Instructive Review of Computation of Electric Fields using Different Numerical Techniques - PDF document

InstructiveReviewofComputationofElectricFieldsusingDifferentNumericalJAWADFAIZDepartmentofElectricalandComputerEngineering,FacultyofEngineering,UniversityofTehran,Tehran,Iran.E-mail:faiz-j@sofe.ece.ut.ac.irM.OJAGHIZanjanRegionalElectricCompany,Zanjan,IranTherearedifferentnumericaltechniquesforcomputingelectricfields.Thesenumericaltechniquesenablethedesignertostudytheproblemsthataredifficulttobesolvedbyanalyticalmethods.Thispaperattemptstogiveaninstructivereviewofdifferentnumericaltechniquesinelectricfieldanalysis.Thesetechniquesinvolvefinitedifference,finiteelement,boundaryelement,chargesimulation,finiteelementwithvariablefieldintensityandMonteCarlomethods.Themeritsandlimitsofthevariousmethodsareoutlined.Someexamplesaregiveninwhichthefieldcomputationsusingdifferentnumericaltechniquesarecompared.Asanexamplethemostconvenienttechniqueapplicabletoelectricfieldcomputationwithinthetankofpowertransformersisintroduced.INTRODUCTIONUNTILNOWelectricandelectromagneticfieldshavebeenregardedasprobablythemostabstractanddifficultpartoftheundergraduateelectricalengineeringcurriculum.ThisislargelyduetothefactthatsuchfieldscannotbevisualizeddirectlydirectlyElectricfieldscanbecomputedusingvariousmethodswithdifferentprecision.However,forinsulationofelectricalequipmentamoreaccurateelectricfieldpredictionisrequired.Withtheadventofcomputingpowerandnumericaltech-niquesinrecentyears,ithasbecomepracticaltousedifferenttechniquestocomputetheelectricfields.Suchnumericaltechniquesenablethedesignertosolveproblemsthataredifficult,anduseofanalyticalapproachwithmanyempiricalfactorsisimpossible.Theaimofthispaperistogiveareviewoftheapplicationofdifferentnumericaltechniquesintheelectricfieldscomputationthatisusefulforanundergraduatecourse.Thismaterialshouldenablethegraduatestudenttousetheseanalysesingraduateresearchaswellaslaterintheworkplace.Theauthorswillalsogivetheirpersonalviewsonthefieldevaluationoftransformers.Intwo-dimensional(2D)analysis,ifthefieldisnottimevarying,theelectricpotentialVintheactualspaceissatisfiedbyPoisson'sequationquationr2Vˆÿ=…r0†…1†whereistheelectricchargedensity,relativeandabsolutepermittivityofthefreespace.InordertosolvepartialdifferentialEqn.1,anumberofboundaryconditionsmustbeimposed.Analyticalsolutionsforpracticalboundarycon-ditionsaredifficult,ifnotimpossibleandtherefore,numericaltechniquesarenecessary.Eachtechniquehasitsownmeritsanddraw-backsandonetechniquecannotbegenerallypreferredtoothers.Basedontheproposedproblem,themostconvenienttechniquemustbeselected.Differentnumericaltechniques,sofarusedfortheelectricfieldanalysis,arebrieflyreviewedandthemostconvenienttechniqueisthensuggestedforelectricfieldevaluationwithintheinteriorspaceofpowertransformers.BOUNDARYCONDITIONSTherearetwofollowingtypesofboundaryconditionsthatareconsideredintheelectricfield1.Boundarybetweenconductinganddielectricmaterials.2.Boundarybetweendifferentdielectricmaterials.Fromtheelectricalpointofview,oneofthefollowingconditionsmaybesatisfiedontheboundariesofthefirsttype:a.Theelectricpotentialsofallpointsontheboundaryareknown.Thisispossibleiftheavailableconductorisconnectedtoafixedpotentialsource(Dirchletcondition). *AcceptedInt.J.EngngEd.Vol.18,No.3,pp.344±356,20020949-149X/91$3.00+0.00PrintedinGreatBritain.2002TEMPUSPublications. Thewholechargeontheboundarysurfaceisknownwhiletheelectricpotentialoverdifferentpointsisunknown.Thisisthecasewhentheavailableconductorisnotconnectedtoafixedpotential(floating).Normallyinsuchacasethewholechargeonthesurfaceboundaryisequaltozero(Neumanncondition).FINITEDIFFERENCEMETHODFinitedifferencemethod(FDM)istheoldesttechniqueinthefieldcomputationsthatwasintro-ducedbyGauss.ThenBoltzmanpublisheditinhisnotesin1892.However,becauseofthelargeamountofcomputationsrequired,theextensiveuseoftheFDMdatesbackonlytotheeventofthecomputer.ElectricpotentialVoveranyregiondependsonthe(x,yandz)coordinatesanditsderivatives.PotentialofanypointmaybegivenversustheelectricpotentialoftheadjacentpointsusingTylor'sexpansion.Forinstance,considerFig.1havingpoint0andsixadjacentpoints.Thecoor-dinatesofthesepointsare:P0…x0;y0;z0†;P1…x0;y0;z0‡h†;P2…x0;y0ÿh;z0†;P3…x0;y0;z0ÿh†;P4…x0;y0‡h;z0†;P5…x0‡h;y0;z0†;P6…x0ÿh;y0;z0†Forsimplicity,thedifferencebetweentheoriginandtheadjacentpointsistakentobeh.PotentialofanarbitrarypointP…x;y;z†maybecalculatedversuspotentialofpointP0asfollows:V…x;y;z†ˆV…x0;y0;z0†‡‰…xÿx0†Vx…x0;y0;z0†‡…yÿy0†Vy…x0;y0;z0†‡…zÿz0†Vz…x0;y0;z0†Š=1!‡‰…xÿx0†2Vxx…x0;y0;z0†‡2…xÿx0†…yÿy0†Vxy…x0;y0;z0†‡2…xÿx0†…zÿz0†Vxz…x0;y0;z0†‡…yÿy0†2Vyy…x0;y0;z0†‡2…yÿy0†…zÿz0†Vyz…x0;y0;z0†‡…zÿz0†2Vzz…x0;y0;z0†Š=2!‡...…2†whereViˆ@V=@i;Viiˆ@2V=@i2;Vijˆ@2V=@i@j;i;jˆx;y;z;hˆ…xÿx0†ˆ…yÿy0†ˆ…zÿz0†Ifhtendstoaverysmallvalue,thetermscontain-ingthethird-andhigher-orderderivativesmaybeignoredandEqn.2canberewrittenasfollows:V1ˆV0‡hVz…P0†‡0:5h2Vzz…P0†V2ˆV0ÿhVy…P0†‡0:5h2Vyy…P0†V3ˆV0ÿhVz…P0†‡0:5h2Vzz…P0†V4ˆV0‡hVy…P0†‡0:5h2Vyy…P0†V5ˆV0‡hVx…P0†‡0:5h2Vxx…P0†V6ˆV0ÿhVx…P0†‡0:5h2Vzz…P0†…3†X6iˆ1Viˆ6V0‡h2‰Vxx…P0†‡Vyy…P0†‡Vzz…P0†Š…4†SolutionofEqn.4andsubstitutingfromEqns.1±2yields:V0ˆ‰V1‡V2‡V3‡V4‡V5‡V6‡h20=Š…5†where0istheelectricchargedensityatpoint0.AsshowninEqn.5,thereisalinearrelationshipbetweenthepotentialofpoint0andthepotentialsoftheadjacentpoints.Forthecaseswherediffer-encebetweenthepoint0andtheadjacentpointsisnotthesameorthesepointsarewithindifferentinsulatingmaterials,andalsoforrotatingfieldsinelectricalmachines,similarequationscanbederived[3,4].IntheFDM,theproposedregionisdiscretizedusingtheequationssimilartoEqn.5.Dimensionsofthemeshesmustbesuchthattheapproximationisacceptable.Thevertexesofthemeshesarenodesontheboundaryoftheregionandtheirpotentialsareknownortheycorrespondtopoint0inFig.1,enclosedbytheothernodes.Forthelatternote(notontheboundary),equationssimilartoEqn.5canbewrittenversusthepotentialsoftheadjacentnodes.Ifthenumberofsuchnodesisequalton,nlinearalgebraicequationswithnunknownvaluesofthenodepotentialsareobtained.Potentialsofthepointsinsidethemeshesmaybedeterminedusingdifferentinterpolationtechniques.TheFDMisnotcapableofcalculatingelectricfielddirectlyatdifferentpointsontheproposedregion.Whenthepotentialsofthenodesareobtained,anumericalderivativeevaluationtech-niqueisusedtocalculatetheelectricfieldintensity:EˆÿrV. Fig.1.Point0andsixadjacentpoints.InstructiveReviewofComputationofElectricFieldusingDifferentNumericalTechniques345 ELEMENTMETHODSAmongthevariousnumericaltechniques,thefiniteelementmethod(FEM)hasadominantpositionbecauseitisversatile,havingastronginterchangeabilityandcanbeincorporatedintostandardprograms[5,6].FEMisbasedonthisfactthatthephysicalsystemsstabilizesattheminimumlevelofenergy.Thegeneralequationofenergyinanelectricfieldis:Wˆ0:5…vrv…rv†dvÿ…vdvÿ…S0svds…6†wherevisthevolumeoftheproposedregionistheelectricchargedensitywithinvolumevS0istheboundarysurfacewithNummanconditionssisthesurfaceelectricchargedensitywithinS0.SinceEqn.1describestheelectricpotentialdis-tributionintherealsystems,basedonthemini-mumenergyleveltheorem,itisconcludedthatEqn.1minimizestheenergypresentedbyEqn.6.IntheFEM,thevolumeoftheproposedregionisdividedintoMsmallpolyhedronelementswheretheirsidesformagridwithNnodes.Thepotentialfunctionisthenapproximatedby:V…r†ˆXNiˆ1fi…r†Vi…7†whererisanypointontheproposedregion.fi…r†iscalledtheshapefunctionhavingthefollowingfeatures:a)fi…r†isequaltozeroanywhere,exceptonthesubregionWi.Thesub-regionWiconsistsoftheelementswherenodeiisoneoftheirvertices.b)fi…r†iscontinuousontheboundariesduetoWi,andapolyhedroninsideeachelement.c)anyfi…r†isequaltounitatthelocationofnodeiandzeroattheothernodes:fi…ri†ˆ1foriˆjfi…rj†ˆ0fori6ˆjViinEqn.7isequaltothepotentialofnodei.SubstitutingEqn.7intoEqn.6,theapproximateenergyispresentedbyWwhichisminimizedunderthefollowingconditions:@W=@Viˆ0;iˆ1;2;...;N…8†SinceWisaquadraticfunctionofVi,applyingconditions8leadstothefollowinglinearalgebraicequations:GVˆA…9†whereVistheknownvectorwithelementsVi;Aistheknownvectorobtainedfromthevolumechargedensityintheproposedregionandbound-aryconditions;Gisthenon-singularsquaresymmetricalmatrix.SolutionofthissystemofequationsgivesthevaluesofVisandhenceanapproximatedistribu-tionofthepotentialcanbedeterminedbasedonEqn.7.Electricfieldintensitywithineachelementisobtainedusingthegradientexpressionasfollows:Emˆ…ÿrV†mˆÿXNiˆ1Vi:fmi…10†Oftenthefirstderivativeoffiisnon-continuous.Therefore,reductionofthemaximumsizeoftheelementsandtendingtozero,leadsEqn.7totherealdistributionofthepotential.Inspiteofthis,nocontinuityofthefieldintensityonthebound-ariesoftheelementsremainsinforce.Iffunctionfiisconsideredasacompleten-orderpolynomial,betterresultscanbeobtained.Ifhpresentsthemaximumsizeoftheelements,reductionofhcanreducethepotentialerrorwithratio…
h†n‡1;andelectricfieldiscontinuousanditserrorisreducedbyratio…
h†n[7].Fig.2showsthemeshingandequipotentiallinesdeterminedusingtheFEM.TheFEMcouldbealsousedwherethepermittivityoftheproposedregionisnotconstant.Insuchacase,itisnecessarytoreplace…r†with,whichshowsthepositiondependency.Methodshavebeenintroducedtoconsiderthefloatingelectrodeswithunknownpotentialordifferentinsulatingmaterials[7].Thereareseveralreportsforautomaticmeshingoftheproposedregion[9±11].BOUNDARYELEMENTMETHODIfdistributionofelectricchargeforeveryregion(includingboundarysurfaces)isknown,electricpotentialandfieldintensityforeachpointcanbecomputedusingCoulomb'slawofGauss'slaw[2].Inpractice,Laplacianequationisnormallyused.Thismeansthatelectricchargeisenclosedonlyinsidetheboundariesoftheproposedregionandvolumechargedensityinsidetheregionisequaltozeroornegligible.Theelectricpotentialandfieldintensityare:V…I†ˆ……ss=…4R†ds…11†E…I†ˆ……ss…4R2†ds:^aR…12†whereIistheproposedpoint,Sisthesummaryofallboundarysurfaces,sisthesurfacechargedensityoversurfaces,RisthedistanceofthedifferentialelementsfrompointIand^aRisunitvectoralongRdirectedfromdifferentialelementdstopointI.InpracticesisunknownanditseemsthatsolvingEqns.11±12isimpossible.However,toJawadFaizandM.Ojaghi346 thisdifficulty,theBEMmaybeemployed.Inthismethod,theboundarysurfaceisdividedintoNelements.Figure3showsatypicalboundaryelementsduetoaflatboundarysurfaceonxy-plane.Thentakingintoaccountthesmalldimensionsoftheboundaryelements,ageneralformmaybeconsid-eredforthesurfacechargedensityonelementj…sj†.Thisgeneralformisoftenapolynomialwithunknowncoefficients.Forinstance,ifitistakentobeaquadraticpolynomialasfollows:sjˆa1j‡a2jx‡a3jy‡a4jxy…13†substitutingsjinEqns.11and12gives:V…I†ˆXNjˆ1……sjs=…4R†dsˆXNjˆ1……sj‰a1j‡a2jx‡a3jy‡a4jxyŠ=…4R†ds…14†E…I†ˆXNjˆ1……sjs=…4R2†ds^aR Fig.2.ComputationofelectricfieldusingFEM:a)meshing,b)equipotentiallines[4].InstructiveReviewofComputationofElectricFieldusingDifferentNumericalTechniques347 XNjˆ1……sj‰a1j‡a2jx‡a3jy‡a4jxyŠ=…4R2†ds^aR…15†ItisclearthatV(I)andE(I)arelinearfunctionsofthecoefficientsofthepolynomials.Byusingthefollowingdefinitions:P1jˆ……sjds=…4R†P2jˆ……sj…xds†=…4R†P3jˆ……sj…yds†=…4R†P4jˆ……sj…xyds†=…4R†f1jˆ……sjds=…4R2†^aRf2jˆ……sj…xds†=…4R2†^aRf3jˆ……sj…yds†=…4R2†^aRf4jˆ……sj…xyds†=…4R2†^aREqns.14and15become:V…I†ˆXNjˆ1X4kˆ1Pkjakj…16†E…I†ˆXNjˆ1X4kˆ1fkjakj…17†Inthenextstage,thenumberofunknownsfunc-tionPsjisselectedoneachboundaryelement.Then,basedontheboundarytypewheresjispartofit,theboundaryconditionsequationforanyselectedpointisformedusingEqns.16and17.Therefore,asystemofalgebraiclinearequationsisobtainedwhichfinallyproducesthecoefficientsofthepolynomial.Pkjandfkjarenumericallyoranalyticallyobtainedbyintegration.Hencethesurfacechargedensitydistributiononallboundarysurfacesisknown.FinallytheelectricpotentialandfieldintensitycanbedeterminedusingEqns.16and17. Fig.3.Aflatquadrangularboundaryelement. Fig.4.ElectricfieldcalculationusingBEM:a)actualelectricalsystem,b)electricfielddensityvectorsonthesphere,c)equipotentialcountors[12].JawadFaizandM.Ojaghi348 4presentsatypicalproblemsolvedbytheBEM,wheretheelectricfieldbetweenthetwosphereshavingthesameradiusrˆ1isobtained.Electricfieldintensityhasbeenshownwithasuit-ablevector.Inaddition,theequipotentialsbetweentwosphereshavebeencalculatedbycomputationoftheelectricpotentialindifferentpoints.Amethodhasbeengivenin[13]forthecurved-shapeboundaryelementsinordertomodelprac-ticalsurfaceswiththedesirableaccuracy.Bound-aryelementswithfullaxialsymmetryhavebeenpresentedintheliterature[14±16].InputandoutputdataprocessingmethodsintheBEMhavebeenintroducedin[12,17,18].CHARGESIMULATIONMETHODThismethodissimilartotheBEM.Thediffer-encebetweenchargesimulationmethod(CSM)andtheBEMisthesimulationofthesurfacechargeexistingontheboundarysurfaces.IntheBEMthesurfacechargedensityfunctiononthedifferentsurfaceboundariesareestimated,whileintheCSMthesurfacechargedensityissubstitutedbyasetofdiscretizedlinearchargedistribution.Thesubstitutedlinearchargedistributionissuchthattheelectricpotentialandfieldintensityversustheirchargesareanalyticknownfunctions.Chargedistributiononaninfinitelengthlinewithaconstantdensity,onafiniteline,onacircleetc.,areexamplesofthechargedistribution.Differentdistributiontypesandtheirequationshavebeengivenin[4]and[19].Electricpotentialandfieldintensityequationsduetotheabovementionedchargedistribution,forthepointsonthecharges,havesingularity.Toovercomethissingularity,thepositionofthereplacedchargesimulationisconsideredoutsidetheproposedspaceandnormallyinsidetheelec-trodes.Theexactpositionandthereplacedchargedistributiontypearearbitrarilyselectedbasedontheexperience.Chargevalueortheirlinearchargedensityiscomputedsuchthattheboundarycon-ditionsaresatisfiedonsomesurfaceboundaries,asdescribedinthefollowingpart.Electricpotentialandfieldintensityduetothereplacedchargesimulationindifferentpointsarelinearfunctionsofthechargevalueorchargedensity:Vj…r†ˆPj…r†qj…18†Ej…r†ˆfj…r†qj…19†wherePj…r†isthepotentialfactor,fj…r†isthefieldintensityfactor,ristherelativepositionoftheproposedpointandqiisthejthreplacedchargedistribution.Pj…r†andfj…r†aredifferentfordiffer-entchargedistributions.Thereisthefollowingrelationshipbetweenthem:fj…r†ˆÿrPj…r†…20†Sincethereisalinearrelationshipbetweentheelectricpotential(andfieldintensity)duetothereplacedchargedistributionindifferentpointsandchargevalueorchargedensity,forasetofsuchchargedistribution,thesuperpositiontheoremcanbeappliedtocalculatetheelectricpotentialandfield:Vj…r†ˆXNjˆ1Pj…r†qj…21†Er…r†ˆXNjˆ1fj…r†qj…22†IntheCSM,thenumberofpointsselectedonthesurfaceboundaryisequaltothereplacedsimula-tioncharges.Dependingontheselectedpoints,Eqns.21and22areusedandtheboundaryconditionsforindividualpointsareconsidered.Theseequationsarelinearfunctionsofqj.Sincetheexactpositionofthechargesandtheselectedpointsareknown,PjsandfjsinEqns.21and22areexactlycalculated.Thenonlyqisaretheunknownvaluesoftheabovelinearequations.Therefore,computationsoftheelectricpotentialandfieldintensityarepossibleusingEqns.21and22.ItisclearthatintheCSM,boundarycondi-tionsaresatisfiedonlyinthepointsselectedtowritetheequations.Beforeusingqjsduetothesolutionoftheequations,itisnecessarytostudytheboundaryconditionsonotherpointsoftheboundarysurfaces.WhenqjsintheCSMiscalculated,theboundaryconditionsondifferentsurfacepointsmustbedetermined.Iftheaccu-racyisnotenough,number,positionandtypeofthereplacedsimulationchargesandalsopositionoftheselectedpointsforwritingtheequationsmustbevariedinordertoobtainsufficientaccuracy.Figure5showsatypicalproblemsolvedusingtheCSM.TheproblemwascomputationoftheelectricfieldbetweentwospheresshowninFig.5a.Figure5bpresentstheequipotentialslinesbysubstitutingthesurfacechargeofeachspherewithtwopoint-charges.Figure5cindicatesthecorrespondingresultwhenthreepoint-chargesaresubstitutedforeachsphere,inwhichtheaccuracyishigher.Inaddition,twopoint-chargeshavebeenusedinFig.5d,butthepositionofchargesisdifferentwiththatofFig.5b.Accuracyofthelattercaseisbetterthantheothertwo.FINITEELEMENTMETHODWITHVARIABLEFIELDDENSITYIntheFEM,themainvariableistheelectricpotentialV,whiletheelectricfieldintensityisnormallytherequiredquantity.Forexample,inthedesignofhighvoltagedeviceinsulation,itisInstructiveReviewofComputationofElectricFieldusingDifferentNumericalTechniques349 rytohavetheamplitudeanddirectionoftheelectricfieldonthesurfacesoftheelectrodesandinsulation.Instudyofthedischargephenom-enon,thepathoftheforcelinesisrequiredwhichcanbedeterminediftheelectricfielddistributionintheproposedspaceisknown.IntheFEManumericalintegrationtechniqueisusedinordertocalculateEthatnormallyhaserror.ThereisamoreaccurateFEMtechniqueinwhichthemainvariablehasbeentakenEinsteadofV[20].Ifthevolumechargedensityintheproposedspaceiszeroand Fig.5.CalculationofelectricfieldusingCSM:a)electricsystem,b)equipotentialwhentwochargepointsusedforeachsphere,c)equipotentialwhenthreechargepointsusedforeachsphere,d)equipotentialwhentwochargepointsindifferentpositionsusedforeachsphere.JawadFaizandM.Ojaghi350 ttivityEisconstantandisotropic,thentheMaxwell'sequationsforelectrostaticfieldsareasfollows:divDˆ0…23†CurlEˆ0…24†whereDistheelectricchargedensityandEistheelectricfieldintensityand:DˆE…25†ApplyingEqn.24andusingEqn.23and25leadsto:r2Eˆ0…26†whichistheLaplacianequation.For2Dfields:…@2Ex=@x2‡@2Ex=@y2†i‡…@2Ey=@x2‡@2Ey=@y2†jˆ0…27†Butavectorwillbezeroifallitscomponentsarezero:@2Ex=@x2‡@2Ex=@y2ˆ0@2Ey=@x2‡@2Ey=@y2ˆ0…28†Foreachcomponentofthefieldintensity,theFEMisusedonceandfinallytheelectricfieldintensityoverthewholeproposedregionisobtained.However,withdefinedboundaryconditions,theequationwillhaveauniquesolution.Inpractice,theboundaryconditionsonthefirst-typeboundaryversuselectricpoten-tialareknownbutthereisnoideaconcerningtheirelectricfieldintensity.In[20],theBEMhasbeenusedinordertoovercomethisdifficult;itmeansthattheBEMisemployedonthementionedboundariesthenfiniteelementmethodwithvariablefielddensity(FEMVFD)isapplied.AnalysishasbeencarriedoutbasedontheFEMandFEMVFDandtheresultshavebeenpresentedinFig.6.Theproblemwascomputationoftheelectricfieldintheregionbetweenthetwocylindershavingpotentialsof100Vand200V.Becauseoftheaxialsymmetryanditsboundariesandinfinitelength,thefieldhasonlyaradialcomponentvaryinginradialdirection.Therefore,analysisofthefieldispossibleintwodimensions.Duetothesymmetryonlyone-quarterofthecylinderisusedforanaly-sis.Figure6bshowsthemeshingforbothtech-niquesandFig.6cpresentstheequi-fieldintensitylines.Asseen,inspiteofthelowernumberofmeshesintheFEMVFD,theequi-fieldintensitylineshaveabettercontinuityandtheiraccuracyisalsohigher.Inthisexample,themaximumerrorinsolutionbytheFEMVFDisabout2%whilethiserroris9.3%whentheFEMisused.MONTECARLOMETHODIntheFDM,thepotentialofeverynodeisequaltothemeanvalueofthepotentialsoftheadjacentnodes.Whenthedistancebetweentheproposednodeandadjacentnodesarethesame,sameweightpotentialsareinvolvedinthecomputationofthemeanvalue.Otherwise,theweightswillnotbeequal[4].Buttheclosernodeswillbeheavier.However,thesumofallweightsisunity.GenerallyintheFDM,potentialofeachnode(V0†versuspotentialofnadjacentnodesisasfollows:V0ˆXNIˆ1WiVi…29†whereWiˆ1,V1isthepotentialofthenodeadjacenttoi-thnodeandWiisitsweight.ValueofWidependsontherelativedistanceofthei-thnodefromtheproposednode.ThebasicequationofMonteCarloMethod(MCM)issimilartoEqn.29.Therefore,thesetwomethodsaresimilar,exceptthatdeterminationoftheadjacentnodesandmethodofcalculationofWidiffers.IntheFDM,allnodesaredefinedaftermeshingprocessandatthesametimeadjacentnodesofeverynodearedetermined.InWicom-putation,analyticalrelationshipsareused.ButintheMCM,adjacentnodesarealwaysontheboundariesandtheirexactpositionsaredeter-minedusingarandomprocessandWisisobtainedusingprobabilitytechniques.InFig.7,Sisshowboundarysurfacesoftheproblem.TheseboundarysurfacesareDrichlettypewithpotentialVi.Calculationofpotentialatanypointsuchasr0isrequiredusingequationssimilartoEqn.29.IntheMCM,simulationofarandommovementisusedinordertodetermineeachinode.Anymovementbeginsfromr0andaftersuccessivejumpswithvariablelengthsandinrandomdirections,leadstoaSj.Conditionsgoverningoneveryrandommove-mentareasfollows:a)Allmovementsbeginfromr0.b)Lengthofanyjumpisequaltotheminimumdistanceofthebeginningpointwithboundarysurfaces(Sis).c)Directionofveryjumpisrandom.d)Theendofanyrandommovementwillreachwhentheminimumdistancementionedinitembissmallerthanthatofthepredefinedvaluesuchas0.Attheendofarandommovement,apointononeoftheboundarieswiththeclosestdistancetotheendpointisselectedasanadjacentpoint.Finally,potentialofpointr0iscalculatedasfollows:V…r0†ˆXNiˆ1V…ri†=N…30†whereNisthetotalnumberoftherandomInstructiveReviewofComputationofElectricFieldusingDifferentNumericalTechniques351 entsandriistheadjacentpointduetothej-thrandommovement.Inordertoeliminatethestatisticalerrors,itisnecessarythatNbecomeslargeenoughandoftenseveralthousands.InEqn.30,allweightsareapparentlythesameandequalto1/N.Infact,itisnotso,becausetakingintoaccountthelargenumberofrandommovements(N),probabilityofseveralreputationsofonepointexistsandforrelativelyshorter Fig.6.ComparisonofcomputedelectricfieldusingFEMandFEMVFD:a)electricsystem,b)meshingbasedonthetwotechniques,c)equi-fieldintensitylinesusingbothmethods.JawadFaizandM.Ojaghi352 cefromr0,thisprobabilityislarger.SupposeconstantpotentialViforanyboundarysurface,Eqn.30canbewrittenasfollows:V…r0†ˆXniˆ1NiVi…31†wherenisthenumberofboundarysurfacesandNiisthenumberofrandommovementsendedtoapointatSi.AlthoughtheMCMwasdescribedusingtheFDM,itisnecessarytonotethattheMCMisitselfanindependentmethodhavingspecial Fig.7.Representationofthepossiblerandommovementpathsforreachingfromtheproposedpoint(r0)tooneoftheboundarysurfaces(Si). Fig.8.Geometryoflaboratoryhighvoltageelectrodes[7].InstructiveReviewofComputationofElectricFieldusingDifferentNumericalTechniques353 ntals[21].Computationofelectricpoten-tialusingMCMissimilartothemicroscopicanalysisofgaspressurewheretherandommove-mentofthemoleculesissimulated.Atechniqueforcomputationoftheelectricfieldoverthespacesconsistingofdifferentisolatingmaterialshasbeenpresentedin[22].In[7,22],anumberoftechniqueshavebeenintroducedtospeedupthecalculationusingtheMCM.Figure8showsthegeometryofthelaboratoryhighvoltageelectrodesinwhichthe Fig.9.ComparisonofthecomputedelectricfieldsofthesystemshowninFig.8usingthreetechniques:a)alongpathAB,b)alongpathCDandc)alongpathEF[7].JawadFaizandM.Ojaghi354 potentialandfieldalonglineAB,CDandEFhavebeenestimatedusingthethreemethods:MCM,CSMandFEM.TheresultsarepresentedinFig.9.ItisclearthattheaccuracyoftheMCMissimilartotheothertechniques.Thereareabout24000randommovementsinthissimulation.COMPARISONOFMETHODSANDCONCLUSIONSAnynumericaltechniqueforelectricfieldevaluationhasitsownmeritsanddrawbacksanditisnotgenerallypossibletopreferonetechniqueovertheothers.Structureofthepowertransformerindicatesthatnormally3Dcomputationoftheelectricfieldisrequiredbecausethereisnosymmetryinordertoignoreonedimension.Inaddition,numer-icaltechniquemustbecapableofdealingwiththenarrowinsulatingorconductinglayers.FDMandFEMintheelectricfieldcomputationhavetwodrawbacks:a)Indeterminationofpotentialdistributionusingtwomethods,numericalderivativetechniquesmustbeusedinordertoobtaintheelectricfieldintensity.Thishasconsiderableerrorthatleadstoalargeerrorintheelectricfieldcomputation.Inmanycases,thefieldisrequiredforthedesignofinsulationofelectricalequipment.Meanwhile,studyofsomephenomenasuchaselectricaldischargeispossiblebyelectricfieldcomputation.b)Inordertopreventalargeelectricfieldanditsdrawbacks,thesharpedgesonthedifferentsurfacesareavoided.Therefore,thecurvedsurfacesareoftenpreferred.FDM,FEMandFEMVFDhavedifficultyinmodelingsuchcurvedsurfaces;butCSMandBEMcanbeeasilyadaptedforsuchcases.Ontheotherhand,althoughFDM,FEMandFEMVFDcantheoreticallycompute3Dfields,therearemanyproblemsindealingwiththismatter.Oneseriousproblemis3Dmeshgenerationanditsmodificationtoapproachtherequiredaccuracy.Generallymanualcalculationiscumbersomeandtimeconsumingandalsocomputerprogrammingisreallycomplicated.Anotherdifficultyisthelargesizeofthecoeffi-cientmatrixofthesystemofequations.Inthesemethodsnumberofequationsisproportionaltothememoryrequiredforcomputationwhileinothermethods(CSMandBEM)thisnumberisproportionalwithareaoftheboundarysurfaces.ItmeansthatinFDM,FEMandFEMVFD,thecoefficientmatrixhasonemoredimensionthanthecoefficientmatrixduetotheotherabove-mentionedmethods.Therefore,morecomputermemoryandlongercomputationtimearerequired.HenceFDM,FEMandFEMVFDmaynotbecon-sideredconvenienttechniquesforelectricfieldcomputation.MCM,atleastinelectricfieldcomputation,isnotsocommonandhasnoconsiderableprogressinrecentyears.Atthepresent,applicationofthismethodinnarrowlayerproblemsisdifficult,ifnotimpossible.Inspiteofthesimplicityofcomputerprogram-mingandhighaccuracyofthemethod,incompu-tationof3DelectricfieldshavingnarrowlayersCSMisconfrontedwithamajordifficulty.Inthismethoditisnecessarytoconsiderchargeswithinthementionedlayerssuchthattheyhaveenoughdistancefromtwosidesofthelayer.Butthethicknessistoonarrowandsuchanassumptionmaynotbecorrect.FinallyBEMiscapableofanalyzing3Dfieldsandtherearesomereportsshowingitsapplicationstonarrowlayers.Inconclusion,BEMmaybeconsideredthemostconvenienttechniqueforelec-tricfieldcomputationwithintheinteriorofthepowertransformertank[23,24].REFERENCES1.N.N.Rao,PC-assistedinstructionintroductoryelectromagnetics,IEEETrans.onEducation,33(1),pp.51±55,Feb.1990.2.W.H.HaytJr,EngineeringElectromagnetics,4thEdition,McGraw-HillBookCo.,1981.3.BharatHeavyElectricals,Transformers,TataMcGraw-HillPublishingCompany,NewDelhi,1987.4.E.KuffelandW.S.Zeungl,HighVoltageEngineeringFundamentals,PergamonPress,1984.5.M.V.K.Chari,G.BedrosianandI.D.Angelo,Finiteelementapplicationsinelectricalengineering,IEEETrans.Magnetics,29,1993,pp.1306±1314.6.C.R.I.Emson,J.SimkinandC.W.Trowbridge,Astatusreportelectromagneticfieldcomputation,IEEETrans.Magnetics,30,1994,pp.1533±1540.7.M.D.R.Beasley,Comparativestudyofthreemethodsforcomputingelectricfields,IEEProc.,126(1),January1979,pp.126±134.8.O.W.Anderson,Finiteelementsolutionofcomplexpotentialelectricfields,IEEETrans.PowerApparatusandSystems,PAS-96(4),July/August1977,pp.1156±1161.9.O.W.Anderson,Laplacianelectrostaticfieldcalculationsbyfiniteelementswithautomaticgridgeneration,IEEETrans.PowerApparatusandSystems,PAS-92,Sept./Oct.1973,pp.1485±1492.10.A.KamitaniandM.Miyauchi,Automaticnumericalgenerationoffiniteelementsfortwo-dimensionalregionofarbitraryshape,Proc.Int.Symp.Electromagnetics(ISEM),Sapporo,Japan,1993,pp.89±92.InstructiveReviewofComputationofElectricFieldusingDifferentNumericalTechniques355 J.FaizandE.Shafagh,AutomaticmeshgenerationusingAuto-Cad,IEEETrans.Education,41(4),November1998,pp.325±330.12.H.Tsuboi,etal.Postprocessingforboundaryelementanalysisinelectromagneticfieldproblems,Proc.Int.Symp.Electromagnetics(ISEM),Sapporo,Japan,1993,pp.117±120.13.T.Misakietal.Computationofthree-dimensionalelectricfieldproblemsbyasurfacechargemethodanditsapplicationtooptimuminsulatordesign,IEEETrans.PowerApparatusandSystems,PAS-101(3),March1982,pp.627±634.14.S.SutoandB.Bachmann,Athreedimensionalhighspeedsurfacechargesimulationmethod(3D-HSSSM),Int.Symp.HighVoltageEngineering,Athens,Greece,Sept.1983,p.11.08.15.S.SatoandW.W.S.Zaengle,Effective3-dimensionalelectricfieldcalculationbysurfacechargesimulationmethod,IEEProc.,133,Pt.a,No.2,March1986,pp.77±83.16.M.ReisterandP.Weib,Computationofelectricfieldsbyuseofsurfacechargesimulationmethod,Int.Symp.HighVoltageEngineering,Athens,Greece,Sept.1983,p.11.06.17.M.NishinoandT.Takedo,Preprocessorforthe3Dboundaryelementmethodbasedontheconstructivesolid-geometry,Proc.Int.Symp.Electromagnetics(ISEM),Sapporo,Japan,1993,pp.93±96.18M.NakahiraandT.Takeda,Ameshrefinementforboundaryelementmethodusinganerrorestimatingindex,Proc.Int.Symp.Electromagnetics(ISEM),Sapporo,Japan,1993,pp.151±154.19.H.Singer,H.SteinbiglerandP.Weise,Achargesimulationmethodforthecalculationofhighvoltagefields,IEEETrans.PowerApparatusandSystems,PAS-93,Sept./Oct.1974,pp.1660±1668.20.H.Yamashita,K.ShinozakiandE.Nakamae,Aboundaryfiniteelementmethodtocomputedirectlyelectricfieldintensitywithhighaccuracy,IEEETrans.PowerDelivery,3(4),October1988,pp.1754±1760.21.R.M.Brvensee,Probabilisticpotentialtheoryappliedtoelectricalengineeringproblems,Proc.IEEE,61(4),April1973,pp.423±437.22.M.KrauseandK.Muller,AMonteCarlomethodfortwo-andthree-dimensionalelectrostaticfieldcalculationinmaterialsofdifferentpermittivity,Int.Symp.HighVoltageEngineering,Athens,Greece,Sept.1983,p.11.04.23.J.FaizandM.Ojaghi,Afastboundaryelementmethodtoelectricfieldcomputationwithinthetankofpowertransformers,Int.J.ComputationandMathematicsinElectricalandElectronicEngineering(COMPEL),17(1/2/3),1998,pp.69±77.24.J.FaizandM.Ojaghi,Noveltechniquesfortreatingsingularityproblemsintheboundaryelementmethodofevaluationwithinthetankofapowertransformers,IEEETrans.PowerDelivery,15(2),April2000,pp.592±598.JawadFaizreceivedtheBachelorandMastersdegreesinElectricalEngineeringfromTabrizUniversityinIranin1974and1975respectivelygraduatingwithFirstClassHonours.HereceivedthePh.D.inElectricalEngineeringfromtheUniversityofNewcastle-upon-Tyne,England,in1988.Earlyinhiscareer,heservedasafacultymemberinTabrizUniversityfor10years.AfterobtaininghisPh.D.degreeherejoinedTabrizUniversitywhereheheldthepositionofAssistantProfessorfrom1988to1992,AssociateProfessorfrom1992to1997,andhasbeenaProfessorsince1998.SinceFebruary1999hehasbeenworkingasaProfessoratDepartmentofElectricalandComputerEngineering,FacultyofEngineering,UniversityofTehran.Heistheauthorof50publicationsininternationaljournalsand60publicationsinconferenceproceedings.Dr.FaizisaSeniorMemberofIEEEandmemberofIranAcademyofSciences.HisteachingandresearchinterestsareswitchedreluctanceandVRmotordesign;designandmodellingofelectricalmachinesanddrives.MansourOjaghireceivedtheB.Sc.degreeinElectricalEngineeringfromtheShahidChamranUniversity,Ahwaz,Iranin1993andtheM.Sc.degreeinElectricalPowerEngineeringfromtheUniversityofTabriz,Iran,in1997.HeisnowworkingasaseniorengineerofZanjanRegionalElectricCompany,Zanjan,Iran.Hisareaofinterestisonthemodelingandfieldanalysisoftransformers,protectiverelayingandelectricalmachinesmodeling.JawadFaizandM.Ojaghi356