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ReactanceofInductionMachinesReactanceofInductionMachine1FundamentalConceptsL=InstantaneousInductanceofCoil(Henries)e=Instantaneousvalueofvoltageinducedinthecoilbythechangeofcurrent(Volts)di/dt=Instantaneousvalueoftherateofchange ofthecurrentwhichflowsthroughthecoilFundamentalConceptsL=Instan2Faraday’sLawepath=inducedvoltagearoundanenclosedpathE=VectorofElectricFieldaroundthepathdl=Vectorofenclosedpathd/dt=RateofChangeofFluxenclosedbythepathFaraday’sLawepath=inducedv3AmoreusefulformofFaraday’slaw:whereIsthesumofallfluxesxlinkingcorrespondingturnsnxofthecoilAmoreusefulformofFaraday’4CombiningEquationsReactanceofaninductivecircuitX=2fL(Ohms)f=freqencyofvoltageacrossinductance(Hz)orCombiningEquationsReactanceo5Ampere’sCircuitLawH=VectorofMagneticFieldIntensity(Amp/meter)dl=VectoralongthelengthofthepathJ=VectorfieldofcurrentdensityenclosedbypathdA=areaenclosedbythepathAmpere’sCircuitLawH=Vector6ToapplyAmpere’sCircuitlaw,considerthesimplemagneticcircuitofatorus

withanon-ferromagneticcoreHr(2r)=NiToapplyAmpere’sCircuitlaw,7B=uruoHB=VectorQuantityofMagneticFluxDensity(Tesla)uo=constantrelatingFluxDensitytoFluxIntensityinfreespaceornon-ferrousmaterials=4x10-7(T?m/A)ur=proportionalityconstantforferromagneticmaterials(unit-less)Rearranging:Also:Amp/meterB=uruoHB=VectorQuantityof8Theintegrationoffluxdensityoveraunitareagivestotalflux.Foratorroid=BAWebersCombiningEquationsforLand,theinductanceofthecoilisshowntobedeterminedentirelybythedimensionsofthecoilandindependentofthecurrentinthewire.ThereactanceofthetorroidisthenTheintegrationoffluxdensit9ForFerromagneticMaterials,avalueofroisreadfromaSaturationCurveForFerromagneticMaterials,a10EffectofSaturationonInductanceEffectofSaturationonInduct11EquivalentCircuitofInductionMotorEquivalentCircuitofInductio12R1PrimaryresistanceX1LeakagereactanceofprimarywindingXMMagnetizingReactancegh=ResistancepathtoaccountforcorelossX2SecondaryleakagereactanceR2/sResistanceofsquirrelcagewindingThereisnostandardmethodforreactancecalculationsassimplifyingassumptionsaremadeatmanystepsintheircalculation.Theformulastofollowhavemanysimplifyingassumptions(suchasnosaturation)thatIwilltrytopointoutaswegoalong.However,thedependencyofreactancetophysicalparametersisthesameforallcalculationmethods.R1PrimaryresistanceThereis13MagnetizingReactanceXMT=ReactanceFactorD=StatorBoreDiameter(I.D.)L=CoreLengthinInchesKW=StatorWindingFactorKdxKp

g=singleairgapinchesKg=Carter’sAirGapCoefficientP=#PolesMagnetizingReactanceXMT=Reac14T=ReactanceFactor=

Wheref=frequencym=#ofphasesZ=#statorconductorsinseriesperphaseT=ReactanceFactor=Where15TheresultantvoltageintheairgapavailabletocreatetherotatingmagneticfieldislessthanthepeakvoltageofthesinewaveatthemotorterminalsBecauseof2factors:Kp:AccountsforthepitchofthewindingKd:AccountsforthedistributionofthewindingToaccountfortheslotopeningsofthestatorandrotorsurface,agapfactorisusedasamultipliertocreatean“effective”airgap.Thisfactoriscalledthe“Carter’sCoefficient”Kg

KW=Kp?KdTheresultantvoltageinthea16TotalLeakage

ReactanceX1+X2ThePrimaryslotreactanceTheSecondaryslotreactanceThezig-zagreactanceThebelt-leakagereactanceTheCoilEndLeakageReactanceTheperipheralleakageBecauserelativelyfewstatorsorrotorsareskewed,theleakagereactancecausedbyeitheraskewedstatororskewedrotorwillnotbeconsidered.Itissufficienttoknowthatanadditionalleakagereactanceexistsforskewedmotors.Reactance3and4aresometimescombinedandcalledDifferentialLeakageTotalLeakageReactanceX1+X217Reactances研發(fā)培訓(xùn),設(shè)計(jì)培訓(xùn)18StatorSlotLeakageReactanceT=ReactanceFactorL=LengthofstatorcoreS=#statorSlotsKs=Factortoaccountforcoilsofdifferentphasesbeinginthesameslot1=PermeanceFactorofStatorSlotBasedonslotgeometryStatorSlotLeakageReactanceT19ForStator:ForRotor:ForStator:ForRotor:20RotorSlotLeakageReactanceXSR=RotorSlotLeakageReactanceReferredtotheStatorT=ReactanceFactorL=LengthofrotorcoreR=#rotorSlotsKw=statorwindingfactor2=PermeanceFactorofrotorslotBasedonslotgeometryRotorSlotLeakageReactanceXS21RotorLeakageReactanceCont.ThevalueofXSRiseffectedbytheoperatingconditionofthemotor.Thevaluecalculatedisfor“running”conditionswhenrotorfrequencyis1-2Hzandrotorcurrentisrelativelylow.Duringrunningconditions,therotorbarcurrentcanbethoughofasdistributingevenlythroughoutthebar.Duringlockedrotor,therotorfrequencyis60Hzandduetothe“DeepBarEffect”thecurrentcrowdstowardthetopofthebar.ThecurrentcrowdingalongwiththesaturationeffectsofthehighinrushcurrentcausethestartingvalueofXSRtobesignificantlyreduced.RotorLeakageReactanceCont.T22DifferentialLeakageBesidesthefundamentalairgapfluxwaveformthereareharmonicwaveformsthataremultiplesofthenumberofpolesandrotateatsub-multiplesofsynchronousspeed.Theseharmonicsinducevoltagesinthewindingsthatproducedthemandthereforeaddtothereactanceofthewinding.Thedifferentialleakageissometimesbrokeintotwocomponents:Thezig-zagandphasebeltleakage.Thefollowingnotationcombinesthebeltleakagewiththezig-zag.DifferentialLeakageBesidesth23XM=MagnetizingReactanceS=#StatorSlotsP=#PolesR+#RotorSlotsKgSCarter’sairgapcoefficientforthestatorKgRCarter’sairgapcoefficientfortherotorKSShortPitchCorrectionFactorKWStatorWindingCorrectionFactorXM=MagnetizingReactance24CoilendLeakageReactanceStatorRotorAS=MeanstatoroverhangAR=MeanRotorOverhangCoilendLeakageReactanceStat25AS=MeanStatoroverhangL.M.H.T.=LengthofmeanhalfturnstatorcoilL=Statorcorelength=perunitcoilpitch=PolePitchKS=ShortpitchcorrectionfactorAS=MeanStatoroverhang26Reactances研發(fā)培訓(xùn),設(shè)計(jì)培訓(xùn)27PeripheralAirGapLeakageLinesoffluxthatentertheairgapbutdonotlinktherotorbutinsteadflowperipherallybackintothestator.Ifweimagineainductionmotorwiththerotorremoved,allthefluxinthestatorborewillbeofthisairleakagecharacter.D=StatorI.D.g=SingleairgapPeripheralAirGapLeakageLine28Peripheralreactanceisnegligibleforinductionmachineswithsmallairgapsbutcanberelevantonhighspeedsynchronousmotorswithverylargeairgaps.Considera4poleinductionmotor,800framewithD=32.75andg=.175:XP/XM=.00091Considera4polesynchronousmotor,800framewithD=33.5andg=1.0:XP/XM=.028Peripheralreactanceisneglig29SummaryofInductionMotorLeakageReactanceXSS=StatorslotleakagereactanceXRS=RotorslotleakagereactanceXZZ=differentialleakagereactance(Zig-Zag+BeltLeakage)XES=StatorEndWindingLeakageReactanceXES=RotorEndWindingLeakageReactanceX2=XSR+?XZZ+XERX1=XSS+?XZZ+XESSummaryofInductionMotorLea30LeakageReactanceSummaryContinuedTheLeakagereactancevariesasthesquareofeffectiveconductorsorturnsperphase.Ifthenumberofstatorslotsandthenumberofparallelcircuitsarenotchanged,itvariesasthesquareofthenumberofturnspercoilInOrdertodecreasetheleakagereactancedooneormoreofthefollowing:Decreasethestatorboreorcorelength.IncreasetheairgapIncreasethenumberofstatororrotorslotsDecreaseslotdepthorincreaseslotwidthReduceStatorCoilPitchTheconverseofalloftheaboveistrueLeakageReactanceSummaryCont31EffectofLeakageReactanceOnMotorPerformance

Startingtorqueincreasesasleakagereactance(X1+X2)decreasesEffectofLeakageReactanceOn32MaximumTorqueorBreakdownTorqueincreasesasleakagereactance(X1+X2)decreasesEffectofLeakageReactanceOnMotorPerformanceCont.OccursatvalueofslipsMaximumTorqueorBreakdownTo33EffectofLeakageReactanceOnMotorPerformanceInrushincreasesasleakagereactance(X1+X2)decreasesEffectofLeakageReactanceOn34QUESTIONSQUESTIONS35EffectofReactanceonTransientShortCircuitCurrentByDefinition:XD=XM+X1EffectofReactanceonTransie36GeneralequationforshortcircuitcurrentWhere

’d>>’’d

Fort=0;I’’=E/XD’’Fort>>’d;I0

For’’d<t<’d;Generalequationforshortcir37ReactanceofInductionMachinesReactanceofInductionMachine38FundamentalConceptsL=InstantaneousInductanceofCoil(Henries)e=Instantaneousvalueofvoltageinducedinthecoilbythechangeofcurrent(Volts)di/dt=Instantaneousvalueoftherateofchange ofthecurrentwhichflowsthroughthecoilFundamentalConceptsL=Instan39Faraday’sLawepath=inducedvoltagearoundanenclosedpathE=VectorofElectricFieldaroundthepathdl=Vectorofenclosedpathd/dt=RateofChangeofFluxenclosedbythepathFaraday’sLawepath=inducedv40AmoreusefulformofFaraday’slaw:whereIsthesumofallfluxesxlinkingcorrespondingturnsnxofthecoilAmoreusefulformofFaraday’41CombiningEquationsReactanceofaninductivecircuitX=2fL(Ohms)f=freqencyofvoltageacrossinductance(Hz)orCombiningEquationsReactanceo42Ampere’sCircuitLawH=VectorofMagneticFieldIntensity(Amp/meter)dl=VectoralongthelengthofthepathJ=VectorfieldofcurrentdensityenclosedbypathdA=areaenclosedbythepathAmpere’sCircuitLawH=Vector43ToapplyAmpere’sCircuitlaw,considerthesimplemagneticcircuitofatorus

withanon-ferromagneticcoreHr(2r)=NiToapplyAmpere’sCircuitlaw,44B=uruoHB=VectorQuantityofMagneticFluxDensity(Tesla)uo=constantrelatingFluxDensitytoFluxIntensityinfreespaceornon-ferrousmaterials=4x10-7(T?m/A)ur=proportionalityconstantforferromagneticmaterials(unit-less)Rearranging:Also:Amp/meterB=uruoHB=VectorQuantityof45Theintegrationoffluxdensityoveraunitareagivestotalflux.Foratorroid=BAWebersCombiningEquationsforLand,theinductanceofthecoilisshowntobedeterminedentirelybythedimensionsofthecoilandindependentofthecurrentinthewire.ThereactanceofthetorroidisthenTheintegrationoffluxdensit46ForFerromagneticMaterials,avalueofroisreadfromaSaturationCurveForFerromagneticMaterials,a47EffectofSaturationonInductanceEffectofSaturationonInduct48EquivalentCircuitofInductionMotorEquivalentCircuitofInductio49R1PrimaryresistanceX1LeakagereactanceofprimarywindingXMMagnetizingReactancegh=ResistancepathtoaccountforcorelossX2SecondaryleakagereactanceR2/sResistanceofsquirrelcagewindingThereisnostandardmethodforreactancecalculationsassimplifyingassumptionsaremadeatmanystepsintheircalculation.Theformulastofollowhavemanysimplifyingassumptions(suchasnosaturation)thatIwilltrytopointoutaswegoalong.However,thedependencyofreactancetophysicalparametersisthesameforallcalculationmethods.R1PrimaryresistanceThereis50MagnetizingReactanceXMT=ReactanceFactorD=StatorBoreDiameter(I.D.)L=CoreLengthinInchesKW=StatorWindingFactorKdxKp

g=singleairgapinchesKg=Carter’sAirGapCoefficientP=#PolesMagnetizingReactanceXMT=Reac51T=ReactanceFactor=

Wheref=frequencym=#ofphasesZ=#statorconductorsinseriesperphaseT=ReactanceFactor=Where52TheresultantvoltageintheairgapavailabletocreatetherotatingmagneticfieldislessthanthepeakvoltageofthesinewaveatthemotorterminalsBecauseof2factors:Kp:AccountsforthepitchofthewindingKd:AccountsforthedistributionofthewindingToaccountfortheslotopeningsofthestatorandrotorsurface,agapfactorisusedasamultipliertocreatean“effective”airgap.Thisfactoriscalledthe“Carter’sCoefficient”Kg

KW=Kp?KdTheresultantvoltageinthea53TotalLeakage

ReactanceX1+X2ThePrimaryslotreactanceTheSecondaryslotreactanceThezig-zagreactanceThebelt-leakagereactanceTheCoilEndLeakageReactanceTheperipheralleakageBecauserelativelyfewstatorsorrotorsareskewed,theleakagereactancecausedbyeitheraskewedstatororskewedrotorwillnotbeconsidered.Itissufficienttoknowthatanadditionalleakagereactanceexistsforskewedmotors.Reactance3and4aresometimescombinedandcalledDifferentialLeakageTotalLeakageReactanceX1+X254Reactances研發(fā)培訓(xùn),設(shè)計(jì)培訓(xùn)55StatorSlotLeakageReactanceT=ReactanceFactorL=LengthofstatorcoreS=#statorSlotsKs=Factortoaccountforcoilsofdifferentphasesbeinginthesameslot1=PermeanceFactorofStatorSlotBasedonslotgeometryStatorSlotLeakageReactanceT56ForStator:ForRotor:ForStator:ForRotor:57RotorSlotLeakageReactanceXSR=RotorSlotLeakageReactanceReferredtotheStatorT=ReactanceFactorL=LengthofrotorcoreR=#rotorSlotsKw=statorwindingfactor2=PermeanceFactorofrotorslotBasedonslotgeometryRotorSlotLeakageReactanceXS58RotorLeakageReactanceCont.ThevalueofXSRiseffectedbytheoperatingconditionofthemotor.Thevaluecalculatedisfor“running”conditionswhenrotorfrequencyis1-2Hzandrotorcurrentisrelativelylow.Duringrunningconditions,therotorbarcurrentcanbethoughofasdistributingevenlythroughoutthebar.Duringlockedrotor,therotorfrequencyis60Hzandduetothe“DeepBarEffect”thecurrentcrowdstowardthetopofthebar.ThecurrentcrowdingalongwiththesaturationeffectsofthehighinrushcurrentcausethestartingvalueofXSRtobesignificantlyreduced.RotorLeakageReactanceCont.T59DifferentialLeakageBesidesthefundamentalairgapfluxwaveformthereareharmonicwaveformsthataremultiplesofthenumberofpolesandrotateatsub-multiplesofsynchronousspeed.Theseharmonicsinducevoltagesinthewindingsthatproducedthemandthereforeaddtothereactanceofthewinding.Thedifferentialleakageissometimesbrokeintotwocomponents:Thezig-zagandphasebeltleakage.Thefollowingnotationcombinesthebeltleakagewiththezig-zag.DifferentialLeakageBesidesth60XM=MagnetizingReactanceS=#StatorSlotsP=#PolesR+#RotorSlotsKgSCarter’sairgapcoefficientforthestatorKgRCarter’sairgapcoefficientfortherotorKSShortPitchCorrectionFactorKWStatorWindingCorrectionFactorXM=MagnetizingReactance61CoilendLeakageReactanceStatorRotorAS=MeanstatoroverhangAR=MeanRotorOverhangCoilendLeakageReactanceStat62AS=MeanStatoroverhangL.M.H.T.=LengthofmeanhalfturnstatorcoilL=Statorcorelength=perunitcoilpitch=PolePitchKS=ShortpitchcorrectionfactorAS=MeanStatoroverhang63Reactances研發(fā)培訓(xùn),設(shè)計(jì)培訓(xùn)64PeripheralAirGapLeakageLinesoffluxthatentertheairgapbutdonotlinktherotorbutinsteadflowperipherallybackintothestator.Ifweimagineainductionmotorwiththerotorremoved,allthefluxinthestatorborewillbeofthisairleakagecharacter.D=StatorI.D.g=SingleairgapPeripheralAirGapLeakageLine65Peripheralreactanceisnegligibleforinductionmachineswithsmallairgapsbutcanberelevantonhighspeedsynchronousmotorswithverylargeairgaps.Considera4poleinductionmotor,800framewithD=32.75andg=.175:XP/XM=.00091Considera4polesynchronousmotor,800framewithD=33.5andg=1.0:X

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