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AccurateMeasurementandDetailedEvaluationofStaticElectromagneticCharacteristicsofSwitchedReluctanceMachines
ShoujunSong,Member,IEEE,LefeiGe,ShaojieMa,ManZhang,andLushengWang
Abstract—Accuratestaticelectromagneticcharacteristicsareessentialbasicdatainthemodelingofswitchedreluctancemachine(SRM)forperformanceanalysisandadvancedcontrolpurposes.Becauseofthedoublysalientstructureanddeepmagneticsaturation,theelectromagneticcharacteristicsofSRMhavehighlynonlinearrelationswithphasecurrentandrotorposition,whichmakethedeterminationofthesecharacteristicsadifficulttask.Thispaperproposesadigitalsignalprocessor-basedmethodtoaccuratelymeasuretheflux-linkageandstatictorquecharacteristicsofSRM.First,theoreticalderivationandpracticalrealizationoftheproposedmethodarepresentedindetail,anditsaccuracyispreliminarilyverifiedbythreeways:coenergymethod,finiteelementmethod,andinductance-capacitance-resistancemeter.Then,theerrorsinmeasurementsareanalyzedandcorrespondingmethodstoreducethemaregiven.Finally,withtheimprovedneuralnetwork,thenonlinearsimulationmodelofaSRMprototypeisbuiltinMATLABwiththemeasuredcharacteristics,andthesimulationresultshavegoodagreementswiththosefromexperiments,whichfurtherverifytheaccuracyoftheproposedmethod.
IndexTerms—Erroranalysis,evaluation,fluxlinkage,measurement,neuralnetwork,switchedreluctancemachine(SRM),torque.
I.INTRODUCTION
B
ECAUSEofitsoutstandingadvantagessuchassimpleandruggedstructure,highefficiencywithwidespeedrange,highreliability,andlowcost,switchedreluctancemachine(SRM)hasattractedmuchattentionforapplicationsinaircraft,electricvehicles,andhouseholdappliances[1],[2].However,becauseofthedoublysalientstructureandmagneticsaturation,theelectromagneticcharacteristicsofSRMarehighlynonlinear,resultingindifficultmodelingandanalysis[3].
TopredicttheperformancesofSRMandimplementadvancedcontrolmethodssuchassensorlesscontrol[4]andtorquerippleminimization[5],itisessentialto
ManuscriptreceivedMarch11,2014;revisedAugust18,2014;acceptedAugust22,2014.ThisworkwassupportedbytheNationalNaturalScienceFoundationofChinaunderGrant51107100,theSpecializedResearchFoun-dationfortheDoctoralProgramofHigherEducationofChinaunderGrant20116102120033,andtheAdvancedProgramFoundationofScienceandTechnologyActivitiesforOverseasChineseScholarsofShaanxiProvince.TheAssociateEditorcoordinatingthereviewprocesswasDr.EdoardoFiorucci.
TheauthorsarewiththeCollegeofAutomation,NorthwesternPolytechnicalUniversity,Xi’an710072,China(e-mail:sunnyway@nwpu.edu.cn).
Colorversionsofoneormoreofthefiguresinthispaperareavailableonlineathttp://ieeexplore.ieee.org.
DigitalObjectIdentifier10.1109/TIM.2014.2358132
obtaintheaccurateelectromagneticcharacteristicsofSRM.Generally,thesecharacteristicscanbeobtainedbyanalyticalmethod,finiteelementmethod(FEM),andexperimentalmea-surement.Intheanalyticalmethod,suchasmagneticequiva-lentcircuit[6],thegenerationofmagneticnetworkisrelativelycomplicatedandtheaccuracycannotbeguaranteedduetoassumptionsandsimplifications.FEMismoreaccuratethananalyticalmethod[7],butitiscomplex,time-consuming,andrequiresexactgeometricaldimensionsandmaterialproperties,whicharesometimesdifficulttobeobtained.Furthermore,theresultsfromFEMarenotalwaysaccurateowingtoend-windingeffects,whicharenoteasytobeconsideredinFEM.Thus,theexperimentalmeasurementmethodispreferred.ThemethodsforexperimentalmeasurementofSRMchar-acteristicscanbeclassifiedintodirectmethodsandindirectmethods.Directmethodsrequiremagneticsensorstomeasurethefluxlinkageinthemachine[8].Becauseofinconvenienceandhighcost,thiskindofmethodisseldomused.
Forindirectmeasurement,manymethodshavebeenproposed.In[9],themagneticcharacteristicsaremeasuredbyasearchcoilmountedonthestatorpoles.Thismethodprovidesanimportantfoundationforfurtherresearchinthisfield.In[10]and[11],theac/dcandpureactestmethodsareproposed.Themeasurementaccuraciesofthesemethodsareaffectedbytheharmonicsintroducedbytheacsourceandincreasedcorelosses,whichneedstobeconsideredcarefully.In[12],adigitalsignalprocessor(DSP)-basedvirtualinstrumentation(VI)testsystemisproposed,whichincludesonlineoffset-errorremovalandwindingresistanceestimation,toautomaticallydeterminethemagneticcharacteristicsofaSRM.In[13],theflux-linkageprofileofaSRMisdeterminedbasedonitsstatictorquemeasurement.Duetothetorqueoffsetcausedbyfrictionandmechanicalmisalignment,thismethodisnotsuitableformachineswithsmalltorquerating.In[14],anewandfullyautomated,computer-basedcharacterizationsystemforSRMispresented.Itcanmeasuretheflux-linkagecharacteristics,statictorquecharacteristics,andback-emfofSRM.ItsoutputmakesadirectinterfacetoaSRM-simulationpackage,sonewlydevelopedSRMcanbesimulatedrapidly.In[15],ameasurementmethodwithoutrotorclampingdevicesandpositionsensorsisproposed.Thismethodreducesthecomplexity,costandtime,andissuitableforproducttestinmassiveproductions.However,itneedsmoresemiconductordevicesandcurrentsensors,anditsmeasurementaccuracywillbeaffectedbyrotormisalignment.
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Furthermore,becauseofmagneticcouplingbetweenphases,thismethodisnotsuitableforapplicationsinSRMwithhighmutualinductanceanddeepmagneticsaturation.
Usually,theaccuracyofcharacteristicmeasurementiseval-uatedbycomparingwiththeresultsfromFEM[16].However,asmentionedpreviously,FEMisnotabsolutelyaccurateorevenimpossibletobecarriedoutwhentheexactinformationofthemachineisunavailable[7].Inthispaper,basedonthepreliminaryresearchofSongetal.[17],anaccuratemethodisproposedtomeasurethestaticelectromagneticcharacteristicsofSRM.Thephasewindingofthemachineisenergizedbyapulseddcvoltage,withitsrotorfixedtoacertainpositionbyasteppingmotor.Thentheflux-linkagecharacteristicsareindirectlycalculatedbasedonthephasecurrent,voltage,andresistance.Thetorquecharacteristicsareobtaineddirectlybystatictorquetransducer.Finally,toverifytheaccuracyoftheproposedmethod,thesimulationmodelofaSRMprototypeisbuiltbasedonthemeasuredcharacteristics,andthesimulationresultsarecomparedwiththosefromexperiments.
Therestofthispaperisorganizedasfollows.ThetheoreticalbasisoftheproposedmethodandarrangementofthemeasurementsystemaredescribedinSectionII.SectionIIIpresentsthemeasurementresultsandpreliminaryverification.InSectionIV,thepossibleerrorsandthemethodstoreducethemarediscussed.ThesimulationmodelisbuiltandtheaccuracyoftheproposedmethodisfurtherverifiedinSectionV.SectionVIdrawstheconclusionofthispaper.
II.MEASUREMENTMETHODANDPLATFORM
ThestaticelectromagneticcharacteristicsofSRMcon-cernedinthispaperincludeflux-linkagecharacteristicsandstatictorquecharacteristics,andbothofthemarenonlinearfunctionsofphasecurrentandrotorposition.
Intheproposedmethod,thestatorwindingisenergizedbyapulseddcvoltage,whentherotorismechanicallyfixedtoadesiredposition,andtheinstantaneousphasecurrentandvoltagearemeasuredandrecorded.Thefluxlinkagecanbecalculatedbythefollowingequation,whichissimplyderivedfromthevoltageequation:
t
ψ(θ,i)=[u(t)−Ri(t)]+ψ(0)
(1)0
θ=const
whereψisthefluxlinkage,u,i,andRarethephasevoltage,current,andresistance,respectively.θistherotorpositionandψ(0)istheinitialfluxlinkage.InSRM,becausethereisnopermanentmagnet,ψ(0)=0.
Torealizeincomputer,thediscreteformof(1)canbewrittenas(2)basedonEuler’sformula,whichapproximatestheintegralbyareasummationofsomerectangles.ImprovedEuler’sandSimpson’sformulaalsocanbeusedψ(n)=
n[12]
[u(k)−Ri(k)]Ts+ψ(0)
(2)k=1
θ=const
wherenisthenumberofsamplingpointundercalculation,kisthenumberofsamplingpointsbeforen,andTsisthesamplingperiod.
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Fig.1.Schematicofthemeasurementplatform.
Fig.2.Photographofthemeasurementplatform.
Themeasurementofstatictorquecharacteristicsisrelativelystraightforward.Weneedtokeepthephasecurrentatacertainvalueandrecordthevaluesoftorquefromstatictorquetransducer.
Figs.1and2showtheschematicdiagramandphotographofthemeasurementplatform.Theplatformmainlyconsistsoftwosubsystems.OneisthemechanicalsubsystemincludingSRM,torquesensor,gearbox,andsteppingmotor.TheotheristheelectricalanddataacquisitionsubsystemincludingPC,DSP,converter,andotherelectronicinterfacecircuitboards.Briefdescriptionsofthecomponentsintheplatformaregivenbelow.
1)SRM:Theratedpower,voltage,current,andspeedoftheSRMare1kW,96V,9.7A,and2000r/min,respectively.Ithasthree-phase12/8-poleconfiguration.Toillustratetheconnectionbetweenwindingandpowerconverterclearly,a6/4-poleconfigurationisadoptedasshowninFig.1.
2)TorqueSensor:Astatictorquesensorisused,anditsfull-scaletorqueratingis10Nm.ThetorquesensorisinstalledbetweentheSRMandthegearbox.
3)GearBoxandSteppingMotor:Thesetwocomponentsareusedtochangetherotorpositionandlockitautomatically.Theholdingtorqueandbasicstepangleofthesteppingmotorare8.4Nmand1.8°,respectively.Thegearboxwithateethratioof1:50isusedtogetsmoothpositionvariationandlargeholdingtorque.4)PC,DSP,andConverter:ThePCisusedasagraphicaluserinterfacedevelopedusingLabVIEWtocontrolandmonitortheoperationofthewholeplatform.AccordingtothecommandfromthePC,DSPcontrolsthephasewindingexcitationoftheSRMthroughtheconverter.DSPalsocollectsthedatafromcurrent,voltage,andtorquesensors,andsendsthemtothecomputerforfurtherprocessing.Theconverterhasasymmetrichalf-bridgeconfiguration.TheMOSFETsinthe
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Fig.3.Flowchartofthemeasurementprocedure.
Fig.4.(a)Waveformofphasevoltageandcurrentatalignedposition.(b)Partialenlargement.
converterareusedtocontroltheexcitationprocessandthediodescanprovidefreewheelingpathforthephasecurrent.
5)OtherInterfaceCircuitBoards:Therearesomeinterfacecircuitboardsthathavebeendesignedanddeveloped,includingpowersupplycircuits,PWMinterfacecircuits,MOSFETdrivercircuits,currentandvoltagesensorandfiltercircuits.III.MEASUREMENTRESULTSANDPRELIMINARYTESTSThephasesinSRMareidenticaltoeachother,soonlyonephaseisselectedformeasurementsfromalignedto
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Fig.5.(a)Measuredflux-linkagecharacteristics.(b)Measuredstatictorquecharacteristics.
unalignedposition,andthecharacteristicsofotherphasescanbeobtainedwithappropriatephaseshifts.Asanexample,Fig.3showstheflowchartofthemeasurementprocedureforflux-linkagecharacteristics.Theinitialrotorpositionisthealignedposition,whichcanbeeasilyfoundbyenergizingonephase.
A.MeasurementResults
Fig.4showsthemeasuredphasevoltageandcurrentwaveformatalignedposition.Itisclearthattheslopeofthecurrentwaveformincreasesalongwiththeincreasingofcurrent,whichimpliesthatthephaseinductancedecreasesundermagneticsaturation.Finally,thecurrentreachesitssteadyvalue,whichisthemaximumcurrentthatthephasewindingcanwithstand.Itshouldbenotedthattheflux-linkagecurveforallcurrentsatonerotorpositionisobtainedbyonevoltagepulse.Therepetitionnumberoftheenergizingstepdependsontherequiredresolutionofrotorposition.
Fig.5showsthemeasurementresultsoftheflux-linkageandstatictorquecharacteristics.Themaximumcurrentadoptedintheflux-linkagemeasurementis20Awiththepositionstepof1°mechanicalangle.Forthetorquemeasurement,themaximumcurrentis24Awithastepof1A.Becausebothoftherotorandstatoraresymmetrical,onlyhalfoftheentireelectricalperiodneedstobemeasured,andthecharacteristicsoftheotherhalfcanbeobtainedeasilybymirroring.
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Fig.6.Comparisonbetweenresultsfrom(4)andtorquecharacteristics.
Fig.7.Coenergydifferencebetweenrealandlinearfluxlinkage.
B.PreliminaryVerification
Theabovemeasuredcharacteristicsarepreliminarilyevalu-atedbythreemethods,includingcoenergymethod,FEM,andinductance-capacitance-resistance(LCR)meter.
1)CoenergyMethod:TheinstantaneoustorqueproducedbyonephaseinaSRMcanbegenerallycalculatedbythespatialderivativeofcoenergy
T(θ,i)=∂W(θ,i)∂θ∂i0ψ(θ,i)di=i=const
∂θ(3)i=const
whereTistheinstantaneoustorque,iisthephasecurrent,
andWisthecoenergy.
WhentheSRMrunswithoutmagneticsaturation,itsflux-linkagecurveswouldbestraightlines.Atanyposition,thecoenergyandthestoredmagneticenergyareequal,thus(3)canbesimplifiedto
T=
1dL2dθ
i2
=ki2(4)whereListhephaseinductanceandkisthecoefficient.AsshowninFig.6,theresultfrom(4)shouldbeapproximatelyequaltothemaximumvalueofthestatictorquecharacteristicswithcorrespondingphasecurrent.
Withmagneticsaturation,therealcoenergywillbesmallerthanthatcalculatedbylinearfluxlinkageasshowninFig.7.Inthefigure,thedifferencebetweenthemisindicatedbyshadowedarea.Itisclearthatthedifferencewillbeenlargedwiththeincreaseofphasecurrent.
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Fig.8.ComparisonbetweenfunctionT=0.022i2andmaximumvalueofmeasuredtorquecharacteristics.
Fig.9.Comparisonbetweencalculatedandmeasuredtorquecharacteristics.
FortheconsideredSRM,Fig.8showsthemaximumvalueofitsmeasuredtorquecharacteristicsalongwiththecurveoffunctionT=0.022i2.Itcanbeseenthattheyhaveagoodagreementwhenthecurrentisunder10A.Duetothesaturation,themaximumvalueofmeasuredtorquewilldivergefromthefunctioncurve,whichagreeswiththeabovetheoreticalanalysis.
BecausetheSRMisintentionallyoperatedwithdeepmagneticsaturationforhigherpowerdensity,itsstatictorquecharacteristicsshouldbecalculatedfromthemeasuredflux-linkagedataby(3).Fig.9showsthecomparisonofthecalculatedresultswithmeasuredtorquecharacteristics,andagoodagreementcanbefound,whichindicatesthatthemeasuredflux-linkageandtorquecharacteristicshavesatisfactoryconsistency.
2)FiniteElementMethod:Asadoptedbymanyresearchers,themeasuredflux-linkagecharacteristicsarealsobrieflycomparedwithresultsfromFEM.
TheFEMmodeloftheexperimentalSRMisbuiltinMagNet.ThemechanicalparametersofthestudiedSRMaresummarizedinTableI,andFig.10showsitsmodelinMagNet.Duetosymmetry,only1/4ofthemachineneedstobeconsidered.
Themeasuredflux-linkagecharacteristicsarebrieflycomparedwiththosefromFEMasshowninFig.11.Inthefigure,onlythefluxlinkageatunaligned,alignedandone
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TABLEI
MECHANICALPARAMETERSOFTHE1kWSRMPROTOTYPE
Fig.10.FEMmodelofexperimentalSRMinMagNet.
Fig.11.Comparisonbetweenmeasuredflux-linkagecharacteristicsandtheresultsfromFEM.
intermediaterotorpositionaregiven,andasatisfactoryagreementcanbefound.ItshouldbenotedthatthereliabilityoftheFEMresultsheavilydependsontheaccuracyofthegeometricdimensionsandmaterialproperties.
3)LCRMeter:Withthemeasuredflux-linkagecharacteristics,theunsaturatedphaseinductanceoftheSRMisobtainedforseveralrotorpositionsbycalculatingψ/i.TheinductanceisalsomeasuredbyaLCRmeter,whichappliessinusoidalvoltageandhasanaccuracyof0.05%.Fig.12showsthecomparisonoftheinductancefromcalculationandmeasurement.Itisclearthattheresultsfromthetwomethodsareingoodagreement.ItshouldbenotedthattheaccuracyoftheLCRmeteralsocontributestotheerrors.
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Fig.12.Comparisonbetweenmeasuredandcalculatedphaseinductance.
Usingtheprecedingthreemethods,theaccuracyofthemeasurementmethodispreliminarilyevaluated.However,thisisnotsoreliable.Thecoenergymethodcanonlyindicatetheconsistencybetweenmeasuredflux-linkageandtorquecharacteristics.AlthoughFEMiswidelyused,itsresultsheavilydependontheaccuracyofthemagnetizationcurve(B–Hcurve)ofthematerialandgeometricdimensionsofthemachine,suchasairgaplength.Furthermore,sometimesthesecriticaldataareprotectedbythemanufacturer,whichmakestheFEMimpossible.TheLCRmetercanonlybeusedatlowcurrentlevels.Accordingtoabovesituations,theaccuracyofthemeasurementmethodisevaluatedindetailbydynamicsimulation,whichispresentedinSectionV.
IV.ERRORANALYSISANDDATAPROCESSING
Theaccuracyofthemeasurementmethodisinevitablyinfluencedbyerrorsandnoiseduringmeasurementandnumer-icalcomputationprocess.Inthissection,theerrorsourcesareanalyzed,andthemethodsforerrorreductionanddataprocessingarepresentedaswell.Theerrorsourcescanbesummarizedasfollows.
A.ParameterErrors
Thiskindoferroralsocanbecalledparameteruncertainty,namely,therealvaluesofparametersarenotthesamewiththeirmeasurements.Inthemeasurementofelectromagneticcharacteristics,theparametererrorsincluderotorpositionandphaseresistance.
Duringthemeasurement,therotorofSRMisrotatedandfixedtoadesiredpositionbyasteppingmotor,andtheexactpositionangleisobtainedbyaresolvermountedontheshaft.Thepositionerrorsmaycomefromthesteppingmotor,gearbox,andresolver.Thebasicstepangleofthesteppingmotoris1.8°,andupto256subdivisionscanbeapplied.Furthermore,theteethratioofthegearboxis1:50.So,theminimumstepangleofthesteppingmotorisassmallas1.8°/(256×50)=0.00014°.Aplanetarygearboxisadoptedtoguaranteesmallbacklash.Ahigh-precisionsine–cosineresolverisselectedtoprovidetheexactrotor
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Fig.13.EquivalentcircuitoftheSRMwhenitisexcitedbyapulseddcvoltagesource.
positionangleforcalibration,anditsaccuracyiswithin±30second.Withtheaforementionedconsiderations,therotorpositionerrorsareverysmall,andcanbeneglected.
Asgiveninsomereferences,Fig.13showstheequivalentcircuitoftheSRMwhenitisexcitedbyapulseddcvoltagesource.Inthefigure,Rcopperisthewindingresistance,andRcoreistheequivalentcore-lossresistance.Lphaseisthephaseinductance,andωistheangularfrequencyoftheaccomponentincludedinthepulseddcvoltage.
InFig.13,Rcoreisrelatedtothecorelossesofthemachinewhichconsistofhysteresisandeddy-currentlossesthatcanbeexpressedas[16]
Pe∝Ke·B2·f2(5)Ph∝Kh·Bmax·f
(6)
wherePeandPharetheeddy-currentandhysteresisloss.BisthefluxdensityandBmaxisthemaximumvalueoffluxdensity.fisthefrequency.KeandKhareconstantparameters.
ItcanbeseenthatPeisproportionaltothesquareoffluxdensityandfrequency,andPhisproportionaltothemaximumvalueoffluxdensityandthefrequency[16].Inthispaper,tominimizethecorelosses,apulseddcvoltagewithlowfrequency(1Hz)andlowamplitudeisappliedtoenergizethephasewinding.Asaresult,RcoreshowninFig.13canbeeliminated.
Duringthemeasurement,duetotheexistenceofwindinginductance,thephasecurrentneedstimetoreachitssteadystateasshowninFig.4.Understeadystate,themagneticfieldcannotchangewithtime,thusωLphaseshowninFig.13canbeshorted.Then,thewindingresistanceRcoppercanbedirectlycalculatedbyterminalvoltageandphasecurrent.
Accordingto(1)and(2),thevalueofwindingresistancewillaffectthecalculatedresultsofthefluxlinkage.Fig.14showstheflux-linkagecharacteristicsoftheSRMatalignedpositionobtainedwithdifferentwindingresistances,andbigdifferencescanbefound.Inthefigure,theresistancevariesfrom0.26to0.32instepsof0.01.Itisobviousthatinaccuratewindingresistancewouldresultinabigerrorinthemeasurement[12].
Inpractice,Rcopperisaffectedbytemperature,andtheincreaseoftemperatureduetocopperlosseswillcausenotableresistancevariation.Inthispaper,theeffectofresistancevariationisminimizedbytwoways.First,thedutyratioofthepulseddcvoltageislimitedtoasmallvalue,suchas1/5.Thedurationofthecurrentflowinthewindingisveryshort,sothetemperatureriseoftheSRMduringthemeasurementis
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Fig.14.Flux-linkagecharacteristicsatalignedpositionobtainedwithdifferentwindingresistances.
notsignificant.Second,thevalueofthewindingresistanceisupdatedforeverymeasurement[12].Additionally,thecontin-ualmeasurementofwindingresistancealsofacilitatesthefaultdetection.Ifwindingresistancechangessignificantlywithinashorttime,theMOSFETs,current,andvoltagesensorsshouldbecheckedforproblems.B.SignalErrors
Inthispaper,thesignalerrorsdenoteserrorsincurrent,voltage,andtorquesignalscausedbysensoroffsetandnonlinearity,electronicnoise,quantizationerror,andsoon.Beforethemeasurement,thesensorsareselected,testedandcalibratedcarefullytoavoiddcoffsetandotherpossibleproblems.Forexample,thecurrentandvoltagesensorsusedinthemeasurementareLA25-NPandLV25-PproducedbyLEM,whichhavehighaccuracyandlinearity.Rail-to-railoperationalamplifierisalsoappliedtoreducetheoffset.Theelectronicnoiseisinherentforallelectronicdevices,anditalsorelatestothecircuit.Quantizationerrorsarecausedbyanalog-to-digital(A/D)conversion.Thenoiseinthesignalwillbeamplifiedgreatlyduringthecalculation,suchasnumericalintegrationanddifferentiation,soitshouldberejected.Inthispaper,thehigh-frequencynoiseinthesampledsignalsissuppressedbyamovingaveragefilter,whichisamostcommonlinearfilter.
Themovingaveragefilterisalow-passfilterbasedontheprincipleofmovingaverage.Awindowwithcertainwidthisassumed,whichmovesalongthedataseries.Foreachmovement,thefirstdatapointinthewindowisexcludedandthenextdatapointfollowingthewindowintheseriesisincluded,andthearithmeticmeanofthedatainthewindowiscalculated.Finally,anewfiltereddataseriesisgenerated.Themovingaveragefiltercaneffectivelyreducerandomnoisewhileretainingasharpstepresponse,whichmakesitthepremierfilterfortimedomainapplications.Itcanbeexpressedas
y[n]=1N
−1N
x[n−i](7)
i=0
wherexisthedataneedtobefiltered,yistheresultsafterfiltering,andNdeterminesthewidthofthemovingwindow.
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Fig.15.
Sampledvoltagesignalbeforefiltering.
Fig.16.
FFTresultsoforiginalandfilteredvoltagewaveform.
Asanexample,Fig.15showsthesampledvoltagesignalbeforefiltering,andFig.16showsthecomparisonofthefastFouriertransform(FFT)resultsoftheoriginalandfilteredvoltagewaveform.BycomparingwiththeresultsshowninFig.4(a),itisfoundthattheperformanceofthefilterissatisfactory.Thehigh-frequencynoiseiseliminated,whiletheamplitudeandshapeofthevoltagewaveformarenotchangedbythefilter.C.CalculationErrors
Numericalintegrationisadoptedtoobtaintheflux-linkagecharacteristicsfromphasevoltageandcurrent.Fortorquecharacteristics,coenergyisfirstlycomputedbynumericalintegration,andthenthetorqueiscomputedfromcoenergybynumericaldifferentiation.Bothnumericalintegrationanddifferentiationwillintroduceerrorstotheresults.Theseerrorsdependonthenumericalmethodsadoptedandcanbequanti-tativelyanalyzedforspecificsituations.
Asanexample,Fig.17showstheflux-linkagecharacteris-ticsobtainedbynumericalintegration.ItspartialenlargementisshowninFig.18,andthefluctuationsareobvious.Thesefluctuationscanaffecttheaccuracyoftorquecharacteristiccalculationandperformancesimulation.Themovingaveragefilterisagainadoptedtoreducethefluctuations,andtheobtainedcharacteristicsaresmoothasshowninFig.5(a).Inadditiontotheaboveanalysisandreductionoferrors,theselectionofsamplingfrequencyiscriticalaswell.
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Fig.17.
Flux-linkagecharacteristicsobtainedbynumericalintegration.
Fig.18.
PartialenlargementofFig.17.
Inthispaper,thenumericalcalculationperiodequalsthesamplingperiod.Ifthesamplingfrequencyistoolow,itcancausenotonlydistortioninsampledcurrentandvoltagewaveform,butalsoincreasesthetruncationerrorofnumericalcalculation.However,ifthesamplingfrequencyistoohigh,itwillresultinalargeamountofcomputationanddata,andincreasestheinfluenceofroundingerror.Furthermore,toohighsamplingfrequencywillintroducehigh-frequencynoiseofanalogsignalintothedigitalsystem.Inthispaper,theDSPprovides12-bA/Dconverters,andtheadoptedsamplingfrequencyis20kHz.
V.DETAILEDEVALUATION
Toevaluatetheaccuracyoftheaforementionedmeasure-mentmethodindetail,asimulationmodeloftheSRMisbuiltinMATLABbasedonthemeasuredflux-linkageandstatictorquecharacteristics,andthesimulationresultsfromthemodelarecomparedwiththosefromexperiment.A.SimulationModel
ThesimulationmodelofSRMisbuiltaccordingtoitsbasicequationsshowninthefollowingequation,whichconsistsofonevoltageequation⎧andtwotorqueequations:
⎨u=Ri+dψ(i,θ)⎩Te=Tdt
a+Tb+Tc
(8)Te=Jddt
+B+TL
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Fig.19.BriefflowchartofHTWNN.
Fig.20.ComparisonbetweentraditionalandhybridtrainedWNN.
whereuisthephasevoltage,Risthephaseresistance,iisthephasecurrent,θistherotorposition,ψisthephasefluxlinkage,Teistheelectromagnetictorque,Ta,Tb,andTcaretheelectromagnetictorqueofphasea,b,andc,respectively,Jisthemomentofinertia,Bisthefrictioncoefficient,istheangularvelocityoftherotor,andTListheloadtorque.ToobtainthesimulationmodeloftheSRM,itisessentialtoestablishtwononlinearrelationships,namelyflux-linkagecharacteristicsψ(i,θ)andstatictorquecharacteristicsT(i,θ).Therearemanymethodstoexpresstheserelationships,suchaslook-uptables,piecewisemethod,curvefitting,artificialintelligencetechnology,andsoon[18].Inthispaper,akindofartificialintelligencemethodcalledwaveletneuralnetwork(WNN)isadoptedandimproved.
TraditionalWNNhashighconvergencespeed,butitisapttofallintothelocaloptimum.Inthispaper,geneticalgorithm(GA)iscombinedwithcommonlyusedgradientdescent(GD)methodtotrainWNN.TheinitialweightsforWNNtrainingaregeneratedbyGAtoensureaccuracyandstabilityofglobalconvergence,andtheWNNwiththeseinitialweightsistrainedbyGDmethodtoguaranteehighlocalsearchingspeed.Fig.19showsthebriefflowchartofthehybridtrainedWNN(HTWNN).
TochecktheperformanceofHTWNN,thegeneralizationerrorsofflux-linkagecharacteristicsgeneratedbytraditional
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Fig.21.(a)Trainingresultsoftorquecharacteristics.(b)Trainingerroroftorquecharacteristics.
Fig.22.Simulationblockdiagramofonephase(phasea).
WNNandHTWNNarecomparedasshowninFig.20.Forcomparison,bothnetworksaretrainedfortentimes,whichmeansthattentrainednetworksareobtainedforeachmethod.Themaximumabsoluteerror(MAE)asdefinedinthefollowingisadoptedasevaluationindex:
MAE=maxHh=1
yh−dh
(9)whereyistheoutputofthenetwork,disthedesiredoutput,
andHisthenumberoftrainingsample.
AsshowninFig.20,itcanbeseenthattheconvergenceaccuracyandstabilityoftraditionalWNNareunsatisfactory,thatis,thetrainingresultsofeachtimearequitedifferent,whileboththeconvergenceaccuracyandstabilityofHTWNNarebetter.
Traditionally,WNNistrainedonlybyGDmethod,andthetimerequiredfortrainingisabout5s.ThetimerequiredtotrainWNNbythecombinationofGAandGDmethodisnearly10min.Thedifferenceoftrainingtimeisbig,butbecauseofunsatisfactoryperformance,traditionalWNNneeds
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Fig.23.Simulationblockdiagramofpowerconverter.
Fig.24.SimulationblockdiagramofthewholeSRM.
Fig.25.Blockdiagramofthetestbenchformotoringmodeexperiment.
alotoftrialanderrortogetrelativelygoodtrainingresults,whichnormallywilltakemuchlongertimethanHTWNN.Furthermore,theresultsfromtraditionalWNNcannotbeguaranteedtobeoptimal,butHTWNNcanobtainoptimaltrainingresultsautomaticallywhentheprogramshowninFig.19isfinished.Thespecificationsofthecomputerusedfortrainingtimeevaluationareasfollows:IntelCorei3,3.3GHz,2GBDDR3,and500GBharddisk.Itshouldbenotedthatifamulticorecomputerisused,therequiredtrainingtimeforHTWNNcanbeeffectivelyreducedbyparallelcomputing.9
Fig.26.Photographofthetestbenchformotoringmodeexperiment.
Fig.27.Comparisonofphasecurrentundermotoringmode.(a)Anglepositioncontrolwithn=2651r/min,θONPWMcontrolwithn=2145r/min,f=4882=0°Hz,anddutyθOFFratio=15°.equals(b)Vtooltageduty78%,θON=0°andθOFF=22.5°.
Fig.28.ComparisonofmechanicalcharacteristicswithθθON=0°andOFF=15°.
Themeasuredflux-linkageandstatictorquecharacteristicsareusedassampledatatotrainHTWNN.Asanexample,Fig.21showsthetrainingresultsoftorquecharacteristics,andsmallerrorscanbefound.
WiththetrainedHTWNNofi(ψ,θ)andT(i,θ),thesimula-tionmodeloftheSRMcanbebuiltinMATLABbasedon(8).Figs.22–24showthesimulationblockdiagramofonephase,powerconverter,andthewholeSRM,respectively.TheSRMconsideredisathree-phase12/8-poleSRM,andthetopologyofitspowerconverterisasymmetricalhalf-bridge.
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Fig.29.ComparisonofenergyloopundermotoringmodewithθONandθ=0°OFF=15°.
Fig.30.Photographofthetestbenchforgeneratingmodeexperiment.
Fig.31.Comparisonofphasecurrentundergeneratingmode.(a)n=300r/min,θON=16°,andθOFF=30°.(b)n=315r/min,θONandθ=17°,OFF=25.5°.
B.ResultsComparison
ThedynamicexperimentalresultsoftheconsideredSRMunderbothmotoringandgeneratingmodesarecomparedwiththosefromthesimulationmodelindetail.
Figs.25and26showtheblockdiagramandphotographofthetestbenchformotoringmodeexperiment,respectively.ItmainlyconsistsoftheSRM,torquetransducer,gearbox,andmagneticpowderbrake(MPB).
IEEETRANSACTIONSONINSTRUMENTATIONANDMEASUREMENT
Thephasecurrent,mechanicalcharacteristics,andenergyloopfromexperimentandsimulationundermotoringmodearecomparedinFigs.27–29,respectively.Itisreadilyobservedthattheyhavegoodagreements.Inthesefigures,nisspeedofSRM,fisfrequencyofPWM,θON,andθOFFareturn-onandturn-offangle,respectively.
Fig.30showsthephotographofthetestbenchforgenerat-ingmodeexperiment.ItmainlyconsistsoftheSRM,torquetransducerandbrushlessdcmachineasaprimemover.
ThephasecurrentfromexperimentandsimulationundergeneratingmodearecomparedasshowninFig.31.Itisreadilyobservedthattheyhavegoodagreements.
Theprecedingdetailedcomparisonsfurtherverifythattheproposedmeasurementmethodhasexcellentaccuracy.
VI.CONCLUSION
AnaccurateDSP-basedmethodformeasuringtheflux-linkageandstatictorquecharacteristicsoftheSRMispresentedinthispaper.Boththetheoreticalbasisandactualimplementationofthemethodareillustrated.
Theerrorsinmeasurementareanalyzedindetailfromthreeaspects:parametererrors,signalerrors,andcalculationerrors.Furthermore,effectivemethodstoreducetheerrorsarepresented.Toavoiderrorsfromthevariationofwindingresistancecausedbytemperaturerise,apulseddcvoltagewithlowamplitudeandlowfrequencyisapplied,andthewindingresistanceisupdatedonline.Themovingaveragefilterisadoptedtoreducetheimpactofelectronicnoiseanderrorsfromnumericalintegrationanddifferentiation.
Theaccuracyofthepresentedmethodisfirstlyverifiedbythreeways:coenergymethod,FEM,andLCRmeter.Forthedetailedverification,asimulationmodeloftheconsideredSRMisbuiltinMATLABbyhybridtrainedWNNbasedonthemeasuredcharacteristics.Severalexperimentsarecarriedoutwithtwotestbenches.Theresultsfromexperimentsandsimulationsunderbothmotoringandgeneratingmodearecomparedindetail,andgoodagreementscanbefound,whichverifiesthehighaccuracyofthepresentedmeasurementmethod.
ThepresentedmethodcanserveasausefulwaytoobtainaccurateelectromagneticcharacteristicsoftheSRM,whicharecriticalfortheperformanceanalysisandadvancedcontrol,suchassensorlesscontrolandtorquerippleminimizationcontrol.
ACKNOWLEDGMENT
TheauthorswouldliketothankProf.C.MifromtheUniversityofMichigan,Dearborn,MI,USA,andW.Li,D.Zhao,andProf.L.LuofromNorthwesternPolytechnicalUniversity,Xi’an,China,forhelpfuldiscussionsandvaluableassistance.
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ShoujunSong(M’08)receivedtheB.S.andM.S.degreesfromNorthwesternPolytechnicalUniversity,Xi’an,China,in2003and2006,respectively,andtheDr.-Ing.degreefromtheTechnicalUniversityofBerlin,Berlin,Germany,in2009,allinelectricalengineering.
HeiscurrentlyanAssociateProfessorwiththeCollegeofAutomation,NorthwesternPolytechnicalUniversity.Hiscurrentresearchinterestsincludeelectricalmachinesanddriveswithanemphasisonswitchedreluctancemachinedesignandcontrol.
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LefeiGereceivedtheB.S.degreeinmeasurementandcontroltechnologyfromNorthwesternPolytech-nicalUniversity,Xi’an,China,in2013,whereheiscurrentlypursuingtheM.S.degreeinelectricalengineering.
Hiscurrentresearchinterestsincludeoptimaldesignandcontrolofelectricalmachines.
ShaojieMareceivedtheB.S.degreeinelectricalengineeringfromNorthwesternPolytechnicalUni-versity,Xi’an,China,in2013,whereheiscurrentlypursuingtheM.S.degreeinelectricalengineering.Hiscurrentresearchinterestsincludemeasurementandcontrolofelectricalmachines.
ManZhangreceivedtheB.S.degreeinelectricalengineeringfromNorthwesternPolytechnicalUni-versity,Xi’an,China,in2012,wheresheiscurrentlypursuingtheM.S.degreeinelectricalengineering.Hercurrentresearchinterestsincludenumericalandanalyticalanalysisofelectromagneticfields.
LushengWangreceivedtheB.S.degreeinmea-surementandcontroltechnologyfromNorthwesternPolytechnicalUniversity,Xi’an,China,in2014.HeiscurrentlywiththeSkyworthDigitalTechnol-ogyCompany,Ltd.,Shenzhen,China.Hiscurrentresearchinterestsincludedigitalsettopbox,router,intelligentgateway,andsmarthomeappliance.
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