IEEE TRANSACTIONS ON MAGNETICS VOL - PDF document

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 1\n\n3,\n%2\n \n%\r\n%,\n?\n,\n\n\r\r\n%\n2\r\n%% 1\r\n 3\n% \r? \r\r2\n\n 1@1\r4\n\r\n%\r\n\n\r\n11\r\nA  \n123 �\n\r\r%3\n\r\n%%\n\n%- IEEETRANSACTIONSONMAGNETICS,VOL.41,NO.10,OCTOBER2005Calculationof-and-AxisInductancesofPMBrushlessacMachinesAccountingforSkewY.S.Chen,Z.Q.Zhu,SeniorMember,IEEE,andD.Howe CollegeofElectricalEngineering,ZhejiangUniversity,Hangzhou,P.R.China310027 DepartmentofElectronicandElectricalEngineering,UniversityofShefÞeld,ShefÞeldS13JD,U.K.Ahybridtwo-dimensional(2-D)Þnite-element/analyticaltechniqueisdescribedforpredictingthe -axisand -axisinductancesofpermanentmagnet(PM)brushlessacmachines,withdueaccountfortheinßuenceofskew.Predictedinductancesarecomparedwithmeasuredvaluesfortwomachineshavingidenticalstators,whichareskewedbyoneslot-pitch,butwhichhavedifferentrotortopologies,onehavingsurface-mountedmagnetsandtheotherhavinginteriormagnets.IndexTerms—Brushlessmachine,permanentmagnet(PM),permanentmagnet(PM)machine,windinginductance.I.INTRODUCTION -axisand -axisinductanceshaveanimportantinßu-enceonboththesteady-stateanddynamicperformanceofpermanentmagnet(PM)brushlessacmachines[1].Thus,theiraccuratepredictionisessential,notonlyforpredictingperfor-manceaspectssuchasthetorqueandßux-weakeningcapabili-ties,butalsofordesigningcontrolsystems,inordertomaximizetheefÞciency,powerfactor,etc.Theinßuenceofmagneticsat-urationisusuallyaccountedforbydeterminingcurrent-depen-dentinductances[1],[2],i.e., and However,sincehighlocalsaturationmayoccurduetothecom-binedinßuenceofthePMsandthearmaturereactionÞeld,espe-ciallyininterior-magnetrotormachines,analyticalmethodsfordeterminingthewindinginductancesareofteninadequate,andanumericalmethodmustbeemployed.Whenthestatorslotsorrotormagnetsareskewed,however,three-dimensional(3-D)calculationsarerequired,whichremaintimeconsuming.Inthispaper,theinßuenceofskewonthe -axisand inductancesisinvestigatedbyemployingahybridtwo-dimen-sional(2-D)Þnite-element/analyticaltechnique,withdueac-countforskewandendeffects.Predictionsarevalidatedbymeasurements,usingadcinductancebridgetechnique[2],ontwobrushlessacmachineshavingidenticalstators,whichareskewedbyoneslot-pitchbutdifferentrotortopologies.II.CALCULATIONOF XISAND NDUCTANCESVarioustechniqueshavebeenemployedtopredictthewindinginductancesusingtheÞnite-elementmethod[3]Ð[5].Whenthemagneticcircuitcanbeassumedtobelinear,theself-andmutual-windinginductancescanbeevaluatedbyenergizingtheappropriatestatorwindings,withthePMsreplacedbyair[5],anddeterminingthestoredenergyorwindingßuxlinkage,suchthatthe -axisand -axisinductancescansubsequentlybededuced.However,ifmagneticsaturationissigniÞcant,boththeself-andmutualinductances,andthe -axisand -axisinductances,arecurrentdependent.Theincrementaland DigitalObjectIdentiÞer10.1109/TMAG.2005.854976apparentself-andmutualinductancesshouldthenbecalcu-latedbyeithertheenergyperturbationmethodorthecurrentperturbationmethod,andtheinßuenceofthePMsshouldbeconsidered[3],[4].However,althoughgenerallyapplicable,themethoddescribedin[4]requiresmultipleÞnite-elementÞeldsolutions,forarangeofcurrentperturbationsineachwinding,and,therefore,demandssigniÞcantcomputationaleffort. -axisand -axisinductances, and ,respectively,whichareusedintheßux-weakeningcontrolofPMbrushlessacmachines[6],canbecalculatedfrom[2],asfollows: (1) (2)where ,and arethe -axiscurrentsandßux-linkages,respectively;and istheßux-linkageduetothePMrotor.However,while2-DÞnite-elementanalysiscanaccountfortheairgap,slotleakage,tooth-tipleakageandmagnetomotiveforce(MMF)harmonicleakagecomponentsofinductancetoahighaccuracy,itcannotaccountfortheend-windingcomponentofleakageinductanceortheinßuenceofskew.Whileskewiseffectiveinreducingtheharmoniccontentintheßuxlinkageandback-electromotiveforce(EMF)wave-forms,aswellasinreducingthecoggingtorque,skewingei-therthestatorteethortherotormagnetsresultsina3-Dmag-netic-Þelddistributionintheactiveregionofamachine.How-ever,sincetheaxialvariationofsaturationduetoskewisgen-erallysmallinmostPMmachines,therelationshipbetweenthe -axisand -axisinductances, and ,whicharecalcu-latedby2-DÞnite-elementanalysisoftheactiveregionofamachinewithoutskewandthe -axisand -axisinductances, and ,forthesamemachinewithskewcanbederivedanalytically,aswillbedescribed.Theself-inductance andthemutualinductance canbeexpressedasfunctionsoftherotorelectricalangle [7],asfollows: (3) 120 0018-9464/$20.00©2005IEEE etal.:CALCULATIONOF -AND -AXISINDUCTANCESOFPMBRUSHLESSacMACHINESACCOUNTINGFORSKEW3941 and arethecomponentsoftheself-andmutualinductancesduetothespace-fundamentalairgapßux, istheadditionalleakageßuxcomponent,andthesecondharmonic and resultfromsaliencyoftherotor.Skewchangestherelativepositionbetweenthestatorandrotoralongtheaxiallengthofamachine.Hence,themagnetic-Þelddistributionalsovaries.However,theself-inductance andthemutualinductance ofamachinewithskewcanbededucedfromtheinductancesofthesamemachinewithoutskewbysubdividingtheactivelengthofthemachineinto slices,witheachslicehavingaskewangleof .Hence, (5) 120 (6)where istheelectricalskewangle,and (7) wheretheskewfactor isgivenby Thus,theconstantcomponentsofthewindinginductancesre-mainunchangedwhenskewisemployed,whilethesecondhar-moniccomponentsarereducedbytheskewfactor ,which ,i.e.,theinductancevariationduetosaliencyisreducedbyskew.Ingeneral,fromtheParktransformation,therelationshipbe-tweenthephaseself-andmutualinductances, and ,andthe -axisand -axisinductances, and ,aregivenby[7] (10) Thus,from(5)Ð(9),the -axisand -axisinductances,ac-countingfortheeffectofskew,canbeobtainedas (12) Combining(11)Ð(13),therelationshipbetweentheinduc- and ,whicharededucedfrom2-DÞnite-ele-mentanalysesoftheactiveregionofamachineneglectingskew,andtheinductances and ,whichaccountforskew,canbederivedas (14) (15) Fig.1.MagneticÞelddistributions.(i)Surface-mountedmagnetrotor:(a)Opencircuit;(b)rated-axiscurrent;and(c)rated-axiscurrent.(ii)Interior-magnetrotor:(a)Opencircuit;(b)rated-axiscurrent;and(c)rated-axiscurrent.Theend-windingcomponentofinductance maybeesti-matedusingawell-provenmethodthathasbeenappliedexten-sivelytoinductionmotors[8].Thus,theresultant -axisand -axisinductancesare (16) III.COMPARISONANDXPERIMENTALALIDATION -axisand -axisinductanceshavebeencomparedwithmeasuredvaluesfortwo1.3-kW,three-phasePMbrushlessacmotorshavingidenticalstators,withsixpoles,18slots,anoverlappingwinding,andaskewofoneslotpitch,butdifferentrotors,viz.onewithsurface-mountedmagnetsandanotherwithinteriormagnets.Themaindesignparametersareasfollows:statoroutsidediameter 106.6mm,activelength 32mm,airgaplength 0.75mm,rotordiameter 60.5mm,andwidthofstatorslotopenings 2.5mm.Bothrotorshavesin-teredNdFeBmagnetswitharemanenceof1.17Tandarelativerecoilpermeabilityof1.07.Itwillbenotedthattherotorback-ironofthemachinewiththesurface-mountedmagnetrotorhasairspacestoreducetheinertiaandminimizetheinßuenceof -axismagneticsaturation,andtherebymaximizetheoverloadtorquecapability.Fig.1showsÞelddistributionswhenthetwomotorsareonopencircuitandonloadwhenratedcurrent(4Aatpeak)isappliedtothe and axes.Fig.2comparesmeasuredandpre- -and -axisinductances,fromwhichtheinßuenceofmagneticsaturationcanbeclearlyobserved.Thepredictionsincludetheend-windingcomponentofinductance,viz.6.9mH IEEETRANSACTIONSONMAGNETICS,VOL.41,NO.10,OCTOBER2005 Fig.2.Measuredandpredicted-axisand-axisinductances:(i)Surface-mountedmagnetrotormotorand(ii)interior-magnetrotormotor.[8].The -axisinductance wasmeasuredbylockingtherotoralongthe axis,whichwasdeterminedbyapplyinga -axiscurrent and andallowingtherotortorotatetothepositionatwhichthetorquewaszero,andusingadcinductancebridge[2].The -axisinductance wassimilarlymeasured.Ingeneral,verygoodagreementisob-tained.Aswillbeseen,duetothedifferentrotortopologies,thevariationof and with and ,respectively,issigniÞ-cantlydifferentforthetwomotors.Fig.3showstheinßuenceofskewonthepredicted -axisand -axisinductancesforbothmotors.Itwillbenoted,bothfromFig.3andtheforegoingequations,thatirrespectiveof and ,thesmallestinductancecomponentisincreasedasaresultofskew,whiletheothercomponentisreduced.Hence,theuseofskewreducestheinßuenceofanyrotorsaliency,andboth and willapproachthevalue astheskewangleisincreased.Foramachinewithoutsaliency,i.e., ,theinßuenceofskewwillgenerallybenegligible. Fig.3.Predicted-axisand-axisinductanceswithandwithoutskew:(i)surface-mountedmagnetrotormotorand(ii)interior-magnetrotormotor.IV.CAhybridtwo-dimensional(2-D)Þnite-element/analyticaltechniquehasbeenproposedforpredictingthe -axisand -axisinductancesofpermanentmagnet(PM)brushlessacmachineswithskew.Predicted -axisand -axisinductanceshavebeencomparedwithmeasurementsintwomachineshavingidenticalstators,whichareskewedbyoneslotpitch,butdifferentrotortopologies,andgoodagreementhasbeenachieved.[1]B.J.Chalmers,S.A.Hamed,andG.D.Baines,ÒParametersandperformanceofahigh-Þeldpermanentmagnetsynchronousmotorforvariable-frequencyoperation,ÓProc.Inst.Elect.Eng.—B,vol.132,pp.117Ð124,1985.[2]P.H.Mellor,F.B.Chaaban,andK.J.Binns,ÒEstimationofparametersandperformanceofrare-earthpermanent-magnetmotorsavoidingmea-surementofloadangle,ÓinProc.Inst.Elect.Eng.—B,vol.138,1991,pp.322Ð330.[3]L.Chang,ÒAnimprovedFEinductancecalculationforelectricalma-chines,ÓIEEETrans.Magn.,vol.32,no.4,pp.3237Ð3245,Jul.1996.[4]T.W.Nehl,F.A.Fouad,andN.A.Demerdash,ÒDeterminationofsatu-ratedvaluesofrotatingmachineryincrementalandapparentinductancesbyanenergyperturbationmethod,ÓIEEETrans.PowerApp.Syst.,vol.101,no.12,pp.4441Ð4451,Dec.1982.[5]D.Pavlik,V.K.Garg,J.R.Repp,andJ.Weiss,ÒAÞniteelementaltechniqueforcalculatingthemagnetsizesandinductancesofperma-nentmagnetmachines,ÓIEEETrans.EnergyConvers.,vol.3,no.1,pp.116Ð122,Mar.1988.[6]S.Morimoto,M.Sanada,andY.Takeda,ÒWide-speedoperationofin-teriorpermanentmagnetsynchronousmotorswithhigh-performancecurrentregulator,ÓIEEETrans.Ind.Appl.,vol.30,no.4,pp.920Ð926,Jul.ÐAug.1994.[7]A.E.Fitzgerald,C.Kingsley,andS.D.Umans,ElectricMa-chinery.NewYork:McGraw-Hill,1989,pp.272Ð276.[8]M.Liwschitz-GarikandC.Whipple,Alternating-CurrentMa-chines.NewYork:VanNostrand,1961,pp.510Ð512.ManuscriptreceivedFebruary7,2005.