《计算电磁学》第十讲

4/11/20209:54AM《计算电磁学》PartII：矩量法Dr.PingDU(杜平)SchoolofElectronicScienceandAppliedPhysics,HefeiUniversityofTechnologyE-mail:[e-mail]3Two-dimensionalElectromagneticFieldsDec.12,20114/11/20209:54AMOutline§3.1TransverseMagneticFields(横磁场)§3.2ConductingCylinders,TMcase(导电柱，TM情形)§3.3VariousApproximations(各种近似)§3.4TransverseElectricFields(横电场)§3.5ConductingCylinders,TEcase(导电柱，TE情形)§3.6AlternativeFormulation(替代表达）4/11/20209:54AM§3.1TransverseMagneticFields(横磁场)Anarbitraryelectromagneticfieldcanbeexpressedasthesumofatransversemagnetic(TM)partandatransverseelectric(TE)part.TheTMparthasonlycomponentsofmagneticfieldHtransversetoz,andtheTEparthasonlycomponentofEtransversetoz.Fortwo-dimensionalfieldsinisotropicmedia,theTMparthasonlyazcomponentofEandtheTEpartonlyazcomponentofH.Inmanycases,theTMandTEpartscanbetreatedseparately,reducingtheproblemtoascalarproblem.Inthissection,weonlyconsidertheTMfields!4/11/20209:54AMThetimeharmonicelectromagneticfield(timevariation)satisfiesthejteMaxwell’sequationsjEHjHE+J(3-1)(3-2)whereJisthevolumedistributionofelectriccurrents.ForTMfields,assumethat,andsimilarlyfor.(,)zzExyEuJTheMaxwell’sequationsthenleadto(3-3)whereisthewavenumber(λisthewavelength).2/kThisisatwo-dimensionalHelmholtzequation(亥姆赫兹方程).22zzzEkEjJ4/11/20209:54AMSolutionsmaybeobtainedbyfirstfindingthefieldfromatwo-dimensionalpointsource,thatis,athree-dimensionallinesource.ThefieldatduetoafilamentofcurrentIatisxyxyρuu''xyxyρ'uu(2)0'4zkEIHkρρ(3-4)whereistheintrinsicimpedanceoffree-spaceand/120(2)0HistheHankelfunction（汉克尔函数）ofthesecondkind,zeroorder.Theof(3-4)istheGreen’sfunctionfortheoperatorof(3-3).zEThesolutionisthesuperpositionofduetoallelementsofsource,orzEzJds4/11/20209:54AM(2)0()(')''4zzkEJHkdsρρρρ(3-5)wheretheintegrationisoverthecrosssectionofthecylinderofcurrents.zJNewwords(生词)：transverseelectric横电的isotropic各向同性scalar标量distribution分布wavenumber波数electriccurrents电流filament细丝,细线zeroorder零阶intrinsicimpedance本征阻抗Green’sfunction格林函数superposition叠加integration积分cylinder柱4/11/20209:54AM§3.2ConductingCylinders,TMcase(导电柱，TM情形)ConsideraperfectlyconductingcylinderexcitedbyanimpressedelectricfieldasrepresentedbyFig.3-1.TheimpressedfieldinducessurfacecurrentsontheizEzJconductingcylinder,whichproduceascatteredfield(散射场).szEXY0ρ'ρndl'ρρ'CFig.3-1.Crosssectionofacylinderandcoordinatesystem.4/11/20209:54AMThefieldduetoisgivenby(3-5)specializedtothecylindersurfaceC.zJTheboundaryconditionis0iszzzEEEonC(3-6)Combining(3-5)and(3-6),wehavetheintegralequation(2)0()(')''4izzCkEJHkdlρρρρ(3-7):known()izEρ:unknown(')zJρ4/11/20209:54AMThesimplestnumericalsolutionof(3-7)consistsofpulsebasisfunctionandpointmatchingprocedure.ThescatterercontourCisdividedintoNsegmentsandpulsefunctionsdefinedasnC1on()0onallothernnmCfCρ(3-8)Lettingtheelectriccurrent,substitutingitin(3-7),andtheresultant1NznnnJfequationatthemidpointofeach,weobtainthematrixequation(,)mmxymCmnnmlg(3-9)wheretheelementsofarethecoefficients.nn4/11/20209:54AMtheelementsofaremg(,)imzmmgExy(3-10)Theelementsofaremnl(2)220()()4nmnmmCklHkxxyydl(3-11)Asolutionforthecurrentisgivenby.1znnmmJflgNosimpleanalyticalexpressionisavailablefortheintegral(3-11).Butwecanevaluateitbyvariousapproximations.ThecrudestapproximationistotreatanelementasafilamentofcurrentznJCwhenthefieldpointisnoton.nC4/11/20209:54AMThatis,(2)220()()4mnnnmnmklCHkxxyy(3-12)when.mnForthediagonalelements(对角元素),theHankelfunctionhasintegralnnlsingularity,andtheintegralmustbeevaluatedanalytically.Forthisproblem,weapproximatebyastraightlineandusethesmallargumentformulanC(2)02()1log2zHzj(3-13)whereisEuler’sconstant.1.7814/11/20209:54AMAnevaluationof(3-11)thengives21log44nnnnkCklkCje(3-14)where2.718eBetterapproximationsforthisproblemwillbediscussedinSection3-3.Example.ConsiderTMplane-wavescatteringbyconductingcylinders.Inthiscase,theimpressedfieldisauniformplanewave.Theanglebetweenthewavevectorandthexaxisis.iTheincidentfieldis(cossin)iijkxyijzEeekr(3-15)4/11/20209:54AMThisdeterminestheexcitationaccordingto(3-10).mgAnapproximationevaluationofisgivenby(3-12)and(3-14).mnlThesolutionforisfoundbymatrixinversionintheusualmanner.zJWecalculatetheradarcrosssection(雷达散射截面),whichis2()()2siEE(3-16)whereisthedistantfieldfrom.()sEzJItcanbefoundbyusingtheasymptoticexpressionforin(3-5).(2)0H4/11/20209:54AMTheresultis('cos'sin)()(',')'sjkxyzCEkKJxyedl(3-17)where(3/4)1()8jkKek(3-18)Substituting(3-15)and(3-17)in(3-16),weobtain22('cos'sin)()(',')'4jkxyzCkJxyedl(3-19)4/11/20209:54AMperfectlyconducting完纯导体induce感应scatteredfield散射场combine组合evaluate计算distantfieldfrom远离integralsingularity积分奇异性straightline直线radarcrosssection雷达散射截面asymptotic渐进的Newwords生词4/11/20209:54AM§3.3VariousApproximations(各种近似)Theaccuracyofasolutionandtheconvergenceratedependontheapproximation.ThesolutionofSection3-2canbeimprovedbymoreaccurateevaluationofthe.mnl(1)Forthe,additionaltermscanbe