天線原理第三章

13RadiationIntegralsandAuxiliaryPotentialFunctions

〔輻射積分和輔助勢(shì)函數(shù)〕1AntennaTheoryandDesign1整理ppt3.1IntroductionAntennaanalysisandsynthesisAnalysisproblem—tospecifythesourcesandthenrequirethefieldsradiatedbythesources.synthesisproblemwheretheradiatedfieldsarespecified,andwearerequiredtodeterminethesources.Vectorpotentials—auxiliaryfunctionsintroducedtoaidinthesolutionoftheproblems.A(magneticvectorpotential)andF(electricvectorpotential)Althoughtheelectricandmagneticfieldintensities(EandH)representphysicallymeasurablequantities,amongmostengineersthepotentialsarestrictlymathematicaltools.2整理pptComputingfieldsradiatedbyelectricandmagneticsources.Theone-stepprocedurerelatestheEandHfieldstoJandMbyintegralrelations,requiringonlyintegration.Thetwo-stepprocedurerelatestheAandFpotentialstoJandMbyintegralrelations.TheEandHarethendeterminedsimplybydifferentiatingAandF.Theintegrandsinthetwo-stepprocedurearemuchsimpler.3整理pptReviewofElectromagneticTheoryMaxwell’s

EquationsQuantitySymbolUnitElectricfieldintensityEV/m(voltpermeter)ElectricfluxdensityDC/m2MagneticfieldintensityHA/m(amperepermeter)MagneticfluxdensityBT(tesla)4整理pptFieldsduetoelectricchargeandcurrentFieldsduetomagneticchargeandcurrentTime-harmonic(ejωt)Fields5整理pptConstitutiveRelationsInanisotropicandlinearmedium,weknowε

isthepermittivityofthemediumμisthepermeabilityofthemediumInfreespaceorair,6整理pptBoundaryConditionsOntheinterfacebetweentwodielectrics,Onthesurfaceofaperfectconductor,Medium1Medium27整理pptElementaryRadiatingElementszxyI0dlHertzianDipole(forwireantennas)Huygens’Element(forapertureantennas)Equivalentmagneticcurrent:8整理pptCoordinatesystemsforcomputingfieldsradiatedbysourcesBoundsofthesources—TheintegrationrequiredtodetermineAandForEandHisrestrictedovertheboundsofthesourcesJandM.Observationpointcoordinates—IntegrationforAandF,anddifferentiationtodetermineEandHmustbedoneintermsoftheobservationpointcoordinates.Sourcepoint(x,

y

,z

)(r

,

,

)Observationpoint(x,

y,z),(r,

,

)R=r-r

9整理pptFar-FieldApproximation0RFieldpointSourcepointFieldpoint:(x,y,z)Sourcepoint:(x’,y’,z’)10整理ppt3.2TheVectorPotentialAforan

ElectricCurrentSourceJMagneticvectorpotentialAConsiderelectricsource(J,ρ)BecauseandusingthevectoridentityThemagneticfieldintensity:(3.2a)11整理pptUsingthefirstMaxwell’sequation,weknowScalarPotential

e

FromthevectoridentityUsingtheLorenzcondition,(3.7a)(3.15)(3.13)12整理pptHelmholtzEquationThecurlofAisdefinedasWeusethevectoridentityForahomogeneousmedium,EquatingMaxwell’sequation,andUsingtheLorenzcondition,(3.10)(3.14)13整理ppt3.3TheVectorPotentialFfora

MagneticCurrentSourceMElectricvectorpotentialFConsiderelectricsource(M,ρm)BecauseUsingIntroducinganarbitrarymagneticscalarpotential(3.16)(3.19)14整理pptTakingthecurlof(3-16)andequatingittoMaxwell’sequationand(3.26)UsingtheLorenzcondition,(3.25)15整理ppt3.4ComputingEandHfrom(J,M)Summary1.SpecifyJandM(electricandmagneticcurrentdensitysources).2.a.FindA(duetoJ)usingTheyaresolutionsoftheinhomogeneousvectorwaveequationof(3-14)and(3-25),respectively.k2=ω2μ

andRisthedistancefromanypointinthesourcetotheobservationpoint.16整理ppt3.a.FindHAandEAb.FindEFandHFwithJ=0withM=0oror4.ThetotalfieldsareororwithJ=0withM=017整理ppt3.5SolutionoftheInhomogeneousVectorWaveEquationWeverifythatthesolutionoftheinhomogeneousvectorwaveequationisaninfinitesimalsourcewithcurrentdensityJz,isplacedattheorigin.AtpointsremovedfromthesourceJz=0,18整理pptAz

isnotafunctionofdirection(θandφ),Az

=Az

(r)

twoindependentsolutions

Inthestaticcase(

=0,k=0),thesolutionsimplifiesasthetime-varyingsolutioncanbeobtainedbymultiplyingthestaticsolutionbye?jkr.19整理pptPoisson’sequationInthepresenceofthesource(Jz

0)andk=0,Thetime-varyingsolutioncanbeobtainedbymultiplyingthestaticsolutionbye?jkr.20整理pptThewaveequationforthecurrentdensitiesinthex-andy-directions(JxandJy),thesolutiontothevectorwaveequationof(3-14)asIfthesourceisremovedfromtheoriginandplacedat(x

,y

,z

),21整理pptForlineardensitiesJandM,ForelectricandmagneticcurrentsIeandIm,

Inasimilarfashion,wecanshowthatthesolutionofis22整理ppt3.6Far-fieldRadiationAgeneralsolutiontothevectorwaveequationof(3-14)insphericalcomponentsTheamplitudevariationsofrineachcomponentareoftheform1/rn.NeglectinghigherordertermsThervariationsareseparablefromthoseofθandφ.(3-55)Using23整理pptNeglectinghigherordertermsof1/rn,theradiatedE-andH-fieldshaveonlyθandφcomponents.TheycanbeexpressedasRadialfieldcomponentsexistonlyforhigherordertermsof1/rn.24整理pptThefar-zonefieldsduetoamagneticsourceM(potentialF)canbewrittenasSimplystated,thecorrespondingfar-zoneE-andH-fieldcomponentsareorthogonaltoeachotherandformTEM(tor)modefields.25整理ppt3.7DualityTheorem〔對(duì)偶定理〕DualitytheoremWhentwoequationsthatdescribethebehavioroftwodifferentvariablesareofthesamemathematicalform,theirsolutionswillalsobeidentical.Thevariablesinthetwoequationsthatoccupyidenticalpositionsareknownasdualquantitiesandasolutionofonecanbeformedbyasystematicinterchangeofsymbolstotheother.26整理ppt27整理pptNotify:Dualityonlyservesasaguidetoformmathematicalsolutions.Therearenomagneticchargesorcurrentsinnature.28整理ppt3.8ReciprocityTheorem〔互易定理〕ReciprocitytheoremappliedtoEMtheoryalinearandisotropicmedium,butnotnecessarilyhomogeneous〞,twosetsofsourcesJ1,M1,andJ2,M2whichareallowedtoradiatesimultaneouslyorindividuallyinsidethesamemediumatthesamefrequencyandproducefieldsE1,H1andE2,H2.ThesourcesandfieldssatisfyLorentzReciprocityTheoremindifferentialformLorentzReciprocityTheoreminintegralform29整理pptLorentzReciprocityTheoremforsource-freeregionsConsiderthatthefields(E1,H1,

E2,H2)

andthesources(J1,M1,J2,

M2)

arewithinamediumthatisenclosedbyasphereofinfiniteradius.Assumethatthesourcesarepositionedwithinafiniteregionandthatthefieldsareobservedinthefarfield(ideallyatinfinity).(3.66)(3.62)(3.63)30整理pptReaction:Eachoftheintegralsin(3-66)canbeinterpretedasacouplingbetweenasetoffieldsandasetofsources,whichproduceanothersetoffields.(3.66)Coupling,notpowerForreciprocitytohold,31整理pptalinearandisotropic(butnotnecessarilyhomogeneous)medium3.8.1ReciprocityforTwoAntennasConjugatematching32整理pptThepowerdeliveredbythegeneratortoantenna#1IfthetransferadmittanceofthecombinednetworkisY21,thecurrentthroughtheloadisVgY21andthepowerdeliveredtotheloadiswhenantenna#2istransmittingand#1isreceivingwhenantenna#1istransmittingand#2isreceivingUnderconditionsofreciprocity(Y12

=Y21),thepowerdeliveredineitherdirectionisthesame.33整理ppt3.8.2ReciprocityforAntennaRadiationPatternsPatterninthereceivingmodeisidentical,becauseofreciprocity,tothatofthetransmittingmode.Reciprocityforantennapatternsisgeneralprovidedthematerialsusedfortheantennasandfeeds,andthemediaofwavepropagationarelinear.thereisadistinctsinglepropagatingmodeateachport.theantennasinthetransmitandreceivemodesarepolarizationmatched,includingthesenseofrotation.Aspecialcase:Measurementsystemusingalinearpolarizedprobetomeasurethecircularlypolarizedpatterno