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FoundationsofAntennaRadiationTheory

IEEEPress

445HoesLane Piscataway,NJ08854

IEEEPressEditorialBoard

SarahSpurgeon, EditorinChief

JónAtliBenediktssonBehzadRazaviJeffreyReed AnjanBoseJimLykeDiomidisSpinellis JamesDuncanHaiLiAdamDrobot AminMoenessBrianJohnsonTomRobertazzi DesineniSubbaramNaiduAhmetMuratTekalp

FoundationsofAntennaRadiationTheory

EigenmodeAnalysis

WenGeyi

Waterloo,Canada

IEEEPressSeriesonElectromagneticWaveTheory

Copyright©2023byTheInstituteofElectricalandElectronicsEngineers,Inc.Allrightsreserved.

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LibraryofCongressCataloging-in-PublicationData

Names:Wen,Geyi,author.

Title:Foundationsofantennaradiationtheory:eigenmodeanalysis/Wen Geyi.

Description:Hoboken,NewJersey:Wiley-IEEEPress,[2023]|Includes index.

Identifiers:LCCN2022059830(print)|LCCN2022059831(ebook)|ISBN 9781394170852(cloth)|ISBN9781394170869(adobepdf)|ISBN 9781394170876(epub)

Subjects:LCSH:Antennas(Electronics)|Antennaradiationpatterns.

Classification:LCCTK7871.6.W462023(print)|LCCTK7871.6(ebook)| DDC621.382/4–dc23/eng/20230111

LCrecordavailableathttps://lccn.loc.gov/2022059830

LCebookrecordavailableathttps://lccn.loc.gov/2022059831

CoverDesignandImage:Wiley

Setin9.5/12.5ptSTIXTwoTextbyStraive,Pondicherry,India

Contents

AbouttheAuthor xi

Preface xiii

1EigenvalueTheory 1

1.1MaxwellEquations 3

1.1.1WaveEquations 3

1.1.2PropertiesofElectromagneticFields 6

1.1.2.1SuperpositionTheorem 7

1.1.2.2ConservationofElectromagneticFieldEnergy 7

1.1.2.3EquivalenceTheorem 12

1.1.2.4Reciprocity 13

1.2MethodsforPartialDifferentialEquations 14

1.2.1MethodofSeparationofVariables 14

1.2.1.1RectangularCoordinateSystem 15

1.2.1.2CylindricalCoordinateSystem 16

1.2.1.3SphericalCoordinateSystem 19

1.2.2MethodofGreen’sFunction 21

1.2.2.1Green’sFunctionsforHelmholtzEquation 22

1.2.2.2DyadicGreen’sFunctionsandIntegral Representations 24

1.2.3VariationalMethod 27

1.3EigenvalueProblemforHermitianMatrix 29

1.3.1Properties 29

1.3.2RayleighQuotient 30

1.4EigenvalueProblemsfortheLaplaceOperatoronScalarField 32

1.4.1RayleighQuotient 32

1.4.2PropertiesofEigenvalues 36

1.4.3CompletenessofEigenfunctions 38

1.4.4DifferentialEquationswithVariableCoefficients 39

1.4.5Green’sFunctionandSpectralRepresentation 41

1.5EigenvalueProblemsfortheLaplaceOperatoronVectorField 44

1.5.1RayleighQuotient 45

1.5.2CompletenessofVectorModalFunctions 48

1.5.3ClassificationofVectorModalFunctions 54

1.6RitzMethodfortheSolutionofEigenvalueProblem 55

1.7HelmholtzTheorems 57

1.7.1HelmholtzTheoremfortheFieldinInfiniteSpace 57

1.7.2HelmholtzTheoremfortheFieldinFiniteRegion 59

1.7.3HelmholtzTheoremforTime-DependentField 60

1.8CurlOperator 61

1.8.1EigenfunctionsofCurlOperator 62

1.8.2Plane-WaveExpansionsfortheFieldsandDyadicGreen’s Functions 64

References 66

2RadiationinWaveguide 69

2.1VectorModalFunctionsforWaveguide 70

2.1.1ClassificationofVectorModalFunctions 71

2.1.2VectorModalFunctionsforTypicalWaveguides 75

2.1.2.1RectangularWaveguide 75

2.1.2.2CircularWaveguide 76

2.1.2.3CoaxialWaveguide 77

2.2RadiatedFieldsinWaveguide 79

2.2.1ModalExpansionsfortheFieldsandDyadicGreen’sFunctions 79

2.2.2DyadicGreen’sFunctionsforSemi-infiniteWaveguide 91

2.3WaveguideDiscontinuities 92

2.3.1ExcitationofWaveguide 92

2.3.2ConductingObstaclesinWaveguide 95

2.3.3CouplingbySmallAperture 97

2.4TransientFieldsinWaveguide 102

References 107

3RadiationinCavityResonator 109

3.1RadiatedFieldsinCavityResonator 110

3.1.1ClassificationofVectorModalFunctionsforCavity Resonator 111

3.1.2ModalExpansionsfortheFieldsandDyadicGreen’s Functions 114

3.2CavitywithOpenings 117

3.2.1CavitywithOnePort 118

3.2.2CavitywithTwoPorts 120

3.3WaveguideCavityResonator 125

3.3.1FieldExpansionsbyVectorModalFunctionsofWaveguide 125

3.3.2ModalRepresentationsofDyadicGreen’sFunctions 133

3.4VectorModalFunctionsforTypicalWaveguideCavityResonators 136

3.4.1RectangularWaveguideCavity 136

3.4.2CircularWaveguideCavity 137

3.4.3CoaxialWaveguideCavity 139

3.5RadiationinWaveguideRevisited 140

3.6TransientFieldsinCavityResonator 141

References 149

4RadiationinFreeSpace(I):GenericProperties 151

4.1AntennaParameters 152

4.1.1Power,Efficiencies,andInputImpedance 152

4.1.2FieldRegions,RadiationPattern,RadiationIntensity,Directivity, andGain 155

4.1.3VectorEffectiveLength,EquivalentArea,andAntenna Factor 158

4.1.4AntennaQualityFactor 163

4.2TheoryofSphericalWaveguide 163

4.2.1VectorModalFunctionsforSphericalWaveguide 164

4.2.2ModalExpansionsofFieldsandDyadicGreen’sFunctions 168

4.2.3PropertiesofSphericalVectorWaveFunctions 180

4.2.4Far-ZoneFields 181

4.3StoredFieldEnergiesandRadiationQualityFactor 182

4.3.1StoredFieldEnergiesinGeneralMaterials 184

4.3.2StoredFieldEnergiesofAntenna 194

4.3.3RadiatedFieldEnergy 199

4.3.4EvaluationofRadiationQualityFactor 202

4.4ModalQualityFactors 206

4.4.1StoredFieldEnergiesOutsidetheCircumscribingSphereof Antenna 206

4.4.2TwoInequalitiesforSphericalHankelFunctions 210

4.4.3PropertiesofModalQualityFactors 212

4.4.3.1ProofofProperties2,4,and7 213

4.4.3.2ProofofProperties1,3,6,8,and9 215

4.4.3.3ProofofProperty5 216

4.4.3.4ProofofProperties10and11 217

4.4.4LowerBoundforAntennaQualityFactor 218

4.5UpperBoundsfortheProductsofGainandBandwidth 220

4.5.1DirectiveAntenna 221

4.5.2OmniDirectionalAntenna 224

4.5.3BestPossibleAntennaPerformance-GuidelinesforSmall AntennaDesign 226

4.6ExpansionsoftheRadiatedFieldsinTimeDomain 230 References 238

5RadiationinFreeSpace(II):ModalAnalysis 243

5.1BasicAntennaTypes 245

5.2EquivalentCurrentDistributionsofAntenna 246

5.3AntennaasaWaveguideJunction 249

5.4IntegralEquationFormulations 250

5.4.1CompensationTheoremforTime-HarmonicFields 251

5.4.2IntegralEquationsforCompositeStructure 252

5.4.3IntegralEquationforWireAntenna 254

5.5VerticalDipole 257

5.5.1FieldsintheRegion r > b258

5.5.2FieldsintheRegion r < b261

5.6HorizontalDipole 261

5.6.1FieldsintheRegion r > b262

5.6.2FieldsintheRegion r < b267

5.7Loop 267

5.8SphericalDipole 269

5.9DipoleNearConductingSphere 271

5.10FiniteLengthWireAntenna 273

5.10.1FieldsintheRegion r > l273

5.10.2TheFieldsintheRegion r < l275

5.11ApertureAntenna 276

5.12MicrostripPatchAntenna 280

5.13ResonantModalTheoryforAntennaDesign 290

5.13.1Formulations 291

5.13.2Applications 293

5.13.2.1Crossed-Dipole 293

5.13.2.2Dual-BandBowtieAntenna 297 References 301

6RadiationinFreeSpace(III):ArrayAnalysisandSynthesis 303

6.1IntroductiontoArrayAnalysis 305

6.1.1ArrayFactor 305

6.1.2LinearArray 307

6.1.2.1LinearArraywithUniformAmplitude 307

6.1.2.2LinearArraywithNonuniformAmplitude 311

6.1.3CircularArray 314

6.1.4PlanarArray 316

6.2IntroductiontoArraySynthesiswithConventionalMethods 318

6.2.1ArrayFactorandSpaceFactorforLineSource 318

6.2.2SchelkunoffUnitCircleMethod 320

6.2.3Dolph–ChebyshevMethod 323

6.2.4FourierTransformMethod 327

6.2.4.1ContinuousLineSource 327

6.2.4.2LinearArray 329

6.3PowerTransmissionBetweenTwoAntennas 330

6.3.1TheGeneralPowerTransmissionFormula 331

6.3.2PowerTransmissionBetweenTwoPlanarApertures 335

6.3.3PowerTransmissionBetweenTwoAntennaswithLarge Separation 341

6.4SynthesisofArrayswithMMPTE 343

6.4.1PowerTransmissionBetweenTwoAntennaArrays 344

6.4.1.1UnconstrainedOptimization 346

6.4.1.2WeightedOptimization 346

6.4.1.3ConstrainedOptimization 347

6.4.2Applications 349

6.5SynthesisofArrayswithEMMPTE 369

6.5.1ArrayswithSpecifiedEnergyDistribution 370

6.5.2ArrayswithSpecifiedPowerDistribution 372

6.5.3Applications 373 References 376

AppendixAVectorAnalysis 381

AppendixBDyadicAnalysis 383

AppendixCSIUnitSystem 385

AppendixDUnifiedTheoryforFields(UTF) 387 Index 417

AbouttheAuthor

WenGeyi(Fellow,IEEE)wasborninPingjiang,Hunan,China,in1963.He receivedtheB.Eng.,M.Eng.,andPh.D.degreesinelectricalengineeringfrom XidianUniversity,Xi’an,China,in1982,1984,and1987,respectively.From 1988to1990,hewasalecturerattheRadioEngineeringDepartment,Southeast University,Nanjing,China.From1990to1992,hewasanassociateprofessorat theInstituteofAppliedPhysics,UniversityofElectronicScienceandTechnology ofChina(UESTC),Chengdu,China.From1992to1993,hewasavisiting researcherattheDepartmentofElectricalandComputerEngineering,University ofCaliforniaatBerkeley,Berkeley,CA,UnitedStates.From1993to1998,hewasa fullprofessorattheInstituteofAppliedPhysics,UESTC.HewasavisitingprofessorattheElectricalEngineeringDepartment,UniversityofWaterloo,Waterloo, ON,Canada,fromFebruary1998toMay1998.From1996to1997,hewasthevice chairmanoftheInstituteofAppliedPhysics,UESTC,wherehewasthechairman oftheinstitutefrom1997to1998.From1998to2007,hewaswithBlackberryLtd., Waterloo,ON,Canada,firstasaseniorscientistwiththeRadioFrequency DepartmentandthenthedirectoroftheAdvancedTechnologyDepartment.Since 2010,hehasbeenaNationalDistinguishedProfessorwithFudanUniversity, Shanghai,China,andtheNanjingUniversityofInformationScienceandTechnology(NUIST),Nanjing,whereheiscurrentlythedirectoroftheResearchCenter ofAppliedElectromagnetics.Hehasauthoredover100journalpublicationsand FoundationsforRadioFrequencyEngineering (WorldScientific,2015), FoundationsofAppliedElectrodynamics (Wiley,2010), AdvancedElectromagneticField Theory (China:NationalDefensePublishingHouse,1999),and ModernMethods forElectromagneticComputations (China:HenanScienceandTechnologyPress, 1994).Heholdsmorethan40patents.

Preface

Wirelesstechnologieshaverevolutionizedmanydifferentfieldsinindustryaswell asinourdailylives.Asavitaldeviceinwirelesssystems,antennasplayanimportantroleinboostingoverallsystemperformance.Thedemandonvarioustypesof antennasfordifferentwirelessapplicationsisgrowingrapidly,whichraisesmany challengesforantennadesigners.Forexample,wirelessterminalshavebecome smaller,andantennasmustbesqueezedintoanevensmallerspace.Atthesame time,multipleantennasystemsandantennascoveringmultiplefrequencybands arebeingdeployedtowirelessterminalstomeettheincreasingdemandfornew servicesandtoimprovethecommunicationquality.Toovercomethesechallenges,antennadesignersneedabetterunderstandingofantennatheory.

Antennatheoryusuallycontainsthreedifferentbutrelatedsubjects:generic propertiesofantenna,antennaanalysis,andantennasynthesis.Thegenericpropertiesofantennaaremeanttobevalidforallantennas,andtheyarethefundamentalsofantennadesign.Forhistoricalortechnicalreasons,manyofthe genericpropertiesofantennadiscoveredinthelastfewdecadeshavenotyetbeen reflectedinmostantennabooks.Toincludethesenewresultsinabook,onehasto introduceanumberofconceptsthatarebarelytouchedinmanyantennabooks, suchasthestoredfieldenergyaroundantenna,theradiationqualityfactor,and thesphericalvectorwavefunctions.Antennaanalysisexaminestheradiation propertiesofantennawithaspecifiedcurrentdistribution,ofwhichtheradiated fieldisconventionallyexpressedasanintegration.Suchaprocessis,however,not alwaysthemostefficientsincetheintegrationmustbecarriedoutforeachobservationpointinordertofindthefielddistributionoutsidethesourceregion.The antennasynthesis,alsocalledpatternsynthesis,istheoppositeprocessofanalysis, inwhichthecurrentdistributionortypeofantenna,includingthegeometryand feedingmechanism,isdeterminedinanoptimalwaysothataprescribedfielddistributioninthefar-ornear-fieldregioncanbeachieved.Sinceacontinuouscurrentdistributionisnoteasytorealizeinpractice,itmustbediscretizedandthen realizedbyanantennaarray.Forthisreason,variousantennasynthesismethods

areprimarilydevelopedforantennaarray.Theconventionalarraysynthesis methodsaredependentonthearrayfactor,whichisnolongereffectivewhen thearrayelementsarenotidentical,thesurroundingenvironmentistoocomplicated,ortheinter-elementspacingbecomesverysmall.Newarraysynthesis methodsbasedoneigenmodeanalysishavebeendevelopedinrecentyearsand canovercometheexistingproblemsassociatedwiththearrayfactor,buthave notyetbeenincorporatedintotextbooksthereforelimitingaccessibilityto studentsandresearchers.

Themainthemeofthisbookiseigenmodeanalysisanditsapplicationsin antennatheoryanddesign.Thefreespace canbeconsideredasasphericalwaveguide.Anantennamaythereforebeviewedasawaveguidejunctionthatconnectsthefeedinglineandthesphericalwaveguide,transformingtheguided modesintosphericalmodesintransmittingmodeorconvertingthespherical modesintoguidedmodesinreceivingmode.Forthisreason,itispossibleto buildatheoryforantennasthatparallelsthetheoryforwaveguides.Theeigenmodeanalysisisthefoundationofwaveguidetheory,anditsimportanceinphysicsandengineeringcannotbeoverstressed.Aneigenmodeisapossiblestateof asystemwhenitisfreeofexcitation,andthecorrespondingeigenvalueoften representsanimportantquantityofthesystem,forexamplethetotalenergy ofthesystem(suchasinquantummechanics)orthenaturaloscillationfrequency(suchasinametalcavityresonator).Anarbitrarystateofthesystem canbeexpressedasalinearcombinati onoftheeigenmodes.Ifonlyoneora feweigenmodesdominateinthelinearcombination,thiswillsignificantly simplifytheanalysisoftheproblem.Intheeigenmodeexpansionofafield, theexpansioncoefficientsareexpressed astheintegralsoverthesourceregion andtheintegrationsareonlyperformedo nce.Aftertheexpansioncoefficients aredetermined,theevaluationofthefielddistributionoutsidethesourceregion onlyinvolvesthesumofseries,whichdecreasesthecomputationalburdenand simplifiesthenumericaltreatmentmostofthetimeascomparedtotheconventionalintegralrepresentation.

Therehavebeenseveralmodaltheoriesforstudyingelectromagnetic(EM)radiationandscatteringproblems.Thesingularityexpansionmethod(SEM)isbased ontheanalysisincomplexfrequencydomainandformulatedbyelectricfieldintegralequation.TheresonantfrequenciesandthemodesinSEMarecomplex,which significantlyincreasesthecomputationaltimeandthedifficultyinnumerical implementations.Theeigenmodeexpansionmethod(EEM)usestheeigenfunctionsofanintegraloperator.SamewiththeSEM,theeigenvaluesandthe eigenmodesinEEMarecomplexnumbers.Inaddition,theEEMlacksasolid mathematicalfoundation.Thecharacteristicmode(CM)analysisisanotherinterestingmodaltheoryandiscarriedoutintherealfrequencydomain,ofwhichthe characteristicvalues(eigenvalues)andCMsareallreal.Itisnotedthatallthe

xiv Preface

modesinvolvedintheCM,SEM,andEEMformulationsdependnotonlyonthe propertiesofthescattererbutalsoontheworkingfrequency.

Thisbookcontainsthenewdevelopmentsinantennatheory,withthegoalto addresstheaforementionedproblemsandchallengesinthebestpossibleway andishopedtobeausefulalternativetothetraditionalapproaches.Theantenna radiationproblemsinbothclosedandopenregionaretreatedinaunifiedmanner intermsoftheeigenmodesavailablefromthesystems.Theeigenmodesare derivedfromwaveguides,cavityresonators,andsphericalwaveguideandareindependentoffrequency,andcanthereforebeusedtoexpandthefieldsineitherfrequencyortimedomain.Theorganizationandtreatmentoftheproposedbookis quitedifferentfromthepreviousbooksonsimilartopics.Themethodofeigenmodes,similartotheFourierseriesexpansioninsignalanalysis,isusedthroughoutthebook.Theantennaanalysisproblemsaretreatedbycombiningthemethod ofseparationofvariables,Green’sfunction,andvariationalmethod.Thevariationalmethodestablishesthecompletesetofeigenmodesandtheirproperties, andthemethodofseparationofvariableisusedtofindtheeigenmodesforsimple geometries.Theradiatedfieldisthenexpandedbyusingtheeigenmodes,from whichdyadicGreen’sfunctionscanbedetermined,avoidingtheproblemcaused bytheinappropriateselectionoftheeigenmodesfortheexpansionofapoint source.WhenthedyadicGreen’sfunctionsareappliedtotheintegralequationformulationforanantenna,asignificantcomputationalburdencanbereducedand thenumericaltreatmentcanbesimplified.Thearraysynthesisproblemsarealso treatedasaneigenvalueproblemwiththemethodofmaximumpowertransmissionefficiency(MMPTE).Thevariationalexpressionisestablishedforthepower transmissionefficiency(PTE)betweentheantennaarrayunderdesignandatestingarray.Analgebraiceigenvalueproblemresultingfromthevariationalprinciple isthensolved,andtheeigenvectorcorrespondingtothemaximumeigenvalueis selectedasthedistributionofexcitationsforthearrayunderdesign.

Thecontentsofthebookareselectedfortheirfundamentalityandimportance, andmanyofthemareformulatedintermsofeigenmodetheoryandappearin bookformforthefirsttime.Thebooknotonlydiscussestheantennaradiation problemsinopenspacebutalsothoseinwaveguideandcavityresonator,and itconsistsofsixchapters.Chapter1describesthebasicsofEMfieldequations andtheirsolutionmethodsandprovidesthenecessarybackgroundinformation forlaterchapters.ItbeginswiththeintroductionofMaxwellequations,thewave equations,andthetheoremsforEMfields.Threeanalyticaltoolsforthesolutionof boundaryvalueproblemsareintroduced,andtheyaretheseparationofvariables, Green’sfunction,andthevariationalmethod.Themainfocusofthischapteristhe treatmentofeigenvalueproblemsarisinginmatrixtheory,scalarandvectorfields, andtheyarefundamentaltoourlaterdiscussions.BymeansoftheRayleighquotient(avariationalexpressionfortheeigenvalueproblem),theeigenmodesofthe

Preface

Laplacianoperatoractingonascalaroravectorfieldaretreatedinasimilar manner,andacompletesetofeigenmodesisconstructedbythevariational analysisoftheRayleighquotient.Inordertounderstandhowavectorfieldisdecomposedintolongitudinal,transverse,andharmoniccomponents,theHelmholtz theoremsforthevectorfieldsdefinedinfiniteorinfiniteregionarepresented. AsageneralizationoftheHelmholtztheorem,theeigenfunctionsofthecurloperatorarealsoexplored,intermsofwhichtheplane-waveexpansionsforthefieldsand thedyadicGreen’sfunctionsareobtained.

Chapter2investigatestheradiationproblemsinwaveguide.Theeigenvalueproblemsinwaveguideareapproachedintransversefieldforitsgenerality.Various dyadicGreen’sfunctionsforwaveguidearederiveddirectlyfromthefieldexpansionsintermsofthevectormodalfunctions,whichavoidsaproblemcausedbythe incompletenessoftheeigenfunctionsselectedtoexpandadyadicpointsourcein theconventionalstudyofdyadicGreen’sfunctionsinwaveguides.Bytheequivalenceprinciple,threecommonwaveguidediscontinuityproblems,theexcitation ofwaveguide,obstaclesinwaveguide,andthecouplingbetweenwaveguides,are analyzedandtreatedasaradiationproblemandcompactlyreformulatedbyusing thedyadicGreen’sfunctions.Theradiatedfieldintimedomainisapproachedby thevectormodalfunctions,andthetransientprocessesinthewaveguideare studied.Forreference,thevectormodalfunctionsintypicalwaveguidesare summarized.

Chapter3dealswiththeradiationproblemsinmetalcavityresonators.Inparticular,thevectormodalfunctionsinthewaveguidecavityresonatorarederived fromthewaveguidemodes.ThedyadicGreen’sfunctionsofelectricandmagnetic typeforacavityresonatorareestablishedfromthemodalexpansionsofthefields. Likethewaveguidetheory,allthecavity-relatedproblemsaretreatedasaradiationproblemthroughtheuseofequivalenceprinciple.Thecircuitparametersfor thecavitywithmultiplewaveguideportsareevaluatedbythemodalanalysis.The vectormodalfunctionsfortypicalwaveguidecavityresonatorsarederived.Itis demonstratedthatthedyadicGreen’sfunctionsforthewaveguidecavityreduce tothoseforthewaveguideifthetwoendsofthewaveguidecavityareextended toinfinity.Thetime-domainfieldsgeneratedbythesourcesinthewaveguidecavityareexpandedintermsofthevectormodalfunctionsinwaveguide,andthetransientresponsesinthecavityresonatorareexamined.

Chapter4discussesthegenericpropertiesofantenna.Typicalantennaparametersaresummarized.Completesetofvectormodalfunctionsforthespherical waveguideisrigorouslyconstructedfromsphericalharmonics,andthemodal expansionsofthedyadicGreen’sfunctionsarederivedfromthefieldexpansions. Ageneraldefinitionofthestoredfieldenergyofantennaisproposedbymeansofa conservationlawforthestoredfieldenergiesinanarbitrarymedium.Twomethodsforevaluatingtheradiationqualityfactorareelucidated.Oneisfromtheinput

impedanceofantennaandtheotherisviathecurrentdistribution.Themodal qualityfactorsarethoroughlyexamined,andtheirfinitepowerseriesexpansions areobtained.Theupperboundsontheproductofgainandbandwidthforboth directionalandomnidirectionalantennaarepresented,andtheirapplications insmallantennadesignaredemonstrated.Theupperboundsansweracommon questionofhowmuchspaceshouldberequestedtoaccommodateanantennato realizeaspecifiedperformance.Theradiatedfieldsfromatransientsourceare studiedthroughthefieldexpansionsintermsofthevectormodalfunctionsfor thesphericalwaveguide.

Inantennaanalysis,theinducedcurrentdistributiononantennaiseithergiven ortobedeterminedfromanimpressedsource.Inmanycases,onemustresortto numericaltechniquestosolveanintegralequationorasetofdifferentialequations derivablefromMaxwellequationswithboundaryconditionstofindtheinduced currentdistribution.Chapter5isdevotedtothemodalanalysisoftypicalantennas.Thefreespaceisconsideredasasphericalwaveguide,andtheradiatedfieldis expressedasalinearcombinationofsphericalvectorwavefunctions.Theintegral equationsforanantennaconsistingofcompositematerialsarederivedbythe modalexpansionsofdyadicGreen’sfunctions.Insteadofusingtheintegralrepresentationofthefieldsinconventionalantennaanalysis,typicalantennas,includingdipole,loop,aperture,andpatchantenna,areallanalyzedbythespherical wavefunctionsortheeigenmodesinbothnear-andfar-fieldregions.Anarbitrary scattererissaidtoberesonantifitsstoredelectricfieldenergyisequaltothestored magneticfieldenergy.Basedonthisdefinition,amethodforcomputingtheresonantmodesisproposedandappliedtotheantennadesign.

Antennasynthesisinvolvesusingwell-organizedoptimizationmethodstofind thecurrentdistributionsoastoachieveaspecifiedfielddistributioninthenear-or far-fieldregion.Asingleantennaisoftenaroundonewavelengthinsize,andits radiationpatterncoversawideangleandthusexhibitspoordirectivity.Inorderto enhancethedirectivityandincreasetheflexibilityofshapingtheradiationpattern, onemustuseanantennaarray.Theperformancesofantennaarrayarecontrolled bytherelativepositioningofelementsandthedistributionofexcitations.To achieveadesiredfieldpattern,aperformanceindex(targetfunction)mustbe properlychosenandoptimized.Forawirelesssystemplannedforthetransmission ofeitherinformationorpower,anaturalperformanceindexisthePTEbetween thetransmitting(Tx)andreceiving(Rx)antennas,whichisdefinedastheratioof thepowerdeliveredtotheloadofthereceivertotheinputpowerofthetransmitter.Toattainthebestpossiblequalityofwirelesscommunicationorpowertransfer,thePTEmustbemaximized.Motivatedbythefactthatantennasmustbe designedtoenhancethePTEforallwirelesssystems,thePTEcanthusbeadopted asaperformanceindexforthedesignofantennas.Theoptimizationprocedure providesapowerfulanduniversaltechniqueforthesynthesisofantennaarrays

ofalltypesandcanovercometheexistingchallengeswiththeconventionalarray synthesismethodsusingarrayfactors.Thetechniqueisbasedoneigenmodeanalysisandcalledthemethodofmaximumpowertransmissionefficiency(MMPTE), whichcanachievevariousfieldpatternsinanycomplicatedenvironmentinthe near-orfar-fieldregion.Theconventionalmethodsofantennasynthesislargely dependonfieldtheory,whiletheMMPTEreducesthefieldsynthesisprobleminto acircuitanalysisproblemsothatcircuittheorymaybeappliedtosolvetheoriginal fieldproblem.ThisfeaturemakesthedesignprocessofantennaarraymoreaccessibleforthosewhoarenotveryfamiliarwithEMfieldtheory.Thecircuitparameterscanbeacquiredbysimulationormeasurement,andthereforethe MMPTEisapplicabletoanycomplicatedproblem.Wheneverthesimulationis beyondthecapabilityofastate-of-the-artcomputer,onecanresorttomeasurementtofindthecircuitparameters.AnotherimportantfeatureoftheMMPTE isthatitcontainstheinformationoftheenvironmentbetweenTxandRxarrays, andthereforecanbemadeadaptivetocomplicatedenvironment,guaranteeingthe bestpossibleperformanceoftheantennaarray.TheMMPTEhasbeenverifiedto besuperiortomostexistingarraydesignmethodsintermsofsimplicity,applicability,generality,anddesignaccuracy.Itgeneratesanoptimizeddistributionof excitationfortheantennaarraytoassurethatthegainandefficiencyofthearray ismaximizedforafixedarrayconfigurationandisequallyapplicableforboth near-andfar-fieldsynthesisproblems.Chapter6summarizesseveraltypicalarray synthesismethodsbasedonthearrayfactors,includingSchelkunoffunitcircle method,Dolph–Chebyshevmethod,andtheFouriertransformmethod.Themain partofthischapteristheformulationsofMMPTE,includingtheunconstrained MMPTE,weightedMMPTE,constrainedMMPTE,andextendedMMPTE (EMMPTE).TheEMMPTEisafieldmethodbutfollowsaproceduresimilarto MMPTE.AnumberofapplicationsofMMPTEandEMMPTEaredemonstrated.

Fortheconvenienceofreaders,threeAppendicesA,B,andCareincludedto providethefundamentalsofvectoranalysis,dyadicanalysis,andtheSIunit system.Aunifiedtheoryforfields(UTF)isexploredinAppendixD.TheUTF unveilsthatanarbitrarystaticfield(eitherscalarorvectorial,calledanontological field)inaninertialsystem(staticsystem)willmergeastwovectorfieldsinaninertialsystemmovingrelativetothestaticsystem,whichsatisfyMaxwell-likeequations.Therefore,theMaxwellequationsarevalidnotonlyfordescribingtheEM fieldsbutalsoforanyphysicalfieldsforwhichanontologicalfieldexists.TheUTF isbasedonthetheoryofspecialrelativityandtheHelmholtztheorem.Inorderto findhowthevectorfieldchangesindifferentinertialsystems,oneonlyneedsto examinehowthecurlanddivergenceofthevectorfieldtransform.Asademonstration,theMaxwell-likeequationsforthegravitationalfieldarederivedfromthe UTF,andtheyarealsoderivedfromtheEinsteinfieldequationsinthetheoryof

generalrelativity.Someuniversallawsofnatureareshowntobederivablefrom theUTF.

Thebookcanbeusedeitherforundergraduateorgraduatecourseson “AdvancedAntennaTheory,” orasareferenceforresearchersandengineersin theareasofmicrowave,antenna,andEMcompatibility.Theprerequisitesfor thebookareadvancedcalculusandlinearalgebra.Afterreadingthebook,the readersshouldbeabletobetterunderstandantennaradiationtheoryandantenna analysisandsynthesisfromadifferentperspectiveintermsofeigenmode analysis.TheSIunitsareusedthroughoutthebook.A e jωt timevariationis assumedfortime-harmonicfields.Aspecialsymbol “□” isusedtoindicatethe endofatheorem,aremark,oranexample.

Theauthorisgratefultohisfamily.Withouttheirconstantsupportand encouragement,thebookwouldneverhavebeencompleted.

Waterloo,Ontario,Canada

Preface xix

EigenvalueTheory

Scienceisspectralanalysis.Artislightsynthesis.

– KarlKraus(Austrianwriterandjournalist,1874–1936)

Thestudyofeigenvalueproblemscanbetracedbacktotheeighteenthcentury, whenSwissmathematicianandphysicistLeonhardEuler(1707–1783)investigatedtherotationalmotionofarigidbody.Theword “eigen” isfromGerman andmeans “own” or “belongingto,” andwasfirstusedbyGermanmathematician DavidHilbert(1862–1943)tocharacterizeeigenvaluesandeigenvectorsin1904. Eigenvalueproblemsoftenariseinmathematics,physics,andengineering sciences.Inlinearalgebra,an eigenvector ofalineartransformationisanonzero vectorthatchangesbyascalarfactorwhenthelineartransformationactsonit. Thescalarfactoriscalledthe eigenvalue correspondingtotheeigenvector.Geometrically,thisimpliesthattheeigenvectorisnotrotatedaftertransformation. Theeigenvalueproblemforadifferentialoperatoroftenresultsfromtheboundary valueproblemsdefinedinafiniteregion.Whenthedefiningregionisunbounded, thediscreteeigenvaluesbecomeacontinuum.AveryusefultechniqueforstudyingtheeigenvalueproblemistoestablishtheRayleighquotientfortheeigenvalues andthenusethecalculusofvariationstoinvestigatethepropertiesofeigenmodes. Inphysics,an eigenmode ofasystemisapossiblestatewhenthesystemisfreeof excitation,whichmightexistinthesystemonitsownundercertainconditions, andisalsocalledan eigenstate ofthesystem.The methodofeigenfunctions isverysimilartotheFourierseriesexpansioninsignalanalysis,andwillbeused throughoutthisbook.Themethodisbasedonthesolutionofaneigenvalueproblemavailablefromthesystem.Anarbitrarystateofthesystemcanbeexpressedas alinearcombinationoftheeigenmodes,andtheexpansioncoefficientscanthen bedeterminedfromthesourceconditionsortheinitialvaluesofthesystem.Ifonly oneorafeweigenmodesdominateinthelinearcombination,thiswillsignificantly simplifytheanalysisoftheproblem.

Themodaltheoryforascattererplaysanimportantroleinantennatheoryand designs.Thebasicideabehindthemodaltheoryistointroducethefundamental fieldpatterns,called modes,sothatthefieldsoutsidethescatterercanbe expandedintoalinearcombinationofthesemodes.Therehavebeenseveral modaltheoriesforstudyingelectromagnetic(EM)radiationandscatteringproblems(exteriorboundaryvalueproblems).The singularityexpansionmethod (SEM)isbasedontheanalysisincomplexfrequencydomainandformulatedby electricfieldintegralequation[1,2].The naturalresonantfrequencies arise fromtherequirementthatanontrivialcurrentdistributionexistsonaconducting scattererfreeofincidentfields.Thecorrespondingfieldpatternsarecalled naturalresonantmodes.ThenaturalresonantfrequenciesandthemodesinSEMare complex,whichsignificantlyincreasesthecomputationaltimeandthedifficulty innumericalimplementations.The eigenmodeexpansionmethod (EEM) expandsthecurrentsandtheradiatedfieldsintermsoftheeigenmodesofanintegraloperator[3,4].SamewiththeSEM,theeigenvaluesandtheeigenmodesin EEMarecomplexnumbers.TheEEMisbasedontheeigenfunctionsofintegral equationsandlacksasolidmathematicalfoundation.Theintegraloperator involvedinEEMisnotsymmetric,anditisthereforehardtoprovetheexistence andcompletenessoftheeigenfunctions.Amoreusefulmethodforthestudyof scatteringproblemisthe singularfunctionexpansion,whichwasfirstintroducedbytheGermanmathematicianErhardSchmidt(1876–1959)in1907[5], andhasbeenappliedtostudyvariousscatteringproblems[6,7].Thetheoryof characteristicmode isanotherinterestingmodalnotionandiscarriedoutin therealfrequencydomain[8–11],ofwhichthecharacteristicvalues(eigenvalues) andthecorrespondingcharacteristicmodesareallreal.Ingeneral,thecharacteristicvaluesrangefrom ∞ to+∞,amongwhichthoseofthesmallestmagnitudes arethemostimportantforradiationandscatteringproblems.Theexternalresonantmodescorrespondtothezerocharacteristicvalues,andcanbedetermined approximatelybysweepingthefrequency.Itisnotedthatalltheabovementioned modalformulationsdependnotonlyonthepropertiesofthescattererbutalsoon theoperatingfrequency.

Theeigenvalueproblemsdiscussedinthisbookarederivedfromwaveguide, cavityresonator,andsphericalwaveguide,whoseeigenfunctionsareindependent offrequencyandcanthusbeusedtoexpandthefieldsineitherfrequencyortime domain.Theimportanceofeigenvaluetheoryinmathematicsandphysicscannot beoverstated.Therehavebeenvariousmethodsdevelopedtocalculateeigenvaluesandeigenfunctions,withthemostimportantonebeingthevariational methodbasedontheRayleighquotient[12].Thischapterprovidesthenecessary backgroundinformationforlaterchapters.TheMaxwellequationsandthesolutionmethodsforpartialdifferentialequations(PDEs)arebrieflyintroduced.The emphasisisupontheeigenvaluetheoryforoperators,includingthematrixandthe

1.1MaxwellEquations 3

Laplacianonscalarandvectorfields.Thepropertiesofeigenfunctionsarederived fromtheRayleighquotient,andtheRitzmethodforthenumericalsolutionofthe Rayleighquotientisdemonstrated.AlsoincludedinthischapteristheHelmholtz theorem,whichstatesthatanyvectorfieldcanbedecomposedintothesumofan irrotationalvectorfieldandasolenoidalvectorfield.Suchadecompositionhas interestingapplicationsinthemodalexpansionoffieldsandisthetheoreticalbasis ofintroducingscalarandvectorpotentials.TheHelmholtztheoremindicatesthat avectorfieldisfullydeterminedbyitsdivergenceandcurl.Indeed,Maxwellequationsarenothingbutacoupleofrulesthatregulatethedivergencesandthecurlsof electricandmagneticfieldsaccordingtoimpressedandinducedsources.Asa generalizationofHelmholtztheorem,theeigenfunctionsofcurloperatorarediscussed,intermsofwhichtheplane-waveexpansionsforthefieldsaswellasthe dyadicGreen’sfunctionscanbeobtained.

1.1MaxwellEquations

MaxwellequationsareasetofPDEsthatunifyelectricityandmagnetismand describehowelectricandmagneticfields,asthefunctionsofspaceandtime, aregeneratedbychargesandcurrentsandalteredbyeachother.Theyhavebeen provedtobeverysuccessfulinexplainingandpredictingavarietyofmacroscopic EMphenomena.

1.1.1WaveEquations

The generalizedMaxwellequations thatincludebothelectricandmagnetic sourcesconsistoftwovectorequationsandtwoscalarequations:

1 1

Intheabove, r istheobservationpointofthefieldsinmeter(m)and t isthetime insecond(s), H isthe magneticfieldintensity measuredinamperespermeter (A/m), B isthe magneticinductionintensity measuredintesla(Wb/m2), E is electricfieldintensity measuredinvoltspermeter(V/m), D isthe electric inductionintensity measuredincoulombspersquaremeter(C/m2), J is electriccurrentdensity measuredinamperespersquaremeter(A/m2), ρ is the electricchargedensity measuredincoulombspercubicmeter(C/m3), Jm

is magneticcurrentdensity involtspersquaremeter(V/m2),and ρm is magneticchargedensity inweberspercubicmeter(Wb/m3).Thefirstequationis Ampère’slaw,anditdescribeshowtheelectricfieldchangesaccordingtothe currentdensityandmagneticfield.Thepositivesigninthefirstequationindicates thatthedirectionsofthemagnetomotiveforceandtheelectriccurrentarerelated bytheright-handrule.Theterm ∂ D/∂ t wasintroducedbyMaxwellin1861andis called displacementcurrent,whichisnecessaryfortheexistenceofwavesolutions.Thesecondequationis Faraday’slaw,anditcharacterizeshowthemagneticfieldvariesaccordingtotheelectricfieldandequivalentmagneticcurrent density.Theminussigninthesecondequationindicatesthatthedirectionsofelectromotiveforceandthemagneticcurrentarerelatedbytheleft-handrule,whichis requiredby Lenz’slaw.Inotherwords,whenanelectromotiveforceisgenerated byachangeofmagneticflux,thepolarityoftheinducedelectromotiveforceis suchthatitproducesacurrentwhosemagneticfieldopposesthechange,which producesit.Thethirdequationis Coulomb’slaw,anditsaysthattheelectricfield dependsonthechargedistributionandobeystheinversesquarelaw.Thelast equationshowsthatthemagneticfieldalsoobeystheinversesquarelawand dependsontheequivalentmagneticchargedistribution.Itshouldbeunderstood thatnoneoftheexperimentshadanythingtodowithwavesatthetimewhen Maxwellderivedhisequations.Maxwellequationsimplymorethantheexperimentalfacts.The continuityequation canbederivedfrom(1.1):

Theelectricchargedensity ρ andtheelectriccurrentdensity J inMaxwellequationsarefreechargedensityandcurrentsandtheyexcludechargesandcurrents formingpartofthestructureofatomsandmolecules.Theboundchargesandcurrentsareregardedasmaterial,whicharenotincludedin ρ and J.Thecurrentdensityusuallyconsistsoftwoparts: J = Jcon + Jimp.Here, Jimp isreferredtoasexternal or impressedcurrentsource,whichisindependentofthefieldsanddelivers energytoelectricchargesinasystem.Theimpressedcurrentsourcecanbeofelectricandmagnetictypeaswellasofnon-electricornonmagneticorigin. Jcon = σ E, where σ isthe conductivity ofthemediuminsiemenspermeter(S/m),denotes the conductioncurrent inducedbytheimpressedsource Jimp.Sometimesitis convenienttointroducean impressedelectricfieldEimp definedby Jimp = σ Eimp Inmoregeneralsituation,onemaywrite J = Jind + Jimp,where Jind isthe induced current bytheimpressedcurrent Jimp.Thecontinuityequationforthemagnetic current Jm andmagneticcharges ρm canbederivedfrom(1.1):

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