Modeling of Permanent Magnet Motor Drives

IEEETRANSACTIONSONINDUSTRIALELECTRONICS,VOL.35,NO.4,NOVEMBER1988"i."537'ii'r,:,,~'::(:''..'"ModelingofPermanentMagnetMotorDrivesPRAGASANPILLAY,MEMBER,IEEE,AN,MEMBER,IEEEAbstract-RecentresearchhasIndicated(hat(hepermanentmagnetmotordrives,whichincludethepermanentmagnetsynchronousmotor(PMSM)andthebrushlessdcmotor(BDCM)couldMhasasinusoidalbackemfandrequiressinusoidalstatorcurrentstoproduceconstanttorquewhiletheBDCMhasatrapezoidalbacMisverysimilartothewoundrotorsynchronousmachineexceptthatthePMSMthatisusedforservoapplicationstendsnottohaveanydamperwindingsandexcitationisprovidedbyapermanentmagnetinsteadofafi~hed,qmodelofthePMSMcanbederivedfromthewell-knownmodelofthesynchronousmachinewieoftbenonsinusoidalvariationofthemutualinductancesbetweenthestatorandrotorintheBDCM,itisalsoshownInthispaperthatnoparticularadvantageexistsintransformingtheabcequationsoftheBCDMtothed,UCTION~(I...J,..:;:~kr,DECENTresearch[1]-[3]tmotordrives,whichincludethepermanentmagnetsynchronousmotor(PMSM)andthebrushlessdcmotor(BDCM)couldMhasasinusoidalbackemfandrequiressinusoidalstatorcurrentstoproduceconstanttorquewhiletheBDCMhasatrapezoidalbacnfusionexists,bothintheindustryandintheuniversityresearchenvironment,MisverysimilartothestandardwoundrotorsynchronousmachineexceptthatthePMSMhasnodamperwindingshed,qmodelofthePMSMcanbederivedfromtheNOMENCLATUREwell-known[4]modelofthesynchronousmachinewiththeequationsofthedamperwindingsandfieldcurrentdynamicsBdampingconstant,Nlradls(persecond)Asiswellknown,thetransformationofthesynchronousea,eb,eca,b,andcphasebackemfs(involts)machineequationsfromtheabcphasevariablestothed,qpeakvalueofbackemf(involts)Epvariablesforcesallsi~usoidallyvaryinginductancesintheabcia,ib,ica,b,andcphasecurrents,(inamperes)frametobecomeconstant-inthed,DCMd,qaxisstatorcurrents(inamperes)id,iq'motor,sincethebackemfisnonsinusoidal,theinductancesdoJmomentofinertia,kg-m2La,Lb,Lcselfinductanceofa,b,andcphases(innotvarysinusoidallyintheabcframeanditdoesnotseemadvantageoustotransformtheequationstothed,qframehenrys)sincetheinductanceswillnotbeconstantaftertransformation.-Labmutualinductancebetweenphasesaandb(inHenceitisproposedtousetheabcphasevariablesmodelforhenrys)tion,thisapproachinthemodelingofthed,qaxisinductances(inhenrys)Ld,LqBDCMallowsadetailedexaminationofthemachine'storquepderivativeoperator-behaviorthatwouldnotrresistance(inohms)Thed,qmodelofthePMSMhasbeenusedtoexaminetheTeelectrictorque(innewtonmeters)transientbehaviorofahigh-performancevectorcontrolledT,loadtorque,(innewtonmeters)PMSMservodrive[5].Inaddition,theabcphasevariableVa,Vb,Vca,b,andcphasevoltages(involts)modelhasbeenusedtoexaminethebehaviorofaBDCMd,.qaxisvoltages(involts)Vd,Vqspeedservodrive[6].ApplicationcharacteristicsofbothVdcdcbusvoltage(involts)machineshavebeenpresentedin[7].Thepurposeofthismutualfluxduetomagnet(inwebers)Aafpaperistopresentthesetwomodelstogetherandtoshowthatd,qaxisfluxlinkages(inwebers)Ad,Aqthed,qmodelissufficienttostudythePMSMindetailwhileWrrotorspeed(inradianspersecond)nchronousspeed,(inradianspersecond)OranglebetweenstatorphaseAandtherotor(inisthereforetutors)*Thepaperisarrangedasfollows:SectionIIpresentstnIVusestheseManuscriptreceivedSeptember4,1987;revisedMarch21,topresentsomekeyresultstoillustratetheuseoftheseTheauthorsarewiththeElectricalEngineeringDepartment,VirginiamodelstostudybothtransientandsteadystatebehaviorofPolytechnicInstituteandStateUniversity,BIacksburg,nIVhastheconclusions.0278-0046/88/1100-0537$01.00@1988IEEE--:,1i,-,I,'r~'j:'!'~'i~.",..,

ERt988ThestatorofthePMSMandthewoundrotorsynchronousmotor(SM)aresimilar.'Inadditionthereisnodifferencebetweenthelowingassumptionsaremadeinthederivation:1)Saturationisneglectedalthoughitcanbetakenintoaccountbyparameterchanges;2)Thebackemfissinusoidal;3)eseassumptionsthestatord,qequationsintherotorreferenceframeofthePMSMare[6],[7]Vd=Rid+PAd-(J),AqVq=Riq+PAq-":rAdi'"~vdeqRLqwe~uivalentcircuitfromdynamicequations.(1)(2)whereAq=LqiqandAd=Ldid+Aaj(3)i~".'q~v(4).ctrictorqueT~=3P[~iq+(Ld-Lq)idiq]12(5)stantfluxoperationwhenidequalszero,theelectrictorqueT,=3Aafiq12=K,iqwhereK,atthistorqueequationforthePMSMresemblesthatoftheregulardcmachineandhenceprovidesPMSMequivalentcircuitfromsteadystateequations.r:=9iq+q'"~eoftheParktransformdefinedbelowcos(0)sin(0)Vb=cos(0-21r/3)sin(0-27r/3)Vc].['cos(0+21r/3)sin(0+27r/3)nstatespaceformpid=(vd-Rid+(J),Lqiq)ILdpiq=(Vq-Riq+(J),Ldid~(J),Aaf)1Lqp(J),=(T~-B(J)r-~)IJpO,=(J),.(6)(7)(8)(9)[m::](11)alinputpowertothemachineintermsofabcvariablesisPowerVdandVqareobtainedfromVa,Vb,andVc-throughtheParktransformdefinedbelow=vaia+vbib+vcic-(12)whileintermsofd,qvariablesThefactor3/edynamicequationsofthePMSM,theequivalent1/2[1/2.1/steadystateoperation,thed,(10)dynamiantageofmodelingthemachineusingthed,qaxisequationsthenisthesubsequentabcvariablesareobtainedfromd,qvawer=3(vdid+vqiq)l2.(13)=213(0)cos(0-21r/3)sin(0)sin(0-27r/3)cos(0+27r/3)sin(0+21r/3)][::]-

PILLAYANDKRISHNAN:ATICALMODELOFTHoththemagnetandthestainlesssteelretainingsleeveshavehighresistivity,hecircuitequationsofthethreewindingsinphasevariablesareVaR00ia[]VbVe=0R0~b00R[][Ie]La+pLbaLbLebLeaLebLeiaea[LbaLea][~b+Ie][ebee(14)]kemfsea,eb'andeehaveatrapezoidal[6].Assumingfurtherthatthereisnochangeintherotorreluctancewithangle,thenLa=Lb=Le=LLab=Lea=Lbc=MVaR000ia~b[VbVe]=[00R][Ie]L+M[M0RMMLMMPL]ia~b+[Ie]eaeb[ee]butia+ib+ie=oreMib+Mie=-MiaVatransformationcan'bemadefromthephasevariablestod,qcoordinateseitherinthestationary,rotor,ancesthatvarysinusoidallyintheabcframebecomeconstantsinthed,hebackemfisnonsinusoidalintheBDCM,itmeansthatthemutualinductancebetweenthestatorandrotorisnonsinu-soidal,hencetransformationtoad,bilityistofindaFourierseriesofthebackemfinwhichcasethebackemfinthed,q(15)consideredtoocumbersomehencetheabcphasevarationofmotionisp(J),=(Te-Tf-B(J),)/J.(21),eb,eofthisnonsinusoidalshapeinthebackemf,S(16)(17)R0R0Oia[Vb=0Ve][00~bR][Ie]L-M+0[00L-M00iaeaL~M]p[Ie]~b+[]ebee(18)Henceinstatespaceformwehavethatp~aIb[ie]--lI(L-M)00[Va0lI(L-M)°lI(L~-M)]iaea(19)R°0~~ThemodelspresentedpreviouslycanbeusedtoexamineboththetransientandsteadystatebehaviorofPM'elsofthecurrentcontrollersandinverterswitcheshavebeenpresentedpreviously[queisheldconstctrictorquusoidalcurrentsrequiredbythePMSMduringthestartrDCM,thecurrent[[~:]-[.~andTt][~:]-[::]]20()Thecurrentsia,ib,ttorquepulsationsareof120'durationineachhalfcycle[6].Withacmachinesthathaveasinusoidalbackemf,a+eeie)/(J),.=(eaia+ebib

ER1988"_~~o~:c80~.,UX0+V5P(ED~REFI'>T,.;(.;!.'X<>-8:J.~CI~1:.I..'..0...TORaUEREFI'>-51~..;o~80§81:.~-.IBRCKEMF'".:'::;''.r.i'~o-8~8:c.~.~0:::RENTI'>CTlA.~~~~CI'0'.251:iTIME!S)Fig.6.0.375.10"0.5"'".;,..:.,,....PMSMsteadystateresults.:1lIJI§'0.00:c0.025TIMEISI0.05VDLT~~N".'""'Fig.4.~o_:§1:8ci51o"Tx0+LPMSMtransientresults.'z0a.°-8:J.~RCTURL..i'.CIOI:U'!.'..SPEEDBRCKEMFXREF<>I'>CTlA.-8:J.a.°"".'.'UTORaUE~CIO--,',.,;'..I:I'!'0.140.32TIMEtS)Fig.7.-10"BDCMsteadystaleresults.0.50REFI'>CTlA..,.g([51:c.9-8~..;~-8~oi~0~o§o:c-:-:'0.00Fig.5.~REF<>I'>~controllersareusedtofoMSMontheotherhand,stateresultsofthecurrentandbackemfforexamplecanalsobestudiedusingthesemodelsasshowninFigs.6and7,thetransientphasecanberemovedinordertofacilitatestudyofthesteadystateorapprSIONSi'.,}'-'i..'.,)..'".'0.05BDCMtransientresults.----Thispaperhhownthatalthoughthe

PILLAYANDKRISHNAN:MODEUNGOFPERMANENTMAGNETMOTORDRIVES541magnetsynchronousmachine.".1A-21,no.2,pp.408-413,.1985.[3],."TheferritePMACmotor-Atechnicalandeconomicassessment,"ryApplications,vol.1A-21,no.4,pp.644-650,Mayllune1985.[4],k:McGraw-Hill,1986.[5]an,"ModelinganalysisandsimulationoCahighperfonnance,vectorcontrolled,pennanentmagnetsynchronousmotordrive,"g,Atlanta,dtheBDCMaresimilarinconstruction,,qmodelofthewoundrotorSMiseasilyadaptedtothePMSMwhileanabcphasevariableNCES[I]r,"Perfonnanceanddesignofanaxialfieldpennanentmagnetsynchronousmotorservodrive,"g,pp.634-640,1985.[2]-Mazenc,ueva,,"Studyandimplementationofahysteresiscontrolledinveneronapennanent[6]-.[7]"Modelingsimulationandanalysisofapermanentmagnetbrushlessdcmotordrive,"presentedattheIEEEIASAnnualMeeting,Atlanta,1987.-,"Applicationcharacteristicsofpennanentmagnetsynchronousandbrushlessdcmotorsforservodrives,"presentedattheIEEEIASAnnualMeeting,Atlanta,1987.