Conformal Array Antenna Theory and Design原版完整文件

CONTENTSPreface xiAbbreviationsandAcronyms xiiiINTRODUCTION 1TheDefinitionofaConformalAntenna 1WhyConformalAntennas? 2History 4MetalRadomes 9SonarArrays 9References 11CIRCULARARRAYTHEORY 15Introduction 15Fundamentals 16LinearArrays 16CircularArrays 19PhaseModeTheory 23Introduction 23DiscreteElements 27DirectionalElements 29TheRippleProbleminOmnidirectionalPatterns 34IsotropicRadiators 37Higher-OrderPhaseModes 40DirectionalRadiators 40ElevationPattern 41FocusedBeamPattern 41References 46THESHAPESOFCONFORMALANTENNAS 49Introduction 49360°Coverage 52vvi CONTENTS360°CoverageUsingPlanarSurfaces 52360°CoverageUsingaCurvedSurface 54HemisphericalCoverage 57Introduction 57HemisphericalCoverageUsingPlanarSurfaces 58HalfSphere 60Cone 60Ellipsoid 62Paraboloid 63ComparingShapes 64MultifacetedSurfaces 67References 70METHODSOFANALYSIS 73Introduction 73TheProblem 75ElectricallySmallSurfaces 76Introduction 76ModalSolutions 76Introduction 76TheCircularCylinder 77AUnitCellApproach 79IntegralEquationsandtheMethodofMoments 80FiniteDifferenceTimeDomainMethods(FDTD) 81Introduction 81ConformalorContour-Patch(CP)FDTD 82FDTDinGlobalCurvilinearCoordinates 83FDTDinCylindricalCoordinates 84FiniteElementMethod(FEM) 87Introduction 87HybridFE-BIMethod 88ElectricallyLargeSurfaces 89Introduction 89High-FrequencyMethodsforPECSurfaces 90High-FrequencyMethodsforDielectricCoatedSurfaces 93TwoExamples 95Introduction 95TheApertureAntenna 95TheMicrostrip-PatchAntenna 98AComparisonofAnalysisMethods 100Appendix4A—Interpretationoftheraytheory 1004A.1WatsonTransformation 1034A.2FockSubstitution 1034A.3SDPIntegration 1044A.4SurfaceWaves 1054A.5Generalization 107References 1085 GEODESICSONCURVEDSURFACES123CONTENTSvii5.1 Introduction1235.1.1DefinitionofaSurfaceandRelatedParameters1255.1.2TheGeodesicEquation1275.1.3SolvingtheGeodesicEquationandtheExistenceofGeodesics1285.2 SinglyCurvedSurfaces1305.3 DoublyCurvedSurfaces1355.3.1Introduction1355.3.2TheCone1355.3.3RotationallySymmetricDoublyCurvedSurfaces1375.3.4PropertiesofGeodesicsonDoublyCurvedSurfaces1395.3.5GeodesicSplitting1415.4 ArbitrarilyShapedSurfaces1495.4.1Hybridsurfaces1505.4.2AnalyticallyDescribedSurfaces152References1526 ANTENNASONSINGLYCURVEDSURFACES1556.1 Introduction1556.2 ApertureAntennasonCircularCylinders1576.2.1Introduction1576.2.2Theory1576.2.3MutualCoupling15IsolatedMutualCoupling15CrossPolarizationCoupling16Arraymutualcoupling1686.2.4RadiationCharacteristics17Isolated-ElementPatterns17Embedded-ElementPatterns1766.3 ApertureAntennasonGeneralConvexCylinders1926.3.1Introduction1926.3.2MutualCoupling19TheEllipticCylinder19TheParabolicCylinder19TheHyperbolicCylinder1946.3.3RadiationCharacteristics19TheEllipticCylinder19EndEffects1966.4 ApertureAntennasonFacetedCylinders2006.4.1 Introduction2006.4.2 MutualCoupling2076.4.3 RadiationCharacteristics2096.5 ApertureAntennasonDielectricCoatedCircularCylinders2096.5.1Introduction2096.5.2MutualCoupling2IsolatedMutualCoupling2ArrayMutualCoupling2156.5.3RadiationCharacteristics221viii CONTENTSIsolated-ElementPatterns 221Embedded-ElementPatterns 223Microstrip-PatchAntennasonCoatedCircularCylinders 230Introduction 230Theory 231MutualCoupling 232Single-ElementCharacteristics 232IsolatedandArrayMutualCoupling 232RadiationCharacteristics 237Isolated-ElementPatterns 237Embedded-ElementPatterns 238TheCone 245Introduction 245MutualCoupling 246ApertureAntennas 246Microstrip-PatchAntennas 247RadiationCharacteristics 248ApertureAntennas 248Microstrip-PatchAntennas 257References 2587ANTENNASONDOUBLYCURVEDSURFACES2657.1 Introduction2657.2 ApertureAntennas2677.2.1Introduction2677.2.2MutualCoupling26IsolatedMutualCoupling26ArrayMutualCoupling2827.2.3RadiationCharacteristics2837.3 Microstrip-PatchAntennas2897.3.1Introduction2897.3.2MutualCoupling28Single-ElementCharacteristics29IsolatedMutualCoupling2947.3.3RadiationCharacteristics297References3028CONFORMALARRAYCHARACTERISTICS3058.1 Introduction3058.2 MechanicalConsiderations3068.2.1ArrayShapes3068.2.2ElementDistributiononaCurvedSurface3088.2.3MultifacetSolutions3088.2.4TileArchitecture3098.2.5StaticandDynamicStress3128.2.6OtherElectromagneticConsiderations3128.3 RadiationPatterns3138.3.1Introduction313CONTENTS ix8.3.2GratingLobes3158.3.3Scan-InvariantPattern3198.3.4Phase-ScannedPattern3198.3.5ASimpleApertureModelforMicrostripArrays3238.4 ArrayImpedance3278.4.1Introduction3278.4.2Phase-ModeImpedance3308.5 Polarization3368.5.1PolarizationDefinitions3368.5.2CylindricalArrays33DipoleElements33Apertureelements3378.5.3PolarizationinDoublyCurvedArrays340AParaboloidalArray3418.5.4PolarizationControl3488.6 CharacteristicsofSelectedConformalArrays3518.6.1NearlyPlanarArrays3518.6.2CircularArrays3518.6.3CylindricalArrays3518.6.4ConicalArrays3518.6.5SphericalArrays3548.6.6ParaboloidalArrays3568.6.7EllipsoidalArrays3588.6.8OtherShapes359References359BEAMFORMING 365Introduction 365ANoteonOrthogonalBeams 366AnalogFeedSystems 366VectorTransferMatrixSystems 367SwitchMatrixSystems 367ButlerMatrixFeedSystems 370RFLensFeedSystems 374TheR-2RLensFeed 374TheR-kRLensFeed 375Mode-ControlledLenses 376TheLuneburgLens 377TheGeodesicLens 378TheDomeAntenna 380DigitalBeamForming 380AdaptiveBeamForming 384Introduction 384TheSampleMatrixInversionMethod 384AnAdaptiveBeamFormingSimulationUsinga 385CircularArrayRemarksonFeedSystems 388References 389x CONTENTSCONFORMALARRAYPATTERNSYNTHESIS 395Introduction 395ShapeOptimization 397FourierMethodsforCircularRingArrays 397Dolph-ChebysjevPatternSynthesis 398IsotropicElements 398DirectiveElements 399AnApertureProjectionMethod 401TheMethodofAlternatingProjections 404AdaptiveArrayMethods 406Least-Mean-SquaresMethods(LMS) 407PolarimetricPatternSynthesis 409OtherOptimizationMethods 409ASynthesisExampleIncludingMutualCoupling 411AComparisonofSynthesisMethods 414References 418SCATTERINGFROMCONFORMALARRAYS 421Introduction 431Definitions 422RadarCrossSectionAnalysis 423General 423AnalysisMethodforanArrayonaConductingCylinder 424AnalysisMethodforanArrayonaConductingCylinder 426withaDielectricCoatingCylindricalArray 427AnalysisandExperiment—RectangularGrid 427Higher-OrderWaveguideModes 430TriangularGrid 432ConclusionsfromthePECConformalArrayAnalysis 433CylindricalArraywithDielectricCoating 435SingleElementwithDielectricCoating 436ArraywithDielectricCoating 437RadiationandScatteringTrade-off 446Introduction 446Single-ElementResults 449ArrayResults 454Discussion 458References 459SubjectIndex 463AbouttheAuthors 471PREFACEThisbookistheresultofclosecooperationbetweenindustryandacademia,notablyssonandtwouniversitiesinSweden:ChalmersUniversityofTechnology(CTH)inGöte-borgandTheRoyalInstituteofTechnology(KTH)inStockholm.In1987,astudentatCTHpresentedherPhDthesisonthetopicofconformalantennas.Itwasthenconsideredaninterestingbutdifficulttechnologywithnoimmediateapplication.About10yearslat-er,manyconformalarrayapplicationswereseriouslyconsideredorindevelopment.thattime,webecameinvolvedinseveralconformalR&Dprogramsthatalsoincludedex-perimentalhardwareformodelverification.Ourintentionwastocompileresultstheseeffortsintoone,thickinternalreport.However,withthesupportandencouragementfrommanycolleagues,wesetoutonthemuchmoredemandingroutetowriteabookonthesubject.Manystandardtextbooksonantennasincludeshortsectionsonconformalarrayanten-nas,butusuallyonlysimplereferencecasesaretreated.Themutualcoupling(whichisanimportantparameter)isoftenjustbrieflymentioned.Examplesofarraycharacteristicswiththemutualcouplingincludedarerare.Thus,webelievethisbookfillsagapintheexistingliterature.Ourpurposeistopresentthefundamentalprinciplesbehindconformalantennas,aswellashands-oninformationnecessaryfortheanalysisanddesignofconformalarrays.Graphicalillustrationsareusedextensively,bothforcalculatedandmeasuredsults,includingresultsnotpublishedbefore.Wedescribetheoreticalmethodsforanalysisanddesign,andincludeexplicitformulaswhereapplicable.Fromapracticalpointofview,mechanicalaspects,beam-formingtechniques,andpackagingofconformalantennasareincluded.Furthermore,scatteringpropertiesarediscussed,whichareofterestinstealthapplications,forexample.Listsofreferencesareprovidedattheendofeachchapterforfurtherstudies.Thus,wehopethatthebookwillbecomeausefultoolforthepracticingantennaandsystemsengineerinunderstandingandworkingwiththeseterestingantennas.Eachchapterstartswithsomeintroductorymaterial,thatis,thebasicconceptsthatessentialtogetanunderstandingofthemoreadvancedaspects.Thefirstthreechapterspresentanoverviewofconformalarrayprinciplesandapplications,includingthetheoryforcirculararraysandphasemodeconcepts,anddiscussionsofvariousshapesofconfor-malarrays.InChapters4and5,theoreticalmethodsforanalysisanddesignaredescribed,in-cludingexplicitformulas;forexample,forgeodesicsonmoregeneralsurfacesthanthecanonicalcircularcylinderandsphere.Doublycurvedsurfacesanddielectriccoveredsurfacesusinghigh-frequencymethodsarealsoincluded.Twocanonicalexamplesarealsodiscussedindetail,thusassistingthereaderinhis/herownconformalantennaanalysis.xixiiPREFACExiiPREFACEChapters6and7dealwithradiatingelementsonsinglycurvedanddoublycurvedfaces.Thefocusisonmutualcouplingcharacteristicsandelementradiationproperties.Elementtypesincludewaveguide-fedaperturesandmicrostrippatches.Forbothmeasureddatasupportsthecalculatedresults.Chapters8and9treatconformalarrayantennacharacteristics—radiation,impedanceandpolarization—aswellasmechanicalandpackagingaspects.Feedingsystemsandbeam-scanningprinciplesarealsoincluded.Chapter10discussesvarioussynthesismethods,withsomeexamples.Also,suchasoptimizingtheshape,distributionofelements,polarization,andbandwidtharecluded.Thefinalchapterdealswithmethodsfortheanalysisofscattering(radarcrosssection)fromconformalarrayantennas;inparticular,waveguide-fedapertureelementswithwithoutadielectriccoating.Weincludealsoadiscussionontheproblemofreducingradarcrosssectionwithoutdecreasingtheantennaperformance.Whilewrittenwithengineeringapplicationsinmind,thisbookcanalsoserveasatextforgraduatecoursesinadvancedantennasandantennasystems.ACKNOWLEDGMENTSAmongthenumerouscolleaguesandfriendswhohavesupportedourworkwiththeirad-vice,encouragement,andcontributions,wecanmentiononlyafew.Inparticular,wewanttothankDr.BjörnThors,RoyalInstituteofTechnology/EricssonAB,Sweden,forhiscontributionsonscatteringandradiationcharacteristics,andforvaluablediscussions.ThanksalsogotoDr.ZvonimirSipus,UniversityofZagreb,Croatia,forcontributingsultsforantennasonsinglyanddoublycurvedsurfaces.Dr.HansSteyskal,U.S.AirForceResearchLab,helpedwithvaluablecomments,especiallyregardingarraytheory,beamforming,andsynthesistechniques.OthercontributorsincludeDr.TorleifSwedishDefenceResearchAgency(FOI);Prof.LarsPettersson,SwedishDefencesearchAgency(FOI);Dr.SilviaRaffaelli,EricssonAB,Sweden;andTechn.Lic.Lanne,EricssonMicrowaveSystemsAB/ChalmersUniversityofTechnology,Sweden.SpecialthanksgotoProf.KjellRosquist,StockholmUniversity,Sweden,forhisguid-anceintotheworldofgeodesics.Wearegratefulforthepermissiontouseresultsfromseveralrecentstudiesonconfor-malantennasinSweden,amongthemworksponsoredbyEricssonMicrowaveSystemsAB,EricssonAB,TheFoundationforStrategicResearch(SSF),andTheSwedishfenseMaterialAdministration(FMV).Lastbutnotleast,weexpressourappreciationandgratitudetoourfamiliesfortheirencouragement,understanding,andpatienceduringthewritingofthisbook. ANDA/D analog/digitalAF arrayfactorBI boundaryintegralBiCG biconjugategradientmethodBOR bodyofrevolutionCAD computeraideddesignCP contourpath;alsocircularpolarization,conformalDBF digitalbeamformingDE differentialequationDOA directionofarrivalEFIE electricfieldintegralequationEGL endfiregratinglobeEM electromagneticEMP electromagneticpulseESM electronicsupportmeasuresEWCA EuropeanWorkshoponConformalAntennasFDTD finite-elementtimedomainFE finiteelementFE-BI finite-elementboundaryintegralFBF fastbeamformingFEM finite-elementmethodFFT fastFouriertransformFMF fastmodeformerFSS frequency-selectivestructureGCM geodesicconstantmethodGCS geodesiccoordinatesystemGaAs galliumarsenideGHOR generalhyperboloidofrevolutionGO geometricalopticsGPOR generalparaboloidofrevolutionGTD geometrictheoryofdiffractionh/D characteristicdimensionofaparaboloid,heightoverdiameterHF highfrequencyHP horizontalpolarizationIBC impedanceboundaryconditionIC integratedcircuitIE integralequationIEP isolatedelementpatternxiiixivABBREVIATIONSANDACRONYMSxivABBREVIATIONSANDACRONYMSIF intermediatefrequencyLEO lowearthorbitLMS leastmeansquareLNA low-noiseamplifierLP linearpolarizationLPI lowprobabilityofinterceptMEMS microelectromechanicalsystemMMIC monolithicmicrowaveintegratedcircuitMOM,MoM methodofmomentsMPIE mixed-potentialintegralequationMTI moving-targetindicationMUSIC multiple-signalclassificationPEC perfectelectricconductorPO physicalopticsPTD physicaltheoryofdiffractionQ qualityfactorRCS radarcrosssectionRF radiofrequencyRX receiveSAR synthetic-apertureradarSBR shootandbounceray(method)SDP steepestdecentpathSMI samplematrixinversionSNIR signal-to-noiseplusinterferenceratioSPNT single-poleN-throwswitchSTAP space–timeadaptiveprocessingTACAN tacticalairnavigationTD timedomainTD-PO timedomainphysicalopticsTD-UTD timedomainuniformtheoryofdiffractionTE transverseelectricTEM transverseelectromagneticTM transversemagneticTX transmitUCA uniformcirculararrayULA uniformlineararrayUPML uniaxialperfectlymatchedlayerUTD uniformtheoryofdiffractionVP verticalpolarizationVPD variablepowerdivider1 INTRODUCTION1.1THEDEFINITIONOFACONFORMALANTENNAAconformalantennaisanantennathatconformstosomething;inourcase,itconformstoaprescribedshape.Theshapecanbesomepartofanairplane,high-speedtrain,orvehicle.Thepurposeistobuildtheantennasothatitbecomesintegratedwiththestruc-tureanddoesnotcauseextradrag.Thepurposecanalsobethattheantennaintegrationmakestheantennalessdisturbing,lessvisibletothehumaneye;forinstance,inanenvironment.Atypicaladditionalrequirementinmoderndefensesystemsisthatthetennanotbackscattermicrowaveradiationwhenilluminatedby,forexample,anenemyradartransmitter(i.e.,ithasstealthproperties).TheIEEEStandardDefinitionofTermsforAntennas(IEEEStd145-1993)givesthefollowingdefinition:conformalantenna[conformalarray].Anantenna[anarray]thatconformstoafacewhoseshapeisdeterminedbyconsiderationsotherthanelectromagnetic;forexample,aerodynamicorhydrodynamic.conformalarray.See:conformalantenna.Strictlyspeaking,thedefinitionincludesalsoplanararraysiftheplanar“shapeisdeter-minedbyconsiderationsotherthanelectromagnetic.”Thisis,however,notcommonpractice.Usually,aconformalantennaiscylindrical,spherical,orsomeothershape,withtheradiatingelementsmountedonorintegratedintothesmoothlycurvedsurface.ConformalArrayAntennaTheoryandDesign.ByLarsJosefssonandPatrikPersson 1©2006InstituteofElectricalandElectronicsEngineers,Inc.2INTRODUCTION2INTRODUCTIONvariationsexist,though,likeapproximatingthesmoothsurfacebyseveralplanarfacets.Thismaybeapracticalsolutioninordertosimplifythepackagingofradiatorstogetherwithactiveandpassivefeedingarrangements.WHYCONFORMALANTENNAS?Amodernaircrafthasmanyantennasprotrudingfromitsstructure,fornavigation,vari-ouscommunicationsystems,instrumentlandingsystems,radaraltimeter,andsoon.Therecanbeasmanyas20differentantennasormore(upto70antennasonatypicalmilitaryaircrafthasbeenquoted[Schneideretal.2001]),causingconsiderabledragandincreasedfuelconsumption.Integratingtheseantennasintotheaircraftskinishighlyde-sirable[Wingert&Howard1996].Preferably,someoftheantennafunctionsshouldbecombinedinthesameunitifthedesigncanbemadebroadbandenough.Theneedforconformalantennasisevenmorepronouncedforthelarge-sizedaperturesthatareneces-saryforfunctionslikesatellitecommunicationandmilitaryairbornesurveillanceradars.Atypicalconformalexperimentalarrayforleading-wing-edgeintegrationisshowninFigure1.2.TheX-bandarrayisconformalwiththeapproximatelyellipticalcrosssectionshapeoftheleadingedgeofanaircraftwing[Kannoetal.1996].Figure1.3showsanevenmorerealisticallywing-shapedC-bandarray(cf.[Steyskal2002]).Arrayantennaswithradiatingelementsonthesurfaceofacylinder,sphere,orandsoon,withouttheshapebeingdictatedby,forexample,aerodynamicorsimilarsons,areusuallyalsocalledconformalarrays.Theantennasmayhavetheirshapedeter-minedbyaparticularelectromagneticrequirementsuchasantennabeamshapeand/oran-gularcoverage.TocallthemconformalarrayantennasisnotstrictlyaccordingtotheIEEEdefinitioncitedabove,butwefollowwhatiscommonpracticetoday.Acylindricalorcirculararrayofelementshasapotentialof360°coverage,eitherwithanomnidirectionalbeam,multiplebeams,oranarrowbeamthatcanbesteeredFigure1.1.Atleast20–30antennasprotrudefromtheskinofamodernaircraft.(From[Hopkinsetal.1997],reprintedbypermissionoftheAmericanInstituteofAeronauticsandAstronautics,Inc.)Figure1.2.Conformalarrayantennaforaircraftwingintegration[Kannoetal.1996].Figure1.3.Amicrostriparrayconformaltoawingprofileinthetestchamber.Seealsocolorinsert,Figure1.(CourtesyofAirForceResearchLab./AntennaTechnologyBranch,HanscomAFB,USA.) 3PAGE10PAGE10INTRODUCTION1.31.3HISTORY PAGE5360°.Atypicalapplicationcouldbeasabasestationantennainamobilecommunicationsystem.Today,thecommonsolutionisthreeseparateantennas,eachcoveringa120°sec-tor.Instead,onecylindricalarraycouldbeused,resultinginamuchmorecompactinstal-lationandlesscost.AnotherexampleofshapebeingdictatedbycoverageisshowninFigure1.4.Thisisasatellite-borneconicalarray(and,hence,thedragproblemiscertainlynotanissuehere).Theargumentsforandagainstconformalarrayscanbediscussedatlength.Theappli-cationsandrequirementsarequitevariable,leadingtodifferentconclusions.Inspiteofthis,andtoencouragefurtherdiscussion,wepresentasummarybasedonreflectionsbyGuy[1999],Guyetal.[1999],Watkins[2001],andothersinTable1.1.HISTORYThefieldofphasedarrayantennaswasaveryactiveareaofresearchintheyearsfromWWIIuptoabout1975.Duringthisperiod,muchpioneeringworkwasdonealsoforconformalarrays.However,electronicallyscanned,phasedarrayantennasdidnotwidespreaduseuntilthenecessarymeansforfeedingandsteeringthearraybecameavail-able.Integratedcircuit(IC)technology,includingmonolithicmicrowaveintegratedcir-cuits(MMIC),filledthisgap,providingreliabletechnicalsolutionswithapotentialforFigure1.4.Aconicalconformalarrayfordatacommunicationfromasatellite[Vourchetal.1998,Cailleetal.2002].Seealsocolorinsert,Figure5.Table1.1.PlanarversusconformalarrayantennasParameterPlanararrayConformalarrayTechnologyMatureNotfullyestablishedAnalysistoolsAvailableIndevelopmentBeamcontrolPhaseonlyusuallysufficient,Amplitudeandphase,morefixedamplitudecomplicatedPolarizationSinglecanbeused(dualoftenPolarizationcontrolrequired,desired)especiallyifdoublycurvedGainDropswithincreasedscanControlled,dependsonshapeFrequencybandwidthTypically20%WiderthanplanarispossibleAngularcoverageLimitedtoroughly±60°Verywide,halfsphereRCSLargespecularRCSLowerthanplanarInstallationonplatformPlanarshapelimitsduetoStructurallyintegrated,leavessweptvolumeextraspace.NodragRadomeAberrationeffectsNoconventionalradome,noboresighterrorPackagingofelectronics Knownmultilayersolutions Sizerestrictioniflargecurvature,facetspossiblelowcost,evenforverycomplexarrayantennas.Animportantfactorwasalsothedevel-opmentofdigitalprocessorsthatcanhandletheenormouslyincreasedrateofinformationprovidedbyphasedarraysystems.Digitalprocessingtechniquesmadephasedarraytennasystemscosteffective,thatis,theyprovidedthecustomersvalueforthemoneyspent.Thisbeingtrueforphasedarraysingeneral,italsoholdsforconformalarrayantennas.However,intheareaofconformalarrays,electromagneticmodelsanddesignknow-howneededextradevelopment.Duringthelast10to20years,numericaltechniques,electro-magneticanalysismethods,andtheunderstandingofantennasoncurvedsurfacesimproved.Importantprogresshasbeenmadeinhigh-frequencytechniques,includinganalysisofsurfacewavediffractionandmodelingofradiatingsourcesoncurvedfaces.Theoriginofconformalarrayscanbetracedatleastbacktothe1930swhenasystemofdipoleelementsarrangedonacircle,thusformingacirculararray,wasanalyzedbyChireix[1936].Later,inthe1950s,severalpublicationsonthesubjectwerepresented;see,forexample,[Knudsen1953a,b].Thecirculararraywasattractivebecauseofitsrota-tionalsymmetry.Properphasingcancreateadirectionalbeam,whichcanbescanned360°inazimuth.Theapplicationswereinbroadcasting,communication,andlaternavigationanddirectionfinding.Anadvanced,morerecentapplicationusingalargeculararrayistheFrenchRIASexperimentalradarsystem[Doreyetal.1989,ColinDuringtheSecondWorldWar,HFcirculararraysweredevelopedforradiosignalin-telligencegatheringanddirectionfindinginGermany.Theseso-calledWullenweber1ar-rays(codewordforthedevelopmentproject)werequitelargewithadiameterofabout100meters.Afterthewar,anexperimentalWullenweberarraywasdevelopedattheUni-versityofIllinois(seeFigure1.5).Thisarrayhad120radiatingelementsinfrontofaflectingscreen.Thediameterwasabout300m;notethesizeofthebuildingsinthecenter[Gething1966].ManysimilarsystemswerebuiltinothercountriesduringtheColdWar.1NamedforJ.Wullenweber,1488–1537,LordMayorofLübeckFigure.1.5.Theexperimental300mdiameterWullenweberantennaattheUniver-sityofIllinois.(CourtesyP.J.D.Gething,“High-FrequencyDirectionFinding,”Pro-ceedingsofIEE,January1966,p.54.)Someofthesehugeantennasmaystillbeoperating.Seealso[IREPGAPNewsletterVol.3,December1960].Duringthisperiod,new,efficientpatternsynthesismethodsandpracticalfeedingbeamcontrolschemeswereinvestigatedbyseveralworkers.Foranoverviewsee1981,1983].Averyusefulapproachinthisworkwasbasedontheconceptofphasemodes.Forthecirculararray,theexcitationcanbeviewedasaperiodicfunctioninazimuth,withperiod21r.TheexcitationcanthereforebeexpressedasaFourierseries.Eachterminthisseriesisaphasemode,whichcanbegeneratedinthepracticalsituationbyconvenientworks,specificallytheButlermatrix.Aphasemodehasconstantamplitudebutalinearphaseprogressionfromoneradiatingelementtothenext,totalingamultipleof21rovercircumference.Thephasemodeconceptprovedtobeanefficienttoolinpatternsynthesis,andwillbedescribedinmoredetailinChapters2and10.Bymeasuringthephasemodesonreception,thesignaldirectionofarrival(DOA)canbedetermined[Rehnmark1980].Figure1.6showsadirection-findingapplicationusingthistechnique.Withomnidirectionalelements,thefullcirclecanbeused.However,constructivead-ditionofsignalsfromboththefrontandtherearpartofthecirculararrayisnoteasilyachieved,inparticularnotoveranextendedbandwidth.Mostcirculararraysthereforeusedirectiveradiatingelements,pointingoutwardfromthecenter.TheWullenweberanten-nashaveusuallyareflectiveelementorscreenbehindeachradiator,makingthemtive.Elementdirectivityhasbeenanalyzedinrelationtothephasemodeconceptandnificantimprovementscomparedtoomnidirectionalelementsweredemonstratedetal.1981].Inordertoincreasethedirectivityandnarrowthebeaminelevation,severalcirculararraysplacedontopofeachothercanbeused.AgoodexampleistheelectronicallyscannedTACAN(tacticalnavigation)antenna[Christopher1974,Shestag1974].TheTACANantennacanbeplacedontheground,radiatingarotatingphase-codedsignalhelpsaircrafttofindtheirpositioninrelationto,forexample,anairfield.JimWaitdidfundamentalworkonradiationfromaperturesinmetalliccircularcylin-ders;see[Wait1959].HisworkhasbeencontinuedbymanyothersemployingeitherFigure1.6.Broadbandcirculararrayforsignal-bearingmeasurements.Seealsocol-orinsert,Figure7.(CourtesyofAnarenInc.,Syracuse,NY,USA.)modalexpansiontechniquesorhigh-frequencydiffractiontechniques[Hessel1970,Pathaketal.1980].Inparticular,mutualcouplingisincludedinthesolutions.Themeth-odswillbedescribedinChapter4.Nose-mountedantennasinmissilesoraircraftareprotectedbyapointedradome.Al-ternatively,theantennaelementscouldbeputontheradomeitself.Thispossibilitycreatedaninterestinconformalarraysoncones[Mungeretal.1974].Theprogr