Fields, Waves and Transmission Lines Fields, Waves and Transmission Lines

F. A. Benson Emeritus Professor Formerly Head of Department Electronic and Electrical Engineering University of Sheffield and T. M. Benson Senior Lecturer Electrical and Electronic Engineering University of Nottingham

0412 363704 o 442 31470 1 (USA)

Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the UK Copyright Designs and Patents Act, 1988, this publication may not be reproduced, stored, or transmitted, in any form or by any means, without the prior permission in writing of the publishers, or in the case of reprographic reproduction only in accordance with the terms of the licences issued by the Copyright Licensing Agency in the UK, or in accordance with the terms of licences issued by the appropriate Reproduction Rights Organization outside the UK. Enquiries concerning reproduction outside the terms stated here should be sent to the publishers at the London address printed on this page. The publisher makes no representation, express or implied, with regard to the accuracy of the informaton contained in this book and cannot accept any legal responsibility or liability for any errors or omissions that may be made. A catalogue record for this book is available from the British Library Library of Congress Cataloging-in-Publication data Benson, F. A. (Frank Atkinson), 1921- Fields, waves, and transmission lines/F. A. Benson and T. M. Benson, - Ist ed. p. cm. Includes bibliographical references and index. ISBN 978-0-412-36370-2 ISBN 978-94-011-2382-2 (eBook) DOI 10.1007/978-94-011-2382-2 1. Electric lines. 2. Wave guides. 3. Antennas (Electronics) 1. Benson, T. M., 1958- . II. Title. TK3201.B46 1991 621.381 '3-dc20 91-18440 CIP Contents

Preface IX List ofprincipal symbols XIII

Part One Theory and Problems 1 1 Electromagnetic theory 3 ~~~oo 3 1.1 Maxwell's equations 3 1.2 Plane waves 9 1.3 Skin depth 9 1.4 Power flow in an electromagnetic wave 9 1.5 Boundary conditions 9 1.6 Behaviour of a plane wave normally incident on a plane perfectly conducting boundary 10 1.7 Behaviour of a plane wave incident obliquely on a plane perfectly conducting boundary 13 1.8 Plane wave incident normally on a plane dielectric· boundary 16 1.9 Multiple dielectric interfaces 19 1.10 Plane wave incident obliquely on a plane interface between two dielectric media 21 Problems 25 Additional problems 29

2 Transmission line theory 31 Introduction 31 2.1 Transmission line equations 31 2.2 Solution for an infinite line 34 2.3 Solutions for a finite line 35 2.4 Solutions for high frequencies 37 vi Contents

2.5 Quarter-wavelength lines 39 2.6 Half-wavelength lines 39 2.7 The distortionless line 40 Problems 40 Additional problems 44

3 Rectangular and circular waveguides and cavity resonators 50 Introduction 50 3.1 Rectangular waveguides 51 3.2 Circular waveguides 62 3.3 Cavity resonators 66 Problems 74 Additional problems 78

4 Miscellaneous waveguiding systems 83 Introduction 83 4.1 Coaxial line 83 4.2 Two-wire line 86 4.3 Parallel-plate transmission line 89 4.4 Microstrip 94 4.5 Stripline 98 4.6 Dielectric waveguides 100 4.7 The optical fibre 105 Problems 110 Additional problems 116

5 Impedance transformation and matching 122 Introduction 122 5.1 Impedance transformation 122 5.2 The Smith chart 123 5.3 Matching 131 Problems 139 Additional problems 144

6 Microwave networks 150 Introduction 150 6.1 Scattering parameters 150 6.2 Matrix forms offour-terminal (two-port) network equations 157 6.3 Signal flow graphs 164 Problems 171 Additional problems 179 Contents vii

7 Antennas and propagation 184 Introduction 184 7.1 Antennas 184 7.2 Sources, potentials and fields 185 7.3 The Hertzian dipole 188 7.4 Antenna properties 190 7.5 Antenna arrays 194 7.6 Receiving antennas and reciprocity 197 7.7 Aperture antennas 198 7.8 Propagation 201 7.9 Refraction in the ionosphere 205 7.10 Antenna measurements 208 Problems 211 Additional problems 218

Part Two Solutions 221 Solutions for Chapter 1 223 Solutions for Chapter 2 238 Solutions for Chapter 3 254 Solutions for Chapter 4 266 Solutions for Chapter 5 284 Solutions for Chapter 6 305 Solutions for Chapter 7 321 References 340 Index 349 Preface

One of us (FAB) published a book Problems in Electronics with Solutions in 1957 which became well established and ran to five editions, the last revised and enlarged edition appearing in 1976. When the first edition was written it covered almost the complete undergraduate electronics courses in engin eering at universities. One book, at a price students can afford, can no longer cover an undergraduate course in electronics. It has therefore been decided to produce a book covering one important section of such a course using the experience gained and a few problems from previous editions of Problems in Electronics with Solutions. The book is based largely on problems collected by us over many years and given to undergraduate electronic and electrical engineers. Its purpose is to present the problems, together with a large number of their solutions, in the hope that it will prove valuable to undergraduates and other teachers. It should also be useful for Master's degree students in electronic and electrical engineering and physics, research workers, engineers and scientists in industry and as a reference source. The material presented will provide a link between the theory of elec tromagnetic fields, waves and transmission lines and its practical applica tion. The book is organized in seven chapters treating electromagnetic theory, transmission line theory, rectangular and circular waveguides and cavity resonators, other waveguiding systems, impedance transformation and matching, microwave networks and antennas and propagation. Suffi cient material and worked examples are included in an introduction to each chapter to cover the essential points of theory and develop necessary formulae. To keep the price at a level that is reasonable for students it was necessary to limit the number of problems. Some topics which readers may expect to find included, e.g. H, trough, groove and dielectric-rod waveguides, filter theory, resonance absorption, tensor permeability, Faraday rotation etc. concerned with ferrite media at microwave frequencies, have had to be x Preface omitted and others have less space devoted to them than one would have liked. Over 300 problems are given of which 82 are worked examples in the introductory sections of chapters and 168 have step-by-step solutions. The solutions are separated from the problems so that students will not see them by accident. The answer is also given at the end of each problem, however, for convenience. A thorough grasp of the principles involved in any particular problem cannot be obtained by merely reading through the solution. Students should therefore not consult the solutions until they have either repeatedly tried hard and failed to obtain the stated answer or successfully solved the problem and wish to compare the method of solution with that given. Some additional problems, totalling 90, with answers but not solutions, are provided in each chapter as student exercises. References to texts and published papers, together with comments on their content, are listed; these will serve for further explanation of key points, enable and encourage independent study of a particular subject area in greater depth and provide background reading for those equations quoted without proof. In conformity with modern practice SI units are used throughout. We cannot possibly claim that all the problems in the collection are original, but it is impossible to acknowledge the sources of those which are not. Most of the problems are new, however, and in many cases they have been formulated to try to encourage thought and understanding; but some which require only numerical substitution in formulae (they are based on practical data wherever possible) are included in the hope that they will develop the student's sense of magnitudes. While great care has been taken to try to eliminate errors some will inevitably have crept in and we shall be glad to have any such brought to our notice so that they can be corrected in subsequent printings or editions. In Chapter 5 Smith Charts have been used of the form given on page 23 of the book Electronic Applications of the Smith Chart-In Waveguide, Circuit and Component Analysis by P. H. Smith and published by McGraw Hill Inc. in 1969. The authors acknowledge the kindness, in granting permission to reproduce these charts, of Anita M. Smith (Mrs Phillip H. Smith) who is the Executrix and Owner of all rights of Phillip Smith, deceased, and her company Analog Instruments Company (PO Box 808, New Providence, NJ 07974, USA). It should be noted that Smith is a Registered Trademark of the Analog Instruments Company. It has been a pleasure working with Chapman & Hall in particular we wish to thank Mr Daniel H. Brown, Commissioning Editor, Electronic Engineering, for his courteous help and co-operation and for useful sugges tions. We express our gratitude and appreciation to Miss Elaine Jessop and Miss Sally Hollingsworth for their excellent typing of the manuscript. Preface xi

Finally, warmest thanks are extended to our wives Kay and Margaret for their valuable support and infinite patience.

F. A. BENSON T. M. BENSON Department of Electronic and Department of Electrical and Electrical Engineering, Electronic Engineering, The University of Sheffield The University of Nottingham LIST OF PRINCIPAL SYMBOLS

A area A constant

[A] = [~ ~J transfer matrix for four-terminal or two-port network A vector A magnetic vector potential A. effective aperture At gain of kth forward path 10 general flow graph equation a constant a dimension of rectangular aperture antenna a long internal dimension of rectangular waveguide a dimension of rectangular cavity resonator a radius of circular waveguide or cavity resonator a radius of inner conductor of coaxial line a radius of each conductor of two-wire line a radius of optical fibre core complex amplitudes of incident waves on an n-port network a( asymmetry factor ax, ay, az vector directions B constant B susceptance B bit rate B magnetic flux density b short internal dimension of rectangular waveguide b dimension of rectangular cavity resonator xiv List of principal symbols b radius of outer conductor of coaxial line b separation of stripline ground planes b normalized mode index bb b2 ,ooo,b" complex amplitudes of reflected waves for an n-port network c constant c leakage capacitance per unit length of transmission line c velocity of light, equal to l/(Jlol'o )1/2 ~ 3 X 108 m s- 1 D constant D detector meter reading D largest dimension of antenna aperture D directive gain D electric displacement d distance d length of cylindrical cavity d spacing of conductors in two-wire line d separation of antennas in an array do diameter of circular cross-section conductor equival ent to a rectangular strip in stripline design dv volume element E magnitude of electric field E electric field vector ET total electric field e base of Naperian logarithms e magnitude of the charge on an electron e" e,l> e", vector directions F force on an electron f frequency fc critical frequency of ionized layer in upper atmos phere fe cut-off frequency fr resonant frequency G constant G leakage conductance per unit length of transmission line G power gam

[H] = [Hll H 12J four-terminal (two-port) matrix H 21 H n H magnetic field strength h separation of plates in parallel-plate transmission line h thickness of guiding layer in dielectric slot waveguide h separation of top conductor and ground plane in microstrip line List of principal symbols xv h length of a Hertzian dipole hCUI cut-off thickness of dielectric slab waveguide I current (I] unit matrix 10 current amplitude i subscript denoting incident J electric current density

Jc conduction current density J d displacement current density In(x) Bessel function of the first kind of order n J~(x) derivative of Bessel function with respect to p K constant K1 constant K2 constant K1 phasor K2 phasor Kn(x) Hankel function of order n k =(w2eJ.1 + y2)112 k constant 2 kJ,k2 given by ki + k~ = k knma nth root of J n(ka) = 0 L inductance per unit length of transmission line L length of cylindrical cavity resonator L distance I length of transmission line I positive integer M constant M material dispersion factor m constant m positive integer m mass of electron N constant N ion density of ionized layer in upper atmosphere N number of propagating modes in an optical fibre N1 group index of optical fibre core n positive integer n turns ratio of transformer n number of elements in an antenna array n index of refraction n = v/2rr. (see t/J) nmin minimum value of index of refraction nmode mode or effective index p point at which field is evaluated p power loss per unit area in cavity resonator xvi List of principal symbols

P time-average power transmitted along waveguide PA power absorbed by an aperture PAL power absorbed in load resistor Pc time-average power dissipated in a cavity

POR power density at a receiving antenna PIN total power accepted by an antenna PI time-average power lost in plates of parallel-plate waveguide

Pr received power PR power received by antenna PI power transmitted down parallel-plate waveguide PT power transmitted by antenna

PTOT total power radiated by an antenna Pw power density in a plane wave p positive real number p see t Ph P3 slab waveguide transverse parameter Pnm mth roots of equation Jn(ka) =0 Pnm' mth roots of equation J~(ka) = 0 Q quality factor q charge q see t q2 slab waveguide transverse parameter R resistance per unit length of transmission line R distance between source and field points Ro radiation resistance Rh axial length of horn antenna RL , R) load resistance R. surface resistance

RT terminating resistance R. receiving antenna r location of field point with respect to some origin r radius r normalized resistance r subscript denoting reflected rj ratio of permittivities r' location of source with respect to some origin [S] scattering matrix S Poynting vector Su, S/2, S21 ,..., Sij, ..., Snn scattering coefficients or scattering parameters for n-port network [S]T transpose of [S] s voltage-standing-wave ratio s separation of antennas List of principal symbols xvii

T thickness T time slot (l/bit rate) [T], [T]' scattering transmission matrices

Tll, T12 , T21>"" Tjj, ... , Tnn scattering transmission coefficients or parameters parameters of scattering transmission matrix with input waves of n-port network as dependent vari ables and output waves as independent variables t time t subscript denoting transmitted t thickness of microstrip top conductor t thickness of central conductor of stripline t = exp( - 2"') = p +jq t21 etc. transmittances tL transit time V radiation intensity U fibre propagation parameter U,UI see '" Uo = !XI V normalized frequency V voltage Vs voltage applied at input to transmission line v velocity of propagation v volume v, VI see '" v, vp phase velocity Vo = jPI vg group velocity VTEM velocity of TEM waves in coaxial line dielectric W. energy stored in electric fields of resonator Wm energy stored in magnetic fields of resonator WT = W. + Wm total energy stored in resonator W fibre propagation parameter W width of parallel-plate transmission line W width of top conductor of microstrip line W width of central conductor of stripline X reactance x distance along transmission line x electron displacement x normalized reactance X rectangular co-ordinate

Xnlll = Pnlll for TM modes in cylindrical cavity

=Pnlll• for TE modes in cylindrical cavity y admittance xviii List of principal symbols

Yo = 1/'10 characteristic admittance Yo = l/zo characteristic admittance YL admittance of load 11 [YL ] = [Y admittance matrix for four-terminal network Y21 Yn(X) Bessel function of the second kind of order n y normalized admittance y rectangular co-ordinate Z impedance [Z] = [Zl1 impedance matrix for four-terminal network Z21 Zo characteristic impedance of transmission line Zoo output impedance of network Zl1 input impedance of network Z21 transmittance Zg internal impedance of generator Zi input impedance of transmission line Zio, Zoe input impedance of open-circuited line Zis input impedance of short-circuited line Z, terminating impedance of transmission line Zs input impedance (sending-end) of transmission line ZTE waveguide impedance for TE waves ZHoI waveguide impedance for TM waves Z rectangular co-ordinate Z distance along transmission line Z normalized impedance Z .;ylindrical polar co-ordinate z, normalized terminating impedance of transmission line attenuation constant, real part of the propagation constant IX current phase angle P phase constant, imaginary part of the propagation constant phase constant in free space phase constant along a rectangular waveguide confinement factor propagation constant profile height parameter of optical fibre terms in the general flow graph equation elementary length of periphery of rectangular waveguide cross-section skin depth loss angle constant List of principal symbols xix bfL bandwidth-length product £ effective permittivity of ionized medium £ permittivity £0 permittivity of free space equal to 8.855 x 1O-12Fm- 1 effective relative permittivity in microstrip relative permittivity '1 intrinsic impedance or wave impedance '1 antenna efficiency '10 impedance of free space, equal to (I.lO/£0)1/2 = 120n or ~ 377Q angle electrical length of transmission line critical angle angle of incidence angle of reflection wavelength wavelength corresponding to resonant frequency of cylindrical cavity ,1,0 free-space wavelength A.c critical wavelength

A.g guide wavelength

A. h wavelength at the mouth of a horn antenna I.l permeabilitY I.lo permeability of free space equal to 4n x 10- 7 H m- 1 I.ll permeability of waveguide wall metal I.lr relative permeability p electric charge density p = Ipl exp(jcp) reflection coefficient p cylindrical co-ordinate (J conductivity (J radar cross-section CP,CPo angle cp phase angle cp magnetic flux cp cylindrical polar co-ordinate cp scalar potential cp symbol which can represent a component of E or H t/J symbol which can represent a component of E or H t/J u + jv = tanh - 1Z :/J Ul + jVl = tanh -lzr t/J progressive phase difference in an antenna array OJ angular frequency angular cut-off frequency