《计算电磁学》第九讲课件

《计算电磁学》PartII：矩量法Dr.PingDU(杜平)SchoolofElectronicScienceandAppliedPhysics,HefeiUniversityofTechnologyE-mail:pdu@Chapter2ElectrostaticFields(静电场)

Dec.5,2011《计算电磁学》PartII：矩量法Dr.PingDUOutline§2.1OperatorFormulation(算子描述)§2.2ChargedConductingPlate(含电荷的导电平板)2Outline§2.1OperatorFormulati§2.1OperatorFormulationThestaticelectricintensityEisconvenientlyfoundfromanelectrostaticpotential,whichis

(2-1)wheredenotesthegradientoperator.Inaregionofconstantpermittivityandvolumechargedensity,theelectrostaticpotentialsatisfies(2-2)istheLaplacianoperator(拉普拉斯算子).3§2.1OperatorFormulationTheForuniquesolution,theboundaryconditionsonareneeded.Inotherwords,thedomainoftheoperatormustbespecified.Fornow,considerfieldsfromchargesinunboundedspace,inwhichcaseconstantas(2-3)whereristhedistancefromthecoordinateorigin(坐标原点),foreveryoffiniteextent.Thedifferentialoperatorformulationis(2-4)where(2-5)4Foruniquesolution,theboundThedomainofListhosefunctionswhoseLaplacianexistsandhaveboundedatinfinityaccordingto(2-3).Thesolutiontothisproblemis(2-6)whereisthedistancebetweenthesourcepoint()andthefieldpoint().Hence,theinverseoperatortoLis(2-7)Notethat(2-7)isinverseto(2-5)onlyforboundaryconditions(2-3).Iftheboundaryconditionsarechanged,changes.5ThedomainofListhoseAsuitableinnerproductforelectrostaticproblems(constant)isThat(2-8)satisfiestherequiredpostulates(1-2),(1-3)and(1-4)iseasilyverified.(2-8)wheretheintegrationisoverallspace.LetusanalyzethepropertiesoftheoperatorL.Forthis,formtheleftsideof(1-5),(2-9)where6AsuitableinnerproductforeGreen’sidentityis(2-10)whereSisthesurfaceboundingthevolumeVandnisoutwarddirectionnormaltoS.LetSbeasphereofradiusr,sothatinthelimitthevolumeVincludesallspace.Forandsatisfyingboundaryconditions(2-3),andas.Henceas.Similarlyfor.Sinceincreasesonlyas,therightsideof(2-10)vanishesas.Equation(2-10)thenreducesto(2-11)7Green’sidentityis(2-10)wherItisevidentthattheadjointoperatoris(2-12)SincethedomainofisthatofL,theoperatorLisself-adjoint(自伴的).Themathematicalconceptofself-adjointnessinthiscaseisrelatedtothephysicalconceptofreciprocity.Itisevidentfrom(2-5)and(2-7)thatLandarerealoperators.Theyarealsopositivedefinitebecausetheysatisfy(1-6).ForL,form(2-13)andusethevectoridentityplusthedivergenceTheorem(散度定理).8ItisevidentthattheadjointTheresultis(2-14)whereSboundsV.AgaintakeSasphereofradiusr.Forsatisfying(2-3),thelasttermof(2-14)vanishesas.Then(2-15)and,forrealand,Lispositivedefinite.Inthiscase,positivedefinitenessofLisrelatedtotheconceptofelectrostaticenergy(静电能）.9Theresultis(2-14)whereSbo§2.2ChargedConductingPlate(含电荷的导电平板)Considerasquareconducting2ameteronasideandlyingontheplanewithcenterattheoriginasshowninFig.2-1.Fig.2-1.Squareconductingplateandsubsections10§2.2ChargedConductingPlateLetrepresentthesurfacechargedensityontheplate.Here,weassumethatthethicknessiszero.Theelectrostaticpotentialatanypointinspaceis(2-16)whereTheboundaryconditionis(constant)ontheplate.Theintegralequationfortheproblemis(2-17)11Letrepresenttwhere,.Theunknowntobedeterminedisthechargedensity.Aparameterofinterestisthecapacitanceoftheplate(2-18)whichisacontinuouslinearfunctionalof.Letusfirstgothroughasimplesubsectionandpoint-matchingsolution,andlaterinterpretitintermsofmoregeneralconcepts.ConsidertheplatedividedintoNsquaresubsections,asshowninFig.2-1.Definebasisfunctions(2-19)12where,Thusthechargedensitycanberepresentedby(2-20)Substituting(2-20)in(2-17),andsatisfyingtheresultantequationatthemid-pointofeach,weobtainthesetofequations(2-21)where(2-22)13ThusthechargedensitycanbeNotethatisthepotentialatthecenterofduetoauniformchargedensityofunitamplitudeover.Asolutiontotheset(2-21)givestheintermsofwhichthechargedensityisapproximatedby(2-20).Thecorrespondingcapacitanceoftheplate,approximating(2-18),is(2-23)Totranslatetheaboveresultsintothelanguageoflinearspacesandthemethodofmoments(MoM),let14Notethatisthepote(2-24),(2-25)(2-26)Thenisequivalentto(2-17).Asuitableinnerproduct,satisfying(1-2)to(1-4),forwhichLisself-adjoint,is(2-27)Wechoosethefunctions(2-19)asasubsectionalbasis.15(2-24),(2-25)(2-26)ThenThetestingfunctionsaredefinedas(2-28)Thisisthetwo-dimensionalDiracdeltafunction.Theelementsofthe[l]matrix(1-25)arethoseof(2-22),andthe[g]matrixof(1-26)is(2-29)Thematrixequationequation(1-24)isidenticaltothesetofequations(2-21).16ThetestingfunctionsaredefiIntermsoftheinnerproduct(2-27),thecapacitance(2-18)canbewrittenFornumericalresults,theof(2-22)mustbeevaluated.Letdenotethesidelengthofeach.The