as (14.17) From Fig. 14.12a , 8vs can be related to 8vso, and by using Eq. (14.15), we obtain the voltage sensitivity as (14.18) where G = r 1 • One sees that it will help in reducing Rs and Be, and by making rj = G-1 large and Rv >> ' i · Hence for the matched case (RF), assuming that Rv >> ri and neglecting Rs one obtains the voltage sensitiv ity as

1

S v,matcbed

=

OVv

Ps

rj

= 4 Vo

(14.19)

µ V / µW

a-

1 If the detector is specified for a current of 50 µA, corresponding to = rj = 17kT/ ql ~ 560 Q >> R s, the voltage sensitivity for a matched detector at room temperature should be about 5 x 103 µ V / µ W . In practice it may not be possible to reach this number. Degradation from this number by a few decibels should be reasonable for a practical detector.

14.3.3 Zero-Bias Detectors For the zero-bias detector, the bias current Ibjas = 0 and the diode output resistance r j = Vo / Io (or r j = V0 / (lo + 8irect) if the rectified current 8irect is of the same order as the saturation current lo).

940

MIXERS AND DETECTORS

The video signal voltage is 8vo

= ! 8v;

Rv 4 Vo Rv + r j

(14.20)

Maximum sensitivity is obtained if the video amplifier input resistance Rv >> ' i · However, for a typical Schottky barrier diode, ri is of the order 30 MQ or larger [yielding a corresponding higher sensitivity according to Eq. (14.19)). Such diodes do require a bias (e.g., ~ 20 µA) for achieving a fair sensitivity with video amplifiers having a reasonable input resistance. Low barrier diodes, however, are available and have a much more reasonable zero-bias resistance, typically 2 kf2. In diode catalogs, the bias current used for tests of the tangential sensitivity and voltage sensitivity should be given. Notice that these catalog data are given for diodes that are matched at the RF input. Since the diode RF impedance as well as the video impedance is high, this means that in practice it may be difficult to achieve high sensitivity over a large bandwidth. Temperature effects may be important in some applications. As discussed in Section 14.2.3, both / 0 and Vo are temperature-dependent parameters.

14.3.4 Tangential Sensitivity and Diode Figure of Merit A tenn often used to define the sensitivity of a video detector is the tangential sensitivity, which is the input power in dBm required to change the de voltage output with an amount equal to the voltage noise fluctuations. This is usually made using an oscilloscope, as shown in Fig. 14.13. The noise fluctuations are caused by the detector diode itself and by the video detector used to amplify the output of the detector. The noise background is, in practice, negligible. The open-circuit noise of the diode is due to three sources: the thermal noise of the series resistance, the shot noise from the junction, and 'low-frequency' 1/ f noise. Forgetting for the moment the latter noise contribution, we can expres the rms noise voltage 8v; at the video amplifier input as 8v;

= 4kT BRs + 2kT Bri + 4kT B Ra

(14.21)

where ri is the junction differential resistance, Ra a fictitious noi e resistance of the video amplifier, and B the video amplifier bandwidth. The standard number for Ra is 1200 Q, and B is typically 10 MHz. The approximate peak-to-peak noise fluctuations are 2.88vn . Hence the power required to change the output voltage by this amount is 2.88vn

Pr s = - Sv

"Tangential" to noise peaks

Figure 14.13 Measurement of the tangential sensitivity.

(14.22)

14.4

MIXERS: SIMPLE THEORY AND BASIC DEFINITIONS

941

U ing Eq . (14.19) and (14.21 ), and a urning the diode bias to be 50 µA and at room temperature. a tangential sensitivity of -55 dBm is obtained for the matched detector case. Typical values een in catalogs are 50 to 56 d Bm, depending on the particular diode and bia point. Notice that matched zero-bias diodes will get a larger tangential en itivity than was just calculated due to the larger junction resistance.

14.3.5 High-Level Diode Detectors For high input powers, the detector no longer responds as a square-law detector (Fig. 14.10). The output voltage will, rather, increase in proportion to the square root of the input power. However, thi is approximate, since in this power regime the impedance of the diode will vary with the power level, making an accurate power measurement quite complicated. See also Section 14.5, where the problem of theoretically calculating the diode response of high power is dealt with in some detail.

14.4 MIXERS: SIMPLE THEORY AND BASIC DEFINITIONS 14.4.1 Introduction

In this section we di cuss fundamental properties of mixers using simplified mixer models, concentrating on single-ended mixers. For further reading, consult Ref. 20, which also deals with multiple-diode mixers. In a mixer, a usually quite weak signal, 8vs cos(a>st), is "mixed" in a nonlinear elemen~ such as a diode, with a strong signal VLo cos{cui.,0 t), called the local oscillator. The resulting current will contain Fourier components of frequencies nww ± ma>s. However. if 8vs « VLo, only the frequency components nww (of comparatively large amplitude) and ncut,o ± a>s (of small amplitude, proportional to 8vs) will be of any significance. Defining the intermediate frequency a>JF as

wn:

= ICut.o - wsl

(14.23)

(where a>s can be either larger or smaller than WLo), the frequency components related to the signal voltage can be expressed as (14.24) where n = 1, 2. 3, .. . , n' = 0, I 2, .... The frequencies n'WLo ± WJF are referred to as the harmonic sidebands. Hence the signal voltage will cause current to flow at all frequencies defined by Eq. (14.24), suggesting that signal power can be transferred to these frequencies. An equivalent multiport circuit of the pumped mixer diode based on these facts is shown in Fig. 14.14. In the most common mixing mode, the signal frequency is close to the local oscillator frequency (i.e., a>n= s ~ cut,o). Notice that we may have either a>s < cut,o or a>s > wt.o (i.e., tbe signal may be in either the lower sideband or the upper sideband). The signal will cause currents and/or voltages at all harmonic sidebands, and power will be transferred if the terminating loads have a resistive part.

942

MIXERS AND DETECTORS I

I

-

Wn

-

~

I

~

o---J I

I

I

Pumped diode

-

WJF

{J)_2 _ I W_n

-

I

b----1 --1 I I

I

Figure 14.14 Multiport representation of the pumped mixer diode. The notations W -n and indicate the harmonic sidebands [Eq. (14.24)].

Wn

Signal W5

Etc. 3wLo Frequency

Figure 14.1S

Signal transfer in a second-harmonic mixer.

In the harmonic mixer mode we let the IF frequency be (14.25)

In Fig. 14.15 the case n

= 2 is illustrated. As for n = 1 (fundamental mixer)

power

is transferred to all harmonic sidebands with (partly) resistive terminations. It is evident that to optimize mixer performance, it is necessary to design the mixer so that a minimum of signal power is lost to the harmonic sidebands. With a proper choice of harmonic sideband impedances, the mixer performance can in fact be considerably enhanced (see Section 14.4.5).

14.4.2 Conversion Loss From the discussion so far, it is obvious that when analyzing mixers one has to consider the response at the harmonic sidebands. In this section we discuss the conversion efficiency of a mixer, which is usually expressed in terms of the conversion loss L, defined as

Ps

signal power available at the input terminal(s)

L=-= ----------------Pw IF power delivered to the output load

(14.26)

14.4

MIXERS: SIMPLE THEORY AND BASIC DEFINITIONS

943

In most mixers the con er ion lo is less than 1. There are a few exceptions. One is when a nonlinear reactance (such a a diode which is backward biased so that the resistance is much larger than the capacitive reactance, Fig. 14.3) is used as a mixing element, and parametric effects may cause gain. Another case i when quantum effects become important~ ucb a in uperconducting quasiparticle mixers (see Section 14.11). The conversion loss may be expre ed as a product of different loss contributions, that i , (14.27)

Lo = los es due to absorption in the (barrier) nonlinear resistance Lh = losses due to power lost to the harmonic sidebands

where

LR,

= lo ses due to absorption in the series resistance

LirmF

= losses due to reflection at the mixer signal

L 1r IFl1

input port L ir,111 2 = 1/ (1 - 1rml2) lo ses due to reflection losses at the mixer output port L ir[F12 = 1/ (1 - IflFl2 )

=

Since the embedding impedances at the various sidebands influence all contributions listed above, the various contributions are not independent of each other. It has been shown that although a conjugate match at the IF port results in maximum power transfer, the signal input port should not, in general, be conjugate matched for minimum conversion loss. We will return to these questions in Section 14.5, where a more general and exact theory is briefly described.

Contribution from Lo- This contribution can be illustrated by evaluating the mixing properties of an idealized mixer, consisting of a purely resistive diode with an exponential relationship between the current and the voltage. We assume that the pumping voltage is purely sinusoidal (which is certainly never the case; see Section 14.5), that is,

I where V0

Vwcos(cuwt)} = Io ( exp { ---Vo

-

i)

(14.28)

k11T = -(~ 28 mV for room temperature) q

(14.29)

The next step is to calculate the small-signal current components caused by a signal, Vs cos(wst). To do this, we first derive the differential conductance of the pumped diode: g(t)

= -di = -I = -Io exp Vo

dV

Vo

(Voc) [Io (VLo) + (VLo) -

Vo

Vo

2/1

-

Vo

cos(Lot) + · · · 00

= g 0 + L cos(ncvwt)

(14.30)

l -

where the In CVLol Vo) are modified Bessel functions of order n and with the argument

Vw / Vo.

944

MIXERS AND DETECTORS i1F

Rg

is

Vs

9o

g,

92

g,

9o

g,

92

g,

9o

R 1F

i; R1

Mixer circuit analyzed in the text.

Figure 14.16

When the signal voltage frequency components (assumed small) interact with the pumped diode, the current can be calculated as

= v(t)g(t)

i (t)

(14.31)

where, as will be shown, v(t) i= Vs cos(wst ). Assume that one may neglect frequency components other than wt:F Ws, 2wt.o Ws = w; (ws and w; correspond to the two sidebands; w; is called the image frequency). Referring to Fig. 14.16 and assuming R; = Rg, the current i (t ) through the diode and the voltage v(t) over the diode are i (t)

= is cos(w t) + i; cos(w; t ) + i u cos(WJ:Ft )

v(t)

= Vs cos(wst) -

5

(14.32)

isRg cos(wst ) - i ; Rg cos(w; t )

(14.33)

- iIFRIF COS(WJ:Ft)

Hence, with Eq. (14.30), we get ls ] [

l~

l;

=

[go

(14.34)

81

82

where Rg is the load impedance at the signal or the image frequencie (here assumed to be equal), and RIF is the load impedance at the output of the mixer. From this expression one can calculate the conversion loss a defined in Eq. (14.26), and one obtains L

= Rg(8o + 82 + l/ Rg) [8o(go + 82 + 1/ Rg) -

8f

2KT]

(14.35)

where Rg is the generator impedance seen by the diode at the signal input port. By differentiating this expression with respect to R g , the minimum conversion loss is obtained as Lrrun = 2 l + Jl - /3 (14.36) 1 - JI - /3 where the pump parameter /3 can be evaluated from Eqs. (14.30)-(14.35). When the pump power increases, /3 also increases, approaching l when the pump amplitude goes

14.4

MIXERS: SIMPLE THEORY AND BASIC DEFINITIO S

945

TABLE 14.3 Approximate Formulas Describing Properties of the Y-Mixer' Short-Circuited Image

I + 2 ( _V,o

Lo (SSB)

) 112

VLo

Vo l oc

(Vw),,, Vo

= R inpu1

2

Broadbandb

Open-Circuited Image

(i + h vo)

l +

VLo

Vo

VLo

f oe Vo✓ 2

= 2Rinpul

CV,)"' _ o VLo

Vo f oe

(Vw)''' 2Vo

= 4Rinpu1

LO unpedance . VLo ~ 2/dc 0

The LO voltage wavefonn is assumed to be perfectly sinusoidal. b£xample, broadband case: Pi.a= l mW; l& = 3 mA yields VLO/ V0 Lo = 2.3 (3.5 dB).

= 12,

Rinput

= 80 Q , R1F = 160 Q ,

toward infinity. Hence the conversion loss then approaches L = 2 (3 dB). The lost half of the signal power is dissipated in the image load. No power at all is dissipated in the diode, which in this case can be considered equal to a perfect switch, operated by the LO between the on and off positions. A similar exercise can be followed for other cases of interest, such as when the image tennination is reactively terminated (see Section 14.4.5 concerning image rejection and image enhancement), short circuited or open circuited, or for a harmonic mixer. Using approximate expressions for the modified Bessel equations, Saleh [20] has evaluated properties for these mixers (also considering the case when the diode is pumped with a sinusoidal voltage). The results, summarized in Table 14.3, are not at all exact but can be used for order-of-magnitude estimations, indicating relative impedance levels, comparing the various cases of image termination, and gaining an understanding of how the LO influences the various properties. The influence of the diode parasitics [Rs and C(vj ) ] can be accounted for approximately as indicated below.

Contribution from Lk. The loss of signal power to the harmonic sidebands can be prevented only by arranging reactive loads at those frequencies.

Contribution from

The series resistance will actually cause losses at all frequencies related to the signal (nwto ± CUJF). A first-order effect is obtained for the signal frequency itself. Assume that when the mixer is operating, we may talk about an "active" junction conductance ( G), which "absorbs" signal power and transfers it to the IF. This conductance (equal to the inverse of the input resistance of Table 14.3) is in parallel with the junction capacitance, which will act as a shunt reactance 1/wsCo) and bypass some of the current. Hence at higher frequencies, the relative power absorbed in the series resistance will become larger and increase the conversion loss. The relative amount of power absorbed in the series resistance is readily calculated as LRr•

c~

(14.37)

946

MIXERS AND DETECTORS

Differentiating this expression with respect to G, one obtains the optimum case, (14.38) Notice that the series resi tance conversion loss contribution is 3 dB for W s

= 0.5 a>cutoff ·

Contribution from Lir1o 12 . As mentioned above, the input impedance is of importance for the mixer operation. A detailed theoretical evaluation is necessary for finding the optimum performance. However, a "reasonable" input match i of course necessary in all practical ca es. Contribution from Lirw12. For the output impedance, a simple general rule is valid: The IF port ha to be matched for optimum performance. r IF can be measured with a network analyzer and subsequently made small using a suitable matching network.

14.4.3

Input and Output Impedance

Table 14.3 summarizes results that can be used for a first approximation of mixer impedances. The influence of the diode capacitance can be accounted for approximately by considering an average junction capacitance {C} (cho en to be ~1.5 time the zero-bias capacitance) in parallel with the input and output impedances e timated from Table 14.3. The series resistance hould, of course. be added in erie to the re ulting impedances.

14.4.4 Noise of a Mixer Receiver In a mixer circuit there are several source of noi e that will be converted to the IF output and hence contribute to the sy tern noise. It hould be empha ized that a mixer in practice is always part of a y tern, and it is alway nece ary to analyze the ystem noi e. The mo t important noise ource contributing to the mixer y tern noi e are: I . Diode Noise: noi e generated within the mixer diode itself at frequencie that can be converted to the IF (see Section 14.2.3) 2. Thermal Noise:

noi e from the embedding circuit converted to the IF

3. LO Noise: noi e from the local o dllator ource, converted to the IF 4. IF Noise:

noi e from the IF amplifier

It i the ability of the mixer to re pond to a number of frequencies, nwt_o ± WiF, that make it nece ary to be quite careful when the noi e of the mixer i analyzed. In particular, engineer have frequently been confu ed when single- ideband (SSB) and double- ideband (DSB) noise i considered. In Fig. 14.17 a mixer receiver is de cribed chematically (omitting LO noise). Using the notations of Fig. 14.17, the equivalent noise temperature of the complete receiver can be calculated. A afe way of doing the calculation i to start calculating the noi e power) at the input of the circuit connected to the temperature (multiplied by k~f

=

14.4

Input load

r,

MlXERS: SIMPLE THEORY AND BASIC DEFINITIONS

--

l

--

Figure 14.17

If

T

Mixer

947

I

amplifier

l

TiF

T

Mixer receiver configuration.

output of the mixer. In Fig. 14 .17 thi circuit is an amplifier with an equivalent input noi e temperature of TIF. A urning that the noise spectrum of the generator impedance (T;) is flat (which is not alway true; e.g., an antenna facing the sky will see a temperature that varies with frequency, since the air and the "empty outer space" have characteristic spectra), we have at the amplifier input:

(14.39)

where L s, L;, Ln+, and Ln- are the conversion losses at the signal and image frequencies and at the upper and lower harmonic sidebands, respectively. T M ,out is the noise generated in the mixer diode itself and seen at the output port of the mixer. Next we define the signal-to-noise power ratio as (14.40)

k !:::.f Tsyst

where Tsyst here is by definition the receiver system temperature for a single-sideband system (useful signal is available at only one sideband). From Eq. (14.40) it is seen that

TsystSSB

= 1].L, =

Ti

(1 + ~; +

r(:.~:.J) +

+ TMXR,SSB + L, TIF (14.41)

where the equivalent temperature of the mixer itself,

TMXR,SSB ,

is defined as (14.42)

The single-sideband noise temperature of the mixer receiver, TM .SSB , is defined as (14.43) The double-sideband (DSB) mixer noise temperature is much easier to measure than the single-sideband noise temperature. The reason is that the noise sources one uses for noise-temperature measurements are broadband, in the sense that they generate noise in both sidebands. The same is true for radiometer-type receivers: noise is received

948

MIXERS AND DETECTORS

in both sidebands. We are thus interested in the case where the signal is received in both sidebands: Ps(l / Ls + 1/ L ;) (14.44) -= kTw6f Hence the double-sideband system noise temperature is

Tsyst,DSB

= l / Ls + l / L; 1 LsL; ~ ( 1 = T, [ 1 + L s + L · ~ -Ln+ + -Ln1

2

)]

LsL;

+ TMXR,DSB + TIP L s + L ·

(14.45)

1

where the double-sideband noise temperature of the mixer itself, TMXR,DSB, is defined as

(14.46) Notice that if Ls = L; , both the single-sideband system noise temperature and mixer noise temperature are two times those for the double-sideband case. For the single-ended mixer, the LO noise can be taken into account in the analyses by adding a relevant amount of noise at the input ports of the mixer. In a practical mixer, it will always help if the LO source is made inherently low noise by filtering and/or phase locking. For multidiode mixers, a more careful analysis is neces ary, and we will return to this question below.

14.4.5

Image Rejection and Image Enhancement

Another important sideband is the image frequency band. The importance of controlling the image frequency becomes obvious when realizing that: a. A considerable amount of the signal power (about 3 dB in a broadband mixer) can be converted to the image frequency band, increasing the conversion lo s. b. Noise from the embedding circuit at the image frequency will be converted to the IF. c. Unwanted signals may enter the receiver in the image band and be mistaken for proper signals in the signal band. It will help in all three respects if the image frequency termination (see Fig. 14.14) is made reactive. Essentially, a filter may do the job, making the mixer receiver properly single sideband by image rejecting. However, the signal power that intended to escape through the image port should be reflected by the proper phase in order to minimize the conversion loss. By doing this, we realize an image-enhanced mixer.

14.4.6 Harmonic Mixers It was mentioned in Section 14.4.1 that it is possible to construct harmonic mixers. Second-harmonic mixers are frequently constructed for millimeter wave applications; that is, a local oscillator (LO) source at approximately half the signal frequency is used.

14.5 THEORETICAL MODELJNG OF SCHOTTKY BARRIER MIXERS

949

The alternative, a fundamental mixer that needs pumping at approximately the signal frequency, may be a more expensive solution due to the local oscillator costs. Harmonic mixers with quite high multiplication factor are frequently u eel in applications where there is no need for any particular sensitivity. Typically, such applications are in phaselocked loops and as mixers in pectrum analyzers. The price one has to pay when u ing a harmonic mixer is often inferior performance. Calculating the conver ion loss for a (single-diode) second harmonic mixer in a way imilar to the approach described in Section 14.4.2 for Lo reveals that for reasonable LO powers the conver ion loss i typically 2 to 3 dB worse than for the fundamental mixer In practice one obtains typically 3 to 5 dB worse than for a fundamental mixer. The difference i partly due to the assumption in the simplified theory that fundamental mixing is not possible between the signal and the LO. This certainly take place unle there are reactive terminations for these mixing products. In general, this may be difficult to arrange. ince the fundamental mixing frequency is close to the LO frequency. In mixers using two diodes in antiparallel, this problem is avoided (see Sections 14.5.6 and 14.8.8) since the fundamental modulation of g(t) is actually at twice the LO frequency. In such a mixer the conversion loss can be made almost as low as for fundamental mixers. Arrangements with multiple diodes have been proposed for fourth harmonic mixers (21] (see also Section 14.8.8).

14.5

THEORETICAL MODELING OF SCHOTTKY BARRIER MIXERS

14.5.1 Introduction

In this section we give information about more accurate methods of analyzing mixer circuits. Toe fact that the nonlinearity of mixer diodes is quite strong makes it a nontrivial problem to analyze mixer devices theoretically. Another reason for the theory being complicated is that for accurate analyses, one must know the tenninating impedances at the harmonics of the LO and at the harmonic sidebands. The number of harmonics that have to be accounted for has to be limited for different practical reasons. However, the required accuracy places a lower limit on how many harmonics one has to include. In short, analyzing a mixer involves the following steps: Nonlinear analysis, finding the gj(t) and the Cj(t ) [compare Eq. (14.30)] of the pumped diode Small-signal analysis, by which the conversion loss between any pair of sidebands and input (output) impedances can be calculated Noise analysis, by which the noise of the mixer can be evaluated

14.5.2 Nonlinear Analysis The equivalent circuit of the Schottky diode was introduced in Section 14.2.3. The analysis outlined below is based on this circuit. The series resistance Rs (which varies slightly with bias and frequency) and the ideality factor 1J (which may increase slightly at high forward biases) are assumed to be constant, which is an excellent approximation

950

MIXERS AND DETECTORS

for most practical cases. From Eqs. (14.3) and (14.4) we obtain the following relations for the time-varying differential resistance and capacitance: g(t )

di q = -dVj = -(/+lo) TJkT

C(t)

= Co ( 1 -

V1· VBi

)-y

(14.47) (14.48)

The current through the resistance is given by Eq. (14.3), and the current through the capacitance is dV · (14.49) ic = C(t) - 1 dt Applying LO power at wp means that all the voltage-dependent variables will become periodic functions that can be expressed using Fourier series. Hence we may write 00

g(t) =

L

Gk exp(j kwpt)

G -k = G;

(14.50)

k=-oo 00

C(t)

=

L

ck exp(jkwp) c _k = c;

(14.51)

k=-oo

There are several related techniques developed for evaluating the waveforms g(t) and C(t): Direct Integration in the Time Domain [22, 23 ]. Thi method is inappropriate if the mixer contains distributed elements. The Harmonic Balance Method [24 ]. This method may have convergence problems when a large number of harmonics are considered. Modification will help but not necessarily eliminate the convergence problem. The p-Factor Method [25). This method may be con idered to be a modified harmonic balance method, with improved convergence properties. The Multiple Reflection Method [26]. Thi method can be considered to be a special case of the p-factor method [25]. Experience shows that it works well in all cases tried so far, but it may converge more slowly than an optimized version of the p-factor method. The Variables Selection Method [27]. By introducing a criterion for selecting which voltages and/or currents should be considered as unknowns, and by using a new and efficient algorithm, very fast convergence has been demonstrated, for large-signal analysis of both Schottky diodes and MESFETs.

The most practical and successful methods are the latter three. They are partly related, and the differences and convergence properties are analyzed in Ref. 25. The multiple reflection method has been found to converge in virtually all practical cases. The p-factor method seems, in general, to be faster than the multiple reflection method.

14.5

THEORETICAL MODELING OF SCHOTTKY BARRIER MIXERS

Linear network

- -

951

Nonlinear network

zL zN Figure 14.18 Divi ion of the mixer circuit into one linear and one nonlinear part.

ext we de cribe briefly the p-factor method [25] applied to the general circuit of Fig. 14.18, which i excited by a inu oidal source E (t ) = VP cos(cvpt). The iteration procedure is as follows: 1.

The analy is starts by guessing an initial voltage y N (t) and calculating the resulting current JN(t ) in the nonlinear element using a fourth-order Runge-Kutta method (or using an analytical expression if possible).

ii. By etting the current into the linear network to JL (t ) = -JN (t ) and using a fast Fourier transform to get JL (cv), one may calculate VL (cv) [= Z (cv) J L(cv)], which, like J L(cv), can be expressed as a Fourier series with harmonics of cvp. 111.

Using a fast inverse Fourier transform, one then calculates VL(t ) and makes a comparison with the initial y N (t ). If they differ by too much, y N (t ) should be modified and a new iteration should take place.

1v. The iteration steps defined in steps ii and iii will continue until the difference between the v N used in step ii and the yL derived in step iii is sufficiently small. Rather than using the derived V L directly as the input voltage V N in step i, a compromise value for V N should be used. Hence for the next (k + 1) iteration, the nth harmonic of y N = Vf+ 1,n of the voltage input (in step ii) is chosen according to (14.52) The way to choose Pn will not be described here; the reader is directed to Ref. 25 for further information. When the waveform of the junction voltage v(t) is known, it can be inserted into Eqs. (14.47) and (14.48) and the Fourier coefficients Gk and Ck of Eqs. (14.50) and (14.51) can be evaluated. Toe outline above of the p-factor method is called the "voltage update" method. Several modifications are possible, and in some cases the "current update" method may prove to be more efficient Applying the algorithm introduced in the variables selection method, an optimum value for the p-factor can be found that yields much improved convergence at moderate current levels in a diode mixer [27]:

_ Pn -

l_ (

v,L _ v,L k,n N

Vk.n -

k-1,n N )

Vk -1,n

- 1

(14.53)

952

MIXERS A D DETECTORS

Thi value of p11 will in general be a complex number, and takes different values for each iteration. 14.5.3

Small-Signal Analysis

The input signal(s) are assumed to have infinitesimally small amplitudes and therefore do not affect g(t ) and C (t). Since the j unction conductance g(t) and capacitance C(t) are in parallel, the total small-signal current through the junction admittance is related to the corresponding small-signal voltage accordingly (28, 30, 31]:

(14.54)

8/_m where Ymn = Gm -n

+ }wmCm-n

(14.55)

It is convenient to form the augmented Y matrix Y' which is defined in Fig. 14.19. Notice that the augmented matrix includes Rs and the external embedding network within the matrix. One has

Y' = Y + diag (

l

Zem + Rsm

)

(14.56)

where Zem is the embedding impedance at the mth harmonic sideband (mwp + cuo), and R sm is the serie re i tance at the same frequency. The ports of the augmented network (Fig. 14.19) are open-circuited. The relation between voltages and currents then is

= Y'8V

(14.57)

8V = Z'8I'

(14.58)

81' Inverting (14.56) yields

where (14.59)

Mixer Port Impedances. The input impedance of mixer port m(Zm) is measured by injecting an infinitesimal current 81,,, into port m, keeping the input impedance Zem = oo and adding the series re istance Rsm, yielding (14.60)

14.5

THEORETICAL MODELfNG OF SCHOTTKY BARRIER MfXERS

953

•

8V0

r--------~--o 1

-

81'

811

-.1.

Z e,+ R51

oV1

-

8/'-

1

8V_1

I

1I

8/'

4 1I

8V_2

1

-

Wo (IF)

Wo + Wp (USB)

Z1

-

I 7

---out

t - - - - - - ' - - --

---'

Zeo IF load

:

1 I I

r--- --- ---- -- ----~

I I I I I I I

w 0 -Wp

z_,

(LSB)

8/_2

Z14+ Rs...2

I I I

R so

Zo

8'-1 Za.,+ Rs..,

1

T

-

810

I

z--2

W0

- 2wp [Y] Intrinsic diode

Augmented network [ Y1

--------- ------ ------------ Figure 14.19 Definition of the augmented network (28]. Du.riag aormal mixer operation the equivalent signal current generator 81' is connected at port 1, the other ports being open circuited. In the aoise analysis, equivalent noise sources 81;111 and ol ~m are connected to all ports. The inset shows the relation between the signal source oV~gm at the mth sideband and its equi valent current source 01:,, . (From Ref. 28~copyright 1978 IEEE, reproduced by permission.)

In particular, the IF output impedance is ZlP

= 2 00. + Rso

(14.61)

00

Conversion Loss. Assuming that the IF port is conjugate matched, the conversion loss between port I and the IF port becomes

L

1 IZeo + Rso1 2 IZe1+ Rs11 2 = -----------2

41Zb11

Re{Zeo}

Re{Zed

(14.62)

The conversion Joss between any two ports i and j can be evaluated using Eq. (14.61) by replacing index O by j and I by i.

14.5.4 Mixer Noise Evaluation According to Section 14.2.3, the junction will contribute shot noise and series resistance thermal noise (and occasionally, hot electron noise) (see also Fig. 14.19).

Ist

Shot Noise. The shot noise rms current fluctuations o of a de-biased diode can be calculated, according to Eq. (14.8), as 2/shq!l.f. However, when the diode is pumped

954

MIXERS AND DETECTORS

with the local oscillator, the shot noise will become modulated. Assume that the pumped diode noise can be considered as consisting of amplitude-modulated pseudosinusoidal current components. Components at frequencies Wm = mwP + u>o(m = 0, ±1, ±2, ... ) can be converted to the various ports of the augmented network, including the IF port. It can be shown [28] that the shot noise rms voltage at the IF port becomes (14.63) where the square matrix (81~8l '!h} is known as the noise correlation matrix, with elements (14.64)

in which l m- n is one of the Fourier coefficients of the current through the nonlinear conductance of the diode.

Thermal Noise. If hot electron noise is generated when the diode is forward biased by the LO, the noise from the series resistance may be modulated [10, 32]. Actually, the series resistance itself will also be modulated, since the contribution from the undepleted part of the epilayer varies with bias. Ignoring both effects, only thermal noise from a constant series resistance has to be considered. In this case there is no correlation of the noise at the various ports of the augmented networ~ and the rms thermal noise voltage at the output IF port becomes [28]

(8 VT2,o )

= Z'o (81'T 8l'tT ) z 'to

(14.65)

where

4kTeqRsm!)..f

IZem + Rsml 2 4kTeqRsm!)..f

1Zol2 and where

Teq

(14.66a) (14.66b)

is the equivalent noise temperature of the series resistance.

Total Mixer Noise. The shot noise and the thermal noise contributions defined at the output IF port are uncorrelated. Therefore, they can be combined to yield the total noise output: 2 (8 VNo ) =Z'o {(8I'sb,o 8l'tsh,o )+(81'T ,o ol'tT ,o )}zto

(14.67)

The equivalent input noise temperature of the mixer, that is, the temperature to which the input impedance has to be heated in order to generate a noise voltage equal to (0Vi0 }, can now be calculated in a straightforward way, yielding

(14.68)

14.5

14.5.5

THEORETICAL MODELING OF SCHOTTKY BARRIER MfXERS

955

Noise of a Purely Resistive Mixer

It is illuminating to con ider the noi e of a purely resistive exponential diode mixer (i.e., a mixer with a diode having the erie re i tance equal to zero), a constant capacitance, and a current- voltage characteristic as determined by Eq. (14.3). U ing the theoretical approach outlined in thi ection, the available noi e power from the mixer diode at the IF output terminal i derived a (33]

(14.69)

where T i the physical temperature of the diode and Te the physical temperature of the embedding network. It is of particular interest to consider the following two special case .

Single-Sideband Mixer. As urning that all ports of the augmented network, except for the signal and the IF ports, are reactively terminated. TMXR.SSB

= -TJT [L i 2

l]

(14.70)

This re ult is identical to the input noise of an ordinary attenuator with the physical temperature equal to TJT / 2 and an attenuation of l0log[Li] decibels.

Double-Sideband Mixer. Assuming that all ports of the augmented network are terminated by reactive impedances, except for the signal ( 1), image (2), and IF (0) ports, then

(14.71)

14.5.6 Multiple-Diode Mixers There are several types of mixers using more than one diode (see Section 14.6). Several theoretical papers have been published on multiple-diode mixers. Extensive analysis has been done on two-diode balanced mixers (31 ,34]. Two-diode subharmonically pumped mixers have also been analyzed (31, 35]. Consider a mixer with M diodes, attached to a linear network having M + 3 ports, as shown in Fig. 14.20. The ports M + 1, M + 2, and M + 3 are connected to the LO pump, the signal mput, and the IF output, respectively. Since the only case considered is that when the signal is much lower in amplitude than the LO pump, the problem of analyzmg the mixer can again be divided into one nonlinear part, one linear part, and one part dealing with the noise. Faber and Gwarek [34] have analyzed this multidiode mixer by essentially using the techniques outlined above for single-diode mixers. In the nonlinear analysis, they used essentially the method described by Held and Kerr for the single-diode case [28] . Subharmonically pumped mixers have been analyzed in a similar way by Kerr [31] and Hicks and Kahn (35].

956

MIXERS AND DETECTORS

1

M+ 1 2

M+2

IF

z0

M+3 M

Figure 14.20 General equivalent circuit of a mixer with M diodes. (From Ref. 34; copyright 1980 IEEE, reproduced by permission.)

14.5.7 Some Further Comments on Mixer Analysis In an interesting paper by Hines it is pointed out that the actual model chosen for analyzing the mixer can affect the result. In the analysis above it is assumed that the bias of the diode is constant in voltage and that the bias supply has zero output impedance. Hence, if the signal changes the de current somewhat, it will not cause any power transfer from the signal to the de load. Hines [36], using a time-domain method, analyzed the situation when the diode was modeled as a perfect switch which is switched between infinite and zero impedance, and the bias was obtained using a "battery" with a certain open-circuited output voltage and an output resistance equal to the IF load. He then found that irrespective of the "diode" being assumed lossless (an on- off switch), and all ports, except for the signal and the IF ports, being terminated by reactive loads, the mixer will have a significant conversion loss which cannot be explained by any mismatch at the input or the output port. A subsequent analysis of the same problem u ing the frequency-domain analysis as discussed above fortunately gave the same result. The explanation of the phenomenon is slightly different depending on the method of analysis. The time-domain analysis suggests that the signal energy is converted to de and then dissipated in the de load. A frequency-domain analysis suggests that the signal energy is converted to an infinite number of high-order modulation products, being dissipated in the infinitesimal resistance of the switch in its low-impedance "on" -state, resulting in a finite loss. The conclusion from this brief discussion is a recommendation that extra thought be given as to how to model the actual mixer being analyzed.

14.6 14.6.1

SINGLE- AND MULTIPLE-DIODE MIXERS Introduction

Below we compare the properties of different mixer circuits using one or several mixer diodes (see Table 14.4). Basically, four types of fundamental frequency mixer circuits have found practical use:

TABLE 14.4 Mixer Comparison Guide Port-to-Port Isolation

Mixer Type Single diode Single balanced, 180° Single balanced, 90° Doubled balanced Double balanced, image rejection Image recovering Double-double balanced mixers SubhannonicaJly pumped 0

Number of Diodes" 1

2 2 4 4 4 8 2

VSWR

Conversion Loss

RF

LO

IF

Good" Good Good Very good Very good

Poor' Fair Good Poor Good

Good" Fair" Good Poor Good

Excellent Good Good

Good Good

Good Good Fair

Fair

LO-RF

LO AM-Noise Rejection

Signal

LO

Moderated Fair Fair Good Good

Moderated Very good Poor Very good Good

None Good Good Good Good

All Odde Odde Odd Odd

AU

AW AIJe Odd Odd

Good Good Depends on filters

Very good Very good Depends on / filters

Good Good Good

Odd Odd All

Odd Odd Even

RF-IF LO-LP

Poor Fair Fair Good Good Good Good Fair

LO power and saturation power is proportional to number of diodes. b[mag~ enhancement possible. cMatching circuit recommended. dDepends on filters. elf signal fed in phase and LO out of phase (compare Fig. 14.22c); if the other way around, reverse the columns (see Section 14.7). fFor RF ➔ LO isolation, filters are necessary.

l,C

tl'I

...:a

Intermodulation Products Generated

958

MIXERS AND DETECTORS

• The single-ended mixer (SE) with one diode • The single balanced mixer (SB) with two diodes • The double balanced mixer (DB) with four diodes • The double-double balanced mixer (DOB) with eight diodes It should be kept in mind that there are alternative circuits possible for most of the mixer configurations discussed below (i.e., the circuits given in the figures should just be considered as examples). Besides fundamental frequency mixers, subhannonically pumped multiple-diode mixers will also be introduced. The discussion in the previous sections essentially applies also to multiple-diode mixers. In fact, multiple-diode mixers can be understood in terms of interconnected single-diode mixers. For further information, consult major suppliers of commercial mixers (e.g., Anaren, Alpha, Hewlett-Packard) and Refs. 37-40. Notice that different multiple-diode configurations are commercially available as monolithic beam lead devices in various types of packages (see Table 14.1).

14.6.2 The Single-Ended Mixer The principal advantage of the single-ended mixer is its simplicity (Fig. 14.21). The signal, local oscillator, and intermediate-frequency ports can be isolated from each other by means of filter circuits. In several common and simple circuit solutions. however, there is poor isolation, if any, between the LO and signal ports (see Section 14.8). Therefore, to achieve broadband coupling of the LO and the signal to the diode requires a broadband coupler (Fig. 14.21a), which will cause increased conversion loss, a worse noise figure, and a multiplied LO power requirement (e .g., a 6-dB directional coupler will increase the conversion loss and the noise figure with 1.25 dB, and require four times more LO power). A narrowband LO injection diplexer (Fig. 14.21b) avoids this problem, but may make tunability difficult. Another important fact is that LO noise is more difficult to avoid, in particular if the IF is a low frequency. In multiple-diode mixer this problem is considerably alleviated, as discussed below. Since the required local o cillator power i proportional to the number of diodes of the mixer, the SE mixer, from that point of view, requires less power than do the other mixer types. However, thi also means that the dynamic range of the single-ended mixer is smaller, since the 1-dB compression point is typically 5 to 10 dB below the LO power level.

(b)

(a)

Local oscillator Single L o a d - - - - - - - - ended Directional mixer coupler

Local oscillator

Load

1--+--..__-~~

Diplexer

Single ended mixer

Examples of single-ended mixer configurations: (a) local oscillator injected through a broadband directional coupler. (b) local oscillator injected via a narrow-band diplexer (e.g., a ring filter).

Figure 14.21

14.6

SINGLE- AND MULTIPLE- DIODE MIXERS

959

SE mixer are u ed in microwave y tern where a imple olution i desirable and/or lower perfonnance level can be tolerated. SE low-noi e mixer arc frequently u ed for the millimeter wave frequency range ( ee Sections 14.9 and 14.10).

14.6.3 Single Balanced Mixers Some of the di ad antage of the SE mi er can be overcome u ing ingle balanced (SB) mixers. Such mi er can be con tructed u ing various types of hybrid circuits and balun for coupling the LO and the ignal to the diodes. Either 90° or 180° 3-dB hybrids can be u ed. each having certain advantage and di advantage , but in both case offering better performance than the SE mixer. Thu they offer reduced purious respon e, cancellation of the de component, uppre sion of local o cillator noise, and isolation between the variou ports. Schematic diagrams of 90° and 180° hybrid SB mixer tructure are hown in Fig. 14.22b. Working out the relative phase of the signal and the LO at the terminals of the diode pair ( ee Fig. 14.22) reveal that in both cases the LO has a 180° phase shift from the diode pair, while the signal ha zero phase hift. Therefore, the origin of the IF ignal is the same for both cases, and can be combined as indicated in fjg_ 14.22b. The 180° hybrid balanced mixer can be described readily using the equivalent circuit of Fig. I ..t..22c. A suming the diode ( D 1 and D 2 ) to be exactly equal, they form a voltage divider, causing a virtual ground (zero phase relative to ground) at point A. The way that ignal current is and LO current iLo is fed to the circuit means that is

(a)

90 degree hybrid (b)

LO Signal

180° or goo

3 dB hybrid

180 degree hybrid

01

02

®

IF

(c) LO

Low pass filter

IF

Figure 14.22 Single balanced mixer configurations: (a) the phase shifts of 90° and 180° hybrids; (b) schematic balanced mixer configurations; (c) equivalent circuH of the 180° hybrid rmxer.

960

MIXERS AND DETECTORS

and iLo add in one diode and subtract in the other. This will cause an imbalance at A, which will slowly cycle with a frequency equal to the IF. Consequently, the IF signal can be extracted between A and ground. Although ideally, there is infinite isolation between the various ports, the practical solutions available may, for various reasons, not be good enough, and therefore filter circuits for improving the isolation may be necessary. Notice that noise from the LO will not cause an imbalance at A, and will con equently not add noise in the IF circuit. Using the I 80° hybrid is equal to combining two SE mixers in parallel, 180° out of phase. Terminating the two output arms of the 180° hybrid with identical impedances will cau e reflected power to go back to the input port. Consequently, there will in thj case be perfect isolation between the signal and the LO port. In practice, with well-matched diodes, the isolation is typically 20 dB or more. However, as in the SE mixer, the VSWR at the RF ports will depend on the match between the diodes and the circuit, typically yielding a VSWR of 2: 1. For optimum performance, filters should be used to separate RF and IF frequencies. With the 180° hybrid, the mixer can be designed to suppress the even harmonics of one of the input signals, usually the LO signal. The amount of suppression depends on how well the diodes are matched and on the balance of the hybrid. Using 90° hybrids yields significantly different properties. The VSWR is excellent over the full performance range of the hybrid. Feeding a signal into one of the RF ports, similar diodes will cause the reflected waves to combine at the other RF port. Hence, either input will have a low VSWR (typically, less than 1.5:1). The isolation between the RF ports will, in this case, depend on the match between the circuit and the diodes, and is typically not better than 7 dB. To obtain optimum performance, filters to separate RF and IF ports have to be included. Since the 90° hybrid is as relatively easy to design and fabricate as either a microstrip, stripline, or coaxial-line circuit, it has been used widely in broadband (octaveband) mixers. Examples of 90° hybrids are (see also Refs. 1 and 41): • Properly designed 3-dB stripline or microstrip couplers [42] • The Lange coupler [43] • The branch-line hybrid [44] Examples of 180° hybrids are: • The waveguide magic T (45] • The ring or "rat race" hybrid [41] • Various combination of microstrip, coplanar, slotline, finline, and so on, junctions (see Section 14.8) Baluns [46] (balanced-to-unbalanced line transfonner; the LO transformer in Fig. 14.22c i a balun) are also used in balanced mixer circuits. A balun evidently has the exact propertie required and can be de igned with decade bandwidth ratios [47]. In Section 14.8, practical balanced mixer circuits are described in some detail.

14.6.4 Double Balanced Mixers Double balanced (DB) mixers are composed of two SB mixers coupled in parallel and 180° out of phase (Fig. 14.23). The diodes can be arranged in either a star or a ring

14.6

SINGLE- AND MULTIPLE-DIODE MlXERS

(a)

961

(b)

Signal

IF IF

~ Signal

l--

j_ --

LO

Figure 14.23 Double balanced mixers: (a) ring mixer; (b) star mixer.

configuration. The ring modulator can be obtained as a very compact monolithic circuit, and i the more commonly used configuration in practical mixers. The symmetry of the circuit ensure complete i olation between the LO and the signal port if the diodes are perfectly matched. Furthermore, the topology of the circuit now yields the suppression of even harmonics of both the signal and the LO frequencies . This fact also means that intermodulation is reduced compared to SE and SB mixers. The DB mixers use two baluns rather than one as in the SB mixer. At low frequencies (a few gigahertz) the baluns can be fabricated from bifilar-wound transformers with central taps for the IF output. The isolation between the three ports becomes independent of frequency and quite large (more than 35 dB). At higher frequencies this type of transformer will not work, and other solutions must be found. Mixing different types of transmission-line circuits and using their symmetry properties is one way (see Section 14.8). Another way is through use of distributed circuits. These types of structures will not achieve the same isolation as will the transformer type; an isolation of about 25 dB may be expected. The RF input ports have the same properties as the 180° hybrid SB mixer (i.e., typically the SVWR is about 2.5:1).

14.6.S Double-Double Balanced Mixers Double-double balanced (DOB) mixers are composed of two DB mixers (an example is shown in Fig. 14.24). Since all three ports (signal, LO, and IF) are balanced, they could also be called "triple balanced." Eight diodes are used, and therefore twice as much LO power is needed as for the DB mixer, and eight times the LO power as for the SE mixer. This means that the dynamic range is increased eightfold compared to the SE mixer, and so on, when compared to the other mixer types. In addition to the larger dynamic range, the intermodulation properties are superior. The drawbacks of the DDB mixer are the requirement for greater LO power and the increased cost due to more diodes being used. In all mixers there must exist ground return paths for both RF and IF currents. This fact results in loss, adding to the conversion loss of the mixer. DOB mixers do not require a return path for the IF, ~Jimioating an additional signal loss, which can be on the order of 1 dB.

962

MIXERS AND DETECTORS C'

A

Y-11, Figure 14.24 Example of a double-double balanced mixer circuit The bold lines connected to A. A' , B. B' . C. C' , D. D' indicate, for example, J.../ 4 lines. necessary not to short circuit the diode quads. (From Ref. 37; reproduced from Microwave Systems News, October 1981, by permission of the publisher, EW Communications, Inc.)

14.6.6 Quadrature IF Mixers and Image-Rejection Mixers The mixer circuits discussed above can be used in various combinations for applications other than simply converting an incoming signal to another frequency. The quadrature IF (QIF) mixer is an example of such a circuit, and can be used, for example, for Doppler systems and network analyzers. The QIF mixer can also be converted to form an image rejection (IR) mixer. The QIF mixer consists basically of two balanced mixers, one 90° hybrid for injecting the signal (LO) and one in-phase power divider for injecting the LO (signal) (Fig. 14.25). Thus a 90° phase difference is introduced between the RF and the LO signals. This will result in two IF output signals (IF 1 and IF2) of equal amplitude but with a phase difference of ±90°. The sign of this phase difference will depend on which frequency is higher, the LO or the RF signal. If the LO is constant, the QIF mixer can distinguish between a signal higher or lower than the LO frequency . Thus the QIF mixer can be used in Doppler systems and in network analyzers as a vector voltmeter. Knowing that the QIF mixer can distinguish between the upper and the lower sidebands, we may combine the two IF outputs in a 90° hybrid, directing the lower sideband into one of the output ports of the IF hybrid, and the upper sideband into the other (Fig. 14.26).

14.6.7 Sobharmonically Pumped Two-Diode Mixers Figure 14.27 shows how two diodes coupled in an antiparallel configuration can be used for frequency conversion. Since the bias voltage is zero volts, the LO voltage will swing over the iv characteristic so as to produce a modulated small-signal conductance g(t) with a modulation rate which is twice the LO frequency. Hence frequency conversion

14.6

SINGLE- AND MULTIPLE-DIODE MIXERS

963

Balanced mixer

t - - - - -- -----4- -l"I IF1 go0 Hybrid

LO

Signal

"---' II

1-----C...........

Load

i - - - - - - - - - - -- ~ IF2 Balanced mixer

Figure 14.25

Quadrature IF mixer outline.

Balanced mixer

goo Hybrid

Signal

LO

-----U-,,,Nv--11 ,

goo

1---.

USB

Hybrid

1------n

LSB

IF

Load Balanced mixer

Figure 14.26

Single-sideband mixer using two balanced mixers and two 90° hybrids.

(a) LO filter

LO

Signal filter

Signal

IF filter

IF (b)

i

(c) g (t )

Figure 14.27 Subbarmonically pumped mixer using an antiparallel diode pair: (a) mixer circuit; (b) de iv characteristic; (c) time dependence of the local oscillator voltage and the differential conductance.

964

MIXERS AND DETECTORS

will occur only for frequencies close to twice the LO frequency, and no fundamental mixing at all will occur near the LO frequency. The concept of using two antiparallel diodes for mixing is of practical importance particularly for millimeter wave mixers, when it is difficult to realize enough LO power near the signal frequency. However, there are also several advantages compared to the one-diode harmonic mixer. One is that fundamental mixing is avoided in a two-diode mixer, while in a one-diode harmonic mixer, fundamental mixing will take place unless there are reactive terminations for these mixing products. This is obviously difficult to arrange unless a considerable LO power loss (mismatch) is accepted, since the fundamental mixing frequency is usually close to the LO frequency. Also, noise from the LO circuit will be converted to the IF in the one-diode mixer. In summary, the advantages of the subharmonically pumped mixer with antiparallel diodes are: • Reduced conversion loss due to suppressed fundamental mixing • Lower noise through suppression of LO noise • Suppression of direct video detection • Inherent self-protection against large peak inverse voltage burnout It is possible to use the planar doped barrier diode, which has a symmetrical iv characteristic described briefly in Section 14.2.2) instead of two diodes in antiparallel. In Section 14.8.8, some practical circuit solutions are described.

14.7 INTERMODULATION IN MICROWAVE MIXERS 14.7.1

Introduction

In many applications, the intermodulation properties are important. The receiver designer should know in advance what spurious responses can occur and what can be done to master intermodulation problems. Intermodulation is discussed briefly here. For further information the reader may consult Refs. 38 and 4. 14.7.2 One-Signal Intermodulation In a fundamental mixer (/IF = I fw - Is I), when the signal power becomes of the same order of magnitude as the LO, undesired intermodulation (IM) products lmfs ± nfwl = /{p (where m and n are integers) will become significant. For single-ended mixers, this type of intermodulation response has been analyzed with the assumption that it uses a purely resistive exponential diode [48] (see Fig. 14.28). The theoretical intermodulation output power is, in this approximate model, proportional to the power m of the signal power (Pt). 14.7.3

Intermodulation with More Than One Signal Present

When more than one signal is mixed with the LO, IM products at the following frequencies will be produced: (14.72)

14.7

0 - 10 ~

m

I

I I

---

CNTERMODULATION IN MICROWAVE MIXERS

I

Theoretical Experimental

-20

0

C

....Q) ~ 0

a.

-

-30

,I

:::,

-50

.:

-60

a.

:::, 0 C 0

.. /4 is a natural way to olve this problem [11 3). An elegant way of introducing the stub required is shown in Fig. 14.50. Exce1lent results have been obtained using thi approach. A conver ion gain (lo s) of about O dB and mixer receiver noise temperatures below 50 K have been demonstrated at 100 GHz.

14.12 FET MIXERS 14.12.1

Introduction

The FET mixer is an active device that combines frequency down-conversion and amplification in one device: the field-effect transistor CFEn. This is its principal advantage over the Schottky mixer. The interest in FET mixers has been relatively low, mainly because most experimental work found that the noise of the mixer is inferior to the noise of the Schottky diode mixer-FET amplifier combination. However,

988

MIXERS AND DETECTORS (a)

Si substrate Pb (Bi) counter electrode

Full height WR-28 WG

I

I

I I I

Open-circuited stub Pb (In Au) base electrode

/

''

SiO

'\ \

\ \

Junction

'

\ \

\

'

/

/

I

I

I

I I

(b)

4250A 3000A - - 2000A Base electrode

Figure 14.50 (a) Open-circuited stub slightly longer than >../ 4 u ed for creating a resonating inductance in parallel with the junction capacitance (from Ref. 113: copyright 1985 IEEE, reproduced by permission); (b) fabricational details. this may be due to the lack of theoretical knowledge concerning how to optimize the

embedding circujt. In this section we de cribe djfferent FET mixer circuit of practical intere t and give an introduction to the analy is of mixer properties. Single- and dual-gate devices can be used in Illlxer applications. It is po sible to apply the LO either between the gate and the source terminals (gate mixer), or between the drain and the ource terminal (drain mixer). Also, balanced and double balanced mixer configurations are possible. For further information, see Refs. 1 and 114. 14.12.2

Mixing in a FET

Several of the circuit element of a FET are bia dependent. Hence, when a low-level signal is applied to a FET pumped with a trong LO signal, the modulated circuit elements will cau e signal power to be converted to other frequencies. Figure 14.51 shows an equivalent circuit of a FET, indicating which circuit parameter are voltage dependent. Most important for the mixing i the pumped transconductance gm (t) . The other bia -dependent circuit parameters play a minor role in the mixing process, but cannot be ignored if an accurate analysis is required. The drain resistance Rdr should always be included, while the gate-source capacitance Csg and

14.12 Gate

Lg

Cdg(Vg, Vd)

Rg

C5 g(V9 , Vd)

+ _ V9 (t)

Rd,

FET MIXERS

989

Drain

Ld

+ Vd(t) Cds

R1

Z9 (nwt_0 )

ld( V9 , Vd)

Zd(flwt_o)

w=- Source

Figure 14.51

Somewhat simplified large-signal equivalent circuit of an FET mixer, indicating the dominating nonlinear circuit elements. R8 , Rs. and Rd, are the parasitic resistances, and L 8 • Ls, and Ld are the parasitic inductances in the gate, source, and drain leads, respectively. Zg(nww) and ZJ(nwt.o) are the embedding impedances at the gate and the drain terminals, respectively. The LO could be applied to the drain terminals (drain mixer) instead of the gate (gate mixer).

the charging resistance R, are of less importance. The equivalent transconductance of an FET for different bias conditions is shown in Fig. 14.52.

14.12.3 Single-Gate FET Mixers More work has been carried out on gate mixers than on drain mixers, although the latter may have some advantages [115]. In this section we discuss some features and design rules for the single-gate FET "gate mixer". Large-Signal Analysis. For a more accurate analysis of the FET mixer, as for the diode mixer, first a large-signal analysis has to be performed to determine the Fourier components of all the voltage-dependent circuit parameters. As for the diode mixer, the behavior of the device itself may be described by a few nonlinear differential equations, which can be integrated numerically, while the response of the embedding circuit may be described in the frequency domain. The two sets of equations can be solved in a way analogous to the diode mixer. However, since the device is active, the large-signal algorithm may not converge if the circuit is unstable. Such a case is, of course., of no practical interest. However, one must keep in mind that failures of the large-signal algorithm to converge may also be due to numerical instabilities in the computer algorithm. The three basic differential equations for analyzing the FET mixer are identified in Fig. 14.53. By analyzing three current loops, one obtains

-2Vii -2V;;

+ R1 (11 + l g1) + V1 + R 2(I1 -

+ R1 (I1 + 18 1) + V8 + I8 1R; + R

(J8 1

+ lJ1) + L

11) + Rs(ld1

+ 18 1) + L

3

-2V0 ;

+ R2(IJ1 -

IJ1 ) + 2Vo; d(lg1

+ IJ1)

dt

d(/8 1 + IJ1)

dt

=0

(14.78)

=0

(14.79)

=0

(14.80)

990

MIXERS AND DETECTORS

---- - 9m - - Cgs

0

Figure 14.52 Typical behavior of the bias-dependent circuit parameters, 8m(Vgs) , Cgs(Vgs, Vds ), and C 8d ( V8 u Vds) .

,, + + - Vg

+

v,

+

+

Vd

3

R,

Rs L

Figure 14.53 Simplified FET equivalent circuit illustrating the voltage and current loops used in the time-domain analysis.

In these equations, relations exist between / 8 1, and V8 , between VI and / 1 , and between l dt and V8 and Vd . The embedding source and load networks are treated in the frequency plane (see Fig. 14.51), that is,

where

n

Vgn -

Z gnlgn -

Vdn -

Zdnldn -

=0 Vdsn = 0

Vgsn

= 0, 1, 2, .. .) (n = 0, 1, 2 ... .)

(n

= nth Fourier component of the different time-dependent currents and voltages

(14.81) (14.82)

14. 12

Vgsn and Vdsn

= gate-source and the drain-

FET MIXERS

991

ource Fourier component ,

re pectively Vgn and Vdn = ource oltage at the gate and drain terminal , re pectively That i for a gate mixer (Fig. 14.51), Vgo = Vgb , Vdo = Vdb, and Vg1 = VLo, while all the other components are zero. For further detail , ee Ref . 1, 27, and 114. Small-Signal A nalysis. Knowing the time dependence of the variou circuit elements from the large- ignal nonlinear analy i , the mall- ignal behavior can be calculated in a way that is analogou to how it was made in the diode case, described in Section 14.5.3. The mall- ignal linear and time- arying equivalent circuit i hown in Fig. 14.54. Details concerning the mathematical approach of the analy is can be found in Ref. l . The analy i allow for calculation of the conver ion gain and the input and output impedance . It is illuminating to make a implified analysi : By a urning that when the signal and the LO are applied to the gate- ource terminal , only wi..o, W s WrF = Ws - wi.o, and w; = ~ o - Ws ha e to be accounted for ( ee Fig. 14.55), and gm(t) is the dominating

Cdg(t)

Rg

Rd Drain

Gate V9 (t)

l

R,

: Csg(t)

Rd(t) gm(t) V9 (t)

eds

Source

Figure 14.54 Small signal linear time-varying equivalent circuit. R8 , Rs, and Rd, are the parasitic resistances in the gate. source, and drain leads, respectively, and R; is the charging resistance for the ource-gate capacitance C sg.

1':

.Flgu.re 14.SS Simplified schematic of FET mixer, including only signal, image, and c~uits (Rd, = Rd of Fig. 14.54). (From Ref. 116; copyright 1976 IEEE, reproduced by pemuss1on.)

992

MIXERS AND DETECTORS

nonlinear circuit element (the time dependence of Cgs and Rt1s is neglected). It follows that the available mixer conversion gain imply is [116]

Ge =

gf

Rd

- 24w2C Rin

(14.83)

1

where g 1 = C = Rd = Rin =

Fourier component at frequency £'VLo of the transconductance time-averaged value of the source-gate capacitance time-averaged value of the drain resistance (Rds) input resistance [Rin = R8 + R; + Rs, where R8 is the gate (parasitic) resistance, R; the charging resistance, and Rs the source (parasitic) resistance]

From Eq. (14.83) it is obvious that the conversion gain can be larger than 1. It is an interesting fact that formula (14.83) is identica] with the expression for the FET amplifier gain, if g 1, C , and Rd are replaced by the corresponding values for the amplifier. In fact, the mixer conversion gain can exceed the amplifier gain. Noise Analysis. Concerning the FET mixer noise, very little has been published until now (117]. Thermal noise emanates from the gate, source, and drain leads, related to R8 , Rs, and Rd. Noise is also caused by fluctuations in the drain current and induced-gate current noise. The latter two contributions are partially correlated [117] . No detailed information exists yet on how to minimize the FET mixer noise. Experimental results yield noise figures at X-band of about 4 dB . Further work using the more elaborate theoretical analysis outlined briefly above, and proper experimental implementation, may lead to significantly improved FET mixer performance. Some Design Rules. The optimum performance of an FET mixer i a compromise between minimum noise figure and conver ion gain. The IF amplifier noi e contributes to the total receiver noise in the same way as for the diode mixer. It seems to be advantageous to make the transconductance dominant in the mixing process and the influence from the other nonlinear impedances negligible. Essentially, this can be achieved by keeping the FET in the current-saturated region throughout the LO cycle. It also seems advantageous to short-circuit the gate to the source at all mixing products other than the LO and the signal frequency [l]. The output impedance of a FET mixer is normally quite high. on the order of 1 kQ, which may cau e ome problems. Another problem with the single-gate mixer is that the LO i amplified and will leak into the IF output port, which implies that the IF circuit mu t include a lowpass filter to prevent the LO from causing problems in the IF circuit. The chematic of a FET gate mixer design is shown in Fig. 14.56.

14.12.4 Drain Mixers As mentioned above, it is also possible to construct a FET mixer such that the LO is applied between the drain and the source terminals. The principal nonlinearities are again the transconductance and the drain re istance. Thus the voltage amplification factorµ,= gmRd becomes a time-varying function. Very few papers have been published describing drain mixers. Experimental evidence indicates that the performance

14.12

F ET MIXERS

993

Drain bias

u

Input

Vas = 3.9V Vos= 4.1 V 105 = 3.7 mA L.C. = +7dBm

6GHz

Figure 14.56 permi ion.)

Input matching circuit

~ C\I

~

Output matching circuit

Gate mixer de ign. (From Ref. 115; copyright 1976 IEEE, reproduced by

Input matching circuit

Drain bias Output 4GHz

Input 6 GHz - - - -

D.C.

D.C.

block

block

Vas= 3.0V V0 s=0.6V

10 s = 1.2 mA LC.= +8 d8m

Figure 14.S7 permission.)

Gate bias

D.C. block

Local oscillator 2 GHz

Drain mixer design. (From Ref. 115; copyright 1976 IEEE, reproduced by

may at least equal the performance of the gate mixer [115]. A simple analysis yields the following formula for the conversion gain [11 8]: (14.84) where µ, 1 is the Fourier component of the voltage amplification factor with the frequency equal to WW, and Rdo is the time average of the drain resistance Rd. Figure 14.57 shows a schematic of a drain mixer design.

14.12.S Dual-Gate Mixers The dual-gate FET has two gate electrodes in parallel, between the source and the drain electrodes. This allows for separate circuits for the LO and the signal, yielding a very

994

MIXERS AND DETECTORS

good LO/signal separation. This is the primary advantage of this mixer. The dual-gate FET can be modeled as two single-gate FETs in series, as shown in Fig. 14.58. Notice, however, that the connection point between the drain of the lower FET and the source of the upper one is not physically accessible. Also note that the operating voltages of the two individual FETs are not applied directly to the dual-gate FET terminals. As for the single-gate mixer, the best operation is obtained when the mixer is operated as a transconductance mixer (i.e. , the unwanted LO and mixing products should be short-circuited at the gate and drain terminals). Since the LO for this mixer is amplified, it is possible to realize a self-oscillating mixer (i.e. , no external LO source is necessary) [119].

14.12.6 Balanced and Double Balanced FET Mixers As for diode mixers, balanced and double balanced mixer configurations can be considered. Either a 90° or 180° hybrid may be used. Essentially, the properties are the same as for the diode mixers, as discussed in Section 14.6. Hence, improved spurious response properties, LO noise rejection, and LO/signal isolation are obtained. As indicated in Fig. 14.59, both the 90° and the 180° hybrid mixer require a 180° output hybrid for combining the Ifs of the two .individual mixers.

Drain

Gate2 o-r---. Source

Two FETs in series is equivalent to a dual-gate PET.

Figure 14.58

(a)

RF/LO RF LO

LO

IF

mixer

I

6

180° 6

180°

Hybrid

(b)

AF

FET

Hybrid

1:

FET

AF/LO

mixer

IF

RF/LO

FET

IF

-vv,.r

mixer

6

90°

180°

Hybrid

Hybrid

I

FET AF/LO

mixer

IF

-vv\,--

~igure 14.59 Balanced PET mixers. (From Stephan A. Maas, Microwave Mixers, p. 306; copynght 1986 Artech House, Inc., reproduced by permission.)

14.13

FURTHER ASPECTS ON MICROWAVE AND MILLIMETER WAVE MIXERS IF filter

RF

goo

0

H brid

IF filter

995

IF USB output LSB output

Power divider

LO

Figure 14.60 Dual-gate-FET image-rejection mixer (From Stephan A. Maas, Microwave Mixers, p. 308; copyright 1986 Artech House, Inc., reproduced by permission.)

In balanced configuration , ingle-gate FETs may be more straightforward to use than dual-gate FETs since the latter requires separate hybrids for the LO and the signal. However, as di cus ed in Section 14.12.5, the dual-gate FET (single-device) mixer has excellent inherent LO/ ignal isolation. The choice among the available configurations will be a matter of performance required. Image rejection mixers and other devices may be realized using a pair of single-gate balanced mixer . However, using two dual-gate FET mixers in a balanced configuration (Fig. 14.60) yield a much simpler circuit with only two hybrids and one power divider. Double-double balanced mixers should also be constructed using dual-gate mixer , to keep the complication of the circuit within reasonable limits. Since the nonlinearities of the FET devices are relatively weak compared to diodes, the balance will usually become very good, resulting in excellent spurious signal rejection and port-to-port isolation.

14.13 FURTHER ASPECTS ON MICROWAVE AND MILLIMETER WAVE MIXERS 14.13.1

Introduction

In this section we discuss mixers using types of nonlinear devices not discussed above, and mixers made using monolithic techniques on GaAs substrates.

14.13.2

Tonel Diode Mixers

Considerable effort in designing tunnel diode frequency converters was put forth in the 1960s [120]. However, the tunnel diode mixer never turned out to be of great practical use, mainly because the noise properties were inferior to the Schottky diode mixers. It is, however, wise to learn the basics about th.is mixer, and keep in mind that a diode which exhibits a negative resistance, like the tunnel diode, has certain very interesting properties, such as available conversion gain. The tunnel diode is discussed briefly in Section 14.2.2. In Fig. 14.61 is shown a small-signal equivalent circuit of the tunnel diode. From this circuit it is evident that above a certain frequency /max, the device will not show

996

MIXERS AND DETECTORS Contact + resistance in semiconductor

Rs Junction neg. diff. resistance

-R Junction capacitance

Figure 14.61

Small-signal impedance of the tunnel diode.

i g (t)

A

o-"'o B

I\

I

I

I

I I I I

O

1\0

I II

I II I I I I II II I

C

go

I

-

-

t

V

g(t)

I

-+------t

I

I

9o

I

g(t)

I

I

-

t V

I C

Figure 14.62 Various bias points for the tunnel diode mixer. [From Ref. 120; copyright 1961 IRE (now IEEE), reproduced by permission.]

any negative resistance. For

/max

we get

1 ~m 1 /max=---- i- n1 2rr 2Rmin Ci

Rs

(14.85)

where Rmin is the smallest value of the differential negative resistance. The iv characteristic of a tunnel diode and the differential conductance versus bias voltage is shown in Figs. 14.2c and 14.62, respectively (120]. In Fig. 14.62 is also shown the time-dependent differential conductances for four bias points, A to D. To prevent oscillations regardless of terminations, g 0 must be positive, a condition that is fulfilled for case A. However, if g0 is negative, LO power can be produced by means of self-oscillations, an arrangement that may cause problems due to instabilities. For the bias point A, an external LO i required, and the operational principle is similar to that of the ordinary mixer, as discussed in Section 14.4. The main differences

14. 13

FURTHER ASPECTS O

MICROWAVE AND MCLLIMETER WAVE MIXERS

997

are that the conversion gain of the mixer can be made larger than unity, a feature wb..ich result from the fact that contrary to the ordinary diode mixer, it i po sible to get 18 1I > 80 [compare Eq. (14.35)]. Bias point C i u eful for econd-harmonic mixer application . The noi e of the tunnel diode mixer can be hown to be approximately equal to the noi e of the tunnel diode amplifier [120]. Typical noi e figure for Ge tunnel diode amplifier are 5 dB at 6 GHz and 6 dB at 14 GHz [3], and imilar number hould be valid for mixer [ 121 ] .

14.13.3

Monolithic Mixers

Microwave monolithic integrated circuits (MMIC) fabricated in GaAs have become rather mature, and everaJ design have been reported concerning both various types of diode mixers and FET mixers. The advantage of monolithic circuits is, in the first place, that they are mall in particular when de igned for millimeter waves, and second, when fabricated in large amounts, they will become quite cheap. Monolithic diode mixers have been succe sfully designed for frequencies up to 100 GHz. Today (1986), at ordinary microwave frequencies it is possible to buy balanced and double balanced MMIC mixer up to at least 15 GHz. Some GaAs foundries offer balanced and double balanced mixers as macrocells from a cell library, to be included in de ign of larger circuits consisting of, for example, low-noise input amplifier tages, followed by a double balanced mixer and further amplification at the IF [122] . Due to the diode parasitics, the cutoff frequency of the monolithic diodes cannot easily be made as b..igh as for the mixers using whiskered honeycomb diode chips. Special design feature such as air bridges or proton bombardment for making the GaAs semi-insulating will isolate the diodes from the circuit parasitics and improve the cutoff frequency. Typical results show that it is possible to fabricate monolithic mi11imeter wave balanced mixers with diodes having a cutoff frequency of about 640 GHz, and a single- ideband conversion loss of about 6 dB at Ka band (123]. Two Ka band designs are shown in Figs. 14.63 and 14.64. In Fig. 14.64, notice that the mixer is integrated with a "bow-tie' dipole antenna. MMIC FET mixers have also been constructed, and as for diode mixers, it is now possible to include such mixers in larger MMICs. A high-volume, low-cost 3- to 6-GHz receiver front-end MMIC on one chip, with three stages of low-noise amplification in

Mixer

cflOde

pa,r IF

Wafer-type

,,

,;

----- ---

waveguide

Ohmic metallizations

mount EPI layers ~ ..... SemHnsolating substrate

~~~="-~.a.,,.;;.;.-+-'

Proton bombarded area

Figure 14.63 Millimeter wave Ka-band GaAs monolithic mixer chip for a crossbar-type waveguide mixer. (From Ref. 124; copyright 1983 IEEE, reproduced by permission.)

998

MIXERS AND DETECTORS

(a)

DC bias line

I I I I

I

"Bowtie" : dipole antenna , ~-.--..--..----,,-,coplanar IinJ 1 coplanar slotline junctfon Coplanar slotline signal input

I1 : , I 1I

Bias filter

- -~

~~

Ground plane present up to broken line

Microstrip-tocoplanar-line transition

~~~~~~~~~~FTii~~=:::~ LOhighpass --LO input Schottky barrier or Mott diodes

filter

IF filter Bias filter

1435/4 !

t

- DC bias line

IF output

(b)

Figure 14.64 A 30 GHz GaAs monolithic single balanced mixer incorporated with a bowtie antenna: (a) circuit; (b) SEM micrograph of one Mott diode. (From Ref. 123; publi hed previously by the Institution of Electrical Engineer in Handbook of Microwave and Optical Components.)

front of a double balanced FET mixer followed by three stages of IF amplification, has been described [ 125].

14.13.4 Other Types of Mixers In this chapter we have described most of the common mixers used in various, mainly low-noise applications. In this section we mention only some less commonly used mixers, both low-noise and "high-noise" types.

REFERENCES

999

Hot electron bolometer mixer are very low noise mixer that are u ed for ubmillimeter application . The mo t commonly u ed detector of this type i the indiumantimonide (InSb) detector, developed initially a a microwave and far-infrared detector. To work properly, the detector ha to be cooled to a few kelvin . The detection i achieved when the conduction band electron are heated by the ab orbed radiation, causing a change in their mobility. The fact that the electron have to transfer their exce energy to the lattice through relatively weak coupling mechanism mean that the detector i relatively low. Hence the maximum IF frequency i only of the order of a few megahertz. The InSb detector can be used with or without a magnetic field . Typically, doubleideband receiver noi e temperature obtained without a magnetic field at 1.6 K are about 180 K at 350 GHz, 350 Kat 492 GHz, and 650 K at 625 GHz (126, 127]. With a magnetic field of about 6 kG, the detector will operate in a cyclotron resonance mode, and impro ed performance i possible. When operated at 1.6 K, the be t receiver noise temperature for the cyclotron resonance detector i 250 K at 495 GHz, 350 K at 625 GHz, and 510 Kat 812 GHz [127]. Self-mixing o cillators have application in low-co t front ends such as doppler radar or burglar alarm . Self-mixing Gunn o cillators have consequently been developed for low-co t application where en itivity is of minor importance. The Gunn diode will erve both a a local o cillator, and because nonlinearities are always pre ent, a a mixing element. A 60-GHz system based on an InP Gunn diode self-mixing oscillator is able to detect ignal levels of -80 dBm (using a 100-MHz bandwidth), a result that i consistent with information reported for lower frequencies [128]. Self-mixing BARITT o cillators: The BARITI diode is a transit-time device and a relative to the IMPAIT diode [3]. However, since the mechanism responsible for the microwave oscillation is thermionic emi sion and diffusion, it will have a much better signal-to-noi e performance than that of the IMPAIT diode. The BARITI diode has also been operated in a self-mixing mode, and experiments at Ka-band indicate the ensitivity to be similar to that of the Gunn diode self-mixing o cillator [l 29].

REFERENCES 1. S. A. Maas, Microwave Mixers. Artech House, Dedham, MA, 1986.

2. M. J. Howes and D. V. Morgan, Eds., Variable Impedance Devices, Wiley, New York, 1978. 3. S. M . Sze, Physics of Semiconductor Devices, 2nd ed., Wiley, New York, 1981. 4. E . L. Kollberg, Ed., Microwave and Millimeter-Wave Mixers, IEEE Press, New York, 1984. 5. R. J. Malilc, "A Subharmonic Mixer Using a Planar Doped Barrier Diode with Symmetric Conductance," IEEE Electron Device Lett., EDL-3, pp. 205 -207, 1982. 6. M. V. Schneider, ''Metal-Semiconductor Junction as Frequency Converters," in K. J. Button, Ed., Infrared and Millimeter Waves, 6, Systems and Components, Academic Press, New York, 1982. 7. R. A. Linke, M . V. Schneider, and A. Y. Cho, "Cryogenic Millimeter-Wave Receiver Using Molecular Beam Epitaxy Diodes," IEEE Trans. Microwave Theory Tech., MTT26(12), pp. 935-938, 1978. 8. G. K. Sherrill, R. J. Mattauch, and T. W. Crowe, "Interfacial Stress and Excess Noise in Schottky-Barrier Mixer Diodes," IEEE Trans. Microwave Theory Tech., MTT-34(3), pp. 342-345, 1986.

1000

MlXERS A D DETECTORS

9. E. L. Kollberg, H. Zirath, and A. Jelenski, •'Temperature-Variable Characteristics and Noi e in Metal-Semiconductor Junction ," IEEE Trans. Microwave Theory Tech., MTT34(9), pp. 913-922, 1986. I 0. H. Zirath, "High-Frequency Noise and Current-Voltage Characteristics of MM-Wave Platinum 11-11 • GaAs Schottky Barrier Diodes," J. Appl. Phys., 60, pp. 1399-1408, August 15, 1986. 11. P. J. Price, "Fluctuations of Hot Electrons," in R. E. Burge s, Ed., Fluctuation Phenomena in Solids, Academic Press, New York, 1965, Chap. 8. 12. W. Baechtold. "Noise Behavior of GaAs Field-Effect Transistors with Short Gate Lengths," IEEE Trans. Electron Devices, ED-19(5), pp. 674-680, 1972. 13. A. Je len ki. E. Kollberg, and H. Z irath, "B roadband Noise Mechanisms and Noise Meaurements of Metal-Semiconductor Junctions," IEEE Trans. Microwave Theory Tech, MTT-34, pp. 1193- 1201 , 1986. 14. T. S. Laverghena, Microwaves Materials and Fabrication Techniques, Artech House, Dedham, MA, 1984. 15. A. E. Fantom, ''Millimeter Wave Power Standard,'' IEE Colloquium on Measurement at Millimetre and Near Millimetre Wavelengths , Institution of Electrical Engineers, Stevenage, Hertford hire, England, 1981. 16. F. L. Warner, "Detection of Millimetre and Suhmillimetre Waves," in F . A . Benson, Ed., Millimetre and Submillimetre Waves, Iliffe B ooks, London, 1969. 17. T. G. Blaney, "Detection Techniques at Short Millimeter and Submillimeter Wavelengths: An Overview," in K. J. Button , Ed., Infrared and Millimeter Waves, Vol. 3, Submillimeter Techniques, Academic Press, New York, 1980. 18. A. Hadni, "Pyroelectricity and Pyroelectric Detectors .', in K. J. Button , Ed., Infrared and Millimeter Waves, Vol. 3, Submillimeter Techniques, Academic Pre , ew York, 1980. 19. P. L. Richards and L. T. Greenberg, "Infrared Detectors for Low-Background Astronomy: Incoherent and Coherent Device from One Micrometer to One Millimeter,'' in K. J. Button, Ed., Infrared and Millimeter Waves, Vol. 6, Systems and Components, Academic Press, New York, 1982. 20. A. A. M . Saleh, Theory of Resistive Mixers, MIT Pre s, Cambridge, MA, 1971. 21. J.-D. Buchs and G. Begemann, "Frequency Conver ion U ing Harmonic Mixers with Resistive Diodes," IEE J. Microwave Opt. Acoust., 2(1 ), -pp. 71-76, 1978. 22. D . A. Fleri and L. D. Cohen. "Nonlinear Analy is of the Schottky-Barrier Mixer Diode." IEEE Trans. Microwave Theory Tech., MTT-21(1), pp. 39- 43, 1973. 23. S. Egarru, "Nonlinear, Linear Analysis and Computer-Aided Design of Re i tive Mixer ," IEEE Trans. Microwave Theory Tech., MTT-22(3), pp. 270- 275, 1974. 24. W. K. Gwarek, •'Nonlinear Analy i of Microwave Mixer ,'' M.S. thesi , Mas acbu etts Institute of Technology, Cambridge, 1974. 25. R. G . Hicks and P. J. Kahn, "Numerical Analysi of Nonlinear Solid-State Device Excitation in Microwave Circuit ," IEEE Trans. Microwave Theory Tech. , MTT-30(3), pp. 251 - 259, 1982. 26. A. R. Kerr, "A Technique for Determining the Local O cillator Waveforms in a Microwave Mixer," IEEE Trans. Microwave Theory Tech. . MTT-23(10). pp. 828-83 1, 1975. 27. C. Camacho-Penalosa, "Numerical Steady-State Analysis of Nonlinear Microwave Circuits with Periodic Excitation," IEEE Trans. Microwave Theory Tech., MTT-31(9), pp. 724- 730, 1983. 28. D. N . Held and A. R. Kerr, ''Conver ion Loss and Noise of Microwave and MillimeterWave Mixers. Part 1. Theory," IEEE Trans. Microwave Theory Tech., MTT-26(2), pp. 49-55. 1978.

REFERENCES

J001

29. D. . Held and A . R. Kerr. "Con ersion Lo and Noi e of Microwave and Mi llimeterWave Mixer . Part 2. Experiment," IEEE Tra11 . Micrm,·ave Theory Tech.. MTT-26(2). pp. 55-61. 1978. 30. P. H. Siegel and A. R. Kerr. A User-Oriem ed Compwer Program for the Analysis of Microwave Mixers. and a Study of the Effects of the Series Inductance and Diode Capacitance on the Perfomwnce of Some Simple Mi.xers, NA A Techn ical Memorandum. e, York. . Y. 1979. Goddard In titute for pace Studie 31. A. R. Kerr. · oi e and Lo in Balanced and Subharmonicall y Pumped Mixers. Part l. Theory," IEEE Trans. ~1icrowm •e Theory Tech.. l.\1TT-27(12), pp. 938 - 943, 1979. 32. T. W . Crowe and R. J. Mattauch, "Analy i, and Optimization of Millimeter and Submillimeter-vVa elength Mi er Diode ," IEEE Trans. Microwave Theory Tech. , MTT35(2). pp. 159- 16 . 19 7. 33. A. R. Kerr, "Shot- oise in Re i tive-Diode M ixer and the Attenuator Noise Model," IEEE Trans. Microwave Theory Tech., MTT-27(2). pp. 135- 140, 1979.

34. M. T. Faber and W . K . Gwarek, " onlinear-Linear Analy is of Microwave Mixer wi th Any umber of Diode ," IEEE Tran s. Microwave Theory Tech.. MTT-28( 11 ). pp. 1174- 1181, 1980. 35. R. G. Hicks and P. J. Kahn, ·• umerical Analysis of Subharmonic Mixers Using Accurate and Approximate Model ," IEEE Trans. Microwave Theory Tech., MTT-30( l l ), pp. 2113-2119, 1982.

36. M. E. Hines, "Inherent Signal Lo es in Resistive-Diode Mixers," IEEE Trans. Microwave Theory Tech., MTT-29(4), pp. 281-292, 1981. 37. B. Henderson., "Mixer De ign Con ideration Improve Performance," MSN Commun. Tech110/., 11(10). pp. 103- 118, 19 I. 38. C. W . Gerst, · ew Mixer De igns Boost D/F Performance," Microwaves, 12( 10), pp. 60-69. 1973. 39. J. F. Reynolds and M. R. Ro enzweig, "Learn the Language of Mixer Specification," Microwaves, 17(6). pp. 72-80, 1978. 40. R. B . Mouw, "A Broad-Band Hybrid Junction and Application to the Star Modulator," IEEE Trans. Microwave Theory Tech., MTI-16(11 ), pp. 911 - 91 8, 1968. 4 1. G. L. Matthai, L. Young, and E. M. T . Jone , Microwave Filters, Impedance Matching Networks, and Coupling Strucw res, Artecb House, Dedham, MA, 1970. 42. S. Rehnmark, "High Directivity CTL-Coupler and a New Technique for the Measurement of CTL-Coupler Parameters," IEEE Trans. Microwave Theory Tech.• MTT25, pp. 1116- 1121, 1977. 43. Wen Pin Ou, " De ign Equation for an Interdigital Directional Coupler," IEEE Trans. Microwave Theory Tech. , MTT-23, pp. 253- 255, 1975. 44. V. K Tripathi, H. B. Lunden, and J. P. Star ki, ''Analy i and Design of BranchLine Hybrids with Coupled Lines," IEEE Trans, Microwa ve 171eory Tech., MTT-32, pp. 427 - 432. 1984. 45. N. Marcuwitz, Waveguide Handbook, Dover, New York, 1951 .

46. B. R. Hallfor~ "A Designer's Guide to Planar Mixer Baluns." Microwaves, 18(12), pp. 52-57, 1979. 47. J. H. Clotte, "Exact Design of the Marchand Balun," Proceedings of the 9th European Microwave Conference, Microwave Exhibitions and Publishers, Tumbridge Wells, Kent. England, 1979, pp. 480- 484. 48. L . M . Orloff, "Intermodulation Analysis of Crystal Mixer," Proc. I EE£, 52(2), pp. 173-179, 1964.

1002

MIXERS AND DETECTORS

49. S. A . Maas, "Two-Tune Intermodulation in Diode Mixers," IEEE Trans. Microwave Theory Tech. , MTT-35, pp. 307-3 14, 1987. 50. J. G. Gardiner, "An Intermodulation Phenomenon in the Ring Modulator," Radio Electron. Eng., 39(4), pp. 193- 197, 1970. 51 . M. A. Maiuzzo and S. H. Cameron, ''Response Coefficients of a Double-Balanced Diode Mixer," IEEE Trans. Electromagn. Compal., EMC-21(4), pp. 316-319, 1979. 52. M. Katoh and Y. Akaiwa, "4-GHz Integrated-Circuit Mixer," IEEE Trans. Microwave Theory Tech., MTT-19(7), pp. 634-637, 1971. 53. M . Akaike and S . Okamura~ "Semiconductor Diode Mixer for Millimeter Wave Region," Trans. Inst. Electron. Commun. Eng. Jpn., 52-B(I0), pp. 601-609, 1969. 54. C. J. Burkley and R. S. O'Brien, "Optimization of an 11 GHz Mixer Circuit Using Image Recovery," Int. J. Electron., 38(6), pp. 777-787, 1975. 55. L. E. Dickens and D . W. Maki, "An Integrated-Circuit B alanced Mixer, Image and Sum Enhanced," IEEE Trans. Microwave Theory Tech., MTI-23(3), pp. 276-281, 1975. 56. H. Ogawa, M . Aikawa, and K. Morita, " K-Band Integrated Double-Balanced Mixer," IEEE Trans. Microwave Theory Tech., MTT-28(3), pp. 180- 185, 1980. 57. T. H. Oxley, "Phasing Type Image Recovery Mixer," IEEE MTI'-S Int. Microwave Symp. Dig., pp. 270-273, 1980. 58. L. E. Dickens and D. W. Maki, "A New 'Phased-Type' Image Enhanced Mixer," IEEE MTT-S Int. Microwave Symp. Dig., pp. 149- 151, 1975. 59. G. P. Kurpi s and J. J. Taub, "Wide-Band X-Band Microstrip Image Rejection Balanced Mixer," IEEE Trans. Microwave Theory Tech., MTT-8(12), pp. 1181-1182, 1970. 60. R. S. Tahim, G. M. Hayashibara, and K. Chang, ''Design and Performance of W-Band Broad-Band Integrated Circuit Mixer," IEEE Trans. Microwave Theory Tech. , MTT-31, pp. 277-283, 1983. 61. G. B. Stracca, F. Aspesi , and T. D ' Arcangelo, "Low-Noise Microwave Down-Converter with Optimum Matching at Idle Frequencies," IEEE Trans. Microwave Theory Tech. , MTT21(9), pp. 544-547. 1973. 62. L. T . Yuan, ''Design and Performance Analysis of an Octave Bandwidth Waveguide Mixer," IEEE Trans. Microwave Theory Tech., MTT-25(12), pp. 1048-1054, 1977.

63. P. J. Meier, "£-Plane Components for a 94-GHz Printed-Circuit Balanced Mixer," IEEE MTT-S Int. Microwave Symp. Dig., pp. 267-269, 1980. 64. R. N. Bates, "Millimeter-Wave Low Noise £-Plane Balanced Mixers Incorporating Planar MBE GaAs Mixer Diodes," IEEE M1T-S Int. Microwave Symp. Dig., pp. 13- 15, 1982. 65. L. Bui and D . Ball, "Broadband Planar Balanced Mixers for Millimeter Wave Applications," IEEE MTT-S Int. Microwave Symp. Dig., pp. 204-205, 1982. 66. A. Blaisdell, R. Geoffroy, and H. H owe, "A Novel Broadband Double Balanced Mixer for the 18-40 GHz Range," IEEE MIT-S Int. Microwave Symp. Dig., pp. 33-35, 1982. 67. B. Henderson, "Full-Range Orthogonal Circuit Mixers R each 2 to 26 GHz," MSN Commun. Technol., 12(9), pp. 122-126, 1982. 68. R. B . Culbertson and A. M . Pavio, "An Analytic De ign Approach for 2-18 GHz Planar Mixer Circuit," IEEE MIT-S Int. Microwave Symp. Dig., pp. 425 -427, 1982. 69. B. R. Hallford, "Single Sideband Mixers for Communications Systems," IEEE M1T-S Int. Microwave Symp. Dig., pp. 30- 32, 1982. 70. B. R. Hallford, "Investigation of a Single-Sideband Mixer Anomaly," IEEE Trans. Microwave Theory Tech., MIT-31( 12), pp. 1030- 1038, 1983. 71. E. R. Carlson, M. V. Schneider, and T. F. McMaster, "Subharmonically Pumped Millimeter-Wave Mixers," IEEE Trans. Microwave Theory Tech., MTI-26(10), pp. 706-715, 1978.

REFERE CES

l 003

72. M. V. Schneider and W . W . Snell, Jr., ''Hannoni ally Pu mped Stripline Down-Converter," IEEE Trans. Microwave Theory Tech., MTT-23(3). pp. 271-275, 1975. 73. J.-D. Buch and G. Begemann, '·Frequency Conver ion U ing Harmonic Mixer Re istive Diode ," IEE J. Microwave Opt. Acoust .. 2, pp. 71 - 76. 1978.

with

74. Hong-Th Cong, A. R. Kerr, and R. J. Mauauch. "The Low- oi e 115-GHz Receiver on the Columbia GISS 4-ft Raclio Tele cope:' IEEE Trans. Microwave Theory Tech. , MTT-27(3), pp. 245-248. 1979. 75. P. F. Gold mith. ''Quasi-opt:icaJ Technique at Millimeter and Submillimeter Wavelength ," in K. J. Button. Ed., Infrared and Millimeter Waves, Vol. 6, Systems and Components , Academic Pre . ew York. 1982. 76. R. L. £j enhart and P. J. Kahn, ·'ToeoreticaJ and Experimental AnaJysis of a Waveguide Mounting Structure,'' IEEE Trans. Microwave Theory Tech., MTT-19(8). pp. 706-7 19, 1971 . 77. A . G. William on, 'AnaJy is and Modelling of a Single-Po t Waveguide Mounting Structure,' IEE Proc. , Pt. H. 129(5 ), pp. 271 - 277, 1982. 78. E. L. Kallberg and H. Zirath, "A Cryogenic Millimeter-Wave Schottky-Diode Mixer," IEEE Trans. Microwave Theory Tech .. MTT-31(2), pp. 230-235, 1983. 79. C. R. Predmore, A. V . Raisanen, . R. Erickson, P. F. Gold mith, and J. L. R. Marrero, '' A Broad-Band, Ultra-Low-Noi e Schottk')' D iode Mixer Receiver from 80 to 115 GHz," IEEE Trans. Microwave Theory Tech., MTI-32(5). pp. 498- 507, 1984. 80. C. E. Hag ~tram and E. L. Ka llberg. "Measurements of Embedding Impedance of Millimeter-Wave Diode Mounts," IEEE Trans. Microwave Theory Tech., MTT-28(8), pp. 899-904, 1984.

81. M . K. B rewer and A. V. Raisanen, "DuaJ-Harmonic Noncontacting Millimeter Waveguide Backshorts: Theory. Design. and Tes 4 ' IEEE Trans. Microwave Theory Tech. , MTT-30(5), pp. 708- 7 14, 1982. 82. S . Weinreb. "Low- oise Cooled GaAs FET Amplifier ," IEEE Trans. Microwave Theory Tech. , MTT-28( 10), pp. 1041- 1054, 1980. 83. J. Callaway, "Model for Lattice Conductivity at Low Temperatures," Phys. Rev., 113, pp. 1046- 1051, 1959. 84. M. Pospie zalskie and S. We inreb, " FET' s and HEMT's at Cryogenic Temperatures-Their Propertie and U e in Low-Noise Amplifiers," IEEE MTI'-S Int. Microwave Symp. Dig., pp. 955-958, 1987. 85. G. K. White, Experimental Techniques in Low-Temperature Physics, Clarendon Press, Oxford, 1979. 86. A . F. Pearce and D. J. Wootton, "Reflex Klystrons," in F. A . Benson, Ed., Millimeter and Submillimeter Waves, Iliffe Books, Lo ndon, 1969. 87. I. G. Edclison, "Indium Phosphide and Gallium Arsenide Transferred-Electron Devices," in K. J. Button, Ed., Infrared and Millimeter Wa ves, 11, Millimeter Components and Techniques , Part 3, Academic Press, New York, 1984. 88. H. J. Kuno, "IMPATI Devices for Generation of Millimeter Waves," in K. J. Button, Ed., Infrared and Millimeter Wa ves , Vol. 1, Sources of Radiation , Academic Press, New York, 1979. 89. J. W. Archer, "Low-Noise Receiver Technology for Near-MiJlimeter Wavelengths," in K. J. Button, Ed., Infrared and Millimeter Waves , 15, Millimeter Components and Techniques, Part 6, Academic Press, New Yor~ 1986. 90. H. P. Roser, E . J. Durwen, R. Wattenbach, and G. V. Schultz, "Investigation of a Heterodyne Receiver with Open Structure Mixer at 324 and 693 GHz," Int. J. Infrared Millim. Waves, 5(3), pp. 301-314, 1984.

1004

MIXERS AND DETECTORS

91. H. P. Roeser, R. Wattenbach, E. J. Durwen, and G. V. Schultz, "A High Resolution Heterodyne Spectrometer from 100 µm to 1000 µm and the Detection of CO (J = 7 6), CO (J = 6 - 5) and 13 CO (J = 3 - 2)," Astron. Astrophys. , 165(1/2), pp. 287-289, 1986. 92. P. F. Goldsmith and N. R. Erickson, "Waveguide Submillimeter Mixers," in E. Kollberg, Ed., Instrumentation for Submillimeter Spectroscopy, Proc. SPIE 598, pp. 52- 59, 1986. 93. W. M. Kelly and G. T. Wrixon, "Optimization of Schottky-Barrier Diodes for Low-noise, Low-Conversion Loss Operation at Near-Millimeter Wavelengths," in K. J. Button, Ed., Infrared and Millimeter Waves, Vol. 3, Submillimeter Techniques, Academic Press, New York, 1980. 94. C. 0. Weiss and A. Godone, "Harmonic Mjxing and Detection with Schottky Diodes Up to the 5 T Hz Range," IEEE J. Quantum Electron. QE-3(2), pp. 67-99, 1984. 95. W. M. Kelly, M. J. Gans, and J. G. Eivers, "Modelling the Response of Quasi-optical Corner Cube Mixers," in E. Kollberg, Ed., Instrumentation for Submillimeter Spectroscopy, Proc. SPIE 598, pp. 72-78, 1986. 96. E. Sauter, G. V. Schultz, and R. Wohlleben, "Antenna Patterns of an Open Structure Mixer at a Submillimeter Wavelength and of Its Scaled Model," Int. J. Infrared Millim. Waves, 5(4), pp. 451 - 463, 1984. 97. J. J. Gustincic, "A Quasi-optical Receiver Design," IEEE MTT-S Int. Microwave Symp. Dig. Tech. Pap., pp. 99- 102, 1977. 98. J. R. Tucker and M . J. Feldman, "Quantum Detection at Millimeter Wavelengths," Rev. Mod. Phys., 57(4), pp. 1055- 1115, 1985. 99. M. R. Beasly and C. J. Kircher, "Josephson Junction Electronics: Materials Issues and Fabrication Techniques," in S. Fonerand and B. B. Schwartz, Eds., Superconductor Materials Science, Plenum Press, New York, 1981, pp. 605- 684. 100. J. M. Lumley, R. E. Somekh, J. E. Evetts, and J. H. James, "High Qualjty All Refractory Josephson Tunnel Junctions for Sqmd Applications," IEEE Trans. Magn., MAG-21(2), pp. 539-542, 1985. 101. E. J. Cucauskas, M . Nisenoff, H. Kroger, D. W. Jillie, and L. R. Smith, "All Refractory, High Tc Josephson Device Technology," in A. F. Clark and R. P. Reed. Eds., Advances in Cryogenic Engineering, 30, Plenum Press, New York, 1984, pp. 547-558. 102. M. K. Wu. J. R. Ashburn, C. J. Tomg, P. H. Hor, R. L. Meng, L. Gao, Z. J. Huang. Y. Q. Wang, and C. W. Chu, "Superconductivity at 93 K in a New Mixed-phase Y-BaCu-O Compound System at Ambient Pres ure," Phys. Rev. Lett., 58(9), pp. 908- 910, 1987. 103. G. J. Dolan, T. G. Phillips, and D. P. Woody, "Low Noise 115 GHz Mrung in Superconducting Oxide-Barrier Tunnel Junctions," Appl. Phys. Lett., 34(5), pp. 347-349, 1979. 104. A. W. Kleinsasser and R. A. Burman, "High-QuaJjty Submicron Niobium Tunnel Junctions with Reactive-Ion-Beam Oxidation," Appl. Phys. Lett., 37(9), pp. 841 - 843, 1980. 105. M. J. Feldman and S. Rudner, "Mixing with SIS Arrays," in K. J. Button, Ed., Reviews in Infrared And Millimeter Waves, 3, Plenum Press, New York, 1983.

106. W. R McGrath, A. V. Raisanen, and P. L. Richards, "Variable-Temperature Loads for Use in Accurate Noise Measurements of Cryogenically Cooled Microwave Amplifiers and Mixers," Int. J. Infrared Millim. Waves, 7(4), pp. 543- 554, 1986. 107. L. Olsson, S. Rudner, E. Kollberg, and C. 0. Lindstrom, "A Low-Noise SIS Array Receiver for Radio Astronomjcal Applications in the 35- 50 GHz Band," Int. J. Infrared Millim. Waves, 4(6), pp. 847- 858, 1983.

REFERENCES

1005

108. M . J. Wengler, D. P. Woody. R . E. Miller, and T . G. Philli p , ·'A Low Noi e Receiver for Millimeter and Submillimeter Wavelength ," /111. J. Infrared Millim. Waves, 6(8), pp. 697-706. 1985. 109. D. Winkler, W . R. McGrath, B. ils on. T. Clae on, J. Johan on, E. Kollberg, and S. Rudner. ·'A Submillimeter Wave Qua iparticle Receiver for 750 GHz - Progress Report,'' in Kollberg Ed., l11strwnentatio11fo r Submillimeter Spectroscopy, P roc. SPIE 598, pp. 33-38. J9 5. JI0. A. V . Raisanen, D. G. Crete, P. L. Richards, and F. L. Lloyd, "Wide Band, Low Noi e MM -Wave SI Mixers with Single Tuning Element," Int. J. Infrared Mi/Lim. Waves, 7 ( 12), pp. 1 34- 1 52. 1986.

111. P. H . Siegel, D. W . Peter on. and A . R. Kerr, ''De ign and Analy i of the Channel Waveguide Tran former," IEEE Trans. Microwave Theory Tech. , MIT-31(6), pp. 473- 484, 1983. 112. L. R. D'Addario. "An SIS Mixer for 90- 120 GHz with Gain and Wide Bandwidth," Int. J. lnfraredMillim. Waves, 5(11), pp. 1419- 1442, 1984. 113. A. V . Rfilsanen. W. R. McGrath. P. L. Richards, and F. L. Lloyd, "Broad-Band RF Match to a Millimeter-Wave SIS-Quasi parricle Mixer," IEEE Trans. Microwave Theory Tech., MTI-33(12). pp. 1495-1500, 1985.

114. R. S. Pengelly. Microwave Field-Effect Transistors-Theory, Design and Applications, 2nd ed., Research Studies Pre

(Wtley), New Yor~ 1986.

115. P. Bura and R. Dikshit, "FET Mixers for Communication Satellite Transponders," IEEE M1T-S Int. Microwave Symp. Dig., pp. 90-92, 1976. 116. R . A. Pucel, D. Mas e, and R . Bera, "Performance of GaAs MESFET Mixers at X-Band," IEEE Trans. Microwave Theory Tech., MTT-24(6), pp. 351 -360, 1976. 117. G. K. Tie and C . S. Aitchison , ' oise Figure and Associated Conversion Gain of a Microwave MESFET Gate Mixer,n Proceedings of the 13th European Microwave Conference, Microwave Exhibitions and Publishers, Tumbridge Wells, Kent, England, 1983, pp. 579-584.

118. G. Begemann and A . Jacob, "Conversion Gain of MESFET Drain Mixers,'' Electron. Lett., 15(18), pp 567- 568, 1979. 119. C. Tsironis, R . Stahlman, and F. Ponse, " A Self-O cillating Dual Gate MESFET X-band Mixer with 12 dB Conversion Gain, Proceedings of the 9th European Microwave Conferen ce. Microwave Exhibitions and Publishers, Tumbridge Wells, Kent, England, 1979, pp. 321 -325. J 20. C . S . Kirn, "Tunnel Diode Converter Analysis," IRE Trans. Electron Devices, ED-8(9), pp. 394-405, 1961. 121. F. Sterzer and A. Presser, " Stable Low-Noise Tunnel Diode Frequency Converters,'' RCA Rev., 23(1). pp. 3- 28, 1962. 122. D. Lockie, A. Podell, and S. Mogbe, "Cell Libraries Provide Stepping-Stone into Monolithic Integration," MSN Commun. Technol., 16(8), pp. 74-85, 1986. 123. U. K. Mishra. S. C. Palmateer, S. J. Nightingale, M. A. G. Upton, and P. M. Smith, 0

"Surface-Oriented Low-Parasitic Mott Diode for EHF Mixer Applications, Electron. Lett.,

21(15), pp. 652-653, 1985. 124. C . Chao, A. Contulatis, S. A. Jamiso~ and P. E. Baudhahn, " Ka-Band Monolithic GaAs Balanced Mixer," IEEE Trans. Microwave Theory Tech., MTI-31(1), pp. 11 - 15, 1983. 125. A. Podell and W. W. Nelson, "High Volume, Low Cost, MMIC Receiver Front Encl,° IEEE Microwave Millim.-Wave Monolithic Circuits Symp., 1986, pp. 57-59.

1006

MIXERS AND DETECTORS

126. T. G. PhiJJips and K B. Jefferts, "A Low Temperature Bolometer Heterodyne Receiver for MilHmeter Wave Astronomy," Rev. Sci. lnstrum., 44, pp. 1009-1014, 1973. 127. E. R. Brown, J. Keene, and T. G. Phi1lips, "A Heterodyne Receiver for the Submillimeter Wavelength Region Based on Cyclotron Resonance in InSb at Low Temperatures," Int. J. Infrared Millim. Waves, 6( 11), pp. 1121-1138, 1985. 128. S. Dixon and H. Jacobs, "Millimeter-Wave InP Image Line Self-Mixing Gunn Oscillator," IEEE Trans. Microwave Theory Tech., MTT-29(9), pp. 958- 961, 1981. 129. P. N. Forg and J. Freyer, "Ka-Band Self-Oscillating Mixer with Schottky BARITI Diodes," Electron. Lett., 16(22), pp. 827-829, 1980.

15 MULTIPLIERS AND PARAMETRIC DEVICES J. W. ARCHER

AND R. A. B ATCHELOR Division of Radiophysics CS/RO. Sydney, Australia

15.l INTRODUCTIO This chapter deals with the theory and practical design of microwave circuits contairung nonlinear reactance devices. Many of the microwave circuits and components discu ed incorporate a semiconductor device known as a varactor djode. A varactor diode is bat one example of a variable-reactance circuit element. The varying reactance is provided by the diode junction capacitance, which changes nonlinearly as a function of the applied voltage. The capacitance variation can be used to produce a number of different effects in a microwave circuit. Use of the varactor diode to generate harmonic of an applied microwave ignal is one of the most important applications of such devjces in modem microwave technology. Very efficient microwave and millimeter wave frequency multipliers can be built using varactors. At lower microwave frequencies a variant of the varactor diode, the step-recovery diode, has become popular because the conversion efficiencies attainable are substantially higher than those of conventional varactor-based circuits. Until recently, a most important use of varactor devices was in parametric amplifiers and up/down-converters. However, the advent of the high-performance field-effect transistor has now displaced the parametric amplifier in low-noise amplifier applications. Other applications of the varactor are: (a) the switching and modulation of a microwave signal through variation of the reactance by means of an externally applied bias, (b) the electronic tuning of resonant structures, and (c) microwave power limiting. In Section 15.2 we discuss theoretical relationships necessary for an understanding of nonlinear reactance devices. Section 15.3 presents a description of practkal variable Handbook of RF/Microwave Components and Engineering, ISBN 0-471-39056-9 © 2003 John WLley & Sons, Inc.

Edited by Kai Chang

1007

1008

MULTIPLIERS AND PARAMETRJC OEVlCES

reactance devices, with emphasis on the widely used varactor ruode. Methods for evaluating the quality of a varactor diode are summarized. Sections 15.4 and 15.5 deal with the de ign of practical microwave frequency multipliers using varactor ruodes. Design examples are given for a millimeter wave varactor frequency tripler and a 2.7 GHz tep-recovery ruode frequency sextupler. In Section 15.6 we ruscuss parametric amplifiers, up-converters, and down-converters, including a design example of a 3.25 GHz nondegenerate parametric amplifier.

15.2 MANLEY-ROWE RELATIONSHIPS Manley and Rowe [l] have derived a set of general equations that relate power flow into and out of a nonlinear reactance. These equations are very useful for undertanding the behavior of harmonic generators, parametric amplifiers, and frequency up/down-converters. They may be used to preruct the ultimate power gain and conversion efficiency that can be obtained. The Manley-Rowe relationships for a nonlinear lossless reactance, which is excited so that the current and voltage have frequency components of the form m/1 + n/2 , where m and n are integers, are given by the following two independent equations: [2] (15.1)

LJLJ ___ '°' '°' m/1 nPm ._ -0 + n/2 n

n

(15.2)

m

is the average power flowing into the nonlinear reactance at the frequencies m/1 + n/2. These equations are a result only of the nonlinear variation of the reactance and are independent of the shape of its characteristic and of the driving power levels. Since the device is considered lossless, the sum of all inward power flow at the different frequencie must be zero. By placing constraints on the impedance presented to the reactance at frequencies m/1 + n/2, it is possible to obtain a variety of useful circuits. If the nonlinear device i excited at /1 and /2 and is presented with an open circuit at all frequencies other than /3 = /1 + /2, then, provided that /1 0), then P3 < 0 and hence P3 is supplied by the reactance-therefore P 1 > 0 . It follows that the device is absolutely stable and has power gain equal to / 3 / / 1• This type of amplifier is called an upper-sideband up-converter. If a small amount of power at the signal frequency (/3) is applied to the varactor, P 1 and P2 are negative. Toe input power P3 is split between P1 and P2 , and if / 3 >> / 1,

15.3

PRACTICAL

ONLINEAR REACTANCE DEVICES

1009

most of the output power i delivered at / 2 • Such a configuration behaves as an upper- ideband down-converter. However, ince P2 is negative, a negative resi tance i pre ented to the pump circuit and the de ice i potentially unstable. If / 3 = /2 - /1. where /2 i again the pump frequency, the Manley- Rowe relation become (15.5)

(15.6)

Since Pi > 0. then P3 < 0 and hence Pt < 0. That is, the reactor emits more power than is upplied by the generator at ft. Thu . with thi embedding configuration, the ignal power can be amplified at the input frequency, in contrast to the previous case. The output power P1 and P3 are dependent on the pump power and the external impedance level . but they are alway related by the equations above. When the input and output frequencie are the ame, power at / 3 is imply dissipated in the circuit and i unused. For thi reason the third frequency is usually called the idler frequency. The idler ignal i an unavoidable result of this type of amplification, and suppressing it would uppre s the amplification of the input signal. The separation of idler and signal frequencies is an important parameter in the design of an amplifier -the closer the idler to the signal. the more difficult it will be to separate them by filtering. If the signal and idler frequencie are eparated sufficiently for the signal circuit to reject the idler. the amplifier is called a nondegenerate amplifier. Conversely, if the signal circuit passes both signal and idler bands and the input termination is common to both of them, the amplifier is called degenerate. If the output power is withdrawn at / 3 and / 3 >> / 1, the device is calJed a lowersideband up-converter. When / 3 < / 1 , the circuit performs the function of a lowersideband down-converter. If the nonlinear reactor is excited only at / 1 , then n = 0. The power flow relationships then become (15.7) m

describing a nonlinear reactor harmonic generator. The total harmonic output power is equal to the fundamental input power. If the circuit is adjusted to reactively terminate all harmonics other than the required output, then for an ideal nonlinear reactor, the conversion efficiency for that harmonic is 100%.

1S.3 PRACTICAL NONLINEAR REACTANCE DEVICES 1S.3.1

Varactor Diodes

The varactor diode (2, 3] is the most convenient nonlinear reactance element available to practical circuit designers. As stated earlier, the varying reactance is provided by the diode junction capacitance, which changes nonlinearly as a function of the applied (reverse) voltage. Varactors may be classified into two broad groups, depending on the method of fabrication: the junction varactor, widely used at microwave frequencies,

1010

MULTIPLIERS AND PARAMETRJC DEVICES Schottky varactor

Junction varactor

Contact whisker

Boron-diffused p-region

Diode metallization (gold on platinum)

Ohmic contact

120µm

n••GaAs

Equivalent circuit

c,

R,

Rs

C1 (0) R1 (0) R5 (0)

Junction

Schottky

2.0 pF > 10 Mn 0.50

0.02 pF > 10 MQ

80

Figure 15.1 Cutaway views of typical varactors, showing approximate dimensions. An equivalent circuit applicable to either Schottky or junction varactors is also shown, along with typical element values.

and the Schottky diode devices, usually used for millimeter wavelength applications. Figure 15.1 shows a cutaway view of typical varactor wafers of each type, as well as giving a typical equivalent circuit for a varactor diode, which applies equally to the two cases. The equivalent circuit for the device includes the following : a. Ci - the junction capacitance, which is a function of the applied voltage. b. Ri - the junction resistance, in shunt with Ci and also a function of bias. c. Rs - the series resistance, which may also be a function of bias. This parasitic resistance includes the resistance of the bulk semiconductor material external to the junction, the resistance of the undepleted epitaxial material, and the resistance of the ohmic contacts to the device. A typical junction varactor is made on an n-type silicon or GaAs wafer, with a heavily doped conducting substrate upon which is grown a lightly doped epitaxial layer. A suitable p-type dopant is then diffu ed into the epitaxial layer to fonn the pn-junction. Ohmic contacts are made to a small circular area on the top of the wafer and to the back of the wafer. Most of the epitaxial layer is then etched away, except in the area of the top contact, forming a mesa of the desired diameter. A Schottky barrier varactor diode consists of a circular metallic contact pad deposited on a lightly doped, epitaxially grown, n-type layer. Once again the substrate is heavily doped GaAs material . The epitaxial material is conductive except in the vicinity of the metal, where an insulating depletion zone is formed. Figure 15.1 shows typical values for the physical dimensions and equivalent circuit parameters for both Schottky and junction GaAs varactors, with a substrate resistivity of 0.004 0-cm and a 1-µm thick epitaxial layer with a donor doping density of 6 x 1016 cm- 3 • The high electron mobility and the inherently low spreading resistances in

15.3

PRACTICAL

ONLINEAR REACTANCE DEVICES

1011

GaA diode make GaA the material of choice for mo t microwave and millimeter wave application . Varactor diode are u ually operated under re er e-bia condition , where the junction re i tance i negligible by compari on with the capacitive reactance of the junction at the operating frequency. Therefore, at microwa e frequencie the equivalent circuit of the rever e-biased aractor diode (either junction or Schottky) may be con idered to be imply a (bia -dependent) capacitor and re i tor in erie . The equivalent circuit of the forward-biased junction varactor at microwave frequencie i generally more comple , ince it must incorporate the diffu ion capacitance of the injected minority carriers as well as the effect of the e carrier on the conductance of the emiconductor material. Minority carrier injection doe not occur for Schottky barrier varactors. Hence if forward-bias operation i employed. the junction varactor will exhibit greater capacitance variation than its Schottky equivalent, becau e of the absence of charge torage of minority carriers in the latter device. At low frequencie the aractor exhibits traightforward behavior in both the forwardand reverse-biased conditions. The forward-bias current increases exponentially with the applied voltage, and for re erse bias a mall aturation current flows. The current-voltage characteri tic i given by ( ee Fig. 15.2) (15.8)

SOmV

10 1

-< -E

10°

C

G) ~ ~

::, 0

,:, ~

co 10-1 ~ ~

1o-3 L---L---L~L-.__ __._ _~ - - - - -

o

0.2

0.4

0.6

0.8

1.0

Forward voltage (V)

Figure 15.l Typical forward current- voltage characteristics for a GaAs varactor diode.

1012

MULTIPLIERS AND PARAMETRIC DEVICES

= Boltzmann, s constant = device physical temperature Is = saturation current q = electronic charge

where k T

Vd y

= applied voltage = measure of the ideality of the device ( y

> 1)

The ideality factor (y) is normally in the range 1 to 2 at 300 K. As the reverse bias is increased, a point is reached where avalanche breakdown occurs in the epitaxial material and the diode current increases very rapidly, since it is limited only by the diode resistance and the external circuit resistance. In general, avalanche breakdown occurs as a result of impact ionization in which a high-energy electron or hole collides with a lattice site and generates a hole-electron pair- hence the reverse current rapidly increases [4-6) . Avalanche breakdown may occur in one of two ways: a bulk-type breakdown or a premature breakdown, commonly called punchthrough. Breakdown voltage as a function of doping for GaAs is given in Fig. 15.3, for both the bulk and punch-through mechanisms (5). The magnitude of the punch-through breakdown is dependent on the epitaxial layer thickness as well as its doping. For

100 -----.-------------~--.----------------.......,,.....,.... Punch-through voltage and breakdown voltage in GaAs

1011

1018

Donor concentration N In cm-3

Figure 15.3 Variation of the onset of bulk and punch-through breakdown in GaAs as functions of doping concentration and thickness for an ideal, uniformly doped sample. (From Ref. 5; reproduced by permission.)

15.3

PRACTICAL NONLINEAR REACTANCE DEVICES

1013

GaAs material doped higher than 2 x 10 17 cm-3, tunneJi ng also becomes an important mechanism in junction breakdown, but this wiJl not be considered here. GaAs exhibits a higher defect density and lower obtainable donor density than silicon. For these reasons it i difficult to obtain high breakdown voltages with GaAs diodes, especially for thin (ca. 1 µm) epitaxial layers. In generaJ , the reverse breakdown in Schottky barrier device is not as sharp as that of a diffused junction of the same material. The dependence of the junction capacitance on the applied reverse voltage for an ideal abrupt junction varactor is given by [6] (see Fig. 15.4)

(15.9) where Jo) varactor diode that has been optimally impedance matched to its embedding circuit at input and output frequencie . Furthermore, all other harmonic mu t be terminated optimally and with low los by the microwa e circuit. For a complete analy i of varactor multiplier performance, the nature of the microwave circuit to which the diode i connected must be considered [18] . Furthermore, the eries re istance must be taken into account when calculating the voltage drop aero the diode. An equivalent-circuit model, which may be used to predict the performance of a varactor diode as a frequency multiplier when mounted in a given microwave circui~ is bown in Fig. 15.7. 1\vo interconnected networks repre ent (a) the lo sles , nonlinear junction capacitor, the dynamic diode resistance, and the eries re i tance, and (b) a multiport embedding network that models the impedance pre ented to the diode by the microwave mounting circuit at all harmonics of the pump. The two networks are generally optimized to obtain maximum power transfer between the embedding network and the time-varying reactance at the input and output harmonics. The conversion efficiency of a given multiplier depends on the large-signal diode current and voltage waveforms induced by the pump signal. These waveforms are, in tum. determined by the impedance presented to the diode by the embedding network at the variou harmonics of the pump frequency. For multipliers operating in the millimeter wavelength range it is difficult to predict the embedding impedances at frequencies substantially greater than the pump frequency, because the mounting nuctures (such as waveguide, stripline, or coaxial lines) become multimoded. One Embedding network

,., -- --- - - - --- - -, I I I

Diode equivalent circuit

.------------

I

nf0

Diode mount I

I

I

I

: Ze(f) !_____________ _ I I I

- - ___, DC bias

Figure 15.7 Conceptual equivalent circuit for a varactor multiplier showing the interconnection of the embedding network and the varactor diode.

1020

MULTIPLTERS AND PARAMETRIC DEVICES

approach to this problem is to measure these impedances on a low-frequency scale model of the diode mount. The measured impedance data and the characteristics of the diode can then be analyzed to determine the diode voltage and current waveforms and hence the conversion efficiency of the multiplier for the chosen output harmonic [19] . In general, the steady-state, large-signal response of the multiplier circuit can be described in tenns of the Fourier coefficients of the voltage, v j , and current, ic [19]:

L Vke jkirtfi t ic(l ) = L

Vj(t ) =

vk=

(V- k)*

(15.21)

k

hejki1rfit

h

= (l -k)*

(1 5.22)

k

where f 1 is the pump frequency. Two sets of boundary conditions must be satisfied simultaneously by these quantities. The first, imposed by the diode, is most easily expressed in the time domain, while the second set, imposed by the embedding network (the embedding impedance Ze), is more conveniently considered in the frequency domain. Assuming that the diode does not conduct in the forward direction during the pump cycle (i.e., v < ), (15.23) where C j(Vj) is given by Eq. (15.9). The embedding network requires that

k

= ±2, ±3, . . .

V±1 = Vp - 1±1 [Ze(±f1) + Rs(±f1 )] Vo

= Vdc -

l o[Ze(O)

(1 5.24)

+ Rs(O)]

where VP and Vdc are the pump signal voltage and dc-bia voltage, respectively. The frequency dependence of Rs was discussed in a previous ection. Except in the case of the doubler, if currents flow only at the input and output frequencies, it is impossible to generate harmonic with the abrupt junction varactor. To generate higher-order harmonics, intermediate harmonic (or idler) currents must flow in the diode [2, 20]. Methods of olving Eqs. (15.21) and (15.22) to obtain the nonlinear, largeignal behavior of specific multipliers have been described by other authors [19, 21]. Figures 15.8 and 15.9 show the results of calculations using idealized models for a frequency doubler and a frequency tripler. These re ults were obtained using a computer-aided technique developed by Gwarek [22] and refined by Kerr and Siegel [21]. Current flow in the diode was permitted only at the pump and output frequencies, and at the econd harmonic idler frequency in the case of the tripler. In the latter case the termination was assumed to be an inductive reactance in shunt, and resonant with, the average diode capacitance at the second harmonic. Conceptual equivalent circuit are hown in Fig. 15.10. The pump signal was assumed to be sufficiently large to swing the diode voltage, vd, between its reverse breakdown limit, Vbr, and the onset of forward conduction (V1w < Vd < ). The plots show how multiplier efficiency, power handling, and input and load resistances vary with

15.4

MULTIPLIER DESIG

USING VARACTOR DIODES

1021

t)

u:: u.

w 104

1.0

103

10- 1

0.8

101

10-3

0.4

10°

10-4

0.2

Wr,lwc

Figure 15.8 Predicted performance of an idealized varactor doubler as a function of frequency (normalized to the dynamic cutoff frequency of the diode). The doubler equivalent circuit used for this analysis is shown in Fig. 15.l0a. PN is given by (V8 - ) 2 / Rs, RL is the load resistance, and. Rm is the diode input resistance.

-rl r?

a:."' ~

14

~

10°

t)

u:: u.

w

1.0 EFFIC

1a3

10-1

0.8

"'rfwc

Figure 15.9

Predicted performance of an idealized frequency tripler as a function of frequency (normalized to the dynamic cutoff frequency of the diode). The tripler equivalent circuit used for this analysis is shown in Fig. 15.l0b.

1022 (a)

MULTIPLIERS AND PARAMETRIC DEVICES Parallel resonant trap at f0

Parallel resonant trap at 2f0

L,

c,

Varactor Output matching transformer

Input matching transformer Doubler (b)

Trap at f0

Trap at 3f0

Series resonant trap at 2(f0)

Varactor Output matching transformer

Input matching transformer Tripler

Figure 15.10 Equivalent circuits for the idealized multiplier analy es presented in Figs. 15.8 and 15.9.

frequency (normalized to the diode dynamic cutoff frequency). The results demonstrate that the dynamic cutoff frequency should be much higher than the output frequency if efficient multiplication is to be obtained. The analy is of more reali tic model structures, outlined in a subsequent section of this chapter, will demonstrate how for a given diode, practical, nonideal mounting tructures result in lower conver ion efficiencies. A number of general guidelines hould be followed in the de ign of any practical multiplier. a. The mount should be designed to enable a good impedance match between the input circuit and the varactor device over the de ired pump frequency range. The same conditions should apply between the varactor and the output circuit at the desired output harmonic. Often power flow is required at a harmonic between the input and output frequencies - an idler frequency. The mount design should allow these component to be reactively terminated with low lo s. b. At frequencies other than the input, output, and idler hannonics, the mount should be poorly matched to the diode to minimize power lo s to these unwanted output components. c. The input and output circuits hould be phy ically and electrically isolated. Although varactor-diode frequency multipliers have long been used at microwave frequencies, they have now been largely superseded in this role by the step-recovery

15.4

MULTIPLIER DESIG

USING VARACTOR DIODES

1023

diode. However. the recent development of highly efficient harmonic generators for millimeter wavelength ha revitalized intere tin the technoJogy [23]. Schottky diodes with diameters of a few micrometer and new mounting tructure have been the major advance that have led to thi renewed intere t. For the mo t part, therefore, in the remainder of thi ection we consider only millimeter wave varactor multiplier design. The de ign technique for varactor multiplier at microwave frequencie have much in common with tho e u ed at millimeter wavelengths. A number of other excellent review of multiplier de ign for microwave frequencie already exist in the literature ( ee, e.g., Ref . 20 and 24). At millimeter wavelengths the best multiplier performance has been obtained with cro ed-waveguide mounts (25, 26], imilar to that shown in Fig. 15.1 1. The whiskercontacted varactor diode chip i mounted in the output guide, which is cut off at the input frequency and u ually of reduced height. Pump power is coupled from the input waveguide to the diode via a low-pass filter, which is cut off at the output frequency. The input circuit usually incorporates a means of de biasing the diode and tuning the mount to optimally couple the pump power to the varactor. Similarly, a movable hackshort in the output guide is used to aid in matching the varactor to the output circuit. For generation of harmonics of third order and higher with high efficiency, low-loss idler terminations must be provided. Optimum idler terminations have been incorporated in mi11imeter-wave multiplier mounts in a number of ways, re ulting es entially in the diode being short circuited or reactively terminated at the idler frequency. Theoretical studies of the performance of crossed-waveguide multipliers require that the diode embedding circuit be determined. The embedding circuit impedances at the harmonics of the pump frequency can be determined by low-frequency scale modeling, or calculated approximately by analytical techniques such as descnoed by Eisenhart and Kahn (27]. Such procedures usually show th.at it is difficult to choose a unique diode capacitance that will enable a simultaneous pump and output frequency impedance match to be obtained over a wide range of frequencies (greater than, say, 5% of a given center frequency). The choice of diode capacitance (i.e., the mean value of the time-varying junction capacitance) must represent a trade-off between several, possibly competing requirements. In general, broadband impedance matching at the input frequency is more easily achieved if the diode reactance at the pump frequency is near resonance with the mount impedance (whisker inductance). Since the output guide is cut off at the pump frequency, the embedding impedance seen by the varactor at this frequency is not greatly influenced by the tuning of the output backshort, except when this backshort is positioned very close to the diode. The whisker length and diode capacitance should also be chosen so that, with the aid of the backshort reactance, a reasonable impedance transformation can be achieved in the output circuit between the low dynamic resistance of the diode (50 to 100 Q) and the guide wave impedance. For mechanical reasons, it is difficult to achieve output guide impedances of much less than 200 Q in the mi11imeter frequency range, so that the impedance transformation required to achieve good match at the output frequency is significant. For a selected mount and diode configuration at a predetermined diode bias, the dynamic, large-signal analysis techniques described by Siegel and Kerr (21] can be used to determine the instantaneous voltage and current waveforms in the mounted varactor diode for one pump cycle. From these waveforms the minimum reverse breakdown voltage required of the diode may be determined, in order for it to handle a given

1024

MULTIPLIERS AND PARAMETRIC DEVICES Bias filter structure (see inset) D.C. bias input

hiddea

Output waveguide (contacting loop type backshort in removed part)

WR-8 output waveguide

flange (on

Input waveguide backshort {contacting type)

WA-15 pump input waveguide flange

Suspended substrate stripline low pass filter and waveguide coupling probe (see inset)

Split block mount

Reduced height output waveguide

Varactor diode chip Quartz bypass capacitor

Low pass filter (quartz substrate)

l 0.25 X 0.28 Input

waveguide coupling probe

Figure 15.11 Isometric sketch of a typical cro sed waveguide frequency doubler, which shows some of the main features important to the de ign of a millimeter wavelength harmonic generator. The inset shows details of the stripline low-pass filter, the diode mounting arrangement, and the de bias input connection. (Dimensions are in millimeters.) (From Ref. 28; copyright 1985 IEEE, reproduced by permission.)

15.4

MULTIPLIER DESIG

USING VARACTOR DIODES

1025

pump power level. Furthennore. by Fourier analy i of the diode current and voltage variation, the conversion efficiency of the multiplier may bee tirnated. The re ult of analy es of simplified model of millimeter wave varactor doubler and tripler mount of the type illu trated in Fig. 15.11 are bown in Table 15.1 to 15.7. The e analyse provide a u eful illu tration of the way in which diode and mount de ign can affect the diode voltage wa eform and the conver ion efficiency of millimeter wave Schottky varactor multiplier . In deriving the data in the table , the diode capacitance was modeled by Eq. (15.9). the forward IV characteristic of the diode by Eq. (15.8), and the frequency dependence of the series re i tance was modeled by (15.25) The following constant value were al o assumed:¢= 0.9 V, q / ykT = 35.0 v- 1, { = 0.5, l sat = 1 X 10- 15 A. R skin = 1 x 10- 6 Q Hz- 112 , and Vdc = - 10 V. The TABLE 15.1 Predicted Performance of Millimeter Wave Frequency Doublers for Various Diode Zero-Bias Capacitances and Series Resistances at a Pump Frequency of 50 GHzD C1 (0)

(fF)

z~t

Rs(O)

( Q)

Ep

€2

1 5 10

15

0.96 0.91 0.84 0.78

0.88 0.80 0.73 0.67

34 58 65 69

20

1 5 10 15

0.95 0.82 0.70 0.61

0.82 0.71 0.60 0.52

40

1 5 10 15

0.82 0.53 0.37 0.27

0.79 0.50 0.35 0.25

10

z 2nd 1

R(Q) X (Q)

y max

Xshort

r

p max

pump

R(Q) X (Q)

(mm)

E3/E2

(V)

(mW)

1010 1012 1000 1000

451 359 359 359

163 347 347 347

1.40 1.35 1.35 1.35

0.054 0.073 0.086 0.092

21.2 21.7 22.1 22.1

17.0 29.3 34.7 37.5

26 30 35 39

479 479 479 479

359 359 359 359

347 347 347 347

1.35 1.35 1.35 1.35

0.039 0.045 0.053 0.061

26.2 26.2 26.l 26.l

10.8 12.4 14.4 16.4

7 11 16 21

239 235 235 236

210 253 253 314

-60 -59 -59 - 42

1.60 1.55 1.55 1.50

0.026 0.054 0.056 0.066

23.9 24.8 24.8 24.9

9.1 17.6 25.4 32.5

0

Toe geometry of the waveguide mounr is illustrated in Figs. J5.11 and 15. l2; for this example, A= 2.03 mm. B = 0.51 mm. C = 0.26 mm.

TABLE 15.2 Predicted Performance of Millimeter Wave Frequency Doublers for Various Pump Frequencies, C j (0) 20 fF, Rs (0) S 2°

=

=

z 2nd

z ~pt

1

/pomp

(GHZ)

40 50 60 0

Xshon

vmax r

p max

(mW)

31.5 12.4 7.7

€p

E2

R(Q)

X (Q)

R(Q)

X (Q)

(mm)

E3/E2

(V)

0.87 0.82 0.70

0.81 0.71 0.68

41 30 18

674 479 387

114 359 80

368 347 18

2.70 1.35 1.40

0.071 0.045 0.029

20.6 26.2 22.0

pump

1be geometry of the waveguide mount is illustrctted in Figs. 15.11 and 15.12; for this example, A = 2.03 mm, B = 0.51 mm, C = 0 .26 mm.

1026

MULTIPLIERS AND PARAMETRIC DEVICES

TABLE 15.3 Predicted Performance of Millimeter Wave Frequency Doublers for Various Diode Zero-Bias Capacitances with Rs(0) 5 Sl, at a Pump Frequency of 50 GHzO

=

z 2nd

z ~Pl

I

C j (0) (fF)

10 20 40

Xshon

y max

p=p

r

Ep

E2

R(Q)

X(Q)

R(Q)

X(Q)

(mm)

E3/E2

(V)

(mW)

0.79 0.75 0.62

0.68 0 .72 0.61

25 21 14

965 477 236

206 206 206

5 5 5

1.80 1.80 1.80

0.107 0 .027 0 .007

20.8 22.6 25.1

16.5 6.3 20.7

The geometry of the waveguide mount is illustrated in Figs. 15.11 and 15.12; for this example, A = 2.03 mm, B = 0.51 mm, C = 0 .34 mm. 0

TABLE 15.4 Predicted Performance of Millimeter Wave Frequency Doublers for Various Diode Zero-Bias Capacitances with Rs(0) 5 Sl, at a Pump Frequency of 50 GH.z6

=

z 2nd

z ~Pl

I

Cj (0) (fF)

10 20

40

X sbort

y max

p max

r

pump

Ep

€2

R(Q)

X(Q)

R(Q)

X(Q)

(mm)

€3/€2

M

(mW)

0.90 0.82 0.71

0.56 0.68 0.65

53 29 18

946 472 239

94 150 119

211 90 61

1.50 1.90 2.10

0.198 0.191 0 .094

19.6 22.5 24.0

27.l 9.0 24.2

0

Toe geometry of the waveguide mount is illustrated in Figs. 15.11 and 15.12 ; for this example, A= 2 .03 nrrn. B = 0 .26 IIlIIl, C = 0-24 mm.

TABLE 15.5 Predicted Performance of Millimeter Wave Frequency Doublers for Various Pump Frequencies, Ci (0) = 20 fF, Rs (0) = 5 2 ° z 2nd l

z ~pl

/ pump

(GHZ) 40 50 60 0

€p

€2

0.91 0 .82 0 .82

0.68 0 .68 0~80

Xsbon

v ma:x r

p=p

R(Q)

X(Q)

R( Q)

X (Q)

(mm)

€3/€-i

(V)

(mW)

60 29

604 472

29

390

17 3 150 120

226 90 104

3.20 1.90 1.40

0.164 0.191 0.012

22.9 22.5 24.1

11.0 9 .0 13.2

The geometry of the waveguide mount is illustrated in Figs. 15.11 and 15.12; for this example, A

2 .03

mm, B = 0 .26 mm, C

= 0.24 mm.

=

TABLE 15.6 Predicted Pedormance of Millimeter Wave Frequency Tripiers for Various Diode Zero-Bias Capacitances with Rs (0) S n, at a Pump Frequency of SO GHz"

=

C1(0) (tF)

10 20 40

z ~PI

z 3rc1

I

X 2nd

I

X-bon

vmax r

p max

pump

Ep

E3

R(Q)

X(Q)

R(Q)

X('2)

(Q)

(mm)

€3/€2

(V)

(mW)

0.71 0.84 0.93

0.67 0.62 0.49

18 34 75

962 456 276

34 68 58

281 123 126

135 135

2.40 1.80 3.00

0 .001 0.229 0 .136

21 .5 18.1 32.3

1.0 7.5 34.l

0

135

=

The geometry of the waveguide mount is illustrated in Figs. 15. lJ and 15.12; for this example. A 2.03 mm, B 0 .26 mm, C 0 .24 mm. The cutoff filter for the second harmonic was spaced 2 .23 mm from the diode.

=

=

15.4

MULTIPLIER DESIGN USING VARACTOR DIODES

1027

TABLE 15.7 Predicted Performance of Millimeter Wave Frequency Tri piers for Various Pump Frequencies, C j(O) = 20 fF, Rs(O} = S Q 0 z 3rd 1

z ~PI

/ pump

(GHz)

40 45 50

55 60

Ep

€3

0.44 0.79 0. 4 0.77 0 .81

0.27 0.63 0.62 0 .36 0 .72

R(Q)

X(Q)

R(Q)

X(Q)

9

584

25

501 456

87 162 68 130 77

85 179 123 134 152

34 23

28

520 375

X 2nd I

X hon

(Q)

(mm)

- 185 147 135 280 113

2.40 3.00 1.80 2 .40 1.40

y ,max

p ma.x

E3/E2

(V)

(mW)

0 .368 0.154 0.229 0.592 0.040

23.2 18.1 18.1 20.0 20.2

2.0 5.2 7.5 2.8 10.7

pump

The geometry of the waveguide mount is illustrated in Figs. 15.l l and 15.12; for this example, A 2.03 mm. B = 0.26 mm. C = 0.24 mm. The filter pacing was 2.23 mm.

0

=

abbreviated column heading are defined a follows: C j (O), diode zero-bias capacitance; Rs(O). diode de series resistance; Ep, pump signal-to-diode coupling efficiency· En, nth harmonic conver ion efficiency; z~P\ optimum pump source impedance; zth, mount embedding impedance at the n th harmonic; Xshon, backshort pacing relative to the diode; v,.max maximum instantaneous diode voltage; p ~P' maximum pump power that can be handled without forward conduction. The following simplifying assumptions were made. 1. The pump circuit can be conjugately matched to the diode impedance at the

pump frequency.

2. The losses associated with waveguide, coaxial, and stripline structures incorporated in the mount are negligible. 3. The output circuit is perfectly decoupled from the pump circuit by a low-pass filter and the filter presents a short circuit to the diode at second and higher h armonics. 4. The diode can be represented by the simple equivalent circuit shown in Fig. 15.7. The series r esistance is assumed bias independent but has a half-power frequency dependence due to skin effect

5. The lossless output waveguide tuning short is adjusted for maximum second- or third-harmonic conversion efficiency. This is achieved when the varactor reactance is resonated as well as possible by the mount at the output and idler frequencies. 6. The reduced height output waveguide is coupled to a broadband termination by a perfec~ broadband impedance transformer. In the case of the tripler, the reduced height guide in the vicinity of the diode is not cut off at the second harmonic. However, the transformer is designed to be cut off at the idler frequency ; it couples all other harmonics losslessly to the termination and is positioned a half guide wavelength from the diode plane at a frequency near the desired mount center frequency. 7. The de bias on the diode is fixed at 10 V reverse bias. The pump power level is adjusted so that the maximum instantaneous forward voltage across the diode is

1028

MULTIPLIERS AND PARAMETRIC DEVICES

equal to ¢ (i.e., no forward current flows in the diode, but the voltage swing is maximized). The mount impedances seen by the diode at the first five pump harmonics were determined using analytical techniques. The current and voltage waveforms were then derived by a nonlinear analysis technique. The analysis was carried out for half-height, third-height, and quarter-height mounts. As stated in assumption 7, the pump drive at this bias voltage is sufficient to maximize the effect of the capacitance nonlinearity of the diode, without causing forward current to flow in the device. For each of these mounts the device zero-bias capacitance was set at 10, 20, and 40 fF, and for the halfheight mount only, the series resistance was varied between 1 and 15 n. The analysis was carried out for both the doubler and tripler configurations at several frequencies in the output waveguide band (80, 100, and 120 GHz foI the doubler; 120, 135, 150, 165, and 180 GHz for the tripler). The conversion efficiencies, peak reverse diode voltage, pump power level, optimized backshort position, and source and load impedances are given in Tables 15.1 to 15.7. Figure 15 .12 illustrates the resultant, theoretical diode voltage waveforms for a number of selected cases. The theoretical efficiency values are not attainable in a practical mount because of the unavoidable ohmic losses that occur in waveguide and stripline structures. These losses can be on the order of 1 to 3 dB in a millimeter wave multiplier block. In addition, where the diode voltage swing is large, additional losses will be incurred, owing to forward current flow in the varactor. However, with these limitations in mind, the results can still provide useful guidelines when designing and optimizing practical millimeter wavelength frequency multipliers. The conversion efficiencies predicted by these analyses of more realistic mi1limeter wavelength mounting structures can be seen to be significantly lower than would be expected from the simplified approach discussed earlier, the results of which were presented in Figs. 15.8 and 15.9. This is because power is lost to the output load at unwanted harmonics and because the waveguide mounting structure cannot always attain the optimum load and source impedances at the input and output frequencies. The choice of diode capacitance and mount geometry for a given application can, to a certain extent, be traded off one against the other. For example, in the case of a doubler with a pump frequency of 50 GHz and target conversion efficiency of 0.7, similar conversion performance can be obtained with the following configurations:

= 0.510 mm, C = 0.255 mm, Ci(0) = 10 fF, Rs = 10 n, Ppump = 34.7 mW B = 0.510 mm, C = 0.255 mm, Cj(0) = 20 fF, Rs = 5 Q, Ppump = 12.4 mW B = 0.255 mm, C = 0.242 mm, Cj(0) = 40 tF, Rs = 5 Q, Ppump = 24.2 mW

(i) B

(ii) (iii)

It is clear from these data that power-handling ability and hence output power are strongly dependent on the choice of diode and mount. Furthermore, the tables show that for a fixed junction capacitance, the diode series resistance has a strong effect on the multiplier performance. It should also be noted from the tables that if the diode capacitance is made too small, it becomes increasingly difficult to conjugately match the pump circuit because of the large inductive source reactance required. Table 15.8 shows the best performance measured with broadly tunable frequency doublers and triplers. Clearly, present multiplier designs are capable of providing adequate power for local oscillator applications in receivers in the millimeter wavelength

2

- -- - - Time

0 -2 ~

~ Q)

g

~~~

L,- / .

.

~

~--

-4 -6

~

0,

IU

_,.

-8

o.c. blasvolatage __ _ -------

•.

- - - - - •• ·- - - - - -

Q)

,

,.,

/ ./_./,_"1/ _- - -- - - ----- ------ ---/_/ ,~

"8 - 10

'o ~

C:

~

as - 14 C

1i,

.E

- 16

A - 2.0 3 -1

- 18

-~.s11-

- 20 - 22

;· /

. '~

g - l2

l ;

-1

C 0.203

1 0.0127 DIAi

,

/4...___ 114 Heigh~ c,(0) ~20 IF

/ /.

-....----~11 ~----,, . "'= .'-.. '

'

/

- ,

1/2 height = 0.508

B

1/3 height = 0.338 1/4 height = 0.254

-----' _I

Diode chip ~~

10n

113 Heigh~

c, (0) ~20 fF

1/2 Height, CJ(0) = 20 fF 1/2 Height, C1 (0) = 15 fF

tow pass filter

....

s '-C

Figure 15.12 Theoretical diode voltage waveforms (one period) for an idealized doubler mount at a pump freq uency of 50 GHz and a pump power of 10 mW. The analysis was carried out for the three different waveguide structures indicated, using the simplifying assumptions noted in the text. For the half-height mount, the curves also show the effect of decreasing the zero-bias capacitance of the diode. (Dimensions modify as shown are in millimeters.) (From Ref. 18; reproduced by permission).

.... C

~

TABLE 15.8 Summary of State-of-the-Art Performance for M illimeter Wave Frequency Multipliers

Mount : Type

'

Doubler

Tripler

x 6 balanced doubler/ tripler

Minimum Output

Tunable Output Operating Band (GHz) 80-120 80- 120 80-120 100 110- 170 140- 150 190- 260 200 400 500- 600 85- 115 96- 120 105 200- 290 190- 240 260- 350 300 450 310- 350

Effie.

1

l

-

10 10 10

8 8 8

-

-

-

-

7

0.7

4 1.8

1.2 1.8

-

-

2.5

2 .0 0.3 1.5

1 1.8

-

-

0.3

0.6

' 1

fl"

- I-

I

" Effie.

(%)

18 16 7

-

~

' 1r.

Power (mW)

(%) 9.5 10.7 10

.

r

I

(

'

,./

Maximum Output

20 12.0 17.6 21.5 18 0 .44

-

-

8 8.2 25 7.5 10 3.75 2 1 0.4

Freq. (GHz)

Power (mW)

14.0 15.5 16 25 15 22 27 19 8.5

26.6 23.2 11

2.4 8.2 18 6 3 3.0 2 0.079 0.75

' • J..

.

88 and 105 100 104 100 'I 120 145 215 200 • 300

106 110 105 225 230 340 300 450 345

Maximum Pump Power (mW)

Notes0

Reference

190 150 70 80 80 80 80 150 5.1 10

2, 3,9 1, 2, 3 1, 4 , 3 6,4 1, 2, 3 1, 2, 3, 5 1, 2, 3 6,4 1, 2, 3, 7 1, 2, 8

28 28 25 29 30 30 30, 31 29 32 33

28 100 72 80 30 80 100 6.3 190

1, 1, 6, 1, 1, 2, 6, 1,

26 34 29 35, 36 26 37 29 32 28

2, 2, 4 2, 2, 3, 4 2, 1, 2,

8 3 3 8 6 3, 7 3, 6, 9

J, Crossed waveguide mount; 2, tuning and bias optimized at each operating frequency; 3, rnicrostrip low-pass filter; 4 , fixed tunfog and bias; 5, narrowbanded version of NRAO 11 0- to 170-GHz doubler; 6, quasi-optical mount; 7, limited pump power available; 8, coaxial low-pass filter; 9, two diode balanced cross guide mounts.

0

15.4

MULTIPLIER DESIGN USING VARACTOR DIODES

1031

range. The varactor diode available today are of ufficiently high quality that higher efficiencies than indicated in Table 15.8 (and multip]ication to higher frequencies) hould, in theory, be po ible. However, increased harmonic efficiency and the extension of multiplier technology to higher output frequencies awaits further improvement in mount design.

15.4.2

oise Characteristics of Varactor Multipliers

The output of any microwave signal source contains unwanted, random frequencymodulation (FM) and amplitude-modulation (AM) of the desired spectral component. The e noi e components may be characterized in terms of their power spectra. It is usually preferable to minimize the energy contained in these unwanted sidebands of the wanted output spectral line. oi eat the output of a varactor frequency multiplier arises from two sources. The primary factor determining the spectral purity of the multiplier output is the output spectrum of the pump ource. It is particularly important that the pump source have low FM noi e, as the frequency multiplication process results in a degradation of the ratio of the signal to FM noi e by n- 2 • where n is the multiplication order [20]. An ideal varactor multiplier, such as discus ed in relation to the data given in Figs. 15.8 and 15.9, does not affect the ratio of the signal to AM noise. However, in a real multiplier the conversion efficiency may vary rapidly with frequency, causing significant FM-to-AM and AM-to-FM noise conversion at frequencies near these excursions. Furthermore, if the bias circuit time constant for the multiplier is too short, the average diode elastance will vary with pump signal amplitude, resulting in a phase modulation of the output signal. Clearly, care is necessary in choice of pump source and in multiplier design to minimize these effects.

15.4.3

Specific Design Example- A 200 to 290 GHz Frequency Tripler

The recent development of efficient frequency triplers for the 1 mm band has made it possible to construct simple, portable all-solid-state receivers in this wavelength range. A 200 to 290 GHz frequency tripler described recently in the literature [36] illustrates one practical approach to the design of an efficient, broadly tunable harmonic generator.

Mount Descriptwn. The tripler employs a split-block construction. Figures 15.13 and 15.14 illustrate the mount design. Power incident in the full-height, 3. 1 by 1.5 mm input guide is fed to the varactor diode via a tunable waveguide to stripline transition and a seven-section , low-pass filter, which passes the pump frequency with low loss but is cut off for higher harmonics. The varactor chip, a 0.1 mm sided cube, is mounted on the filter substrate adjacent to the reduced-height, 1.14 by 0.93 mm output waveguide. One of the many diodes on the chip is contacted and coupled to the output guide with a postmounte~ 0.0125 mm diameter by 0.15 mm long, gold-plated, phosphor-bronze whisker, which has been suitably pointed and prebent. Output tuning is accomplished with the aid of an adjustable backsbort in this guide. The bias circuit comprises a 140 Q transmission line, consisting of a 0.025 mm diameter gold wire center conductor bonded at one end to a low-impedance section of the low-pass filter and at the other end to a I 00 iF quartz dielectric bypass capacitor. The outer shield is a slot of rectangular cross section milled into the face of one of the blocks. At 95 GHz the bias line

1032

MULTIPLIERS AND PARAMETRIC DEVICES

Figure 15.13 Photograph of the 200 to 290 GHz frequency tripler designed by Archer [36).

approximates a quarter-wave short-circuited stub, thus minimizing its effect on the performance of the low-pass filter near cutoff. A quarter-wave, two-section impedance transformer, with dimensions shown in Fig. 15.14b, couples the 1.14 by 0.23 mm reduced height guide to the 0.76 by 0.38mm output guide. Power can flow in the wider guide at the second harmonic, whereas the output guide is cut off at this frequency. The transformer is thus used to implement a reactive second-harmonic idler termination in the manner described in the preceding section. The transformer is spaced 0.352 mm from the plane of the diode (approximately >..8 /2 at the second harmonic, where the guide wavelength equals >..g), The varactor diode used in this multiplier (type 5M2) was a Schottky diode device fabricated under the supervision of R. J. Mattauch at the University of Virginia. The zero-bias capacitance was 21 fF, the de series resistance was 8.5 Q , and the reverse breakdown voltage was 14 V at 1 µA . These devices have a highly nonlinear capacitance versos voltage law, which approximates the inver e half-power behavior of an ideal abrupt junction varactor to within about 2 V of the breakdown limit (see Fig. 15.4). The length of the contact whisker is chosen so that its inductance approximately series resonates the average capacitance of the pumped diode at the input frequency . Furthermore, this choice of whisker length provides, with the aid of the tuning short, a convenient transformation between the diode impedance and the waveguide impedance at the output frequency . At the pump frequency the low-pass filter, which is about a half wavelength in total length, transforms the approximately real-valued impedance of the whisker/varactor combination (of the order of 20 to 50 Q) to a similar realvalued impedance at the plane of the waveguide-to-stripline transition. Pump circuit impedance matching is achieved using two adjustable waveguide stubs with sliding contacting shorts. One stub acts as a backshort for the probe type waveguide-to-stripline transition and a second as an E-plane series stub located >..~/2 (at the pump wavelength

15.4 MULTIPLJER DESIGN USING VARACTOR DIODES

1033

(a)

Contacting backshort

(b)

Section AA C\l C')

~ ~

,.....

1-~-,~F~I Whisker pin

-

~ Waveguide transformer Diode chip.....-4-V,'"''17:~~~ .,.,.,"'P

Quartz stripline substrate Input waveguide

Transformer dimensions

Lt)

Stee

Width

Height

0

1.143 1.054 0.851 0.762

0.229 0.279 0.356 0.381

1

2 3

Output backshort detail

o.25¼C = = = = =

T

1.o{I-

-1 - - - - -

Figure 15.14 (a) View of the 200 to 290 GHz trip1er block split along the partition between blocJe-.s, showing the input guide, stripline filter, bias circuit, and output waveguide. (Dimensions are in millimeters.) (b) Section through the block detailing the waveguide transformer and diode mounting arrangement. The output backshort design is also shown. (From Ref. 36; copyright 1984 IEEE, reproduced by permission.)

A~) toward the source from the plane of the transition. This tuning configuration, with two degrees of freedom, facilitates the matching of the guide impedance to a wide range of impedances at the input to the low-pass filter. Mechanical adjustment of these tuners typically enables the input to be matched to the diode impedance with a VSWR of 2:1 or less at any frequency within the operating bandwidth of the WR-12 pump waveguide.

1034

MULTIPLIERS AND PARAMETRIC DEVICES

Low-Pass Filter Design. A special feature of this multiplier mount was the novel stripline structure used to implement the low-pass filter. The seven-section low-pass filter was a quasi-lumped element, 0.2-dB ripple Chebycheff design, implemented using high/low-impedance stripline sections on a crystalline quartz substrate. The stripline geometry and design data are given in the original paper [36]. A significant advantage of the stripline structure used in the design is that it allows the channel to be milled in only one of the pair of split blocks, while maintaining a large ratio between high and low impedances. For broadband multiplier performance, the low-pass filter should present, at the very minimum, a short circuit in its stopband to all expected second- and third-harmonic frequencies . In this design the channel and substrate dimensions were carefully chosen so that the moding cutoff frequency of the channel was above the stopband limit imposed by spurious resonances in the transmission-line sections, thus maximizing the useful stopband (36]. Figure 15.15 shows the line dimensions for the filter used in the multiplier and compares its predicted frequency response to an earlier filter design using conventional suspended substrate technology. The input frequency band is 67 to 97 GHz and the maximum -20 dB stopband width achieved is 130 to 350 GHz, providing a reactive termination for the diode at second-, third-, and most fourthharmonic frequencies. A computer analysis predicts that the filter should appear as short circuit to the diode at 265 GHz and that at 200 GHz it should exhibit a capacitive reactance of 10 n. When used in the multiplier mount the length of the low-impedance section to which the varactor diode is mounted is shortened by about 0.25 mm to compensate for the stray capacitance between the diode and the channel walls. Multiplier Performance. The frequency tripler exhibits significantly wider tuning bandwidth than do previously reported designs (26, 35, 29]. As shown in Fig. 15.16, 0

••

-. - 10

al

"O C 0

:.;

cu

cii

••••••• I I I

a: -40

~~ ~~ _._

'\ \ \

New , filter ,

' '

_____ .,,,,

I ,/

~ --

I I

.

~

I\;I

I I I I I

:-r:,3\ ?"21

\ :t• i

\

,:.'='~ l\ I I :: •

?is ,\

(/)

-

•• •• •• ••• Prototype \ •• • • • \ •• filter •• \ •• • • •• \ •• • • •••• •••

§\ \

I

I I

/

- 50 ______,__.,__......_---.IL..-....L.--ll--..L..__,J-..J....-' 1.0 2.0 3.0 4.0 5.0 6.0 7.0 (62.5) (125.0) (187.5) (250.0) {312.5) (375.0) (437.5) Frequency (GHz)

Figure 15.15 Measured transmission response of 62.5 x scale models of the filter and an earlier design that used suspended-substrate stripline. The frequencies in parentheses are the corresponding millimeter wave frequencies. The inset shows the metallization pattern for the millimeter wave version of the filter. (From Ref. 36; copyright 1984 IEEE, reproduced by permission.)

15.5

MULTIPLIERS USING STEP-RECOVERY DIODES

1035

6

-s ~

E

Input power = 80 mW

ci> 4 3: 0

~ 3 ::,

a. 3

0

2 1 0 ...___.~'--'--__.____,__..___.___._--'-_..___,____._____.___.__.L.-L......JL........1,-J 200 210 220 230 240 250 260 270 280 290 300 Output frequency (GHz)

Figure 15.16 Output power versu frequency for the 200 to 290 GHz tripler. Bias and tuning were optimized at each measurement frequency. The points have been arbitrarily interconnected with straight-line egments for clarity. (From Ref. 36; copyright 1984 IEEE, reproduced by permission.)

between 200 and 290 GHz the device provides more than 2 mW output power for 80 mW in. Backshon tuning and de bias were optimized at each measurement frequency. The typical rever e de bias voltage was 5 V, with forward currents between 0.1 and 0.5 m A. The peak power output of 4.6 mW occurs at 220 GHz, with a corresponding conversion efficiency of 5.7%. As shown in Fig. 15.16, higher conversion efficiencies may be obtained at lower pump powers. At 35 mW input power, the maximum efficiency obtained was 8 % at 222 GHz.

15.5 15.5.1

MULTIPLIERS USING STEP-RECOVERY DIODES Multiplier Design

A conceptual block diagram of a step-recovery diode multiplier is shown in Fig. 15.17. A signal ource at the input frequency delivers power to the step-recovery diode, which is used as an impulse generator, converting the energy in each input cycle into a narrow, large-amplitude voltage pulse. The pulse excites a resonant output circuit, converting the impulse into a damped ringing waveform. This signal is then filtered to select the desired CW output component. To form the impulse generator the diode is embedded in the circuit shown in Fig. 15.18. The high-amplitude, short impulse is formed by storing energy in the drive inductance just prior to the transition of the diode from forward to reverse bias. Since the step-recovery diode is driven hard into forward and then into reverse bias, its equivalent circuit may simply be represented as a small resistor, approximately equal to the series resistance, in the forward-biased state, and as a capacitor, Cr, in the reverse-biased state. The stored energy appears across C,, after switching, as a negative, half-sine pulse with a peak voltage given by [17]

(15.26)

1036

MULTIPLIERS AND PARAMETRIC DEVICES

Source

f f\ I\

Impulse - - Resonator generator

f

L _ __ _____.

I

=iltR

---:j~~

Output fitter

Load

To

Figure 15.17 Simplified conceptual block diagram of a step-recovery diode multiplier. (Adapted from Ref. 38, courtesy of Hewlett-Packard.)

Matching Drive network Inductance r ------ 1

I _______I

'---

Block

I

R;n

Vo

Figure 15.18 Circuit diagram for a practical impulse generator in a step-recovery multiplier and its output voltage waveform. (Adapted from Ref. 38, courtesy of Hewlett-Packard)

where IP is the peak current flowing in the diode. The impulse width is determined by the resonant frequency of the LC combination and is given by (15 .27) where

ff

~-1 2R1

Cr

(15.28)

is a damping factor determined by the loaded Q of the circuit. The total power in the impulse train is given by

(15.29) The input circuit of the impulse generator must include a means for de biasing the diode and a reactive network to match the impedance of the step-recovery diode

15.5

1037

MULTIPLIERS USING STEP-RECOVERY DIODES

to the source impedance of the generator. A typical (narrowband) L-C network that can be u ed to achieve these function is hown in Fig. 15.18. The tuning capacitor resonates the drive inductance at the input frequency. Since this capacitor carries the RF current at all harmonics of the input ignal (at least up to f = l.5 / tp) it must be a very high quality capacitor with a high elf-resonant frequency (certainly greater than l / 2tp)- The input impedance at the terminals of CT i resistive (at /m) and is given approximately by RIN = 2anfmL, where a:::'. 1 + l.6r For Rs/ RIN < 10, where Rs is the source re i tance the matching circuit element values are given by [38] _ ✓RsRIN

L

CM

(15.30)

2Jrfin

M -

1 = -----;:===::

(15.31)

2nfm ✓Rs RTN

The element values for the bias network are chosen so that it forms a maximally flat high-pass filter with a cutoff frequency of 0.8/m· It is particularly important for adequate tability of the multiplier that no high- Q series resonances be present in the bias network. especially at low frequencies . Bias circuit components should be chosen with thi in mind. Component values are given by [38] 24.4 LB=--

fm

8.85

X

10- 3

CB=----

(15.32)

fm

The bias network must be located at a distance of much less than Am / 4 from the diode, to avoid unsuitable driving impedances for the diode at frequencies less than fin• The equivalent frequency-domain spectrum of the pulse train is shown in Fig. 15.19a . The impulse generator can clearly be considered to behave as a "comb" -generating device in the frequency domain. The pulse width determines the variation in power between any two adjacent frequencies of the comb, with a narrower pulse resulting in a flatter amplitude line spectrum and a higher frequency for the first zero crossing. Pulses of as short as 70 ps can be produced with practical step-recovery diodes.

(a)

Vo

Vo

Co

L

m

--

........

...

' , ... ...

Vo Ep

1

Vo

(b)

2

3 .......

Vo

•

•• •• ••

,I

Damped waveform Resonant output network

,• •• •.. .. n

fl~n

Q 1 ~ mr/2

Figure 15.19 Form of the diode waveform and the approximate frequency spectrum of the impulse for (a) no output filtering and (b) output filtering with Qn = rrn/ 2. (Adapted from Ref. 38, courtesy of Hewlett-Packard.)

1038

MULTIPLIERS AND PARAMETRIC DEVICES

The pulse width is usually chosen to lie between l / 2lout and 1/ l out for multiplier applications, thus optimizing the output spectrum of the comb generator near the desired output frequency . The output from the impulse circuit is fed to a resonant output network with loaded Q adjusted so that most of the energy in the impulse is delivered to the network during one cycle of the input signal. The loaded Q required to achieve this is approximately nn/ 2, where n is the multiplication order. Most of the energy of the output spectrum of the impulse generator (nearly 75%, if the diode series resistance can be neglected) bas now been concentrated around lout, as shown in Fig. 15.19b. The output network is most easily implemented as a quarter-wave transmission-line resonator or as a series L-C resonant network, where the required output voltage is developed across a capacitor to ground. Design information for these two types of resonators is given in Fig. 15.20. In most applications additional filtering of the output signal will be required, in order to reduce the amplitude of the unwanted residual components of the comb spectrum to an acceptable level. Various high-Q filter designs are available for this task, including cavity filters and structures implemented in stripline and microstrip. The reader is referred to the references for further information on the design of such filters [39] . The selection of an appropriate diode for a particular multiplier application is a most important step in multiplier design. The important parameters of the diode are as follows . Power-Handling Ability. The power-handling capability of a step-recovery diode in a .multiplier circuit is determined by one of two limitations. The first is a constraint

• x2

Rett=2nZo= Zo + _c

Zo

1 Xe= - -= Zo J 2n-1 wCc

~ A'

+cN) - +w 2

Cc

2 c N2R 2

fo=1 2Jr

Figure 15.20 Design data for two possible resonant output networks that would be suitable for filtering the diode output.

15.5

MULTIPLIERS USING STEP-RECOVERY DIODES

1039

impo ed by the maximum power di ipation allowable in the device. The second is a con traint impo ed by the rever e breakdown voltage limit for the diode, which limits the maximum amplitude of the oltage impul e. Both of the e limitation are trongly influenced by the con er ion efficiency of the multiplier and limit the maximum output power attainable with a given de ign. The impul e generator efficiency i determined by four parameter : recombination lo in the diode, finite matching circuit lo , forward-to-rever e bias transition loss, and erie re i ranee lo . Of the e. only tran ition lo and eries resistance loss are of great ignificance. Recombination lo s can be minimized by keeping w-r >> 1; circuit lo i mall becau e Q 1 i - 1. In term of ju t the two important effects, the conversion efficiency 77cw-unp of the power in the input CW signal to the power in the impulse is given by [17. 3 ] (15.33) where ln p = Ip x 10 12 , fc1 = 159/ RsCr. fout = nfin, and N = 1/(2/mfp)The impul e to CW reconversion efficiency between the impulse power and the power in the damped ringing waveform i given by [38]

1Jimp- CW

=

4 XCOS(Jrx/2) 1 ] [ ,r 1 - x2 1 + (2 Rs/ Zo)

2

(15.34)

v.rhere x = 2/001 1p, and 2 0 i the characteristic impedance of the resonant output quarter-wave transmission line. Figure 15.21 hows a plot of these efficiencies for several typical step-recovery diodes. The power output limits related to the diode parameters may now be expressed as follows. If the diode is not to overhea~ we must have [38]

P.out < l

77tot P. d1ss, max - 77Lot

(15.35)

where 1Jtot = 0.75Lru7JCW- imp1Junp-CW and L@ = output filter loss. The diode will not exceed the reverse breakdown limit if [38] (15.36) The factor 0.75 in Eqs. (15.34) and (15.35) arises from the assumption that the loaded Q factor of the resonant output circuit is nrr /2, meaning that 75% of the total damped waveform power is then present in the nth harmonic of the input signal. For a given diode the restriction on output power changes from a breakdown limit to a dissipation limit with increasing frequency. The frequency at which the breakpoint occurs is given by (15.37)

Reverse-Biased Capacitance. The magnitude of the reverse-biased capacitance determines the energy in the impulse as well as the impedance level of the output resonator.

1040

MULTIPLIERS AND PARAMETRIC DEVICES

80

(a)

HP 0300 70 1-----.....,ji------+----+-------+-----t--~........, HP 0310 60 1-----+----+----.+----+------r-----, n=5 HP 0320 >,

g 40'----+-~~..--~---+----+----t------1

n = 10

Q)

·5

~ 30 i----__:~~--+-~~----!:~::----+------,------, n= 15

n= =..::2.:::..0..li,....~--~~---+-----+--__;;;---=t:::------1 20 L--!..:.

0

2

0

4

6

8

10

12

Output frequency, f0 (GHz)

90

(b)

-

~

n =5

80 .___ _ _µ,..._ _ _4--_ _ _+----...------+-------4

I

~ 70 i ~

n= 10 n = 20

601----~~~~-+---~"'k--....;::,,,,-...i::::-+----,----+----t

10.._---"---------~---~--------0 2 4 6 8 10 12 Output frequency, f0 (GHz)

Theoretical conversion efficiencies for use in the design of multipliers constructed using Hewlett-Packard step-recovery diodes. The diode part numbers are indicated on the curves. (a) Overall efficiency from input CW to damped output waveform. (b) Efficiency of converting impulse to damped waveform. (Adapted from Ref. 38, courtesy of Hewlett-Packard). Figure 15.21

This capacitance is specified at - 10 V reverse bias, since the capacitance of steprecovery diodes is independent of voltage at this level. A rule of thumb determining the value of Zo is 10 n < Z0 < 20 r2 (50-n system (15.38) where Zo

= l / 2rrfoutC,.

15.5

MULTIPLIERS USING STEP-RECOVERY DIODES

1041

Fundamental Time Constants. Two fundamental time con tant are of importance in the selection of an appropriate tep-recovery diode for a given application. The fir tis the minority carrier lifetime (r), which determine the lo that occurs during forward charge torage due to carrier recombination as well a the value of the elf-bias re istance which develops the diode bia due to rectification current. The lifetime effects are minimized if 2rrlior > 10. The econd intrin ic time constant i the tran ition time (t, ), which determines the ability of the diode to achieve the required impul e width and set the maximum output frequency limit as t, ~ l /l out. Package Parasitics. The package inductance ( L p) is in eries with the drive inductance. Its magnitude relati e to that of the external drive inductance determines the proportion of the energy in the total effective dri e inductance that is coupled to the output re onator. Typically, if Lp < Zo/ (2:rclou1), the effect may be neglected. The package capacitance (Cp) appear in shunt with the diode capacitance and is unde irable in that it i not active in the impulse-generating process. The package capacitance hould be small compared to the reverse-biased junction capacitance.

15.5.2

oise in Step-Recovery Diode Multipliers

As with the varactor diode frequency multiplier, the multiplication process in a step-

recovery diode results in a degradation of the ratio of the signal to FM noise at the output, relative to that of the pump input, of 6 dB for each doubling of the frequency. In addition to the effects on the input FM noise spectrum, a step-recovery diode multiplier can generate noi e internally [16, 40]. In particular, the effective loaded Q of the output filter has a significant impact on the level of internally generated AM noise. Modulation of the source impedance to the filter, due to the intermittent diode conduction, can also produce FM noi e in the output spectrum. There is also the possibility of phase noise being introduced by effects associated with the storage of charge when the diode is overdriven. The reader is referred to Refs. 16 and 40 for further discu sion of these effect and for design techniques that can be used to reduce their impact on multiplier perfonnance.

15.5.3 Specific Design Example - A 2.7 GHz Sextupler The performance requirements for the practical multiplier, designed by Cooper and Wells [41], which is to be used to demonstrate the implementation of the design techniques outlined above, were

/in = 450 MHz lout = 2700 MHz Pout,min = 2 W The diode chosen for this application was the HP0300 device. The characteristics of this diode are

Cr= 4 pF

Rs= 0.1 2 Q

r > 100 ns

t, < 600 ps

Lp = 0.3 nH

Cp = 0.1 pF

For an output frequency of2700 MHz, the impulse length was constrained as follows : 185 ps < tp < 370 p s Hence t P

= 210 ps was chosen.

1042

MULTIPLIERS AND PARAMETRIC DEVICES

The theoretical values for the circuit elements of the impulse generator were determined for this value of t p . Referring to Fig. 3.18, they are

L = 1.84 nH

Cr = 67 .7 pF

R£N (diode)

~

7 Q

LM = 6.6 nH

CM= 18.9 pF

The resonant output network in this example is a quarter-wave slab line with a characteristic impedance of 15 Q, design details for this type of network are given in Fig. 15.20. The loaded Q of the output resonator should be approximately 9.5 to optimize the concentration of energy in the sixth harmonic of the input signal. The equivalent Zo of the output line was chosen so that it was equal to 1/(2rrf0 0 1Cr) . The effective load resistance presented to the diode is equal to 2nZo = 180 Q. The reverse breakdown voltage for the 0300 diode is 75 V and the maximum allowable power dissipation is 9 W. Assuming a filter loss of 0.2 dB gives the theoretical overall efficiency for this multiplier as 0.3, resulting in a maximum output power of 3.85 W if the maximum dissipation rating of the diode is not to be exceeded. If the voltage of the impulse is not to exceed the breakdown voltage, the maximum output power must not exceed 5 .4 W. Hence the maximum output power is constrained to 3.85 W by the requirement that the dissipation in the diode not exceed 9 W. A photograph of the multiplier is shown in Fig. 15.22. Figure 15.23 gives a schematic diagram of the unit. The optimum value of the drive inductance, L,

Diode

Figure 15.22 Photograph of the 6 x multiplier designed by Cooper and Wells. (From Ref. 41, reproduced by permission.)

15.5

MULTIPLIERS USING STEP-RECOVERY DIODES

1043

Monitor

RFC

C1 4-30 pF

hp 0300

Schematic diagram of the 6 x multiplier circuit designed by Cooper and Wells. (From Ref. 41, reproduced by permission.)

Figure 15.23

TABLE 15.9 Performance Data for the 2.7 GHz Step Recovery Diode Sextupler Designed by Cooper and Wells Pump Power (W)

Output Power (W)

Efficiency (%)

1.8

0.65 1.40 2.00

37 41 40

3.4

5.0

Source: Ref. 41, reproduced by permission.

corresponds to a few millimeters of 50-Q line. The tuning capacitor Cr, with a value of about 70 pF, was designed to exhibit very low parasitic inductance, since it serves as a bypass for currents at the output frequency. It was constructed by wrapping polythene film around a brass block, with a 0.05-mm clearance from the top and bottom ground planes. Input matching was acrueved using a conventional LC network, with LM implemented as a length of 50-n transmission line. The diode was self-biased through an 8-kQ resistor when operating at 2 W output. It was found to be important to eliminate unnecessary shunt capacitance in the bias circuit to avoid parametric oscillations in the multiplier. The output circuit of the multiplier is composed of the nominal quarter-wave resonator, which is coupled to the output through a three-section mterdigital filter. The filter attenuates the adjacent harmonics at 2250 and 3150 MHz by 50 dB relative to the 2700-MHz output. Performance data for the multiplier are summarized in Table 15.9. The multiplier was designed for use as a local oscillator for a radiometer receiver, and therefore the level of close-in sideband noise was an important consideration. Figure 15.24 shows the approximate form_ of the noise spectrum of this multiplier.

1044

MULTIPLIERS AND PARAMETRIC DEVICES

-130

'i

.c

- 140

"O "O C

ns

..0 N

:r: -150 ,.... C

.....

(l>

1: .....

ns

0

0

- 160

Q)

>

:;::::;

~

ns

Q) .....

co

Cl

-170

- 180 L---__.~___._____._________ 10-3 101 102 Modulation frequency (MHz)

Figure 15.24 Approximate form of the output noise spectrum of the example multiplier. (From Ref. 41 , reproduced by permission.)

15.6 PARAMETRIC AMPLIFIERS AND VARACTOR FREQUENCY CONVERTERS Pumped varactor diodes found early widespread acceptance in low-noise amplifiers. The first parametric a1nplifiers (or paramps as they are popularly known) were produced at a time when the alternative for sensitive microwave reception was a silicon pointcontact mixer followed by an intermediate-frequency amplifier of moderate performance by today' s standards [42]. Radio astronomy, with its requirement for the utmo t sensitivity, and in keeping with its experimental status at the time, provided the first significant application for parametric amplifiers [43]. The radio a tronomy microwave receiver of the early 1960s consisted of a mixer followed by a vacuum-tube amplifier. The addition of a paramp ahead of the mixer gave an immediate improvement in sensitivity (a factor of 3 or 4). Paramps gained acceptance because the improvement in sensitivity they offered outweighed the complications they introduced (the requirement for a low-loss ferrite circulator and a high-power, high-frequency pump oscillator-coupled with a reputation for being touchy to operate). As experience was gained, and with their adoption by the satellite industry [44], paramp operation became a relatively routine procedure. In applications where a greater sensitivity requirement justified the additional complication, the paramp could be cooled to take advantage of the direct dependence of noise on the varactor' s physical temperature [45]. The earliest cooled paramps had the varactor and its associated microwave circuitry immersed in a bath of liquid nitrogen while the input circulator was outside the dewar at room temperature (46] . This

15.6

PARAMETRIC AMPLlFIERS AND VARACTOR FREQUENCY CONVERTERS

104S

arrangement rapidly gave way to y tern in which the circulator and amplifier were cooled to temperature as low a 15 K in closed-cycle refrigerators employing helium gas as the refrigerant. Paramps have recently been di placed by gallium arsenide MESFET amplifiers which yield comparable noise performance with le complication. Figure 15.25 shows noi e temperature versus frequency for FET amplifiers and paramps, both cooled and at room temperature. Cbaracteri 'tic of both devices (advantages and disadvantages) are listed for compari on in Table 15.10.

Room temp FET

Room temp paramp

-

Q' 100 CJ) ,_

::,

Cooled (20K) FET

~

CJ)

Q.

E

-"' CD

CJ)

0

z

10

10

1

100

Frequency (GHz)

Figure 15.25 of frequency.

Comparative noise performance for FET and parametric amplifiers as a function

TABLE 15.10 Comparative Characteristics of FET and Parametric Amplifiers: Advantages and Disadvantages Paramps

FET Amps

Low noise temperature Lower noise cooled One circulator per stage

Low noise temperature Lower noise cooled One isolator per amp

High-frequency pump Bulky Expensive

De supply Compact Comparatively cheap More stable

Comments Comparable Comparable FET amp often needs an isolator for stability

1046

MULTIPLIERS AND PARAMETRIC DEVICES

Varactor frequency converters have not found widespread application outside radio astronomy. The most notable example of their use in this field was in the VLA (an array of 27 antennas located at Socorro in New Mexico). Each antenna was equipped with dual-channel up-converters, converting signals between 1.35 and 1.75 GHz to 4.5 to 5.0 GHz for amplification in a three-stage parametric amplifier [47]. Such a system requires separate pump sources for the up-converter and parametric amplifiers. FET amplifiers have recently been retrofitted to this instrument.

15.6.1

Frequency Converters

Varactor frequency converters are characterized by the frequency relationships between the input and output and the pump supply. Four basic types are distinguished: the uppersideband up-converter (USUC), the lower-sideband up-converter (LSUC), the uppersideband down-converter (USDC), and the lower-sideband down-converter (LSDC). In their simplest form only three frequencies are active: the pump, input, and output. Additional frequencies (idlers) can be allowed to exist. In some cases this can lead to improved performance, such as in a four-frequency USUC [48], which can have gain greater than the upper limit for a three-frequency circuit (the ratio of output to input frequency) . The two lower-sideband, three-frequency converters present negative resistances at the input and output frequencies, and while this allows the possibility of achieving significant gain, care has to be taken to ensure stability. The upper-sideband converters, on the other hand, are both stable and have a conversion loss or gain that is less than the ratio of output to input frequency (equal to it for a lossless varactor). The performance of the USDC rapidly deteriorates as the separation between input and output frequency increases. When varactor up-converters first came to prominence, they offered a means of obtaining low-noise performance at frequencies in the hundreds of megahertz region, where their high gain and low noise could overcome the noise contributed by a relatively poor microwave receiver [20]. For example, an up-converter with 15 dB gain and 50 K noise would give rise to an overall noise temperature of about 80 K (noise figure of 1 dB) when followed by a mixer receiver with a 6-dB noi e figure. When this technology was in its infancy, this represented very good performance in the VHF and UHF region of the spectrum. Such performance was possible with the USUC when the frequency conversion range was sufficiently high (e.g .. input at less than 100 MHz and output at 10 GHz). For input frequencies between 100 and 1000 MHz, a LSUC would be required to achieve sufficient gain [20]. The development of low-noi e bipolar transistors and field-effect transistor provided a simpler means of achieving high sensitivity in this spectral range. Another way of using an up-converter was to follow it with a low-noise parametric amplifier. In th.is case a modest gain was all that was necessary to achieve good overall performance. An early example of this application consisted of an up-converter followed by a degenerate paramp, which gave an overall system noise temperature of 160 K [43]. A more recent example is the previously mentioned case of the receivers in the VLA [47]. Each antenna in the array was fitted with receivers covering the range 4.5 to 5.0 GHz. These consisted of cooled parametric amplifiers and each could be preceded by a switch-selectable varactor USUC converting from 1.35 to 1.75 GHz. The gain of these up-converters was only 2 to 2.5 dB, but the overall system noise temperature was only 49 K. GaAsFET amplifiers have taken over from such complicated

15.6

PARAMETRIC AMPLIFIERS AND VARACTOR FREQUENCY CONVERTERS

1047

systems, providing comparable noi e performance and greater operational simplicity and tability. The dj cu ion above ha been concerned with the u e of varactor converters in low-noise application . Large-signal converters are also of interest. For information on these and detail of the de ign of all type of varactor converters. the reader is referred to Refs. 2. 20, and 49.

15.6.2

Parametric Amplifiers

In a parametric amplifier the output frequency i the same as the input. Normally, three frequencies are pre ent the pump Ip, the signal Is and an idler 11 = IP ± Is, but as i the case with the converters, additional idlers can be incorporated. The three-frequency configuration i analogou to the lower-sideband converters, with output being taken at the ignal frequency in tead of at the difference frequency, which in this case plays the role of an idler. A a con equence of current flowing at the idler frequency, a negative resistance is presented to the external circuit at the signal frequency. This negative resistance, when correctly terminated by the signal circuit, gives rise to a reflection coefficient greater than unity and is the source of the device' s amplification.

Degenerate Parametric Amplifiers. There are two basic classes of paramp, degenerate and nondegenerate. In the degenerate paramp the signal frequency (Is) is close to the idler (fp - ls), and in the limiting case, when the signal and idler bands overlap, the pump frequency is twice the signal band center frequency. The nondegenerate amplifier has the signal and idler bands separated. The latter type of amplifier is the mo t common, with degenerate amplifiers mainly finding application in some of the early broadband radiometers and more recently in a number of millimeter wave paramps. The degenerate amplifier's output contains noise contributions from both the signal and idler bands, which in the case when they coincide are of equal magnitude. This is analogous to the double-sideband, variable-resistance mixer in which the signal and image bands contribute equally. In radiometer applications, where the signal is broadband noise, th.is type of amplifier has good sensitivity, since the signal is received equally in both the signal and idler bands. An effective input noise temperature for degenerate paramps is [20] (15.39) where Td is the varactor temperature, and m 1 f ed, the dynamic figure of merit, is equal to 2rcfoQd [see Eqs. (3. 14) and (3.15)] (m 1 is a modulation index which indicates the degree of pumping and is equal to the ratio of the first Fourier coefficient of the pumped elastance to the maximum range of elastance available from the varactor).

!Sil

m1=-- - -

Smax - Smin

(15.40)

This effective noise temperature can be used to compare degenerate paramps for the same type of use. However, for a more general comparison of degenerate and nondegenerate amplifiers, account must be taken of the nature of the signal; whether it is contained in a band that does not include Jp /2 (single-sideband operation), or includes

1048

MULTrPLIERS AND PARAMETRIC DEVICES

f,., / 2 (double-sideband operation), or whether it is noiselike, as in radio astronomy. In the double- ideband case, account must also be taken of the nature of the detector. A detailed discussion of this subject can be found in Ref. 2. Because its signal and idler bands are coincident, the degenerate paramp comprises a comparatively simple circuit. A means must be provided for coupling in the pump power efficiently and for preventing its flow in the signal line. A short length of highimpedance trans1nission line resonates the varactor at the signal frequency and, at the same time, provides an idler circuit. The circuit is coupled to the input signal line by a quarter-wave .impedance transformer to achieve the required condition for gain [50]. The condition for gain of a degenerate paramp is given by [2]

[p 2

= m1fcd

(15.41 )

To achieve gain the negative resistance must be transformed to a value that is close to, but less than, the resistance presented by the input circuit. As the magnitude of the transformed negative resistance approaches that of the input transmission line, the gain increases, going to infinity when the magnitudes are equal and satisfying the condition for oscillation when the net resistance is negative. The circuit described so far is a one-port device. To make an amplifier out of it, a means of separating the amplified output from the input must be provided. A ferrite circulator is the usual way of satisfying this requirement, and this is a feature of virtually all paramps, both degenerate and nondegenerate. Early paramps used threeport circulators, but most recent designs incorporated ft ve-port circulators to provide increased immunity from the effects of source impedance variations and to provide greater isolation between the first and second stages.

Nondegenerate Parametric Amplifiers. The most common form of parametric amplifier is the nondegenerate type, in which the signal and idler bands are clearly separated. In most recent designs the pump frequency is usually greater by a factor of 5 or more than the signal frequency. In this type of amplifier a separate circuit mu t be provided to support the idler. The basic nondegenerate parametric amplifier i illustrated in Fig. 15.26. This shows schematically the varactor diode connected as a common element between signal, idler, and pump ports. Ideally, current should not 'flow at any other frequencies (i.e .• all other sidebands should be terminated in an open circuit). If this condition i atisfied, the only noise ources internal to the amplifier are the varactor series re i tance and any resistance in the idler circuit. Assuming this condition to be satisfied. we see that the requirement for gain in such an amplifier i (15.42) which, when the idler re istance R; = 0, reduce to [2] (15.43) When R; = 0, the noise temperature i [2] (15.44)

15.6

PARA 1ETRJC AMPLIFIER

D YARACTOR FREQUENCY CONYERTl::RS

1049

Pump coupling Pump oscillator

> - - ---4

Signal filter and .,__-~~-~ transformer

Signal (input and output)

Idler filter

Varactor

Figure 15.26

Schematic diagram of a parametric amplifier.

When the paramp i pumped at an optimum pump frequency, a minimum noise temperature is obtained [2]:

Tm.in=

Td

_2_f~- [

m 1fcd

t

+

m 1fcd

(15.45)

The corre ponding optimum pump frequency i [2] (15.46) To obtain the best result from a given varactor it should be fully pumped, which means that the pump energy should be ufficient to swing the varactor' elastance over its maximum range. If a varactor is fu]]y pumped, and if the pumped ela tance waveform is sinusoidal, m I ha a value of 0.25. For arbitrary clastance waveform , m 1 ha an upper limit of 0.3 18. In practice the degree of pumping is u ually le s than the maximum. Thi is discus ed in Section 15.6.3. In Ref. 51 . ix useful formulas have been collected and examined to determine their accuracy. Good agreement was obtained between calculated parameters and mea urements on an actuaJ paramp. The formulas presented are for noi e temperature, gain tability, phase stability, pump power, aturation, and bandwidth.

15.6.3

Practical Considerations in Parametric Amplifier Design

The real limitation on the performance of parametric amplifier are the result of the nature of practical varactor diode , pump source . ferrite circulators, and the physical realizability of circuit elements. Amplifiers have been constructed to operate at frequencies of some hundreds of megahertz to just below I 00 GHz, but most have operated in the range of frequencies from 1 to 10 GHz. Table 15.11 lists example of

j,oo,l

= UI =

TABLE 15.11

Characteristics of Some Representative Parametric AmpHfiers 0 ·b

Frequency (GHz)

Type

1.3

Nondegenerate

2.7

BW (MHz)

Gain (dB)

Pump (GHz)

Idler (GHz)

Year

Reference

Comments

70 (room temp.) 29 (Ta= 77 K)

1964

51

Broadbanding stubs

60DSB

1968

49

197 1

52

1972

53

Broadbanding stubs, noise includes all input loss Two stages, broadband idler, balanced varactors, dual quarter-wave transformers Wafer- mounted balanced Schottky varactors Two stages, broadband idler, balanced varactors, 20 K ambient dual quarter-wave transformers, input losses included Two stages DSB noise temperature, including circulator loss and second stage contribution

110 180 300 400

11.3

Degenerate

20 13 7 15

5.4

2.7

3.95

Nondegenerate

26

500

34.7

31.2

74- 85

7.6

Nondegenerate

15

500

70

62.4

63

11.6

Nondegenerate

20

600

42.5

30.9

50

1974

54

14.95

Nondegenerate Degenerate

26

95

80.05 24

75 362 (room temp.)

1976 1970

55 56

64.4 46

150 (Ta= 20 K) 404 220 (room temp.)

1974 1973

57 58

240 K for amplifier alone DSB noise temperature referred to waveguide window

76

40 (T0 = 20 K) 1000 (target)

1973

59

Measured data not given

24

37

46

94

Nondegenerate Degenerate

Nondegenerate

cf

14

500 > 100

15 18 19

> 85 100 180

22

200

--

48

101.4 92

170

10

Noise (K)

•

aThis table presents a sample of amplifiers. It is not intended to be exhaustive, nor do the amplifiers listed necessari ly represent the best performance at their respective

frequencies. bNot all authors indicate whether their noise temperature includes circulator and other losses.

15.6

PARAMETRIC AMPLIFIERS AND VARACTOR FREQUE CY CO VERTERS

1051

amplifier operating over a wide frequency range. At the 100-GHz end of the pectrum, ferrite circulator lo e , the need for very high-power, high-frequency pump ources, and the difficulty of con tructing the idler circuit have made the technology unattractive compared with GaA Schottky mixer , which are comparatively imple and capable of quite low noi e performance when cooled. Almo t all parametric amplifier have used packaged varactor diode . A sociated with the package are para itic reactance which form part of the circuit at each of the frequencie taking part in the amplification proce [60]. The e parasitics play a particularly important role in the idler circuit. The reactance that the idler circuit i required to pre ent aero the varactor package depend on the relation between the idler frequency and the natural re onance of the mounted varactor. Optimum idler performance i achieved if the idler current can be confined to the diode and its package reactance . Thi mode of operation re ult in maximum idler bandwidth and minimize idler lo e . Placing a hort or open circuit aero the package at the idler frequency achieve thi goal [61] . Referring to Fig. 15.5, when the package is hort circuited, L s, CI . and C1 form a re onant circuit. For the type of package con idered in Ref. 62, thi re onance i in the vicinity of 10 GHz. In the open-circui t case, for thi package, all the reactance combine to produce a resonance at around 20 GHz. The e termination can be provided by external circui t elements, such as a lumped LC circuit or a length of tran mi ion line, but the added elements will degrade the idler performance. An alternative i to employ balanced circui ts and make use of ymmetry. One uch arrangement con i ts of mounting a pair of varactors side by side with oppo ite polarity and to upply the pump and signal voltage in uch a way that the idler current are in antiphase [63]. Thi results in the idler current flowing in a loop through the two varactor , so that each effectively short circuits the other. Another approach is to mount the diodes end to end acros the pump waveguide with the ignal line entering through the idewall and contacting the junction between them [61 ]. To achieve the optimum parametric amplifier design, microwave engineers should have at their disposal a variety of varactor chips so that varying requirements of pump frequency and idler circuit can be met. If engineers are able to de ign the varactors as well, they are in even better position. The idler circuit design has an important bearing on two of the most important characteristics of a parametric amplifier, the noise performance and the bandwidth. The equation given for noise temperature [Eq. (15.43)] assumes R; to be zero. In fact, R; will be nonzero, and in general at the same physical temperature as the varactor. Confining the idler to the varactor minimizes R; and thereby its contribution to the noise. The bandwidth that can be achieved is a function of both the signal and idler bandwidths. If the idler circ uit can be made sufficiently broad, the problem of incr easing the overall bandwidth reduces to the problem of designing a wideband signal circuit. One way of accomplishing this is to use a pair of broadbanding stubs forming a shunt resonant circuit, which combines with the series resonant circuit at the varactor input to provide a double-tuned response [52]. Schemes of this type can be extended to multiple-element configurations. Another approach is to use a two-stage quarter-wave transformer to achieve the condition for gain over a wider range of frequencies [64]. Bandwidth is also increased when the operating gain is reduced, and recent paramp systems have usually consisted of two or three low-gain stages, the low noise of the

1052

MULTIPLIERS AND PARAMETRIC DEVlCES

following stages compensating for the reduced gain of the first stage. While early amplifiers were capable of around 50 MHz bandwidth at frequencies of several gigahertz, large numbers of paramps were later made to cover the satellite band from 3.7 to 4.2 GHz. Where wider-frequency coverage is required but instantaneous bandwidth is less important, tuned systems can be used. One way this can be achieved is by mechanically (a)

V-Varactor

Reduced height and width waveguide

(b)

Figure 15.27 (a) Section diagram of a 3.25 GHz parametric amplifier; (b) photograph of the amplifier with its four-port circulator.

15.6

PARAMETRIC AMPLIFIERS AND VARACTOR FREQUENCY CONVERTERS

1053

tuning the idler frequency. Alternatively, the idler could be de igned to be very broad and the varactor bias varied to change the mean capacitance of the varactor junction, thereby tuning the ignaJ circuit. Thi method implie that the varactor wi]] not be fully pumped. A combination of both tuning method could aJ o be u ed. The gain tability i an important con ideration. e pecially in ome applications, ucb as radio astronom . Thi i greatly dependent on the tability of the pump ource. for both power output and frequency. Thi wa a particular problem with early paramp u ing ldy tron pump ource . To minimize the effect of power variations, ervo loop were introduced to tabilize the pump power level incident on the varactor. This could be accompli hed by ampling the pump level with a coupler and monitoring detector and controlling it by arying an attenuator in the pump line. The more popular approach was to control the attenuator to maintain a con tant varactor bias. Frequency fluctuation could be taken care of in critical application by locking the pump to a table reference. The introduction of Gunn o cillator as pump sources re ulted in a con iderable improvement in tability, making tabilization schemes unnece ary.

15.6.4

Specific Design Example -A 3.25 GHz Parametric Amplifier

Figure 15.27a how a ection drawing of a nondegenerate parametric amplifier covering the frequency range 3.1 to 3.4 GHz with an instantaneous bandwidth of 40 MHz. This amplifier was designed by M. W. Sinclair of the CSIRO, Division of Radiophysics, for pectraJ-line radio astronomy. The varactor diode is mounted in the £-plane of a reduced-height wa eguide which couples pump power from a 22 GHz reflex klystron to the varactor. A short length of high-impedance coaxial line series resonates the diode mean capacitance at the ignal frequency. A three-element low-pass filter isolate the pump and idler from the input line while the pump waveguide is cut off at the idler frequency, confining the idler to the vicinity of the varactor and the idler cavity. The idler circuit consists of a tunable cavity coupled to the varactor by an iris. The position of this iri was chosen to optimize the pump coupling to the varactor. The idler frequency is determined by the combination of the package parasitic reactance , the coupling, and the tunable cavity, which consists of a micrometer-adjustable

TABLE 15.12 Performance Data for a 3.25 GHz Parametric Amplifier Signal frequency Pump frequency Idler frequency Gain Instantaneous bandwidth iteasured receiver noise0 Estimated amplifier noise0 Calculated amplifier noise

3.1-3.4 GHz 22 GHz

18.9-18.6 GHz 20 dB 40 MHz

118 K 60 K

59 K

lncludcs l 0 K from second stage. 0.15 dB circulator loss, and 0.4 dB switch and coupler loss. Estimated noise is obtained by removing the effect of these losses from 118 K. The degree of agreement is fortuitous. 0

1054

MULTIPLIERS AND PARAMETRIC DEVICES

noncontacting short circuit in a cylindrical tube. The pump and idler blocking filter forms part of a quarter-wave desired gain. No external bias is provided, the varactor being pumped until self-bias is developed. Figure 15.27b shows a photograph of the amplifier together with its four-port circulator. Its performance is summarized in Table 15.12.

REFERENCES 1. J.M. Manley and H . E . Rowe, "Some General Properties of Nonlinear Elements. Part 1.

General Energy Relations," Proc. IRE , 44(7), pp. 904-913, July 1956. 2. P. Penfield, Jr. and R P. Rafuse, Varactor Applications , MIT Press, Cambridge, MA, 1962. 3. J. C. Irvin, T . P. Lee, and D . R. Decker, "Varactor Diodest in H . A. Watson, Ed., Microwave Semiconductor Devices and Their Circuit Applications, McGraw-Hill, New York, 1969, pp. 149-193. 4. S. M. Sze, Physics of Semiconductor Devices, 2nd ed., Wiley, New York, 1981, Chap. 3. 5. M. V. Schneider, "Metal-Semiconductor Junctions as Frequency Converters," in K. J. Button, Ed., Infrared and Millimeter Waves , Vol. 6, Systems and Components, Academic Press, New York, 1982, pp. 209-275. 6. T. P. Lee, " p-n Junction Theory," in H. A. Watson, Ed., Microwave Semiconductor Devices and Their Circuit Applications, McGraw-Hill, New York, 1969, pp. 95-125. 7. L. E. Dickens, "Spreading Resistance as a Function of Frequency," IEEE Trans. Microwave Theory Tech., M'IT-15(2), pp. 101-109, February 1967. 8. E. R. Carlson, M . V . Schneider, and T. F. McM aster, "Subhannonically Pumped Millimeter Wave Mixers," IEEE Trans. Microwave Theory Tech., MTI-26(10), pp. 706-715, October 1978. 9. K. S. Champlin and G. Eisenstein. "Cutoff Frequency of Submillimeter Schottky Barrier Diodes," IEEE Trans. Microwave Theory Tech., M'IT-26(1), pp. 31-34, January 1978. 10. J. W. Archer, B. B. Cregger, R. J. Mattauch, and J. D . Oliver, "Harmonic Generators Have High Efficiency," Microwaves, 21(3), pp. 84-88, March 1982. 11. B. C. DeLoach, Jr., "A New Microwave Measurement to Characterize Diodes and an 800 Ge Cutoff Frequency Varactor at Zero Volts Bias," IEEE Tra11s. Microwave Theory Tech. MTT-12( 1), pp. 15- 20, January 1964. 12. T. P. Lee, "Evaluation of Voltage Dependent Series Resistance of Epitaxial Varactor Diodes at Microwave Frequencies," IEEE Trans. Electron Devices, ED-12(8), pp. 457 -470, August 1965. 13. N. Houlding, "Measurement of Varactor Quality," Microwave J., 3(1). pp. 40-45, January 1960. 14. K. Kurokawa, "On the Use of Passive Circuit Measurements for the Adjustment of Variable Capacitance Amplifiers," Bell Syst. Tech. J., 41(1), pp. 361-381, January 1962. 15. J. L . Moll, S. Krakauer, and R. Shen, "P-N Junction Charge-Storage Diodes," Proc. IRE. 50(1), pp. 43- 53, January 1962. 16. S. M. Krakauer, "Harmonic Generation, Rectification, and Lifetime Evaluation with the Step Recovery Diode," Proc. IRE, 50(7), pp. 1665-1676, July 1962. 17. S. Hamilton and R. Hall, "Shunt-Mode Harmonic Generation Using Step Recovery Diodes," Microwave J., 10(4), pp. 69- 78, April 1967. 18. J. W . Archer, ''Low-Noise Receiver Technology for Near-Millimeter Wavelengths," in K. J. Button, Ed., Infrared and Millimeter Waves, Vol. 15, Millimeter Components and Techniques, Part 6, Academic Press, New York, 1986, pp. 1-86.

REFERE CES

1055

. H eld and A. R. Kerr. ··Conver ion Lo • and oi~e of Microwave and Millimeter Wave Mixer. ." IEEE Trans. Microwave Theory Tt.>ch. MTT-26(2), pp. 49- 55, February 1978.

19. D.

20. M. Uenohara and J. W . Gewartow~ki. '"Varactor ApplicaLion :· in H. A. Wat.on. Ed .. Microware Semiconductor Devices and Their Circuir Applications. McGraw-Hill. ew York. 1969. pp. 194-269. 21. P. H. Siegel and A. R. Kerr, "Computer Analy i of Microwave and Millimeter Wave Mix ers.'' IEEE Trans. Aficrowm·e Theory Tech. 1TT-28(3). pp. 275-276, March 1980. 22. W. K. Gwarek. ·· o nlinear Analy i of Microwave Mixers:· M.S. the i , Mas achusetts In titute of Technology. Cambridge. September 1974. 23. J. W . Archer. "Lo\\-- oi e Receiver Technology for Near-Millimeter Wave Radio Astronomy:· Proc. IEEE . 73(1). pp. 109-130, January 1985.

24. 0. P. Gandhi. ~ficrowave Engineering and Applications. Pergamon Pre s, Elmsford, 1981. Chap. 13. 25. J. W. Archer, "A High Performance Frequency Doubler for 80-120 GHz: ' IEEE Trans. Microwa,·e Theory Tech. ~ITr-30(5). pp. 824-825, May 1982. 26.

.,_ R. Erickson. "A High Efficiency Frequency Tripler for 230 GHz.'' Dig. 12th Eur. Microwai·e Conf.. Hel inki Finland. pp. 288-292, September 1982. 27. R L. Eisenhart and P. J. Kahn. ·Theoretical and Experimental Analysis of a Waveguide Mounting Structure;' IEEE Trans. Microwm·e Theory Tech. MTT-19(8), pp. 706- 719, August 1971. 28. J. W. Archer and M . T. Faber, ·•High Power 80- 120 GHz Doublers for a 310-345 GHz x 6 Multiplier Chain:· TEEE Trans. Microwave Theory Tech., MTT-33(6), pp. 533 - 539, June 1985. 29. J. A . Calviello, ·'Advanced Devices and Components for Millimeter and Submillimeter Sy terns," IEEE Trans. Electron D evices. ED-26(9), pp. 1273-1281, September 1979. 30. J. W. Archer, ..Millimeter Wavelength Frequency Multipliers," IEEE Trans. Microwave Theory Tech., 1\lTT-29(6), pp. 552-557, June 1981 . 31. K . Lundien, R. J. Mattauch, J. W. Archer, and R~ Majjk, "Hyperabrupt Junction Varactor Diodes for M illimeter Wavelength Harmonic Generation," IEEE Trans. Microwave Theory Tech., :MTT-31(2), pp. 235-238, February 1983. 32. T. Takada and M. Hirayama. '"Hybrid Integrated Frequency Multipliers at 300 and 450 GHz,'"' IEEE Trans. Mic rowave Theory Tech., MIT-26( 10), pp. 733-737, October 1978. 33. 34.

35.

36.

37.

38.

. R. Erickson and H. R. Fetterman. "Single Mode Waveguide Submillimeter Frequency Multiplication and Mixing," Bull. Am. Phys. Soc. , 27, p. 836 (abstract only), 1982. J. W . Archer and M . ]. Crawford, " A Synthesized 90-120 GHz Signal Source," Microwave J., 28(5), pp. 227- 250, May 1985. J. W. Archer, ''An All Solid-State Receiver for 210- 240 GHz," IEEE Trans. Microwave Theory Tech., MTT-30(8). pp. 1247-1252, August 1982. J. W. Archer. " An Efficient 200- 290 GHz Frequency Tripler Incorporating a Novel Striplfoe Structure," IEEE Trans. Microwave Theory Tech , MTT-32(4), pp. 416- 421, April 1984. J. W . Archer, "A Novel Quasi-optical Frequency Multiplier Design for Mi1limeter and Submj]limeter Wavelengths," IEEE Trans. Microwave Theory Tech. , MTT-32(4), pp. 421-427, April 1984. Harmonic Generation Using Step Recovery Diodes and SRD Modules , Hewlett-Packard Application Note 920, Hewlett-Packard, Palo Alto, CA.

1056

MULTIPLIERS AND PARAMETRIC DEVICES

39. G. L. Matthaei, L. Young, and E. M. T. Jones, Microwave Filters, Impedance-Matching Networks, and Coupling Structures, McGraw-Hill, New York, 1964. 40. J. C. McDade, "Measurements of Additive Phase Noise Contributed by the Step Recovery Diode in a Frequency Multiplier," Proc. IEEE , 54(2), pp. 292-294, February 1966. 4 1. B. F. C. Cooper and G. A. Wells, "Six-Times Multiplier with Two Watts Output at 2700 MHz," Proc. !REE (Aust. ), 30(10), pp. 340- 341 , October 1969. 42. C. T . McCoy, "Present and Future Capabilities of Microwave Crystal Receivers," Proc. IRE , 46(1), pp. 61- 66, January 1958. 43. B. J . Robinson, "Development of Parametric Amplifiers for Radio Astronomy," Proc. IRE (Aust.), 24(2), pp. 119- 127, February 1963. 44. C. L. Cuccia, "Ultralow-Noise Parametric Amplifiers in Communication Satellite Earth Stations," in L. Young, Ed., Advances in Microwaves, Vol. 7, Academic Press, New York, 1971. 45. R. C. Knechtli and R. D . Weglein, ''Low Noise Parametric Amplifier," Proc. IRE , 47(4), pp. 584- 585, April 1959. 46. F. F. Gardner and D . K Milne, "A 1400 Mc/S Continuum Radiometer," Proc. IRE (Aust.), 24(2), pp. 127-132, February 1963. 47. S. Weinreb, M . Balister, S. Maas, and P. J. Napier, "Multiband Low-Noise Receivers for a Very Large Array," IEEE Trans. Microwave Theory Tech. , MTT-25(4), pp. 243-248, April 1977. 48. J. A. Luksch, E. W. Matthews, and G. A. VerWys, "Design and Operation of FourFrequency Parametric Upconverters," IRE Trans. Microwave Theory Tech. , MTT-9(1 ), pp. 44-52, January 1961. 49. H. C. Okean and L. J. Steffek, "Octave Input S to Ka-Band Large Signal Up-Converter," IEEE MIT-S Int. Microwave Symp. Dig., pp. 218-220, June 1974. 50. RA. Batchelor, J. W. Brooks, and B. F. C . Cooper, "Eleven Centimeter Broadband Correlation Radiometer," IEEE Trans. Antennas Propag .• AP-16(2), pp. 228-234, March 1968. 51. P . J. Moogk and U . Rutulis, ''Six Formulas Simplify Paramp Design,'' Microwaves , 5(5), pp. 36- 42, May 1966. 52. J. T. DeJager, "Maximum Bandwidth Performance of a Non-degenerate Parametric Amplifier with Single-Tuned Idler Circuit," IEEE Trans. Microwave Theory Tech., MTT-12(7). pp. 459-467, July 1964. 53. J. C. Vokes, J. R. Dawsey, and H. A. Deadman, ''Low-Noise Room-Temperature Parametric Amplifiers," Electron. Lett., 7(22), pp. 657-658, November 4, 1971. 54. L. E. Dickens, " A Millimeter-Wave Pumped X-Band Uncooled Parametric Amplifier," Proc. IEEE , 60(3), pp. 328-329, March 1972. 55. J. Thirlwell, J. McPherson, and R. R. Bell, "Broadband Cryogenic Parametric Amplifier Operating at 11.6 GHz," Electron. Lett., 10(16), pp. 329- 330, August 8, 1974. 56. H . C. Okean, J. A. DeGruyl, and E. Ng, "Ultra Low Noise Ku-Band Parametric Amplifier Assembly," IEEE MIT-S Int. Microwave Symp. Dig., pp. 82-84, June 1976. 57. J. Ed.rich, ''Parametric Amplification of Millimeter Wave Using Wafer Diodes: Results, Potentials and Limitations," IEEE Trans. Microwave Theory Tech., MTI-18(12), pp. 1173- 1175, December 1970. 58. M . A Balfour, A. Larson, S. Nausbaum, and J. Whelahan, "Miniaturized Nondegenerate Ka-Band Paramp for Earth to Satellite Communications," IEEE 'MTI'-S InJ. Microwave Symp. Dig., pp. 225-227, June 1974. 59. J. Edrich, "20 K Cooled Parametric Amplifier for 46 GHz with Less Than 60 K Noise Temperature," IEEE MTI'-S Int. Microwave Symp. Dig., pp. 72-74, June 1973.

REFERE CES

1057

60. W. J. Get inger. 'The Pa kaged and Mounted Diode a a Microwave Circuit." IEEE Trans. Micro..,,:ave Theory Tech., t TT-14(2). pp. 5 - 69. February 1966. 61. J. Edrich, .. Rau channe Parametri che Eigenre onanzver tarker mil gro . er Bandbreite"

(Low- oi e Parametric Self-Re onance Amplifiers with Large Bandwidth), Frequenz, 20( 10). pp. 337-343. 1966. 62. H. C. Okean. J. R. Asmus. and L. J. Steffek, ''Low- oi e 94 GHz Parametric Amplifier De\elopmem:· IEEE MIT-S Int. Microwa,·e Symp. Dig. , pp. 78- 79, June 1973. 63. J. D. Pearson and K. S. Lunt. "A Broadband Balanced Idler Circuit for Parametric Amplifiers.'' Radio Electron. Eng. 27(5), pp. 331 - 335. May 1964. 64. C. S. Aitchi on. R. Da\ie , and C. D. Payne, " Bandwidth of a Balanced Micropill -Diode Parametric Amplifier:· IEEE Tra11s. A-ficrowa\le n,eory Tech .. MTT-16(1), pp. 46-47, Jan-

uary 196 .

16 SEMICONDUCTOR CONTROL DEVICES: PIN DIODES* JOSEPH F. WHITE JFW Technology, Inc. Orleans, Massachuseus

16.1 THE PIN DIODE-AN EXTENSION OF THE PN JUNCTION 16.1.1 Structure The PIN diode should not be thought of as something physically different from the PN junction, but rather different in a sense of degree. With the abrupt junction the width of the dep]etion zone is inversely proportional to the resistivity of the P or N region, whichever bas the lesser impurity doping concentration. As the width of the depletion zone increases, the capacitance per unit area of the junction decreases. This effect is very beneficial for a diode which is intended for use as a microwave switch because the Jower the capacitance, the higher the impedance of the diode under reverse bias and the more effective the device is as an "open circuit." The limiting case of high-resistivity material is undoped (or "intrinsic") I silicon. In practice, of course, no silicon material is without some impurities. A practical PIN diode, then, consists of an extremely high resistivity P or N zone between low resistivity (highly doped) P and N zones at its boundaries, as shown in Fig. 16.1. To distinguish unusually heavily or lightly doped material, special nomenclature has evolved. Heavily doped P and N materials are referred to as P+ and N+ , respectively. To identify very lightly doped, high-resistivity P and N materiaL the Greek letters are used; thus highresistivity P material is called ]'(-type and high-resistivity N material is called v-type. Recognizing that perfectly intrinsic material is not practically obtainable, the I region

* This

chapter is reproduced in its entirety, with slight modifications, from Joseph F . White, Microwave Semiconductor Engineering, Noble Publishing, Norcross, Georgia, 1995, where it appears as Chapter 2.

Ha1Uibook of RF/Microwave Components and Engineering, Edited by Kai Chang ISBN 0-471-39056-9

© 2003 John Wiley & SonsJ Inc.

1059

1060

SEMICONDUCTOR CONTROL DEVICES: PIN DIODES (a)

r -=-

(b)

C , "" constant "" C (0)

1 ~ - - - -- - - - j \ 1 ~ -- - - - - -0 V

7

Fixed capacitance approx. Equiv. CKT. for PIN diode

r

I (TJ)

P+ Doped crystal (P+,v, N+)

- -o v

N+

+ + + ___ + +

+

+ + + P'""!""----.----,.--,----,--r- + (c)

-- 0 Ionized impurity profile (P+, v, N+)

(d)

r

X

' - _ P-N junction

P+

/ {,r)

N+

Doped crystal (P+, 1r, N+)

- -o

++

t

!! ++

p

++ ++

PN junction "-....

!!

" ' ++

(e)

X

Ionized impurity profile (P+, ,r, N+)

Figure 16.1

Profiles for the two PIN diode types.

of a PIN diode can consist of either v- or rr-type material. The resulting diodes are indistinguishable from a microwave point of view; however, the actual junction forms at opposite ends of the intrinsic zone depending on the choice. This distinction is diagrammed for both cases in Fig. 16.1. The first type shown in Fig. 16.lb shows a P+, v, N+ diode structure. H the I region is of sufficiently high resistivity, what few impurity atoms it has will be ionized and the depletion zone will extend throughout the I region and include a small penetration into both the P and N regions. Because of the heavy doping in the P+ and N+ zones the depletion zone will not extend very far into them, and the depletion

16. 1 THE PCN DIODE-

EXTE SIO

OF THE P JUNCTIO

1061

zone will be e entially equal to the I-layer width, W1• The alternate diode tructure, P+. rr, + i hown chematicaUy in Fig. 16.ld. Here the depletion zone width i likewise approximately equal to the width of the intrin ic layer, but the junction i formed at the + interface rather than that of P+. Controlling the location of the junction ha important con equence from the standpoint of pa ivating the diode chip, but no impact on performance. Mo t PIN diode u e v material for the I region and the junction i formed at the P+ interface.

16.1.2

C (V) Law and Punch-through Voltage

In the preceding ection it was a urned that the I layer is of such high re 1st1viry that even with no applied bia , the depletion zone extend across the I )ayer to the P+ and - + zone . Under uch circurn tance C 1 is practically independent of applied voltage. At zero voltage the depletion zone has already extended through the I region; a further reverse bias is applied to the diode , little further widening of the depletion zone proceeds because of very high impurity concentrations and corre pondingly large availability of ionizable donor and acceptor in the P+ and N+ region . The PIN diode wmch actually does have so high a resistivity I layer that it is depleted at zero bias is called a zero-punch-through diode, because the depletion zone has .. punched through" to the high-conductivity zones even before bias is applied. Such a situation, however, represents an idealization. Not all practical diodes are zero punchthrough. A more general definition of the PIN is a semiconductor diode which consists of two heavily doped P and N regions separated by a substantially higher-resistivity P or N region. Figure 16.2 shows chematically a practical PIN diode with ionized impurity profiles at zero bias and at punch-through. At zero bias a large portion, but not necessarily all, of the I-region impurities have been ionized and the depletion zone, W(O), may be somewhat less than the I-layer width, W . As reverse-bias voltage is applied to this diode, depletion layer spreading occurs, and the capacitance, hown in Fig. 16.2b , decreases until the depletion layer has spread definitely to the N+ region, as shown in Fig. l 6.2c. At this voltage the depletion layer width, W ( Vp-r), is approximately equal to W1. Further spreading of the depletion layer into the low-resistivity P+ and N + regions is, for most applications, negligible. The voltage at which the depletion zone just reaches the + contact is the punch-through voltage, Vp-r . Becau e in practice the resistivity levels in the P+, I, and N+ regions do not change abruptly, the resulting capacitance versus voltage characteristics have a soft knee. Therefore, the punch-through voltage is not directly measurable with precision. However, the practical diode usually does have two definable slopes in its C( V ) characteristic, when plotted using semilog paper as shown in Fig. 16.2d. By convention, the voltage intersection of these two straight-line projected slopes is called the punchthrough voltage. It is to be emphasized that this C(V) characteristic is what one obtains when the measurements are made at relatively low frequencies, typically 1 MHz. At microwave frequencies, the dielectric susceptibility of silicon is much larger than the conductivity of v or rr material; thus the capacitance is effectively equal to the minimum capacitance for all values of reverse bias, as is shown in the following discussion of dielectric

relaxation.

1062

SEMICONDUCTOR CONTROL DEVICES : PIN DIODES Practical PIN diode

(a)

w,

J (b}

0

I (v)

P+

Zero bias ionized impurity profile

W (O )

N+

J

~

+ ++++++

-

0

X

I I I I I I I I

n;=; I

(c)

Ionized impurity profile at punch thru

-

(d)

C-V Characteristic measured at 1 MHz.

W ( Vpn

I I

ii I

+ + + + X

0

\

\ \

C(V)

4-..---)

-o V

1 I - -

C \

\

\

\ \ \

--------- ----~---~ ------ ----~---\

' I I I

0.1

1

VPT

10

100

1000

( V- c/>) (Volts)

Figure 16.2 Practical PIN and reversed punch-through characteristics.

16.1.3 Capacitance Measurements and Dielectric Relaxation If the capacitance of a PIN diode that does not punch through at zero bias is measured at zero bias, a larger value of capacitance will be measured at a low frequency (such as 1 MHz) than would be measured at microwave frequencies (such as with a network analyzer measurement at 1 GHz). The reason is that silicon, in addition to being a variable conductor, also has a high dielectric constant. Therefore, its bulk differential

16.1 THE PIN DIODE- AN EXTE SIO

OP THE PN JU CTION

1063

equivalent circuit appear as a parallel combination of conductance and capacitance. The relative current di i ion between the e two equivalent-circuit parameters varies with the frequency of the applied ignal. higher-frequency currents being carried mostly by the capaciti e path. To illu trate thi point, con ider Fig. 16.3, which how a PIN diode below punchthrough. The portion of the P+ and the I regions that are depleted repre ented the depletion zone, or '' wept region." The remainder of the I region is "unswept" and can be modeled, as hown in Fig. 16.3c. as a parallel re i tance-capacitance circuit, represented by the equi alent-circuit elements. Cus and RusThe divi ion of current through Cus and Rus depend on the ratio of the susceptance of Cus to the conductance (1 / Rus). This ratio in turn depends on the dielectric constant of ilicon to its bulk resi tivity. The frequency at which the current division between the e two elements i equal (i.e .. when the usceptance is equal to the conductance) is defined as the dielectric relaxation frequencies, f R, of the material. When the operating frequency, f, i equal to or greater than 3/R, the total capacitance repre ented by the erie combination of Csw and Cus is approximately equ al

(a) PIN model

°71

P+

I

I I

I

I

I

I {v)

I

[:Swept] I

I

I

••

•.&

Re,

Rp..-

(c ) Detailed equivalent circuit

I

I 1 Cus1

/ 1I '\M,

I

I

I I I

X

I

I I I I I

Depletion zone before punchthru

•0 •

I

I I

•• • • •

(b)

Unswept region

I

region

•

~

N+

I • I

\

I

I [ 1 GHz major : _ current pa:h I

I

'

Csw

Rr Rus

/.RN+

l . Rc2

1 MHz major current path

--.c

Cus

C sw 0--J\/-"'-'-A...,._I

:

- - f0

Rus

(d) Low freq. simplified equiv. CKT.

(e) Microwave equiv. CKT.

o--""'..f'"""""'·---111----0 CJ

Figure 16.3 Reverse-biased PIN equiv alent circuit.

1064

SEMICONDUCTOR CONTROL DEVICES : PIN DIODES

to C 1 (within 10%), the parallel-plate capacitance of the totally depleted I region. Th.is value corresponds to the minimum capacity CMIN measured beyond punch-through at low frequency . This point is a major one in the practical characterization of PIN diodes intended for microwave switching applications. It means that practical measurements of the capacitance of a PIN junction can be made at 1 MHz, and the values so attained will represent a good approximation to the actual capacitance applicable at microwave frequencies. This test only requires that sufficient bias voltage is used during the low-frequency measurement to ensure that the I region is fully depleted. A check to determine whether the I region is in fact fully depleted can be made simply by plotting the C(V) characteristic for a few representative diodes from the production lot to determine at what minimum bias voltage the measured capacitance reaches what i s essentially its minimum value. The remaining required quantity to determine the applicability of the low-frequency CMIN as a representation for the microwave capacitance, C 1 , is an estimate of the relaxation frequency for the I region of the diodes being measured. High-purity silicon material used to make PIN diodes typically has resistivity in the range 500 to 10,000 O-cm prior to the diffusion and/or epitaxial growth steps used to achieve the low-resistivity P+ and N+ regions. However, after the high-temperature processing needed to realize these regions, the resistivity of the I region is always less than that of the starting crystal. Typical values for I-region resistivity are in the range 100 to 1000 O-cm. The dielectric relaxation frequency for the unswept portion of the I region can be written in terms of the equivalent-circuit parameters, directly from the definition, which requires that the conductance and capacitive susceptance be equal at JR. The result is

1

fR=--2rr Rus Cus

(16.1)

In turn, the specific values for Rus and C us can be written in terms of the length, L , and the area, A, of this unswept region together with the bulk resistivity, p, and the absolute dielectric constant, EoER, as follows : pL

(16.2)

Rus=A C us =

EoE RA

L

(16.3)

Substituting these expressions into Eq. (16.1) together with the value ER = 11.8 for silicon yields Eq. (16.4), which gives the dielectric relaxation frequency directly in gigahertz when the resistivity, p, is known.

I

f R = - --

2rrEoERP

fR

153

= p (n~,-cm)

. gigahertz

(16.4)

This expression is shown graphically in Fig. 16.4. Strictly speaking, since the final resistivity of the I layer of a practical diode depends on the actual processing steps used to fabricate the diode, one could not know beforehand what dielectric relaxation

1065

16. 1 THE PIN DIODE - AN EXTENSION OF THE PN JUNCTION

N

J:

-

(9

1

>, 0

C

a> ::,

CT

-... a>

C

0

.::i

N = 1014

2,000

N

E

0

~

::i

~ :.0 0

E

1,000 800 600

C

e

0

400 N= 1018

Q,)

w

200 N = 1019

100 - 50

0

50

100

150

200

Temperature (°C)

1,000 800 600 400 c,j Q,) fl')

... 0 I

> ;;E 0 it

::i

~ :.0 0

N = 1017

200

N = 1018

100

80 60

E Q,)

0

40

:t

N = 1019

20 10 ._____________...__ _ _ __.__ _ _ _ _ _ _ _ __

-50

0

50

100

150

200

Temperature (°C)

Figure 16.8 Hole and electron mobilities versus temperature for various impurity densities in section. (After Ref. 1).

16.2

-E

:::1.

MICROWAVE EQUIVALENT CIRCUIT

1071

I ',=Tp

100

.s:::

'O

~

C 0 0,

about - 100 V, is sufficient to hold off conduction of the diode under the application of an RF voltage whose peak voltage amplitude is as large as 1000 V. Again, the brief duration of the half-period of the RF cycle is not sufficient to cause appreciable modulation of the I region of the diode, and the diode appears as a high impedance event with this large voltage magnitude applied. One might ask why any reverse bias is necessary at all if the diode is nearly nonconducting at zero bias. First, reverse bias fully depletes the I region and its boundaries of charge. Thus the diode has a higher microwave Q with reverse bias. Secon~ the role of a reverse bias is to maintain an average field which tends to prevent the accumulation of significant amounts of charge in the I region. The presence of excessive charge in the space, under high RF fields, can produce impact ionization, with a "runaway" current rise and resultant diode destruction. Nevertheless, under large RF excitation, impact • R. Ryder (BelJ Telephone Laboratories, Murray Hill, New Jersey) in a talk given at the NEREM Conference

in Boston, circa 1970.

16.2

11CROWAVE EQUIVALENT ClRCUlT

1077

ionization effect are often ob erved, re ulting in a pulse leakage current, ince it occur only under the combined action of RF and rever e-bia excitation. le i nece ary that the driver circuit have sufficiently low impedance to be capable of providing thi ' pul e leakage current (u ually 1 to 5 mA) in a high-power control device without cau ing an appreciable drop in the bia voltage upplied, if de tructive diode conduction in the rever e-bia tate with high RF applied voltage i to be avoided.

16.2.2

Forward-Biased I-Region Resistance

Having demonstrated the uitability of the charge control approach for determining microwave propertie . let u use it to calculate the conductivity and re i tance of the I region under forward bias. Conductivity. a, i a bulk property equal to the ratio of current den ity. J. to applied electric field trength, E : J

(J

= -E

(16.23)

But J is the directed average rate of flow of electric charge. In terms of I-region holes and electrons, J (Vp·p VN· n) (16.24) a=-=e

E

-E- + -E-

Al o, by definition, mobility, µ, is the average carrier velocity per unit of applied electric field: thu (16.25) where

e

= +1.6 x 10- 19 C = magnitude of electron'

charge µ,p N = mobility of holes and electrons, respectively p. n = re pective, injected hole and electron densities in I region

The formula for the resistance of a cylindrical conductor of electrical conductivity, a, length W along the current path, and cro s-sectional area A is [8]

w

R= aA

(16.26)

Using the dimensional notation of Fig. 16.10, the I-region resi tance i then

w

R1 ~ - - - - - eA(µ,p p + J.LNn )

(16.27)

Three main assumptions* have been made in this derivation of R1: I. The I region as a whole is electrically neutral. 2. The bias current, Io, injects holes and electrons that recombine with each other in the I region; the limitations of this assumption are discussed later.

• N. H. Fletcher, "The High Current Limit for Semiconductor Junction Devices," Proc. fRE, Vol. 45, pp. 862- 872, Jun 1957.

1078

SEMJCONDUCTOR CONTROL DEVICES: PIN DIODES

3. The carrier lifetime is sufficiently long that both the holes and electrons are uniformly distributed within the I region. Another way of stating this point is that the average hole and electron diffusion lengths, L p and L N , are much longer than the I-region width, W . This condition is usually valid for weU-designed PIN diodes and can always be verified by using the following relation for diffusion length: (16.28) where DAP the ambipolar diffusion constant= 2DpDN/ (Dp life-time within the I region.

+ DN) and r: is the

In silicon, DAP has an effective average value for boles and electrons, the ambipolar diffusion constant, of 15.6 cm2 /s [9]. Thus

L- { -

40✓r (µs)

µm

l.7✓r (µs)

mils

(16.29)

For example, if the bulk lifetime is 10 µs the diffusion length is about 133 µm (5 mils).* Under these combined assumptions, it follows that the injected bole and electron densities are equal and uniform: p=n (16.30) and, furthermore, since they recombine with one another directly, (16.31) Then

w

R1=---2eAµ,APp

(16.32)

where /J,AP = 2µ,pµ,N / (µ,p + µN ), 610 cm2 / V-s in silicon [9], is the ambipolar mobility (i.e., the effective average of the hole and electron mobilities). But the injected charge is directly proportional to the biase current: Qp = epAW = for:

(16.33)

Combining the last two equations gives

w2

Ri=-- 2 µAPr: lo

(16.34)

This expression is applied frequently. We note from it that R1 is theoretically independent of the I-region area, being proportional to the square of the I-region width and varying inversely with mobility, lifetime, and bias current. However, care must

*Foran analysis of the case where this assumption is not made, see Leenov's paper, Ref. 9.

16.2

MlCROWAVE EQU JVALE T C lRCUlT

1079

be taken in the application of Eq. (16.34) to practicaJ situation . In particular, the following generalization bouJd be quaJjfied: 1. Holding all process steps the same except for va0 •ing A produces a selection of diodes with different capacitances but the same R 1 for a given bias current. This ituation i true onJy if r remain con tant; but generally, r decrea e with a decrea e in A. ince I-region carrier are then nearer the periphery, where recombination can occur more rapidly.

2. R , decreases as (1 / Io). Again, thi tatement hold true only a long a r remain con tant. Howe er, a Io increa es. carrier den ity increa e , and the recombination probability increase , decrea ing r. Furthermore, aturation is reached when p and 11 increase ufficienUy that ub tantial injection (holes into the N+ region and electron into the P+ region) become ignificant, in violation of the econd as umption u ed to deri e Eq. ( 16.34). Put imply, if there are high densities of electrons and hole in the I region, their chance for recombining increase , decreasing the a erage lifetime, r . 3. Above the transit-time frequency, R1 is essentially independent offrequency. This tipulation i only approximately true for most microwave PIN applicatio ns. Skin effect cau e both the contact and I-region re i tances to increase somewhat with frequency . De pite the e limitations, Eq. (16.34) i very useful and is typically invoked to estimate I-region re istance at microwave frequencies. For example, consider a PIN with a 100-µm (4-mil) I regjon and a 5-µ s lifetime operated with 100 mA bias current. U ing µ ~ 610 cm2 / V- g1ve u 10- 4

Ri

=

(2)(0.1 A)(S x I0- 6 )(610 cm2 / V-s)

= 0.16 Q

(16.35)

This result is in reasonable agreement with the measured value of 0.3 Q for a 1.56mm (61-mil) diameter, when one considers that the mea ured vaJue include resistive contributions of the ohmic contacts as well as those of the P+ and N+ region . Furthermore, the lifetime at 100 mA is likely to be less than the 5-µs vaJue which is measured at 10 mA - an additional factor contributory to a higher measured resistance than that calculated. Using this example, let us examine the role of skin effect in the forward-biased I region. Using the parameters of the example above and solving Eq. (16.26) gives a = 3 (Q-cm)- 1• The skin depth, 8, in a conductor is given by [8] (16.36)

= operating frequency (hertz) µ 0 = 41r x 10- 9 H/cm = free-space penneability a = conductivity (Q-cm) - 1 From Eq. (16.36), the skin depth for a = 3 (O-cm)- 1 at 1 GHz is 0.09 cm, about equal where f

to the diode radius. This diode example has a junction capacitance of about 2 pF and

1080

SEMICONDUCTOR CONTROL DEVICES: PIN DIODES

would not usually be used at frequencies much above 1 GHz. At higher frequencies a lower capacitance, and hence reduced diameter, would be employed. Thus it can be seen that I-region* skin effect usually has but a moderate effect in PIN control devices in the frequency range 0.1 to 10 GHz. Before leaving the subject of I-region conductivity it is interesting to note what level of carrier density, p, was injected into the I region of this sample diode to produce R1 = 0.16 Q. An estimate can be made using Eq. (4.32) andµ, ~ 610 cm-2 /V-s; thus (16.37) Since there is an approximately equal electron density, n, in the I region, the total free carrier density required to produce R1 = 0.16 n is 3.4 x 1016 cm- 3 . Recalling that the atom density is about 1023 cm-3, this figure represents less than one carrier per million atoms. It is therefore easy to see why the skin depth, so significant with metallic conductors at microwave frequencies, has only a moderate effect even under "high injection" levels in the I region of the PIN diode. 16.2.3 RR and C1 Reverse-Biased Circuit Model

Under reverse bias the I region is depleted of carriers and the PIN appears as an essentially constant capacitance to a microwave signal. The presence of dissipative losses can be taken into account by either a series or a parallel resistance element in the equivalent circuit. In a well-made PIN, the I region has sufficiently high resistivity that most of the dissipation under low RF power conditions occurs in the ohmic contacts made to the diode and in the resistances of the P+ and N+ regions. Accordingly, a fixed series resistance, RR, used to represent these losses can be expected to offer an equivalent-circuit model that is ~applicable over a broader bandwidth than a parallel conductance. In any event, due to the ratio of diode capacitive reactance to practical RF circuit impedances, the dissipative losses of the PIN under reverse bias are usually much smaller than those under forward bias; thus the choice of series or parallel R- C equivalent circuit under reverse bias usually can be made according to whichever offers greater computational convenience. Because of the high relative dielectric constant for silicon (ER = 11.8), the fringing capacitance (in air) around the I region is relatively small and the capacitance calculated using the parallel-plate capacitance formula given below provides a useful estimate of junction capacitance, C 1 . Thus (16.38) where

Eo €R

D W

= 8.85 x 10- 14 Flem= free-space permittivity = 11. 8 = relative dielectric constant for silicon = junction diameter

= I-region thickness

• Skin resistance may be more important in the P and N regions and in the leads attached to them because it affects how the currents enter the I region.

16.2

MICROWAVE EQUIVALENT CfRCUTT

1081

1.0

G:'

/ 2)

(17.9c)

Z

B2

(17.9a)

=0

(17.9d)

Class 3 (CC): Z

= Z 0 cos(/ 2)

(17.l0a)

0 =rr/ 2 B1

17.2.3

=-

B2

(17.l0b)

= Yo tan(/ 2)

(17.lOc)

Theoretical Performance under Ideal Switching Conditions

Class 2 (LIU) Phase Shifters. Figure 17 .2 shows the calculated performance characteristics of class 2 (LIU) loaded-line phase shifters based on Eq. (17.9a)-(17.9d). The phase error is calculated as the absolute deviation from constant phase. It must be noted that the phase error, with this design, will be significantly lower if the requirement is for true time delay rather than constant phase. Table 17 .1 lists the bandwidth of lossless 45°, 22.5°, and 11.25° class 2 phase shifters with maximum allowable VSWR and phase error as parameters. The bandwidth is defined as percentage bandwidth = 200.0 x

(/2 - /1)

(f

l

+

f:

2)

(17.11)

1102

SEMICONDUCTOR CONTROL DEVICES: PHASE SHIFTERS AND SWITCHES

1.5

12.5

I &1 < 5°, VSWR ~ 1.5

I I I I I I I

Z c= 45fl, 01 = 70°, 02 = 42° 10.0

PS2 optimized for l8tJ>I < 2°, VSWR ~ 1.2

Z c= 46fl, 01 = 68°, 0 2 = 44°

I

1.5

1.4

a:

~ 1.3 en

>

/ ps1

/

I

/1 / /ps1

5.0

1.2

2.5

1.1

0.85

0.90

0.95

1.00

1.05

1.10

Normalized frequency

Figure 17.4

Performance characteristics of two class 3 (CC) 45° type 2 phase shifters.

(b)

(a)

Cp

Figure 17.S PIN diode equivalent circuit: (a) reverse bias; (b) forward bias.

17.2.4 Design Considerations Using PIN Diodes The ability of a PIN diode to change from a low resistance under forward bias to a low-loss capacitor with reverse bias makes it attractive for phase shifting and switching applications. Figure 17 .5 shows the equivalent circuit of a packaged PIN diode in the forward- and reverse-biased states. When reverse biased the junction capacitance, Ci, is given by (17.12)

17.2

LOADED-LINE PHASE SHIFfERS

1107

where W1 is the width of the I-layer, E the permeability, and A the cro s- ectional area. In the forward-biased tate the erie resi tance, Rs, i the um of all the re i tances of the undepleted ilicon region and the contact re i tance of both ohmic contact . The parasitic , as ociated with the packaging, contribute to C P and Ls. For well-designed diodes Rs range from a fraction of an ohm to 3.0 n, depending on junction ize. Since the reactance as ociated with Ls i generally mall, the PIN diode can be repre ented as a fixed-value low-lo capacitor in the re er e-biased tate. Thi is particularly true in the low-frequency microwa e region. In the forward-biased tate the re i tance, RI , i given by the um (17.13) where Rst i a con tant re i tance. which i generally the same as Rs for abrupt junction diode and R1 i the resi tance a ociated with the intrin ic region, given by (17.14) with JI being the forward-biased current, µ, the average carrier mobility, and TL the carrier lifetime in the I -region. For ufficiently large / 1 , the intrinsic resistance R 1 can be made negligibly mall so that R I approache Rst. However, this is accompanied by increased de power di ipation and decreased switching speed. For low-voltage PIN diodes a bias current of IO mA i uffic:ient to make RI ~ Rst.

Design Neglecting Losses. When the diode series ON resistance is 2 n or less, the initial de ign i determined by neglecting the series resistance and using the simplified equivalent circuits hown in Fig. 17 .6, for the diode, consisting of j X f in the ON state and - j X R in the OFF state. Class 2 (LIU). The main tran mission-line parameters (Zand 0) and the loading stub parameters (Z s and Bs) completely define UU-type phase shifter. The stub parameters are not unique and given by (17.15a) where

Ys = 1/ Zs Bs =

ON

(17.15b)

b + b(l + 8) 112

(17.15c)

2 (1 + 82)

OFF

Figure 17.6 PIN diode simplified equivalent circuit.

1108

SEMJCONDUCTOR CONTROL DEVICES: PHASE SHIFfERS AND SWITCHES

b

= Y0 (1- X J BR)(2 tan / 2)

(17.15d)

BR= - 1/ XR

= 4BR(l + 82)(2 tan(/ 2)]/ b 2 82 = X1 [2tan( / 2) + BR]

(l 7.15e)

81

(17.15f)

The main transmission-line parameters are given by

Bn tan(/ 2)

= 0 = (n + )/2

Z

+ [1 + B; sec2(/ 2)] 112 1 B2 + n

(17.16a) (17.16b)

where (17.16c)

Class 3 (CC). The design equations defining the matching section parameters Zm, 0m and the stub parameters Zs, 0s are given in Table 17.3 for four important stub configurations. The main-line parameters Z , 0 are assumed to satisfy Eqs. (17. lOa)-(17.lOc). Again, the design parameters are not unique.

17.2.5 Design with GaAs FET Switches GaAs FETs are three-terminal devices, as shown in Fig. 17.7. In general, they are characterized by higher ON resistance and larger OFF capacitance compared to PIN diodes. As a result, phase shifter designs must take the finite ON resistance into account, and closed-form design equations are cumbersome. Although the physical nature of the GaAs FET device warrants a complex equivalent-circuit representation, for almost all practical design purposes, the GaAs FET may be represented by a parallel RC circuit with a separate set of R and C values in the ON and OFF states. Under lossy conditions, a perfect match in both phase states is generally not possible. The following equations serve as a starting point for the design under lossy conditions.

= Zo[cos(/ 2)/ sin0]/(1 - t:..2 ) 1/ 2 B; = Yo[cos0 sec(/ 2)(1 - t:..2) 112 ± tan(/ 2)] t:.. = G ;Zo cos( / 2) Z

(17.17a) (17.17b) (17.17c)

In the design of loaded-line phase shifters, with lossy switching devices, the designer is faced with the problem of finding an impedance transforming network that transforms respectively. The the two impedances Z 1 and Z2 of the switching device to Z~ and abed parameters of such a lossless transforming network are given by the following equations [9]:

z;,

a =ad

(17.18a)

= {3d

(17.18b)

c=yd

(17.18c)

b

d

= l / (a + /Jy)112

(17.18d)

17.3

REFLECTlO

PHASE SHIFTER

1109

TABLE 17.3 Design Equations Using Nonideal but Lossless Switching Elements Stub Configuration Zm, Bm

De ign Equation° Zs, 85

--c:::r-r. Senes . sw,tc ~ h

Series witch

Zm =

-

short-circuit stub

Xtm(X R - XF)l /2 [X R - X p - 2 X (1 + t,;,)]1 / 2

For hort-circuit tub

For open-circuit stub

Series switch open-circuit stub

_+_Z_ s _ Z; tm + Z m(XR - X ) Zm + Xtm tan Os

+ XX Rtm

Shunt switch

[( BR - BF) - 2B(1 + tm) 2]l/ 2 Zm = ______ ..:,___;_~:_ Btm( BR - BF) 112

1

For short-circuit stub

T

Shunt switch short-circuit stub

_

+zs tan 0s -

Ym + Btm Y~lm + Ym( BR - B ) + BBRtm

~---------

For open-circuit stub

~'---------.J

i

Shunt switch open-ci rcuit stub

where et= R~ - y(Ri X1

+ X ~R1)/ R1

= x ~ + y(R1R; - X1XD - aX1 y = (R~R2 - R~R1)/[R2(R~X 1 + x;R 1) Z1.2 = R1 .2 + j X1,2 Zi,2 = R~.2 + j X~.2 /3

(l 7. 18e) ( l 7. 18f) R1(R; x 2 + x ; R2)] (17.18g)

(17.18h) (17.18i)

17.3 REFLECTION PHASE SHIFTER The basic reflection-type phase shifter [10-15], also known as a hybrid coupler phase shifter, consists of variable or switched impedance elements Za and Zb terminating

1110

SEMICONDUCTOR CONTROL DEVICES: PHASE SHIFfERS AND SWITCHES

Figure 17.7 GaAs PET switch with resonating loop inductor.

(a)

v,~

□

Za

-

(b)

v, ~ M

Za

Figure 17.8

~

□

~

Vo

zb

Vo

M

Zb

Reflection phase shifter: (a) basic configuration; (b) configuration with match-

ing network.

the coupled ports of a 3-dB quadrature hybrid. The output is from the isolated port of the hybrid, as shown in Fig. 17.8a. The reflection design is most appropriate in applications where the desired phase hifts exceed of 45°. In a multibit digital phase shifter, the larger (90° and 180°) bits typically employ a reflection-type design.

17 .3

17.3.1

REFLECTION PHASE SHIFfER

1111

Principle of Operation

In Fig. 17 .8a the output V0 from the hybrid is given by (17.19) where V; is the input signal and r O and r b are the complex reflection coefficients corresponding to the terminating impedances Z0 and Zb, respectively. In practice Z 0 = Zb = Z, where Z is made up of a witching device connected to an open- or shortcircuited stub. If Z I and Z2 represent the impedances of the combination (of the switching device and tub) in the ON and OFF tates, the foUowing equality must be satisfied for proper phase shift operation: (17.20) where 6.. is the desired pha e shift. Neglecting the losses in the hybrid, the phase shifter loss is then given by loss (dB) = 20 log 1r 1

(17.21)

where 1r 1= lf'd = II'2IWhen the switching device is ideal, the desired phase shift is realized simply by switching a pair of open-circuited stubs of electrical length bi. / 2 in and out of the circuit. However, in practice Eq. (17.20) is not satisfied without introducing an impedance transforming circuit as shown in Fig. 17.8b . The purpose of this network, M, is to transform Z I and 2 2 to Z~ and such that

z;

(17.22)

When Eq. (17 .22) is satisfied, the reflection coefficient at the input is the same as that of the hybrid alone. The principal advantages of the reflection phase shifter are: a. Only two switching devices per bit are required. b. The input match is dependent only on the design of the hybrid, thereby allowing independent optimization of the phase shifting terminations for best phase shift and insertion loss characteristics over the desired bandwidth. However, in practice, the nonideal characteristics of the hybrid complicate the optimization process. c. Unlike the loaded-Jine phase shifter, the incremental phase shift of a reflection phase shifter is not limited to small values.

17.3.2 Design with Ideal Switches Table 17 .4 shows common stub configurations along with the required stub lengths, for proper phase shift, under ideal switching conditions.

17.3.3 Nonideal but Lossless Switch PIN diodes with very low ON resistance fall under this class. The design formulas for the basic configurations are listed in Table 17.5.

llU

SEMICONDUCTOR CONTROL DEVICES: PHASE SHIFfERS AND SWITCHES

TABLE 17.4 Common Stub Configurations for Reflection Phase Shifter with Ideal Switching Devices0 6/2

1

Series switch open-circuit stub

__.....-I.._____~ 90 ° - !).Cl>/2

2

Series switch short-circuit stub

-41

____.,--1....._ _ _ 90 ° -6/2

3

1 1

Shunt switch open-circuit stub

...

4

0

6/2

Shunt switch short-circuit stub

l

6 is the smallest number representing the absolute phase shift

TABLE 17.5 Design Equations for Reflection Phase Shifters Using Nonideal Lossless Switc.hing Devices

No.

Stub Configuration

1

_,---1'-------flZs, Os Series switch short-circuit stub

2

Defining Equations

...

I Zs, 6s

-f

.,_....-

X s = Zs tan 0s

X; +bXs +c = 0 b = (XF + X R) Zo

Xs = Zs cot 0s

Series switch open-circuit stub

3

Xs

= Zs tan 0s

B; +bXs +c = 0 b

=

Zo XF

Shunt switch short-circuit stub

4

Shunt switch open-circuit stub

+

Zo ZR

17.3

17.3.4

REFLECTION PHASE S HIFTER

1113

Nonideal and Lossy Switching Devices

As mentioned in Section 17 .3. 1, a tran forming network is generally required for reflection phase hifter u ing nonideal switching device . By means of an elegant bilinear tran formation Atwater [ 13] tran lated the problem of the reflection phase shifter design into that of an impedance-matching problem. In Ref. 13 an expre sion is deri ved for a fictitious impedance, Zm, uniquely characterized by the phase shifter parameters Z 1, Z2. ; therefore, when a matching network tran forms Zm to Zo (the characteristic impedance of the hybrid), it al o transforms 2 1 and 2 2 to Z ~ and satisfying Eq. (17.20). A graphical method of determining Zm has been described by Watanabe et aJ. (14]. The matching cheme would be impractical if the solution for Zm does not exist or if the real part of Zm i negative. Although any network that matches Zm to Zo will result in proper pha e hift operation at the design frequency, the bandwidth depends on the characteristic of the witching device, matching networ~ and the hybrid itself. Thus a olution satisfying Eq. (17.20) hould be considered as only a starting point for further optimization using computer-aided de ign (CAD) techniques. Switching devices with a large o, re i tance and/or a large OFF capacitance generally result in poor bandwidth characteristics. In the case of GaA FETs, this problem may be alleviated by using an inductor to re onate the OFF capacitance. At millimeter wave frequencies, the Qfactor of a line inductor may be tailored to balance the ON and OFF insertion loss characteristic.., [15] . Figure 17.9 is a photograph of an X-band 5-bit monolithic digital phase hifter incorporating resonating line inductors. This phase shifter incorporates a loaded-line design for the 11 .25° 22.5°, and 45° bits and reflection designs for the 90° and 180° bits. The re onating inductors also keep the source and the drain terminals of the switching FETs at ground potential without any vias. This is a very important yield consideration in the fabrication of monolithic phase shifters. In the loaded-line bits, the source and drain are tied together with a short section of very lossy line fabricated as part of the monolithic process. The purpose of the lossy lines is to equalize the insertion loss in the ON and OFF states. The design equations for these line inductors are presented next.

z;,

17.3.5

Design of Resonating Inductors

The design equations for wire, ribbon, and spiral inductors are given in Ref. 16. The application of such inductors is generally limited to the lower microwave region. For monolithic millimeter wave applications the resonating inductors are conveniently made in the form of high-impedance microstrip transmission lines. Figure 17.10 shows such a U-sbaped line inductor along with two equivalent representations. In the first representation, the curved part of the inductor is treated as an uncoupled line of length rrr, with r being the mean radius. The curvature effects are neglected in this representation. In the second representation, the curved portion is replaced by an equivalent short-circuited coupled section of length rr r / 2 and spacing 2,. This leads to an overall simplified representation in terms of a single short-circuited coupled transmission line. The input impedance of such a transmission line is given by (17.23) where z00 and Z is a constant potential of the diode. Hyper-abrupt varactor diodes may be fabricated with a constant gamma over a limited voltage range with a high-capacitance variation. If a reflection phase shifter is built using two such diodes, terminating the direct and coupled ports of a quadrature coupler with characteristic impedance, Zo, then the phase shift, ct>, is given by (17 .38)

where Xd is the normalized diode reactance given by

Xd=

I (l+V/ cJ>)Y wCZo wCjoZo

(17.39)

and w is the radian frequency. Diodes with gamma values greater than unity have nearly linear phase shift characteristics as a function of applied bias voltage. Thus the problem reduces to that of fabricating diodes with a large gamma that remains constant over the desired range of applied voltage.

1126

SEMICONDUCTOR CONTROL DEVICES: PHASE SHIFfERS AND SWITCHES

17.7 MISCELLANEOUS CONSIDERATIONS 17.7.1

Biasing Schemes

The switching devices require de bias for turning the device ON and OFF. The bias must be applied such that the de path does not short circuit the RF signal or otherwise result in degraded RF performance. In practice the bias is applied through a choke inductor that blocks RF while acting as a de short. Lumped inductors are also used at low frequencies. A short-circuited quarter-wavelength stub is convenient means of implementing the biasing choke at microwave frequencies. This could take either stripline or a microstrip form. Figure 17.16 is a loaded-line phase shifter bit demonstrating the biasing arrangement commonly used in microstrip implementations. The de ground return is provided through a plated-through hole at the end of a quarter-wave stub AB. CD and C' D ' are matching stubs which in combination with the open-circuited stubs DE and D' E ' provide proper phase shift. PQ and QR, which are both a quarter-wave long, are used to provide proper biasing without affecting the RF performance. Since PQ and QR are a quarter-wavelength, the open circuit at R is reflected as an RF short circuit at Q and an RF open at P . Thus the biasing sections PQ and QR do not present any load at P . The location of point P can be anywhere along DE as long as PQ is a quarter-wave long. One limitation of this biasing scheme is its inherent narrow bandwidth. However, the bandwidth can be increased by making QR a radial line stub. Biasing of switching FETs is somewhat simpler since RF isolation can be provided by placing a large resistor between the gate terminal and the de bias control line. This is practical because the voltage-controlled FETs draw neglegible current, on the order of a few tens of microamperes at the most. The monolithic 5-bit phase shifter shown in Fig. 17. 9 makes use of such a biasing scheme. Figure 17 .17 is a photograph of a 3-bit mi11imeter wave monolithic phase shifter ultilizing similar biasing scheme. Another

A

C

C' R

Q D

D'

P'

p De ground return

E

E' To bias control

Figure 17.16

A typical phase shifter biasing network.

17.7

1fl CELLANEOUS CO S IDERATIO S

1127

Figure 17.17 M onoLilhic millimeter wave 3-bit pha e hifter.

feature to be noted in this photograph i that the de return to ource and drain, which are tied together by means of re onating inductor , is provided through a quarter-wave tub grounded through a via hole, hown black in the figure. 17. 7 .2

Power Handling

The maximum RF power that a PIN diode can handle is }jm.ited by either the diode breakdown voltage or it therma] di ipation capability, in most cases the latter. In a good de ign, a phase shifter can handle RF power levels many times greater than the power di ipated in the diode. The maximum power, Pd, that a PIN diode can dissipate is given by (17.40) where

Tjm T0

= maximum operating diode junction temperature = ambient temperature

e jc , 0 ca = thermal resistances from

the junction to case and case to ambient,

re pectively The diode package design and diode mounting determine 0jc and E>ca, respectively. The ratio of incident power to dissipated power is a function of the pha e shifter design and the diode ON/OFF impedance characteristics. A simplified expression for this ratio [57] applicable to a double-pole double-throw switch is given by (17.41)

1128

SEMICONDUCTOR CONTROL DEVICES : PHASE SHIFfERS AND SWITCHES

where Pi Pd R1

Zo

= average incident power = average dissipated power = ON resistance of the diode = characteristic impedance of the line

The peak power, Pmax, that a shunt diode can control is given by 2

_

P max -

( VBR -

VBIAS )

8Zo

(17.42)

where V8 R is the breakdown voltage of the diode and VsIAs is the magnitude of the reverse bias applied to the diode. An applied bias of zero volts results in maximum controlled power. However, this may not be practical in fast switching applications where reverse bias is required to reduce the switching time. Further, a zero reverse bias may result in excessive current through the diode during positive RF voltage swings. The expressions given in Eqs. (17.41) and (17.42) are only approximate. It is best to simulate and analyze the complete phase shifter circuit, or at least one complete bit, to determine the power dissipation in, and voltage swings across, the diodes under worst-case conditions. Loaded-line and reflection-type phase shifters can generally control much higher power compared to switched-line phase shifters. Power-handling considerations for reflection-type phase shifters have been treated by Burns and Stark [58]. The Sparameters of loaded-line phase shifters can be conveniently determined in terms of odd- and even-mode impedances, and the dissipated power in the diodes may then be obtained, by neglecting the line losses, as (17.43) where Pi is the incident power.

17.7.3 Switching Speed The switching speed of semiconductor-controlled phase shifters is generally much higher than those of ferrite phase shifters. GaAs FET devices can switch much faster than PIN diodes, due to the very high mobility of the carriers in a GaAs FET device. In determining the operational speed of a phase shifter, one must consider various delays - driver delay, driver rise time, and PIN diode delay-as well as the diode switching time, which is the time required to change the state of the I-region from "no stored charge" high impedance to "large stored charge" low impedance, and vice versa. A typical PIN diode switching waveform is illustrated in Fig. 17 .18. It shows the various contributions [59, 60] to the delay time. The reverse to forward switching time of a PIN diode is typically a few percent of the specified lifetime of the diode, and is much smaller than switching time from forward to reverse. When in the high-impedance state the 1-V characteristics of the diode are inductive and the driver circuit must deliver a current spike, with substantial overvoltage, in order to reduce the switching time. This is generally accomplished by incorporating a "speed-up" capacitor in parallel with the dropping resistor at the output of the driver circuit. When switching from forward to reverse mode, the switching time is minimized by providing a reverse current on the order of 10 to 20 times the forward

17.7

1129

Diode Impedance Rs

t

High Rs

t Forward current

MlSCELLANEOUS CONS lDERATIONS

/

/

/~,

\,

,,,,/

_,, "'

Bias current

r=O

Figure 17.18

Time

..

Typical PIN diode switching waveforms.

bias with a moderately high reverse-biased voltage. The actual "RF switching time" will be minimized by a larger negative bias and/or by a low forward bias. It should be noted, however, that a low forward bia will also mean somewhat increased insertion loss in the ON tate. It i preferable to design the bias circuit to have the same characteristic impedance as the RF line, to minimize the reflection and ringing effects. Extraneous capacitance, in the form of blocking and bypass elements, must be kept to a minimum in applications requiring maximum switching speeds. Again, a trade may exist between RF performance and the switching time. To minimize the delays associated with shunt capacitance in the control lines between the driver circuit and the diodes, it is best to build the driver circuit on the back side of the phase shifter board and make the bias connection in the form of simple feed-througbs.

17.7.4 Bandwidth Considerations The switched line and low-pass/high-pass phase shifter are most suited for broadband application . Reflection- and loaded-line phase shifters are inherently narrow-band. However, loaded-line bits with small phase shifts, 22° and less can be designed to have broader bandwidths. The bandwidth of a true time-delay switched-line phase shifter is limited by the bia circuitry and by the resonances in the phase shifter. Despite these limitations, switched-line phase shifters can be designed with bandwidths in excess of one octave. However, in any broadband design some RF performance must be compromised in one or more of the following areas: • Insertion loss • Insertion loss flatness • Input mismatch • Phase versus frequency characteristics • Circuit complexity

1130

SEMICONDUCTOR CONTROL DEVICES: PHASE SHIFTERS AND SWITCHES

Switched-line phase shifters can also be designed with constant phase shift versus frequency response over broadband by using dispersive compensating schemes. These compensating schemes may take the form of a Schiffman section [40, 41] or a combination of shunt stubs in the reference path. Burns et al. [4] reported that addition of two shorted )../4 stubs, )../4 apart, to the reference path, results in equalized phase versus frequency response for the two switched paths over a limited bandwidth of 12%. Wilds [61) reported a simple, octave band 90° constant phase shift bit. It operates by switching between a straight 50 Ohms 3).. / 4 long transmission line and a 31 Ohms ).. / 2 transmission line, with two 31 Ohms )../ 8 shunt stubs at mid point, one of which is short circuited and the other open circuited. Boire et al. [62) reported a particularly interesting 180° constant phase shift circuit, that operates by switching between a parallel coupled line, characterized by (0, Zoe, Z00 ) and a J'{ -network with shunt and series arms characterized by (0, Zoe) and (0, [2*Z0 / Z00 /(Z 0 e-Z 00 )]) respectively. High-pass/low-pass type phase shifters can be designed for broader bandwidths by a judicious choice of switching element characteristics [49] . This usually entails in a larger number of diodes (or GaAs FETs), typically 6 or more per bit, which are not necessarily identical. Phillippe et.al. [63] reported GaAs monolithic quadrature phase shifter based on a phase locked loop system covering 0.1 to 4.5 GHz frequency range. Broadbanding of analog phase shifters is generally accomplished by cascading several identical broadband phase shifters, each capable of providing a variable phase shift of limited magnitude. A decade bandwidth hybrid analog cascadematch reflection phase shifter using MMIC reflection tenninations has been reported by Lucyszyn and Robertson [64].

REFERENCES 1. J. F. White, "Diode Phase Shifters for Array Antennas," IEEE Trans. Microwave Theory Tech. , MTT-22, pp. 658-674, June 1974. 2. M. E. Davis, "Integrated Diode Phase Shifter Elements for an X-Band Phased Array Antenna," IEEE Trans. Microwave Theory Tech. , MTT-23, pp. 1080- 1084, December 1975. 3. F. G. Terrio, R. J. Stockton, and W. D. Sato, "A Low Cost p-i-n Diode Phase Shifter for Air-borne Phased-Array Antennas," IEEE Trans. Microwave Theory Tech., MTT-22, pp. 688-692, June 1974. 4. R. W. Bums et al, "Low Cost Design Techniques for Semiconductor Phase Shifter,'' IEEE Trans. Microwave Theory Tech., MTT-22(6). pp. 675-688, June 1974.

5. F. L. Opp. and W. F. Hoffman, "Design of Digital Loaded-Line Phase-Shift Networks for Microwave Thin-Film Applications," IEEE J. Solid-State Circuits, SC-3, pp. 124- 130, June 1968. 6. T. Yahara, "A Note on Designing Digital Diode-Loaded Line Phase Shifter," IEEE Trans. Microwave Theory Tech., M'IT-20, pp. 703-704, October 1972. 7. W. A. Davis, "Design Equations and Bandwidth of Loaded Line Phase Shifters," IEEE Trans. Microwave Theory Tech., MTT-22, pp. 561-563, May 1974. 8. I. J. Bahl and K. C. Gupta, "Design of Loaded-Line p-i-n Diode Phase Shifter Circuits," IEEE Trans. Microwave Theory Tech., M'IT-28, pp. 219-224, March 1980. 9. H. A. Atwater, "Circuit Design of the Loaded-Line Phase Shifter," IEEE Trans. Microwave Theory Tech. , MTT-33(7), pp. 626-634, July 1985.

REFER£ CES

1131

10. R. E. Fi her et al., '•Digital-Reflection-Type Microwave Pha. e Shifters," Microwave J.. 12( 12). pp. 63 -68. May 1969. 11 . K. Kurokawa and W. 0 . chlo er. "Quality Factor of Switching Diodes for Digital Modulation." Proc. I£EE. 38, pp. 1 0-181 , January 19 0.

12. P. Wahl and K. C. Gupta. "Effe t of Diode Parameter on Reflection-Type Pha e Shifters." IEEE Trans. Microwave Theory Tech.. MTT-24. pp. 619-621, September 1976. 13. Harry L. Atwater, ··Reflection Coefficient Tran formation for Phase-S hift Circuit ," IEEE Trans. Microwave Theory Tech. , MTT-28(6). pp. 563-568. June J 980. 14. K. Watanabe et al .. ·'Graphical De ign of p-i-11 Diode Pha e Shifter ," IEEE Trans. Microwave Theory Tech.. 29( ). pp. 29-831, Augu t 198 I. 15. A. Sreeni a , Use of Shunt Resistors and Lossy Line Inductors to Improve the Perfom,ance of Reflection and Loaded Line Type Phase Shifters. BASD SER. o. 3319- 15, No ember 1986. 16. K. C. Gupta et al .. Computer Aided Design of Microwave Circuits. Artech Hou e, Dedham, MA. 1981, pp. 207- 211. 17. B. Schiek, ''Hybrid Branchline Coupler : A U eful New Cla s of Directional Coupler ," IEEE Trans. Microwave Theory Tech., 22(10), pp. 865-869, October 1974. 18. T. Okashi et al .. "Computer Oriented Synthesis of Optimum Circuit Pattern of 3-dB Hybrid Ring by Planar Circuit Approach," IEEE Trans. Microwave Theory Tech. , MTT-29. pp. 194-202, March 1981. 19. V. K. Tripathi et al .. "Analy i and De ign of Branchline Hybrids with Coupled Lines,·• IEEE Tra11s. Microwave Theory Tech.. ~fTT-32(4), pp. 427- 432, April 1984. 20. M . K.ir ching and R. H . Jansen, "Accurate Wide Range Design Equations for the Frequencydependent Characteri tic of Parallel Coupled Micro trip Line ," IEEE Trans. Microwave Theory Tech. , MTT-32, pp. 83-90, January 1984; corrections, MTT-33, p. 288. March 1985. 21. B. Petrovic, "A ewTypeofMicrostrip Directional Coupler," Microwave ]., 29, pp. 197-201, ApriJ 19 6. 22. H . Howe, Jr., Strip/ine Circuit Design , Artech Hou e, Dedham, MA 1974, Chap. 3.

23. Ibid., pp. 153- 159. 24. F. C. de Ronde, '·Wide-Band High Directivjty in MIC Proximity Coupler by Planar Means," IEEE MIT Symp. Dig.• pp. 480- 482, May 1980, Washington, D.C. 25. J. Lange, ·Toterdigitated Stripline Quadrature Hybrid," IEEE Trans. Microwave Theory Tech , MTI-17, pp. 1150- 1151 , December 1969. 26. R. Waugh and D . La Combe, "Unfolding the Lange Coupler," IEEE Tran s. Microwave Theory Tech., MTT-20, pp. 777-779, November 1972. 27. W. P. Ou. "Design Equations for an Interdigitated Directional Coupler," IEEE Trans. Microwave Theory Tech., MTT-23, pp. 253- 255, February 1975. 28. V. Rizzoli, "Stripline lnterdigitated Couplers: Analysis and Design Considerations," Electron. Lett. , 11, pp. 392- 393, August 1975. 29. S. J. Hewitt and R. S. Pengelly, ..Design Data for Interdigital Directional Couplers," Electron. Lett., 12, pp. 86-87, February 1976. 30. D . D . Paolino, "Design More A ccurate Interdigitated Coupler," Microwaves, 15, pp. 34- 38, May 1976. 31. J. A. G . Malherbe, ''Interdigital Directional Couplers with an Odd or Even Number of Lines and Unequal Characteristic Impedances,,, Electron. Lett., 12, pp. 464-465, September 1976. 32. A. Presser, ''Interdigitated Microstrip Coupler Design," IEEE Trans. Microwave Theory Tech. , MTT-26, pp. 801 -805, October 1978.

1132

SEMICOND UCTOR CONTROL DEVICES: PHASE SHIFTERS AND SWITCHES

33. D. Kaj fez et al., "Simplified Design of Lange Coupler," IEEE Trans. Microwave Theory Tech., MTT-26, pp. 806-808, October 1978. 34. V. Rizzoli and A. Lipparini, 'The Design of Interdigitated Couplers for MIC Applications," IEEE Trans. Microwave Theory Tech., MTT-26, pp. 7 - 15, January 1978. 35. R. C. Waterman, Jr., W. Fabian, R. A. Pucel, Y. Tajima, and J. L. Vorhaus, "GaAs Monolithic Lange and Wilkinson Couplers," IEEE Trans. Electron Devices , ED-28, pp. 212-216, February 1981. 36. R. M. Osmani, "Synthesis of Lange Couplers," IEEE Trans. Microwave Theory Tech., MTT29(2), pp. 168-170, February 1981. 37. E. M. Rutz and J. E. Dye, "Frequency Translation by Phase Modulation," IRE WESCON Conv. Rec., Pt. 1, pp. 201 -207, 1957. 38. E. J. Wilkinson, L. I. Parad, and W. R. Connemey, "An X-Band Electronically Steerable Phased Array," Microwave J. , 7, pp. 43 - 48, February 1964. 39. R. V. Garver, "Broad-Band Diode Phase Shifters," IEEE Trans. Microwave Theory Tech. , MTT-20, pp. 314-323, May 1972. 40. B. M. Schiffman, "A New Class of Broadband Microwave 90-Degree Phase Shifters," IRE Trans. Microwave Theory Tech., MTT-6, pp. 232-237, April 1958. 4 1. R. P. Coats, "An Octave Band Switched Line Microstrip 3-Bit Diode Phase Shifter," IEEE Trans. Microwave Theory Tech. , MTT-21, pp. 444-449, July 1973. 42. C. M. Gravling, Jr., and B. D. Geller, "A Broad-Band Frequency Translator with 30-dB Suppression of Spurious Sidebands," IEEE Trans. Microwave Theory Tech., MTT-18, pp. 651 - 652, September 1970. 43. K. C. Gupta, R. Garg, and I. J. Bahl, Microstrip Lines and Slot Lines, Artech House, Dedham, MA, 1979, pp. 189-192. 44. William A. Suter, "Active Two-Port Program Speeds Amplifier Design," Microwaves, 19, pp. 79-83, March 1980. 45. S-Parameter Techniques for Faster, More Accurate Network Design , Hewlett-Packard Application Note 95-7, Hewlett-Packard, Palo Alto, CA. 46. P. Onno and A. Plitkins, "Lumped Constant, Hard Substrate, High Power Phase Shifters," presented at the IEEE MIC (Materials and Design) Seminar, Monmouth College, West Long Branch, NJ, June 1970. 47. P. Onno and A. Plitkins, "Miniature Multi-kilowatt PIN Diode MIC Digital Phase Shifters," IEEE MIT-s Int. Microwave Symp. Dig., pp. 22- 23, 197 1. 48. P. Stabile et al., EHF Phase Shifter, RADC-TR-84- 169, August 1984. 49. Y. Ayasli et al., "Wide-Band Monolithic Phase Shifter," IEEE Trans. Electron Devices, ED-31( 12), pp. 1943 - 1947, December 1984. 50. R. V. Garver, "360 Degree Varactor Linear Phase Modulator," IEEE Trans. Microwave Theory. Tech., MTT-17, pp. 137- 147, March I 969. 51. S. Hopfer, "Analog Phase Shifter for 8- 18 GHz," Microwave J., 22(3), pp. 48-50. March 1979. 52. R. K . Mains, G. I. Haddad, and D . F. Peterson, ''Investigation of Broadband, Linear Phase Shifters Using Optimum Varactor Diode Doping Profiles," IEEE Trans. Microwave Theory. Tech., MTT-29, pp. 1158- 11 64, November 1981. 53. B. T . Henoch and P. Tamm, "A 360° Reflection Type Diode Modulator," IEEE Trans. Microwave Theory Tech., MIT-19, pp. 103- 105, January 1971. 54. B. Ulriksson, "Continuous Varactor-Diode Phase Shifter with Optimized Frequency Response," IEEE Trans. Microwave Theory Tech., MTT-27, pp. 650- 654. July 1979.

REFERENCES

1133

55. C. L. Chen. W. E. Courtney, L. J. Mahoney, M. J. Manfra. A. Chu, and H. A. Atwater, "A Low-Loss Ku-Band Monolithic Analog Pha e Shifter," IEEE Trans. Microwave Theory Tech., MTT-35, pp. 315-320, March 1987. 56. D. M . Krafcsik. S. A. Imhoff, D. E . Dawson. and A. L. Conti, "A Dual-Varactor Analogue Phase Shifter Operating at 6 to 18 GHz," IEEE Trans. Microwave Theory Tech. , MTT-36, pp. 1938 - 1941 , December 1988. 57. R. V. Garver. 'Theory of TEM Diode Switching," IRE Trans. Microwave Theory Tech., MTT-9. pp. 224- 238, May 1961.

58. R. W . Burn and L. Stark, "PIN Diodes Advance High-Power Phase Shifting," Microwaves, 4(11 ), pp. 38-48. November 1965.

59. PIN Diode Basics. Hewlett-Packard Application Note 80200, Hewlett-Packard, Palo Alto, CA. 60. J. C. McDade and F. Schiavone, "Switching Tune Performance of Microwave PIN Diodes," Microwave J. , 17, Se1>tember 1974. 61. R. B. Wtlds. ''Try )... / 8 Stubs for Fast Fixed Phase Shift," Microwaves , 18, pp. 67-68, December 1979. 62. D . C. Boire, J. E. Degenford and M. Cohn, "A 4.5 to 18 GHz Phase Shifter," IEEE MIT Int. Microwave Symp. Dig., pp. 601-604, 1985. 63. P. Phillippe, P. Coget, V. Pauker, P. Dautriche, and P. Jean, "A Multi-Octave Active GaAs MMIC Quadrature Phase Shifter," IEEE Trans. Microwave Theory Tech., MIT-37, pp. 2119-2124, December 1989. 64. S. Lucyszyn and l. D. Robertson, ''Decade Bandwidth Hybrid Analogue Phase Shifter Using MM Reflection Terminations," Electron. Lett. , 28(11 ), pp. 1064-1065, May 1992.

18 TRANSFERRED ELECTRON DEVICES CHENG SUN Califon1ia Polyreclmic State University San Luis Obispo, California

18.1

INTRODUCTION

The transferred electron effect, also referred to as the Gunn effect, was discovered by J. B . Gunn [1] in 1963. Gunn observed current oscillations as the applied voltage exceeded a certain threshold voltage on the n-type GaAs and InP samples. The frequency of these oscillations could be made to lie in the microwave range by choosing the proper sample doping level and thickness. His measurements showed that a highfield domain formed near the cathode, propagated across the sample, and disappeared at the anode. As the domain was absorbed at the anode, the current increased to the threshold value and a new domain nucleated at the cathode contact. The current would drop and the process would be repeated. The transferred electron effect was explained in 1964 by Kroemer [2] to be the negative differential mobility mechanism described by Ridley and Watkins [3] in 1961 . The theory of achieving negative differential mobility was further advanced by Hilsum [4]. The theory shown by Ridley and Watkins [3], and Hilsum (4) indicated that the negative differential mobility of electrons could be obtained in GaAs by exciting high-mobility electrons from a lower conduction valley to a low-mobility, upper conduction valley. While GaAs has dominated the development work for the transferred electron in recent years because of its maturity, materials such as InP, ZnSe, CdTe, and GaAsP have also demonstrated the same effect. By applying pressure to certain crystal samples, the energy bands of lnAs, for example, can be changed so that the transferred electron effect exhibiting the negative differential mobility mechanism was observed. In fact, GaAs samples subjected to hydrostatic pressure caused a reduction in the threshold voltage, as discovered by Hutson et al. [5] and Allen et al. [6]. Handbook of RF/Microwave Componems and Engineering,

Edited by Kai Chang

ISBN 0-471-39056-9 © 2003 John Wiley & Sons, Inc.

113S

1136

TRANSFERRED ELECTRON DEVICES

Transferred electron devices have been used extensively for low-noise local oscillators, low-power transmitters, and wideband tunable sources. For a single device, continuous-wave (CW) power levels of several hundred milliwatts in X-, Ku-, and Kaband frequency ranges and about 75 mW in W-band frequencies are commercially available. Higher power levels have also been achieved with power combining techniques by combining multiple devices. In this chapter we present an overview on transferred electron effect devices. In Section 18.2 we review the negative differential mobility mechanism. The modes of operation of transferred electron oscillators and amplifiers are described in Sections 18.3 and 18.4, respectively. In Section 18.5 we discuss the equal-area rule and in Section 18.6, present equivalent-circuit models. Microwave circuits and combining techniques are covered in Sections 18.7 and 18.8. In Section 18.9 we summarize epitaxial growth of the material and device fabrication. Finally, recent developments and future trends are given in Section 18.10.

18.2 NEGATIVE DIFFERENTIAL MOBILITY The existence of the negative differential mobility effect associated with the transferred electron devices was explained by Ridley and Watkins [3] and Hilsum [4], and is often called the Ridley-Watkins-Hilsum mechanism. For n-type GaAs, this effect is due to its energy band structure, which allows the transfer of conduction electrons from a higher-mobility, lower-energy valley to lower-mobility, higher-energy satellite valleys along the {100) crystallographic directions. A simple two-valley band structure model of n-type GaAs is shown in Fig. 18.1. ~ l.2me, m ~ The effective masses of electrons in the lower and upper valleys are 2 0.068me, respectively, where me is the mass of a free electron. The electron mobility in the lower valley /J,1 is approximately 50 times greater than the mobility in the upper valley JJ,2. The energy separation between the two valleys is 0.36 eV. The steady-state

mr

e

n1

"2

JJ-1 Lower

~

valley

Upper

valley

Q)

C

Q)

C

e u Q)

w 1.4eV

Valence band

Wave number

Figure 18.1

Simplified band structure of GaAs.

18.2

NEGATNE DIFFERENTIAL MOBILITY

1137

conductivity of the n-type GaAs i (18. 1)

where n 1 and n 2 are electron densities in the lower and upper valley bands, re pectively. The total electron den ity n is given by

= n 1 + n 2 = con

n

tant

(18.2)

In Eq. (18.1), /1-1, 11 1 and /1-2 n2 are functions of the applied electric field£ . Differentiation of Eq . (18.1) and (18.2) with respect to E reduces to da

dE dn1

-

dE

=q

(

dn1

/1-i dE

dn2)

+ µ 2 dE + q

( ni

dµ,1 dE

dµ, 2)

+ n 2 dE

(18.3)

dn2 dE

= --

If we assume that f.1,1 and µ, 2 are proportional to EP, where p is a constant [7], Eqs. (18.3) and (18.4) can be combined to (18.5)

From Ohm's law J and rearrangement:

= a E,

one obtains the following equation by differentiation

1 dl Eda - - -=I+-a dE a dE

(18.6)

To provide negative conductivity, dJ/dE must be negative in Eq. (18.6). This condition can be solved by combining Eqs. (18.1), (18.5), and (18.6): µ,1 - /1-2 ) ( E dn 1) ] [ ( µ,1 + f /1-2 - ~ dE - p > I

(18.7)

where f = n2/ n 1 · dn 1/dE in Eq. (18.7) is negative because more electrons will be transferred from the lower to the upper valley. Equation (18.7) is satisfied if the following requirements are met 1. µ1 > µ,2 or electrons must start in a lower valley and transfer to an upper valley when an electric field is applied. 2. The exponent p should be negative and large. If lattice scattering is dominant, p is negative. When the impurity scattering is dominant, p becomes positive and is therefore not desirable for the transferred electron effect. A more detailed discussion on scattering mechanisms is given by Bulman et al. [8]. For a material to give rise to a negative differential mobility, its band structure must satisfy the following criteria based on the discussion above:

l. Two conduction subbands must wst. The energy difference between the lower and upper valley bands must be much greater than the thermal energy.

1138

TRANSFERRED ELECTRON DEVICES

2. The electron mobiljty µ 2 for the upper valley must be lower than the electron mobility µ 1 for the lower valley. The effective mass m 2 for the electrons in the upper valley must be higher than that for electrons in the lower valley m !. 3. The energy difference between the upper and lower valleys should be much smaller than the semkonductor bandgap (1.43 eV for GaAs); otherwise, the semiconductor would break down and become highly conductive before the electrons can be transferred to the upper valleys.

Because of its advanced technology, n-type GaAs is the most frequently used negative differential mobility material. Other compound semiconductors, such as InP, CdTe, ZnSe, GaAsP, and InAs, also meet the requrrements above. A simple graphical representation for the electron concentrations of a two-valley semiconductor as a function of electric field is illustrated in Fig. 18 .2. For Fig. 18.2a:

n1 ~ n

For Fig. 18.2b:

n1

and n 2 ~ 0

+ n2 ~ n

For Fig. 18.2c: n1 ~ 0 and n 2 ~ n

for O < E < Ea

(18.8)

for Ea < E < Eb

(18.9)

for E > Eb

(18.10)

and the current density takes on the asymptotic values: qnµ1E l= { qnµ2E

for O < E < Ea for E > Eb

(18.11) (18.12)

If JJ,1 Ea is greater than JJ,2Eb, a region of negative differential mobility will exist between Ea and E b, as shown in Fig. 18.3. JT corresponds to the threshold current den-

sity and Er is the threshold field at the onset of the negative differential conductivity. Er is 3200 V/crn for GaAs and 10,500 V/cm for InP. Experimental measurements of the drift velocity ver u electric field for GaAs were made by Ruch and Kino [9] to show the existence of the negative differential mobility, as shown in Fig. 18.4. Butcher and Fawcett [10] made a numerical calculation which showed excellent correlation with the measurements, as indicated in Fig. 18.4.

(a)

>-

e> Q) C

Q)

C

e

u Q)

w

(b)

Lower

Upper

valley

~

(c)

-- ,-------'....

=1o.36eV

1.4 eV

I\ ( 000 )

( 100 )

I\ ( 000 )

I\ ( 100 )

( 000 )

( 100 )

Wave number

Figure 18.2 Electron di tributions for a two-valley semiconductor under different electric fields.

18.3

MODES OF OPERATION FOR OSCILLATORS

1139

qµ 1nE I

...,

Jr

I

~ ·u;

I

I

C

I

a>

-........

I

-0 C

I I I

----...,._----------- --

a>

:)

u

I

--

Ea Er

Figure 18.3

-~ (.)

Eb

Electric field

Current density versus electric field of a two-valley semiconductor.

2 X 107

~

0

0

a, >

,t:

.::::

1 X 107

C)

0 _ _ _..._____....___.,_ _._____._ _.___._................ 0

Figure 18.4

1

2

3

4 5 6 7 8 Electric field (kV/cm)

9

10

11

Theoretical and experimental velocity-electric field characteristic of GaAs.

McCumber and Chynoweth [11] determined that the magnitude of negative differential mobility decreases with increasing temperature and vanishes around 800 K.

18.3 MODES OF OPERATION FOR OSCILLATORS Since the discovery of the transferred electron effects, many modes of oscillation dependent on material and circuit parameters have been unveiled. A summary of these oscillating modes is outlined in this section.

18.3.1

Gunn Mode or Dipole Mode

The Gunn mode is characterized by the formation of a domain or dipole first observed by Gunn [l]. In a low-impedance circui~ the oscillation frequency was related to Vs/ L, where Vs is the domain velocity and L the device length. From the current continuity equation and Poisson's equation,

8p

V•J= - -

at

(18.13)

1140

TRANSFERRED ELECTRON DEVICES

where (18.14) (18.15) By combining Eqs. (18.13)-(18 .15), one can solve the macroscopic space charge density p, (18.16) where rd = E/ ao = E/qnoµ,n = dielectric relaxation time E = semiconductor dielectric permittivity no = doping concentration µn = electron mobility q = electron charge a0 = conductivity of the sample For GaAs semiconductor material with a0 = 1 (O-cm)- 1 and E = 13. l Eo, Eo is the permittivity of air, rd is approximately 10- 12 s, and macroscopic space-charge neutrality is quickly satisfied at every point in the material. Therefore, any deviations or fluctuations in carrier density will quickly disappear. For a material with negative differential mobility, on the other hand, rd is negative and space-charge fluctuations build up in time rather than being neutralized. Figure 18.5 (a)

E

-+ -+ -+

+

Gunn sample

-+

ltlllll:4t-----L------~1 Space charge

(b}

E t2

r , -I I

0

I

---~-------~--~x L

Figure 18.S (a) Charge dipole at a local nonunifonnity; (b) electric field buildup due to charge ti and t2 with the device biased beyond £ 7 •

dipole in time

18.3

MODES OF OPERATIO

FOR OSCil..LATORS

1141

shows an example of an in tability building up in the emiconductor sample. If a small nonunifonnity or fluctuation in the electron density occur at the cathode of the device, a charge dipole will form locally as shown in Fig. 18.5a . If the Gunn device is biased in the negative differential mobility region, the localized electric field will build up as shown in Fig. 18.Sb . Thi dipole or domain will grow and drift to the anode. When the domain reaches the anode, a current pul e will appear in the external circuit. Another domain now nucleates at the cathode and the proce is repeated. Therefore, the pulse frequency i inver ely proportional to the drift length L . After the initial growth of the domain, a stable condition is eventually realized. At this condition the electrons drift at a constant velocity vd everywhere in the sample, and the domain travel down the ample without further growth. The table condition may be explained with the velocity versus field diagram for n-type GaA as hown in Fig. 18.6. A sume that the sample is biased at E 1• Since the field in the dipole region i higher than £ 1, the electrons will move slower than those in the neutral regions. Hence the electron density will vary, causing accumulation of electrons on the left ide and depletion on the right side of the dipole layer, causing the field in the dipole layer to rise until it reaches £ 2 • Since the total voltage applied to the sample remain the same, an increa e of the electric field in the dipole layer will result in a decrease of the electric field outside the domain. A stable condition is finally reached when the field outside the domain drops to Es, with the same velocity Vs as that for the domain shown in Fig. 18.6. For the Gunn mode of operation, the space-charge domain must grow to its stable condition before it reaches the ohmic contact at the anode of the device. This implies that the dielectric relaxation time r d must be less than the transit time through the sample length L , E

E

L

q lµ,nl no

Vs

( 18.17)

--- q Iµ,Tl- I For n-type GaAs, Vs ~ 107 emfs, E = 13.lEo, and must be approximately > 10 12 cm- 2 .

-~

2.2

-·g

X

JJ-n

~ - 100 cm2 / V-s; noL product

107

0

~

- ,---T---

107

Q)

>

€

0

0

0Es

I I I

I

Er

E1

-vs

I

I

E2

E

Electric field

Figure 18.6

Electron drift velocity versus electric field for GaAs.

1142

TRANSFERRED ELECTRON DEVICES

The frequency of oscillation f is related to the transit time T, of carriers through the device: I Vs (18.18) f= -T, =L or f L

= 107 emfs

The Gunn mode of operation is simple to operate since only a de bias is required. However, the efficiency is low, only a few percent. The operation of a Gunn mode is indicated in Fig. 18.7b, where ET is the threshold electric field, £ 5 the sustaining electric field, and T the period of the oscillation (12].

18.3.2 Resonant Gunn Mode For the Gunn or dipole mode discussed in Section 18.3.1, the applied electric field is above the threshold value and is constant because the RF circuit used bas a low impedance. For the resonant Gunn mode, the device is operated in a resonant circuit whose resonant frequency is selected to be approximately equal to the transit-time frequency. If the field never drops below the sustaining field E s, a domain will be

T1= transit time

(a)

V

T =..1 =oscillation f 'd

period

=dielectric relaxation time in the negative mobility region

-L--

Vs

I

I I

Es= sustaining field Er= threshold field

I

Es Er

E

(c) E

(b) E

(d) E t-----i~---

- - - - - - Er

De bias

-Er

-----E

------E

s

t

T1

(e) E

~----'---s 1i

t

t

(f) E

"""--1•--

- - Er

- - - Es

1i

Er

- - Es

t

1i

t

Figure 18.7 Modes of operation: (a) velocity-field characteristic for GaAs; (b) Gunn mode; (c) resonant Gunn Mode; (d) inhibited or delayed mode; (e) quenched domain mode; (f) LSA mode.

18.3

MODES OF OPERATIO

FOR OSCILLATORS

1143

formed and can propagate to the anode during each cycle, as hown in Fig. 18.7c. For 12 this mode, noL ~ 10 cm- 2 and f L = 107 cm/ . The efficiency i till low, typically le than 10%.

18.3.3 Inhibited Mode or Delayed Mode When the re onant circuit impedance i increased o that the electric field drop below the thre hold alue but remain above the ustaining electric field. a new domain may be delayed until the field ri e abo e the threshold ET a indicated in Fig. 18.7d . For thi mode, 106 < f L < 107 cm/ , and T > Tr. Since the time between output current pulse has been increased, the efficiency i improved over the Gunn and re onant Gunn mode . The o cillating frequency can al o be lightly altered by the re onant circuit. Theoretical calculation have hown that the efficiency of the delayed-domain in mode can reach 27% (13].

18.3.4 Quenched Domain ~1ode If the bia field drops below the ustaining field Es for the negative part of the cycle, the domain may be quenched before it reaches the anode, as shown in Fig. 18.7e. In this quenched domain mode, another domain is nucleated when the bia field goes above ET and the proce s repeats again. Consequently, the frequency of operation i rai ed ince the transit time of electrons ha been effectively shortened. The oscillation frequency of the resonant circuit can be several times the tran it-time frequency. A theoretical efficiency of 13% for the quenched domain mode has been calculated [1 3].

18.3.5 LSA Mode The limited space-charge accumulation (LSA) mode [14] operate in a ample where only a mall accumulation layer is formed before it is quenched by the RF voltage as shown in Fig. 18.7f. For the LSA mode, the frequency is so high that domain do not have enough time to form~ and the entire length of the sample is subjected to an electric field greater than the threshold field Er . Therefore, a large part of the sample length is maintained in the negative conductance state during most of the RF cycle. The requirements for the LSA mode are that the period of the oscillating frequency be no more than a few times the dielectric relaxation time rd in the negative conductance region, but must be larger than the rd+ for the low-field po itive conductance region to dissipate any accumulation of electrons while the signal is below the threshold field Er. For µ,n = -100 cm2 / V-s, rd= E/qnolµ,n l, T r:d , 2 3 no/I= 7 x Ia4 scm - is calculated. For µ,n = 8000 cm / V-s in the positive conductance region, rd. = E/ qn 0 µ,,,, T = r dT, and no/I= HP is calculated. Selecting the low-frequency limit such that T ~ 3rd(no/f = 2 x lo-5) and the high-frequency limit so that T = 20rd~(no/f = 2 x 104 ) will provide the requirement for the LSA mode,

=

2 x 104 < no < 2 x 1()5 cm- 3 s

I

These limits have been experimentally verified by Copeland [15]. Maximum efficiencies in the range of 18 to 23% [14, 15] have been calculated for this LSA mode.

1144

TRANSFERRED ELECTRON DEVICES

A distinct advantage for the operation of the LSA mode over other modes is its high efficiency and high output power capability. Since the entire LSA sample is biased in the negative conductance region rather than a narrow region, as in the domain modes, a larger percentage of the sample is useful for generating microwave power, resulting in an improvement of efficiency. The operating voltage for the LSA mode can be applied considerably higher before the impact ionization occurs because the peak field in the LSA mode is less than that in the domain modes. Consequently, the input power to the device and the output power are higher. The major problem with the LSA mode is its sensitivity to circuit tuning and doping fluctuations [16]. A dipole domain might form to create a high dielectric field to damage the device.

18.3.6 Hybrid Mode The hybrid mode [1 7, 18] operates between the LSA and domain modes. This mode is characterized by a space-charge growth that occurs during each cycle of oscillation but does not reach the stable domain condition of the domain modes. During the time in which the applied voltage is above threshold for the hybrid mode, the presence of the domain causes the current in the external circuit to be less than that of the LSA mode but greater than that of domain modes. As a result, the efficiency of the hybrid mode is also a compromise between them. The hybrid mode operation is easier to achieve than the LSA mode because of its less stringent n 0 / f conditions (17] and doping homogeneity requirements. Furthermore, the hybrid mode is les sensitive to the circuit conditions. To summarize all the modes for transferred electron oscillators. Copeland [ 15] has shown a mode chart as depicted in Fig. 18.8. The ordinate and ab ci a of the chart are the fL and noL products of the device, respectively. The locus of a con tant no/ f ratio is a straight line at 45° with the axes.

no= 2 x 105 f

~

E u ::; 10 s Negative conductance .c mode C)

I

-

Quenched domain

C

~ X

Gunn or dipole Delayed 106 ________________..._ _ _ ___.__ _ _ _...J

1011

1012

1013

1014

101s

Carrier concentration x length, n0 L (cm- 2)

Figure 18.8 Mode chart for transferred electron devices. (From Ref. 15.)

I .4

18.4 MODES OF OPERATIO

tODE OF OPERATIO

FOR AMPLIFIERS

1145

FOR AMPLIFIERS

Tran ferred electron de i e can be u ed as negative re i tance reflection amplifier in two basic type . One type i to u e an o cillator with the pre ence of a traveling domain a a negati e re i tance element [19]. Thi arrangement ha not been very u eful ince purious ignal due to the domain o cillation are not de irable fo r many Y tern application . The econd type i to tabilize a tran ferrcd electron device o that o cillation are uppre ed, and the negati e re i tance i achieved by biasing the device in the negati ve differential mobility region. Thi type of amplifier i of more intere t becau e of its implicity. Four method of tabilizing the amplifier ha e been ugge ted: 1. Subcritical noL. GaA i gt\'CO by

ection 18.3. 1 ha di cu ed that the tability condition fo r (18 .19)

De•Jice that ati fy inequality in Eq. (18. 19) are calJed subcritical, while devices with n 0 L > 10 11 cm- 2 are called upercritical. 2. Circuit Stabilization. McCumber and Chynoweth [ 11 ] have hown that all tran ferred electron device are unconditionally table under con tant current condition . Under con tant voltage condition , devices can be made stable with a eries po itive re i tor [20). 3. Diffusion Stabilization. Jeppe en and Jeppsson [21) realized the importance of diffusion effects in stabilizing the transferred electrcn devices. Diffu ion tend to balance the space-charge growth; thu the device can be stabilized. 4. lnjection-Limiring Cathode. Thim [22) reported that a uniform electric field from cathode to anode inside a transferred electron device i the optimum field configuration. A stable and optimum operation of device can be achieved by introducing a doping notch near the cathode contact. The width and depth of the notch can be made to provide a uniform electric field di tribution [23). Other methods of injection limiting, such as using a p-n junction at the cathode or a low-height Schottky barrier. have al o been ugge ted [24, 25]. Two types of stable amplification mode are available for Gunn devices: the subcritically doped mode and the supercritically doped mode. For the subcritically doped mode, the n 0 L product of the GaAs device is less than about 1ot2 cm- 2 and the dipole domain for the device will not be formed, as indicated in the mode chart of Fig. 18.8. However a negative resistance stiU exi ts in the device at microwave frequencies, and the device can be used as an amplifier in the vicinity of the tran it-time frequency and its harmonics without oscillation [26]. If the device is incorporated in a circuit with adequate positive feedback, the circuit can be used as an oscillator. Halcki [27] has demonstrated that a Gunn device can amplify at microwave frequencies or it can be used simultaneously as an amplifier and local o cillator. Becau e of the low value of n0 L, the output power and efficiency are quite low compared with the upercritically doped mode of amplification. For the supercritically doped mode of amplification, the value of noL of the device must be greater than 10 12 cm- 2 • With this n 0 L product the dipole domain is formed. The device oscillates at the transit-time frequency, whereas it can be amplified at ome

1146

TRANSFERRED ELECTRON DEVICES

other frequencies. Walsh et al. [28] have demonstrated this mode of amplification. Broadband amplification performances were achieved for 4.5 to 8.0 GHz, 8 to 12 GHz, and 12 to 15 GHz [29], respectively. Linear gains of 6 to 12 dB and saturated power up to 500 mW have been obtained. All the performances were realized by providing a broadband circuit for the amplification frequencies and a suitable stabilizing circuit to prevent the Gunn oscillation frequencies from entering the amplifying frequency band. Although devices operated in the supercritically doped amplifier mode provided much better power and efficiency performance than the devices operated in the subcritically doped mode, typical efficiencies still range from 2 to 3%. These are significantly lower than the efficiencies that can be achieved by FET amplifiers now available for the same frequency ranges. The main applications of Gunn amplifiers appear to lie in the high-millimeter-wave frequency bands, where FET devices are not mature or available.

18.5 EQUAL-AREA RULE As discussed in previous sections, the stable domain within the device travels at a constant velocity. On the left and right sides of the domain, the electric field is below threshold. Inside the domain, the field rises quickly through the accumulation layer to a peak value in excess of Er and decreases through the depletion layer. Somewhere inside the domain the field will be maximum at E = Ed, and at this point the carrier concentration will equal Nd, the ionized donor concentration. An exact analysis of the domain physics requires a nonlinear solution of Poisson's equation and current continuity equation (30, 31], (18.20) where n is the instantaneous carrier concentration and Na is the ionized donor concentration. dn aE J = qnv(E)- qD - + E (18.21) dx

at

where J is total current density from a Gunn device and is the sum of the drift, diffusion, and displacement current densities, v(E) is the field-dependent carrier velocity, and D is the diffusion constant. While the current versus voltage behavior of the domain physics for the Gunn device has been solved by computer analysis, Butcher (32, 33] has analyzed this problem as the equal-area mies to be described below. In addition to the Poisson equation and the current continuity equation given in Eqs. (18.20) and (18.21), respectively, the boundary conditions satisfied by a stable domain propagation are

E = E0

= constant at X = ±oo

(18.22)

As a IesuJt,

cJE

-

ar

=0

atX=±oo

(18.23)

18.5

EQ UAL-AREA RULE

1147

and Eq. (18.2 1) reduce to (18 .24) where vo = v(Eo)

(18 .25)

Inside, the domain mu t have the fol1owing fo rm of solution:

where

ud

= E (x -

vdt )

( 18.26)

n =n (x- vdt )

(18.27)

E

is the drifting velocity of the domain. From Eq. (18.26) one obtains

aE

aE

=-Ud81 ax

(18.28)

Al o, fro'll Eq. (18.27) we have

an ax

an -aE

= -

(1 8.29)

aE ax

Since the total current density in the device must be constant, this implies that Eq. (18.21) can be written as

qNdvo

= qn v(E ) -

an + E -aE ax at

qD-

(18.30)

Substituting Eqs. ( 18.28) and (1 8.29) into (18.30) yields q (Ndvo - nv (E ) )

=-

dn ) aE ( qD -dE - Evd -

ax

(18.31)

Substituting 8 E / ax from Eq. (1 8.20) and dividing Eq. (18.31) by nNd and integrating both sides from infinity to any point in the domain, we have the following result: (18.32) When the diffusion constant is assumed to be independent of electric field, Eq. (18.32) c-an be expressed as

!:_ - 1- ln !:_ Nd Nd

=

E

qNdD

1£ Eo

{cv(E ) - Vd] - Nd (vo - vd ) J dE n

(18.33)

In Eq. (18.33), if E = Ed, then n = Nd and the left side of the equation must vanish. As seen in Fig. 18.9, the integration of F.q. (18.33) can be made from E 0 to Ed through the accumulation layer, or through the depletion layer, where n < Nd . Both results must zero according to Eq. (18.33). Since the first term in the integral is independent of n, one must have (18.34)

1148

TRANSFERRED ELECTRON DEVICES

-:i-

(ii

, , Dynamic characteristic

:::,.

·o

'

Vo

0

a5

>

Static characteristic

Electric field, E

Figure 18.9

Equal-areas rule for transferred electron devices.

and (18.35) From Eq. (18.35), one can obtain the equal-area rule that the shaded areas in the two regions of Fig. 18.9 must be equal. Two important conclusions can be derived from the equal-area rule: 1. If the diffusion constant D of the electrons is not a function of electric fiel~ the domain travels at the same velocity v 0 as the electrons in the uniform region outside the domain. 2. A dynamic characteristic curve, as shown by the dashed line in Fig. 18.9, relates vo to the peak field Ed . If the variation of the diffusion constant with electric field is included, most behaviors of the domain remain the same, except that the domain travels at a velocity which is slightly different from the velocity of the electrons drifting outside the domain [34] . The total voltage V applied to the sample can be written V

= Vx + EoL

(18.36)

where Vx is the domain potential defined as the voltage across the domain, and L is the length of the sample. Vx

=

1L

[E(x ) - Eo] dx

(18 .37)

Assuming a triangular-shaped field domain with the peak field Ed and the uniform field outside domain Eo, the width W of the domain calculated from Poisson's equation given by Eq. (18.20) is €

W= -(Ed-Eo)

(18.38)

qNd

The domain voltage Vx is approximately the area under the triangular shape: 1 Vx ~ - W(Ed - Eo) 2

€ = --(Ed -

2qNd

Eo)

2

(18.39)

18.6

EQUIVALENT-CIRCUIT MODELS

1149

Consider a GaA Gunn diode with a dipole domain operating with the following parameter :

L µn

= 10 µm = 6000 cm2 / V-s

Nd= 2 x 10 15 cm- 3

area= 3 x 10- 4 cm2 in tantaneou current /

= l. 152 A

The domain velocity uo can be calculated from vo

I

=- = 1.2 x qANd

107 emfs

One can find Eo, corre ponding to vo from a given velocity versus electric field characteristic for GaAs, Eo = 2 x 103 V/cm and Ed = 10.5 x Ia3 V/cm from the equal-area rule. The domain voltage Vx calculated from Eq. (18.39) is

Vx

= 2qNd E (Ed -

Eo)2

= 0.13 V

The total voltage V applied to the device from Eq. (18.36) is found to be V =

EoL

+ Yx = (2 x

1a3)(10 x 10-4)

+ 0.13 =

2.1 3 V

and the frequency of oscillation for the Gunn mode (transit time mode) is Vo 1.2 X 107 f = - = - - - = 12 GHz L 10 X 10- 4

18.6 EQUIVALENT-CIRCUIT MODELS For a circuit designer, the equivalent-circuit models for Gunn devices are useful to serve as a starting p oint for the circuit design. Equivalent circuit models may be grouped for devjces operating in domain modes or LSA modes.

18.6.1 Domain Modes Although many researchers working in this field have analyzed the domain modes of transferred electron devices with detailed computer calculations, the results are difficult to use because the device parameters employed in the computer simulations are not always applicable. Instead, simpler equivalent circuits with lumped elements are devised to represent the operation of the device. Hobson [35] suggested that the diode can be expressed as a series combination of a low-field and high-field equivalent circuit. As shown in Fig. 18.10. Cn and 1/G are the active components in the circuit. CD is the domain capacitance and G. is the negative conductance responsible for the power-generating part of the device. The components C and R are related to

1150

TRANSFERRED ELECTRON DEVICES

R

1 G

C

Figure 18.10 Equivalent circuit for Gunn devices by Hobson. (From Ref. 35; previously published by the Institution of Electrical Engineers in Electronic Letters. Vol. 3, June 1966, reproduced by permission.)

R

C

Figure 18.11 Equivalent circuit for Gunn devices by Robrock. (From Ref. 37; copyright 1970 IEEE, reproduced by permission.)

the dielectric relaxation time and low-field resistance of the sample, respectively. For a device with 1-Q-cm GaAs with Nd = 10 15 cm- 3 , an area of 100 µm square and 10 µm long, CD~ 0.30 pF, 1/ G ~ -500 Q, R ~ 10 Q, and C ~ 0.10 pF. Carroll and Giblin [36] used the analog model for the Gunn device operation. By incorporating ideal switches and transistors, the model can calculate voltage and current waveforms associated for different loading requirements. The operating modes of the Gunn device can be identified by comparing the waveforms calculated from the analog model with those occurring in the actual sample. Robrock [37] devised a lumped-circuit model to describe the phenomena of domain nucleation, modulation, and quenching in GaAs samples of uniform doping. As shown in Fig. 18.11, a voltage-dependent current source ID is used to represent the propagating high-field domain, which is characterized by its excess voltage Vx . Dynamic effects are included in the differential domain capacitance CD . The remaining part of the bulk sample is represented by a parallel circuit formed by the ohmic resistance R and the low-field capacitance C . Since EoL is the voltage drop across the low-field ohmic resistance R of the bulk material of the device, EoL = RI (18.40) Under quasi-static conditions, the current I flowing through the bulk device of donor density Nd and area A is given by (18.41) where v(Eo), the velocity of carriers outside the domain, is available from velocity versus electrical field characteristics of the device. Thus R can be solved from Eqs. (18.40)

18.6

EQUCVALE T-CTRCUIT MODELS

1151

and (18.41).

EL

R = -- - qNdA v( Eo)

( 18.42)

The low-field capacitance C can be e pre ed a

EA

C= -

L

( 18.43)

To generate the quasi- tatic current ver u voltage characteri tic that will be u ed to detem1ine the current ource / 0 in the equivalent circuit, the following teps are involved: 1. A curve of v( £ 0 ) ver u E 0 L i fir t produced u ing the velocity electric field characteri tic for the ample u ed by a scaling factor L to provide £ 0 L . 2. The appropriate V ersu £ 0 curve will then be made by mapping the £ 0 value to the corre ponding velocity u( Eo). Tbj curve is al o computer generated by Copeland [3 ] a hown in Fig. 18.12.

3. The total voltage V i obtained by Eq. (18.36) producing a plot of v(Eo) ver u V. The I v - V characteri tic i then obtained by u ing Eq. (18.41 ). The differential capacitance C O of a high-field domain hunting the current ource in the equivalent circuit i defined as C D

_ dQv dVx

(18.44)

10

9 8

-~

~ )(

7 6

ai C)

c,s

0 > (/) en a> u >< w

5 4

3

2

nd = 1015 1 0

1.5

2.5 3.0 2.0 Outside electric field E0 (KV/CM)

Figure 18.12 Excess voltage Vx versus electric field Eo outside the domain by Copeland. (From Ref. 38; copyright 1966 IEEE, reproduced by permission.)

1152

TRANSFERRED ELECTRON DEVICES

Again we assume that the effects of diffusion are neglected. The resulting electric field distribution associated with the dipole domain provides a triangular shape, with width W and domain voltage Vx given in Eqs. ( 18.38) and ( 18.39), respectively. Combining Eqs. (18.38) and (18.39) yields ( 18.45) The charge Q v contained within the depletion region of the domain is ( 18.46) Cv can then be obtained by using Eqs. ( 18.44) and (18.45)

EA

(18.47)

CD=-

w

The equivalent circuits provided by Hobson and Robrock were compared with fair agreement [39]. A domain width of 2 µmused in Robson's simulation corresponds to a Vx value of 2.76 V according to Eq. (18.45), where Er 13.1. With a doping 15 3 of Nd= 10 cm- and Vx = 2.16 V, and Eo value of 1.8 kV/cm i s obtained from Copeland's curve shown in Fig. 18.12. For this value of Eo, v(Eo) is 1.4 x 107 emfs from Eq. (18.41). R is calculated to be 10.2 n using Eq. (18.42), C is 0.077 pF by Eq. (18.43), I = 116 mA by Eq. (18.41), and CD= 0.448 pF by Eq. (18.47). The e values agree approximately with Hobson's model, Cv = 0.30 pF, R = IO n, and C ~ 0.10 pF.

=

18.6.2 LSA Mode For the LSA modes, the key features that must be included in the equivalent circuit are the lack of domain formation and the long, uniform high-field region in the sample. These are two features that allow a high RF voltage to be develope~ thereby producing a high power- impedance product. The equivalent circuit used to represent the LSA mode is given in Fig. 18.13 [(14,) (36) (40)], which is a negative resistance (-R) in parallel with the low-field capacitance C. The negative re istance (- R) calculated by Copeland [15) is dependent on RF electric field E 1 applied to the ample, the low-field mobility JJ,o, RF power output PR F from the sample, and low-field positive re istance Ro, R qEf µ,o (18.48) Ro 2PRF

- = _.;;...__

C

Figure 18.13

-R

Equivalent circuit for LSA mode.

18.7

OSCILLATOR/AMPLIFIER CIRCUlTS AND MO OUTHIC DEVlCE DEVELOPMENT

1153

For a fixed de bia fi eld of 104 V/crn, and for £ 1 = 2 x 103 to 9 x 103 V/cm, R/ Ro= - 12 to - 20 (15]. In designing an LSA o cillator, circuits mu t be designed to induce the device operating in the LSA mode and to prevent domain formation. A de]ay-line loading circuit [4 I] and a fa t ri e-time bias puJ e [40] for pulsed LSA o cillator have proven succe ful .

18.7 OSCILLATOR/AMPLIFIER CIRCUITS AND MONOLITHIC DEVICE DEVELOPMENT Three type of microwave circuits are commonly u ed to accommodate the transferred electron o cillator and amplifier : waveguide, coaxial, and microstrip. In addition, monolithic Gunn device were al o developed.

18. 7 .1

Waveguide Circuits

At high frequencies, from X-band to millimeter wave ranges, a common method of coupling the Gunn devices into a waveguide cavity is to mount the packaged diode under a po t as hown in Fig. 18.14. The circuit provides high stability, low FM noise, and good mechanical tuning range by adjusting a movable short in the circuit. The additional impedance matching for transferred electron devices may be achieved by using the reduced height waveguide to lower the impedance, by incorporating coaxial spacers around devices, and by providing matching impedance transformers at the output as shown in Fig. 18.14a, b, and c. Another waveguide circuit suggested by Harkless [43] is shown in Fig. 18. 14d, where a lossy absorber is inserted to eliminate aU undesirable oscillating modes. Figure 18.14e is a full-height waveguide circui t with diode mounted under a radial cap on the center post. The frequency of the oscillator is controlled mainly by the dimensions of the cap. The circuit has very low losses and is frequently used in millimeter wave applications. Althoug h waveguide circuits for transferred electron devices have many advantages, they do have di sadvantages over coaxial and microstrip circuits. They are bulky, costly to machine, and likely to experience mode jumping and spectral purity problems. The diode mounting structure shown in Fig. 18. 14a to d has been used extensively for solid-state devices such Gunn and IMPATT devjces for high frequencies at X-band and beyond. Many theoretical analyses are available to predict the impedance presented by the circuit to the packaged diode terminals. Eisenhart et al. [44] suggested an equivalence between a coaxial entry and a gap in the post. Based on this equivalence and the equivalent-circuit model developed by Eisenhart and Khan [45, 46], the theoretical analysis for the coaxial-waveguide junctions shown in Fig. 18.14a to d can be obtained. A dyadic Green's function is provided for a rectangular waveguide using the Lorentz reciprocity theorem, which determines the electric field and current density at two locations in the waveguide. This approach permits the calculation of the impedance seen by the device placed on a post located anywhere in the waveguide. To facilitate mathematical computations, Eisenhart and Khan [45] assumed that the post may be replaced by an infinitesimally thin strip of width W . They found that a post with diameter d can be replaced by a strip with W = 1.8d. Using this circuit

11S4

TRANSFERRED ELECTRON DEVICES {a)

Impedance transformer

{b) Bias

To ~ load

Device

Tuning short

(d)

{c)

Coaxial section

Coaxial section

(e)

Cap resonator

Figure 18.14 Waveguide circuits for transferred electron devices. (From Ref. 42, reproduced by permission.)

approach along with the previo\}sly described model for a Gunn device, Eisenhart and Khan [47] have developed a Gunn oscillator where jump in the oscillation frequency were predicted according to their theory. Another approach to analyze the cross-coupled coaxial-waveguide diode mounting structure is given by Lewin [48, 49] . Based on his analysis, an equivalent circuit was developed for this mounting structure. Chang and Ebert [50] modified the equivalent circuit and verified experimentally with reasonably good agreement. The equivalent circuit was also used for the individual IMPATf module design for a W-band power combiner. A general cross-coupled coaxial-waveguide mounting structure is shown in Fig. 18.15 and its equivalent circuit in Fig. 18.16. The coaxial line in the upper and lower sections has different diameters d1 and d2. Z1, Z2 , Z 3 , and Z4 are the load impedances at each port, respectively. Zo is the characteristic impedance of the waveguide, and Z01 and Zo2 are those of the coaxial lines. Zop is an inductive component due to the post in

18.7

OSCILLATOR/AMPLIFIER CIRCUITS AND MO OLITHJC DEVICE DEVELOPMENT

1155

(a) ~A

d1

t

b

+ ~ A

d

(b )r l•

a

1 3 1 ZT·r1.

0

__,..

z. -

2r

!1

0

Za 0

2

~

Figure 18.15 (a) Coaxial-waveguide diode mounting structure; (b) side view on AA plane. (From Ref. 50; copyright 1980 IEEE, reproduced by permission.) -jXb

-jXb

z, l,

ZIN1

Zo1

Zop

1 3

Z4

Y'

Zo

Zo

Z3

1:N

y2P

2'o2 ZIN2

Z2

ZIN4

ZIN3

Figure 18.16 Equivalent circuit for a coaxial-waveguide diode mounting structure. (From Ref. 50; copyright 1980 IEEE, reproduced by permission.)

1156

TRANSFERRED ELECTRON DEVICES

waveguide excited by TEno modes. Y', Y1p, and Yi p are the a~mit~ces due . to the effects of waveguide-coaxial junctions. The circuit elements given 1n the equivalent diagram shown in Fig. 18. 16 are repeated here for clarity. 00

~

cos mrc - mrrr/ b y =JL.,.;--e Xm m=l I

Y Ip

•

oo

I

~ = J. L,_; - e - mrrr/ b Xm

Loo cos -mrc - e- mrrr/b _ y. - 2p

•

J

m=l

_ . Zo [ 2 Zop - J 4 ko

(rc)2] a

_

m=l

112

Xm

~cos(nrc r / a)-cos[nrc(2d±r)/ a] L.,.; (n2rc2 Ja 2 - k2)1 / 2 n= l

0

-- -k5 -rea ) ([Ko(rrm)- K o(2dr m)D

b2 -rJ- (mire 2 Xm= 4 koab b2

xb=

a ( 2rc r ) Zo- - )..g

a

2

)

(

. 2 red SIIl -

a

red rc(d ± r ) N= csc-csc--a a

where rm= Zo

and

m 2 re 2 (

b2

2 ) 1/ 2

- ko

b)..g

= 217 a )..

2rc Ag=-;:====

j k5- (rc / a)2

11

= 120rc

ko = 2rc j )..

ZINl is the input impedance locking into the circuit at the coaxial end with the other three ports terminated by Z2, Z3, and Z 4 • ZIN2, ZIN3, and ZIN4 are defined the same way, respectively. R 1 and R3 are the real parts of the impedance locking into ports 1-1 and 3-3, as shown in Fig. 18.16. If a Gunn device is placed at Z2 to produce microwave power, the power generated will be split into two parts. One goes to the waveguide load and the other reaches the absorber located at Z 1 , as suggested by Harkless, to suppress undesirable oscillation modes. The ratio R of the power, which provides the coupling between the coaxial line section and waveguide cavity, is then equal to R3/ R1. The objective is to maximize R by using optimum circuit parameters, while the circuit is stabilized to prevent all spurious oscillations at the same time. The accuracy of this equivalent circuit model has been checked with X-band experimental results made by Eisenhart and Khan [45]. Figures 18.17 and 18.18 show the comparison between them. The discrepancy between the experimental results and

18.7

OSCILLATOR/AMPLIFIER CIRCUITS AND MO OLITHIC DEV ICE DEVELOPMENT

60

7-mm coaxial airline, matches on all ports

40 )(

~-

R jX}

Z,n= +

>
_ _ _ _-,--_ _ _ _,

Gallium zone 775°C

l Reaction IDeposition I

I

zone I 850°C

I

zone 750°C

Continues to exhaust and stopcock

Figure 18.29 Schematic diagram of a furnace for vapor-phase epitaxial growth with ar ine system. (From Ref. 83; reproduced by permission of the publisher, The Electrochemical Society, Inc.)

Finally, As4 reacts with GaCl at about 750°C to form the epitaxial layer: 6GaCl + As4 - - 4GaAs + 2GaCh The source saturation step is not required in the arsine system. The critical temperature control of the Ga source is also unnecessary ince the reaction of HCl with Ga is almost complete at temperatures exceeding 800°C. Similar carrier concentrations and mobilities have been obtained for the arsine and arsenic trichloride sy terns.

18.9.2 Liquid-Phase Epitaxial Growth Liquid-phase epitaxial growth is achieved on a GaAs ubstrate by recrystallization of a suitable solution at the liquid-solid interface. A typical solution composition is 10 atom% As in 90 atom% molten Ga. Liquid-phase epitaxy of GaAs depends on the solubility of As in Ga-rich solutions, which decreases with decreasing temperature, as described by the Ga-As phase diagram in Fig. 18.30. At point a in the figure, the solution is saturated with As at temperature T0 • As the temperature decreases to Tb, the state of the system will provide the precipitation of GaAs until the new saturation condition at point c is reached. Thi precipitate may be arranged to deposit on a GaAs substrate as an epitaxial layer. Liquid-pha e epitaxy growth of GaAs wa first reported by Nelson [85]. Figure 18.31 shows the apparatus of a liquid epitaxial reactor in which the substrate and a melt are placed at opposite ends of a graphite boat in a tilt system. The growth of an n-GaAs epitaxial layer involves the following steps: 1. The melt consisting of tin and GaAs is heated to about 650°C to saturate the tin with GaAs.

EPITAXIAL MATERLAL GROWTH

18.9

D DE ICE FABRlCATlO

1169

1200

-6

a

0

~

:::,

800

Ta

, ----------b -----------

C

~ ,_

I I

(1)

I I

a. E

~

Tb

I I 400

I I I I I I

I I I I

1l

2s.0 ° 50

100

Atomic percent arsenic

Figure 18.30 Schematic of GaA pha e diagram. (From M. J. Howes and D . V. Morgan, Gallium Arsenide Materials, Devices, and Circuits; copyright 1985, reproduced by permission of John Wiley & Sons, Ltd.)

Figure 18.31

Schematic diagram of a tjpping liquid-phase epi taxy system. (From Ref. 85.)

2. The system is tilted so that the melt will be in contact with the substrate. After the system is cooled, GaAs initially dissolves from the substrate surface until a solution equilibrium is obtained. With further cooling, epitaxial growth on the GaAs occurs. 3. The system is tilted to its original position to end growth. Two common· growth techniques are used for the liquid-phase epitaxy: (1) the transient method and (2) the steady-state method. The transient method keeps the temperature of the liquid-solid-vapor system uniform. The temperature is then uniformly

1170

TRANSFERRED ELECTRON DEVICES

decreased so that recrystallization of the solute occurs. The steady-state method keeps the system in steady state with a fixed temperature difference between the solution and substrate so that solute crystallizes on the cooler substrate. Both vertical and horizontal systems have been employed [8]. Epitaxial layers grown below about 800°C without intentional doping are usually n-type and in the range 1014 cm- 3 . Room-temperature mobilities between 7500 and 9300 cm2 / V-s are routinely obtained. To produce the desired electron density for transferred electron devices, it is usually necessary to introduce suitable levels of impurity elements into the molten solution. Sn, Se, or Te are commonly used dopants. Good doping uniformity may be achieved over lengths of 100 µm, making liquid-phase epitaxy attractive for low-frequency or LSA devices.

18.9.3 Molecular Beam Epitaxy Transferred electron devices have recently been fabricated by molecular beam epitaxy (MBE) [86] with good results. Molecular beam epitaxy is basically a sophisticated extension of the vacuum evaporation technique that grows elemental, compound, and alloy semiconductor films by impinging directed thennal-energy atomic or molecular beams on a crystalline surface under ultrahigh-vacuum conditions. A simplified system of MBE is shown in Fig. 18.32 for GaAs. Separate effusion ovens are used for Ga, As, and the dopants. All the effusion ovens are enclosed in an ultrahigh vacuum chamber with pressure around 10- 10 torr. The temperature of each oven is adjusted to provide the desired evaporation rate. The substrate holder rotates continuously to achieve unifonn epitaxial layers.

Effusion

In shutter Rotating substrate holder

As

I

Gate vaJve

}=

~Effusio oven shutters

Sample exchange load lock

....__. J

To variable-speed motor and substrate heater supply

Figure 18.32 Schematic diagram of a molecular beam epitaxial system. (From A. y . Cho and J. R . Arthur, "Molecular Beam Epitaxy," Prog. Solid-State Chern., Vol. 10; copyright 1975 Pergamon Journals, Ltd., reproduced by permission.)

18. 10

CO CLUSIO SA D RECENT DEVELOPMENTS

1171

Haydl et al. [87] have made Gunn de ice for 50 to 110 GHz using MBE. The material wa grown at a temperature of about 580°C, which is low compared with the 700 to 800°C for LPE and VPE technique . The low-temperature operation i advantageou ince the outdiffu ion from the ub trate i minimized. Millimeter wave Gunn diode require an active layer thick.ne of 1 to 3 µm with a doping concentration between 0.5 and 2 x 1016 cm- 3 . Tin wa u ed a the dopant. With these diode , 20 mW at 90 GHz, 10 mW at 100 GHz, and 4 mW at 110 GHz have been achieved. The e re ult are comparable with tho e obtained from device u ing GaA grown by the conventional LPE and VPE method [8 ].

18.9.4 Buffer Layers For ideal tran ferred electron device , the epitaxial process should form a smooth graded n-11+ interface when grown on an n+ ub trate. In practice, however, highre i tance layer often exi t at the n-n + interface. To prevent this problem, an n + buffer epitaxial layer of a few micronrneter is grown between the substrate and the active 11-1egion to produce a reasonable yield for the device. Also, a contact buffer layer i frequently grown on top of the n-layer to ensure a reliable operation of the Gunn device.

18.9.5

Metal Contact and Diode Fabrication

After epitaxial growth, metal contacts are applied to both the epitaxial contact layer and the back of the highly doped ub trate. Electrical contact to the device usually requires low-resi tance ohmic contacts, even though it was suggested that a barrier contact at the cathode for ome devices may provide superior performance [89]. The ohmic contacts can be formed by a photolithographic pattern of their configuration and evaporation of suitable metal , followed by an alloying process. Gennaniumgold alloy is commonly used with a small amount of Ni. Alloying is performed in a bort heating cycle in the furnace at about 450°C. Contact is finally made to the alloyed metal with a thermocompression bonding of gold wire or ribbon. The wafers are then diced by wire aw or cleaving. Alternatively, the wafers may be etched into me as before the devices are separated for mounting.

18.10

CONCLUSIONS AND RECENT DEVELOPMENTS

Significant accomplishments have been made in the past four decades on transferred electron devices in the areas of material growth, device fabrication, circuit development, and systems. Although transferred electron devices are useful in certain applications, field-effect transistor can fulfill many requirements better in microwave frequencies . Therefore, recent developments on transferred electron devices are mainly directed to millimeter wave and submillimeter wave frequency applications using lnP devices. In addition, GaN devices have also shown some promises. The details of the recent developments are discussed below:

Indium Phosphide (lnP) Trans/e"ed Electron Devices Development. The steady improvement of growing high-purity InP material required for Gunn devices has led to

1172

TRANSFERRED ELECTRON DEVICES

some significant advances in the performance of these devices. A detailed discussion on InP Gunn devices compared with GaAs has been described elsewhere (90]. Table 18.2 compares the important semiconductor characteristics of InP and GaAs. The key characteristics for its high efficiency, high power, and high-frequency response are the high peak-to-peak valley ratio and fast energy-transfer-time constant. Other important factors affecting output power for InP are its high threshold field, higher electrical breakdown field, and higher thermal conductivity. Table 18.3 summarizes the InP Gunn oscillator and stable reflection amplifier performance for frequencies up to 94 GHz (91]. These efficiency and output power results are at least a factor of 2 better than the equivalent GaAs Gunn devices. The devices used in Table 18.2 are made from VPE materials and are operated in the fundamental frequency mode. For frequencies beyond 94 GHz, InP devices mounted on diamond heat sink and operating at fundamental frequency mode, RF power levels exceeding 130 mW around 132 GHz, 80 mW around 152 GHz, and 25 mW around 162 GHz were reported (92] . TABLE 18.2 Semiconductor Characteristics for GaAs and InP Gunn Devices

Material Properties

InP

Low field mobility (cm 2 /V-s at 500 K) Energy gap (ev at 300 K ) Thermal conductivity (W/cm-°C at 300 K) Breakdown field (kV/cm at N O = 106) Peak-to-valley ratio (300 K ) Threshold field (kV/cm) Effective transit velocity (emfs) Temperature dependence of electron velocity Inertial energy time constant (ps) Energy relaxation time (ps)

TABLE 18.3

Oscillators

3000

5000

1.34 0.68 500 3.5 10.0 1.2 X 107 -0.1%/°C

1.43 0.54 400 2.2 3.2 0.7 X 107 - 0.5%/°C

0.75 0.2

1.5

0.4

Performance of Present InP Gunn Oscillators and Amplifiers

Frequency (GHz)

Power (mW)

Efficiency

35 56

500 385 125

15 11 6

94

Amplifiers

GaAs

(%)

Frequency (GHz)

P sawrated

Power Added Efficiency

(mW)

(%)

35 56

1000 250 50

15

94

11 6

REFERE CES

1173

Material from different epitaxial growth y tern uch a metalorganic chemical vapor deposition (MOCVD) and chemical beam epitaxy (CBE), and molecular beam epitaxy (MBE) wa u ed with ucce [93). With InP Gunn device mounted on diamond heat ink and operated in a econd harmonic mode, RF power level of 0.7 mW al 269 GHz, 1.2 mW at 280 GHz , 1 mW at 300 GHz and 1.1 mW at 315 GHz (94- 95] were achieved. A a re ult, the lnP Gunn devices on diamond beat sink are the most powerful olid tate fundamental frequency RF ource operated at room temperature and for frequencie above 300 GHz. Doping profile for InP are critical to RF performance. The graded doping profile (92]. as oppo ed to the conventional flat doping profile , have been successfull y tested around 130 GHz. A theoretical analysis ha moilified the graded doping profile with a doping notch at the cathode and/or a doping mesa located in the center of the active region [96]. Improved performance i demonstrated over the graded doping profiles for both the fundamental frequency mode and the econd harmonic frequency operations.

GaN -Based Transfe"ed Electron Devices Developme1tt. Due to the higher electron velocitie and reduced time constants computer simulation have indicated that GaN transferred electron devices offered twice the frequency capability of the GaAs transferred electron devices, while their output power density was 2 x 1a5 W /cm 2 compared 2 with 103 W/cm [97] for the GaAs device , indicating a trong potential of the potential of GaN as a microwave oscillator. The improvements offered by the wide-gap semiconductor uch as GaN are due to a significantly higher electrical field strengths which allows an operation with higher doping levels and at a higher bias, than in a conventional narrow gap ill- V semiconductor. Using the Monte Carlo technique, Gruzinskis et al. [98] made a study of 200-300 GHz microwave power generation in GaN transferred electron devices. It was shown that at 300 Kin the 230-250 GHz range over 350 mW in cw mode can be delivered by the (n+ )-p-n-(n-)-(n+) Wurztie GaN structure and over 1.3 W in pulsed mode. However, no experimental results on GaN transferred electron devices have been reported at this time. REFERENCES 1. J. B. Gunn, ''Microwave Oscillations of Current in ill- V Semiconductors," Solid State

Commun., 1, p. 88, September 1963.

2. H. Kroemer, 'Theory of Gunn Effect/' Proc. IEEE , 52, p. 1736, December l 964. 3. B. K. Ridley and T. B. Watkins, 'The Possibility of Negative Resistance Effects in Semiconductor," Proc. Phys. Soc. London , 78, pp. 294-304, August 1961. 4. C. Hilsum ''Transferred Electron Amplifiers and Oscillators," Proc. IRE , SO, pp. 185- 189, February 1962. 5. A. R . Hutson. A . Jayaraman, AG. Chynoweth, A . S. Coriell, and W. L. Feldman, "Mechanism of the Gunn Effect from a Pressure Ex periment," Phy. Rev. Lett., 14, p. 639, 1965. 6. J. W. Allen, M . Shy~ Y. S. Che~ and G. L. Pearson, ''Microwave Oscillations in GaAsxPi-x Alloys," Appl Phys. Lett., 7, p. 78, August 1965. 7. F. Soohoo, Microwave Electronics , Addison-Wesley, Reading, MA, 1971. 8. P. J. Bulman, G. S. Hobs~ and B. C. Taylor. Transferred Electron Devices, Academic Press, London, 1972.

1174

TRANSFERRED ELECTRON DEVICES

9. J. R. Ruch and G. S. IGno, " Measurement of the Velocity-Field Characteristic of Gallium Arsenide," Appl. Phys. Lett., 10, pp. 40- 42, January 15, 1967. 10. P. N. Butcher and W. Fawcett. "Calculation of the Velocity-Field Characteristic of Gallium Arsenide;' Phys. Lett., 21, pp. 489-490, June 15, 1966. l l. D. E. McCurnber and A. a. Chynoweth, "Theory of Negative-Conductance Amplification and of Gun Instabilities in Two-Valley Semiconductors," IEEE Trans. Electron Devices. ED-13. pp. 4-21 , January 1966. 12. B. G. Streetman, Solid State Electronic Devices, 2nd ed., Prentice-Hall, Englewood Cliffs, NJ, 1980. 13. H. W. Thim, "Computer Study of Bulle GaAs Devices with Random One-Dimensional Doping Fluctuations," J. Appl. Phys., 39, p. 3897, 1968. 14. J. A. Copeland, "A New Mode of Operation for Bulle Negative Resistance Oscillators," Proc. IEEE. 54, pp. 1479-1480, October 1966. 15. J. A. Copeland, "LSA Oscillator Diode Theory," J. Appl. Phys., 38, pp. 3096-3101, July 1967. 16. J. A. Copeland, "Doping Uniformity and Geometry of LSA Oscillator Diodes," IEEE Trans. Electron Devices, ED-14, p. 497, 1967. 17. H. C. Huang and L. A. Mackenzie, "A Gunn Diode Operated in the Hybrid Mode," Proc. IEEE , 57, p. 261, 1969. 18. G. S. Hobson, The Gunn Effect , Clarendon Press, Oxford, 1974. 19. H. W. Thim, "Linear Microwave Amplification with Gunn Oscillators." IEEE Trans. Electron Devices, ED-14, pp. 517- 522, September 1967. 20. F. Sterzer, "Stabilization of Supercritical Transferred Electron Amplifiers,.. Proc. IEEE, 57, pp. 1781- 1783, October 1969. 21. P. Jeppesen and B. I. Jeppsson, ''The Influence of Diffusion on the Stability of the Supercritical Transferred Electron Amplifier," Proc. IEEE 60, p. 452, 1972. 22. H. W. Thim, "Noise Reduction in Bulk Negative Resistance Amplifiers," Electron. Lett. , 7. pp. 106-108, 1971. 23. J. Magarshak, A. Rabier, and R. Spitalnik, "Optimum De ign of Transferred Electron Amplifier Devices in GaAs," IEEE Trans. Electron Devices, ED-21, pp. 652-654, October 1974. 24. M . M . Atalla and J. L. Moll, "Emitter Controlled Negative Re i tance in GaAs:' Solid State Electron., 12, pp. 119-129, 1969. 25. T. Hariu, S. Ono, and Y. Shibata, "Wideband Performance of the Injection Limited Gunn Diode," Electron. Lett. , 6, pp. 666- 667, 1970. 26. H. W. Thim and M. R. Barber, "Microwave Amplification in GaA Bull( Semiconductor," IEEE Trans. Electron Devices, ED-13, pp. 110- 114, January 1966. 27. B. W. Hak.ki, "Amplification in Two-Valley Semiconductors, I. Appl. Phys., 38, pp. 808-818, February 1967. 28. T. E. Walsh, B. S. Perlman, and R. E. Enstrom, "Stabilized Supercritical Transferred Electron Amplifiers,'' IEEE J. Solid-State Circuits, SC-S, pp. 374-376, December 1969. 29. B. S. Perlman, C. L. Updahyayula, and R. E. Marx. "Wideband Reflection-Type Transferred Electron Amplifiers," IEEE Trans. Microwave Theory Tech., MTT-18, pp. 911-921 , November 1970. 30. M. R. Lakshminarayana and L . D. Partian, "Numerical Simulation and Measurement of Gunn Device Dynamic Microwave Characteristics," IEEE Trans. Electron Devices, ED-27, pp. 546- 552, March 1980. 31. J. A. Copeland, "Stable Space-Charge Layers in Two-Valley Semiconductors," J. Appl. Phys. , 37, pp. 3602- 3609, August 1966.

REFER£ CES

1175

32. P . N . Butcher, "Theory of Stable Domain Propagation in the Gunn Effect.'' Ph )'s. Leu., 19, pp. 546-547, December 1965. · 33. P. N . Butcher, W . Fawcett. and C. Hilsum. ''A Simple Analy i of Stab]e Domain Propagation in the Gunn EffecL"' Br. J. Appl. Phys.. 7, pp. 841- 850, 1966. 34. I. B. Bott, and W . Fawcett ...The Gunn Effect in Gallium Ar enide.'' in L. Young, Ed., Advances in Micro·waves , Academic Pre . ew York. 1968. p. 251. 35. G. S. Hob on. "Small-Signal Admittance of a Gunn Effect Device," Electron. Lett., 2. pp. 207- 208. June 1966. 36. 1. E. Carroll and R. A. Giblin, "A Low Frequency Analog for a Gunn Effect Oscillator," IEEE Trans. Electron De,·ices. ED-14, pp. 640-656. October 1967.

37. R . B. Robrock II. ·'A Lumped Model for Characterizing Single and Multiple Domai n Propagation in Bulk GaAs.n fEEE Trans. Electron Devices. ED-17, pp. 93- 102, February 1970. 38. J . A. Copeland~ ''Electro tatic Domain in Two-Valley Semiconductor ,' IEEE Trans. Electron Devices. ED-13, pp. 189- 192, January 1966. 39. W. A. Davi . Microwave Semiconductor Design. Van Nostrand Reinhold, New York, 1984. 40. S. Y. arayan and F . Sterzer, ..Transferred Electron Amplifiers and Oscillators," IEEE Trans. Microwave Theory TeclL, MIT-18, pp. 773- 783. ovember 1970. 41. J. A. Copeland and R. R. Spiw~ "LSA Operation of Bulk GaAs Diodes," Int. Solid-State Circuit Conj. Dig. Tech. Pap., pp. 26-27, 1967. 42. K. J. Button. Infrared and Millimeter Waves, Vol. 1 Academic Press, New York, 1979. 43. E . T . Harkless. U.S . Patent 3,534.293, October 13. 1970. 44. R. L. Eisenhart et al., "A Useful Equivalence for a Coaxial-Waveguide Junction," IEEE Trans. Microwave Theory Tech.. MTI-26, pp. 172-174, March 1978. 45. R. L. Eisenhart and P. J . Khan. 'Theoretical and Experimental Analysis of a Waveguide Mol!Ilt:ing Structure,'· IEEE Trans. Microwave Theory Tech., MTT-19, pp. 706-719, August 1971. 46. R. L. Eisenhart. "Discussion of a 2-Gap Waveguide Mount," IEEE Trans. Microwave Theory Tech., MTI-24, pp. 987-990, December 1976. 47. R. L. Eisenhart and P. J . Khan. "Some Tunning Characteristics and Oscillation Conditions of a Waveguide-Mounted Transferred Electron Oscillator," IEEE Trans. Electron Devices, ED-19, pp. 1050- 1055, September 1972.

48. L. Lewin, "A Contribution to the Theory of Probes in Waveguides,' Proc. Inst. Elec. Eng. , M onogr. 259lt pp. 109- 116 October 1957. 49. L. Lewin, Theory of Waveguides , Wtley, New York, 1975, Chap. 5. 50. K. Chang and R. L. Ebert, "W-Band P ower Combiner Design," IEEE Trans. Microwave Theory Tech., MTT-28, pp. 295-305, April 1980. 51. P. J. Allen. B . D . B ates, and P. J. Khan, "Analysis and Use of Harkless Di ode Mount for IMPAIT Oscillators," IEEE MTT-S Int. Microwave Symp. Dig., pp. 138-141, 198 1.

52. M. E. Bialkowski. "Modeling of a Coaxial-Waveguide Power-Combining Structure," IEEE Trans. Microwave Theory Tech., MTI-34, pp . 937-942, September 1986. 53. K. Kurokawa, "Some Basic Characteristics of Broadband Negative Resistance Oscillator Circuits," Bell Syst. Tech. J., pp. 1937-1955, July 1969. 54. H . J . Kuno, J. F. Reynolds, and B. E . Berson, ''Push-Pull Operation of Transferred Electron Oscillators," Electron. utt., 5, pp. 178-179, 1969. 55. M. Omori, "Octave Tunning of a CW Gunn Diode Using a YIG Sphere," Proc. IEEE , 51, p. 97, January 1969. 56. Y. S. Lau and D. Nicholson, "Barium Ferrite Tuned Indium Phosphide Gunn Millimeter Wave Oscillators," IEEE MTI-S Int. Microwave Symp. Dig. , pp. 183- 186, 1986.

1176

TRANSFERRED ELECTRON DEVICES

57. D. Cawsey, "Wide Range Tuning of Solid-State Microwave Oscillators," IEEE J. Solid-State Circuits, SC-5, pp. 82-84, April 1970. 58. C. S. Aitchison and R. V. Celsthorpe, "A Circuit Technique for Broadbancling the Electronic Tunning Range of Gunn Oscillators," IEEE J. Solid-State Circuits, SC-12, p. 21, February 1977. 59. C. D. Corbey et al., "Wideband Varactor Tuned Coaxial Oscillators," IEEE Trans. Microwave Theory Tech. , MTT-24, p. 31, January 1976. 60. B. J. Downing and F. A. Myers, "Broadband Varactor Tuned X-Band Gunn Oscillator," Electron. Lett., 1, p. 407, July 1971. 61. G. E. Brehm and S. Mai, "Varactor-tuned Integrated Gunn Oscillators," presented at the International Solid-State Circuits Conference, Philadelphia~ 1968. 62. L. D. Cohen and E. Sard, ''Recent Advances in the Modelling and Performance of Millimeter Wave InP and GaAs VCOs and Oscillators," IEEE MTT-5 Int. Microwave Symp. Dig., pp. 429-432, 1987. 63. A. Chu et al., ''Low Cost Millimeter Wave Monolithic Receivers," IEEE Microwave Millim. Wave Monolithic Circuits Symp., 1987, pp. 63-67. 64. N. Wang, S. W. Schwartz, and T. Hierl, ''Monolithically Integrated Gunn Oscillator at 35 GHz," Electron. Lett., 20, p. 603, July 1984. 65. J. C. Chen, C. K. Pao, and D. W. Wong, "Millimeter-Wave Monolithic Gunn Oscillators,U IEEE Microwave Millim.-Wave Monolithic Circuits Symp., 1987, pp. 11-13. 66. T. Weng et al., ''Two and Three Tenninal Planar InP TEDs," IEEE Electron Device Lett. , EDL-1, pp. 69-71, May 1980. 67. S. C. Binari, P. E. Thompson, and H. L. Grubin, "Self-Aligned Notched Planar InP Transferred-Electron Oscillators," IEEE Electron Device Lett., EDL-6, pp. 22-24, January 1985. 68. K. J. Russell, "Microwave Power Combining Techniques," IEEE Trans. Microwave Theory Tech. , MTI-27, pp. 472- 478, May 1979. 69. K. Chang and C. Sun, "Millimeter-Wave Power Combining Techniques," IEEE Trans. Microwave Theory Tech., MTT-31, pp. 92-153, February 1983. 70. K . Kurokawa and F. M. Magalhaes, "An X-band 10-Wan Multiple-lMPATI Oscillator," Proc. IEEE , pp. 102-103, January 1971. 71. R. S. Harp and H. L. Stover, ''Power Combining of X-band IMPAIT Circuit Modules," IEEE-ISSCC Dig. Tech. Pap., 16. pp. 118- 119, February 1973. 72. Y. Ma and C. Sun, "1-W Millimeter-Wave Gunn Diode Combiner:• IEEE Trans. Microwave Theory Tech., MTT-28, pp. 1460- 1463, December 1980. 73. K. R. Varian, "Power Combining in a Single Multiple-Diode Cavity." IEEE MTI-S Int. Microwave Symp. Dig., pp. 344-345, June 1978. 74. C. Sun, E. Benko, and J. W. Tully, "A Tunable High Power V-Band Gunn Oscillator," IEEE Trans. Microwave Theory Tech., MTT-27. pp. 512-514, May 1979. 75. A. K. Talwar, "A Dual-Diode 73 GHz Gunn Oscillator," IEEE Trans. Microwave Theory Tech., MTT-27, pp. 510- 512, May 1979. 76. J. J. Sowers, J. D . Crowley, and F. B . Fanlc, "CW InP Gunn Diode Power Combining at 90 GHz." IEEE MIT-S Int. Microwave Symp. Dig. , pp. 503-505, June 1982. 77. T . G. Ruttan, "42 GHz Push-pull Gunn Oscillator," IEEE Proc. , 60, pp. 1441-1442, November 1972. 78. J. J. Potoczniak, H . Jacobs, C. L. Casio, and G. Novick, ''Power Combiners with Gunn Diode Oscillator," IEEE Trans. Microwave Theory Tech., M'IT-30, pp. 724-728, May 1982. 79. L. Wandinger and V. Nalbandian, ''Millimeter-Wave Power Combiner Using Quasi-optical Techniques," IEEE Trans. Microwave Theory Tech., MTI-31, p . 189, February 1983.

REFERENCES

1177

80. M . A. di Forte-P oi on et al., ''High-Power High-Efficiency LP-MOCVD Gunn Diodes for 94 GHz." Electron. Leu.. 20. pp. 1061 - 1062, December 1984.

81. M . J. Howe and D. V. Morgan, Gallium Arsenide Materials, Devices, and Circuits, Wiley, ew York, 1985. 82. J. R. Knight et al.. "The Preparation of High Purity GaA by Vapor Pha e Epitaxial Growth,'' Solid Stare E lecrron., 8. pp. 178- 180, 1965. 83. J. J. Tietjen and J. A . Amick. 'The Preparation and Propertie of Vapor-Deposited Epitaxial GaA 1- f P , U ing Arsine and Pho phine;• J. Elecrrochem. Soc., 113. pp. 724- 772, July 1966. 84. J. V. DiLorenzo. ·'Vapor Growth of Epitaxial GaA : A ummary of Parameters Which Influence Purity and Morphology of Epitaxial Layers," J. Cryst. Growth , 17, pp. 184- 206, 1972.

85. H.

el on. "Epitaxial Growth from the Liquid State and It Application to the Fabrication of Tunnel and Laser Diodes.'' RCA Rev.. 24, pp. 603- 615. December 1963.

86. A. Y. ChoandJ. R. Arthur. "Molecular BeamEpitaxy,"Prog.Solid-StateChem. , 10, p. 157, 1975. 87. W . H . Haydl. R. S. Smita and R. Bosch, ''50-110 GHz Bunn Diodes Using Molecular Beam Epitaxy:' IEEE Electron Device Lett., EDL-1(10), pp. 224-226, October 1980. 88. T. G. Ruttan , ·'Gunn-Diode O cillator at 95 GHz." Electron. Lett., 11, pp. 293-294, July 1975.

89. S. P. Yu et al., ''Tran it-Tlille Negative Conductance in GaAs Bulk-Effect Diodes," IEEE Trans. Electron Devices . ED-18, pp. 88-93, 1971. 90. F . B. Frank et al., "High Efficiency InP Millimeter-Wave Oscillators and Amplifiers," Eur. Microwaves Conf Proc., September 1984, pp. 575- 580. 91. F. B . Fank, Private communication. 92. H . Eisele and G. I. Haddad, ''High-Performance InP Gunn Devices for FundamentalMode Operation in D-Band ( 110-170 GHz)," TEEE Microwave and Guided Wave Lett., 5, pp. 385-387 ov. 1995. 93. H. Eisele, G. 0 . unns, and 0 . I. Haddad, "RF Performance Characteristics of InP Mim meter-Wave (N+)-(N-)-(N+) Gunn Devices," IEEE Int. Microwave Symp. Dig., Denver, Co, June 17- 21, 1997. 94. H. Eisele, " Second-Harmonic Power Extraction from lnP Gunn Devices with More Than 1 mW in the 260-320 GHz Frequency Range," Electron. Lett., 34, pp. 2412-2413, 1998. 95. H. Eisele, A. Rydberg, and G. L Haddad, "R ecent Advances in the Performance of InP Gunn Devices and GaAs TUNNETT Diodes for the 100-300 GHz Frequency Range and Above;~ IEEE Trans. Microwave Theory Tech., MTT-48, pp. 626- 631 , April 2000. 96. R. Judaschke, "Comparison of Modulated Impurity-Concentration InP Transferred Electron Devices for Power Generation at Frequencies Above 130 GHz," IEEE Trans. Microwave Theory Tech. , MTT-48, pp. 719-723, April 2000. 97. E . Alekseev and D. Pavlidis, ''Large-Signal Microwave Performance of GaN-Based NDR Diode Oscillators," Solid-State Electronics, 44, pp. 941 - 947, 2000. 98. V . Gruzinskis et al., "Comparative Study of 200-300 GHz Microwave Power Generation in GaN TEDs by the Monte Carlo Technique," Semiconductor Science & Technology, 16, pp. 798- 805, September 2001.

19 IMPATT AND RELATED TRANSIT-TIME DEVICES KAI

CHANG Texas A &M Universiry College Station, Texas

H. JOHN K UNO QuinStar Technology Inc. Torrance. Califo rnia

19.1

INTRODUCTION

The terrn 'iMPAIT' stands for "impact-ionization avalanche transit time." The IMPATT device is one of the most powerful solid-state microwave sources. The IMPATT ruode u es impact-ionization and transit-time properties to produce the negative resistance required for oscillation and amplification of microwave signals. The concept of the IMPATI diode was first proposed by Read in 1958 for a relatively complex diode structure n+p;p +, which is now commonly called the Read diode [I]. It was not until 1965 that the experimental observation of IMPATT oscillation from a p -n junction diode was reported by Johnston et al. (2). Since then a large amount of effort bas been directed toward a theoretical understanding of the physical mechanism and the development of practical microwave and mi11imeter wave IMPAIT devices, oscillators, and amplifiers. IMPATT devices have been fabricated with Si, GaAs, InP, and other semiconductor materials. The Si-IMPATT devices have been operated at frequencies up to 394 GHz [3] . Although three-terminal devices are now more commonly used at microwaves, the IMPATf diode is still very useful as a power source at miJlimeter wave frequencies. At present, it generates the highest power output among all solid-state devices at millimeter wave frequencies. IMPATI devices are particularly effective as high peak power, short pulse mi11imeter-wave sources generating several tens of watts peak power with a few hundreds nano second pulse width. At lower microwave frequencies, FETs and Handbook of RF/Microwave Components and Engineering,

Edited by Kai Chang

ISBN 0-471-39056-9 © 2003 John Wtley & Sons, Inc.

1179

1180

IMPATT AND RELATED TRANSIT-TIME DEVICES

bipolar transistors, which are three-terminal devices, have replaced IMPAITs for many applications. For this reason, we focus our discussion on mi11imeter wave applications. In this chapter we present an overview of the IMPAIT device and its applications. The device physics of the IMPATT diode is described in Section 19.2. Included are both small-signal and large-signal models. In Section 19.3 we describe the typical device design and fabrication. In Sections 19.4 and 19.5 we discuss the applications of the IMPAIT diode in oscillator and amplifier circuits. The use of power combining techniques to combine several IMPATT diodes in circuits is given in Section 19.6. In Section 19.7 we present possible applications of the IMPAIT diode as a nonlinear reactor or a frequency multiplier. Noise properties and techniques to improve noise characteristics are discussed in Section 19.8, followed by a description in Section 19.9 of related transit-time devices, such as TRAPATI, MITATT, TUNNETT, and BARITI diodes. Section 19.10 is devoted to developments in InP IMPATI diodes, travelingwave IMPATT devices, heterojunction IMPATT diodes, monolithic IMPATT devices, active arrays, and other topics. Finally, Section 19 .11 describes the recent research efforts in SiC and GaN IMPATT devices, monolithic millimeter-wave integrated circuits, and the performance improvements of Si, GaAs, lnP, and heterojunction IMPATT diodes.

19.2 DEVICE PHYSICS AND MODELING Microwave oscillation and amplification are due to the frequency-dependent negative resistance arising from the phase delay between current and voltage waveforms due to avalanche breakdown and transit-time effects. Since Read first proposed a negative resistance diode consisting of an n+pip+ (or p+nin+) structure, many other con.figurations have been developed. For silicon diodes, the single-drift diode (p+nn+) and the double-drift diode (p+ pnn+) are the two most popular structures. For GaAs diodes, single-drift Read diodes (p+n+nn+ or p +n - n+nn+), double-drift Read diodes (p+ pp+n+nn+ or p+pp+pnn+nn+). and hybrid Read diodes (p+ pn+nn+ or p +pnn+nn+) have been fabricated. In this section the physics of avalanche breakdown in a p-n junction will first be discussed. A Read diode will be used for an analytical description of the operation theory of an IMPATI diode. For complicated structures, computer programs are generally used to generate small-signal and largesignal device parameters.

19.2.1 Avalanche Breakdown in p-n Junction Avalanche breakdown normally imposes an upper limit on the reverse voltage applied to most p-n junction diodes. In an IMPATT device, the same avalanche breakdown, which is caused by avalanche multiplication (or impact ionization), can be used effectively to generate microwave power. A p-n junction is shown in Fig. 19.1 together with its 1-V characteristics. As the reverse-biased voltage is near V8 , the electric field around the p-n junction reaches a very high value, and avalanche breakdown occurs. The ionization rate ex for holes and electrons can be represented by (19.1)

19.2

DEVICE PHYSICS AND MODELING

1181

n

p

I

V

Figure 19.1

p-n junction and its 1- V curve.

where m, b, and A are constants depending on the material, and E is the electric field. In general, the ionization rates for holes (ap) and electrons (an ) are not equal (i.e., an :ft a p). F or a p-n junction, the multiplication factor M p of holes can be written

as [4] (19.2) where w is the width of depletion region. A similar result applies to the multiplication factor Mn of electrons (19.3) The avalanche breakdown occurs as Mp or Mn approaches infinity. The breakdown condition is thus given by (19.4) or (19.5) Solving Eq. (19.4) and (19.5) will give avalanche breakdown voltages. For a semiconductor with equal ionization rates for electrons and holes, Eg. (19.4) and (19.5) reduce to (19.6)

1182

IMPATT AND RELATED TRANSIT-TIME DEVICES

:II p+

;

n

n+

E

I I

I I

Avalanche _ .,_,. 1.- .. 1,_,._ 1 1 region

I I

X

Drift _ _, 1 region

a

X

Figure 19.2

Read diode (p+n;n+ ) and its £-field distribution.

19.2.2 Physical Explanation of Operation Theory [5] The operation theory can best be understood by considering a Read diode a hown in Fig. 19.2. The electric field distribution inside the device and the ionization rate are shown in the same figure. The device can be divided into two regions: avalanche region and drift region. The avalanche region is a small high-field region inside the n layer near the p-n junction, where impact ionization occurs and electrons and holes are generated. The drift region is the low-field region where carriers drift at saturated velocity. Due to the time delay in avalanche and drift, there is a phase shift between the voltage and current. The total phase shift is equal to

(19.7) where rA is the time delay attributed to avalanche multiplication and rd is due to finite drift time. For negative resistance to occur, 9 needs to be greater than 90°. Consider

19.2

DEVICE PHYSICS AND MODELING

1183

the ionization rate a

ex= A exp [ - (

b Ede + Erf

)m]

(19.8)

Here the total field i equal to E de + En and let E de ~ breakdown field . Then the addition of an ac field will cau e avalanche breakdown due to the exponential increase in electron-bole pairs. Thi additional ac field could be introduced due to a noise ignal in the oscillation tartup. Thus the current generated by the avalanche process will have its maximum when the RF field goe through zero. Therefore, as shown in Fig. 19.3, the avalanche proces contributes a 90° inductive lag to the current generated in the avalanche region. Thi current is then injected into the drift region of the diode. The current induced in the external circuit by thi change is shown in the bottom of Fig. 19.3. It is obvious that the external current is more than 90° out of phase with re pect to the RF voltage. A negative resistance is therefore created.

J\

J\

Time

C

~

:::, 0

...~

Cl)

)(

w

Time

Figure 19.3

Current waveforms.

IMPATI AND RELATED TRANSIT-TIME DEVICES

1184

Metal contact Double drift

Single drift

~

Metal contact

p+

r=:-1~ N EPI

diff.

ee e(±) e (±) © ee + +

rr

©e +©0 (±)

w

p+

p

substrate

diff.

EPI

N+ substrate

N EPI

Metal contact

C)~

C ··-a. C

(/)

0

Q)

0-o

"O

Q) ;;::

UJ

------- ..... Electron

E~ a> ·-

...

(/)

.._ C

' \\

\L~~~---

::, Q)

() "O

Figure 19.4

Electron

\ , L_~~~-

IMPATT diode model. (From Ref. 6, reproduced by permission.)

19.2.3 Generalized Model Consider a p-n junction under reverse avalanche breakdown conditions as shown in Fig. 19.4 for a single-drift or double-drift diode [6]. Assuming a one-dimensional analysis and neglecting the diffusion effects, a et of equations governing the dynamic of electrons and holes are given by [7] Poisson's equation: (19.9) and by the continuity equations for electrons and holes: (19.10) (19.11)

= doping concentration in the n and p materials p, n = hole and electron carrier densities, respectively E = dielectric constant l p, In = hole and electron current densities, respectively

where ND, NA

t

= time

19.2

x q

1185

DEVICE PHYSICS AND MODELING

= distance from the junction = electron charge

The total current density is

8E

= qn vn +qpvp +Ear

J

where

Vp

and

Vn

(19.12)

are drift velocities of holes and electrons, re pectively.

19.2.4 Small-Signal Analysis For small-signal analysis, it is assumed that the ac signal is small compared with the de component. Small-signal analysis provides useful information on the impedance and frequency re ponses of the device. The electric field and current density J can be written as a de component superimposed by an ac signal, that is, (19.1 3)

= l o+ l 1e1.w t ~E = E iejwt J

(19.14) (19.15)

8ct a= ao + -~E 8E

= ao +a E 1e1'wt I

(19.16)

. t

n =no+ n 1e1 p

=Po+

(19.17)

cu C ._

·5. c

0 Q) C) O C 0 0

n

Distance

Distance

Figure 19.12

Doping and electric-field profiles for a single-drift Read low- high- low structure.

advantages. The hole mobility in GaAs is about 20 times smaller than the electron mobility. The additional series resistance introduced in the p region of the DOR Read tructure reduces the power and efficiency. The increase in thermal resistance due to the additional p-layer of GaAs material also places a limitation on the available efficiency of a DDR Read diode.

Hybrid Read Profile. The hybrid Read structure is a combination of single-drift flat tructure and single-drift Read structure. The device resembles a p-type single-drift flat structure in series with an n-type single-drift Read diode. As shown in Figs. 19.15 and 19.16, two types of hybrid profiles can be realized, one is a flat-high-low structure, the other is a flat- low-high-low structure. The RF performances of these two structures are similar. Since the electron mobility is much higher than the hole mobility in GaAs, the single-drift Read side is always an n-type and the p region in the flat side is designed to be punchthrougb at operating current to reduce the series resistance. Similar to the double-drift Read diode, a larger-area diode can be used to increase the power output. The same thermal limitation applied to the double-drift Read structure

19.3

DEVlCE DESIGN AND FABRICATION

1195

( C

0 O>:.:;

Ce

·Qc 0 Q) 0

(.) C 0

(.)

- l- - - . J I

p

p+

n+ - - - -

n

Distance

(.)

·5(.) -"'O

-Q)

~

w

Distance

Figure 19.13 Doping and electric-field profiles for a double-drift Read low- high structure.

a1so places an upper limit on the efficiency of the hybrid Read diode. Consequently, the hybrid Read diode should have the same efficiency but slightly higher power compared to the single-drift Read diode. Compared with the double-drift Read ruode, this structure is easier to fabricate and the power output capability is similar.

Schottky-Barrier Diode. For a GaAs device, a Schottky-barrier junction formed by a metal-semiconductor junction can be used to replace the p +n junction. IMPATI diodes made with a Schottky barrier have been reported with good output power and efficiency [24, 25].

19.3.3 Device Design Example for a Silicon 94-GHz IMPATT Double-Drift Pulse Diode [8, 26] The device design is based on an iterative process shown in Fig. 19.17. The theoretical value for diode parameters such as doping densities and epi-layer thickness is first obtained via a small-signal computer calculation. IMPATI diode wafers are then fabricated using these diode parameters. After diode fabrication, the diode profile is

1196

IMPATI AND RELATED TRANSIT-TIME DEVICES

C 0

o o C).::

p ++

c~

·-a. C, 0

Q)

C

8

p

n

P

n

Distance

Distance

Figure 19.14 Doping and electric-field profiles for a double-drift Read low- high- low structure.

characterized by C-V measurement or SIMS analysis. Finally, RF testing of diodes will yield information and correlate performance with device parameters. For pulsed diode design, the primary consideration is the impedance-frequency characteristic of the diode as a function of current density. Because IMPAIT operation is strongly dependent on the bias current density, the frequency for peak negative conductance is a function of the operating current. As the current density increases, the optimum frequency and the diode output power also increase. For CW diodes the maximum current density is limited thermally, but for pulsed diodes this limit is extended many times, depending on the pulse width and duty factor. For an extremely narrow pulse, low-duty operation, the diode is no longer thermally limited. The current density can be extended further until space-charge effects cause power saturation and efficiency reduction. In other words, the ultimate diode output power is limited electronically. The theoretical design of silicon IMPATT diodes for a given frequency is normally carried out with a small-signal computer analysis. Strictly speaking, optimum diode design requires knowledge of the large-signal characteristics of the devices, and its performance is strongly dependent on the circuit parameters as well. Since an exact analysis and accurate prediction of the circuit response is difficult at millimeter

19.3

DEVICE DESIGN AND FAB RICATION

1197

..,

p ++

n ++

C 0

-

C); C ~

·a. E 0

Q)

a u C 0

(.)

p

.,

n+

n

Distance

Distance

Figure 19.15

Doping and electric-field profiles for a hybrid Read flat-high - low structure.

wave frequencies, the diode design is based on the small-signal analysis, morufying the design subsequently according to the experimental results. For a specified current density, junction temperature, and junction doping profile, the small-signal computer program calculates and plots the de electric field as a function of distance. The computer program then uses this de solution to calculate the small-signal RF conductance and susceptance per unit area, and the device Q, as a function of frequency for a specified frequency range. The designer uses this program by running it iteratively for different values of input parameters, such as doping density, until a condition is reached for which the plot of device Q versus frequency displays its maximum near the desired frequency of operation. The parameters of the device that produce this are then taken as the design values. Note that the optimum frequencies cover a relatively wide range (i.e., the device Q varies slowly with frequency around its maximum point). To design a pulsed diode properly, it is necessary to predetermine the operating current density. This value of current density that is used as an input parameter in the computer program must be determined independently, such as by theoretical means, which would generally include a thermal analysis, or by an estimate derived from the

1198

IMPATI AND RELATED TRANSIT-TIME DEVICES

C 0

C

.:: ns ,_

0

Q)

-

p ++

oo c,

·-

.c I-

20

Diamond heat sink

10

o-----------"------~----'-----' 60 20

30

40

70

50

80

90

Diode diameter (µm)

Figure 19.30 Thermal resistance versus diode diameter. (x, experimental values for diamond heat sink). 100

90 80

Breakdown voltages

70

-~

60

()

0

G)

(.)

-... C

50

ca Cl) ·u; (J)

co

40

...

E G)

~ 30

1.0

1.5

2.0

2.5

3.0

Zero bias junction capacitance C0 (pF)

Figure 19.31 Thermal resistance versus zero-bias junction capacitance for millimeter wave IMPATT diodes (silicon diode on copper beat sink).

1216

IMPATT AND RELATED TRANSIT-TIME DEVICES

2.0 ---------------------------Junction diameter = 87 .5 µm

1.6

~

()

0

'-' 1.4 Q) (.)

C

as

u, .(n

~

1.2

m

... .c E E

Q)

o.8

Q)

u, C

cu

i=

0.4

Duty = 0.5% copper heat sink

0

50

100

150

200

250

Pulse width (NSEC)

Figure 19.32 Transient thermal resistance of a typical W-band pulsed IMPATI diode plotted as a function of pulse width for several diode junction cliameters. (From Ref. 47; copyright 1979 IEEE, reproduced by permission.)

plotted in Fig. 19.33 as a function of bias current for four diode junction diameters. It is seen that for a given bias current pul e, the chirp frequency (~/) decrease with increasing junction diameter. Thi is due to the increase in the diode thermal time constant with the increasing junction diameter. For a fixed diode diameter, the chirp frequency increases with increasing bias current Since the diode impedance i al o dependent on bias curren4 the amount of frequency chirp and rate of chirp can be controlled by the current pul e waveform. The frequency variation caused by the thermal effect can, therefore, be compensated by changing the operating current density. The method is illu trated in Fig. 19.34. For a flat current pulse, the oscillator frequency decrease (i.e., downward chirp ). By providing an upward ramp on the current pul e, the amount of chirp can be decreased. A continuous increa e in ramp slope will reach a point where the thermal and current effects cancel each other and little chirp is pre ent. Further increase in the ramp slope beyond this point will cause frequency to chirp upward. Thus by controlling the current waveform, the frequency chirp characteri tic can be controlled.

19.4.6 CW Oscillator Performance CW IMPATT o cillators have been built from 7 to 400 GHz for silicon diodes and up to 130 GHz for GaAs diode . The results are ummarized here.

CW Silicon IMPATI' Oscillators. At 10 GHz, a silicon IMPATT oscillator can produce 5 to 10 W of output power [6]. At millimeter wave, power levels of 2 to

19 4

OSClLLATORS

1217

3.0 r - - - - - - - - - - - - - - - - - - - - - - - Junction diameter = 38 µm 94-GHz pulsed impatts pulse width = 100 nsec Rep. rate = 50 kHz

-

~ 2.0

(!)

.c.

i5

·~

-0 C

n,

.c

e-

.c.

(.)

1.0

00' - - - - - - _ _ , 1 1 . . . . - - - - - - - " - - - - - - ' - - - - ' 1.0 2.0 3.0 Bias current {A)

Figure 19.33 Typical chirp bandwidth as a function of bia current for everal diode junction diameters. Pul e width = 100 ns: repetition rate = 50 kHz. (From Ref. 4 7 ~ copyright 1979 IEEE. reproduced by permi ion. )

C Upward chirp

t A Downward chirp

t

C

Q)

~

:, (.)

Time

Figure 19.34 Frequency chirp characteristics of a double-drift IMPATI diode responding to a current ramp. (From Ref. 47; copyright 1979 IEEE, reproduced by permission.)

1218

IMPATI AND RELATED TRANSIT-TIME DEVICES

3 W at 35 GHz [6], 1 to 2 W at 60 GHz [23], 500 mW to 1 W at 94 GHz (48], 100 mW at 140 GHz (27], 60 mW at 170 GHz (27], 78 mW at 185 GHz [3], 25 mW at 217 GHz (27], 7.5 mW at 285 GHz [3], and 200 µ W at 361 GHz [3] have been achieved. The power variation as a function of frequency follows a relationship Pf = constant for frequencies below 100 GHz and P/ 2 for frequencies above 100 GHz. This indicates that the power is determined by thermal limitation for frequencies below 100 GHz and by circuit impedance limitation for frequencies above 100 GHz. The steep power falloff at high frequencies is due mainly to the increased adverse effects of diode, package, and mounting parasitics. Since an IMPATT diode has a negative resistance over a very wide frequency band, the oscillator can be tuned over a wide frequency range using mechanical tuning or bias current tuning techniques. Figure 19 .35 shows the mechanical tuning characteristics of a 60-GHz IMPATT oscillator [6] as an example. It can be seen that the oscillator can be tuned over 10 GHz. Figure 19.36 shows the bias tuning characteristics of a 140-GHz oscillator. The efficiency of a silicon IMPAIT diode oscillator is normally below 10% for a junction temperature of 250°C. Some examples are: 10% at 40 GHz, 6% at 60 GHz, and 5.8% at 94 GHz (48].

CW GaAs IMPATI' Oscillators. For frequencies below 50 GHz GaAs IMPAIT diode offer higher efficiency and power output compared with silicon diodes. At 10 GHz output power of 20 to 30 W has been reported [49] . At higher frequencies, power levels reported are 5 W at 20 GHz with 15 to 19% efficiency [50], 2.8 W at 44 GHz with 18% efficiency (51], and 5 mW at 130 GHz with 0.5% efficiency (25]. 1

19.4.7 Pulsed Oscillator Performance

In many system applications, high-peak-power pulsed oscillators are required. IMPATI devices can be operated as pulsed power sources to achieve a high peak power output

200 r--""'T"""--r-r--~-,---,---~~~-----------,...---.

150

§'

E

15

~ 0

100

....

a,

l

50

0 5~0---------~--'--"---......_--'----IL..-..&.....-J...--',___,J 55 60 65 Frequency (GHz)

Figure 19.35

Mechanical tuning characteristics of a millimeter wave oscillator.

19.4

OSCILLATORS

1219

2

-~ 0

1

>, 0

C

Q)

·c:; :E

w

150

0

~

-

E 100 ~

CD

~

0

Q.

3

.9:::,

50

0

0

140

135

N

J: C,

>, 0

C CD

130

:::, CF CD ~

u.

_ _ _ _ _ _ _ __.__ _ _..__ _--JL..------' 125 400 300 500 600 Bias current (mA)

Figure 19.36 Output power. frequency, and efficiency for a CW diode. (From Ref. 27; copyright 1981 IEEE, reproduced by permission.)

over a relatively short pulse width (about 100 ns) with a low duty cycle. Figure 19.37 shows a pulsed IMPATT oscillator with its associated pulse modulator.

Pulsed Silicon IMPAIT Oscillators. In operating pulsed oscillators, the maximum pulse width as well as the pulse duty factor is one of the most important parameters that determine the achievable peak output power. Since IMPATT devices have small thermal time constants, as discussed in the preceding section, the junction temperature rises rapidly within a pulse. To achieve high peak power output from IMPATT oscillators, the pulse width should be kept below 100 ns and the pulse duty factor below 1%. For longer pulse widths or a higher pulse duty factor, the peak power output will decrease

1220

IMPATT AND RELATED TRANSIT-TIME DEVICES

Figure 19.37

Pulsed IMPATT oscillator and modulator.

due to the reduced input power to keep the peak junction temperature below the upper limit consistent with the reliability. Another important property associated with the pul ed operation of an IMPATI oscillator is the frequency chirping effect. As the junction temperature increases according to the transient thermal impedance change with a pulse cycle, the diode impedance (or admittance) changes. Typically, frequency chirping greater than 1 GHz can be obtained with pulsed IMPATI oscillators. Noting that the frequency of oscillation i also dependent on the bias current, we can control the amount of frequency chirping to meet specific system requirements by shaping the bias pulse current waveform, as shown in Fig. 19.34. The peak power output from a silicon IMPAIT o cillator i 50 Wat 10 GHz [6], 30 Wat 35 GHz, 13 Wat 94 GHz [52), 3 Wat 140 GHz [27] , 1.3 Wat 170 GHz [27), and 0.7 W at 217 GHz [27). A pulse width of 100 ns and a 25- to 50-kHz pulse repetition rate were generally used. Figure 19.38 shows the output power, frequency, and efficiency versus bias current. An oscillo cope picture of input bias current and output video pulse is given in Fig. 19.39. The efficiency i generally below 10%.

Pulsed GaAs IMPAIT Oscillators. Pulsed GaAs IMPATI diodes provide a higher efficiency for frequencies below 50 GHz. One example is 16 W peak power at efficiencies up to 15% with a 5% duty cycle at 40 GH.z [51).

19.S

POWER AMPLIFIERS

IMPATI devices have been used effectively as microwave/millimeter wave power amplifiers. For power amplification both stabilized amplifier and injection-locked amplifiers have been developed. While injection-locked amplifiers (or oscillators) are suited

19.5

POWER AMPLIFIERS

18

10 9

6

--

5

C

4 3

·u

8 7

> 16

2

1221

~ 0

>u

Q)

:E

w

1 0

14

12

-

~

:i a. 10 :i 0

~

CD

~ 0 0. ..:ic::

as Q)

8

98

6

94

~

N

I

~ >-

(.)

C

90

4

Q)

:J CT

--........ Q)

Q)

C

86

2

0

Q)

()

_ _ _ _ _ _ _ _..__ _ _ _ _ _ _.___ _....__ _-J82

7.2

7.6

8.0

8.4 8.8 Bias current (A)

9.2

9.6

Figure 19.38 Output power, frequency. and efficiency versus bias current. (From Ref. 52; copyright 1979 IEEE, reproduced by permission.)

for high gain (>20 dB) and narrow bandwidth ( < l GHz) operation, stabilized amplifiers are for low gain (about 10 dB/stage) and broader bandwidth (> 1 GHz) applications. The maximum power achievable from an amplifier is approrimate]y the ame as that obtainable from the same device operated as an oscillator.

19.5.1

Reflection-Type Stable Amplifiers [53, 54]

Two-tenninal devices with negative resistance can be employed as reflection-type stable amplifiers. A circulator is used to separate the input and output ports. Figure 19.40 shows a general reflection-type amplifier. The equivalent circuit for such a system can be represented in terms of a Norton or Thevenin equivalent circuit as shown in Fig. 19.41. Zc and Ye are the equivalent-circuit impedance and admittance, respectively,

1222

IMPATI AND RELATED TRANSIT-TIME DEVICES

Figure 19.39 Oscilloscope picture of input bias current (top trace) and output video pulse (bottom trace). Horizontal: 20 as/division; vertical: 2 A/division (current), 20 mV/division (video). (From Ref. 52; copyright 1979 IEEE, reproduced by permission.)

Circulator

t Device

Figure 19.40

General reflection-type amplifier.

Figure 19.41 Equivalent circuit for the reflection amplifier.

19.5

POWER AMPLIFIERS

1223

een at the device terminal and thu include device package and tran forming circuits. The gain for this type of amplifier i the reflection coefficient at the plane of the device terminal. The reflection coefficient i given by

r = zD -

Zc Zv + Zc

=

Ye - y D Y, + Yv

(19.56)

The power gain is given by

. gam

Po

.,

= - = 1r1- = P;

Ye - Y0 Ye+ Yv

2

(19.57)

The power generation efficiency i defined by 17

=

( Po - P;)/(Pdc

+ P,)

(19.58)

YD and Ye can be expres ed as

Yv=-Gv+j Bv

(19.59)

and (19.60) Sub tiruting into Eq. (19.57) and assuming that

Bc + Bv= O

(19.61)

Eq. (19.57) becomes (19.62) It i therefore obvious that any amount of gain can be obtained by proper choice of Ge. ote that when Ge = Gn, the gain becomes infinite and the device would oscillate. Therefore, for stable amplification, G c > GD. Any amount of small-signal gain can be obtained by choosing Ge properly.

19.5.2

Injection-Locked Amplifiers/Oscillators

If an external signal at frequency f; and of power P; is injected into a free-running oscillator whose frequency is Jo and whose power output is P0 , then if/,· comes close

to Jo, the free-running oscillator will be injection-locked by this external input signal, and all the output power will appear at f, . The locking range (~J) depends on the external Q of the oscillator and the power gain of the system as given by Adler [55] : (19.63) where t:::.J is the one-sided locking bandwidth and Qe is the external Q of the oscillator. This phenomenon can be used to reduce the noise of an oscillator by locking it to an external low-noise source and can ajso be used as an amplifier, since within the locking range !!,.J, a small input signal at /; appears as a large output signal at J;.

1224

IMPATI AND RELATED TRANSIT-TIME DEVICES

Of course, when power at f; is removed, the power will continue to appear at Jo . The oscillator then works as an injection-locked amplifier. The gain of the system is given by (19.64) gain= Po/ P; = (1 / Q;)(fo/ 6./)2 The gain of this type of amplifier is generally high. Locking gains of 10 to 30 dB have been reported. Equation (19.63) can be used for extemal-Q measurement for an oscillator since all other quantities in the equation can be measured.

12

dB Pout• out

Pin• in

10

lmpatt diode

Pit/Pop

8 C

CC" ~

::,

0.05

0..0

c

'«i

6

0) ~

Q)

~

0

a. 4

0

+1ao

'S

r----r----,---~--~--~,-----,----~--

+90°

P1J P09

~

c

~

>(U

0

0.1

a; "O a,

en (U s=.

a.

0.5

_900

~-~~---:-:-::---~--+---~--~--_._ __+o.4 _j +o.3 -0.3 -0.2 -0.1 0 +o.1 +o.2

- 180° -0.4

Normalized frequency, (f - f0Vf0

Figure 19.42 Large-signal effects on gain-bandwidth and phase-shift characteristics of stabilized IMPATI amplifier. (From Ref. 17; copyright 1973 IEEE, reproduced by permission.)

19.5

POWER AMPLlFIERS

1225

Injection-locking phenomena have been studied exten ively by Kurokawa [56]. The injection locking can be u ed to ynchronize one or more o cillators to a lower power ma ter or reference o cillator and al o reduce part of the FM noi e [57]. Subharmonical injection locking wa al o po ible u ing a low-frequency injection signal [58] .

19.53 Amplifier Performance Both injection-locked and table amplifier have been demon trated for CW and pulsed application . A few e ample are de cribed in this section. A 50-GHz ilicon IMPATT djode amplifier was reported in 1968 by Lee et al. [59]. Both injection-locked and table amplifier were reported. The injection-locked amplifier ha 20 dB of locking gain over a 500-MHz bandwidth and stable amplifier with 13 dB of gain and a 3-d.B bandwidth of 1 GHz. The circuit uses a cap resonator circuit. Scherer reported a three- tage X-band injection-locked IMPATT amplifier with a total of 36 dB gain and a power output of 0.2 W over a 200-MHz bandwidth [60]. Kuno tudied the nonlinear effects and large signal effects of a stable or injectionlocked amplifier [17. 18]. The effects of bandwidth on the transient response of the IMPATT amplifiers as applied to phase-modulated signals and amplitude-modulated signal were inve ligated in detail. Figure 19.42 hows the calculated effects of the input signal level on the bandpass and phase characteristics of a stable IMPAIT amplifier tuned to a mall-signal gain of 10 dB. It can be seen that as the input signal level increase the gain decrease and the bandwidth increases. A similar calculation for an injection-locked amplifier is given in Fig. 19.43. Peterson [61] reported a

1.s -------------.....,..---"'""T"---"""T""------, ~

-

Cl.o

:i

Cl.o ..:

P0 sc= 70 mW}

a> ~ 0

a.

:::::,

1mW

1.0

/

a. 3

0 ,:,Cl)

-~ ns § 0

z

P1n=20 mW 0.5

1

fosc= BO GHz

- 1.5

- 1.0

-0.5

0

+0.5

+1.0

+1.5

t- f0 sc (GHz) Figure 19.43 Large-signal effects on locking characteristics of an injection-locked IMPATT oscillator. (From Ref. 18; copyright 1973 IEEE, reproduced by permission.)

1226

IMPAIT AND RELATED TRANSIT-TIME DEVICES

measurement and characterization technique that allows the design of IMPAIT amplifiers operating at maximum efficiency. A 60-GHz stable amplifier with 6.9 dB of gain and a 1.9-GHz bandwidth was reported by Weller et al. [62]. The bandpass characteristics of this amplifier is shown in Fig. 19.44. A two-stage 86-GHz high-power IMPATT stable amplifier was reported with 10 dB of gain and 18 dBm of output power (63). Recently, most high-power IMPATT amplifiers were developed with a power combiner as the output stage. Details of these amplifiers are presented in the next section, on power combiners. Since a circulator is always needed for an IMPATT amplifier construction, the effects of a nonideal circulator on the amplifier performance are important information. The effects have been studied by Bates and Khan (64]. Microstrip IMPATT amplifiers have also been developed using a microstrip circulator (43, 65]. Figure 19.45 shows a W-band all-rnicrostrip IMPATI amplifier [43]. At the top is the microstrip circulator, with the input and output ports coupled to the transitions. At the bottom is the microstrip IMPAIT oscillator. Using the measurement setup shown in Fig. 19.46, the injection-locking gain as a function of power gain can be measured (Fig. 19.47). The external Q can be calculated using Eq. (19.63). For this particular circuit, the Qe is approximately 37. Figure 19.48a shows the free-running

11.0

10.0

9.0

m a.o

-0 C

·a;

CJ

7.0 ••••

6.0

•• ••

5.0

4.0 3.0

•••• ••

••••

• ••• ••

•••••

··,

···················- "'~ Pin =- 100 mW •••••• 125 mW ··, PinPin= = 150 mW

~-:---------'----'----'-----J--.....1--........JL...,__..J 58.0

59.0

60.0

61 .0

62.0

Frequency (GHz)

Figure ~9.44 ~mall- and large-signal bandpass characteristics for circulator-coupled amplifier. The de bias requirements are 475 mA current at 27 .95 V. The measured diode thermal resistance is l8.6°C/W. (From Ref. 62; copyright 1978 IEEE, reproduced by permission.)

19.6

POWER COMBINERS

1227

Circulator

lmpatt amplifier

Figure 19.45 W-band microstrip IMPATT amplifier. (From Ref. 43; copyright 1985 IEEE, reproduced by permission.)

spectrum of the IMPATI oscillator. After the application of the locking signal, the spectrum exhibits the low-noise characteristic shown in Fig. 19.48b.

19.6 POWER COMBINERS Although the IMPATT diode is the mo~ powerful millimeter wave solid-state device, the output power from a single diode is limited by fundamental thermal and impedance

1228

LMPATI AND RELATED TRANSIT-TIME DEVICES Microstrip impatt amplifier 92 to 98 GHz klystron

I 1

Frequency meter 10 dB coupler

:

M.

.

,crostnp circulator

I

Variable attenuator

I 1

: I

Variable attenuator

lmpatt MIC oscillator

Thermistor mount

_ _ ___ _ _ _ _ _ J

Frequency meter

Power meter Harmonic

10 dB coupler

Thermistor mount

Power

Spectrum analyzer

meter

Figure 19.46 Injection-locking measurement system. (From Ref. 43; copyright 1985 IEEE, reproduced by permission.)

3500 , - - - - - - - - - - -- - - - - - - - - - - - - - - - 3000 ,_

-....

2500 ...

N

I

~

2000 ...

~

1500 -

1000

®

i-

500 ...

I

I

5

10

I

15

I

20

Gain (dB)

Figure 19.47 Injection-locking bandwidth (26/) as a function of power gain (Po/ Pi). (From Ref. 43; copyright 1985 IEEE, reproduced by permission.)

19.6

POWER COMB£NERS

1229

(a)

(b)

Figure 19.48 Spectra of locked and unlocked IMPATf amplifier: (a) free-running spectrum; (b) injection-locked spectrum. (From Ref. 43; copyright 1985 IEEE, reproduced by pennission.)

problems. To meet many system requirements, it is necessary to combine several diodes to achieve high-power levels. Many power combining techniques have been developed in the microwave and millimeter wave frequency range during the past 20 years. These techniques have been reviewed by Russell [66] and by Chang and Sun [67]. The methods of power combining fall mainly into four categories, as shown in Fig. 19.49 [67]: chip-level combiners, circuit-level combiners, spatial combiners, and combinations of these three. The circuit-level combiners can be further divided into resonant and nonresonant combiners. Resonant combiners include rectangular and cylindrical-waveguide resonant-cavity combining techniques. The

1230

IMPATT AND RELATED TRANSIT-TIME DEVICES

Combining techniques

l Chip-level combiners

Circuit-level combiners

Spatial combiners

Other combiners (dielectric guide, push-pull, cap resonator, distributive circuit, and harmonic combiners)

Multiple-level combiners (combinations of chjplevel, circuit-level and spatial combiners) Resonant cavity combiners

Rectangular waveguide resonant cavity combiners

Cylindrical resonant cavity combiners

Nonresonant combiners l

N-way

Corporate combiners

combiners

Conical

Radial

waveguide

line combiners

combiners

Wilkinson combiners

Hybrid coupled combiners

Chain coupled combiners

Figure 19.49 Different power combining technique . (From Ref. 67; copyright 1983 IEEE. reproduced by permission.)

nonresonant combiners include hybrid-couple~ conical waveguide. radial-line, and Wilkinson-type combiners. For IMPAIT diodes, the three mo t commonly used combining techniques are rectangular resonator, cylindrical resonator, and hybridcoupled combiners. IMPATI combiners have been built from 10 to 220 GHz.

19.6.1 Resonant Cavity Combiners A resonant-cavity combiner was first proposed and demonstrated by Kurokawa and Magalhaes in 1971 [68] with a 12-diode power combiner that operated at X-band. The circuit

19.6

POWER COMBINERS

1231

consisted of a rectangular-waveguide cavity with diode mounted in cro s-coupled coaxial waveguide diode mounting module in the waveguide wall . Kurokawa [69] also developed the oscillator circuit theory, which indicated why his circuit configuration gave table o cillation, free from the multiple-diode moding problem. Later, Harp and Stover (70] modified the combiner configuration by replacing the rectangular resonant waveguide cavity with a cylindrical re onant cavity for increased packaging density to accommodate a large number of diode in a mall volume. Thi technique has been used to con truct power combiner for variou application . The combiner can be used as an oscillator, injection-locking amplifier. or stable amplifier. The resonant cavity combiner has the following advantages: a. Combining efficiency i generally high becau e the power outputs of the devices combine directly without any path los . b. The cheme i capable of combining a number of diodes up to 300 GHz. c. It has a compact size and can be u ed as a building block for multiple-level combining. d. Built-in isolation exist between diode to avoid mutual impedance variations by

coupling to the cavity mode. The disadvantages of resonant combiner are: a. Bandwidth is limited to le s than a few percent, although some techniques have been proposed to reduce the circuit Q and thus slightly increase the bandwidth (71. 72]. b. Number of diodes to be combined in a cavity is limited by moding problems since the number of modes increases with the cavity dimensions. c. Electrical or mechanical tuning is difficult. The rectangular and cylindrical waveguide resonant cavity combiners are shown in Fig. 19.50.

Resonant Rectang,ilar Cavity Combiners. For a rectangular-waveguide resonant cavity combiner , each diode is mounted at one end of a coaxial line terminated by a tapered absorb, which serves to stabilize the oscillation. To couple properly to the waveguide cavity, the coaxial circuits mu t be located at the magnetic field maxima of the cavity; therefore, the diode pairs must be spaced one-half wavelength ()..g / 2) apart along the waveguide (Fig. 19.50). The cavity is formed by the iris and a sliding short. Using this circuit, 10.5-W CW power at 9.1 GHz was achieved with 6.2% efficiency by combining 12 IMPATT diodes. To increase the diode capacity, two or more diodes can be positioned on either side of the peak magnetic field [73, 74]. Since the inception of Kurokawa' s combiner, many researchers have attempted to improve and apply the circuit in microwave and millimeter wave frequency ranges. For CW applications, a three-stage amplifier with an output stage using a 12-diode combiner was reported with 39 dB of gain and 16 W of output power at 20 GHz [75]. A two-stage amplifier using a two-diode combiner as the output stage was reported with 11 dB of gain and 3 W of output power at 44 GHz (76]. At 41 GHz, a 10-W output power was achieved with a two-stage amplifier with a 12-diode resonant rectangular cavity combiner as the output stage [77]. The power gain is 30 dB and efficiency is

1232

IMPAIT AND RELATED TRANSIT-TIME DEVICES Waveguide cavity

(a)

t To bias supply

Tapered termination ~ - >J4 transformer

Bias

(b)

Coaxial lines

Eccosorb terminating ~ element - - - U Combiner cavity Transformer

Coaxial lines Magnetic field

Coaxial

• · 1t===:;:::::l.1.:::=E:I-

Input-output _ ___,•. ~ • circulator

coupling link

Waveguide-to-coaxial transition

Figure 19.50 Re onant cavity power combiner : (a) Kurokawa waveguide combiner configuration and cross ections; (b) cylindrical re onant cavity combiner. (From Ref. 67; copyright 1983 IEEE, reproduced by penni ion.)

10% over a 100-MHz bandwidth. The 43.5- to 45.5 GHz amplifier hown in Fig. 19.51 wa developed with 5 W of output power and 24 dB of gain. The output tage is a fourdiode combiner [78]. At 60 GHz, a two-diode combiner with 1.4 W of output power and a four diode combiner with 2.1 W of output power have been reported [79]. For IMPATT diodes in pulsed operation, a W-band two-diode combiner ha been developed to generate 20.5 W at 92.4 GHz with 82% combining efficiency [8] . The diodes were operated with a 100-ns pulse width and 0 .5% duty cycle. Each diode generated approximately 12 W in an optimized single-diode o cillator circuit. Later, the design was extended to a four-diode combiner to achieve 40 W of peak output power with 80% combining efficiency by combining four 10- to 13-W diodes [8]. The combining circuit is shown in Fig. 19.52. At 140 GHz, a two-diode combiner was developed using a lightly oversized waveguide cavity. A peak output power of 3 W has been achieved at 145 GHz by combining two diodes with a 1- and 2.5-W power outpu~ respectively [80]. The two-diode

Preamplifier

HPA

r-----------------------------------------------------T---------------------7

l I

20 mW

1 1 I

(2 passes)

L =0.25 dB

,------.

...1

L = 0.1 dB

, - -- - - - .

t--

t-

~

L = 0.5 dB

.------,

l

(2 passes)

L =0.25 dB

,----.

~ t--

L =0.5 dB

.-----,

t-

L =0.25 dB

....- - - .

I--

20dB coupler

-

1 • I

ILO

IG=

10dB

ILO IG = 9dB

.------.....,

l

L = 0.25 dB

I I I I I I I

I

I

15

: watts

4 diode combtner

1 watt I

Detector

'

I

G = 7.5 dB

I I I I I I I I I I I I

------------ ~-- ------ ~---- -- -------- ~--------------L-----~---------------' Optional control

Bias& control circuit

I SOVDC Figure 19.S1 Schematic diagram of the 5-W solid-state amplifier assembly. (From Ref. 78; copyright 1985 IEEE, reproduced by permission.)

~ (M (M

1234

IMPAIT AND RELATED TRANSIT-TIME DEVICES (a)

(b)

Bias slab

Sliding short

Figure 19.52 W-band. 40-W, four-diode combiner: (a) a embled; (b) disas embled. (From Ref. 8; copyright 1980 IEEE, reproduced by permission.)

combiner wa later ·modified and optimized to generate 5.2 Wat 142.2 GHz from two 3-W diodes [27]. A four-diode combiner was also developed to produce 9.2 W of peak output power with a 100-n pulse width and a 25-kHz pul e repetition rate. The combining efficiencie are about 80 to 90%. The ame de ign wa caled up to 217 GHz in the development of a two-diode combiner with 1.05 W of output power (27].

Cylindrical Resonant Cavity Combiners. The cylindrical resonant-cavity combiner was fir t proposed by Harp and Stover [70]. The combiner shown in Fig. 19.50 consists of a number of identical coaxial modules on the periphery of a cylindrical resonator.

19.6

POWER COMBINERS

1235

The output power of each diode i combined by the cro -coupling of coaxial module and cylindrical re onator. The combined power is coupled to the load through a coaxial probe in the re onator and a coax-to-waveguide transition. This combining cheme ha been e tablished and i very ucce ful in the microwave frequency range for its mall ize and ymmetrical geometry (a few example can be found in [71 - 72, 81-90]). However. at higher frequencie the cylindrical re onator is less de irable because of the moding problem and the requirement of an input/output coaxial probe. A circuit de ign for bandwidth increase can be found in Ref. 84. Many cylindrical re onant cavity combiners have been developed at microwave frequencies. For example. a 16-diode combiner was built with 135 W peak and 45 W aver age output power at 10 GHz (86]. A 60-W CW six-diode combiner wa developed at 5 GHz [82]. The higbe t-frequency cylindrical combiner built con i ts of a four-diode and an eight-diode combiner developed at 37 GHz with 3.6 and 5 W of CW output power, respectively [90] . One-wan ilicon double-drift diode were used in these combiners, which form the last two stage of a five-stage communication amplifier (Fig. 19.53).

------------------------, , Driver module 1

-t4 dBm

: G = 12 dB I

G = 8 dB

G = 3 dB

t I

G = 7 dB 27 dBm

G = 3dB

34dBm

37dBm

I I I I

I I I I I_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ JI

4-diode combiner

8-diode combiner

Figure 19.53 A 37-GHz, 5-W, five-stage amplifier using an eight-diode cylindrical resonant cavity combiner as output power stage. (From Ref. 67; copyright 1983 IEEE, reproduced by permission.)

1236

IMPATT AND RELATED TRANSIT-TIME DEVICES

19.6.2 Hybrid-Coupled Combiners Unlike resonant combiners, the hybrid-coupled combiner has a wide bandwidth capability (larger than 5%); thus the combiner design can be achieved independent of the hybrid characteristics. The hybrid coupler also provides isolation between sources, so that the device interaction and instability problems associated with multidevice operation are minimized. The design approach therefore reduces to that of the hybrid circuit and the diode module. The diode module can be a single-diode circuit or a combiner circuit. The schematic diagram of a 3-dB hybrid-coupled combiner is shown in Fig. 19.54. The combiner is usually operated in the injection-locked mode for phase alignment. When input power is applied to port 1, the power is evenly coupled for ports 2 and 3, and port 4 is isolated from port 1. If ports 2 and 3 are terminated by a pair of matched amplifiers, a signal applied at port 1 is amplified and reflected from ports 2 and 3. The reflected waves are added at port 4 but canceled at port 1, due to the phase relationships between the two reflected waves. Thus the power applied to port 1 is amplified and accumulated at port 4. An analysis by Nevarez and Herokowiz [91] provides the design guideline for this type of combiner. It was found that the amplitude balance and proper phase relationship must be achieved among individual sources at the same frequency. Therefore, in combiner development the individual cavity or module configuration is required to obtain proper tracking in amplitude, phase, and frequency. As the number of devices increases, the difficulty in achieving the required relationship among sources increase . The hybrid coupler insertion loss also poses an upper limit on the number of amplifiers that can be combined; unfortunately, this loss increases with frequency. Therefore, the hybrid approach is not attractive for combining a large number of devices, e pecially at higher frequencies. Use of the hybrid-coupled combining scheme ha been demonstrated in the millimeter wave frequency at V- and W-band. At 60 GHz, a four-diode hybrid-coupled combiner was first reported by Kuno and English [92]. A two-stage amplifier was developed using a two-diode combiner at the first tage and a four-diode combiner at the second stage. A CW output power of I W has been achieved, with a small-signal gain of 22 dB and a bandwidth of 6 GHz in the 60-GHz range. The hardware of thi amplifier is shown in Fig. 19.55. The amplifier was later improved to generate 2.5 W of CW output power at about 61 GHz by using four higher power double-drift IMPATI diodes [93] . At W-band, a combination of hybrid-coupled and Kurokawa combining schemes was first used by Yen and Chang to combine four two-diode combiners to Input 3dB ' 1

2

Output Coupler

Hybrid

X

Source (I)

4 3

Source (II)

Figure 19.54 Hybrid-coupled combiner schematic diagram. (From Ref. 67; copyright 1983 IEEE, reproduced by permission.)

19.6 Hybrid coupling adapter

POWER COMBINERS

1237

lmpatt amplifiers (2)

Termination

Short slot hybrids (3 dB) Isolator

RF input

RF output Short slot hybrid (3 dB) Termination

lmpatt amplifiers (2)

lmpatt amplifiers (2) (DD 70)

Figure 19.55 V-band two-stage IMPATT amplifier/combiner block diagram and hardware. (From Ref. 67; copyright 1983 IEEE, reproduced by permission.)

generate 63 W of peak output power using eight 10- to 13-W diodes (94] . A three-stage injection-locked amplifier was developed using this eight diode combiner as the output stage (Fig. 19.56). A hybrid-coupled combiner/amplifier using a branch-line coupler in a microstrip medium has been established at microwave frequencies (95).

19.6.3 Other Combining Schemes In addition to the commonly used combining techniques described above, several other combining schemes have been proposed-and demonstrated for IMPATI diodes. A few examples of these methods are discussed here.

1238

IMPATT AND RELATED TRANSIT-TIME DEVICES Termination

Input

(a)

.___ 3 dB hybrid coupler

2-diode combiner

2-diode combiner

4-way adaptor

2-diode combiner

2-diode combiner

3 dB hybrid coupler

Termination

..___ _

Ouput

>-------'

(b)

Figure 19.56

(a) Hybrid coupling for four two-diode combiners; (b) W-band 63-W peak output

power three-stage injection-locked transmitter. (From Ref. 67; copyright 1983 IEEE, reproduced by permission.)

Josenhans fir 't proposed combining IMPATT diodes electrically in eries and thermally in parallel on a diamond heat sink (96]. An output power of 4.5 Wat 13 GHz with an efficiency of 6.4% wa achleved. The fundamental limitations of chip-level combining are the circuit impedance-matching and device interaction . The number of diodes is limited at high frequency, due to the mall dimen ion and thermal interactions. Perhaps the most successful chip-level combining was developed by Rucker et al. (97 - 99]. The combining geometrie are hown in Fig. 19.57a. Quartz capacitors were placed in parallel with each diode chip to avoid the in tability problems associated with multichip interactions. Most results reported are at X-band. An article summarizing the analytical and experimental results can be found in [99]. Recent experiments have extended this technique to 40 GHz (100] .

19.6 (a)

Diode

POWER COMBINERS

1239

Capacitor

I /(

?5)

I

(b)

Figure 19.57 Chip-level combiner. (a) chip-level power combining geometries; (b) power combining through diode array. (From Ref. 67; copyright 1983 IEEE. reproduced by permission.)

A similar technique using the parallel diode array method was demonstrated by Swan et al. [101]. In this technique, diodes arranged in a small area are considered as a single diode from the RF point of view (Fig. 19.57b). Consequently, a single tuning circuit is sufficient for operation. However, as frequency increases, the lateral dimensions of the diode array are no longer small compared to the wavelength, and each diode does not share the same electromagnetic environment. Impedance matching between the array and circuit thus becomes more difficult due to the low impedance of the multidiode array. Spatial or quasi-optical combiners are promising for applications at high mi11imeter wave frequency since the size restriction and moding problems are less severe. Examples of these combiners are a 35-GHz active array developed by Durkin et al. (Fig. 19.58) [102] and a 10-GHz space power combiner by Dinger et al. [l 03]. The technique proposed by Hummer and Chang [104] for a spatial combiner with Gunn diodes mounted directly on microstrip patch antennas can also be used for IMPAIT diodes. Quasi-optical combining techniques proposed by Mink [105], Wandiger and

1240

IMPATT AND RELATED TRANSIT-TIME DEVICES Radiating elements

Exciter

I: D.t,;Z_ 6EL To receiver

Oscillator coupling · slots

)< Radiating slots

Comparator coupling slots

Figure 19.58 35-GHz spatial combiner block diagram and antenna array layout. (From Ref. 67; copyright 1983 IEEE, reproduced by permission.)

Nalbandian [106], and Young and Stephan (107] are al o prorrn mg for IMPATT diodes. Other combining techniques that have been u ed for IMPATT diode are conical waveguide power combiners, radial-Une power combiner , Rucker's combiners, pushpull combiners, and harmonic power combiner [67] .

19.7 IMPATT DIODE FOR FREQUENCY MULTIPLICATION AND CONVERSION . The nonlinear properties of IMPATT diodes can be used for frequency multiplication and conversion [108]. From the small-signal analysis described in Section 19.2, it can

NOISE CHARACTERISTICS AND SPURIOUS OSClLLATIONS

19.8

Input f1 and DC bias

100

A

4

E

1241

transformer f1

90

80 ~ E

.Q)...

70

60

~

~

transformer f 11

0

a.

:::,

a. 0 :::,

50 40 30

20 10

10

100

1000

Input power (mW)

Figure 19.59 Experimental output power variations of x 11 multiplier as a fu nction of drive le el (From Ref. J08: copyright 1976 IEEE, reproduced by permission.)

be seen that the avalanche zone behaves as a nonlinear inductor. The ''Manley Rowe" power relations show that the avalanche zone can operate as a parametric amplifier, a harmonic generator. a frequency multiplier, or a frequency up-converter or downconverter (109]. The nonlinear behavior of an IMPATT diode is due primarily to the nonlinearity of the avalanche zone. The influence of the transit zone is quite different, and its dimensions must be optimized [ l 08]. For better nonlinearity and high-order harmonic operation, a punch-through diode at breakdown is preferred for its short transit time (108]. To eliminate series resistance, it is also desirable to have the p-n junction bounded by heavily doped p+ and n+ regions. Experimental output power results for an x 11 multiplier versus input power are shown in Fig. 19.59. The diode used is a silicon p +nn+ diode with breakdown voltage V8 = 20 V and drift region width W = 0.6 µm . The output frequency is 38.8 GHz. It can be seen that a conversion efficiency of 30% was achieved for a 20-dBm 1nput power. The use of IMPATT diodes for up-converters was also reported [108].

19.8 NOISE CHARACTERISTICS AND SPURIOUS OSCILLATIONS Noise characteristics of oscillators are important properties for system applications. The AM noise of the IMPAIT diode is about 10 dB higher than in the Gunn oscillator. This makes the Gunn diode more suitable than the IMPATT diode for local oscillator

1242

-

.c

IMPATT AND RELATED TRANSIT-TIME DEVICES

- 130 ---"'T""'ll~"lr'T'--r--,--,----,---,---,--,-~----,---,---r-~-.-, dBc

"O

.§

"O C

cu .o - 140 N

94GHz 92 GHz

I

.,... C

0

~

:: - 150 Q)

·;::

....

cu (.) 6

-

~ 59 GHz

' - 160 0 Q)

en C

~ ~

h~~~~

ffi

o -1701.._ _ _ _K_.,H_z__.___1_0_ _ _ _ _ _ _ _1_,0_o'""""~---1_ _G_H._z_ _._...___ 10 Frequency away from carrier (FM)

Figure 19.60 AM noise characteristics of mil1imeter wave oscillators. Solid line, IMPATf; dashed lines, Gunn; dashed- dotted line, Klystron. (From Ref. 6, reproduced by permission.)

application. The IMPATI oscillator also suffers from parametric and bias oscillation, which eventually leads to diode burnout if care is not taken to prevent it.

19.8.1

AM and FM Noise Characteristics [6]

Figure 19.60 shows the measured AM noise characteristics of typical mi11i:meter wave oscillators. For comparison, those of Gunn oscillators and klystrons are also shown. It is interesting to note that the AM noise characteristics of IMPAIT oscillators near the carrier are similar to those of Gunn oscillators and klystrons. At higher modulation frequencies (higher than several hundred megahertz), however, the IMPATI oscillator noise is higher than that of a Gunn oscillator or a klystron. For this reason IMPATT oscillators are in general difficult to use as local oscillators for mixers in receiver applications. Shown in Fig. 19.61 is the calculated effect of local oscillator (LO) on receiver noise figures for various values of an LO noise suppression factor. For a typical IMPATT local oscillator with carrier-to-noise ratio of 150 dB/Hz, LO noise suppression of 30 dB is required to keep the adverse effect of the LO on the receiver noise figure negligible. The required LO noise suppression can be accomplished either by a bandpass filter for the LO frequency or by a balanced mixer configuration. With proper LO noise suppression techniques, IMPATT oscillators can be used effectively as local oscillators for low-noise mi1limeter wave mixers. Shown in Fig. 19.62 are measured FM noise characteristics of a mi11imeter wave IMPATT oscillator. FM noise characteristics on the oscillator depend strongly on the circuit Q. Values of the circuit Q for typical millimeter wave IMPATT oscillators range between 20 to 100. These values are based on the injection-locking gain-bandwidth characteristics measurements.

19.8

OISE CHARACTERISTICS AND SPURIOUS OSCILLATIONS

1243

-CD "C U)

CD

'-

:::, 0)

20

:.:: CD

U)

0

C

'-

CD

>

«> (.) Q)

10

'-

co ... 0

~

160 Signal-to-noise ratio of LO (dB/Hz) At IF

Figure 19.61 Effects of local oscillator noise on mixer noise figure. NF= Lc(NrF + NR - 1) + PLo(S/ N)Lo / KToa ), Lc(NrF + NR - l ) = 7 dB, and PLo = 5 dBm. (From Ref. 6, reproduced by permis ion.)

40

-N

I

60

CD

"C

0

.::

e

80

CD U)

0

CI 0

-

100

I

coC

0)

(f)

120

140

10 M

1M Modulation frequency (Hz)

Figure 19.62 Measured FM noise characteristics of millimeter wave IMPAIT oscillator. (From Ref. 6, reproduced by permission.)

19.8.2 Frequency and Phase Stabilization Techniques [6] A typical IMPATT oscillator has a frequency stability of - 0.5 x 10- 4 c- 1• This means that the frequency drift rate is approximately -2 MHz/°C at 40 GHz and -5 MHz/°C 0

1244

IMPAIT AND RELATED TRANSIT-TIME DEVICES

at 100 GHz, for example. For applications where temperature variations cause excessive frequency drifts, a number of techniques have been developed for controlling the frequency. The simplest method is to control the oscillator cavity temperature by means of a small heater and a control circuit (see Fig. 19.63). Since solid-state millimeter wave oscillator cavities have small masses, it is relatively easy to control their temperature in this way. The temperature variation can be kept to less than 1°C. The frequency drift of an IMPATI oscillator can be reduced significantly by means of a high-Q cavity. Shown in Fig. 19.64 is a schematic diagram for the high- Q -cavity frequency stabilization technique [110] . Another approach to frequency stability is to use a frequency discriminator such as an Invar cavity filter with an AFC loop (see Fig. 19.65). This method does not require heater power but does require a more complex control circuitry.

Oscillator cavity

Cont CKT

I - -- - - - - '

Ou1put

Heater

Figure 19.63 Cavity temperature-controlled frequency stabilization. (From Ref. 6; reproduced by permission.)

Bias pin

Tuning short

To load •

;....:=~~~'-

L Figure 19.64 sion.)

Diode

High Q cavity

High-Q cavity for frequency stabilization. (From Ref. 6; reprodu_ced by permis-

AFC loop

DET lnvar cavity .__-r-_, filter

osc

r---~xx~--- Outp~

Figure 19.65 AFC loop-controlled frequency stabilization. (From Ref. 6; reproduced by permission.)

19.8

OISE CHARACTERISTICS AND SPURJOUS OSCILLATIONS

1245

In addition to long-term frequency tability, many y tern require phase stabilitie of cry tal quality. For uch application two ba ic approache have recently been developed for millimeter wa e ource . One is a phase-locked-loop approach, and the other i an injection-locking approach u ing a multiplier chain. In the phase-locked o cillator (Fig. 19.66) the ample power of the millimeter wave frequency is converted down to an IF by a harmonic m.ixer the pha e i compared with the phase of the reference cry tal o cillator. and the phase error i then corrected by mean of a feedback loop to the millimeter wave o cillator. Thi technique ha been applied successfully to mil1imeter wave IMPAIT o cillator up to 217 GHz [1 11]. Shown in Fig. 19.67 is a compari on of phase noi e of free-running and pba e-locking millimeter wave IMPATT o cillators. A ign.ificant reduction in phase noise can be seen within the locking band. The locking bandwidth i limited by the locking loop bandwidth, which is typically 1 to 10 MHz. Similar improvement in phase stability can be achieved by means of an injectionlocked o cillator with a frequency multiplier chain as shown in Fig. 19.68. Since tradeoff between locking gain and bandwidth can be made, a broader locking bandwidth can be achieved in an injection-locked than in a phase-Jocked oscillator. However, an injection-locked o cillator using a multiplier chain is considerably more complex than a phase-locked o cillator at mi11imeter-wave frequencies.

19.8.3 Spurious Oscillations Nonlinear effects within the IMPATT diode allow it to support spurious oscillations at any frequency. As diodes are operated for even greater power capacity, the control of purious oscillations becomes more difficult.

20mW Tunable 217 GHz im£att 0 C

5.4mW

3.SmW Crosscoupler (13.8 dB)

Isolator

0.2mW at 217 GHz

Spectrum analyzer

' Phase-lock loop (electronics)

-

RF output

I

Harmonic mixer (x 12)

IF amplifier

'

Reference

osc

-

() 140 (theory) 0.68

1.11

1.43

1.34

8.5

X

106

Source: Ref. 142, reproduced by permission.

Little work has been devoted to InP IMPATI development. Fank et al. [142) reported a p +nn+ IMPATI with a CW power of 1.6 Wand 11.1 % efficiency at 9.78 GHz. With a 10% duty cyc1e and 500-ns pulse width, the best perlormance obtained was 6.1 W of peak output power with 13.7% efficiency at 10.8 GHz. The operating voltage is generally higher for InP IMPATI diodes, due to the higher peak-to-valley ratio.

19.10.2 Traveling-Wave IMPATT Devices A traveling-wave IMPATI device was first proposed by Midford and Bowers [143] in 1968. An analysis was provided by Hambleton and Robson in 1973 [144]. The structure is shown in Fig. 19.75, where region 1 is the active device region and region 2 is the substrate region. Analyses of similar structures have also been carried out by various researchers, including Franz and Beyer (145, 146], Fukuoka and Ioth [147], and Mains and Haddad (148]. The analyses are based on solving Maxwell's equation,

p+

Figure 19.75 Distributed IMPAIT structure. Region 1 is the active device, region 2 is the substrate, and the shaded areas are the metal contacts. (From Ref. 144, reproduced by permission from International Journal of Electronics.)

19. 10 OTHER DEVICES

1255

di tributed small-signaJ model, and transmission-Une formulation. Experimental results were reported by Bayraktaroglu and Shih [149] in 1983 for di tributed GaAs IMPATT oscillator with output power level 1.5 W at 22 GHz, 0.5 W at 50 GHz, and 7 mW at 89 GHz.

19.10.3 Monolithic IMPATT Devices Monolithic implementation of olid-state device provides the potential for small size, light weight, low-co t production, improved reliability and reproducibility, and easy assembly. The monolithic realization of IMPATT diodes i lagging behind that of FETs and mixer diodes. Recently. Luy et al. reported a monolithically integrated coplanar 75-GHz ilicon IMPATT o cillator (150]. Figure 19.76 shows the circuit and device (a)

-

1as network -

Bias network

H

10µm n•Sb3x1019

(b)

n

Sb 0 .7 x 10

17

n Ga 1 x 1017

Figure 19.76 Monolithic IMPATI' diode: (a) Circuit structure (b) device configuration. (From Ref. 150.)

1256

IMPATI AND RELATED TRANSIT-TIME DEVJCES

~~1: ~~~ ~ ~ ;;;.;_ '/&.~~i'iil~Ciiii;;;i~iii-

-

--~ ~~;::::: - ~~~

Figure 19.77

SEM picture of the monolithic IMPATT diode. (From Ref. 150.)

(a)

SI GaAs substrate AIGaAs layer GaAs active layers Plated heat sink

(b)

(c )

r22Z2Z221 ~ o p contact metallization

f:72Z2ZJ (d)

IMPATT

Capacitors

First-level Pl

(e)

L1

C1

L2

C2 Second level Pl

Figure 19.78 Processing sequence of monolithic IMPATT o cillator. (From Ref. 151 ; copyright 1985 IEEE, reproduced by permission.)

19.lO

OTHER DEVICES

1257

structures. The active layer were grown by ilicon molecular beam epitaxy. A disk resonator was used to control the frequency, and oscillation of 76 GHz with a CW power of 1 mW was detected. Figure 19.77 show a SEM ( canning electron microscopy) picture of the diode. GaA monolithically compatible IMPATT oscillators have also been developed [151]. The proce ing tep are hown in Fig. 19.78. The best performance with output power of 1.25 W at 32.5 GHz wa achieved with 27% efficiency.

19.10.4

Heterojunction IMPATT and MITATT Devices

Heterojunction structures have been used for FET:, HEMTs, and many Optoelectronics device . The u e of a beterojunction for IMPATT or MITATT is relatively new [1 52). Re ults from a large- ignal analysis show that significant improvements in efficiency can be achieved u ing heterojunction structure . Both single heterojunctions (GaAlAs-GaAs) and double beterojunctions (GaA1As- GaAs-GaA1As) have been studied [152]. Figure 19.79 shows GaAIAs-GaAs-GaAIAs double-heterojunction tran it-time device. Device and oscillators using heterojunction structures were

(a)

Drift region

•

I I

GaAIAs GaAs Region (j) _(2) 0

I•

_(3)

I

GaAIAs l1-_

Xg

"I •

Generation region

{b)

Terminal voltage

0w

Injected current

1--t

0m

Induced current

0o 12 I I

I

11 0h1

1r/2

7r

I

0w

0h2

I I

~ w 3,r/2

2Jr

Figure 19.79 (a) GaAIAs-GaAs- GaAIAs double-heterojunction two-te~al transit-time device; (b) terminal voltage, injected current, and induced current for the device. (From Ref. 152; copyright 1987 IEEE, reproduced by permission.)

1258

fMPATI AND RELATED TRANSTT-TIME DEVICES

fabricated and tested [ 152]. Preliminary results show a typical power of 45 mW at 72 GHz for a 1% duty cycle and 1-µ s pul e width.

19.10.5

Active Antenna or Array Elements Using IMPATT Diodes

An active antenna element can formed by integrating the solid-state devices directly on the antenna. Many elements can be combined to build an active array or a spatial power combiner. Recent development in microwave/millimeter wave integrated circuit has made it possible to integrate the active solid-state devices with planar antennas. Perkins [153] ha mounted an IMPATT diodes on a circular microstrip patch antenna. Camilleri and Bayraktaroglu have built IMPATI diodes and resonator/antenna circuits monolithically [154] . Similar circuits have also been developed u ing Gunn diodes [155, 156].

19.11

RECENT DEVELOPMENT

In the past 10 years, research and development efforts have been devoted to Silicon Carbide (SiC) and Gallium Nitride (GaN) IMPAIT device . monolithlc millimeterwave integrated circuits, and the performance improvement of Si, GaA . InP, and heterojunction IMPAIT devices. A summary of these development i given below.

19.11.1

SiC and GaN IMPATT Devices

For high-power and high-temperature application , wideband-gap material such a SiC and GaN have many potential advantages. The e material have a relatively high thermal conductivity, high critical field, and high aturation velocity. Theoretical simulations for SiC IMPATT were reported in [157-159]. A largeignal simulation study has been performed on 4H-SiC IMPATT o cillators. Peak power capability of more than 1 kW with greater than 15% power efficiency was predicted for both Ka-band and X-band SiC IMPATT diode [157]. Millimeter-wa e SiC IMPATT oscillator was analyzed at 500 K and 800 K with temperature-dependent ionization rate and aturation velocity [158] . The large signal imulation demon trate the fact that SiC IMPATI device have efficiency and power advantage over Si and GaA IMPAIT device at millimeter-wave frequencie . The potential of SiC and diamond for producing IMPATT device wa al o reviewed [159] . Theoretical tudies for GaN IMPATT were al o reported [160- 162]. DC and dynamic characteri tic have hown GaN-ba ed IMPATI and potential candidate for replacing traditional Si and GaA -ba ed IMPAIT device at D-band operation [ 160]. The noi e of GaN-based IMPATI i found to be higher than that of GaA -based IMPATTs but equivalent to Si-based IMPATfs [161). In additional to the theoretical simulation , SiC IMPATI oscillator were experimentally demon trated. A prototype single-drift IMPATT diode with a high-low doping profile wa tested in a reduced-height waveguide cavity. 0 cillations were ob erved at 7.75 GHz with a power output of 1 mW [163]. A pul ed SiC IMPATT oscillator was also built with a peak power of 300 mW at X-band [164).

19.11

RECE T DEVELOPMENT

1259

19.11.2 IMPATT Diodes Integrated with Monolithic Millimeter-Wave Integrated Circuits IMPATT diodes integrated with monolithic millimeter-wave integrated circuits have been studied [165 , 166]. Two method have been developed to characterize the monolithic IMPAIT re onator u ed in the de ign of the millimeter-wave oscillator [165]. They are the tran mission re onance method and the varactor method. Although they are only good for di crete frequency point , the accuracy i much better than the method of de-embedding. Presting et al. [ 166] reported ome experimental results of silicon IMPAIT monolithic integrated circuits used as o cillators in radar y tern application in the 60-80-GHz frequency range. The device was mounted up ide-down on a copper heat sink integral with cavity re onator. Figure 19.80 show the cro - ectional view of the processed IM.PATT diode fabricated in monolithic technology. The IMPATI diode was grown by MBE with a double drift doping profile. The diode has a diameter ranging from 24 to 30 µm. Figure 19.81 how a schematic view of the chip mounted on the heat sink with integrated cavity re onator consisting of a thin carrier plate and a thick Cu block. A slit with corresponding dimensions to the lot antenna is inserted in the carrier plate to provide radiation coupling into the resonance cavity. Experimental results showed a maximum radiated power of -13 dBm at frequencies around 7 1.4 GHz. IMPATT diodes also play a key role in the Si/SiGe MMIC's [167, 168]. Siliconbased millimeter-wave integrated circuits can provide new solutions for near-range sensor and communication applications above 50 GHz. The key devices are IMPAIT diodes for mi1limeter-wave power generation and detection in the self-oscillating mixer mode. IMPATI diodes integrated with GaAs monolithic circuits have also been reported (169-172]. Chips produced 100-mW CW output power in the 58-65-GHz range with 14.5% efficiency [170], 27-mW CW output power at 43 GHz with 7.2% efficiency [171], and 120-mW peak power at 48 GHz [172].

19.11.3 Performance Improvements of Si, GaAs, InP, and Heterojunction IMPATI Devices Recent developments in Si IMPAIT diodes have resulted in 30O-m W CW output power at 140 GHz using a double-drift double-Read profile (173]. An integrated planar oscillator with 100-mW CW output power at 140 GHz with 4.5% efficiency was also reported (174]. Double-drift Si IMPAIT diodes with hybrid Read profiles were

Air bridge metallization

Figure 19.80 Cross-sectional view of processed IMPATT diode fabricated in monolithic technology (166]. (Permission by IEEE.)

1260

IMPATT ANO RELATED TRANSIT-TIME DEVICES

,..

1.6mm

mm wave chip on Si substrate

Bias connectors

AIN (Cu)- plate

3mm

Resonator cavity (Cu) 2.8 mm (76 GHz) - 3.4 mm (62 GHz)

it 0.1 mm Bottom foil (Au, Cu)

Figure 19.81 Scheme of up ide-down mount of slot tran mitter chip on heat sink with integrated resonator [166]. (Permi ion by IEEE.)

fabricated u ing vapor pha e epitaxy growth. The following results were achieved: CW power output of 1.95 W with 11.7% efficiency at 40 GHz, 1.05 W with 13.6% efficiency at 61 GHz, and 612 mW with 5.7% efficiency at 93 GHz [175]. Signjficant progress has been made in high-frequency operation of GaAs IMPATI diodes. For CW operatio~ GaAs double-Read IMPATI diodes have produced 100 mW at 144 GHz with 5% efficiency [176]. GaAs single-drift flat-profile IMPATI diodes fabricated from MBE grown material was reported with a CW output power of 15 mW at 135 GHz with 1.5% efficiency [177]. GaAs W-band IMPATI diodes fabricated form MBE material has demonstrated an output power of 270 mW for very low noise operation [ 178]. Quasi-optical power combining array employing GaAs IMPATI diodes in a weakly coupled arrangement was demonstrated at 60 GHz [179]. A 2 x 4 array generated a total radiated power in excess of 2 W. Some simulations have been performed to compare the GaAs and lnP IMPATI diodes [I 80, 181 ]. It shows that the RF power levels of more than 1 W as well as

REFERENCES

1261

de-to-RF conver ion efficiencie of more than 18% around I 00 GHz can be expected from optimized diode . In the beterojunction IMPATT diode , Ku-band experimental devices exhibit up to 2-dB higher power and 3- 6-dB le phase noise content as compared to a conventional p-n junction device (182). Some imulated results for GalnP IMPAIT ha e been reported for millimeter-wave applications. [183).

19.12

SUMMARY AND FUTURE TRENDS

After over 30 years of re earch and development, IMPATT diode technology has gradually reached it maturity. Currently, IMPATT diodes can be produced routinely, operating from 10 to 300 GHz. IMPATT diodes are still the most powerful solid-state power source at mi)limeter wave frequencie . At frequencies above 140 GHz, IMPATI or transit-time device are the u efuJ solid- tate power sources operating at the fundamental frequency. With increasing future demands on miUimeter wave systems in radar, ensor and communication applications, IMPATT and transit-time devices will play an iraportant role in the ucces ful development of these systems.

REFERENCES 1. W . T. Read, ..A Propo ed High-Frequency Negative Resistance Diode," Bell Syst. Tech.

J., 37, pp. 401-446, March 1958. 2. R. L. Johnsto~ B. C. DeLoach, Jr., and B. G. Cohen, "A Silicon Diode Microwave Oscillator.n Bell Syst. Tech. J.. 44, pp. 369-372, February 1965.

3. M. !no. T. Ishtl>ashi, and M. Ohmori, "CW Oscillation with p+ p n+ Silicon IMPATI Diodes in 200 GHz and 300 GHz Bands," Electron. Lett., 12(6), pp. 148-149, March 18, 1976. 4. S. M . Sze, Physics of Semiconductor Devices, 2nd ed., Wiley, New York, 1981. 5. G. I. Had~ P. T . Greiling, and W. E. Schroeder, "Basfo Principles and Properties of Avalanche Transit-Tune Devices," .IEEE Trans. Microwave Theory Tech., MTI'-18, pp. 752-772, November 1970. 6. H . J. Kuno, ''IMPAIT Devices for Generation of Millimeter-Waves," in K. Button, Ed., Infrared and Millimeter-Waves, 1, Sources ofRadiation , Academic Press, New York, 1979, Chap. 2. 7. T. Misawa,' egative Resistance in p -n Junctions under Avalanche Breakdown Conditions. Parts I and II," IEEE Trans. Electron Devices , ED-13, pp. 137-151, January 1966. 8. K. Chang and R. L . Ebert, "W-Band Power Combiner Design," IEEE Trans. Microwave Theory Tech., MTI-28, pp. 295-305. April 1980. 9. M. Gilden and M. E. Hines, ''Electronic Tuning Effects in the Read Microwave Avalanche Diode," TEEE Trans. Electron Devices, ED-13, pp. 169- 175, January 1966. 10. W . J. Evans and G. L Haddad. "A Large-Signal Analysis of IMPAIT Diodes," IEEE Trans. Electron Devices , ED-15, pp. 708-717, October 1968. 11. I. L . Blue, "Approximate Large-Signal Analysis of IMPATI Oscillators," Bell Syst. Tech. J., 48, pp. 383- 396, February 1969. 12. D. L. Scharfetter and H . K. Gummel, ''Large-Signal Analysis of a Silicon Read Diode Oscillator," IEEE Trans. Electron Devices, ED-16, pp. 64-77, January 1969. 13. K. Mouthaan, "Nonlinear Analysis of the Avalanche Transit-Time Oscillator," IEEE Trans. Electron Devices, ED-16, pp. 935- 945, November 1969.

1262

IMPATT AND RELATED TRANSIT-TIME DEVICES

14. P. T. Greiling and G. 1 Haddad, ''Large-Signal Equivalent Circuits of Avalanche TransitTime Devices," IEEE Trans. Microwave Theory Tech. , MTT-18, pp. 842-853, November 1970. 15. D. R. Decker, C. N. Dunn, and R. L. Frank. "Large-Signal Silicon and Germanium Avalanche-Diode Characteristics," IEEE Trans. Microwave Theory Tech. , MTT-18, pp. 872-876, November 1970. 16. E . F. Scherer, "Large-Signal Operation of Avalanche-Diode Amplifiers." IEEE Trans. Microwave Theory Tech. , MTT-18, pp. 922-932, November 1970. 17. H. J. Ku.no, "Analysis of Nonlinear Characteristics and Transient Response of IMPATT AmpHfiers," IEEE Trans. Microwave Theory Tech., MTT-21, pp. 694-702, November 1973. 18. H. J. Kuoo and D . L. English, HNontinear and Large-Signal Characteristics of MillimeterWave IMPATT Amplifiers," IEEE Trans. Microwave Theory Tech. , MTT-21, pp. 703-706, November 1973. 19. N. B. Kramer, "Characterization and Modeling of IMPATT Oscillators," IEEE Trans. Electron Devices, ED-15, pp. 838-846, November 1968. 20. R. S. Ying, "X-Band n+pp+ IMPATT Diodes," Electron. Lett., 8, pp. 297-298, June 15, 1972. 21. D. L. Scharfetter, W . J. Evans, and R. L. Johnston, "Double-Drift-Region (p - pnn+ ) Avalanche Diode Oscillators," Proc. IEEE , 58, pp. 1131 - 1133, July 1970. 22. T. E. Seidel and D. L. Scharfetter, "High-Power Millimeter-Wave lMPAIT Oscillator with Both Hole and Electron Drift Spaces Made by Ion Implantation/' Proc. IEEE , 58, pp. 1135-1136, July 1970. 23. Y. E. Ma, E. M. Nakaji, and W. F. Thrower, "V-Band Double-Drift Read Silicon IMPATis," IEEE MIT-S Int. Microwave Symp. Dig., pp. 167- 168, 1984. 24. J. R. Grierson et al., "High-Power 11 GHz GaAs Hi-Lo IMPATT Diodes with Titanium Schottky Barriers," Electron. Lett., 15, pp. 13-14, January 4, 1979. 25. K. Chang, J. K. Kung, P. G . Asher, G. M . Hayashibara. and R. S. Ymg, "GaAs ReadType IMPATT Diode for 130 GHz CW Operation," Electron. Lett. , 17, pp. 471-473 June 25, 1981.

26. K Chang, H. Yen, and E. M. Nakaji, 94 GHz High Power Solid State Trwzsmitter, Final report to BMDSC, DASG 60-78-C-0148, 1981 . 27. K.. Chang, W. F. Thrower, and G . M . Hayashibara, "Millimeter-Wave Silicon IMPATT Sources and Combiner for the 110-260 GHz Range," IEEE Trans. Microwave Theory Tech., MTT-29, pp. 1278- 1284, December 1981. 28. W. N. Grant, "Electron and Hole Ionization Rates in Epitaxial Silicon at High Electric Fields," Solid State Electron., 16, pp. 1189- 1203. 1973. 29. C. Canali, G. Majni, R. Minder, and G . Ottaviani, "Electron and Hole Drift Velocity Measurements on Silicon and Their Empirical Relation to Electric Field and Temperature," IEEE Trans. Electron Devices. ED-22, pp. 1045 - 1046, November 1975.

30. J. Freyer, E. Kasper, and H. Barth. "Pulsed V-Band M. B . E. Si IMPATI Diodes/' Electron. Lett., 16, pp. 865-866, November 6, 1980. 31. A. Y. Cho, C. N. Dunn, R. L. Kuua , and W. E. Schroeder, uGaAs IMPATT Diodes Prepared by Molecular Beam Epitaxy," Appl. Phys. Lett., 25, pp. 224-226, August 15, 1974. 32. R. S . Ying, D . L. English, K . P. Weller, E . M. Nakaji, D. H. Lee, and R L. Bernick, "Millimeter-Wave Pulsed IMPATT Diode Oscillators," IEEE J. Solid-State Circuits, SC-11, pp. 279-285, April 1976.

REFERENCES

1263

33. K. P . WeUer, R. S. Ying, and D . H . Lee, "Millimeter-Wave IMPATT Sources fo r the 130- 170 GHz Range," IEEE Trans. Micro\i1ave TheOI)' Tech., MTT-24, pp. 738-743, ovember 1976. 34. C. Chao, R. L. Bernick. E. M . akaji, R . S. Ying, K. P. Weller, and D. H. Lee, "Y-Band ( 170-260 GHz) Tunable CW IMPA1T Diode Oscillators;• IEEE Trans. Microwave Theory Tech .. MTT-25, pp. 985-991. December 1977. 35. R. L. Eisenhart and P. J . Khan, ' Theoretical and Experimental Analysis of a Waveguide Mounting Structure," IEEE Trans. Microwave Theory Tech. , MTI-19, pp. 706- 719, August 1971 . 36. T. T. Fong. K . P . Weller. and D. L. English, ''Circuit Characterization of V-Band IMPATT 0 cillators and Amplifiers," IEEE Trans. Microwave Theory Tech., MIT-24, pp. 752-758, ovember 1976.

37. A. G. Williamson. ''Analy is and Modeling of Two-Gap Coaxial Line Rectangular Waveguide Junctions," IEEE Tra11s. Microwave Theory Tech., MTT-31, pp. 295-302, March 1983.

3 . B. D. Bates, ''Analysis of Multiple-Step Radial-Resonator Waveguide Diode Mounts with Application to IMPATr Oscillator Circuits:' IEEE MTI'-S Int. Microwave Symp. Dig., pp. 669--672. 1987. 39. G. B. Morgan, ''Microstrip IMPATI-Diode Oscillator for 100 GHz," Electron. Lett., 17, pp. 570- 571, August 6, 1981.

40. P. Yen, D. English. C. Ito, and K. Chang, "Millimeter-Wave IMPATI Microstrip Oscillators," IEEE MTT-S Int. Microwave Symp. Dig., pp. 139-140, 1983. 41. B . S. Glance, "Low-Q Microstrip IMPATT Oscillator at 30 GHz," Proc. IEEE , 60, pp. 1105-1106, September 1972. 42. B. S. Glance and M. V . Schneider, ''Millimeter-Wave Microstrip Oscillators," IEEE Trans. Microwave Theory Tech., MTf.22, pp. 1281 -1283, December 1974.

43. K. Chang, D. M. English, R . S. Tabim, A. J. Grote, T. Pham, C. Sun, G. M . Hayashibara, P. Ye~ and W. P iotrowski, "W-Band (75-110 GHz) Microstrip Components," IEEE Trans. Microwave Theory Tech., MTT-33, pp. 1375-1382, D ecember 1985. 44. M. D ydyk, ''EHF Planar Module for Spatial Combjning,'' Microwave J., 26, pp. 157 - 176, May 1983. 45. G. B. Morgan, "'Stabilization of a W-Band Microstrip Oscillator by a Dielectric Resonator," Electron. Lett., 18, pp. 556-558 June 24, 1982.

46. K. Kurokawa, "Some Basic Characteristics of Broadband Negative Resistance Oscillator Circuits," Bell Syst. Tech. J., 48, pp. 1937- 1955, July 1969. 47. T. T. Fong and H. J. Knno, "Millimeter-Wave Pulsed IMPATI Sources," IEEE Trans. Microwave Theory Tech., M'IT-27, pp. 492-499, May 1979. 48. T. A . Midford and R L. Bernie~ ''Millimeter-Wave CW IMPATI Diodes and Oscillators," IEEE Trans. Microwave Theory Tech., MIT-27, pp. 483- 492, May 1979. 49. R . Mallavarpu and G. MacMaster, "500 GHz, 100 W X -Band Solid State Amplifier," IEEE MIT-S Int. Microwave Symp. Dig., pp. 387- 390, 1985. 50. M. J. Delaney, M . H . Jones, and C. Sun, "20 GHz GaAs IMPATI Diode Development for Solid State Transmitter," IEEE MIT-S Int. Microwave Symp. Dig., pp. 525-527, 1985. 51. M. G. Adlerstein and S. L. G . Chu, "GaAs IMPAIT Diodes Pulsed at 40 GHz," IEEE M1T-S Int. Microwave Symp. Dig., pp. 481-485, 1984. 52. K. Chang, C. Sun, D. L. English, and E. M. Nakaji, ''High Power 94 GHz Pulsed IMPATT Oscillators," IEEE MIT-S Int. Microwave Symp. Dig., pp. 71-72, 1979.

53. K. Chang, ''Millimeter-Wave Planar Integrated Circuits and Subsystems," in K. Button, Ed., Infrared and Millimeter-Wave, 14, Academic Press, New York, 1985.

1264

IMPATT AND RELATED TRANSIT-TIME DEVICES

54. K. Chang, ·'Millimeter-Wave Microstrip Solid-State Sources and Amplifiers," Int. l. Infrared Millim. Wave , 7, pp. 729-747, May 1986. 55. R. Adler. "A Study of Locking Phenomena in Oscillators," Proc. IRE., 34, pp. 351-357, June 1946. 56. K . Kurokawa, "Injecti on Locking of Microwave Solid-State Oscillators," Proc. IEEE, 61, pp. 1386-1409, October 1973. 57. M . E. Hines, J. Collinet, and J. Ondria, "FM Noise Suppression of an Injection PhasedLocked Oscillator," IEEE Trans. Microwave Theory Tech., MIT-16, pp. 738-742, September 1968. 58. C. H. Chien and G. C. Dalman, "Subharmonic Injected Phaselock IMPATI-Oscillator Experiments," Electron. Lett., 6, pp. 240-241 , April 16, 1970. 59. T. P. Lee, R. D. Standley, and T. Misawa, "A 50 GHz Silicon IMPATI Diode Oscillator and Amplifier," IEEE Trans. Electron Devices, ED-15, pp. 74 1-747, October 1968. 60. E. F. Scherer, "A Multistage High-Power Avalanche Amplifier at X-Band." IEEE I . SolidState Circuits, SC-4, pp. 396-399, December 1969. 61. D. F. P eterso~ "A Device Characterization and Circuit Design Procedure for Realizing High Power Millimeter-Wave IMPATT Diode Amplifiers," IEEE Trans. Microwave Theory Tech., MTT-21, pp. 681 -689, November 1973. 62. K. P. WelJer, D. L. English,andE. M . Nakaji, "HighPowerV-BandDouble DriftIMPATI Amplifier," IEEE MTT-S Int. Microwave Symp. D ig., pp. 369-371, ] 978. 63. M. Ando et al., "86 GHz High Power IMPATT Negative Resistance Amplifier," IEEE MIT-S Int. Microwave Symp. Dig., pp. 312-314, 1978. 64. B. D. Bates and P. J. Khan, "Influence of Non-ideal Circulator Effects on NegativeResistance Amplifier Design.» IEEE MIT-S Int. Microwave Symp. Dig., pp. 174-176, 1980. 65. V. Sokolov, M. R. Namordi, and F. H. Doerbeck, "A 4 W 56 dB Gain Microstrip Amplifier at 15 GHz Utilizing GaAs FET's and IMPATI Diodes," IEEE Trans. Microwave Theory Tech. , MTT-27, pp. 1058- 1065, December 1979. 66. K. J. Russell, "Microwave Power Combining Techniques/' IEEE Trans Microwave Theo,·y Tech. , MTT-27, pp. 472-478, May 1979. 67. K. Chang and C. Sun, "Millimeter-Wave Power Combining Techniques," IEEE Trans. Microwave Theory Tech., MTT-31, pp. 91-107, February 1983. 68. K . Kurokawa and F. M. Magalhaes, "An X-Band 10-Watt Multiple-IMPATI Oscillator." Proc. IEEE , 59, pp. 102- 103, January 1971. 69. K. Kurokawa, 'The Single-Cavity Multiple Device Oscillator," IEEE Trans Microwave Theory Tech., MTT-19, pp. 793- 801, October 1971. 70. R S. Harp and H. L. Stover, "Power Combining of X -Band IMPATT Circuit Module ," IEEE-JSSCC Dig. Tech Pap., 16, pp. 118- 119, February 1973. 71. R. Aston, "Technique for Increasi ng the Bandwidth of a TMo 10-Mode Power Combiner," IEEE Trans. Microwave Theory Tech., MTT-27, pp. 479- 482, May 1979. 72. R. S. Harp and K. J. Russell. "Improvement in Bandwidth and Frequency Capability of Microwave Power Combinational Techniques," IEEE-ISSCC Dig. Tech. Pap.. 11, pp. 94- 95, February 1974. 73. S. E. Hamilton, "32-diode Waveguide Power Combiner," IEEE MIT-S Int. Microwave Symp. Dig., pp. 183- 185, May 1980. 74. S. E. Hamilton and B. M . Fish, "Multidiode Waveguide Power Combiners," IEEE MTT-S Int. Microwave Symp. Dig.. pp. 132- 134, June 1982. 75. Y. C. Ngan, J. Chan, and C. S~ "20 GHz High Power IMPAIT Transmitter," IEEE MIT'S Int. Microwave Symp. Dig., pp. 487-488, 1983.

REFERENCES

1265

76. G. Jerinic, J. Fine , and M . Schindler. "3 W . Q-Band Solid-State Amplifier," IEEE MIT-S Int. Microwave Symp. Dig. , pp. 48 l - 483, l983. 77. D . W. Mooney and F. J. Bayuk. •'Injection Locking Performance of a 4 l GHz 10 W Power Combining Amplifier." IEEE Tra11s. M icrowave Theory and Tech.. MTT-31 , pp. J7 J - 177, February 1983.

78. G. H.

e bit and W. H. Leighton. ''EHF Solid-State Amplifier:· IEEE MIT-S Int. Microwave Symp. Dig.. pp. 391- 394. L985.

79. Y. Ma and C. Sun ...Millimeter-Wave Power Combiner at V-Band." Proc. 7th Cornell E lectr. Eng. Con/., Augu t 1979. pp. 299-308. 80. Y. C. gan. "'Two-Diode Power Combining Near l40 GHz.'' Electron. Lett.. 15(13), pp. 376-377. June 21. 1979.

81. K J. Russell and R. S. Harp. ··Power Combiner Operation with Pulsed IMPATTs," !EEEISSCC D ig. Tech. Pap.. 18, February 1975. 82. R. M. Wallace. M. G. Adlerstein, and S. R. Steele, "A 60-W CW Solid State Oscillator at C-Band." IEEE Trans. M icro-wave Theory Tech .• MTT-24, pp. 483- 485, July 1976. 83. K. Rus ell and R S. Harp. "A Multistage High-Power Solid-State X-Band Amplifier," IEEE-ISSCC Dig. Tech. Papers, 21. pp. 166- 167, February 1978. 84. J. Obregon. J. P. B alabaud. P. Guillon, and Y. GarauJt, ''New Radial Power Combiner U ing Tubular Dielectric Cavities.'' Electron. Lett.. 18(18), pp. 771-772, September 2 1982. 85. M . Dydyk "Efficient Power Combining," IEEE Trans. Microwave Theory Tech. , MTT-28. pp. 755 - 762, July 1980.

86. S. E. Hamilton, R. S. R obert on, F. A. Wtlhelmi, and M . E. Dick, "X-Band Pulsed SolidState Transmitter;• IEEE MTT-S /111. Microwave Symp. Dig., pp. 162- 164, May 1980. 87. R. J. Pankow and R G. Mastroianni, ''A High Power X-Band Diode Amplifier," IEEE M1T-S Int. Microwave Symp. Dig., pp. 151 - 161 . May 1980. 88. C. A. Drubin, A. L . Hieber. G. Jerinic, and A . S. Mariuilli, '·A 1 KW Peak. 300 W avg IMPATT Diode Injection Locked Oscillator." IEEE M1T-S Int. Microwave Symp. Dig., pp. 126- 128, June 1982.

89. R. Laton , S. Simoes, and L. Wagner, "A Dual TMo2o Cavity for IMPATT Diode Power Combining," IEEE M1T-S lnT. Microwave Symp. Dig., pp. 129- 131, June 1982. 90. F. J. BayukandJ. Raue, "K-BandSoUd-StatePower Amplifier," IEEEMIT-Slnt.Microwave Symp. D ig., pp. 21 -31. May 1977. 91. J. R. Nevarez and G. J. Herokowiz, " Output Power and Los Ana1ysis of 2n InjectionLocked Oscillators Combined through an Ideal and Symmetric Hybrid Combiner," IEEE Trans. Microwave Theory Tech. , MTT-17, pp. 2-J0. January 1969. 92. H. J. Kuno and D. L. English~ "Millimeter-Wave IMPATT Power Amplifier/Combiner/' IEEE Trans. Microwave Theory Tech.. MTT-24, pp. 758-767, November 1976. 93. Y. Ma, C. Sun. and E. M. Nakaji. " V-Band High Power IMPATT Amplifier," IEEE MIT-S Int. Microwave Symp. Dig., pp. 73-74, May 1980. 94. H . C. Yen and K. Chang. "A 63-W W-Band Injection-Locking Pulsed Solid-State Transmitter," IEEE Trans. Microwave Theory Tech.. MTT-29, pp. 1292- 1297, December 1981. 95. R. E. Lee, U. H . Gysel. and D . Parker. "High-Power C-Band Multiple-IM.PATT-Diode Amplifiers," TEE£ Trans. Microwave Theory Tech., M'IT-24, pp. 249-253, May 1976.

96. J. G. Josenhans, "Diamond as an Insulating Heat Sink for a Series Combination of IMPATT Diodes," Proc. IEEE, S6, pp. 762-763. April 1968. 97. C. T. Rucker, G. N. Hill~ N. W. Cox, and J. W . Amoss, "Symmetry Experiments with Four-Mesa IMPATT Diodes," IEEE Trans. Microwave Theory Tech. , MTT-2S, pp. 75- 76, January I 977.

1266

IMPAIT AND RELATED TRANSIT-TIME DEVICES

98. C. T. Rucker et al., " Series-Connected GaAs and Si Diode Chips: Some New Results," Electron. Lett., 13(11), pp. 331-332, May 26, 1977. 99. C. T. Rucker, J. W . Amoss, G. N. Hill, and N. W. Cox, "Multichip IMPATT Power Combining, a Summary with New Analytical and Experimental Results," IEEE Trans. Microwave Theory Tech., MTT-27, pp. 951-957, December 1979. 100. C. T. Rucker, J. W . Amoss, and G. N. Hill, "Chip-Level IMPATT Combining at 40 GHz," IEEE MTT-S Inst. Microwave Symp. Dig., pp. 347-348, June 1981. 101. C. B . Swan, T. Misawa, and L. Marinaccio, "Composite Avalanche Diode Structures for Increased Power Capability," IEEE Trans. Electron Devices, ED-14, pp. 584-589, September 1967. 102. M . F. Durkin, "35 GHz Active Aperture," IEEE MTI'-S Int. Microwave Symp. Dig., pp. 425-427, 1981. 103. R. J. Dinger, D. J. White, and D. Bowling, "A 10 GHz Space Power Combiner with Parasitic Injection-Locking," IEEE MTT-S Int. Microwave Symp. Digest, pp. 163-166, 1986. 104. K. A. Hummer and K. Chang, "Spatial Power Combining Using Active Microstrip Patch Antennas," Microwave Opt. Technol. Lett., 1, pp. 8-9, March 1988. 105. J. M. Mink, "Quasi-optical Power Combining of Solid-State Millimeter-Wave Sources," IEEE Trans. Microwave Theory Tech., MTT-34, pp. 273-279, February 1986. 106. L. Wandinger and V. Nalbandian, "Millimeter-Wave Power Combiner Using Quasi-optical Techniques," IEEE Trans. Microwave Theory Tech., MTT-31, pp. 189-193, February 1983. 107. S. Young and K. D. Stephan, "Stabilization and Power Combining of Planar Microwave Oscillators with an Open Resonator," IEEE MTT-S Int. Microwave Symp. Dig., pp. 185-188, 1987. 108. P. Rolland, J. L. Vaterkowski, E. Constant, and G. Salmer, "New Modes of Operation for Avalanche Diodes: Frequency Multiplication and Upconversion," IEEE Trans. Microwave Theory Tech., MTT-24, pp. 768-775, November 1976. 109. R. E. Collin, Foundations for Microwave Engineering, McGraw-Hill, New York. 1966. 110. W. W. Gray, L. Kikushima, N. P. Morenc, and R. J. Wagner, "Applying IMPATf Sources to Modem Microwave Systems," IEEE-ISSCC Meeting, 1969. 111 . K. Chang, M . Morishita, and C. Sun, "High Frequency Silicon Impact Ionization Avalanche Transit-Time (IMPATT) Sources beyond 100 GHz," Soc. Photo-Opt. Instrum. Engi. Meet., 259, pp. 46- 50. 112. C. A. Brackett, "The Elimination of Tuning-Induced Burnout and Bias-Circuit Oscillations in IMPATT Oscillators," Bell Syst. Tech. J. , 52, pp. 271-306, March 1973. 113. Y. Hirachi, T. Nakagami, Y. Toyama, and Y. Fukukawa, "High-Power 50 GHz DoubleDrift-Region IMPAIT Oscillators with Improved Bias Circuits for Eliminating LowFrequency Instabilities," IEEE Trans. Microwave Theory Tech., MTT-24, pp. 731-737, November 1976. 114. M. E. Hines, "Large-Signal Noise, Frequency Conversion, and Parametric Instabilities in IMPATI Diode Networks," Proc. IEEE , 60, pp. 1534- 1548, December 1972. 115. W. E. Schroeder, "Spurious Parametric Oscillations in IMPATr Diode Circuits," Bell Syst. Tech. J., 53, pp. 1187-1209, September 1974. 116. D. J. Coleman, Jr., and S. M. Sze, "A Low-Noise Metal-Semiconductor-Metal (MSM) Microwave Oscillator," Bell Syst. Tech. J., SO, pp. 1695-1699, May/June 1971. 117. C. P. Snapp and P. Weissglas, "On the Microwave Activity of Punch-Through Injection Transit-Tune Structure," IEEE Trans. Electron Devices, ED-19, pp. 1109-1118, October 1972.

118. D. J. Coleman, Jr., "Transit-Time Oscillations in BARIIT Diodes," J. Appl. Phys., 43, pp. 1812- 1819, April 1972.

REFERENCES

1267

119. G. T. Wright, ..Small-Signal Characteristic of Semiconductor Punch-Through Injection and Tran it-Time Diodes," Solid-Stare Electron., 16. pp. 903- 912. August 1973. 120. J. R Ea t, H. guyen-Ba. and G. I. Haddad, "De ign, Fabrication, and Evaluation of BARITI Device for Doppler System Application ," IEEE Trans. Microwave Theory Tech., M'IT-24. pp. 9-4J-948. December 1976. 121. J. Chen, J. R. Ea t. and G. I. Haddad. Development of Innovative Millimeter-Wave Broadband Mixers, Final report to aval Re earch Laboratory, N 00173-80-C-0 11 8, 1983.

122. 0. Eknoyan. S. M. Sze. and E. S. Yang, "Mjcrowave BARITT Diode with Retarding Field -An In e tigation:· Solid State Electron., 19, p. 795, 1976. 123. H. J. Prager, K. K. . Chang. and S. Wei brod, ''High-Power, High-Efficiency Silicon Avalanche Diodes at Ultra High Frequencie ,'' Proc. IEEE, 55. pp. 586-587, Apri l 1967. 124. B. C. DeLoach. Jr .. and D. L. Scharfetter. ·'Device Physics of TRAPAIT O cillators," IEEE Trans. Electron Del'ices. E D-17, pp. 9-21. January 1970. 125. W. J. EYans, ··cw TRAPATT Amplification," IEEE Trans. Microwave Theory Tech., MTT18. pp. 986-988. ovember 1970. 126. W. J. Evans~ "Circuits fo r High-Efficiency Avalanche-Diode O cillators," IEEE Trans. Micron·ave Theory Tech., MTT-17, pp. 1060-1067, December 1969. 127. W. J. Evan . ··computer Experiments on TRAPATT Diodes," IEEE Trans. Microwave Theory Tech. l\r[TT-18, pp. 862-871, November 1970. 128. J. R. Ea ~ . A. Masnari, and G. I. Haddad, " Experimental Investigation of TRAPATT Diode Trigger Conditions." IEEE J. Solid-State Circuits. SC-12, pp. 14- 20, February 1977. 129. R. J. Trew. . A. Masnari, and G. I. Haddad, "Optimization of S-band TR.APATT Oscillators;· IEEE Trans. Microwave Theory Tech., MIT-22, pp. 1166-1170, December 1974. 130. S. G. Liu and J. J. Risko, "Fabrication and Performance of Kilowatt L-Band Avalanche Diode :• RCA Rev., 31. p. 3, 1970. 131. D. F. Kostichack, 'llHF Avalanche Diode Oscillator Providing 400 Watts Peak Power and 75 Percent Efficiency." Proc. IEEE , 58, p. 1282, 1970. 132.

. W . Cox, C. T. Rucker, G. . Hill, and K. E. Gsteiger , ''X-Band TRAPATT Amplifier," IEEE Trans. Microwave Theory Tech., MTT-22, pp. 1325- 1328, December 1974.

133. J. Nishizawa, K. Motoya. and Y. Okuno, '200 GHz TUNNETT Diodes," Jpn. J. Appl. Phys., 17, pp. 167-172, 1978. 134. J. ishizawa. K. Motoya. and Y. Okuno, "The GaAs TUNNETI Diodes," IEEE-MIT Microwave Symp. Dig. Tech. Pap., pp. 159-161, 1978. 135. M . E. Elta. H . R . Fetterman. W. V. Macropoulos, and J. J. Lambert "150 GHz GaAs MITA1T Sources," IEEE Electron Device Lett. EDL-1, pp. 115 - 116, June 1980. 136. J. ishizawa. K. Motoya, and Y. Okuno, "The GaAs TUNNEIT Diodes," IEEE MTT-S Int. Microwave Symp. Dig., pp. 159- 161, 1978. 137. M. E. Elta and G. I. Haddad, "Mixed Tunneling and Avalanche Mechanisms in p-n Junctions and Their Effects on Microwave Transit-Time Devices," IEEE Trans. Electron Devices, ED-25, pp. 694- 702, June 1978. 138. M. E. Elta and G. I. Haddad, "High-Frequency Limitations of IMPATT, MITATI, and TUNNETI Mode Devices," IEEE Trans. Microwave Theory Tech. , 27, pp. 442-449, May 1979. 139. S. P. Kwok and G. I. Haddad, "Effects of Tunneling on an IMPA1T Oscillator," J. Appl. Phys.. 43, pp. 3824-3830, September 1972. 140. W . A. Lukaszek, A. Van Der Diel, and E. R. Chenett, "Investigation of the Transition from Tunneling to Impact Ionization Multiplication in Silicon p-n Junctions," Solid-State Electron. , 19, pp. 57-71, January 1976.

1268

IMPAIT AND RELATED TRANSIT-TIME DEVICES

141 . T. Ishibasi et al., "400 GHz Band Operation of Cooled Silicon IMPATT Diodes," Jpn. J. Appl. Phys., 17, 1978. pp. 173- 178. 142. F. B. Fanlc, J. D. Crowley, and J. J. Berenz, "InP Material and Device Development for Millimeter-Waves," Microwave J., 22, pp. 86-91 , June 1979. 143. T. A. Midfors and H. C. Bowers, "A Two-Port IMPATT Diode Travelling Wave Amplifier," Proc. IEEE, 56, pp. 1724 -1725, October 1968. 144. K. G. Hambleton and P. N. Robson, "Design Considerations for Resonant Traveling Wave IMPATI Oscillators," Int. J. Electron. , 35, pp. 225-244, 1973. 145. M. Franz and J.B. Beyer, ''The Traveling-Wave IMPATT Mode," IEEE Trans. Microwave Theory Tech. , MTT-26, pp. 861-865, November 1978. 146. M. Franz and J.B . Beyer, "The Traveling-Wave IMPAIT Mode. Part Il. The Effective Wave Impedance and Equivalent Transmission Line," IEEE Trans. Microwave Theory Tech. , MTT-28, pp. 215-218, March 1980. 147. Y. Fukuoka and T. Itoh, ( BJP

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.

!.

.

=--- --~r----~--~:-~~-;~ - --?-c-- -ll-~ +-~J,--F r,ET ;a:,:1

: ! •

10

!::_

- -

'

··----+--··-+-+-i-1_._--+--~"-- -~ ~

1-----1_

l'~G

-~ ~~ . ...., .

~ ~ i __..1_..;..1--4-i_.__ _~ - - ! - -:-",i'.-T:""'- : - --;

1 _ ____;.2:..,__ _i.4_....i..6_ __:~1--...:... 2- -4-'---6,:__,:__10---.___, '-20 0.L Frequency (GHz)

5

(b)

-

-

4

al

-0 CJ)

3

~

:,

O>

~

CJ)

2

u,

\ \

·5

z

\

1 0

.1

.2

.4

.6

1

2

4

6

10

20

Frequency (GHz)

Figure 20.4 Performance comparison of unpackaged 750-µ,m-gate-width GaAs FET with 2-µ,m emitter- emitter pitch silicon bipolar discrete and monolithic Darlington tran i tors: (a) S21 and maximum available gain (MAG) versus frequency; (b) minimum noi e figure ver us frequency.

MESFET has a 0.25-µm gate with a width of 750 µm and the ilicon bipolar transistor has a 0.5-µm emitter with 2-µm emitter-emitter pitch and is fabricated with a 10-GHz f, process. The significantly higher untuned S21 gain of the bipolar transistor at the lower microwave frequencies is a direct con equence of it higher tran conductance. A further increase in S21 gain can easily be achieved with properly sized bipolar transistors by using the merged two-transistor Darlington configuration. For many microwave applications the increase in device junction temperature over ambient is of critical importance for maximizing performance, maintaining performance stability over temperature, and ensuring acceptable reliability. Minimizing junction temperature rise is obviously critical for reliable power amplification but can also be a major factor in more subtle performance parameter such as post-tuning drift in varactor-tuned oscillators. The higher thermal conductivity (Fig. 20.5) of silicon compared to GaAs is therefore an important intrinsic material advantage. Technology factors other than basic material and device properties which affect the potential performance contribution of any high-speed semiconductor in practical production systems include circuit design sophistication and the ability to interconnect

20.2

BIPOLAR TRANSISTOR STRUCTURE AND MODELL G Thermal conductivity W/cm

0

1275

c

2.00 r - - - - - - - - - - - - - -- - - -- -

Silicon

1.50

1.00

1j max GaAs

0.50 .........

......... ------

50

--- ----------T1 max --------

100

150

200

250

Temperature (°C)

Figure 20.5 Thermal conductivity of silicon and GaAs versus temperature.

to the external world without signal loss or distortion. It is also essential to be able to fabricate the desired device predictably in a production environment with a minimum of

unidentified process variables. In these respects silicon bipolar transistors and MMICs are generally uperior to alternative high-speed devices. In the remaining sections of this chapter we review briefly the basis bipolar proce structures, together with small- and large-signal modeling approaches (Section 20.2) and then provide an update on the present capabilities and performance of discrete microwave transistors (Section 20.3) and MMICs (Section 20.4). In Section 20.5 we make projections for future performance and practical levels of integration.

20.2 BIPOLAR TRANSISTOR STRUCTURE AND MODELING The basic device physics and fundamental performance limitations for microwave silicon bipolar transistors are well established and have been thoroughly reviewed [1 -4]. Increases in microwave performance are continuing to occur through the straightforward scaling of critical lateral geometries to reduce parasitic capacitances and resistances, and by the more complex scaling of the emitter-to-collector impurity profiles to reduce the fundamental time delays that are the ultimate limitation on performance. The time delays between emitter and collector determine the frequency where the common-emitter current gain becomes unity. This frequency, fr, is one of the basic bipolar transistor figures of merit, particularly for digital or current switching applications. A figure of merit that is more relevant for hybrid microwave circuits is f max, defined as the frequency where the unilateral power gain (UG) goes to unity. A simple analytical approximation for /max includes f,, total base resistance (Rb), and total

1276

MICROWAVE SILICON BIPOLAR TRANSISTORS

collector- base capacitance (Ccb) :

(20.1) Another, more complex figure of merit bas recently been proposed for bipolar transistors in integrated circuits where external matching circuit networks are usually not possible [ 11].

20.2.1

Structure and Process Technology

Improvements in the intrinsic silicon bipolar transistor figures of merit have occurred and are continuing to occur as the result of dramatic advances in semiconductor process technology. The process architectures and techniques used to fabricate modem microwave silicon bipolar transistors and integrated circuits with submicrometer critical dimension bear little resemblance to those used to manufacture the first microwave transistors with f max greater than 10 GHz. Some of the features of modem silicon bipolar processes include: a. Simple process architectures with as few as four masks required to fabricate 25-GHz f max discrete transistors b. Either nitride self-aligned or polysilicon self-aligned emitter and base c. Extensive use of dry etching, including sputter etching, plasma etching, and reactive ion etching d. Fully ion implanted with arsenic emitter e. Recessed oxide or trench isolation for integrated circuits

f. Thin-film polysilicon resistors g. Gold-based metal systems The cross sections of discrete interdigitated silicon bipolar transi tors fabricated with nitride self-aligned and polysilicon self-aligned emitter processes are shown in Fig. 20.6. Key features of a nitride self-aligned process are: a. Simplest process architecture with minimum number of masks b. One critical mask alignment step c. Minimum emitter and base contact resistance d. Maximum performance achieved for multifinger transistors Key features of a polysilicon self-aligned process are: a. No critical alignment steps b. Highest de current gain c. Susceptible to erratic emitter or base contact resistance d. Maximum performance achieved for single-finger transistors

20.2 BIPOLAR TRANSISTOR STRUCTURE AND MODELING (a)

B

E

B

E

1277

B

NEPI N-+ collector

{b)

B

E

B

E

B

NEPI

N4collector

.____.IMetal

-

Poly Si

-

Oxide/nitride

FJgUre 20.6 Cross sections of modem discrete silicon bipolar transistors: (a) nitride selfaligned emitter proces ; (b) polysilicon elf-aligned emitter process.

Discrete bipolar transistors generally have a performance advantage over integrated transistors in part because of the lower parasitics and greater epi-layer flexibility inherent in a structure having the collector contact on the bottom of the chip. Most monolithic integraled circuits require the complete electrical isolation of adjacent transistors, which usually cannot be accomplished without some negative impact on performance. Two approaches that accomplish isolation with a minimum of performance sacrifice are recessed oxide isolation and deep trench isolation. Cross sections of nitride self-aligned transistors isolated with these two different approaches are shown in Fig. 20~7. Advances in bipolar process technology have greatly improved the ability to scale and control the very shallow emitter-to-collector impurity profiles required to achieve high J,. Ion implantation, diffusion from _polysilicon, and rapid thermal annealing are used extensively in the fabrication of microwave transistors and integrated circuits. An example of a typicaJ emitter-to-collector vertical impurity profile is shown in Fig. 20.8 for a 10-GHz fr small-signal discrete transistor. Key features include a very highly doped arsenic emitter, base width less than 0.1 µm, and collector doping and thickness elected to optimize f mn. Future innovations in basic bipolar structures and materials are also possible beyond the evolutionary scaling of vertical and horizontal dimensions. Structures can be conceived that would dramatically increase /max by entirely eliminating the parasitic base resistances and capacitances that are not associated with the intrinsic transistor directly under the emitter contact Heterojunction structures with emitter and base having different bandgaps are also being explored.

20.2.2 Modeling and Equivalent Circuits The ability to model the characteristics of individual transistors accurately, quickly, and inexpensively is crucial for circuit designers to exploit the full potential of any

1278

MICROWAVE SILICON BIPOLAR TRANSISTORS

(a)

B

E

B

E

B

C

B

C

Local N ...,.. buried layer

p - substrate (b)

E

B

B

E

Global N ++ buried layer

p - substrate

.______IMetal

-

Trench

-

Oxide/nitride

Cross sections of modern nitride self-aligned monolithic integrated circuit silicon bipolar transistors: (a) recessed oxide-isolated process with local n "T"- buried layer; (b) trench-isolated process with global n -4 buried layer.

Figure 20.7

1021

N ++ emitter (arsenic) 1020

-

7 E u C

N ++ collector (antimony) 1019

0 .:;

...co

C

(I)

u C 0

1018

P +active base (boron)

u

....>·.:: ::,

a.

.s

-

1017

( I)

z

1016

N EPI

Figure 20.8 Emitter-to-collector doping profile of LO-GHz/, discrete silicon bipolar transistor versus depth from the emitter surface. Approximate physical location of major time delays are shown.

20.2

BIPOLAR 1RANSISTOR STRUCTURE AND MODELING

1279

Base

Re Collector

Figure 20.9 Small-signal equivale nt circuit of microwave bipolar transistor chip excluding bond wire inductances and package paras_itic .

TABLE 20.2 Definition of Small-Signal Equivalent-Circuit Elements for Microwave Bipolar Transistors Symbol

Definition Emitter bond pad capacitance Base bond pad capacitance Emitter contact resistance Base contact resistance Early effect resistance

Umts pF pF

n n n

~

Ro = Ic:

Collector resistance Distributed base resistance

Distributed base resistance Emitter resistance

kI Re = qle Emitter- base junction capacitance Common-base current gain

pF

ao a= 1 + jf/ / bexp(-j2rrfr:d) Low frequency common-base current gain Collector depletion region delay time Base region delay time Base cutoff frequency

ps ps GHz

1 fb= 2rrr:b f

Operating frequency

GHz

1280

MICROWAVE $[LICON BIPOLAR TRANSISTORS

technology. Silicon bipolar transistors have well-established models and equivalent circuits that have been proven effective for both small-signal and large-signal operation at microwave frequencies. The T-equivaJent circuit shown in Fig. 20.9 is based on a regional physical model of the bipolar transistor chip structure and has been found to be very effective in modeling small-signal performance for fixed-bias conditions [2, 3, 5]. Accurate microwave noise modeling capability can be achieved using this equivalent circuit if both the fixed-base and current-dependent emitter delays are properly accounted for [12, 13]. Definitions of the circuit elements, time delays, and single-pole current generator for the T-equivalent circuit are given in Table 20.2. S-paramecers calculated using this equivalent circuit can agree very closely with measured results even up to 20 GHz when correct element values are used and unavoidable chip bond wire and carrier parasitics are properly accounted for. For discrete microwave transistors it is particularly important to provide an accurate equivalent

(a)

(b)

,--··~---··-, i

! j

Figure 20.10 Minimum-size packages suitable for small-signal bipolar transistors and MMICs: (a) p~otograp~s of hennetic high-reliability 70-mil gold ceramic, hermetically glass-sealed 85-mJJ ceramic, a_nd low-cost 85-mil plastic microstrip packages; {b) equivalent circuit that can be used effecnvely to model small-signal bipolar transistor packages.

20.2

BIPOLAR TRANSISTOR STRUCTURE AND MODELING

1281

circuit for the package or hybrid carrier. Examples of low-para itic packages suitable for small-signal bipolar tran i tors and MMICs are shown in Fig. 20.lOa . The package equivalent circuit shown in Fig. 20.1Ob accounts for parasitic lead inductance and transmission-line effect as well a bond wire inductance and hunt capacitance . For comprehen i ve large- ignal modeling it is necessary to account for device characteristics over the full operation range of current, voltage, frequency, and temperature. The complexity of thi ta k can partly be appreciated by considering the generic common-emitter bipolar transi tor current-voltage characteri tic hown in Fig. 20.1 la. The central linear region i bounded by non linear regions which must be properly accounted for in order to accurately simulate large- ignal operation. Normal bipolar transistor tum-on non linearitie occur at low current and voltages (regions 2 and 3). The Kirk effect [3] limit operation at high currents (region 4). Other nonlinearitie occur at high collector-emitter voltages due to avalanche breakdown (region 1) and at high bias power d.i sipation due to temperature effects such as thermal runaway (region 5). The quiescent bias point, ignal amplitude, and frequency of operation determine the degree of nonlinearity and signal distortion. The major modes for bipolar transistors are defined by the alternative bias point shown in Fig. 20.11 b. Class A bias results in the most linear amplification, lowest noise figure, and the largest dynamic range; class C

(a)

.. ················· ..

2

.•

Voe collector - emitter voltage (b)

.:

;

:

!

: : I :

··· ········.·.·.·:.·.·....... / I 1111

i ~

~

00

~

;;:::, where Vt1ssa1 ) ; and the pinchoff voltage Vp, which is the values of negative gate voltage required to reduce the drain current to a small value (typically 1-10 µA). Also shown in Fig. 22.37 are the regions of operation for both ]ow-noise ( A) and high-gain (B ) devices. Both operate in the aturated region of the transistor. The lownoise bias region occurs at low current (typically 0. l - 0.21dss) while bias for high gain is at current levels much closer to l t1ss . Both modes of operation are employed in multistage low-noise amplifier . The de FET parameter are easily measured at the wafer level using autoprobe techniques in conjunction with computer -aided data reduction. Since low-noise MESFET chips are usually small. a 2- or 3-in.-diameter wafer may contain several thou and devices. Thus, because of the sheer numbers, automated measurements are necessary.

RF Characteristics of Low-Noise MESFETs. Microwave GaAs MESFETs intended for low-noise applications are usually characterized by two-port, small-signal S-parameters [98]. This characterization is preferred because (1) a standard impedance (500) is easily provided at the input and output; (2) such terminations usually result in stable operation at all frequencies; (3) extensive computer software packages for device modeling and amplifier design are readily available; and (4) instrumentation and onwafer probing systems are availab]e for rapid RF characterization of devices from their two-port S-parameters. Two-port common source S-parameters are usually measured for low-noise amplifier applications. This configuration~ shown in Fig. 22.38, also provides the maximum

1436

FETS: LOW-NOISE APPLICATIONS

0 ----

-0

G

s

s Ground

s Microstrip form

I

I

I I I I

I I I

I

Input Output Reference plane

Figure 22.38

Common source MESFET configuration.

small-signal gain. Multistage low-noise amplifiers usually employ gain stages after the first or second stage of low-noise amplification. With the exception of on-wafer measurements, S-parameters are normally measured in a 50-0 transmission line, usually microstrip. This configuration is shown in Fig. 22.38. In this geometry, the position of the reference planes which define input gate inductance and the output drain inductance must be accurately determined by through-line or short circuit measurements. In this way, the measured S-parameters will include the parasitic inductances of the bonding leads which will also be pre ent when the chip is bonded into an amplifier. Bonding leads are kept as hort as possible to minimize the inductances so as to retain circuit bandwidth [98] . On-wafer probing systems capable of accurate, small-signal S-parameter characterization are now commonly used [9]. These probing systems are based on a tapered coplanar transmission line system which is used to make signal and ground contact to transistors at the wafer level. This technique offers a number of advantages: 1. It allows device evaluation before backside wafer processing. 2. It eliminates the need for device mounting. 3. It allows nondestructive device evaluation. 4. It greatly improves device evaluation throughput so that 100% RF evaluation is possible. Figure 22.39 shows an RF wafer probe together with a typical probe footprint layout. In order to ensure accurate measurements, the device pad layout must be matched to the probe. Shown in the figure is a ground- signal-ground layout which requires three contacts at both the input and output of the transistor. This configuration is favored in most applications. To account for wafer probe loss, an accurate S-parameter characterization of the probe must be obtained. The wafer probe is a noninsertable device which has a

22.5

LOW- O CSE AMPLIFIER DES IGN

1437

(a)

Source

Signal

Dram

Source

(b)

Figure 22.39 hardware.

On-wafer RF probe: (a) ground-signaJ- ground probe footprint; (b) probe

miniature coaxial connector at one end and a coplanar footprint on the other. The probe S-parameters are obtained using standard spectrum analysis two-port calibration procedures in conjunction with sets of on-wafer coplanar standards (open, hort, and load). Although the setup and calibration time for these measurements is significant, once completed, large numbers of devices can be characterized quickly, with little human labor and with computer-formatted output data.

Noise Characterization Parameters Influencing the MESFET Amplifier Noise Performance. From the earlier portions of this chapter, it is apparent that a number of MESFET design variables as well as other parameters directly influence the noise figure F of a low-noise microwave amplifier. Fullowing Gupta et al. [99], the "other variables'' include six parameters in

1438

FETS: LOW- OISE APPLICATIONS

F

(a)

(c)

(b)

--....--~...----4► Bg

' '-J.. __

L-

,_____ (d) Fmin

~------t► / o

--411►

T

(e) F min

-------411► Vos

Parameters influencing the noise figure of a submicrometer gate MESFET amplifier (after Gupta [92]): (a) generator admittance G 8 + j B8 , (b) operating frequency Jo, (c) temperature T , (d) de drain current 10 , (e) de drain-source voltage Vns-

Figure 22.40

the case of a single-stage common source stage without feedback. Three of these are related to the circuit: the operating frequency f and the real and imaginary part of the source or generator admittance Y8 = G 8 + j B8 • The other three device-related parameters include the temperature T, the de drain-to- ource voltage Vos, and the de drain current los. The dependency of F on these parameter has been extensively investigated by mean of theoretical model [4, 5], empirical formu1as [10, 46], and experimental measurements [100]. Figure 22.40 illu trates the dependency of F on these six parameters. The admittance Y8 of the generator which drives the low-noi e amplifier influences the noise figure F (Y8 ) as for any linear two-port network [34] o that (22.22) where Y8o = G8 o + j B8 o is the optimum ource admittance required for F to be equal to Frrun and rn is the equivalent noise re istance which detennine the increase in noi e figure resulting from a nonoptimum noise match. Equation (22.22) i the admittance counterpart of Eq. (22. 17). Equation (22.22) define a et of four noi e parameters: Fmio, G 8 0, B8 o, and rn . Note that each i a function of frequency. Figure 22.40a illustrates the dependence of F(Y8 ) on Y8 . The dependency of Fmin on the other parameter - the operating frequency f, the drain current Io, the temperature T, and the drain source voltage VDs - is also shown in Fig. 22.40. The monotonic decrease of Fmin (to O dB), a shown by the dashed line in Fig. 22.40b, results from the decreasing input conductance of the MESFET device with decreasing frequency [IO 1, I 02]. In practical ]ow-noise amplifiers, circuit loss

22 .5

LO\ - 01 EA tPl 11--l~R OtSIG '

1439

re ult in Fmin approaching a value (>0 dB) as hown by the , olid line in the figure. The behavior of Fmm with drain urrent / D (Fig. 22A0d ) is di cus cd in Section 22.3.2. A hown in Fig. 22A0c. Fnun increases monotonicall 1 wilh increasing temperature, as de cribed by Weinreb [ I 03 ]. Finally, as ·hown in Fig. 22.40e. the drain voltage VDs h little effect on Fnun \\ hen the device i , bta. ed in the current ~aturation region. ln the linear region below the knee of the 1-V characteristic, the minimum noi e figure i larger be au ·e of IO\\ er gain. t higher drain \ oltages. the minimum noi e figure al o increase be au e of increa ing / n and the po ibility of Gunn domai n fo rmation and a\alanching in the channel (104].

oise Characteri-;.ation Atethod . An accurate detennination of ME FET de ice noi e parameters i required to perform low-noi e amplifier de ign a well as to evaluate device de ign and compare relative device performance. H i tor ically. two method ha\e been u ed. The fir t method, hown chcmatically in Fig. 22.41. illu trate the conventional approach [45). The automatic noi e figure meter dctenninc the noi e figure Fm and gain Gm cf the complete amplifier te t etup, which include -, in addition to the device under te t (\\ith noi e figure F). input and output tuner and bia network . Treating

Bias

Bias

Noise source

Out

Tuner

Noise figure meter

F1 , G1 .

S,]

Input tuner + bias

In a

F2 , G2 ,

F, G

A

I

Tuner

\ Half of test fixture

I\

I

FET chip

S,t

A

Half of test fixture

\ Output tuner + bias

Two-port characterized by a through-short-delay (TSO) technique Measurable two-port

Figure 22.41

Conventional method of MESFET noise characterization [42].

1440

FETS: LOW-NOISE APPLICATIONS

these elements as passive, noisy two ports, the expression for Fm is (22.23)

where F 1 and G I are the noise figure and available gain of the input matching circuit which includes the input tuner, bias circuit, and half the test fixture containing the test chips, while F2 denotes similar components in the output circuit. Because the input and output matching networks are passive, F 1 = 1/ G 1 and F2 = 1/ G2, so that Eq. (22.23) can be rewritten as (22.24) where Gm = GG 1 G 2 is the total measured gain. An accurate determination of the device noise figure thus requires an accurate measurement of G 1 which can be most accurately detennined from the S-parameters of the input network [105] and the test fixture [I 06]. The disadvantages of this technique include: (1) an extensive amount of time is required to find the minimum noise (including the correct bias); (2) a minimum values of Fm may not always provide a minimum device noise figure after loss corrections; and (3) an additional measurement is required to determine the noise resistance (rn ) or conductance (g,,). A more systematic method of determining noise parameters has been proposed by Lane [107]. In this approach, an alternate method is used to determine the noise figure for different input matching conditions. Equation (22.22) may be written alternatively in terms of input reflection coefficients:

(22.25) where ro is the input reflection coefficient and r opt is the reflection coefficient corresponding to the optimum source impedance. At least four measurements are required to determine the four unknowns Fcrun, rn, and the real and imaginary parts of r opt• In practice, more measurements are performed and the four noise coefficients determined by a least-squares fit of Eq. (22.25) [108] or directly from noise figure measurements (109]. The measurements may in addition be performed on the wafer. The major advantage of this technique is the possibility of fully automated device noise and gain characterization. Disadvantages are primarily related to accuracy (45] and include: (1) following computation, the results are very sensitive to measurement errors if r,, is large; (2) the measurements are sensitive to low-frequency oscillations to which short-gate-length, high- fr devices are prone; (3) the matrix of the four-equation system can become singular for some values of input termination; and (4) there is a problem in dealing with input network losses in the case of very low noise figures . Because of the increasing availability of automated instrumentation, on-wafer RF probing systems operating at higher frequencies and with improved accuracy, and the desire to reduce testing and characterization costs, on-wafer noise characterization is of increasing interest for both production and research requirements.

22.5

22.5.3

LOW-NOISE AMPUFJER DES IGN

1441

Design Example: K-Band Three-Stage Low-Noise Amplifier

A a mean of illu trating ome of the principle. of low-noi e amplifier de ign and fabrication. thi ection describe the detail of a three-stage hybrid ]ow-noise amplifier designed to operate in lhe K-band over a frequency range from 21.5 to 26.5 GHz.

MESFET Devices. The amplifier u c Lwo device type which have the 'arnc geometricaJ layout and imilar epitaxial material tructure and are designated a DFET-5 and DFET-6. The e device both have a 0.25-µm gate length and a gate width of 60 µm. The device layout i hown in Fig. 22.42. Two 30-µm gate finger are employed. It was determined through earlier evaluation that the DFET-5 lot ha a lower noise figure, o thi device wa u ed in the fir t tage of the amplifier. while DFET-6 devices were u ed in the econd and third ta~e .... . Device Characteri:atio11. Both device types were characterized by S-parameter meaurement from 2 to 26.5 GHz and 14-element equi valent circuit models developed from the measured data The e are shown for both device type in Figs. 22.43a and 22.43b. The imulated S-parameters for the e two models are hown in Figs. 22.44a and 22.44b. oi e modeling was next performed. The noi e parameters Fmin. Rn and the real and imaginary parts of r opt were determined and are hown in Table 22.5 for both device . A implified noi e model developed by Podell et a1. [110] was used. Amplifier Design. Using the equivalent circuit and noise models, the amplifier design was developed following a procedure imilar to that outlined in Section 22.5.1 using a commercial CAD software package. Several iteration were carried out to optimize the final de ign, which i shown in Fig. 22.45. Amplifier Construction. The three-stage amplifier was fabricated in micro trip on multiple 25-mil-thick quartz substrates with TiWAu metallization. WR-4 waveguide input and output ports are used in the amplifier. To effectively coup]e the microstrip amplifier circujt to the waveguide ports. a rrricrostrip-to-waveguide transition is employed. The transitions are also fabricated on 25-mil-thick sub trate . The insertion lo of a single transition is approximately 0.3 dB over a full 18-26.5-GHz waveguide bandwidth. The amplifier consists of three cascaded stages. Each stage contains input and output matching circuits. Between the stages are de blocking circuits. Input matching circuits

Figure 22.42

Layout of the MESFET used in the K-band amplifier.

1442

FETS: LOW-NOISE APPLICATIONS

(a)

0.0 nH

2.80

0.001 pF

5.60

0.0 nH

12

11

4

13

14

0.065 pF

2 3

5 9.0

0.007 pF 1.90

5 = 9m (mS)

6 = r (pS)

6 1.2

a

1

7

co 0 co

2.80

8

0.035 pF

(b)

0.0 nH

1.450

0.001 pF

6.720

0.0 nH

12

11

4

13

14

0.065 pF

2 3

5 12.1 5 =gm(mS)

0.012 pF 2.660

1

6= r (pS)

6 1.2 7

10 0.019 nH

0.028 pF

9

Figure 22.43 Small-signal equivalent circuit models for the MESFETs u ed in the K-band amplifier: (a) model for DFET-5, (b) model for DFET-6.

consist of impedance transformers, high-impedance inductances, and hunt resistors, which serve as out-of-band terminations. The entire set of cascaded amplifier circuits is mounted in a carrier located in a channel operating below waveguide cutoff. The carrier is andwiched between the microstrip-to-waveguide transitions. A photograph of an amplifier circuit is shown in Fig. 22.46. Amplifier Performance. The measured amplifier noise figure and gain are shown in Fig. 22.47. Shown also is the simulated performance based on the final design. Agreement is good. Note that the experimental data have not been corrected for approximately 0.6 dB of input circulator and waveguide-to-microstrip transition loss.

12.6 (a) 0

CO CLUS JO S AND OUTLOOK

1443

1

S11 DFET5 S22

+ DFETS

o

S12 DFETS

x

S21 OFET5

f1: 2.00000 f2 : 40.0000

0.2

0.5

(b)

1

2

= f'

1 0

S11 DFET6

-r--

S22 DFET6

o

S12 DFET6 S21

x OFET6

f1 : 2.00000 12: 40.0000

0.2

0.5

1

2

Figure 22.44 SimuJated S-parameters for the two MESFETs used in the K-band amplifier: (a) DFET-5, (b) DFET-6.

22.6

CONCLUSIO S AND OUTLOOK

The GaAs MESFET is a widely used so]jd state low-noise amplifying device. It is used from below I to well above 30 GHz. As discussed earlier in this chapter, these devices are capable of excellent low noise performance at twice this frequency using stateof-the-art materials and processing technologies. The low noise MESFET has been in the field in a wide variety of applications, including military, space, and commercial sockets, for nearly 30 years. During the last 20 years, MESFET processes have become increasingly mature, although not yet comparable with silicon technology. During the decade of the 90s,

1444

FETS: LOW-NOISE APPLICATIONS

TABLE 22.5 Frequency (GHz)

Noise Parameters for the MESFETS Used in the K-Band Amplifier Minimum Noise Figure

Magnitude

r op1

Angle

RN (Normalized to 50 Q)

DFET-5 18.000 20.000 21.000 22.000 23.000 24.000 25.000 26.000 27.000 28.000 29.000 30.000

1.800 1.996 2.093 2.191 2.287 2.384 2.480 2.576 2.671 2.766 2.860 2.955

0 .81 8 0.804 0.798 0.792 0.787 0 .781 0.776 0.772 0.767 0 .763 0 .760 0.756

RN (Normalized)

43.510 47.893 50.037 52.148 54.226 56.269 58.277 60.251 62.189 64.091 65.958 67.789

1.835 1.835 1.835 1.835 1.835 1.835 1.835 1.835 1.835 1.835 1.835 1.835

47.119 49.468 51 .779 54.050 56.280 58.469 60.616 62.721 6-J.783 66.803 68.780 70.714 72.605

1.951 1.951 1.951 1.951 1.951 1.951 1.951 1.951 1.951 1.951 1.951 1.951 1.951

DFET-6 18.000 19.000 20.000 21.000 22.000 23.000 24.000 25.000 26.000 27.000 28.000 29.000 30.000

2.146 2 .262 2.377 2.492 2.607 2 .720 2.834 2.946 3.058 3.170 3.28 1 3.39 1 3.500

0 .800 0.793 0.786 0.780 0.774 0.768 0.763 0.758 0.753 0.749 0.745 0.741 0.738

low noise MESFETs, with performance comparable to heterojunction tran i tors uch a HEMTs, have been produced using all ion-implanted proce e and dry etch delineation of critical features. These device no longer depend on wet chemical gate rece ing and specialized liftoff technique with their inherent variability and proce s control issues. This new breed of MESFETs i well uited for production of not only di crete devices, but also of MMIC , high-speed digital and mixed integrated circuit . The low noise GaAs MESFET i a highly reliable device (111. 11 2]. Extensive data from accelerated testing programs as well a field ervice data upport this contention. As it is a majority carrier device, the MESFET is al o radiation hard [113-115], an important property for space and certain military applications. For the e attributes, as well as its excellent performance, the MESFET will continue to play an important role in low noise applications. There are two important i sue that will impact the future of low noise GaAs MESFET devices: (1 ) competition with a variety of heterojunction field-effect transistors such as GaAs and InP HEMTs and PHEMTs, and (2) the trong trend toward monolithic

03 02 01

133 380 14 38

goo

13

r

03 183 1330

02

01

goo

90°

01 = 0FETS O2 = 0FET6 O3 = 0FET6

I G1

G2

Figure 22.45

~

£

35 !2 90

1330

K -band MESFET amplifier design.

I G3

1446

FETS: LOW-NOISE APPLICATIONS

atching

Input matching circuit

Ork

• Input waveguide to microstrip transition

First transistor

Figure 22.46

Second transistor

Third tcansistor

Output microstrip to waveguide transition

Photograph of the K-band MESFET amplifier.

10.00 ------..---,.---.,.-,-----,--~-~~--,-.,..--, 30.00 Simulated

,.,,,,,, /

.... - ............. ---

Measured

in -0

-

C

~

::, 0>

..::

(I) (/)

·5

z

5.000

----~-------...._ ___ ___ Measured

____

---

~

O>

15.00 ',

~

ai

u0

(/) (/)

Simulated

50 GHz) are available commercially from several companies at a very reasonable cost. The superiority of the HEMT over a MESFET with the same geometry is twofold. First, due to the confinement of the electron channel layer in the undoped GaAs side of the heterojunction, the two parameters forming the current (i.e., the electron density and mobility) are optimized such that at a given drain voltage, the transconductance in the HEMT is higher than in the MESFET, thu providing a better cutoff frequency. Second, a number of experimental data clearly bow that HEMTs have a better noise figure than MESFETs. This may be due partially to a better source resjstance, which is lower in the HEMT because of the better electron mobility in the channel near th.e source and drain contacts. The HEMT is also potentially better with regard to logic application . Actually, ince high saturation drain currents can easily be obtained, delay time per gate is minimized. It can be shown that tlris time is shorter when transconductance and average electron drift velocity under the gate are higher. We will ee 1ater that thi i realized with the HEMT. Switching delay times per gate of less than 10 ps are obtained, dropping to 5 ps at low temperature. These figures are very competitive with those of other technologie : MOSFET or ECL bipolar. Indeed, one find from ring o cillator measurements that a 1-µm-gate HEMT is as fast as a ½-µ m-gate standard MESFET [4, 5]. Thu integration of HEMTs could be easier and the proce s could yield better than with MESFETs. Apparent average electron velocities approaching 2 x 107 emfs are found in HEMTs (see Section 23.3). Concerning di ipation in relation to switching delay times, values as ]ow as 10 to 20 fJ are estimated for the dissipated energy, making HEMT logic an attractive low-consuming technology for integrated circuits [6]. In the re t of this chapter we review fir t the most elementary theoretical approach of HEMT functioning, the small-signal equivalent circuit useful in microwave applications, and some technological process principles and improvements for the near future. Then a more physical description of the device is presented. Following that, practical applications are reviewed concerning low-temperature behavior, low-noise amplification, and high-power use. We finish the chapter with a summary of the new structures deriving from the basic HEMT structure, using multiple heterolayers and/or new ID- V semiconductor compounds, including pseudomorphic structures.

23.1

PRIN lPLE AND MODEL OF THE HEMT

1457

23.2 PRINCIPLE AND MODEL OF THE HEI\IIT The mo t elementary general tructure of a HEMT i ketched in Fig. 23.l. A more elaborate tructure of a real device i hown in Fig. 23.3 . Looking at Fig. 23 .1, one ee that a HEMT i fo rmed by a thin layer of doped AlGaA epitaxially grown on a GaA undoped buffer layer depo ited o er a GaA emi-in ul ating ub trate. At not too low temperature , one e ·pect that mo t of donor tate will deliver a free e1cctron whk h then diffu e free I in the doped material . We know from A nderson' theory that the empty electron tate in the undopcd GaA layer are much lower in energy than the electron tate in the AlGaA layer. We will . ee in the next paragraph that the difference in energy i in practice in the range 0.2 to 0.3 eV . In thi proce s a large number of electron accumulate in the GaA layer and cannot tran fer back to the AlGaAs layer, due LO the barrier at the heterojunction. At the rune time, since the local neutrality 1 de troyed, a pace-charge reaction tend to cau e the electrons to accumulate again t the interface between the two material . The ituation obtained is de picted in Fig. 23.2a, where the band cheme i hown a a fu nction of the distance x of Fig. 23. l. In Fig. 23.2a we make the distingui h among the neutral zones, the depleted zone, and the accumulation zone. ow, we uppo e that a metallic (Schottky) contact ha been made over the AlGaAs layer (the gate) and a bia voltage i applied to thi contact the other contacts being at zero. This situation i depk ted in Fig. 23.2b . If the AlGaAs layer i not too wide (in practice, about 500 A when ND ,,..., 10 18 cm- 3 ) , a moderate negative bias will deplete the totality of the AlGaA electrons into the GaAs layer, uch that the areal electron density ns in Ga.A may be in the range of about 1012 cm- 2 . One then obtains a large amount of electron moving in a high-quality material, and the areal den ity ns can ea ily be controlled by the gate bias voltage . Next, if one of the ohmic contacts is positively bia ed (drain) while the other (source) is grounded (technologically, one trie to have direct acce to the channel electron layer by minimizing access contact resistances) the electrons will move, ubject to the electric field between source and drain. under control of the gate bias. A field-effect transistor is obtained. As in the conventional FET, normally-on or normally-off structures can be obtained depending

(b)

{a) Vacuum level

I•

I

}1_E£, !

X2:

Ec2- ·

I

- •-•ft.. : : llEv ! I I

•

•

---u_

Eg2:

I I I I i.

}i-

I

I I

Ev2 N

..lol. ~

.

A

. I.

a

• •

N

Figure 23.2 Band scheme of the heterojunction: (a) without contact, where N is the neutral zone, A the accumulation zone, and D the depleted zone; (b) with a biased Schottky contact at a distance a from the interface.

1458

HIGH-ELECTRON-MOBILITY TRANSISTORS: PRINCIPLES AND APPLICATIONS Drain

Source Gate

-sooA

n+·GaAs

R

n+·AIGaAs

-UndopedAiGaAs-

-

-

-

-

-

-

-

Spacer -

w;- - - - -

•••• .•••••••• .............. ........................ . ••••• ••••• ••••••••••••• •••••••••••••••••••••••••••• •••••••••••••• •• 2 DEG channel ~ ~

Undoped GaAs

•• ••••••• •••••••••• - 1 µm

S-1 GaAs

Typical structure of a real HEMT. Note that the gate metal is deposited in the recessed doped GaAs layer at the top; also, note the position of the undoped spacer layer next the heterojunction. Figure 23.3

on whether or not the AlGaAs layer is thick. The barrier height of the metal contact is about 1 V (increasing slightly with the Al mole fraction xAI), and for the total depletion voltage given by VP = q N va 2 / 2€ at ND = 1018 cm- 3 we find a ~ 400 A using € = 12.9€0. Figure 23.3 shows a real structure of the HEMT. Compared to Fig. 23.1, two important details appear which indeed improve substantially the functioning of the device in current applications. Firs4 to improve the quality of the source and drain access zones and make the access resistances to the electron channel as low as possible, a doped GaAs layer (Nv ,...., 3 x 1018 cm- 3 ) is grown over the AlGaAs layer. A second improvement comes from the presence of the undoped spacer layer next to the interface. As the electrons move in the channel, they stay in close contact with the heterojunction interface, and thus in the vicinity (although outside) of the positive ionic layer. If one wishes to obtain a very good electron mobility (and/or drift velocity), it is necessary to separate the electron layer from the Si ions. It is well known that the coulombic field due to ions on the electrons strongly degrades the mobility by deviating the trajectories of the electrons continuously as they move along the channel. It is obvious that the deviations are smaller when the electrons are farther away from the ions. Unfortunately, at the same time, if the spacer is too wide, it will prevent electrons from transferring correctly to the GaAs layer. A number of microscopic simulations and experiments have shown that a good trade-off between these two conflicting effects is a spacer in the range 20 to 40 A in usual structures. Now we turn to a more quantitative, although still elementary analysis of the HEMT.

23.2.1 Anderson Theory The Anderson theory [7], which was originally developed for Ge/GaAs heterojunctions, applies readily to GaAs/AlGaAs heterojunctions. This theory postulates that the vacuum level is continuous between the two materials and that no interface defects exist. Then

23 2

PR1 ClPLE AND MODEL OF THE HEMT

1459

the Fermi level on each ide line up when no current i flowing through the interface. Thu , depending on the type of doping o f each material (n or p ), e cral configuratio n for the band cheme may o cur. Here, with the HEMT, we are more intere ted in a p - -GaA / 11 +-AIGaA y tern. Indeed. e perience how that undopcd GaA , a grown by molecular beam epitaxy. for in tance. mo t often i of p - type, due to the pre ence of variou · contaminant which are not all well identified. The electron affinity x i defined a the e nergy to be pent to e. tract an electron from the conduction band edge to the vacuum level: thu (23. l ) where 1 tand for GaAs and _ for AlGaA . E perience how that Eg, < Eg2 and x1 > x2: talcing into account that the po ition of the Fermi level in the two materials may be very different becau e of the type . according to the Ander on theory the band cheme of Fig. 23 .2a i obtained, in which the conduction band state are lower in GaA than in AlGaA . When the type are rever ed, o ne can obtain an accumulation of holes in GaA rather than of electron . Since the bandgap are ea ily mea ured with photo-lumin cence experiments in AIGaA as a function of the Al content XAI, one need a fonnula for l),.£c and 6.Et in term of the bandgap rather than the affinities. Thi i not an easy problem. but here again, a number of experiments how that the mo t probable aloe are (23.2) where E82 (xAJ) and Eg1 are now well known [8] . XAJ, in practice, lies in the range 0.2 to 0.4. A ~ a result the band off et l),.£c lies in the range 0.2 to 0.3 eV.

23.2.2

Charge Control Law by the Gate

Thi i one of the mo t important point in the analy i of the HEMT mode of functioning [9]. Charge control law by the gate determines for the most part the performance to be expected from a device. We refer to Fig. 23 .2b as the basis for the calculation preented here. We assume that the bias applied to the gate i ufficient to totally deplete the doped AlGaAs layer. Let ND and a be the doping density and the thickne s of thi layer, re pecti vely. The origin x = 0 is taken at the interface. Assuming that the impurities are fully ionized, one can readily integrate Poisson's equation between 0 and -a and obtain

(23.3) where Es is the electric field at the interface on side 2. As urning that the doping concentration in GaAs is negligible, we also have _ qns _ Qs Es E2

(23.4)

€2

Next, from Fig. 23~2h we note that V (- a ) is also given by (23.5)

1460

HIGH-ELECTRON-MOBILITY TRANSISTORS: PRINCIPLES AND APPLICATIONS

where M is the metal barrier height. Combining Eqs. (23.3) and (23.5) gives (23.6) Using Eq. (23 .4) for Es, we then have (23.7) As we will see later, the Fermi level EF is a complicated function of n 5 • Worse, this dependence can only be evaluated in an electron system at thermal equilibrium. Then Eq. (23.7) is transcendental and cannot be solved analytically. However, in most cases E F is small compared with the other quantities, and we can write qns

~

E2

- (V8 a

-

Vr ) ,

where

(23.8)

is the threshold voltage, which depends only on technological parameters. When V8 = Vr one obtains the flat conduction band condition in which the electron accumulation no longer exists. When there is a spacer layer of width Ws in the AlGaAs layer at the interface, a must be replaced by a - W s in Eq. (23.8). If an interface state charge Qss is also present, the threshold voltage takes the form (23.9) Equation (23.8) predicts a linear dependence to hold between the accumulated charge ns and the gate voltage V8 . Although this law is approximate, it is a good basis for the calculation of the HEMT characteristics together with the gradual channel approximation. The result obtained is generally in good agreement with experiment. Equation (23.8) also predicts that the source-gate capacitance (per unit area) C gs = d(qns)/dV8 is a constant. In practice, however, this is never the case because the charge control by the gate is more complicated than described here. Together with the free-electron contribution in the source-drain channel, there is also a contribution from the free electrons remaining in the AlGaAs layer (total depletion is not always realized), and also a contribution from the ionic charge because the Si ions create several donor levels (some of them are deep DX centers) which interact with the conduction electron states. In Fig. 23.4, for the same structure, we compare the results of an accurate calculation of Cgs with the value predicted by Eq. (23.8). Of course, the difference is significant. At present in submicrometer HEMTs, gate capacitances as low as 0.2 to 0.5 pF/mm are commonly realized. In Fig. 23.5 we compare the results of a careful analysis of ns(V8 ) with the result of Eq. (23.8). One notes that there exists a nearly quadratic part when V8 ~ Vr, which indeed must be considered in low-current, low-noise applications, and at V8 >> Vr a sublinear part, a trend to saturation of ns(V8 ), which indeed is a principal limitation for power applications. The linear part of ns(Vg) is observed between these two extremes. It is important to note, however, that Eq. (23.8) is extremely useful and is used extensively in the elaboration of CAD models of the HEMT.

23.2

PRI C lPLE A D MODEL OF THE HEMT

1461

4 105 pF/ cm 2

3

2

1

-1

1

0

Vg (V)

Figure 23.4 Gare- ource capacitance as a function of gate bia vo ltage. The horizontal dashed line is the result of Eq. (23.8). The olid line i the totaJ capacitance Cx.r =Co+ C 1 + C2, where Co i the contribution from the electrons in the channel, C 1 the contribution from the electrons in the AlGaA la_er. and C 2 the contribution of the electron trapped on the deep donor level in AJGaA . x.\1 = 0 .3. a = 600 . 0 = 10 1 cm-3, T = 300 K.

I

I I I I I I I I I I I

1012

l

I I I I I I

5 I I

I I I I I I

..

I I I I I

.. 0 L__.J..__J...-.a:.:..Ji,__...L..L--'----L---L--.....__..___-'--' 0 -1 Vg(V)

Figure 23.5 Compari on of an accurate calculation of n 5 ( V8 ) (solid line) with the prediction of 3 Eq. (23.8) (dashed line). XAJ = 0.28, a = 370 A, w_f = 40 A, No = 2 x 10 18 cm- , T = 300 K.

1462

23.2.3

HIGH-ELECTRON-MOBILITY TRANSISTORS: PRINCIPLES AND APPLICATIONS

Effect of a Source-Drain Bias Voltage

When a drain voltage Vd is applied, the potential (and the electric field) applied to the electrons changes all along the channel. The source-drain abscissa is referred to here as z. If the potential at z is V (z), the charge in the channel can be written as (23. 10) and the drain current is given, using the gradual channel approximation, by (23. 11) where Z is the width of the source and drain contacts and vd(Z) is the electron drift velocity due to the electric field at z. Equation (23.11) assumes that no gate current is flowing. We assume a vd( E ) dependence of the form (23.12) where E e is a critical field and µ o is the ohmic (low-field) mobility. Using Eq. (23. 12), to be justified later, we have (23.13) which can readily be integrated between V(O) = R sl d and V ( L g) = Va - Rd l a, where Lg is the gate length and Rs and Rd are the source and drain access resistances, respectively. Assuming that R s = Ra, we can express the drain current as a function of Vd and V8 as

Id = B

- B - (B 2

4AC) 112

-------2A

=-

with A

+ Va + 2ZE2JJ,0Rs(V8 -

[L g

c=

-

a

Ee

Z E2JJ,o(V8

-

2 Rs Ee

=-

Vr - Vd/2) ]

(9.14)

Vr - Vd/2) vd

a From Eq. (23.14) expressions for the transconductance gm(Vd, V8 ) = dld / dV and 8 the output drain conductance Gd(Va, V8 ) = dld/ dVd are easily obtained. These, together with C8s , are essential to the development of an equivalent electric circuit of the HEMT. Next, the integration of Eq. (23.13) over z from O to z gives an expression for V(z) that can be put in Eq. (23.10) to obtain the space dependence of ns(z) as a function of bias voltages. Interested readers can derive this expiession for themselves and will observe that ns (z) decreases monotonously from the source to the drain and as a consequence the electron drift velocity and the electric field strength increase from the source to the drain accordingly. Equation (23.14) describes the so-called ohmic part of the static characteristics. Drain current saturation can be treated using other

23.3 30

SMALL-Sl GNAL EQUlVALE T ClRCUlT

1463

Id (mA) Vg

,,,--

---------- ------ 0.5

20

----=-=-=-=-=-=-:..:-:.-:..:.: :..=.a.....---...-..- 0 10

--- - ---- - 0.5

......,,,=::b======::t::::========i-1

ll___ _

0

1

2

3

Vd( V )

Figure 23.6 Static characteri tic of a large-gate-length HEMT: La = 5 µm . Solid lines are the experimental data. da hed line are the re ults of Eq. (23.14) with µ 0 = 6200 cm 2Nls, Ee = 4000 /cm. (l = 450 A, W.s = 40 A, ND= 8 X 10 17 cm- 3 , Rs = Rd= IO n, z = 150 µm, T = 300 K.

a sumption . If the current saturation is the con equence of drift velocity saturation, due to hot electron effects, a oon as E (z) ~ Ee in the channe] , one assumes that vd(E ~ Ee)= Vs = JLoEc/2. The point z = L where E (z) = Ee can be called the aturation point where velocity aturation begins. Accordingly, the saturation drain current is obtained by saying that at a given V8 , E (L g) = Ee, which also gives the value of the corresponding Vd. Beyond this point the ns remains equal to ns(L ) and the saturation zone can be treated using Poisson equation. If access resistances are con idered, one ob erves a very sJow rise of the saturation drain current as Vd increa es. An example of application and a comparison with experimental data are shown jn Fig. 23.6. The two fitting parameters needed are µ, 0 and Ee- Although the value used for these parameters are physically justified the resulting saturation drift velocity Vs i not (vs ~ 1.2 x 107 emfs). In micrometer and submicrometer structures~ the ituation can be even worse since Vs as much as 2 to 2.5 x 107 cm/sis necessary to obtain the fit, vaJues in total contradiction to the va]ues obtained for microscopic p hysical theories or from experiment (vs ~ 0.8 x 107 cm/s). At any rate, a fitting of the experimental Id( Vd, V8 ) with the results of the present model is by no means able to deliver a correct eva]uation of Vs . This is particularly true in submicro1neter structures. Here Vs is simply taken as an empirica1 parameter and must not be considered as a physical parameter.

23.3 SMALL-SIGNAL EQUIVALENT CIRCUIT The small-signal equivalent circuit (SSEC) of a HEMT closely resembles that of the conventional MESFET. It is described at the top of Fig. 23.7. Only the intrinsic device circuit can be studied using physical models of the HEMT [10]. This is represented in the dashed rectangle of Fig. 23.7. The par(i$itic access elements are represented outside the dashed rectangle; they do not depend, as a first assumption, on the bias voltages

1464

lllGH-ELECTRON-MOBILITY TRANSISTORS: PRINCIPLES AND APPLICATIONS Intrinsic device r ________

Lg

Ag

1

Cgd

l ______ _ :

Ad

Ld

/\---.-411!11!!!11!~....

G --4t11RIOIRR!1...---r-✓

I

:vt I

I I

Cgs _ 1m Ai

D

I

gd

ds

I I I

I I I

I I I

________ I

im = gm.exp (-jwr). v Rs Ls

9m (mS)

0.4

Cgs (pF)

80 0.3 60

0.2

40

0.1

20 Vgs (V)

0 ..__ _____._ _ __.__ _ ___, -1

60

-0.5

0

0.5

Vgs (V) 0 ..__ ___.______ __ _

-1

-0.5

0

0.5

5

fc (GHz)

4

40

3 2

20 1 Vgs (V)

0_1

-0.5

0

0.5

0_1

-0.5

0

0.5

Figure 23.7 Small-signal equivalent circuit of the HEMT (top of figure), including the extrinsic access elements represented outside the dashed rectangle. Below we repre ent typical experimentally measured values of the circuit: transconductance, gate capacitance, and output conductance of a 0.5-µm-gate HEMT (Z = 200 µm) at room temperature as functions of the gate voltage at Vd = 2 V. We also represent the variation of the cutoff frequency calculated from Eq. (23.15).

applied to the device. The experimental analysis of the SSEC can be accomplished by adjusting the various lumped-element values in order to fit closely the otherwise measured broad-band S-matrix parameters of the transistor in the fixture or through wafer-probing systems. These are illustrated at the bottom of Fig. 23.7, where we display the transconductance, the gate-source capacitance, the output conductance, and

23.4

TECHNOLOGY OF THE HEMT

1465

the cutoff frequency, all a function, of the gate-to- ource voltage. One note , for in tance, that the tran conductance. which increa e. initially with the gate voltage a a re ult of the increase of ns. at po iti e gate voltage begin to drop as a result of the appearance of electron in the AlGaA layer when the gate voltage i o high that the conduction band there approache the Fermi level. A the e electron move in a highly doped layer, where the mobility i very low, the drain current cannot continue to increa e: thu the tran conductance drop rapidl y. Thj phenomenon i ometimes called the " parallel MESFET." A new method for the determination of the element of the SSEC has recently been developed [11] which no longer nece itate broad-band S-parameter measurements. It has already been applied with ucce to the SSEC of conventional FETs; it is also u able for the determination of the SSEC of HEMTs. To ab tract the Z-matrix parameter of the intrin ic FET. the para itic element of the extrinsic FET must be eliminated. U ing S-matrix mea urement at Vd = 0 and forward gate bias V8 on one ide, and Va = 0 and subthre hold V8 on the other side, all eight parasitic elements can be calculated from the S-matrix data. Then. u ing successive transformation from Z to Y matrice and conversely, the Z-matrix of the intrin ic transistor is arrived at for any values of Vd and Vg. An intere ting point is that all the parasitic elements can be obtained with measurements at relatively low frequencies only (F ~ 5 GHz). Of principal imponance is the determination of the intrin ic cutoff frequency of the transistor. which i defined a (23.15) and can be related to the electron transit time under the gate r = L 8/ {vd} , where (· • •) indicates the average value along the channel. If we write Id = qZ {ns}{vd) , we have gm:::: (E2/ a)Z{vd ) and Cgs = Zl 8 d (qns)/ dV8 = Zlg(E2 / a), so that C8s/ g,,,:::: L 8 / { ud ) = r, or fe = l / 2n r . The measurement of l e thus gives the average electron drift velocity during transit under the gate. In submicrometer HEMTs, {vd) values in exce s of 1.5 x 107 emfs are commonly obtained at ambient temperature, corresponding to cutoff frequencies higher than 50 GHz. Tlris is the case for the transistor illustrated in Fig. 23.7, where the maximum f c is 54 GHz and the average velocity i 1.7 x 107 emfs, since L 8 = 0.5 µm . These average velocities can be substantially rugher than tho e obtained in a MESFET having the same geometry. With regard to Gd, the output drain conductance it is essentially linked to injection conditions at the contacts and to the short-channel effect in very short structures. Its numerical value depends strongly on the technology and on the geometry of the device. In general, it goes through a maximum when the gate forward regime is approached.

23.4 TECHNOLOGY OF THE HEMT The fabrication of good-quality heterojunctions was rendered possible recently with the development of very sophisticated techniques for growing epitaxial monocrystal layers. At present, the two techniques used most often are molecular beam epitaxy (MBE) and metalorganic vapor-phase epitaxy (MOVPE) (12]. Of course, the quality of the epitaxial layers is strongly linked to the quality of the semi-insulating substrates usually obtained from Czochralski or Bridgeman grown ingots. Semi-insulating GaAs substrates still contain rather large densities of localized and extended defects, making

1466

HJGH-ELECTRON-MOBrLITY TRANSISTORS: PRINCIPLES AND APPLICATIONS

large-scale integration more difficult than in silicon technology. However, at present, the improvement in quality of substrates allows processing of 3-in. wafers with good uniformity of threshold voltages and yield, at least for discrete microwave transistors. The problem of the influence of defects at the surface of semi-insulating substrates is resolved in part by careful cleaning and etching, and by growing a buffer layer about 1 µm thick before fabrication of the active layers, which are much thinner. This, of course, requires more time for the growing process and the cost of devices is therefore higher. AIGaAs doping, usually in the range of 10 18 cm-3, is generally done with silicon atoms in very well controlled conditions, such that the atoms are in the position of donor states. Moreover, silicon does not induce serious pollution in the growth chambers and can be handled easily in the technological process of the material. With regard to device processing, gates as short as 0.3 µm are currently fabricated using electron-beam lithography. More and more in use now are plasma-enhanced etching and deposition techniques, which might lead to much better results than chemical processes because of better accuracy in the control of the process kinetics and more regular delineation of the various patterns. This kind of technological process might become an important interesting alternative for etching in the near future. A current process for the fabrication of a HEMT is described in Fig. 23.8. On doped GaAs layers, very high quality ohmic contacts can be made with the present technology. They are much more difficult to make directly on a AIGaAs layer, especially when the Al content is high. Thus, before gate deposition, the GaAs layer must be recessed such that the gate metal is against (or very near) the AlGaAs layer. The problem of access resistances is extremely important in these devices, becau e of their effect on microwave and noise performance, as we will see later. The trend in HEMT technology at present is toward the development of transistors with cutoff frequencies higher than 100 GHz, to obtain the large t po sible bandwidths

esist -- -- - - -t---~---¥--+-n+ GaAs

~

-

~

AIGaAs

GaAs

GaAs

1

2 Source

3

Drain

4

Figure 23.8 Elementary HEMT technological process. (1) In ulation by mesa chemical etching (orion implantation). (2) Ohmic contacts patterns and metallization: lift-off and annealing. (3) Rece s ~d g~te pattern; recess etching and gate metallization. (4) Gate lift-off and annealing. Surface passivation and contact pads plating are not represented.

23 4

TECH OLOGY OF THE HE 1T

1467

for telecommunication application . u ing 9-l- and 140-GH, atmo. pheric window . for in tance. It i clear that in thi domain \ery hort (i.e.. ubmicrometer) gate length are needed. But to keep good control of the current channel over o small di5tanccs. the thickne a of the active layer mu t be dimini ·hcd in proportion. and the width of the tran i tor mu t be uch that impedance matching to propagation network is till po ible. At the ame time, a. a re ult of a mall a, doping concentration mu t be increa ed to or above 10 1 cm 3 • With high doping in the AIGaA layer. evere limitation ari e. be au e highly doped layer where XAJ > 0.2 contain deep donor level . limiting the areal den ity 11 t of the electron channel. The c deep level are called DX centers [13. 14]. Another problem re ulting from hrinking the gate length i known a the hortchannel effect. In the calculation of Section 23 .2 we uppo ed that the driving field applied to the electron wa e entially directed along the ource-drain axi5. If Lg is very mall. thi i no longer true. Even near the ohmic contact the electric field may ha ea ignificant component in the direction tran ver e to the heterojunction plane, and complete imulation of the tran i tor in two dimen ion become nece ary. We tudy thi que tion in more detail in the next ection. Moreover. at pinch-off in particular, the channel electron may diffuse into the ub trate near the drain-end side of the gate, where the electron accelerated by the field can be very energetic. thu increasing the output drain conductance. To overcome the e problem , everal olution are po ible. For in tance. by grov.,ing a p-GaA layer beneath the undoped layer where the electron move, one can prevent them from diffusing into the ub trate ince they are repulsed by the diffusion barrier of the p layer. Thi elution has already been applied with succe s to the case of MESFETs (15]. An alternative elution can al o be tried by growing an AlGaA barrier beneath the undoped layer or u e a o-called inverted structure [ 16, 17]. Unfortunately. this solution. which i much impler than the previous one, is more difficult to apply becau e the quality of the interface obtained in thi way i not very good compared with that at the top heterojunction. The exi tence of a gate access resistance Rg and ource acce re istance Rs also degrade the transconductance and the available gain. For in tance, when Rs :f= 0. Em talces on the value , Em (23 .16) Em = 2irCgs(l + Em Rs) At very high frequencie . however. the structure of the ource acce zone 1s more complicated than a imple ohmic resi tance, ince displacement current through the heterojunction layers (via a R-C transmi ion-line scheme) must be taken into account [10]. A a result, one can show that the real part Rs of the acces impedance Zs m ay drop by a factor of 3 between the de regime and the millimeter wave range. At any rate, it also depends on the beet resistance of the doped contact layer. on the ohmic contact resistivity, and on the distance between the contact metal and the edge of the gate. We have hown previou ly bow a n + cap layer may improve the quality of the ohmic contact . However, it is clear from Fig. 23.3 that the electrons coming from the source metal do not have direct acce to the channel under the gate. To minimize the distance between the ource contact and the gate edge, elf-aligned deeply ion-implanted contacts can be made (managing for gaps, such that the gate cannot short the source- drain channel), a process in which the metal of the gate serve as a protecting mask for the ion implantation. Unfortunately, ion implantation is always

1468

HJGH-ELECTRON-MOBILITY TRANSISTORS: PRINCIPLES AND APPLICATIONS

followed by high-temperature annealing for crystal reconstruction and ion activation, and refractory metals must be employed for constructing the gate. This may lead in turn to a degradation of the gate resistance which is rather high with refractory metals (W or Mo). This solution is, however, good for logic applications where the gate width is generally small (Z ~ 10 µm), whereas in microwave applications Z ~ 100 µm. The cross-sectional shape of submicrometer metal gates is triangular rather than rectangular, making the gate access resistance higher that expected. To improve gate access resistance, mu hroom-shaped or multidigit gates must be made.

23.5 PHYSICAL ANALYSIS OF THE HEMT Despite its apparent simplicity, the HEMT is the seat of many interesting physical phenomena. A good understanding of the essential physical phenomena occurring in this device is necessary if we wish to develop accurate models in order to undertake the optimization of a transistor. In this section we describe several of them. We finish with a brief description of some sophisticated methods used for accurate modeling of HEMTs. A more detailed account and an abundant bibliography giving the present state of the art on all these questions are given in Refs. 6 and 10.

23.5.1

Low-Field Mobility and Charge Control Law by the Gate

Figure 23.9 shows the temperature dependence of the experimental low-field electron Hall mobility in a high-quality AlGaAs/GaAs heterojunction [18] compared with that measured in undoped bulk GaAs [19]. If the bulk GaAs were doped at 1017 cm- 3 , the mobility would drop to the range 2500 to 3500 cm2 fV/s between 77 and 300 K . The difference between the two curves of Fig. 23.9 is striking at low temperature. Here the spacer of the heterolayer (about 300 A) completely prevents ionized impurity scattering, while even in undoped bulk GaAs, the residual background impurities are sufficiently effective for the mobility to drop strongly below 40 K. At low temperature, mobility is dominated by lattice phonon interactions whose probability evolves as Ta (a ~ 1), a depending on the type of interaction. On the contrary, ionized impurity cattering probability evolves as T - 3/ 2 . As a result, very high electron mobility can be found, in excess of 106 cm2 / V/s in heterojunctions, as shown in Fig. 23.9. Values as high as 3 x 106 are commonly reported now at 4 K. This high mobility makes heterojunction devices very attractive for low-temperature applications (see Section 23 .6). Next, as mentioned earlier, the electron space charge in the heterojunction is usually very thin. As a result, the conduction band i also very narrow and is called a "well." Accurate calculations of the band structure ba ed on quantum mechanical treatments show that the width of the well near the Fermi level is on the order of 100 A, which i al o the thickness of the electron layer. This number is comparable (even at high temperature) with the thermal wavelength of the electrons perpendicular to the heterojunction. The motion is thus impeded along x from the point of view of quantum mechanics. As a consequence, the electrons are free to move along the plane of the heterojunction (y and z) only. The electron system, which is then called a quasi twodimensional electron gas (or simply Q2DEG or 2DEG), evolves on a ladder of energy subbands of which only the lowest few are occupied. The conduction band of a 2DEG

23.5

10 6

PHYSICAL ANALYSJS OF THE HEMT

1469

02DEG

f

;;E

0

Hall mobility

T(K) 1

10

100

Figure 23.9 Experimental low-field mobility measured in a high-quality heterojunction (solid line) and in undoped bulk GaAs (dashed line), as a function of temperature.

then takes the form

Ee =

tz2 En_.. + -2m• (_k_2_+_k_2_) y

(23 .17)

z

where the series of En., . (n x is a subband index) can be calculated using Scbrodinger's equation (and Poisson s equation for a self-consistent treatment), and ky and kz are the quasi continuous-wave vector components in the other directions. A detailed account of this question is given in Refs. 20 and 21. At the same time the envelope wave functions in each subband for a given ns in the well, the Fermi level EF, and the conduction band are calculated (via Poisson's equation), from which the potential at the gate can be deduced. Thus the control law ns(Vg) can be calculated exactly. An example was illustrated in Fig. 23.5. In this process the ion charge density N't, in the AlGaAs layers can be introduced in the calculation since E F and the donor levels are known. Of course, the calculation proceeds iteratively. In this way the effect of the doping density ND and XAJ in the AlGaAs layers can be taken into account. In particular, the areal density ns the maximum ns that can be obtained in the heterojunction and its dependence with Nv and XAJ, is one of the essential parameters entering into optimization of the device. 0

,

1470

HIGH-ELECTRON-MOBILITY TRANSISTORS: PRINCIPLES AND APPLICATIONS

23.5.2 Hot-Electron Effects From the point of view of the transistor, high low-field mobility is a good criterion only when the material is being tested before device processing. As a matter of fact, in the device the electric field is rather high and of complicated geometry. As a consequence, knowledge of hot-electron drift velocity as a function of field is more important than ohmic mobility in order to study the physics of electron transit under the gate. Figure 23.10 compares the electron drift velocity vd(E) in bulk undoped GaAs (22] with that of a 2DEG in a heterojunction based on Monte Carlo simulations (23], and experimental results (24] . At room temperature, the difference between the curves is not exceedingly large. This may not be the case at lower temperatures. The physics of charge transport in heterojunctions is more complicated than in bulk GaAs. As the driving field increases, electrons absorb energy, and their equivalent temperature may increase well above the lattice temperature. In these conditions higherenergy subbands can be occupied. This enhances thermal diffusion, and the carriers tend to disperse more easily into space than they do at equilibrium. As a consequence, the change in shape of the electron layer all along the channel, as the electrons move along the gate, in turn leads to a change in the conduction band from place to place. One can then wonder to what extent the two-dimensional character of the electron channel layer is preserved in the case of a high field. Moreover, the energy absorbed by the electrons may be large enough for them to transfer back to the AlGaAs layer much more easily than they do at equilibrium. This phenomenon, known as "real space transfer," may lead to transconductance degradation because the electron mobility in AlGaAs is low due to the ionized impurity scattering. As a consequence, a high value of ~Ee is desirable, which is obtained with a high value of XAf. Unfortunately, several phenomena appear when XAJ is too high: r -L-X valley crossover in AIGaAs when

2

1 T = 300 K

0

1

2

3

4

5 Field (kV/cm)

6

7

8

9

10

Figure 23.10 Experimental heterojunction drift velocity as a function of the electric field (solid line) with XAJ = 0.3 [24]. Simulation results (Monte Carlo technique) in undoped bulk GaAs (dashed-dotted line) [22] and in a heterojunction with XAJ = 0.3 (dashed line) (23].

23.5

PHYSICAL ANALYSIS OF THE HEMT

1471

> 0.4; enhancement of deep level in the doped AlGaAs, which trongly limits the ns and degradation in the quality of interface stoichiometry. A in the nonnal MESFET, non tationary tran port can be obtained in submicrometer HEMTs. Roughly peaking, the high field zone exists only under the gate. If the gate length is comparable with the carrier mean free path (nearly 1000 A at room temperature). the carriers coming from the ource near equilibrium may not have enough time to e tabli h an equilibrium elocity or energy di tribution in the pre ence of the field changing over very hort di tance . Then the transit i accompli hed in essentially tran ient conditions before the carrier reach the drain. In that case, transient drift velocitie can be achieved which are much higher than the usual maximum velocities attainable in bulk GaA . In other words, if the transit time under the gate is too short, the electrons do not have time to reach the energy of about 0.3 e V that is necessary for them to transfer to a satellite valley of GaA , even though they drift in a high electric field. Thi ituation i illustrated in Fig. 23.1 1 in which results of HEMT static simulations are displayed. One note that the drift velocity may approach 4 x 107 cm/s, and the average velocity under the gate (ud ) is about 3 x 107 emfs. We have shown earlier that the latter quantity is directly related to the cutoff frequency . Although there is no direct experimental ob ervation of this phenomenon (called "velocity overshoot,,), it is striking that simulation results are in fair agreement with evaluations of velocities from l e measurements. A a resul~ in submicrometer structures, the drift velocity is no longer dependent on the local electric field, but rather, on the energies of the particles, or to first order on the average energy of the electron system. This idea is the source of a number of sub.micrometer HEMT simulations. Thus the vd(E) of Eq. (23.12) is p robably wrong for a submicrometer structure. It is not surprising that the good agreement obtained between the Ia(Va, Vg) derived in Section 23.2 and experimental data of subnricrometer structures leads to a va(E ) that is not physical [e.g., this dependence is silicon-like, although the true vd( E ) has a negative differential mobility region]. Actually~ Eq. (23.12) incorporates in an empirical manner a number of physical phenomena, the most important being velocity overshoot. X.AJ

0 ;

23.5.3

HEMT PhysicaJ Simulations

Several kinds of simulation models have been developed. Here we mention the Monte Carlo model and the solution of fundamental semiconductor equations. The basic

Vz

10

107 cm/s

8 6

4 2

0

s

G

D

Figure 23.11 Monte Carlo simulations: influence of the lattice temperature on the space variation of the electron drift velocity in the HEMT channel under the gate with L 8 = 0.3 µ.m. Solid line, T = 77 K; dashed line, T = 300 K.

1472

HIGH-ELECTRON-MOBILITY TRANSISTORS: PRINCIPLES AND APPLICATIONS

principle of Monte Carlo simulations [25] is to follow the motion of carriers rep~e~e~tati ve points in reciprocal and direct spaces, taking into account both the detenrurustic effect of the driving electric field according to Newton 's law and the stochastic effect of the various types of scattering event the carriers may undergo in the course of time.

(a)

VOS == 2.0 V; VGS

=- 0.8 V; WY = 0.32 micron; WZ = 0.96 micron

s

G

D

cc::::_:::::::=..., - -·:::;25::..-

(b)

0.3

s

0.3

0.3

• I•

0.1

•I

G

0

(0 C")

0 -.:t

CJ Above 40.00 □ 20.00-40.00 - 10.00--20.00 - 7.50-10.00 D 5.00-7.50 1.00-5.00 - 0.05-1.00 • Below0.05

0 0 0

.....

CJ Above 0.70 D 0.65-0.70 0.60-0.65 - 0.55-0.60 - 0.50-0.55 - 0.45-0.50 - 0.40-0.45 CJ 0.35-0.40 Cl 0.30-0.35 - 0.25-0.30 - 0.20-0.25 - 0.15-0.20 - 0.10- 0.15 • Below0.10

(a) Electron equidensity contours in a planar HEMT simulated with a Monte Carlo model, A typical Monte Carlo uncertainty i clearly shown. XAJ = 0.3, a = 40 A, w 5 = 0. Figure 23.12 18

3

No= 10 cm- , L 8 = 0.3 µ.m (all distances are at scale). All densities are in 10 17 cm- 3 • External V8 = 0.2 V, Vd 2 V. (b) Electron equideosity contours (top) and the equi-energy lines (bottom) in a pJanar HEMT calculated after the solution of the semiconductor balance equations. They are labeled in 10 16 cm- 3 and eV. respectively. The horizontal distances are in micrometers, the vertical in angstrom. A spacer of 40 Ai included here. N O = 10 18 cm-3, XA1 = 0.3. A depleted zone extending on each ide of the gate appears because a surface potential has been included in the simulations. External V8 = 0.4 V, Vd = 2.5 V.

=

23.6

LOW-TEMPERATURE BEHAVIOR

1473

The bias voltage applied at the contact are introduced by mean of boundary conditions, and Poi on ' equation can be solved in two dimen ion ince the distribution of the carriers in space i known at all time . From this one gets a map of the potential everywhere in the imulated trucrure. Then external currents and device characteri tic can be computed. Figure 23. 12a repre ent the carrier equidensity contours in a sub.micrometer gate HEMT. This clearly show , for instance, carrier injection into the buffer layer near the ex.it of the gate. Another method is based on the o1ution of balance equations of the carrier mean energy and momentum [26]. These equation are deduced from the moments of the Boltzmann tran port equation and use the relaxation time approximation. The ba ic postulate is to as llIIle that all the moments of the distribution function are a function of the average energy only. which depend on pace and time. The equations are

J = µ (W ) • [qnE - V (nkT (W)) ] qd(n W ) dt

= J. E -

(23.18)

V(J. W ) - V (JkT (W)) - n (W r

Wo)

(23.19)

plus the usual -current continuity and Poisson equations. n is the electron density, J the current density, µ an equivalent mobility, E the electric field, W the average electron energy. T the electron temperature, and r an energy relaxation time. These quantities are all dependent on x and z. k is Boltzmann's constant. A typical result of this technique is shown in Fig. 23.12b in case of a sub.micrometer HEMT. Of course, these methods of simulation are computer time consuming, but they are able to predict in detail what should occur in a device with a high degree of accuracy.

23.6 LOW-TEMPERATURE BEHAVIOR One of the most important advantage of HEMTs is that the 2DEG mobility increase strongly as temperature decreases. Figure 23.9 showed the typical variation of lowfield mobility with temperature compared with that observed in bulk GaAs. As a consequence, we can expect a significant improvement of device characteristics at low temperature such as transconductance gm and current gain cutoff frequency f c, and as a result of the main device performance, such as propagation time, noise figure, and associated gain. However , the gm enhancement observed when the temperature drops from the ambient down to 77 K as predicted, for example, by Monte Carlo simulations (see Fig. 23.13) is less important than what could be expected from the mobi lity rise of Fig. 23.9. This is due to the fact that gm and so on are roughly proportional to the average velocity under the gate rather than to the low-field mobility. However, it is not easy to profit from the potential advantages of HEMTs at low temperature, due to the existence of parasitic phenomena which change the expected behavior considerably. Firs~ at low temperature, the HEMT static characteristics is very sensitive to light illumination [27, 28]. Even under low-intensity light, the l d(Vd, Vg) are excellent and exhibit no anomaly. On the contrary, in the dark they are strongly degraded; one observes a shift of the threshold voltage and a collapse of the curves at low drain voltage. When the HEMTs are cooled down in the dark, in most cases, the shift 8VT of Vr is positive (see Fig. 23.14). With normally-on devices, the shift is much larger than with normally-off devices. With light illumination, the shift can be reduced considerably.

1474

HIGH-ELECTRON-MOB ILITY TRANSISTORS: PRlNClPLES AND APPLICATIONS Gm mS/mm

1250 1000 750 500

,,,

_____ - - - - - - - 293 K

250 0 .._____.,_______..___ _._ _ _ _ _ _ _ Vgsv

- 1.6

- .8

- 1.2

- .4

0

Figure 23.13 Monte Carlo theoretical prediction of the transconductance as a function of gate voltage at room temperature (dashed line) and 77 K ( olid line), in a 0.3-µ, rn-gate HEMT.

Id mA/mm

200

.-·- , ---~

I I

,1 00 I

I I I / /

Vgsv

/

- 1.5

-1

- .5

0

-.5

-1

Figure 23.14 Typical transfer characteristics of a HEMT. Solid line, at 300 K; dashed line, at 77 K in dark; da hed-dotted line, at 77 K with light illumination.

The typical degradation of the tatic characteri tic , a-called collap e, i illu trated in Fig. 23.15. These characteristic are measured under light and in the dark after cooling down to 77 K. They are obtained u ing ordinary te t mea urement setups. Thi degradation, however, wa found to be critically dependent on the Va and V8 bias ranges of operation, as shown by Ka tal ky et al. [29]. For in tance, a pronounced degradation i observed only if Va and V8 are higher than l V and 0.5 V. re pectively. In addition, it i ob erved that once degradation has begun, it per i ts even though Vt1 is reduced below 1 V. Thi degradation i characterized by a trong increase in the access drain resi tance. reaching value a high as several kilohm [30]. Note that although light illumination ha an instantaneou effect, relaxation is very long once light ha been witched off. Thi rather complicated behavior can be explained by considering the existence of deep trap levels in the AlGaA layer, called DX center . A controver y still exists as to the exact nature of the DX centers. but their influence on phenomena occurring in HEM1: at low temperature i now generally admitted. If we represent the position of the

23.6

LOW-TEMPERATURE BEHAVIOR

1475

Ids

mNmm Vgs ♦

- .8 V

100

-- --,,,,"

50

,,, ,,, _,.

.,,✓... -­ /

,,,"' -----------,.--

V

Ov -------~ ------ ---

Vdsv

.5

0

Figure 23.15

~

- .4

1

1.5

2

HEMT tatic characteristics at 77 K . Dashed lines, in dark; solid lines, with light

illumination.

2.4

2.2

>Cl) >,

2.0

e Cl)

C

w

1.8

1.6 1.4

0

.2

.4

.6

.8

1

Al mole fraction

Figure 23.16 Lower-conduction band edges of AlxAI Ga 1_ ..(AI As as functions of the Al mole fraction XAJ. Note the band crossover at nearly 0.45. The DX center deep level is also represented.

DX center energy level as a function of XAJ (see Fig. 23. 16) with respect to the positions of the r-L-X band edges, we can observe that the DX center level approximately follows the L band edge and becomes resonant with the r valley at XAJ < 0.22 (31]. As a consequence, at XAJ > 0.2 a good part of the electrons are trapped in DX centers and cannot take part in the drain current. This proportion increases quickly with doping concentration and may reach 20 and 95% for XAJ equal to 0.2 and 0.3, respectively, at ND near 1018 cm- 3 (32). This scheme explains why the potential barrier height for emission from a deep level is independent of XAJ and why it decreases with XAJ for capture into it. The shift of Vr can also be explained by electron trapping in deep DX centers in the undepleted part of the AlGaAs layer. Because of the exceedingly long emission and capture times at low temperature, equilibrium cannot be reached after the perturbation has been applied to the system, unless a very long time has elapsed. Collapse can also be explained by hot electron trapping in the AlGaAs layer between gate and drain. The result is a degradation of the drain access resistance, due to the reduction of the free electron density in both the AlGaAs layer and the channel quantum well.

1476

HIGH-ELECTRON-MOBILITY TRANSISTORS: PR1NCIPLES AND APPLICATIONS

Several authors have shown that it is possible to suppress the collapse with a proper design of the recessed zone, depending on the gate length. For instance, collapse disappears if the distance between the end of the gate metal and the edge of the recessed zone is made shorter than 0.4 µm. This effect can be explained by considering the localization of the high-field domain between gate and drain and the electron transfer from the channel to the drain contact through the AlGaAs layer. However, it is always possible to avoid these parasitic effects simply by using XAI lower than 0.15 to 0.17 in normal, uniform doping levels (108 cm- 3 ) . Unfortunately, in this case, the height of the potential barrier llEc and the available ns0 are strongly reduced compared with conventional compositions (xAJ > 0.2). For these reasons, specific devices for low-temperature applications have been conceived and realized. They are described in Section 23.9.

23.7 LOW-NOISE AMPLIFICATION HEMTs have demonstrated better noise performance than conventional MESFETs. For instance, Duh et al. (33] report noise performance of 0.25-µm HEMTs and MESFETs fabricated at General Electric Laboratory, with the same gate topology. They are compared in Fig. 23.17 in the range 8 to 60 GHz at room temperature. In the entire frequency range, the HEMT has a lower noise figure, making this device very attractive for low-noise applications, up to millimeter wave range. Now, we will describe briefly some of the physical aspects of HEMT noise behavior and describe how device and circuits can be optimized.

23.7.1 Physical Aspects of Noise

In semiconductor devices, four kinds of noise can be ob erved: thermal or diffusion noise; shot noise; generation- recombination noise; 1/ f noise. The last three appear only in amplifiers at low frequency, and as a consequence, they do not have an influence on microwave low-noise amplification. Diffusion noise is due to carrier velocity fluctuations, increasing with electronic temperature. From a macroscopic point of view, this results in local fluctuations of the conduction current. These fluctuations occur not only in the channel, but also in the access zones and at the electrodes. As to their influence on the microwave signal1 they are considered to generate thermal noise and

NFmln dB 4 3 2

1

F GHz 0 .......___..______,ji.......,_.__...i.......,......._ . _...... i........_

5

10

20

40

60

100

Figure 23.17 Comparison of noise figures for 0.25-µ m-gate MESFETs and HEMTs at room temperature between 8 and 60 GHz. (After Ref. 33.)

23.7

LOW-NOISE AMPLIFICATION

1477

their contribution to the total noise are ea ily evaluated. Each can be represented as a noise generator with the amplitude

(23.20) Source and drain access resistance have the large t influence on noise performance because they are located in series with the input of the amplifier. In the channel, the noi e ource i a bit more complicated. Fluctuations in the conduction current can be repre ented by uncorrelated current noise sources distributed in series all along the channel. If we as ume a one-dimensional model of the device along the ource-drain axis, we can divide the channel into elementary sections of equal length 6.z. The mean- quare value of the noise current source between z and z + 6.z i given by (23.21) where {vi} is the average quadratic fluctuation of the drift velocity component parallel to the interface. The noise generators can also be characterized in terms of current pectral density given by (23.22) where D11 is the component of the diffusion coefficient parallel to the axis of the channel. These coefficients are usually considered as dependent on the electric field, but in fact, it is more realistic to assume that they are related to the average energy of the carrier, in analogy to what was said at the end of Section 23.5. Although D 11 is now relatively well known in bulk GaAs, the field dependence in the case of a 2DEG is still rather unclear. Some results (obtained with Monte Carlo simulations) are presented in Fig. 23.18, indicating that the D u(E ) obtained in the 2DEG are nearly similar to the bulk GaAs. This is not the case with the diffusion coefficient perpendicular to the interface, which is much smaller than in the bulk. This results from the fact that velocity fluctuations in this direction are reduced due to the existence of a potential barrier at the interface. This might partially explain why noise is reduced in HEMTs. Gate and drain current noise fluctuations provoke gate and drain voltage fluctuations by coupling through the gate and drain external circuits. This process of noise generation is somewhat complicated, and as a result, it is not easy to say what part of the device really contributes the main part of the total noise. First, as the Du decreases almost monotonously with the average energy of the carriers, Eq. (23.22) shows that the amplitude of the noise source decreases from the input to the output of the electron channel. Second, the influence coefficients relating the noise sources to the corresponding contributions to the gate and drain noise currents change all along the source-drain axis. These coefficients can be calculated by using the impedance field method [34-36], consisting in a calculation of the gate and drain current variations resulting from a vanishingly small current fluctuation in a section of the channel. The equivalent gate and drain noise sources can be calculated by a summation of the

1478

HIGH-ELECTRO -MOBILITY TRANSISTORS: PRINCIPLES AND APPLICATJO S 300

en

;;--

---5 ·o

200

C

Q)

:E

Q)

0

C

0

"::, iii

100

:t:

i5

E (kV/cm)

40

30

20

10

0

Figure 23.18 Evolution of diffusion coefficients as functj on of the electric field a l 300 K. Solid line, D ( £ ) in bulk GaA ; dashed line, D (E) in a AlGaAs/GaAs beterojunction; dotted line, D1. (£ ) in bulk GaAs; dashed-dotted line, D1- (E) in a AlGaAs/GaAs heterojunction.

quadratic fluctuation as (VJ) = 4q Z6. f o

(ii,> = 4q

2

1

L~

2

Z1",f

o

d Z (z, eu) Qs(z) D u(.:) - - dz

fl

lo ' Q, (z) Du(z)

dA

dz

2

d ::.

2

dz

(23.23) (23.24)

The terms Z (z, w) and A are determined by application of the impedance field method. dZ (z , eu) / d z is the impedance field [37) and A i an influence coefficient relating the noi e current in two adjacent section of the channel [36]. Moreover, a gate and drain noi e fluctuations are correlated. the correlation coefficient entering the expre ion of the noise figure (a we will see later) i calculated assuming that the transit time under the gate i negligible compared with the period of the microwave ignal. In thi case, the correlation coefficient i purely imaginary: (23 .25)

23.7.2

Calculation of the Noise Figure

The noise figure can be evaluated by a method imilar to that of Rothe and Dahlke [38]. All the noi e ourccs are referenced at the input of the FET equivalent circuit and represented by three elements: voltage and current source · which are uncorrelated, and a correlation impedance Z cor = R cor + j X cor (see Fig. 23.19). All the elements are related in terms of drain and gate noise sources [35] correlation coefficients, and gate and source acces resistance . Finally, the noise figure F can be expressed in terms

23.7

LOW-NOlSE AMPLlACATIO

G

1479

D

i - - - 1- -t - - - -

Noiseless

FET

s G

D Noiseless

FET

s- - - - - -- ...L------1

s

Figure 23.19 Representation of a noisy PET as a noiseless FET in series with a noise equivalent network at the input.

of a noise resistance Rn, a noi e conductance Gn, and the correlation impedance Zcor, which depends only on the de bias voltages:

F

I

2

= 1 + Ro (Rn+ Gn IZo + Zcorl )

(23.26)

Here Rn= (U; )/4kT~f, Gn = (i; )/4kT~f, and Zo =Ro+ }Xo is the internal impedance of the "equivalent generator" connected to the FET. Equation (23.26) shows that a proper matching can minimize the noise figure. The matching conditions are 112

Xo = Xm = -Xccr,

Ro = Rm= ( R~, + ~: )

(23.27)

The minimum noise figure is given by (23.28) and the dependence with the generator impedance Zo is given by (23.29)

23.7.3

Gain Calculation and Main Dependences

For low noise as well as power amplification, the available power gain is an important device characteristics. It can easily be calculated from the equi~alent circuit elements as

1-iso

HIGH-ELECTRO -MOBILITY TRANSISTORS: PR1NCIPLES AND APPLICATIO S

where Ls i the para itic acce source inductance, and We = 2rrle- Note the role of le• which directly determines the frequency and amplitude dependence of the available power gain. The gain is directly proportional to I/ , and a maximum value of l e mu t be re earched in device optimization. On the other hand, in the denominator of Eq. (23.30). we note the influence of the intrin ic elements Gd and C gd and the access re i tance Rs and R 8 which must be reduced all together a much a po sible in order to improve the available power gain. However, the mo t important parameter remain the current gain cutoff frequency hown in Fig. 23.20, representing the variations of G and le a function of the de bia current in a typical submicrometer HEMT. Note the important degradation of both G and le at low or high current. The maximum i obtained at medium current near 25 mA in the HEMT illustrated in the figure, a value not too far from /dss/2.

23.7.4

Main Dependences of Noise Figure

We have hown that the noi e figure depend trongly on the generator impedance and matching circuit. Moreover, its minimum value al o depends on operating conditions (de biases and frequency) and on technological parameters of the device. Figure 23.21 repre ents typical variation of the minimum noise figure as a function of the de bia current. We observe a minimum value of noi e figure for drain current in the range 40 to 60 mA/mm. This dependence can be explained by considering the corresponding variations of the noi e source (iJ }. (i; ) and the correlation coefficient with the drain current (see Fig. 23.21). Since the noi e figure decrease with (iJ) (while C increase lightly), it is obvious that F decrease with Id. At low Id. gm and l e decreases, involving a strong increa e in the noise figure. The minimum of F i le pronounced a the gate length is shorter [39]. In a 0 .25-µm gate length the variation of F with Id is very weak and related only to the pre ence of ource and drain re i tance . If they can be ignore~ one find that the noise figure become independent of Id. Thi i very interesting becau e the a ociated gain trongly decrease near pinchoff regime. We can also note that the optimum value of the drain current (the one

G(dB) fc (GHz)

12

60

G (15 GHz)

10

50

8

40

6

30

4

20

2

10

0

10

20

30

40

50

60

Figure 23.20 Evolution of the cutoff frequency le and the available gain as functions of the drain current. The tructure i the same as in Fig. 23. 7.

23.7

-

1.5

Cl)

'O

:E en

Cl) '-

0 0

::,

:.::

C

·5

CD

0)

-

1

C

.2 1

'

cu

Q) (/)

a5 ....

0

....

z

---

__;, 2.15 V because of the long stress time (> 105 s) required at lower stress voltages. This OC TTF increases rapidly beyond reasonable time as the stress voltage is lowered below 2.75 V. The current-accelerated FC data (filled circles and triangles}, however, gave TfF as high as 2.3 x 109 s (--745 years). From Fig. 24.7, the 10-year TTF was attained when VEB-stress ~ 2.2 V in the Si BIT and VEB-srress ~ 2.26 V in the SiGe HBT.

24.2.2 Hot Carrier Behavior As the dimensions of semiconductor devices brink and the internal field ri e, a large fraction of carriers in the active regions of the device are in states of high kinetic energy. At a given point in space and time, the velocity distribution of carriers may be narrowly peaked, in which case one speaks about "ballistic" electron packets. The term "hot electron" purports a nonequilibrium ensemble of high-energy carriers that have effective electron temperatures higher than the lattice temperature. The hot electrons, which introduce the current gain degradation, are those injected into the Si-SiO2 interface of Si and SiGe bipolar transistors. Degradation of the bipolar current gain under emitter-base junction reverse-bias stress bas been investigated extensively in the past 10 years [28-33]. It was believed that the current gain degradation was due to stress-generated interface traps at the oxide-covered emitterbase junction space-charge region. These interfacial traps were thought to be due to the dangling bonds from ruptured weak bonds during hot electron impact. Higher

24.2

TRANSISTOR RELIABILITY

1507

interface trap den ity give higher e lectron-hole recombination rate, larger ba e current. and lower current gain via the SRH recombination current in the p-n j unction pace-charge region. Figure 24.8 how an n +-p junction and it energy band diagram under rever ebia tre . . Under the o ide. there are interface trap. Drr (open triangle ) at weak bond (filled triangles) and charging of oxide electron (open quare ) and hole (open hexagon ) trap Q OT by hot electron and bot hole . Becau e the only direct ource of hot electron in n-p-n bipolar tran i tor under emitter-ba e rever e-bias tress is the thermal emi ion of trapped electron . the e thermal electron are accelerated to high kinetic energie by the applied emitter-ba e oltage. The thennal emi ion rate increa es rapidly with increasing temperature. Examine the mea ured base currents taken at 77 K and 298 K hown in Fig. 24.9. The tre s-induced ba e current increments are not a trong function of temperature. Thi indicates that thermally activated hot electrons are not the dominant hot carriers in a rever e-bia emitter-base junction. Becau e of the very high ba e and emitter doping concentration , the emitter reverse current should be dominated by tunneling. uch as tunneling of the valence-band electrons in the p-base into the unoccupied conduction-band tates. This tunneling occur predominately at the highe t electric field location near the n +Ip dopant-compensation boundary of the eminer-base junction. The tunneling is consi tent with the small temperature dependence ob erved in Fig. 24.8. The tunneled valence-band electron labeled by the circled dots next to (T) leave a thermal hole in the valence-band of the emitter pace-charge region. Thi thermal hole (the circle with the dot removed or electron dot tunneling

--·n 1- Si emitter

(b)

* Ekinebc=

q V EB-stress

+ qV EB-bl

Ekinetic=

q V EB-stress

+ VEB-Sl (.)

70 60

C

Q)

"c:5 :E w

50 40 30 20 10 00

15

30

45

60

75

01(degrees)

Figure 24.26 (a) collector current and collector-emitter voltage as a function of time and (b) power efficiency ver u

wi tching time delay 0 1 (after Blanchard and Yuan. Ref. 55 © IEEE).

The voltage and current waveform derived are computed to demon trate the utility of the analytical equation . Figure 24.26a how the collector current nonnalized to/cc versus time at 0f = 0, 30, and 60°. For a higher 0f , the collector current has a longer exponential decay when the transistor i off and the collector-emitter voltage is shifted to higher degree . The power efficiency a a function of 01 i depicted in Fig. 24.26b. In this figure, the olid-line repre ent the analytical prediction using an exponential collector falling waveform and the square represent the theoretical result using a linear collector falling waveform [56] . U ing r = 0.50I in the exponential collector

24A

c rRC IT APPLICATIONS

1529

falling waveform, good agreement between the pre ent analytical result and the re ult calculated from [56] i obtained. For 01 = 0. 30, and 60°, Lhe collector efficiency pred-icted by the analytical model i 100%, 96.8%. and 86.6%, and calculated from [56] i 100%. 97 .7%. and 90.8%. The power efficiency decrease with an increa e in 0 I · Thi i due to a larger tran i tor power lo when 01 increa e . To improve power efficiency of the clas E power amplifier at high frequencie , high- peed device · uch a heterojunction bipolar transi tor are needed. The high cutoff frequency and maximum o cillation frequency of the HBT improve the collector current fall time and power amplifier power efficiency. The main requirements for power amplifier u ed in mobile tran ceiver are hjghpower efficiency and the ability to work at low-power upplie . A cla s E power amplifier for low-Yoltage mobile communications was evaluated [57]. A fu}]y integrated clas E power amplifier module operating at 835 MHz i de igned, fabricated , and te ted. The amplifier deli er 24 dBm of power to the 50-Q load with a power added efficiency greater than 50% at a upply voltage of 2.5 V. The power dissipation in the integrated matching networks i 1.5 time the power dissipated in the transistors.

24.4.2.1 Third-Order l11tennodulation. Communication y tern require linear power amplifier with high efficiency and very low intermodulation distortion. Intermodulation (IM) distortion often defines the upper limit to the ignal-handling capability of a microwave receiver. It i particularly erious in broadband receivers de igned for communication or pectral urveillance. Consider a cage having a tran fer function given by the power series expansion (24.34)

Two input ignals at freq uencies 6./ and 26./ from the on-channel input frequency are involved in producing a third-order Intermodulation respon e in an amplifier. Let thee two signal be designated by A and B, where A= f + 6.f B = f + 26.f, and x = A + B . The power erie expansion becomes (24.35)

Evaluating the subexpansion of the third-order term in (10.24) gives a3A 3 + 3a3A 2 B

+ 3a3AB 2 + a3B 3

The underlined term in the above expression produces the IM response. To determine the slope m of thi response, express this term in decibel . IM (dB) = 20log(3a3 )

+ 20log(A2 ) + 20 log(B)

(24.3 6)

As signals A and B are both allowed to vary, a I -dB increase in A and B results in a 3dB increase in undes ired IM response. Hence, for a third-order IM in an amplifier stage, m = 3. Knowing the slope is useful when evaluating the intermodulation distortion because for every I-dB change in the two input signals, the IM response seen on the spectrum analyzer should change by 3 dB . The typical output power versus input power plot is discussed. The curve with slope I in the P0 - P;n plot is the fundamental response. The curve with slope 3 is the thirdorder response. The third-order Intermodulation intercept point (IP3) is the extrapolated

1530

HETEROJUNCTION BfPOLAR TRANSISTORS AND APPLICATIONS

intersection of the third-order Intermodulation product and the fundamental output power versu the input power. The high-output impedance of the HBT associated with the high Early voltage, high transconductance, and slow variation of current gain with collector current can be used to achieve high linearity capabilities in device and circuit applications. For comparable devices and biases, the HBT is observed to have an IP3/Pde power ratio of 5 to 10 times higher than typical MESFET and HEMT devices. Additional insight into the HBT nonlinearities was gained by using the Volterra series expansion method [58-60). The calculation of the nonlinear transfer functions is performed sequentially, from lowest to highest order, by solving a linear system of equations. The first-order transfer functions determine the response of the linear circuit, while the second- and higher order functions account for the device's nonlinear behavior. The second (or third) order current is proportional to the second (or third) derivative of the charge of a particular element with respect to the base-emitter voltage and collector-emitter voltage. The analysis is carried out at the frequency of hannonics. It is concluded that the nonlinear current generated by the emitter-base junction capacitance turns out to be the strongest of all elements (CJE CJe gie, and gm) followed by the transconductance. At different bias or frequency conditions, emitter-base conductance g;e and emitter-base capacitance CJE show a more pronounced canceling effect. Compared to CJE and gm, co1lector-base junction capacitance C1 e generates a weaker current. However, the interaction of C1c with other elements results in more pronounced nonlinear characteristics. Furthermore, the presence of the nonlinearities due to C1e, gje, and gm, together improve the third-order Intermodulation. Recently, Niu et al. (61] presented a systematic analysis of-intermodulation in SiGe HBTs using the Volterra-series approach. Avalanche multiplication and collector-base capacitance are shown to be the dominant nonlinearities in a single-stage commonemitter amplifier. At a given l e, an optimum collector-emitter voltage exists for a maximum third-order intercept point (IIP3). The IlP3 is limited by the avalanche multiplication nonlinearities at low l e , and limited by the Ce8 nonlinearities at high le. The decrease of the avalanche multiplication rate at high l e is beneficial to linearity in SiGe HBTs. The IlP3 is sensitive to the biasing condition because of strong dependence of the avalanche multiplication current and base-collector capacitance on l e and Ve£• The load dependence of linearity is attributed to the feedback through the CCB and avalanche multiplication in the collector-base junction.

24.4.3 Low Noise Amplifiers In a microwave amplifier, even when there is no input signal, a small output voltage can be measured. This small output power is referred to as the amplifier noise power. The total noise output power is composed of the amplified noise input power plus the noise output power produced by the amplifier. The noise figure NF describes quantitatively the performance of a noisy microwave amplifier. The noise figure of a microwave amplifier is defined as the ratio of the total available noise power at the output of the amplifier to the available noise power at the output due to thermal noise. The noise figure of a two-port amplifier is given by

rn gs

NF= NF min+ -IYs -Yol

2

(24.37)

where rn is the equivalent normalized noise resistance of the two-po~ Ys = gs + jbs represents the source admittance, and Y0 = g0 + jb0 represents the source admittance, which results in the minimum noise figure NF min·

24.4

CIRCUIT APPLJCATIO S

1531

U ing the equivalent noi e and network parameters of the HBT, the minimum noise figure a a function of de ice parameter i obtained [11]

(24.38) where R = Rb + Re. Equation (24.38) indicate that as the device i operated with a high collector current, F min monotonically increa e due to a decrease of f3 and an increase in grr. However, reduction of l e to a low level will reduce fr and increase NF min· For a given l e, when f ·• fr. the noise figure will increase with frequency, and it can be approximated as (24.39)

In portable coo urner applications where conserving battery life directly impacts co t, obtaining high gain and low noise figure under low de bias operation is highly de irable. HBTs are more attractive than FETs for these applications because of their high device transconductance and linearity under low de bias operation, small size, and low device noi e figure capability at L- S-, and C-band frequencies. For HBT devices, lower current generally means lower minimum noise. For lownoi e amplifier design, the source impedance required to achieve minimum noise needs to be considered over bias (62]. Figure 24.27 shows a plot of the optimum noise source impedance (gamma opt.) from 300 MHz to 3 GHz of a 3 x 10-µm2 quad-emitter HBT as a function of collector current. The impedance plot shows that at lower currents, the optimum impedance bas a large real part, greater than 50 Q , with a significant inductive HBT 3

x 1O µm2 quad-emitter gamma opt.

1

0 Ice= 1 mA

+ lcc=2 mA ◊

lcc = 4 mA

x Ice = 16 mA

.2

.5

1

2

=f

f1 =0.3 GHz

f2=3.0GHz

Figure 24.27 Optimum noise source impedance (gamma opt) loci from 300 MHz to 3 GHz (after Kobayashi et al., Ref. 62 © IEEE).

1S32

HETEROJUNCTION BIPOLAR TRANSISTORS AND APPLICATIONS

reactance. For higher collector currents, gamma opt. is mostly real and decreases below 50 n. At 16 mA, the impedance is close to 50 Q over the frequency range. At this bias poin~ the gamma opt. is nearly coincident with 50 n, which makes it easier to design both good input return-loss and low noise figures over a broad band. However, the minimum achievable noise figure at this bias will be higher than at lower bias currents. For low noise and low de bias operation, a design employing series inductive matching at the input of the HBT device will result in optimum noise match over a narrow band. A schematic of a 2-GHz low de power, low-noise HBT amplifier is shown in Fig. 24.28a. The LNA is a one-stage narrowband design that is matched for a center frequency of 1.9-2.0 GHz. A 3.8-nH input series inductor, Lb, is used to match the input of a 3 x 10-µm2 quad-emitter HBT for minimum noise. A collector bias current

(a)

1.9 GHz HBT LNA

Vee

Cbypass I

TLINo

-

-

~

Out

Gout

In

(b)

3 x 10 µm 2 quad-emitter

2-stage 5.7 GHz HBT LNA

Vcc2 R~

I Cbypass2

I Cbypass1

L CCi TUN

2

u --. ---. ~

Lee,

TUN,

RF0 ut

Cout2

3 x 10µm2 quad-emitter

1

10µm2 ad-emitter

Figure 24.28 Schematic of the (a) one-stage and (b) two-stage low-power, low-noise amplifiers (after Kobayashi et al., Ref. 62 © IEEE).

24.4

CIRCUIT APPLICATIO S

1533

of l mA and a Ve£ = 2.0 V wa cho en in order to realize a gain greater than 8 dB and a minimum noi e figure le than 2.0 dB with a total power con umption of 2 mW at 2 GHz. A 0.5-nF piral inductor. Le, in erie with the emitter of the HBT i u ed to tune gamma optimum o that it coincide more clo ely to the 50-Q ource impedance in order to achieve optimum noi e and input return-loss match. The output of the amplifier i matched u ing a erie L-C matching network comprising C oia and TLIN0 • An inductive choke Le. in eries with a mall load re i tance, Ri, provide both a high-pas ac load and a mean for bia ing the collector of the HBT. A chematic of a t\ o- tage C-band low-noi e amplifier u ing the same narrowband noi e matched topology is shown in Fig. 24.28b . Two stages are used in order to obtain reasonable gain at thi C-band frequency . In addition, resistive self-biasing comprised by re i tors R 1 • R2. and RLdc i u ed to implify the biasing of the circuit, which is convenient for on-wafer evaluation. The de current flowing through the resistive bias network i on the order of 0.2 mA per tage to en ure proper operation over variations in de current gain from wafer to wafer. The HBTs of each stage of the amplifier are bia ed at 10 mA and Ve£ = 2 V . At this bias, a nominal gain greater than 15 dB and a minimum noise figure of 2.52 dB are obtained at 5.7 GHz. The losses of the spiral inductors in erie with the base of the HBT: attribute to the ~ 0.4-dB noise figure. oise figures and gain were measured for the S-band low-noise amplifier at different Vcc in order to find the optimum noi e bias of the amplifier. Figure 9 .29a shows gain and noi e figure performance at 2 GHz as a function of biases at I cc = 5 mA. This plot sho~ that at a Vee ~ 2.0 V, the amplifier achieves a minimum noise figure and a maximum gain performance. This may be explained by the fact that the collector-base capacjtance is fully depleted under revere biases of greater than about 0.6 V, which corresponds to a Vee ~ Vee = 2.0 V. At this optimum collector voltage, the gain and noi e figures were measured as a function of l cc at 2 GHz, as shown in Fig. 24.29a. Thls figure shows that the optimum low noise bias current is between 2 mA and 4 mA for the LNA. At currents less than 2 mA, the noise figure increases quickly due to a rapid change in the gamma opt over bias. A SiGe HBT L A was de igned with a single stage [63]. The bias point of the LNA was fixed at 2.0 V and 8 mA, in consideration of collector-emitter breakdown voltage (BV CEO = 2.5 V), power consumption, gain, and noise figure. The resistor was added in the bias network to prevent low-frequency oscillation, and the emitter degeneration technique was used for stability. Additional series inductance for emitter degeneration technique was realized by microstrip line. An input-matching network that terminates the transistor with a conjugate of the optimum reflection coefficient was used to achieve the best noise match. In general, the input return loss was sacrificed for the noise match. Conjugate matching is used in the output-matching network to maximize the gain out of the circuit. A simulated small-signal gain of 15 dB, a gain flatness of ± 0.3 dB, an input return loss of 13 dB, an output return loss of 18 dB, and a noise figure of 0.8 dB for the desired frequency bandwidth were achieved. This LNA generated 1-dB compression input power of -9 dBm and 1-dB compression output power of 5 dBm. Additional bypassing capacitors Cd were added to improve the IP3 characteristics. The designed LNA was fabricated and mounted on a Teflon substrate using a packaged SiGe HBT and chip-type passive components. The out-of-band termination technique was used to improve linearity without significant gain reduction or additional current consumption. This SiGe LNA resulted in an input return loss of 11 dBm, an output return loss of 14.3 dB, a noise figure of 1.2 dB, and a 16-dB gain over the

1534

HETEROJUNCTION BIPOLAR TRANSISTORS AND APPLICATIO S (a)

Optimum Vee (Ice= 5 mA) 14 . - - - - - - - - - - - - - - , 3 . 0

2.8 13

-

2.6

C

CXl

"'O (1)

CXl

"'O

-.... ::,

12

C)

;.:

n;

2.4

(!)

(1)

"' ·5 z

11

10

2.2

4

3

2 Vee ( V)

1

0

2.0

Optimum Ice (Vee = 2.0 V)

(b)

3.2

14

3.0 12

"'O

-....

C

.;

2.8

--

CXl

CXl

"'O

10

2.6

C)

·«;

(!)

Q)

::,

2.4

8

2.2

Q)

0"'

z

6 2.0 4

0

2

4

6

8

10

121 .8

/cc (mA)

Figure 24.29 Measured gain and noise figure as a function of (a) bias and (b) collector current at 2 GHz (after Kobayashi et al., Ref. 62 © IEEE).

desired frequency range (2.11- 2.17 GHz). The upply voltage was 3 V, and the current consumption was 8 mA. The LNA generated the 1-dB compression input power of -10 dBm and the 1-dB compression output power of 5.2 dBm. When the RF two-tone input power was -23 dBm, a ~IM3 of 62.83 dBc, a IlP3 of 8.415 dBm, and an OIP3 of 24.415 dBm were achieved for a tone spacing of I MHz at 2.14 GHz.

24.4.4 Voltage-Controlled Oscillators Many microwave systems require the use of oscillators to generate the reference signal. A voltage-controlled oscillator (VCO) is central to the operation of many important

24..l

C IRCUIT APPLICATlO S

1535

electronic y tern . It i u ed in a pha e locked loop (PLL). fo r demodul ation of FM ignal , and in frequency ynthe izer that are particularly iinponant fo r wircle. application . Furthermo re. VCO are the critical component o f FMCW radars used for range and velocity mea urement . A uch, CO mu t have ery low pha e noi e and be highly linear. For example. for an accurate FMCW radar. the VCO mu t have a very linear weep. typicall y between 0.5% for mo t application as well a have ultra-low pha e noi e. In o cillators. the low-frequency fli cker no i e i mixed with the tughfrequency o cillation and re ult in phase noise around the carrier frequency, which degrade y tern perfo rmance. In the FMCW radar, pha e noi e in the VCO broaden the pectrum o f the detected ignal and make target that are phy ically clo e together In a microwa, •e o cillator. t\J o type of noi e are concerned. The fir t is the background no i e at the o cillation frequency band caused by thermall y generated current and voltage in the emiconductor device. Thi noi e i "white" in character and sets the noise floor for the o cillator . The magnitude of thi no i e i u ually very mall. The other noi e ource i the 1/ f noi e, which has phase and amplitude componen ts. When the noi e of an o cillator i added to the carrier and fed back through the resonator, the re ulting ignal i both amplitude and pha e modulated. The limiting amplifier will remove the amplitude modulated component. but it does not affect the pha e modulation. Phase noise dominates the near-carrier pectrum of the oscillator and often determine one endpo int of the dynamic range in a y tern. The maximum undi torted ignal depend on y tern linearity. while the minimum i et by noise. Phase noise i crucial to the performance of many wireless communication systems. In cellular communication systems, for example, the phase noise of the local o cillator determines the minimum spacing between adjacent channel in order to avoid interference. In Doppler radar sy terns phase ooi e can disgui e weak, reflected incoming signals. Pha e noi e pectrum of bipolar o cillator is dominated by their 1/ f noise [64. 65). Recent experimental data bow that both phase noise and low-frequency 1/ f noi e originate from diffu ion coefficient fluctuation in the base (66]. The intrinsic baseband 1/ f noise is up-converted to the o cillation frequency through nonlinearities in the device (67, 68]. Although the power in the ideband is not very high, its presence degrades the performance of microwave ystems. For example, in a coherent radar system. the return ignal is mixed with a portion of the transmi tter signal for detection. If the noi e in the sidebands is too hig~ the return signal magnitude fall below the noi e level and the detection sensitivity of the radar i reduced accordi ngly. For the ca e where upconversion of ideal, l/ f noise dominate ; within the resonator bandwidth, the phase noise varies as 1/ f ~ , where fm is the offset frequency. The phase noise spectral density of an 11 -GH z HBT dielectric resonator oscillator was investigated [69]. The schematic diagram of the dielectric resonator oscillator i shown in Fig. 24.30. Parallel feedback is u ed with the dielectric resonator providing the neces ary inductive feedback between the collector and the base. Series feedback in the emitter is u ed to improve output matching. The collector and base are biased from a single variable de voltage source that is connected to the circuit through high-frequency chokes. The base current is controlled via a 10-Q wire wound potentiometer. The lowfrequency base termination Rb,, can be switched from a value R set by the potentiometer to approximately O n. The high-frequency choke allows signaJ transmission below

100 kHz. The measured spectrum for the low-frequency short-circuit base termination case, with VcE 5 V and le= 50 mA is shown in Fig. 24.31. The oscillation frequency

=

1536

HETEROJUNCTTON BIPOLAR TRANSISTORS AND APPLICATIONS 50 pF

50 Q, 01

Vs ias

ORO

R Rb,I:

A.C. short

_l

l

Out

High freq. choke

50 Q , 02 50 Q , 03

Vsias

Figure 24.30 © IEEE).

Schematic diagram of a dielectric resonator oscillator (after Tutt et al., Ref. 69

LP1 1.56 MKR: 1.500 000 0 GHz RLV: - 10.0 dBm 0

. . .

-

. .

.. .

- 30

..

.

-40

.

- 10 - 20

.

. .

- 50

.

-60

. .

.. . .

.. .

\1

ti '-'" .

I\ r· \ 1

r~

..

.. .. ..

Center: 1.500 000 0 GHz

Figure 24.31 © IEEE).

ADJ CH MEAS

L 1: - 49.2 dB L2: - 63.2 dB U1 : - 49.2 dB U2: - 63.8 dB

.. ..

I.II.

:/ .

~ .

.

... . .. ..

SGLSWP A: POS B: POS

AT 0 dB# RB 300 Hz# ST 20 s# VB 300 Hz

... ..- ~ .... .... l :1 .. .. .. . . :_.J .... . l .. I : , ... u .... ,,,-: .. a...""'

-80

- 100

.

.. ..

J "'... ~·r:-·.

- 70

- 90

.

-30.69 dBm

- I

~

1

.......

·,:.

~

. ... ..

.=··M .:

. .. .

.

...

[Band]

...

i~ .. ..

... .. . ...

.. . ... .

. .. ...

,~ \

~

• J

. ...

-

.. =' ...

. ... ... .

.

,

.

..

..

..

.

. .

. . . .

.

.

.·"""""' \..:._.

~, I

..

a I

. ..

.

I J.j

.~

[1W .•

Span: 250 kHz

Measured spectrum of the dielectric re onator o cillator (after Tun et al., Ref. 69

is 11 .02 GHz, and the output power is 5.6 dBm. No tuning of the emitter stub was attempted to maximize the output power. The external measured quality factor Q ranges from 1200 to 2200 as the collector current decreases from 50 mA to 20 mA. The phase noise of the o cillator was measured from 10 to 100 kHz using an HP3048A pha e noi e measurement system. A plot of £(fm) versus frequency for l e = 50 mA and VcE = 5.0 V for different base tenninations is shown in Fig. 24.32. This figure clearly shows the impact of the base termination on £(/m). The noise was reduced by about 4 - 7 dB over the entire measurement band when the base

24.4

CIRCUIT APPLICATIONS

1537

0.--------------------20

-40

--u N

I

-60

CD

"O

.._E

Rb,t= 0

--80

Rt,,,= R

t ,.4

- 100 VcE=S V

-120

le= 50 mA Frequency = 11 GHz

1000

10000

100000

Offset frequency (Hz)

Phase noi e versus offset frequency (after Tutt et al., Ref. 69 © IEEE).

Figure 24.32

Le

son

Cp

(ext.)

Va Q

"~.. ,

'A-"\""

Ve

0

T

V0 =-5.5 V

Figure 24.33

Circuit schematic of a differential VCO.

was ac short-circuited at baseband frequencies. At 10 kHz, £(fm) = -101 dBc/Hz for R b,r = 0 and -95 dBc/Hz for R b,r = R . On-chip VCO using SiGe technologies have been reported recently (70- 72]. The low-phase-noise and low-cost millimeter-wave VCO bas been fully integrated in commercial SiGe bipolar technologies (72]. Circuit diagram of the differential VCO is shown in Fig. 24.33. The fully differential configuration was chosen because of onchip noise problems, which are substantially reduced, and high-frequency grounding

1538

HETEROJUNCTION BIPOLAR TRANSISTORS AND APPLICATIONS

as well as on- and off-chip decoupling of supply and bias voltage are less critical due to the virtual ground nodes. The virtual ground nodes allow us to realize easily adjustable IC for different applications in a wide frequency range. The circuit was fabricated using Infineon B&HF Si Ge technology. Both f T and f max are between 70 and 75 GHz. The circuit was measured mainly on wafer using RF probes and a 50-GHz spectrum analyzer. The oscillation frequency can continuously be tuned from 43.6 to 47.3 GHz when the corresponding bias voltage V8 is varied from -2.6 to 0.7 V as shown in Fig. 24.34. In this frequency range, the single-sideband phase noise varies between -103 and - 108.5 dBc/Hz at I-MHz offset from the carrier. The differential output voltage swing is quite high and changes between 1.6 and 1.8 V p - p for 50-n loading of each node. This corresponds to an output power of about 5.6 dBm for the differential and 2.6 dBm for the single-ended output, respectively. The spectrum of the (uncorrected) single-ended output power at a center frequency of about 45 GHz is shown in Fig. 24.35. The total power consumption of the oscillator is 280 mW at -5.5 V.

24.4.5 Multipliers/Mixers From de to RF frequency, Gilbert cell analog multipliers can be u sed for doublebalanced active mixers and up-converters in microwave applications, and for highly sensitive detection mixers in coherent optical heterodyne receivers. Comparing with conventional diode double-balanced mixers, the active double balanced mixer has the advantages of low LO driver power with high linearity, the ability to provide positive conversion gain, and elimination of the need for bulky hybrid balun circuitry. The circuit configuration on an analog multiplier is shown in Fig. 24.36. The Gilbert cell configuration is selected for its double-balanced implementation, which offers high conversion gain and improved spur performance in a very compact size. The RF and

48 ,--,----,---y---,.---,--r-,r--,---r--.----r~T"'""T'-.--T--1---r""-r-,.......,..--P--......-.--.-...,.......,~~-- -90

47

- 95 N

N'

J:

(!)

co

J:

D 2.5 kHz/

.

~, '

~Ii. la ....

Span 25 kHz

~ig~re 25.31 ~easured spectrum of a synthesizer where the loop filter is underdamped, resultmg m ~ l 0-dB mcrease of the phase noise at the loop-filter bandwidth. In this case, we either do not meet the 45° phase margin criterion or the filter is too wide, so it shows the effect of the up-converted reference frequency.

25.5

PHASE LOCKED LOOP DESIG S

1579

40

-CD

(a) Second order PLL, d.f. =0.45 (b) Second order PLL, d.f. =0.1 (c) Third order PLL, 0 = 45° (d) Third order PLL, 8 =10°

30

"O

U)

20

0:(1)

VJ C

0 ci

10

(/)

...

Q)

0

"O

.... Q) (13 '-

- 10

....C

- 20

O>

Q)

- 30

Figure 25.32

10

1

100 1000 Frequency (Hz)

10000

Integrated re ponse for various loops as a function of the phase margin.

+20 Phase margin = 10°

+10

CD

0

u

Cl) (/)

- 10

/

C

0 ci (/)

Phase margin = 45°

- 20

Q)

C[

-30

-40 - 50

Figure 25.33

10

1

1000 100 Frequency (Hz)

10 000

Closed-loop response of a Type 2. third-order PLL having a pha e margin of 10°.

We will now determine the transient behavior of this loop. Figure 25.34 shows the block diagram. Very rarely in the literature is a clear distinction between pull-in and lock-in characteristics or frequency and phase acquisition made as a function of the digital phase/frequency detector. Somehow, all the approximations or linearizations refer to a sinusoidal phase/frequency comparator or its digital equivalent, the exclusive-OR gate. The tristate phase/frequency comparator follows slightly different mathematical principles. The phase detector gain is

, Va phase detector supply voltage K - - - ----------d -

Ct.Jo -

loop idling frequency

and is valid only in the out-of-lock state and is a somewhat coarse approximation to the real gain, which, due to nonlinear differential equations, is very difficult to calculate.

1580

OSCILLATORS AND FREQUENCY SYNTHESIZERS

K'c,

n1(5) +

F(s)

Phase frequency detector

A(s)

B(s) Filter

VCO

0 2(s)

0 (s)

-2 N

Divider +N

-

Note: The frequency transfer const. of the VCO = K0 ( not :

0

,

which is valid for phase transfer only.)

Figure 25.34

Block diagram of a digital PLL before lock is acquired.

However, practical tests show that this approximation is still fairly accurate. Definitions:

= $ [L\w1 (t)] f22(s) = ..zf[L\a>i(t) ]

n1 (s)

Reference input to 8/ w detector Signal VCO output frequency

f2e(s)

= ..zf[we(t)]

Qe(s)

= n1(s) -

n2(s)

= [01 (s) -

Qe(s)]N

f22(s)

Error frequency at 8/ w detector N

From the circuit above,

= Oe(s )K~ B(s) = A(s)F(s) 0 2(s) = B(s)K A(s)

0

The error frequency at the detector is

ne (s) = 01(s)N N + K 1K~F(s)

(25 .52)

0

The signal is stepped in frequency: L\w1

01 (s) = - -

s

(L\w1 = magnitude of frequency step)

(25.53)

Active Filter of First Order. If we u se an active filter F(s)

= 1 + si-2 St'1

(25 .54)

25.5

PHASE LOC KED LOOP DES lGNS

1581

and in ert thj in Equation (25.51), the erro r frequency i (25 .55)

Utilizing the Laplace tran formation. we obtain (25.56)

and lim We (f ) r-0

lim We(t) t-+

D.w1 N

_

- -- - - - -

N

+K

0

Kd(r2/ r1 )

=0

(25.57) (25.58)

Passive Filter of First Order. If we use a pa ive fi lter

,-lim We(t ) = 0

(25.59)

for the frequency step (25.60) the error frequency at the input becomes

(25.61)

For the first term, we will use the abbreviation A , and for the second term, we will use the abbreviation B . (25.62)

(25.63)

After the inverse Laplace transformation, our final result becomes (25.64)

1582

OSCILLATORS AND FREQUENCY SYNTHESIZERS

(25 .65) and finally (25.66) What does the equation mean? We really want to know how long it takes to pull the VCO frequency to the reference. Therefore, we want to know the value oft, the time it talces to be within 2rr or less of lock-in range. The PLL can, at the beginning, have a phase error from -2rr to +2rr, and the loop, by accomplishing lock, then takes care of this phase error. We can malce the reverse assumption for a moment and ask ourselves, as we have done earlier, how long the loop stays in phase lock. This is called the pull-out range. Again, we apply signals to the input of the PLL as long as the loop can follow and the phase error does not become larger than 2rr . Once the error is larger than 2rr , the loop jumps out of lock. When the loop is out of lock, a beat note occurs at the output of the loop filter following the phase/frequency detector. The tristate phase/frequency comparator, however, works on a different principle, and the pulses generated and supplied to the charge pump do not allow the generation of an ac voltage. The output of such a phase/frequency detector is always unipolar, but relative to the value of Vbau/2, the integrator voltage can be either positive or negative. If we assume for a moment that this voltage should be the final voltage under a locked condition, we will observe that the resulting de voltage is either more negative or more positive relative to this value, and because of this, the VCO will be "pulled in" to this final frequency rather than swept in. The swept-in technique applies only in cases of phase/frequency comparators, where this beat note is being generated. A typical case would be the exclusive-OR gate or even a sample/hold comparator. This phenomenon is rarely covered in the literature and is probably discussed in detail for the first time in the book by Roland Best [9] . Let us assume now that the VCO bas been pulled in to final frequency to be within 2rr of the final frequency, and the time t is known. The next step is to determine the lock-in characteristic.

25.5.2.2 Lock-in Characteristic We will now determine the lock-in characteristic, and this requires the use of a different block diagram. Figure 25.5, of the Frequency Synthesizer Fundamental section, shows the familiar block diagram of the PLL, and we will use the following definitions: 01 (s)

=$

[A81 (t)]

= Z [A82(t)] Be(s) = $ [8e(t)] 82 Be(s) = Bt(S) (s) 02(s)

N

Reference input to l, / w detector Signal VCO output phase Phase error atl,/ w detector

25.5

PHAS E LOCKED LOOP DES rGNS

1583

From the block diagram, the following i apparent:

B (s)

= 0c(s) Kd = A (s) F (s)

02(s)

= B (s) -Ko

A (s)

s

The phase error at the detector i (25.67) A tep in phase at the input. with the worst-case error being 2rr, re ults in

01 (s)

1

= 2rr -

s

(25.68)

We will now treat the two case u ing an active or pas ive filter.

Active Filter. The transfer cbaracteri tic of the active filter is (25.69)

This results in the formula for the phase error at the detector, 0~(s)

= 2rr - - - - - - -s- - - - - - s2 + (s K Kdr2/r1)/N + (KaKd/ r1 )/N

(25 .70)

0

The polynomial coefficients for the denominator are

=1 a1 = (K0Kdr2/ r:1)/ N ao = (K0Kd / r1)/N a2

and we have to find the roots W1 and W2 • Expressed in the form of a polynomial coefficient, the phase error is Be(s)

=

2

,r

s

(s

+ Wi)(s + W2)

(25.71)

After the inverse Laplace transformation has been performed, the result can be written in the form (25.72) ,.

1584

OSCILLATORS AND FREQUENCY SYNTHESIZERS

with

and

=0

lim 8e(t )

t ➔ OO

The ame can be done using a passive filter. Passive Filter. The transfer function of the passive filter is

(25.73)

If we apply the same phase step of 2rr as before, the resulting phase error is (25.74)

Again, we have to find the polynomial coefficients, which are a2

=1

a1

=

N + KoKdr:2 N (r 1 + r:2)

ao =

K 0 Kd

N (r:1

+ r:2)

and finally find the roots for W1 and W2 • This can be written in the form (25 .75) Now we perform the inverse Laplace transformation and obtain our result: (25.76) with lim 8e(t )

t➔ O

= 21r

with

Jim 8e(t)

t ➔ oo

=0

When analyzing the frequency response for the various types and orders of PLLs, the phase margin played an important role. For the transient time, the Type 2, second-order

25.5 PHASE LOCKED LOOP DES IGNS

1585

loop can be repre ented with a damping factor or. for higher orders, with the phase margin. Figure 25.35 how the normalized output respon e for a damping factor of 0.1 and 0.47. The ideal Butterworth re pon e would be a damping factor of 0.7, which correlate with a phase margin of 45°.

25.$.3

Loop Gain/I'ransient Response Examples

Given the imple filter hown in Fig. 25.36 and the parameters as listed, the Bode plot is hown in Fig. 2S.37. Thi approach can al o be tran lated from a Type I into a

2.0 1.8

Damping factor = 0.1 Q)

1.6

(/)

C

0

Q.

1.4

(/)

~

-

1.2

::, 0

1.0

::,

Q.

"'O

-~ 0.8 CK

QN

Figure 25.57 Custom-built phase detector with a noise floor of better than - 168 dBc/Hz. This phase detector shows extremely low phase jitter.

1596

25.6

OSCTLLATORS AND FREQUENCY SYNTHESIZERS

THE FRACTIONAL-N PRINCIPLE

The principle of the fractional-N PLL synthesizer was touched previously m Section 25.2. The following is a numerical example for better understanding. Example. Considering the problem of generating 899.8 MHz using a fractional-N loop

with a 50-MHz reference frequency,

899.8 MHz= 50 MHz ( N

+ ~)

K The integral part of the division N has to be set to 17, the fractional part F needs to be

996

(the fractional part K is not a integer), and the VCO output has to be divided by

JBfxevery 1000 cycles. ~is can easily be implemented by adding the number 0.996 to the contents of an accumulator every cycle. Every time the accumulator overflows, the divider divides by 18 rather than by 17. Only the fractional value of the addition is retained in the phase accumulator. If we move to the lower band or try to generate 4 850.2 MHz, N remains 17 and K becomes - -. This method of using fractional F 1000 division was fi rst introduced by using analog implementation and noise cancellation, but today, it is implemented totally as a digital approach. The necessary resolution is obtained from the dual modulus prescaling, which allows for a well-established method for achieving a high-performance frequency synthesizer operating at UHF and higher frequencies. Dual modulus prescaling avoids the loss of resolution in a system compared to a simple prescaler; it allows a VCO step equal to the value of the reference frequency to be obtained. This method needs an additional counter, and the dual modulus prescaler then divides one or two values depending on the state of its control. The only drawback of prescalers is the minimum division ratio of the prescaler for approximately N 2 . The dual modulus divider is the key to implementing the fractional-N synthesizer principle. Although the fractional-N technique appear to have a good potential of solving the resolution limitation, it is not free of having its own complications. Typically, an overflow from the phase accumulator, which is the adder with the output feedback to the input after being latched, is used to change the instantaneous division ratio. Each overflow produces a jitter at the output frequency, caused by the fractional di vision, and is limited to the fractional portion of the desired division ratio. In our case, we had chosen a step size of 200 kHz, and yet the discrete sidebands K 4 K996 vary from 200 kHz for - = - - to 49.8 MHz for - = - -.It will become the F 1000 f 1000 task of the loop fi]ter to remove those discrete spunous components. Although in the past the removal of the di crete spurs has been accomplished by using analog techniques, various digital methods are now available. The microprocessor has to solve the following equation:

N*

= ( N + ~) = [N(F -

K )+ (N + l)K]

(25.77)

25.6 THE FRAC BONAL-N PRINCIPLE

1597

Example. For F o

= 850.2 MHz, we obtain :

N* _ 850.2 MHz _ 50 MHz - 17.004

Following the formula above:

N"

=

(N+ K) = F

= F

[17(1000 - 4) + (17 1000 [16932 + 72] 1000

= 50 MHz x

Olli

+ 1)

x 4]

= 17.004

(16932 + 72] 1000

= 846.6 MHz +

3.6 MHz

= 850.2 MHz By increa ing the number of accumulators, frequency resolution much below 1-Hz step size is pos ible with the same switching speed. There is an interesting, generic problem associated with all fractional-N synthesizers. Assume for a moment that we use our SO-MHz reference and generate a 550-MHz output frequency. This means our division factor is 11. Aside from reference-frequency sidebands (±50 MHz) and harmonics, there will be no unwanted spurious frequencies. Of course, the reference sidebands will be suppressed by the loop filter by more than 90 dB. For reasons of phase noise and switching speed, a loop bandwidth of 100 kHz has been considered. Now, taking advantage of the fractional-N principle, say we want to operate at an offset of 30 kHz (550.03 MHz). With this new output frequency, the inherent spurious-signal reduction mechanism in the fractional-N chip limits the reduction to about 55 dB. Part of the reason why the spurious-signal suppression is less in this case is that the phase-frequency detector acts as a mixer, collecting both the 50-MHz reference (and its harmonics) and 550.03 MHz. Mixing the 11th reference harmonic (550 MHz) and the output frequency (550.03 MHz) results in output at 30 kHz; as the loop bandwidth is 100 kHz, it adds nothing to the suppression of this signal. To solve this, we could consider narrowing the loop bandwidth to 10% of the offset. A 30-kHz offset would equate to a loop bandwidth of 3 kHz, at which the loop speed might still be acceptable, but for a 1-kHz offset, the necessary loop bandwidth of 100 Hz would make the loop too slow. A better way is to use a different reference frequency- one that would place the resulting spurious product considerably outside the 100-k:Hz loop-filter window. H1 for instance, we used a 49-MHz reference, multiplication by 11 would result in 539 MHz. Mixing this with 550.03 MHz would result in spurious signals at ± 11.03 MHz, a frequency so far outside the loop bandwidth that it would essentially disappear. Starting with a VHF, low-phase-noise crystal oscillator, such as 130 MHz, one can implement an intelligent reference-frequency selection to avoid these discrete spurious signals. An additional method of reducing the spurious contents is maintaining a division ratio greater than 12 in all cases. Actual tests have

1598

OSCILLATORS AND FREQUENCY SYNTHESIZERS

shown that these reference-based spurious frequencies can be repeatedly suppressed by 80 to 90 dB .

25.6.1

Spur-Suppression Techniques

Although several methods have been proposed in the literature, the method of reducing the noise by using a sigma-delta modulator has shown to be most promising. The concept is to get rid of the low-frequency phase error by rapidly switching the division ratio to eliminate the gradual phase error at the discriminator input. By changing the division ratio rapidly between different values, the phase errors occur in both polarities, positive as well as negative, and at an accelerated rate that explains the phenomenon of high-frequency noise push-up. This noise, which is converted to a voltage by the phase/frequency discriminator and loop filter, is filtered out by the ]ow-pass filter. The main problem associated with this noise shaping technique is that the noise power rises rapidly with frequency. Figure 25.58 shows noise contributions with such a sigma-delta modulator in place. On the other hand, we can now, for the first time, build a single-loop synthesizer with switching times as fast as 6 µ s and very little phase-noise deterioration inside the loop bandwidth, as seen in Fig. 25.58. As this system maintains the good phase noise of the ceramic-resonator-based oscillator, the resulting performance is significantly better than the phase noise expected from high-end signal generators. However, this method does not allow us to increase the loop bandwidth beyond the I 00-kHz limit, where the noise contribution of the sigma-delta modulator takes over.

Filter frequency response/predicted SSB modulator noise Required filter attenuation

0

!g - 100 Reference noise floor (-127 dBc)

- 150

- 200 5 kHz

50 kHz 500 kHz Bandwidth =- 100 kHz for r= 5 µs

5MHz

50 kHz

Figure ~5.58 The filter frequency response/phase noi e analysis graph shows the required attenuation for the reference frequency of 50 MHz and the noise generated by the sigma-delta converter (three steps) as a function of the offset frequency. It becomes apparent that the sigma-delta converter noise dominates above 80 kHz unless attenuated.

TABLE 25.2

1599

THE FRACTlONAL-N PRINCIPLE

25.6

Modern Spur-Suppression Methods

Technique DAC Phase Estimation Pulse Generation Phase Interpolation Random littering Sigma-Delta Modulation

Feature

Problem

Cancel Spur by DAC Insert Pulse Inherent Fractional Divider Randomize Divider Modulate Division Ratio

Analog Mismatch Interpolation Jitter Interpolation Jitter Frequency Jitter Quantization Noise

Table 25 .2 shows some of the modern spur-suppression methods. These three-stage sigma-delta methods with larger accumulator have the most potential. The power spectral re pon e of the phase noise for the three-stage sigma-delta modulator is calculated from : L (f)

=

(2rr)2 12

Ii

X

[

(rr/

x 2sin -

ref

)]2(n-l) rad /Hz 2

(25.78)

fref

where n is the number of the stage of the cascaded sigma-delta modulator. Equation (78) shows that the phase noise resulting from the fractional controller is attenuated to negligible levels close to the center frequency and further from the center frequency, the phase noise is increased rapidly and must be filtered out prior to the tuning input of the VCO to prevent unacceptable degradation of spectral purity. A loop filter must be used to filter the noise in the PLL loop. Figure 25.58 showed the plot of the phase noise versus the

fre

50MHz Frequency reference

LPF

..__ _ Dual-modulus prescaler & programmable divider

Buffer

Output

Bit manipulator and decoder Data input k -~

3-stage L6 converter

Figure 25.59 Block diagram of the fractional-N synthesizer built using a custom IC capable of operation at reference frequencies up to 150 MHz. The frequency is extensible up to 3 GHz using binary (-:-2, -:-4, +8, etc.) and fixed-division counters.

i

Modulation/sweep

Frequency divider ACTFAEO

WOBMOOE-,..- - - - - - - - - - ,

i - -- - - - - - - t - - - ' " 1 '

I\ 0

C") Q) V.:,

I

I- ai

, - - - - - - - - - - , Sweep mux

a: > w a.

!

3: l!l ::x: en (.) en

·c

£ en

55 55

4

Step width ~ SCHWEIT 1 x ::,

E Q)

:>

0 1 Step clock STEPCLK-,

X

CLK AES

s·

I

z-, 15

., •

ID

0

SB15 LSB40

. 1s

Compenstation adder A

40

I

PAOGMUX 3

1 n~ ...

LO

2 1 CLEAR Compenstatlon value calculation

L

s

~ h TF

/

i.1 5

B CIN

_J

Output register

CLKEN CLK RES

AUFAB 11 ~ M ULT1

15

I I

FASTEN PARSER CENB WAB ROB ADR

Mo d u l ~ register

,.1s

Modulation signal AOIN C>----J

• 10

55

' .26

Multiplier

CLKEN=:l

I

+/-

Compenstation register CLKEN CLK RES

I•,--

Lla I

Start frequency Frequency STFREO redlster

A Modulation

L

55

OATA

J

INPOAT

OUTPORT

Figure 25.60

er:1

..... DI -·

MULT1 SOHWERT WOBMOOE SCHWEIT STRFEO PROGMUX AEFCLKGENEN OETECTEN FAEO_ON MCO_fNV SOFTRES KALSPEEO ACOC FMOCKALEN RESETKAL ACOC FMDCKALEN AESETKAL KALSPEED FASTF FASTEN DELTAIN C>

FASTTF

_I

Divider

2

~

z en

::.Jw w

(.) !I(

a:

d FMDCKAL

DIVPROG

5

MCO_INV C LK I ..--.... RESET

~

- RES

DELTAOUT

Detai led block diagram of the inner workings of the :fractional-N-division synthesizer chip.

EE

CLKO MCO CLKO2

[ Load file: ]

Pll order

d:\CVl\pll\flles\ocxo1 0.syg

INT SYNCON4 RFDIV

Reference source/MHz

ffq:qo" : ]

I

rr1 ~ I

d:\CVl\pU\hles\voo200 leeson.syg

EJ

Phase detector

I

91,1••5 V/cm. The breakdown voltage of the p-i-n diode can be computed by knowing the width of the intrinsic region. When the reverse bias is applied, the depletion region widens and at a certain voltage the depletion region covers the entire I -region. This bias voltage is called the puncbthrough voltage.

p-i-n Diode Equivalent Circuit. A simplified lumped element equivalent circuit of a p-i-n diode is shown in Fig 26.15a for reverse-biased state, and in Fig. 26.15b for forward-biased state. Ls and Cp represent the parasitic inductance and capacitance of the package. Rs is the series resistance of the ohmic contacts, and Cj is the junction capacitance of the diode. R I is the total resistance in the forward-biased condition, and it consists of the intrinsic layer resistance and ohmic contact resistance. 'fypical equivalent circuit values for a packaged p-i-n diode are Ls = 0.3 pf, Rs = 0.2 Q , Ci = 0.02 pf, and R 1 = 1 n. The forward current bias current is about 10 mA, and the breakdown voltage is about 50 Vat a current level of 10 µA [1]. NDR Diodes. Two-terminal devices that exhibit negative differential resistance under certain bias conditions are classified as negative differential resistance (NOR) diodes.

1626

RF COMPONENTS

100

-........

~

10

C: Q)

::,

(.)

-etU ~

1 mA

.E (.)

C

500

600

700

800

900

1000

v,, 25 ° C, millivolts Figure 26.13 1-V characteristics of a p-i-n diode (Ref. 2, with permission from John Wtley & Sons.)

1k

g C:C 100

10

10

100

Forward bias current (mA)

Figure 26.14

Series resistance of a p-i-n diode as a function of device bias.

26.1 ACTTVE DEVICES (a)

p

L5

1627

(b)

i

n

I

I

I

I

p

i

I

n

I

I '

C1

o----'000\~----1fr---vVv----r--o

Figure 26.15 A implified lumped element equivalent circuit of a p-i-n diode: (a) for reverse-biased rate and (b) for forward-biased state.

When the ,·oJtage applied across the e devices is increased, a decrease in current is ob erved, re uJting in a negative differential resistance. These devices can oscillate and produce gain at RF frequencie . Few of the special type of diode that belong to du group are IMPATT (IMPact Avalanche Transit Time) diodes, Gunn diodes, and Tunneling diodes. 1MPATT and Gunn diode produce oscillations and hence generate power by creating a phase delay greater than 90° between the RF current and the voltage in the device through a combination of internal feedback mechani m and transit time effects. Tunneling diodes, which have negative differential resi tance region under certain bias conditions, can generate power at high frequencies. Gunn diodes have produced o cillations up to about 150 GHz, whereas IMPATT diodes and tunneling diodes have produced o cillations up to 400 GHz.

IMPATI' Diodes. The impact-ionization and tran it-time properties of semiconductor devices are utilized in the design of IMPAIT devices. A negative resistance regime is created at RF frequencies by proper choice of the device bias to generate power at high frequencies. A p-n junction diode can be operated as an IMPAIT diode by biasing beyond the reverse avalanche breakdown voltage. Figure 26.16 shows a sketch of p+n-n- IMPAIT diode and its electric field distribution. The doping profile of the diode is tailed in such a way that when it is reverse biased the n-region is depleted of free electrons. When this happens, the electric field at the p+ -n junction exceeds the field required for avalanche breakdown and the field required for maintaining the velocity saturation throughout then-region. The carrier population builds to a higher level with a delay time r0 that is characteristic of the avalanche. At the external terminal, the current is further delayed due to the transit time r:,, during which the ca..rriers are collected by the electrodes. When the total delay of the temrinal current exceeds the terminal voltage by more than 90°, the in-phase component of the current becomes negative and the device exhibits a negative differential conductance characteristic. A device operating in this mode can be made to oscillate and generate power at RF frequencies. The equivalent circuit that represents the diode under this operating condition consists of a negative resistance in series with an inductor, as shown in Fig. 26.17. In addition, this figure also shows IMPAIT diode capacitance, series parasitic resistance, and inductances

1628

RF COMPONENTS V-bias .I

'I

-

n+

n

p+

-

.w

0

I I I I I

I I

«s

z -0 z en ....----, I

C

·o.. 0

-

-0 Q)

z

Distance

(.)

·c::

u Q)

iii Distance

Figure 26.16

A sketch of p+ -n-n+ IMPATT diode and its electric field distribution.

Cd Figure 26.17 regime.

Equivalent circuit of ™PAIT diode is operating in the negative resistance

associated with it. It is important to note that NOR behavior is exhibited only under RF conditions. This means there is a lower frequency below which the diode does not generate the negative resistance and hence cannot generate power. The negative resistance is typically small in the range of -1 Q to -10 Q. The transit time of the IMPA1TT diode is given by the equation (26.48)

26. 1 ACTIVE DEVICES

1629

where W i the total width of the depletion layer and Vd is the drift velocity in the cattering limited condition. The ac impedance of the device can be repre ented a

Z

.

l

= R+JX =G +j B

(26.49)

where R and X are the real and imaginary part . re pectively, of the impedance; G and B are the real and imaginary parts, re pectively, of the admittance. The re onance frequency. fr. i defined a the frequency at which the imaginary part (B) of the admittance change from inductive to capacitive. The cutoff frequency, f c, i defined as the frequency at which the real part (G) of the admittance changes from po itive to negati e. Typically. fr i higher than f c for IMPATT diodes. These devices are capable of producing RF power in the range of few milliwatts to tens of watts up to millimeter-wave frequencie . DC-RF efficiencies are in the range of 10-20%.

Gunn Diodes. The Gunn diode, like the IMPATT diode, has characteristics that can be repre ented by a negative conductance in parallel with capacitance and series resistance. However. the operation of the Gunn diode is based on an entirely different mechanism than that of the IMAPTr diode. In this case, the transferred electron characteristic is a bulk material property of certain compound emiconductors such as GaAs and InP, and not as ociated with p-n or Schottky junction properties. Gunn [7] in 1963 observed microwave power generation when an electric field in excess of a certain critical field was applied acros an n-type GaAs material. The frequency of oscillation was approximately equal to the inverse of the transit time of carriers across the short sample of the material. Later, Kroemer [8] pointed out that the properties of the oscillation were consi tent with the negative differential properties due to the field-induced transfer of conduction band electron from low-energy, high-mobility valley (the r valley) to higher energy. low-mobility atellite valleys (X and L valleys) of the GaAs material. In compound semiconductor materials like GaAs and InP, the r valley conduction band i in clo e energy-momentum proximity to higher order satellite valleys. The effective mass is less or "light'' in the r valley, and higher or "heavy" in X and L valleys. When an electric field is applied across thi type of semiconductor , initially the current flow is due to the electrons in the valley. When the electric field is increased, an increasing number of electrons are transferred to satellite valleys where the electrons have much heavier effective mass. These heavier electrons move slower through the electric field, thus decrea ing the conduction current through the crystal. The net result is that during a certain range of bias aero s the crystal, there is a decrease in the current as the voltage is increased, thereby creating a negative resistance region. A charge dipole domain is formed in the semiconductor, and this domain travels through the device creating a transit time effect. The combination of this transit time effect and the transferred electron effect produce a phase delay between the RF current and voltage. When the phase delay is greater than 90°, the device is unstable and it begins to oscillate. Gunn diodes are used to generate microwave power up to mi11imeter-wave frequencies. They can be manufactured at very low cost in GaAs and InP materials. (Silicon Gunn diodes are not possible, because silicon is not a direct band-gap material.) Gunn diodes have produced output power in the range of few milliwatts to watts. The DC-RF conversion c;fficiencies are typically below 20%.

1630

RF COMPONENTS

Tunnel Diodes. Tunnel diodes exhibit negative differential resistance region in their forward-current voltage characteristics. Tunnel diodes have been used to demonstrate numerous applications and potential market opportunities, including digital-to-analog converters, clock quantizers, shift registers, and ultra low-power SRAMs [9]. All of these applications derive benefit from the inherent bistable behavior of the tunnel diode and utilize the NDR to increase the transition speed between the two stable states. Manes et al. (10] have discussed applications of NDR devices for designing RF circuits, such as VCOs, self-oscillating mixers, and bidirectional amplifiers. The first explanation of the negative differential resistance behavior was presented by Leo Esaki (11] in 1958 by using the quantum tunneling concept. The tunnel diode consists of a p-n junction in which both p- and n-sides are very heavily doped. When the impurity concentration is higher than the effective density of states, the semiconductor is said to be degenerate. In this case, the Fermi level resides within the conduction band or valence bands. In order to realize the tunneling phenomenon, both n- and p-type semiconductors must be degenerate. Energy band diagram of a tunneling diode is shown in Fig. 26.18, and a typical I-V characteristic is shown in Fig. 26.19. The energy band diagram and the corresponding I-V characteristics of the tunnel diode for various bias voltages are shown in Fig. 26.20a through f [4]. In the reverse-biased direction, the current increases as the reverse bias voltage is increased (Fig. 26.20a). The Fermi level on the n-side is higher than the p-side, and the electrons flow from then- to p-side because there are available energy states on the p-side. When the device is in thermal equilibrium, the Fermi levels on the n- and p-side are aligned. There are no filled states above the Fermi level and empty states below the Fermi level on either side of the junction. Hence, there is no current flow through the device (Fig. 26.20b ). When a forward bias voltage is applied, a band of energy levels exists for which there is filled states on the n-side.

- ~ Depletion region

.......__________ Ev Figure 26.18 Energy band diagram of a tunneling diode.

"

26. 1 ACTIVE DEVJCES

1631

(a)

(Peak)

Ip

_,I__ V

Vp

(b)

I Band-to-band tunnel current

.I V

Thermal current

Figure 26.19 Typical I-V characteristics of a tunneling diode.

In addition, corresponding energy states that are available and empty exjt on the pside. This allows tunneling current to flow through the junction. As the forward bias increases, tunneling current keeps on increasing until the Fermi level of the p-side aligns with the bottom edge of the conduction band of the n-side. This point corresponds to

1632

RF COMPONENTS

the maximum current flow through the device (Fig. 26.20c). This value of the current is noted as / P and the corresponding voltage as VP in the 1-V graph (Fig. 26.19a). With further increase in the applied voltage, fewer and fewer unoccupied states are available on the p-side. This continues until the edge of the conduction band is exactly aligned with the top of the valence band, and there are no available states opposite the filled states. Thus, the current decreases as the voltage is increased and reaches a low value, called the valley current Iv, when applied voltage reaches the valley voltage Vv (Figs. 26.19a and 26.20d). When the applied voltage is further increased, no tunneling current can flow because the conduction band on the n-side is well above the valence band of the p-side. The normal diode thermal current flow occurs beyond this point (Fig. 26.20e). The equivalent circuit of a tunnel diode is shown in Fig. 26.21. The circuit consists of a series parasitic resistance Rs, series parasitic resistance L s, the diode capacitance CD, and the negative diode resistance -RD. The series resistance is mainly due to the lead resistance and ohmic contact resistance. The series inductance is due to the parasitic ohmic leads and metal layouts. The intrinsic diode consists of its capacitance CD and negative resistance -RD . The differential dynamic resistance dV / di is minimum

(a)

I

p +I I

'

(b)

(c)

(e}

(d)

In+

I I I I I I I

L--vi

I

Figure 26.20 Energy band diagram and 1-V characteristics at different bias conditions.

Rs

Ls

Figure 26.21 Equivalent circuit of a tunnel diode.

26.1

ACTIVE DEVICES

1633

at the inflection point of the 1-V curve. Its value i

VP lp

Rmm ~ 2 -

(26.50)

The input impedance of the tunnel diode equi valent circuit is

(26.5 1) The cutoff frequency, defined a the frequency when the real part of the input impedance goe to zero. i given by

Fi D

1 f f c0 and f c0 > Jo where Jo is the operating frequency. In order to satisfy these conditions, the series inductance L s should be lowered to very small values. The noise figure of a tunnel diode is defined as (26.56)

where IR/lmin is the minimum value of the negative resistance-current product. The product IRI Imm is called noise constant Kand is a material constant. Tunnel diodes have been demonstrated up to 100s of GHz and primarily limited by the packaging parasitics. RF power available from tunnel diode is somewhat limited because the RF voltage swing is restricted to the forward turn-on characteristics of the

1634

RF COMPONENTS

diode. Typical range of power output is in the range of few hundreds of milliwatts. DC-RF conversion efficiency is about 10%. It has significantly low noise figures. In addition to tunneling diode described so far, there is another class of diodes known. as interband resonant tunneling diodes (HITDs). These diodes have much higher cutoff frequencies and resonant frequencies. These tunnel diodes have been used for designing mixers, amplifiers, and VCOs [12] . Enhancement in semiconductor growth techniques, in particular MBE, has lead to an improvement in device quality and performances, and it has introduced new families of tunneling devices. One of them is a family of the heterojunction resonant interband tunneling diodes (HITDs or RITDs), which make use of resonant interband tunneling through potential barriers. These devices are interesting because of their high peak-to-valley current ratio and the large voltage span of the negative differential resistance region. The main differences between Si-based and III-V-based tunnel diodes lie in the inherent higher peak-valley current ratio (PVCR) obtained in the ill-V semiconductor structures. Even though an absolute maximum ratio of 144 has been reported [13] for III-V-based materials, values in the range of 30-50 are more typical for III-V interband semiconductor structures [14]. On the other hand, silicon devices hardly reach the value of 4 for interband structures and even worse for the intraband TDs. The higher PVCR of III-V tunnel diodes results in higher output power and higher frequencies of operation. The point of strength for Si tunnel diodes is the compatibility with the standard CMOS processes and the adaptability for mixed signal circuits. Applications in this area are expected to emerge in the next few years. As seen in Eq. (26.56), the noise figure is proportional to the factor RI, which is material related. Common values for that parameter are 2.4 for GaAs and 3 for Si [14].

26.1.2 Three-Terminal Devices Three terminal devices have a third terminal to control the flow of electrons from the terminal that generates electrons to the terminal that collects electrons. These types of devices, commonly known as transistors, have the ability to amplify the signal applied at the control terminal. In this section, various types of transistors will be discussed. Several solid-state transistors are being used to develop wireless circuits and include silicon bipolar junction transistors (BJTs), silicon metal oxide semiconductor field effect transistors (MOSFETs), laterally diffused metal-oxide semiconductor (LDMOS) transistors, GaAs metal semiconductor field effect transistors (MESFETs), or simply FETs, both GaAs- and InP-based high electron mobility transistors (HEMTs), and both silicon germanium (SiGe)- and GaAs-based heterojunction bipolar transistors (HBTs). Each device technology has its own merits, and an optimum technology choice for wireless applications depends on not only technical issues, but also on economic issues such as cost, power supply requirements, time to develop a product, time to market a product, existing or new markets, and so on. The purpose of this section is to give an overview of the various device technologies described above and compare their advantages and disadvantages. Different RF/microwave circuits require different transistor parameters. For example, power amplifiers use transistors with higher power densities, low noise amplifiers employ transistors with low noise characteristics, and switches use transistors having low "on-resistance" and small "off-capacitance" features. Various figure-of-merit

26. 1 ACTIVE DEV ICES

TABLE 26.1 Fabrication Property

1635

Compari on of Substrate Materials Used for Monolithic Integrated-Circuit Silicon

SiC

GaA

lnP

GaN

Semi-in uJating 0 Yes Yes Ye Yes 3 - 105 Resi tivity (Q cm) 7 9 7 > 1010 10 > 1010 10 - 10 "'-' 10 Dielectric con tant 11.7 40 12.9 14 8.9 Electron mobility (cm1 / V sec) 1450 500 8500 6000 800 7 7 Saturation electricaJ velocity (cm/ ec) 9 X 106 7 2 X 10 1.3 X 10 J.9 X 10 2.3 X 107 Radiation hardne Poor Excellent Very good Good Excellent Den ity (g/cm3) 2.3 3. 1 5.3 4.8 ThermaJ conductivity (W /cm°C) 1.45 4.3 0.46 0.68 1.3 Operating temperature (° C) 250 > 500 350 300 > 500 Energy gap (e 1. 12 2.86 1.42 1.34 3.39 Breakdown field (kV/cm ) ~ 300 ~ 2000 400 500 ~ 5000

terms are u ed to evaluate and compare transistor characteri tics, including maximum available gain. cutoff frequency (fr), maximum frequency of oscillations (/max), minimum noi e figure (Frrun) , output power density, and power-added efficiency (PAE). Fabrication of any solid-state device starts with the selection of a wafer type or sub trate. Variou ubstrate materials used for active devices are silicon, silicon carbide. GaAs, InP, and GaN. Their electrical and physical properties are compared in Table 26. 1. E xcept Si. all other substrate materials are called compound semiconductors. Silicon dominates the marketplace. GaAs is a distant second with less mature technologies, such as InP, SiC, and GaN only now emerging. The semj-insulating property of the ubstrate material is crucial to providing higher device isolation and lower dielectric los for MMIC . For example, although bipolar silicon devices are capable of operating up to about 10 GHz, the relatively low resistivity of bulk silicon precludes monolithic integration for frequencies above S band (2- 4 GHz). GaAs semi-insulating substrates provide isolation up to about 100 GHz. InP has been used for millimeter-wave HEMTs. Pseudomorpruc HEMTs fabricated on InP substrate exhibit much higher performance in terms of gain, noise figure, and power than does a GaAsbased PHEMT of similar geometry. In this case, the InP substrate supports higher two-dimensional electron gas densities, resulting in high current and transconductance values. The high value of transconductance in InP-based PHEMTs is responsible for ultra low noise figure, high gain, and high frequency of operation. For high-power and high-temperature applications, wide band-gap materials with relatively high thermal conductivity, such as SiC and GaN, play a significant role. Recent advancements in the epitaxial techniques have made it possible to develop active devices on these substrates. Table 26.2 compares various active device technologies [l]. GaAs FETs and HEMTs have the highest frequency of operation, lowest noise figure, excellent switch characteristics, and have high power and PAE capability at lower operating voltages. These salient features of GaAs FETs have enabled the introduction of power amplifiers in portable communication products such as cellular phones. They have high 1/ / noise comer frequency (10 to 100 MHz), which precludes their use in ultra low phase noise oscillators. Due to the semi-insulating property of GaAs substrates, the matching networks and passive components fabricated on GaAs have

1636

RF COMPONENTS

TABLE 26.2

Comparison of Various Semiconductor Device Technologies

Capability

Linearity Noise Figure Power PAE Control Circuits Mixers/Oscillators Passive Components Integration on a single chip Single Polarity Supply Turn-on Voltage Control Multiple Thresholds Low Voltage Operation Maximum Operating Temperature Tj (max) (°C) Wafer Size (mm) Technology Maturity Cost

E-Mode

D-Mode

GaAs HBT

FET

HEMT

FET

HEMT

+ + ++ ++ ++

+ ++ ++ ++ ++

0

0

+ +

++ +

0

0

++ +

+

+

0

+

+

+

+

+

++

+

No

No

Yes

0

0

0

0 0

++ ++ + + +

++ +

0

0

0

0

Yes

Yes

Yes

Yes

Yes

0

++

++ ++

NIA

NIA

+

++

++

Si SiGe Si HBT BIT MOSFET

IA

+ ++

++

++

+

+

+

0

0

++

++

++

++

+

0

0

0

150 100

150 100

125 200

125 200

125 200

+

150 100 0

125 100

+

150 100 0

++

++

Moderate

High

Moderate

High

Moderate- Low Low

Low

+ High

Maximum

+

++

+

++

+

+

0

0

Frequency of Operation

++, Excellent; +, Good; 0, Fair; -

, Poor

lower loss than on Si. The GaAs FET as a single discrete transistor has been widely used in hybrid amplifiers (low noise, broadband, medium power, high power, high efficiency}, mixers, multipliers, switching circuits, and gain control circuits. This wide utilization of GaAs FETs can be attributed to their high frequency of operation and versatility. However, increasing emphasis is being placed on new devices for better performance and higher frequency operation. HEMT and HBT devices offer potential advantages in microwave and millimeter-wave IC applications, arising from the use of heterojunctions to improve charge transport properties (as in HEMTs) or p-n junction injection characteristics (as in HBTs). HEMTs appear to have a performance edge in ultra low-noise, low-loss switches and high-linearity, high-frequency applications. The MMICs produced using novel structures such as pseudomorphic and lattice-matched HEMTs have significantly improved the noise performance and high-frequency (up to 200 GHz) operation. The p-HEMTs that utilize multiple epitaxial ill-V compound

26.1 ACTI E DEVlCES

Source

1637

Drain Ohmic contact

Active layer

\

Gate

n

Buffer layer

n· layer

Semi-insulating GaAs substrate

(b)

Source

Drain

Gate

n-.. GaAs

n~ GaAs

n+ AIGaAs Undoped AIGaAs spacer

··································, Undoped GaAs buffer

'-.. 2 dimension electron gas

Semi-insulating GaAs substrate

GaAs

(c)

p+ GaAs nGaAs fr

implanted region

n+ GaAs

Semi-insulating GaAs substrate

Figure 26.22 A cross-sectional view of the three basic device types: (a) MESFET (b) HEMT, and (c) HBT.

layers have shown excellent milJimeter-wave power performance from Ku- through Wbands. HBTs are vertically oriented heterostructure devices and are gaining popularity as power devices when operated using single power supply. They offer better linearity and lower phase noise than do FETs and HEMTs. A cross-sectional view of the three basic device types (MESFET, HEMT, and HBT) is shown in Fig. 26.22. On the other hand, bipolar transistors require only a single power supply, have low leakage, low l / f

1638

RF COMPONENTS

noise, and are produced much cheaper on Si. The SiGe HBTs have low-cost potential of Si BJTs and electrical performance similar to GaAs HBTs. Thus, discrete silicon BJTs, SiGe HBTs, and MOSFETs have an edge over GaAs FETs, HEMTs, and HBTs in terms of cost at low microwave frequencies. For highly integrated RF front-ends, GaAs FETs and HEMTs are superior to bipolar transistors and Si substrate-based devices due to high-performance multifunction devices and lower capacitive loss, respectively. The electrical performance and cost tradeoffs between Si and GaAs generally favors silicon devices below 2 GHz due to single power supply operation and lower cost, whereas above 3 GHz, GaAs-based devices are preferred due to high performance at low voltage operation.

26.1.3 Si Bipolar Transistor A silicon bipolar transistor is a current-driven device in which the base current modulates the collector current of the transistor. The three configurations of the bipolar transistors are (1) common-base, (2) common-emitter, and (3) common-collector. For power gain applications, common-emitter and common-base configurations are often employed. In a typical common-emitter configuration, the emitter-base junction is forward biased and the collector-base junction is reverse biased using the same polarity power supply. The emitter current IE of a bipolar transistor is given by (26.57) The transconductance gm is defined as g

ale

m

q

IE(in mA )

-- - - -aol E -- - -26- avBE - kT

(26.58)

where l s is VBE is V; is 1c is ao is

the the the the the

surface combination depletion current base-emitter voltage builtin potential co11ector current low-frequency common base current gain

The physical cross section of a single celJ silicon bipolar transistor is shown in Fig. 26.23. When multiple cells are combined in parallel, the most important device dimensions that determine the frequency responses are the emitter pitch and the emitter area. Distributed T-equivalent circuit, shown in Fig. 26.24, has been found to be an effective small-signal model at a fixed bia condition. In thi figure, R 1 , R 2 , ... Rn are the distributed base resistances; C 1 , C2 , ... C11 are the distributed base-to-collector capacitances; Cbp is the base bond pad capacitance; C t!p is the emitter bond pad capacitance; C be is the base-to-emitter junction capacitance; rc is the collector resistance; rbe is the base-to-emitter resistance, re is the emitter resistance; rec is the emitter contact resistance; ie is the emitter current; and a is common base current gain. Hybrid -rr equivalent circuit shown in Fig. 26.25 is also used extensively in RF circuit designs [15]. In this figure, Lb is the base inductance, Le is the emitter inductance, r b is the base resistancet rbe is the base-to-emitter resistance, C cbx is the extrinsic

26. 1 ACTIVE DEVICES

Base

Emitter

Base

p-base

Figure 26.23

Phy ical cro s section of a BIT.

le.., le, lei

I

B

'c

•••

C

Cn

c;

a~

R3 ••

c ep

Figure 26.24 Distributed T-equivalent circuit of a BIT.

Ccbx B

4

ccbl

'i,

vbe

cbe

' be

C

gmVt,e

Le

E Figure 26.25 Hybrid 1r-equivalent circuit of a BIT.

E

1639

1640

RF COMPONENTS

base-to-emitter capacitance, C cbi is the intrinsic base-to-collector capacitance, C be is the base to emitter capacitance, and Vbe is the voltage across that capacitor. The bonding inductances at the base, Lb, and at the emitter, L e, must be included in RF design for accurate prediction of the circuit performance. The figure of merit of a bipolar transistor can be expressed as (26.59) where /max is the maximum frequency, at which the unilateral gain becomes unity; rb is the base resistance; C c is the collector base capacitance; and f r is related to the transit time by the expression 1 (26.60) fr= - 2nrec Here

rec

is the transit time, (the delay time from the emitter to collector), given by (26.61)

where r e is emitter delay time due to excessive carriers, r b is base transit time, re is base-collector capacitance charging time through collector, t eb is emitter-base capacitance charging time, and t he is base-collector charging time through emitter. The frequency fr is also defined as the frequency at which common-emitter current gain reduces to unity.

Low-Frequency Noise Properties of BJT. The I/ /-noise characteri tics of bipolar transistors were extensively discussed by Van der Zeil (16]. An equivalent circuit for analyzing the bipolar transistor low-frequency noise is shown in Figure 26.26. At the input of the device, two noise sources are present: the voltage noise source, en, due to the thermal noise of the resistance, rs, and the current noise source, in; , primarily due to the minority carrier recombination in the emitter region (17) . The dominant contributor to the flicker noise is the current generator. The noi e power generated due to the resistor is often negligible in comparison to the recombination noi e.

t-+-----,-----,--< > V;

t

.-------.Po

son

Figure 26.26 Low-frequency noise equivalent circuit of a BIT.

26.l

ACTIVE DEVICES

1641

An expre sion for the noi e power referred to the input can be written as

(26.62a)

.,

~ e;; (

P0 rs + 50) 2 = 200 G

Pnr = -

(26.62b)

where

e, is 4kTrs6 / which i the thermal noise (a sumed to be zero in the approximate expre ion) en is the equivalent input noi e voltage (as urned to be zero in the approximate expre ion) C is the correlation factor (assumed to be zero in the approximate expression) Po is the noise output power to son load G is low frequency gain As the noise in the device is mostly due to the current transport, the data are usually plotted in dBA/Hz. The ilicon bipolar transistor has very low comer frequency for flicker noise. The low comer frequency of the transistor allows the design of low-phase noise VCOs. High-Frequency Noise Performance of Silicon BJT. The high-frequency noise properties of silicon BIT can be characterized by three noise sources. These dominant noise generators are represented in the noise model of BIT (Fig. 26.27). The base thermal noise, eb, due to the base resistor, rb, shot noise, ee, due to the forward-biased emitterbase junction and the collector partition noise, icp, are the dominant noise sources. The parameter, Zg, is the impedance of the signal source, eg, and re is the emitter resistance. ;; is the forward-biased emitter current, and ii is the current thr-0ugh the load impedance. The noise generators of the BIT can be expressed as [18]

e;= 4kT!llRg

(26 .63a)

= 4kT 6.lrb e; = 2kT 6.lre e~

i!P = 2kT IV ( ao ~,la1 kT re = -

(26.63b) (26 .63c) 2 )

(26.63d) (26.63e)

qle

ao

a=---(1 jf/lb)

+

(26.63f)

where a 0 is low-frequency common-base current gain, Rg is the real part of the impedance, Zg, lb is the base cutoff frequency, and Ill is the bandwidth. The device achieves minimum noise figure when it is operated at the optimum input impedance given by Zopt

= Rop, + iXopt

(26.64)

1642

RF COMPONENTS icp eb

....

rb

B fe ie

cbe

Zg

C

le= a l8

iL

ig

t

i

i

ie

E

Figure 26.27

High-frequency noise equivalent circuit of a BIT.

Following the analysis of Nielsen [19], expressions for the impedance and minimum noise figure can be written as

(26.65a) 2

rb -

Fmm

2

X opt

ao re(2rb + re)

+ lal2

rb + R opt ao = a -re- - + -I ct 12

a

(26.65b) (26 .66)

where (26.67) The equivalent noise resistance, Rn , is given by [20].

(26.68)

There are two limiting cases of operation for the bipolar transistor: The base-limited case when the base time constant, r b, is most dominant, and the emitter-limited case when the emitter time constant, re, is most dominant. The minimum noise figure of the transistor in a base-limited case can be expressed as

(26.69)

26.1 ACTl VE DE ICES

When BJT expre ed a

1643

operated in emitter-limited condition, the mm1mum noi e can be

(26.70) where l e i the emitter cutoff frequency . The minimum noi e figure increase more rapidly in the base-limited ca e than in the emitter-limited case. Al o, notice that Frrun increa e by / 2 above ome com er frequency gi en by l e-

26.1.4

GaAs MESFET

A MESFET con i ts of a highly conducting layer of high-quality n-type em.iconducto r layer placed on a semi-in ulating (i.e., high-re i tivity) ub trate. The conducting channel i interfaced with external circuitry through two ohmic contact (called the ource and drain), eparated by 3 to 4 µm di tance. The gate electrode i constructed by forming a rectifying (Schottky barrier) contact between the two ohmic contacts. The conducting channel depth i on the order of 0.1-0.3 µm , and it i optimized o that the depletion region that forms under the Schottky contact (gate) can efficiently control the flow of current in the active layer. Actual depth depends on the application: thinner for Jow-noi e applications and thicker for power and witching applications. The device. therefore behave as a voltage-controlled current source witch, capable of very high modulation rates. A phy ic of device operation and the device de ign of MESFETs are discus ed elsewhere in thi book, only a brief description wiU be included in this chapter.

MESFET Small-Signal Equivalent Circuit. The small- ignal equivalent circuit i useful for circuit designs at a lower power level. Figure 26.28 shows the mal1-signal equivalent circuit and the location of the circuit element in the FET tructure. The various components in the model are defined in the followi ng:

Intrinsic Elements R; Input (channel) resi tance C gs Gate-to-source capacitance Cgd Gate-drain feedback capacitance Rds Drain-source resistance gm Transconductance r

Phase delay due to carrier transit in channel

Extrinsic Elements Cds Drain-source capacitance Rd Drain-to-channel re istance, including contact resistance Rs Source-to-channel resistance, including contact resistance R8 Gate-metal resistance Parasitic inductances Ld, Ls, and L 8 could be added in series with Rd, Rs, and R8 in the equivalent circuit to account for the effects of device pads and bonding wires in the

1644

RF COM PONENTS

(a)

Lg

Rg

Ld

Rd

Cgd

D

G

Cgs gme-Jwr

+

Rds

Cds

R;

s (b)

Drain

Figure 26.28 Small-signal equivalent circuit of a MESFET showing the location of the circuit elements.

case of discrete devices. The values of intrinsic elements depend on the channel doping, channel type, material, and dimensions. The large values of the extrinsic resistances will seriously decrease power gain and efficiency and increase the noise figure of the device.

Figure of Merit. High-frequency operation of an PET requires high gain. The maximum available gain (MAG) is given by

MAG=

('T)2 f

1 4R/ Rds + 41rfrC8d(R + Rg + 1rfrLs)

(26.71)

26.1 ACTIVE DEVICES

1645

where

R

= Rg + R; + Rs +rrfr L s and

fr=

gm 2rrCgs

fr i known as the cutoff frequency at which the device has unity current gain. The maximum frequency of operation i given by

(26.72) Thi equation illu trate the importance of Rg + R; + Rs, C gd and L s, in addition to fr , to extend the operation of FET: to higher frequencies. The output power depends on the maximum voltage and current swings for variou value of gate length and maximum-channel current that the device can provide before the breakdown. Shorter gate Jength implies higher channel doping due to thinner channel thickne becau e gate length/channel thickness ratio is almost constant. The maximum-current wing limit i about 10-20% (depending on the pinch-off voltage) greater than the drain-source saturation current / dss for power devices. This occurs when the gate- ource bias is positive and lower than the builtin voltage. The maximum current will be attained when the difference between the voltage drop across the thickest part of the depletion layer and Vgs reaches zero, i.e., (26.73) where Es is the saturation electric field (3.5 kV /cm) and ls is the distance carriers travel at the saturation velocity, which is approximately equal to gate length. The minimumcurrent swing level is zero, because negative current is impossible for positive drain bias with respect to source and gate. The maximum voltage swing between the gate and the drain is limited by avalanche breakdown, due to intense local-field strength at the drain edge of the gate metal.

26.1.5 Heterojunction Field-Effect Transistor The performance improvement of heterojunction field-effect transistor (HFET) over the GaAs MESFET can be understood by comparing the device material structures (see Figs. 26.22a and b) . In HFET devices, different materials are grown one on top of the other using epitaxial growth technology, like molecular beam epitaxy (MBE) or molecular organic chemical vapor deposition (MOCVD). HFET is also called modulated doped FET (MODFET). There are different variations of HFETs. One of the most commonly used versions is called a high-electron mobility transistor (HEMT). This device structure, shown in Fig. 26.29a has a layer of aluminum gallium arsenide (AlGaAs), which has a larger energy-band gap than GaAs, grown on top of a GaAs layer. The difference in the Fermi energy band between two materials would cause band bending at the heterojunction interface (Fig. 26.29b). This band bending results in a quantum well, where a large population of electrons forms a two-dimension gas very close to the interface of the two materials. The AlGaAs layer is doped to provide electrons for the current conduction, and the GaAs layer is undoped. The electrons from the donor atoms in the AJGaAs move to the low-energy level on the undoped GaAs. This effectively separates the

1646 (a)

RF COMPONENTS

n+ GaAs contact layer 500 A n+ AIGaAs donor layer 500

(b)

A

Undoped AIGaAs spacer layer 50 A

---------------- - - 7 -- - - 2-DEG electron gas Undoped GaAs buffer "" 1 µm

GaAs

GaAs

___

....._

Semi-insulating GaAs substrate

Figure 26.29

___ __ _

(a) HEMT device structure and (b) HEMT band diagram.

donors residing in the AlGaAs from the electrons residing in the GaAs layer. As donors and free electrons are in two different media, chances of collision between the donors and electrons are minimized and the drift velocity of the electrons is increased. These electrons can be very easily modulated by the application of a voltage at the gate terminal. The HEMT devices exhibit higher transconductance and lower noise-figure properties and hlgher frequency operation compared to GaAs MESFETs. Another variation of HFET is called pseudomorphic HEMT or p-HEMT. The device structure and band diagram of p-HEMT is shown in Fig. 26.30a and b, respectively. The p-HEMT device has superior microwave properties than does the HEMT. This device is rapidly replacing the GaAs HEMT device in many applications. In this structure, a thin layer of indium gallium arsenide (InGaAs) is introduced between the undoped GaAs and the doped AlGaAs layers. The InGaAs material ha energy band gap lower than the AlGaAs and GaAs. When this layer is sandwiched between the two layers of higher band-gap (lnGaAs and GaAs) materials, the lowest energy quantumwell states would reside in the lnGaAs layer. Therefore, the free electrons provided by the donor atoms would move to the energy levels within the thin layer of InGaAs. These electrons are confined to this layer because of the presence of higher energy

(a)

d · GaAs contact layer 500 A n➔ AIGaAs donor layer 500

{b)

A

Undoped AIGaAs spacer layer 50 A Undoped lnGaAs channel l!}yer 200 A 2-DEG electron gas / I

I

Undoped GaAs buffer ,., 1 µm

I I

Ee

I

GaAs Semi-Insulating GaAs substrate

I

lnGaAs •I GaAs

----- -- - - --- - - - -- -- -EF Figure 26.30

(a) p-HEMT device structure and (b) p-HEMT band diagram.

26.1 ACTIVE DEVICES

1647

band-gap material on both side . The e electron have much higher drift velocity and can be modulated by the application of a voltage at the gate tenninal. These p-HEMT devices exhibit even higher tran conductance and superior RF properties than do GaAs HEMT and GaAs MESFET. The mall- ignal equivalent circuit of the HEMT and p-HEMT are very similar to the GaAs MESFET. The circuit designs often u e the GaA MESFET equivalent-circuit configuration with proper value for the parameter . Low-Frequency Noise Properties of MESFETs and HEMTs. The origin of the 1//noi e of field-effect transi tor i attributed to the existence of multiple deep-level trap that gives rise to noise with 1/ f spectral distribution (21]. Another source of noise that ha a l / f re ponse was suggested, and theoretically established, by Handel [22]. According to Handel' theory, the current carried by a beam of electrons, when scattered from an arbitrary potential barrier, would exhibit fluctuations that have l / / noi e characteristics. Based on Handel's theory, Hooge [23] developed the following expre sions for the drain noise current of heterostructure material :

(26.74) where ll.f is the bandwidth, N is the total number of carries, and aH is the Hooge parameter (aH = 7.1 x 10- 6 for GaAs at 300 K). For a short gate-length device, the total number of electrons in the channel when the device is operated in a currentsaturated condition can be estimated as (26.75)

1;

In this, v011 is the average electron velocity and is the effective channel length. By combining Eqs. (26.74) and (26.75), we can obtain the expression for the drain- noise current as (26.76) The above expression has been experimentally validated, and the value of aH in the range of 3 - 4.5 x 10- 5 has been obtained for a conventional HEMT structure with ranging from 0.2 to 11 µm [24]. effective gate length, The drain generation-recombination noise current generator, ig-r, consists of many components due to multiple deep-level traps located between the gate and 2-DEG channel of a HEMT device. This noise can be represented as an equivalent gate noisevoltage generator, Vng. As the frequency increases, the 1/ f noise and the generationrecombination noise spectral density decrease and eventually fall below the thermal noise associated with the FET channel. Figure 26.31 shows the equivalent circuit that describes the low-frequency noise behavior of a HEMT device. It consists of a noiseless equivalent circuit of a FET and the noise sources at the drain and the gate terminals. The noise currents at the drain are ig-r due to the generation recombination, i 1/ / due to the 1/ f noise, and i,. due to the thermal noise. The noise current, i8 , at the gate is due to gate thermal noise and

1;,

1648

RF COMPONENTS

1------.---......------r--o D

Noise free device

t

t

s Figure 26.31

Low-frequency noise equivalent circuit of a HEMT.

channel noise. e8 n is the thermal noise due to the gate resistance, Rg, and esn represents the thermal noise due to the source resistance, Rs.

High-Frequency Noise Properties of MESFETs and HEMTs. When a MESFET, HEMT, or p- HEMT is used as an amplifying device, it exhibits superior noise-figure and power-gain performance. Noise generators for a FET can represented by the following expressions [25, 26]: (26.77) (26.78) where P and R are numerical factors associated with the drain- and gate-thermal noise generators. In addition, there are two more noise-voltage generators: one due to the gate resistance, R 8 , and the other due to source resi tance, Rs. These noise sources can be expressed as V~s

= 4kT l:lf Rs

(26.79)

v~8

= 4kT l:lf R8

(26.80)

Pucel et al. (27] introduced the concept of carrier diffusion noise in 1975. They were able to predict the noise performance of MESFETs more accurately using this approach. The high-frequency noise-equivalent circuit of a FET can be represented as shown in Fig. 26.32. Following the analysis of Puce] et al., it can be shown that the minimum noise figure can be expressed as

(26.81)

26.1

ACTIVE DE ICES

1649

i---- - ----- - ------ - -------, io ~

I

G

D

Zg + "----- -

--- ------- ---- -- --

s Figure 26.32

High-frequency noi e equivalent circuit of a FET.

where Cu = C8 s + C gsl , C gs l i the parasitic capacitance from gate to ource and C gs i the intrin ic gate capacitance. Fukui (28] measured the MESFET de and mall- ignal parameters and derived an empirical relation hip for the MESFET noise parameters

Fmu,

=1

+(fr)

k1Jcm(R,

k2

+ Rg)

(26.82) (26.83a)

Rn = -

Km

1 Ropt = k3 ( - 4gm

+ Rs + Rg)

(26.83b) (26.83c)

where k 1 k2 , k3 , k4 are empirical fitting factor and f is the operating frequency. By comparing Eq. 26.81 with Eq. 26.82, the Fukui factor, k1, can be written as (26.84)

As (26.85) the fukuj equations become (26.86)

This Fukuis expression is extensively used to predict noise-figure performance of FETs.

1650

RF COMPONENTS

An expression for k 1 , as a function of the bias current, as follows:

I ds ,

1s given by [29] (26.87)

where I ds is the drain-source current (mA) at which the noise perlonnance is measured, gm is the transconductance in mS , and L is the gate length measured in µm . The above equation, along with Eq. 26.86, calculates Frrun with reasonably good accuracy for several FET types, including MES FETs and HEMTs [29] . Indium Phosphide (lnP) HEMTs. Indium phosphide technology developed more slowly than GaAs technology due to the difficulty in the growth of the material. InP substrate has a lattice constant that is very close to that of alloy composed of 50% GaAs and 50% InAs. Ga o.47 Ino_s3As has an energy band gap of 0.77 eV, Afo.4slno.s2As material has a band gap of 1.48 eV, and InP has a band gap of 1.358 eV. A cross section of an InP HEMT structure is shown in Fig. 26.33. The high band-gap material AllnAs is used as the donor layer, and the low band-gap material is the channel layer. The free electrons supplied by the donor layer reside in the low band-gap channel layer. The GalnAs/AllnAs/InP system offers many advantages over GaAs HEMT structures. The high-conduction band discontinuity of the structure allows hlgh, two-dimensional electron gas concentration. The high mobility of electrons in GalnAs, coupled with the high density of electrons in the channel, leads to higher conductivity in the active channel. Because of these superior material characteristic , InP-based HEMTs exhibit very high transconductance, very low noise figure, and hlgh gain compared to conventional HEMTs fabricated on GaAs material. All of the noise sources that contribute to the device noise figure are lower in GalnAs/AllnAs HEMTs than in

Source

lnGaAs cap layer

Gate

\

'

Drain

, lnAIAs schottk.y

I

.

(

•••••••••••••••••••••••••••••••••••••••••••••••••••••••••• lnAIAs spacer lnGaAs channel lnAIAs buffer

lnP substrate

Figure 26.33

InP HEMT device structure.

Si 8 dopping

26.1 ACTIVE DEYLCES

1651

3r-----------------------0.15 µm GaAs PHEMT

co

-

2

X

"O

/

x

0.15 µm lnP HEMT

0

-~ 1

o ~ ~ m lnP

0

z

o

- x -

0

0 '-----:---__.__ __ 10

HEMT

20

, 1 __

____1___

40

__.J____,L__j_L.....J-L_ _ __

60

80

100

___J

200

Frequency (GHz)

Figure 26.34

oi e figure of InP HEMT device.

GaAs HEMTs. The extremely high conductivity of the two-dimensional electron gas lowers the ource re istance and associated thermal noise. Due to lower inter-valley electron cran fer probability the velocity of electrons in the channel is higher, leading to higher fr of the device. A capacitive coupling of the channel to the gate electrode is lower, coupling of noise sources to th~ gate is minimized. Figure 26.34 shows the noi e figure of HEMTs as a function of frequency [30]. The superior performance of InP based HEMT at millimeter-wave frequencies is very evident from the figure.

26.1.6 Heterojunction Bipolar Transistors Shockley first proposed the concept of heterojunction bipolar transistor in 1948. Kroemer [31] developed the theoretical analysis of this device. The advancement of MBE and MOCVD epitaxial techniques enabled the fabrication of AlGaAs/GaAs heterostructure HBTs. Most of the initial developments of HBTs have been using the AlGaAs/GaAs material tructure. HBTs of excellent performances have also been demonstrated on IoP substrate . A cross section of an HBT is shown in Fig. 26.22c. The doping concentrations on different regions of the device and the band diagram of an HBT are shown in Fig. 26.35a and b, respectively. When the emitter-base junction is forward biased, carriers are injected from the n-type emitter to the p+ base region. These injected electrons are swept across the base region by drift and diffusion process, and ultimately collected by the reverse-biased collector-base junction. The electrons are minority carries in the p+ base region. Due to the short lifetime of electrons, they recombine with the majority carrier holes in the base region, resulting in a current through the base terminal. The transport of electrons across the collector-base space-charge region takes place due to the high-field electron saturation velocity. The primary advantage of HBT is its high-emitter efficiency. The forward-biased emitter injection efficiency is very high for HBT because wider band gap material (AlGaAs) emitter injects electrons into the GaAs base at a lower energy level. At the same time, the holes are prevented from flowing into the emitter by the energy barrier. The base can be doped heavily to reduce -the base resistance, and the implant damage can be used to reduce the parasitic collector-base capacitance.

1652

RF COMPONENTS

(a) Base : region

Emitter region

Collector region

1

1

► I

I C

- - -- - --ri'

.... ' fl---- - - - -~

n+ cap II n+ layer I

I I I I I

I

I I

AIGcfAs:

I

I I

n-

I

' GaAs I I I

GaAs

GaAs

I I

(b)

I

I I I I I

n-AIGaAs Conduction band

EF

Emitter

+

P

I I

: GaAs : 1 base 1I I

___ , I

n

I

:GaAs: n+ GaAs

- ~--.... -I

....

~

I

I

Valence band

Figure 26.35 (a) The doping profile of emitter, base, and collector regions of HBT and (b) energy band diagram of HBT.

The most important performance parameters in HBT design are the de current gain, /J, and emitter-base doping concentrations ratio. These parameters can be expressed as [32] (26.88) (26.89)

26. 1 ACTNE DEVICES

1653

where

le h

1

Ip

I,

1

ls

In f3mtLr

1

Nl' / Pb

l

6Eg

J

Vnb / Vpe

l

the collector current the base current the re er e hole injection current the bulk recombination current lhe urface depletion region recombination current the injected electron current the maximum vaJue of /3 in the ab ence of recombination current the emitter-ha e doping concentration ratio the e nergy band gap ratio between the emitter and base material the ratio of the electron and hole elocitie

The high /3max allow a bipolar tran i tor de ign with low emitter doping to achieve a low emitter capacitance and high base doping to achieve low ba e re istance. These two parameters are critical in reducing the RC time constant, which, in tum, leads to improved peed . In ilicon bipolar tran i tor , peed and linearity are relatively ]ow due to relatively low base doping that can be achieved while maintaining the good alue of /3. Compared to ilicon bipolar transistors, HBT has higher gain-bandwidth product with relaxed geometries due to the higher mobility in GaAs/InP and reduced paraitic . Higher common-emitter output impedance re ulting from the high base doping permitted by the heterojunction, minimizes the base-width modulation. This leads to higher linearity and lower harmonic distortion in HBTs. The semi-insulating sub trate of GaAs or InP HBTs allows fabrication of miniature MMIC circuits, with better RF performance due to lower dielectric substrate loss compared to silicon-based technologie .

MESFET: and HEMT: are majority carrier devices with lateral-current conduction, whereas HBT i a vertical device that allows the electron and hole-current conduction. The speed of the HBT device is determined by the transit time through the thin-vertical base-collector layer . The maximum speed of the FET is determined by a transit time and i controlled by the gate length, defined by the lithographic techniques. The HBTs have higher tran conductance than FET due to the exponential output current to input voltage variation. Other advantages of HBT are the high-output current per device-unit gate width high current g~ and lower 1/ /-noise properties. The power-handling capability of this device is much higher because the entire emitter area can carry the current, as a resu]t of the lower emitter resistance. The high-output power-handling capability along with the single power supply operation have made thi the most suitable candidate for power amplifiers used in handheld portable communication.

Low-Frequency Noise Properties of HBT. The 1/ / noise of HBTs is much lower than that of GaAs MESFETs because the surface states of GaAs no longer contribute significant noise to the emitter current. A comer frequency below I MHz has been observed for the HBT, which is comparable to silicon bipolar devices. This enables the design of monolithic VCOs in GaAs with low phase noise characteristics. The three noise sources that contribute to the low-frequency noise in an HBT are generation-recombination noise in the emitter-base space charge region, the surface recombination related noise at the base surface near the emitter-base junction, and the

1654

RF COMPONENTS

quantum 1/ f noi e of the emitter-base junction. All of these noises are associated with the hole current in the base region. The quantum 1/ / noise generation is due to the fluctuation in the current produced by the beam of electrons that are scattered by an arbitrary potential. This quantum noise is also known as diffusion noise. All of these noise contributions can be represented as noise-current generators in conjunction with a noiseless transistor. Figure 26.36 shows the noise equivalent circttit of an HBT. The noise-current generator associated with I / f diffusion current can be expressed as (26.90) where Po is the characteristic parameter representing the background 1/ f noise and g;r i the small-signal base-emitter conductance. An expression for the composite current generator due to g - r noise, surface recombination, and bulk recombination noise can be written as n -:-2 ~ Pr Tr 2 (26.91a) ' g-r = 4kT 6.f ~ ( 2 2 -grr r=l 1 + 100 (Pulse Tube and Stirling) TRW, Raytheon, Lockheed-Martin, Leybold, Aisin, Hymatic, Oxford, BAE, AIM, Ricor, DRS, Aisin, LGE, Stirling Technology Company

"'15 years

Not Available (Not enough units)

""'5 + years

Low

Moderate

High

Moderate

High

High

Low

Moderate

Moderate

1-5 years (adsorber replacement)

None Until Failure

None Until Failure

5°C to 40°C

5°C to 40°C

5°C to 40°C

Air or water cooled

Air or water cooled

Air or water cooled

5°C to 40°C

- 40°C to 60QC 18

- 40° C to 60° C

Air cooled

Air cooled Moderate

Air cooled Moderate

Manufacturers

Reliability Expected Lifetime (80% Confidence Level) Susceptibility to Wear Susceptibility to Contamination Susceptibility to Gas Leakage Maintenance Interval

Indoor Installation Temperature Range Cooling Outdoor Installation Temperature

> 1000 Sunpower, Superconductor Technologies, Helix

Range Cooling

EMI Noise

Low 63- 67 dBA@ 4 ft

40- 50 dBA@ 4 ft

50-56 dBA@ 4 ft

Vibration

Moderate

Moderate Negligible

Moderate (Uncompensated) NegligibJe

(Compensated) Compact ( < .50 ft3)

(Compensated) Compact ( < .50 ft3 )

(Uncompensated)

Envelope (with drive electronics)

Moderate ( Cl)

-300

rJl

co .c a.

-400

Frequency (GHz)

Figure 28.25 Micromacbined liquid crystal (LC) phase shifter [94]. Copyright © 2002 IEEE. Reprinted with permission.

1.9 pF, which provides a tuning range of 3.4% for the VCO. The VCO provides a phase noise of -122 dBc/Hz at 1 MHz from the carrier.

28.4.4 Reconfigurable Antennas Reconfigurable antennas, sometimes referred to in the literature as smart antennas, are an evolving technology that is unique to every application (98-101]. MEMS technology has an enabling impact in this area jn that actuators may be exploited to effect structure reconfiguration while maintaining high performance. The simplest application of MEMS to antennas is using the phase-shifters previously described to implement high-performance phased-array antenna . An intriguing concept published on thi subject i the Vee-antenna (Fig. 28.28). Micromachined electrostatic actuators are used to alter the polarization sensitivity of the antenna by changing the angular separation of the antenna elements forming the Vee. This can a]so be used for direct beam steering. This unique application can be extended one step further to include micromechanical impedance matching as shown in Fig. 28.29. A phase-shifter is electro tatically moved along the transmission line to change the circuit impedance as the Vee-antenna is reconfigured. Similarly, shunt switches can be u ed to switch antenna polarizations a shown in Fig. 28.30. This concept uses microfabricated switches to add or remove elements from an antenna. Switching can also be used to create high-perfonnance agile antenna systems through

Resonator on membrane

0 r-r-T'""I"-.--.-,--,-,.-,-,.._.....-........-,-,.....,....,....,..-,.-,.....,.....,......-.---.-,......,..~-.--~-

- 20

e co

f0 = 28.6536 GHz P0 u1 ::,: - 3.5 dBm (at specimen analyzer) Resolution bandwidth = 3 kHz

:!:!. -40 Power out

l

S -60

~

0

- 80

- 100 28.632

28.634

28.636

28.638

28.640

28.642

28.644

28.646

Frequency (GHz) -40r---,--,--,.-,-.....,.........,---,---r-----.----,-.......-.........--,,--,---.-..-.-....--.-~

t

port

I

Q)

' ... ...

J:

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~ !g -80

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