Fields, Waves andTransmission 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

IUI11 SPRINGER-SCIENCE+BUSINESS MEDIA, B. V.

First edition 1991

© 1991 F. A. Benson and T. M. Benson Originally published by Chapman & Hali in 1991

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.

1. Benson, T. M., 1958- . II. Title. TK3201.B46 1991 621.381 '3-dc20

3. Antennas (Electronics)

91-18440 CIP

Contents

Preface IX

List of principal symbols XIII

Part One Theory and Problems 11 Electromagnetic theory 3~~~oo 31.1 Maxwell's equations 31.2 Plane waves 91.3 Skin depth 91.4 Power flow in an electromagnetic wave 91.5 Boundary conditions 91.6 Behaviour of a plane wave normally incident on a plane

perfectly conducting boundary 101.7 Behaviour of a plane wave incident obliquely on a plane

perfectly conducting boundary 131.8 Plane wave incident normally on a plane dielectric·

boundary 161.9 Multiple dielectric interfaces 191.10 Plane wave incident obliquely on a plane interface between

two dielectric media 21Problems 25Additional problems 29

2 Transmission line theory 31Introduction 312.1 Transmission line equations 312.2 Solution for an infinite line 342.3 Solutions for a finite line 352.4 Solutions for high frequencies 37

vi Contents

2.5 Quarter-wavelength lines 392.6 Half-wavelength lines 392.7 The distortionless line 40Problems 40Additional problems 44

3 Rectangular and circular waveguides and cavity resonators 50Introduction 503.1 Rectangular waveguides 513.2 Circular waveguides 623.3 Cavity resonators 66Problems 74Additional problems 78

4 Miscellaneous waveguiding systems 83Introduction 834.1 Coaxial line 834.2 Two-wire line 864.3 Parallel-plate transmission line 894.4 Microstrip 944.5 Stripline 984.6 Dielectric waveguides 1004.7 The optical fibre 105Problems 110Additional problems 116

5 Impedance transformation and matching 122Introduction 1225.1 Impedance transformation 1225.2 The Smith chart 1235.3 Matching 131Problems 139Additional problems 144

6 Microwave networks 150Introduction 1506.1 Scattering parameters 1506.2 Matrix forms of four-terminal (two-port) network equations 1576.3 Signal flow graphs 164Problems 171Additional problems 179

7 Antennas and propagationIntroduction7.1 Antennas7.2 Sources, potentials and fields7.3 The Hertzian dipole7.4 Antenna properties7.5 Antenna arrays7.6 Receiving antennas and reciprocity7.7 Aperture antennas7.8 Propagation7.9 Refraction in the ionosphere7.10 Antenna measurementsProblemsAdditional problems

Part Two SolutionsSolutions for Chapter 1Solutions for Chapter 2Solutions for Chapter 3Solutions for Chapter 4Solutions for Chapter 5Solutions for Chapter 6Solutions for Chapter 7

References

Index

Contents vii

184184184185188190194197198201205208211218

221223238254266284305321

340

349

Preface

One of us (FAB) published a book Problems in Electronics with Solutions in1957 which became well established and ran to five editions, the last revisedand enlarged edition appearing in 1976. When the first edition was writtenit covered almost the complete undergraduate electronics courses in engin­eering at universities. One book, at a price students can afford, can no longercover an undergraduate course in electronics. It has therefore been decidedto produce a book covering one important section of such a course usingthe experience gained and a few problems from previous editions ofProblems in Electronics with Solutions.

The book is based largely on problems collected by us over many yearsand given to undergraduate electronic and electrical engineers. Its purposeis 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 andelectrical engineering and physics, research workers, engineers and scientistsin industry and as a reference source.

The material presented will provide a link between the theory of elec­tromagnetic fields, waves and transmission lines and its practical applica­tion. The book is organized in seven chapters treating electromagnetictheory, transmission line theory, rectangular and circular waveguides andcavity resonators, other waveguiding systems, impedance transformationand matching, microwave networks and antennas and propagation. Suffi­cient material and worked examples are included in an introduction to eachchapter to cover the essential points of theory and develop necessaryformulae. To keep the price at a level that is reasonable for students it wasnecessary to limit the number of problems. Some topics which readers mayexpect to find included, e.g. H, trough, groove and dielectric-rod waveguides,filter theory, resonance absorption, tensor permeability, Faraday rotationetc. 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 haveliked.

Over 300 problems are given of which 82 are worked examples in theintroductory sections of chapters and 168 have step-by-step solutions. Thesolutions are separated from the problems so that students will not see themby accident. The answer is also given at the end of each problem, however,for convenience. A thorough grasp of the principles involved in anyparticular problem cannot be obtained by merely reading through thesolution. Students should therefore not consult the solutions until they haveeither repeatedly tried hard and failed to obtain the stated answer orsuccessfully solved the problem and wish to compare the method of solutionwith that given. Some additional problems, totalling 90, with answers butnot solutions, are provided in each chapter as student exercises. Referencesto texts and published papers, together with comments on their content, arelisted; these will serve for further explanation of key points, enable andencourage independent study of a particular subject area in greater depthand 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 areoriginal, but it is impossible to acknowledge the sources of those which arenot. Most of the problems are new, however, and in many cases they havebeen formulated to try to encourage thought and understanding; but somewhich require only numerical substitution in formulae (they are based onpractical data wherever possible) are included in the hope that they willdevelop the student's sense of magnitudes.

While great care has been taken to try to eliminate errors some willinevitably have crept in and we shall be glad to have any such brought toour 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 23of the book Electronic Applications of the Smith Chart-In Waveguide,Circuit and Component Analysis by P. H. Smith and published by McGraw­Hill Inc. in 1969. The authors acknowledge the kindness, in grantingpermission 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 aRegistered Trademark of the Analog Instruments Company.

It has been a pleasure working with Chapman & Hall in particular wewish to thank Mr Daniel H. Brown, Commissioning Editor, ElectronicEngineering, for his courteous help and co-operation and for useful sugges­tions.

We express our gratitude and appreciation to Miss Elaine Jessop andMiss Sally Hollingsworth for their excellent typing of the manuscript.

Preface xi

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

F. A. BENSONDepartment of Electronic andElectrical Engineering,The University of Sheffield

T. M. BENSONDepartment of Electrical andElectronic Engineering,The University of Nottingham

LIST OFPRINCIPAL SYMBOLS

AA

[A] = [~ ~JAAA.At

aaaaa

aaa

a(ax, ay, az

BBBBbb

areaconstant

transfer matrix for four-terminal or two-port network

vectormagnetic vector potentialeffective aperturegain of kth forward path 10 general flow graphequationconstantdimension of rectangular aperture antennalong internal dimension of rectangular waveguidedimension of rectangular cavity resonatorradius of circular waveguide or cavity resonatorradius of inner conductor of coaxial lineradius of each conductor of two-wire lineradius of optical fibre corecomplex amplitudes of incident waves on an n-portnetworkasymmetry factorvector directionsconstantsusceptancebit ratemagnetic flux densityshort internal dimension of rectangular waveguidedimension of rectangular cavity resonator

xiv List of principal symbols

bbbbb b2 ,ooo,b"

cc

cDDDDDdddddo

dvEEETee

e" e,l> e",F

ffc

fefrGG

G

[H] = [HllH 21

Hhhh

radius of outer conductor of coaxial lineseparation of stripline ground planesnormalized mode indexcomplex amplitudes of reflected waves for an n-portnetworkconstantleakage capacitance per unit length of transmissionlinevelocity of light, equal to l/(Jlol'o )1/2 ~ 3 X 108 m s- 1

constantdetector meter readinglargest dimension of antenna aperturedirective gainelectric displacementdistancelength of cylindrical cavityspacing of conductors in two-wire lineseparation of antennas in an arraydiameter of circular cross-section conductor equival­ent to a rectangular strip in stripline designvolume elementmagnitude of electric fieldelectric field vectortotal electric fieldbase of Naperian logarithmsmagnitude of the charge on an electronvector directionsforce on an electronfrequencycritical frequency of ionized layer in upper atmos­pherecut-off frequencyresonant frequencyconstantleakage conductance per unit length of transmissionlinepower gam

H 12J four-terminal (two-port) matrixH n

magnetic field strengthseparation of plates in parallel-plate transmission linethickness of guiding layer in dielectric slot waveguideseparation of top conductor and ground plane inmicrostrip line

hhCUI

I(I]10

iJJc

J d

In(x)J~(x)

KK1

K2

K1

K2

Kn(x)kkkJ,k2

knmaLLLIIMMmmmNNNN1

nnnnn = v/2rr.nmin

nmodepp

List of principal symbols xv

length of a Hertzian dipolecut-off thickness of dielectric slab waveguide

currentunit matrixcurrent amplitudesubscript denoting incidentelectric current densityconduction current densitydisplacement current densityBessel function of the first kind of order nderivative of Bessel function with respect to pconstantconstantconstantphasorphasorHankel function of order n=(w2eJ.1 + y2)112

constantgiven by ki + k~ = k2

nth root of J n(ka) = 0inductance per unit length of transmission linelength of cylindrical cavity resonatordistancelength of transmission linepositive integerconstantmaterial dispersion factorconstantpositive integermass of electronconstantion density of ionized layer in upper atmospherenumber of propagating modes in an optical fibregroup index of optical fibre corepositive integerturns ratio of transformernumber of elements in an antenna arrayindex of refraction(see t/J)minimum value of index of refractionmode or effective indexpoint at which field is evaluatedpower loss per unit area in cavity resonator

xvi List of principal symbols

P time-average power transmitted along waveguidePA power absorbed by an aperturePAL power absorbed in load resistorPc time-average power dissipated in a cavityPOR power density at a receiving antennaPIN total power accepted by an antennaPI time-average power lost in plates of parallel-plate

waveguidePr received powerPR power received by antennaPI power transmitted down parallel-plate waveguidePT power transmitted by antennaPTOT total power radiated by an antennaPw power density in a plane wavep positive real numberp see t

Ph P3 slab waveguide transverse parameterPnm mth roots of equation Jn(ka) =0Pnm' mth roots of equation J~(ka) = 0Q quality factorq chargeq see tq2 slab waveguide transverse parameterR resistance per unit length of transmission lineR distance between source and field pointsRo radiation resistanceRh axial length of horn antennaRL , R) load resistanceR. surface resistanceRT terminating resistanceR. receiving antennar location of field point with respect to some originr radiusr normalized resistancer subscript denoting reflectedrj ratio of permittivitiesr' location of source with respect to some origin[S] scattering matrixS Poynting vectorSu, S/2, S21 ,..., Sij, ..., Snn scattering coefficients or scattering parameters

for n-port network[S]T transpose of [S]s voltage-standing-wave ratios separation of antennas

List of principal symbols xvii

group velocityvelocity of TEM waves in coaxial line dielectricenergy stored in electric fields of resonatorenergy stored in magnetic fields of resonator= W. + Wm total energy stored in resonatorfibre propagation parameterwidth of parallel-plate transmission linewidth of top conductor of microstrip linewidth of central conductor of striplinereactancedistance along transmission lineelectron displacementnormalized reactancerectangular co-ordinatefor TM modes in cylindrical cavityfor TE modes in cylindrical cavityadmittance

normalized frequencyvoltagevoltage applied at input to transmission linevelocity of propagationvolume

see '"phase velocity

y

xxxX

Xnlll = Pnlll

=Pnlll •

v, VI

v, vp

Vo = jPIvg

VTEM

W.Wm

WT

W

W

W

W

X

U

U,UI

Uo = !XIVVVs

vv

tttttt21 etc.tL

V

T thicknessT time slot (l/bit rate)[T], [T]' scattering transmission matricesTll , T12 , T21>"" Tjj , ... , Tnn scattering transmission coefficients or

parametersparameters of scattering transmission matrix withinput waves of n-port network as dependent vari­ables and output waves as independent variablestimesubscript denoting transmittedthickness of microstrip top conductorthickness of central conductor of stripline= exp( - 2"') = p + jqtransmittancestransit timeradiation intensityfibre propagation parameter

see '"

xviii List of principal symbols

Yo = 1/'10Yo = l/zoYL

[YL ] = [Y11

Y21

Yn(X)yyZ

[Z] = [Zl1Z21

ZoZooZl1Z21

ZgZiZio, ZoeZisZ,ZsZTEZHoI

Z

Z

Z

Z

z,

IX

P

characteristic admittancecharacteristic admittanceadmittance of load

admittance matrix for four-terminal network

Bessel function of the second kind of order nnormalized admittancerectangular co-ordinateimpedance

impedance matrix for four-terminal network

characteristic impedance of transmission lineoutput impedance of networkinput impedance of networktransmittanceinternal impedance of generatorinput impedance of transmission lineinput impedance of open-circuited lineinput impedance of short-circuited lineterminating impedance of transmission lineinput impedance (sending-end) of transmission linewaveguide impedance for TE waveswaveguide impedance for TM wavesrectangular co-ordinatedistance along transmission linenormalized impedance.;ylindrical polar co-ordinatenormalized terminating impedance of transmissionlineattenuation constant, real part of the propagationconstantcurrent phase anglephase constant, imaginary part of the propagationconstantphase constant in free spacephase constant along a rectangular waveguideconfinement factorpropagation constantprofile height parameter of optical fibreterms in the general flow graph equationelementary length of periphery of rectangularwaveguide cross-sectionskin depthloss angleconstant

bfL£

£

£0

'1'1'10

,1,0A.cA.gA.h

I.lI.loI.llI.lrpp = Ipl exp(jcp)p(J

(J

CP,CPocpcpcpcpcp

t/Jt/J:/Jt/JOJ

List of principal symbols xix

bandwidth-length producteffective permittivity of ionized mediumpermittivitypermittivity of free space equal to 8.855 x1O-12Fm- 1

effective relative permittivity in microstriprelative permittivityintrinsic impedance or wave impedanceantenna efficiencyimpedance of free space, equal to (I.lO/£0)1/2 = 120nor ~ 377Qangleelectrical length of transmission linecritical angleangle of incidenceangle of reflectionwavelengthwavelength corresponding to resonant frequency ofcylindrical cavityfree-space wavelengthcritical wavelengthguide wavelengthwavelength at the mouth of a horn antennapermeabilitYpermeability of free space equal to 4n x 10- 7 H m- 1permeability of waveguide wall metalrelative permeabilityelectric charge densityreflection coefficientcylindrical co-ordinateconductivityradar cross-sectionanglephase anglemagnetic fluxcylindrical polar co-ordinatescalar potentialsymbol which can represent a component of E or Hsymbol which can represent a component of E or Hu + jv = tanh - 1Z

Ul + jVl = tanh -lzr

progressive phase difference in an antenna arrayangular frequencyangular cut-off frequency