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SynchronousMachines-ReactanceandExcitationCalculationMarkFanslowTecoWestinghouseEngineeringTrainingNov.2005SynchronousMachines-Reactan1StatorRevolvingMagneticFieldMagneticPolePairsrotateatSynchronousSpeedThreePhaseACVoltageStatorRevolvingMagneticFiel2SynchronousRotorDCinputtotherotorcreateselectromagneticpolespairsPolesofdifferentpolarityarecreatedbywindingaroundthepoleindifferentdirectionsSynchronousRotorDCinputtot3SynchronousMotorRotor“Locked”intopowersupply,rovlovesatsynchronousspeedSynchronousMotorRotor“Locked4PhasorsandPhasorDiagramsTheconceptofPhasorsisrelatedtosinusoidalwaveformsthataredistributedinspaceandvarywithtime.Phasorsarecomplex(asincomplexnumbers)quantitiesusedforsimplifiedcalculationoftimevaryingandtravelingwaveforms.Themagnitudeandphaserelationshipbetweenthevectorsandcanbeshownsimplyinphasordiagrams.PhasorsandPhasorDiagramsThe5PhasorDiagramExampleConsiderthesimpleRL(resistiveinductive)circuitV=Vmsin(ωt)Voltagefunctionoftime.CurrentfunctionoftimeanddisplacedbyangleVILMagneticFluxinInductor90°outofphasewithvoltagewaveformi90°PhasorDiagramExampleConsider6PhasorDiagramCont.VIFLФLRecall:

B:MagneticFluxDensitydenoteФ

H:MagneticFluxIntensityormmfdenoteFo:Constantofpermeabilityoffreespacer:Constantofpermeabilityofelectricalsteel

PhasorDiagramCont.VIFLФLRec7CylindricalRotorSynchronousMachine:Duetotheevenairgap,theformulationsforbasicmachinequantitiesissimplified.Wewillconsiderthecylindricalmachinetheoryfirstandthenextendtheanalysistothesalientpolecase.CylindricalRotorSynchronous8AxisofFieldAxisofPhaseA90°lagArmaturecurrentmmfFarResultantmmfFrFieldmmfFfRotorRotationStatorRotorNSStatorI.D.RotorO.D.AxisofFieldAxisofPhaseA909IaVtIaraEintEaΦar-ΦarfΦfRjIaxAjIax1Φr-ararErIaVtIaraEintEaΦar-ΦarfΦfRjI10CylindricalRotorPhasorDiagramForSynchronousGeneratorVt: TerminalVoltageIa: StatorCurrentLagsterminalVoltagebyAngleEa: AirGapVoltage,thevoltageinducedintothearmaturebythe fieldvoltageactingalone,the“opencircuit”voltageEr: ReactivevoltageproducedbyarmaturefluxEint: ResultantVoltageintheAirGapar: FluxfromArmaturecurrent(ArmatureReaction)f: Fluxduetofieldcurrentr: Resultantfluxar+fFf: mmfofFieldAar: mmfofArmatureCurrent(ArmatureReaction)FR: Resultantmmf,thesumofF+AIara: Voltagedropofarmatureresistance Iaxl: ReactivevoltagedropofArmatureleakagereactanceIaxA: ReactivevoltagedropofreactanceofArmaturereactionCylindricalRotorPhasorDiagr11FromthediagramitisevidentthatEr,thevoltageinducedinthestatorbytheeffectofthearmaturereactionflux,isinphasewithIaXl.ThesummationofErandIaXlgivesthetotalreactivevoltageproducedinthearmaturecircuitbythearmatureflux.TheratioofthistotalvoltagetoarmaturecurrentisdefinedasXsorthesynchronousreactance.Fromthediagramitisevident12SalientPoleSynchronousMachinesSalientPoleSynchronousMachi13SalientPoleSynchronousMachineSalientPoleSynchronousMachi14TwoReactionTheoryOneoftheconceptsusedincylindricalrotortheorywasthesummationofbothfluxandmmfwaveforms.r=f+arandFr=Ff+FarThisisallowedsince

B=orHandthemagneticpermeabilityoftheairgapisconstantaroundtherotorHoweverforsalientpolemachines,thepermeabilityofthefluxpathvariessignificantlyastheratioofairgaptosteelchanges.Thereforerf+ar

TwoReactionTheoryOneofthe15SalientPoleSynchronousMachineWithasalientpolemachine,asinusoidaldistributionoffluxcannotbeassumedintheairgapduetothevariationinmagneticpermeancealongtheairgap.However,owingtothesymmetryofasalientpolemachinealongthedirectandquadratureaxis,asinusoidaldistributionofmmfcanbeassumedalongeachaxis.BreakthemmfofthearmaturecurrentAarintotwocomponents,AdandAq.AdisthecomponentofAarthatworksalongthedirectaxisandAqisthecomponentofAarcenteredonthequadratureaxis.SalientPoleSynchronousMachi16TwoReactionTheory:IntroductionIfyouacceptthatAarcanbebrokenintotwocomponentsAdandAq,itfollowsthatforeachmmfwaveform,aemfwaveformexists90degreesoutofphasewithit.SoAdhasanassociatedEd,andAqhasanassociatedEq.ThesevoltagedropscanbethoughofhasbeingcreatedbyafictitiousreactancedropEd=XadIdandEq=XaqIq

,whereId: DirectaxiscomponentofarmaturecurrentIq:QuadratureaxiscomponentofarmaturecurrentXad:DirectaxisarmaturereactionreactanceXaq:QuadratureaxisarmaturereactionreactanceTwoReactionTheory:Introduct17IaVtIaraEintEaΦarθjIaxAAjIax1IdxldIqxlqIdxAdIqxAqForCylindricalRotor:Ia2=Id2+Iq2Xld=XlqXAd=XAqIaVtIaraEintEaΦarθjIaxAAjIax1I18TwoReactionDiagramSinceforaCylindricalRotor:Ia2=Id2+Iq2and(IaXs)2=(IdXd)2+(IqXq)2Xld=XlqXAd=XAqXs=Xd=XqItisnotnecessarytousetworeactiontheorytodescribethequantitiesofacylindricalrotormachine.TwoReactionDiagramSincefor19EaIqIaIdΦfΦrΦaqΦadΦarPhasorDiagramofaSalientPoleSynchronousGeneratorEaIqIaIdΦfΦrΦaqΦadΦarPhasorDi20IaVtIdEintIqK1EintIdxadIqxaqEaIaraIaxlIa:StatorcurrentVt:StatorterminalVoltageIra:VoltageofstatorresistanceIxl:VoltageofstatorleakagereactanceEint:InternalairgapvoltageK1Eint:Extrammfrequiredtoovercomestatorsaturation,K1issaturationfactorIdxad:reactivevoltagedropdirectaxisIdxaq:ReactivevoltagedropquadratureaxisEa:Totalsumofdirectaxisvoltage,airgap voltage,opencircuitvoltage?εδId=Iasin(+δ)Iq=Iasin(+δ)IaVtIdEintIqK1EintIdxadIqxaqE21PoleFaceDesign-MagneticFieldsThemmfwaveofarmaturereactionandthemmfwaveofthepolearecreatedontwodifferentsidesoftheairgapbutmustbecombinedtodeterminearesultantmmf.Todothiswemustdetermineconversionfactorstoconvertanstatorsidemmftoandequivalentrotorsidemmf.90°lagArmaturecurrentmmfFarResultantmmfFrFieldmmfFfRotorRotationStatorRotorNSPoleFaceDesign-MagneticFi22PoleFaceDesign–MagneticFieldsC1:Ifpeakofthefundamentalisunity,thenC1ispeakofacutalwaveform.NotethatWiesemancallsthisA1No-Load:motorisexcitedbythefieldwindingonlyPoleFaceDesign–MagneticFi23PoleFaceDesign–MagneticFieldCd1:Ifpeakofthefundamentalisunity,thenCd1ispeakofacutalwaveform.NotethatWiesemancallsthisAd1ArmaturemmfSinewavewhoseaxiscoincideswiththepolecenterPoleFaceDesign–MagneticFi24PoleFaceDesign–MagneticFieldCq1:Ifpeakofthefundamentalisunity,thenCq1ispeakofacutalwaveform.NotethatWiesemancallsthisAq1ArmaturemmfSinewavewhoseaxiscoincideswiththegapbetweenpolesPoleFaceDesign–MagneticFi25ListofPoleConstantsCd1–RatioofthefundamentaloftheairgapfluxproducedbythedirectaxisarmaturecurrenttothatwhichwouldbeproducedwithauniformgapequaltotheeffectivegapatthepolecenterCq1–RatioofthefundamentaloftheairgapfluxproducedbythequadratureaxisarmaturecurrenttothatwhichwouldbeproducedwithauniformgapequaltotheeffectivegapatthepolecenterC1–Theratioofthefundamentaltotheactualmaximumvalueofthefieldformwhenexcitedbythefieldonly(no-load)Cm–Ratiooffundamentalairgapfluxproducedbythefundamentalofarmaturemmftothatproducedbythefieldforthesamemaximummmf.Thisisthearmaturereactionconversionfactorforthedirectaxis.Cm=Cd1/C1K–Fluxdistributioncoefficient;theratiooftheareaoftheactualnoloadfluxwavetotheareaofitsfundamentalListofPoleConstantsCd1–R26PoleConstantsWhatfollowsaregraphsthatrelatethephysicalgeometryofthepoletothepoleconstants.ThesegraphscanbefoundintheappendixofEngineeringNote106.Thegraphsintheengineeringnoteareidenticaltographsthatfirstappearedina1927AIEEpapertitledGraphicalDeterminationofMagneticFieldsbyRobertWieseman.PoleConstantsWhatfollowsare27PoleConstantsWiesemanusedhandplottingtechniquestoplotthefluxfieldsofseveralhundredsofpoleshapestocomeupwiththegraphs.Duetotheintensivenatureofthework,thegraphsareplottedforalimitedrangeofpolegeometry:Polearc/Polepitch=0.5to.75 Gmax/Gmin=1.0to3.0 Minimumgap/polepitch=.005to.05 SincethesecurvesareusedbySMDStocalculatemotorperformance,SMDSwillnotrunwithanyoneofthesethreevariablesoutsideofthegivenrange.Thereisnoreason,besidesthelimitationsoftheoriginalcurves,whyvariablesoutsidetherangeslistedabovecouldn’tbeused.PoleConstantsWiesemanusedha28PoleFaceDesign–MagneticFieldsDeterminationofKPoleFaceDesign–MagneticFi29PoleFaceDesign–MagneticFieldsDeterminationofC1PoleFaceDesign–MagneticFi30PoleFaceDesign–MagneticFieldsDeterminationofCq1PoleFaceDesign–MagneticFi31PoleFaceDesign–MagneticFieldsDeterminationofCd1PoleFaceDesign–MagneticFi32PoleFaceDesign–MagneticFieldsPolefacedesignscomeintwoflavors,singleradiusanddoubleradius.Thereasonforthisistheshapeofthepoleheadrelativetothestatorboreradiushasalargeinfluenceoftheshapeofthefieldfluxwaveform.PoleFaceDesign–MagneticFi33ReactanceCalculationsXad=ReactanceofarmaturereactiondirectedalongthedirectaxisXaq=ReactanceofarmaturereactiondirectedalongthequadratureaxisT=CommonReactanceFactora=PermeanceFactorCd1=PoleConstantCq1=PoleConstantReactanceCalculationsXad=Re34T=CommonReactanceFactorm=#phasesL=StatorCoreLengthf=frequencyZ=SeriesConductorsperPhaseKw=WindingfactorStatorP=#PolesReactanceCalculationsT=CommonReactanceFactorRea35a=PermeanceFactorD=DiameterofStatorBoreP=#PolesKg=Carter’sGapCoefficeintgmin=MinimumairgapatcenterofpoleReactanceCalculationsa=PermeanceFactorReactance36ArmatureLeakageReactanceisdeterminedusingamethodologyidenticaltotheinductionmachine.SynchronousReactancesXd=Xl+XadXq=Xl+XaqArmatureLeakageReactanceis37ExcitationCalculations1.CalculatetotalMagneticFlux2.ConvertarmaturemagneticFluxtoFieldEquivalent3.CalculateEintbyaddingarmatureresistanceandleakagereactancedropstoterminalvoltage.4.Usingstep2,calculatetheamphereturnsrequiredtomagnitizetheairgap.5.UsingstepEintfromstep3andelectricalsteelmagnitizationcurvescalculatetheampereturnsrequiredtomagnitzethestatorcoreandairgap.6.Usingtheresultsof4and5calculatethesaturationfactor.7.Usingtheresultsof3and5calculatethedirectaxiscomponentofmmf.8.Calculatethedirectaxiscomponentofarmaturereaction9.Usingtheresultsof6and7calculatethemmfrequirementsatthepoleface10.Calculatethepolesaturationmmf11.Totaldirectaxisexcitationisthesumof8and9ExcitationCalculations1.Cal38ExcitationCalculationsStepOne:TotalfundamentalfluxperpoleEph: PhasevoltageatstatorterminalsKp: PitchFactorKd: DistributionFactorFreq: FrequencyNSPC: ArmatureseriesturnsperphasepercircuitExcitationCalculationsStepOn39ExcitationCalculation Step2:CalculatetotalfluxperpoleonopencircuitKrelatesfundamentalfluxperpoletototalfluxperpoleusingfactors.ExcitationCalculation Step240ExcitationCalculationCont.Step3:Calculatetotalairgapfluxperpoleatthespecificvoltageandloadofinterest.ThisvoltagewasshownonthepreviousphasordiagramasEint.Step3a:CalculatethestatorleakagereactanceusingsameformulasderivedfortheInductionmotorstator.ExcitationCalculationCont.St41FieldExcitationCont.PortionofthepreviousphasordiagramisredrawnFromdiagramitisevidentthat:IaxlIaraIaVtIdEintIqFieldExcitationCont.Portion42IaVtEintIqK1EintIdxadIqxaqIaxaqIaraIaxlαεδIdK1Eintsin(α)IaVtEintIqK1EintIdxadIqxaqIax43FieldExcitationCont.Step4:Calculatetheampereturnsneededtomagnetizetheairgapatratedvoltage:Fg.Samegapfactorusedfrominductionmotortheory(i.e.Carter’scoefficient)FieldExcitationCont.Step4:44FieldExcitationCont.Step5:Calculatethestatorcore+statorteethampereturnsatthevalueoffluxcorrespondingtoEint.Ac:AreaofstatorcoreAt:AreaofstatorteethBCmax:maximumvalueoffluxdensityinstatorcoreBTmax:maximumvalueoffluxdensityinstatorteethFieldExcitationCont.Step5:45FieldExcitationCont.UsingB-Hcurvesformagneticsteelusedforstatorlaminations,readoffavalueofmmfinamphereturnsforthevalueofBCmaxandBTmaxcalculatedinprevioussteps.Makesureunitsmatch.FieldExcitationCont.UsingB-46FieldExcitationCont. Step6:CalculatethecomponentmmfinthedirectaxiscorrespondingtoK1eintStep6a:CalculatesaturationfactorK1fromB-Hcurveforelectricalsteel.Statormmf:ActualvalueofampereturnscorrespondingtoEintvalueoffluxfromstep5.FgEint:ValueofmmfthatwouldresultbyextendingthestraitportionofB-Hcurve.Fgwascalculatedinstep4.FieldExcitationCont. Step6:47mmfeintmmfeint48FieldExcitationCont.Step7:K1eld=K1Eintcos?K1eld:Componentofmmfinthedirectaxis correspondingtoK1EintK1:SaturationFactorfromstep6Eint:Voltage

intheairgapfromstep4?:AnglebetweenFieldExcitationVoltageand Eint.:

?=

-FieldExcitationCont.Step7:49FieldExcitationCont.Step8:Calculatethedirectaxiscomponentofarmaturereaction:IdxadXadcalculatedinreactancesectionId=Iasin(ε)FieldExcitationCont.Step8:50FieldExcitationCont.Step9:Addingtheresultsfromstep7andstep6,yougetthemmfrequirementsatthepolefaceFPF.FPF=K1Eintcos(?)+IdxadTheunitsofFPFareampereturns.FieldExcitationCont.Step9:51FieldExcitationCont.Step10:Calculatethepolesaturationmmf.Usingtheresultsofthenextlecture,calculatethepoleleakagefactorKLTherotorfluxperpoleisthenKLxEintxFReadthevalueofmmfperpolefromthepolesteelB-HCurve.FieldExcitationCont.Step10:52FieldExcitationFinishThetotalexcitationrequiredalongthedirectaxisisthesumofsteps9and10.Theresultisinunitsofampere-turns.Dividetheampere-turnsbythenumberofturnsperpoletoarriveatFullLoadFieldAmps.FieldExcitationFinishThetot53THEEND Questions?THEEND Questions?54SynchronousMachines-ReactanceandExcitationCalculationMarkFanslowTecoWestinghouseEngineeringTrainingNov.2005SynchronousMachines-Reactan55StatorRevolvingMagneticFieldMagneticPolePairsrotateatSynchronousSpeedThreePhaseACVoltageStatorRevolvingMagneticFiel56SynchronousRotorDCinputtotherotorcreateselectromagneticpolespairsPolesofdifferentpolarityarecreatedbywindingaroundthepoleindifferentdirectionsSynchronousRotorDCinputtot57SynchronousMotorRotor“Locked”intopowersupply,rovlovesatsynchronousspeedSynchronousMotorRotor“Locked58PhasorsandPhasorDiagramsTheconceptofPhasorsisrelatedtosinusoidalwaveformsthataredistributedinspaceandvarywithtime.Phasorsarecomplex(asincomplexnumbers)quantitiesusedforsimplifiedcalculationoftimevaryingandtravelingwaveforms.Themagnitudeandphaserelationshipbetweenthevectorsandcanbeshownsimplyinphasordiagrams.PhasorsandPhasorDiagramsThe59PhasorDiagramExampleConsiderthesimpleRL(resistiveinductive)circuitV=Vmsin(ωt)Voltagefunctionoftime.CurrentfunctionoftimeanddisplacedbyangleVILMagneticFluxinInductor90°outofphasewithvoltagewaveformi90°PhasorDiagramExampleConsider60PhasorDiagramCont.VIFLФLRecall:

B:MagneticFluxDensitydenoteФ

H:MagneticFluxIntensityormmfdenoteFo:Constantofpermeabilityoffreespacer:Constantofpermeabilityofelectricalsteel

PhasorDiagramCont.VIFLФLRec61CylindricalRotorSynchronousMachine:Duetotheevenairgap,theformulationsforbasicmachinequantitiesissimplified.Wewillconsiderthecylindricalmachinetheoryfirstandthenextendtheanalysistothesalientpolecase.CylindricalRotorSynchronous62AxisofFieldAxisofPhaseA90°lagArmaturecurrentmmfFarResultantmmfFrFieldmmfFfRotorRotationStatorRotorNSStatorI.D.RotorO.D.AxisofFieldAxisofPhaseA9063IaVtIaraEintEaΦar-ΦarfΦfRjIaxAjIax1Φr-ararErIaVtIaraEintEaΦar-ΦarfΦfRjI64CylindricalRotorPhasorDiagramForSynchronousGeneratorVt: TerminalVoltageIa: StatorCurrentLagsterminalVoltagebyAngleEa: AirGapVoltage,thevoltageinducedintothearmaturebythe fieldvoltageactingalone,the“opencircuit”voltageEr: ReactivevoltageproducedbyarmaturefluxEint: ResultantVoltageintheAirGapar: FluxfromArmaturecurrent(ArmatureReaction)f: Fluxduetofieldcurrentr: Resultantfluxar+fFf: mmfofFieldAar: mmfofArmatureCurrent(ArmatureReaction)FR: Resultantmmf,thesumofF+AIara: Voltagedropofarmatureresistance Iaxl: ReactivevoltagedropofArmatureleakagereactanceIaxA: ReactivevoltagedropofreactanceofArmaturereactionCylindricalRotorPhasorDiagr65FromthediagramitisevidentthatEr,thevoltageinducedinthestatorbytheeffectofthearmaturereactionflux,isinphasewithIaXl.ThesummationofErandIaXlgivesthetotalreactivevoltageproducedinthearmaturecircuitbythearmatureflux.TheratioofthistotalvoltagetoarmaturecurrentisdefinedasXsorthesynchronousreactance.Fromthediagramitisevident66SalientPoleSynchronousMachinesSalientPoleSynchronousMachi67SalientPoleSynchronousMachineSalientPoleSynchronousMachi68TwoReactionTheoryOneoftheconceptsusedincylindricalrotortheorywasthesummationofbothfluxandmmfwaveforms.r=f+arandFr=Ff+FarThisisallowedsince

B=orHandthemagneticpermeabilityoftheairgapisconstantaroundtherotorHoweverforsalientpolemachines,thepermeabilityofthefluxpathvariessignificantlyastheratioofairgaptosteelchanges.Thereforerf+ar

TwoReactionTheoryOneofthe69SalientPoleSynchronousMachineWithasalientpolemachine,asinusoidaldistributionoffluxcannotbeassumedintheairgapduetothevariationinmagneticpermeancealongtheairgap.However,owingtothesymmetryofasalientpolemachinealongthedirectandquadratureaxis,asinusoidaldistributionofmmfcanbeassumedalongeachaxis.BreakthemmfofthearmaturecurrentAarintotwocomponents,AdandAq.AdisthecomponentofAarthatworksalongthedirectaxisandAqisthecomponentofAarcenteredonthequadratureaxis.SalientPoleSynchronousMachi70TwoReactionTheory:IntroductionIfyouacceptthatAarcanbebrokenintotwocomponentsAdandAq,itfollowsthatforeachmmfwaveform,aemfwaveformexists90degreesoutofphasewithit.SoAdhasanassociatedEd,andAqhasanassociatedEq.ThesevoltagedropscanbethoughofhasbeingcreatedbyafictitiousreactancedropEd=XadIdandEq=XaqIq

,whereId: DirectaxiscomponentofarmaturecurrentIq:QuadratureaxiscomponentofarmaturecurrentXad:DirectaxisarmaturereactionreactanceXaq:QuadratureaxisarmaturereactionreactanceTwoReactionTheory:Introduct71IaVtIaraEintEaΦarθjIaxAAjIax1IdxldIqxlqIdxAdIqxAqForCylindricalRotor:Ia2=Id2+Iq2Xld=XlqXAd=XAqIaVtIaraEintEaΦarθjIaxAAjIax1I72TwoReactionDiagramSinceforaCylindricalRotor:Ia2=Id2+Iq2and(IaXs)2=(IdXd)2+(IqXq)2Xld=XlqXAd=XAqXs=Xd=XqItisnotnecessarytousetworeactiontheorytodescribethequantitiesofacylindricalrotormachine.TwoReactionDiagramSincefor73EaIqIaIdΦfΦrΦaqΦadΦarPhasorDiagramofaSalientPoleSynchronousGeneratorEaIqIaIdΦfΦrΦaqΦadΦarPhasorDi74IaVtIdEintIqK1EintIdxadIqxaqEaIaraIaxlIa:StatorcurrentVt:StatorterminalVoltageIra:VoltageofstatorresistanceIxl:VoltageofstatorleakagereactanceEint:InternalairgapvoltageK1Eint:Extrammfrequiredtoovercomestatorsaturation,K1issaturationfactorIdxad:reactivevoltagedropdirectaxisIdxaq:ReactivevoltagedropquadratureaxisEa:Totalsumofdirectaxisvoltage,airgap voltage,opencircuitvoltage?εδId=Iasin(+δ)Iq=Iasin(+δ)IaVtIdEintIqK1EintIdxadIqxaqE75PoleFaceDesign-MagneticFieldsThemmfwaveofarmaturereactionandthemmfwaveofthepolearecreatedontwodifferentsidesoftheairgapbutmustbecombinedtodeterminearesultantmmf.Todothiswemustdetermineconversionfactorstoconvertanstatorsidemmftoandequivalentrotorsidemmf.90°lagArmaturecurrentmmfFarResultantmmfFrFieldmmfFfRotorRotationStatorRotorNSPoleFaceDesign-MagneticFi76PoleFaceDesign–MagneticFieldsC1:Ifpeakofthefundamentalisunity,thenC1ispeakofacutalwaveform.NotethatWiesemancallsthisA1No-Load:motorisexcitedbythefieldwindingonlyPoleFaceDesign–MagneticFi77PoleFaceDesign–MagneticFieldCd1:Ifpeakofthefundamentalisunity,thenCd1ispeakofacutalwaveform.NotethatWiesemancallsthisAd1ArmaturemmfSinewavewhoseaxiscoincideswiththepolecenterPoleFaceDesign–MagneticFi78PoleFaceDesign–MagneticFieldCq1:Ifpeakofthefundamentalisunity,thenCq1ispeakofacutalwaveform.NotethatWiesemancallsthisAq1ArmaturemmfSinewavewhoseaxiscoincideswiththegapbetweenpolesPoleFaceDesign–MagneticFi79ListofPoleConstantsCd1–RatioofthefundamentaloftheairgapfluxproducedbythedirectaxisarmaturecurrenttothatwhichwouldbeproducedwithauniformgapequaltotheeffectivegapatthepolecenterCq1–RatioofthefundamentaloftheairgapfluxproducedbythequadratureaxisarmaturecurrenttothatwhichwouldbeproducedwithauniformgapequaltotheeffectivegapatthepolecenterC1–Theratioofthefundamentaltotheactualmaximumvalueofthefieldformwhenexcitedbythefieldonly(no-load)Cm–Ratiooffundamentalairgapfluxproducedbythefundamentalofarmaturemmftothatproducedbythefieldforthesamemaximummmf.Thisisthearmaturereactionconversionfactorforthedirectaxis.Cm=Cd1/C1K–Fluxdistributioncoefficient;theratiooftheareaoftheactualnoloadfluxwavetotheareaofitsfundamentalListofPoleConstantsCd1–R80PoleConstantsWhatfollowsaregraphsthatrelatethephysicalgeometryofthepoletothepoleconstants.ThesegraphscanbefoundintheappendixofEngineeringNote106.Thegraphsintheengineeringnoteareidenticaltographsthatfirstappearedina1927AIEEpapertitledGraphicalDeterminationofMagneticFieldsbyRobertWieseman.PoleConstantsWhatfollowsare81PoleConstantsWiesemanusedhandplottingtechniquestoplotthefluxfieldsofseveralhundredsofpoleshapestocomeupwiththegraphs.Duetotheintensivenatureofthework,thegraphsareplottedforalimitedrangeofpolegeometry:Polearc/Polepitch=0.5to.75 Gmax/Gmin=1.0to3.0 Minimumgap/polepitch=.005to.05 SincethesecurvesareusedbySMDStocalculatemotorperformance,SMDSwillnotrunwithanyoneofthesethreevariablesoutsideofthegivenrange.Thereisnoreason,besidesthelimitationsoftheoriginalcurves,whyvariablesoutsidetherangeslistedabovecouldn’tbeused.PoleConstantsWiesemanusedha82PoleFaceDesign–MagneticFieldsDeterminationofKPoleFaceDesign–MagneticFi83PoleFaceDesign–MagneticFieldsDeterminationofC1PoleFaceDesign–MagneticFi84PoleFaceDesign–MagneticFieldsDeterminationofCq1PoleFaceDesign–MagneticFi85PoleFaceDesign–MagneticFieldsDeterminationofCd1PoleFaceDesign–MagneticFi86PoleFaceDesign–MagneticFieldsPolefacedesignscomeintwoflavors,singleradiusanddoubleradius.Thereasonforthisistheshapeofthepoleheadrelativetothestatorboreradiushasalargeinfluenceoftheshapeofthefieldfluxwaveform.PoleFaceDesign–MagneticFi87ReactanceCalculationsXad=ReactanceofarmaturereactiondirectedalongthedirectaxisXaq=ReactanceofarmaturereactiondirectedalongthequadratureaxisT=CommonReactanceFactora=PermeanceFactorCd1=PoleConstantCq1=PoleConstantReactanceCalculationsXad=Re88T=CommonReactanceFactorm=#phasesL=StatorCoreLengthf=frequencyZ=SeriesConductorsperPhaseKw=WindingfactorStatorP=#PolesReactanceCalculationsT=CommonReactanceFactorRea89a=PermeanceFactorD=DiameterofStatorBoreP=#PolesKg=Carter’sGapCoefficeintgmin=MinimumairgapatcenterofpoleReactanceCalculationsa=PermeanceFactorReactance90ArmatureLeakageReactanceisdeterminedusingamethodologyidenticaltotheinductionmachine.SynchronousReactancesXd=Xl+XadXq=Xl+XaqArmatureLeakageReactanceis91ExcitationCalculations1.CalculatetotalMagneticFlux2.ConvertarmaturemagneticFluxtoFieldEquivalent3.CalculateEintbyaddingarmatureresistanceandleakagereactance

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