fundamentals of aperture antennas and arrays

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FUNDAMENTALS OFAPERTURE ANTENNASAND ARRAYS

FUNDAMENTALS OFAPERTURE ANTENNASAND ARRAYSFROM THEORY TO DESIGNFABRICATION AND TESTING

Trevor S Bird PhD FTSEPrincipal Antengenuity Adjunct Professor Macquarie University ampHonorary CSIRO Fellow Australia

This edition first published 2016copy 2016 John Wiley amp Sons Ltd

Registered OfficeJohn Wiley amp Sons Ltd The Atrium Southern Gate Chichester West Sussex PO19 8SQ United Kingdom

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The advice and strategies contained herein may not be suitable for every situation In view of ongoing researchequipment modifications changes in governmental regulations and the constant flow of information relating to theuse of experimental reagents equipment and devices the reader is urged to review and evaluate the informationprovided in the package insert or instructions for each chemical piece of equipment reagent or device for amongother things any changes in the instructions or indication of usage and for addedwarnings and precautions The fact thatan organization or Website is referred to in this work as a citation andor a potential source of further information doesnot mean that the author or the publisher endorses the information the organization or Website may provide orrecommendations it may make Further readers should be aware that Internet Websites listed in this work may havechanged or disappeared between when this work was written and when it is read No warranty may be created orextended by any promotional statements for this work Neither the publisher nor the author shall be liable for anydamages arising herefrom

Library of Congress Cataloging-in-Publication Data

Bird Trevor S 1949ndashFundamentals of aperture antennas and arrays from theory to design fabrication and testing Trevor S Bird

pages cmIncludes bibliographical references and indexISBN 978-1-118-92356-6 (cloth alk paper) 1 Aperture antennas I TitleTK78716B49 20166213841 35ndashdc23

2015021598

A catalogue record for this book is available from the British Library

Set in 1012pt Times by SPi Global Pondicherry India

1 2016

This book is dedicated to my mentors and collaborators particularlyPeter JB ClarricoatsGeorge A Hockham

Peter L ArlettGeoffery T PoultonDavid G Hurst

Contents

Preface xiii

Acknowledgement xv

1 Introduction 1References 6

2 Background Theory 721 Maxwellrsquos Equations for Time-Harmonic Fields 7

211 Field Representation in Terms of Axial Field Components inSource-Free Regions 9

212 Boundary Conditions 10213 Poyntingrsquos Theorem 11214 Reciprocity 11215 Duality 13216 Method of Images 13217 Geometric Optics 13

22 Equivalent Sources 15221 Aperture in a Ground Plane 17222 Conformal Surfaces 17

23 Radiation 18231 Near-Field 21232 Far-Field 21233 Mutual Coupling Between Infinitesimal Current Elements 23

24 Problems 26References 27

3 Fields Radiated by an Aperture 2931 Radiation Equations 2932 Near-Field Region 3233 Fresnel Zone 3234 Far-Field Region 33

341 Example of a Uniformly Illuminated Rectangular Aperture 3835 Radiation Characteristics 40

351 Radiation Pattern 41352 Half-Power Beamwidth 42353 Front-to-Back Ratio 42354 Polarization 42355 Phase Centre 44356 Antenna Gain and Directivity 44357 Effective Aperture 46358 Radiation Resistance 47359 Input Impedance 473510 Antenna as a Receiver 48

36 Aberrations 4837 Power Coupling Theorem 5038 Field Analysis by High-Frequency Methods 52

381 Asymptotic Physical Optics 533811 Example Scattering Radiation from Large Conducting

Wire Loop 553812 Special Case APO in Two Dimensions 57

382 Geometrical Theory of Diffraction 6139 Problems 67References 70

4 Waveguide and Horn Antennas 7141 Introduction 7142 Radiation from Rectangular Waveguide 7243 Pyramidal Horn 74

431 Design of a Standard Gain Pyramidal Horn 79432 Dielectric-Loaded Rectangular Horn 81

44 Circular Waveguides and Horns 85441 Circular Waveguide 86

4411 Matching at a Circular Aperture 90442 Coaxial Waveguide 91

4421 Matching of a Coaxial Aperture 954422 Coaxial Apertures with an Extended Central Conductor 97

443 Conical Horn 101444 Corrugated Radiators 105445 Cross-Polarization 110

45 Advanced Horn Analysis Topics 114451 Flange Effects 114452 Mode Matching in Horns 115

viii Contents

453 Profiled Horns 1234531 Optimization 1264532 Parametric Profiles 126


5 Microstrip Patch Antenna 13751 Introduction 13752 Microstrip Patch Aperture Model 13853 Microstrip Patch on a Cylinder 14354 Problems 146References 147

6 Reflector Antennas 14961 Introduction 14962 Radiation from a Paraboloidal Reflector 150

621 Geometric Optics Method for a Reflector 1526211 Dipole Feed 1546212 Circular Waveguides and Horn Feeds 157

622 Edge Taper and Edge Illumination 160623 Induced Current Method 162

6231 Radiation from Symmetrical Reflectors with General Profile 1646232 Spherical Reflector 167

624 Receive-Mode Method 16863 Focal Region Fields of a Paraboloidal Reflector 172

631 Asymptotic Representation of the Scattered Field 17664 Blockage 18165 Reflector Antenna Efficiency 18366 Reflector Surface Errors 18867 Offset-fed Parabolic Reflector 18968 Cassegrain Antenna 196

681 Classical Cassegrain 196682 Offset Cassegrain Antenna 198

69 Shaped Reflectors 202691 Reflector Synthesis by Geometric Optics 203692 Reflector Synthesis by Numerical Optimization 209


7 Arrays of Aperture Antennas 21971 Introduction 21972 Two-Dimensional Planar Arrays 219

721 Rectangular Planar Array 221722 Hexagonal Array 223

73 Mutual Coupling in Aperture Antennas 228731 Infinite Periodic Arrays 230

ixContents

732 Finite Arrays 235733 Mutual Impedance and Scattering Matrix Representation 239734 Analysis of Arrays of Aperture Antennas by Integral Equation Methods 242

7341 Moment Method Approach 2457342 Mode Matching in Arrays 247

735 Mutual Coupling Analysis in Waveguide Apertures 2497351 Rectangular Waveguide Arrays 2497352 Self-Admittance of TE10 Mode 2537353 Arrays of Circular and Coaxial Waveguides 2577354 Self-Admittance of TE11 Mode in Circular Waveguide 2627355 Mutual Coupling in Other Geometries 2667356 Waveguide-Fed Slot Arrays 2697357 Arrays of Microstrip Patches 2737358 A Numerical Formulation of Coupling in Arbitrary

Shaped Apertures 278736 An Asymptotic Expression for Mutual Admittance 281737 Radiation from Finite Arrays with Mutual Coupling 284

74 Techniques for Minimizing Effects of Mutual Coupling 286741 Element Spacing 286742 Aperture Field Taper 287743 Electromagnetic Fences 287744 Mutual Coupling Compensation 287745 Power Pattern Synthesis Including the Effect of Mutual Coupling 289

75 Low-Sidelobe Arrays and Shaped Beams 28976 Problems 300References 302

8 Conformal Arrays of Aperture Antennas 30781 Introduction 30782 Radiation from a Conformal Aperture Array 308

821 Waveguide with E-Field Polarized in Circumferential Direction 308822 Waveguide with E-Polarized in Axial Direction 315823 Historical Overview of Asymptotic Solutions for Conformal Surfaces 317

83 Mutual Coupling in Conformal Arrays 319831 Asymptotic Solution for Surface Dyadic 322

84 Coupling in a Concave Array Periodic Solution 32585 Problems 331References 331

9 Reflectarrays and Other Aperture Antennas 33591 Introduction 33592 Basic Theory of Reflectarrays 33793 Extensions to the Basic Theory 34194 Other Aperture Antennas 344

941 Lenses 344942 FabryndashPeacuterot Resonator Antennas 352

x Contents


10 Aperture Antennas in Application 357101 Fabrication 357

1011 Machining 3571012 Printing 3581013 Mould Formation 3581014 Electroforming 3581015 Lightweight Construction 3581016 Pressing and Stretch Forming of Reflector Surfaces 3591017 Assembly and Alignment 360

102 Measurement and Testing 3611021 Far-Field Measurement 3611022 Near-Field Measurement 3641023 Intermediate-Field Measurement 369

103 Modern Aperture Antennas 3711031 Compact Low-Sidelobe Horns 3711032 Multibeam Earth Station 3751033 Radio Telescopes 379


Appendix A Useful Identities 391A1 Vector Identities 391A2 Geometric Identities 392A3 Transverse Representation of the Electromagnetic Field 393A4 Useful Functions 394References 394

Appendix B Bessel Functions 395B1 Properties 395B2 Computation of Bessel Functions 400References 401

Appendix C Proof of Stationary Behaviour of Mutual Impedance 403

Appendix D Free-Space Dyadic Magnetic Greenrsquos Function 405Reference 406

Appendix E Complex Fresnel Integrals 407References 409

Appendix F Properties of Hankel Transform Functions 411References 412

xiContents

Appendix G Properties of Fock Functions for Convex Surfaces 413G1 Surface Fock Functions 413

G11 Soft Surface Functions (m gt 0) 414G12 Hard Surface Fock Functions (m lt 0) 415

G2 Acoustic Fock Functions 417G21 Soft Acoustic Fock Function 418G22 Hard Acoustic Fock Function 419

References 421

Index 423

xii Contents

Preface

Aperture antennas are a class of antennas in common daily use and some have even becomesynonymous with areas of science and technology Typical examples include reflectors hornslenses waveguides slits and slots Other antennas can be conveniently described by means ofaperture concepts Some of these include microstrip patches and reflectarrays In this book wedescribe the underlying theory and application of these antennas as well as their use in arraysThe history of aperture antennas is inextricably linked with historical developments in wire-

less and also the verification of Maxwellrsquos equations The very first waveguide was demon-strated by Lodge in 1894 and in 1895 Bose used circular waveguides as an antenna alongwith pyramidal horns for experiments on the polarization properties of crystals About thirtyyears later a 10 m diameter reflector became the first radiotelescope when it detected emissionsfrom electrons in interstellar space In the 1960s aperture antennas accompanied the firsthumans on the moon and more recently they have contributed to the wireless revolution thatis presently underwayAperture antennas are normally associated with directional beams and indeed this is their

role in many applications They can also occur on non-planar or curved surfaces such as onaircraft or ground-based vehicles These antennas may consist of a single radiator or in arraysIn this form they are often used to provide directional or shaped beamsDirectional beams are needed in terrestrial and satellite microwave links to efficiently use the

available power as well as to reduce interference and noise Radar systems also require direc-tional antennas to identify targets As well arrays of aperture antennas can produce almostomnidirectional radiationA limitation of a directional planar antenna is that when it is scanned from broadside the beam

broadens and the pattern deterioratesWhen the antenna is conformal to a convex surface such as acylinder or a cone the beam can be scanned in discrete steps through an arc while maintaining aconstant pattern Of importance in the design of low sidelobe antenna arrays both planar and con-formal is predicting the effect of mutual coupling between the array elements Maximum perfor-mance is achieved from arrays when the coupling between elements is fully taken into accountThis book gives an introduction to the techniques that are used to design common aperture

antennas as well as some approaches to their fabrication and testing The intention is for it to bea single textbook for a course in antennas in the final year undergraduate or in a masterrsquos degreeby coursework It assumes that the reader has undertaken a course on Maxwellrsquos equationsfields and waves Some of these topics are reviewed in the early few chapters to provide con-tinuity and background for the remainder of the book The antennas covered in later chapters

include horns reflectors and arrays Some examples are pyramidal and corrugated horns par-abolic and spherical reflectors reflectarrays planar lenses and coaxial waveguide array feedsTo provide more than a simplified treatment of arrays the topic of mutual coupling is coveredin more detail than most similar books on this topic Also included is an introduction to sourcesand arrays on non-planar surfaces which is of importance for applications involving aerody-namic surfaces and for making aperture antennas unobtrusive A chapter is included on modernaperture antennas that extend the concepts introduced in earlier chapters This is to show whereadvances have been made in the past and how they could be made in the future Also includedare some topics of a practical nature detailing some techniques for fabrication of aperture anten-nas and their measurement

xiv Preface

Acknowledgement

The author thanks his many colleagues who have provided comments on some of the materialand for their contribution to the projects described herein In particular for this book thanks areextended to colleagues Drs Stuart Hay Doug Hayman Nasiha Nikolic Geoff James Stepha-nie Smith and Andrew Weily who readily gave up time to providing comments on early draftsof the chapters Thanks are extended to members of the editorial staff at Wiley in particularAnna Smart Sandra Grayson and Teresa Nezler and also to the SPi publications content man-ager Shiji Sreejish The kind permission to use pictures by CSIRO Australia the Institution ofEngineering and Technology (IET) Engineers Australia and SES ASTRA is gratefullyacknowledgedThe author also wishes to acknowledge in particular several folk who provided mentorship

and gave great support and encouragement during his early research career To mention a fewsignificant mentors by name Prof Peter Clarricoats FRS gave the author a great start in thearea of aperture antennas and arrays the late Dr George Hockham FRS inspired the author anddemonstrated the importance of some of the techniques described herein for applications andhow practical antennas could be realised Dr Geoff Poulton provided a number of importantopportunities for the author and long-term friendship and the many technical discussions duringour working careers and finally the late Mr David Hurst helped the author in his first attemptsin the wireless area graciously and shared his wide knowledge and practical expertise Theauthor wishes to thank Ms Dallas Rolph for assisting greatly in editing the text as well as gen-eral assistance when they worked together at CSIRO Finally the author is very grateful to hiswife Val for her love and strong support over 40 years when many of the topics discussedherein were investigated

1Introduction

The topic of this book covers a class of antenna in common use today as well as a way ofdescribing many others Examples include waveguides horns reflectors lenses slits slotsand printed antennas Some examples are illustrated in Figure 11 In the following chaptersthe background theory and application of some basic forms of these antennas are described aswell as how they can be designed fabricated and tested Additionally detail will be provided onsome of the individual antennas pictured in Figure 11Aperture antennas are normally associated with directional radiation beams and indeed this

is their purpose in many applications They can also create other types of beams such as shapedor contoured beams either separately or combined as arrays as will be shown Aperture anten-nas can also occur on non-planar or conformal surfaces such as on aircraft or missile bodieswhere airflow and aerodynamic performance are paramount Conformal antennas can consistof a single radiator or arrays in the surface where they can be used to provide directional andshaped beamsAperture antennas can be used to produce omnidirectional radiation patterns which are

important if the antenna platform is unstable or the user direction is unknown for all-roundelectronic surveillance and monitoring or where the location of another user cannot be guar-anteed such as in mobile radio systems A 360-degree coverage can be achieved with a con-formal antenna or with electronic switching between planar elementsDirectional beams are required in terrestrial and satellite microwave links to efficiently use

the available power as well as to reduce interference and noise Directional antennas are alsorequired in radar systems to identify targets A limitation of a directional planar antenna is thatwhen it is scanned from broadside (typically boresight) the beam broadens and the pattern dete-riorates When the antenna is conformal to a convex surface such as a cylinder or a cone the

Fundamentals of Aperture Antennas and Arrays From Theory to Design Fabrication and TestingFirst Edition Trevor S Birdcopy 2016 John Wiley amp Sons Ltd Published 2016 by John Wiley amp Sons LtdCompanion website wwwwileycomgobird448

(a) (b)

(e) (f)

(c) (d)

Figure 11 Examples of aperture antennas (a) Open-ended waveguide antennas (right to left) coaxialcircular and rectangular (b) Circular waveguide (diameter 327 mm) with three ring-slots designed foroperation at 9 GHz (c) Feed array of pyramidal horns for 1225ndash1275 GHz (d) 11ndash145 GHz high-performance circular corrugated feed horn diameter 273 mm and flare angle 118 (e) Smallparaboloidal reflector and rear waveguide feed designed for a 15 GHz microwave link (f ) 64 m Parkesradio telescope is a front-fed paraboloid (fD = 0408) This versatile instrument has been used forfrequencies from 30MHz to gt90 GHz Source Reproduced with permission from CSIRO (andashf)

2 Fundamentals of Aperture Antennas and Arrays

(g) (h)

(l)

(i) (j)

(k)

Figure 11 (continued) (g) Two multibeam earth station antennas at Danish Radiorsquos multimediahouse in restad in Copenhagen Denmark covering different segments of the geostationary satellite arc(h) Multibeam feed system for the Parkes radio telescope Source Reproduced with permission fromCSIRO (i) On-board Ku-band satellite antennas under test on an outdoor test range prior to launch(j) Dual-offset Cassegrain antenna with a waveguide array feed cluster under test in anechoic chamber(Bird amp Boomars 1980) (k) Series-fed microstrip patch array for a microwave landing systemSource Reproduced from INTERSCAN International Ltd (l) Conformal array of rectangularwaveguides (2286 times 1016 mm) on a cylinder of radius 12624 mm Source Picture courtesy of PlesseyElectronic Systems

3Introduction

beam can be scanned in discrete steps through an arc while maintaining a constant patternRecent developments in microwave and optical components have simplified the design of feednetworks thereby making conformal antennas and arrays attractive alternatives for directiveapplications as well as for scanned beam and in ultra-low sidelobe antennas Of importancein the design of the latter both planar and conformal antenna arrays are often employedand in this application predicting the effect of mutual coupling between the array elementsshould be undertaken Maximum performance is achieved from arrays when the effects of cou-pling are known and included in the design Otherwise the full potential of the array flexibilitymay not be realizedAperture antennas may be analysed in much the same way as the conceptually simpler wire

antennas First the designer needs to find the currents on the conductors or in other materialsfrom which the antenna is constructed To do this exactly is usually impossible except in a fewidealized cases and numerical methods are required to obtain approximate solutions After thecurrents are known the radiated fields are obtained from Maxwellrsquos equations Sometimeshowever adequate design information may be obtained from simplified approximations tothe current similar in some regards to adopting a sinusoidal current approximation on a linearwire antenna This approach is especially valuable for analysing the far-field radiation charac-teristics which are relatively insensitive to second-order variations in the current distributionHowever for more detailed information or quantities such as the input impedance reflectioncoefficient at the input of horns or the effects of mutual coupling from nearby antennas anaccurate representation of the currents is usually required to properly take account of the currentvariations and near-field behaviourThe representation of actual currents on the antenna structure may be difficult or impossible

to achieve analytically because of the geometry and materials involved It is convenient andalso physically allowable to replace the actual sources by equivalent sources at the radiatingsurface the antenna lsquoaperturersquo which need not lie on the actual antenna surface but on anotheroften fictitious surface close by For example the aperture of a paraboloid reflector may be theprojection of the rim onto a suitable plane These equivalent sources are used in the same way asactual sources to find the radiated fields Once these fields are known an assessment of theantennarsquos performance can be madeFor the engineer wishing to specialize in the area of communications systems some knowl-

edge is needed of the theory and design of aperture antennas The intention of this book is toprovide some of this basic information Today compared with prior to the 1980s and even ear-lier a variety of full wave computer solvers are now available and are particularly valuable forfinal design and analysis The fundamental material available in this book is important as astarting point and for understanding the physical nature of the antenna structure before moredetailed design is undertaken It is intended that readers should be able to move from the presentmaterial to more specialized topics and to the research literature In addition the details pro-vided herein should help the non-specialist in antennas to critically assess aperture antenna spe-cifications Where possible useful design information has also been included An underlyingassumption is that the reader is familiar with the basic concepts of electromagnetic fields wavesand radiation as presented in a variety of excellent textbooks (Harrington 1961 Jones 1964Jordan amp Balmain 1968 Kraus amp Carver 1973 Johnk 1975) Some topics of a moreadvanced nature have also been included here beyond those of a typical introductory courseThese are indicated by an asterisk (lowast) after the section heading They have been included as


possible extensions from standard material for more specialized courses research or possiblypart of a projectThe material included here is based on notes for several courses in antennas given to fourth

year students in Electrical Engineering at James Cook University of North Queensland and alsoat the University of Queensland in the 1980s At that time there was no suitable modern text-book available on antennas for undergraduate teaching Since then several excellent textbookshave appeared (Balanis 1982) In addition the notes were found useful over the years by mem-bers of my research group at CSIRO Other relevant material had been developed on mutualcoupling for presentation at several symposia held in the 1990s and some of this informationhas been included here As might be anticipated practical topics of relevance that were encoun-tered during my research career have been included as wellThe purpose of this book is to provide a stand-alone textbook for a course in antennas pos-

sibly in the final undergraduate years or in a masterrsquos degree by coursework It should also beuseful for PhD candidates and practising engineers For continuity some background electro-magnetics fields and waves are includedThe antennas described in detail include horns reflectors lenses patch radiators and arrays

of some of these antennas Because of its importance and to provide more than a superficialtreatment of arrays the topic of mutual coupling is covered in greater detail than most similarbooks in the area Also included is an introduction to sources and arrays on non-planar surfaceswhich is important for applications involving aerodynamic surfaces and for making apertureantennas unobtrusive An introduction to the fabrication and test of aperture antennas isincluded as well as some recent examples of themThe theory needed for analysing aperture antennas is given in Chapter 3 Material is also

included for handling conformal aperture antennas Starting with the concept of equivalentsources the equations for radiation from an aperture are developed from the fields radiated bya small electric dipole and a small loop of current The basic theory that is needed for moredetailed development is also provided This includes details of the far-field radiation fromuniformly illuminated rectangular and circular apertures and also how phase aberrations on theaperture impact the far-fields The radiation from waveguide and horn aperture antennas aredescribed in Chapter 4 and material is included for the radiation from rectangular waveguideantenna Thismodel is used as a basis for detailed description of the pyramidal horn The radiationproperties of circular waveguides and horns are reviewed in this chapter and details are providedon the corrugated horn A simple model of the microstrip patch antenna is given in Chapter 5along with details of the radiation properties of these antennas The purpose is to describe anotherform of aperture antenna and as background for reflectarrays The properties of reflector antennasin common use are described in Chapter 6 including the paraboloid the Cassegrain and spheroidgeometries as well as some offset counterparts Planar arrays of aperture antennas and mutualcoupling in arrays are detailed in Chapter 7 This is followed in Chapter 8 by similar detailsfor apertures on conformal surfaces The areas of arrays and reflectors come together in thereflectarray antenna which is introduced in Chapter 9 This chapter also includes details of someother aperture antennas not treated elsewhere in particular lenses and the Fabry-Peacuterot cavityantennas Finally some possible approaches for the fabrication and testing of aperture antennasare described in Chapter 10 In addition it includes examples of some aperture antennas that makeuse ofmany of the techniques covered earlier in the bookAt all times the intention is an emphasison fundamentals and where possible practical information for design is also included

5Introduction

ReferencesBalanis CA (1982) lsquoAntenna theory analysis and designrsquo Harper and Row New YorkBird TS and Boomars JL (1980) lsquoEvaluation of focal fields and radiation characteristics of a dual-offset reflector

antennarsquo IEE Proc (Pt H) Microwav Optics Antennas Vol 127 pp 209ndash218 Erratum IEE Proc (Pt H) Vol128 1981 p 68

Harrington RF (1961) lsquoTime-harmonic electromagnetic fieldsrsquo McGraw-Hill New YorkJohnk CTA (1975) lsquoEngineering electromagnetic fields and wavesrsquo John Wiley amp Sons Inc New YorkJones DS (1964) lsquoThe theory of electromagnetismrsquo Pergamon Press London UKJordan EC and Balmain KG (1968) lsquoElectromagnetic waves and radiating systemsrsquo 2nd ed Prentice-Hall Eagle-

wood Cliffs NJKraus JD and Carver KR (1973) lsquoElectromagneticsrsquo 2nd ed McGraw-Hill International Student Edition Kaga-

kuska Ltd Tokyo Japan


2Background Theory

In this chapter some background theory is provided and notation is introduced in preparationfor use throughout the remainder of this text The equations that were devised by Maxwell andplaced in differential form by Heaviside and Hertz are introduced Throughout this book allfield and sources are assumed to be time harmonic and the formulation of the field equationsand their consequences will be explored under this limitation The important concepts of fieldduality equivalent sources and image theory are summarized Finally radiation from elemen-tary sources is investigated and this allows a description of some basic radiation parameters aswell as an introduction to mutual coupling

21 Maxwellrsquos Equations for Time-Harmonic Fields

The instantaneous vector field quantity r t may be expressed in terms of a complex vectorfield A(r) where all fields and sources have a time-harmonic dependence as follows

r t =Re A r exp jωt 2 1

where bold type face indicates vector quantities ω= 2πf is the angular frequency (rads)t denotes time (s) and f is the frequency (Hz) of the harmonic oscillationField and source quantities are defined as follows (MKS units given in square brackets)

E(r) = Electric field intensity [V mminus1]H(r) = Magnetic field intensity [A mminus1]J(r) = Electric current density [A mminus2]


M(r) =Magnetic current density [V mminus2]ρe(r) = Electric charge density [C mminus3]ρm(r) = Magnetic charge density [Wbmminus3]

where V is the volt A is the Ampere m is metre Wb is Weber and C is the CoulombThe equations governing the interaction of these fields and sources are known as Maxwellrsquos

equations after James Clerk Maxwell (1831ndash1879) who first presented them in componentform and in terms of potentials from the earlier results of Faraday Oumlersted Ampere Weberand others Oliver Heaviside (1850ndash1925) and independently Heinrich Hertz (1857ndash1894)reduced these 20 equations to the four vector field equations that are essentially used today(Sarkar et al 2006) For Heaviside the concepts of fields symmetry and vector notation werevital With the present assumption that fields and sources vary harmonically with timeMaxwellrsquos equations are expressed as follows

nablatimesE=Mminus jωμH 2 2a

nablatimesH= J+ jωεE 2 2b

nabla E=ρeε

2 2c

nabla H=ρmμ 2 2d

where nabla is the gradient operator nablatimes is the curl operation nabla denotes divergence ε is theelectric permittivity [F mminus1] and μ is the magnetic permeability [H mminus1]A general field may be considered as the superposition of the fields due to two types of

sources respectively electric (e) and magnetic (m) as follows

E=Ee +Em and H =He +Hm

The partial field pairs (EeHe) and (EmHm) satisfy separate sets of Maxwellrsquos equations asshown in Table 21 and originate from electric or magnetic sources The former is due to phys-ical electric currents and charges while the latter is due to magnetic currents and charges whichare of an equivalent type andwere introduced to maintain the symmetry ofMaxwellrsquos equations(Harrington 1961 Jones 1964) More will be said about equivalent sources in the followingsections

Table 21 Maxwellrsquos equations for electric and magnetic sources

Electric J 0 M= 0 Magnetic J= 0 M 0

nablatimesEe = minus jωμHe nablatimesHm = jωεEm

nablatimesHE = J+ jωεEe nablatimesEm =Mminus jωμHm

nabla Ee = ρe ε nabla Hm = ρm μnabla He = 0 nabla Em = 0


211 Field Representation in Terms of Axial Field Componentsin Source-Free Regions

In problems involving sections of uniform structures that guide electromagnetic waves suchas waveguides and transmission lines when J and M are absent it is convenient to representall field components in terms of the field components in the direction of propagation that is inthe direction of uniformity By convention this direction is usually taken as the z-component ina cylindrical co-ordinate system with directions denoted by (uvz) It is recognized that the fieldcomponents Ez(uvz) and Hz(uvz) satisfy Helmholtz wave equations where (uv) are trans-verse co-ordinates The guiding structures are assumed to exhibit reflection symmetry andtherefore it is sufficient to represent the total field as the superposition of forward and reversetravelling wave solutions in the z-direction For time harmonic fields of the type defined byEq 21 a forward travelling wave (in the +z-direction) has the following fields

E uvz = Et uv + zEz uv exp minus jγz 2 3a

H uvz = Ht uv + zHz uv exp minus jγz 2 3b

where Et(u v) andHt(u v) are the transverse electric and magnetic field vectors and γ = βminus jα isthe complex propagation constant β is the phase shift per unit length and α is the attenuationconstant For lossless structures α= 0 With the field represented by Eqs 23 the transversefield components can be obtained from Maxwellrsquos equations in the following form

k2zEt uv = jωμz timesnablatHzminus jγnablatEz 2 4a

k2zHt uv = minus jωεztimesnablatEzminus jγnablatHz 2 4b

where k2z =ω2εμminusγ2 is the axial wave number and nablat is the transverse gradient operator

For homogeneous materials the permittivity and permeability are ε = εrεo and μ= μrμorespectively where εr is the relative permittivity μr is the relative permeability εo = 8 854 times10minus12 Fm and μo = 4π times 10

minus7 Hm are the permittivity and permeability of free-space The firstterm on the right-hand side of k2z namely k =ω εμ is the propagation constant of a planewave in the homogeneous medium In free-space εr = 1 and k = 2π λ = ko =ω μoεo =ω c

where c= 1 εoμo is the free-space wave velocity and equals c = 2 99859 times 108msminus1 Ratiosof components of E andH in Eqs 24 have dimensions of impedance and are referred to as the

wave impedance In a general medium the intrinsic impedance is η= μ ε Ω By substitutingEqs 24 into Eqs 22b and 22d it can be shown that the axial field components satisfy thefollowing scalar wave equations

nabla2t + k

2z

Ez uv

Hz uv= 0 2 5

It is seen from Eq 25 that k2z is also the transverse wavenumber If the co-ordinates u andv are separable there will be separation constants in these directions as well For fields thatare TE to the propagation direction Ez = 0 and the simultaneous pair of wave equations

9Background Theory

simplify to a solution of the wave equation in Hz only and Eqs 24a and 24b becomeas follows

k2zEt uv = jωμz timesnablatHz 2 6a

k2zHt uv = minus jγnablatHz 2 6b

where now k2z is constant for a fixed geometry and is the cut-off wavenumber of the guidingstructure For example for the TE modes of circular cylindrical waveguide of radius a kz is aroot of the derivative of the Bessel function (see Appendix B) namely Jn kza = 0 where n isthe azimuthal period that arises in the solution of the wave equation in the azimuthal co-ordinateΦ in the transverse plane Similarly a field that is transverse magnetic (TM) to the propagationdirection is obtained by setting Hz = 0 and an equivalent simplification occurs in Eqs 24namely

k2zEt uv = minus jγnablatEz 2 7a

k2zHt uv = minus jωεztimesnablatEz 2 7b

212 Boundary Conditions

Consider a volume that is divided into two regions 1 and 2 by a surface S as shown in Figure 21There are currents on this surface namely an electric surface current Js [A mminus1] and a magneticsurface currentMs [V mminus1] On either side of a surface of discontinuity the field pairs (E1H1)and (E2H2) satisfy the following continuity conditions

Js = ntimes H2minusH1 2 8a

Ms = minusn times E2minusE1 2 8b

Thus the tangential components are discontinuous by an amount equal to the current at thesurface The associated boundary conditions for the normal components to the surface are asfollows

n

E2 H2

E1 H1

S

Figure 21 Fields on either side of a boundary surface S


ρes = n ε2E2minusε1E1 2 9a

ρms = nH2

μ2minusH1

μ1 2 9b

where ρes is the electric surface charge in [C mminus2] and ρms is the magnetic surface charge[Wb mminus2] In many physical problems ρes = 0 and so n ε2E2minusε1E1 = 0 No physicalmagnetic surface charges have been detected and therefore Eq 29b is usuallyexpressed n H2 μ2minusH1 μ1 = 0

213 Poyntingrsquos Theorem

The time-averaged conservation of energy in the electromagnetic field is given by Poyntingrsquostheorem In an isotropic medium of volume V with permeability μ permittivity ε and conduc-tivity σ this is

12 Σ

EtimesHlowast ndS =12 V

jω μH HminusεE E + σE E dV 2 10

where Σ is the surface bounding V On left-hand side n is the normal to the surface and is direc-ted into Σ The quantity P= 1 2 EtimesHlowast is the power density entering V that is called the com-plex Poynting vector The integral of this vector over the closed surface Σ is the power input PIOn the right-hand side the three terms are related to from left to right the energy stored inthe magnetic field Wm the energy stored in the electric field We and the power lost due toconduction loss PL Expressed succinctly Eq 210 is PI = 2jω WmminusWe +PL where

Wm =14Re

VμH HdV

We =14Re

VεE EdV and

PL =12σ

VμE EdV

In an ideal lossless medium σ = 0 and as a consequence PL = 0 Therefore PI = 2jωWmminusWe This says that the input power converts totally to energy in the fields which istotally reactive and is the difference of the energies stored in the magnetic and electric fields

214 Reciprocity

Of importance in all types of antenna systems is the relationship between the receiving andtransmitting fields In more general terms the response in the vicinity of one source due tofields from a second source and the relationships when the roles are reversed are of particular

11Background Theory

significance Suppose there are two sources in a region denoted by a and b Thus the sourcepairs are Jas M

as and Jbs M

bs These produce fields (EaHa) and (EbHb) respectively that

satisfy their own sets of Maxwellrsquos equations as shown in Table 21 Making use of the vectoridentity

nabla Ea timesHb =Hb nablatimesEaminusEa nablatimesHb

and the Maxwell curl relations it follows that

nabla Ea timesHb =Hb Mas minus jωμH

a HbminusEa Jbs minus jωεEa Eb 2 11a

Similarly

nabla Eb timesHa =Ha Mbs minus jωμH

a HbminusEb Jas minus jωεEa Eb 2 11b

Subtraction of Eq 211b from Eq 211a results in

nabla Ea timesHbminusEb timesHa = minusEa Jbs +Eb Jas +H

a Mbs minusH

b Mas

At any point within the region where the sources are not present the right-hand side is zero

nabla Ea timesHbminusEb timesHa = 0 2 12

This result is called the Lorentz reciprocity theorem When Eq 212 is integrated throughoutthe source-free region Σ the divergence theorem allows it to be expressed as follows

ΣEa timesHbminusEb timesHa ndS= 0 2 13

where the integral sign refers to a closed surface with volume V When sources are containedwithin the surface the result is

ΣEa timesHbminusEb timesHa ndS=

VminusEa Jbs +E

b Jas +Ha Mb

s minusHb Ma

s dV 2 14

If the surface is a sphere with a very large radius as the fields decay as 1r the integral on theleft-side limits to zero As a result the right-side of Eq 214 gives

VEa Jbs minusH

a Mbs dV =

VEb Jas minusH

b Mas dV 2 15

where V is now all space The two integrals on the left and right side of Eq 215 are termedreaction integrals Eq 215 is sometimes expressed as the reaction of field a on source b is thesame as the reaction of field b on source a When a and b are the same the integral is called self-reaction Although not immediately obvious Eq 213 is also applicable when the volume Vcontains all sources


FUNDAMENTALS OFAPERTURE ANTENNASAND ARRAYS















2015021598



1 2016



Contents

Preface xiii

Acknowledgement xv






















viii Contents
















ixContents













x Contents













xiContents




References 421

Index 423

xii Contents

Preface








xiv Preface

Acknowledgement



1Introduction






(a) (b)

(e) (f)

(c) (d)



(g) (h)

(l)

(i) (j)

(k)


3Introduction











5Introduction







2Background Theory













nabla E=ρeε

2 2c

nabla H=ρmμ 2 2d























nabla2t + k

2z

Ez uv

Hz uv= 0 2 5


9Background Theory












n

E2 H2

E1 H1

S




ρms = nH2

μ2minusH1

μ1 2 9b




12 Σ




Wm =14Re

VμH HdV

We =14Re

VεE EdV and

PL =12σ

VμE EdV


214 Reciprocity


11Background Theory


as and Jbs M







Similarly





a Mbs minusH

b Mas







VminusEa Jbs +E

b Jas +Ha Mb

s minusHb Ma

s dV 2 14


VEa Jbs minusH

a Mbs dV =

VEb Jas minusH

b Mas dV 2 15

















2015021598



1 2016



Contents

Preface xiii

Acknowledgement xv






















viii Contents
















ixContents













x Contents













xiContents




References 421

Index 423

xii Contents

Preface








xiv Preface

Acknowledgement



1Introduction






(a) (b)

(e) (f)

(c) (d)



(g) (h)

(l)

(i) (j)

(k)


3Introduction











5Introduction







2Background Theory













nabla E=ρeε

2 2c

nabla H=ρmμ 2 2d























nabla2t + k

2z

Ez uv

Hz uv= 0 2 5


9Background Theory












n

E2 H2

E1 H1

S




ρms = nH2

μ2minusH1

μ1 2 9b




12 Σ




Wm =14Re

VμH HdV

We =14Re

VεE EdV and

PL =12σ

VμE EdV


214 Reciprocity


11Background Theory


as and Jbs M







Similarly





a Mbs minusH

b Mas







VminusEa Jbs +E

b Jas +Ha Mb

s minusHb Ma

s dV 2 14


VEa Jbs minusH

a Mbs dV =

VEb Jas minusH

b Mas dV 2 15





Contents

Preface xiii

Acknowledgement xv






















viii Contents
















ixContents













x Contents













xiContents




References 421

Index 423

xii Contents

Preface








xiv Preface

Acknowledgement



1Introduction






(a) (b)

(e) (f)

(c) (d)



(g) (h)

(l)

(i) (j)

(k)


3Introduction











5Introduction







2Background Theory













nabla E=ρeε

2 2c

nabla H=ρmμ 2 2d























nabla2t + k

2z

Ez uv

Hz uv= 0 2 5


9Background Theory












n

E2 H2

E1 H1

S




ρms = nH2

μ2minusH1

μ1 2 9b




12 Σ




Wm =14Re

VμH HdV

We =14Re

VεE EdV and

PL =12σ

VμE EdV


214 Reciprocity


11Background Theory


as and Jbs M







Similarly





a Mbs minusH

b Mas







VminusEa Jbs +E

b Jas +Ha Mb

s minusHb Ma

s dV 2 14


VEa Jbs minusH

a Mbs dV =

VEb Jas minusH

b Mas dV 2 15

















viii Contents
















ixContents













x Contents













xiContents




References 421

Index 423

xii Contents

Preface








xiv Preface

Acknowledgement



1Introduction






(a) (b)

(e) (f)

(c) (d)



(g) (h)

(l)

(i) (j)

(k)


3Introduction











5Introduction







2Background Theory













nabla E=ρeε

2 2c

nabla H=ρmμ 2 2d























nabla2t + k

2z

Ez uv

Hz uv= 0 2 5


9Background Theory












n

E2 H2

E1 H1

S




ρms = nH2

μ2minusH1

μ1 2 9b




12 Σ




Wm =14Re

VμH HdV

We =14Re

VεE EdV and

PL =12σ

VμE EdV


214 Reciprocity


11Background Theory


as and Jbs M







Similarly





a Mbs minusH

b Mas







VminusEa Jbs +E

b Jas +Ha Mb

s minusHb Ma

s dV 2 14


VEa Jbs minusH

a Mbs dV =

VEb Jas minusH

b Mas dV 2 15


















ixContents













x Contents













xiContents




References 421

Index 423

xii Contents

Preface








xiv Preface

Acknowledgement



1Introduction






(a) (b)

(e) (f)

(c) (d)



(g) (h)

(l)

(i) (j)

(k)


3Introduction











5Introduction







2Background Theory













nabla E=ρeε

2 2c

nabla H=ρmμ 2 2d























nabla2t + k

2z

Ez uv

Hz uv= 0 2 5


9Background Theory












n

E2 H2

E1 H1

S




ρms = nH2

μ2minusH1

μ1 2 9b




12 Σ




Wm =14Re

VμH HdV

We =14Re

VεE EdV and

PL =12σ

VμE EdV


214 Reciprocity


11Background Theory


as and Jbs M







Similarly





a Mbs minusH

b Mas







VminusEa Jbs +E

b Jas +Ha Mb

s minusHb Ma

s dV 2 14


VEa Jbs minusH

a Mbs dV =

VEb Jas minusH

b Mas dV 2 15















x Contents













xiContents




References 421

Index 423

xii Contents

Preface








xiv Preface

Acknowledgement



1Introduction






(a) (b)

(e) (f)

(c) (d)



(g) (h)

(l)

(i) (j)

(k)


3Introduction











5Introduction







2Background Theory













nabla E=ρeε

2 2c

nabla H=ρmμ 2 2d























nabla2t + k

2z

Ez uv

Hz uv= 0 2 5


9Background Theory












n

E2 H2

E1 H1

S




ρms = nH2

μ2minusH1

μ1 2 9b




12 Σ




Wm =14Re

VμH HdV

We =14Re

VεE EdV and

PL =12σ

VμE EdV


214 Reciprocity


11Background Theory


as and Jbs M







Similarly





a Mbs minusH

b Mas







VminusEa Jbs +E

b Jas +Ha Mb

s minusHb Ma

s dV 2 14


VEa Jbs minusH

a Mbs dV =

VEb Jas minusH

b Mas dV 2 15















xiContents




References 421

Index 423

xii Contents

Preface








xiv Preface

Acknowledgement



1Introduction






(a) (b)

(e) (f)

(c) (d)



(g) (h)

(l)

(i) (j)

(k)


3Introduction











5Introduction







2Background Theory













nabla E=ρeε

2 2c

nabla H=ρmμ 2 2d























nabla2t + k

2z

Ez uv

Hz uv= 0 2 5


9Background Theory












n

E2 H2

E1 H1

S




ρms = nH2

μ2minusH1

μ1 2 9b




12 Σ




Wm =14Re

VμH HdV

We =14Re

VεE EdV and

PL =12σ

VμE EdV


214 Reciprocity


11Background Theory


as and Jbs M







Similarly





a Mbs minusH

b Mas







VminusEa Jbs +E

b Jas +Ha Mb

s minusHb Ma

s dV 2 14


VEa Jbs minusH

a Mbs dV =

VEb Jas minusH

b Mas dV 2 15






References 421

Index 423

xii Contents

Preface








xiv Preface

Acknowledgement



1Introduction






(a) (b)

(e) (f)

(c) (d)



(g) (h)

(l)

(i) (j)

(k)


3Introduction











5Introduction







2Background Theory













nabla E=ρeε

2 2c

nabla H=ρmμ 2 2d























nabla2t + k

2z

Ez uv

Hz uv= 0 2 5


9Background Theory












n

E2 H2

E1 H1

S




ρms = nH2

μ2minusH1

μ1 2 9b




12 Σ




Wm =14Re

VμH HdV

We =14Re

VεE EdV and

PL =12σ

VμE EdV


214 Reciprocity


11Background Theory


as and Jbs M







Similarly





a Mbs minusH

b Mas







VminusEa Jbs +E

b Jas +Ha Mb

s minusHb Ma

s dV 2 14


VEa Jbs minusH

a Mbs dV =

VEb Jas minusH

b Mas dV 2 15



Preface








xiv Preface

Acknowledgement



1Introduction






(a) (b)

(e) (f)

(c) (d)



(g) (h)

(l)

(i) (j)

(k)


3Introduction











5Introduction







2Background Theory













nabla E=ρeε

2 2c

nabla H=ρmμ 2 2d























nabla2t + k

2z

Ez uv

Hz uv= 0 2 5


9Background Theory












n

E2 H2

E1 H1

S




ρms = nH2

μ2minusH1

μ1 2 9b




12 Σ




Wm =14Re

VμH HdV

We =14Re

VεE EdV and

PL =12σ

VμE EdV


214 Reciprocity


11Background Theory


as and Jbs M







Similarly





a Mbs minusH

b Mas







VminusEa Jbs +E

b Jas +Ha Mb

s minusHb Ma

s dV 2 14


VEa Jbs minusH

a Mbs dV =

VEb Jas minusH

b Mas dV 2 15




xiv Preface

Acknowledgement



1Introduction






(a) (b)

(e) (f)

(c) (d)



(g) (h)

(l)

(i) (j)

(k)


3Introduction











5Introduction







2Background Theory













nabla E=ρeε

2 2c

nabla H=ρmμ 2 2d























nabla2t + k

2z

Ez uv

Hz uv= 0 2 5


9Background Theory












n

E2 H2

E1 H1

S




ρms = nH2

μ2minusH1

μ1 2 9b




12 Σ




Wm =14Re

VμH HdV

We =14Re

VεE EdV and

PL =12σ

VμE EdV


214 Reciprocity


11Background Theory


as and Jbs M







Similarly





a Mbs minusH

b Mas







VminusEa Jbs +E

b Jas +Ha Mb

s minusHb Ma

s dV 2 14


VEa Jbs minusH

a Mbs dV =

VEb Jas minusH

b Mas dV 2 15



Acknowledgement



1Introduction






(a) (b)

(e) (f)

(c) (d)



(g) (h)

(l)

(i) (j)

(k)


3Introduction











5Introduction







2Background Theory













nabla E=ρeε

2 2c

nabla H=ρmμ 2 2d























nabla2t + k

2z

Ez uv

Hz uv= 0 2 5


9Background Theory












n

E2 H2

E1 H1

S




ρms = nH2

μ2minusH1

μ1 2 9b




12 Σ




Wm =14Re

VμH HdV

We =14Re

VεE EdV and

PL =12σ

VμE EdV


214 Reciprocity


11Background Theory


as and Jbs M







Similarly





a Mbs minusH

b Mas







VminusEa Jbs +E

b Jas +Ha Mb

s minusHb Ma

s dV 2 14


VEa Jbs minusH

a Mbs dV =

VEb Jas minusH

b Mas dV 2 15



1Introduction






(a) (b)

(e) (f)

(c) (d)



(g) (h)

(l)

(i) (j)

(k)


3Introduction











5Introduction







2Background Theory













nabla E=ρeε

2 2c

nabla H=ρmμ 2 2d























nabla2t + k

2z

Ez uv

Hz uv= 0 2 5


9Background Theory












n

E2 H2

E1 H1

S




ρms = nH2

μ2minusH1

μ1 2 9b




12 Σ




Wm =14Re

VμH HdV

We =14Re

VεE EdV and

PL =12σ

VμE EdV


214 Reciprocity


11Background Theory


as and Jbs M







Similarly





a Mbs minusH

b Mas







VminusEa Jbs +E

b Jas +Ha Mb

s minusHb Ma

s dV 2 14


VEa Jbs minusH

a Mbs dV =

VEb Jas minusH

b Mas dV 2 15



(a) (b)

(e) (f)

(c) (d)



(g) (h)

(l)

(i) (j)

(k)


3Introduction











5Introduction







2Background Theory













nabla E=ρeε

2 2c

nabla H=ρmμ 2 2d























nabla2t + k

2z

Ez uv

Hz uv= 0 2 5


9Background Theory












n

E2 H2

E1 H1

S




ρms = nH2

μ2minusH1

μ1 2 9b




12 Σ




Wm =14Re

VμH HdV

We =14Re

VεE EdV and

PL =12σ

VμE EdV


214 Reciprocity


11Background Theory


as and Jbs M







Similarly





a Mbs minusH

b Mas







VminusEa Jbs +E

b Jas +Ha Mb

s minusHb Ma

s dV 2 14


VEa Jbs minusH

a Mbs dV =

VEb Jas minusH

b Mas dV 2 15



(g) (h)

(l)

(i) (j)

(k)


3Introduction











5Introduction







2Background Theory













nabla E=ρeε

2 2c

nabla H=ρmμ 2 2d























nabla2t + k

2z

Ez uv

Hz uv= 0 2 5


9Background Theory












n

E2 H2

E1 H1

S




ρms = nH2

μ2minusH1

μ1 2 9b




12 Σ




Wm =14Re

VμH HdV

We =14Re

VεE EdV and

PL =12σ

VμE EdV


214 Reciprocity


11Background Theory


as and Jbs M







Similarly





a Mbs minusH

b Mas







VminusEa Jbs +E

b Jas +Ha Mb

s minusHb Ma

s dV 2 14


VEa Jbs minusH

a Mbs dV =

VEb Jas minusH

b Mas dV 2 15













5Introduction







2Background Theory













nabla E=ρeε

2 2c

nabla H=ρmμ 2 2d























nabla2t + k

2z

Ez uv

Hz uv= 0 2 5


9Background Theory












n

E2 H2

E1 H1

S




ρms = nH2

μ2minusH1

μ1 2 9b




12 Σ




Wm =14Re

VμH HdV

We =14Re

VεE EdV and

PL =12σ

VμE EdV


214 Reciprocity


11Background Theory


as and Jbs M







Similarly





a Mbs minusH

b Mas







VminusEa Jbs +E

b Jas +Ha Mb

s minusHb Ma

s dV 2 14


VEa Jbs minusH

a Mbs dV =

VEb Jas minusH

b Mas dV 2 15









2Background Theory













nabla E=ρeε

2 2c

nabla H=ρmμ 2 2d























nabla2t + k

2z

Ez uv

Hz uv= 0 2 5


9Background Theory












n

E2 H2

E1 H1

S




ρms = nH2

μ2minusH1

μ1 2 9b




12 Σ




Wm =14Re

VμH HdV

We =14Re

VεE EdV and

PL =12σ

VμE EdV


214 Reciprocity


11Background Theory


as and Jbs M







Similarly





a Mbs minusH

b Mas







VminusEa Jbs +E

b Jas +Ha Mb

s minusHb Ma

s dV 2 14


VEa Jbs minusH

a Mbs dV =

VEb Jas minusH

b Mas dV 2 15








nabla E=ρeε

2 2c

nabla H=ρmμ 2 2d























nabla2t + k

2z

Ez uv

Hz uv= 0 2 5


9Background Theory












n

E2 H2

E1 H1

S




ρms = nH2

μ2minusH1

μ1 2 9b




12 Σ




Wm =14Re

VμH HdV

We =14Re

VεE EdV and

PL =12σ

VμE EdV


214 Reciprocity


11Background Theory


as and Jbs M







Similarly





a Mbs minusH

b Mas







VminusEa Jbs +E

b Jas +Ha Mb

s minusHb Ma

s dV 2 14


VEa Jbs minusH

a Mbs dV =

VEb Jas minusH

b Mas dV 2 15















nabla2t + k

2z

Ez uv

Hz uv= 0 2 5


9Background Theory












n

E2 H2

E1 H1

S




ρms = nH2

μ2minusH1

μ1 2 9b




12 Σ




Wm =14Re

VμH HdV

We =14Re

VεE EdV and

PL =12σ

VμE EdV


214 Reciprocity


11Background Theory


as and Jbs M







Similarly





a Mbs minusH

b Mas







VminusEa Jbs +E

b Jas +Ha Mb

s minusHb Ma

s dV 2 14


VEa Jbs minusH

a Mbs dV =

VEb Jas minusH

b Mas dV 2 15














n

E2 H2

E1 H1

S




ρms = nH2

μ2minusH1

μ1 2 9b




12 Σ




Wm =14Re

VμH HdV

We =14Re

VεE EdV and

PL =12σ

VμE EdV


214 Reciprocity


11Background Theory


as and Jbs M







Similarly





a Mbs minusH

b Mas







VminusEa Jbs +E

b Jas +Ha Mb

s minusHb Ma

s dV 2 14


VEa Jbs minusH

a Mbs dV =

VEb Jas minusH

b Mas dV 2 15




ρms = nH2

μ2minusH1

μ1 2 9b




12 Σ




Wm =14Re

VμH HdV

We =14Re

VεE EdV and

PL =12σ

VμE EdV


214 Reciprocity


11Background Theory


as and Jbs M







Similarly





a Mbs minusH

b Mas







VminusEa Jbs +E

b Jas +Ha Mb

s minusHb Ma

s dV 2 14


VEa Jbs minusH

a Mbs dV =

VEb Jas minusH

b Mas dV 2 15




as and Jbs M







Similarly





a Mbs minusH

b Mas







VminusEa Jbs +E

b Jas +Ha Mb

s minusHb Ma

s dV 2 14


VEa Jbs minusH

a Mbs dV =

VEb Jas minusH

b Mas dV 2 15



fundamentals of aperture antennas and arrays

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Transcript of fundamentals of aperture antennas and arrays