Variational principles and distributed circuits [1st ed.] 9780863802560, 0863802567

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Variational Principles and Distributed Circuits

OPTOELECTRONICS AND MICROWAVES SERIES Series Editor:

Professor T Benson and Professor P Sewell The University of Nottingham, UK

1.

Variational Principles and Distributed Circuits A. Vander Vorst and I. Huynen

2.

Propagating Beam Analysis of Optical Waveguides J. Yamauchi *

* forthcoming

Variational Principles and Distributed Circuits A. Vander Vorst Université catholique de Louvain and I. Huynen National Fund for Scientific Research, Université catholique de Louvain

RESEARCH STUDIES PRESS LTD. Baldock, Hertfordshire, England

RESEARCH STUDIES PRESS LTD. 16 Coach House Cloisters, 10 Hitchin Street, Baldock, Hertfordshire, England, SG7 6AE and 325 Chestnut Street, Philadelphia, PA 19106, USA

Copyright © 2002, by Research Studies Press Ltd. All rights reserved. No part of this book may be reproduced by any means, nor transmitted, nor translated into a machine language without the written permission of the publisher.

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Library of Congress Cataloguing-in-Publication Data Vorst, A. vander (André), 1935Variational principles and distributed circuits / A. Vander Vorst and I. Huynen. p. cm. – (Optoelectronic and microwaves series) Includes bibliographical references and index. ISBN 0-86380-256-7 1 Microwave transmission lines—Mathematical models. 2. Electromagnetism—Mathematics. 3. Variational principles. I. Huynen, I. (Isabelle), 1965-II. Title. III. Series. TK7876 .V6324 2000 621.381’31—dc21

British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library.

ISBN 0 86380 256 7 Printed in Great Britain by SRP Ltd., Exeter

00-039022

Editorial Foreword Variational prin iples have a long history of providing insight and pra ti al solutions to problems in a wide range of s ienti and engineering dis iplines. In parti ular, their substantial ontribution toward the development of high frequen y te hnologies during and after World War II, by underpinning the equivalent network approa h for the design and analysis of mi rowave ir uits, is learly re ognised. At a time before modern omputers, variational prin iples provided solutions to immediate engineering problems with unparalleled a

ura y. However, even in the modern era, as the power of omputers has in reased at an astonishing pa e, variational methods have evolved and remain an invaluably elegant and powerful tool for resear hers in a variety of elds. Therefore, we are very pleased to wel ome this book, whi h provides both a

omprehensive and in-depth treatment of all aspe ts of variational prin iples applied to mi rowave, millimetre-wave and opti al ir uits, to this series. Starting with a thorough review of the eld, the book learly establishes the form and physi al signi an e of variational prin iples as well as their relationship with other approa hes. A detailed dis ussion of the appli ation of variational prin iples to planar mi rowave geometries is followed by the establishment of a general variational prin iple for ele tromagneti s and its appli ation to a wide variety of pra ti ally important on gurations, in luding transmission lines, avities, jun tion problems as well as gyrotropi devi es. Throughout, the mathemati al basis and pra ti al implementation of the variational approa hes is learly and elegantly presented and evidently bene ts from the authors' extensive knowledge and original ontributions to the eld. Therefore, this book is likely to be of interest both to experien ed resear hers in the eld as well as those wishing to explore the stimulating and pra ti ally valuable topi of variational prin iples for the rst time.

TM Benson P Sewell University of Nottingham July, 2001

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Contents Editorial Foreword

v

Prefa e

xiii

1 Fundamentals

1

1.1

Histori al ba kground

. . . . . . . . . . . . . . . . . . . . . . .

1.2

1

Extremum prin iple

. . . . . . . . . . . . . . . . . . . . . . . .

3

1.3

Variational prin iple

. . . . . . . . . . . . . . . . . . . . . . . .

5

1.4

Mi rowave network theory . . . . . . . . . . . . . . . . . . . . .

6

1.5

Variational expressions for waveguides and avities

. . . . . . .

11

1.6

New theoreti al developments . . . . . . . . . . . . . . . . . . .

12

1.7

Con lusions

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

16

1.8

Referen es . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

16

2 Variational prin iples in ele tromagneti s 2.1

2.2

2.3

Basi variational quantities

19

. . . . . . . . . . . . . . . . . . . .

19

. . . . . . . . . . . . . . . . . . . . . . . .

19

2.1.1

Introdu tion

2.1.2

Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . .

20

2.1.3

Eigenvalues

22

2.1.4

Iterating for improving the trial fun tion . . . . . . . . .

23

Methods for establishing variational prin iples . . . . . . . . . .

26

. . . . . . . . . . . . . . . . . . . . . . . . .

2.2.1

Adequate rearrangement of Maxwell's equations

. . . .

26

2.2.2

Expli it versus impli it expressions . . . . . . . . . . . .

30

2.2.3

Fields

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

Comparison with other methods 2.3.1

2.3.2

31

. . . . . . . . . . . . . . . . .

31

Rea tion . . . . . . . . . . . . . . . . . . . . . . . . . . .

31

2.3.1.1

Variational behavior of rea tion

34

2.3.1.2

Self-rea tion and impli it variational prin iples

Method of moments

. . . . . . . .

. . . . . . . . . . . . . . . . . . . .

2.3.2.1

Variational hara ter of error made

2.3.2.2

Variational fun tionals asso iated with the MoM

. . . . . .

2.3.2.3

The MoM and the rea tion on ept

2.3.2.4

MoM and perturbation

34 35 36 37

. . . . . .

42

. . . . . . . . . . . . .

44

CONTENTS

viii

2.4 2.5

2.6 2.7

2.3.3 Variational prin iples for solving perturbation problems 2.3.4 Mode-mat hing te hnique . . . . . . . . . . . . . . . . . Trial eld distribution . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Rayleigh-Ritz pro edure . . . . . . . . . . . . . . . . . . 2.4.2 A

ura y on elds . . . . . . . . . . . . . . . . . . . . . Variational prin iples for ir uit parameters . . . . . . . . . . . 2.5.1 Based on ele tromagneti energy . . . . . . . . . . . . . 2.5.2 Based on eigenvalue approa h . . . . . . . . . . . . . . . 2.5.3 Based on rea tion . . . . . . . . . . . . . . . . . . . . . 2.5.3.1 Input impedan e of an antenna . . . . . . . . . 2.5.3.2 Stationary formula for s attering . . . . . . . . 2.5.3.3 Stationary formula for transmission through apertures . . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Referen es . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45 46 49 49 50 51 51 52 56 56 58 59 61 61

3 Analysis of planar transmission lines 65 3.1 Topologies used in MICs . . . . . . . . . . . . . . . . . . . . . . 65 3.2 Quasi-stati methods . . . . . . . . . . . . . . . . . . . . . . . 67 3.2.1 Modi ed onformal mapping . . . . . . . . . . . . . . . 69 3.2.2 Planar waveguide model . . . . . . . . . . . . . . . . . . 70 3.2.3 Finite-dieren e method for solving Lapla e's equation 71 3.2.4 Integral formulations using Green's fun tions . . . . . . 74 3.3 Full-wave dynami methods . . . . . . . . . . . . . . . . . . . . 77 3.3.1 Formulation of integral equation for boundary onditions 78 3.3.2 Solution of the integral equation in spa e domain . . . . 82 3.3.3 Moment method . . . . . . . . . . . . . . . . . . . . . . 85 3.3.4 Spe tral domain Galerkin's method . . . . . . . . . . . . 89 3.3.5 Approximate modeling for slot-lines: equivalent magneti urrent . . . . . . . . . . . . . . . . . . . . . . . . 92 3.3.6 Modeling slot-lines: transverse resonan e method . . . 92 3.3.7 Transmission Line Matrix (TLM) method . . . . . . . . 92 3.4 Analyti al variational formulations . . . . . . . . . . . . . . . . 94 3.4.1 Expli it variational form for a quasi-stati analysis . . . 95 3.4.1.1 Quasi-TEM apa itan e based on Green's formalism . . . . . . . . . . . . . . . . . . . . . . 95 3.4.1.2 Quasi-TEM indu tan e based on Green's formalism . . . . . . . . . . . . . . . . . . . . . . 99 3.4.1.3 Capa itan e based on energy . . . . . . . . . . 106 3.4.1.4 Indu tan e based on energy . . . . . . . . . . . 110 3.4.1.5 Generalization to non-isotropi media . . . . . 113 3.4.1.6 Link with eigenvalue formalism . . . . . . . . . 117 3.4.2 Impli it variational methods for a full-wave analysis . . 118 3.4.2.1 Galerkin's determinantal equation . . . . . . . 118

CONTENTS

ix 3.4.2.2 3.4.2.3 3.4.2.4

Equivalent approa h based on rea tion on ept 121 Conditions for the Green's fun tion . . . . . . 123 Link with expli it variational eigenvalue problems . . . . . . . . . . . . . . . . . . . . . . . . 125 3.4.3 Variational methods for obtaining a

urate elds . . . . 127 3.5 Dis retized formulations . . . . . . . . . . . . . . . . . . . . . . 129 3.5.1 Finite-element method (FEM) . . . . . . . . . . . . . . 129 3.5.2 Finite-dieren e method (FDM) . . . . . . . . . . . . . 133 3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 3.7 Referen es . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 4 General variational prin iple 143 4.1 Generalized topology . . . . . . . . . . . . . . . . . . . . . . . . 143 4.2 Derivation in the spatial domain . . . . . . . . . . . . . . . . . 145 4.2.1 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . 146 4.2.2 Proof of variational behavior . . . . . . . . . . . . . . . 148 4.2.3 Spe i ities of the general variational prin iple . . . . . 150 4.3 Derivation in the spe tral domain . . . . . . . . . . . . . . . . . 151 4.3.1 De nition of spe tral domain and spe tral elds . . . . 152 4.3.2 Formulation of variational prin iple in spe tral domain . 152 4.4 Methodology for the hoi e of trial elds . . . . . . . . . . . . . 154 4.4.1 Constraints and degrees of freedom . . . . . . . . . . . . 154 4.4.2 Choi e of integration domain . . . . . . . . . . . . . . . 154 4.4.3 Spatial domain: propagation in YIG- lm of nite width 155 4.4.4 Trial elds in terms of Hertzian potentials . . . . . . . . 163 4.4.5 Trial elds in terms of ele tri eld omponents . . . . . 172 4.4.6 Link with the Green's formalism . . . . . . . . . . . . . 175 4.5 Trial elds at dis ontinuous ondu tive interfa es . . . . . . . . 177 4.5.1 Conformal mapping . . . . . . . . . . . . . . . . . . . . 177 4.5.2 Adequate trial omponents for slot-like stru tures . . . 178 4.5.3 Adequate trial omponents for strip-like stru tures . . . 183 4.5.4 Rayleigh-Ritz pro edure for general variational formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 4.5.5 EÆ ien y of Mathieu fun tions as trial omponents . . . 190 4.5.6 Ratio between longitudinal and transverse omponents . 191 4.6 Resulting advantages of variational behavior . . . . . . . . . . 192 4.6.1 Error due to approximation made on in trial elds . . 194 4.6.2 In uen e of shape of trial elds at interfa es . . . . . . . 198 4.6.3 Trun ation error of the integration over spe tral domain 200 4.7 Chara terization of the al ulation eort . . . . . . . . . . . . . 204 4.7.1 Comparison with Spe tral Galerkin's pro edure . . . . . 204 4.7.2 Comparison with nite-elements solvers . . . . . . . . . 208 4.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 4.9 Referen es . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

x

CONTENTS

5 Appli ations 215 5.1 Transmission lines . . . . . . . . . . . . . . . . . . . . . . . . . 215 5.1.1 Transmission line parameters . . . . . . . . . . . . . . . 215 5.1.1.1 Propagation onstant . . . . . . . . . . . . . . 216 5.1.1.2 Power ow . . . . . . . . . . . . . . . . . . . . 216 5.1.1.3 Attenuation oeÆ ient . . . . . . . . . . . . . 216 5.1.1.4 Chara teristi impedan e . . . . . . . . . . . . 219 5.1.2 Mi rostrip . . . . . . . . . . . . . . . . . . . . . . . . . . 220 5.1.3 Slot-line . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 5.1.4 Coplanar waveguide . . . . . . . . . . . . . . . . . . . . 222 5.1.5 Finline . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 5.1.6 Planar lines on lossy semi ondu tor substrates . . . . . 223 5.1.7 Planar lines on magneti nanostru tured substrates . . . 225 5.2 Jun tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228 5.2.1 Mi rostrip-to-slot-line transition . . . . . . . . . . . . . 228 5.2.1.1 Indu tan e . . . . . . . . . . . . . . . . . . . . 228 5.2.1.2 Capa itan es C1 and C2 . . . . . . . . . . . . . 229 5.2.1.3 The ratio n of the transformer . . . . . . . . . 229 5.2.1.4 VSWR of jun tion . . . . . . . . . . . . . . . . 230 5.2.1.5 Results . . . . . . . . . . . . . . . . . . . . . . 232 5.2.2 Planar hybrid T-jun tion . . . . . . . . . . . . . . . . . 232 5.3 Gyrotropi devi es . . . . . . . . . . . . . . . . . . . . . . . . . 233 5.3.1 State-of-the-art . . . . . . . . . . . . . . . . . . . . . . . 233 5.3.1.1 Planar lines on magneti and anisotropi substrates . . . . . . . . . . . . . . . . . . . . . . 234 5.3.1.2 YIG-tuned planar MSW devi es . . . . . . . . 234 5.3.2 Under oupled topologies . . . . . . . . . . . . . . . . . . 236 5.3.3 One-port under oupled topology . . . . . . . . . . . . . 239 5.3.4 Two-port under oupled topology . . . . . . . . . . . . . 240 5.3.5 One-port over oupled topology . . . . . . . . . . . . . . 241 5.4 Optoele troni devi es . . . . . . . . . . . . . . . . . . . . . . . 244 5.4.1 Topologies . . . . . . . . . . . . . . . . . . . . . . . . . . 245 5.4.2 Modeling mesa p-i-n stru tures . . . . . . . . . . . . . . 245 5.4.2.1 Simulation using a parallel plate model . . . . 247 5.4.2.2 Measured transmission line parameters . . . . 247 5.4.2.3 Simulation using a mi rostrip model . . . . . . 247 5.4.2.4 Simulation using extended formulation . . . . 248 5.4.3 Modeling oplanar p-i-n stru tures . . . . . . . . . . . . 250 5.5 Measurement methods using distributed ir uits . . . . . . . . 252 5.5.1 L-R-L alibration method . . . . . . . . . . . . . . . . . 253 5.5.2 L-L alibration method . . . . . . . . . . . . . . . . . . 255 5.5.3 Redu tion to one-line alibration method . . . . . . . . 256 5.5.4 Extra tion method of onstitutive parameters . . . . . . 256 5.5.4.1 Overview of the existing standards . . . . . . . 256

CONTENTS

xi

5.5.4.2 Des ription of L-L planar diele trometer . . 5.5.4.3 Proposed new standard . . . . . . . . . . . . 5.5.5 Other diele trometri appli ations of the L-L method 5.5.5.1 Soils . . . . . . . . . . . . . . . . . . . . . . . 5.5.5.2 Bioliquids . . . . . . . . . . . . . . . . . . . . 5.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7 Referen es . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Green's formalism

A.1 A.2 A.3 A.4 A.5 A.6 A.7

B

C

De nition and physi al interpretation . . . . . . . Simpli ed notations . . . . . . . . . . . . . . . . . Green's fun tions and partial dierential equations Green's formulation in the spe tral domain . . . . Re ipro ity . . . . . . . . . . . . . . . . . . . . . . Distributed systems . . . . . . . . . . . . . . . . . Referen es . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

257 258 259 259 260 261 261 269

269 274 274 276 277 278 278

Green's identities and theorem

279

Fourier transformation and Parseval's theorem

281

B.1 S alar identities and theorem . . . . . . . . . . . . . . . . . . . 279 B.2 Ve tor identities . . . . . . . . . . . . . . . . . . . . . . . . . . 280 B.3 Referen es . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280 C.1 Fourier transformation into spe tral domain . . . . . . . . . . . 281 C.2 Parseval's theorem . . . . . . . . . . . . . . . . . . . . . . . . . 283 C.3 Referen es . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284

D Ferrites and Magnetostati Waves

D.1 Permeability tensor of a Ferrite YIG- lm D.2 Demagnetizing ee t . . . . . . . . . . . . D.3 Magnetostati Wave Theory . . . . . . . . D.3.1 Magnetostati assumption . . . . . D.3.2 Types of Magnetostati Waves . . D.4 Perfe t Magneti Wall Assumption . . . . D.5 Referen es . . . . . . . . . . . . . . . . . .

E

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

285

285 285 286 286 288 288 288

Floquet's theorem and Mathieu fun tions

291

Index

294

E.1 Floquet's theorem . . . . . . . . . . . . . . . . . . . . . . . . . 291 E.2 Mathieu's equation and fun tions . . . . . . . . . . . . . . . . . 292 E.3 Referen es . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293

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Prefa e When, in 1888, Hertz dedu ed the on ept of propagation from Maxwell's theories and when, a few years later, in 1895, Mar oni made his rst propagation experiments, nobody would have expe ted the extraordinary development of radio that was to ome. In those days, transmission lines were already present, su h as one or two wires above the ground, transmitting telegraphy. Consideration at the time and distan e dependen e of the signal resulted into what was alled the telegraphist's equations. It did not take long to establish the link between these two-variable equations and the four-variable Maxwell's equations. Simultaneously, ir uit theory evolved from lumped ir uits to distributed ir uits. The two developments merged, and transmission line theory was born. The rst transmission lines were analyzed using a quasi-stati approa h. Homogeneously lled waveguides ne essitated the use of transverse ele tri (TE) and transverse magneti (TM) waves whi h were solutions developed in the 19th entury. It is quite surprising to see how mu h s ienti material was already available at that time, with solutions developed by Lapla e, Poisson, Helmholtz, Bessel, Legendre, Lame, Floquet, Hill, Mathieu, and others. During World War II, mi rowave network theory be ame an re ognized subje t, mainly due to the eorts developed at the Massa husetts Institute of Te hnology (MIT) Radiation Laboratory, USA. Computers were not available, fortunately one might say, enabling eorts to be on entrated on theory. It is extremely interesting to observe that advan es ame out of the fruitful intera tion of dierent s ienti dis iplines: methods hara teristi of quantum me hani s and nu lear physi s were fo used on the appli ation of ele tromagneti theory to pra ti al mi rowave radar problems. Out of these rather spe ial ir umstan es emerged a strategi lesson. In seeking to apply a fundamental theory at the observational level, it was found very advantageous to

onstru t an intermediate theoreti al stru ture, a phenomenologi al theory,

apable of organizing the body of experimental data into a set of relatively few numeri al parameters, and using on epts that fa ilitate onta t with the fundamental theory. The task of the latter then be ame the explanation of the parameters of the phenomenologi al theory, rather than the dire t onfrontation with raw data. Right after the war ame the wonderful s attering

xiv

PREFACE

matrix tool, simultaneously adapted to mi rowave ir uits from s attering in physi s, by Belevit h in Belgium and Carlin in the US. S hwinger played a entral role in the development of mi rowave network theory at the MIT Radiation Laboratory. He was interested in developing \a systemati analyti al pro edure for redu ing ompli ated problems to the fundamental elements of whi h they are omposed by the appli ation of generalized transmission line theory, symmetry onsiderations, and Babinet's prin iple". He used essentially two methods. One is the Integral Equation Method, in whi h imposing the boundary onditions on a dis ontinuity in a waveguide leads dire tly to integral equations from whi h the eld is determined. He repla ed the dis ontinuity by a lumped parameter network in a set of transmission lines. The ir uit parameters were expressed dire tly in terms of the elds, so that the solution of the equations leads to the elements of the equivalent ir uit. The other method he alled the Variational Method, where the impedan e elements are expressed in su h a form that they are stationary with respe t to arbitrary variations of the eld about its true value. By judi iously hoosing a trial eld, he obtained remarkably a

urate results. Moreover, these ould be improved by a systemati pro ess of improving the trial eld, whi h, if

arried far enough, will lead to a rigorous result. Furthermore, in solving a given problem, several methods an be used in onjun tion. For example, one might solve an integral equation approximately by a stati method and then use this approximate solution as a good trial eld in the variational expression. These developments resulted in a rigorous and general theory of mi rowave stru tures in whi h onventional low-frequen y ele tri al ir uits appeared as a spe ial ase. Parti ular use was made of the distin tion made between the theoreti al evaluation of mi rowave network parameters, whi h entails the solution of three-dimensional boundary-value problems and belongs to ele tromagneti theory, and, on the other hand, the network al ulations of power distribution, frequen y response, resonan e properties, et ., hara teristi of the \far- eld" behavior in mi rowave stru tures, involving mostly algebrai problems and belonging to mi rowave network theory. In the 1950s, there was a tremendous breakthrough in transmission line te hnology. It started with the request for higher bandwidth and hen e higher frequen ies. The stripline, and more obviously the mi rostrip line, opened the era of planar hybrid distributed ele troni s. They brought up a new problem, that of an inhomogeneous transverse se tion. Stripline and mi rostrip were rapidly followed by the slot-line, the ele tromagneti omplement of mi rostrip. Then ame the oplanar waveguide. At the lowest frequen ies, a quasi-stati approa h was still usable although not at the higher frequen ies, and new approa hes and methods be ame ne essary. Quite naturally, quasi-stati methods led to full-wave dynami methods. New approa hes were developed, su h as the moment method. Old strategies were improved, among whi h were the Rayleigh-Ritz and Galerkin pro e-

PREFACE

xv

dures. With the in reasing use and apabilities of the omputer, dis retized formulations were developed, su h as nite-dieren es, nite-elements, and time-domain nite-dieren es. Te hnology ontinuously evolved, not only in terms of topologies, but also toward smaller size and higher frequen ies, with mi rowave integrated ir uits, both hybrid and monolithi . Today the variety of stru tures is quite large: mi rostrip, mi roslot, and oplanar wave guide, on diele tri substrates possibly with ferromagneti inserts, as well as on semi ondu ting substrates. The frequen y range is enormous: it goes from the lowest range of mi rowaves, with mobile telephony ir uits at 900 MHz as a main appli ation, up to millimeter waves, with a view to automotive appli ations at 100 GHz. Communi ations with opti al arriers, in parti ular for ultra-high bandwidth data transmission, require solutions for optomi rowave devi es. Today, the pra ti ing engineer has at his disposal a number of methods for analyzing lines and resonators, as well as professional software. So, his question is: what method shall I use? The purpose of this book explore the methods for al ulating ir uit parameters, omplex propagation onstants and resonant frequen ies, for the whole variety of on gurations and over the whole frequen y range. We will show that variational methods an do that. They are analyti al methods and we strongly believe in the value of pushing analyti al methods as far as possible when al ulating ir uit parameters. The reason is simple. In general, we are not interested in al ulating the parameters of just one on guration. What we are mostly interested in is ir uit synthesis and, with this in mind, we are eager to know how hanges in geometri al and physi al parameters an ae t the ir uit's performan e. This is pre isely what analyti al methods an do. Perturbational methods have also been used for de ades. The variational approa h, however, an be applied to a wider variety of problems. Indeed it is not ne essary, as with perturbation theory, that the departure from a simple known ase is small. The variational approa h an be used su

essfully even when the departure is large. As an example, it an be applied to deformed lines and resonators, possibly inhomogeneously loaded, and to singularities su h as orners of re-entrant ondu ting surfa es or of blo ks of diele tri or magneti material. So, the main subje t of the present book is learly that of mi rowave network theory. It is interesting to observe that the general variational prin iple developed in Chapter 4, for solving problems of shielded and open multilayered lines with gyrotropi non-Hermitian lossy media, exa tly follows the guidelines des ribed by S hwinger: the proof of the prin iple is established; an equation is solved by a stati method, using onformal mapping; the approximate solution is used as a good trial eld in the variational expression; the trial eld is improved by a systemati pro ess, and very a

urate results are obtained with a remarkably small number of iterations.

xvi

PREFACE

However, the eld used to obtain results, may not be a good approximation of the eld in the exa t stru ture; in parti ular this is the ase near abrupt dis ontinuities. It is however, important that the integral of the eld is stationary. These eld values are an ex ellent starting point for iterative pro edures in whi h the eld is the quantity of interest. The reader will nd in Chapter 1 a review of the areas in whi h variational methods have been used, in some ases for enturies. The importan e of variational methods and prin iples for solving mi rowave engineering problems is also highlighted. In Chapter 2, variational prin iples are introdu ed in general terms, for

al ulating impedan es and propagation onstants. They are rst developed so that the value of the variational quantity has a dire t physi al signi an e, su h as energy and eigenvalues. Then, by re-arranging Maxwell's equations, it is shown that expli it expressions an be obtained as well as impli it expressions. Variational prin iples are nally ompared with the rea tion on ept, moment method, Galerkin pro edure, Rayleigh-Ritz method, mode mat hing te hnique, and perturbational methods. Quasi-stati and full-wave dynami methods for analyzing planar lines are reviewed in Chapter 3. The quasi-stati methods usually provide a straightforward expression for a line parameter. On the other hand, full-wave methods do not provide a simple expli it des ription of the behaviour of a line parameter over a range of frequen ies. The main part of the hapter is devoted to detailed variational formulations, appli able to a variety of on gurations, in luding anisotropi media. Dis retized formulations are ompared with analyti al formulations. The ombination of variational prin iples with other methods is also dis ussed. The eÆ ien y of variational prin iples is illustrated in Chapter 4 by a general prin iple developed by the authors. It is appli able to planar multilayered lossy lines, in luding those involving gyrotropi lossy materials. In ombination with onformal mapping, it drasti ally redu es the omplexity and duration of numeri al al ulations. Both spatial and spe tral domain approa hes are derived. The mathemati al and numeri al eÆ ien y are demonstrated, while the hoi e of the method for deriving trial quantities is dis ussed. Mathieu fun tions are shown to be very eÆ ient expressions for trial elds in slots. The method is validated by measurements on topologies su h as slot-lines,

oupled slot-lines, nlines, and oplanar waveguides, with YIG-layers. Finally, Chapter 5 is devoted to appli ations operating up to opti al frequen ies, in luding multilayered lines with diele tri and semi ondu ting layers, lines oupled to gyrotropi resonators, uniplanar and mi rostrip-to-slotline jun tions, and optomi rowave devi es. It des ribes an original measurement method, appli able to a variety of substan es, in luding bioliquids su h as blood and axoplasm, for investigating mi rowave bioele tromagneti s, and soil hara teristi s for demining operations. Combined with a variational prin iple, it is used for determining the omplex permittivity of a planar substrate.

PREFACE

xvii

The rst author started using variational prin iples in 1967, extending the variation-iteration method des ribed in Chapter 2. This led to several do toral theses investigating waveguides and avities loaded with two- and three-dimensional diele tri and ferrimagneti inserts. Later, during her do toral thesis, the se ond author developed the general variational prin iple demonstrated in Chapter 4, whi h has sin e been used in a variety of on gurations. This book, however, would not have been written without the stimulating environment oered by Mi rowaves UCL, Louvain-la-Neuve, Belgium. Dis ussions were held ontinuously with lose olleagues, whi h were tremendously fruitful. We owe spe ial thanks to Prof. D. Vanhoena kerJanvier, who has been involved in all what happened in our Laboratory for fteen years. She was the key person in setting up the high performan e mi rowave measurement fa ilities and had a entral role in developing mi rowave passive and a tive ir uits on sili on-on-insulator te hnology. Our thanks also go to Dr. T. Kezai, who analyzed millimeter wave nlines using the Galerkin pro edure, Prof. J.-P. Raskin and Dr. R. Gillon, who brought in new problems with their resear h on sili on-on-insulator te hnology, raising a need for analysis of planar lines on semi ondu ting substrates, and Dr. M. Serres, who presented us with the problem of analyzing optomi rowave devi es. Our most sin ere thanks go to Dr. M. Serres, who kindly a

epted to devote time in professionally editing our manus ript, as well as to M. Dehan, Ph.D. student, who also was of great assistan e in this task. We are most grateful to both of them. We wish to thank the series editors, Prof. T. Benson and Dr. P. Sewell, for their en ouragement and their very useful

riti al remarks. We also wish to thank the Publishing Editor of RSP who was very helpful in sorting out the questions about the presentation of the manus ript. Despite the help we have had, any mistakes or ina

ura ies are, of ourse, the sole responsibility of the authors. Writing this book has been a pleasure. We sin erely hope that the reader will nd some pleasure in reading it. Andre Vander Vorst Isabelle Huynen

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hapter 1

Fundamentals

1.1

Histori al ba kground

Cal ulus of variations, variational equations, variational methods, variational prin iples, variational systems, least a tion, minimum energy, stability - all these on epts relate to the subje t of this book. They have been used for years, and some for enturies. They have not the same meaning, however, and we shall larify at least some of those on epts to better de ne our entral topi s. With this in view, some histori al ba kground is ne essary. Plato, des ribing the reation of the osmos, said that the reator \made it spheri al, equidistant from enter to end, the most perfe t and uniform of all forms; be ause he found uniformity immensely better than its opposite" [1.1℄. Aristotle made another omment, about the alleged ir ular hara ter of the planetary movements: \if the sky movement is the measure of all movements be ause it is the only one to be ontinuous and regular and eternal, and if, for any spe ies, the measure is the minimum, and the minimum motion is the fastest, then, learly, the sky movement must be the fastest of all. Among the lines whi h lose themselves however, the ir le line is the shortest, and the fastest movement is that whi h follows the shortest urve. Hen e, if the sky moves itself into a ir le and is fastest than any other, then it must be ne essarily spheri al" [1.1℄. As we see, the on ept of maximum and minimum was already present in me hani s at the time of Plato and Aristotle. In the rst entury A.D., Hero of Alexandria des ribed the rst minimum prin iple, in geometri al opti s:\all what moves with no velo ity hange, moves along a straight line, (: : :) the obje t tries to move along the shortest distan e, be ause it has no time for a slower movement, i.e. for a movement along a longest path. And the shortest line between two points is the straight line" [1.1℄. Hen e, the on ept of extremum in s ien e was already present 2000 years ago. Mu h later, in the 17th entury, Pierre de Fermat expressed the law of refra tion as a onsequen e of a minimum prin iple, assuming that the velo ity of light de reases when the density of the medium in reases. Des artes, in ontrast, had proposed as \ onforming to experiment" the law based on the assumption that light propagates easier and faster in media with higher density. Fermat writes: \Our postulate is based on the fa t that nature operates by the easiest ways"; ontrary to Des artes, \we do not onsider the

2

CHAPTER 1.

FUNDAMENTALS

shortest spa es or lines, we do onsider those whi h an be followed the most easily and in the shortest time" [1.2℄. Hen e Fermat used a minimum time prin iple. Fermat's questions were extended to the investigation of fun tions maximizing or minimizing a quantity, in general an integral, like establishing the minimum length of a urve joining two points on a surfa e, the minimum time required for a mass to shift from one point to another, the minimum value of an area limited by a given urve, et . In 1744, Maupertuis wrote: \Light, when propagating through dierent media, does not use either the shortest path or the smallest time, (: : :) it follows a path whi h has a more a tual advantage: the path on whi h the quantity of a tion is smallest. (: : :) When a body is shifted from one point to another, a ertain a tion is needed: it depends on the velo ity of the body and of the spa e it des ribes, it is however neither velo ity nor spa e taken independently. The quantity of a tion is the larger as the speed is high and the path is long; it is proportional to the sum of the spa es ea h multiplied by the velo ity at whi h the body des ribes them. This quantity of a tion is the a tual expense of nature, whi h redu es it as far as possible in the movement of light" [1.1℄. The theory, however, was approximate and suering from an unne essary metaphysi al ba kground. At about the same time, the on ept of maximum and minimum was extended by the Swiss mathemati ian Euler, under the name of al ulus of variations, to a mu h more advan ed level. The prin iple of least a tion was de ned with a

ura y. Theoreti al and on eptual developments then ontinued, with the work of d'Alembert, Lagrange, Hamilton, Ja obi, Gauss, Thompson, Kelvin, Weierstrass, Hilbert, Sommerfeld, S hrodinger, Dira , Feynman, Volterra, and others. Extrema were of ourse dire tly linked with stability and, with no surprise, one dis overs that the subje t of stability has grown onsiderably over the years. Some bases are the variational equations of Poin are and the methods of Liapounov. The analyti al approa h to the theory of stability develops from the so- alled variational equations of Poin are, with some ambiguity, however, between variationality and perturbation theory [1.3℄: variational equations, variational equations of singular points, variational systems with onstant oeÆ ients, et . Various methods were developed to redu e variational systems based on dierential equations with periodi oeÆ ients to those based on the dierential equations with onstants oeÆ ients. Sommerfeld extended the theory of Bohr's atom by observing that there had to be a ertain onne tion between Plan k's onstant and the a tion integral of Hamilton's prin iple. De Broglie was in uen ed by the prin iple of least a tion in establishing the basis of wave me hani s, and he brought together opti s and me hani s through variational prin iples. S hrodinger wrote his equation of quantum me hani s as an extremum of an integral obtained from Hamilton-Ja obi's theory. Dira wondered about the link between quantum

1.2.

EXTREMUM PRINCIPLE

3

me hani s and the Lagrangian formalism, and the do toral thesis of Feynman in 1942 was entitled \The prin iple of least a tion in quantum me hani s", paving the road towards his ontributions in quantum ele trodynami s for whi h he obtained the Nobel prize. 1.2

Extremum prin iple

To the best of our knowledge, the rst time a variational approa h was used in engineering was by the Bernoulli brothers, in the middle of the 18th entury, in hydrauli s. There had been quite a orresponden e between them and Euler on the subje t. A number of problems however, in the area ommon to mathemati s and me hani s, were solved using the al ulus of variations. The ore of the methods using su h al ulus is based on variations, whi h means investigating an extremum by attributing variations to the arguments. The lo al aspe t is dominant. We shall point out later in this hapter that this subje t is not to be onfused with variational prin iples in general. The ambiguity, however, is present in many books, su h as [1.4℄, whi h states: \Many variational problems do appear be ause the laws of nature allow interpretations as variational prin iples ". It must be observed that the examples

ited in those referen es are, stri tly speaking, related to the stationarity of one integral around a given point, yielding an extremum, either a maximum or a minimum, for that integral. We shall ome ba k to this later. Many problems re eive their initial formulation dire tly in the domain of the al ulus of variations. A number of examples of the stationarity of an integral an be found, su h as the evaluation of: - the shortest path from one point to another (geometri al opti s), or from a point to a urve, or between two urves - the fastest des ent of a parti le in movement, submitted to fri tionless gravitation (bra hysto hrone) - the minimum area des ribed by a urve rotating around an axis (isoperimetri problem and geodesy). The problem we are onfronted with bears some resemblan e to the one of nding the minimum or the maximum of a simple fun tion. The problem en ountered in the al ulus of variations, however, is mu h more diÆ ult, be ause instead of nding a point where a simple fun tion f (x) has a minimum, we are required to determine an argument fun tion y (x), out of an in nity of possible fun tions de ned over an interval of x, whi h makes a de nite integral J of the fun tional F (y ) a minimum. A further very fundamental dieren e between the maximum and minimum problem and the

al ulus of variations problem lies in the fa t that a theorem (attributed to Weierstrass) always ensures the solution of the straight maximum-minimum problem, while no su h general fundamental existen e theorem exists for the variational al ulus problem. Consequently, in the al ulus of variations, the

4

CHAPTER 1.

FUNDAMENTALS

existen e of a solution to any given extremum problem requires a spe ial proof. In those ases where a solution exists, it turns out that a ne essary ondition for an extremum is that the rst variation ÆJ of the fun tional must be zero. This requirement is formally analogous to the requirement for the straight minimum problem that the dierential df vanish. Hen e the meaning of the rst variation of an integral has to be de ned arefully. An Euler dierential equation is obtained when the total derivative d=dx of the partial derivative F=y is taken [1.5℄. That this equation must be satis ed by y (x) onstitutes a ne essary ondition for the existen e of an extremum. The dierential equation arising out of the rst variation of the fun tional J plays the same role in the al ulus of variations minimum problem, as the ordinary dierential quotient plays in the simple minimum problem in ordinary al ulus. There are several ir umstan es in whi h it is simple to obtain the solution of the Euler equation. So alled variational methods have been used abundantly in physi s [1.6℄, be ause many laws of physi s an be expressed in terms of variationality. For example, in lassi al me hani s, it has been shown that the motion of a me hanized system is always su h that the time integral of the Lagrangian fun tion taken between two on gurations of the system has an extreme value, a tually a minimum. This prin iple is today widely known as that of least a tion. In its present form, another su h prin iple is that of Fermat, whi h states that the propagation of light always takes pla e in su h a way that the a tual opti al path, i.e. the length of geometri path multiplied by the index of refra tion of the medium, is an extreme value, usually a minimum. If the index of refra tion varies ontinuously from point to point in the medium, the appropriate integral must be applied. In the ase of automati ontrol [1.7℄, the names of Pontrjagin and his

olleagues are well known for what is alled the prin iple of Pontrjagin, whi h is a prin iple for determining a maximum [1.8℄. They investigated the ontrol problem, whi h involves nding an extremum for a fun tional where the fun tion is submitted to a onstraint, by determining whi h aspe ts, hara terizing the simpler problem of nding an extremum for the fun tional, remain valid for the ontrol problem and an be used as a basis for solving it. For many boundary-value problems of even order, it is possible to spe ify an integral expression whi h an be formed for all fun tions of a ertain

lass and whi h has a minimum value for just that fun tion whi h solves the boundary-value problem [1.9℄. Consequently, solution of the boundary-value problem is equivalent to minimizing the integral. In the Ritz method the solution of this variational problem is approximated by a linear ombination of suitably hosen fun tions. This method shares, along with the nite-dieren e and the nite-element methods, a very favored position among the methods for the approximate solution of boundary-value problems. An appropriate expression for the integral an be obtained by trying to write the dierential equation of the given boundary-value problem as the Euler equation of some

1.3.

VARIATIONAL PRINCIPLE

5

variational problem. In the above dis ussion, the problem involving only one unknown fun tion of a single independent variable was onsidered. The required generalization to over problems wherein more than one fun tion of a single independent variable is involved an readily be made [1.4℄. The same is true for the many physi al problems requiring the determination of a fun tion of several independent variables, giving rise to an extremum in the ase of a multiple integral. In this ase, the Euler partial dierential equation is found for the problem involving several independent variables. A ne essary ondition for an extremum of J is that the fun tion of the variables satis es this equation. If the fun tion is required by onditions of the original problem to have spe i ed values on a boundary, a solution of Euler's equation must be obtained whi h satis es the given boundary onditions. Instan es of physi al examples whi h

an be repla ed by a variational problem are: - the vibrating string, by using the prin iple of least a tion - potential problems involving Lapla e equation, for whi h the potential fun tion makes an integral an extremum. For variational problems in whi h it is required to determine the fun tion u(x) of a variable whi h makes a spe i integral F an extremum subje t to an auxiliary ondition, it may be shown that the problem is generally, but not always, equivalent to nding another fun tion y (x) of the same variable. The result will yield an extremum for the same integral, of a modi ed fun tion F  , without regard to the auxiliary ondition. The modi ed fun tion F  is a spe i linear ombination of the fun tion F and of the auxiliary ondition [1.5℄. As the reader an see, the above presentation is limited to nding fun tions whi h make one integral extremum. By doing so, a stationary value is obtained whi h is insensitive to a rst-order variation of the variables, be ause su h a rst-order variation indu es a se ond-order variation, whi h is negligible, in the extremum value of the integral. Variational problems, variational methods, variational prin iples, stationarity, least a tion, and perturbation methods are expressions whi h have been used extensively in the literature. From now on, we shall address those problems as depending upon an extremum prin iple [1.10℄. 1.3

Variational prin iple

Throughout this book, we intend to reserve the appellation variational prin iple to problems where the unknown quantity is expressed as a ratio of integrals. The value of the ratio will be said to be variational if there exists an extremum value for this ratio, as for the ases illustrated in the rst part of this hapter. There is a fundamental dieren e, however, between them. If indeed the numerator and the denominator of the ratio are both variational for the same value of the fun tion or if they both vary in the same way around

6

CHAPTER 1.

FUNDAMENTALS

the value yielding stationarity, then one feels that the stationarity might be better than in the other ases. To illustrate our point, let us onsider variational prin iples for eigenvalue problems. The eigenvalue equation relates dierential or integral operators a ting on the fun tion: L( ) = M( )

(1.1)

and it an be proved [1.6℄ that the following expression is a variational prin iple for : R L( )dS ℄ = Æ[℄ = 0 (1.2) Æ[ R M( )dS where the fun tion  is also determined by the variational prin iple. It satis es indeed the equation and boundary onditions whi h are the adjoints of those satis ed by the fun tion : it is the adjoint solution. When the operators L and M are self-adjoint, one has  = and the variational prin iple assumes the simpler form R L( )dS R Æ [℄ = Æ [ L and M self-adjoint (1.3) M( )dS ℄ = 0; The important point here is to observe that indeed variational prin iple (1.2) is expressed as the ratio of two integrals. This is the ase for a number of eigenvalue problems, like S hrodinger's equation and the vibration of a membrane, subje t to the Helmholtz equation. These are problems for whi h the energy eigenvalues form a dis rete spe trum. They deal with either a nite domain upon whose surfa e the wave equation satis es boundary onditions, or an in nite domain in whi h ase the wave fun tion must go to zero at in nity. There are also situations in whi h the eigenvalues form a ontinuous spe trum. Solutions satisfying the appropriate boundary onditions exist for all values of the eigenvalue within some range of values. The variational prin iples des ribed above are not dire tly appli able in this ase, be ause the wave fun tions are no longer quadrati ally integrable, and suitable modi ations must be introdu ed. In su h ases variational prin iples an be found, for instan e, for phase shift, transmission and re e tion of waves by a potential barrier, s attering of waves by variations of index of refra tion, s attering from surfa es, radiation problems, and variation-iteration s hemes [1.6℄,[1.11℄. 1.4

Mi rowave network theory

One feature brought to the fore by World War II was the ne essity to solve rapidly new and diÆ ult engineering problems. The MIT Radiation Laboratory grouped a number of s ientists and engineers, from various dis iplines of s ien e and te hnology. The group was very prestigious indeed and nine

1.4.

MICROWAVE NETWORK THEORY

7

members of the Laboratory later obtained a Nobel prize. In the eld of variational prin iples, members of the group started using su h prin iples to solve engineering mi rowave problems. This really was a breakthrough. One should remember that omputers were not available at that time, and analyti al methods, even approximate, were essential. After World War II, Prof J. S hwinger, future Nobel prize winner, used to le ture to a small group of olleagues at the MIT Radiation Laboratory. Notes were dupli ated after ea h le ture and handed out to the parti ipants. The war ended before the series had nished, leaving le tures undelivered and some le ture notes unwritten. By the end of 1945, the notes were dupli ated again and sent out to an a

umulated mailing list. Interest in these le tures has never waned. Some of the notes were nally published more than twenty years later. They on ern waveguide dis ontinuities [1.12℄. It is quite appropriate to quote the authors' prefa e. \These notes are an interesting do ument of the fruitful intera tion of different s ienti dis iplines. Attitudes and methods hara teristi of quantum me hani s and nu lear physi s were fo used on the appli ation of ele tromagneti theory to pra ti al mi rowave radar problems. And out of these rather spe ial ir umstan es emerged a strategi lesson of wide impa t. In seeking to apply a fundamental theory at the observational level, it is very advantageous to onstru t an intermediate theoreti al stru ture, a phenomenologi al theory, whi h is apable on the one hand of organizing the body of experimental data into a relatively few numeri al parameters, and on the other hand employs

on epts that fa ilitate onta t with the fundamental theory. The task of the latter be omes the explanation of the parameters of the phenomenologi al theory, rather than the dire t onfrontation with raw data. The ee tive range formulation of low energy nu lear physi s was an early postwar appli ation of this lesson. It was a substantial return on the initial investment, for now mathemati al te hniques developed for waveguide problems were applied to nu lear physi s. There have been other appli ations of these methods, to neutron transport phenomena, to sound s attering problems. And it may be that there is still fertile ground for applying the basi lesson in high energy parti les physi s". S hwinger was interested in developing sound \engineering methods", whi h he de ned as \a systemati analyti al pro edure for redu ing ompli ated problems to the fundamental elements of whi h they are omposed by the appli ation of generalized transmission line theory, symmetry onsiderations, and the prin iple of Babinet". He used essentially two methods. One is the Integral Equation Method, in whi h the imposition of the parti ular boundary onditions whi h the ele tromagneti eld must satisfy - be ause of the existen e of the dis ontinuity - leads dire tly to one or more integral equations for the determination of the eld. He therefore obtains equivalent

ir uits, by means of whi h a dis ontinuity in a waveguide is repla ed by a lumped parameter network in a set of transmission lines. These ir uit pa-

8

CHAPTER 1.

FUNDAMENTALS

rameters an be expressed dire tly in terms of the elds, so that the solution of the integral equations leads almost at on e to the impedan e (or admittan e) elements of the equivalent ir uit. He points out that a stati method, with the help of onformal mapping, an solve many dynami problems. S hwinger alled the Variational Method the means of solution in whi h \the impedan e elements are expressed in su h a form that they are stationary with respe t to arbitrary small variations of the eld about its true value. With the aid of su h a method, one an, by judi iously hoosing a trial eld, obtain remarkably a

urate results. Moreover, these results an be improved by a systemati pro ess of improving the trial eld, whi h, if

arried far enough, will lead to a rigorous result. Furthermore, in solving a given problem one often uses two or more of these methods in onjun tion. For example, one might solve an integral equation approximately by a stati method and then use this approximate solution as a good trial eld in the variational expression"[1.12℄. In his well-known Waveguide Handbook [1.13℄, one of the twenty-eight volumes des ribing the resear h developed by the MIT Radiation Laboratory and published after World War II, Mar uvitz mentions the dominant role played by S hwinger in formulating eld problems using the integral equation approa h, pointing the way forward both in the setting up and the solution of a wide variety of mi rowave problems. These developments resulted in a rigorous and general theory of mi rowave stru tures in whi h onventional low-frequen y ele tri al ir uits appeared as a spe ial ase. One extremely useful aspe t was the distin tion made between the theoreti al evaluation of mi rowave network parameters, whi h usually entails in general the solution of three-dimensional boundary-value problems and belongs in the domain of ele tromagneti theory and, on the other hand, the network al ulations of power distribution, frequen y response, resonan e properties, et ., hara teristi of the \far- eld" behavior in mi rowave stru tures, involving mostly algebrai problems and belonging in the domain of mi rowave network theory. The main subje t of the present book is learly that of mi rowave network theory. It is interesting to observe that the very general variational prin iple demonstrated in Chapter 4 in view of solving problems of shielded and open multilayered transmission lines with gyrotropi non-Hermitian lossy media and lossless ondu tors [1.14℄ exa tly follows the guidelines des ribed by S hwinger: the proof of the variational prin iple is established; an equation is solved by a stati method, using onformal mapping; the approximate solution is used as a good trial eld in the variational expression; the trial eld is improved by a systemati pro ess, and very a

urate results are obtained with a remarkably small number of iterations. Hen e the ne essary omputing time is drasti ally redu ed, as will be shown. The variational approa h an be applied to a wider variety of problems than an perturbation theory. It is indeed not ne essary, as it is with perturbation theory, for the departure from a simple known ase to be small. As

1.4.

MICROWAVE NETWORK THEORY

9

an example, the variational approa h an be applied to deformed waveguides and avities, and to avities and waveguides ontaining various diele tri s. It

an be used su

essfully, unlike perturbation theory, even when the departure from a simple ase is large [1.15℄. Compli ated waveguides and avities an be treated, and singularities su h as those o

urring at the orners of re-entrant

ondu ting surfa es, or at orners of blo ks of diele tri , an be dealt with. The variational approa h does have its limitations. It an only be applied to the evaluation of one eigenvalue at a time, unlike an exa t method yielding a hara teristi equation giving impli itly all the eigenvalues, or the perturbation method whi h an give general formulae for eigenvalue shifts appli able to all the modes, or at least to all the modes pertaining to a given

lass. Furthermore, it should never be forgotten that, pre isely be ause of the variationality of the eigenvalue, whi h is a variationality with respe t to the elds, the values of the elds in the integrals whose ratio is variational an be rather dierent from the unknown a tual values, at least in some parts of the domain of interest. It is now time to show the dieren e between perturbation theory and variational theory. Perturbation theory is valid only when a small hange is introdu ed in a system. It evaluates the modi ation in the parameter of interest due to the perturbation with respe t to the value of that parameter in the absen e of perturbation. Taking avities as an example, and onsidering the resonant frequen y as the parameter of interest, perturbation theory an be used to al ulate the resonant frequen y of a avity lled with gas ompared to that of an empty avity, whi h implies a very small hange over a large volume. It an also be used to al ulate the variation in resonan e frequen y when a small solid body is introdu ed, whi h is a large hange over a small volume, or when part of a perfe t ondu tor is repla ed by a metal with a large but nite ondu tivity, whi h means a hange not over a volume but over a surfa e, or when the avity boundary is slightly modi ed [1.16℄. The reason why perturbation theory an be applied only for small hanges, whi h is not the ase for variation theory, is a fundamental question. In perturbation theory an approximation is always made: a term is negle ted, be ause it is small due to the fa t that the hange is small. This term would not be small if the hange was not just a perturbation. Usually the negle ted term is the integral of a quantity whi h may be of importan e only in a small part of the domain of interest, so that its integration over the whole domain is small and an be negle ted to the rst order. As an example, when establishing the value of the rst-order variation Æ!=! of the resonant frequen y of a avity [1.15℄, the negle ted term is an integral over the whole volume of the avity of a quantity whi h is signi ant only in the vi inity of, for instan e, a small solid sample introdu ed in the avity, or a small deformation of the surfa e of the avity. As a onsequen e, the equation whi h is solved for al ulating the parameter of interest is not exa t : it is valid only to the rst-order. Hen e the small size of the hange is essential for the pra ti al use

10

CHAPTER 1.

FUNDAMENTALS

of perturbational methods. The perturbation method yields adequate values be ause it al ulates the ee t of a perturbation on the unperturbed known system. This is not the ase when using a variational approa h: the equation solved for al ulating the parameter of interest is exa t. Only the exa t values of the elds throughout the surfa e or the volume are unknown. The variationality ensures a very a

urate value for a ir uit parameter, like the omplex propagation onstant and impedan e of a transmission line, or the omplex resonant frequen y of a avity. This high degree of a

ura y is obtained by using eld values whi h may be ina

urate; this is due to two reasons. One is be ause the integrations are made over the whole surfa e, or volume, or

ontour. The other is be ause the result is expressed as a ratio of integrals, with numerator and denominator varying the same way under the in uen e of a hange of the elds, for a rather wide variation of those elds. There is a very fundamental reason why the quality of the two methods dier: there is an energy relation underneath the variational approa h, whi h is not the ase for the perturbation method. Rumsey, another member of the MIT Radiation Laboratory who later obtained a Nobel prize, de ned in 1954 a physi al observable, the rea tion, to simplify the formulation of boundary value problems in ele tromagneti problems [1.17℄. To illustrate the value of this new on ept, he used it to obtain formulas for s attering oeÆ ients, transmission oeÆ ients, and aperture impedan es. The formulas so obtained have a stationary hara ter and thus the results ould also be obtained from a variational approa h. Rumsey pointed out that this physi al approa h, on eptually simple, is general while the variational approa h has to be worked out for ea h problem. His point of view is that of an experimenter, whose obje tive is to use the theory to orrelate his measurements. He pointed out the fa t that an expression is stationary for variations of an assumed distribution about the orre t distribution does not justify the assumption that it will yield the best approximation when the assumed distribution is ompletely arbitrary. It should be noted however that there is no way of de iding whi h approximation is \best". Harrington [1.18℄ showed that many of the parameters of interest in ele tromagneti engineering are proportional to rea tions, for instan e, the impedan e parameters of a multiport network. The paper by Rumsey limits the rea tion on ept to isotropi media and elds ontained in a nite volume. In the last part of his paper, however, and in an addendum published a little later, he shows that the rea tion on ept an be extended to anisotropi media by using a more general form of the re ipro ity theorem. An argument developed later, however, [1.19℄ questions the use of the rea tion method for gyrotropi media su h as magnetized ferrites, and magneto-ioni media, but states it is well suited to isotropi materials and materials with rystalline anisotropy, lossy or lossless. The diele tri onstant and the permeability are, at most, symmetri tensors. Today, there is still pla e for resear h on the limitations of

1.5.

VARIATIONAL EXPRESSIONS FOR WAVEGUIDES AND CAVITIES

11

the rea tion on ept. 1.5

Variational expressions for waveguides and avities

The next milestone is the ex ellent paper published by Berk in 1956 [1.19℄. The author presented variational expressions for propagation onstants of a waveguide and for resonant frequen ies of a avity, dire tly in terms of the eld ve tors. Those situations o

ur when the ele tromagneti problem annot depend from a s alar formulation satisfying Helmholtz equation and are typi ed by the presen e of inhomogeneous or anisotropi matter. In these

ases the need for ve tor variational prin iples is apparent. Su h variational formulas are presented and derived dire tly in terms of the eld ve tors. They are valid for media whose diele tri onstant and permeability are Hermitian tensors, restri ted, however, to lossless media. It should be underlined that the pra ti al use of a formula developed for lossless gyroptropi media is quite limited, be ause gyrotropy implies losses, ex ept in very narrowband ir umstan es. Furthermore, all the formulations are limited to losed waveguides and resonators. Several examples illustrated the advantages of approximations based on the variational expressions of the paper. They in luded the ut-o frequen y of the fundamental mode of a re tangular waveguide in whi h a verti al diele tri slab is pla ed respe tively adja ent to one of the walls and symmetri ally in the middle of the guide, as well as a verti al ferrite slab pla ed o enter. In 1971, Gardiol and Vander Vorst [1.20℄, analyzing E-plane resonan e isolators, used a variational prin iple for the propagation onstant in lossy gyrotropi ferrites derived by Eidson in an internal report. The formulation involves six s alar quantities orresponding to the ele tri and magneti eld

omponents in the stru ture. In 1994, a general variational prin iple had been established by Huynen and Vander Vorst, valid for planar lines, open or shielded, ontaining an arbitrary number of layers and of ondu ting sheets [1.14℄. It yields an expli it variational expression for the propagation onstant as a fun tion of the three s alar omponents of only one eld. The materials may be gyrotropi and lossy. The formulation is general enough to hara terize, for instan e, magnetostati modes in Yttrium-Iron-Garnet (YIG) lms used as gyrotropi substrates. The eÆ ien y of the method is due to both the expli it formulation and the fa t that the error made on the trial eld is ompensated by an exa t analyti al integration of the elds over the whole spa e. It yields results whi h agree very well with new experimental data on slot-lines and YIG-resonators in a mi rostrip on guration. The ondu tors, however, do have to be lossless. The ee t of ondu tor losses must be al ulated by another method, su h as a perturbation method. A spe tral domain formulation of the prin iple [1.14℄ has also been established [1.21℄. In ombination with onformal mapping, it drasti ally redu es the omplexity of the numeri al omputation and leads to rapidly onvergent results even when higher order modes are

12

CHAPTER 1.

FUNDAMENTALS

onsidered. The formulation applies to open and shielded multilayered lines with oplanar ondu tors lying at dierent interfa es between the layers. It is also valid for gyrotropi non-Hermitian lossy media. Results are shown to be in ex ellent agreement with previously published results, al ulated by other methods, as well as with new measurements made by the authors. The propagation onstant is variational with respe t to the elds and not to the material properties. Hen e, a new method for hara terizing the properties of planar materials with a very good a

ura y, over a broad frequen y range at frequen ies up to 100 GHz, was established [1.22℄. Chapter 4 is essentially devoted to the demonstration of this prin iple, validated by some appli ations, while Chapter 5 des ribes a number of variational appli ations. Finally, in 1998, Huynen et al. proved the stationary hara ter of the magneti energy in the ase of a resonator ontaining lossy gyrotropi media and supporting mi rowave magnetostati waves [1.23℄. A variational expression is obtained for the input impedan e of the ir uit. It is used for al ulating the input re e tion oeÆ ient of a planar multilayered magnetostati wave straight-edge resonator. The model is an eÆ ient tool for designing low-noise wide-band YIG-tuned os illators. 1.6

New theoreti al developments

Appli ations of the variational approa h are numerous. A number of them an be found in the two editions of the book by Collin [1.24℄[1.25℄, with new material in the se ond edition. They essentially relate to waveguide dis ontinuities, su h as diaphragms, various dis ontinuities, jun tions, diele tri steps, input impedan e of waveguide, indu tive posts, ferrite slabs, et . The author uses the Hertzian potential approa h, leading to just two s alar eld omponents, appli able only to the most elementary problems. For more ompli ated problems, the resulting eigenvalue equations are not of the standard type, be ause the eigenvalue either does not appear in a linear form or is present in the boundary or interfa e onditions. The author also oers an ex ellent outline of the method for that lass of problems. For mi rostrip-like transmission lines, the rst expli it variational prin iples, developed by Yamashita, were quasi-stati ones, providing values for quasi-transverse-ele tromagneti (TEM) apa itan es [1.26℄. In 1971, English and Young [1.27℄ obtained a variational formulation for waveguide problems in terms of three s alar eld omponents (the E formalism), while the expressions developed by Berk [1.19℄ were in terms of six s alar eld omponents (the EH formalism). Be ause the parameter of interest - the propagation onstant - appears in their fun tional equation in quadrati form with rst- and se ond-order powers, they have to apply the variational method in a reverse way: solve for the frequen y, whi h is normally known, in terms of the propagation fa tor, whi h is normally unknown. Lindell [1.28℄ de ned the eigenvalue problem in a less restri tive manner so that dierent parameters involved in the problem an be interpreted as

1.6.

NEW THEORETICAL DEVELOPMENTS

13

eigenvalues. These general eigenvalues are alled non-standard eigenvalues. Spe i ally, the basis of the method is that it uses any existing stationary fun tional for a standard eigenvalue, solves it for any parameter in the fun tional, and obtains a stationary fun tional for that parameter, whi h is by de nition a non-standard eigenvalue of the problem. A uni ed theory for obtaining stationary fun tionals for dierent non-standard eigenvalue problems, based on a mathemati al prin iple, is presented. The problems are lassi ed in terms of the omplexity of their fun tional equation. Be ause there may exist many parameters ea h re ognizable as a nonstandard eigenvalue of the problem, there may exist dierent fun tionals giving a hoi e of methods of dierent omplexity in solving the same problem. Several examples are presented, like the uto frequen y problem of a waveguide with rea tan e boundaries,

omparing dierent formulations of the problem, the azimuthally magnetized ferrite- lled waveguide propagation problem, the orrugated waveguide, and the avity with a homogeneous insert. (A orresponden e on the possible falla y of the proposed prin iple lasted for two years in the IEEE Transa tions of the So iety Mi rowave Theory and Te hniques (MTT), in September 1983 and April 1984). Later, the method was applied to the al ulation of attenuation in opti al bers [1.29℄. In 1990, Baldomir and Hammond [1.30℄ showed that a geometri al approa h to ele tromagneti s uni es the subje t of using methods whi h are

oordinate-free and based on geometry rather than algebrai equations, oering guidan e in the hoi e of simple numeri al methods of al ulation. They show that the prin iple of least a tion is a parti ular ase of the geometri al symmetry of the on guration spa e. Appli ations are illustrated in ele trostati s, magnetostati s, and ele trodynami s. Ri hmond dis ussed the interesting subje t of the variational aspe ts of the moment method [1.31℄. Variational te hniques were in use prior to the introdu tion of the moment method. Sin e then, the moment method has been widely adopted for solving ele tromagneti problems via the integral-equation approa h. At the same time, interest in variational methods almost vanished. Ri hmond raises several very good questions. Under what onditions does the moment method possess the variational property? When the moment method is variational, pre isely whi h quantities are stationary ? Do the variational moment methods oer any signi ant advantage over the non-variational moment methods? A limitation of the paper is that it is restri ted to timeharmoni problems involving perfe tly ondu ting antennas and s atterers, formulated with the ele tri eld integral equation. The author also points out that his paper does not onsider other important te hniques su h as nitedieren es, nite-elements, T-matrix, onjugate gradient, or least squares. He uses the terms \moment method" and \Galerkin's method" as they are de ned by Harrington [1.32℄. Ri hmond shows that admittan e, impedan e, gain and radar ross se tion an be expressed in terms of the self-rea tion or the mutual rea tion, in Rumsey's sense [1.17℄, of the ele tri urrent dis-

14

CHAPTER 1.

FUNDAMENTALS

tribution(s) indu ed on the body. Thus the question of the stationarity of these important quantities redu es to the question of the stationarity of the rea tion. Subje t to the above restri tions and assuming the usual re ipro ity theorems are appli able, it is shown that the rea tion is stationary with Galerkin's method, while it is not when the non-Galerkin moment method is applied in the ustomary manner. A few numeri al results are in luded to

ompare the performan e of the variational and non-variational moment methods. A simple example is that of a two-dimensional problem involving a perfe tly ondu ting ir ular ylinder with transverse-magneti polarization. The self-rea tion per unit length is al ulated as a fun tion of a parameter p whi h vanishes for the exa t solution, via the Galerkin and the non-Galerkin moment method, respe tively. Both methods yield pre isely the orre t rea tion when parameter p is zero. The Galerkin result, however, displays zero slope at p = 0 and thus maintains good a

ura y even when the basis fun tion takes on a rather large error. On the other hand, the non-Galerkin result displays a large nonzero slope when p = 0, and therefore qui kly loses a

ura y as the basis fun tion departs from the orre t fun tion. Furthermore Wandzura [1.33℄ proved that the Galerkin method, when applied to s attering problems, gives more rapid solution onvergen e than more general moment methods. As said before, most of the results obtained when using a variational approa h are based on a mi rowave network point of view, for quantities su h as propagation onstant, resonant frequen y, and input impedan e. Few onsider eld omputations, and this is parti ularly true for problems involving anisotropi media. Jin and Chew [1.34℄ have presented a formulation with spe i fun tionals for general ele tromagneti problems involving anisotropi media. Su h a formulation provides a foundation for the development of the nite-element method for the analysis of su h problems. Spe i fun tionals are stated and their validity is proven, rather than onstru ting them from the original boundary-value problems. The property of the asso iated numeri al system is dis ussed for the following three ases: (i) lossy problems involving anisotropi media having symmetri permittivity and permeability; the resultant numeri al system obtained from the nite-element dis retization is symmetri (ii) lossless problems involving anisotropi media having Hermitian permittivity and permeability; the resulting numeri al system is Hermitian (iii) general problems involving general anisotropi problems; the resulting numeri al system is neither symmetri nor Hermitian. The variational aspe ts of the rea tion in the method of moments have been dis ussed again by Mautz in 1994 [1.35℄. He refers to Ri hmond [1.31℄, who pointed out that the rea tion between two ele tri surfa e urrent densities J1 and J2 (the integral of the dot produ t of J1 with the ele tri eld produ ed by J2 ) is variational with Galerkin's method but is not ne essarily variational when the non-Galerkin moment method is applied in the manner where one set of fun tions is used to expand both J1 and J2 and another

1.6.

NEW THEORETICAL DEVELOPMENTS

15

set of fun tions is used to test both the integral equation for J1 and that for J2 . Mautz shows that the same rea tion is variational when the non-Galerkin method is applied in the manner of operation where the set of fun tions used to expand J2 is the set of fun tions used to test the integral equation for J1 , and the set of fun tions used to test the integral equation for J2 is the set of fun tions used to expand J1 . Hen e a non-Galerkin formulation may be variational, whi h is illustrated by al ulating the rea tion with four non-Galerkin methods, in luding two variational and two non-variational results. This operating way may be more appropriate than Ri hmond's if a variational result is desired. The paper establishes the onditions under whi h a symmetri produ t is variational. Liu and Webb derived a variational formulation for the propagation onstant satisfying the divergen e-free ondition in lossy inhomogeneous anisotropi re ipro al or nonre ipro al waveguides whose media tensors have all nine

omponents [1.36℄. Their formulation is implemented using the nite-element method. The variational expressions are in the form of standard generalized eigenvalue equations, where the propagation onstant appears expli itly as the eigenvalue. It is also shown that for a general lossy nonre ipro al problem the variational fun tional exists only if the original and adjoint waveguide are mutually bi-dire tional, i.e., for ea h mode with the propagation onstant in the original waveguide there exists a mode with propagation onstant for the adjoint waveguide. On the other hand, for a general lossy re ipro al problem the variational fun tional exists only if the waveguide is bi-dire tional, i.e., if modes with propagation onstant and exist simultaneously for the same waveguide. Finally, Huynen and Raida re ently ompared nite-element methods with variational analyti al methods for planar guiding stru tures [1.37℄. They ompared an expli it variational prin iple for the propagation onstant of guiding stru tures with the nite-element method using a fun tional involved in the expli it variational prin iple formalism. They show that ombining the niteelement method with this fun tional provides stationary values of the propagation onstant with respe t to trial-dis retized elds, provided, however, that the media are either isotropi or lossless gyrotropi . Comparative results show how the onvergen e is in uen ed by the diering nature of the expli it unknown ( eld for nite elements, propagation onstant for the expli it variational prin iple). The performan es of both methods are ompared in terms of CPU time, and a

ura y. It is on luded that the variational prin iple has to be preferred for onventional ondu tor shapes, when analyti al expressions in the spatial or the spe tral an be derived a priori for trial quantities. Exa t elds an be derived by using the stationarity of the propagation onstant to obtain su

essive improvements of eld expressions.

16 1.7

CHAPTER 1.

FUNDAMENTALS

Con lusions

From all those onsiderations, it appears wise to onsider that a resolution method should be formulated using one of the two approa hes: the variational approa h and Galerkin's approa h. Although Galerkin's approa h is

on eptually simpler, the authors re ommend using the variational approa h be ause, as well as yielding a more elegant formulation, it has a solid foundation in physi s and mathemati s and an provide physi al insights to some diÆ ult on epts su h as essential and natural onditions or the hoi e of expansion (basis) fun tions. However, unlike Galerkin's approa h, whi h starts dire tly with dierential equations, the variational approa h starts from a variational formulation. The appli ability of the approa h depends, of ourse, dire tly on the availability of su h a variational formulation. On the other hand, the link between the use of a variational prin iple based on an analyti al fun tion and the use of a dire t numeri al al ulation still oers some unexplored elds of investigation. 1.8

Referen es

[1.1℄ J. Mawhin, \Le prin ipe de moindre a tion et la nalite", Revue d'ethique et de theologie morale, Le supplement, Editions du Cerf, Cahier no. 205, pp. 49-82, Jun. 1998. [1.2℄ P. de Fermat, \Methode pour la re her he du maximum et du minimum", Oeuvres. Paris: Gauthier-Villars, 1891-1922, Tome 3, pp.121156. [1.3℄ N. Minorski, Nonlinear Os illators. Prin eton, N.J.: Van Nostrand, 1962. [1.4℄ E. Roubine, Mathemati s Applied to Physi s. Berlin: Springer-Verlag, 1970. [1.5℄ L.P. Smith, Mathemati al Methods for S ientists and Engineers. New York: Dover, 1961, Ch. 13, pp. 404-409. [1.6℄ P.M. Morse, H. Feshba h, Methods of Theoreti al Physi s. New York: M Graw-Hill, 1953. [1.7℄ L.D. Berkovitz, \Variational methods in problems of ontrol and programming", J. Math. Analysis and Appli ations, Vol. 3, no. 1, pp. 145169, Aug. 1961. [1.8℄ L.S. Pontrjagin, V.G. Boltyanskii, R.V. Gamkrelidze, E.F. Mis henko, The Mathemati al Theory of Optimal Pro esses (translated from Russian). New York: Inters ien e Publishers, 1962. [1.9℄ L. Collatz, The Numeri al Treatment of Dierential Equations. Berlin: Springer-Verlag, 1966. [1.10℄ R.P. Feynman, R.B. Leighton, M. Sands, The Feynman Le tures in Physi s, Vol. II. Reading, Mass.: Addison-Wesley, 1964.

1.8.

REFERENCES

17

[1.11℄ A. Vander Vorst, R. Govaerts, \Appli ation of a variation-iteration method to inhomogeneously loaded waveguides", IEEE Trans. Mi rowave Theory Te h., vol. MTT-18, pp. 468-475, Aug. 1970. [1.12℄ J. S hwinger, D.S. Saxon, Dis ontinuities in Waveguides. Notes on Le tures by Julian S hwinger. New York: Gordon and Brea h, 1968. [1.13℄ N. Mar uvitz, Waveguide Handbook. Lexington, Mass.: Boston Te hni al Publishers, 1964. [1.14℄ I. Huynen, A. Vander Vorst, \A new variational formulation, appli able to shielded and open multilayered transmission lines with gyrotropi non-hermitian lossy media and lossless ondu tors", IEEE Trans. Mi rowave Theory Te h., vol. MTT-42, pp. 2107-2111, Nov. 1994. [1.15℄ R.A. Waldron, Theory of Guided Ele tromagneti Waves. London: Van Nostrand, 1969. [1.16℄ J.C. Slater, Mi rowave Ele troni s. Prin eton, N.J.: Van Nostrand, 1950. [1.17℄ V.H. Rumsey, "Rea tion on ept in ele tromagneti theory", Phys. Rev., vol. 94, no 6, pp. 1483-1491, June 15, 1954. [1.18℄ R.F. Harrington, Time-Harmoni Ele tromagneti Fields. New York: M Graw-Hill, 1961. [1.19℄ A.D. Berk, \Variational prin iples for ele tromagneti resonators and waveguides", IRE Trans. Antennas Propagat., vol. AP-4, no 2, pp. 104111, Apr. 1956. [1.20℄ F.E. Gardiol, A. Vander Vorst, \Computer analysis of E-plane resonan e isolators", IEEE Trans. Mi rowave Theory Te h., vol. MTT-19, pp. 315-322, Mar. 1971. [1.21℄ I. Huynen, D. Vanhoena ker-Janvier, A. Vander Vorst, \Spe tral domain form of new variational expression for very fast al ulation of multilayered lossy planar line parameters", IEEE Trans. Mi rowave Theory Te h., vol. MTT-42, pp. 2009-2106, Nov. 1994. [1.22℄ M. Fossion, I. Huynen, D. Vanhoena ker, A. Vander Vorst, \A new and simple alibration method for measuring planar lines parameters up to 40 GHz", Pro . 22nd Eur. Mi row. Conf., Helsinki, pp. 180-185, Sept. 1992. [1.23℄ I. Huynen, B. Sto kbroe kx, G.Verstraeten, \An eÆ ient energeti variational prin iple for modelling one-port lossy gyrotropi YIG straightedge resonators", IEEE Trans. Mi rowave Theory Te h., vol. MTT-46, pp. 932-939, July 1998. [1.24℄ R.E. Collin, Field Theory of Guided Waves. New York: M Graw-Hill, 1960. [1.25℄ R.E. Collin, Field Theory of Guided Waves, 2nd edition. NewYork: IEEE Press, 1991.

18

CHAPTER 1.

FUNDAMENTALS

[1.26℄ E. Yamashita, \Variational method for the analysis of mi rostrip-like transmission lines", IEEE Trans. Mi rowave Theory Te h, vol. MTT16, pp. 529-535, Aug. 1968. [1.27℄ W. English, F. Young, \An E ve tor variational formulation of the Maxwell equations for ylindri al waveguide problems", IEEE Trans. Mi rowave Theory Te h., vol. MTT-19, pp. 40-46, Jan. 1971. [1.28℄ I.V. Lindell, \Variational methods for nonstandard eigenvalue problems in waveguide and resonator analysis", IEEE Trans. Mi rowave Theory Te h., vol. MTT-30, pp. 1194-1204, Aug. 1982. [1.29℄ M. Oksanen, H. Maki, I.V. Lindell, \Nonstandard variational method for al ulating attenuation in opti al bers", Mi rowave Opti al Te h. Lett., vol. 3, pp. 160-164, May 1990. [1.30℄ D. Baldomir, P. Hammond, \Geometri al formulation for variational ele tromagneti s", IEE Pro ., vol. 137, PtA, pp. 321-330, Nov. 1990. [1.31℄ J.H. Ri hmond, \On the variational aspe ts of the moment method", IEEE Trans. Mi rowave Theory Te h., vol. MTT-39, pp. 473-479, April 1991. [1.32℄ R.F. Harrington, Field Computation by Moment Methods. New York: Ma Millan, 1968. [1.33℄ S. Wandzura, \Optimality of Galerkin method for s attering omputations", Mi rowave Opti al Te h. Lett., vol. 4, pp. 199-200, Apr. 1991. [1.34℄ J. Jin, W.C. Chew, \Variational formulation of ele tromagneti boundary-value problems involving anisotropi media", Mi rowave Opti al Te h. Lett., vol. 7, pp. 348-351, June 1994. [1.35℄ J.R. Mautz, \Variational aspe ts of the rea tion in the method of moments", IEEE Trans. Antennas Propagat., vol. 42, pp. 1631-1638, De . 1994. [1.36℄ Y. Liu, K.J. Webb, \Variational propagation onstant expressions for lossy inhomogeneous anisotropi waveguides", IEEE Trans. Mi rowave Theory Te h., vol. 43, pp. 1765-1772, Aug. 1995. [1.37℄ I. Huynen, Z. Raida, \Comparison of nite-element method with variational analyti al methods for planar guiding stru tures", Mi rowave and Opti al Te h. Lett., vol. 18, no. 4, pp. 252-258, July 1998.

hapter 2

Variational prin iples in ele tromagneti s 2.1 2.1.1

Basi variational quantities Introdu tion

It is well known that exa t solutions of the equations of physi s may be obtained only for a limited lass of problems. This is true whether the formulation of the equation is dierential or integral. So, we are fa ed with the task of developing approximate te hniques of suÆ ient power to handle most of the problems. In this book, we are interested with distributed ir uits, operating mostly at entimeter and millimeter wavelength. These ir uits are found in high frequen y or high speed ele troni ir uits used in ommuni ations and in omputer systems, where they need to be synthesized. Hen e, the approximate te hniques have to be powerful indeed, with respe t to both a

ura y and speed of al ulation. Perturbation methods are ommonly used. The general theory is well des ribed in the literature, as well as a variety of its appli ations. Perturbations are deviations from exa tly soluble situations. They may be surfa e perturbations, referring to deviations in the boundary surfa e or boundary onditions, or both, from the exa tly soluble ase. Su h te hniques may obviously be used to solve a number of transmission line or avity problems. They may also be volume perturbations, whi h destroy the separability of the problem. In the perturbation method, the volume or surfa e perturbations are assumed to be small and expansions in powers of a parameter measuring the size of the perturbation may be made, the leading term being the solution in the absen e of any perturbation. Perturbation methods are espe ially appropriate when the problem losely resembles one whi h is exa tly solvable. It presumes that one may hange from the exa tly solvable situation to the problem under onsideration in a gradual fashion - the dieren e is not singular in hara ter. This requires the perturbation to be a ontinuous fun tion of a parameter, measuring the importan e of the perturbation. When this is the ase, it is possible to develop formulas whi h des ribe the hange in the physi al situation as the parameter varies from zero. Perturbation formulas may be shown to be the onsequen e of the appli ation of an iterative pro edure to the integral formulation of the

20

CHAPTER 2.

VARIATIONAL PRINCIPLES

IN ELECTROMAGNETICS

problem. They have been developed for the determination of eigenvalues and eigenfun tions, and for problems in whi h the eigenvalues form a ontinuous spe trum, whi h is typi al of s attering and dira tion, in a variety of situations [2.1℄. When the perturbation is large, perturbation methods be ome tedious and the expressions whi h are developed so omplex that the results lose their physi al meaning. It this ase the variational method may be more appropriate [2.2℄. For this method, the equations have to be put in a variational form, usually a variational integral, whi h implies nding a quantity involving the unknown fun tion whi h is to be stationary upon variation of the fun tion. This quantity is a s alar number and is given by the ratio of two integrals. In pra ti e, a fun tion will be used - alled the trial fun tion - involving one or more parameters. It is inserted for the unknown fun tion into the variational prin iple. The fun tion may be varied by hanging the value of the parameters and the pro edure may be improved by introdu ing additional parameters. Also, the original trial fun tion may be improved by making the trial fun tion the rst term in an expansion in a omplete set of fun tions whi h are not ne essarily mutually orthogonal. When the trial fun tion diers from the orre t fun tion by a given quantity, the variational form of the equations diers from the true value by an amount proportional to the se ond-order, and, in the neighborhood of the stationary values, the variational form is less sensitive to the details of the trial fun tion than it is elsewhere. Be ause of the stationary hara ter of the variational expression and its rst-order insensitivity to the errors in the trial fun tions, it is often possible to obtain ex ellent estimates of the unknown with a relatively rude trial fun tion. This property is of great pra ti al importan e. The quantity of interest will be, in our ase, the propagation onstant of a transmission line, the re e tion oeÆ ient of a ir uit or the resonant frequen y of a avity. In ele tromagneti s, there are essentially two physi al quantities whi h an be extremely useful as variational entities: energy and eigenvalues. In this se tion we shall introdu e their variational properties in general terms. Details will be available later in this hapter, and espe ially in Chapter 3 and 4. 2.1.2

Energy

In this book, we are essentially interested with transmission lines and resonators. They are indeed the basi elements of distributed ir uits. Hen e, we are going to develop variational methods to al ulate omplex impedan es and propagation onstants, in a variety of on gurations. It is often possible to arrange the form of a variational prin iple so that the value of the variational quantity for the exa t unknown has a physi al signi an e, for instan e energy. This will be illustrated by the very simple example of the apa itan e of a transmission line per unit length [2.3℄. The ele trostati energy per unit length stored in the eld surrounding

2.1.

21

BASIC VARIATIONAL QUANTITIES

the two ondu ting surfa es of a two- ondu tor transmission line is given by Z Z h    2    2 i 1 " dx dy = C0 V 2 + y (2.1) We = 2 x 2 where V is the potential dieren e between the two ondu tors and " is assumed to be onstant. It an easily be proved that, when al ulating the rst-order variation in We to a rst-order variation in the fun tional form of , subje t to the ondition that the potential dieren e between the two

ondu tors is V , the rst-order variation vanishes, provided  satis es the equation 2 2 +   = 2 = 0 (2.2) x2

y 2

rt

Sin e we know that  is a solution of Lapla e's equation, we obtain the following variational expression for the apa itan e: Z Z 1 C0 = (2.3) t t  dx dy "V 2 r


r

The integral is always positive and, hen e, the stationary value is an absolute minimum. So, if we substitute an approximate value for t  into the integral, the al ulated value C for the apa itan e will always be too large, and the value for the hara teristi impedan e, equal to 1=C0v where v is the speed of light in ", will always be too small. In pra ti e, if the apa itan e is a fun tion of a number of variational parameters, the best possible solution for C is obtained by hoosing the minimum value of C that an be produ ed by varying the parameters. This is obtained by treating the parameters as independent variables and equating all the derivatives of C with respe t to the parameters to zero. This yields a set of homogeneous equations whi h, together with the boundary onditions, gives a solution for the parameters and, as a onsequen e, for the minimum value of C . In this way, what has been demonstrated is a variational prin iple for an upper bound on the apa itan e and, hen e, a lower bound for the hara teristi impedan e. For the same on guration, energeti onsiderations an also yield a varir

ational prin iple for the lower bound of the apa itan e and, hen e, an upper

bound for the hara teristi impedan e. As a matter of fa t, a variational prin iple is easily developed for an upper bound of 1=C , whi h yields a lower bound for the apa itan e. To do so, instead of al ulating dire tly the energy, one solves the problem in terms of the unknown harge distribution on the

ondu tors. The problem is then to nd the potential fun tion whi h satis es Poisson's equation. The method utilizes the Green's fun tion te hnique for solving boundary-value problems, by solving the parti ular expression of Poisson's equation for a unit harge lo ated on a ondu tor, in su h a way that the boundary onditions for the potential are satis ed. It an be shown

22

CHAPTER 2.

VARIATIONAL PRINCIPLES

IN ELECTROMAGNETICS

[2.4℄, as detailed in Chapter 3, that a variational prin iple is obtained for 1=C by multiplying the solution of Poisson's equation, expressed as a fun tion of the Green's fun tion integrated over one ondu tor, by the unknown harge density and integrating over the ondu tor. This is easily done. It yields a variational prin iple for 1=C , in the form of a ratio of integrals in whi h both numerator and denominator are always positive, and the stationary value is an absolute minimum. Hen e it is an upper bound for the apa itan e and a lower bound for the hara teristi impedan e. Obtaining a variational prin iple for an upper bound of a physi al quantity and one for a lower bound for the same quantity is extremely interesting:

omparing the upper and lower bounds yields the maximum possible error in

.

the approximate values

2.1.3

Eigenvalues

The general operator notation will be used here, be ause it reveals most learly the te hnique employed in forming the variational prin iple. As said in Chapter 1, the eigenvalue equation relates dierential or integral operators a ting on the fun tion: (1.1) L( ) = M( ) We shall now prove that the following expression is a variational prin iple for  [2.5℄: R Æ[ R

L( )dS ℄ = Æ[℄ = 0 (1.2) M( )dS where fun tion  is also determined by the variational prin iple. It will be de ned in Subse tion 2.1.4. A square bra ket is pla ed around  to indi ate that the quantity to be varied is not , whi h is the unknown exa t eigenvalue. The integration is over all the volume determined by the independent variable upon whi h and also  depend. Equation (1.2) is easily obtained by multiplying (1.1) by , as yet arbitrary, integrating and solving for . It is obvious that, if the exa t  is inserted into (1.2), the exa t  is obtained. To show that (1.1) follows from (1.2), we onsider the equation Z

[℄ M( ) dV =

Z Z

L( ) dV

(2.4)

Varying  and , i.e. performing the variation, yields Z

[ ℄ M( ) dV + [℄

Æ

Z

Æ

M( ) dV =

Z

Æ

L( ) dV

(2.5)

Inserting the ondition Æ[℄ = 0 and repla ing [℄ by  elsewhere, sin e the ee t of the variation is only al ulated to rst-order, we obtain Z

Æ

 [L( )



M( )℄ dV = 0

(2.6)

2.1.

23

BASIC VARIATIONAL QUANTITIES

Sin e Æ is arbitrary, equation (1.1) follows. One may then wonder about the equation satis ed by , whi h is also determined by the variational prin iple. It may be shown easily [2.2℄ that  satis es the equation and boundary onditions whi h are the adjoints of those satis ed by , hen e  is the solution adjoint to . When the operators L and M are self-adjoint, one has  = and the variational prin iple assumes the simpler form R R Æ [℄ = Æ [

L( )dS M( )dS ℄ = 0

(1.3)

One important point here is to observe that the variational prin iple (1.2) is a ratio of two integrals. Other variational prin iples for  may be obtained: there may indeed be many ways to formulate a problem. A number of examples are given in [2.2℄, to be used in a variety of physi al problems. The formulation used here, as well as in the referen e just mentioned, is very abstra t. The advantage is that it reveals the te hnique from whi h variational prin iples are formed. Spe i examples will follow later in this hapter, and in Chapter 3. 2.1.4

Iterating for improving the trial fun tion

Generally, for estimating the a

ura y of the results obtained by a variational method, the user simply inserts additional variational parameters and observes the onvergen e of the quantity of interest with the number of su h parameters. In prin iple, su h a method must involve an in nite number of parameters, for one is ertain of the answer only when all of a omplete set of fun tions has been employed as trial fun tions in the variational integrals. The variation-iteration method, developed by Morse and Feshba h [2.5℄, is a superb te hnique whi h not only provides an estimate of the error, by giving both an upper and a lower bound to quantities being varied, but also results in a method for systemati ally improving upon the trial fun tion. We have used the method with su

ess in the past, for al ulating modes, in luding higher-order modes, of propagation in waveguides loaded with twodimensional inserts, as well as for resonant frequen ies in avities loaded with three-dimensional inserts, lossy or not [2.6℄-[2.8℄. Our rst presentation already illustrated the fa t that the method was quite powerful. Indeed the

hairman of the session at whi h we presented our method [2.6℄, A. Wexler, pointed out that this was the rst numeri al solution for the ve tor se ondorder eigenvalue equation with partial derivatives, in a two-dimensional inhomogeneous medium. It should be noti ed that those results were obtained more than 30 years ago, at a time when omputers were not at all as powerful and friendly as they are today. A reviewer of one of our papers at that time mentioned that he believed that S hwinger had used the method at the MIT Radiation Laboratory, during World War II. The method an be rapidly outlined. We limit ourselves to positive-

24

CHAPTER 2.

VARIATIONAL PRINCIPLES

IN ELECTROMAGNETICS

de nite self-adjoint operators. Using a formal operator language, the eigenvalue problem is hara terized by

Lp = p Mp (p = 0; 1; 2; : : :) where L and M are positive-de nite self-adjoint operators

(2.7)

p is an unknown eigenfun tion p is the orresponding unknown eigenvalue (0  1  2  : : :)

When the operators are not positive-de nite, the method may still be used after some adjustments are made, however with a slower onvergen e. To solve the eigenvalue equation, a method like the Rayleigh-Ritz pro edure (see Se tion 2.4) uses a limited series expansion in terms of the known eigenfun tions of another eigenvalue equation. When the dieren e between the two equations is not too signi ant, the Rayleigh-Ritz method shows a reasonable onvergen e. However it is often used in other ases, where the dieren e is quite signi ant. The onvergen e of this method is then very slow, whi h is a major drawba k. On the other hand, starting with an initial trial fun tion, the variationiteration method provides a pro ess of iteration, whi h improves this fun tion until the required a

ura y is obtained. The a

ura y is he ked by omparing the upper and lower bounds of the eigenvalue until they are lose enough. We assume here non-degenerate eigenvalues. Let 0 be the initial trial wave fun tion whi h, of ourse, does not satisfy (2.7). The unknown eigenfun tions form a omplete, orthogonal set in terms of whi h the trial fun tion an be expanded: 0 =

1 X p=0

ap p

(2.8)

We then de ne the nth iterate (n is an integer) n by n = L 1 Mn 1 Hen e, from (2.7) to (2.9) n =

1 a X p

n p p=0 p

(2.9) (2.10)

whi h shows that the set n onverges to 0 by elimination of the unwanted

omponents ontained in the trial fun tion, if 0 is smaller than p+1 . When L and M are self-adjoint, the eigenfun tions are given by p =

R R

p Lp dV p Mp dV

(2.11)

2.1.

25

BASIC VARIATIONAL QUANTITIES

whi h an also be written

p = R

R

p Mp dV p ML 1 Mp dV

(2.12)

by noting that

p = p L 1 Mp

(2.13)

It is shown in [2.5℄ that (2.12) and (2.13) express a variational prin iple for

.

Inserting the iterates into (2.12) and (2.13) leads to the following approx-

0 , the lowest eigenvalue: R n 1 Mn 1 dV n 1=2 = R 0  ML 1 Mn 1 dV R n 1 n 1 Mn 1 dV = R n 1 Mn dV R R n Ln dV n Mn 1 dV n 0 = R = R n Mn dV n Mn dV

imations for

(2.14)

(2.15)

The half-integral value of the supers ript is made lear when noting that the set of approximate eigenvalues, in luding both integral and half-integral supers ripts, forms a monotoni de reasing sequen e, approa hing the exa t value

0 from above, if

some amount of

0

was present in the trial fun tions:

n0 1=2  n0  0n+1=2  : : :  0 and

n ; n+1=2

0 0

n!1

!

Z

0 ;

if

(2.16)

0 M0 dV

6= 0

(2.17)

The formal proof of this statement has been given earlier [2.7℄. be noti ed here that

one

iterate leads to the evaluation of

two

It should

su

essive ap-

proximate eigenvalues: one (half-integral supers ript) by introdu ing on e the new iterate into the se ond expression (2.15), and one (integral supers ript) by introdu ing it three times into the same se ond expression (2.15). An extrapolation method an be used even after only one iterate from a given trial fun tion. Furthermore a

lower bound

pro eeded far enough so that

n+1

0

n+1

 0  0

" 1

an also be found for

n0 +1  1

0 .

If the iterations have

then the following inequality holds:

n0 +1=2 n0 +1 1 n0 +1

# (2.18)

26

CHAPTER 2.

VARIATIONAL PRINCIPLES

IN ELECTROMAGNETICS

Only two su

essive iterates are required for this lower bound. If three su

essive iterates are al ulated and if n0 +1 n0 +3=2  21 , then # " n+1=2 n0 +3=2 n+3=2 n+1 0 (2.19) 1 0 0  0 21 0n+3=2 n0 +1 For this method to be used, an estimation of 1 is ne essary. In many transmission lines and avities problems, this estimation is available. Otherwise, a pro edure for nding an approximate value of 1 is outlined in [2.5℄. As mentioned, an extrapolation method is available to obtain a more a

urate estimation of the exa t answer. Assuming that, after some iterations, the only ontamination of the eigenfun tions 0 is the next eigenfun tion 1 , one lets

n = 0 + b1

(2.20)

where b2 is small. Hen e, one has

  n+1 = 0 + b 1 0 1

and

  n+2 = 20 + b 21 0 1

(2.21)

These three iterates lead to expressions of 0n+1=2 , n0 +1 , and 0n+3=2 in terms of the unknown quantities 0 , b2 , and 0 =1 . Hen e, al ulating three su

essive approximate eigenvalues by (2.14) and (2.15), the extrapolated value 0 (together with b2 , and 0 =1 ) an be al ulated. It has been shown [2.5℄ that the ondition b2 0), the elds are des ribed by 1 X
(2.96a) Eyr = B1 e 1 z 1 + Bm m e mz m=2

Hyr = Yr1 B1 e

1 X

1 z


1

m=2

Bm Yrm m e

m z

(2.96b)

Subs ripts l and r refer to areas respe tively to the left and right of the dis ontinuity, respe tively, while Yln and Yrm denote the admittan e of mode n to the left of the dis ontinuity and m to the right of the dis ontinuity, respe tively. The next step is to impose the two fundamental ontinuity equations in the plane of the dis ontinuity z = 0: 1 1 X X A1 (1 + 1 ) 1 + Bn n = B1 1 + Bm m (2.97a) n=2

Yl1 A1 (1

1 ) 1

=

1 X

n=2

m=2

Bn Yln n + Yr1 B1 1 +

1 X

Bm Yrm m

m=2

(2.97b)

This is the exa t solution. Next, we use the mode orthogonality in the two waveguides. Multiplying the two sides of equations (2.97a,b) by the dominant mode of input waveguide 1 , and integrating over the ross-se tion we obtain Z Z 1 X A1 (1 + 1 ) = B1 1 1 dS + Bm m1 dS (2.98a) S

Yl1 A1 (1

1)

= Yr1 B1

m=2

Z S

1 1 dS +

S

1 X m=2

Bm Yrm

Z S

m 1 dS

(2.98b)

48

CHAPTER 2.

VARIATIONAL PRINCIPLES

IN ELECTROMAGNETICS

Use has been made of the fa t that the fun tions fng form a set of orthogonal fun tions on the input waveguide aperture. The pro edure leading to equations (2.98a,b) is alled the mode-mat hing te hnique. The problem is now that we have to repeat the pro edure several times (theoreti ally for a double in nity of modes) with all orthogonal modes n and m in the two waveguides, in order to generate a suÆ ient number of equations relating the Bm and Bn to 1 , and then eliminate them to solve the problem for the re e tion oeÆ ient 1 of the dominant mode only. A variational approa h may be helpful to solve the problem [2.4℄. Assuming that in the jun tion aperture the ele tri eld has the (unknown) distribution E (x; y), equation (2.97a) an be rewritten as A1

(1 +

1 ) 1 +

1 X

n n = B1 1 +

1 X

B

n=2

B

m=2

m m

= E (x; y)

(2.99)

This means that rst, the unknown distribution an be expanded in terms of the modal solutions in ea h of the two waveguides, and se ond, that ea h

oeÆ ient of eld expansions (2.95a,b) and (2.96a,b) an be derived from this distribution, by applying the orthogonality relationship between modes: n

B

B

=

m=

A1

Z

n E (x ; y ) dS 0

S0 Z

0

mE (x ; y ) dS 0

S

(1 +

0

1)

=

Z

0

(2.100b)

0

1 E (x ; y ) dS 0

S0

(2.100a)

0

0

(2.100 )

0

When doing this, we perform a mode-mat hing between the waveguide modes in the left and right areas and the unknown eld distribution in the aperture. The input admittan e in presen e of the dis ontinuity is in

Y

=

1 1+

1 1

Y

(2.101)

l1

Introdu ing (2.100a,b, ) into (2.97b), and ombining the result with (2.100 ) yields R Y

l1 A1 (1

=

1 X

1 ) 1

ln n

Y

n=2

S 0 1 E (x ; y

Z

) dS A1 (1 + 1) 0

0

n E (x ; y ) dS + 0

S0

0

0

0

1 X

Y

m=1

rm m

Z

m E (x ; y ) dS 0

S0

0

0

(2.102)

2.4.

49

TRIAL FIELD DISTRIBUTION

whi h nally provides a relationship for the input admittan e of the dis ontinuity in 1

Y

Z

S0

1 (

0

E x ;y

0

)

dS

0

Z

=

(

0

E x ;y

S0

0

) (

0

G x ;y

0

jx; y) dS

(2.103a)

0

provided that the Green's fun tion (Appendix A) is de ned by (

j

0

G x; y x ; y

0

)=

X 1

Y

n=2

ln n (x ; y 0

0

)n (

x; y

)+

X 1

rm m (x ; y 0

Y

m=1

0

) m (

x; y

)

(2.103b)

To obtain a onvenient expression for the input admittan e, it is now suÆ ient to take the inner produ t of both sides of (2.103a) with the trial fun tion ( ): E x; y

in

Y

Z

S

(

E x; y

Z ) Z = 1

S0

S

1 (

Z ) 0

E x ;y

(

E x; y

0

S0

)

dS

(

0

dS

0

E x ;y

0

) (

0

G x ;y

0

jx; y) dS

0

(2.104) dS

whi h nally yields in as the ratio of two integrals. We nally obtain for the input admittan e of the dis ontinuity Y

ZZ

Y

in =

S S0

(

nZ  ) (

0

E x; y E x ; y

0

(

) (

1 E x; y

S

0

)

jx; y) dS

o

G x ;y

0

0

dS

(2.105)

2

dS

Comparing this expression with equation (1.2), it is obvious that it is an eigenvalue problem, with Green's fun tion (2.103b) as linear operator. Hen e, the eigenvalue in is variational about the eld distribution ( ). This mode-mat hing approa h has been used by some authors [2.28℄ to al ulate the input admittan e of a planar mi rostrip antenna, fed by a oaxial able. 2.4 Trial eld distribution Y

2.4.1

E x; y

Rayleigh-Ritz pro edure

For planar transmission lines, the quasi-stati analysis is based on the omputation of a quasi-TEM lumped- ir uit element, apa itan e or indu tan e, noted . For a number of variational prin iples found in the literature, the expression obtained for is expli it, like p

p

p

= L1 [ i ( L2 [ i (

n )F j (x1 ; x2 ; : : : ; xn )℄ n )F j (x1 ; x2 ; : : : ; xn )℄

F

x1 ; x2 ; : : : ; x

F

x1 ; x2 ; : : : ; x

(2.106)

50

CHAPTER 2.

VARIATIONAL PRINCIPLES

IN ELECTROMAGNETICS

where x1 ; x2 ; : : : ; xn are the variables des ribing the domain of the problem F i and F j the s alar or ve tor quantities asso iated to the problem L1 , L2 denote linear operators. The variational behavior of the s alar quantity p holds only when spe i

onditions are veri ed by trial expressions F t . In ele tromagneti s, these trials are potentials or elds, and they have to satisfy Maxwell's equations and boundary onditions. This is usually not suÆ ient to determine eÆ ient shapes of the trial eld, so that the stationarity of p is frequently used to improve the des ription of trial quantities F t . Methods using the stationarity of p, are referred to as variational methods in [2.29℄. In these methods, the trial expressions are expanded into a set of suitable fun tions weighted by unknown

oeÆ ients. These oeÆ ients, alled variational parameters, are then found as rendering p extremum. When the expansion is linear, the method is known as the Rayleigh-Ritz method. Assuming that F t is des ribed by

F

0

t i;j

= F i;j +

X N

k

k i;j

F

k i;j

(2.107)

=1

the Rayleigh-Ritz method imposes

p =0  k i;j

for k = 1; : : : ; N

(2.108)

The oeÆ ient 0i;j has been normalized to 1, be ause p is al ulated as a ratio. Sin e only produ ts of oeÆ ients ki;j appear in the expression of p, by virtue of (2.106), we obtain from (2.108) a set of N linear equations to be solved for the unknowns oeÆ ients ki;j . Hen e, having obtained the values of the oeÆ ients ki;j , we nd p by using (2.107). It should be underlined that the value obtained is the best one for the number N and set of fun tions k F i;j that have been hosen. If we hange either the number of terms in the k serial expansion or the shape of the fun tions F i;j , we do not know a priori if we nd a better value for p (higher or lower, depending on whether we are looking for a maximum or a minimum, respe tively). So, with a given expansion (2.107) we are never ertain to have found the \best" extremum. Finally, it is now obvious that solving equations (2.52b) and (2.60 ) of the MoM presented in Se tion 2.3, is equivalent to applying the RayleighRitz method to the fun tional  (2.58) asso iated with the problem or to the error fun tional (2.54a).

2.4.2 A

ura y on elds

The fa t that some fun tionals are stationary with respe t to some eld distributions or parameters gives an insight into the a

ura y of the elds. We

hoose trial fun tions and adjust their parameters until the value of the fun tional, expressed in terms of the trial fun tion, is minimized (or maximized).

2.5.

VARIATIONAL PRINCIPLES

FOR CIRCUIT PARAMETERS

51

A tually, by doing this, it is never possible to be sure that the estimate obtained for the fun tional is the lowest (or the highest) one. All that an be said is that it is the best estimate that an be obtained with su h a lass of trial fun tions. It is not possible to determine whether the trial distribution whi h minimizes (or maximizes) the fun tional approa hes the exa t solution for the distribution. This will be true only if the trial distribution forms a

omplete set of eigenfun tions apable of des ribing all distributions, and if this is proven, when the fun tional is extremum, the orresponding trial fun tions are solutions of the equation des ribing the problem. What an be said, however, is that the exa t value will always be lower (or higher) than the best estimate found, and that, if the estimates of the eigenvalues appear to

onverge to a limit when improving the trial fun tions, that limit is probably an eigenvalue of the system [2.26℄. Also, this suggests a way to solve for higher-order modes parameters, using trial fun tions, trial elds and variational prin iples. On e we have identi ed one modal solution (dominant mode, for example), with its own set of trial fun tions obtained from the Rayleigh-Ritz method, it is suÆ ient, for solving for a higher-order mode, to remove from the set of trial fun tions used those already involved in the des ription of the dominant mode. This will be illustrated in an example in the next se tion. 2.5

Variational prin iples for ir uit parameters

Various appli ations of variational prin iples to ele tromagneti problems are found in literature. Many authors provide a review of some typi al problems solvable with variational prin iples. They an be lassi ed as follows: variational prin iples based on energy, variational prin iples asso iated with an eigenvalue problem (expli it or impli it), and variational expressions based on a rea tion. Some examples are presented in this se tion, with omments based on the on epts reviewed throughout this hapter. 2.5.1

Based on ele tromagneti energy

Ele trostati energy is a stationary quantity. Hen e, expressions of ir uit parameters depending linearly on energy guarantee the stationarity of those parameters. In Se tion 2.1.2, this was introdu ed for the ase of the apa itan e per unit length of a transmission line. A rigorous proof will be presented in Chapter 3, for the ase of the apa itan e per unit length, and for its dual parameter, the indu tan e per unit length, that we will show to be related to the magnetostati energy. Baldomir and Hammond [2.30℄ present a geometri al formulation for variational ele tromagneti s based on energy, for both stati and dynami ases. The stationarity of ele trostati energy is helpful for solving in terms of ele trostati potential. It an be demonstrated (Se tion 2.1.2) that under spe i onditions the potential distribution whi h minimizes the ele trostati energy is also the distribution whi h is the solution of Lapla e's equation. This feature is similar to that observed for eld distributions, where the stationarity of a given fun tional is often used to de-

52

CHAPTER 2.

VARIATIONAL PRINCIPLES

IN ELECTROMAGNETICS

rive the solution of a problem, instead of solving exa tly the equations of this problem. 2.5.2

Based on eigenvalue approa h

Waldron [2.26℄ proposes a series of variational prin iples based on an eigenvalue approa h for waveguide omponents, su h as ut-o frequen y, propagation onstant, and resonant frequen y of a waveguide resonator. He bases his approa h on the fa t that \In the ase of a system whi h is arrying waves, the prin iple tells us that of all on eivable wave fun tions, the one a tually obtained will be the one whi h minimizes an appropriate eigenvalue". This suggests of ourse the Rayleigh-Ritz method, whi h we have seen to be eÆ ient for hoosing trial eld expressions. Waldron, however, postulates that the eigenvalue of a waveguide problem is stationary be ause he relates it to the prin iple of least a tion. With this in mind, the resonant frequen y of an os illating system, for example, will always be as low as possible. We will show in this se tion that the proof of the stationarity of waveguide eigenvalues is immediate. For the resonant frequen y of a waveguide resonator, Waldron starts from Helmoltz equation: r2 + !2 "0 0 = 0 (2.109) where ! is the unknown resonant frequen y, and is the z - omponent of either the E or H eld. Multiplying both sides of (2.109) by , integrating on the volume of the avity, and rearranging, yields Z

2

! "0 0

=

VZ

r2 dV

2

V

(2.110)

dV

whi h forms an eigenvalue problem similar to (1.1) and (2.4), with  = !2 "0 0 as eigenfun tion and as eigensolution. Waldron does not prove this stationarity. It is immediate however, sin e varying  and in (2.110) and negle ting se ond-order variations yields Z

Æ

V

2 dV + 2 +

Z

V

Z

V

Æ

Z

dV +

r Æ dV +  2

Z

V



2

V

Æ

dV

r2 dV +

Z

V

r2 dV = 0

(2.111)

The invariant terms are those of equation (2.110), whi h is satis ed by the exa t eld. Next, applying the Green's theorem (Appendix B) Z

V

r

A 2 B dV

=

Z

V

r

B 2 A dV

+

Z

S

r


A B n dS

Z

S

r


B A n dS

(2.112)

2.5.

VARIATIONAL PRINCIPLES

(2.111) be omes Æ

Z

V

2 dV + 2

Z

FOR CIRCUIT PARAMETERS

Æ f + r2 g dV Z rÆ
n dS Æ r
n dS = 0

Z V

+

S

53

(2.113)

S

Sin e (2.109) is satis ed, the only way to an el the rst-order error Æ is to assume that one has Z

S

rÆ
n dS

Z

S

Æ r
n dS = 0

(2.114)

On the boundary of the ross-se tion of an empty waveguide resonator, either or the normal omponent of r vanishes. The only way to satisfy (2.114) is to assume that the trial distribution ( + Æ ) satis es the same boundary

onditions on the surfa e S as the exa t solution does. For the ut-o frequen y of a waveguide, noted ! , the formulas proposed are similar to (2.109) and (2.110), but sin e a waveguide at ut-o an be viewed as a resonator having no variation along the z -axis, the integration is limited to the ross-se tion S of the waveguide: Z

! 2 "0 0 =

ZS

r2 dS

S

"r 2 dS

(2.115)

Away from ut-o, the wave equation be omes r2 + f!2"r "0 0 2 g = 0 (2.116) yielding dire tly the variational expression for the propagation onstant Z



2

=

S

r2 dS + Z

S

Z

S

! 2 "r "0 0 2 dS

2 dS

(2.117)

Equations (2.115) and (2.117) are valid even if the permittivity varies with the position in the waveguide: the relative diele tri onstant "r is left inside the integrals. Applying su h a formulation to waveguides is easy, be ause the wave fun tion is not ae ted by medium inhomogeneities in the transverse se tion: the fun tion is related to the z - omponent of either the ele tri or the magneti eld in the waveguide stru ture, whi h are not subje t to diele tri dependent boundary onditions in the transverse se tion. Also, Green's theorem used for the proof does not involve the diele tri onstants, so that the derivation of the proof arried out for the resonant frequen y of an empty

54

CHAPTER 2.

VARIATIONAL PRINCIPLES

IN ELECTROMAGNETICS

waveguide resonator remains valid for the ut-o frequen y of the diele tri loaded waveguide. It an also be veri ed that the proof for the propagation

onstant, even for an inhomogeneous ase, yields an equation very similar to (2.113) with the same nal onstraint (2.114). To on lude this subse tion, we illustrate the eÆ ien y of the RayleighRitz method, by al ulating the ut-o frequen y of a hollow re tangular waveguide of width 2 ( = 1 m), using variational prin iple (2.115). We assume as rst estimate that the longitudinal omponent z -dependen e has the general polynomial form = 5+ 4+ 3+ 2+ + (2.118) From the general theory of re tangular waveguides, we know that the z

omponent of the dominant mode has an odd -dependen e, and that all eld omponents have no variation along the -axis. Also, the boundary

onditions require that the z omponent vanishes at the lateral boundaries of the waveguide. Hen e, we may a priori simplify the trial fun tion for the dominant mode as dom = 5 + 3 + (2.119a) and impose for =  dom = 5 4 + 3 2 + 1 = 0 (2.119b) a

a

H

Ax

Bx

Cx

Dx

Ex

x

F

H

x

y

H

0

0

B x

A x

x



a

0

0

B a

x

x

A a

whi h yields 2 = 1 +53 4 and the nal form for the trial dominant mode as 2 dom = 1 +53 4 5 + 3 + 0

B

A a

0

a

0

A a

a

x

0

A x

x

(2.119 ) (2.120)

Introdu ing this trial in the variational prin iple (2.115) provides an expression for ut-o frequen y as a fun tion of the unknown parameter . Figure 2.4a shows the dependen e of the resulting ut-o frequen y on the value of . It exhibits a minimum for = 5000. For this parti ular value, the resonant frequen y yielded by the variational prin iple is 7.51 GHz, whi h

orresponds losely to the exa t theoreti al value (dashed line) 0

f

0

A

0

A

A

= 20 2 = 7 5 GHz (2.121) For this value of , Figure 2.4b ompares the trial dom obtained from (2.120) to the exa t solution z = sin( x 2a ). The trial solution is a very good f



a

:

A

0

H

2.5.

VARIATIONAL PRINCIPLES

FOR CIRCUIT PARAMETERS

Cut−off frequency

−3

x 10

12 variational exact

trial exact

10

2

9

0

8

−2

7

−4

GHz

4

0 2 parameter A’ x 104

field Hz

6

11

6 −2

−6 −0.01 0 0.01 −1 spatial variable x in m

(a)

Fig. 2.4

of width and trial

55

(b)

Rayleigh-Ritz pro edure for dominant mode of re tangular waveguide

2a = 1 m (a) fun tional (2.115) versus A ; (b) exa t omponent Hz dom for A = 5000 0

0

approximation of the true eld. Hen e, applying the Rayleigh-Ritz pro edure to variational prin iple (2.115) expressed as a fun tion of the unknown parameter A provides the minimum value of 7.51 GHz for the resonant frequen y, lo ated at the abs issa A = 5000. The resulting trial distribution asso iated to this extremum is shown in Figure 2.4b. A similar reasoning is made for the rst higher-order mode, assuming that its Hz x-dependen e is even: higher = B x4 + A x2 + 1 (2.122a)  higher = 4B a3 + 2A a = 0 (2.122b) 0

0

00

x

00

00

00

yielding

(2.122 ) higher = 2Aa2 x4 + A x2 + 1 Results similar to Figure 2.4 are obtained in Figure 2.5. The extremum value of the ut-o frequen y is now about 15 GHz. It is obtained for A" = 42500, while the asso iated trial distribution slightly diers from the exa t one (Hz =

os( 22xa )). The exa t ut-o frequen y for this higher-order mode is f = 15 GHz. 00

00

56

CHAPTER 2.

VARIATIONAL PRINCIPLES

IN ELECTROMAGNETICS

field Hz

Cut−off frequency 16 1

14

0.8

12

0.6 0.4

GHz

10

0.2

8

0 −0.2

6

−0.4

4

−0.6

0 −10

−0.8

variational exact

2

trial exact

−1

−5 0 parameter A’’ x 104

−0.01 0 0.01 −1 spatial variable x in m

(a)

(b)

Rayleigh-Ritz pro edure for rst-higher order mode of re tangular waveguide of width 2a = 1 m (a) fun tional (2.115) versus A"; (b) exa t

omponent Hz and trial higher for A" = 42500 Fig. 2.5

Both gures illustrate that a

urate values of the fun tional ! 2 an be obtained on e dierent distributions are hosen for the dominant (odd xdependen e) and the higher-order modes (even x-dependen e). We remove the odd ee t of the dominant mode from the general trial fun tion (2.118) when we sear h for the rst-higher order mode, yielding a

urate results for this mode. In Chapters 3 and 4, other appli ations of the Rayleigh-Ritz method will be given. 2.5.3

Based on rea tion

There are numerous formulations for ir uit and system parameters whi h are based on rea tion in ele tromagneti s. The reader will nd in this se tion the most popular ones: input impedan e of an antenna, e ho of an obsta le, and transmission oeÆ ient through an aperture. 2.5.3.1

Input impedan e of an antenna

Harrington [2.17℄ proposes the following formula for the input impedan e of an antenna, supposed to be perfe tly ondu ting: hA; ai = 1 Z E a
K a dS (2.123) Z = in

I2

I2

S

s

2.5.

VARIATIONAL PRINCIPLES

FOR CIRCUIT PARAMETERS

57

where K s is a surfa e urrent (A=m). The antenna is fed by a urrent sour e and the urrent distributes itself on the surfa e S in su h a manner that the resulting ele tri eld has vanishing tangential omponents on the ondu tors

of the antenna. Hen e, the produ t E
K s is zero, ex ept at the feeder point on the surfa e of the antenna, where the total urrent I in ident from the feeding sour e spreads onto the ondu tor, while the voltage between the two terminals of the feeder reates the ele tri eld normal to the antenna ondu tor at the feeding point. Hen e, the rea tion hA0 ; a0 i between orre t eld and sour e redu es to the integration of their produ t between the terminals of the antenna, supposed to be lose together, as shown by:

I,

Zin

=

hA0 ; a0 i = I2

V0 I

(2.124)

where V0 is the orre t voltage a ross the terminals. When a trial surfa e a

urrent K s is assumed, it an be hosen su h that its integration gives the same value of feeding urrent I (this is just a matter of normalization). Hen e, the rea tion hA0 ; ai (where a orre t urrent distribution is assumed) is still equal to the produ t V0 I , whi h validates the equality

hA0 ; ai = hA0 ; a0 i

(2.125)

Next, the nature of the problem (antenna on guration in isotropi medium) ensures that re ipro ity o

urs :

hA0 ; ai = hA; a0 i

(2.126)

Using the notations of Subse tion 2.3.1, the rea tions in (2.126) are rewritten as

hA0 ; ai = hA0 ; a0 i + hA0 ; Æai hA; a0 i = hA0 ; a0 i + hÆA; a0 i

(2.127a) (2.127b)

whi h imposes

hA0 ; Æai = hÆA; a0 i = 0

(2.128)

On the other hand, the trial self-rea tion dedu ed from (2.39) is

hA; ai = hA0 ; a0 i + hÆA; a0 i + hA0 ; Æai + hÆA; Æai

(2.129)

whi h, using (2.128), redu es to

hA; ai = hA0 ; a0 i + hÆA; Æai

(2.130)

It demonstrates that the impedan e of a perfe t ondu ting antenna omputed using (2.123) is variational about the urrent density and asso iated ele tri eld.

58

CHAPTER 2.

VARIATIONAL PRINCIPLES

IN ELECTROMAGNETICS

The formula is widely used for antennas al ulations. Again, it has to be underlined that the trial urrent and eld are not required to satisfy exa t boundary onditions on the ondu tor of the antenna. Identity (2.126) only imposes that the produ t of the exa t eld by the trial urrent, integrated on the surfa e of the antenna, is equal to the produ t of the trial eld by the exa t urrent, integrated on the same surfa e. This indeed does not ensure that the tangential omponent of the trial eld vanishes on ea h point of the surfa e, although it is the ase for the exa t eld. Obviously, it an be easily demonstrated that the following is a variational formula for the mutual impedan e between two antennas, supporting surfa e a b

urrent densities K s and K s : Z 1 h B; ai b a Zm = E
K s dS (2.131) a b = a b I

2.5.3.2

I

I

I

S

Stationary formula for s attering

The ele tri eld s attered by an obsta le is obtainable by a variational rea tion. We assume that the obsta le is a perfe t ele tri ondu tor. Assuming i that the sour e is a urrents element I l, we denote by E the eld generated by this urrent, and by E the eld s attered by the obsta le. The sum of those two elds form the total eld in the whole spa e. The rea tion of the s attered eld on the urrent element is noted hS; ii:

h i= S; i

s l =

I lE

IV

s

(2.132)

where V s is the voltage appearing a ross l. The e ho is then de ned as the ratio between Els and the urrent element I l:




e ho =

E

s l =

I l

h i S; i

h i I; s

= (I l)2 (I l)2 Z 1 i s E
K s dS = (I l)2 S Z 1 s s E
K s dS = = (I l)2 S

(2.133a) (2.133b)

h i S; s

(I l)2

(2.133 )

s

where re ipro ity has been assumed, and K s is the surfa e urrent indu ed on the perfe t ondu ting obsta le. On the surfa e, the total ele tri eld vanishes, yielding equation (2.133 ), relating the e ho to the self-rea tion hS; si. Equations (2.133) are valid for rea tions written with the exa t elds, so they must be written with the subs ript 0. Repla ing hS0 ; s0 i by the trial hS; si yields a variational formula for the e ho, provided that (2.38) is satis ed:

h i=h 0 i = h0 i S; s

S ;s

I ;s

(2.134a) (2.134b)

2.5.

VARIATIONAL PRINCIPLES

59

FOR CIRCUIT PARAMETERS

s

To render rea tion hS; si insensitive to the unknown magnitude of K s , the rea tion is nally expressed, imposing (2.134b), as 2 h i = hh 0 ii I ;s

S; s

(2.135)

S; s

yielding the nal variational form for the e ho as

e ho =

h0 i I ;s

Z 2

E

(I l)2 hS; si

=

i

SZ

(I l)2

S




K

E

s

s 2 s dS




K

(2.136)

s s dS

It has to be noted that the in ident eld is assumed to be derived from the

urrent element, so that no error results on I . 2.5.3.3

Stationary formula for transmission through apertures

As shown in [2.31℄, the dire t appli ation of Babinet's prin iple to the problem of transmission through apertures is illustrated in Figure 2.6: the eld transmitted by an aperture in a plane ondu ting s reen is equal to the opposite of the eld s attered by the omplementary obsta le. The omplementary

complete electric conductor

electric conductor E t , Ht

x

E t , Ht x x x x x

incident plane wave

K

n

y=0 (a)

m

incident plane wave n

y=0 (b)

Transmission through aperture (a) initial on guration; (b) appli ation of Babinet's prin iple

Fig. 2.6

obsta le is a sheet of perfe t ele tri ondu tor, with a perfe t surfa e magneti urrent, denoted by K m , owing on it. The transmission oeÆ ient T of the aperture is ommonly de ned as the ratio between power transmitted

60

CHAPTER 2.

VARIATIONAL PRINCIPLES

IN ELECTROMAGNETICS

through the aperture and power in ident at the aperture:


Re(Pt ) = =

Taperture

Re(Pi )

Re Re

Z

Zaperture aperture

E E

t



t (H ) dS

i



i (H ) dS





(2.137)

The in ident power is known, sin e it depends only on in ident sour e elds. In the aperture, upon appli ation of Babinet's prin iple, the transmitted magneti eld is related to the in ident eld by nH =nH t

i

(2.138)

Assuming that the magneti eld is adjusted to be real in the aperture, then the onjugate an be omitted in (2.137). The equivalent magneti urrent in Figure 2.6b is: Km =

nE

t

(2.139)

yielding the rea tion formulation of the transmitted power: Re(Pt ) =

Re

Z

K m
H dS i

aperture



= Re(hI; i)

(2.140)

where denotes that the orre t magneti urrent distribution is assumed. We denote by a the rea tions involving trial magneti urrents. By virtue of (2.138), we also have

hI; ai = hA; ai

(2.141)

and by re ipro ity, hI; ai = hA; ii. Hen e, we have the onditions for a stationary rea tion. As for e ho al ulations, we normalize the rea tion and obtain the nal variational prin iple for the real part of transmitted power as

 Z

Re(Pt ) =

hI; ai2 = hA; ai

Re

Re

Zaperture aperture




a i K m H dS

2




a a K m H dS



(2.142)

For a plane wave normally in ident to the aperture, with a magneti eld of unit magnitude oriented along dire tion u and a wave impedan e  , the in ident power is given by the produ t Re(Pi ) = Saperture

(2.143)

and the nal variational prin iple for the transmission oeÆ ient is

 Z

Re(Pt ) =

hI; ai2 = hA; ai S

Re

u
n  E dS a

Z

aperture

aperture Re

aperture

H

a

2


n  E a dS



(2.144)

2.6.

2.6

SUMMARY

61

Summary

Variational prin iples have been introdu ed in this hapter in general terms to al ulate impedan es and propagation onstants as, in this book, we are essentially interested with transmission lines and resonators. Variational prin iples have rst been developed so that the value of the variational quantity for the exa t unknown has a physi al signi an e, like energy and eigenvalues. It has been shown that a method exists for systemati ally improving upon the trial fun tion by iteration, thus providing an estimate of the error, by giving both an upper and a lower bound to those quantities whi h are being varied. Then,we developed and illustrated methods for establishing variational prin iples by adequately re-arranging Maxwell's equations. It has been shown that expli it as well as impli it expressions an be obtained. The main part of the hapter has been devoted to omparing variational methods with other on epts and methods. The rea tion on ept has been demonstrated to be eÆ ient in obtaining variational prin iples for a wide variety of ele tromagneti parameters, and self-rea tion has been shown to have some additional interesting properties. The method of moments (MoM) has been developed and widely ommented upon and it has been shown that it is possible to minimize the error made. Variational fun tionals asso iated with the MoM have been dis ussed. The Galerkin method has also been dis ussed in detail and we have presented where and when the MoM and Galerkin method are variational or not. Links have been established between the MoM and the rea tion on ept as well as with the perturbation method. The differen es between perturbational and variational methods have been arefully detailed. The mode-mat hing te hnique has been brie y summarized, as well as the Rayleigh-Ritz method. Finally, examples of variational prin iples for ir uit parameters have been presented and on epts reviewed throughout the hapter, su h as waveguide

ut-o frequen ies as eigenvalues, input impedan e of an antenna, e ho from an obsta le, and transmission oeÆ ient through an aperture. 2.7

Referen es

[2.1℄ P.M. Morse, H. Feshba h, Methods of Theoreti al Physi s. New York: M Graw-Hill, 1953, pp. 1001-1106. [2.2℄ P.M. Morse, H. Feshba h, Methods of Theoreti al Physi s. New York: M Graw-Hill, 1953, pp. 1106-1158. [2.3℄ R.E. Collin, Field Theory of Guided Waves. New York: M Graw-Hill, 1960, pp. 148-150. [2.4℄ R.E. Collin, Field Theory of Guided Waves, 2nd ed. New York: IEEE Press, 1991, Ch. 8, pp. 547-552. [2.5℄ P.M. Morse, H. Feshba h, Methods of Theoreti al Physi s, pp. 10261030 and 1137-1158. New York: M Graw-Hill, 1953.

62

CHAPTER 2.

VARIATIONAL PRINCIPLES

IN ELECTROMAGNETICS

[2.6℄ A. Vander Vorst, R. Govaerts, F. Gardiol, \Appli ation of the variationiteration method to inhomogeneously loaded waveguides", Pro . 1st Eur. Mi rowave Conf., pp. 105-109, Sept. 1969. [2.7℄ A. Vander Vorst, R. Govaerts, \Appli ation of a variation-iteration method to inhomogeneously loaded waveguides", IEEE Trans. Mi rowave Theory Te h., vol. MTT-18, pp. 468-475, Aug. 1970. [2.8℄ A. Laloux, R. Govaerts, A. Vander Vorst, \Appli ation of a variationiteration method to waveguides with inhomogeneous lossy loads", IEEE Trans. Mi rowave Theory Te h., vol. MTT-22, pp. 229-236, Mar. 1974. [2.9℄ A.D. Berk, \Variational prin iples for ele tromagneti resonators and waveguides", IRE Trans. Antennas Propagat., vol. AP-4, no. 2, pp. 104111, Apr. 1956. [2.10℄ I.V. Lindell, \Variational methods for nonstandard eigenvalue problems in waveguide and resonator analysis", IEEE Trans. Mi rowave Theory Te h., vol. MTT-30, pp. 1194-1204, Aug. 1982. [2.11℄ M. Oksanen, H. Maki, I.V. Lindell, \Nonstandard variational method for al ulating attenuation in opti al bers", Mi rowave Opt. Te hnol. Lett., vol. 3, pp. 160-164, May 1990. [2.12℄ W. English, F. Young, \An E ve tor variational formulation of the Maxwell equations for ylindri al waveguide problems", IEEE Trans. Mi rowave Theory Te h., vol. MTT-19, pp. 40-46, Jan. 1971. [2.13℄ A. Konrad, \Ve tor variational formulation of ele tromagneti elds in anisotropi media", IEEE Trans. Mi rowave Theory Te h., vol. MTT24, pp. 553-559, Sept. 1976. [2.14℄ K. Morishita, N. Kumagai, \Uni ed approa h to the derivation of variational expressions for ele tromagneti elds ", IEEE Trans. Mi rowave Theory Te h., vol. MTT-25, pp. 34-40, Jan. 1977. [2.15℄ V.H. Rumsey, \Rea tion on ept in ele tromagneti theory", Phys.Rev., vol. 94, pp. 1483-1491, June 1954, and vol. 95, p. 1705, Sept. 1954. [2.16℄ R.F. Harrington, A. T. Villeneuve, \Re ipro ity relationships for gyrotropi media", IRE Trans. Mi rowave Theory Te h., pp. 308-310, July 1958. [2.17℄ R.F. Harrington, Time-Harmoni Ele tromagneti Fields. New York : M Graw-Hill, 1961, Ch. 7, p. 341. [2.18℄ R.F. Harrington, Field Computation by Moment Methods. New York: Ma millan, 1968. [2.19℄ E.K. Miller, L. Medguesi-Mits hang, E.H. Newman (Editors), Computational Ele tromagneti s - Frequen y-Domain Method of Moments. New York: IEEE Press, 1992. [2.20℄ I. Huynen, Modelling planar ir uits at entimeter and millimeter wavelengths by using a new variational prin iple, Ph.D. dissertation. Louvain-la-Neuve, Belgium: Mi rowaves UCL, 1994.

2.7.

REFERENCES

63

[2.21℄ R.F. Harrington, \Origin and development of the method of moments for eld omputations", in Computational Ele tromagneti s Frequen y-Domain Method of Moments (edited by E. K. Miller, L. Medguesi-Mits hang, E.H. Newman). New York: IEEE Press, 1992, pp. 43-47. [2.22℄ A.F. Peterson, D.R. Wilton, R.E. Jorgenson, \Variational Nature of Galerkin and Non-Galerkin Moment Method Solutions", IEEE Trans. Antennas Propagat., vol. 44, no. 4, pp. 500-503, Apr. 1996. [2.23℄ R.H. Jansen, \The spe tral domain approa h for mi rowave integrated

ir uits", IEEE Trans. Mi rowave Theory Te h., vol. MTT-33, no. 10, pp. 1043-1056, O t. 1985. [2.24℄ T. Itoh, \An overview on numeri al te hniques for modelling miniaturized passive omponents", Ann. Tele ommun., vol. 41, pp. 449-462, Sept.-O t. 1986. [2.25℄ J.H. Ri hmond, \On the variational aspe ts of the Moment Method", IEEE Trans. Antennas Propagat., vol. 39, no. 4, pp. 473-479, Apr. 1991. [2.26℄ R.A. Waldron, Theory of Guided Ele tromagneti Waves. London: Van Nostrand Reinhold Company, 1969. [2.27℄ T. Itoh, Numeri al Methods for Mi rowave and Millimeter Wave Passive stru tures. New York: John Wiley&Sons, 1989, Ch. 5. [2.28℄ W. C. Chew, Z. Nie, and T.T. Lo, \The ee t of feed on the input impedan e of a mi rostrip antenna", Mi rowave Opt. Te hnol. Lett., vol. 3, no. 3, Mar h 1990, pp. 79-82. [2.29℄ P.M. Morse, H. Feshba h, Methods of Theoreti al Physi s, Volume II. New York: M Graw-Hill, 1953, Ch. 11. [2.30℄ D. Baldomir, P. Hammond, \Geometri al formulation for variational ele tromagneti s", IEE Pro eedings, vol. 137, Pt. A, no. 6, pp. 321-330, Nov. 1990. [2.31℄ R.F. Harrington, Time-Harmoni Ele tromagneti Fields. New York : M Graw-Hill, 1961, Ch. 7, p. 365.

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hapter 3

Analysis of planar transmission lines This hapter details the various forms of variational prin iples presented in Chapter 2 for the analysis of planar transmission lines and ir uits. The formalism is based on the use of Green's fun tions and Fourier transforms. After a brief overview of the various analysis te hniques, the emphasis is put on the variational methods and their variants. They will be ompared with a general variational model in Chapter 4. First the three basi line topologies used in mi rowave integrated ir uits (MICs) are presented, namely the mi rostrip line, the slot-line and the oplanar waveguide, as well as their shielded variants. Se ondly, the various analysis methods of planar lines are des ribed. They are divided into two lasses: approximate quasi-stati methods and full-wave methods. The advantages and drawba ks of ea h method are highlighted. Thirdly, an in-depth analysis of the variational behavior underlying ea h method is proposed. 3.1

Topologies used in MICs

The analysis methods for planar lines at mi rowave and millimeter wave frequen ies an be divided into two lasses: the approximate quasi-stati methods and the full-wave dynami methods. The rst ategory is based on the assumption that the dominant mode, propagating along the z -axis of the line, is well approximated by a TEM des ription. This is the ase when the a tual mode is of a TM kind, with the TEM mode being a parti ular ase of a TM mode. The se ond ategory uses a hybrid TE+TM representation of the elds in the guiding stru ture [3.1℄, yielding an a

urate frequen y-dependent des ription of the propagation hara teristi s. The basi line topologies used in mi rowave integrated ir uits are represented in Figure 3.1. They are respe tively: the mi rostrip line (a), the slot-line ( ), and the oplanar waveguide (e). Various ombinations of these lines may be onsidered with a view to obtaining oupled stru tures. The basi on gurations for oupled lines are depi ted in Figure 3.1b, d, and f. It should be noted that the oplanar waveguide topology is in fa t the odd- oupled version of the slot-line (with a

hange of sign of the ele tri eld omponent a ross the slot), while the even oupled version of the slot-line is referred to as oupled slots in the literature (Fig. 3.1f). The basi line topologies of Figure 3.1 are open. The shielded versions are used at high frequen ies to avoid radiation losses: they onsist in

66

CHAPTER 3.

ANALYSIS OF PLANAR TRANSMISSION LINES

inserting the planar line into a metalli en losure. In parti ular, the shielded version of the slot-line, more ommonly denoted nline, is widely used for appli ations at millimeter waves.

W layer 1

layer 1 az

ax az

layer 2

ay

(a)

ax

layer 2

ay

(b)

W

layer 1 layer 1

az

ax

ax

layer 2 ay

layer 3

az

layer 2 ay

layer 3 (d)

(c) layer 1 az

ay ax

layer 1 az

layer 2 layer 3

ay

ax

layer 2 layer 3

(e)

Fig. 3.1 Basi line topologies for MICs (a) open mi rostrip; (b) oupled mi rostrips; ( ) open slot-line; (d) oupled slot-lines; (e) open oplanar waveguide; (f ) oupled oplanar waveguides

From these basi topologies a number of ombinations are possible, involving stru tures having metallizations in dierent planes and layers of dierent width. Two parti ular stru tures are shown in Figure 3.2a, b [3.2℄. They are the antipodal nline and the p-i-n travelling wave photodete tor respe tively. These two stru tures a t like transmission lines at mi rowave and millimeter wave frequen ies. Hen e, they an be analysed using onventional transmission lines analysis te hniques.

3.2.

67

QUASI-STATIC METHODS

air W = 25 µm

metal (Au) 0.5µm

p+ InP

1µm 6.4 S/mm

n- InGaAs 2µm 0.0 S/mm

az

ax

n+ InP 2µm

p+InGaAs 0.1µm 13.4 S/mm 128.0 S/mm

ay s.i. substrate 500 µm

(a)

0.0 S/mm

(b)

Examples of ombinations of basi line topologies (a) antipodal nline; (b) p-i-n photodete tor

Fig. 3.2

3.2

Quasi-stati methods

The quasi-stati methods are based on the fa t that the TEM approximation holds. This is the ase when the line has at least two ondu tors. Under the TEM assumption, the transverse elds are very lose to the stati ele tri and magneti elds. They are derived from the ele trostati s alar or magnetostati ve tor potentials respe tively, whi h are both solution of Lapla e's equation. The main advantage of the quasi-stati approa h is that the equations for the ele tri and magneti elds are totally de oupled and hen e mu h simpler to handle: we are left with two stati problems, ele tri and magneti respe tively. The propagation onstant and the hara teristi impedan e of the line are expressed as [3.3℄[3.4℄

p

= ! LC Z =

rL

(3.1a)

(3.1b) C where L and C are the indu tan e and apa itan e per unit length respe tively, dedu ed from the stati elds. When the two ondu tors of the line are surrounded by a homogeneous medium having the onstitutive parameters ", , the phase velo ity on the line is given by

1 1 (3.2) =p =p 0 vph = p " "r r LC with 0 phase velo ity in va uum "r , r relative permittivity and permeability respe tively. When the stru ture is multilayered, relations (3.2) are no longer valid, be ause the onstitutive parameters are dierent in the two material layers. The

68

CHAPTER 3.

ANALYSIS OF PLANAR TRANSMISSION LINES

main feature of the quasi-stati method is that one an derive the equivalent parameters of a unique ee tive homogeneous medium from the parameters of the two layers. They provide a single ee tive phase velo ity, whi h is

omputed from the value of the ee tive stati parameters L or C of the multilayered stru ture:

vph =

p1

LC

=

p"eff1 eff

=

p"reff 0reff

(3.3)

For multilayered diele tri stru tures, where all the layers have the same relative magneti permeability, the stati apa itan e C of the stru ture is

al ulated and ompared to the apa itan e C0 of the same stru ture for whi h all layers are assumed to have the parameters "0 , 0 of va uum. This yields an ee tive relative permittivity

"reff =

C C0

(3.4a)

L0 is obtained from C0 as L0 =

1 C0 20

(3.4b)

and the line is des ribed as an equivalent TEM-line having the parameters L0 , C . For multilayered magneti stru tures, where all layers have the same relative diele tri permittivity, the indu tan e L of the stru ture is al ulated and ompared to the indu tan e L0 - the same stru ture for whi h all the layers are assumed to have the parameters "0 , 0 of va uum. This yields an ee tive relative permeability

reff =

L L0

(3.5a)

C0 is obtained from L0 as C0 =

1 L0 20

(3.5b)

and the line is des ribed as an equivalent TEM-line with parameters L, C0 . When the layers are both ele tri ally and magneti ally inhomogeneous, the two previous methods are ombined in the following manner: the indu tan e L is omputed for an ele tri ally homogeneous medium using (3.5a) and the

apa itan e C is omputed for a magneti ally homogeneous medium using (3.4b). Those two stati parameters are ombined to form the quasi-TEM transmission line parameters (3.1b). The quasi-stati methods only dier on how the stati apa itan e C or indu tan e L are obtained. This is explained below. It should be noted that quasi-stati methods annot be applied to slot lines, be ause these support a dominant mode whi h is fundamentally TE.

3.2.

69

QUASI-STATIC METHODS

layer 1 az ay (a)

ax layer 2

ε ra2

εr1

x

(b)

aε r 2 x

εr1

(c)

Fig. 3.3

Conformal mapping for mi rostrip line (a) original stru ture; (b) rigorous mapping of right-hand side of original stru ture; ( ) approximate

apa itive stru ture

3.2.1 Modi ed onformal mapping

The modi ed onformal mapping method has been widely used to al ulate the equivalent stati apa itan e of mi rostrip and oplanar waveguide lines. It onsists of a transformation of the original stru ture into a parallel plate

apa itor in a transformed domain, for whi h the al ulation of the apa itan e or indu tan e is straightforward. Conformal mapping is applied rst to the homogeneous line, and then to the multilayered line. The appli ation of this method is illustrated in Figure 3.3 for mi rostrip lines and in Figure 3.4 for oplanar waveguides respe tively. In the ase of original mi rostrip topologies (Fig. 3.3a), an exa t and simple analyti al expression an always be found for the apa itan e C0 ("r1 = "r2 = 1). This is not true for the apa itan e C , be ause the interfa e y = 0 is transformed into an ellipti al-looking urve (Fig. 3.3b). Wheeler [3.5℄[3.6℄ proposed to approximate this transformed boundary by a re tangular boundary, as shown at Figure 3.3 . Hen e, an equivalent lling fa tor is de ned for the parallel plate apa itan e, and an equivalent ee tive diele tri onstant is obtained for the line. Unfortunately, the transformation of the diele tri -air boundary results in simple expressions only when this boundary

orresponds to a line of ele tri eld, whi h is not the ase for the mi rostrip introdu ed in Figure 3.1a. Moreover the al ulation of the onformal mapping requires the evaluation of ellipti fun tions, whi h onverge slowly. In the ase of oplanar lines (Fig. 3.1e), the onformal mapping is more rigorous, sin e the ele tri eld a ross the slot is tangential to the diele tri interfa e. Hen e, when layer 2 is of in nite thi kness, an exa t solution for the apa itan e of the transformed stru ture of Figure 3.4b is obtained using a S hwarz-Christoel onformal mapping [3.7℄. Due to the obvious symmetry of the stru ture, the ee tive diele tri onstant is the mean value of the diele tri onstants of the two layers as shown by: "eff

=

"1 + "2

2

(3.6)

70

CHAPTER 3.

W

S

ANALYSIS OF PLANAR TRANSMISSION LINES

layer 1

εr2

+

εr1

layer 2 (a)

(b) layer 1

εr3

layer 2

εr2

+

εr1

layer 3 (c)

(d)

Conformal mapping for oplanar waveguide (a) original two-layered stru ture; (b) partial apa itan es orresponding to two-layered stru ture; ( ) original three-layered stru ture; (d) partial apa itan es orresponding to three-layered stru ture

Fig. 3.4

while the total apa itan e C0 is obtained as a fun tion of the omplete ellipti integral of the rst kind K (k ): C0

= 4"0

K (k ) K (k 0 )

(3.7)

p

with k = S=(S + 2W ) k = 1 0

k2

When layer 2 has a nite thi kness (Fig. 3.4 ), the stru ture in the transformed domain has to be modi ed as shown in Figure 3.4d: layer 3 is mapped into an area of ellipti shape. In this ase the apa itan e has to be omputed via a distributed shunt-series apa itan es modeling, as done by Davis et al. [3.8℄. In addition to the stati assumption made for the al ulation of the elds, the onformal mapping method has important limitations. First, it is only appli able to open stru tures (Fig. 3.1a, and e). Se ondly, the ee t of the nite thi kness of the metallization is never taken into a

ount. Thirdly, the nite thi kness of the layers has to be evaluated by numeri al dis retizations or urve ttings on a presumed transformed boundary, without any a priori knowledge of the in uen e of the approximation on the result. 3.2.2

Planar waveguide model

The planar waveguide model is appli able to open mi rostrip lines. It has been developed by Kompa [3.9℄ and Mehran [3.10℄. It onsists of a waveguide having perfe t ondu tive top and bottom shieldings, while lateral limits are perfe t

3.2.

71

QUASI-STATIC METHODS

W

Weff(f)

layer 1

az

ax

H

εeff(f)

H

layer 2

ay (a)

(b) perfect magnetic conductor

perfect electric conductor

Fig. 3.5

Planar waveguide model (a) original mi rostrip stru ture; (b) equivalent waveguide

magneti walls. The guide is lled by the ee tive diele tri onstant of the mi rostrip line (Fig. 3.5), al ulated by another method [3.11℄. The ee tive diele tri onstant and the width of the waveguide are frequen y-dependent, to ensure that the impedan e of the TEM mode supported by this waveguide is a good representation of the a tual impedan e of the dominant mode on the mi rostrip. By extrapolation, this stru ture is expe ted to be eÆ ient for the al ulation of the parameters of higher-order modes supported by the mi rostrip line. The model is mainly applied to model planar mi rostrip dis ontinuities. In this ase, mode mat hing requires the knowledge of the higher-order modes supported by the mi rostrip line. 3.2.3 Finite-dieren e method for solving Lapla e's equation

The transverse dependen e of the ele tri eld of the TEM mode is derived from the stati potential , whi h has to satisfy the two-dimensional Lapla e's equation 2 2 (3.8) r2 ( ) =  +  = 0 x; y





2 x

y

2

The equation is dis retized by the nite-dieren e method. Hen e, the method is only appli able to shielded lines [3.11℄, as illustrated in Figure 3.6. The potential is assumed to be zero on one ondu tor and 0 on the other. The appli ation of the nite-dieren e method, taking into a

ount the boundary

ondition at the diele tri interfa e, is detailed in [3.12℄. Basi ally, the solution for the potential in the neighborhood of point (points , , , ) is expanded into a Taylor's series around the solution at point , noted A : 3 3 2 2  4 (3.9a) B = A + + 2 + 3! 3 + ( ) A A A 2! V

A

B

A

h



x

h



x

h



x

O h

C

D

E

72

CHAPTER 3.

ANALYSIS OF PLANAR TRANSMISSION LINES

B E

A

layer 1

C

Φ = Vo

Γ

D

layer 2

Φ=0

Dis retized domain for analysis of shielded mi rostrip line by using the nite-dieren e method

Fig. 3.6







3 3 2 2 D = A h x + h2! x2 h3! x3 + O(h4 ) A A A 2 3 3 2 C = A + h y + h2! y2 + h3! y3 + O(h4 ) A A A

E = A







  h2  2  h + y A 2! y 2 A

h3  3  4 3! y3 A + O(h )

(3.9b) (3.9 ) (3.9d)

where subs ript A means that quantities are evaluated at point A and where O(h4 ) is a quantity proportional to h4 . Adding these four equations yields: B + C + D + E = 4A + h

2



2 x2

+

2 y 2



A

+ O(h4 )

(3.10)

Sin e  has to satisfy (3.8) everywhere, the following is a good approximation for the potential solution of (3.8): A = B + C +4 D + E (3.11) whi h an be repeated at ea h mesh point. It has a somewhat dierent form if point A is lo ated at the interfa e between two dierent media. At the boundary points,  or its derivative, or a linear ombination of both, is spe i ed. Hen e, (3.11) written for ea h mesh point plus the set of boundary onditions may be rewritten using a matrix formalism as M

=b

(3.12)

3.2.

QUASI-STATIC METHODS

73

where the right-hand side is a olumn ve tor ontaining information provided by the boundary points, and  is a olumn ve tor for the unknown values of potential at the various mesh points. Solving (3.12) for  is equivalent, to the h4 order, to obtaining a solution for (3.8) satisfying some imposed boundary

onditions. Hen e, the method is eÆ ient for small values of h, whi h implies that the mesh has to be suÆ iently small. On e the potential is obtained, the ele tri eld normal to the strip ondu tor of Figure 3.6 is al ulated and the total harge Ql per unit length on the strip is obtained as a line integral Ql

=

I

"i E i
dn

(3.13)

where is the ontour of the strip. Hen e, the apa itan e per unit length of the line is expressed as C

=

Ql V0

(3.14)

The mathemati al prepro essing is minimal, and the method an be applied to a wide range of stru tures but a pri e to pay is its numeri al ineÆ ien y. Open stru tures have to be trun ated to a nite size, and the method requires that mesh points lie on the boundary. The main drawba k of the method is that an a

urate expression of the apa itan e is obtained only when the dis retization of the domain of interest is small enough. This may require

onsiderable omputation time and memory spa e. The method, however, is used for hara terizing transmission lines ontaining layers with highly inhomogeneous onstitutive parameters as is the ase for inter onne tions on doped semi ondu ting layers. The nite-dieren e method is also used in the time domain as a full-wave dynami method. What has been des ribed is the original pro edure for dis retizing the stru ture, where square - or ubi - mesh elements are used, with a uniform mesh parameter throughout. This pro edure has been modi ed over the years with the aim to redu e errors, memory requirements, and numeri al expenditure, as well as toward improving the eÆ ien y of programming te hniques. Regular and irregular graded meshes have been used, for adapting the density of the network to lo al non-uniformities of the elds, resulting from sharp

orners of ns, for instan e. If dierent mesh sizes are used along the dierent

oordinate axes, then the indu tan e and apa itan e per unit length as well as the length of the bran hes, are dierent a

ording to the dire tions, and the equations have to take these dieren es into a

ount. So- alled ondensed node s hemes have led to onsiderable savings in omputer resour es, parti ularly when they are ombined with a graded mesh te hnique in whi h the density of the mesh an be adapted to the degree of eld non-uniformity. However, it is not our intent to give a thorough presentation of the numeri al methods used for solving ele tromagneti problems. The interested reader will nd a number of good books on the subje t.

74

CHAPTER 3.

3.2.4

ANALYSIS OF PLANAR TRANSMISSION LINES

Integral formulations using Green's fun tions

The Green's formalism (Appendix A) is interesting for quasi-stati formulations, be ause it oers a elegant way to obtain solutions for Poisson's equation

r2 (x; y; z ) =

(A.23)

(x; y; z )

This equation is the basis of most ele trostati problems, whi h usually require nding the harge and potential distributions present in a given stru ture in order to al ulate the equivalent apa itan e of the system. A previous se tion illustrated this approa h, when using Lapla e's equation. Lapla e's equation is a parti ular ase of the Poisson's equation, where the sour e harge density  is assumed to be zero. A. Equation (3.8) may also be solved using a Green's formalism whi h imposes boundary onditions on the two line ontours modeling the surfa e of the two ondu tors in the transverse se tion of the line (Fig. 3.6). The general Green's fun tion asso iated with Diri hlet boundary onditions on a surfa e expressed as P (r; t) =

Z

Z

x

G(r; tjr 0 ; t0 ) 
P (r0 ; t0 ) x dr0 dt0 n Tx

(A.20d)

is formulated in two dimensions. The potential is the solution of a homogeneous Poisson's equation whi h satis es inhomogeneous boundary onditions on the ontours limiting the two ondu tors in the transverse se tion of the transmission line (Fig. 3.6): (r) =

Z

1

GP (rjr1 )(r1 )d 1 + n

Z

2

GP (rjr2 )(r2 )d 2 n

(3.15)

where the spa e ve tors are two-dimensional, GP is the s alar Green's fun tion asso iated to the two-dimensional Poisson's equation derived from (A.23), and n is the normal to the line ontour onsidered. Assuming that (r 1 ) = 0 on 1 and (r2 ) = V0 on 2 , the rst term of the sum vanishes. Then, equations (3.13) and (3.14) with = 2 yield the apa itan e C . B. Collin [3.13℄ has given another simple expression for the apa itan e of a TEM transmission line, also based on a Green's fun tion. The formulation relates the apa itan e to the inner produ t between a harge distribution in the ross-se tion of the line and the orresponding ele trostati potential: I 1 1 = 2 (x; y)(x; y)d (3.16) C

Ql

where the line path is taken on the surfa e of the inner ondu tor in the transverse se tion (x; y) is the stati potential solution of the two-dimensional Poisson's equation

3.2.

75

QUASI-STATIC METHODS

(x; y ) is the distribution of harge density on the surfa e of the ondu tor Ql is the integral of the harge density on the inner ondu tor. The potential is then related to the harge density through the two-dimensional s alar Green's fun tion asso iated to Poisson's equation, S being the rossse tion of the transmission line:

(x; y) =

Z

S

GP (x; y jx0 ; y 0 )(x0 ; y 0 )dx0 dy 0

(3.17)

This equation simpli es when the inner ondu tor is assumed to be of zero thi kness in plane y = 0 and the harge density  is lo alized at its surfa e. There is then a surfa e harge density s (x) distributed on the ondu ting strip only. The total harge on the inner ondu tor per unit length is given by the line integral over the width W of the inner ondu tor Ql

=

Z

W2

W2 s (x)dx

(3.18)

Sin e the normal derivative of the potential is proportional to the harge density on the ondu tor, the Green's formalism asso iated to Neumann boundary

onditions is appli able: P (r 0 ; t0 ) 0 0 P (r; t) = G(r; tjr ; t )
dr0 dt0 n x

x Tx Z

Z

(A.20b)

The potential an be rewritten as a fun tion of s (x) using the same Green's fun tion asso iated with Poisson's equation: (x; y) =

Z

W2

0 0 0 W2 GP (x; yjx ; 0)s (x )dx

(3.19)

Equation (3.17) illustrates the pra ti al use of two-dimensional Green's fun tions. Sin e the transverse se tion of the line is not homogeneous, the orre t

ontinuity has to be imposed between the solutions found for ea h layer. This

an be done either in the spa e or in the spe tral domain along the x-variable. The hoi e will depend on the omplexity of the geometry and, in parti ular, the lo ation of the interfa es between the dierent layers, as explained in Se tion 3.4. The main problem in using (3.17) and (3.19) is that there is no a priori knowledge of the harge distribution. The previous formalism requires the knowledge of the sour e of ex itation. The a tual harge distribution, however, results from the boundary onditions for the potential on the strip and at the interfa es between the layers. Hen e, the problem seems to have no pra ti al solution. The rst way to solve this diÆ ulty is to assume a reasonably

76

CHAPTER 3.

ANALYSIS OF PLANAR TRANSMISSION LINES

adequate distribution for the surfa e harge  . Intuitively, the distribution will be relevant if the variational behavior of expression (3.16) about  has been proven. The se ond way is to solve the integral equations (3.17) or (3.19) using the moment method, applying a known boundary ondition. This is in fa t an alternate way to solve an inhomogeneous problem subje t to inhomogeneous boundary onditions. For our parti ular ase, the linear integral equation to be onsidered is s

s

(x; y)j(j j W2 x

;y

=0)

= V0 =

Z

W2



(3.20)

0 0 0 W2 G (x; yjx ; 0) (x )dx P

s

(jxj W2 ;y=0)

It has to be solved for the line path des ribing the ondu ting strip, supposed to be at potential V0 . On e the Green's fun tion has been found, it an be

onsidered as a linear integral operator for the boundary ondition problem (3.20). Using the notations of Chapter 2, the problem is formulated as L(f ) = g

on

(3.21a)

with f (x; y ) = (x; y ) Z

(3.21b)

GP (x; y jx0 ; y 0 )(x0 ; y 0 )dx0 dy 0

(3.21 )

(x; y)j( )2 = V0 whi h redu es in this ase to a one-dimensional problem:

(3.21d)

L(f ) =

S

g (x; y ) =

x;y

f (x) = s (x) L(f ) =

(3.22a)

W2

Z

0 0 0 W2 G (x; 0jx ; 0) (x )dx P

(3.22b)

s

(x; 0)jj j W2 = V0 (3.22 ) The harge density is then expanded into a set of basis fun tions  weighted by unknown oeÆ ients a : g (x) =

x

sn

n

s (x) =

N X

=1

(3.23)

an sn

n

The basis fun tions are taken su h as sn (x) = 0

for jxj >

W

2

(3.24)

3.3.

77

FULL-WAVE DYNAMIC METHODS

Applying Galerkin's pro edure, the oeÆ ients an be determined. Equation (3.18) then yields the total harge Q on the inner ondu tor, and the

apa itan e is obtained from de nition (3.14). A third pro edure has been presented by Gupta et al. [3.11℄. Equations (3.22) are dis retized in order to provide the orresponding matrix formulation: l

v

= pq

(3.25)

where v and q are olumn matri es representing the potential and the density of surfa e harge on the ondu tors respe tively, while p is a matrix ontaining the ee t of the Green's fun tion. Sin e the potential is usually known on the

ondu tor, system (3.25) may be inverted to nd the harge ve tor q. The

apa itan e is then obtained using (3.14) as the integral of the harge density on the ondu tor: C

=

XX j

(3.26)

pjk1

k

where p 1 is the jkth- omponent of the inverse of matrix p. To on lude this subse tion, it has to be pointed out that the main advantage of the ombination Green's fun tion - integral formulation is that the integration is usually performed on a redu ed domain of the ross-se tion, as is illustrated by the transformation of (3.21) into (3.22). Sin e the Green's fun tion ontains information about the entire stru ture, the potential satis es the physi al laws des ribing the system (Poisson's equation, for instan e). Parti ular boundary onditions, ne essary to ensure a omplete determination of the solution, are applied at spe i interfa es only (surfa es or lines). jk

3.3

Full-wave dynami methods

At high frequen ies the quasi-stati approximation is no longer valid for multilayered stru tures, be ause the ele tromagneti elds supported by the stru ture have longitudinal omponents whi h are no longer negligible. These omponents are ne essary to ensure that the boundary onditions are satis ed at the interfa es. The elds annot be derived anymore from stati potential solutions of Lapla e's or Poisson's equation. The elds must be expressed as a ombination of TE and TM modes whose parameters dier from one layer to the other, ea h mode being asso iated with a s alar potential solution of Helmholtz equation [3.4℄:

r2 TE + (k2 + 2)TE = 0 i

i

i

r2 TM + (k2 + 2)TM = 0 with k = !p"  and where subs ript i refers to layer i. i

i

i

i

i

i

(3.27a) (3.27b)

78

CHAPTER 3.

ANALYSIS OF PLANAR TRANSMISSION LINES

A omplex e z dependen e is assumed along the propagation axis. The propagation onstant is unknown and has to be al ulated. Sin e the physi al elds in the stru ture are ombinations of TE and TM modes, they are usually asso iated with the on ept of \hybrid modes". The most popular method for the full-wave analysis of planar transmission lines is the integral equation expressed as a line boundary ondition and solved by Galerkin's pro edure. It has been widely used for many years and almost all the methods found in the literature derive from it. It is des ribed in the next subse tion. Another method is also presently used for the dynami analysis of planar lines: the transmission line matrix method (TLM), ombined with the nite-dieren e time domain method (FDTD). It should be mentioned, however, that a parti ular method has been developed by Cohn for slot-lines [3.14℄, in order to obtain simple expressions for the parameters of these lines, as has been done for mi rostrips and oplanar waveguides in the stati ase. It will be presented in Se tion 3.3.6. 3.3.1

Formulation of integral equation for boundary onditions

As for the quasi-stati appli ations based on an integral equation, the formulation starts from a relationship involving a Green's fun tion. The Green's fun tion may be derived in two ways, as mentioned by Itoh [3.15℄. First it an be derived from the Green's fun tion established for the ve tor potential solution of the ve tor wave equation introdu ed in Appendix A in the following way. For strip-like problems, the strip is repla ed by an in nitely thin sheet of urrent, hen e removing the metallization. In ea h layer, the ele tri eld is derived from a ve tor wave potential satisfying the ve tor wave equation

r2 A(r ) + !2 "A(r) =

(A.27)

J (r )

The Green's fun tion asso iated with this equation is G0 (r jr 0 ) = I

e j jkj
jr r0 j jr r0 j

(3.28)

It an be shown [3.16℄ that equation (A.27) for the potential has an equivalent form for the ele tri eld E :

r  r  E jkj2 E =

(3.29)

j!0 J

A Green's fun tion GE may be dire tly asso iated with the ele tri eld solution of this equation, by repla ing E by GE and j!0 J by the spa e impulse unit fun tion Æ (r 0

r0 ) = Æ (x0

x0 )Æ (y 0

y0 )Æ (z 0

z0 )

(A.18d)

3.3.

79

FULL-WAVE DYNAMIC METHODS

The solution of (3.29) then be omes [3.16℄ 1 rr) e jjkj
jr r0 j ki2 jr r0 j from whi h a Green's relationship between E and J applies: GE (r jr0 ) = (I +

E (r ) =

Z

V

GE (r jr 0 )
J (r 0 )dr0

(3.30) (3.31)

where V is the volume wherein the ele tri eld has to be determined. It should be noted that expression (3.30) has to be modi ed when the medium is nite; thus when spe i boundary onditions apply to the ele tri eld at the boundary surfa e S en losing the volume V . It has been demonstrated that in this ase a se ond term must be added to the solution [3.17℄: E (r ) =

Z

ZV

GE (r jr 0 )
J (r 0 )dr0

Z

+ (n  r0  E )
GE dS 0 + (r0  GE )
(n  E )dS 0 S

(3.32)

S

where r0 means derivatives with respe t to the sour e oordinates r0 . The

two surfa e integrals an be related to equivalent ele tri and magneti \polarization" urrents owing on the interfa e planes, taking into a

ount the boundary onditions at those interfa es. When the medium is multilayered, parti ular solutions must be added to GE , so that the boundary onditions at ea h interfa e are satis ed [3.18℄. For a general ase, the solution be omes quite intri ate, be ause the spatial form of Green's fun tions for a strati ed medium is obtained as a serial expansion involving residues integrals. The method is detailed in Wait [3.19℄. The se ond way to derive the Green's fun tion is more onvenient to use. It starts from the sour e-free Helmholtz equations (3.33a,b), for whi h a general solution may be found in the spe tral domain for both TE and TM sour e-free Hertzian potentials:

r2 Ei + (ki2 + 2 )Ei = 0

(3.33a)

r2 Mi + (ki2 + 2 )Mi = 0 (3.33b) Applying Maxwell's equations, the ele tri and magneti elds in ea h layer of the stru ture (Fig. 3.1) are derived in the spe tral domain from the general solution of the respe tive potentials (3.33a,b). From the derivation of the spe tral form of Green's fun tions (Appendix A), it follows that the partial derivatives in the spatial form of Maxwell's equations are repla ed by a produ t of the orresponding spe tral omponents in the spe tral domain. As a result, simple algebrai expressions are obtained between elds and potentials

80

CHAPTER 3.

ANALYSIS OF PLANAR TRANSMISSION LINES

in the spe tral domain. The details of the elds derivation will be illustrated in Chapter 4. As another result, ele tri and magneti elds do depend upon ea h other sin e they derive from the same potentials. An equivalent tangential urrent density J t is de ned as the dieren e between the tangential magneti elds at the interfa e between two layers where there is a planar

ondu tor (plane y = 0 in Figure 3.1). Using the dependen e between the ele tri and the magneti elds, the following equations are obtained in this plane (subs ript t holds for tangential), in spe tral form: ~

~ (k ; 0; z ) = G (k ; 0; )
J~ (k ; 0; z ) for strip-like problems E t x st x t x ~

~ (k ; 0; z ) for slot-like problems J~ t (kx ; 0; z ) = Gsl (kx ; 0; )
E t x or, after inversion, in the spa e domain: Z 1 E t (x; 0; z ) = Gst (x; 0jx0 ; 0; )
J t (x0 ; 0; z 0 )dx0

1

= Lst (J t ) J t (x; 0; z ) =

Z 1

1

for strip-like problems

Gsl (x; 0jx0 ; 0; )
E t (x0 ; 0; z 0 )dx0

= Lsl (E t )

for slot-like problems

(3.34a) (3.34b)

(3.35a)

(3.35b)

In these expressions, G is the inverse Fourier-transform along the x-variable ~ in the spa e domain of the spe tral dependen e G. The spe tral quantities are de ned by the x-Fourier-transform (Appendix C). It should now be obvious that the G fun tions are Green's fun tions in the sense of (3.31) and (3.32). First, there is a orresponden e between the algebrai dyadi relation (3.34) and the spe tral de nition of the Green's fun tion: ~

~ (k ; k ; k ) P~ (kx ; ky ; kz ; ! ) = G(kx ; ky ; kz ; ! )
X x y z Se ondly, the potential forms satisfying (A.27) or (3.33a,b) are equivalent sin e they both derive from Maxwell's equations. Thirdly, the linearity of the Fourier-transform ensures that the homogeneous problem with inhomogeneous boundary onditions solved in the spe tral domain for ea h layer is equivalent to the spatial form (3.32). Equations (3.32) and (3.34a,b) derive indeed from (A.27) and (3.33a,b), respe tively. The potentials are a fun tion of the propagation onstant by virtue of (3.27) and (3.33a,b). Hen e, the spe tral and spatial notations of the Green's fun tions involve the unknown propagation onstant along the z -axis. This is in a

ordan e with the general form for elds and potentials having a propagation dependen e e jkz0 z along the z 0 -dire tion: 0

3.3.

81

FULL-WAVE DYNAMIC METHODS

P (r; t) =

Z

Z

Vx Tx

~

G(r; tjr 0 ; t0 ; kz0 )
X (r 0 ; t0 )dr0 dt0

with ~

G(r; tjr0 ; t0 ; kz0 ) =

Z

Z

Vx Tx

(A.40a)

0 G(x; y; z; tjx0 ; y 0 ; z 0 ; t0 )e jkz0 (z z) d(z 0

z)

For a distributed stru ture, the z -dependen e an be dropped in equations (3.34a,b) and (3.35a,b), by de ning the elds and urrents as


E t (x; y; z ) = et (x; y )e z

(3.36a)


J t (x; y; z ) = j t (x; y )e z

(3.36b)

Introdu ing (3.36) into (3.34) and (3.35) and dividing both sides of ea h equation by e z yields ~ ~ ~et (kx ; 0) = G st (kx ; 0; )
j t (kx ; 0) for strip-like problems

(3.37a)

~ ~j (k ; 0) = G et (kx ; 0) for slot-like problems sl (kx ; 0; )
~ t x

(3.37b)

or, in the spa e domain, Z 1 et (x; 0) = Gst (x; 0jx0 ; 0; )
j t (x0 ; 0)dx0

1

(3.38a)

1

(3.38b)

= Lst (j t ) for strip-like problems Z 1 j t (x; 0) = Gsl (x; 0jx0 ; 0; )
et (x0 ; 0)dx0 = Lsl (et )

for slot-like problems

Expressions (3.35a,b) and (3.38a,b) are integral equations where the Green's fun tion is a linear integral operator L. They are basi ally obtained in a similar way, only the hoi e of the nal expression hanges: for strip-like problems, the Green's formulation yields the tangential ele tri eld at the interfa e of the strip as a fun tion of the urrent on the strip, while for slot-like problems it yields the urrent distribution at the interfa e of the metallization as a fun tion of the ele tri eld in the slot. One may also say that the sour e for strip-like problems is the urrent density owing on the strip, while for slot-like problems it is the ele tri eld in the plane of the slot. Throughout this se tion, the term \sour e" will now be used for either E t or J t . Using equations (3.33a,b) takes advantage from the fa t that elds are al ulated in the spe tral domain for a homogeneous sour e-free equation (3.33a,b), and that the integral equation involves the sour e at the interfa e only. This is not the ase when using (3.32), be ause sour es have to be integrated on the

82

CHAPTER 3.

ANALYSIS OF PLANAR TRANSMISSION LINES

surfa e limiting the volume of interest for adequately taking into a

ount the boundary onditions. One may also say that, when using equation (3.32), the integral equation has to be applied to the whole spa e while, when using equations (3.33a,b), sour e ee ts apply only in the metalli interfa e. Formulation (3.35a,b) presents the same diÆ ulty as the quasi-stati problem (3.19), sin e the sour e distribution is unknown. If the ondu tors are perfe t, however, it is known that the sour e has to vanish on the whole interfa e y = 0, ex ept on the line path des ribing the strip for strip-like problems or the slot for slot-like problems. Hen e, formulations (3.35a,b) are most appropriate for an integral boundary ondition formulation. The equivalent sour e in the right-hand side of (3.35a,b) is indeed lo ated on a redu ed segment of the interfa e, whi h limits the integration on the strip or slot area (Fig. 3.7): ( )=0

Et x

for jxj >

W

2

( )=0

for jxj >

W

( )=0

for jxj
W2

and u = x; z

(3.61b)

3.3.

91

FULL-WAVE DYNAMIC METHODS

For slot-like problems, a similar formulation is applied to the sour e et : et (x; 0) = alx elx (x)ax + alz elz (x)az

(3.62a)

with elu (x) = 0 for jxj >

W

and u = x; z

2

(3.62b)

In this ase, notations (3.60a,b) be ome ~ det(Lst ( )) = 0

for strip-like problems

(3.63a)

~ det(Lsl ( )) = 0

for slot-like problems

(3.63b)

with

L~

st;sl

2 L~ ( ) = 4 ~ L

~

( )xx Lst;sl ( )xz ~ st;sl ( )zx Lst;sl ( )zz

st;sl

3 5

(3.63 )

where L~st;sl ( )uv are now s alar integral expressions involving the Green's operator:

L~ ( ) = st

uv

L~ ( ) = sl

uv

Z1

1

Z1 1

~jlu (kx )G~ st;uv (kx ; 0; )~jlv (kx )dkx for strip-like problems ~ sl;uv (kx ; 0; )~elv (kx )dkx e~lu (kx )G for slot-like problems

(3.63d)

(3.63e)

with k; n = 1; :::; N and u; v = x; z . This simpli ation is referred to, in the literature, as the rst-order approximation. The zero-order approximation

onsists of negle ting the z - omponent of the sour e ele tri eld in the ase of slot-like lines, or the x- omponent of the sour e urrent density for strip-like lines. Under those assumptions, onditions (3.63d,e) simplify into

Z1

1 Z1 1

~jlz (kx )G~ st;zz (kx ; 0; )~jlz (kx )dkx = 0 for strip-like problems (3.64a) ~ sl;xx (kx ; 0; )~elx (kx )dkx = 0 for slot-like problems (3.64b) e~lx (kx )G

~ sin e the matrix Lst;sl ( ) now redu es to a s alar. The zero-order approximation has been widely used for omputing the propagation onstant of planar transmission lines, be ause it is the less time onsuming. However, it still requires the solution of an impli it equation for the propagation onstant .

92

CHAPTER 3.

ANALYSIS OF PLANAR TRANSMISSION LINES

3.3.5 Approximate modeling for slot-lines: equivalent magneti

urrent

In his famous paper Cohn [3.14℄ presented the rst attempt to model the transmission line behavior of a slot-line. His quasi-stati approa h is based on the de nition of an equivalent magneti urrent dedu ed from the transverse tangential ele tri eld in the slot aperture. If the slot width W is mu h smaller than the free-spa e wavelength, this approximation is valid. This equivalent line urrent M is embedded in a equivalent homogeneous medium with a permittivity equal to ("r + 1)=2. Then M is onsidered as a sour e in Maxwell's equations, from whi h the magneti eld H is dedu ed. This model is useful for evaluating the far- eld behavior of the ele tri and magneti eld. The line approximation made about the aperture urrent of ourse limits the eÆ ien y of this model, whi h is relevant for narrow slots only. 3.3.6

Modeling slot-lines: transverse resonan e method

The transverse resonan e method was developed for slot-lines by Cohn [3.14℄ and applied by Mariani [3.34℄. In this method the slot-line is modeled as a re tangular waveguide, as illustrated in Figure 3.8. In Figure 3.8a the slotline is bounded by ele tri walls pla ed perpendi ular to the z-propagation axis, spa ed by a distan e a = s =2. This ombination will of ourse not disturb the elds of the stru ture. Next, in Figure 3.8b, ele tri or magneti walls are inserted in planes parallel to the slot and perpendi ular to the substrate at planes x = b=2, where b is hosen large enough to assume that the walls have no ee t on the elds. The introdu tion of ele tri walls generates the on guration of a apa itive iris (Figure 3.8 )in a waveguide having the y -propagation axis and is usually preferred. The transverse resonan e method imposes the sum of the sus eptan es along the y-axis to be zero. This sum in ludes the sus eptan es of the TE mode looking in the +y and y dire tions and the apa itive iris sus eptan e due to higher order modes on both sides of the iris. >From this equation, the value of a is extra ted, whi h provides the slot-line wavelength. The main drawba ks of the method are that the al ulation of the sus eptan e involves series expansions that onverge slowly and are ompli ated fun tions of a, and it is not valid for ratios W=H greater than unity. et al.

3.3.7

Transmission Line Matrix (TLM) method

In the TLM method, invented by Johns and developed by Hoefer [3.35℄[3.36℄, the eld problem is onverted into a three-dimensional network problem. In its simplest form, the spa e is dis retized into a three-dimensional latti e with period
l, to des ribe six eld omponents by a hybrid TLM ell obtained as a

ombination of shunt and series nodes (Fig. 3.9a, ) and modeled as equivalent transmission lines (Fig. 3.9b,d). Some equivalen ies an be found between the eld omponents and the voltages, as well as urrents and lumped elements of the ell. This method has been developed for modeling time-domain propagation. It has also been improved by a number of authors [3.15℄ and has

3.3.

93

FULL-WAVE DYNAMIC METHODS

PEC

(a) a

ay

b

az

ay

ax

ax

PEW or PMW

(b)

(a)

b

az

ay

az ax

a (c)

Equivalent waveguide for modeling slot-lines by transverse resonan e te hnique (a) limitation of slot along z -axis by using perfe t ele tri walls; (b) limitation of slot along x-axis by using perfe t ele tri or magneti walls; ( ) equivalent apa itive iris Fig. 3.8

be ome a subje t in itself whi h we do not over here in detail. Choi and Hoefer [3.37℄ have shown that the eÆ ien y of the TLM method an be enhan ed by using a nite-dieren e method in the time domain (FDTD): the

ombination of TLM and FDTD redu es by half the omputation time and memory spa e needed for the TLM method only. The time-domain response

94

CHAPTER 3.

ANALYSIS OF PLANAR TRANSMISSION LINES

L∆l/ 2 L∆l/ 2

Iz Ix

L∆l/ 2

L∆l/ 2

2C ∆l/2

Vy

∆l

(b)

(a)

2C ∆l/2

L∆l/ 2

∆l/ 2

Vz

L∆l/ 2 ∆l/ 2

2C ∆l/2

Iy

L∆l/ 2 Vx

2C ∆l/2

L∆l/ 2

2C ∆l/2

(c)

(d)

Equivalent nodes for Transmission Line Matrix method [3.38℄ (a) shunt node; (b) equivalent lumped element model of shunt node; ( ) series node; (d) equivalent lumped element model of series node

Fig. 3.9

of the ell is omputed, from whi h the frequen y response is obtained by Fourier transform. From our point of view, it is important to note that the drawba k of the TLM method is that a ltering ee t is introdu ed in the frequen y response, be ause of the spatial dis retization. It is, however useful, for analyzing planar dis ontinuities. 3.4

Analyti al variational formulations

The previous se tions have presented a wide variety of methods for analyzing planar multilayered transmission lines. This se tion on entrates on variational methods. They are lassi ed as expli it or impli it variational forms

3.4.

ANALYTICAL VARIATIONAL FORMULATIONS

95

(Chapter 2). As introdu ed in Chapter 1, a variational prin iple is usually formulated as a ratio of integral forms, yielding an expli it expression for a s alar parameter of an equivalent ir uit or transmission line, while variational methods use the well-known variational hara ter of a fun tional in order to ompute nearly exa t fun tions or eld distributions. Those elds, however, may be a fun tion of the propagation onstant , so that rendering the fun tional extremum must also yield the a tual value for the propagation

onstant. Hen e, the possible variational hara ter of the propagation onstant has to be investigated in onne tion with the possible variational hara ter of the hara teristi equation in the ase of non-standard, impli it, eigenvalues problems des ribed in the previous se tion. We rst on entrate on variational prin iples providing an expli it expression for quasi-stati equivalent transmission line parameters. We shall then larify the variational hara ter of Galerkin's pro edure as impli it non-standard eigenvalue problem for the propagation onstant of planar lines. 3.4.1

Expli it variational form for a quasi-stati analysis

For planar transmission lines, the quasi-stati analysis is based on the omputation of a quasi-TEM lumped- ir uit omponent, apa itan e or indu tan e. 3.4.1.1

Quasi-TEM apa itan e based on Green's formalism

Formulations (3.16) to (3.19) presented in Se tion 3.3 is a well-known variational prin iple for the apa itan e. Hen e, the harge density on the ondu tor is onsidered as the trial quantity, and the trial potential is dedu ed via (3.17) on e the Green's fun tion asso iated to the problem has been found. For the present purpose, the variational prin iple is rewritten using (3.16) to (3.19) and (3.22a) as I 1
1 (3.65) = 2 (x; y)(x; y)d Y = C

whi h yields Z W2 Yf

W2

Ql

s (x)dxg2

=

Z W2

Z W2

W2 s (x)f

0 0 0 W2 GP (x; 0jx ; 0)s (x )dx gdx

(3.66)

where GP is the s alar Green's fun tion asso iated to the problem des ribed by Poisson's equation. The proof of the variational behavior of Y with respe t to distribution of surfa e harge s (x) is given by Collin [3.39℄. It is brie y reported here. We use a simpli ed notation for the integration on the strip

ondu tor: Z Z W2 : : : dx : : : dx = W2 strip

96

CHAPTER 3.

ANALYSIS OF PLANAR TRANSMISSION LINES

Varying Y and s in (3.66) and negle ting se ond-order variations, we obtain Z Z Z s (x)dxg2 s (x)dxg + ÆY f Æs (x)dxgf 2Y f strip strip strip Z Z GP (x; 0jx0 ; 0)s (x0 )dx0 gdx Æs (x)f = (3.67) strip strip Z Z + s (x)f GP (x; 0jx0 ; 0)Æs (x0 )dx0 gdx strip

strip

Assuming that the Green's fun tion is symmetri al in variables x and x0 , so that the re ipro ity theorem holds, right-hand side of (3.67) is re ombined into Z Z Z 0 0 0 V0 Æs (x)dx (3.68) GP (x; 0jx ; 0)s (x )dx gdx = Æs (x)f strip

strip

strip

sin e the Green's integral in (3.68) is pre isely the de nition of (3.19), (3.20) and (3.22b, ) of the a tual potential on the strip whi h is equal to V0 : Z W2 0 0 0 V0 = (3.69) W GP (x; 0jx ; 0)s (x )dx

2

Equation (3.67) is nally written as Z Z Æs (x)dxg[V0 s (x)dxg2 = 2f ÆY f strip

strip

Y

f

Z strip

s (x)dx

g℄

(3.70a)

where Y and s are the exa t quantities. By virtue of (3.18), the right-hand side of (3.70a) is transformed into Z 2f Æs (x)dxg(V0 Y Ql ) (3.70b) strip

where (3.14) appears. Hen e, expression (3.70a) vanishes. So, instead of solving Poisson's equation for a parti ular harge distribution, it is possible to obtain a value of the apa itan e per unit length whi h is orre t to the se ond-order, by taking a trial harge distribution and ombining it with the exa t Green's fun tion asso iated to Poisson's equation. Sin e the Green's fun tion is assumed to be symmetri al, Collin states that the integral in (3.66) is always positive and that the variational value obtained for the inverse of the apa itan e is a minimum [3.40℄. Hen e, the value provided by (3.66) is always smaller than the a tual one. Se tion 3.3.4 brie y outlined that the spe tral domain formulation oers an easy way to express Green's fun tions for planar ir uits. Yamashita and Mittra have investigated this approa h when using variational prin iple (3.16) applied to a multilayered mi rostrip line [3.41℄[3.42℄. This is a good illustration

3.4.

97

ANALYTICAL VARIATIONAL FORMULATIONS

strip layer 3

az ax

H2

layer 2

ay

H1

layer 1

Fig. 3.10 Mi rostrip line investigated by Yamashita and Mittra [3.41℄[3.42℄

of various on epts and methods presented in Se tions 3.2 and 3.3. It is reported here as an example. Yamashita and Mittra onsider a multilayered mi rostrip (Fig. 3.10) and assume that the metallizations have a zero thi kness. The density of surfa e

harge on the strip satis es (3.18) and (3.22a), is lo ated at the interfa e between layer 1 and 2, and an be des ribed as a fun tion of the -variable only. Hen e, the stati potential in the three layers satis es the harge-free Poisson's equation, namely the two-dimensional Lapla e's equation. An inhomogeneous boundary ondition applies at interfa e between layers 1 and 2 to the normal omponent of the ele tri eld. The -Fourier transform of Lapla e's equation (3.8) yields a se ond-order ordinary dierential equation for the spe tral form of the potential: 2 ~ ( ) 2~ ( )=0 (3.71) x

x



i kx ; y

y

2

kx

i kx ; y

where the subs ript refers to layer , layer 1 is the ondu tor-ba ked layer, layer 3 the semi-in nite air layer, and ~ ( ) the spe tral potential. Equation (3.71) has the well-known general solution in ea h layer : ~ ( ) = ( ) x + ( ) x (3.72) On the other hand, the transverse ele trostati eld derives from the potential in ea h layer:   ( )  ( ) (3.73a) [ ( ) ( )℄ = i

i

i kx ; y

i

i kx ; y

Ai kx e

k y

Exi x; y ; Eyi x; y

Bi kx e



k y

i x; y x

;



i x; y y

and transforms in the spe tral -domain into  ~ ~ ~ ( ) f ( ) [ ( ) ( )℄ =  x ( ) g kx

Exi kx ; y ; Eyi kx ; y

j kx

i kx ; y ; kx

Bi kx e

kx y A i kx e

k y

(3.73b)

98

CHAPTER 3.

ANALYSIS OF PLANAR TRANSMISSION LINES

The knowledge of the six oeÆ ients fAi (kx ); Bi (kx )g is suÆ ient to des ribe the potential and the ele tri eld. These elds have to satisfy the following Fourier-transformed boundary onditions at the various interfa es: a. on the ground plane y = H1 + H2   (3.74a) E~x3 kx ; (H1 + H2 ) = 0 b. on the diele tri interfa e y = H2 E~x2 (kx ; H2 ) = E~x3 (kx ; H2 ) (3.74b) (3.74 ) "2 E~y2 (kx ; H2 ) = "3 E~y3 (kx ; H2 )

. on the diele tri interfa e y = 0 E~x2 (kx ; 0) = E~x1 (kx ; 0) (3.74d) (3.74e) "1 E~y1 (kx ; 0) "2 E~y2 (kx ; 0) = ~s (kx ) where ~s (kx ) is the x-Fourier transform of the harge distribution s (x) modeling the strip d. at y = 1 the ele tri eld and potential have to vanish, whi h imposes A1 (kx ) = 0 (3.74f) The six equations (3.74a-f) form an inhomogeneous system of six linear equations to be solved for the six unknown oeÆ ients fAi (kx ); Bi (kx )g. Sin e only the right-hand side of (3.74 ) is dierent from zero and equal to ~s (kx ), ea h oeÆ ient fAi (kx ); Bi (kx )g is proportional to ~s (kx ): s (kx ) (3.75a) Ai (kx ) = KA (kx )~ s (kx ) (3.75b) Bi (kx ) = KB (kx )~ with, of ourse, KA1 (kx ) = 0. Hen e, the potential in ea h layer is expressed as (3.76) ~ i (kx ; y) = fKA (kx )ek y + KB (kx )e k y g~s (kx ) i

i

x

i

x

i

whi h yields the spe tral form of the Green's fun tion relating the potential to the harge in ea h layer: ~ P (kx ; y) = fKA (kx )ek y + KB (kx )e k y g (3.77) G i

x

i

x

As already mentioned, at this stage of the development the user has the

hoi e either to invert the spe tral Green's fun tion to obtain its orresponding form in the spa e domain, or to omplete the solution of the problem in the spe tral domain. Yamashita etal: has hosen the se ond approa h. He expresses the basi relation (3.16) in the spe tral domain, using Parseval's

3.4.

99

ANALYTICAL VARIATIONAL FORMULATIONS

theorem. Keeping in mind that the harge distribution s (x) satis es (3.65) and vanishes outside the strip yields 1 = 1 Z  (x)(x; y)dx = 1 Z 1  (x)(x; y)dx (3.78a) C

Q2l strip s Q2l Z 1 1 ~ 2 (kx ; 0)dkx ~s (kx ) = 2Q 2 l 1

1

s

(3.78b)

Introdu ing (3.66) and (3.76) in (3.78b), then yields 1 = 1 Z 1 ~ (k )fK (k ) + K (k )g~ (k )dk B2 x s x x C 2Q2l 1 s x A2 x

(3.78 )

where Ql is obtained from (3.18). Expression (3.78 ) has the advantage that the tedious inversion of the Green's fun tion (3.77) is avoided and that the integrands result in a produ t of simple algebrai expressions. Yamashita has also investigated various shapes for the trial harge density and determined the most adequate by using the Rayleigh-Ritz method. His results agree with experiments up to 4 GHz. Obviously the model fails at higher frequen ies be ause the stati assumption is no longer valid. 3.4.1.2

Quasi-TEM indu tan e based on Green's formalism

Most of the quasi-stati appli ations of variational prin iples for modeling quasi-TEM planar lines are based on the omputation of the apa itan e per unit length. This approa h is not valid when magneti materials are involved in the ross-se tion of the line, be ause the indu tan e of the line is no longer equal to the indu tan e in air. As shown in Se tion 3.2, instead of omputing the pair (C; C0 ) from whi h indu tan e L0 is dedu ed, the pair (L; L0 ) must be

omputed whi h yields C0 . Hen e, a variational prin iple for the indu tan e is useful. It is expe ted that, by duality, the variational form for the indu tan e will be similar to (3.16). In magnetostati s, the elds are derived from a magneti ve tor potential A satisfying the ve tor equation

r2 A(x; y) =

0 r
J (x; y )

(3.79)

The stati ele tri and magneti elds and the magneti ux density satisfy respe tively

r  A = B (x; y)

(3.80a)

r  H = J (x; y) r
B =0 B = 0 r
H

(3.80b) (3.80 ) (3.80d)

100

CHAPTER 3.

ANALYSIS OF PLANAR TRANSMISSION LINES

By virtue of (3.80b) the magneti eld an be derived from a s alar potential in the areas where the urrent density is zero. The rst variational form presented for the indu tan e is mentioned by Collin [3.43℄: Z

1

L=

I2

strip

Jz (x; 0)0 (x; 0)dx

(3.81)

where J (x; 0) is the distribution of the longitudinal urrent density owing on the strip: Z W2 Z
J (x; 0)dx = I = (3.82) W2 J (x; 0)dx z

z

z

strip

and 0 (x) is the integrated indu tion ux per unit length between the ground plane and the strip (Fig 3.11). Indeed one has 0 (x)
z = =

Z z+
z Z 0 I

z

Bx (x; y )dydz

(3.83a)

H




(3.83b)

A d

= [A (x; 0) z

=
z

Z

Az (x;

H )℄
z

(3.83 )

j

(3.83d)

GW;zz (x; 0 x0 ; 0)Jz (x0 ; 0)dx0

strip

using Stoke's theorem and noting that the integrals on the two verti al paths of at z and z +
z an el. The ve tor potential is de ned to be zero on the ground plane. The proof of the variational behavior of (3.81) is similar to that of equation (3.16) in the previous Se tion. Introdu ing (3.83d) in (3.81), varying L and J , and negle ting se ond-order variations, we have z

Z

2Lf

gf

ÆJz (x; 0)dx

Z

strip

= +

Z

ÆJz (x; 0)

strip

Jz (x; 0)

strip

f

Z

Z

f

Z

g + ÆLf

Jz (x; 0)dx

strip

Z

g

Jz (x; 0)dx 2

strip

GW;zz (x; 0 x0 ; 0)Jz (x0 ; 0)dx0 dx

j

strip

g

(3.84)

GW;zz (x; 0 x0 ; 0)Jz (x0 ; 0)dx0 dx

strip

j

g

Assuming again that the Green's fun tion satis es re ipro ity, the right-hand side of (3.84) is re ombined into Z

Z

strip

f

Z

GW;zz (x; 0 x0 ; 0)Jz (x0 ; 0)dx0 ÆJz (x; 0)dx

strip

=

j

0 (x)ÆJ (x; 0)dx z

strip

g

(3.85)

3.4.

101

ANALYTICAL VARIATIONAL FORMULATIONS

Wc layer 1 strip

dz az

ax

x2 ax

x1

H

Γ

ay

layer 2

V

perfect electric conductor

Fig. 3.11 Mi rostrip line with integration paths for al ulating quasi-TEM

indu tan e

Equation (3.84) is nally written as Z

f

ÆL

( 0)dxg = 2 2

Jz x; strip

Z

f

"Z

0 (x)ÆJ (x; 0)dx z

strip

( 0)dxgf

Z

strip

( 0)dxg

(3.86)

Jz x;

ÆJz x;

L

#

strip

As 0 (x) remains onstant on the strip area, it may be extra ted from the integral. Using (3.82), equation (3.86) nally redu es to Z

f

Z

( 0)dxg = 2f

Jz x;

ÆL

strip

2

( 0)dxg(0

ÆJz x; strip

LI

)

(3.87)

Sin e L and J in the right-hand side of equation (3.86) are exa t quantities, they satisfy the ommonly stated de nition for the indu tan e per unit length: 0 L= (3.88) z

I

whi h auses the rst-order error made on the indu tan e in (3.87) to vanish. That 0 remains onstant on the strip area is demonstrated by applying the integral formulation of divergen e equation (3.80 ) on volume V (Fig. 3.11). From the divergen e theorem, the integral of the left-hand side of (3.80 ) is equal to the ux integral of the indu tion eld on the surfa e en losing volume V . Sin e no variation of elds o

urs along the z -axis, the ontribution to the integral of the two planar fa es perpendi ular to this axis is an eled. On

102

CHAPTER 3.

ANALYSIS OF PLANAR TRANSMISSION LINES

the other hand, the omponent of the a tual magneti ux density normal to the ondu tor vanishes on the ondu tor, and the ontributions to the surfa e integral of the top and bottom fa es of V are zero. The ux integrals between the ground plane and the strip on the two planes x = x1 and x = x2 are identi al, be ause the integral of the right-hand side of (3.80 ) remains equal to zero. Another parti ular hoi e for the integration of the ux per unit length is depi ted in Figure 3.12. Here 0 (x) is the integrated indu tion ux per unit length owing through the surfa e at plane y = 0 limited by ontour (Fig. 3.12a,b): 0 (x)
z = =

Z z+
z Z 0

( 0 0)dx0 dz

(3.89a)

By x ;

I

z

H




A

(3.89b)

d

= [A (x; 0) z

=
z

Z



Az

(

(

GW;zz x;

1 0)℄
0j 0 0) ( ;

x ;

z

0 0)dx0

Jz x ;

(3.89 ) (3.89d)

using Stoke's theorem, and noting that the integrals on two of the paths of , respe tively BA in z + dz and DC in z , are zero, while the ve tor potential vanishes at x = 1. For oplanar waveguides (CPW) (Fig. 3.12b), 0 (x)

orresponds to the a tual ux. For strips, it is equivalent to the previous definition (3.83b), by applying the divergen e theorem to volume V 0 (Fig. 3.12a) where surfa e integrals vanish at in nity. Hen e, the ontribution of surfa es ABC D and BC F E to the integrals ompensate themselves, whi h demonstrates that the two ontours of Figures 3.11 and 3.12a are equivalent for the strip problem. De nition (3.89a) implies that no ontribution to 0 (x) o

urs on the strip, be ause the y- omponent of the magneti ux density vanishes on a perfe t ondu tor. As a onsequen e, 0 (x) remains onstant during the integration on the strip and an be extra ted from the integral. As for the quasi-TEM apa itan e, the knowledge of the Green's fun tion on the strip area is required. Its derivation is very lose to that dis ussed when presenting Yamashita's method. Now it is only brie y outlined. Sin e the urrent distribution is lo ated on a strip of zero-thi kness, the various layers are urrent sour e-free. Hen e, a stati magneti s alar potential  is onsidered in ea h layer, from whi h the magneti eld: M i

Hi

r

=

(3.90)

M i

When the media are isotropi , ombining (3.90) with (3.80 ) yields Lapla e's equation for  and the general solution (3.72) is appli able in ea h layer: M i

~ (k M i

x; y

) = A (k )e M i

x

x

k y

+ B (k )e M i

x

x

k y

(3.91)

3.4.

103

ANALYTICAL VARIATIONAL FORMULATIONS

Wc layer 1 Γ

A

B

dz D

az a x ax

C x

Λ

strip

ay

layer 2

E

H

F V’

(a) Wc Ws

layer 1

strip

Γ az

dz Λ

ax

x ay

perfect electric conductor (b)

layer 2 H

layer 3

Integration paths for energeti variational prin iple (a) mi rostrip; (b) oplanar waveguide Fig. 3.12

104

CHAPTER 3.

ANALYSIS OF PLANAR TRANSMISSION LINES

The spatial and spe tral magneti elds in ea h layer are obtained by repla ing i and i in (3.73a,b) by i and M i , respe tively:   M i( ) i ( ) M (3.92a) [ xi ( ) yi ( )℄ = E

H

H

x; y ; H



x; y

x; y

x

;



x; y

y

whi h transform in the spe tral domain into  ~ ~ [ xi ( x ) yi ( x )℄ = x ~ M i ( x ) xf M i ( x ) kx y  M k y x g i ( x) H

k ;y ;H

k ;y

k ;y ;k

jk

B

k

A

k

e

(3.92b)

e

As previously for the ele trostati ase, the supers ript ~ holds for spe tral quantities. When anisotropy or gyrotropy o

urs, M i still remains the solution of a se ond-order dierential equation. The dependent oeÆ ients in the general solution have a dierent expression, provided as an example in the next hapter. From the spe tral magneti eld, the spatial and spe tral magneti ux densities are al ulated and the following boundary onditions are applied to these spe tral elds: a. on the ground plane = ~y2 ( x )=0 (3.93a) y

y

B

k ;

H

H

b. on the interfa e = 0 ~y2 ( x 0) = ~y1 ( x 0) y

B

k ;

B

(3.93b)

k ;

~ y1 ( x 0) ~ y2 ( x 0) = ~z ( x ) (3.93 ) where ~z ( x ) is the -Fourier transform of the urrent density distribution z ( 0) modeling the strip

. at = 1, the magneti eld and potential have to vanish, whi h provides the same ondition as (3.74f) H

k ;

J

J

H

k

k ;

J

k

x

x;

y

M ( x) = 0 (3.93d) Four relations are obtained, relating the four oeÆ ients f M i ( x ) iM ( x ) des ribing the magnetostati potential in the two layers. Hen e, the spe tral magnetostati potential in ea h layer is related to the spe tral urrent distribution ~z ( x ) when solving the resulting system. The spe tral form of magneti ux density is derived from (3.80d) and yields the spe tral form of the indu tion ux per unit length ~ 0 ( x ) = y2 ( x ) (3.94) x A1

k

A

J

k

k

B

k ;y

jk

k

;B

k

3.4.

105

ANALYTICAL VARIATIONAL FORMULATIONS

The Fourier-transform of (3.89a) - equation (3.94) - yields the Green's fun tion relating the spe tral ve tor potential A~ (k ; 0) to J~ (k ). An alternate way would be to express all eld omponents from a s alar expression for the A~

omponent of the stati ve tor potential. The appli ation of this variational prin iple mentioned by Collin has been

arried out for a mi rostrip line on guration. Kitazawa [3.44℄ used expression (3.81) to al ulate the quasi-TEM parameters of oplanar waveguides and mi rostrips on a magneti substrate, possibly anisotropi . The derivation of the Green's fun tion in the CPW ase follows the method presented above, with the two integration ontours and  (Fig. 3.12). The results were

ompared to those obtained theoreti ally [3.45℄ and experimentally [3.46℄ by Pu el and Masse. Kitazawa also proposed another variational prin iple, expressed as z

x

z

x

z

Z

1 L

U

W2

= Z

=

By (x; 0)

f

U

Z

W2

g

Jz (x0 ; 0)dx0 dx

x

U

g

U

By (x; 0)

W2

f

Z

By (x; 0)dx 2

f

Z

Z 1

x

1

U

Z

f

j

g

GM;zy (x0 ; 0 x00 ; 0)By (x00 ; 0)dx00 dx0 dx U

W2

g

By (x; 0)dx 2

(3.95)

where U = W =2 + W for oplanar waveguides, and U = 1 for mi rostrips. Expression (3.95) assumes that a suitable Green's fun tion G an be de ned between the urrent density J owing on the strip and the magneti ux density B . The spe tral ux density B~ an be dire tly related to the spe tral magneti eld via (3.80d). It is related to the spe tral potential by (3.92b), and nally to the spe tral oeÆ ients fA (k ); B (k )g whi h have been shown to be proportional to the urrent density. So, the required G is obtained by inverting the relation found between the spe tral magneti ux density B~ and J~ . Varying (1=L) and B into (3.95) yields

s

M

z

M i

M i

x

x

M

z

Æ(

Z

)f W

L 2 1

U

Z

g

By (x; 0)dx U

= W

2 Z

y

U

+ W

2

ÆBy (x; 0)

f

By (x; 0)

f

Z

Z

+ 2( )f W

L 2 Z Z 1 1

2

x

1 Z 1

U

gf

By (x; 0)dx

j

U

1

U

W2

g

ÆBy (x; 0)dx

g

GM;zy (x0 ; 0 x00 ; 0)By (x00 ; 0)dx00 dx0 dx

U x

Z

j

g

GM;zy (x0 ; 0 x00 ; 0)ÆBy (x00 ; 0)dx00 dx0 dx

106

CHAPTER 3.

ANALYSIS OF PLANAR TRANSMISSION LINES

(3.96) Under the same assumptions of the Green's fun tion used before, the rstorder error in (1=L) is Z 1 B (x; 0)dxg2 Æ ( )f U

L

=2 =2 =2

Z

W2

x

ÆBy x;

2( L1 )f

Z

U

( 0)f

( 0jx00 ; 0)B (x00 ; 0)dx00 dx0 gdx

Z

By x;

Z

x

Z

U

( 0)Idx

W2

( 0)dxg

(3.97a)

( 0)dxg

(3.97b)

( 0)dxg

(3.97 )

ÆBy x;

( 0)dx0gdx

( 0)dxgf

U

U

y

Jz x0 ;

U

2( L1 )f

W2

1

GM;zy x0 ;

( 0)dxgf

U

W2

Z 1

U

ÆBy x;

W2 Z

Z

( 0)f

U

W2 Z

y

Z

By x;

U

W2

ÆBy x;

ÆBy x;

W2

Z 1 2( )f L

( 0)dxgf

U

W2

Z

By x;

U

W2

ÆBy x;

The rst term of the right-hand side of (3.97b) ontains an integral over x0 whi h remains onstant and equal to I during the integration over the spe i ed domain for the x variable. Hen e, (3.97 ) is nally equivalent to: Z Z [I L1 f W B (x; 0)dxg℄f W ÆB (x; 0)dxg 2 2 (3.98) Z 1 = [I L 0℄f W ÆB (x; 0)dxg = 0 U

U

y

y

U

y

2

and by virtue of equation (3.88), equation (3.95) is indeed variational. 3.4.1.3

Capa itan e based on energy

The other variational prin iple for a apa itan e proposed by Collin provides an upper bound for the apa itan e. It is based on the assumption that the ele trostati energy W stored in a length
z is stationary. It is
z Z E
("E )dA (3.99a) W = 2 Z =
2z " ( r)
( r)dA (3.99b) e

e

A

A

3.4.

107

ANALYTICAL VARIATIONAL FORMULATIONS

∞

Γ

∞

Γ

(a)

(b)

Integration paths over transverse se tion A for energeti variational prin iple (a) mi rostrip; (b) oplanar waveguide

Fig. 3.13

It is easily demonstrated that the rst-order variation of We vanishes when  satis es Lapla e's equation. Varying indeed We and  in (3.99b) and applying Green's rst identity (Appendix B) we obtain ÆWe

=
z "

Z

=
z "f

ZA

(

A

r)
( rÆ)dA Æ (

r2 )dA +

I

r)
nd g

Æ (

(3.100)

Contour may be hosen su h that it surrounds the various boundaries of transverse se tion A for a mi rostrip line (Fig. 3.13a) and for a oplanar waveguide (Fig. 3.13b). The line integrals at in nity vanish be ause of the de ay of the elds and potentials at in nity. The ontribution of the bran h uts is an eled, be ause the same values of potential and eld are integrated in opposite dire tions and these bran hes are taken in nitely lose to ea h other. On the ondu tor, only the tangential omponent of the exa t ele trostati eld ( r) vanishes. Hen e, the only way to an el the ontribution of the total eld is to impose Æ being zero on the ondu tors. This is equivalent to imposing that the approximate solution satis es the spe i ed boundary onditions on the ondu tor. Under this assumption, the ontribution of the line integral on ontour in (3.100) is zero. On the other hand, the a tual ele trostati potential satis es Lapla e's equation in ea h layer, whi h demonstrates that the rst-order error made on We vanishes, i.e. that We is stationary about the ele trostati potential. This proof is valid only when all the layers have the same permittivity, as assumed by Collin, who proposed the proof for a homogeneous medium. It is, however, possible to extend the proof to a non-homogeneous ase, in the following manner. The ele trostati energy and its variation are still given by (3.99b) and (3.100) rewritten for ea h layer. Green's rst identity is applied at the boundaries of ea h layer, for whi h a

108

CHAPTER 3.

Γ1

ANALYSIS OF PLANAR TRANSMISSION LINES

H1 ∞

Γ2

H2

Γ3

Λ1

Λ2

H3

Γ1

H1

Γ2

H2

Γ3

H3

Λ1 ∞ Λ2

Λ3

(a)

Λ3

(b)

Integration paths for inhomogeneous energeti variational prin iple (a) mi rostrip; (b) oplanar waveguide

Fig. 3.14

line ontour i is de ned (Fig. 3.14):

Wei

ÆWei

Z


z E
(" E )dA 2 Ai i i i i Z
z = "i ( ri )
( ri )dAi 2 Ai

=

=
z "i f

Z

Ai

Æ i (

r2 i )dAi +

(3.101a) (3.101b)

I i

ri )
ni d i g

Æ i (

(3.101 )

The total variation of energy is the sum of the variations of energies ontained in ea h layer, whi h yields

ÆWe

= =

N X i=1 N X i=1

ÆWei


z "i

Z

Ai

Æ i (

r2 i )dAi +

N X i=1


z "i

I i

Æ i (

ri )
ni d

i

(3.102)

The line integrals at in nity vanish be ause of the de ay of the elds and potentials at in nity. The ontribution of the line integrals at the interfa e between two adja ent layers i and i + 1 may be rewritten as a line integral on

3.4.

109

ANALYTICAL VARIATIONAL FORMULATIONS

this interfa e, with an asso iated line path denoted by i: ÆWe

XN ei i Z N X =
i i ( r i ) i A i N X Z +
f i( i ) i ri(

=

=1

=1

ÆW

z "

Æ

2

dA

x; H

"

i

i=1

z

i

Æ

x; Hi

)
ni

+ Æi+1(x; Hi ) "i+1ri+1 (x; Hi )
ni+1 gdi

=

XN
i Z i( r i) i A i Z N h i( i)f iri( X +
=1

z "

2

Æ

(3.103)

dA

i

i=1

z

i

Æ

x; H

"

x; Hi

)g

i

i+1(x; Hi )f"i+1ri+1 (x; Hi )g
nidi The behavior of the integrands of the line integral has to be onsidered over two dierent areas. If no ondu tors are present between layers i and j , the normal omponent of the exa t displa ement eld and the exa t ele trostati potential are ontinuous on the two sides of this ondu tor-free area: f"i ri (x; Hi )g
ni = f"i+1 ri+1 (x; Hi )g
ni (3.104a) and i (x; Hi ) = i+1 (x; Hi ) (3.104b) In this ase the line integral is then rewritten as Æ

XN
Z i=1

z

i

[Æi (x; Hi )

Æ

i+1 (x; Hi )℄ f"iri (x; Hi )g
nidi

(3.105)

If a ondu tor is present on a portion of the interfa e between layers i and j , only the tangential omponent of the ele trostati eld vanishes on it. Hen e, no simpli ation of (3.103) o

urs on the ondu tor. The only way to an el the error arising from the line integral is to impose Æ i (x; Hi ) = Æ i+1 (x; Hi ) (3.106) whi h implies the ontinuity of the trial potential on the ondu tor-free part of the interfa e. The exa t value is ontinuous (3.104b). Equations. (3.105) and (3.106) also imply that the trial solution has to be equal to the value

110

CHAPTER 3.

ANALYSIS OF PLANAR TRANSMISSION LINES

of the exa t potential on the ondu tor. Under these assumptions the ontribution of the line integral on ontour in (3.100) is equal to zero. On the other hand, the exa t ele trostati potential satis es Lapla e's equation in ea h layer, whi h demonstrates that the rst-order error made on We vanishes, i.e. that We is stationary about the ele trostati potential. It must be emphasized that the stationary hara ter of We is related to a fundamental

on ept of ele trostati s. Thomson's theorem [3.47℄ states, indeed, that the

harges present on ondu ting stru tures distribute themselves in su h a way that the ele trostati energy is minimal. Hen e, in some ases the variational prin iple is related to physi al laws of the real world. One simply has to nd a trial distribution for the potential, imposing that it has the value V0 on one ondu tor and zero on the other, and that it is ontinuous at diele tri interfa es. Using then (3.99) and (3.101), a variational prin iple for the

apa itan e per unit length is obtained: 1 2 1 N " ( ri )
( ri )dAi (3.107) CV = 2 0 2 i=1 i Ai Sin e the integral of the ele trostati energy density is always positive, Collin similarly states that (3.107) will always provide too large a value, i:e: an upper bound for the apa itan e [3.48℄. When the medium is homogeneous, the trial potential has only to be imposed equal to its exa t value on the ondu tors.

XZ

3.4.1.4

Indu tan e based on energy

There is a duality between the equations of ele trostati s and magnetostati s. It is interesting to investigate under whi h assumptions an expression similar to (3.99a) an be derived for the magnetostati energy and proven to be variational. It is well-known that the ele tromagneti power owing into a volume V is related to the dieren e between two fundamental quantities, the time-average ele tri and magneti energies ontained in the volume: 1 We = E
(D) dV (3.108a) 2 V 1 B
(H ) dV (3.108b) Wm = 2 V Collin [3.47℄ states that these two quantities are positive fun tions, and proves it for the ase of the ele trostati energy in the ase of an isotropi homogeneous medium. For isotropi media, the onstitutive parameters are s alar:

Z Z

D = "E

(3.109a)

B = H (3.109b) Hen e, the magnetostati energy an be expressed either from a sour e-free s alar potential (3.90) or using the z -dire ted magneti ve tor potential Az A = az Az (x; y )

(3.110)

3.4.

111

ANALYTICAL VARIATIONAL FORMULATIONS

yielding the magneti ux density B via (3.80a). The magneti ux density is rewritten as

B

B

=

az

 rAz

(3.111)

and the magneti energy Wm in a se tion of line having a length
z is Z

1 ( faz  rAz g)
f  1faz  rAz ggdA (3.112a) 2 ZA 1 ( rAz )
f 1 ( rAz )gdA (3.112b) =
z 2 A where A is the ross-se tion of the line. Expression (3.112b) is valid be ause the elds and potentials are assumed to have no variation along the z -axis. Hen e, if the medium is isotropi , the proof of the variational behavior leads to the following equation, obtained by repla ing  by Az in (3.100): Wm

=
z

ÆWm

=
z

1

=
z

1

Z

ZA

(

rAz )
( rÆAz )dA

ÆAz (

r2 Az )dA +

I

A Equation (3.111) is equivalent to

(3.113a)

r




ÆAz ( Az ) nd

rAz = az  B

(3.113b)

(3.114)

Hen e, the s alar produ t in (3.113b) is rewritten as (rAz )
n = (az  B )
n = (n  B )
az

(3.115)

The a tual magnetostati ve tor potential Az satis es the sour e-free form of (3.79), and the surfa e integral in (3.113b) vanishes. Contour may again be

hosen as in Figure 3.13a,b, for a mi rostrip line and a oplanar waveguide. The line integrals at in nity vanish be ause of the de ay of elds and potential at in nity. The ontribution of the bran h uts an el, be ause the same values of potential and eld are integrated in opposite dire tions on bran hes whi h are lose to ea h other. On the ondu tor, the exa t a tual magneti

ux density satis es




n B

=0

(3.116)

whi h implies that the only way to an el the rst-order error in the magneti energy is to impose that the magneti ve tor potential is equal to its exa t value on the ondu tors. This is not a problem when remembering de nition (3.83 ), whi h shows that the dieren e of magneti ve tor potential between the ondu tors is equal to the magneti ux per unit length. Hen e, sin e the magneti energy is ontained in the indu tan e, a variational prin iple for the

112

CHAPTER 3.

ANALYSIS OF PLANAR TRANSMISSION LINES

inverse of the indu tan e of the line is obtained by imposing a value for the

ux inside of the line: 2 1 (3.117) Wm = L
zI 2 =
z 0 2 2L whi h nally yields 1=

Zr




Az 

1 rA dA z

(3.118) 20 by virtue of (3.112b). Sin e the integral of the magneti energy Wm is always positive, (3.118) provides a lower bound for the indu tan e per unit length. When the medium is not magneti ally homogeneous, the proof of stationarity is obtained similarly to the ele trostati inhomogeneous ase. It an easily be shown that the variation (3.113b) of magneti energy in ea h layer be omes A

L

ÆWm

XN mi i Z N X =
i zi ( r zi ) i A i XN Z +
f zi( i ) i r zi (

=

=1

=1

ÆW

z

1

2A

ÆA

dA

i

i=1

z

i

ÆA

x; H 

1 A

x; Hi

)
ni

+ ÆAzi+1 (x; Hi )i+11 rAzi+1 (x; Hi )
ni+1gdi

=

XN
i Z zi( r A i Z N X h zi( +
=1

z

1

ÆA

zi )dAi

2A

i

i=1

z

i

ÆA

(3.119)

)

i  B i (x; Hi )gz

1 fn

x; Hi i

i

( ) f  B i+1(x; Hi )gz di where (3.115) has been used. Where no ondu tor exists, the tangential magneti eld is ontinuous: 1 fn  B (x; H )g i 1 fni  B i (x; Hi )g = i+1 (3.120) i i+1 i and the line integral in (3.119) be omes

XN
Z i=1

z

i

1 n ÆAzi+1 x; Hi i+1 i

[ÆAzi (x; Hi )

(

)℄

ÆAzi+1 x; Hi i

i  B i (x; Hi )gz di

1 fn

(3.121)

3.4.

ANALYTICAL VARIATIONAL FORMULATIONS

113

This integral vanishes when the Az - omponent of the trial ve tor potential is ontinuous at the ondu tor-free part of the interfa e. On the ondu tive part of the interfa e, the tangential magneti eld is no longer ontinuous. The only way to an el the error produ ed by the line integral in (3.119) is to impose that the ve tor potential is onstant and equal to its exa t value on the ondu tor. This is equivalent to imposing that the Az - omponent of the trial ve tor is ontinuous at any interfa e between layers. Its value on a

ondu tor is put equal to a onstant, whi h xes again (by virtue of equation 3.83 ) the value of the magneti ux per unit length inside the line. Under su h onstraints, and keeping in mind that the exa t ve tor potential satis es Lapla e's equation in ea h layer, the following variational prin iple is obtained for the inhomogeneous ase:

XN Z

1 = i=1

L

3.4.1.5

Ai

rAzi
i 1 rAzi dAi

20

(3.122)

Generalization to non-isotropi media

Up to now the various expli it quasi-stati prin iples have been proven variational under the assumption that the materials are hara terized by s alar

onstitutive relations (3.109a,b). As a matter of fa t, the various variational prin iples developed in the literature for planar lines are usually restri ted to isotropi layers. More general onstitutive relations, however, are B = 0 r
H (3.123a) D = "0 "r
E (3.123b) and the two stati energeti formulations be ome, for homogeneous media:
z E
("r
E )dA We = "0 (3.124a) 2 A = "0
2z ( r)
f"r
( r)gdA (3.124b)

Z Z A Z
z Wm = 2 AZB
(r
B )dA = 
z ( rM )
f
( rM )gdA 1

0

(3.125a)

(3.125b) r 2 A Expression (3.125b) involves the s alar magneti potential, be ause this is more onvenient for dealing with the dyadi formulation. It only holds in areas with no urrent sour e. As in the previous examples, the various ondu tors are assumed to be of zero thi kness, so that any urrent ow redu es to a

urrent sheet at the interfa e between two layers. The layers are onsidered 0

114

CHAPTER 3.

ANALYSIS OF PLANAR TRANSMISSION LINES

to be sour e-free. This is exa tly the same on ept as developed earlier for the Green's formalism, for whi h the integral equation may be solved either for a given sour e distribution and zero boundary onditions, or for a sour e-free situation with inhomogeneous boundary onditions. In the present ase, the existen e of urrent on the ondu tor is imposed as a boundary ondition on a sour e-free form of the magnetostati potential M . For both ases, the rst-order variation yields = "0

ÆWe

ÆWm

Z


z h ( rÆ)
f"r
( 2 A + ( r)
f"r
(

= 0

Z

r)g rÆ)g


z h ( rÆM )
fr
( 2 A + ( rM )
fr
(

i

(3.126) dA

rM )g

(3.127)

i

rÆM )g dA

In ea h ase, there is a s alar produ t involving a tensor dyadi , whi h satis es the following identity: T a
(u
b) = b
(u
a) (3.128) where supers ript T denotes the transpose of the dyadi . Using this identity, (3.126) and (3.127) are rewritten as ÆWe

= "0

Z


z h ( 2 A

rÆ)
f("r + "Tr )
( r)g

Z


z h ( ÆWm = 0 2 A Making use of identity a


rf = r
(f a)

i

rÆM )
f(r + Tr )
( rM )g f

(3.129)

dA

r
a

i

dA

(3.130) (3.131)

(3.129) and (3.130) transform respe tively into ÆWe

ÆWm

Z

= "0
z [ (Æ)r
f("r + "Tr )
( r)gdA A I (Æ)f("r + "Tr )
( r)g
nd ℄

(3.132)

Z

= 0
z [ (ÆM )r
f(r + Tr )
( rM )gdA A I M (Æ )f(r + Tr )
( rM )g
nd ℄

(3.133)

3.4.

ANALYTICAL VARIATIONAL FORMULATIONS

115

When the tensors are simply symmetri al, that is anisotropy without gyrotropy, one has T u=u (3.134) and (3.132) and (3.133) respe tively simplify into Z

= 2"0
z [ (Æ)r
f"r
rgdA AI Æ f"r
rg
nd ℄

ÆWe

Z

= 2"0
z [ (Æ)(r
D)dA AI Æ D
nd ℄ ÆWm

(3.135)

Z

= 20
z [ (ÆM )r
fr (rM )gdA AI (ÆM )fr
rM g
nd ℄ = 20
z [

Z

(ÆM )r
BdA

AI

(3.136)

(ÆM )B
nd ℄

where relationships (3.123a,b) have been used. The exa t displa ement eld satis es the harge-free divergen e equation

D

r
D =0

(3.137)

and its normal omponent does not vanish on the ondu tors. It is found again that the only way to an el the rst-order error made on We in the ase of nongyrotropi anisotropi diele tri media is to impose an exa t, onstant value for the potential on the ondu tor, as detailed previously. As a onsequen e, the following expression is a variational prin iple for the apa itan e per unit length in the ase of anisotropi media whose permittivity tensor satis es (3.134): Z

1 2 "0 CV = ( 2 0 2 A

r)
f"r
( r)gdA

(3.138)

For the magnetostati energy, the magneti ux density B satis es the hargefree divergen e equation

r
B =0

(3.139)

116

CHAPTER 3.

ANALYSIS OF PLANAR TRANSMISSION LINES

and its normal omponent vanishes on the ondu tor. This is why the magnetostati energy (3.125b) has been proven to be variational about the s alar magneti potential M . Keeping in mind the previous development leading to expressions (3.90) and (3.93 ), it is possible to relate the potentials to an equivalent urrent density Jz owing on the strip. Assuming a given urrent I on the strip and using (3.82) makes Wm proportional to I . Hen e, a variational expression is dedu ed for the indu tan e per unit length: R

( L = 0 A

rM )
fr
( rM )gdA I2

(3.140)

This variational prin iple provides a lower bound for the indu tan e. Obviously, the two derivations (3.138) and (3.140) may be extended to the ase of multilayered anisotropi and inhomogeneous stru tures, following the reasoning of the previous subse tion. We on lude this dis ussion by underlining that variational prin iples an usually be easily derived when symmetri al tensors hara terize the materials involved or when the Green's fun tion has spe i symmetry properties. When this is not the ase, that is when the medium is gyrotropi or nonHermitian, the derivation by assuming simple trial elds or potentials may be impossible. This has been observed by a number of authors. Formulas derived from Rumsey [3.49℄ for losed waveguides are variational, provided the permittivity and permeability are, at most, symmetri tensors. This means that isotropi materials and materials with rystalline anisotropy, lossy or lossless, are allowed while gyrotropi media su h as magnetized devi es and magneto-ioni media are ex luded. On the other hand, Berk [3.50℄ developed variational formulas appli able to gyrotropi media or, in general, media whose diele tri onstant and permeability are Hermitian tensors, restri ted, however, to lossless materials. All those formulations were limited to losed waveguides and resonators. More re ently, Jin and Chew [3.51℄ have studied the onditions required for a variational formulation of ele tromagneti boundary onditions involving anisotropi media. They found that the formulation is variational and an be expressed as a fun tion of ele tri eld and its omplex onjugate. This is possible only in the presen e of lossy anisotropi media having symmetri permittivity and permeability tensors, or in the presen e of lossless anisotropi media having Hermitian permittivities and permeabilities. The fun tional is interesting for the appli ation of the nite-element method to stru tures ontaining anisotropi materials. When the medium is non-Hermitian, however, the problem must be solved both for the ele tri eld and for an adjoint eld satisfying Maxwell's equations involving the transpose onjugate of the onstitutive tensors. This is be ause the fun tional is a fun tion of the eld and of its adjoint. A similar on lusion will be obtained in the following se tion when investigating the variational behavior of impli it variational methods.

3.4.

ANALYTICAL VARIATIONAL FORMULATIONS

117

3.4.1.6 Link with eigenvalue formalism

Expressions (3.16) and (3.66), and (3.81) and (3.95), are variational prin iples for the apa itan e and the indu tan e of the line, respe tively. This an be shown quite easily by observing that the problem they solve is in fa t losely related to a linear expression involving an integral operator. The de nition of apa itan e is Q = CV0

(3.141)

If harge density and potential are related by the Green's formalism (3.20), (3.141) is rewritten using (3.20), (3.18) and (3.22b) as Ql =

W2

Z

W2 s (x)dx = C

Z

W2

0 0 0 W2 G (x; 0jx ; 0) (x )dx P

s

(3.142)

whi h is equivalent to the following eigenvalue formalism with two linear operators L and M:

L( ) = M( )

(3.143)

As introdu ed in Chapters 1 and 2, there is a general variational prin iple for the eigenvalue  in the ase of self-adjoint operators L and M [3.52℄: R L( )dS R = (3.144) M( ) dS Repla ing M in this expression by integral (3.18) and L by the Green's formalism (3.20) and (3.22b), the result will yield the previous expressions (3.16) and (3.66). For the variational prin iples (3.81) and (3.95) for indu tan es, the orresponding eigenvalue problems are, respe tively 0 =

Z

GW;zz (x; 0jx0 ; 0)Jz (x0 ; 0)dx0 Z

= I =  J (x; 0)dx with  = L

(3.145)

z

I

=

Z

x

A

Z

1 1

GM;zy (x0 ; 0jx00 ; 0)By (x00 ; 0)dx00 dx0

Z

A

= 0 =  W B (x; 0)dx 2

(3.146a)

y

with

=

1 and x  W 2

L

(3.146b)

118

CHAPTER 3.

3.4.2

ANALYSIS OF PLANAR TRANSMISSION LINES

Impli it variational methods for a full-wave analysis

The possible variational hara ter of full-wave dynami methods is losely related to their impli it nature. Most of them are impli it non-standard eigenvalue problems, related to the rea tion on ept and to the general theory of eigenvalue problems. The dis ussion on entrates rst on the moment method, whose variational behavior may be studied using these three formalisms: impli it non-standard eigenvalue problems, rea tion on ept, and general eigenvalue problems. Sin e the impli it method yields a trans endental equation to be solved for the unknown propagation onstant, it an rst be studied following the non-standard eigenvalue formalism presented by Lindell [3.21℄. 3.4.2.1

Galerkin's determinantal equation

The general presentation of the moment method in Chapter 2 established that (3.54a,b) minimizes the error " made on Lst (j t (x; 0)) (3.54a) or Lsl (et (x; 0)) (3.54b). It is interesting to determine what error is made on when the determinantal equation (3.57a,b) is solved for a given set of basis fun tions satisfying (3.50b) or (3.51b). The dis ussion is adapted to the ase of the slot-line (3.57b), a similar reasoning holding for other lines. Harrington [3.53℄ states that using Galerkin's pro edure is equivalent to applying the RayleighRitz pro edure to the error " aused by applying the linear integral operator to the trial quantity instead of applying it to the exa t quantity. Harrington expresses this error as " = hf ; L(f )i

hf ; g i

(3.147)

where the inner produ t is de ned a

ording to (3.52a). Hen e, applying Galerkin's pro edure is equivalent to minimizing the error, by virtue of the de nition of the Rayleigh-Ritz pro edure. This means that when the exa t value f 0 is repla ed by a trial one, su h as f = f 0 + Æf

(3.148)

the rst-order error made on " is zero. For the exa t solution (f 0 ; 0 ) orresponding to the exa t elds, these elds satisfy the boundary onditions expressed as integral equation (3.52a,b):

hf 0 ; gi = hf 0 ; L0 (f 0 )i = he 0 (x; 0); L t

sl0

(et0 (x; 0))i = 0

(3.149a)

so that the orresponding exa t value for " is zero: "0 = 0

(3.149b)

The subs ript 0 for Lsl or L indi ates that values of the Green's fun tion are taken for the exa t value 0 . This is equivalent to the notation Lsl (et (x; 0); 0 ) in expressions (3.58a,b). Sin e the error " is made stationary about f by

3.4.

119

ANALYTICAL VARIATIONAL FORMULATIONS

Galerkin's pro edure, it also vanishes (to the se ond-order) in the presen e of the trial f , whi h imposes " = hf ; L0 (f )i hf; g i = hf ; L0 (f )i hf ; L0 (f0 )i = het (x; 0); Lsl0 (et (x; 0))i het (x; 0); Lsl0 (et0 (x; 0))i = 0

(3.150a)

Expanding the trial eld and negle ting se ond-order variations yields " = het0 (x; 0); Lsl0 fÆet (x; 0)gi

(3.150b)

So that (3.150a) nally redu es to

he 0 (x; 0); L 0 fÆe (x; 0)gi = 0 t

sl

(3.151)

t

Considering the basis of Galerkin's pro edure (3.52) applied to the integral equation for the boundary onditions, and developing its left-hand side around the exa t solution, negle ting se ond-order variations, yields:

he (x; 0); L

sl (et (x; 0))i = het0 (x; 0); Lsl0 (et0 (x; 0))i + hÆet (x; 0); Lsl0 (et0 (x; 0))i + het0 (x; 0); Lsl0 (Æet (x; 0))i

t

L + (Æ )het0 (x; 0); sl0 j 0 (et0 (x; 0))i = 0 

L is self-adjoint, one has he 0 (x; 0); L 0 (Æe (x; 0))i = hÆe (x; 0); L

(3.152)

If operator t

sl

t

t

sl0

(et0 (x; 0))i

(3.153)

whi h, ombined with (3.151), implies that only the term ontaining the rst variation (Æ ) is a priori non-zero in the right-hand side of (3.152). Sin e the left-hand side of this equation is imposed to be zero by the formulation, it is

on luded that the rst-order error made on using Galerkin's pro edure with a self-adjoint operator is zero. This also means that, to the se ond-order, the fun tional  that Harrington asso iates to the moment method and de nes in the ase of a self-adjoint operator as [3.54℄ =

hf ; gihf ; gi hL(f ); f i

(3.154a)

vanishes, whi h means

he (x; 0); L t

sl0

(et0 (x; 0))i = 0

(3.154b)

This is equivalent to say that  is stationary about et (x; 0). Equation (3.154b) also implies, by virtue of (3.150a),

he (x; 0); L t

sl0

(et (x; 0))i = 0

(3.154 )

120

CHAPTER 3.

ANALYSIS OF PLANAR TRANSMISSION LINES

det(γ,ft)

det(γ,ft2) det(γ,ft1)

γ

det(γ,ft3)

Fig. 3.15 Possible behavior of determinantal equation as a fun tion of prop-

agation onstant

for various hoi es of trial quantity

The self-adjoint hara ter of the integral operator L - see equations (3.38a,b) - is losely related to the re ipro ity and symmetry properties of the Green's dyadi fun tion used for formulating the integral equation for boundary onditions. This on ept will be lari ed in Se tion 3.4.2.3. It should be underlined that the previous development is an attempt to show that the solution found with Galerkin's pro edure is variational about a variation of the basis fun tions. The method, however, still needs to solve the impli it equations (3.57a,b), whi h requires evaluation of the determinantal equation several times. This is be ause the method provides no information about the behavior of the impli it equation with respe t to . At this stage of the dis ussion, there is no expli it variational prin iple available whi h would express as a ratio of integrals. The only feature highlighted by the previous dis ussion is that, whatever trial elds are present in the line, the produ t hf ; gi rosses the axis at about the value 0 (whi h is the orre t one, to the se ond-order) (Fig. 3.15). This, however, does provide neither information about the slope of the produ t hf ; g i with respe t to nor a guideline about the most eÆ ient way to solve equation (3.58b):

hf; gi( ) = hL (e (x; 0); ); e (x; 0)i = 0 sl

t

t

(3.155)

Solving this kind of equation may be very diÆ ult, as mentioned by Davies

3.4.

121

ANALYTICAL VARIATIONAL FORMULATIONS

[3.55℄, parti ularly when spurious solutions may o

ur [3.56℄. It requires spe i numeri al te hniques, espe ially when losses are present and evanes ent modes are onsidered [3.57℄[3.58℄. For these reasons, the method has usually been limited to lossless appli ations on diele tri multilayered media, as mentioned by Jansen [3.59℄. 3.4.2.2

Equivalent approa h based on rea tion on ept

In Chapter 2, the rea tion on ept introdu ed by Rumsey has been demonstrated to be eÆ ient for obtaining variational prin iples on a wide variety of ele tromagneti parameters. Using the notations of Chapter 2, two sour es lo ated in dierent areas of spa e and noted a and b generate their asso iated elds denoted by A and B , respe tively. Hen e, the rea tion of eld A on sour e b has the form Z (2.31a) hA; bi = fE a
dJ b H a
dM b g Lorentz re ipro ity prin iple is then rewritten using the rea tion on ept for isotropi media:

h

i=h

A; b

B; a

i

(2.32)

Equation (2.32) states that the rea tion of eld A on sour e b is equivalent to the rea tion of eld B on sour e a. When the medium is anisotropi , Rumsey proposes a orre tion to (2.32), attributed to Cohen [3.49℄:

h

(2.33) i=h^ i where h ^ i is written for the ase orresponding to the same sour es as h A; b

B; a

B; a

i

A; b

but where the elds generated by sour e b are solution of Maxwell's equations for a medium with transposed permittivity and permeability tensors. It is demonstrated in Chapter 2 that the rea tion hA; bi is variational about A and b provided that

h

A; b

i=h

i=h

A0 ; b

A; b0

i

(2.38)

In the ase of self-rea tion, assuming that the trial sour e and asso iated eld generated by the sour e are fun tions of a parameter p and knowing a priori that the value of the a tual rea tion hA0 ; a0 i is zero, it is demonstrated that by removing the trial rea tion hA; ai parameter p be omes stationary about the trial sour e and asso iated eld:

h

A; a

i=h

A0 ; a0

i=0

(2.43)

Referring to the hara teristi equations of the moment method using Galerkin's pro edure, it is now obvious that the left-hand side of equations (3.52a,b)

122

CHAPTER 3.

ANALYSIS OF PLANAR TRANSMISSION LINES

are indeed self-rea tions of the trial sour e of urrent and asso iated ele tri eld, whi h are bounded by the Green's fun tion. Sin e the exa t value of the rea tion is known a priori to be zero, (2.43) is appli able: imposing (2.43) on the trial rea tion ensures that the rst-order error made on the parameter p = is zero, provided that rea tions formed using both trial and exa t quantities satisfy (2.38). In the ase of Galerkin's pro edure, an elling (3.58b) ensures rst that (2.43) is satis ed by the trials, while the exa t elds are known to exist only on omplementary areas of the domain x of the stru ture:

hA; ai = hL (e (x; 0); ); e (x; 0)i = 0

(3.156a)

hA0 ; a0 i = hL 0 (e 0 (x; 0); 0 ); e 0 (x; 0)i = 0

(3.156b)

sl

t

t

t

sl

t

Condition (2.38), rewritten for self-rea tion, is partially ensured by the variational hara ter of fun tional ", whi h indu es (3.151) by virtue of (3.156b):

hA; ai = hA0 ; ai = he 0 (x; 0); L 0 (e (x; 0))i = 0 t

sl

t

(3.157)

To omplete the relationship between the rea tion formalism and Galerkin's impli it pro edure, Galerkin's formalism has to satisfy the right-hand part of equation (2.38):

hA0 ; ai = he 0 (x; 0); L 0 (e (x; 0))i = hA; a0 i = he (x; 0); L 0 (e 0 (x; 0))i t

sl

t

sl

(3.158a) (3.158b) (3.158 )

t

t

The right-hand sides of (3.158a) and (3.158 ) are equal if the linear integral operator is self-adjoint, as obtained in (3.153):

he (x; 0); L 0 (e 0 (x; 0))i = he 0 (x; 0); L 0 (e (x; 0))i t

sl

t

t

sl

t

(3.159)

If onditions (3.40a) and (3.51b) for the trial eld and asso iated sour e in the ase of slot-like lines are satis ed by the exa t eld, the right-hand side of (3.158 ) vanishes. Hen e, sin e the an ellation of the right-hand side of (3.157) has been demonstrated to be equivalent to stating that error " is stationary, it follows that " is stationary if the operator is self-adjoint. This was not expli itly stated by Harrington [3.53℄. Equation (3.159) also means that a re ipro ity relationship must exist between the set fLsl0 (et0 (x; 0); et (x; 0))g and the set fLsl0 (et (x; 0); et0 (x; 0))g. Hen e, the symmetry and re ipro ity properties of the Green's fun tion a ting as an integral linear operator nally determines the possible variational hara ter of the spe tral domain Galerkin's pro edure. This depends, of ourse, on the (non-)isotropi hara ter of the media. The properties of the Green's fun tion are investigated in the next se tion.

3.4.

123

ANALYTICAL VARIATIONAL FORMULATIONS

3.4.2.3

Conditions for the Green's fun tion

As we have seen, the variational hara ter of both the propagation onstant obtained using Galerkin's pro edure and of the quasi-stati parameters depends on the re ipro ity properties of the Green's fun tion relating the tangential urrent to the ele tri eld at an interfa e. An interesting review of these properties is presented by Barkeshli and Pathak [3.60℄. They report that the free-spa e dyadi Green's fun tion has been investigated by Collin [3.61℄, Collin and Zu her [3.62℄, Tai [3.63℄ and others. It has been shown that the free-spa e dyadi Green's fun tion satis es re ipro ity as well as symmetry relationships (Appendix A): (A.37a)

( j ) = G(r 0 jr )

G r r0

with =G

G

T

(A.37b)

The additional symmetry property in free-spa e (i.e., free-spa e dyadi equal to its transpose) derives from the fa t that for an unbounded medium inter hanging the lo ations of sour e and eld points does not modify any existing boundary ondition problem, be ause of the symmetry of the va uum or free-spa e unbounded medium. In this simpli ed ase, relation (A.37a) is valid. For other on gurations, however, new boundary onditions problems are introdu ed, generally when sour e and eld lo ations are inter hanged. Overall, the Green's dyadi does not satisfy the symmetry relation (A.37a), even if it satis es ertain re ipro ity relations. One of them is derived by Collin [3.64℄ from the re ipro ity theorem (2.32) - in the ase of a bounded homogeneous medium (assuming that no magneti sour es are present). Introdu ing the Green's dyadi operator as ) = GfX (r 0 ; t0 )g

(A.17b)

u=

X Gf

(A.17 )

u=

XZ Z

(

P r; t

with P

P

v

v

[

(

0

X r ;t

Vx Tx

0

)g℄uv

uv (r; tjr

G

0

0

;t

)Xv (r 0 ; t0 )dr0 dt0

(A.17d)

124

CHAPTER 3.

ANALYSIS OF PLANAR TRANSMISSION LINES

into (2.32) yields Z b a dJ (r 0 )
[GfJ (r )g℄(r 0 ) V Z a b dJ (r )
[GfJ (r 0 )g℄(r ) =

(3.160)

V

in whi h both sides are general forms for the right-hand sides of equations (3.52a,b), for instan e. This equation has to be satis ed whatever sour es a and b are, but one parti ular ase is for unit point sour es a and b lo ated respe tively at r 0 and r 00 and having an arbitrary orientation. Assuming the presen e of sour e points J

a

a

(r ) = j Æ (r

b

J (r )

r0 )

b

= j Æ (r

r 00 )

(3.160) is rewritten as j

a


fG(r 0 jr0 )
j b g = j b
fG(r0 jr0 )
j a g 0

0

(3.161)

whi h implies, by appli ation of the dyadi identity (3.128),

j

G(r 0 r 00 )

T

= G (r 00 jr 0 )

(3.162)

Similarly Tai [3.63℄ presented re ipro ity relations for the isotropi homogeneous half-spa e dyadi Green's fun tion. On the other hand, Felsen and Mar uvitz [3.65℄ have investigated the re ipro ity relationships for timeharmoni ele tromagneti elds in spatially varying anisotropi media: they established a set of general re ipro ity relationships for anisotropi and its asso iated transposed medium. Moreover Papayannakis et al. [3.66℄ have shown a more general form of the Love equivalen e theorem applied to a nite-size spa e. It an be expe ted that, for media with anisotropi properties, a similar development is appli able, starting from the modi ed re ipro ity theorem (2.33): j

a


fG(r 0 jr0 )
j b g = j b
fG (r0 jr0 )
j a g 0

0

0

(3.163)

whi h implies

j

G(r 0 r 00 )

T

= G0 (r 00 jr 0 )

(3.164)

where the prime notation for the dyadi on the right-hand side of (3.163) and (3.164) refers to the Green's fun tion established between the urrent sour e and the ele tri eld in the transposed medium. Up to now, and to the best of our knowledge, there is no satisfa tory

ondition proving that the 2x2 dyadi Green's fun tion involved in Galerkin's

3.4.

125

ANALYTICAL VARIATIONAL FORMULATIONS

method used for planar line analysis satis es the usual re ipro ity theorem for arbitrary ve tor elds A and B : hA; Lsl0 (B )i = hB; Lsl0 (A)i (3.165) and for any medium involved in the planar layers. It is suspe ted that when lossy gyrotropi media are involved, equation (3.162) may not be valid anymore, be ause the Green's fun tion a ting as linear integral operator no longer satis es (A.37a) or (3.160), and the variational hara ter of the solution of (3.52) with respe t to trials in the partially metallized interfa e of the planar lines disappears.

3.4.2.4 Link with expli it variational eigenvalue problems

The expli it quasi-stati variational prin iples (3.16), (3.81) and (3.95) are related to an expli it eigenvalue formalism. They have a orresponding determinantal form, as explained by S hwinger [3.67℄. Starting again from the general form (3.144), the Rayleigh-Ritz pro edure is applied to the fun tion expanded into N = n n n=1 Then (3.144) is rewritten for the three ases: N M n m n L( m )dS n =1 m=1 (3.166) = N 2 ( n n dS ) n=1 Sin e  al ulated by (3.144) is variational, taking the partial derivative with respe t to ea h n and an elling it yields a homogeneous system of N equations for the m : M N m k L( m )dS ( k dS )( n n dS )  m =1 n=1 =2 N k (3.167) ( n n dS )2 n=1 =0 for k = 1; : : : ; N . This system has a non-trivial solution if, and only if, the determinant of its hara teristi matrix vanishes, whi h yields the hara teristi determinantal equation asso iated to those expli it variational prin iples:

X

XX Z X Z X Z

det

X Z

Z

X Z

h Z nL( m)dS (Z ndS)(Z mdS)i = 0

126

CHAPTER 3.

ANALYSIS OF PLANAR TRANSMISSION LINES

whi h is equivalent to Z

det

h

Z

n L( m )dS Z

( n dS )( m dS )



i

=0

(3.168)

Hen e, it is possible to obtain a determinantal expression for the parameter under onsideration. As underlined by S hwinger [3.67℄, however, the equation that is required is neither a polynomial expression of degree N in , nor an eigenvalue problem asso iated with the matrix. Indeed subtra ting the rst row from all the rest yields a matrix with  present only in the rst row. Hen e the result of the determinant omputation is a linear expression in . \It is also evident", says S hwinger, \that one annot solve (3.168) rigorously. However, in view of the stationary nature of  we do know that the error in the value of  produ ed by an in orre t eld or urrent fun tion will be proportional not to the rst power of the deviation of the eld for the orre t value, but to the square of the deviation. Thus, if a eld or urrent fun tion is hosen judi iously, the variational prin iple an yield remarkably a

urate results with relatively little labor. Unfortunately, the ability to hoose good trial fun tions generally omes only with experien e" [3.67℄. It is also to be pointed out that applying the Rayleigh-Ritz pro edure to the general eigenvalue variational prin iple for al ulating quasi-stati parameters is equivalent to applying Galerkin's pro edure of the moment method to the de nitions (3.142), (3.145), and (3.146a) orresponding to these quasistati parameters. Multiplying both sides of these equations by the appropriate form of the fun tion and integrating the results is indeed applying Galerkin's pro edure, but the result of this operation pre isely yields the general variational prin iple (3.144). Hen e, in the ase of the standard expli it eigenvalue problem, applying Galerkin's pro edure to the de nition of the parameter under onsideration is a variational method, be ause it is equivalent to applying the Rayleigh-Ritz pro edure to a variational prin iple established for this parameter. However, the determinantal equation to be solved will provide a linear equation to be solved for the parameter, hen e an expli it form for this parameter. This point has been mentioned by Collin for the quasi-stati apa itan e [3.43℄. To on lude this se tion about quasi-stati formulations, the omment by S hwinger also implies that, when hoosing a judi ious in nite series for the trial quantity, the Rayleigh-Ritz pro edure nds an exa t distribution for the quantity, provided that the trial elds derived from this trial quantity ful ll all the other boundary onditions satis ed by the a tual exa t elds and potentials. Sin e the proof of the variational behavior of the quasi-stati formulations derived in the previous se tion usually does not imply spe i boundary onditions on the trials, the trial elds used in the quasi-stati variational prin iples usually do not satisfy all the boundary onditions of the

3.4.

ANALYTICAL VARIATIONAL FORMULATIONS

127

a tual eld, and even an in nite summation for the trial quantity does not guarantee that exa t elds are obtained.

3.4.3

Variational methods for obtaining a

urate elds

Galerkin's pro edure minimizes the error fun tional " on the trial f with respe t to the a tual distribution solution of the problem (f 0 ) = g

L

(3.169)

As seen previously, this is equivalent to saying that Gakerkin's pro edure minimizes the fun tional  (3.154) and the self-rea tion (2.43). This has been

on rmed by Ri hmond [3.68℄ and Harrington [3.53℄. Peterson et al. [3.69℄, however, have shown that non-Galerkin's moment methods are also able to minimize this error, even when the linear integral operator is not self-adjoint. This means that Galerkin's method may exhibit no signi ant advantages when ompared to non-Galerkin's moment methods. Our attention, however, is fo used on the ommonly used analysis te hniques for planar lines, so we start from Galerkin's pro edure and the assumption of Ri hmond [3.68℄. For planar lines, equation (3.169) has to be solved for a very redu ed portion of the spa e, namely the area of the strip or of the slot jxj  W2 , where L(f ) has to vanish. The proper hoi e of the basis fun tions ensures that integrating the produ t f
L(f ) on the area (jxj  W2 ) and making it vanish is equivalent to integrating it over the whole x-domain. In doing this, however, the user is only ertain, after solving for the approximate eld or

urrent distribution f , that the integral of the produ t f
L(f ) on the area W vanishes. Nothing guarantees that the quantity provided by the jx j  2 Green's relationship vanishes, whi h means that (3.40) is satis ed for every value of x. It will in fa t be the ase if the Green's relationships (3.34a,b) and (3.35a,b), yielding the ele tri eld (for strip) or the urrent density (for slot), preserve the symmetry about the x-variable of the sour e of urrent (for strip) or of ele tri eld (for slot). Sin e the Green's fun tion ontains the unknown propagation onstant , it is a priori possible that some parti ular values of this propagation onstant provide a Green's fun tion su h that the integral of the resulting eld (3.40a,b) vanishes on the aperture, while the lo al value of the eld varies in the aperture. This behavior is s hemati ally depi ted in Figure 3.16 for the strip and slot ases. If a symmetri (even) slot eld or strip urrent distribution, noted f (x), is assumed over the area of interest, and if for parti ular values or ranges of values for the Green's fun tion generates an odd behavior for the resulting quantity, the produ t of the two will of ourse vanish when integrated over the area. The lo al value of the produ t, however, may not be zero. Hen e, in this instan e, the value of provided by the determinantal equation does not orrespond to an a tual or physi al eld distribution. On the other hand, it is quite obvious that a tual physi al solutions are obtained if the basis fun tions hosen for expanding the unknown trial eld

128

CHAPTER 3.

ANALYSIS OF PLANAR TRANSMISSION LINES

f (x)

x

⇓

⇓ L2( γ2,f(x) )

L1( γ1, f(x) )

x

x

⇓

⇓

L( γ,f(x) ) . f (x)

x

Possible behavior of produ t L(f )
f as a fun tion of the value of the propagation onstant (even slot eld or strip urrent distribution, noted f) Fig. 3.16

represent rigorously the physi al solution [3.68℄. In this ase, and in this ase only, it an be expe ted that onsidering an in nite number of terms in the expansion will provide a solution whi h orresponds to physi al elds, provided that all the boundary onditions for the elds are satis ed by a proper expression of the Green's fun tion. This also means that, if the trial and Green's fun tions for the problem are su h that all the onventional boundary onditions are satis ed - ex ept the onditions in the slot or on the strip (3.40) - then the problem is fully resolved on e these onditions are satis ed.

3.5.

DISCRETIZED FORMULATIONS

129

As on luded before, some doubt may remain, sin e only an integral produ t

ontaining (3.40) is for ed to vanish. Moreover, it is impossible to take an in nite number of terms in the expansion of the trial - a trun ation error will always be present. But as S hwinger mentioned for the Rayleigh-Ritz pro edure, be ause of the stationary hara ter of Galerkin's pro edure with respe t to under the assumption (3.165), the trun ation error will have no ee t to the rst order on the value obtained for the propagation onstant. To on lude the dis ussion about the hoi e of the trial and test elds, it has to be underlined, following the theory presented in Chapter 2, that the stationarity of " and , and of the rea tion al ulated by the moment method, is proven only when Galerkin's pro edure is used - that is when the basis and test fun tions belong to the same set. This has been on rmed by Ri hmond [3.68℄. In the ase of a mutual rea tion, however, Mautz [3.70℄ has shown that another hoi e of basis and test fun tions yields a stationary expression for the mutual rea tion hA; bi. It onsists of: taking as test fun tions for the integral dealing with the b sour e the basis fun tions used for expanding the a sour e, while the test fun tions for the integral ontaining the a sour e are the ones used for expanding the b sour e. This, however, is not dire tly part of the present dis ussion, sin e the spe tral domain Galerkin's method works with a self-rea tion only. As a matter of fa t, Denlinger's method presented in Se tion 3.3 is a moment method using a Dira fun tion as a test fun tion for a self-rea tion. Taking indeed the inner produ t between the aperture eld (3.44) in the slot or the urrent density (3.45) on the strip and the Dira fun tion in x-domain, is equivalent to system (3.46) and (3.47) or to equation (3.48). Hen e, this method will not, to the best of our knowledge, provide a variational result for the propagation onstant. 3.5

3.5.1

Dis retized formulations

Finite-element method (FEM)

At the end of the previous se tion, the possibility of obtaining exa t eld distributions from variational prin iples has been dis ussed. The nite-element method (FEM) takes advantage of a variational quantity to solve dierential equations for a spe i ed s alar or ve tor quantity. The purpose of this subse tion is not to des ribe extensively the FEM, but to point out where a variational behavior is invoked and what exa tly are the possible links with our previous developments. An ex ellent review of the nite-element method is provided by Davies [3.55℄. Basi ally, the FEM is used to nd a s alar or ve tor eld distribution satisfying a given dierential equation. It dis retizes a trial distribution and takes advantage of a variational form for adjusting the

oeÆ ients of the distribution in order to satisfy the equation. The most simple example is the stati Lapla e's equation (3.8) that we previously used for quasi-stati appli ations. Instead of dire tly solving Lapla e's equation, the nite-element method uses a fun tional of the unknown potential, denoted by J (), whi h has the interesting property of being stationary about the

130

CHAPTER 3.

ANALYSIS OF PLANAR TRANSMISSION LINES

unknown quantity, provided that the exa t unknown satis es the equation to be solved. Hen e, the Rayleigh-Ritz pro edure is applied: it imposes that the rst-order derivative of () is zero. The resulting equation provides the

oeÆ ients yielding an exa t des ription of the unknown . As a rst step, the domain of interest is dis retized into ells (surfa es for 2-D or volumes for 3-D problems). For 2-D problems, re tangles or triangles are used. The quantity  is approximated in ea h of the ells as a polynomial expression of the oordinates. The simplest expression is a linear fun tion: ( ) = 0 + x + y (3.170) whi h means that ve tor may be expressed at ea h vertex of the triangle as a fun tion of the value of the potential at this vertex (or node), denoted by k: ( k k ) = 0 + x k + y k = k with = 1 2 3 (3.171) Ea h ell 2 has 3its own ve tor J

N

x; y

p

p x

p y

p

k

U

x ;y

p

p x

p y

U

k

;

;

n

n p

=4

n 0n px n py p

5

(3.172)

From the set of three equations written at ea h vertex of ea h ell n+ n + n = k (3.173) 0 x k y k the potential at the three nodes of the ell is rewritten in matrix form. Inverting the resulting system yields the set of oeÆ ients n asso iated with ea h ell. Introdu ing them in (3.170) yields 2 3T 2 3T 1 1 1 k n ( ) = 4 5 n = 4 5 (3.174) p

p x

p y

U

n

p

x

x; y

x

p

y

A

U

y

where 2 3 1 1 1
=4 1 2 2 5 1 3 3 Hen e, the potential in ea h ell is onsidered as an interpolation polynomial between the values at the three verti es of the ell. Finally the fun tional to be minimized has to be omputed. For Lapla e's solution, it an be shown [3.71℄ that minimizing the following fun tional -domain () yields the solution : Z Z 2 2 () = r2  = ( 2 + 2 ) SZ S (3.175)  2 [( ) + (  )2 ℄ = A

x

y

x

y

x

y

x

J

dS

S





x

y





x

y

dS

J

dS

3.5.

131

DISCRETIZED FORMULATIONS

The right-hand side is related to the ele trostati energy (3.99a), whi h has been demonstrated to be variational with respe t to trial elds satisfying Lapla e's equation. As explained in [3.71℄, expression (3.100) also means that when ( + Æ) and  are two distributions satisfying the same boundary

onditions on the ondu tor, that is Æ = 0 on the ondu tor, then the fun tion whi h an els ÆWe is for ed to satisfy Lapla e's equation. Using (3.174) the two partial derivatives are rewritten as 2

3

T  = 4 01 5 A x



1

U

k

(3.176a)

y

2

3

T  = 4 x0 5 A 1 U k y 1 and their squared power has the matrix form 

 T ( x )2 = U k fA 

 T ( y )2 = U k fA 

1

2 3 2 3T 0 0 gT 4 1 5 4 1 5 A

1

2 3 2 3T 0 0 gT 4 x 5 4 x 5 A

1

y

1

(3.176b)

U

k

(3.177a)

k

(3.177b)

y

U

1 1 Introdu ing (3.177a,b) into (3.175) yields the nal expression for J : J

() =



U

k T

GU

k

(3.178)

where G is a global matrix resulting from the integration over ea h ell of the produ t of the oordinate matri es by their transpose, their summation over all the ells, and a rearrangement as a fun tion of the various produ ts U k U j . Equation (3.178) is then rewritten using as boundary onditions, those nodes where the potential or its derivative is a priori known. These nodes are de ned by ve tor U k0 , while nodes where the potential is unknown are in ve tor U ku . The matri es are then rearranged and partitioned as J

() =

"

U U

ku k0

#T "

Guu

Gu0

G0u

G00

#"

U U

ku k0

#

(3.179)

Applying the Rayleigh-Ritz method to (3.179) is equivalent to taking its rstderivative with respe t to ea h omponent of U ku : J () (3.180a) = G U ku + G U k0 = 0 U ku

uu

u0

132

CHAPTER 3.

ANALYSIS OF PLANAR TRANSMISSION LINES

whi h yields the nal equation for FEM = Gu0 U k0 (3.180b) The above developments need some further omments. First, the FEM is learly most appli able to en losed stru tures only, sin e it requires des ription of the spa e or the transverse se tion by de ning a mesh and asso iated nodes. Open stru tures have to be approximated by

losed ones, with end boundaries lo ated far away so that elds and potentials have a negligible value. Su h boundaries are usually diÆ ult to predi t. At these boundaries an arti ial boundary ondition has to be spe i ed. Initially they were perfe t magneti or ele tri walls. Sin e then, however, various lo al absorbing boundary onditions have been implemented in the TLM method. As an example, for using the method in open problems, like antenna and s attering problems, it is ne essary to terminate the mesh in su h a way that it simulates free spa e. The perfe t absorbing boundary ondition should perfe tly absorb in ident waves with arbitrary angles and at all frequen ies. Sin e su h walls do not exist, a number of attempts have been made with a reasonable su

ess for de ning very good absorbing walls. Se ondly, the potential or eld is still in uen ed by the hoi e of the polynomial expansion used to des ribe it in ea h ell. As we have previously seen, however, an exa t solution will be obtained by the Rayleigh-Ritz method if the basis fun tions form a omplete set. In the ase of FEM, this is equivalent to saying that the polynomial expansion (3.170) des ribes the exa t eld when the order of the basis polynomial used for ea h ell goes to in nity. It should be noted that mesh re nement te hniques do presently exist in the FEM and have already been su

essfully applied. Thirdly, there is a ru ial question after having solved equation (3.180b): what an we do with the result? We hope to have obtained the exa t potential or eld in the stru ture, but we have no idea about the way to use it for transmission lines. The result has to be introdu ed in another formulation yielding a parameter of interest (radar ross-se tion, hara teristi or input impedan e, propagation onstant, ut-o frequen y) as a fun tion of the potential or eld that we have just al ulated. A judi ious solution is to ombine the FEM with a variational prin iple for a parameter of interest, su h as a quasi-stati apa itan e or indu tan e, or a propagation onstant. For example, we have seen in Chapter 2 that there is a variational prin iple for the ut-o frequen y of a waveguide involved in the Helmholtz equation: Guu U

ku

Z

! 2 "0 0 =

ZS

r2 dS

S

"r 2 dS

(2.115)

Letting p2 = ! 2 "0 0 , it an be shown [3.72℄ that equation (3.180b) has to

3.5.

DISCRETIZED FORMULATIONS

133

be rewritten for this ase as XU

ku

= p2 Y U

ku

(3.181)

whi h is a matrix formulation for an eigenvalue problem, dis ussed in Se tion 3.4.2.2 and hara terized by the determinantal equation (3.168). In both equations (3.168) and (3.181), the minimizing pro edure makes the unknown appear as an eigenvalue for the nal matrix or as a determinantal equation to be solved. This is a onsequen e of the formulation involving a ratio, and not a spe i feature of the variational theory. In fa t  in (3.168) or p2 in (3.181) may be repla ed by their expressions given by (2.115) or (3.166) respe tively. This yields a possibly more ompli ated equation. After solving it, (2.115) or (3.166) should be used for obtaining the required parameter. Hen e, equations (3.168) and (3.181) are a shorter way for nding the solution. The example above shows that FEM just performs a dis retization of the trial elds. The trial elds are then adjusted using the Rayleigh-Ritz pro edure in order to possibly yield at the same time exa t elds asso iated with a parameter of interest. To summarize, the interest of FEM ombined with a variational form is strongly related to the physi al meaning of the fun tional J to be minimized. In the rst ase, it is just used to nd exa t elds (their a

ura y being subje t to the interpolation approximation, hen e to the number of ells), while in the se ond ase FEM is used to obtain a parameter of interest. To omplete this brief outline of FEM, it has to be mentioned that the method is also used to dis retize equations for boundary onditions, su h as ondition (3.40). It provides, for example, a matrix formulation of those equations: ZU

ku

=0

(3.182a)

whi h yields a determinantal equation: det(Z) = 0

(3.182b)

where ea h omponent of Z is a fun tion of the unknown propagation onstant.

3.5.2 Finite-dieren e method (FDM)

The omments made about the FEM are also valid for the nite-dieren e method (FDM). At the start of this hapter, we have seen that nite dieren es may be useful for solving Lapla e's equation for quasi-stati problems. Dis retizing the potential by using a Taylor's expansion results in a system (3.12) whi h yields a solution of Lapla e's equation whi h is orre t to the h4 order. The same dis retization te hnique may be used for solving Helmholtz equations (3.27) for TE and TM potentials. Rearranging (3.10) and negle ting

134

CHAPTER 3.

ANALYSIS OF PLANAR TRANSMISSION LINES

O(h4 ) yields:

2 2 ( x2 + x2 ) = A

( +  +  +  B

C

D

4 )

E

h2

(3.183)

A

Using this approximation in (3.27) yields TE + TE + TE + TE + h2 (k2 + 2 ) TE = 0 i

i

B

i

C

TM

TM

i

D

TM

i

E

TM

i

(3.184)

A

TM

+ + + h2 ( k 2 + 2 )  = 0 (3.185)  + or in matrix form: M = h2 (k2 + 2) (3.186) Up to now, no variational assumption has been made. We obtain using the nite dieren es an eigenvalue problem in ea h layer. The quantity h2(k2 +

2 ), however, has a dierent value in ea h layer. Also, the resulting matrix M is asymmetri . To over ome those drawba ks, Corr and Davies [3.73℄ propose to use a variational expression suitable for the problem, whi h ontains only the longitudinal omponents of the ele tri and magneti elds: Z 1 1 (!" " E r2 E + ! H r2 H ) + !" " E 2 + ! H 2 dA J= i

i

B

i

C

T E;T M

i

i

D

i

E

i

A

T E;T M

i

i

i

i

A

0

"r k02

r

z

t

z

0

z

t

z

0

r

z

0

z

(3.187) Corr and Davies have rewritten this expression as a fun tion of the Helmholtz potentials, sin e the longitudinal omponents of the elds are proportional to these potentials: J (

TM

;

TE

)=

Z

A

A"r jrt TM j2 +  jrt TE j2

TE TE TM TM +  2A(  x  y  y  x ) k02 f(TE )2 + A" (TM )2 gdA

(3.188)

r

where A=( 

0 2 ) ! 2

= !!2 """0 0 0

r

0

2 2

Using then, for example, de nitions (3.9a-d) yields an approximation for the rst derivative of the potential, orre t to the h3 order:   (3.189a) =  x A

B

h

D

3.6.

135

SUMMARY



  y B

= C h E

(3.189b)

Introdu ing those approximations into fun tional (3.188) yields a matrix formulation similar to (3.182) "

Z

#

TE = 0 TM

(3.190)

where the eigenvalues of Z are fun tions of A and  , hen e of the unknown propagation onstant. In the present ase, we obtain an impli it eigenvalue problem, by imposing the rst-derivative of the variational form to be zero. This approa h is quite dierent from the impli it determinantal equation asso iated to Galerkin's pro edure, presented in the previous se tions. In Galerkin's ase, we have shown that the eigenvalue obtained is variational about the trial potentials or elds, but this is obtained be ause we know a priori the exa t value of the variational form, whi h was zero for our ase. In the present ase, we do not know the exa t value of the fun tional J and we only impose its rst-order derivative to be zero. Hen e, we annot on lude about any variational behavior of the obtained eigenvalue with respe t to the TE and TM dis retized potentials. For homogeneously- lled waveguides, it is lear that simpler forms like Z

J () = k + = AZ 2

2

r2 dA

A

2 dA

(3.191)

may be used, in ombination with (3.10) where the O(h4 ) term is negle ted. The obtained matrix equation is then similar to (3.181). As illustrated by the abundant literature, FDM and FEM are very powerful general purpose methods in their own right. In this ontext, however, we onsider them mainly useful for obtaining trial elds. We do not wish to underrate them, we are simply limiting their appli ation. With this in mind, it an be on luded that FDM and FEM are indeed eÆ ient variational methods when ombined with a variational prin iple yielding a ir uit parameter of interest for the analysis of distributed ir uits. In this ase, applying one of these two methods is equivalent to dis retizing in the spa e domain the trial eld to put in the variational prin iple. They are of great interest for nding trial eld distributions for stru tures whi h have an intri ate geometry. 3.6

Summary

In the rst two se tions in this hapter we have reviewed quasi-stati methods and full-wave dynami methods. The quasi-stati methods usually provide a straightforward expression for a line parameter. They are valid, however, in

136

CHAPTER 3.

ANALYSIS OF PLANAR TRANSMISSION LINES

the low frequen y range only, typi ally below 5 GHz. On the other hand, full-wave methods have as their main drawba k that they provide no simple expli it behavior of a line parameter, despite their wider band appli ation. The sear h for the root of the impli it equation is usually tedious, so that the use of those methods is preferably limited to lossless lines. The third se tion was devoted to detailed analyti al formulations based on variational prin iples, appli able to a variety of on gurations. A general variational prin iple will be presented in detail in Chapter 4, and ompared with the other methods. Chapter 5 will be devoted to various appli ations of the variational prin iple. They will in lude theoreti al and experimental hara terizations of multilayered lossy planar lines parameters, gyrotropi planar devi es, and planar jun tions and transitions involved in the design of MICs and monolithi mi rowave integrated ir uits (MMICs) subsystems. 3.7

Referen es

[3.1℄ R.E. Collin, Field Theory of Guided Waves. New York: M Graw-Hill, 1960, Ch. 2, p. 25. [3.2℄ Z. Zhu, I. Huynen, A. Vander Vorst, \An eÆ ient mi rowave hara terization of PIN photodiodes", Pro . 26th European Mi rowave Conferen e, Prague, vol. 2, pp. 1010-1014, Sept. 1996. [3.3℄ A. Vander Vorst, Ele tromagnetisme. Brussels: De Boe k-Wesmael, 1994, Ch. 1. [3.4℄ A. Vander Vorst, D. Vanhoena ker-Janvier, Bases de l'ingenierie mi roonde. Brussels: De Boe k-Wesmael, 1996, Ch. 2. [3.5℄ H.A. Wheeler, \Transmission line properties of parallel wide strips by

onformal mapping approximation", IEEE Trans. Mi rowave Theory Te h., vol. MTT-12, no. 3, pp. 280-289, Mar. 1964. [3.6℄ H.A. Wheeler, \Transmission line properties of parallel wide strips separated by a diele tri sheet", IEEE Trans. Mi rowave Theory Te h., vol. MTT-13, no. 2, pp. 172-185, Feb. 1965. [3.7℄ C.P. Wen, \Coplanar waveguide: A surfa e strip transmission line suitable for nonre ipro al gyromagneti devi e appli ation", IEEE Trans. Mi rowave Theory Te h., vol. MTT-16, no. 12, pp. 1087-1090, De . 1969. [3.8℄ M.E. Davis et al., \Finite-boundary orre tions to the oplanar waveguide analysis", IEEE Trans. Mi rowave Theory Te h., vol. MTT-21, no. 6, pp. 594-596, June 1973. [3.9℄ G. Kompa, \Dispersion measurements of the rst two higher-order   modes in open mi rostrip", Ar h. Elek. Ubertragung ., vol. AEU-27, no. 4, pp.182-184, Apr. 1973. [3.10℄ R. Mehran, \The frequen y-dependent s attering matrix of mi rostrip   right angle bends", Ar h. Elek. Ubertragung ., vol. AEU-29, no. 11, pp. 454-460, Nov. 1975.

3.7.

REFERENCES

137

[3.11℄ K.C. Gupta, R. Garg, I. Bahl, P. Bhartia, Mi rostrip Lines and Slotlines, 2nd ed. Boston: Arte h House, 1996. [3.12℄ A. Vander Vorst, Ele tromagnetisme. Brussels: De Boe k-Wesmael, 1994, Ch. 2. [3.13℄ R.E. Collin, Field Theory of Guided Waves. New York: M Graw-Hill, 1960, Ch. 4. [3.14℄ S.B. Cohn, \Slot-line on a diele tri substrate", IEEE Trans. Mi rowave Theory Te h., vol. MTT-30, no. 10, pp. 259-269, O t. 1969. [3.15℄ T. Itoh, Numeri al Te hniques for Mi rowave and Millimeter-Wave Passive Stru tures. New York: John Wiley&Sons, 1989, Ch. 3, p. 139. [3.16℄ R.E. Collin, Field Theory of Guided Waves, 2nd ed. New York: IEEE Press, 1991, Ch. 2, pp. 91-96. [3.17℄ R.E. Collin, Field Theory of Guided Waves, 2nd ed. New York: IEEE Press, 1991, Ch. 2, pp. 95. [3.18℄ R.E. Collin, Field Theory of Guided Waves, 2nd ed. New York: IEEE Press, 1991, Ch. 3, p. 230. [3.19℄ J.R. Wait, Ele tromagneti Waves in Strati ed Media, 2nd ed. Oxford: Pergamon Press, 1970. [3.20℄ E.J. Denlinger, \A frequen y dependent solution for mi rostrip transmission lines", IEEE Trans. Mi rowave Theory Te h., vol. MTT-19, no. 1, pp. 30-39, Jan. 1971. [3.21℄ I.V. Lindell, \Variational methods for nonstandard eigenvalue problems in waveguide and resonator analysis", IEEE Trans. Mi rowave Theory Te h., vol. MTT-30, no. 8, pp. 1194-1204, Aug. 1982. [3.22℄ T. Itoh, R. Mittra, \Spe tral-domain approa h for al ulating the dispersion hara teristi s of mi rostrip lines", IEEE Trans. Mi rowave Theory Te h., vol. MTT-21, no. 7, pp. 496-498, July 1973. [3.23℄ J.B. Knorr, K. D. Ku hler, \Analysis of oupled slots and oplanar strips on diele tri substrate", IEEE Trans. Mi rowave Theory Te h., vol. MTT-23, no. 7, pp. 541-548, July 1975. [3.24℄ T. Itoh, R. Mittra, \Dispersion hara teristi s of slot-line", Ele tron. Lett., vol. 7, pp. 364-365, Jan. 1971. [3.25℄ R. Janaswamy, D. S haubert, \Dispersion hara teristi s for wide slotlines on low permittivity substrate", IEEE Trans. Mi rowave Theory Te h., vol. MTT-33, pp. 723-726, July 1985. [3.26℄ R. Janaswamy, D. S haubert, \Chara teristi impedan e of a wide slot on low permittivity substrates", IEEE Trans. Mi rowave Theory Te h., vol. MTT-34, pp. 900-902, July 1986. [3.27℄ T. Itoh, R. Mittra, \A te hnique for omputing dispersion hara teristi s of shielded mi rostrip lines", IEEE Trans. Mi rowave Theory Te h., vol. MTT-22, no. 1, pp. 896-898, Jan. 1974.

138

CHAPTER 3.

ANALYSIS OF PLANAR TRANSMISSION LINES

[3.28℄ L.P. S hmidt, T. Itoh, \Spe tral domain analysis of dominant and higher order modes in nlines", IEEE Trans. Mi rowave Theory Te h., vol. MTT-28, no. 9, pp. 981-985, Sept. 1980. [3.29℄ J.B. Knorr, P. Shayda, \Millimeter wave nline hara teristi s", IEEE Trans. Mi rowave Theory Te h., vol. MTT-28, no. 7, pp. 737-743, July 1980. [3.30℄ T. Itoh, \Spe tral-domain immitan e approa h for dispersion hara teristi s of generalized printed transmission lines", IEEE Trans. Mi rowave Theory Te h., vol. MTT-28, no. 7, pp. 733-736, July 1980. [3.31℄ T. Uwano, T. Itoh, \Spe tral domain analysis", in Numeri al Te hniques for Mi rowave and Millimeter Wave Passive Stru tures (Ed. T. Itoh). New York: John Wiley&Sons, 1989, Ch. 5. [3.32℄ M. Kirs hning, R. Jansen, \A

urate wide-range design of the frequen y-dependent hara teristi s of parallel oupled mi rostrips", IEEE Trans. Mi rowave Theory Te h., vol. MTT-32, no. 1, pp. 83-90, Jan. 1984. [3.33℄ R. Garg, K.C. Gupta, \Expressions for wavelength and impedan e of a slot-line", IEEE Trans. Mi rowave Theory Te h., vol. MTT-24, no. 8, pp. 532, Aug. 1976. [3.34℄ E.A. Mariani, C.P. Heinzman, J.P. Agrios, S.B. Cohn, \Slot-line hara teristi s", IEEE Trans. Mi rowave Theory Te h., vol. MTT-17, no. 12, pp. 1091-1096, De . 1969. [3.35℄ W.J. Hoefer, A. Ros, \Fin line parameters al ulated with the TLMmethod", IEEE MTT-S Int. Mi rowave Symp. Dig., pp. 341-343, Orlando, Apr. 1979. [3.36℄ S. Akhtarzad, P.B. Johns, \Three-dimensional transmission-line matrix

omputer analysis of mi rostrip resonators", IEEE Trans. Mi rowave Theory Te h., vol. MTT-23, no. 12, pp. 990-997, De . 1975. [3.37℄ D.H. Choi, J.R. Hoefer, \The nite-dieren e-time-domain method and its appli ation to eigenvalue problems", IEEE Trans. Mi rowave Theory Te h., vol. MTT-34, no.12, pp. 1464-1470, De . 1986. [3.38℄ J.R. Hoefer, \The transmission line matrix method - Theory and appli ations", IEEE Trans. Mi rowave Theory Te h., vol. MTT-33, no.10, pp. 882-893, O t. 1985. [3.39℄ R.E. Collin, Field Theory of Guided Waves. New York: M Graw-Hill, 1960, Ch. 4, pp. 152-155. [3.40℄ R.E. Collin, Field Theory of Guided Waves, 2nd ed. New York: IEEE Press, 1991, Ch. 4, p. 279. [3.41℄ E. Yamashita, R. Mittra, \Variational method for the analysis of mi rostrip lines", IEEE Trans. Mi rowave Theory Te h., vol. MTT-16, no. 8, pp. 251-256, Apr. 1968.

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REFERENCES

139

[3.42℄ E. Yamashita, \Variational method for the analysis of mi rostrip-like transmission lines", IEEE Trans. Mi rowave Theory Te h., vol. MTT16, no. 8, pp. 529-535, Aug. 1968. [3.43℄ R.E. Collin, Field Theory of Guided Waves, 2nd ed. New York: IEEE Press, 1991, Ch. 4, p. 328, ex. 4.17. [3.44℄ T. Kitazawa, \Variational method for planar transmission lines with anisotropi magneti materials", IEEE Trans. Mi rowave Theory Te h., vol. 37, no. 11, pp. 1749-1754, Nov. 1989. [3.45℄ R.A. Pu el, D.J. Masse, \Mi rostrip propagation on magneti substrates - Part I: Design theory", IEEE Trans. Mi rowave Theory Te h., vol. MTT-20, no. 5, pp. 304-308, May 1972. [3.46℄ R.A. Pu el, D.J. Masse, \Mi rostrip propagation on magneti substrates - Part II: Experiment", IEEE Trans. Mi rowave Theory Te h., vol. MTT-20, no. 5, pp. 309-313, May 1972. [3.47℄ R.E. Collin, Field Theory of Guided Waves. New York: M Graw-Hill, 1960, Ch. 1, pp. 11-13. [3.48℄ R.E. Collin, Field Theory of Guided Waves, 2nd ed. New York: IEEE Press, 1991, Ch. 4, p. 274-275. [3.49℄ V.H. Rumsey, \Rea tion on ept in ele tromagneti theory", Phys. Rev., vol. 94, pp. 1483-1491, June 1954, and vol. 95, p. 1705, Sept. 1954. [3.50℄ A.D. Berk, \Variational prin iples for ele tromagneti resonators and waveguides", IEEE Trans. Antennas Propagat., vol. AP-4, no. 2, pp. 104-111, Apr. 1956. [3.51℄ J.Jin, W.C. Chew, \Variational formulation of ele tromagneti boundary-value problems involving anisotropi media", Mi rowave and Opt. Te hnol. Lett., vol. 7, no. 8, pp. 348-351, June 1994. [3.52℄ P.M. Morse, H. Feshba h, Methods of Theoreti al Physi s. New York: M Graw-Hill, 1953, vol. 2 , Ch. 9. [3.53℄ R.F. Harrington, Field Computation by Moment Methods. New York: Ma millan, 1968, Ch. 1. [3.54℄ R.F. Harrington, Field Computation by Moment Methods. New York: Ma millan, 1968, Ch. 1, p. 19. [3.55℄ J.B. Davies in T. Itoh, Numeri al Te hniques for Mi rowave and Millimeter Wave Passive Stru tures. New York: John Wiley&Sons, 1989, Ch. 2. [3.56℄ M. Essaaidi, M. Boussouis, \Comparative study of the Galerkin and the least squares boundary residual method for the analysis of mi rowave integrated ir uits", Mi rowave and Opt. Te hnol. Lett., vol. 7, no. 3, pp. 141-146, Mar. 1992.

140

CHAPTER 3.

ANALYSIS OF PLANAR TRANSMISSION LINES

[3.57℄ F.L. Mesa, M. Horno, \Quasi-TEM and full wave approa h for oplanar multistrip lines in luding gyromagneti media longitudinally magnetized", Mi rowave and Opt. Te hnol. Lett., vol. 4, no. 12, pp. 531-534, Nov. 1991. [3.58℄ N.K. Das, D.M. Pozar, \Full-wave spe tral-domain omputation of material, radiation, and guided wave losses in in nite multilayered printed transmission lines", IEEE Trans. Mi rowave Theory Te h., vol. MTT39, no. 1, pp. 54-63, Jan. 1991. [3.59℄ R.H. Jansen, \The spe tral domain approa h for mi rowave integrated

ir uits", IEEE Trans. Mi rowave Theory Te h., vol. MTT-33, no. 10, pp. 1043-1056, O t. 1985. [3.60℄ S. Barkehli, P.H. Pathak, \Re ipro al properties of the dyadi Green's fun tion for planar multilayered diele tri /magneti media", Mi rowave and Opt. Te hnol. Lett., vol. 4, no. 9, pp. 333-335, Aug. 1991. [3.61℄ R.E. Collin, Field Theory of Guided Waves. New York: M Graw-Hill, 1960, Ch. 2. [3.62℄ R.E. Collin, F.J. Zu her, Antenna Theory. New York: M Graw-Hill, 1969, Ch. 2. [3.63℄ C.T. Tai, Dyadi Green Fun tions in Ele tromagneti Theory, 2nd ed. New York: IEEE Press, 1994. [3.64℄ R.E. Collin, Field Theory of Guided Waves, 2nd ed. New York: IEEE Press, 1991, Ch. 2, p. 102. [3.65℄ L.B. Felsen, N. Mar uvitz, Radiation and S attering of Waves. New York: Prenti e-Hall, 1973, Ch. 1. [3.66℄ A. Papayannakis, T.D. Tsiboukis, E.E. Kriezis, \An investigation of the theory and the properties of the dyadi Green's fun tion for the ve tor Helmholtz equation", Ele tromagneti s, vol. 7, pp. 91-100, 1987. [3.67℄ J. S hwinger, D.S. Saxon, Dis ontinuities in Waveguides - Notes on Le tures by J. S hwinger. London: Gordon and Brea h, p. 36-37, 1968. [3.68℄ J.H. Ri hmond, \On the variational aspe ts of the moment method", IEEE Trans. Antennas Propagat., vol. 39, no. 4, pp. 473-479, Apr. 1991. [3.69℄ A.F. Peterson, D.R. Wilton, R.E. Jorgenson, \Variational nature of Galerkin and non-Galerkin moment method solutions", IEEE Trans. Antennas Propagat., vol. 44, no. 4, pp. 500-503, Apr. 1996. [3.70℄ J.R. Mautz, \Variational aspe ts of the rea tion in the method of moments", IEEE Trans. Antennas Propagat., vol. 42, no. 12, pp. 16311638, De . 1994. [3.71℄ R.A. Waldron, Theory of Guided Ele tromagneti Waves. London: Van Nostrand Reinhold Company, 1969. [3.72℄ J.B. Davies in T. Itoh, Numeri al Te hniques for Mi rowave and Millimeter Wave Passive stru tures. New York: John Wiley&Sons, 1989, Ch. 2, pp. 79-80.

3.7.

REFERENCES

141

[3.73℄ D.J. Corr, J.B. Davies, \Computer analysis of the fundamental and higher order modes in single and oupled mi rostrip lines - Finite differen e methods", IEEE Trans. Mi rowave Theory Te h., vol. MTT-20, no. 10, pp. 669-678, O t. 1972.

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hapter 4

General variational prin iple 4.1

Generalized topology

Variational prin iples were used by S hwinger to solve dis ontinuity problems in waveguides [4.1℄. Formulas from Rumsey [4.2℄ were shown to be variational for losed waveguides provided that the permittitivity and permeability are, at most, symmetri tensors. As a result, isotropi materials and materials with rystalline anisotropy, lossy or lossless, are allowed while gyrotropi media su h as magnetized devi es and magneto-ioni media are ex luded. On the other hand, Berk [4.3℄ developed variational formulas appli able to gyrotropi media or, in general, media whose diele tri onstant and permeability are Hermitian tensors, restri ted to lossless substan es. The pra ti al use, however, of a formula developed for lossless gyrotropi media is quite limited, be ause gyrotropy implies losses, ex ept in very narrow-band ir umstan es. All these formulations were limited to losed waveguides and resonators. The general variational formulation developed here is appli able to losed as well as open planar and oplanar line stru tures, whi h an be multilayered, with gyrotropi lossy inhomogeneous media. The ondu tors must be lossless. The validity of the formulation is demonstrated by original measurements on transmission lines, as well as on more ompli ated stru tures. Figure 4.1 shows a s hemati representation of a general transmission line analyzed by using this variational prin iple [4.4℄. It onsists of plane ondu tive layers inserted between planar layers, ea h of them hara terized by permittivity and permeability tensors. There are no limitations on the tensors. Furthermore, ea h ondu tive layer may have several ondu tors. It should be noted that a distin tion is usually made between slot-like ondu tors and strip-like ondu tors, based on the amount of metallization extending in the xdire tion. For strip-like ondu tors, the metal is on ned to nite areas along the x-axis, while slot-like ondu tors ontain semi-in nite metal sheets and

an be viewed as ondu tive layers with slots of nite extent. This distin tion is of prime interest for the hoi e of trial elds, be ause their formulation is based on a trial quantity whi h is non-zero only on an area of nite extent along the x-axis, namely a urrent for strip-like problems and an ele tri eld for slot-like problems. This will be detailed in Se tion 4.5. The whole multilayered stru ture may be fully open, or en losed in a waveguide, or partially bounded by ele tri or magneti shielding. When using the moment method,

144

CHAPTER 4.

GENERAL

VARIATIONAL PRINCIPLE

HNn

HN

H2

W2

T

HN1

S

W1

ax HM1

az

W3 ay

A

HM

Wg

HMm

dielectric or gyrotropic lossy substrate slot-like conductive layer

strip-like conductive layer

perfect magnetic or electric shielding perfect magnetic or electric shielding

Fig. 4.1

prin iple

Geometry of transmission line analyzed by using general variational (0  + M + N + 6= N M 1 2  1 with 6 0) 2+  g  1 and 1 + 2 =

0; W1 + W

H; H

S

W

;H

; T ; S; W ; W W

W

H

H

H

T

4.2.

DERIVATION IN THE SPATIAL DOMAIN

145

fully open stru tures are usually onsidered as shielded lines whose shielding are far away from the a tive part of the line [4.5℄[4.6℄, but not at in nity. By doing so, advantage is taken of the simpli ation of integrations into summations resulting from the parti ular formulation of the problem for shielded lines. This artefa t is not required with the formulation developed here, be ause of its onvenient onvergen e properties even for fully open lines, as has been shown in [4.7℄, and will be explained in Se tion 4.6. The parti ular ase of diele tri [4.8℄ or gyrotropi -loaded waveguides may also be al ulated. It

onsists of a multilayered stru ture, without planar ondu tive layers, and en losed into a waveguide. In this hapter, the general form of a variational prin iple is derived in the spatial domain in Se tion 4.2 and in the spe tral domain in Se tion 4.3. In Se tion 4.4, the methodology for formulating suitable expressions for trial elds is developed. Constraints, degrees of freedom, and formulations in terms of Hertzian potentials and of ele tri eld omponents are presented and dis ussed. The hoi e of adequate trial omponents at dis ontinuous ondu tive interfa es is developed in Se tion 4.5, like onformal mapping for slot- and strip-like stru tures. The Rayleigh-Ritz pro edure is shown to be ompatible with the general variational prin iple, while Mathieu fun tions are demonstrated to be extremely eÆ ient as trial omponents. The importan e of the in uen e of the ratio between the longitudinal and transverse omponents of the trials is dis ussed. The advantages of the variational behavior are demonstrated and illustrated in Se tion 4.6. The in uen e of the value hosen for the propagation onstant in the trial elds, the shape of trial omponents, and the number of Mathieu fun tions are al ulated, as well as the trun ation error of integrations in the spe tral domain. Some illustrative ases are al ulated in Se tion 4.7 where the eÆ ien y of the variational al ulation is demonstrated. On-line results are obtained with a regular PC, in a few se onds for the whole frequen y range, in the ase of lossy multilayered planar transmission lines. The spe tral domain approa h using Galerkin's pro edure is ompared with the variational prin iple. It will be shown that the numeri al omplexity of the rst pro edure is mu h larger than that indu ed by the variational prin iple. Finite-elements solvers are also ompared, with the same advantage in favor of the variational pro edure, with an extra-advantage in the ase of very lossy stru tures. 4.2

Derivation in the spatial domain

The elds in the general stru ture of Figure 4.1 are de ned as


e(x; y)e E (x; y; z ) =

z

(4.1a)


h(x; y)e z H (x; y; z ) = (4.1b) with =  + j and where a omplex e z dependen e along the propagation axis az is assumed. It will now be shown that the solution of the following

146

CHAPTER 4.

GENERAL

VARIATIONAL PRINCIPLE

omplex equation is a variational formulation of the omplex propagation

onstant [4.9℄:

2

Z

(az  e )
[

AZ

+

1
(a  e)℄ dA z

f(r  e )
[ 1
(az  e)℄

(az  e )
(

1
r  e)g dA

Z Z A 1 (r  e )
(
r  e) dA + ! 2 e
("
e) dA = 0

(4.2)

A

A

In this notation e(x; y ) and h(x; y ) have both transverse and longitudinal

omponents, while e denotes the omplex onjugate of e, A is the area of the

ross-se tion, and r is the lassi al 3-dimensional del-operator. 4.2.1

Derivation

Using the notation of (4.1), Maxwell's equations are written as

r  e + j!
h = az  e r  h = j!"
e + az  h

(4.3) (4.4)

with no assumptions about the permittivity and permeability tensors. Entering (4.3) into (4.4) yields the se ond-order equation for the elds

r  [ 1
( az  e r  e)℄ 1 1 = 2 az  [
(az  e)℄ az  (
r  e)

!2 "
e

(4.5)

Multiplying by the omplex onjugate e and rearranging yields

r  [



= 2 az

1
(a  e)℄
e r  ( 1
r  e)
e z  1  [
(a  e)℄
e

az  (

z

1
r  e)
e

(4.6)

! 2 ("
e)
e

written for onvenien e as

A

B = 2C

D

E

(4.7)

where oeÆ ients A, B , C , D, and E are obtained by identi ation with (4.6). Using ve tor identity X
rY =Y


r  X r
(X  Y )

the oeÆ ients an be written as




A = [

1
(a  e)℄
r  e r
fe  [ 1
(a  e)℄g z z

(4.8)

4.2.

147

DERIVATION IN THE SPATIAL DOMAIN

B


= (r  e )
[

C


= (az  e )
[

1


(az  e)℄

(az  e )
(

1


r  e)


D=

1


(r  e)℄ r
[e  ( 1
r  e)℄

After introdu ing into (4.7) and rearranging, the equation be omes

2 (az  e )
[

1


(az  e)℄ 1 1 

f(az  e )
(
r  e) [
(az  e)℄
r  e g (r  e )
( 1
r  e) + !2 ("
e)
e + r
fe  [ 1
r  e  1
(az  e )℄g = 0

(4.9)

Integrating in the transverse plane yields surfa e integrals for ea h medium and line integrals at the boundaries:

2

Z

+

(az  e )
[

AZ

Z A

+

IA

1


(az  e)℄ dA

f(r  e )
[ 1
(az  e)℄

(r  e )
(

1

(az  e )
(


r  e) dA + !2

fe  [
r  e 1



Z

A

1


r  e)g dA

e
("
e) dA

1


(az  e)℄g
n d

1


(az  e)℄ d

(4.10)

=0

The line integral an be written as I

(n  e )
[

1


re



(4.11)

It vanishes when e is the exa t eld and if there are no losses at the ondu tive boundaries, be ause:



at ondu ting planes n  e vanishes provided no losses are present in the ondu tor



at interfa es between media e and h have ontinuous tangential omponents, and the ee t of integration on both sides of the interfa e vanishes.

So (4.10) redu es to (4.2) and is satis ed by the exa t eld.

148

CHAPTER 4.

4.2.2

GENERAL

VARIATIONAL PRINCIPLE

Proof of variational behavior

Repla ing the exa t eld into equations (4.2) and (4.10) by a trial eld etrial = e + Æe

(4.12)

indu es a variation on . Negle ting the se ond-order variations, the equation be omes

Z ( + 2 Æ ) (az  e )
[ A Z 2

+

A

(az  Æe )
[

1

Z

1


(az  e)℄ dA


(az  e)℄ dA +

Z

A

(az  e )
[

1




(az  Æe)℄ dA

 1 1 + ( + Æ ) (r  e )
[
(az  e)℄ (az  e )
(
r  e) dA A Z  1 1  (r  e )
[
(az  Æe)℄ (az  Æe )
(
r  e) dA +  ZA  1 1 (r  Æe )
[
(az  e)℄ (az  e )
(
r  Æe) dA + AZ Z Z + ! 2 f e
("
e) dA + Æe
("
e) dA + e
("
Æe) dAg A A Z A 1 [(r  e )
(
r  e)

A

+ (r  Æe )
(

1


r  e) + (r  e )
( 1
r  Æe)℄ dA = 0

(4.13)

It an easily be seen that the terms without variation form the left-hand side of equation (4.10), hen e the sum of those terms vanishes. The rst-order variations also vanish, under spe i onditions. Re-arranging equation (4.13) yields indeed the following results. 1. The term ontaining the rst-order variation Æe of the eld is

Z

(az  e )
[
(az  Æe)℄ dA AZ 1 1 + f(r  e )
[
(az  Æe)℄ (az  e )
(
r  Æe)g dA ZA Z 1 2 +! (r  e )
(
r  Æe) dA e
("
Æe) dA

2

1

A

A

4.2.

DERIVATION IN THE SPATIAL DOMAIN

149

whi h an be written as

2

Z

n

 ZA

e
az  [


(az  Æe)℄

o

dA

o r  [ 1
(az  Æe)℄ dA I nA o 1 + e  [
(az  Æe)℄
n d  Z + e
[az  ( 1
r  Æe)℄ dA AZ Z 1 + !2 e
("
Æe) dA e
[r  (
r  Æe)℄ dA A I A 1 e  [
(r  Æe)℄
n d +

e


n

1

(4.14)

where ve tor identity (4.8) has been used. Re ombining the integrands of the various surfa e integrals yields the surfa e integral of e dot-multiplied by equation (4.5), written in terms of Æe instead of the exa t eld. It vanishes if Æe satis es Maxwell's equations (4.3) and (4.4). Hen e, the trial elds have to be hosen su h that they satisfy those two equations. On the other hand, the

ondition to an el the sum of the line integrals in (4.14) is that the quantity 

1


[ az  Æe r  Æe℄

has ontinuous tangential omponents so that the ee t of integration on both sides of an interfa e vanishes. This is equivalent to saying that the trial magneti eld has ontinuous tangential omponents at any interfa e. Finally, on

ondu ting planes, n  e vanishes provided the ondu tor is lossless. 2. Similarly, it an be shown that the term ontaining the onjugate of the rst-order eld variation vanishes provided the tangential omponents of the trial ele tri eld are made to vanish at the perfe t ondu ting boundaries, and are ontinuous at any interfa e. The term ontaining Æe is

2

Z

(az  Æe )
[

AZ

1


(az  e)℄ dA

f(r  Æe )
[ 1
(az  e)℄ (az  Æe )
( 1
r  e)g dA ZA Z 2 +! Æe
("
e) dA (r  Æe )
( 1
r  e) dA

+

A

A

150

CHAPTER 4.

whi h an be written as

2

Z

 ZA

n

Æe
az  [

1

Æe
[az  (

1


(az  e)℄

GENERAL

o

VARIATIONAL PRINCIPLE

dA


r  e)℄ dA o 1 Æe  [
(az  e)℄
n d +  Z + Æe
r  [ 1
(az  e)℄ dA AZ Z 1 + !2 Æe
("
e) dA Æe
[r  (
r  e)℄ dA A I A 1 Æe  [
(r  e)℄
n d

+

I nA 

(4.15)

where ve tor identity (4.8) has been used again. Re ombining the integrands of the various surfa e integrals yields the surfa e integral of Æe dot-multiplied by equation (4.5), written in terms of the exa t eld so that the sum of the surfa e integrals vanishes. Similarly, the ondition suÆ ient to an el the sum of the line integrals in the term in Æe is that the tangential omponents of etrial are ontinuous at any interfa e and vanish at ondu ting planes, so that Æe also vanishes with e. Hen e the error Æ vanishes, be ause: 1. the term without variation has been shown to vanish 2. the terms ontaining variations of Æe and Æe have also been shown to vanish too, by equations (4.14) and (4.15), so that the remaining term of the left hand of equation (4.13) ontaining Æ is for ed to vanish. 4.2.3 Spe i ities of the general variational prin iple The solution of equation (4.2) has been proved to be variational with respe t to the ele tri eld e and to its omplex onjugate e . From the proof just given, it is on luded that, when the exa t ele tri eld in expression (4.2) is repla ed by a trial eld denoted etrial , no rst-order error is made on the solution of equation (4.2), provided the trial elds meet the following onditions: 1. etrial satis es Maxwell's equations (4.3) and (4.4) 2. etrial and htrial have ontinuous tangential omponents at any surfa e of dis ontinuity of the media in the transverse plane 3. etrial has vanishing tangential omponents at ondu ting planes,

4.3.

DERIVATION IN THE SPECTRAL DOMAIN

151

and provided the ondu tors are lossless. It should be emphasized that, in the derivation, no assumption is made on the diele tri and magneti properties of the materials of the line. The derivation does not require omplex onjugates of the onstitutive tensors of the layers. Hen e the Hermitian nature of " and  is not required here. Furthermore, the expression is variational even when losses are present in the media. Formulation (4.2) has a number of advantages. 1. It implies that the error on the value of al ulated by the equation is small if the trial ele tri eld is an adequate approximation. Its eÆ ien y is due to the fa t that the error made on any omponent of the eld is

ompensated by an exa t analyti al spatial integration performed over the whole ross-se tion, and not only over a line boundary at an interfa e

ontaining ondu tive layers, as was the ase for the integral equation approa h introdu ed in Chapters 2 and 3. 2. It is a dynami formulation, whilst usually expli it variational formulations are quasi-stati [4.10℄, providing values for quasi-TEM parameters su h as apa itan es or indu tan es. 3. It provides a straightforward evaluation of the omplex propagation onstant by solving a se ond-order equation, whi h is mu h simpler and faster than the lassi al extra tion of the root of the determinantal equation in the Galerkin's pro edure [4.11℄[4.12℄. Davies mentions in [4.13℄ that the solution of the determinantal equation is usually diÆ ult to obtain. 4. The solution is variational even when the medium is inhomogeneous, lossy, and gyrotropi . Only the ondu tors have to be lossless. This is a major advantage when ompared to the moment method used in the spe tral domain. Indeed in this ase, the problem is usually simpli ed by assuming lossless diele tri layers with a view to sear hing for a real root, as mentioned by Jansen [4.14℄. The variational formulation will also be useful for modeling stru tures supporting leaky modes or slow waves, hara terized by non-negligible values of both real and imaginary parts of the propagation onstant. These stru tures are usually only

al ulated by using perturbational methods, as dis ussed by Das and Pozar [4.15℄. 5. The onstitutive tensors of the layers are involved in the surfa e integrals so that non-uniformities in a layer, if any, an be taken into a

ount. Those advantages will be illustrated by examples in Se tions 4.6 and 4.7. 4.3

Derivation in the spe tral domain

The spatial form of equation (4.2) is diÆ ult to use for open and shielded planar lines. These present dis ontinuities of the fun tion des ribing the eld

152

CHAPTER 4.

GENERAL

VARIATIONAL PRINCIPLE

omponents along the x-axis at the interfa es in luding the slots. The major

onsequen e is that the x- and y -dependen ies of the eld annot be easily separated, and the integration over the ross-se tions in (4.2) be omes diÆ ult. The spe tral domain te hnique over omes this problem, as will now be shown.

4.3.1 De nition of spe tral domain and spe tral elds

The spe tral domain te hnique simply onsists of a Fourier transform performed along the x-axis on ea h omponent of the ele tri and magneti elds supported by the multilayered stru ture (Appendix C): e~v (kx ; y ) =

~hv (kx ; y ) =

Z1 v 1 Z1 1

e (x; y )e jkx x dx

(4.16a)

hv (x; y )e jkx x dx where v = x; y; z

(4.16b)

In all future dis ussions, these Fourier transforms of the spatial elds will be referred to as spe tral elds. Trial quantities (ele tri eld for slot-like lines and dieren e of magneti elds for strip-like lines) are then hosen su h that their Fourier transform is a ontinuous fun tion of the spe tral variable kx . The Fourier transform of Maxwell's equations (4.3) and (4.4) provides a general solution for the ele tromagneti elds. These are usually separable into a hyperboli y -dependen e multiplied by spe tral oeÆ ients, fun tions of the spe tral variable kx . Hen e, the Fourier transform of the boundary

onditions spe i ed in Se tion 2 provides boundary onditions at the y -plane interfa es whi h result in a set of simple algebrai relations between the spe tral oeÆ ients ae ting the y -dependen e and the trial spe tral quantities. Be ause of the linearity properties of the Fourier transform, this is equivalent to saying that the solutions of the Fourier-transformed Maxwell's equations are the Fourier transform of the trial elds, namely the trial spe tral elds. Dierent ways to obtain the trial spe tral elds will be illustrated in the next se tion.

4.3.2 Formulation of variational prin iple in spe tral domain

The basi variational equation (4.2) an be rewritten in the spe tral domain as a fun tion of the spe tral trial elds. It is rewritten for onvenien e as

2

with

X i

Ai

X i

(Bi

Z 
z A Z r 




Ai = Bi =

X i

Di

!2

X i

Ei = 0

(4.17a)

(a

e ) [

1
(a  e)℄ dA z

(4.17b)

(

e ) [

1
(a  e)℄ dA z

(4.17 )

i

Ai

Ci ) +

4.3.

153

DERIVATION IN THE SPECTRAL DOMAIN




Ci =




Di =




Ei =

Z ZAi

(az  e )
[

1
(r  e)℄ dA

(4.17d)

(r  e )
[

1
(r  e)℄ dA

(4.17e)

Z Ai

Ai

e
("i
e) dA

(4.17f)

where subs ript i holds for layer i. Use is made of Parseval's theorem [4.16℄ (Appendix C), whi h establishes the equality Z Z1 1 1 ~  f (k )f~ (k )dk (4.18) f1 (x)f2 (x)dx = 2 1 1 x 2 x x 1 where the fun tions in the integrands are related by the Fourier transform (4.16). Hen e, assuming that the media of ea h layer is uniform, a new set of

oeÆ ients is de ned, ea h of them being equal to the orresponding oeÆ ient in set (4.17) by virtue of (4.18): Z Z1  1 1
~ Ai = [a  ~ei (kx ; y ) ℄
i
[az  e~i (kx ; y )℄ dkx dy (4.19a) 2 yi 1 z Z Z1 1
~ (k ; y ) ℄
 1
[a  ~e (k ; y )℄ dk dy ~ [ url (4.19b) Bi = i x z i x x i 2 yi 1 Z Z1 1 ~ (k ; y)℄
[a  ~e (k ; y) ℄ dk dy
1 (4.19 ) C~i = [i
url i x z i x x 2  yi 1 Z Z1
1 ~ (k ; y )℄ dk dy ~ (k ; y ) ℄
[ 1
url ~i = (4.19d) D [ url i x i x x i 2 yi 1 Z Z1
1 ~e (k ; y )
["i
~ei (kx ; y )℄ dkx dy E~i = (4.19e) 2 yi 1 i x ~ (k ; y ), ~ = D ; E~ = E , and where url with A~ = A ; B~ = B ; C~ = C ; D

i

i

i

i

i

i

i

i

i

i

de ned as the Fourier transform of r  e, is expressed as  ~ex (kx ; y ) ~ (k ; y ) =  ~ez (kx ; y ) a

url )az i x x jkx~ez ay + (jkx~ey y

y

i x

(4.20)

The integrals along the y -axis an be performed analyti ally, before the integration along the kx variable, be ause the trial spe tral elds are usually

hosen to have a hyperboli y -dependen e, as will be shown later. Be ause of the equality between ea h of the oeÆ ients in set (4.19) with the orresponding oeÆ ients in set (4.17), the variational equation (4.17a) an be written as X X X X ~ i !2 E~i = 0 (4.21) D (B~i C~i ) + A~i

2

i

i

i

i

whi h is the variational spe tral equation, used as the basis of the method in the spe tral domain.

154

CHAPTER 4.

GENERAL

VARIATIONAL PRINCIPLE

4.4 Methodology for the hoi e of trial elds 4.4.1

Constraints and degrees of freedom

It has been shown in Se tion 4.2 that adequate spatial trial elds must satisfy Maxwell's equations (4.3) and (4.4) and have ontinuous tangential omponents at interfa es between layers, ex ept on ondu ting parts where only the trial tangential ele tri eld has to vanish. By virtue of the linearity properties of the Fourier transform, the spe tral trial elds are a solution of the Fourier transform of Maxwell's equations, and satisfy the Fourier transforms of the spatial boundary onditions at y -planes orresponding to interfa es between two layers. There are two degrees of freedom for determining the trial eld. First, trials may be expressed in the spatial or spe tral domain. Se ondly, adequate trial elds may be omposed either from a well-known solution of Maxwell's equations in ea h layer of the stru ture under onsideration, or from a reasonable guess at a solution to Maxwell's equations. In the rst ase, it will be shown that eld solutions of Helmoltz equations rewritten in ea h layer are adequate trials satisfying (4.3) and (4.4), so that one only has to impose the

ontinuity onstraint on those elds. In the se ond ase, the reasonable guess is dedu ed from physi al onsiderations and a rigorous appli ation of (4.3) and (4.4), and of the boundary onditions. In the following subse tion, the two approa hes are developed extensively in the spe tral domain. The same reasoning, however, may be applied to trial elds in the spatial domain, as exempli ed in the next subse tion, where the elds in a YIG- lm are obtained from a s alar potential. 4.4.2

Choi e of integration domain

In Se tions 4.2 and 4.3 the spatial and spe tral forms of the variational prin iple were introdu ed. They dier with regard to the integration domain related to the x-axis, respe tively over the x-domain or over its Fourier transform, the spe tral kx -domain. The hoi e depends essentially on the topology of the stru ture (presen e or absen e of dis ontinuous ondu tive layers in y -planes), and on the x-dependen e of the onstitutive parameters of ea h layer. The spe tral form may indeed be diÆ ult to use when the onstitutive parameters of one of the layers are fun tions of the x-variable. Parseval's identity (4.18) must be applied with are for ea h term of the s alar produ t of elds appearing in oeÆ ients (4.17b-f) when the onstitutive permeability tensor is a fun tion of the position. For oeÆ ient Ai for instan e, holds:

f1(x; y) = [az  e(x; y ) ℄v

(4.22a)

f~1(kx ; y) = [az  ~ei (kx ; y) ℄v

(4.22b)

f2(x; y ) = fi 1 (x; y )
[az  ei (x; y )℄gv

(4.22 )

4.4.

METHODOLOGY FOR THE CHOICE OF

f~2 (kx ; y) = =

Z1

1 Z1 1

f

1

f~

1

i

i

(x; y )
[az  ei (x; y )℄gv e (kx

155

TRIAL FIELDS

jkx x

dx

u; y )
[az  ~ei (u; y )℄gv du

(4.22d) (4.22e)

where subs ript i is for layer i and where v = x; y; z . As is well known, the Fourier transform of the produ t of two fun tions is not equal to the produ t of the Fourier transforms of the two fun tions, but to its onvolution produ t. Hen e, triple integrals appear in the expression of Z Z1 1 [a  ~ei (kx ; y ) ℄ A~i = 2 yi 1 z (4.23) Z 1 1
i (kx u; y)
[az  e~i (u; y)℄ du dkx dy

1

~ i respe tively, and for The same remark is valid for the al ulation of B~i ; C~i ; D ~

oeÆ ient Ei if the onstitutive permittivity tensor is also a fun tion of the position. As a on lusion, the spatial form (4.2) is preferred when layer i is des ribed by a non-uniform permeability or permittivity onstitutive tensor. Various appli ations overing the spatial- and spe tral-domain use of the variational formulation will be illustrated in Chapter 5. The next se tions, devoted to the hoi e of spe tral trial elds, will extensively evaluate two topologies in the spe tral domain, illustrating the use of the spe tral form of the variational formulation. Before using the spe tral domain te hnique however, the hoi e of a spatial formulation will be illustrated for the trial elds by an example requiring the use of the spatial form of the variational formulation. The methodology used for the sear h of trial elds is similar in the spatial and spe tral domains.

4.4.3 Spatial domain: propagation in YIG- lm of nite width

Yttrium-Iron-Garnet (YIG) is onsidered in the lm on guration illustrated in Figure 4.2. The magnet poles generate a DC-magneti eld along the y axis. The lm is assumed to be in nite along the z -axis and has a nite width W along the x-axis. Taking advantage of the ferrite nature of the lm, the propagation onstant of this guiding stru ture an be modi ed by varying the applied DC magneti eld. These lms nd appli ations in planar mi rowave devi es operating in the magnetostati wave range. The reader will nd a des ription of these waves in Appendix D as well as a rigorous derivation of the approximations. A simpli ed analysis of the stru ture of Figure 4.2 is available [4.17℄[4.18℄, based on the following assumptions: 1. The DC-magneti eld inside of the lm is uniform, whi h implies that the demagnetizing ee t and the external DC-magneti eld are uniform. 2. The edges of the lm an be approximated by Perfe t Magneti Walls (PMW).

156

CHAPTER 4.

GENERAL

VARIATIONAL PRINCIPLE

upper pole

ay

H1

left area az .

H

layer 1

W right area

ax microstrip dielectric substrate

H2

layer 3 layer 2

lower pole planar dielectric layer

Fig. 4.2

YIG film

magnet poles

metal

Con guration of YIG- lm for YIG-tuned devi es

3. Low losses o

ur in the lm, so that they an be obtained by a perturbational method. These hypotheses are usually made for the pra ti al design of MagnetoStati Waves (MSW) devi es, be ause of the la k of simple models taking into a

ount the non-uniformity of the internal eld, the losses in the lm, and the nite width ee t. For the present example only, the se ond assumption is retained. The purpose is to show that the variational equation (4.2) is eÆ ient for the al ulation of the propagation onstant of MSW in YIG- lms in the presen e of losses and of a non-uniform DC-magneti eld. The following parameters are de ned (Fig. 4.2):

x y H Hi W

horizontal oordinate, entered at one edge of YIG- lm verti al oordinate, entered at interfa e between YIG and layer 2 thi kness of substrate, alled layer 3 distan e between the interfa e layer i - layer 3 and the metalli housing, if any width of YIG- lm.

Maxwell's equations are rewritten under the magnetostati assumption in all the three media of Figure 4.2: r

E i + j!i
H i = 0

H i  0 (magnetostati assumption) r
Bi = 0 r

(4.24) (4.25) (4.26)

4.4.

METHODOLOGY FOR THE CHOICE OF

TRIAL FIELDS

157

where  is the permeability tensor of layer i, de ned as i

i

2 11 =4 0

;i

3 0 5

0

21;i

0

22;i

0

12;i

(4.27)

The expressions for 11 3 , 21 3 , 12 3 in layer 3 are given in Appendix D, whi h des ribes the permeability tensor of a YIG- lm. They depend on the value of the internal DC magneti eld. As detailed in the appendix the internal DC magneti eld in a YIG- lm having a nite width is spatially non-uniform, be ause of a non-uniform demagnetizing ee t [4.19℄. Hen e, the permeability tensor 3 is spatially non-uniform. In layers 1 and 2, the medium is assumed to be isotropi : ;

11;i

;

= 22 = 0 and ;i

;

12;i

= 21 = 0 for ;i

i = 1; 2

(4.28)

Using the notations of (4.1), equations (4.24) and (4.25) may be rewritten as the orresponding magnetostati form of equations (4.3) and (4.4):

r  e + j!
h = a  e r  h = a  h i

i

i

z

i

(4.29a)

(4.29b) From (4.25), it an be stated that the magneti eld is derived from a s alar potential : i

Hi

z

= r

i

(4.30)

i

The magneti ux density B is easily dedu ed in ea h layer from the magneti eld as Bi i.e.

=H i

i

: B 1;2

= 0 H 1 2

(4.31a)

;

= 0 H 3 (4.31b) B 3 = 11 3 H 3 + 21 3 H 3 (4.31 ) B 3 = 12 3 H 3 + 11 3 H 3 (4.31d) Combining (4.30) and (4.31a-d) into (4.26) yields in ea h region after simpli ation 2 2 2 +   ℄+   =0 (4.32)  [ By3

y

x

;

x

;

z

z

;

x

;

z

11;i

i

x2

i

z 2

0

i

y 2

158

CHAPTER 4.

GENERAL

VARIATIONAL PRINCIPLE

whi h has the following general solution, obtained by separating the three variables x, y and z : i (x; y; z ) = [A sin kx x + B os kx x℄[Ci es0 y + Di e i

with s0i

q

= (11;i =0 )(kx2

2)

s0i y

℄e

wavenumber along y-axis

z

(4.33a) (4.33b)

kx wavenumber

along x-axis MagnetoStati Forward Volume Waves (MSFVW) o

ur in the ferrite lm when 11;3 is negative, i.e. when s03 is purely imaginary (no losses are onsidered) [4.20℄. The next step is the determination of magneti elds and magneti uxes in ea h layer. Equation (4.30) is used in ea h layer to obtain Hxi = kx [A os kx x B sin kx x℄[Ci es0 y + Di e s0 y ℄e z (4.34a) = hxi e z i

i

= s0i [A sin kx x + B os kx x℄[Ci es0 y Di e s0 y ℄e z (4.34b) = hyi e z Hzi = [A sin kx x + B os kx x℄[Ci es0 y + Di e s0 y ℄e z (4.34 ) = hzi e z The ele tri eld in ea h layer is obtained by a straightforward appli ation of equations (4.24) and (4.29a). Taking into a

ount (4.25), MSFVW are modeled as TE-waves with respe t to the z -axis, whi h implies Ezi = ezi e z = 0 for i = 1; 2; 3 (4.35) Hyi

i

i

i

i

This assumption is on rmed in the literature [4.20℄[4.21℄. Hen e, entering (4.35) and de nition (4.1) into equation (4.29a) results in eyi

)[11;i hxi + 21;i hzi ℄ = ( j!

(4.36a)

exi

)0 hyi = +( j!

(4.36b)

=0 (4.36 ) Up to now, the ele tri eld e obtained by (4.36a- ) and the magneti eld h de ned by (4.34a- ) satisfy the magnetostati form (4.29a,b) of Maxwell's equations (4.3) and (4.4), sin e (4.32) is derived from (4.25), whi h is equivalent to (4.29b). The x-dependen e of the elds is governed by the edge boundary ondition at planes x = 0 and x = W . The perfe t magneti wall assumption ezi

4.4.

METHODOLOGY FOR THE CHOICE OF

TRIAL FIELDS

159

(PMW), introdu ed by O'Keefe and Paterson [4.22℄, is widely used for the analysis of MSW devi es [4.23℄[4.24℄, and will be used here for simpli ity. In the next hapter however, a signi ant theoreti al improvement of this approximation will be developed whi h, ombined with variational formulation (4.2), provides a better agreement with experiment. As a matter of fa t, the stru ture of Figure 4.2 analyzed here under the PMW assumption onsists of a gyrotropi -loaded waveguide with lateral PMW shielding, as was introdu ed in Se tion 4.1. The PMW ondition is expressed as

8 y; z Hyi (W; y; z ) = Hzi (W; y; z ) = 0 8 y; z Hyi (0; y; z ) = Hzi (0; y; z ) = 0

(4.37a) (4.37b)

whi h, ombined with (4.34b- ), results in

B=0 (4.37 ) m with m arbitrary (4.37d) kx = W Hen e, the stru ture supports an in nite number of propagating modes, hara terized by their width-harmoni number m. CoeÆ ients Ci and Di of (4.34a- ) an now be al ulated in ea h layer. They are obtained by applying the following boundary onditions at respe tive interfa es y = H2 , y = 0, y = H and y = H1 + H . 1. In the plane y = H2 . The tangential ele tri eld has to vanish:

ex2 (x; y ) = 0

8x

(4.38a)

ez2 = 0 is satis ed by virtue of (4.36 ). Equations (4.34b), (4.36b), and (4.38a) yield D2 = e 2s02 H2 C2

(4.38b)

2. In the plane y = H1 + H . The tangential ele tri eld has to vanish:

ex1 (x; y ) = 0

8x

(4.39a)

ez1 = 0 is satis ed by virtue of (4.36 ). Equations (4.34b), (4.36b), and (4.39a) yield D1 = e2s01 (H1 +H ) C1

(4.39b)

3. In the plane y = 0. The ontinuity of hx and hz is imposed:

hx2 = hx3

8x

(4.40a)

160

CHAPTER 4.

GENERAL

VARIATIONAL PRINCIPLE

8x

hz2 = hz3

(4.40b)

Equations (4.34a) and (4.40a) yield after simpli ation C3 + D3 = C2 + D2

(4.40 )

The ontinuity of ex is imposed:

8x

ex2 = ex3

(4.41a)

Equations (4.34b), (4.36b), and (4.41a) yield after simpli ation s02 (C2

D2 ) = s03 (C3

D3 )

(4.41b)

4. In the plane y = H . The ontinuity of hx and hz is imposed:

8x 8x

hx1 = hx3 hz 1 = hz 3

(4.42a) (4.42b)

Equations (4.34a) and (4.42a) yield after simpli ation C3 es03 H + D3 e

s03 H

= C1 es01 H + D1 e

(4.42 )

s01 H

The ontinuity of ex is imposed: ex 1 = ex 3

8x

(4.43a)

Equations (4.34b), (4.36b), and (4.43a) yield after simpli ation s01 (C1 es01 H

D1 e

s01 H

) = s03 (C3 es03 H

D3 e

s03 H

)

(4.43b)

Finally, the variational equation (4.2) is used. The solution is based on a ratio of eld quantities, so that the knowledge of the absolute value of the ele tri eld to put in the equation is not required. From the whole set of equations (4.38) to (4.43), relationships between C1 , C2 , C3 , D1 , D2 , and D3 are obtained. They an be expressed as D3 =

j tanh s02 H2 C3 j + tanh s02 H2

C3 es03 H + D3 e s03 H es01 H + es01 (2H1 +H ) C3 + D3 C2 = e 2s02 H2 + 1 C1 =

with

=

q

11;3 =0

(4.44a) (4.44b) (4.44 )

(4.44d)

4.4.

METHODOLOGY FOR THE CHOICE OF

161

TRIAL FIELDS

while introdu ing (4.44) in (4.38) and (4.39) yields D1 and D2 . Using the same equations, the unknown oeÆ ients Ci and Di an be eliminated, whi h yields a trans endental equation ontaining the required propagation exponent tanh s01 H1 = jn

Ke s03 H Ke s03 H

es03 H es03 H

(4.45)

with K=

j tanh s02 H2 j + tanh s02 H2

This is the usual simple equation widely used for the analysis of MSFVW in YIG- lms [4.17℄. Its solution, however, is only valid for a uniform permeability tensor, whi h is not the ase here. However, inserting this solution in the trial eld expressions (4.33a,b), (4.34a- ), and (4.36a- ), provides trial elds whi h satisfy all the boundary onditions and are therefore a

eptable for the variational equation (4.2). The trial ele tri eld in ea h layer is nally expressed as the produ t of an x-dependen e, obtained by ombining relations (4.34) and (4.36), by a hyperboli y -dependen e whose oeÆ ients are given by (4.44) and are proportional to the same undetermined fa tor expressed as the produ t (4.46)

AC3

On e the x- and y -dependen e of the trial elds have been determined, the various integrands of oeÆ ients (4.17) are expressed using expressions (4.36) for the trial ele tri eld. The resulting formulations are (az  e )
i (az  e) 1

= e2z

2  ! 111;i

j11;i hxi + 21;i hzi j2 + 0 jhyi j2



re
(i 1 az  e) = e2z

!2 1  [11;i hzi + 12;i hxi ℄ [11;i hxi + 21;i hzi ℄

21;i

(az  e )
(i

1

!2

r  e)

= e2z  121;i [11;i hxi + 21;i hzi ℄ [11;i hzi + 12;i hxi ℄

r  e
(i 1 r  e) = e2z !2111;i j11;i hzi + 12;i hxi j2 e "i
e = "i e2z

2  !

j11;i hxi + 21;i hzi j2 + 20 jhyi j2



(4.47a)

(4.47b)

(4.47 ) (4.47d) (4.47e)

162

CHAPTER 4.

GENERAL

VARIATIONAL PRINCIPLE

be ause the following onditions are satis ed:

az  e = ( eyi ; exi ; 0)

r  e = (0; 0;

j!Bzi ) e z

i 1
(az  e) = ( 111;i eyi ; exi ; 211;i eyi ) i 1
r  e = (

j!121;i Bzi j!111;iBzi ; 0 ; ) e z e z

(4.48a) (4.48b) (4.48 ) (4.48d)

The last step is to evaluate the integrals in (4.17b-f). The y -dependen e of the elds onsists of simple exponentials and the integration along the y axis an be performed analyti ally. The x-dependen e of the integrands is the produ t of the x-dependen e of the inverted non-uniform permeability tensor by simple sinusoidal fun tions. This integral is evaluated numeri ally by a simple trapezoidal rule algorithm. Figure 4.3 shows the results obtained when ombining the spatial trial elds obtained in this paragraph with variational equation (4.2). The real and imaginary parts of the propagation onstant solution of (4.2) are al ulated for the rst width-harmoni mode (solid line) - non-uniform ase - and ompared to the solution provided by equation (4.45) (dashed line), valid under the assumption of a uniform internal eld. A signi ant shift of the dispersion relation is observed when the non-uniformity of the internal eld is taken into a

ount. This is of prime interest for the design of YIG-tuned devi es, for whi h the enter operating frequen y is a fun tion of the size of the YIG lm used as resonator. Sin e magnetostati waves are slow waves, the size of the resonator to be used may be onsiderably redu ed. For sizes of the order of 1 mm along the z -axis, the propagation onstant at the resonan e is  103 rad/m. Figure 4.3 shows that negle ting the non-uniformity of the eld in the sample yields an error of about 1 % in the resonant frequen y. It an be seen that variational equation (4.2) in the spatial domain yields better results when modeling propagation ee ts in a non-uniformly biased gyrotropi YIG- lm. Two methods for deriving the trial elds will now be des ribed in the spe tral domain. The rst deals with Hertzian potentials obtained from Maxwell's equations. The se ond dedu es the elds from physi al onsiderations and rigorous appli ation of the boundary onditions required by the appli ation of variational formulation (4.2). The two approa hes are developed extensively in the spe tral domain. The same reasoning, however, may be applied to trial elds in the spatial domain. This is exempli ed for the last example, where the elds in a YIG- lm are obtained from a s alar potential. Ea h approa h will be illustrated by a pra ti al example.

4.4.

METHODOLOGY FOR THE CHOICE OF

TRIAL FIELDS

163

propagation coefficient β [rad/m]

attenuation coefficient α [ Np/m]

20 18 16 14 10 9 8 6

7000 6000 5000 4000 3000 2000

4 6.0

6.1

6.2

6.3

Frequency (GHz)

6.4

6.0

6.3 6.1 6.2 Frequency (GHz)

(a)

6.4

(b)

Fig. 4.3 Propagation onstant of YIG- lm al ulated for non-uniform inter-

nal DC- eld with variational prin iple (solid line) and for uniform DC- eld (dashed line) (a) attenuation oeÆ ient; (b) propagation oeÆ ient

4.4.4 Trial elds in terms of Hertzian potentials

The lassi al ele tri and magneti Hertzian potentials, solutions of Helmoltz equation in the various layers, are derived from the sour e-free formulation of Maxwell's equations [4.25℄. For lassi al waveguides, they are expressed in ea h layer as a fun tion of modes whi h are respe tively TE (or H) and TM (or E) with respe t to the z -axis of propagation:

Hi (x; y; z) = Hi (x; y)e z az Ei (x; y; z) = Ei (x; y)e z az

(4.49a)

(4.49b) where Hi and Ei are s alar potentials. Hen e the ele tri and magneti elds in (4.1) an be derived in ea h layer from expressions in [4.25℄ as Ei

=e

z

[ rEi + (ki2 + 2)Ei az + j!i az  rHi ℄

(4.50a)

= e z [ rHi + (ki2 + 2 )Hi az j!"i az  rEi ℄ (4.50b) with ki2 = !2i "i . The potentials are solutions of the following Helmoltz equations, obtained from Maxwell's equations (4.3) and (4.4): Hi

r2 Hi + (ki2 + 2 )Hi = 0

(4.51a)

r2 Ei + (ki2 + 2 )Ei = 0

(4.51b)

164

CHAPTER 4.

GENERAL

VARIATIONAL PRINCIPLE

The ele tri and magneti elds an be dedu ed in ea h layer from the s alar potentials  (x; y) and  (x; y):   e = j! (4.52a) H i

E i

E i

xi

i

x

H i

y

 + j! x

 E i y

H i

eyi

=

ezi hxi

= (k2 + 2) =   + j!"  

hyi

=

i

(4.52 )

E i

i

H i

i

x

(4.52b)

 H i y

j!"i

E i

(4.53a)

y

 E i x

(4.53b)

= (k2 + 2 ) (4.53 ) The x-Fourier transform of equations (4.51a,b) results in the spe tral equation ~ (k ; y) 2 + ( 2 k2 + k2 )~ (k ; y) = 0 (4.54a) 2y hzi

H i

i

H;E

x

i

H;E

x

i

x

i

whi h has the general solution ~ (k ; y) = X~ (k ) sinh(s0 y) + Y~ with H;E

H;E

x

i

s0i

q

=

kx2

x

i

ki2

2

i

(k ) osh(s0 y)

H;E

x

i

i

= jk

(4.54b) (4.54 )

yi

wavenumber along y-axis ~ X and Y~ spe tral oeÆ ients (4.54d) In the following dis ussion the oeÆ ients X~ and Y~ ae ting the hyperboli dependen ies of the spe tral potentials will be referred to as spe tral oeÆ ients. The spe tral elds asso iated with the spe tral potentials are: ~ (k ; y) j!  ~ (k ; y) (4.55a) e~ (k ; y ) = jk  kyi

H;E

H;E

i

i

H;E

i

xi

x

x

e~yi (kx ; y ) = e~zi

i

x

; y) =

x

~ Ei (kx ; y)  y

= (k2 + 2)~ (k

~h (k xi

E i

E i

x

H i

i

x

y

+ j! jk ~ (k i

; y)

~ Hi (kx ; y) + j!"i

jkx 

H;E

i

H i

x

~ 

E i

x

(k

x

y

; y)

; y)

(4.55b) (4.55 ) (4.56a)

4.4.

METHODOLOGY FOR THE CHOICE OF

~ yi (kx ; y) = h

~ Hi (kx ; y)  y

165

TRIAL FIELDS

~ Ei (kx ; y) j!"i jkx 

(4.56b)

(4.56 ) = (k2 + 2 )~ (k ; y) Considering the (m + n) layer stru ture of Figure 4.1, and looking for simpli ations of the general solution of equation (4.54) in the ase of shielded and open lines, one de nes (Fig. 4.1): x horizontal oordinate, entered in middle of the waveguide in ase of shielded lines y verti al oordinate, entered in the plane ontaining the

ondu ting layer between two layers, alled respe tively layer N1 and layer M1 thi kness of layer M , N H W width of waveguide in the ase of shielded lines
P H = H =1
P H = H =1 The general form of solution (4.54) in layer M (y positive) is ~ (k ; y) = X~ (k ) sinh[s0 (y a )℄ + Y~ (k ) osh[s 0 (y a )℄ (4.57a) with ~ zi h

H i

i

x

i

Mi ;Ni

i

g

m

M

Mj

j

n

N

Nj

j

i

H;E

ai

=

H;E

x

Mi

Mi

X1 H

x

Mi

H;E

i

Mi

x

M i

i

j

=1

i

(4.57b)

Mj

The general form of solution (4.54) in layer N (y negative) is i

~

H;E Ni

with bi

=

(k

x

~NH;E ; y) = X (kx ) sinh[s0N i (y + bi )℄ + Y~NH;E (kx ) osh[sN 0i (y + bi )℄ i i

X1 H

(4.58a)

i

j

=1

(4.58b)

Nj

This general solution may be ustomized to various topologies in the following manner. 1. Covered lines are de ned by 0 < H < 1 and 0 < H < 1 The tangential ele tri eld has to vanish in planes y = H and y = H , and one dedu es from (4.55a) and (4.55 ) the equivalent onditions (4.59a) ~ (k ; H ) = 0 and ~ (k ; H ) = 0 Mm

Nn

M

E Mm

xn

M

E Nn

xn

N

N

166

CHAPTER 4.

~ HMm (kxn ; y)  y

GENERAL

VARIATIONAL PRINCIPLE

~ HNn (kxn ; y)  y y = HN

y=HM

= 0 and

=0

(4.59b)

whi h yield the parti ular solution for the spe tral s alar potentials in layers Mm and Nn : (4.60a) ~ EMm (kx ; y) = X~ME m (kx ) sinh[s0Mm (y HM )℄ (4.60b) ~ HMm (kx ; y) = Y~MHm (kx ) osh[s0Mm (y HM )℄ (4.61a) ~ ENn (kx ; y) = X~NEn (kx ) sinh[s0Nn (y + HN )℄ H H (4.61b) ~ Nn (kx ; y) = Y~Nn (kx ) osh[s0Nn (y + HN )℄ 2. Open lines are de ned by HMm ! 1 and HNn ! 1. All the elds have to vanish at in nity, so that the parti ular solution for the spe tral s alar potentials in layers Mm and Nn is (4.62a) ~ EMm (kx ; y) = X~ME m (kx )e [s0Mm (y am )℄ (4.62b) ~ HMm (kx ; y) = Y~MHm (kx )e [s0Mm (y am )℄ (4.63a) ~ ENn (kx ; y) = X~NEn (kx )e[s0Nn (y+bn)℄ [ s ( y + b )℄ H H (4.63b) ~ Nn (kx ; y) = Y~Nn (kx )e 0Nn n 3. Shielded lines are hara terized by ele tri or magneti lateral shielding at planes x = Wg =2. Some parti ularities in the formulations of these lines are used in the literature, without any satisfa tory mention of their origin. In the following dis ussion, a rigorous derivation of the simpli ations is presented. In the ase of perfe t ele tri shielding, the tangential ele tri eld has to vanish on lateral planes, while for perfe t magneti shielding, the tangential magneti eld has to vanish at those planes. Hen e, the spatial eld may be des ribed as a summation of periodi fun tions of period n=Wg . As a

onsequen e, the spe tral elds and potentials exist only for dis rete values kxn of the kx variable kxn =

n Wg

The hoi e of n is related to the symmetry of the s alar potentials des ribing the parti ular mode under investigation, the nature of shielding and the relations existing between elds and potentials by virtue of equations (4.52a- ) and (4.53a- ). The odd part of Hi (x; y) is des ribed by sine fun tions of x while the even part of Ei (x; y) is des ribed by osine fun tions of x. Both parts ontribute to the sine omponent of the tangential magneti eld and the osine omponent of the tangential ele tri eld. On the other hand, the even part of Hi (x; y) is des ribed by osine fun tions of x while the odd part of Ei (x; y) is des ribed by sine fun tions of x. Both parts ontribute to the

4.4.

METHODOLOGY FOR THE CHOICE OF

TRIAL FIELDS

167

osine omponent of the tangential magneti eld and the sine omponent of the tangential ele tri eld. Hen e, when looking for a parti ular mode having an even symmetry of eyi and hxi , we impose Hi (x; y) even and Ei (x; y) odd with kxn = (2n + 1)=Wg for perfe t ele tri lateral shieldings kxn = 2n=Wg for perfe t magneti lateral shielding while, when looking for a parti ular mode having an odd symmetry of eyi and E hxi , we impose H i (x; y ) odd and i (x; y ) even with kxn = (2n + 1)=Wg for perfe t magneti lateral shieldings kxn = 2n=Wg for perfe t ele tri lateral shielding The fundamentals of the spe tral domain te hnique have been des ribed. The method has been widely used many years. It originated in a paper by Yamashita [4.10℄. Itoh and Mittra ombined it with Galerkin's pro edure to solve an integral equation by the moment method [4.5℄,[4.26℄, as introdu ed in Chapter 2 and developed in Chapter 3 for planar lines. As seen in Chapter 3, the main advantage of the method is that the Green's fun tion relating the

urrents and the ele tri elds in the stru ture is an algebrai expression in the spe tral domain. The next example will illustrate the point in the ase of the variational prin iple. An important feature of this new formulation is that the spe tral potentials, hen e the spe tral elds, are expressed as the produ t of a simple hyperboli y-dependen e with spe tral oeÆ ients as des ribed in Se tion 4.3. Hen e the integrations (4.19a-e) along the y-axis are performed analyti ally before the integration along the spe tral kx -axis. A omment is ne essary about the variational spe tral equation (4.21) when shielded lines are onsidered. The analysis of shielded lines is hara terized by the use of values of spe tral potentials asso iated with dis rete values of the spe tral variable kx , as explained above. Moreover, these potentials have a physi al meaning only over the area of the waveguide. As a onsequen e, the integrals based on Parseval's equality (4.18) are repla ed by an in nite summation over the integrands evaluated in kx = kxn . From now on, the distin tion between shielded and open lines will be omitted in the text. In the ase of open or overed lines, integrals over the whole kx spe trum are

onsidered in expressions (4.19), while the integrals are repla ed by dis rete summations in the ase of shielded lines. Two stru tures diering only by the absen e or presen e of lateral shielding will have exa tly the same expressions for the potentials, elds and integrands of (4.19). Those quantities will simply be evaluated respe tively over the whole spe trum kx or at dis rete values kxn . As dis ussed at the beginning of this se tion, the development will now be entered on an example, to ensure a better illustration of the method. A mi rostrip line is onsidered, as illustrated in Figure 4.4 [4.4℄. Sin e the line has dis ontinuities at its ondu tive layer in plane y = 0, the spe tral domain

168

CHAPTER 4.

GENERAL

VARIATIONAL PRINCIPLE

Wg

H1

layer 1 W ax az layer 2

H ay

dielectric substrate conducting planes conducting shielding

Fig. 4.4 Geometry of shielded mi rostrip line analyzed by general variational prin iple (Reprinted by kind permission of GET/Hermes-S ien e [4.4℄)

formulation of the variational prin iple will be used here. A

ording to the above notations, the following parameters are de ned: H thi kness of substrate, alled layer 2 H1 distan e between interfa e layer 1 - layer 2 and metalli housing Wg width of waveguide W width of strip The stru ture of Figure 4.4 has only two layers and is fully shielded. The dominant mode of this shielded stripline obviously has an even x-symmetry of the ey and hx eld omponents. So, the following parti ular solution is

onsidered for the potentials, derived from (4.60a,b) and (4.61a,b): ~ E2 (kx ; y) = X~ 2E (kx ) sinh[s02 (y H )℄ (4.64a)

~ H2 (kx ; y) = Y~2H (kx ) osh[s02 (y H )℄ ~ E1 (kx ; y) = X~ 1E (kx ) sinh[s01 (y + H1 )℄ ~ H1 (kx ; y) = Y~1H (kx ) osh[s01 (y + H1 )℄ with (2n + 1) k =k = x

xn

Wg

(4.64b) (4.64 ) (4.64d) (4.64e)

4.4.

METHODOLOGY FOR THE CHOICE OF

169

TRIAL FIELDS

The spe tral elds satisfy Maxwell's equations (4.3) and (4.4), rewritten in the spe tral domain in terms of Helmoltz equations applied to the spe tral s alar potentials (4.54b). The next step is the appli ation of the boundary onditions at the various y -interfa es, as for the pre eding example. Be ause of the linearity of the Fourier transform and of the required boundary onditions on the trial elds, the boundary onditions on the spatial elds are equivalent to the same boundary onditions expressed on the spe tral elds. 1. In the plane y = H1 . The tangential ele tri eld has to vanish. Combining the spe tral de nition (4.55a- ) for the spe tral ele tri eld with expressions (4.64a-e) yields ~ 1E (kx ) e~x1 (kx ; y ) = [ jkx X

j!1 Y~1H (kx )s01 ℄ sinh[s01 (y + H1 )℄ (4.65a)

~ 1E (kx ) sinh[s01 (y + H1 )℄ e~z1 (kx ; y ) = (k12 + 2 )X

(4.65b)

The two expressions vanish for y = H1 . 2. In the plane y = H . The tangential ele tri eld has to vanish. Combining again the spe tral de nition (4.55a-e) for the spe tral ele tri eld with expressions (4.64a,b) yields ~ 2E (kx ) e~x2 (kx ; y ) = [ jkx X

j!2 Y~2H (kx )s02 ℄ sinh[s02 (y

~ 2E (kx ) sinh[s02 (y e~z2 (kx ; y ) = (k22 + 2 )X

H )℄

H )℄

(4.66a) (4.66b)

These two expressions vanish for y = H . 3. In the plane y = 0. The ontinuity of ex and ez is imposed. Using expressions (4.65a,b) and (4.66a,b) for the trial ele tri eld in ea h layer, the boundary onditions are rewritten as [ jkx X~ 1E (kx ) j!1 Y~1H (kx )s01 ℄ sinh(s01 H1 ) (4.67) ~ 2E (kx ) + j!2 Y~2H (kx )s02 ℄ sinh(s02 H ) = [ jkx X (k12 + 2 )X~ 1E (kx ) sinh(s01 H1 ) = (k22 + 2 )X~ 2E (kx ) sinh(s02 H ) (4.68) The hx and hz omponents of the magneti eld are related to the urrent densities owing on the strip in the x- and z -dire tions: hx1 (x; 0)

~ x1 (kxn ; 0) h hz2 (x; 0)

~ z2 (kxn ; 0) h

hx2 (x; 0) = Jz (x; 0)

m

8x

~ x2 (kxn ; 0) = J~z (kxn ; 0) h hz1 (x; 0) = Jx (x; 0)

m

8n

(4.69a)

8x

~ z1 (kxn ; 0) = J~x (kxn ; 0) h

8n

(4.69b)

170

CHAPTER 4.

GENERAL

VARIATIONAL PRINCIPLE

with Jz (x; 0) and Jx (x; 0) non-zero only in the range x = W=2; x = W=2. Combining the spe tral de nition (4.56a- ) for the spe tral magneti eld with expressions (4.64a-d) yields in layers 1 and 2 ~hx1 (kx ; y ) = [ jkx Y~1H (kx ) + j!"1 s01 X~ 1E (kx )℄ osh[s01 (y + H1 )℄ (4.70a) ~hz1 (kx ; y ) = (k12 + 2 )Y~1H (kx ) osh[s01 (y + H1 )℄

~hx2 (kx ; y ) = [ jkx Y~2H (kx ) + j!"2 s02 X~ 2E (kx )℄ osh[s02 (y ~hz2 (kx ; y ) = (k22 + 2 )Y~2H (kx ) osh[s02 (y H )℄

(4.70b) H )℄

(4.70 ) (4.70d)

Hen e the boundary onditions (4.69a,b) are rewritten as [ jkx Y~1H (kx ) + j!"1 s01 X~ 1E (kx )℄ osh(s01 H1 ) (4.71) [ jkx Y~2H (kx ) + j!"2 s02 X~ 2E (kx )℄ osh(s02 H ) = J~z (kx ) (k22 + 2 )Y~2H (kx ) osh(s02 H ) = J~x (kxn ; 0)

(k12 + 2 )Y~1H (kx ) osh(s01 H1 )

(4.72)

Equations (4.67), (4.68), (4.71), and (4.72) form a set of four linear equations ~ 1E (kx ), X~ 2E (kx ), Y~1H (kx ) and Y~2H . This in terms of the spe tral oeÆ ients X set an easily be solved for these oeÆ ients, whi h are nally expressed as algebrai expressions of the spe tral variable kx and a linear ombination of the spe tral quantities J~z (kxn ; 0) and J~x (kxn ; 0): Y~1H (kx ) =

[a22 J~z (kxn ; 0) (a11 a22

a21 J~x (kxn ; 0)℄ a12 a21 )

[a11 J~z (kxn ; 0) a12 J~x (kxn ; 0)℄ (a11 a22 a12 a21 ) ~ 1E (kx ) = a1 Y~1H (kx ) + a2 Y~2H (kx ) X ~ 2E (kx ) = K X~ 1E (kx ) X

Y~2H (kx ) =

(4.73a) (4.73b) (4.73 ) (4.73d)

with K=

(k12 + 2 ) sinh(s01 H1 ) (k22 + 2 ) sinh(s02 H )

(4.73e)

a1 =

j!1 s01 sinh(s01 H1 )

jkx [sinh(s01 H1 ) + K sinh(s02 H )℄

(4.73f)

a2 =

j!2 s02 sinh(s02 H )

jkx [sinh(s01 H1 ) + K sinh(s02 H )℄

(4.73g)

a11 = [( jkx + j!"1 s01 a1 ) osh(s01 H1 ) + j!"2 s02 a1 K osh(s02 H )℄

(4.73h)

4.4.

METHODOLOGY FOR THE CHOICE OF

a12 =

TRIAL FIELDS

[( jkx + j!"2 s02 Ka2 ) osh(s02 H ) + j!"1 s01 a2 K osh(s01 H1 )℄

171 (4.73i)

(k12 + 2 ) osh(s01 H1 ) (4.73j) 2 2 a22 = (k2 + ) osh(s02 H ) (4.73k) Hen e the potentials and the elds are readily expressed as a linear ombination of the spe tral urrent density. The shape of the various omponents of the urrent density remains to be determined. As a rst test, the transverse

urrent density is negle ted: Jx (x; 0) = 0 8 x whi h yields J~x (kxn ; 0) = 0 8 n a21 =

Be ause of the variational nature of the solution, only an approximation of the

urrent density on the strip is needed. The error made on the urrent indu es an error on the trial ele tri eld (4.65a,b) and (4.66a,b), be ause this eld is obtained from the potential (4.64a-d) whose spe tral oeÆ ients (4.73a-d) are linear ombinations of the spe tral urrent density. But, as a result of the variational hara ter, this trial spe tral eld introdu ed in expressions (4.19ae) does not indu e a rst-order error on the propagation onstant obtained with (4.21). The longitudinal omponent of the urrent density is assumed to be onstant on the strip, and imposed to be zero outside of the strip (unit step fun tion): Jz (z; 0) = I0 =W

for

W=2 < x < W=2

Jz (z; 0) = 0 for x < W=2 or x > W=2 The Fourier transform of this fun tion is well known. It is sin(kxn W=2) J~z (kxn ; 0) = I0 2 (kxn W )

(4.74a) (4.74b) (4.74 )

The validity of the variational spe tral equation (4.21) ombined with the spe tral elds derived from (4.64a-d) and satisfying boundary onditions (4.67) to (4.69a,b) is illustrated in Figure 4.5, where results obtained by Mittra and Itoh ( ir les) [4.27℄ are ompared with results obtained with this formulation (solid line). They agree very well. It should be underlined, however, that the model of Mittra and Itoh solves the integral equation by Galerkin's method in the spe tral domain. They report a solution at one frequen y in 3 se onds on an IBM 360 omputer. However, when the variational formulation is implemented on a regular PC, the al ulation does not ex eed a few se onds for the total of twenty frequen y points involved in Figure 4.5. This

omparison illustrates the validity of the variational formulation as well as its numeri al eÆ ien y. Hen e, the variational method an be used for on-line designs up to millimeter waves. As an example, the propagation onstant of a millimeter wave shielded line was al ulated up to 200 GHz [4.28℄. Results are presented in Figure 4.6. The omputation time is also of a few se onds on a regular PC.

172

CHAPTER 4.

GENERAL

VARIATIONAL PRINCIPLE

40

frequency (GHz)

30

20

10

400

0

800

1200

1600

propagation coefficient β [rad/m ]

Fig. 4.5 Propagation oeÆ ient of shielded mi rostrip (Fig. 4.4) al ulated with variational formulation (|) and by Mittra and Itoh [4.27℄ (o o o) (W = 1:27 mm, H1 = 11:43 mm, H = 1:27 mm, Wg = 12:7 mm, "r1 = 1, "r2 = 4:2) 4.4.5 Trial elds in terms of ele tri eld omponents

The variational formulation (4.21) only involves the spe tral ele tri elds

omponents. Hen e, using a suitable form for the ele tri eld omponents and applying the onditions required by the variational formulation, it is possible to determine the unknown amplitude oeÆ ients. This is applied to the open slot-line represented in Figure 4.7, with the de nitions horizontal oordinate, entered in middle of slot verti al oordinate, entered in plane of interfa e ontaining the slot H thi kness of substrate, alled layer 2 W width of slot layer 1 in nite area orresponding to y < 0 layer 3 in nite area orresponding to y > H . x y

Only simple y -dependen ies of the elds are of interest be ause of the integration along the y -axis in (4.19a-e). Hen e the following expressions are

hosen for the spe tral trial ele tri eld in the three layers: ~v (kx )es01 y e~v1 (kx ; y ) = A

(4.75a)

~v (kx ) sinh(s02 y ) + C~v (kx ) osh(s02 y ) e~v2 (kx ; y ) = B

(4.75b)

4.4.

METHODOLOGY FOR THE CHOICE OF

173

TRIAL FIELDS

200

frequency (GHz)

150

100

50

0

10000

5000

15000

20000

propagation coefficient β [rad/m ]

Propagation oeÆ ient of shielded mi rostrip (Fig. 4.4) at millimeter waves, al ulated with variational formulation (W = 0:5 mm, H1 = 10:43 mm, H = 0:254 mm, Wg = 2:0 mm, "r1 = 1, "r2 = 25)

Fig. 4.6

W

t

layer 1

az ax

εo, µo, σ

layer 2

ay

H

layer 3

conducting planes (conductivity σ) dielectric or gyrotropic lossy substrates

Fig. 4.7

Geometry of open slot-line analyzed by using the variational prin iple

174

CHAPTER 4.

~ v (kx )e e~v3 (kx ; y ) = D

GENERAL

VARIATIONAL PRINCIPLE

s01 (y H )

(4.75 )

The ontinuity of the ele tri eld at the interfa es is rst imposed. Then, the spe tral trial magneti elds are dedu ed from Maxwell's equation (4.3) rewritten in the spe tral domain and the ontinuity of the magneti eld at the interfa es is imposed. By doing so, six equations are obtained, in terms of the 14 unknowns, whi h are ~ v where v = x; y; z (12 unknowns) A~v ; B~v ; C~v and D

(4.76a)

s01 and s02 (2 unknowns)

(4.76b)

The variational hara ter of equation (4.21) only holds when Maxwell's equation (4.4) is satis ed. Rewriting (4.4) in ea h layer in terms of the trial spe tral ele tri eld (4.75) and spe tral magneti eld obtained from the spe tral form of (4.3) provides a set of 9 equations whi h must be ful lled for ea h value of y . It an easily be shown that this transforms the set of 9 equations into a homogeneous system of 12 equations to be solved for the 14 unknowns. With the parti ular hoi e s01 = s02 =

q kx q 2

2

k12

(4.77a)

kx2

2

k22

(4.77b)

only four of them are independent. They are given by s01 [jkx A~y (kx )

s01 A~x (kx )℄ + [ jkx A~z (kx )

~ z (kx ) ~ x (kx )℄ + [ jkx D ~ y (kx ) + s01 D s01 [jkx D

A~x (kx )℄ = k12 A~x (kx )

(4.78a)

~ x (kx )℄ = k12 D ~ x (kx )

D

(4.78b)

~x (kx ) (4.78 ) s02 [jkx B~y (kx ) s02 C~x (kx )℄ [jkx C~z (kx ) + C~x (kx )℄ = k22 B

~z (kx )+ B~x (kx )℄ = k22 B~x (kx ) (4.78d) s02 [jkx C~y (kx ) s02 B~x (kx )℄ [jkx B These four independent homogeneous equations relate the 14 unknowns. Hen e, ombining equations (4.77a,b) with the six equations (4.65a,b), (4.66 a,b), (4.67), and (4.68) resulting from boundary onditions, into the equations (4.78a-d) yields 10 equations in terms of the 12 spe tral oeÆ ients involved in (4.75a- ). Assuming a trial tangential ele tri eld at the interfa e ontaining the slot, two additional equations may be used: A~x (kx ) = e~x (kx ; 0)

(4.79)

A~z (kx ) = e~z (kx ; 0)

(4.80)

4.4.

METHODOLOGY FOR THE CHOICE OF

TRIAL FIELDS

175

where e~x (kx ; 0) and e~z (kx ; 0) are the known trial ele tri eld omponents at the interfa e. Finally, a system of 12 linear equations is obtained: the six equations resulting from boundary onditions, (4.78a-d), (4.79) and (4.80), to ~ v . As for the mi rostrip be solved for the 12 spe tral oeÆ ients A~v ; B~v ; C~v ; D

ase treated in the former subse tion, the spe tral oeÆ ients - hen e the spe tral trial elds - are related to the trial quantities at the interfa e ontaining a dis ontinuous ondu tive layer, namely for slot-like lines the tangential omponents of the ele tri eld at this interfa e. The shape of the ele tri eld omponents at the interfa e remains to be determined. Be ause of the variational nature of the solution, only approximations of these omponents are needed. As a rst test, the longitudinal

omponent is negle ted: ( 0) = 0

ez x;

8

x

whi h yields e~z (kx ; 0) = 0

(4.81)

The transverse omponent of the ele tri eld is assumed to be onstant a ross the slot, and imposed to be zero outside of the slot: ( 0) = V0 =W for

ex x;

( 0) = 0 for

ex x;

x
W=

2

(4.82b)

whi h yields ~(

ex kx ;

0) = V0 2

sin(kx W=2) (kx W )

(4.83)

Hen e, the problem is totally hara terized and an be solved. 4.4.6

Link with the Green's formalism

Two ways for obtaining trial elds, for strip-like and slot-like lines respe tively, have been des ribed in the spe tral domain. It is underlined that both formulations, in terms of potentials or elds, respe tively, an be used for either slot-like or strip-like stru tures. As an example, Table 4.1 ompares results obtained with the se ond formulation for narrow and wide open slotlines to results obtained by Janaswamy and S haubert [4.29℄. It appears that the dieren e is negligible. The formulation in terms of elds, however, oers some advantages when the in uen e of ea h omponent of the ele tri eld has to be investigated. Green's fun tions are present in the variational formalism, be ause the spe tral elds are written as a fun tion of spe tral oeÆ ients (4.54d) or (4.76a), expressed as linear ombinations of the spe tral urrent density or tangential ele tri eld in the aperture. As a onsequen e, the spe tral ele tri and magneti elds an be written as linear algebrai ombinations of the spe tral urrent density for striplines, and of the spe tral tangential ele tri eld in the slot for slot-lines, respe tively. Hen e, in these ases the spe tral domain formalism transforms the spatial integral de nition of Green's

176 W=H 1.335

10.71

CHAPTER 4.

Frequen y in GHz 2.0 2.5 3.0 3.5 4.0 2.0 2.5 3.0 3.5 4.0 5.0 6.0

GENERAL

VARIATIONAL PRINCIPLE

( =0 ) [Jan.-S haub.℄ 0.8885 0.8834 0.879 0.875 0.871 0.958 0.954 0.951 0.948 0.943 0.939 0.933 0

( =0 ) [Huynen℄ 0.8890 0.8841 0.8798 0.8758 0.8708 0.9581 0.9545 0.9515 0.9488 0.9460 0.9409 0.9360 0

Comparison of results obtained with variational formulation (Huynen) and by Janaswamy and S haubert [4.29℄, for narrow and wide slots

Table 4.1

fun tion [4.30℄[4.31℄ into a set of algebrai linear equations in the spe tral domain. This algebrai formulation is a major advantage of the spe tral domain method. As explained in Chapter 3, Green's fun tions are frequently involved as an operator in the linear integral equation being solved by the moment method. In this ase, however, they are only used to nd suitable trial elds to introdu e into oeÆ ients (4.19a-e) of the variational spe tral equation (4.21). Green's fun tions appear in a quadrati form in equation (4.21) where squared powers of the elds are required. For slot-like problems, the urrent density is obtained as the dieren e between the tangential magneti elds in plane y = 0. Using the dependen e between the ele tri eld and the magneti eld given by Maxwell's equations, it is possible to express the ele tri eld in the whole stru ture as a fun tion of the tangential ele tri eld in the slot, via the spe tral oeÆ ients, ommon to both elds. A new Green's fun tion is then established between the ele tri elds in the whole spa e and in the slot. Hen e, for both formulations, potentials or elds, the nal result is an expression of the ele tri eld in terms of trial quantities at an interfa e with a dis ontinuous ondu tive layer. The trial quantities are the tangential urrent density owing on the ondu ting layer for strip-like lines and the tangential ele tri eld a ross the interfa e for slot-like lines. Sin e the trial elds satisfy Maxwell's equations, they dier from the exa t (unknown) elds only by their shape. This is dis ussed in the next se tion.

4.5.

TRIAL FIELDS AT DISCONTINUOUS CONDUCTIVE INTERFACES

177

4.5 Trial elds at dis ontinuous ondu tive interfa es It has been shown that Mathieu fun tions (Appendix E), eigenfun tions of Lapla e's equation in ellipti al and hyperboli oordinates, form a very eÆ ient set of trial expressions for the tangential trial ele tri eld omponents for slot-like stru tures [4.7℄. In this se tion, the fundamentals of the derivation of these fun tions for slot-like stru tures are detailed. It will be shown that the same fun tions may be used to des ribe the urrent densities owing on the strips in strip-like problems. Due to the formulation developed in Se tion 4.4, the trial quantity redu es to omponents of the ele tri eld or urrent density, tangential to the interfa e ontaining the dis ontinuous metallization. The analyti al expression of the trial quantity is determined here by onformal mapping. 4.5.1

Conformal mapping

The hange of oordinates x = a os u osh v y = a sin u sinh v

(4.84)

where a = W=2 transforms the (x; y) plane of Figures 4.1, 4.4 and 4.7 into the (u; v) plane. The

on guration will be detailed when examining strip-like stru tures in Se tion 4.5.3. It is onsidered that the slot is pla ed in a homogenous medium of permittivity ("1 + "2 )=2, whi h is the mean value of the permittivities of the two layers adja ent to the slot. The Helmoltz equations (4.51a,b) to be solved in the (x; y) plane for TE/TM modes an be transformed into the (u; v) plane as

r H;E (u; v) + (kef f + )( a2 )( osh 2v os 2u)H;E (u; v) = 0 (4.85) 2

2

2

2

where kef2 f = !2 0 ("1 +2 "2 ) by using the following relations, derived from the hange of oordinates:  H;E  H;E  H;E = a sin u osh v a os u sinh v u x y (4.86) H;E H;E H;E    = a os u sinh v a sin u osh v v x y Letting 2q = (kef2 f + 2 )(a2 =2) yields Mathieu's equation

r H;E (u; v) + 2q( osh 2v os 2u)H;E (u; v) = 0 2

(4.87)

whose general solutions are obtained by separating the variables: H;E (u; v) = U H;E (u)V H;E (v)

(4.88)

178

CHAPTER 4.

GENERAL

VARIATIONAL PRINCIPLE

The solutions of U (u) are alled Mathieu fun tions, while those of V (v) are alled the modi ed Mathieu fun tions. Only the solutions of U (u) will be of interest here. Those solutions are lassi ally noted e (q; u) and se (q; u), respe tively. They an be simply developed into osine fun tions weighted by a power expansion of the q parameter as H;E

H;E

H;E

n

n

e (q; u) =

1

X i

se (q; u) =

a (q ) os(iu)

(4.89)

ni

n

=1

1

X

b (q) sin(iu)

(4.90)

mi

m

=1

i

where n and m are arbitrary integers, ex ept that m 6= 0. Suitable expressions for the oeÆ ients a (q) and b (q) an be found in Appendix E and in [4.32℄. ni

4.5.2

mi

Adequate trial omponents for slot-like stru tures

Be ause of this formulation, the trial quantity for slot-like stru tures redu es to omponents of the ele tri eld, tangential to the interfa e between two diele tri layers adja ent to a dis ontinuous ondu tive interfa e. Due to the boundary ondition, it must have the same x-variation a ross the slot on the two sides of the interfa e, and vanish on the metallization. The analyti al expression of this trial tangential eld is determined here from onformal mapping (4.84), for whi h it is onsidered that the slot is pla ed in a homogenous medium of permittivity ("1 + "2)=2, whi h is the mean value of the permittivities of the two layers adja ent to the slot. Sear hing for a solution in plane y = 0 and for 1 < x=a < 1, the equivalent domain to be onsidered in the (u; v) plane is de ned by 0 < u <  and v = 0 whi h yields u = ar

os(

2x )

(4.91)

W

For the slot-line in the homogeneous medium onsidered in Figure 4.8, the ar

osine onformal mapping (4.91) transforms the ondu ting planes of the slot into a parallel plate waveguide (stru ture 3, Fig. 4.8). The transverse omponent of the tangential ele tri eld a ross the slot under TE assumption an be expressed as   j!0 dV e  j!0 = (4.92) U ( u ) y = =0 a sin u dv =0 H

H

H

x

y

v

v

It an be seen that only the solutions for U (u) are needed, be ause the x-dependen e of the eld a ross the slot is entirely determined by U (u) at H;E

H;E

4.5.

TRIAL FIELDS AT DISCONTINUOUS CONDUCTIVE INTERFACES

179

3 2

az ( ε1, µo) 1 ax ay

( ε2, µo) ( ε1, µo)

W

Transformation of ondu tors of slot-line when applying ar

osine

onformal mapping, from the ondu ting planes of the slot (1) to the parallel plate waveguide (3)

Fig. 4.8

v = y = 0. For v = 0, and to properly des ribe the ex omponent in the slot, the potential H may not vanish in planes u = 0 and u = . Hen e, sen (q; u) are avoided here. The even (n even) and odd (n odd) x-dependen ies of ex a ross the slot an be expressed, using (4.89) to (4.92), as Kx U H (u) sin u )℄ = Kx en [q; ar

os(2x=W 2 1 (2x=W ) = Kx ani (q) os[i ar

os(2x=W2 )℄ 1 (2x=W ) i

exn (x; 0) =

p X

p

(4.93)

with i even for n even and i odd for n odd. It appears that this equation

ombine Mathieu fun tions with the orre t singularity in the eld at the edge of an in nitesimally thin metal sheet known to be of order 1=2, be ause of equation (4.91), due to the onformal mapping. Hen e, one may expe t a rapid onvergen e. On the other hand, the longitudinal omponent ez tangential to the in-

180

CHAPTER 4.

GENERAL

VARIATIONAL PRINCIPLE

terfa e is obtained from equations (4.52 ) as ez

= (k2 + 2 )E y=v=0 = (k2 + 2 )U E (u)V E (v)

(4.94)

Similarly, for v = 0, the potential E has to vanish in planes u = 0 and u =  to properly des ribe the ez omponent in the slot. Hen e, ei (q; u) have to be avoided here. The even (m odd) and odd (m even) x-dependen ies of ez a ross the slot are nally expressed, using (4.90), (4.91), and (4.94), as ezm (x; 0) = Kz U E (u) = Kz sem [q; ar

os(2x=W )℄

= Kz

X

i

bmi (q ) sin[i ar

os(2x=W )℄

(4.95)

with i odd for m odd and i even for m even. The reader re ognizes in expansions (4.93) and (4.95) the T heby he polynomial basis fun tions modi ed by an edge ondition, used by some authors [4.33℄[4.34℄ as basis fun tions in Galerkin's pro edure. Be ause of the physi al point of view ( onformal mapping of the stru ture) adopted here, the series expansion obtained is a very good formulation of the eld a ross the slot: the oeÆ ients ae ting the basis fun tions are determined dire tly from physi al onsiderations without solving any eigenvalue problem. The Fourier transform of any desired mode an be found as the Fourier transform of expressions (4.93) and (4.95) [4.35℄: e~xn (kx ) = Kx

X

e~zm (kx ) = Kz

X

i i

ani (q )(j )i Ji (kx W=2) bmi (q )(j )i

1

W=2) (i) J(ik(kxW= 2)

x

(4.96a) (4.96b)

where Ji is the Bessel fun tion of rst kind and order i. Any term e~xn and

an be taken as a useful expression for higher order modes in slot-like stru tures. The relationship between n and m is readily dedu ed from (4.52a- ) where it is shown that the x-dependen e of the ex omponent is proportional to the rst x-derivative of the ez omponent. Hen e, if an even dependen e of the ex omponent is imposed by the hoi e of n even, ez must have an odd dependen e, whi h means that m is even. It is therefore on luded that n and m in expressions (4.93) to (4.96a,b) have the same even or odd parity. For the dominant mode in the parallel plate waveguide (stru ture 3, Fig. 4.8), the x- omponent of the potential does not depend on the u-variable. Hen e, a reasonable assumption for ex in the slot is obtained by onsidering the image of expressions (4.89) and (4.90) taken for n = 0 and m = 2 when

e~zm

4.5.

181

TRIAL FIELDS AT DISCONTINUOUS CONDUCTIVE INTERFACES

applying the ar

osine onformal mapping, whi h yields (4.93) to (4.96a,b) rewritten for n = 0 and m = 2. Moreover, for moderate slot widths, the value of q remains negligible, so that only the rst term in the serial expansions (4.93) to (4.96a,b) has to be onsidered. Hen e the omponents of the dominant mode may be approximated by

ex0

8 >< Kx p = >: 0 1

1 (2x=W )2

if 0 < jxj
< +2N R eq n [q; ar

os( W )℄ n x )2 1 ( W= ex (x; 0) = 2 >: n=N 0 1

1

if 0 < jxj < W2

(4.99a)

otherwise

while the longitudinal omponent is obtained from Mathieu fun tions sem :

8 MX >< +2M 2x Zm sem [q; ar

os( )℄ W ez (x; 0) = >: m=M 0 1

1

if 0 < jxj < W2

(4.99b)

otherwise

where Rn and Zm are the unknown oeÆ ients. The Fourier transform of those expressions are e~x (kx ) =

e~z (kx ) =

with

1 X

X

N1 +2N

Rn [

n=N1

X

M1 +2M m=M1

2 + 2) q = (kef f

r=1

anr (q )(j )r Jr (kx W=2)℄

1 X

Zm [

W2

16

Jr (kx W=2) bmr (q )(j )r 1 r ℄ (kx W=2) r=1

(4.99 ) (4.99d)

(4.99e)

4.5.

TRIAL FIELDS AT DISCONTINUOUS CONDUCTIVE INTERFACES

183

("1 + "2 ) (4.99f) 2 The expression is valid for a single slot. For oupled slot-lines a formulation similar to (4.98) established for the dominant mode may be derived. The

oeÆ ients Rn and Zm are obtained by applying the Rayleigh-Ritz pro edure. The advantages of the formulation using Mathieu fun tions are twofold. First, the ombination of the basis fun tions is known a priori, sin e q depends only on the geometri al and physi al parameters of the line, and on the frequen y. This is not the ase for Galerkin's pro edure where the oeÆ ients weighting the basis fun tions are the eigenve tors asso iated with an eigenvalue problem. Se ondly, the ombination of basis fun tions is frequen y-dependent, whi h ensures a dynami formulation of the trial quantities ounterbalan ing the quasi-stati hypothesis underlying the use of a onformal mapping. This is be ause the Helmoltz equation (4.85) being solved in the transformed domain involves the in uen e of frequen y.

k2

ef f

4.5.3

= ! 2 0

Adequate trial omponents for strip-like stru tures

Be ause of the formulation, the trial ele tri eld for strip-like stru tures is related to the omponents of the urrent density owing on strips in the interfa e between two diele tri medias. The analyti al expression of these omponents is determined here by the same onformal mapping (4.84) as for slot-like lines, for whi h it is onsidered that the strip is pla ed in a homogenous medium of permittivity ("1 + "2 )=2. The ondu ting strip in the homogeneous medium is shown on Figure 4.10. Solving (4.85) for the potential around the strip is equivalent to solving for the potential between two homofo al ellipti al

ylinders, when the inner ylinder has bi = 0 and the outer ylinder goes to in nity. The hange of oordinates of (4.84) transforms the (x; y ) plane of Figure 4.10a into the (u; v ) plane. Looking for a solution in the whole spa e bounded by the two homofo al ellipti al ylinders of parameters (ai ; bi ) and (ae ; be ), the equivalent domain to be onsidered in the (u; v ) plane is de ned by

 < u <  and osh 1 (a ) < v < osh 1 (a ) i

e

When the inner ylinder is degenerate and the outer ylinder goes to in nity (Fig. 4.10b), the area between the two ylinders is transformed into a semiin nite spa e bounded in the u-dire tion by the planes u =  (Fig. 4.10 ):

 < u <  and 0 < v < 1

(4.100)

The strip area orresponds to v = 0, so that equation (4.91) still holds. The general solution (4.88) of Mathieu's equation (4.87) is hosen in the domain given in (4.100) so that V H;E (v ) vanishes at in nity, taking advantage of the serial expansion of the modi ed Mathieu fun tions into hyperboli fun tions of the v -variable [4.32℄. Only solutions of U H;E (u) are onsidered here, be ause

184

CHAPTER 4.

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VARIATIONAL PRINCIPLE

2 1

1

εeff,µo ai

az

ae ax

bi bW e W ay (a)

−π

εeff,µo

2 1

az

ai

ax

π

az

2

au av

ae

∞ εeff,µo

ay (b)

( )

Transformation of ondu tors of mi rostrip when applying ar

osine

onformal mapping (a) original ellipti al ylinders; (b) mi rostrip on guration; ( ) equivalent stru ture in transformed domain

Fig. 4.10

4.5.

TRIAL FIELDS AT DISCONTINUOUS CONDUCTIVE INTERFACES

185

one is looking for the surfa e harge density on the strip, whi h is proportional to the ey - omponent of the eld, normal to the strip at plane y = v = 0. The

omponent of the ele tri eld normal to the strip under the TM assumption

an be expressed (4.52) as dV E  E

E U (u) = (4.101) ey  y y=v=0 a sin u dv v=0 From this ey - omponent, the even (n even) and odd (n odd) x-dependen e of the surfa e harge density s using (4.89), (4.91), and (4.101) is sn (x; 0) = eyn (x; 0) U E (u) = Kssin u

en [q; ar

os(2x=W )℄ (4.102) = Ks p 1 (2x=W )2 X = Ks ani (q) os[pi ar

os(2x=W2 )℄ 1 (2x=W ) i with i even for n even and i odd for n odd. Following the stati approximation of Denlinger [4.37℄, the longitudinal

urrent density on the strip is related to the surfa e harge density by ! (4.103a) jz = s

Hen e, the even and odd x-dependen ies of the longitudinal urrent density on the strip are obtained as X

os[i ar

os(2x=W )℄ (4.103b) ani (q ) p jzn (x; 0) = Ks 1 (2x=W )2 i As for slot-like stru tures, the shape of the transverse urrent omponent is dedu ed from the relationship between the potential and the hz omponent responsible for the jx urrent. Equations (4.53a- ) yield indeed jx = hz1 hz2 = (k2 + 2) [H1 H2 ℄ y=v=0 (4.104) H H 2 2 H H = (k + ) [U2 (u)V2 (0) U1 (u)V1 (0)℄ y=v=0 U H (u) is a Mathieu fun tion, be ause it is a solution of (4.87). Hen e, jx has the form jxm (x; 0) = Ks U H (u) = Ks sem [q; ar

os(2x=W )℄ (4.105) X = Ks bmi (q) sin[i ar

os(2x=W )℄ 0

00

00

00

i

186

CHAPTER 4.

GENERAL

VARIATIONAL PRINCIPLE

with i odd for m odd and i even for m even. Be ause at v = 0, jxm has to vanish at planes u = 0 and u =  to des ribe properly the nite width of the strip, all ei (q; u) are avoided in (4.105). The Fourier transform of the urrent densities (4.103b) and (4.105) are of ourse similar to those of (4.96a,b): ~jzn (kx ) = Ks0

X

~jxm (kx ) = Kz00

i

ani (q )(j )i Ji (kx W=2)

X i

bmi (q )(j )i 1 (i)

(4.106a)

Ji (kx W=2) (kx W=2)

(4.106b)

Similarly, the relation between n and m is obtained as for slot-like lines by inspe tion of equations (4.53a- ), whi h imply that the x-dependen e of the hx omponent is proportional to the rst x-derivative of hz . The jz - omponent is related to the transverse tangential magneti eld by

jz = hx1

hx2

(4.107)

and equation (4.104) provides a similar relationship between jx and hz . It

an therefore be on luded that jx must have an even/odd x-dependen e to preserve the odd/even dependen e of jz , so that m and n must have the same even or odd parity. Any term ~jxn and ~jzm an be taken as a useful expression for higher-order modes in strip-like stru tures. Hen e, it has been demonstrated that Mathieu fun tions used as trial expressions for the tangential ex and ez - omponents are also adequate trial expressions for the urrent densities jz and jx , respe tively. As for slot-lines, the potential orresponding to the dominant mode in the parallel plate stru ture of Figure 4.10 does not depend on the u-variable. Hen e, by virtue of (4.101) to (4.103b), a reasonable assumption for jz on the strip is the image of expressions (4.89) and (4.90) taken for n = 0 and m = 2 when applying the onformal mapping of (4.84), namely (4.103b) with n = 0 and m = 2. The approximations made for moderate slot width are readily rewritten for moderate strip width, whi h yields the following approximations for the omponents of the dominant mode

8 >< Ks0 p jz (x; 0) = >: 0 1

1 (2x=W )2

if 0 < jxj < elsewhere

W2

(4.108a)

and

jx (x; 0) = Ks00 sin[2 ar

os(

2x )℄ W

(4.108b)

whi h, respe tively, have as Fourier transform ~jz (kx ) = Ks0 W J0 (kx W=2) 2

(4.108 )

4.5.

187

TRIAL FIELDS AT DISCONTINUOUS CONDUCTIVE INTERFACES

and ~jx (kx ) = Ks00 2j J2 (kx W=2) (kx W=2)

(4.108d)

Of ourse, the z - omponent of the urrent density an be related to the total

urrent I0 owing on the strip by integrating jz over the strip area, whi h yields

Ks0 = 2

I0 W

(4.108e)

As for slot-like lines, the dependen e of the longitudinal urrent density an be used to des ribe oupled strip-lines (Fig. 4.11), by modifying the Fourier transform of the longitudinal urrent to take into a

ount the presen e of two strips, entered at planes x = (W1 + S )=2 and x = (W2 + S )=2, respe tively: ~jz (kx ) = 2 [J0 (kx W1 =2)ejkx (W1 +S )=2 + Ci J0 (kx W2 =2)e

jkx (W2 +S )=2 ℄

(4.109) where Ci des ribes the ratio of longitudinal urrents owing on the two strips of respe tive widths W1 and W2 and S is the spa ing between the strips. In the ase of symmetri lines, the ratio Ci is made equal to 1 for odd oupled strip lines, and to +1 for even oupled strip-lines. In the ase of asymmetri lines, the ratio Ci is al ulated by the Rayleigh-Ritz pro edure, as mentioned before for slot-like lines. layer N1 W2 S

W1

az

ax ay

layer M1

Fig. 4.11 Geometry of oupled mi rostrip lines for determining trial quanti-

ties

It has been seen that the trial expressions obtained for the jx - and jz omponents owing on a strip are similar to those obtained respe tively for the trial ez - and ex - omponents in slot-like stru tures. The general expressions

188

CHAPTER 4.

GENERAL

VARIATIONAL PRINCIPLE

for the trial urrent densities on a strip for higher-order modes are therefore readily dedu ed from equations (4.99a,b) as

8 N1 +2N X >

en [q; ar

os( 2Wx )℄ > < Sn q x )2 1 ( W= jz (x; 0) = n=N1 2 > > : 0

if 0 < jxj < W2

(4.110a)

elsewhere

and jx (x; 0) =

M1X +2M m=M1

Tm sem [q; ar

os(

2x )℄ W

(4.110b)

where Sn and Tm are the unknown oeÆ ients. The Fourier transform of these expressions are: ~jz (kx ) =

~jx (kx ) =

N1X +2N n=N1

Sn [

M1X +2M m=M1

1 X r=1

Tm [

anr (q )(j )r Jr (kx W=2)℄

1 X

Jr (kx W=2) bmr (q )(j )r 1 r ℄ (kx W=2) r=1

(4.110 )

(4.110d)

2 given by (4.99e,f). For oupled strip-lines, it an be modiwith q and kef f ed as for the dominant mode. The advantages resulting from the a priori expression of the oeÆ ient q have been mentioned above and are still valid. Similarly, oeÆ ients Sn and Tm are obtained from Rayleigh-Ritz pro edure, whi h will be applied in the following se tion. 4.5.4

Rayleigh-Ritz pro edure for general variational formulation

Sin e a variational prin iple is the basis of the al ulation, the Rayleigh-Ritz pro edure [4.1℄ and [4.38℄ an be applied to obtain the values of oeÆ ients Rn and Zm in (4.99a, ), or Sn and Tm in (4.110a-d). This method simply makes use of the variational nature of equation (4.21): be ause it is variational,

oeÆ ients Rn are determined by imposing the ondition that the value of extra ted from (4.21) is extremum. When evanes ent modes are onsidered, namely when is real, equation (4.21) simpli es be ause in this ase the al ulation of (4.19b, ) shows that one has B~i = C~i , and an be rewritten as

X

2 =

i

!2

~i D

X i

A~i

X i

E~i



(4.111)

4.5.

TRIAL FIELDS AT DISCONTINUOUS CONDUCTIVE INTERFACES

189

For onvenien e it is assumed that the trial elds are des ribed by the series expansion e~x (kx ) =

X

N1 +2N n=N1

Rn e~xn (kx )

(4.112)

Expression (4.112) is a simpli ed notation of the series expansion (4.99 ) where the left-hand side of expression (4.96a) has been used. For slot-like stru tures the integrand of ea h term of (4.111) is proportional to je~x (kx )j2 when e~z (kx ) is negle ted, so that ea h term an be rewritten when N = 1 as Y~i =

X2 X2

m=1 n=1

Rm Rn Y~imn

(4.113)

~ i , and E~i , and where Y~imn is al ulated as in (4.19a-e), with Y~i = A~i ; B~i ; C~i ; D with je~x (kx )j2 repla ed by [~exm (kx )~exn (kx )℄ . Entering (4.113) into (4.111) yields

2 =

R12 N~11 + 2R1 R2 N~12 + R22 N~22 ~ 22 ~ 12 + R22 D ~ 11 + 2R1 R2 D R12 D

(4.114a)

!2

(4.114b)

with N~mn =

~ mn = D

X Xi i

~ imn D

X i

E~imn

A~imn

(4.114 )

Taking advantage of the variational nature of (4.111) and (4.114a), the Rayleigh-Ritz pro edure imposes that the solution is extremum. Normalizing the numerator and denominator of (4.114a) to R12 and de ning X = R2 =R1 yields

2 =

~12 + X 2 N~22 N~11 + 2X N ~ 22 ~ ~ D11 + 2X D12 + X 2 D

(4.115)

Taking then the rst derivative of the right-hand side of this equation with respe t to X , the se ond-order equation in (R2 =R1 ) is obtained: ~ 11 + 2X D ~ 12 + X 2 D ~ 22 ) (2X N~22 + 2N~12 )(D ~ 22 + 2D ~ 12 )(N~11 + 2X N~12 + X 2 N~22 ) = 0 (2X D

(4.116)

Only the solution for X = (R2 =R1 ) yielding a positive value for 2 is used. It must be emphasized that the use of the Rayleigh-Ritz pro edure for more than one term in development (4.99a-d) requires the solution of a determinantal equation, whi h leads to a set of linear equations in terms of the

190

CHAPTER 4.

GENERAL

VARIATIONAL PRINCIPLE

ratios Rn =RN1 . In this ase, the method results in solving a system of equations related to an eigenvalue problem [4.1℄, having about the same omplexity as the Galerkin's pro edure. Mathieu fun tions, however, are eÆ ient enough to provide a

urate trial elds when expansion (4.99) is limited to N = 0. This will be illustrated in the next subse tion. 5

1600

| γ | [ rad/m ]

1400

4

3

1200 1000

0

2

800 600 400 1

200 26

28

30

32 34 36 38 frequency (GHz)

40

42

44

Fig. 4.12 Real and imaginary parts of propagation onstant al ulated with N = 0 for nline (Helard et al. [4.33℄), for two dierent slot widths (| 1 mm, - - - 2 mm ) (numbering refers to order of mode, 0 denoting dominant propagating mode)

4.5.5 EÆ ien y of Mathieu fun tions as trial omponents

Mathieu fun tions have been used as trial elds in [4.7℄ for al ulating higherorder modes in a shielded slot-like line, namely a nline operating in the Ku-band. The eÆ ien y of the method ombined with the series expansion of (4.99 ,d) limited to only one Mathieu fun tion (N = 0) is highlighted when

al ulating the propagation onstant of several higher-order modes of the nline analyzed by Helard et al. in Figure 3 of [4.33℄. The advantage of the formulation of (4.99 ,d) limited to one term, however, is that the ombination of the basis fun tions is known a priori. This is not the ase for the results presented at Figure 3 of Helard et al., where Galerkin's pro edure is used with at least two basis fun tions and requires the solution of a determinantal eigenvalue equation. Figure 4.12 shows the real () and imaginary ( ) part of the propagation

onstant in the ! plane for the dominant mode ( urve 0), and for the rst ve higher order modes ( urves 1 to 5), al ulated with only one term in (4.99 ,d) in the range 26 to 44 GHz, for two dierent slots as onsidered by Helard et al. The omparison with their results shows a very good agreement,

4.5.

191

TRIAL FIELDS AT DISCONTINUOUS CONDUCTIVE INTERFACES

and the behavior of the dominant and rst higher-order modes is orre tly predi ted by our model. The redu ed mathemati al omplexity of expression (4.99a-d) ombined with the expli it formulation (4.21) is obvious: it does not require the solution of an eigenvalue problem to obtain the amplitude of the basis fun tions and the value of whi h sets the determinantal equation equal to zero. 4.5.6

Ratio between longitudinal and transverse omponents

It has been shown that one Mathieu fun tion is suÆ ient for des ribing the shape of any omponent of the trial quantity. The ratio between the magnitudes of the transverse and longitudinal omponents, however, remains unknown. For all the results presented up to now, only the transverse omponent of the trial ele tri eld was onsidered for slot-like lines, while for strip-like lines the transverse urrent density was negle ted. It has been observed, however, that both the transverse and longitudinal omponents are important when modeling higher-order modes in open lines. The knowledge of these modes is ne essary for modeling planar dis ontinuities, for example when using a mode mat hing te hnique. For mi rostrip lines, it is possible to dedu e the hara teristi s of higher-order modes from a planar equivalent waveguide [4.39℄-[4.41℄ modeling the mi rostrip. For slot-like lines, the moment method is usually applied to determine the ratio between the transverse and the longitudinal omponents. It will now be shown that a fairly good estimate of this ratio an be obtained by imposing an additional boundary ondition on the ele tri eld at the diele tri interfa e. This is of prime interest, sin e the variational equation involves the ele tri eld. One simply imposes the ontinuity of the mean value of the displa ement eld at the interfa e between two diele tri layers. For the slot-line of the previous se tion, this is equivalent to imposing "2 e~y2 (kx ; H ) = "1 e~y3 (kx ; H )

(4.117)

Be ause the ase is lossless, we do not need to are about the phase, so that the previous ondition is equivalent to imposing the ontinuity of the squared magnitude of the displa ement eld "2 e~y2 (kx ; H )j2 = j"1 e~y3 (kx ; H )j2

j

Integrating over the whole kx -spe trum yields: Z 1 Z 1 j"1 e ~y3 (kx ; H )j2 dkx j"2 e ~y2 (kx ; H )j2 dkx =

1

1

(4.118)

(4.119)

Sin e it has been shown that the trial elds in ea h layer are linear ombinations of the omponents of the trial quantity, a simple analyti al expression for the ratio Rn =Zm may be obtained from (4.119). Few expli it results are found in the literature for higher-order modes on open slot-like lines. They are usually omputed for shielded lines. To our

192

CHAPTER 4.

GENERAL

VARIATIONAL PRINCIPLE

knowledge, only Fedorov et al. [4.42℄ provide su h results for wide slots. Figure 4.13 ompares their results (symbols  and o), for the rst higher-order mode, with those obtained by the variational formulation (4.21) (lines), used with one Mathieu fun tion and both e~x - and e~z - omponents in the slot. The

omparison is made for two on gurations on the same substrate orresponding to two dierent slot widths, W = 4 mm and 8 mm. The agreement between the two results is ex ellent for ea h of the two slot widths onsidered. It must be emphasized, however, that Fedorov et al. use the Galerkin's pro edure and several basis fun tions, so that the order of the system to be solved is important. Sin e the formulation used here is expli it and analyti al, no tedious iterations are ne essary, ex ept for the integration along the kx -axis. So, the variational method is an eÆ ient tool for al ulating planar dis ontinuities by mode-mat hing methods. 2.5

2.0 β/ko

1.5

1.0 0.5

1.5

2.5

3.5

λo [cm]

Propagation oeÆ ient for rst higher-order mode of open slot-line (Fig. 4.7, with H = 1 mm, "r1 = "r2 = 1; "r3 = 9), al ulated respe tively with variational prin iple (4.21) (lines) and by Fedorov et al. [4.42℄ (symbols), for two dierent slot widths (   : W = 8 mm; o o o: W = 4 mm)

Fig. 4.13

4.6

Resulting advantages of variational behavior

To start the al ulations, trials have to be introdu ed for the omplex propagation onstant as well as for the elds. The hoi e made for the trials, espe ially for the elds, may strongly ae t the a

ura y of the result, and the omputation time.

4.6.

RESULTING ADVANTAGES OF

VARIATIONAL BEHAVIOR

193

1. The unknown value appears in variational equations (4.2) and (4.21) whi h are solved for . It may also appear in the spatial or spe tral formulation of the trial elds, as shown for instan e by (4.55a,b), (4.56a,b) and (4.78a-d). A reasonable initial value for , noted 0 , is introdu ed in the expression of the elds. 2. Applying the boundary onditions to the trial elds results in a set of homogeneous linear equations relating the spe tral oeÆ ients (4.54d) or (4.76a) to the tangential ele tri eld a ross the slot for slot-like lines, and to the urrent densities owing on the strip for strip-like lines. Only an approximation of these quantities is ne essary be ause of the variational hara ter. 3. The integration over the kx spe trum for open lines, or the in nite summation for shielded lines, is stopped when the onvergen e is onsidered as satisfa tory; that is when the addition of the last term of the summation or of the ontribution of the last subinterval of integration does not signi antly modify the solution of (4.21). Be ause of the linearity of the Fourier transform, any trun ation of the x-Fourier transform of the elds yields an error over the eld in the spatial domain. However, due to the variational formulation, this error may also exist in the trial eld used in (4.19a-e), without signi antly ae ting the result. The variational nature of equation (4.2) yields rapidly onvergent results on e a trial eld has been hosen. It an also improve the quality of the approximation made on the trial eld. Three major advantages of variational formulation (4.2) are summarized here. Ea h of them is detailed in the following subse tions. 1. The al ulation of the propagation onstant does not require solving either a determinantal equation or a high number of iterations, as in other pro edures [4.1℄,[4.5℄,[4.14℄ The propagation onstant is simply obtained by solving a se ond-order equation, with an approximate value of inserted in the trial elds if ne essary. 2. The tangential spe tral quantity required at an interfa e ontaining a dis ontinuous ondu tive layer an be approximated with a strongly redu ed number of basis fun tions. These are best derived by a onformal mapping and adequate boundary onditions at the interfa e. The Rayleigh-Ritz pro edure may be applied in ase of asymmetri oupled lines for whi h the a tual ratio between the spe tral quantities on the two lines is unknown. 3. The integration or series expansion is limited to a small number of terms be ause the integrands of expressions (4.19a-e) for al ulating the oeÆ ients of the se ond-order equation (4.2) or (4.21) are rapidly de reasing

194

CHAPTER 4.

GENERAL

VARIATIONAL PRINCIPLE

fun tions of the spe tral variable kx , as will be shown in a further subse tion. Moreover, the propagation onstant is expli it in equation (4.21). It is al ulated at ea h step of the summation, whi h dire tly indi ates when the onvergen e is onsidered satisfa tory, that is when the addition of one more term does not modify the value of the solution.

4.6.1 Error due to approximation made on in trial elds A suitable approximation of is ne essary in the expression of the trial elds. It is denoted t = t + j t in the following dis ussion. A good hoi e for t is [4.7℄

t = j t = j

r"

( 1 + "2 ) k0 2

(4.120)

The solution of variational equation (4.21), obtained using approximation (4.120), is introdu ed as a new value for t in the spe tral elds, whi h yields a new solution for the equation. The quality of the approximation is then evaluated, and improved if ne essary. Iterations are stopped when the dieren e between and t is found negligible. Sin e the formulation is variational with respe t to the eld, the error introdu ed by an approximation made on the value t involved in the eld expression is only of the se ond-order, whi h strongly redu es the number of iterations. This is illustrated in Figure 4.14 for a lossless slot-line on a diele tri substrate [4.4℄. The solid line depi ts the solution of equation (4.21) obtained for a wide range of values of the imaginary part t . Sin e there are no losses, purely imaginary and t are onsidered for the solution of (4.21) and for the approximation for the eld are onsidered. Starting with the value provided by (4.120) for t in the trial elds (A in Fig. 4.14), the rst solution (denoted by 0 ) of (4.21) is point B. This value is then used as a new approximation, denoted by t1 , for the trial elds (point C). After solving again (4.21), it leads to an updated value for , denoted by 1 (point D). It is observed that this solution is within 0.067 % of the nal solution, whi h is rea hed when n = n 1 = tn = 527 rad/m (point E). This explains why the results of Figures 4.3, 4.5, and 4.6, as well as other results presented earlier [4.7℄, have ne essitated no more that 3 iterations (n = 2) to obtain a nal value of within a 0.05 % un ertainty. The onvergen e s heme may thus be formalized as follows. 1. Equation (4.21) is rewritten as

2

XA u XB u XD u ~i [ ~ t ℄

~i [ ~ t ℄ +

i

i

~ i [ ~t ℄ + ! 2

i

X"  E u i

i

i

~i [ ~ t ℄ = 0

(4.121)

4.6.

RESULTING ADVANTAGES OF

195

VARIATIONAL BEHAVIOR

550

propagation coefficient β [rad/m] solution of (4.21) (4.128c)

540 β = βt β1

530

D E B 520

β0

510

C

500 400

450

A

500

550

600 βt0

βt1 trial propagation coefficient βt [rad/m]

Fig. 4.14 Convergen e properties of variational prin iple for lossless open

slot-line

where u~t = ~ei ( t ). It has the general solution

P B~ [u~ ℄

=  + j = P ~ ~ rn P 2 o A [uP℄ n P ~~ P~~o "  E [u ℄ D [u ℄ 4 A~ [u~ ℄ ! B~ [u~ ℄  P 2 A~ [u~ ℄ i

i

t

i

i

t

2

i

i

t

i

i

2

t

i

i

i

i

i

i

t

i

i

t

t

(4.122)

For a lossless ase, in (4.121) one has = j and t = j t . The imaginary part of solution (4.122) is represented by the solid urve in Figure 4.14. 2. The solution at iteration n is obtained by solving (4.21) with tn equal to the solution obtained at iteration n 1, that is with

tn = n

1

(4.123)

in the oeÆ ients of (4.121) and in (4.122). For n = 0, t0 is given by (4.120). 3. The onvergen e is obtained when the relative dieren e between results

196

CHAPTER 4.

GENERAL

VARIATIONAL PRINCIPLE

after two su

essive iterations is lower than a given value x, that is

n



n

1

n



< x%

(4.124)

whi h, by virtue of (4.123), yields approximately

n = n

1

= tn

(4.125)

With onvergen e riterion (4.124), the nal solution is lo ated at the interse tion between the solid urve depi ting solution (4.122) as a fun tion of the trial value t and the linear fun tion

( t ) = t

(4.126)

represented by the dashed line in Figure 4.14. It is underlined that the variational behavior of (4.21) is related to a stationarity on ept. Indeed starting with an approximate value t0 of about 600 rad/m (point A) orresponding to an error of about 14 % on , point B yields a value of 522 for 0 orresponding to an error of about 1 % on . Hen e, the rst-order error is redu ed to a se ond-order one. So, a very limited number of iterations is ne essary. It has also to be observed that the solution de ned by (4.124) is lo ated exa tly at the extremum value of the

urve representing the solution (4.122). When lossy layers are onsidered, a most important feature of the variational hara ter is that the solution obtained with equation (4.21) is omplex,

orresponding to a stationary value for both the real and imaginary parts of the propagation onstant. This is observed at Figure 4.15, where the real (a) and imaginary (b) parts of the solution for a oplanar waveguide lying on a doped semi ondu tor substrate is represented as a fun tion of a wide range of values for (t ; t ) (t in the range 100-300 Np/m, t in the range 300700 rad/m). The resistivity of the substrate, due to the presen e of arriers in the doped semi ondu tor, is introdu ed in the variational formulation via a frequen y-dependent loss tangent asso iated with ea h layer and yielding the imaginary part of its relative permittivity. A frequen y-dependent omplex value of the permittivity is thus onsidered, in the trial elds as well. The real and imaginary parts of solution (4.122) written as a fun tion of the real and imaginary parts of the trial t are ea h represented by a surfa e above the plane (t ; t ). A number of features may be pointed out in Figure 4.15. First, the variational behavior of the solution is obvious: the real and imaginary parts of the solution remain quite onstant over a very wide range of values for both the real and imaginary parts of t . For a relative variation of 300 % for t and of 230 % for t , the variations of the real part  and of the imaginary part of the solution from (4.122) remain within 10 % and 5 %, respe tively. Next, in the ase of iterations arried out to improve the trial elds and hen e the results on , the onvergen e is quite immediate. As a rst step,

4.6.

RESULTING ADVANTAGES OF

197

VARIATIONAL BEHAVIOR

Solution of variational principle − Real part

Solution of variational principle − Imaginary part

300

580 560

200

540 100

520

0 300

500 300 200 α 100 300

(a)

400

500

600 β

700 200 α 100 300

400

500 β

600

700

(b)

Variational solution as a fun tion of real and imaginary parts of trial propagation onstant introdu ed in trial elds for a CPW (20

m resistivity substrate) (a) real part; (b) imaginary part

Fig. 4.15

t is kept onstant. Again solving equation (4.121) yields a new value for noted 0 . The pro ess is repeated n times with t kept onstant, until

the imaginary part of the solution (4.122) be omes equal to the imaginary part of the trial t , that is when riterion (4.124) is satis ed. Pra ti ally, as said before, iterations are stopped when the relative dieren e between results obtained after two su

essive iterations is lower than a given value x, that is

n n n



1

< x%

(4.124)

Be ause of the very small variation of the solution with respe t to t whi h is observed, (Fig. 4.15b), a very limited number of iterations is ne essary. This number of iterations is noted NI . The onvergen e ondition (4.124) written for every value of t forms the interse tion of the surfa e representing the imaginary part of solution (4.122) (Fig. 4.15b) with plane (t ; t ) = t . This interse tion is obtained for the set of values in plane (t ; t ) whi h are represented by the dashed urve. Using in a se ond step a similar onvergen e riterion for the real part of solution (4.122), that is, for a xed value of t n = n

1

= tn

(4.127)

and writing the onvergen e ondition (4.127) for every value of t yields the solid urve in Fig. 4.15. It orresponds to the interse tion between the surfa e representing the real part of the solution (4.122) (Fig. 4.15a) and plane (t ; t ) = t . The solid and dashed urves are ea h parallel to a oordinate axis. This property expresses the fa t that the real and imaginary parts of the

198

CHAPTER 4.

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VARIATIONAL PRINCIPLE

solution of (4.122), obtained after some iterations, be ome independent of the imaginary and real parts of the trial propagation onstant introdu ed in the trial elds, respe tively. Hen e, iterating on the imaginary part while xing the real part at an arbitrary value, then iterating on the real part while retaining the solution found for the imaginary part, onverges towards a solution lo ated at the interse tion between the solid and dashed urves (Fig. 4.15). Moreover, at ea h step of the iterative pro ess, the real and imaginary parts of the trial an be simultaneously updated without ae ting the onvergen e, be ause of the orthogonality of the two urves. Hen e, when new iterations are ne essary, updating at the same iteration the values of t and t also redu es the numeri al omplexity without signi antly damaging the onvergen e. The

onvergen e s heme is nally equivalent to solving equation (4.121) n times with, at iteration n, n obtained as solution of (4.121)

tn = n

1

(4.128a)

whi h means

tn = n

1

(4.128b)

tn = n

1

(4.128 )

The onvergen e is obtained when the two onditions (4.124) and (4.127) are satis ed at the same iteration. The number of iterations is very limited, be ause of the atness of the surfa es representing both the real and imaginary parts of the solution. It has been observed that the two dashed and solid urves interse t ea h other, so that a solution exists. The same property is found for gyrotropi layers. The formulation is also eÆ ient in the ase of leaky planar lines or slowwave devi es. The propagation onstants of these lines indeed exhibit both real and imaginary parts, even in the ase of lossless media, and a solution is diÆ ult to obtain from an impli it method, as mentioned by Das [4.15℄. Moreover, the results in Figure 4.15 are omputed for a high-loss ase, and the solution found exhibits this feature, be ause the real and imaginary parts of the solution are of the same order. Su h a ase is thus parti ularly suitable for omparing the eÆ ien y of the variational solution (4.122) with that of the impli it Galerkin's pro edure. This will be presented in the next se tion.

4.6.2 In uen e of shape of trial elds at interfa es

The formulation is variational with respe t to the ele tri eld introdu ed in equation (4.2). This has parti ular appli ations in the ase of oupled lines, where the ratio between the trial quantities onsidered for ea h line has to be determined. A ratio of voltages Cv has been de ned for oupled slot-like lines (4.98), while a ratio of urrents Ci is onsidered for strip-like lines (4.109). Figure 4.16 shows, for lossless oupled symmetri al slot-lines having a width ratio W2 =W1 equal to 1, the value of obtained from equation (4.21) as a

4.6.

RESULTING ADVANTAGES OF

199

VARIATIONAL BEHAVIOR

2.1 2.0 β/ ko 1.9 1.8 1.7 1.6

-2

-1

0 C

1

2

v

Fig. 4.16 Propagation oeÆ ient of oupled slot-lines (Fig. 4.9) al ulated with variational formulation for dierent values of ratio between trial ele tri elds in the two slots (W1 = W2 = 1 mm, H = 1:2 mm)

1

2.1

3 4

2.0 β/ ko

6

1.9

5

1.8 1.7 2

1.6

-2

-1

0 C

1

2

v

Fig. 4.17 Propagation oeÆ ient of lossless oupled slot-lines al ulated with variational formulation as a fun tion of ratio between trial ele tri elds in the two slots, with ratio W2 =W1 = 0:5 (1), 0.75 (2), 1.0 (3), 1.25 (4), 1.5 (5), and 2.0 (6)

200

CHAPTER 4.

GENERAL

VARIATIONAL PRINCIPLE

fun tion of ratio Cv . It is observed that two extrema of o

ur, respe tively a maximum for Cv = 1, and a minimum for Cv = +1. This on rms that equation (4.2), rewritten in the spe tral domain as (4.21), is variational with respe t to the ele tri eld, sin e the solution exhibits extrema for values of Cv whi h are indeed the a tual ratios of the transverse tangential ele tri elds in this symmetri al stru ture. Hen e, the appli ation of the Rayleigh-Ritz pro edure in this ase will provide the a tual ratios, sin e they orrespond to extrema of the solution. In the ase of asymmetri lines, the ratio Cv or Ci annot be determined a priori, be ause of the la k of x-symmetry of the tangential ele tri eld in the stru ture. However, a

ording to the theory of asymmetri oupled lines [4.43℄, there are two solutions for , orresponding to two values of Cv or Ci . The rst is asso iated with the in-phase behavior of the trial quantity on the two lines (Cv , Ci positive) and is alled the  -mode. The se ond is asso iated with the out-of-phase behavior of the trial quantity (Cv , Ci negative) and is alled the C-mode. One an expe t that the Rayleigh-Ritz pro edure in the ase of asymmetri lines will determine the a tual value of by sear hing the values of Cv whi h render extremum. Figure 4.17 shows the variation of the solution of (4.21) as a fun tion of Cv for oupled slot lines having dierent widths (W2 =W1 varying between 0.5 and 2.0). In this asymmetri ase, extrema now o

ur for values of Cv whi h vary with ratio W2 =W1 . Applying the Rayleigh-Ritz pro edure, as explained for the higherorder modes in a pre eding se tion, yields the value of Cv whi h maximizes the solution of (4.21), as well as the value of this extremum. The variational pro edure is applied to the lossless asymmetri al oupled slot-like line previously studied with Galerkin's method by Kitlinski and Janisa z [4.44℄. Results are ompared in Tables 4.2 (C-mode) and 4.3 ( mode) for the ase W2 =W1 = 2. The values of the normalized propagation

onstant obtained with this variational formulation (Huynen) are ompared to results obtained by Kitlinski et al. over the frequen y range 2-20 GHz. The

orresponding values obtained for the ratio Cv are also given. The agreement between the two methods is better than one per ent over the whole frequen y range. 4.6.3

Trun ation error of the integration over spe tral domain

Figure 4.18 shows a typi al variation of the integrands of the oeÆ ients de ned by (4.19) as a fun tion of the spe tral variable kx , for a slot-line at xband. As explained in Se tion 4, these integrands are similar for laterally-open or - shielded slot-lines, be ause the spe tral elds are the same for a given value of kx . If the line has lateral shielding, the integration is simply repla ed by a summation over dis rete values of the integrands orresponding to dis rete values of kx . The gure shows that all the integrands are rapidly de reasing as the spe tral variable kx in reases. In this ase it indi ates learly that the integration or summation an be stopped when kx equals about 8000 m 1 . If the slot-line is shielded at least in the x-dire tion, and for Wg taken 10

4.6.

RESULTING ADVANTAGES OF

VARIATIONAL BEHAVIOR

201

C-mode Frequen y

Cv

(0 =0 )

(0 =0 )

(GHz)

[Huynen℄

[Kit.-Jan.℄

[Huynen℄

2

-0.975

1.970

1.9436

4

-0.975

1.960

1.9542

6

-0.975

1.978

1.9680

8

-0.925

1.985

1.9842

10

-0.925

2.007

2.0031

12

-0.925

2.025

2.0200

14

-0.925

2.040

2.0376

16

-0.925

2.066

2.0604

18

-0.925

2.093

2.0829

20

-0.925

2.120

2.1000

Comparison of results obtained with this formulation (Huynen) and by Kitlinski and Janisa z [4.44℄ for C-mode of asymmetri al oupled slot-lines

Table 4.2

 -mode Frequen y

Cv

(0 =0 )

(0 =0 )

(GHz)

[Huynen℄

[Kit.-Jan.℄

[Huynen℄

2

1.700

1.330

1.2350

4

1.725

1.370

1.3650

6

1.725

1.425

1.4226

8

1.775

1.470

1.4731

10

1.775

1.540

1.5447

12

1.725

1.580

1.5838

14

1.825

1.640

1.6349

16

1.725

1.690

1.6810

18

1.775

1.730

1.7255

20

1.875

1.775

1.7709

Comparison of results obtained with this formulation (Huynen) and by Kitlinski and Janisa z [4.44℄ for  -mode of asymmetri al oupled slot-lines

Table 4.3

202

CHAPTER 4.

GENERAL

VARIATIONAL PRINCIPLE

~ ~ ~ ~ Integrands of Ai, Bi, Di, Ei

800 4 [ (Volts / m) 2 ]

600 400 1 200 0

5 6 2 3

-200

2000

6000

10000

14000

spectral variable kx [m -1]

Integrands of set of the oeÆ ients (4.19) for a slot-line (RTduroid 6010, thi kness 0.635 mm, manufa turer's relative diele tri onstant 10.8, slot width = 0.390 mm); urve2 1: k02A~1; urve 2: k02A~3; urve 3 : C~1 + C~3 ;

urve 4: D~ 1 + D~ 3 ; urve 5: k0 E~1 (indistinguishable from urve 1); urve 6: k02 E~3 (indistinguishable from urve 2) Fig. 4.18

times larger than the slot width (W = 0.390 mm in the present ase), then the maximum number nmax of terms needed for the summation is obtained from relationship kxn = 2nmax=Wg whi h yields 8000 10 0:39 10 3 = 5 nmax = (4.129) 2 If the slot-line is open, then the Romberg integration method [4.45℄ is su

essfully applied to the integrands, be ause a large integration step size is suÆ ient for most part of the integration range (interval 2000-8000 in this

ase). Hen e, only a few evaluations of the integrands are needed in this interval. Only the interval 0-2000 has to be strongly dis retized by the Romberg algorithm to take into a

ount the presen e of a maximum in the integrands. A redu ed number of evaluations of the integrands is needed when they have a monotoni behavior in some subintervals, whi h ensures a rapid onvergen e of the Romberg algorithm in these subintervals. Hen e, the integration along kx an be performed with a very few number of terms in the ase of shielded lines, or very few steps in the ase of open lines. This is a major advantage





RESULTING ADVANTAGES OF

2

700 characteristic impedance Zc (Ω)

203

VARIATIONAL BEHAVIOR

650

1

600 550

2 1

500

( β / ko ) 2

4.6.

450 400 3

350 300

0 10

20

30 40 50 frequency (GHz)

60

70

Chara teristi impedan e of the dominant mode ( urve 1), and ee tive diele tri onstant of the dominant mode ( urve 2) and rst higherorder mode ( urve 3) of nline with 20 spe tral terms

Fig. 4.19

of the formulation: the spe tral elds, hen e the integrands of (4.19), have to be evaluated for a very limited number of values for kx . Figure 4.19 shows results obtained when al ulating the parameters of the dominant and rst higher-order mode of the shielded line analyzed by S hmidt and Itoh (Fig. 3b of [4.46℄). They report that 200 spe tral terms and appropriate basis fun tions led to a onvergen e within 0.2 % for the ratio ( =k0 )2 . Using the variational approa h, however, the hara teristi impedan e ( urve 1) and the propagation onstant ( urve 2) of the dominant mode are al ulated with equation (4.21) while expression (4.111) is used for the propagation onstant of the rst higher-order mode ( urve 3). The number of spe tral terms used in (4.19) never ex eeds 20. Hen e, the numeri al

omplexity is strongly redu ed, although the omparison with Figure 3b of [4.46℄ shows su h a perfe t agreement that it is impossible to see the dieren e between the two results on a gure of normal size. Applying Galerkin's pro edure in the spe tral domain results in a determinantal equation involving and the spe tral variable kx , whi h is therefore an impli it equation for . As reported by Jansen [4.14℄, a typi al number of spe tral terms ne essary to obtain a good onvergen e is between 100 and 500 terms for ea h al ulation. Moreover, a number of evaluations of the determinantal equation is ne essary to nd the roots. A

ording to Jansen, a typi al

204

CHAPTER 4.

GENERAL

VARIATIONAL PRINCIPLE

number of evaluations is 10. As a rule of thumb, the numeri al omplexity of Galerkin's pro edure an be evaluated by multiplying the number of terms by the number of evaluations, the result being of the order of a few thousands. Sin e no expli it formulation of exists in this ase, the tedious pro edure for nding the root of the determinantal equation requires that a number of integrations over the whole kx -spe trum have to be performed ea h time the determinantal equation is solved. Hen e, to evaluate the improvement in a

ura y when in reasing the integration domain, the determinantal equation has to be solved ea h time the integration domain is hanged. Moreover, the number of terms ne essary may vary with the geometry of the stru ture and the physi al parameters of the layers. Hen e, an a priori knowledge of an upper limit for the integration is not available. For this reason, some authors [4.6℄ hoose an arbitrary very large limiting value for the integration, to ensure suÆ iently a

urate results for a wide variety of topologies. They also assume a virtual lateral shielding pla ed at a distan e of about 10 times the slot width and transform the integration pro ess into a dis rete summation [4.5℄. Choosing a large limit of integration, they in rease the omputation time for most of the ases of pra ti al interest. On the ontrary, the expli it variational formulation dire tly evaluates the ee t of a modi ation of the size of the integration domain on the result: the integration an be stopped when the onvergen e is found to be satisfa tory, whi h minimizes the omputation time for ea h spe i topology.

4.7 Chara terization of the al ulation eort

All the results presented in this hapter were obtained on a 25 MHz IBM PC 80386. The propagation onstant (a), attenuation (b), and hara teristi impedan e ( ) of a slot-line are represented in Figure 4.20, in the frequen y range 4-24 GHz. At ea h frequen y, the al ulation is stopped when the values of the three parameters omputed from the solution of (4.21) ea h remain within 0.1 % , when the size of the integration domain is in reased. Figure 4.20d shows that this result is a hieved in less than 1 se ond for ea h frequen y point. It illustrates that on-line results an be obtained with this expli it variational formulation in the ase of lossy multilayered planar transmission lines. In the following dis ussion, results provided by the variational prin iple are ompared with those from other methods, in terms of number of iterations, a

ura y of the results obtained, and omputation time, taking into a

ount the omputational environment. 4.7.1

Comparison with Spe tral Galerkin's pro edure

As already dis ussed in Chapter 3, the Spe tral Domain Approa h (SDA) uses the Galerkin's pro edure in the Fourier-transform domain in order to nd the propagation onstant. A review of this te hnique is presented in [4.14℄ and [4.47℄. The SDA an be tedious for on-line designs of planar lines [4.14℄. When

ompared with variational equation (4.121), Galerkin's pro edure sear hes for

4.7.

205

CHARACTERIZATION OF THE CALCULATION EFFORT

( β / ko ) 2

6

5

2

4

1

3

4

120

9 14 19 frequency (GHz) (a)

α [Np/m]

3

24

0 4

9

14

19

24

frequency (GHz) (b)

characteristic impedance Zc (Ω)

Computation time (Seconds)

1.0 0.8

110

0.6

100

0.4 90

0.2 80 4

9 14 19 frequency (GHz)

24

0.0

4

(c)

9 14 19 frequency (GHz)

24

(d)

Cal ulation of line parameters of slot-line (a) ee tive diele tri

onstant; (b) attenuation oeÆ ient; ( ) hara teristi impedan e, (d) omputation time at ea h frequen y Fig. 4.20

the omplex root of the following impli it determinantal equation, obtained in Chapter 3 (3.60): ~ det(Lst ( )) = 0

for strip-like problems

(4.130a)

~ det(Lsl ( )) = 0

for slot-like problems

(4.130b)

Solving the se ond-order variational equation for diers from Galerkin's pro edure. The general solution (4.122) of the equation is indeed known a priori

206

CHAPTER 4.

GENERAL

VARIATIONAL PRINCIPLE

while Galerkin's method repla es the equation by the impli it determinantal equation (4.130a,b) to be solved for . Solving this equation requires the evaluation of the determinantal equation several times, and may be very diÆ ult, as mentioned by a number of authors [4.48℄. This is be ause the method provides no information about the behavior of the impli it equation (4.130a,b) with respe t to . In parti ular spurious solutions may o

ur [4.11℄. Solving (4.130a,b) requires spe i numeri al te hniques, espe ially when losses are present and evanes ent modes are onsidered [4.12℄ and [4.15℄. For these reasons, the method is usually limited to lossless appli ations on diele tri multilayered media, as mentioned by Jansen [4.14℄. Under low-loss assumptions, Das and Pozar [4.15℄ propose to use \the spe tral-domain perturbation method" but, as they mention, it is only valid for small loss appli ations. Su h a method is not suited for general uses. Indeed Figure 4.15 shows values of the attenuation and propagation oeÆ ients having the same order of magnitude. When using the SDA, the number NS of spe tral terms ne essary to obtain a good a

ura y for ea h al ulation of the determinant is typi ally between 100 and 500 [4.12℄. It has already been ommented that a number of evaluations of the determinant are ne essary to nd the omplex roots. A typi al number of evaluations is NE = 10. The numeri al omplexity of SDA

an be quanti ed by NS  NE , whi h is of the order of a few thousands [4.12℄. On the other hand, for the variational equation (4.21), the oeÆ ients A~i , B~i , D~ i , and E~i are obtained by summing NS spe tral terms, while NE has to be repla ed by NI , the number of iterations ne essary to obtain a satisfa tory onvergen e on . More pre isely, NI is de ned as the number of iterations, arried out simultaneously on the real and imaginary parts of (4.21) or (4.121), whi h are ne essary to obtain su

essive values within x % for both the real and imaginary parts of the solution:



n n n n

n

n



1



1

< x%

(4.131a)

< x%

(4.131b)

Figure 4.21 shows several parameters omparing the eÆ ien y of the variational prin iple (VP) (4.21), with that of the SDA using Galerkin's pro edure, on a oplanar waveguide up to 20 GHz. Both methods are applied to the same low-resistivity Sili on-on-Insulator (SOI) stru ture as shown in Figure 4.15 (resistivity 20

m). When omputing the propagation oeÆ ient, the iteration pro edure is stopped when the onvergen e to x = 0:5 % is simultaneously obtained on both the real and imaginary parts of the solution. Figure 4.21a shows that VP (-) agrees very well with SDA (o). From Figure 4.21b,d, it is on luded that the spe tral variational prin iple is mu h more eÆ ient than SDA. It needs far fewer spe tral terms and iterations per frequen y point: the numeri al omplexity of the SDA (NS  NE ) is between 8000 and 12000 whilst that of the VP (NS  NI ) is between 100 and 200.

4.7.

207

CHARACTERIZATION OF THE CALCULATION EFFORT

(a)

(b)

NS for VP (+) and SDA (o)

Results from VP (-) and SDA (o) 1200

1000 Imaginary part

1000

800

800

600

600 400

400

Real part 200

200

GHz

GHz 0 0

5

10

15

20

CPU time per frequency using VP(+) and SDA(o) 2000 seconds

0 0

5

10

15

20

NI for VP (+) and NE for SDA (o) 30 25

1500 20 1000

15 10

500 5 0 0

GHz 5

10 ( )

15

20

0 0

GHz 5

10

15

20

(d)

Comparison of the eÆ ien y of the variational formulation (VP) and of the SDA for a CPW transmission line (standard 20

m resistivity) (a) real and imaginary parts of the propagation onstant; (b) number of spe tral terms needed; ( ) CPU times; (d) number of iterations for VP (+) and number of evaluations of the determinantal equation for the SDA (o) Fig. 4.21

The very low values of NI are explained by the properties of the variational solution, pointed out in the omments dealing with Figure 4.15. The redu ed number of spe tral terms NS is due rst to the variational nature of the solution, whi h ensures that trun ation ee ts on the spe tral eld have only a small in uen e on the result. Se ondly, it is also due to the fa t that a ratio of spe tral terms is used in the variational expression (4.122), so that errors made on terms involved in the numerator and denominator vanish. It has to be underlined that the results provided by (4.21) and by Galerkin's method, respe tively, have been obtained using the same omputer and programming language (Matlab 4.0). As a onsequen e, the omparison between CPU times is valid (Figure 4.21 ). Hen e, the variational prin iple (4.21) is

208

CHAPTER 4.

GENERAL

VARIATIONAL PRINCIPLE

mu h less time- onsuming: it is less than 50 s for VP, while it is of the order of 1,000 s for the SDA using Galerkin.

4.7.2 Comparison with nite-elements solvers

The High Frequen y Stru ture Simulator (HFSS) from Hewlett-Pa kard is a 3-D ele tromagneti wave simulator, omputing ele tromagneti elds and s attering parameters of a 3-D stru ture, geometri ally and physi ally de ned by the user and en losed in a box. The ports of the equivalent N -port are spe i ed as limiting plane surfa es of the box. The user has to spe ify all the materials inside the box, and boundary onditions on its surfa es. The 3-D mesh to be solved by the nite-element method is built from a 2-D port analysis: the ve tor wave equation is rst solved in 2-D for ea h port using a triangular adaptive mesh (iterative s heme) and yields the transmission line parameters of the onsidered mode in ea h port. Hen e, the 2-D port analysis provides the transmission line parameters of any stru ture having the same transverse se tion as the port. Figure 4.22 ompares results obtained from Version 3.0 of HFSS and from the variational spe tral form, for the same oplanar waveguide shown in Figures 4.15 and 4.21, whose transmission line parameters have been measured [4.49℄. The VP transmission line parameters (- - -) agree perfe tly with the experiment (|), while the results obtained with HFSS (o) are less a

urate. HFSS seems unable to take into a

ount materials having high-loss frequen y dependent onstitutive parameters. Furthermore, a 2-D omputation with HFSS takes about 7 minutes per frequen y point, whereas only a few se onds per frequen y are needed using (4.21) on the same omputer. Hen e, the variational prin iple (4.21) proves to be more eÆ ient for high-loss stru tures than this example of a popular 3-D tool. Similar results are observed on strip lines, as shown in Figure 4.23. The variational prin iple (4.21) and the nite-element method are applied to the shielded mi rostrip line investigated both in Se tion 4.4 and by Itoh and Mittra [4.5℄,[4.26℄. Attention is drawn to the fa t that the stru ture is now lossy. Figure 4.23a,b shows the ee tive diele tri onstant and attenuation oeÆ ient, obtained using the variational prin iple (solid line) and a nite-element algorithm ( ir led line). In these gures, it is observed that even after re ning the mesh from 180 elements to 448 elements ( rossed line) dis repan ies remain between VP and FEM values. Figure 4.23 illustrates on e more the very limited numeri al omplexity of the VP in terms of number of spe tral terms and iterations, while Figure 4.23d ompares the omputation time for FEM (in minutes) with that of the VP (in se onds). Applying FEM ode to the stru ture of Figure 4.4 requires about 3 and 9 minutes per frequen y for a 180 and 448 element-mesh respe tively, while only a few se onds are ne essary when using the VP on the same omputer. The CPU time in FEM is ru ially in uen ed by the number of elements of the mesh and by the eÆ ien y of the eigenvalue solver required by the FEM algorithm, whi h in turn strongly ae ts the a

ura y of the solution obtained [4.50℄.

4.8.

209

SUMMARY

(a)

(b)

Effective dielectric constant

Attenuation coefficient (Np/m) 250

15

200 150 10 100 50 GHz

0 0

5

10

15

20

Real part of impedance (Ohms)

5 0

5

10

GHz 15

20

Imaginary part of impedance (Ohms) 20

60 50

15 40 30

10

20 5 10 0 0

5

10

GHz 15

( )

20

0 0

5

10

15

20

(d)

|

Measured results ( ), simulated with (VP) (- - -), and simulated with HFSS (o) for transmission line parameters of multilayered lossy CPW (a) attenuation oeÆ ient; (b) ee tive diele tri onstant; ( ) real part of impedan e; (d) imaginary part of impedan e (Reprodu ed from Pro . NUMELEC'97 [4.49℄)

Fig. 4.22

4.8

Summary

A general variational formulation for planar multilayered lossy transmission lines has been established in this hapter. The formulation is very general. It has a number of advantages, related to its variational hara ter and the expli it formulation. It an be used in both the spatial and the spe tral domain, depending upon the ee ts it is sought to highlight. In ombination with onformal mapping, it drasti ally redu es the omplexity and the time of numeri al omputation. It leads to rapidly onvergent results even when

210

CHAPTER 4.

GENERAL

VARIATIONAL PRINCIPLE

(a)

(b)

Effective dielectric constant

Attenuation coefficient (Np/m)

5

3 2.5

4 2 3

1.5 1

2 0.5 1 0

10

20 GHz

30

40

0 0

10

NS (-) and NI (--) for VP

20 GHz

30

40

30

40

CPU time

10

10

8

8 6

6

VP in seconds FEM in minutes FEM in minutes

4

4

2 2 0

10

20 GHz ( )

30

40

0 0

10

20 GHz (d)

Comparative results using variational prin iple (VP) and niteelement method (FEM) for shielded mi rostrip (a) ee tive diele tri onstant; (b) attenuation oeÆ ient; ( ) number of spe tral terms (NS ) and of iterations (NI ) required by VP; (d) CPU time in se onds for VP (|), and in minutes for FEM

Fig. 4.23

higher-order modes are onsidered. Mathieu fun tions are shown to be very ef ient expressions for trial elds of the dominant and the higher-order modes in slots. The al ulation is fast: it is performed on-line on a regular PC. Results obtained on open and shielded lines have been su

essfully ompared with previously published data. The method is general enough to a

omodate gyrotropi lossy substrates. The method has been validated by a tual measurements on various types of lines like slot-lines, oupled slot-lines, nlines, and oplanar waveguides, in whi h YIG-layers may be in luded. It has also been validated on various stru tures involving lines and transitions between them. These experimental results are presented in the next hapter.

4.9.

4.9

REFERENCES

211

Referen es

[4.1℄ J. S hwinger, D. S. Saxon, Dis ontinuities in Waveguides - Notes on Le tures by J. S hwinger. London: Gordon and Brea h, 1968. [4.2℄ V.H. Rumsey, \Rea tion on ept in ele tromagneti theory", Phys. Rev., vol. 94, pp. 1483-1491, June 1954, and vol. 95, p. 1705, Sept. 1954. [4.3℄ A.D. Berk, \Variational prin iples for ele tromagneti resonators and waveguides", IEEE Trans. Antennas Propagat., vol. AP-4, no. 2, pp. 104-111, Apr. 1956. [4.4℄ I. Huynen, B. Sto kbroe kx, \Variational prin iples ompete with numeri al iterative methods for analyzing distributed ele tromagneti stru tures", Annals of Tele ommuni ations, vol. 53, no. 3-4, pp. 95103, Mar h-April 1998. [4.5℄ T. Itoh, Numeri al Te hniques for Mi rowave and Millimeter Wave Passive Stru tures. New York: John Wiley&Sons, 1989, Ch. 5. [4.6℄ S.S. Bedair, I. Wol, \Fast, a

urate and simple approximate analyti formulas for al ulating the parameters of supported oplanar waveguides for (M)MIC's", IEEE Trans. Mi rowave Theory Te h., vol. MTT40, no. 1, pp. 41-48, Jan. 1992. [4.7℄ I. Huynen, D. Vanhoena ker-Janvier, A. Vander Vorst, \Spe tral domain form of new variational expression for very fast al ulation of multilayered lossy planar line parameters", IEEE Trans. Mi rowave Theory Te h., vol. 42, no. 11, pp. 2099-2106, Nov. 1994. [4.8℄ J.B. Knorr, P. Shayda, \Millimeter wave nline hara teristi s", IEEE Trans. Mi rowave Theory Te h., vol. MTT-28, no. 7, pp. 737-743, July 1980. [4.9℄ I. Huynen, A. Vander Vorst, \A new variational formulation, appli able to shielded and open multilayered transmission lines with gyrotropi non-hermitian lossy media and lossless ondu tors", IEEE Trans. Mi rowave Theory Te h., vol. 42, no. 11, pp. pp. 2107-2111, Nov. 1994. [4.10℄ E. Yamashita, \Variational method for the analysis of mi rostrip-like transmission lines", IEEE Trans. Mi rowave Theory Te h., vol. MTT16, no. 8, pp. 529-535, Aug. 1968. [4.11℄ M. Essaaidi, M. Boussouis, \Comparative study of the Galerkin and the least squares boundary residual method for the analysis of mi rowave integrated ir uits", Mi rowave and Opt. Te hnol. Lett., vol. 7, no. 3, pp. 141-146, Mar. 1992. [4.12℄ F.L. Mesa, M. Horno, \Quasi-TEM and full wave approa h for oplanar multistrip lines in luding gyromagneti media longitudinally magnetized", Mi rowave and Opt. Te hnol. Lett., vol. 4, no. 12, pp. 531-534, Nov. 1991. [4.13℄ J.B. Davies in T. Itoh, Numeri al Te hniques for Mi rowave and Millimeter Wave Passive Stru tures. New York: John Wiley&Sons, 1989, Ch. 2, p. 91.

212

CHAPTER 4.

GENERAL

VARIATIONAL PRINCIPLE

[4.14℄ R.H. Jansen, \The spe tral domain approa h for mi rowave integrated

ir uits", IEEE Trans. Mi rowave Theory Te h., vol. MTT-33, no. 10, pp. 1043-1056, O t. 1985. [4.15℄ N.K. Das, D.M. Pozar, \Full-wave spe tral-domain omputation of material, radiation, and guided wave losses in in nite multilayered printed transmission lines", IEEE Trans. Mi rowave Theory Te h., vol. MTT39, no. 1, pp. 54-63, Jan. 1991. [4.16℄ D.C. Champeney, A Handbook of Fourier Theorems. Cambridge: Cambridge University Press, 1987. [4.17℄ J.P. Parekh, K.W. Chang, H.S. Tuan, \Propagation hara teristi s of magnetostati waves", Cir uits, Systems and Signal Pro eedings, vol. 4, no. 1-2, pp. 9-38, 1985. [4.18℄ K. Yashiro, S. Ohkawa, \A new development of an equivalent ir uit model for magnetostati forward volume wave transdu ers", IEEE Trans. Mi rowave Theory Te h., vol. MTT-36, no. 6, pp. 952-960, June 1988. [4.19℄ R.I. Joseph, E. S hlomann, \Demagnetizing eld in nonellipsoidal bodies", J. Applied Physi s, vol. 36, no. 5, pp. 1579-1593, May 1965. [4.20℄ P. Kabos, V.S. Stalma hov, Magnetostati Waves and their Appli ations. London: Chapman&Hall, 1994. [4.21℄ S.N. Bajpai, \Insertion losses of ele troni ally variable magnetostati wave delay lines", IEEE Trans. Mi rowave Theory Te h., vol. MTT-37, no. 10, pp. 1529-1535, O t. 1989. [4.22℄ I.W. O'Keefe, R.W. Paterson, \Magnetostati surfa e waves propagation in nite samples", J. App. Phys., vol. 49, pp. 4886-4895, 1978. [4.23℄ J.D. Adam, S.N. Bajpai, \Magnetostati forward volume wave propagation in YIG strips", IEEE Trans. Magneti s, vol. MAG-18, no. 6, pp. 1598-1600, Nov. 1982. [4.24℄ P.R. Emtage, \Intera tion of magnetostati waves with a urrent", J. App. Phys., vol. 49, no. 8, pp. 4475-4484, Aug. 1978. [4.25℄ R.E. Collin, Field Theory of Guided Waves. New York: M Graw-Hill, 1960, Ch. 1. [4.26℄ T. Itoh, R. Mittra, \Spe tral-domain approa h for al ulating the dispersion hara teristi s of mi rostrip lines", IEEE Trans. Mi rowave Theory Te h., vol. MTT-21, no. 7, pp. 496-499, July 1973. [4.27℄ R. Mittra, T. Itoh, \A new te hnique for the analysis of the dispersion hara teristi s of mi rostrip lines", IEEE Trans. Mi rowave Theory Te h., vol. MTT-19, no. 1, pp. 47-56, Jan. 1971. [4.28℄ I. Huynen, Modelling planar ir uits at entimeter and millimeter wavelengths by using a new variational prin iple, Ph.D. dissertation. Louvain-la-Neuve, Belgium: Mi rowaves UCL, 1994.

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REFERENCES

213

[4.29℄ R. Janaswamy, D.H. S haubert, \Dispersion hara teristi s for wide slot-lines on low-permittivities substrates", IEEE Trans. Mi rowave Theory Te h., vol. MTT-33, no. 8, pp. 723-726, Aug. 1985. [4.30℄ R.E. Collin, Field Theory of Guided Waves. New York: M Graw-Hill, 1960, Ch. 2. [4.31℄ R.E. Collin, Field Theory of Guided Waves, 2nd ed. New York: IEEE Press, 1991, Ch. 4. [4.32℄ N. M La hlan, Theory and Appli ations of Mathieu Fun tions. Oxford, UK: Oxford University Press, 1951. [4.33℄ M. Helard, J. Citerne, O. Pi on, V. Fouad Hanna, \Theoreti al and experimental investigation of nline dis ontinuities", IEEE Trans. Mi rowave Theory Te h., vol. MTT-33, no. 10, pp. 994-1003, O t. 1985. [4.34℄ E.B. El-Sharawy, R. W. Ja kson, \Coplanar waveguide and slot line on magneti substrates: analysis and experiment", IEEE Trans. Mi rowave Theory Te h., vol. MTT-36, no. 6, pp. 1071-1079, June 1988. [4.35℄ G.N. Watson, A Treatise on the Theory of Bessel Fun tions. Cambridge: Cambridge University Press, 1966. [4.36℄ M. Abramowitz, I. Stegun, Handbook of Mathemati al Fun tions. New York: Dover Publi ations, 1965. [4.37℄ E.J. Denlinger, \A frequen y dependent solution for mi rostrip transmission lines", IEEE Trans. Mi rowave Theory Te h., vol. MTT-19, no. 1, pp. 30-39, Jan. 1971. [4.38℄ R.F. Harrington, Time-Harmoni Ele tromagneti Fields. New York: M Graw-Hill, 1961, Ch. 7. [4.39℄ E. Kuhn, \A mode mat hing method for solving elds problems in   waveguide and resonator ir uits", Ar h. Elek. Ubertragung. , vol. AEU27, no. 12, pp. 511-518, De . 1973. [4.40℄ G. Kompa, \Dispersion measurements of the rst two higher-order   modes in open mi rostrip", Ar h. Elek. Ubertragung. , vol. AEU-27, no. 4, pp. 182-184, Apr. 1973. [4.41℄ R. Mehran, \The frequen y-dependent s attering matrix of mi rostrip   right angle bends", Ar h. Elek. Ubertragung. , vol. AEU-29, no. 11, pp. 454-460, Nov. 1975. [4.42℄ A.N. Fedorov, N.N. Levina, N.A. Khametova, \Some results of a numeri al investigation of slot and strip-slot lines", Radio Engin. and Ele tron. Physi s (USA), vol. 28, no. 7, pp. 42-49, July 1983. [4.43℄ I. Huynen, \Modelisation d'interferen es sur ir uits mi roruban hyperfrequen es", Revue E, 107, no. 3, pp. 40-45, Mar. 1991. [4.44℄ M. Kitlinski, B. Jani zak, \Dispersion hara teristi s of asymmetri

oupled slot lines on diele tri substrates", Ele troni Lett., vol. 19, no. 3, pp. 91-92, Feb. 1983. [4.45℄ G. Dahlquist, A. Bjor k, Numeri al Methods. New Jersey: Prenti eHall, 1974, Ch. 7.

214

CHAPTER 4.

GENERAL

VARIATIONAL PRINCIPLE

[4.46℄ L.P. S hmidt, T. Itoh, \Spe tral domain analysis of dominant and higher order modes in nlines", IEEE Trans. Mi rowave Theory Te h., vol. MTT-28, no. 9, pp. 981-985, Sept. 1980. [4.47℄ T. Itoh (Editor), Numeri al Te hniques for Mi rowave and Millimeter Wave Passive Stru tures. New York: John Wiley&Sons, 1989, Ch. 2 and 5. [4.48℄ J.B. Davies in T. Itoh, Numeri al Te hniques for Mi rowave and Millimeter Wave Passive Stru tures. New York: John Wiley&Sons, 1989, Ch. 2. [4.49℄ I. Huynen, B. Sto kbroe kx, \Variational prin iples ompete with numeri al iterative methods for analyzing distributed ele tromagneti stru tures", Pro eedings of the NUMELEC'97 Conferen e, Lyon, Fran e, pp. 76-77, 19-21 Mar h 1997. [4.50℄ Z. Raida, \Comparative eÆ ien y of semi-analyti al and nite-element methods for analyzing shielded planar lines", Internal Report, Mi rowaves Lab., UCL, Feb. 1997.

hapter 5

Appli ations 5.1 5.1.1

Transmission lines Transmission line parameters

Transmission line theory usually uses three parameters to des ribe the propagation along a guiding stru ture [5.1℄: the propagation onstant taking into a

ount the velo ity of propagation, the attenuation oeÆ ient  des ribing the attenuation along the line, and the hara teristi impedan e Z , the ratio between the forward voltage and urrent waves. The impedan e is usually

hosen in su h a way that the power ow on the equivalent line equals the power ow on the physi al guiding stru ture, obtained by integrating the Poynting ve tor over the ross-se tion of the stru ture. De ning the z -axis as propagation axis, the transmission line equations are

V (z ) = V+ e z + V e+ z I (z ) =

1h

V e z Z +

V e+ z

(5.1a) i

(5.1b)

where we have de ned as in (4.1):

=  + j

(5.1 )

The power ow along the z -axis is expressed as  1  P (z ) = Re V (z )I (z ) = P0 e 2z

2

(5.2a)

with the de nition

jV j2 jV j2 Z jI j2 P0 = + = 0 = 0 2Z 2Z 2

(5.2b)

Requiring P0 to equal the ele tromagneti power ow yields: Z

1 
P0 = e  h
dS 2 S

(5.3)

216

CHAPTER 5.

APPLICATIONS

5.1.1.1 Propagation onstant

In Chapter 4 we derived spatial and spe tral forms of an expli it variational prin iple for the propagation onstant . They onsist of se ond-order equations for

2

XA X B

( i

Ci ) +

XA X B

C~i ) +

i

i

i

in the spatial domain, and

2

i

~i

i

( ~i

XD ! XE

(4.17a)

XD ! XE

(4.21)

i

i

i

~i

2

i

2

i

i=0

~i = 0

in the spe tral domain, equivalent to (4.2), where the spatial or spe tral oef ients are fun tions of the spatial or spe tral transverse dependen ies of the trial ele tri eld omponents, as shown by expressions (4.17b-f) and (4.19ae), respe tively.

5.1.1.2 Power ow

A simple al ulation dire tly shows that P0 (5.3) an be expressed as a fun tion of oeÆ ients involved in the variational equations of (4.17a-f) and (4.21):

P0 = =

h X A X C i j! h X A X C i 1

0

1

j!0

i i

i

~i

i

i

i

for spatial trial elds

(5.4a)

~i

for spe tral trial elds

(5.4b)

where the oeÆ ients Ai , Ci ,A~i ,C~i are de ned, respe tively, by expressions (4.17b,d) and (4.19a, ). On e the propagation onstant has been al ulated by (4.17a) or (4.21), we dire tly obtain the value of P0 without any additional omputation, sin e (5.4a,b) only uses oeÆ ients already al ulated for obtaining . This is another major advantage of our formulation. The power lost per unit length on the equivalent line is then obtained as the rst derivative of the power ow along the z -axis:

P (z ) = 2P0 e 2z z 5.1.1.3 Attenuation oeÆ ient Pdiss =

(5.5)

A number of phenomena ontribute to a non-zero value for the attenuation

oeÆ ient . Two of them have already been onsidered when using (4.17a).

A.

Evanes ent modes

These are readily obtained by solving the variational equation (4.17a), sin e this equation may yield a omplex solution, depending on the hoi e of trial

5.1.

TRANSMISSION LINES

217

elds. If the trial elds asso iated with a mode are properly hosen, then the behavior of the propagation onstant will give the orre t evanes ent behavior. This has already been illustrated in Chapter 4 where higher-order mode propagation onstants were al ulated for shielded lines below ut-o, yielding a real value for the propagation onstant (Figs. 4.12 and 4.19). B.

Losses in the layers

As has been shown in Chapter 4, substrate losses are al ulated by using

omplex expressions for " and  in the variational equation, yielding a omplex solution for equations (4.2) and (4.21):

= s + j

where s takes into a

ount substrate losses.

Hen e the substrate loss onstant is dire tly obtained from the solution of (4.2) al ulated when " and  are lossy. This has already been illustrated for a gyrotropi YIG- lm, where real and imaginary parts of the propagation

onstant were found simultaneously. Two other ategories of losses may also be present on planar lines, namely

ondu tor losses and radiation losses. The modeling of ondu tor losses is performed as follows. Sin e equation (4.17a) is variational in the absen e of

ondu tor losses, these are al ulated by using a perturbational expression, based on a skin-ee t formulation [5.2℄. Indeed in Chapter 4, we have developed expressions for trial elds, assuming that the thi kness of the perfe tly

ondu ting layer is negligible, so that surfa e urrent densities are owing on these layers. The perturbational approa h that will now be used is valid when the ondu tors are good, although not perfe t, in other words under a low loss assumption. The perturbational approa h onsiders that the value of the tangential magneti eld at the ondu ting interfa es under low loss does not dier mu h from its lossless value. This lossless value may be ombined with a skin-ee t formulation to obtain an estimation of the ondu tor losses. The skin-ee t formulation has already been used for planar lines analyzed in the spatial domain by Rozzi et al. [5.3℄. In [5.4℄ we presented a new spe tral domain formulation of the skin ee t, taking into a

ount the variation along the x- and z -axis of the magneti eld. It is summarized below. A good ondu tor is hara terized by the ondition

 >> !"0

(5.6)

Under this assumption, Maxwell's equations result in diusion equations in terms of spe tral ele tri eld, magneti eld or urrent density. They have as a general spe tral solution

x~(kx ; y) = x~(kx ; 0) e Ky

p where K = j!  0

(5.7)

x = e; h; j for the ele tri eld, magneti eld, and urrent density, respe tively.

218

CHAPTER 5.

APPLICATIONS

By virtue of the perturbational approa h, h~(kx ; 0) is equal to the magneti eld at the ondu ting interfa e assuming losslessness. Hen e it an be al ulated as shown in Chapter 4. The surfa e impedan e Zm of the ondu tor is related to the tangential ele tri eld at the surfa e of the ondu tor to the tangential magneti eld as: ~e(kx ; 0) = Zm h~ (kx ; 0) where

r

Zm =

(5.8a)

j!0 

(5.8b)

Using Parseval's relations (Appendix C), the power dissipated on the two sides of any ondu tive layer of thi kness t is obtained as a fun tion of the spe tral elds as

Pdiss = e 2z = e 2z

Ii =

Z

+

1 0 +1 Z t

Z

1

jZm j2

= with

1Z t

Z

1Z t

+

j

0

~e(kx ; y )j~ (kx ; y ) dy dkx


IN 1 + IM 1 e 2z

2

1

0

~e(x; y )j~ (x; y ) dy dx



h~xi (kx ; 0)j2 + jh~zi (kx ; 0)j2 e

K )y dy dk x

2 Re(

(5.9)

where the subs ript i = N 1; M 1 refers to the two layers adja ent to the

ondu tor, a

ording to a previous de nition (Chapter 4, Fig. 4.1). The attenuation onstant due to ondu tor losses is then obtained from (5.5) as

P j  = diss =0 2P0 j=0

(5.10)

It should be noted that (5.10) is equivalent to (5.5) rewritten using the perturbational approa h, that is for Pdiss , P and P0 al ulated for  = 0 using (5.9), (5.2a) and (5.4) respe tively. An important omment has to be made about the shape of the eld used for the skin-depth ee t. This eld is al ulated not only without losses but also for zero-thi kness ondu tors: it is not the a tual eld for the lossless nite thi kness ase. However, the error made on the ondu tor losses due to a nite thi kness of ondu ting layer is assumed to be small. Some authors have studied theoreti ally [5.5℄-[5.7℄ and experimentally [5.8℄-[5.10℄ the limitations

5.1.

TRANSMISSION LINES

219

of existing modeling te hniques when a nite thi kness is onsidered. In fa t, as mentioned by Itoh [5.11℄, the simpli ity of the spe tral domain te hnique related to a Green's formalism fails when the zero-thi kness assumption is not valid. For these reasons, the zero-thi kness assumption is usually maintained but orre ted by a perturbational al ulation of ondu tor losses, based on a skin-ee t formulation whi h involves the thi kness of the ondu tor. Hen e the total attenuation oeÆ ient is the sum of two terms; s due to the substrate, obtained by the variational pro edure, and  due to the

ondu tors, obtained by a skin-depth perturbational formulation. 5.1.1.4

Chara teristi impedan e

When the guiding stru ture supports a pure TEM propagation mode, a unique

orresponden e is found between the urrent and voltage of the equivalent line and the physi al magneti and ele tri elds existing in the stru ture [5.2℄ and [5.12℄. In the ase of TEM lines, the hara teristi impedan e is equal to the wave impedan e, whi h depends only on the onstitutive parameters of the homogeneous medium surrounding the ondu tors of the TEM line. The de nition of a hara teristi impedan e for a multilayered guiding stru ture (Fig. 4.1), supporting TE or TM modes is not unique, sin e a wave impedan e is de ned for ea h TE or TM mode onsidered and also for ea h of the layers onsidered. We only may use a relation between V+ and Z when the power ow P0 is assumed to be known. Hen e we have to relate the ele tromagneti elds in (5.3) to the voltage or urrent waves of the equivalent transmission line. Using (5.2b) we relate V+ to either a urrent or a voltage, as:

V2 Z P V = 0 2P0 2P Z P I = 20 I0

A.

Power-Voltage de nition

(5.11)

Power-Current de nition

(5.12)

Chara teristi impedan e for slot-like lines

As many authors [5.13℄-[5.15℄, we hoose for the hara teristi impedan e of the dominant mode of slot-like lines the de nition (5.11). It is easy to relate the transverse omponent of the trial quantity de ned in Chapter 4 for slot-like lines to a voltage on ept, by integration over the nite area of the slot. The normalization fa tor for the trial transverse ele tri eld a ross the slot is hosen su h that its integral a ross the slot is equal to the voltage V0 arbitrarily imposed a ross the slot. As a onsequen e, the spe tral elds and hen e the spe tral quantities in (4.19a-e), are proportional to V0 . When introdu ed into (5.4), the power ow P0 be omes proportional to the squared power of V0 , whi h renders expression (5.11) for the impedan e independent of V0 . This illustrates another advantage of the method presented here: no additional omputation is ne essary to al ulate the hara teristi impedan e of

220

CHAPTER 5.

APPLICATIONS

the stru ture, be ause the power ow and hen e the impedan e, is dire tly expressed as a fun tion of the oeÆ ients already used in the al ulation of the propagation onstant. B.

Chara teristi impedan e for strip-like lines

The longitudinal omponent of the urrent density de ned as the trial quantity for strip-like problems is related to the total urrent 0 owing on the strip, using de nition (5.12). This urrent is simply obtained as the integration of the longitudinal urrent density over the strip area. Some authors [5.16℄, however, propose another de nition of the hara teristi impedan e, based on a power-voltage de nition, where the voltage is obtained from the line integration of the ele tri eld between the enter of the strip and the ground plane. Jansen and Kirs hning have pointed out in [5.17℄ that the power urrent de nition has various theoreti al advantages, be ause it is related to a urrent, hen e to a magneti eld, whi h is not responsible for the dispersion due to the diele tri interfa e. As a onsequen e, the behavior of a al ulation based on a power- urrent de nition will be less frequen y-dependent than that with a power-voltage de nition, as observed from results presented in [5.16℄. Hen e, the power- urrent de nition of the hara teristi impedan e will better agree with the pure TEM de nition of . I

Z

C. Other de nitions for the hara teristi impedan e

It is important to note that the problem of a orre t de nition of hara teristi impedan e remains unsolved, and is still the subje t of dis ussions in the literature. Williams, for example, attempts to present a uniform de nition for planar lines, based on ausality onsiderations [5.18℄. Also, measuring this parameter is parti ularly diÆ ult [5.19℄. It requires spe i alibration te hniques, su h as those developed by the National Institute of Standards and Te hnology (NIST) [5.20℄. On the other hand, to the best of our knowledge, it is not proven that expression (5.4) ombined with either de nition (5.11) or (5.12) yields a variational expression for the hara teristi impedan e and hen e that the resulting value of is orre t to the se ond-order. We will nevertheless illustrate in this hapter that de nitions (5.11) (5.12) are very well validated by experiments arried out on a number of on gurations, be ause they are al ulated using the solution of variational prin iple (4.17a) and (4.21) for the propagation onstant involved in their expressions [5.21℄. Z

5.1.2 Mi rostrip

The validity of the variational approa h (4.21) is rst tested on a quasi-TEM mi rostrip line, et hed on a diele tri substrate with a omplex permittivity 2 36(1 0038). The line width is 1.5 mm and has been designed to present a 50 hara teristi impedan e on the substrate onsidered in this example . Figure 5.1a shows the losses per unit length, obtained as the real part of the :

j:

5.1.

221

TRANSMISSION LINES

omplex propagation onstant , while Figure 5.1b shows the real part of the

omplex ee tive diele tri onstant, obtained from the imaginary part of

as  2 (5.13) "eff = 0 ! Solid lines are for parameters extra ted from measurements, while dashed ones show values of parameters al ulated using the variational prin iple (4.21). 0

2.2 −0.02

2.1

−0.04

2 dB/mm

0

var. meas.

20 Frequency in GHz

(a) 40

0

20 Frequency in GHz

(b) 40

Transmission parameters of mi rostrip line et hed on a substrate plate of thi kness 0.508 mm and relative permittivity 2:36(1 j 0:0038) (a) losses; (b) ee tive diele tri onstant; solid lines are for measurements, dashed ones for variational prin iple (4.21)

Fig. 5.1

A good agreement is obtained for both losses and ee tive diele tri onstant. The measurement method is a

urate and will be des ribed in Se tion 5.5. 5.1.3

Slot-line

Next, the validation is extended to a slot-line, whose dominant mode is of TEtype. The line onsidered is et hed on the same substrate as for the mi rostrip in Figure 5.1, but its width is designed to be 0.37 mm in order to obtain a good mat hing with the slot-to-mi rostrip transitions used for the measurement. Su h transitions will be des ribed in Se tion 5.2. After measurements and alibration, the ee ts of the mi rostrip transition to the slot-line are removed, and the experimental values for the transmission parameters presented in Figure 5.2 are only those of the slot-line. Again, a good agreement is observed for both losses and ee tive diele tri onstant. This illustrates that our formulation is eÆ ient for dynami (non quasi-TEM) situations. It has to be noted, however, that the measuring frequen y range is limited by the bandwidth of the mi rostrip-to-slot-line transition (2-25 GHz). The outof-band mismat h is too high and the alibration pro edure an no longer

ompensate for the transition.

222

CHAPTER 5.

0

1.6

−0.02

var meas.

1.4

−0.04 −0.06

APPLICATIONS

1.2 db/mm 5 10 15 20 Frequency in GHz

(a) 25

1

5

10 15 20 Frequency in GHz

(b) 25

Fig. 5.2 Transmission parameters of slot-line et hed on a substrate plate of thi kness 0.508 mm and relative permittivity 2:36(1 j 0:0038) (a) losses; (b) ee tive diele tri onstant; solid lines are for measurements, dashed ones for variational prin iple (4.21)

5.1.4

Coplanar waveguide

As another example, Figure 5.3 ompares the measured and al ulated terms of the s attering matrix of a oplanar waveguide impedan e step. The simulation for this stru ture involves the attenuation oeÆ ient, the propagation onstant, and the hara teristi impedan e of the lines. Only the dominant modes on the two lines are onsidered. The hara teristi impedan e of the a

ess line is taken as referen e impedan e for the s attering matrix of the jun tion, as the measurement te hnique uses a de-embedding algorithm whi h moves the referen e planes for jun tions between the two lines of dierent width. It an be seen in Fig. (5.2a,b) that a good agreement is obtained for both the re e tion and transmission parameters. Hen e, al ulated impedan e agrees with measurement. Measured and omputed phases also agree very well, espe ially in the ase of transmission oeÆ ient (Fig. 5.3d). This validates the al ulated propagation onstant. The al ulated diele tri and ondu tor losses also agree with the measured ones: al ulated and measured transmission urves are indeed in oin iden e when the ele tri length of the step line is a multiple of a half wavelength. The degradation of the measured s attering terms around 18 GHz for the oplanar line is attributed to surfa e wave losses due to a mismat h of the oaxial-to- oplanar laun her used for de-embedding. 5.1.5

Finline

Finally, a validation is performed for a boxed (or shielded) line, namely a nline. It an be viewed as a slot-line on a diele tri substrate, surrounded by a waveguide en losure. Hen e, the variational prin iple (4.21) an be used, but the spe tral integrals yielding the oeÆ ients of its se ond-order equation have to be repla ed by dis rete summations over spe tral variable kx . The slot-line inside the waveguide is et hed on a substrate having a thi kness and permit-

5.1.

223

TRANSMISSION LINES

−20

0

−40

−1

−60

−2

−80 0

dB

(a)

10 GHz

20

−3 0

degrees

dB

(b)

10 GHz

20

degrees

100

100

0

0

−100

−100 (c)

0

10 GHz

20

(d)

0

10 GHz

20

Fig. 5.3 Modeled (- -) and measured (|) s attering terms of oplanar waveguide step (a) magnitude of re e tion oeÆ ient; (b) magnitude of transmission oeÆ ient; ( ) phase of re e tion oeÆ ient; (d) phase of transmission

oeÆ ient

tivity dierent from Figure 5.2; its width is designed to be 0.45 mm in order to obtain a good mat hing with the slot-to-waveguide transitions used for the measurement. The parti ular measurement method is des ribed in Se tion 4.5 [5.22℄. Figure 5.4 shows results obtained for the ee tive diele tri onstant. Again, a very good agreement is observed between the experimental values (lines) and the al ulations using the variational prin iple (4.21) ( ir les). 5.1.6

Planar lines on lossy semi ondu tor substrates

Planar transmission lines are frequently used as inter onne ts on multilayered low-resistivity substrates in high-frequen y MMICs, su h as bulk sili on wafers, doped areas in gallium arsenide (GaAs), or photo-illuminated layers in optoele troni s. These substrates are modeled by using an equivalent loss tangent, obtained from their resistivity : j!"

= j!"0 "r + 1= 

= j!"0 "r 1 = j!"0 "r 1

j !"0 "r

(5.14a)



j tan Æequ

(5.14b)


(5.14 )

Figure 4.22 ompares the transmission line parameters obtained from the variational spe tral forms (4.21), (5.4) and (5.11), with measured transmission line parameters, for the same oplanar waveguide as in Figures 4.15 and

224

CHAPTER 5.

APPLICATIONS

1.2 1.15

var. meas.

1.1 1.05 1 0.95

30 35 Frequency in GHz

40

Fig. 5.4 Ee tive diele tri onstant of nline et hed on a substrate plate of thi kness 0.272 mm and relative permittivity 2:22(1 j 0:0018), waveguide se tion 2a2 with a = 3:556 mm; solid lines are for measurements, symbols o for the variational prin iple (4.21)

(a)

(b) 55

2000

45

β [rad/m]

Re(Zc) [Ω]

2500

1500

35

1000

25

500 0 0

10 20 30 Frequency [GHz]

40

35

15 0

10 20 30 Frequency [GHz]

40

10 20 30 Frequency [GHz]

40

20

α [Np/m]

Im(Zc) [Ω]

15

25

10

15 5 0

10 20 30 Frequency [GHz]

( )

40

5 0 0

(d)

Results measured (solid), simulated with (VP) (- -), for transmission line parameters of mi rostrip line on low-resistivity sili on substrate of resistivity 225

m (a) propagation onstant; (b) real part of impedan e; ( ) attenuation oeÆ ient; (d) imaginary part of impedan e

Fig. 5.5

5.1.

225

TRANSMISSION LINES

3

dB/mm

(a)

4000

m−1

(b)

3500 2.5 3000 2 50 55

60 GHz

2500 50

70

Ohms

(c)

50

60 GHz

70

(d)

6.2 6 5.8

45

5.6 40 50

60 GHz

70

50

60 GHz

70

Results measured (solid), simulated with (VP) (- -), for transmission line parameters of oplanar waveguide on low-resistivity SOI substrate of resistivity 20

m (a) attenuation oeÆ ient; (b) propagation onstant; ( ) real part of impedan e; (d) ee tive diele tri onstant

Fig. 5.6

4.21. The VP transmission line parameters (- - -) perfe tly agree with experimental results (|). It should be observed that propagation on high-loss substrates is highly-dispersive in the low frequen y range. A slow-wave mode of propagation o

urs, typi al of integrated ir uits (ICs) on low resistivity substrates. The variational prin iple (4.21) yields a similar agreement in the ase of stripline topologies on low-resistivity substrates. As an example, Figure 5.5 shows the transmission line parameters of a mi rostrip line on a lossy sili on substrate, using (4.21). Again, perfe t agreement o

urs between the VP transmission line parameters (- -) and measured ones (|) [5.23℄. At higher frequen ies, substrate losses have no ee t on dispersion, but the variational modeling still holds, as shown in Figure 5.6. Values of attenuation, ee tive diele tri onstant and hara teristi impedan e extra ted from measurements agree with simulated ones (Fig. 5.6a, ,d) in the millimeter wave frequen y range (50-75 GHz). The peaks observed in the measurement around 70 GHz are due to a problem during alibration and not to the devi e. 5.1.7

Planar lines on magneti nanostru tured substrates

Variational formulas established for perturbation problems are parti ularly eÆ ient for modeling the behavior of magneti ally-nanostru tured planar ir uits and devi es over a wide frequen y range, as shown in Figure 5.7. The devi e onsists of an array of magneti metalli nanowires embedded in a

226

CHAPTER 5.

APPLICATIONS

Fig. 5.7 Mi rostrip line on nanostru tured substrate, in luding nanowires (Reprodu ed from Pro . PIERS'2000 [5.25℄)

porous diele tri substrate of thi kness H [5.24℄[5.25℄. The metalli wires are perpendi ular to the diele tri plate, all with diameter D. Typi al values of D are in the nanometer range, between 50 and 500 nm. This substrate was rst developed for mi rostrip appli ations: a mi rostrip is deposited on the top side of the omposite substrate, while the bottom is metallized to serve as ground plane. The magneti wire is made of obalt, ni kel or iron. The experimental results presented in Figure 5.8 (solid lines) show that the devi e has interesting tuning properties at mi rowave frequen ies. Transmission measurements between the input and output of mi rostrip exhibit a stopband behavior in a ertain frequen y range, whi h is shifted to higher frequen ies when a DC-magneti eld is applied parallel to the ni kel nanowires (Fig. 5.8a, ). A similar shifting behavior is obtained with the other ferromagneti metals onsidered ( obalt and iron), but at xed or zero DC- eld the

enter frequen y of stopband diers with the kind of wire material. We have su

essfully related these experimental relationships to ferrimagneti theory [5.24℄[5.25℄. This theory predi ts a resonan e behavior at mi rowaves for ferrimagneti materials, tunable when applying a DC-magneti eld. Hen e, ferromagneti nanowires a t like thin ferrite implants in the diele tri substrate, and an be treated by variational prin iples parti ularized for material perturbation problems. In Chapter 2, a formulation orre t to se ond-order is obtained for the shift indu ed on the propagation onstant, in the ase of material perturbation: i.e.

Z

j ( 1

) 

j! fe
(
"
e) + h
(

h)g dS S Z Z  h
(az  e) dS e
(az  h) dS S

with
" = "1
 = 1

" 



S

(2.94b)

5.1.

227

TRANSMISSION LINES

0

dB

−10 −20 −30 0

5

10

15 20 25 Frequency in GHz

30

35

40

5

10

15 20 25 Frequency in GHz

30

35

40

4 radians

2 0 −2 −4 0

(a) 0

dB

−10 −20 −30 −40 −50 0

5

10

15 20 25 Frequency in GHz

30

35

40

5

10

15 20 25 Frequency in GHz

30

35

40

4 radians

2 0 −2 −4 0

(b)

Fig. 5.8 Transmission model (2.94a) (dashed line) and experiment(solid line) on mi rostrip on nanostru tured substrate (D = 500 nm, H = 20 m), with

obalt nanowires; magnitude [dB℄ and phase [radians℄ for (a) Hd = 350 Oe; (b) Hd = 3650 Oe

Based on the ferromagneti resonan e hypothesis, a full analyti model is derived using the variational prin iple (2.94a,b). We take as trial elds the ele tri and magneti elds between the two ondu tors of a parallel plate transmission line, assuming that a uniform urrent density is owing on ea h

ondu tor. This assumption is valid for a wide mi rostrip line, that is for geometries with high W=H ratios (typi al values for Fig. 5.7: W=H = 20). The permeability tensor 1 is taken equal to the gyrotropi tensor modeling the ferromagneti resonan e ee t, as introdu ed in Appendix D. Other

228

CHAPTER 5.

APPLICATIONS

permeabilities and permittivities are s alar. Assuming that the propagation

onstant in the absen e of nanowires is al ulated exa tly, formula (2.94a) yields the propagation onstant of the mi rostrip line represented in Figure 5.7. The dashed urves in Figure 5.8 are al ulated with the resulting model, obtained by omputing the transmission fa tor e L where L is the length of the mi rostrip. An ex ellent agreement with experiment is observed, for both magnitude and phase and for ea h value of magneti eld onsidered. 5.2

Jun tions

5.2.1

Mi rostrip-to-slot-line transition

Simple transitions between slot lines and measuring equipment are ne essary for testing and designing slot-line ir uits. The rst attempt to measure slotline hara teristi s was made by Cohn [5.26℄, followed by Mariani et al. [5.27℄ and Robinson and Allen [5.28℄. They used a dire t transition between a oaxial able and the slot-line. The drawba ks of this transition are the poor agreement between modeling and experiment, and the diÆ ulty of manufa turing repeatable transitions. The transition has, however, been used by Lee [5.29℄ to he k the validity of his slot-line impedan e model. The mi rostrip-to-slot-line transition shown in Figure 5.9 oers an elegant way to measure and hara terize slot-like devi es. The frequen y range over whi h this hara terization is eÆ ient is related to the bandwidth of the mi rostrip-to-slot-line transition. Thus we need an a

urate model for this transition, in order to optimize its design for wideband appli ations. This transition was introdu ed by Cohn [5.26℄ and used by Mariani et al. [5.27℄, and Robinson and Allen [5.28℄. It has been used for the propagation onstant measurements presented in the previous se tion. We model the transition as two transmission lines ended by quarterwavelength stubs (Fig. 5.9a) and inter onne ted via a oupling network (Fig. 5.9b). The network represents the oupling area shown in Figure 5.9 . It

onsists of: 1. a transformer of ratio [5.5℄,[5.26℄, where n is a fun tion of frequen y: it depends on the on guration of the magneti elds around the strip and slot, whose shape is a fun tion of wavelength and hen e of frequen y (Fig. 5.9 ) 2. two fringing apa itan es C1 modeling the intera tion of the ele tri eld between the strip and slot in the oupling area 3. an equivalent apa itan e C2 des ribing the ele tri eld a ross the slot in

oupling area 4. a series indu tan e L modeling the strip rossing the slot in oupling area. 5.2.1.1

Indu tan e

The value of L is taken equal to the internal indu tan e of a wire of length equal to the width Ws of the slot in the oupling area: 7 L = 10 2 Ws

[H℄

5.2.

229

JUNCTIONS

λs/ 4

(a)

λm / 4

coaxial connector

ground plane

slotlike-circuit port

microstrip line

λm / 4

L

microstrip port 1

C1 I’1

open ended microstrip stub

C1 V’1

(b)

n:1

I’2

V’2

short-circuited slotline stub

C2

λs/ 4

slotline port 2

propagation along strip

p slo rop t aga

tio

n

al

on

g

(c)

electric field

magnetic field

Mi rostrip-to-slot-line transition (a) topology; (b) equivalent ir uit for modeling oupling area; ( ) oupling area

Fig. 5.9

5.2.1.2

Capa itan es

5.2.1.3

The ratio

C1

and

C2

We dedu e the equivalent apa itan es C1 and C2 from the line parameters of the mode propagating along the pie e of ondu tor-ba ked slot-line (Fig. 5.10). It is obvious that the ele tri eld behavior in this stru ture indu es a apa itive ee t.

n of the transformer

Knorr [5.30℄ proposes the following formulas for the transformer ratio n:

V1 = nV2 0

0

(5.15a)

230

CHAPTER 5.

APPLICATIONS

layer 1

az

ε o, µ o , σ

ax

layer 2

ay conducting planes (conductivity σ) dielectric or gyrotropic lossy substrates electric field

Fig. 5.10 Condu tor-ba ked slot-line

I2 = nI1 0

(5.15b)

0

where

n=

1

os()

=

2Huf 3

sin  tan q

q =  + ar tan p

u

v = "eff s 1 p u = "r "eff s

v

(5.15 ) (5.15d) (5.15e) (5.15f) (5.15g)

Sin e the slot-line is ondu tor-ba ked in the oupling area, we improve Knorr's formula by introdu ing into v and u the ee tive diele tri onstant of the

ondu tor-ba ked slot-line of Figure 5.10 omputed using variational prin iple (4.21) instead of the value orresponding to a simple slot-line. 5.2.1.4

VSWR of jun tion

The input impedan es of the quarter-wavelength stubs are

Zs = Z s tanh( s Ls )

(5.16a)

for the slot stub, where Ls is the ee tive length of the stub, taking end ee ts into a

ount, and s the propagation onstant of slot, and

Zm = Z m oth( m Lm )

(5.16b)

for the mi rostrip stub, where Lm is the ee tive length of the stub, taking end ee ts into a

ount, and m is the propagation onstant of mi rostrip. Zs and

5.2.

231

JUNCTIONS

0

0 measurement Knorr

our modelling measurement

−20

−20 dB

−10

dB

−10

−30

−30

−40

−40 (a)

−50

5

10 15 Frequency in GHz

(b) −50

5

10 15 Frequency in GHz

Chara terization of the mi rostrip-to-slot-line transition (a) omparison between Knorr's model and measured re e tion oeÆ ient; (b) omparison between ondu tor-ba ked slot-line model and measured re e tion oeÆ ient Fig. 5.11

Zm are al ulated using (5.11) and (5.12) for the hara teristi impedan es and variational prin iple (4.21) for the omplex propagation onstants. Using the intermediate variables 1 1 (5.17a) Y = j!C2 + + Zs Z s A1 = 1

2

!2 LC1

1 nA1 A2 = Y A1 1 n + nj!L 2n 2Y )A + j!C1 2 j!C1 the VSWR of the equivalent ir uit of Figure 5.9b is de ned as A3 = Zm + (1 n +

VSWR =

1+ 1

1 1

(5.17b) (5.17 ) (5.17d)

(5.18a)

where 1

=

A3 Z m A3 + Z m

(5.18b)

232

CHAPTER 5.

APPLICATIONS

I

4

1

IV

6

5

2

3 III

II

metallization of uniplanar line metallization of microstrip line

Fig. 5.12 Topology of planar hybrid T-jun tion

5.2.1.5

Results

Figure 5.11a ompares our measurement of the re e tion oeÆ ient of a mi rostrip-to-slot-line jun tion and Knorr's model evaluated with only a transformer ratio. It is observed that the bandwidth of the measured and simulated re e tion oeÆ ients do not mat h well. A signi ant improvement is obtained when introdu ing the modi ed value for the transformer ratio, apa itan es C1 and C2 and indu tan e L, as observed at Figure 5.11b. In parti ular, the lo ation of the zeros and the magnitude of the re e tion oeÆ ient in the bandpass agree well. The observed improvement is due to a

urate modeling of the ee tive diele tri onstant formula (5.15a-g). 5.2.2

Planar hybrid T-jun tion

Figure 5.12 shows a planar hybrid T-jun tion. It has the following phase-shift properties: 1. When port IV is fed, the waves at ports I and II are in phase. 2. When port III is fed, the waves at ports I and II are out-of-phase. The stru ture of Figure 5.12 has been designed at 25 GHz by using the frequen y-dependent models developed for: a. propagation oeÆ ient and impedan e of single (1) and oupled (3) slotlines, in luding diele tri and ohmi losses b. uniplanar jun tions (4)(5)

. mi rostrip-to- oupled slot-line jun tion (2). All jun tions were optimized to be as wide-band as possible. Figure 5.13 shows the phases measured at the dierent ports. A good behavior is observed, sin e paths I-IV and II-IV are in phase (Fig. 5.13a,b), while paths I-III and II-III are out-of-phase (Fig. 5.13 ). This illustrates that variational models are a

urate, espe ially for the line parameters and the slot-line-to-mi rostrip transition.

5.3.

233

GYROTROPIC DEVICES

0.040

GHz

40.00 (a)

(b)

15.04

GHz

30.00

( )

Fig. 5.13 Phase measurements on planar hybrid T-jun tion (a) phase of path

I-IV; (b) phase of path II-IV; ( ) phases of paths I-III and II-III

5.3

5.3.1

Gyrotropi devi es

State-of-the-art

Be ause of the in reasing development of wideband ommuni ation systems, there is a growing need for tunable omponents ompatible with planar integrated on gurations. Su h elements are used as phase-shifters or tunable resonators and lters in os illators and frequen y synthesizers, for on-board and automotive systems operating at mi rowave and millimetre-wave frequen ies. For this reason, resear h involving planar lines on anisotropi or gyrotropi substrates for the 20 past years has on entrated on two areas: the rst is modeling and designing planar lines on magneti substrates that may serve as ferrite phase shifters or planar ir ulators. The se ond is designing magnetostati wave (MSW, Appendix D) devi es, in luding YIG- lms operating as frequen y-tunable devi es in frequen y synthesizers, hannel- lters, delay lines, and tuned os illators. In the rst ategory, the DC-biasing magneti eld is usually kept onstant, while tuned devi es use the variation of DC- eld

234

CHAPTER 5.

APPLICATIONS

to modify the operating frequen y. 5.3.1.1

Planar lines on magneti and anisotropi substrates

Various on gurations have been investigated for planar phase shifters. In all

ases, the propagation onstant has to be al ulated. Wen [5.31℄ proposed using a oplanar waveguide supporting a pie e of YIG- lm and experimentally tested the phase-shift properties of the stru ture. No attempt to al ulate this phase-shift was made. The same year, Robinson and Allen [5.28℄ experimentally tested a slot-line phase shifter in the same on guration as Wen. A number of quasi-TEM modeling methods have been developed, based on variational formulas for indu tan e or apa itan e per unit length. Masse and Pu el [5.32℄[5.33℄ de ned an equivalent lling fa tor for the ee tive permeability of the line and al ulated the apa itan e per unit length of a mi rostrip on a magneti substrate to dedu e its dispersion and phase behavior. Other on gurations supporting quasi-TEM modes on magneti substrates were investigated with a similar approa h by Kitazawa [5.34℄ and Horno et al. [5.35℄. On the other hand, lines supporting non-TEM modes are usually analyzed by the moment method, and the impli it Galerkin pro edure is used to nd the dispersion of the gyrotropi layer. This was done by Ja kson for slot-lines and

oplanar waveguides [5.36℄, and by Mesa et al. [5.37℄ for oplanar multistrip lines. Usually, the magnetostati range is never taken into a

ount when

al ulating propagation onstants, and the losses of the layer are negle ted [5.38℄. 5.3.1.2

YIG-tuned planar MSW devi es

For many years, gyrotropi passive planar omponents have been used as wide-band tunable resonators or phase-shifters for spa e quali ed YIG-tuned os illators. They ompare advantageously with YIG-sphere resonators and lters, be ause of their planar geometry and two-dimensional oupling me hanism, yielding a redu tion of mass and size. The desired ee t results from a judi ious ombination of planar transmission lines with planar gyrotropi YIG-ferrite lms whi h operate in their magnetostati wave frequen y range [5.39℄. Various on gurations are summarized in Figure 5.14. They all onsist of planar YIG-ferrite layers ut as resonators and oupled to planar transdu ers et hed on a diele tri substrate. The magnet poles generate a DC-magneti eld. Taking advantage of the ferrite nature of the lm, the resonant frequen y of the stru ture is varied by hanging the applied DC-magneti eld. The most signi ant part of the theoreti al work performed on planar MSW devi es deals with MSW delay-lines (Fig. 5.14f). It onsists of a planar YIG- lm epitaxially grown on a Gadolinium-Gallium-Garnet (GGG) rystal, oupled to mi rostrip transdu ers. The insertion loss and delay between ports 1 and 2 are varied with frequen y by hanging the applied DC-magneti eld. The model is based on the following assumptions:

5.3.

235

GYROTROPIC DEVICES

Upper pole H1

A

1L

H

A

3L

Left area

az H2

W

ay

A 2L

Right area

ax

.

A

1R

A

3R

layer 3

A

2R

layer 2

layer 1

microstrip dielectric substrate Lower pole

az

ay

ay

YIG Port 1

L

L

Wm

ay az

1'

1 Port 2

ay

ay

az

ax L

az ax YIG

ax

Port 1

(a)

ax

az

(b)

ax (d)

(c)

port 2

port 1 Hdc ay

ay az

D

ax

W

az ax

S

GGG substrate YIG

Ws 1'

1

(e)

(f) MSW delay line planar dielectric layer metallic conductor RF magnetic field

YIG film magnet poles

Con gurations for YIG-tuned devi es (a) geometry of YIG- lm for MSFVW; (b) one-port under oupled YIG-to-mi rostrip on guration; ( ) twoport under oupled YIG-to-mi rostrip on guration; (d) one-port over oupled YIG-to-mi rostrip on guration; (e) MSFVW two-port under oupled YIG-toslot on guration; (f) MSFVW delay-line

Fig. 5.14

236

CHAPTER 5.

1. 2. 3. 4. 5. 6.

APPLICATIONS

magnetostati approximation non-linear ee ts are negle ted uniformity of elds along the length of ondu tion strips good ondu tors are assumed and thi kness of strips is small nite length (along z -axis) of YIG- lm is not onsidered non-uniformity of DC-magneti eld is not onsidered.

The theoreti al model evaluates the part of power whi h is \radiated" from the mi rostrip into the YIG- lm. The result is an equivalent radiation impedan e loading the mi rostrip transdu er. It is obtained by integrating over the ontinuous spe trum of the various propagating modes along the z -axis in the YIG- lm. As a onsequen e, tedious integrations in the omplex plane using the residue method are ne essary [5.40℄. This te hnique has been applied by a number of authors [5.41℄-[5.44℄. Then, a rough evaluation of the power transmitted between ports 1 and 2 is obtained. To our knowledge, no attempt has been made to model the various YIGresonator on gurations for magnetostati forward volume waves (MSFVW) depi ted in Figure 5.14b-e, used for YIG-tuned os illators. In parti ular, the on gurations for whi h no physi al onta t exists between the planar transdu er and the YIG- lm have never been modeled. Also, the stru tures of Figure 5.14 annot be easily analyzed by onventional variational prin iples [5.45℄ or transmission line analysis te hniques, sin e these methods only are valid provided the present media are either isotropi or lossless. The following se tion illustrates the eÆ ien y of various variational formulations that we have developed for planar multilayered lossy gyrotropi stru tures used for MSW devi es. The approa h departs from the impli it full-wave methods for modeling multilayered magneti planar lines, and from the radiation impedan e on ept, be ause we onsider the resonator as a resonant transmission line whose parameters are al ulated from expli it variational formulations. Hen e, this approa h advantageously ompares with impli it methods sin e on-line results an be obtained with a regular PC in a few se onds. The formulation in ludes any spatial non-uniformity of the non-Hermitian tensor des ribing the gyrotropi layer. It an be used both in the spatial and the spe tral domains. This will be illustrated when al ulating an under oupled MSW-resonator modeled with the spatial domain formulation, and an over oupled MSW-resonator modeled with the spe tral domain formulation. In both ases, trial potentials and elds are derived under the magnetostati assumption (Appendix D, Chapter 4). 5.3.2

Under oupled topologies

The on gurations depi ted in Figure 5.14 an be divided into two lasses: under- and over oupled topologies, depending on the proximity between the YIG sample and the oupling planar transdu er. Con gurations b, ,d involve mi rostrip transdu ers. A s hemati representation of the magneti eld patterns inside the YIG- lm is shown for ea h

5.3.

237

GYROTROPIC DEVICES

on guration in Figure 5.14b-d. These should be ompared with the eld representation in an isolated YIG- lm of nite width (Fig. 5.14a). As we see,

on gurations b and , for whi h the mi rostrip has no onta t with the YIGlayer (side- oupling), exhibit a magneti eld pattern whi h slightly diers from the pattern of an isolated lm. As a onsequen e, the eld on guration in the YIG- lm an be approximated by that of the isolated YIG- lm of Chapter 4. We model the lm as an equivalent transmission line (Fig. 5.15a,b) along the z -axis (Fig. 5.14a), with forward and reverse propagation onstants

omputed from MSW assumption as follows. In Chapter 4, the Perfe t Magneti Wall (PMW) assumption was used to derive the dispersion relationship for MSW inside the planar YIG- lm. We have shown in [5.46℄ that a more suitable des ription, taking into a

ount the non-uniform demagnetizing ee t (Appendix D), is obtained when trial magnetostati potential (4.33) inside the YIG- lm is repla ed by the following:   YIG (x; y; z ) = A sin kx x + B os kx x Y (y)e z for 0  x  W (5.19a) L(x; y; z ) = Le nxY (y)e z for x  0 (5.19b)

x W

z R n (x; y; z ) = Re Y (y )e for x  W (5.19 ) where W is the width of the YIG- lm and Y the y-dependen e (Chapter 4). The x-dependen e of the potential is left unknown in (5.19a- ) in order to allow use of a more rigorous boundary ondition than the PMW- ondition. On the left- and right-sides of the YIG- lm in layer 3 the potential is assumed to exponentially de rease from the edges (5.19b, ). Imposing boundary onditions at planes x = 0 and x = W yields a relationship between kx and , involving values at the edge of the lm be ause of the non-uniform demagnetizing ee t, and an expression for the ratio B=A: 2 n  xx; kx (5.20) tan kx W =

( n ) zx; xx; kx 3

3

+

3

3

+

3

(

)

3

3

0

0

B=A =

2

xx;3 kx

zx;3 + n 0

3

3

2

3

2

(5.21)

Assuming that kx is the unknown in (5.20), the kx -solution is independent of the sign of , that is from the forward or reverse nature of propagation. However, this is not the ase for oeÆ ient B whose value diers with the sign of , as shown by (5.21). Hen e, we have theoreti ally demonstrated that when taking into a

ount an edge ee t related to a non-uniform demagnetizing ee t for a nite-width YIG- lm, it is impossible, with the same pattern of eld, to obtain identi al forward and reverse propagation oeÆ ients for a unique physi al value of kx . This is only possible for isotropi layers, having no zx tensor omponents. Having obtained these dependen ies, the various integrands of oeÆ ients of equation (4.2) are expressed using (4.34a- ) and (4.36a- ) for the trial

238

CHAPTER 5.

YIG

YIG

L

Port 1 e

L

Port 1

-γ L +

e

e

+ γ- L

K coupling 2-ports

(a)

Port 2

-γ L + K2

K1

K

APPLICATIONS

e

+γ-L

K1,K2 coupling 2-ports

(b)

Equivalent ir uits modeling various YIG-devi es topologies (a) one-port under oupled YIG-to-mi rostrip on guration; (b) two-port under oupled YIG-to-mi rostrip on guration

Fig. 5.15

ele tri eld, derived from the magnetostati potential. The x-dependen e of the integrands is the produ t of the x-dependen e of the inverted nonuniform permeability tensor with simple sinusoidal fun tions. This integral is evaluated numeri ally by a simple trapezoidal rule algorithm. CoeÆ ients A, B , C , and E in ea h layer are summed and equation (4.2) is solved for . The nal result onsists of two omplex propagation onstants noted f and r , asso iated respe tively with a forward and reverse propagation dire tion. They are obtained when solving equation (4.2) with the set of values (kx ; B ) from (5.20) (5.21) whi h best mat hes the outside sour e eld on the mi rostrip transdu er. Figure 5.16 ( on guration of Fig. 5.14a) shows the results obtained for

f and r . A signi ant dieren e is observed between the forward (solid) and reverse (dashed) propagation oeÆ ients at the same frequen y. Hen e non-re ipro al ee ts are possible. This is in ontradi tion with the assertions found in literature [5.47℄[5.48℄, where the propagation of magnetostati forward volume waves (MSFVW) is stated to be re ipro al. This is be ause the analysis methods found in literature are based on the PMW-assumption, and do not take into a

ount the edge ee t due to the demagnetizing ee t. In our formulation, we involve the onstitutive parameters xx;3 and zx;3 in

al ulating the kx - onstant. Under oupled resonators are hen e des ribed by an equivalent transmission line modeling the YIG- lm, having dierent forward and reverse propagation onstants, and oupled to mi rostrip-transdu ers by oupling networks K (Fig. 5.15). The z -dependen e of the magnetostati magneti elds inside

5.3.

239

GYROTROPIC DEVICES

8000

−1

m 7000 6000 5000 4000

forward reverse

3000 2000 1000

6

6.1

6.2

Frequency in GHz

6.3

6.4

Fig. 5.16 Forward (solid line) and reverse (dashed line) propagation onstants in YIG- lm al ulated with variational prin iple using non-uniform internal DC- eld and rigorous boundary onditions at edge of lm

the YIG- lm thus has the general form: Zx;y (z ) = Vf e f z Vr e+ r z for x- and y - omponents

z + r z f Zz (z ) = f Vf e + r Vr e for z - omponent

5.3.3

(5.22a) (5.22b)

One-port under oupled topology

For a one-port on guration, the al ulation of the input impedan e is suÆ ient. We onsider port 1 (Fig. 5.14b). From Poynting's theorem, the one-port equivalent impedan e Zr1 of the resonator is related to the energy ontained in volume V [5.49℄:  1  1 j V1 j2 = 2j! 2 Zr1

Z V

B
H



E
" E




dV

(5.23)

The elds in (5.23) are derived from the magnetostati assumptions (Chapter 4), but with a z -dependen e (5.22a,b) and transverse dependen ies (5.19a ) for the magnetostati potential in the YIG- lm. The boundary onditions at z = 0 and z = L are approximated by a PMW, together with an exponentially-de aying Bz2 eld in the left and right areas outside the YIG-sample, yielding Vr = e( f + r )L Vf

(5.24)

240

CHAPTER 5.

0

−90

dB

degrees

−2 −4 −6 11.4

APPLICATIONS

−100 −110 −120

(a) 11.5 11.6 Frequency in GHz

−130 11.4

(b) 11.5 11.6 Frequency in GHz

Comparison between modeled (dashed) and measured (solid) re e tion oeÆ ient of one-port under oupled mi rostrip-to-YIG on guration (a) magnitude; (b) phase

Fig. 5.17

and the voltage indu ed on the mi rostrip line Z WZ 0 BzL2 (x; y; S + L + Wmi =2) dy dx V1 = j! 0

H2

(5.25)

where Wmi is the width of mi rostrip transdu er and S the spa er between transdu er and YIG- lm. Equation (5.24) ombined with eld des riptions (5.19a- ) to (5.22a,b) yield ele tri trial elds under magnetostati assumption, with a z -dependen e derived from (5.22a,b). These trials are introdu ed in expression (5.23) for the input impedan e. The modeled re e tion oeÆ ient (dashed) is ompared with the measured one (solid) in Figure 5.17. Model and experiment agree very well. This demonstrates the eÆ ien y of the formulation for lossy gyrotropi media. As expe ted, the on guration is under oupled, as an be seen from the phase of re e tion oeÆ ient (Fig. 5.17b): the phase variation does not ex eed 180 degrees in the resonant range. 5.3.4

Two-port under oupled topology

The s attering matrix of the two-port on guration is obtained from its impedan e matrix. We de ne two impedan e matri es Z r and Z m asso iated respe tively with the YIG-resonator and the oupled mi rostrip transdu ers (without YIG-sample). Terms Z11r and Z22r have already been al ulated:

Z11r = Zr1

(5.26a)

Z22r = Zr2

(5.26b)

The two mutual terms are dedu ed from the self-impedan es as follows. Looking at port 1 (Fig. 5.14 ), the total ratio V2 =V1 derived from (5.22b) and (5.24) must preserve the phase delay from one port to the other, so that the mean

5.3.

GYROTROPIC DEVICES

241

delay fa tor Im 0:5( f + r )L is added:

e f L + e r L V2 = e0:5 Im( f + r )L f f L r r L V1

f e + r e

(5.27)

From ratio V2 =V1 we easily dedu e the oupled terms of the impedan e matrix of the resonator: Z21r =

V2 Zr1 V1

(5.28a)

Z12r =

Zr1 V2 =V1

(5.28b)

The impedan e matrix of the YIG-mi rostrip transdu er on guration is the sum of the two impedan e matri es Z = Zr + Zm

(5.29)

where Z m is the impedan e of the oupled mi rostrip transdu ers without the YIG- lm. De ning a referen e impedan e Z R , the s attering matrix S is derived from the impedan e matrix Z . We validate the model by designing a two-port mi rostrip-to-YIG MSFVW Stray-Edge Resonator (SER) (Fig. 5.14 ). The phases of the re e tion terms, not shown here, exhibit an under oupled behavior similar to Figure 5.17b. Hen e, the modal propagation onstants are al ulated without taking into a

ount the vi inity ee t of the transdu ers, be ause of the weak

oupling. Due to demagnetizing ee ts, the internal eld is not uniform along the width of the sample, and the permeability tensor is not uniform over the

ross-se tion. As a onsequen e, the two forward and reverse solutions omputed for the propagation onstant are appli able. The measured and modeled magnitudes of both the reverse and forward transmission terms show a good agreement (Fig. 5.18a,b). However, Figure 5.18 ,d highlight a signi ant nonre ipro al ee t: the modeled phase (dashed line) of the reverse and forward transmission terms are markedly dierent. This disparity has also been on rmed experimentally (solid lines) [5.46℄,[5.50℄[5.51℄. It should be noted that the designs of tunable os illators using MSFVW SERs found in the literature use one-port on gurations [5.52℄[5.53℄. 5.3.5

One-port over oupled topology

A large strip lo ated between the YIG-sample and the diele tri layer strongly modi es the pattern of the magneti eld, sin e the urrent owing on the mi rostrip for es the magneti eld to surround the strip. Hen e, the magnetostati elds in the resonator have to be al ulated in the presen e of the

ondu ting strip. We expe t the resonator in on guration Figure 5.14d to be over oupled.

242

CHAPTER 5.

0

APPLICATIONS

0

(a)

dB

−10

dB

−10

(b)

−20 −30

−20

6 6.2 Frequency in GHz

−30

6.4

6 6.2 Frequency in GHz

(c)

(d) 100 degrees

100 degrees

6.4

0 −100

0 −100

6 6.2 Frequency in GHz

6.4

6 6.2 Frequency in GHz

6.4

Fig. 5.18 Comparison between modeled (dashed) and measured (solid) transmission s attering terms of two-port under oupled mi rostrip-to-YIG on guration at 6 GHz (a) magnitude of reverse transmission; (b) magnitude of forward transmission; ( ) phase of reverse transmission; (d) phase of forward transmission

Two models have been tested for the one-port over oupled topology. The rst is the transmission-line approa h depi ted in Figure 5.19, showing a top view of the topology (a), its equivalent ir uit (b) and the ross-se tion of its equivalent transmisson line ( ). One simply al ulates the propagation

onstant and hara teristi impedan e of a boxed mi rostrip line in the presen e of a YIG-layer and with lateral perfe t magneti walls. Hen e, the input impedan e of the line and the resulting re e tion oeÆ ient are obtained, for a parti ular rea tive load at the end of the mi rostrip transdu er. The al ulations use the variational prin iple (4.21) in the spe tral domain, be ause of the presen e of a strip ondu tor of nite extent in the lose vi inity of the YIG-layer. As a onsequen e, trial spe tral elds are derived from the spe tral magnetostati potential solution of a Fourier-transformed magnetostati equation [5.54℄. The spe tral eld omponents are related to surfa e

urrent densities on the strip, as previously done in Chapter 4 for the boxed mi rostrip line. This enables the omputation the trial spe tral eld in the whole transverse se tion. The se ond approa h uses the fa t that the following expression for the

5.3.

243

GYROTROPIC DEVICES

L layer 1

H1 ay

Wm

ax ay

layer 3 YIG

H

Wm ax

az

layer 2

H2

(a)

-γL e e (b)

+γL

dielectric substrate shielding modelling the poles conducting strip YIG film perfect magnetic shielding (c)

Equivalent transmission line model of one-port over oupled YIG-tomi rostrip on guration (a) top view; (b) equivalent ir uit; ( ) ross-se tion in z = L/2 of equivalent line

Fig. 5.19

impedan e, similar to (5.23), is based on squared powers of elds, hen e on energy, so that it an be expe ted that its right-hand side exhibits a stationary behavior: Z 1  
2 B
H E
" E dV (5.30) Zr1 jI1 j = 2j! 2 V Davies [5.55℄ mentions the use of the stationarity of the right-hand of equation (5.30) to dedu e some variational expressions for the resonant frequen y of

avities. Collin [5.56℄ makes use of the same stationarity assumption to dedu e stationary formulas for the equivalent rea tive load of diele tri obsta les in waveguides. We have proven [5.54℄ the stationarity of the right-hand side of equation (5.30) under magnetostati wave assumption when the volume of integration ontains gyrotropi lossy media. We applied it to the topover oupled on guration of Figure 5.14d, modeled as an equivalent avity instead of a transmission line. Basi ally, the trial transverse magnetostati eld on gurations are identi al for the two approa hes, but their integration is made in a transverse se tion only for transmission line formulation (4.21) while it is omputed in the whole avity volume for the energeti formulations (5.30). Hen e, the total trial eld for the se ond approa h involves a trial longitudinal z -dependen e similar to (5.24) but with f = r and PMWboundary onditions at z = 0 and z = L. Figure 5.20a,b ompares the two variational approa hes, i.e. the equivalent transmission line modeled by (4.21) (solid), and the avity model using energeti formula (5.30) (dashed). Both formulations predi t the over oupled behavior whi h is experimentally observed [5.54℄. The al ulated magnitudes of re e tion oeÆ ient dier by

244

CHAPTER 5.

APPLICATIONS

less than 0.5 dB. 0

100

−2

0 −4 −6

−100 (a) 10.2 10.3 10.4 Frequency in GHz

(b) 10.2 10.3 10.4 Frequency in GHz

Comparison between re e tion oeÆ ient of one-port top over oupled modeled either by energeti approa h (- -) or transmission line approa h (|) (a) modeled magnitude; (b) modeled phase

Fig. 5.20

5.4

Optoele troni devi es

Multilayered p-i-n photoni devi es are developed for ultra wideband optoele troni appli ations, with several advantages. First, ultra-large intrinsi bandwidths an be obtained by redu ing the thi kness of the intrinsi layer and hen e the arrier transit times inside the devi e. Se ondly, suitable values for the thi kness and diele tri onstant of some layers an be sele ted in order to on ne the opti al beam in the vi inity of the absorbing intrinsi layer, and to guide it in the ase of travelling wave operation. Finally, highdoping levels are imposed on some layers to obtain the desired opto-ele tri al (O/E) onversion, implying that the propagation me hanism is usually of a slow wave type. This means that even for very short guiding stru tures the propagation ee ts are important at mi rowave frequen ies. Several papers have been published about modeling slow wave propagation in multilayered semi ondu tor transmission lines. To our best knowledge they deal with metal-insulator-semi ondu tor stru tures studied by Gu kel et al. [5.57℄, or S hottky stru tures on Si or GaAs semi ondu ting layers, as presented by Jager [5.58℄-[5.60℄. The methods found in literature for analyzing su h stru tures (quasi-TEM or spe tral domain analysis, mode-mat hing method) assume that the semi ondu tor is divided into several homogeneous layers of in nite extent [5.61℄-[5.63℄. This se tion presents a variational approa h for modeling the transmission line behavior in Travelling Wave Photodete tors (TWPDs). Ex ellent reviews of the TWPD on ept are given in [5.61℄ and [5.64℄. Waveguide photodete tors are devi es where the ele trodes olle ting the photogenerated

urrent a t as a transmission line. Hen e, the photogenerated arriers a t as

5.4.

OPTOELECTRONIC DEVICES

245

a distributed radio-frequen y (RF) sour e of urrent indu ing voltages and

urrents travelling on the ele trodes towards their ends. TWPDs are waveguide photodete tors in whi h the opti al signal is guided along the length of the devi e as well as the RF photogenerated signal, so that the RF modulating envelope of the opti al arrier travels towards the mat hed output of the devi e at the group velo ity, undergoing no re e tions. Also, the trade-o between maximal internal eÆ ien y and transit-time bandwidth limitation is avoided, be ause the dire tions of opti al beam propagation and arrier drift are orthogonal. Under su h onditions, and when the group velo ity of the opti al signal mat hes the velo ity of the mi rowave signal, the bandwidth of TWPDs is limited only by the nite arrier transit times, and by mi rowave losses and the resulting unwanted mismat h of its ele tri al ports [5.65℄. Hen e, it is of prime interest to have a

urate transmission line models in order to optimize the TWPD bandwidth. 5.4.1

Topologies

Most of TWPDs presented in literature use mesa-type p-i-n stru tures (striplike transmission line as in Fig. 5.21a), in whi h the ele trodes olle ting the mi rowave photo urrents lay on a multilayered substrate with ea h layer having a uniform arrier on entration. Hen e, both the opti al and RF propagation an be modeled by simple equivalent transmission lines having a TEM behavior. In [5.66℄ a oplanar waveguide TWPD topology is proposed for Sili on-on-Insulator (SOI) te hnology, where CPW ele trodes lay on P and N diusion areas (Fig. 5.21b). Both mesa and CPW topologies presented in Figure 5.21 exhibit two parti ular features: some of the layers have a signi ant ondu tivity due to the high doping, responsible for the slow-wave phenomena, and some of the layers are not homogeneous or of nite extent along the x-axis. The variational approa h proposed in this se tion takes these two features into a

ount simultaneously. The eÆ ien y of the variational prin iple (4.21) for slow wave phenomena has already been illustrated in Se tion 5.1, where CPW and mi rostrips on low resistivity substrates have been su

essfully modeled in the slow wave range. 5.4.2

Modeling mesa p-i-n stru tures

The geometry of the transverse se tion of the mesa p-i-n photodete tor is shown in Figure 5.21a. Mi rowave and opti al propagation o

urs along the z -axis. The p-i-n jun tion is reverse biased with an adequate saturating voltage between top strip ondu tor and bottom grounding plane (Au), in su h a manner that the intrinsi layer (third layer) is fully depleted, with an equivalent ondu tivity equal to zero. The RF-modulated part of the opti al power intensity indu es an RF-variation of the photogenerated urrent in the intrinsi area, whi h is olle ted at the top and bottom ele trodes and propagates on the equivalent RF-transmission line. The opti al beam is on ned between the layers having the highest diele tri onstants. In order to obtain a photodete tor devi e, the doping of the third layer (n-InGaAs) is designed to

246

CHAPTER 5.

APPLICATIONS

air W = 25 µm layer 1 layer 2

metal (Au) 0.5µm

layer 3: intrinsic area ox oz oy

−∞

layer 4 layer 5 ∞

(a)

Wdiff

Wintr az

P+ ρ = 3e-4

ax Si3N4 I (P-) ay ρdepl = 3e8

Wdiff N+ ρ = 3e-4

SiO2 Si ρ =1.5e1

(b)

Travelling wave photodete tors topologies (a) mesa p-i-n GaAs stru ture (layer 1: p+ InGaAs, "r = 15, H = 0:1 m,  = 13:4 S/mm; layer 2: p+ InP, "r = 14:5, H = 1 m,  = 6:4 S/mm; layer 3: n- InGaAs, "r = 14:5, H = 2 m,  = 0:0 S/mm; layer 4: n+ InP, "r = 15, H = 2 m,  = 128:0 S/mm; layer 5: InP, "r = 14:5, H = 500 m,  = 0:0 S/mm); (b)

oplanar p-i-n SOI stru ture Fig. 5.21

5.4.

247

OPTOELECTRONIC DEVICES

PMW W

air oz oy

-∞

oz

ox

(a)

∞ -∞

oz

ox oy

∞ -∞

(b)

ox oy

∞

(c)

Approximate topologies for mesa p-i-n (a) parallel plate; (b) mi rostrip; ( ) hybrid model

Fig. 5.22

absorb the light. The highest doping level o

urs in the fourth layer whi h is neither a good ondu tor nor a good diele tri . 5.4.2.1

Simulation using a parallel plate model

The geometry of Figure 5.21a is rst simpli ed into a parallel plate waveguide (Fig. 5.22a), by Jager [5.67℄ and Gu kel et al. [5.57℄. For this TEM stru ture, however, a spe i numeri al algorithm is required be ause the propagation

onstant is the omplex root of a determinantal equation asso iated with the

orresponding eigenvalue problem. Tedious numeri al iterations are ne essary be ause of the multilayered hara ter and the high ondu tivity of some layers. Curves marked  in Figure 5.23a,b represent the results obtained using this model. A slow wave phenomenon is learly predi ted, whi h illustrates the importan e of a good model for designing broadband TWPDs. 5.4.2.2

Measured transmission line parameters

We have measured the omplex propagation onstant of the stru ture shown in Figure 5.21a, applying a two-line alibration te hnique [5.68℄ for planar lines, des ribed in the next se tion. It yields the omplex propagation onstant without using the inverse Fourier transforms proposed by Giboney et al. [5.65℄, be ause the measurement is arried out in the frequen y domain. Measured results are reported in Figure 5.23a,b (solid lines). The results obtained using the parallel plate model () of Figure 5.22a do not agree with the measurements. The parallel plate model indeed does not take into a

ount the air area above the strip and the nite width of the strip. 5.4.2.3

Simulation using a mi rostrip model

Hen e, we introdu e the ee t of the strip by using the variational approa h (4.21) applied to the mi rostrip on guration shown in Figure 5.22b: the various layers are in nite in the x-dire tion, and a strip ondu tor of nite width lies between the air and rst layer. Ea h layer i is hara terized by its omplex permittivity. The imaginary part of the omplex permittivity is derived from the layer ondu tivity i using formula (5.14a). The appli ation of the variational prin iple (4.21) is similar to that performed for the mi rostrip line

248

CHAPTER 5.

40

50

measured parallel plate microstrip hybrid

dB/mm 40

30

APPLICATIONS

30 20 20 10

(b)

10 (a)

0 0

10

20

30

GHz

0 0

10

GHz

20

30

Fig. 5.23 Comparison between measurement and simulations (a) losses; (b) ee tive refra tive index

on low-resistivity sili on, ex ept for the number of layers and strip geometry. Simulated results using the mi rostrip stru ture of Figure 5.22b are reported on Figure 5.23 ( urves marked o). The simulated slow wave phenomenon is still more important than the measured ee t: the mi rostrip quasi-TEM stru ture does not take into a

ount the nite width of some layers. 5.4.2.4

Simulation using extended formulation

Keeping in mind the high diele tri onstant in the layers (values between 14 and 15 in Fig. 5.21a), we assume that Perfe t Magneti Walls (PMW) limit the layers of nite extent at planes x = W=2, to nd trials in those layers (Fig. 5.22 ) [5.69℄. A

ording to the spe tral domain te hnique, applying this PMW-boundary ondition means that the spatial eld may be des ribed by a summation of periodi fun tions of period n=W . As a onsequen e, only the values of the spe tral elds for dis rete values kxn of the kx -variable are added up:

X1

~ei (kx ; y ) =

= 1

~ei (kxn ; y )Æ (kx

kxn )

(5.31a)

n

X1

and ei (x; y )

= n

= 1

~ei (kxn ; y )e

xn x

jk

(5.31b)

where kxn

=

n W

(5.31 )

where W is the width of layers of nite extent. The hoi e of n is related to the symmetry of trial elds des ribing the parti ular mode under investigation,

5.4.

249

OPTOELECTRONIC DEVICES

and to the nature of the shielding. For the dominant mode of a shielded stripline with PMW as onsidered here, n is taken as even. Sin e all the other boundary onditions imposed on the elds in a layer of nite extent are identi al to those applied in the same layer of in nite extent along the xaxis, the kx - and y -dependen ies of fun tion ~ei (kx ; y ) are taken to be identi al for a layer of nite or in nite extent. Adequate trial fun tions ~ei (kx ; y ) for ea h layer are thus al ulated as in the previous se tion, following the method detailed in Chapter 4. Simply, the whole kx -spe trum is onsidered for in nite layers, while only a set of dis rete values (5.31 ) is taken in layers of nite extent along the x-axis. As a result, applying Parseval's theorem together with the spe tral form (5.31a) yields expressions for the spe tral oeÆ ients of equation (4.21), whi h are summations of dis rete values of their integrands, evaluated at dis rete values of the in nite set (5.31 ). For example, oeÆ ient A~i for layer i of nite extent be omes

1 1 X A~i (~e) = 2 n= 1

Z yi



az  ~ei (kxn ; y )

 




az  ~ei (kxn ; y ) kx1 dy

(5.32)

A similar expression is easily dedu ed for oeÆ ients B to E . To summarize, we determine rst trial spe tral elds in ea h layer for the stru ture having all layers of in nite extent along the x-axis (identi al to those found in the previous se tion). Next, we impose a PMW-boundary

ondition at the left- and right-hand sides of the layers of nite extent. We express this ondition by extending the formulation of the spe tral oeÆ ients for ea h layer i in equation (4.21): for in nite layers of the mesa stru ture in Figure 5.22 , expressions (4.19b) to (4.19e) are used, while for layers of nite extent, only dis rete values (5.31a) of the spe tral trial eld are onsidered, leading to the series expression (5.32). The Spe tral Index method was introdu ed in 1989 for nding the guided modes of semi ondu tor rib waveguides [5.70℄. The trans endental equations developed by the authors are variational in nature. The method posseses the useful variational property that ea h value of \beta" is a lower bound. It improves the al ulation by ompensating the penetration into the air by slightly displa ing the boundaries, taking advantage of the fa t that guided waves in semi ondu tors have a low penetration depth into air. The on ept of ee tive depth is used to a

omodate the large jump in diele tri onstant at the air/semi ondu tor interfa e. Figure 5.23 shows that this extended variational formulation signi antly improves the agreement between theory ( urves ) and experiment (solid). We demonstrate with this analysis the possibility of using variational formulation (4.21) for stru tures ombining layers of various ondu tivity and of nite and in nite extent, while onsidering simple trial elds. It has to be mentioned that simulation () is obtained on-line on a regular PC. With our approa h, we totally negle t the in uen e of the air regions to

250

CHAPTER 5.

APPLICATIONS

the left and right of the layers of nite extent. To validate this assumption, the method of lines [5.71℄[5.72℄ was used for omputing the shape of the transverse ele tri eld in the various layers of Figure 5.21a. Results presented in [5.71℄ show that in the three top layers, and in parti ular in the intrinsi area, the eld is on ned in the layers and vanishes rapidly in air, so that negle ting the elds in the air areas x < W=2 and x > W=2 outside of the layers of nite extent is relevant. On the other hand, in the fourth layer having a high

ondu tivity (n+ InP), the longitudinal urrent density is spread outside the area under the strip, so that elds and urrents annot be onsidered on ned in area jxj  W=2. Those ee ts are observed at relatively low (15 GHz) and high frequen ies (50 GHz). 5.4.3

Modeling oplanar p-i-n stru tures

The layout of the SOI p-i-n photodete tor is shown in Figure 5.21b. Values of resistivity are in m. Figure 5.24a shows the basi p-i-n stru ture, whi h

onsists of highly doped P+ and N+ zones, separated by an intrinsi zone. Ohmi onta ts are onsidered to exist between the P+ and N+ zones and the two metal ele trodes. The stru ture is reverse biased with an adequate saturating voltage, so that the intrinsi zone is depleted. The transmission line parameters along the z -axis are obtained by generalizing the previous approa h, whi h has already su

essfully been applied to mesa InP/GaAs p-i-n photodete tors. The stru tures of Figure 5.21b are divided into n layers (perpendi ular to y-axis). In ea h layer i; j (i) sub-layers are de ned, with their number depending on the variation of arrier on entrations along the x-axis. For instan e, the p-i-n layers of Fig. 5.24a and of Fig. 5.21b ontain j (2) = 3 sub-layers. Ea h sub-layer is hara terized by its omplex permittivity. The imaginary part of the omplex permittivity of the sub-layer is derived from its resistivity, while the oxide layers have of ourse a purely real permittivity: h

"i;j (i)

= " 1 2!"

"SiO2

= " SiO2

ri

i j ri i;j (i)

(5.33a) (5.33b)

r

(5.33 ) = " Si4 N3 where j (i) is the number of sub-layers in layer i. The present ase onsists of a new appli ation of the method, sin e for the rst time planar layers having a strong variation of resistivity along the x-axis are treated. In the p-i-n layer, we have "Si4 N3

r

pin;1

=

pin;2

= 3 10 8 m

pin;3

= 3 10 4 m

5.4.

251

OPTOELECTRONIC DEVICES

Wdiff

Wdiff

Wintr

P+ ρ = 3e-4

az

ax

N+ ρ = 3e-4

I (P-) ay ρdepl = 3e8 (a)

0

S/m

10

−10

(b)

10

5

10

10

15

10 Frequency in Hz

10

−10

10

F/m −11

10

(c) −12

10

5

10

10

15

10 Frequency in Hz

10

CPW SOI p-i-n stru ture (a) simpli ed p-i-n jun tion; (b) modeling p-i-n ondu tan e; ( ) modeling p-i-n apa itan e; solid urves are for variational prin iples and symbols o for MEDICI simulations

Fig. 5.24

Hen e, the variational prin iple yielding the omplex propagation onstant

of the stru ture along the z -axis is rewritten as

2

X

()

i;j i

A~i;j (i) (~e) +

X

C~i;j (i) (~e)

X

~i;j (i) (~e) B

() () X 2 ~ " ( ) E ( ) (~e) = 0 + ! "0 0 () i;j i

i;j i

i;j i

(5.34)

i;j i

i;j i

where the trial eld (5.31b) an be used in ea h sub-layer. For the hara teristi impedan e, a power- urrent de nition (5.12) is used:




Z P I =

1 P = I2 2j!I 2

X

()

i;j i

C~i;j (i) (~e)

X ()

i;j i



A~i;j (i) (~e)

(5.35)

252

CHAPTER 5.

APPLICATIONS

where I is the magnitude of the urrent owing on one of the ele trodes, and P is the power owing through the ross-se tion of the p-i-n stru ture.

Results are ompared with simulations using MEDICI software [5.66℄. Figure 5.24b, shows the frequen y response of the ondu tan e and apa itan e per unit length of the p-i-n stru ture, omputed using MEDICI software ( urve o) and our variational approa h ( urve |). From the omplex propagation onstant and hara teristi impedan e, the equivalent ondu tan e G and apa itan e C per unit line length are derived as G = Re

C = Im

  Z

  !Z

(5.36a) (5.36b)

The agreement is ex ellent up to opti al frequen ies. The advantage of the variational approa h is that results an be obtained for the whole frequen y range (about 21 frequen ies) within a few se onds, while several minutes are ne essary when using MEDICI Version 4.0 with the same number of frequen y points. Also, our model yields ele tromagneti parameters su h as propagation onstant and hara teristi impedan e, while MEDICI solves equations involving ele tri harges, potentials and elds, with no a

ess to magneti parameters and propagation hara teristi s. Hen e, the eÆ ien y of the variational approa h yields an a

urate model of TWPD equivalent transmission line parameters for both RF and opti al frequen y ranges. Based on this formalism, an extensive omparative study of the bandwidth of SOI p-i-n TWPD stru tures has been presented, in terms of load mat hing, velo ity mat hing, type of semi ondu tor material used, and a helpful s attering matrix formalism has been established [5.73℄. 5.5

Measurement methods using distributed ir uits

In [5.74℄ an a

urate method is presented for determining the omplex permittivity of planar substrates, at frequen ies up to 40 GHz. It is based on the

omparison between al ulated and measured values of the ee tive diele tri

onstant and losses on two omplementary geometries, namely a mi rostrip and a slot-line. The omplex permittivity is determined with an a

ura y of 1 %, over the whole frequen y range. A major advantage of the method is that, as it is based upon transmission lines and not on resonant stru tures, it oers a wideband hara terization with a redu ed number of test boards. The method requires an a

urate measurement of the omplex propagation

onstant of transmission lines. The measurement pro edure is des ribed in Subse tions 5.5.1 to 5.5.3, while Subse tions 5.5.4 and 5.5.5 dis uss diele trometri standards and present a new standard based on the measurement pro edure, for a wide variety of substan es.

5.5.

MEASUREMENT METHODS USING DISTRIBUTED CIRCUITS

5.5.1

253

L-R-L alibration method

The measurement method for the propagation onstant is derived from a simpli ation of the Line-Return-Line (L-R-L) alibration method. Measurements on planar lines an indeed be a hieved with the lassi al L-R-L

alibration method des ribed extensively in [5.75℄-[5.80℄. It requires two identi al lines of dierent length and a re e tive devi e (short or open ir uit) (Fig. 5.25a). The magnitude and phase of the s attering parameters of these elements are measured, using a Ve tor Network Analyzer (VNA). This method allows the movement of the referen e plane after alibration to the middle plane of the shortest line (plane PP', Fig. 5.25b). This is of signi ant interest for hara terizing a transmission line whi h does not have simple and good transitions to the oaxial onne tors of the measurement setup, su h as a slot-line, for example. A simple explanation of the method an be found when looking at the s hemati representation of the devi e to be measured and of its two-ports A' and B', with the following de nitions: - X: two-port devi e to be measured - A' and B': two-ports hara terizing the transition between the two-port and

oaxial ports of VNA - A and B: two-port networks hara terizing the transition between referen e plane PP' and the oaxial ports of VNA, for ea h element to be measured. The method assumes that two-port networks A' and B' are the same for all the measurements illustrated in Figure 5.25a,b, and that the two lines are identi al, ex ept for their length. This is the ase if the transition between

oaxial ports and lines allows reprodu ible measurements (using the VNA's broadband test xture) and if the et hing of the planar lines an be well reprodu ed. Under these assumptions, the s attering matri es of two-port networks hara terizing the transition between the left-hand side oaxial port of VNA and referen e plane PP' are identi al, as well as those of two-port networks hara terizing the transition between the referen e plane PP' and right-hand side oaxial port of VNA. As a onsequen e, the same two-ports A and B are onsidered for ea h element to be measured. Hen e, only twoports A and B have to be hara terized. Considering the set of measurements illustrated in Figure 5.25b, we have as unknowns the two s attering matri es A B of the two-port networks S , S , and the omplex terms e
L and rea t , yielding 10 omplex unknowns. On the other hand, the measurement of the four on gurations (Fig. 5.25b) provides the two s attering matri es of the short and long lines, and the two re e tion oeÆ ients of on gurations 3 and 4, resulting in 10 omplex quantities. The real and imaginary parts of the s attering matri es and re e tion oeÆ ients of the four on gurations are expressed as a fun tion of the real and imaginary parts of the four omplex unknowns, providing a system of 20 equations for 20 real unknowns. Hen e, the s attering matri es of two-ports A and B may be determined and introdu ed as orre ting two-ports in pro essing the measurement setup. Finally, the measurement of the devi e X (Fig. 5.25a) provides a s attering

254

CHAPTER 5.

APPLICATIONS

X

matrix whi h an be expressed as aBfun tion of s attering matrix S of X A and of s attering matri es S and S whi h have been determined from the X

alibration measurements, from whi h S is extra ted. This method has been used for measuring the oplanar waveguide impedan e step presented in Figure 5.3. An important omment has to be made about the L-R-L algorithm. The
L line is modeled as a single transmission term, whi h implies that the referen e impedan e at plane PP' for the s attering matri es is the

hara teristi X impedan e of the line. Hen e, the referen e impedan e for S is the hara teristi impedan e of the line, whi h still remains unknown after alibration be ause it is not needed in the alibration algorithm. As a onsequen e, measurements of hara teristi impedan e are possible after alibration if, and only if, the line alibration standards are pre isely known over the whole

alibrating frequen y range. (a) Measurement of the device X

BB

AA

X

A’ A'

B B’ B' A ' '

(b) Calibration measurements P Configuration 1 C A line 2

A' A’

B B’ B' A' '

Configuration 2

AE

P'

∆L

BF line 1

A’ A'

B B’ B' A' '

Configuration 3

Configuration 4

BH

A G A’ A'

BD

reactive reactive termination termination

B' B’

Measurement pro edure using the Line-Return-Line alibration method (a) measurement of the unknown devi e; (b) alibration pro edure

Fig. 5.25

5.5.

255

MEASUREMENT METHODS USING DISTRIBUTED CIRCUITS

5.5.2

L-L alibration method

When looking into the algorithm of the L-R-L alibration method, there is one pie e of information involved in the pro ess whi h is never extra ted and used: the transmission line matrix orresponding to the length dieren e
between the two alibration lines. This matrix ould be used as a rst step for the extra tion of the propagation onstant from the measurement of the two

alibration lines [5.81℄. The major problem, however, is the instability when the phase of the
line approa hes 0 or 180 degrees. Usually this phase has to be kept around 90 degrees (a dieren e of a quarter-wavelength) to ensure

omputational stability of the L-R-L alibration. Su h an ele tri al length is not suÆ ient to extra t insertion losses with an a

eptable pre ision. A third line of suÆ ient length, typi ally three or four wavelengths at the lowest frequen y of the operating band, must be used to obtain a

urate results. Hen e, four elements (two lines and one re e tive devi e for alibration, and a very long line used as devi e X for wavelength and losses measurements) are required to hara terize the propagation onstant of one line with the lassi al L-R-L alibration method. The method we proposed in [5.68℄ redu es the number of ne essary elements to only two. It is based on the assumption that the two-ports des ribing the transition from ea h oaxial onne tor to the measured
line are identi al from me hani al and ele tri al points of view (Fig. 5.25b). Two-ports A and B are identi al and have re ipro ity properties, i.e. A B for = 1 2 (5.37a) ii = ii X = 21Y for = (5.37b) 12 Hen e only four omplex variables have to be determined:


L and (5.37 ) 11 22 12 (= 21 ) From the two line ir uits, one an measure eight omplex quantities (the four s attering parameters provided by the VNA for ea h ir uit), but owing to the symmetry properties of these ir uits, only four of them are signi ant. These measurements are performed after a oaxial alibration (using throughline and mat hed, short and open ir uit loads) of the VNA, be ause the twoports modeling the internal behavior of VNA from its sour e to its oaxial

onne tors do not satisfy (5.37a,b). The measured S-parameters of the two lines are expressed as a fun tion of the 4 omplex unknowns, yielding a system of 4 non-linear equations. Solving this system yields the term
L, hen e the wavelength and losses orresponding to the length dieren e
between the two lines. The Line-Line (L-L) method has been used for extra ting the measured omplex propagation onstant for all transmission line results presented in Se tion 5.1. It has to be mentioned that Janezi [5.82℄ has proposed an equivalent L-L pro edure whi h does not assume that two-ports A and B are identi al. The L

L

L

S

S

S

S

S

;S

i

;S

X; Y

S

;

A; B

e

e

L

256

CHAPTER 5.

APPLICATIONS

disadvantage of this pro edure is that eight omplex quantities have to be measured and manipulated for ea h hara terization. 5.5.3

Redu tion to one-line alibration method

One an a hieve a alibration with a single line if the two-port networks are not only symmetri but also lossless. Lossless two-ports satisfy 4 s alar equations relating their 3 unknown S-parameters (re ipro ity is assumed), so that nally one of these omplex S-parameters remains unknown. Hen e, only two omplex quantities, provided by re e tion and transmission measurements on a single line, are needed to determine one of the S-parameters of the twoport networks and the unknown term e
L . This has been investigated by T. Kezai et al. [5.22℄ for shielded low-loss lines ( nlines). They extra ted the losses and propagation onstant of a nline from a single line measurement. This measurement method has been used for the experimental results on nline presented in Se tion 5.1 (Fig. 5.4). 5.5.4 5.5.4.1

Extra tion method of onstitutive parameters Overview of the existing standards

At low frequen ies (up to 300 MHz), the omplex permittivity of a planar substrate used in MIC stru tures is dedu ed from the measurement of the omplex

apa itan e of on entri ele trodes et hed on the diele tri test sample [5.83℄. At mi rowave frequen ies, the methods are divided into two lasses: metal waveguide or avity methods, and stripline resonator methods. In the rst one, the omplex permittivity is derived from the measured variation of the re e tion oeÆ ient at the input of a short- ir uited waveguide [5.84℄ or of a resonant avity [5.85℄, loaded by a diele tri sample. The drawba ks of this method is the amount of time ne essary to prepare, adjust and make measurements, as well as the rather extreme pre ision with whi h the position or frequen y hanges have to be measured. For these reasons, the stripline resonator method seems more attra tive for MICs designers : the omplex permittivity of a planar substrate is derived from the measured resonant frequen y and loaded-Q of four boards et hed on the substrate, ea h of them onsisting of a stripline or mi rostrip line resonator loosely oupled to two a

ess lines. The stripline resonator method, however, presents some disadvantages related to the use of a measured resonant urve for determining the unknown permittivity: allowan e should be made for the radiation losses, shape ee ts of the resonator, insertion losses of the oaxial laun hing onne tors [5.86℄, and that the dimensions of the resonator be ome too large below 3 GHz. Finally, manufa turing and measuring four ir uits is ne essary at ea h frequen y where the hara terization is needed, be ause a linear regression is performed over the measured four resonant frequen ies to eliminate the shape ee t of the resonator at the frequen y of interest. The method we present in the next se tion extra ts the omplex permittivity of the planar substrate by omparing measurements arried out on two lines with very dierent topologies et hed on the plate and al ulations of the

5.5.

MEASUREMENT METHODS USING DISTRIBUTED CIRCUITS

257

ee tive diele tri onstant and the losses of the lines, based on our variational model. It is valid up to 40 GHz. The main advantage is that it oers a wideband hara terization, be ause transmission lines are used instead of resonant elements and dis ontinuities and radiating sour es are removed by the two-line alibration method. Only three ir uits are needed to obtain a hara terization valid over the frequen y range 0 to 40 GHz, be ause we make use of the broadband two-line alibration method for measuring losses and ee tive diele tri onstants. 5.5.4.2

Des ription of L-L planar diele trometer

Previous se tions have demonstrated the eÆ ien y of the variational model developed for planar lines on diele tri or gyrotropi lossy substrates. It takes into a

ount the geometri al parameters of the line and the substrate, together with the ele tri al parameters of the stru ture: ondu tivity of the metallization, and omplex permittivity and/or permeability of the substrate. The omplex propagation onstant of the dominant mode of a planar line is obtained on-line as a result of the variational formulation: the real part of the propagation onstant provides the losses per unit length, while the imaginary part yields the ee tive diele tri onstant expressed as (Im( ) 0 =! )2 . It should be emphasized again that equation (4.21) has not been proven to be variational with respe t to the onstitutive parameters of the layers. The sensitivity of the value of the propagation onstant to a variation of these parameters is preserved. Upon performing the experimental validation of the variational formulation for planar lines on diele tri substrates, a signi ant dieren e was observed between the measured (Fig. 5.26a, urve |) and al ulated (Fig. 5.26 a, urve - -) values of the ee tive diele tri onstant and losses of a slot-line when using the value of the omplex permittivity given by the manufa turer. A signi ant dieren e was also observed between the measured (Fig. 5.26b,d

urve |) and al ulated (Fig. 5.26b,d urve - -) values of the ee tive diele tri onstant and losses of a mi rostrip et hed on the same substrate when using the same manufa turer's value. It is observed on Figure 5.26 that the relative dieren e between urves - - and | is of the same order for the mi rostrip (b,d) and the slot-line (a, ). The two topologies are ele tromagneti ally omplementary of ea h other, whi h ensures that the observed dis repan y between modeling and measurement in the two ases is mainly due to a dieren e on a parameter whi h is ommon to the two topologies, i:e: the omplex permittivity of the substrate. Furthermore it is well known that manufa turer's values have to be used with are: at X-band and higher one annot guarantee an a

ura y better than 10 % for permittivity obtained with the existing standards. Hen e, adjusting the value of the permittivity to eliminate the dieren e between al ulations and measurements seems a pertinent way to obtain an improved value. When in reasing the real part of the omplex permittivity by 5 % above the manufa turer's value and the diele tri loss tangent from 0.0028 to 0.0038, the al ulated urves (Fig. 5.26

258

CHAPTER 5.

APPLICATIONS

10 6

manuf. actual meas.

5

9 8

4 (a)

3

5

10 GHz

15

0

20

(c)

−0.05

7 0

10

20 GHz

manuf. actual meas. (b) 30 40

0 (d) −0.1

−0.1

−0.2

−0.15 −0.3

−0.2

−0.4

−0.25

dB/cm 5

10 GHz

15

20

−0.5 0

dB/cm 10

20 GHz

30

40

Comparison of measured and al ulated ee tive diele tri onstant and losses of two omplementary geometries et hed on a RT duroid 6010 diele tri substrate plate of thi kness 0.635 mm (a),( ) slot-line (width 0.389 mm); (b),(d) mi rostrip line (width 0.580 mm): urve - -: al ulated with manufa turer's omplex relative permittivity 10:8(1 j 0:0028);

urve |: measured; urve -
-: al ulated with modi ed omplex relative permittivity 11:4(1 j 0:0038)

Fig. 5.26

urve -
-) yield ex ellent agreement with the measured values over the wide frequen y range for the two lines. The resulting un ertainty on the permittivity is less than 1 %, yielding 11:4(1 j 0:0038) over the frequen y range 0-40 GHz, instead of 10:8(1 j 0:0028) given by the manufa turer at X-band. 5.5.4.3

Proposed new standard

The relative permittivity of a planar substrate is obtained as the omplex value whi h, when using a

urate wideband models, provides the best agreement over the 0-40 GHz band between al ulated and measured ee tive diele tri onstant and losses of two lines with very dierent topologies (one mi rostrip line and one slot-line), et hed on the same substrate. The two-line alibration method is used to measure e
L, i:e: the ee tive diele tri onstant and the losses of the lines. Hen e, only two lines of dierent

5.5.

259

MEASUREMENT METHODS USING DISTRIBUTED CIRCUITS

lengths are needed for ea h of the two topologies onsidered. Three topologies are in ompetition to serve as a standard: mi rostrip, slot-line and oplanar waveguide. The mi rostrip-slot-line ombination is the best hoi e for the following reason: the oplanar waveguide is too similar to the mi rostrip, with a quasi-TEM propagation on both topologies. This redu es its eÆ ien y with respe t to the mi rostrip-slot-line ombination. Hen e, three ir uits have to be et hed, respe tively two slot-lines, with their transitions to mi rostrip lines, and one mi rostrip line. Using this arrangement, the omplex permittivity of a layer is determined with only three

ir uits over the whole band, and not only at dis rete frequen ies. The proposed method ombines rapidity and a

ura y of both theoreti al and experimental hara terization of any planar line. Be ause planar lines are used instead of resonators, a number of spurious limiting ee ts are avoided. Also, the method removes the ee ts of the onne tors and of the transitions between the oaxial plane of measurement of the VNA and ea h of the a

ess planes of the
L line, provided that two-port networks A and B (Fig. 5.25a,b) are identi al (reprodu ible), for ea h line and from line to line. Hen e, a high a

ura y is obtained over a wide frequen y range with a redu ed number of measurements. Su h results have been obtained for a wide variety of planar substrates, with a diele tri permittivity range from 2.33 to 10.8 and a substrate thi kness from 0.254 mm to 2 mm. 5.5.5

Other diele trometri appli ations of the L-L method

5.5.5.1

Soils

The L-L diele trometri method is also appli able to the ele tromagneti hara terization of soils. We have extensively developed the method in the frequen y range 0-18 GHz in order to extra t the omplex permittivity, from whi h the equivalent ondu tivity is obtained. This is ne essary to evaluate the performan es of mi rowave te hnologies in landmine dete tion for humanitarian purposes [5.87℄, like ground penetrating radars and radiometers. Measurements of various soils, both sandy and silty, have been arried out, using the L-L measurement method with two re tangular waveguides of width a, diering only by their length, and homogeneously lled with the soil to be

hara terized. Assuming, as previously, that the me hani al and ele tri al identity of the transitions between the VNA oaxial output and the lines, and an identi al material distribution in the two guides, the propagation onstant related to length
L of lled waveguide is extra ted. Then, instead of using a variational prin iple as for planar lines, the omplex diele tri onstant extra ted from measured propagation onstant is given by "0r

= Re

n 2

"00r

= Im

n 2

0

!2 0

!2

o

(5.38a)

o

(5.38b)

2

+ (=a)2


2

+ (=a)2


260

CHAPTER 5.

APPLICATIONS

Measurements were arried out at room temperature on four soils with different textural ompositions: two pure sands and two silty soils. For ea h soil, measurements were done for dierent water ontents, up to 20 %. From the extra ted permittivities of those soils, a model has been elaborated by Storme etal: [5.88℄, yielding the expression of the omplex permittivity as a fun tion of easily available soil parameters su h as textural omposition and density. Using volume fra tions and permittivities of bound water, free water, air and dry soil, the formula obtained for the diele tri onstant of a water-soil mixture is "m

=

3"s + 2Vfw ("fw 3 + Vfw ""fws

"s ) + Vbw ("bw
1 + Vbw ""bws

"s ) + 2Va ("a "s )

1 + Va ""as 1

(5.38 )

where subs ripts bw, f w, a, and s refer to bound water, free water, air and dry soil, respe tively, and the following hypotheses have been made: 1. Vbw = 0:06774 0:00064  SAND + 0:00478  CLAY where CLAY and SAND are lay and sand ontents, respe tively, in per ent of weight of dry soil 2. Permittivity of bound water is put equal to 35 j 15 3. Contribution of bulk water takes into a

ount the diele tri onstant of pure water, Debye model [5.89℄ at room temperature 4. Volume of free water Vf w is taken as the dieren e between total water volume and Vbw . 5.5.5.2

Bioliquids

A good knowledge of the omplex permittivity of biologi al media is ne essary for adequately determining their response to ele tromagneti elds, both for the study of biologi al ee ts as well as for medi al appli ations. Based on the L-L method, we have set up a new pro edure for measuring the permittivity of liquids [5.89℄. The measurement pro edure uses waveguide spa ers of dierent thi kness, pla ed between the two waveguide ports of VNA. A syntheti lm (Para lm), is pla ed at both ends between waveguides and spa er, ontaining the liquid in a known volume. Be ause of the L-L method, the lm has no ee t on the measurements, provided that the transition is reprodu ible when using both spa ers. A preliminary measurement on dioxane, having a known onstant permittivity, is used to validate the measurement set-up. Measurements have been arried out for the omplex permittivity of biologi al and organi liquids at frequen ies above 20 GHz up to 110 GHz, on methanol, axoplasm, and beef blood. The values obtained have been ompared with Debye's law. We observed, as presented in [5.90℄, that for biologi al liquids the rst-order Debye model using only one relaxation time is not suÆ ient. Several relaxation phenomena o

ur, requiring the re al ulation of higherorder relaxation terms. Finally, we found that the Cole-Cole diagram [5.91℄, representing " as a fun tion of " with frequen y as a parameter, is very eÆ ient for improving models for diele tri relaxation. 00

0

5.6.

5.6

SUMMARY

261

Summary

Chapter 5 is devoted to a number of appli ations of the variational prin iple to al ulate transmission line parameters. First, expressions for the parameters are re alled. Then, they are al ulated for simple line on gurations on diele tri substrates: mi rostrip, slot-line, oplanar waveguide, and nline. Lossy semi ondu ting substrates are onsidered for mi rostrip and oplanar waveguide. A re ent appli ation has been des ribed: the al ulation of parameters of a mi rostrip on a magneti nanostru tured substrate, onsisting of an array of magneti metalli nanowires embedded in a thin porous diele tri substrate. Another lass of examples were jun tions: mi rostrip-to-slot-line jun tion and planar T-jun tion. Gyrotropi devi es were also onsidered, su h as YIG-tuned planar MSW devi es, under oupled as well as over oupled, for one-port and two-port on gurations. Optomi rowave devi es have also been analyzed, using a variational prin iple, illustrating how to model mesa p-i-n stru tures and how important the dieren es an be when using dierent topology models, and oplanar p-i-n stru tures. The hapter ends with the des ription of a measurement method based on a variational prin iple to determine the omplex permittivity of a planar substrate with an ex ellent a

ura y. The measurement method is then illustrated with examples for hara terizing soils for demining operations, and bioliquids for medi al purposes (blood and axoplasm). 5.7

Referen es

[5.1℄ A. Vander Vorst, Transmission, propagation et rayonnement. Brussels: De Boe k-Wesmael, 1995. [5.2℄ A. Vander Vorst, Ele tromagnetisme. Brussels: De Boe k-Wesmael, 1994, Ch. 5. [5.3℄ T. Rozzi, F. Moglie, A. Morini, E. Mar hionna, and M. Politi, \Hybrid modes, substrate leakage and losses of slot-lines at millimetre wave frequen ies", IEEE Trans. Mi rowave Theory Te h., vol. MTT-38, no. 8, pp. 1069-1078, Aug. 1990. [5.4℄ I. Huynen, D. Vanhoena ker, A. Vander Vorst, \Spe tral Domain Form of New Variational Expression for Very Fast Cal ulation of Multilayered Lossy Planar Line Parameters", IEEE Trans. Mi rowave Theory Te h., vol. 42, no. 11, pp. 2099-2106, Nov. 1994. [5.5℄ R. Faraji-Dana, Y. Leonard Chow, \The Current Distribution and AC Resistan e of a Mi rostrip Stru ture", IEEE Trans. Mi rowave Theory Te h., vol. MTT-38, no. 9, pp. 1268-1277, Sep. 1990. [5.6℄ E. Yamashita, \Variational method for the analysis of mi rostrip-like transmission lines", IEEE Trans. Mi rowave Theory Te h., vol. MTT16, no. 8, pp. 529-535, Aug. 1968.

262

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[5.7℄ T. Kitazawa, T. Itoh, \Asymmetri al Coplanar Waveguide with Finite Metallization Thi kness Containing Anisotropi Media", IEEE Trans. Mi rowave Theory Te h., Vol. MTT-39, no. 8, pp. 1426-1433, Aug. 1991. [5.8℄ L. Zhu, E. Yamashita, "New method for the Analysis of Dispersion Chara teristi s of Various Planar Transmission Lines with Finite Metallization Thi kness", IEEE Mi rowave Guided Wave Letters, vol. 1, no. 7, pp. 164-166, July 1991. [5.9℄ W.H. Haydl, J. Braunstein, T. Kitazawa, M. S hle htweg, P. Tasker, L.F. Eastman, "Attenuation of Millimeter Wave Coplanar Lines on Gallium Arsenide and Indium Phosphide over the range 1-60 GHz", IEEE MTT-S Int. Mi rowave Symp. Dig., pp. 349-352, June 1992. [5.10℄ W.H. Haydl, "Experimentally Observed Frequen y Variation of the Attenuation of Millimeter-Wave Coplanar Transmission Lines with Thin Metallization", IEEE Mi rowave Guided Wave Letters, vol. 2, no. 8, pp. 164-166, August 1992. [5.11℄ T. Itoh, Numeri al methods for Mi rowave and Millimeter Wave Passive stru tures. New York: John Wiley&Sons, 1989, Ch. 5. [5.12℄ A. Vander Vorst, D. Vanhoena ker-Janvier, Bases de l'ingenierie mi roonde. Brussels: De Boe k&Lar ier, 1996, Ch. 2. [5.13℄ J.B. Knorr, P. Shayda, "Millimeter wave n line hara teristi s", IEEE Trans. Mi rowave Theory Te h., vol. MTT-28, no. 7, pp. 737-743, July 1980. [5.14℄ L.P. S hmidt, T. Itoh, "Spe tral domain analysis of dominant and higher order modes in n lines", IEEE Trans. Mi rowave Theory Te h., vol. MTT-28, no. 9, pp. 981-985, Sept. 1980. [5.15℄ J.B. Knorr, K.D. Ku hler, "Analysis of oupled slots and oplanar strips on diele tri substrate", IEEE Trans. Mi rowave Theory Te h., vol. MTT-23, no. 7, pp. 541-548, July 1975. [5.16℄ J.B. Knorr, A. Tufek ioglu, "Spe tral-Domain Cal ulation of Mi rostrip Chara teristi Impedan e", IEEE Trans. Mi rowave Theory Te h., vol. MTT-23, no. 9, pp. 725-728, Sept. 1975. [5.17℄ R.H. Jansen, M. Kirs hning, \Arguments and an A

urate Model for the Power-Current Formulation of Mi rostrip Chara teristi   Impedan e", Ar h. Elek. Ubertragung. , vol. AEU-37, no. 3-4, pp. 108112, Mar h-April 1982. [5.18℄ D.F. Williams, \Chara teristi impedan e, power and ausality", IEEE Mi rowave Guided Wave Lett., vol 9, no. 5, pp. 181-182, May 1999. [5.19℄ R. Gillon, J.P. Raskin, D. Vanhoena ker, J.P. Colinge, \Determining the referen e impedan e of on-wafer TRL alibrations on lossy substrates", Pro . 26th Eur. Mi rowave Conf., Prague, pp. 170-176, Sept. 1996.

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[5.20℄ R.B. Marks, D.F. Williams, \Chara teristi Impedan e Determination Using Propagation Constant Measurement", IEEE Mi rowave Guided Wave Lett., vol. 1, no. 6, pp. 141-143, June 1991. [5.21℄ I. Huynen, A. Vander Vorst, \Variational non quasi-stati formulations for the impedan e of planar transmission lines", IEEE Trans. Mi rowave Theory Te h., vol. 47, no. 7, pp. 995-1003, July 1999. [5.22℄ T. Kezai, R. S iuto, A. Vander Vorst, \Methode d'etalonnage et de mesure pour des lignes a ailettes", Journees Nationales Mi roondes (JNM 93), Pro ., pp. D6.6-7, Brest, May 1993. [5.23℄ M. Serres, Opti al ontrol of mi rowave planar devi es on semi ondu tor substrate, Ph.D. dissertation. Louvain-la-Neuve, Belgium: Mi rowaves UCL, 1999. [5.24℄ I. Huynen, G. Goglio, D. Vanhoena ker, A. Vander Vorst, \A novel nanostru tured mi rostrip devi e for tunable stopband ltering appli ations at mi rowaves", IEEE Mi rowave Guided Wave Lett., vol. 9, no. 10, pp. 401-403, O t. 1999. [5.25℄ I. Huynen, L. Piraux, D. Vanhoena ker, A. Vander Vorst, \Ferromagneti resonan e in metalli nanowires for tunable mi rowave planar ir uits", Pro . Progr. Ele tromagneti Res. Symp. (PIERS 2000), Cambridge, pp. 320, July 2000. [5.26℄ S.B. Cohn, \Slot Line on a diele tri substrate", IEEE Trans. Mi rowave Theory Te h., vol. MTT-30, no. 10, pp. 259-269, O t. 1969. [5.27℄ E.A. Mariani, C.P. Heinzman, J.P. Agrios, S.B. Cohn, \Slot Line

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[5.34℄ T. Kitazawa, \Variational Method for Planar Transmission Lines with Anisotropi Magneti Media", IEEE Trans. Mi rowave Theory Te h., Vol. MTT-37, no. 11, pp. 1749-1754, Nov. 1989. [5.35℄ M. Horno, F.L. Mesa, F. Medina, R. Marques, \Quasi-TEM Analysis of Multilayered Multi ondu tor Coplanar Stru tures with Diele tri and Magneti Anisotropy In luding Substrate Losses", IEEE Trans. Mi rowave Theory Te h., vol. MTT-40, no. 3, pp. 524-531, Mar h 1992. [5.36℄ E.B. El-Sharawy, R.W. Ja kson, \Coplanar Waveguide and Slot Line on Magneti Substrates: Analysis and Experiment", IEEE Trans. Mi rowave Theory Te h., vol. MTT-36, no. 6, pp. 1071-1079, June 1988. [5.37℄ F.L. Mesa, M. Horno, \Quasi-TEM and full wave approa h for oplanar multistrip lines in luding gyromagneti media longitudinally magnetized", Mi rowave Opt. Te hnol. Lett., vol. 4, no. 12, pp. 531-534, Nov. 1991. [5.38℄ M. Tsutsumi, T. Asahara, \Mi rostrip Lines Using Yttrium Iron Garnet Films", IEEE Trans. Mi rowave Theory Te h., vol. MTT-38, no. 10, pp. 1461-1470, O t. 1990. [5.39℄ I. Huynen, B. Sto kbroe kx, A. Vander Vorst, \Variational Prin iples are eÆ ient CAD tools for planar tunable MICs involving lossy gyrotropi layers", International Journal of Numeri al Modelling : Ele troni networks, devi es and elds, vol. 12, no. 5, pp. 417-440, Sept.O t. 1999. [5.40℄ S.N. Bajpai, \Insertion losses of ele troni ally variable magnetostati wave delay lines", IEEE Trans. Mi rowave Theory Te h., vol. MTT-37, no. 10, pp. 1529-1535, O t. 1989. [5.41℄ A.K. Ganguly, D.C. Webb, \Mi rostrip ex itation of Magnetostati Surfa e Waves : Theory and Experiment", IEEE Trans. Mi rowave Theory Te h., vol. MTT-23, no. 12, pp. 998-1006, De . 1975. [5.42℄ P.R. Emtage, \Intera tion of magnetostati waves with a urrent", J. Appl. Physi s, vol. 49, no. 8, pp. 4475-4484, Aug. 1978. [5.43℄ J.D. Adam, S.N. Bajpai, \Magnetostati Forward Volume Wave Propagation in YIG strips", IEEE Trans. Magneti s, vol. MAG-18, no. 6, pp. 1598-1600, Nov. 1982. [5.44℄ I.J. Weinberg, \Insertion Loss for Magnetostati Volume Waves", IEEE Trans. Magneti s, vol. MAG-18, no. 6, pp. 1607-1609, Nov. 1982. [5.45℄ A.D. Berk, \Variational prin iples for ele tromagneti resonators and waveguides", IEEE Trans. Antennas Propagat., vol. AP-4, no. 2, pp. 104-111, Apr. 1956. [5.46℄ I. Huynen, G. Verstraeten, A. Vander Vorst, \Theoreti al and experimental eviden e of non-re ipro al ee ts on magnetostati forward volume wave resonators", IEEE Mi rowave Guided Waves Lett., vol. 5, no. 6, pp. 195-197, June 1995.

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[5.47℄ J.P. Parekh, K.W. Chang, H.S. Tuan, \Propagation hara teristi s of magnetostati waves", Cir uits, Systems and Signal Pro eedings, vol. 4, pp. 9-38, 1985. [5.48℄ P. Kabos and V.S. Stalma hov, Magnetostati waves and their appli ations. London: Chapman&Hall, 1994. [5.49℄ L. Kajfez , P. Guillon, Diele tri resonators. London: Arte h House, 1986. [5.50℄ I. Huynen, B. Sto kbroe kx, G. Verstraeten, D. Vanhoena ker, A. Vander Vorst, \Planar gyrotropi devi es analyzed by using a variational prin iple", Pro . 24th Eur. Mi row. Conf., Cannes, pp. 309-314, Sept. 1994. [5.51℄ I. Huynen, A. Vander Vorst, \A new variational formulation, appli able to shielded and open multilayered transmission lines with gyrotropi non-hermitian lossy media and lossless ondu tors", IEEE Trans. Mi rowave Theory Te h., vol. 42, no. 11, pp. 2107-2111, Nov. 1994. [5.52℄ D.W. Van Der Weide, \Thin- lm os illators with low phase noise and high spe tral purity", IEEE MTT-S Int. Mi rowave Symp. Dig., vol. 1, pp. 313-316, 1992. [5.53℄ W. Kunz, K.W. Chang, W. Ishak, \MSW-SER based tunable os illators", Pro . Ultrasoni s Symposium, pp. 187-190, 1986. [5.54℄ I. Huynen, B. Sto kbroe kx, G. Verstraeten, \An eÆ ient energeti variational prin iple for modelling one-port lossy gyrotropi YIG Straight Edge Resonators", IEEE Trans. Mi rowave Theory Te h., vol. 46, no. 7, pp. 932-939, July 1998. [5.55℄ J.B. Davies in T. Itoh, Numeri al methods for Mi rowave and Millimeter Wave Passive stru tures, Chapter 2. New York: John Wiley&Sons, 1989. [5.56℄ R.E. Collin, Field theory of guided waves (se ond edition). New York: IEEE Press, 1991. [5.57℄ H. Gu kel, P. Brennan I. Palo z, \A parallel-plate waveguide approa h to mi rominiaturized planar transmission lines for integrated ir uits", IEEE Trans. Mi rowave Theory Te h., vol. MTT-11, no. 8, pp. 468-476, Aug. 1967. [5.58℄ D. Jager, \Slow-wave Propagation along variable S hottky-Conta t Mi rostrip Line", IEEE Trans. Mi rowave Theory Te h., vol. MTT-24, no. 9, pp. 566-573, Sept. 1976. [5.59℄ K. Wu, R. Vahldiek, \Hybrid-Mode Analysis of Homogeneously and Inhomogeneously Doped Low-Loss Slow-Wave Coplanar Transmission Lines", IEEE Trans. Mi rowave Theory Te h., vol. MTT-39, no. 8, pp. 1348-1360, Aug. 1991.

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[5.60℄ J.K. Liou, K.M. Lau, \Analysis of Slow Wave Transmission Lines on Multi-Layered Semi ondu tor Stru tures In luding Condu tor Losses", IEEE Trans. Mi rowave Theory Te h., vol. MTT-41, no. 5, pp. 824-829, May 1993. [5.61℄ K.S. Giboney, M.J.W. Rodwell, J.E. Bowers, \Travelling-Wave Photodete tor Theory", IEEE Trans. Mi rowave Theory Te h., vol. 45, no. 8, pp. 1310-1319, Aug. 1997. [5.62℄ V. M. Hielata, G.A. Vawter, T.M. Brennan, B.E. Hammons, \Traveling-Wave Photodete tors for High-Power, Large-Bandwidth Appli ations", IEEE Trans. Mi rowave Theory Te h., vol. 43, no. 9, pp. 2291-2298, Sept. 1995. [5.63℄ L.Y Lin, M.C. Wu, T. Itoh, T.A. Vang, R.E. Muller, D.L. Siv o, A.Y. Cho, \High-Power High-Speed Photodete tors - Design, Analysis, and Experimental Demonstration", IEEE Trans. Mi rowave Theory Te h., vol. 45, no. 8, pp. 1320-1331, Aug. 1997. [5.64℄ K. Giboney, J. Bowers, M. Rodwell, \Travelling-wave photodete tors", 1995 IEEE MTT-S Symp. Dig., pp. 159-162, 1995. [5.65℄ K. Giboney, R. Nagarajan, T. Reynolds, S. Allen, R. Mirin, M. Rodwell \Travelling-wave photodete tors with 172-GHz bandwidth and 76-GHz bandwidth-eÆ ien y produ t", IEEE Photon. Te hnol. Let., vol. 7, no. 4, pp. 412-414, Apr. 1995. [5.66℄ I. Huynen, A. Salamone, M. Serres, \A traveling wave model for optimizing the bandwidth of p-i-n photodete tors in Sili on-On-Insulator te hnology", IEEE J. Sel. Top. Quantum Ele troni s, vol. 4, no. 6, pp. 953-963, Nov.-De . 1998. [5.67℄ D. Jager, R. Kremer, A. Stohr, \Travelling-wave optoele troni devi es for mi rowave appli ations", 1995 IEEE MTT-S Symp. Dig., pp. 163166, 1995. [5.68℄ M. Fossion, I. Huynen, D. Vanhoena ker, A. Vander Vorst, \A new and simple alibration method for measuring planar lines parameters up to 40 GHz", Pro . 22nd Eur. Mi row. Conf., Helsinki, pp. 180-185, Sept. 1992. [5.69℄ Z. Zhu, I. Huynen, A. Vander Vorst, \An eÆ ient mi rowave hara terization of PIN photodiodes", Pro . 26th European Mi rowave Conferen e, Prague, vol. 2, pp. 1010-1014, Sept. 1996. [5.70℄ W.A. M Ilroy, M.S. Stern, P.C. Kendall, \Fast and a

urate method for al ulation of polarised modes in semi ondu tor rib waveguides", Ele troni s Lett., vol. 25, no. 23, pp. 1586-1587, 9th Nov. 1989. [5.71℄ Z. Zhu, A. Vander Vorst, \Mi rowave Propagation in p-i-n Transmission Lines", IEEE Mi rowave Guided Wave Lett., vol. 7, no. 6, pp. 159-161, June 1997.

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[5.84℄ G. Kent, \A diele trometer for the measurement of substrate permittivity", Mi rowave J., vol. 34, pp. 72-82, De . 1991. [5.85℄ M. Olyphant, Jr., \Pra ti al diele tri measurement of CLAD mi rowave substrates", Cu-Tips Note no. 8, 3M Co. Publi ation ELCT/8(70.6)R.

268

CHAPTER 5.

APPLICATIONS

[5.86℄ Appendix 1, MIL-P-13949G, Military spe i ation, Plasti Sheet, Laminated, Metal-Clad, U.S. Army Ele troni s Command, Fort Montmouth, N.J., Feb. 1987. [5.87℄ C. Steukers, M. Storme, I. Huynen, \Modelisation of ele tri al properties of soils", in Chapter 4 of the book \Sustainable Humanitarian Demining: Trends, Te hniques, and Te hnologies", edited by Humanitarian Demining Information Center, James Madison University (CoEditors: D. Barlow, C. Bowness, A. Craib, G. Gately, J.-D. Ni oud, J. Trevelyan). Verona, Virginia, USA: Mid Valley Press, pp. 305-313, 1998. [5.88℄ M. Storme, I. Huynen, A. Vander Vorst, \Chara terisation of wet soils in the 2-18 GHz frequen y range", Mi rowave Opt. Te hnol. Lett., vol. 21, no. 5, pp. 333-335, June 1999. [5.89℄ F. Duhamel, I. Huynen, A. Vander Vorst, \Measurements of omplex permittivity of biologi al and organi liquids up to 110 GHz", 1997 IEEE MTT-S Symp. Dig., Denver, pp. 107-110, June 1997. [5.90℄ H.P. S hwan, K.R. Forster, \RF- eld with biologi al systems: ele tri al properties and biophysi al me hanisms", Pro . of the IEEE, vol. 68, no. 1, pp. 104-113, Jan. 1981. [5.91℄ V. Daniel, Diele tri relaxation. London: A ademi Press, 1967, Ch. 7.

appendix A Green's formalism

A.1 De nition and physi al interpretation

Green's fun tions have been used for many years in ele tromagneti s and physi s. A review of the most useful appli ations of Green's fun tions is given in Morse and Feshba h [A.1℄. Only the fundamentals are summarized in this appendix, referring rst to an analogy with ordinary dierential equations. A linear system is des ribed by an ordinary dierential equation with

onstant oeÆ ients. As an example, the general form of the equation for a se ond-order system is A

d2 f (t) dt2

+B

df (t) dt

+ Cf (t) = x(t)

(A.1)

where the unknown response f (t) is, for instan e, a voltage or a urrent, while the ex itation x(t) omes from independent sour es. The equation is often solved using a transformation. Using the Fourier transformation (Appendix C) yields (j! )2 AF (! ) + j!BF (! ) + CF (! ) = X (! )

(A.2)

from whi h is obtained F (! )

= H (! )X (! )

(A.3)

with H (! )



= (j! )2 A + j!B + C

 1

(A.4)

The fun tion H (! ) is alled the transfer fun tion of the system. It is obviously the response of the system to an ex itation with uniform spe trum: F (! )

= H (! )

if X (! ) = 1

(A.5)

Noting that the inverse transform of the fun tion with uniform spe trum is the Dira impulse Æ (t), the inverse transform h(t) of the transfer fun tion H (! ) is obviously the impulse response of the system. Hen e, the solution of (A.1)

270

APPENDIX A.

GREEN'S FORMALISM

is obtained by al ulating the inverse transform of the produ t H (! )X (! ). It is well known that this transform is the onvolution integral Z +1 x(t  )h( )d (A.6) f (t) =

1

This physi ally means that the response f (t) to an ex itation x(t) is the superposition of a ontinuous sequen e of weighted impulse responses. In pra ti e, however, the integral an only be al ulated in a rather small number of ases. What has been presented for ordinary dierential equations an be extended to equations involving partial derivatives, su h as Maxwell's equations in ele tromagneti s, with their boundary onditions and onstitutive relations, and where sour es are in general harge and urrent densities. In this ase, s alar and ve tor potentials satisfy an equation su h as

r F + a tF + b tF = X 2

2

(A.7)

2

whi h is the ase for Poisson's equation, the wave equation, and the diusion equation for instan e. The multi-dimensional Fourier transformation (C.7) is applied to this ve tor equation, transforming the spa e domain into the spe tral domain: F (k ) =

Z V

jk
r dV (r )

F (r)e

F (r) = (21 )

Z

3

F (k )ejk
r dV (k )

V

(A.8a) (A.8b)

When the problem is time-variant, the orresponding equations have, of

ourse, to be extended to 4 dimensions: F (k; ! ) =

Z

Z

V (r) t

F (r; t) = (21 )

Z

4

F (r; t)e Z

V (k) !

j (k
r+!t) dV (r )dt

F (k; ! )ej (k
r+!t) dV (k )d!

(A.9a) (A.9b)

We an now extend to partial derivative equations what has been shown to be valid for ordinary dierential equations. To simplify the notation, we assume that there is no time dependen y. By doing so, we shall illustrate the orresponden e from the time-to-frequen y transformation in problems des ribed by ordinary dierential equations, and the spa e-to-state spa e (spe tral domain) transformation in problems des ribed by partial derivative equations. There should be no diÆ ulty in writing this transformation

A.1.

DEFINITION AND PHYSICAL INTERPRETATION

271

in 4 dimensions, transforming the equation from a spa e-time volume into a state spa e-frequen y volume. Assuming that the equation is (A.10) L(r )F (r) = X (r) where L is a dierential operator and F the system response to the ex itation X , its 3-D Fourier transform is L(k )F (k )

= X (k )

(A.11)

where L, F , and X are the transforms of operator, response, and ex itation, respe tively. The solution is F (k )

= G(k )X (k)

with G(k ) = 1=L(k)

(A.12)

where G(k ) is the \transfer fun tion", i:e: the Fourier-transform of the \impulse response" of the system. It is indeed the response of the system submitted to an ex itation with \uniform spe trum": F (k )

= G(k )

if X (k) = 1

(A.13)

The inverse transform of the uniform spe trum fun tion is the generalization of the Dira impulse Z (A.14) Æ (r )e jk
r dV (r ) = 1 V

Hen e, the solution of (A.11) is the inverse transform of produ t (A.12), whi h is obviously the generalization of the onvolution integral: Z L(r ) = G(r s)p(s) dV (s) (A.15) V

The fun tion G(r s), the inverse transform of the \transfer fun tion", is the impulse response of the system, so that

L(r )G(r) = Æ(r)

(A.16)

It is the response at observation point r of a point sour e lo ated at the enter of oordinates. It is alled the Green's fun tion. Extending this 3-D transformation result into the orresponding 4-D transformation result shows that Green's fun tions des ribe basi ally the spa e and time response of a physi al system at point r at time t when a point sour e of ex itation is applied to the system at point r 0 at time t0 . By response, one means the spa e and time-des ription of a parti ular s alar or ve tor quantity P hara terizing the system. A general ex itation X an be either a s alar or a ve tor. There are eight variables involved in the des ription of the physi al system: the six spa e oordinates x, y , z , x0 , y 0 , z 0 represented by the ve tors

272

APPENDIX A.

GREEN'S FORMALISM

and r 0 , and the time variables t, t0 . The response of the system to a given ex itation X is totally determined on e Green's dyadi G has been found for ea h pair of points (r; r 0 ) of domain Vx . De ning Vx and Tx as the spa e and time domains of the ex itation, respe tively, Green's dyadi notation yields response r

P (r; t)

=

ZZ

Vx Tx

j




G(r; t r 0 ; t0 ) X (r 0 ; t0 )dr0 dt0

(A.17a)

Intuitively, the response of the system to ex itation X at any observation point is obtained by superimposing the ontributions at the observation point of in nitesimal unit point sour es of ex itation lo ated at the various points (r 0 ; t0 ) of domain (Vx ; Tx ) and having the magnitude and orientation of X at these points. From de nition (A.17a), the Green's fun tion an also be viewed as a linear integral dyadi operator applied to ex itation X with the view of nding the orresponding distribution of the parameter P . It is denoted by G : = GfX (r 0 ; t0 )g

P (r; t)

with Pu

= =

X hGf XZ Z v

v

g

X (r 0 ; t0 )

i

(A.17b) (A.17 )

uv

j

Guv (r; t r 0 ; t0 )Xv (r 0 ; t0 )dr0 dt0

Vx Tx

(A.17d)

For a unit point sour e oriented along the u-axis and applied at (r 0 ; t0 ) X (r; t)

= Æ (r

r 0 )Æ (t

t0 )au (u

= x; y; z )

(A.18a)

the three omponents of the u-th olumn of Green's dyadi are obtained as the three omponents of ve tor when an ex itation Æ (r 0 r0 )Æ (t0 t0 )au is applied: P (r; t)

=

ZZ

Vx Tx

j

G(r; t r 0 ; t0 )


fÆ(r

0

r 0 )Æ (t0

g

t0 )au dr0 dt0

= Gxu (r; tjr0 ; t0 )ax + Gyu (r; tjr0 ; t0 )ay + Gzu (r; tjr 0 ; t0 )az

(A.18b) (A.18 )

In expressions (A.18a,b), the notation Æ (r 0 r0 ) represents for the triple produ t Æ (x0

x0 )Æ (y 0

y0 )Æ (z 0

z0 )

(A.18d)

expressing the presen e of a point sour e with unit magnitude, lo ated at point r 0 at time t0 . The same formalism an also be used for applying spe i boundary onditions to a system, that is sear hing for the spa e and

A.1.

273

DEFINITION AND PHYSICAL INTERPRETATION

time distribution of the response asso iated with the system whi h satis es spe i boundary onditions on a surfa e x . A suitable expression for the Green's dyadi s is obtained by al ulating at the points of x whi h are not on the boundary x while imposing a homogeneous zero boundary

ondition at every point lo ated on x , ex ept for one point 0 on x where a unit boundary ondition is imposed. The boundary ondition is imposed on (Diri hlet ondition) or on its gradient (Neumann ondition). Hen e, the required solution is obtained when summing over x the partial solutions orresponding to a point boundary ondition imposed on an in nitesimal portion of x around : P

G

P

V

r

P

r

( )=

0

Z

P r; t

Z

x Tx

s (r; tjr

G

0

;t

0

)

(

0

0


X r ;t

)

0

dr dt

(A.19)

0

An important feature of the Green's formalism is that the Green's fun tion , providing for a given system the distribution of orresponding to any sour e distribution (A.17), is essentially the same as Green's fun tion s (A.19) yielding spe i boundary onditions on a surfa e for the same system: G

P

G

1. for Neumann boundary onditions s

G

=

and

G

yielding ( )=

Z

P r; t

(

X r ;t

Z

x Tx

0

(

0

0

)=

0

G r; tjr ; t

n

)




(A.20a)

P

x

(

0

P r ; t

0

= yielding s

(

and

G n

)=

P x; y; z; t

Z

(

0

X r ;t

Z

x Tx

0

)=

(

0

 G r; tjr ; t n

0

dr dt



x

n

2. for Diri hlet boundary onditions G



)

(A.20b)

0



(A.20 )

P

x

0

)

(

0

0


P r ;t



) x

0

dr dt

0

(A.20d)

where is the normal at the surfa e [A.1℄. The Green's fun tion is in fa t the solution for a situation whi h is homogeneous everywhere ex ept at one point. The solution of a homogeneous equation satisfying inhomogeneous boundary

onditions is equivalent, using the Green's formalism, to the solution of an inhomogeneous equation satisfying homogeneous boundary onditions, with the inhomogeneous part being a surfa e layer of harge. More ompli ated situations are then handled as linear ombinations of the solutions of a homogeneous problem with inhomogeneous boundary onditions, and of an inhomogeneous problem with homogeneous boundary onditions, whi h is equivalent to inhomogeneous problems with homogeneous boundary onditions. n

274

APPENDIX A.

GREEN'S FORMALISM

A.2 Simpli ed notations

Generally speaking, the notation G(r; tjr ; t ) holds for a 3  3 tensor whose elements are fun tions of the variables r , t, r , t . Simpler Green's fun tions may be de ned when the geometry of the problem yields a onstant or a a priori -known dependen e of the problem along a parti ular dire tion, or when its time-dependen e is well known or equal to a onstant (steady-state systems). For instan e, 2-D Green's formulations similar to (A.18b) are de ned as Z P (x; y; t) = G(x; y; tjx ; y ; t )Æ (x x0 )Æ (y y0 )Æ (t t0 )dr dt 0

0

0

0

0

0

0

0

0

0

= G(x; y; tjx0 ; y0 ; t0 )

0

0

(A.21a)

when no variation of the quantities of interest o

urs along the z -axis. The 2-D Green's fun tion is then denoted by

j

G(r; t r 0 ; t0 )

= G(x; y; tjx0 ; y0 ; t0 )

(A.21b)

with = xax + yay r = x ax + y ay r0 = x0 ax + y0 ay

(A.21 ) (A.21d) (A.21e)

r

0

0

0

The one-dimensional Green's fun tion is similarly noted

j

G(x; t x0 ; t0 )

(A.21f)

When the ex itation X is des ribed by a s alar, the dyadi redu es to a ve tor Green's fun tion of the eight variables (r; r ; t; t ), and denoted by G(r; tjr ; t ). When both P and X are s alar, the dyadi redu es to an algebrai Green's fun tion of the eight variables (r; r ; t; t ), noted G(r; tjr ; t ). For ve tor and algebrai Green's formulations, 1- and 2-D forms are readily dedu ed by substituting respe tively the adequate s alar oordinate or r ve tor to the 3-D ve tor in the previous notations. 0

0

0

0

0

0

0

0

A.3 Green's fun tions and partial dierential equations

The Green's formalism has been developed in onne tion with problems whi h

an be des ribed by inhomogeneous partial dierential equations and suitable boundary onditions. It oers an elegant way of handling these problems, be ause instead of solving the partial dierential equation for the quantity P in the presen e of ex itation X or for spe i boundary onditions, the

orresponding equation for the Green's fun tion is solved in the presen e of the unit point sour e of ex itation, and (A.17) or (A.20) is then applied. It is

A.3.

GREEN'S FUNCTIONS AND PARTIAL DIFFERENTIAL EQUATIONS

275

expe ted that solving the equation for G is mu h easier than solving it for P in the presen e of X and/or spe i boundary onditions. Denoting L the dierential operator asso iated with the partial dierential equation, the problem be omes (A.22a) LfG(r; tjr ; t )g
au = Æ(r r )Æ(t t )au instead of (A.22b) LfP (r; t)g = X (r; t) In ele tromagneti s, ve equations are of main interest. 0

0

0

0

1. Poisson's equation: (A.23) r2 (r) = (r) where LP = r2 The ele trostati eld distribution is obtained as the gradient of ele trostati potential . 2. Homogeneous Helmholtz equation: (A.24) r2 (r) + k2 (r) = 0 where LH = r2 + k2 The harmoni TE and TM elds are obtained as parti ular ombinations of potential  solution of the Helmholtz equation and its partial derivatives. A time dependen e ej!t is assumed for both potentials and elds, so that one has k 2 = ! 2 " (A.25) 3. S alar wave equation: 2 r2 (r) (1= )2  (r) =

 r; t

4. Ve tor wave equation: r2 A(r) + !2 "A(r) =

()

t2

5. Diusion equation: r2 (r) a2  (r) = 0 t

( )

J r

2

 (A.26) where L0 = r2 (1= )2 t 2

where Ld = r2 + !2"

where Ld = r2

a2

 t

(A.27)

(A.28)

For a ve tor wave equation, a Green's fun tion an also be derived, whi h relates the ve tor eld to the ve tor sour e. Usually, those Green's fun tions are derived from the Green's fun tions relating the sour e to a s alar or ve tor potential.

276

APPENDIX A.

A.4

GREEN'S FORMALISM

Green's formulation in the spe tral domain

The Green's formulation in the spe tral domain (Appendix C) is obtained by taking the Fourier transform along the three spa e variables, yielding the spe tral domain, and along the time variable, yielding the harmoni domain. When the medium is multilayered, parti ular solutions must be taken into a

ount, so that the boundary onditions at ea h interfa e are satis ed. This has been onsidered in Subse tion 3.3.1. The dierential operator r is transformed into jk and the partial time derivative into j!t, whi h yields 1 ZZ ZZ P (x; y; z; t)ej!t ejk r dkd! P~ (kx ; ky ; kz ; ! ) = (A.29) (2)4 and the orresponding forms of the various equations in the spe tral domain, for homogeneous layers. We obtain for Poisson's equation: jkj2~ (kx ; ky ; kz ) = ~(kx ; ky ; kz ) where L~ d = jkj2 (A.30)


The orresponding spe tral s alar Green's fun tion G~ P satis es jkj2G~ P (kx ; ky ; kz ) = 1

(A.31)

and the general form of the spe tral Green's fun tion asso iated with Poisson's equation is ~ (k ; k ; k ) = 1 (A.32) G P

x

y

z

jk j2

Equation (A.30) ombined with (A.32) provides the spe tral form of the potential ~ (kx ; ky ; kz ) = G~ P (kx ; ky ; kz )~(kx ; ky ; kz ) (A.33) Its inverse Fourier-transform is (r; t) =

Z

Z

Vx Tx

GP (r

r0 ; t

t0 )(r 0 ; t0 )dr0 dt0

(A.34)

The Green's formulation in the spe tral domain has the advantage that the dierential or integral operators are repla ed by simple algebrai expressions, so that the derivation of the spe tral Green's fun tion is very simple (A.31). Also, the solution is obtained by taking the produ t of the spe tral Green's fun tion by the Fourier-transform of the ex itation (A.33), instead of using the Green's fun tion as a linear integral operator for elaborating the solution. The user has the hoi e either to invert the spe tral Green's fun tion and apply (A.17) in the spa e domain, or to invert the obtained spe tral solution (A.33). A Green's fun tion may of ourse be dedu ed for relating the ele trostati eld gradient of the potential to the ex itation distribution (r).

A.5.

277

RECIPROCITY

Green's fun tions oer an elegant way of hara terizing a physi al system, for any sour e or boundary onditions, sin e the knowledge of the Green's fun tion is suÆ ient to al ulate the response of the system to any ex itation. Hen e, only the Green's fun tion is usually mentioned for the ve ategories of equations introdu ed above. For example, we have: a. for the Helmholtz equation:

G~ H (kx ; ky ; kz ; !) =

1 jkj + !2 "

(A.35)

2

b. for the diusion equation:

G~ D (kx ; ky ; kz ; !) = A.5

1 jkj + j!a2

(A.36)

2

Re ipro ity

Green's fun tions are subje t to the fundamental theorems of ele tromagneti s. The most important for the present purpose is the re ipro ity theorem: for steady-state systems in an unbounded free-spa e, the ee t at the observation point r of a sour e lo ated at point r is identi al to the ee t observed at point r when the same sour e is lo ated at point r: 0

0

G(r jr0 ) = G(r 0 jr ) with

T

G=G

(A.37a)

(A.37b)

This property is satis ed by the s alar Green's fun tion asso iated to Poisson's equation for quasi-stati ele tromagneti problems and by the Green's dyadi asso iated with the ve tor Helmoltz equation solved in free-spa e [A.2℄. Re ipro ity in a free-spa e unbounded medium is thus expressed by a symmetry property of the orresponding Green fun tion in the position ve tors r0 and r. Property (A.37a,b) is valid for most ele tromagneti problems, provided they are des ribed in a unbounded free-spa e medium. For more pra ti al situations (inhomogeneous and/or bounded media), Green's fun tions satisfy some re ipro ity properties, however they are no longer symmetri al dyadi s. When time is introdu ed, identity (A.37a) is no longer valid. It has to be repla ed by

G(r; tjr0 ; t0 ) = G(r 0 ; t0 jr; t)

(A.38)

The sign inversion for the time variable ensures the ausality of the physi al system des ribed by the Green's fun tion.

278

APPENDIX A.

A.6

GREEN'S FORMALISM

Distributed systems

This review of the main hara teristi s of the Green's formalism is now adapted to distributed systems, where propagation o

urs along a parti ular dire tion, taken as the z -axis. Assuming kzo as the propagation onstant, the sour e distribution an be written ( 0 0 ) = x(x0 ; y 0 ; t0 )e jkz0 z

0

(A.39)

X r ;t

where the lower ase x holds for transverse dependen e of elds for whi h dependen e along the longitudinal dire tion may be separated. Introdu ing this expression into general de nition (A.18) yields the relationship between the response P and the transverse dependen e of the ex itation: Z Z 0 0 0 0 0 jkz0 z dr0 dt0 P (r; t) = G(r; tjr ; t )
x(x ; y ; t )e (A.40a) Vx Tx Z Z 0 0 0 0 0 jkz0 (z z) G(r; tjr ; t )
x(x ; y ; t )e = (A.40b) Vx Tx jk z 0 0 0 0 z 0 e dx dy d(z z )dt Z Z ~ 0 0 0 0 0 0 0 0 0 = e jkz0 z G(x; y; tjx ; y ; t ; kz 0 )
x(x ; y ; t )dx dy dt 0

0

Vx Tx

(A.40 )

whi h is a Green's formulation with six variables. A.7

Referen es

[A.1℄ P. M. Morse and H. Feshba h, Methods of Theoreti al Physi s. New York: M Graw-Hill, 1953, Ch. 7. [A.2℄ R. E. Collin, Field theory of guided waves, 2nd edn. New York: IEEE Press, 1991, Ch. 2.

appendix B Green's identities and theorem

B.1

S alar identities and theorem

Let V be a losed region of spa e bounded by a regular surfa e S , and let a and b be two s alar fun tions of position whi h together with their rst and se ond derivative are ontinuous throughout V and on surfa e S . (It is mentioned by Stratton that this ondition is more stringent than is ne essary, and that the se ond derivative of one fun tion b need not be ontinuous [B.1℄). Then the divergen e theorem applied to the ve tor bra gives Z Z (bra)
n dS r
(bra) dV = (B.1) S

V

Expanding the divergen e to r


(bra) = rb
ra + br
ra = rb
ra + br2 a

(B.2)

and noting that r

a
n=

a n

(B.3)

where a=n is the derivative in the dire tion of the positive normal, yields what is known as Green's rst s alar identity [B.1℄[B.2℄: Z Z Z a b br2 a dV = rb
ra dV + dS (B.4) V

V

S n

If, in parti ular, we pla e b = a and let a be a solution of Lapla e's equation, (B.4) redu es to Z Z a 2 (B.5) (ra) dV = a dS V

S n

If the roles of the fun tions a and b are inter hanged, i.e. applying the divergen e theorem to the ve tor arb, one obtains Z Z Z b dS (B.6) a ar2 b dV = ra
rb dV + V

V

S n

280

APPENDIX B.

GREEN'S IDENTITIES AND

THEOREM

Upon subtra ting (B.6) from (B.4) a relation between a volume integral and a surfa e integral is obtained of the form Z

V

br2 a




ar2 b dV =

known as Green's se ond [B.1℄. B.2

Z 

S

b

a n

s alar identity

a

b n



(B.7)

dS

or also frequently as Green's theorem

Ve tor identities

It is possible to establish a set of ve tor identities wholly analogous to those of Green for s alar fun tions. Let P and Q be two fun tions of position whi h together with their rst and se ond derivatives are ontinuous throughout V and on the surfa e S . Then, if the divergen e theorem is applied to the ve tor P  r  Q , one has Z

V

r
(P  r  Q) dV

=

Z

S

(P

 r  Q)
n dS

(B.8)

Upon expanding the integrand of the volume integral one obtains the ve tor analogue of Green's rst identity Z

V

(r  P


rQ


r  r  Q) dV

P

=

Z

S

(P

 r  Q)
n dS

(B.9)

The analogue of Green's ve tor se ond identity is obtained by an inter hange of the roles of P and Q in (B.9) followed by subtra ting from (B.9). As a result Z

V

(Q
r  r  P

P

= B.3

Z


r  r  Q) dV

S

(P

rQ

Q  r  P )
n dS

(B.10)

Referen es

[B.1℄ J.A. Stratton, Ele tromagneti Theory. New York: M Graw-Hill, 1941, Chs. 3 and 4. [B.2℄ E. Roubine, Mathemati s applied to physi s, 1970. Berlin: SpringerVerlag, Ch. V.

appendix C Fourier transformation and Parseval's theorem

C.1

Fourier transformation into spe tral domain

A number of mathemati al texts have been published on the Fourier transformation. Some others, like [C.1℄, are intended for those who are on erned with applying Fourier transforms to physi al situations rather than with furthering the mathemati al subje t as su h. The purpose of this appendix is not to review or summarize the subje t. Quite spe i ally, it is intended to show how to extend the lassi al Fourier transform, relating time and frequen y domain, to the relation between spa e and spe tral domain, for s alar as for ve tor quantities. Most usually, the Fourier transformation is applied to a fun tion f (t), des ribed in the time domain. It is de ned by Z +1 F (! ) = f (t)e j!t dt (C.1)

1

whi h is a fun tion des ribed in what is alled the frequen y domain. Using a similar transformation, whi h is alled the inverse transformation, yields Z 1 +1 f ( t) = F (! )ej!t d! (C.2) 2 1 Cal ulating the Fourier transform of an ordinary dierential equation oers the enormous advantage of transforming the operators ontained in the equation into terms of a polynomial expression. It must be observed that the se ond transformation (C.2) is not exa tly the same as the rst. On the other hand, the Fourier transformation an be applied to any pair of quantities, relating the rst quantity to its spe tral transform. Regarding (x, s) as su h a pair, and applying the Fourier transformation twi e yields Z +1 F (s) = f (x)e j 2xs dx (C.3)

1

and f (w) =

Z +1

1

F (s)e j 2ws ds

(C.4)

282

APPENDIX C.

FOURIER TRANSFORMATION AND

PARSEVAL'S THEOREM

This time, the same transformation has been applied twi e. When f (x) is an even fun tion of x, that is when f (x) = f ( x), the repeated transformation yields the initial fun tion. This is the y li al property of the Fourier transformation, and sin e the y le is in two steps, the re ipro al property is implied: if F (s) is the Fourier transform of f (x), then f (x) is the Fourier transform of F (s). The y li al and re ipro al properties are imperfe t, however, be ause when the initial fun tion is odd, the repeated transformation yields f ( w). When f (x) is neither even nor odd, the imperfe tion is still more pronoun ed. The ustomary formulas exhibiting the reversibility of the Fourier transformation are Z +1 F (s) = f (x)e j 2xs dx (C.5)

f (x) =

1 Z +1 1

F (s)ej 2xs ds

(C.6)

When variable x is time t, then variable 2s is frequen y ! , whi h yields expressions (C.1) and (C.2). Fa tor 2 is often taken out of the integral in the expressions. Fourier transformation an easily be applied to fun tions with several variables. In ele tromagneti s, the pairs of variables will often be the spa e variables (x, y , z ) on one side and the spe tral variables (kx , ky , kz ) on the other, to obtain Z +1 Z +1 Z +1 f (x; y; z )e j (xkx +yky +zkz ) dxdydz (C.7) F (kx ; ky ; kz ) =

f (x; y; z ) =

1 (2 )3

1

1

1

Z +1 Z +1 Z +1

1

1

1

F (kx ; ky ; kz )ej (xkx +yky +zkz ) dkx dky dkz (C.8)

Similarly, the Fourier transformation an be used to transform the four variables of spa e-time domain into the four spe tral variables kx , ky , kz and ! . The extension to ve tor elds is immediate: the omponent of the transform is the transform of the orresponding omponent, and the transform of a ve tor eld is also a ve tor eld. Transforming, for instan e, from the time domain into the frequen y domain for an ele tri eld, yields Z
+1 E (r; t)e j!t dt E (r; ! ) = (C.9)

1

Z 1 +1
E (r; t) = 2 E (r; ! )ej!t d! 1

(C.10)

C.2.

283

PARSEVAL'S THEOREM

while transforming from the spa e domain into the spe tral domain yields expressions whi h are identi al to (C.7)(C.8) ex ept for repla ing F (kx ; ky ; kz ) and f (x; y; z ) by the orrespondent ve tor elds. It is interesting to observe that the sum xkx + yky + zkz is the s alar produ t of the ve tors position in spa e domain and in spe tral domain, respe tively. It is rather ustomary to express this sum as a dot produ t in the integrals, whi h renders the expressions more ompa t. A further improvement is to express dx dy dz and dkx dky dkz as unit volumes dV (r) and dV (k ) in spa e domain and spe tral domain, respe tively. C.2

Parseval's theorem

There is some ambiguity about what is alled Parseval's theorem. Rigorously speaking, Parseval's theorem is related to Fourier series, while its equivalent for Fourier transforms is sometimes alled Rayleigh's theorem [C.1℄. Calling F (u; v ) the two-dimensional Fourier transform of the two-dimensional fun tion f (x; y ), Parseval's theorem for Fourier series is written [C.1℄ Z

+1 Z +1

1

1

jf (x; y)j2 =

XX

a2mn

(C.11)

where F

2



(u; v ) =

XX

a2mn [Æ (u

m; v

n)℄

When related to Fourier transforms, Parseval's theorem is sometimes alled Rayleigh's theorem. The reason, reported in [C.1℄, is that the theorem was rst used by Rayleigh in his study of bla k-body radiation, published in 1899. It states that the integral of the squared modulus of a fun tion is equal to the integral of the squared modulus of its spe trum; that is Z

+1

1 or Z

+1

1

jf (x)j2 dx = 21 f (x)f  (x) dx

Z

1 Z

=

+1

+1

jF (s)j2 ds 0 f (x)f  (x)e j 2xs dx

1 = F (s0 )  F  ( s0 ) Z +1 = F (s)F  (s s0 ) ds 1 Z +1 F (s)F  (s) ds = 1

(C.12)

s0

=0

s0

=0

s0

=0

(C.13)

Ea h integral represents the amount of energy in a system, one integral being taken over all values of a oordinate, the other over all spe tral omponents.

284 APPENDIX C.

FOURIER TRANSFORMATION AND

The theorem is true if one of the integrals exists.

PARSEVAL'S THEOREM

Form (C.13) is used in

Chapters 3 and 4 of this book. It has be ome ustomary to repla e ontinuous Fourier transforms by an equivalent dis rete Fourier transform (DFT) and then evaluate the DFT using the dis rete data. Parseval's theorem is then usually written as [C.2℄:

Xf

N

1

n=0

2

(

n) =

XFk

N

1

1

N k=0

j

(

)j

2

(C.14)

where

F (k)j2 = F (k)F  (k)

j

C.3

Referen es

[C.1℄

R.N. Bra ewell,

The Fourier Transform and Its Appli ations

. New

York: M Graw-Hill, 1965. [C.2℄

R.A. Dorf,

The Ele tri al Engineering Handbook

Press, 1993.

. Bo a Raton:

CRC

appendix D

Ferrites and Magnetostati Waves D.1 Permeability tensor of a Ferrite YIG- lm

Assuming a DC eld along the y-axis, the magneti permeability tensor of a YIG- lm is de ned as [D.1℄ 2 3 11 0 21 =4 0 1 0 5 (D.1a) 12 0 11 where  (f=! )(! gM + jfa)  0 h s (D.1b) 11 = 0 1 + (!hgMs + jfa)2 f 2 12 = 0

= with a !h !0 H Hint Ms


H g



f 2 =!0 j (!hgMs + jfa)2

21

 f2

(D.1 ) (D.1d)

=
H=(2Hint) = Hint=Ms = f=(gMs) thi kness of lm [m℄ DC-magneti eld inside YIG- lm [Oe℄ when a eld H0 is reated by magneti poles saturation magnetization of the lm [G℄ (usually noted 4Ms in data sheets) linewidth of YIG- lm [G℄ gyromagneti ratio = 2.8 [MHz/Oe℄

D.2 Demagnetizing ee t

The YIG- lm is pla ed in a uniform DC-magneti eld applied along the yaxis (Fig. 4.2). The demagnetizing ee t exists be ause a YIG-sample pla ed into a uniform DC-magneti eld H0, generates an internal omponent of

286

APPENDIX D.

FERRITES AND MAGNETOSTATIC WAVES

eld ne essary to ensure the boundary onditions at the interfa es between YIG and air. The result is that the total internal magneti eld inside the YIG-sample, denoted by Hint , is lower than the external applied eld, hen e a demagnetizing ee t o

urs. It is usually expressed via a demagnetizing fa tor Ny :

Hint = H0

Ny M s

(D.2)

For ellipsoidal bodies, the demagnetizing ee t is easy to al ulate [D.2℄. For a YIG- lm in nite along the x- and z -axis, one has

Ny = 1

(D.3a)

so that

Hint = H0

Ms

(D.3b)

For a YIG- lm of nite width, the exa t al ulation of the demagnetizing fa tor is diÆ ult. One solution proposed by Joseph and S hlomann [D.3℄ develops the total internal magneti eld as a series of as ending powers in (H0 =Ms ). The rst-order term is used in our models and is suÆ ient to predi t the internal magneti eld. The main hara teristi s of the demagnetizing ee t obtained when using those expressions is that the resulting internal eld is nonuniform over the sample width. The demagnetizing fa tor is about 1 at the enter of the lm, while it de reases to 1=2 at its edges. Hen e, the permeability tensor to be onsidered in the lm varies with the x-position in the sample, sin e the internal eld varies. The edge values of the permeability tensor are al ulated by using the value of Hint at the edges of the lm. The position-dependent inverted permeability tensor required in (4.2) is al ulated for ea h x-position and the integral of ea h term of (4.2) over the lm width is performed numeri ally (simple trapezoidal method). D.3 D.3.1

Magnetostati Wave Theory Magnetostati assumption

Magnetostati waves are derived from the solution of Maxwell's equations for magneti ally ordered matter, ompleted with the onstitutive equations:

r  E = j!B r  H = J  + j!D D ="
E

B =
H

r
B =0 r
D =  J  = E

(D.4a) (D.4b)

In this appendix we only deal with magneti diele tri s ( ferrites) in whi h free

harge arriers do not exist and the urrent density is negligible. In addition, we assume that this matter is ele tri ally isotropi , so that the permittivity

D.3.

287

MAGNETOSTATIC WAVE THEORY

tensor redu es to the s alar ". If we separate the quantities in (D.4a) into DC and RF omponents: H = H 0 + h ej!t B = B 0 + b ej!t we obtain r  e = j!b r  h = j!d

E = E 0 + e ej!t D = D0 + d ej!t

r
b=0 r
d=0

(D.5a)

d = "e b=
h (D.5b) We now simplify system (D.5a) [D.4℄, taking advantage of the fa t that we are only p interested in slow ele tromagneti waves in ferrimagneti media (i.e. k >> k0 "="0 ), so that the following ondition is met: ! 2 2 >> or !