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SynchronousMachines-ReactanceandExcitationCalculationMarkFanslowTecoWestinghouseEngineeringTrainingNov.2005StatorRevolvingMagneticFieldMagneticPolePairsrotateatSynchronousSpeedThreePhaseACVoltageSynchronousRotorDCinputtotherotorcreateselectromagneticpolespairsPolesofdifferentpolarityarecreatedbywindingaroundthepoleindifferentdirectionsSynchronousMotorRotor“Locked”intopowersupply,rovlovesatsynchronousspeedPhasorsandPhasorDiagramsTheconceptofPhasorsisrelatedtosinusoidalwaveformsthataredistributedinspaceandvarywithtime.Phasorsarecomplex(asincomplexnumbers)quantitiesusedforsimplifiedcalculationoftimevaryingandtravelingwaveforms.Themagnitudeandphaserelationshipbetweenthevectorsandcanbeshownsimplyinphasordiagrams.PhasorDiagramExampleConsiderthesimpleRL(resistiveinductive)circuitV=Vmsin(ωt)Voltagefunctionoftime.CurrentfunctionoftimeanddisplacedbyangleVILMagneticFluxinInductor90°outofphasewithvoltagewaveformi90°PhasorDiagramCont.VIFLФLRecall:

B:MagneticFluxDensitydenoteФ

H:MagneticFluxIntensityormmfdenoteFo:Constantofpermeabilityoffreespacer:Constantofpermeabilityofelectricalsteel

CylindricalRotorSynchronousMachine:Duetotheevenairgap,theformulationsforbasicmachinequantitiesissimplified.Wewillconsiderthecylindricalmachinetheoryfirstandthenextendtheanalysistothesalientpolecase.AxisofFieldAxisofPhaseA90°lagArmaturecurrentmmfFarResultantmmfFrFieldmmfFfRotorRotationStatorRotorNSStatorI.D.RotorO.D.IaVtIaraEintEaΦar-ΦarfΦfRjIaxAjIax1Φr-ararErCylindricalRotorPhasorDiagramForSynchronousGeneratorVt: TerminalVoltageIa: StatorCurrentLagsterminalVoltagebyAngleEa: AirGapVoltage,thevoltageinducedintothearmaturebythefieldvoltageactingalone,the“opencircuit””voltageEr: ReactivevoltageproducedbyarmaturefluxEint: ResultantVoltageintheAirGapar: FluxfromArmaturecurrent(ArmatureReaction)f: Fluxduetofieldcurrentr: Resultantfluxar+fFf: mmfofFieldAar: mmfofArmatureCurrent(ArmatureReaction)FR: Resultantmmf,thesumofF+AIara: VoltagedropofarmatureresistanceIaxl: ReactivevoltagedropofArmatureleakagereactanceIaxA: ReactivevoltagedropofreactanceofArmaturereactionFromthediagramitisevidentthatEr,thevoltageinducedinthestatorbytheeffectofthearmaturereactionflux,isinphasewithIaXl.ThesummationofErandIaXlgivesthetotalreactivevoltageproducedinthearmaturecircuitbythearmatureflux.TheratioofthistotalvoltagetoarmaturecurrentisdefinedasXsorthesynchronousreactance.SalientPoleSynchronousMachinesSalientPoleSynchronousMachineTwoReactionTheoryOneoftheconceptsusedincylindricalrotortheorywasthesummationofbothfluxandmmfwaveforms.r=f+arandFr=Ff+FarThisisallowedsinceB=orHandthemagneticpermeabilityoftheairgapisconstantaroundtherotorHoweverforsalientpolemachines,thepermeabilityofthefluxpathvariessignificantlyastheratioofairgaptosteelchanges.Thereforerf+arSalientPoleSynchronousMachineWithasalientpolemachine,asinusoidaldistributionoffluxcannotbeassumedintheairgapduetothevariationinmagneticpermeancealongtheairgap.However,owingtothesymmetryofasalientpolemachinealongthedirectandquadratureaxis,asinusoidaldistributionofmmfcanbeassumedalongeachaxis.BreakthemmfofthearmaturecurrentAarintotwocomponents,AdandAq.AdisthecomponentofAarthatworksalongthedirectaxisandAqisthecomponentofAarcenteredonthequadratureaxis.TwoReactionTheory:IntroductionIfyouacceptthatAarcanbebrokenintotwocomponentsAdandAq,itfollowsthatforeachmmfwaveform,aemfwaveformexists90degreesoutofphasewithit.SoAdhasanassociatedEd,andAqhasanassociatedEq.ThesevoltagedropscanbethoughofhasbeingcreatedbyafictitiousreactancedropEd=XadIdandEq=XaqIq,whereId:DirectaxiscomponentofarmaturecurrentIq:QuadratureaxiscomponentofarmaturecurrentXad:DirectaxisarmaturereactionreactanceXaq:QuadratureaxisarmaturereactionreactanceIaVtIaraEintEaΦarθjIaxAAjIax1IdxldIqxlqIdxAdIqxAqForCylindricalRotor:Ia2=Id2+Iq2Xld=XlqXAd=XAqTwoReactionDiagramSinceforaCylindricalRotor:Ia2=Id2+Iq2and(IaXs)2=(IdXd)2+(IqXq)2Xld=XlqXAd=XAqXs=Xd=XqItisnotnecessarytousetworeactiontheorytodescribethequantitiesofacylindricalrotormachine.EaIqIaIdΦfΦrΦaqΦadΦarPhasorDiagramofaSalientPoleSynchronousGeneratorIaVtIdEintIqK1EintIdxadIqxaqEaIaraIaxlIa:StatorcurrentVt:StatorterminalVoltageIra:VoltageofstatorresistanceIxl:VoltageofstatorleakagereactanceEint:InternalairgapvoltageK1Eint:Extrammfrequiredtoovercomestatorsaturation,K1issaturationfactorIdxad:reactivevoltagedropdirectaxisIdxaq:ReactivevoltagedropquadratureaxisEa:Totalsumofdirectaxisvoltage,airgapvoltage,opencircuitvoltageßεδId=Iasin(+δ)Iq=Iasin(+δ)PoleFaceDesign-MagneticFieldsThemmfwaveofarmaturereactionandthemmfwaveofthepolearecreatedontwodifferentsidesoftheairgapbutmustbecombinedtodeterminearesultantmmf.Todothiswemustdetermineconversionfactorstoconvertanstatorsidemmftoandequivalentrotorsidemmf.90°lagArmaturecurrentmmfFarResultantmmfFrFieldmmfFfRotorRotationStatorRotorNSPoleFaceDesign––MagneticFieldsC1:Ifpeakofthefundamentalisunity,thenC1ispeakofacutalwaveform.NotethatWiesemancallsthisA1No-Load:motorisexcitedbythefieldwindingonlyPoleFaceDesign––MagneticFieldCd1:Ifpeakofthefundamentalisunity,thenCd1ispeakofacutalwaveform.NotethatWiesemancallsthisAd1ArmaturemmfSinewavewhoseaxiscoincideswiththepolecenterPoleFaceDesign––MagneticFieldCq1:Ifpeakofthefundamentalisunity,thenCq1ispeakofacutalwaveform.NotethatWiesemancallsthisAq1ArmaturemmfSinewavewhoseaxiscoincideswiththegapbetweenpolesListofPoleConstantsCd1–RatioofthefundamentaloftheairgapfluxproducedbythedirectaxisarmaturecurrenttothatwhichwouldbeproducedwithauniformgapequaltotheeffectivegapatthepolecenterCq1–RatioofthefundamentaloftheairgapfluxproducedbythequadratureaxisarmaturecurrenttothatwhichwouldbeproducedwithauniformgapequaltotheeffectivegapatthepolecenterC1–Theratioofthefundamentaltotheactualmaximumvalueofthefieldformwhenexcitedbythefieldonly(no-load)Cm–Ratiooffundamentalairgapfluxproducedbythefundamentalofarmaturemmftothatproducedbythefieldforthesamemaximummmf.Thisisthearmaturereactionconversionfactorforthedirectaxis.Cm=Cd1/C1K–Fluxdistributioncoefficient;theratiooftheareaoftheactualnoloadfluxwavetotheareaofitsfundamentalPoleConstantsWhatfollowsaregraphsthatrelatethephysicalgeometryofthepoletothepoleconstants.ThesegraphscanbefoundintheappendixofEngineeringNote106.Thegraphsintheengineeringnoteareidenticaltographsthatfirstappearedina1927AIEEpapertitledGraphicalDeterminationofMagneticFieldsbyRobertWieseman.PoleConstantsWiesemanusedhandplottingtechniquestoplotthefluxfieldsofseveralhundredsofpoleshapestocomeupwiththegraphs.Duetotheintensivenatureofthework,thegraphsareplottedforalimitedrangeofpolegeometry:Polearc/Polepitch=0.5to.75Gmax/Gmin=1.0to3.0Minimumgap/polepitch=.005to.05SincethesecurvesareusedbySMDStocalculatemotorperformance,SMDSwillnotrunwithanyoneofthesethreevariablesoutsideofthegivenrange.Thereisnoreason,besidesthelimitationsoftheoriginalcurves,whyvariablesoutsidetherangeslistedabovecouldn’tbeused.PoleFaceDesign––MagneticFieldsDeterminationofKPoleFaceDesign–MagneticFieldsDeterminationofC1PoleFaceDesign–MagneticFieldsDeterminationofCq1PoleFaceDesign–MagneticFieldsDeterminationofCd1PoleFaceDesign–MagneticFieldsPolefacedesignscomeintwoflavors,singleradiusanddoubleradius.Thereasonforthisistheshapeofthepoleheadrelativetothestatorboreradiushasalargeinfluenceoftheshapeofthefieldfluxwaveform.ReactanceCalculationsXad=ReactanceofarmaturereactiondirectedalongthedirectaxisXaq=ReactanceofarmaturereactiondirectedalongthequadratureaxisT=CommonReactanceFactora=PermeanceFactorCd1=PoleConstantCq1=PoleConstantT=CommonReactanceFactorm=#phasesL=StatorCoreLengthf=frequencyZ=SeriesConductorsperPhaseKw=WindingfactorStatorP=#PolesReactanceCalculationsa=PermeanceFactorD=DiameterofStatorBoreP=#PolesKg=Carter’sGapCoefficeintgmin=MinimumairgapatcenterofpoleReactanceCalculationsArmatureLeakageReactanceisdeterminedusingamethodologyidenticaltotheinductionmachine.SynchronousReactancesXd=Xl+XadXq=Xl+XaqExcitationCalculations1.CalculatetotalMagneticFlux2.ConvertarmaturemagneticFluxtoFieldEquivalent3.CalculateEintbyaddingarmatureresistanceandleakagereactancedropstoterminalvoltage.4.Usingstep2,calculatetheamphereturnsrequiredtomagnitizetheairgap.5.UsingstepEintfromstep3andelectricalsteelmagnitizationcurvescalculatetheampereturnsrequiredtomagnitzethestatorcoreandairgap.6.Usingtheresultsof4and5calculatethesaturationfactor.7.Usingtheresultsof3and5calculatethedirectaxiscomponentofmmf.8.Calculatethedirectaxiscomponentofarmaturereaction9.Usingtheresultsof6and7calculatethemmfrequirementsatthepoleface10.Calculatethepolesaturationmmf11.Totaldirectaxisexcitationisthesumof8and9ExcitationCalculationsStepOne:TotalfundamentalfluxperpoleEph:PhasevoltageatstatorterminalsKp: PitchFactorKd: DistributionFactorFreq:FrequencyNSPC: ArmatureseriesturnsperphasepercircuitExcitationCalculationStep2:CalculatetotalfluxperpoleonopencircuitKrelatesfundamentalfluxperpoletototalfluxperpoleusingfactors.ExcitationCalculationCont.Step3:Calculatetotalairgapfluxperpoleatthespecificvoltageandloadofinterest.ThisvoltagewasshownonthepreviousphasordiagramasEint.Step3a:CalculatethestatorleakagereactanceusingsameformulasderivedfortheInductionmotorstator.FieldExcitationCont.PortionofthepreviousphasordiagramisredrawnFromdiagramitisevidentthat:IaxlIaraIaVtIdEintIqIaVtEintIqK1EintIdxadIqxaqIaxaqIaraIaxlαεδIdK1Eintsin(α)FieldExcitationCont.Step4:Calculatetheampereturnsneededtomagnetizetheairgapatratedvoltage:Fg.Samegapfactorusedfrominductionmotortheory(i.e.Carter’’scoefficient)FieldExcitationCont.Step5:Calculatethestatorcore+statorteethampereturnsatthevalueoffluxcorrespondingtoEint.Ac:AreaofstatorcoreAt:AreaofstatorteethBCmax:maximumvalueoffluxdensityinstatorcoreBTmax:maximumvalueoffluxdensityinstatorteethFieldExcitationCont.UsingB-Hcurvesformagneticsteelusedforstatorlaminations,readoffavalueofmmfinamphereturnsforthevalueofBCmaxandBTmaxcalculatedinprevioussteps.Makesureunitsmatch.FieldExcitationCont.Step6:CalculatethecomponentmmfinthedirectaxiscorrespondingtoK1eintStep6a:CalculatesaturationfactorK1fromB-Hcurveforelectricalsteel.Statormmf:ActualvalueofampereturnscorrespondingtoEintvalueoffluxfromstep5.FgEint:ValueofmmfthatwouldresultbyextendingthestraitportionofB-Hcurve.Fgwascalculatedinstep4.mmfeintFieldExcitationCont.Step7:K1eld=K1EintcosßK1eld:ComponentofmmfinthedirectaxiscorrespondingtoK1EintK1:SaturationFactorfromstep6Eint:Voltageintheairgapfromstep4ß:AnglebetweenFieldExcitationVoltageand Eint.:ß=-FieldExcitationCont.Step8:Calculatethedirectaxiscomponentofarmaturereaction:IdxadXadcalculatedinreactancesectionId=Iasin(ε)FieldExcitationCont.Step9:Addingtheresultsfromstep7andstep6,yougetthemmfrequirementsatthepolefaceFPF.FPF=K1Eintcos(ß)+IdxadTheunitsofFPFareampereturns.FieldExcitationCont.Step10:Calculatethepolesaturationmmf.Usingtheresultsofthenextlecture,calculatethepoleleakagefactorKLTherotorfluxperpoleisthenKLxEintxFReadthevalueofmmfperpolefromthepolesteelB-HCurve.FieldExcitationFinishThetotalexcitationrequiredalongthedirectaxisisthesumofsteps9and10.Theresultisinunitsofampere-turns.Dividetheampere-turnsbythenumberofturnsperpoletoarriveatFullLoadFieldAmps.THEENDQuestions?9、静夜四四无邻，，荒居旧旧业贫。。。1月-231月-23Sunday,January1,202310、雨中黄黄叶树，，灯下白白头人。。。22:58:1522:58:1522:581/1/202310:58:15PM11、以我独沈沈久，愧君君相见频。。。1月-2322:58:1522:58Jan-2301-Jan-2312、故人人江海海别，，几度度隔山山川。。。22:58:1522:58:1522:58Sunday,January1,202313、乍见见翻疑疑梦，，相悲悲各问问年。。。1月-231月-2322:58:1522:58:15January1,202314、他乡生生白发，，旧国见见青山。。。01一一月202310:58:15下下午22:58:151月-2315、比不了得得就不比，，得不到的的就不要。。。。一月2310:58下午1月-2322:58January1,202316、行行动动出出成成果果，，工工作作出出财财富富。。。。2023/1/122:58:1622:58:1601January202317、做做前前，，能能够够环环视视四四周周；；做做时时，，你你只只能能或或者者最最好好沿沿着着以以脚脚为为起起点点的的射射线线向向前前。。。。10:58:16下下午午10:58下下午午22:58:161月月-239、没有失失败，只只有暂时时停止成成功！。。1月-231月-23Sunday,January1,202310、很多多事情情努力力了未未必有有结果果，但但是不不努力力却什什么改改变也也没有有。。。22:58:1622:58:1622:581/1/202310:58:16PM11、成功就是日日复一日那一一点点小小努努力的积累。。。1月-2322:58:1622:58Jan-2301-Jan-2312