電機(jī)學(xué)（第七版）（修訂版）課件全套 劉新正 Chapter-1-Magnetic-Circuits-and-Magnetic-Materials -Chapter-10-Speed-and-Torque-Control

Chapter1MagneticCircuitsandMagneticMaterials7/10/2024MagneticCircuitsandMagneticMaterials1IntroductionThemagneto-quasi-staticformofMaxwell’sequationsareusedatlowfrequenciestypicalofenergy-conversiondevicesAmpere’slawandGauss’slaw:7/10/2024MagneticCircuitsandMagneticMaterials2CoilorwindingofNturnsonamagneticcoreofhighpermeabilitym>>moFluxismostlyconfinedtothecoreSimplifyAmpere’slaw:coilMMF=Ni7/10/2024MagneticCircuitsandMagneticMaterials3Gauss’slaw:fluxlinesareclosed,sonetfluxisfluxdensityBctimecross-sectionalareaAc Fc=BcAcSinceweareusingaveragefluxdensityBcthenthecorrespondingaveragemagneticfieldintensityHcwillbeapproximately7/10/2024MagneticCircuitsandMagneticMaterials4lcistheaverage(mean)corelengthwhichisperpendiculartothecross-sectionalareaAcThematerialpropertiesgivetherelationbetweenBandH,suchasB=mHforalinearmaterial7/10/2024MagneticCircuitsandMagneticMaterials5Magneticcircuitwithairgap7/10/2024MagneticCircuitsandMagneticMaterials6AnalogybetweenDCelectriccircuit(a)andmagneticcircuit(b):VoltageVisanalogoustoMMFFCurrentIisanalogoustofluxfResistanceRisanalogoustoreluctanceRFluxlinkage,inductanceandenergyFluxlinkageofacoilisl=NfwherethefluxfisassumedtolinkallNturns,thenAmpere’slawgivesthecoilinducedvoltagee=dl/dtCoilonprecedingslidehasNturnsandinductance

L=l/i=N2/Rtot

whereRtot=Rc+RgPowertothecoilp=ei=idl/dtEnergystoredinmagneticfieldW=pdt7/10/2024MagneticCircuitsandMagneticMaterials7PropertiesofmagneticmaterialsFerromagneticmaterialssuchasironhaveveryhighpermeabilitym,

idealforfocusingmagneticfluxinmachineryalsohighlynonlinearifdrivenintosaturation7/10/2024MagneticCircuitsandMagneticMaterials8HysteresisinBversusHforferromagneticmaterialACoperationMagneticcoreunderACexcitation,havingsinusoidalflux,willhaveanon-sinusoidalexcitationcurrentif7/10/2024MagneticCircuitsandMagneticMaterials9Hysteresisandeddy-currentlossesTheenergyperunitvolumelostpercycleistheareaenclosedbythehysteresisloopThereisadditionalenergylostduetoeddycurrentsinducedintheironbythetime-varyingfluxdensityEddy-currentlossisminimizedbylaminatedcores,builtupfromsheetselectricalsheetsteelFiguresinthetextillustratepropertiesofM-5grainorientedsheetsteel(propertiesareobtainedbytherollingprocessesusedinthemill)7/10/2024MagneticCircuitsandMagneticMaterials10PermanentmagnetsPermanentmagnetresidualmagnetismBrandcoercivityHcareshowninthefigureBrgivesthefluxdensitywithnodemagnetizingmmfHcgivesthedemagnetizingfieldneededtoreduceBtozeroTheloadlineshownillustratestheeffectofcuttinganairgapinthemagneticcore(seeExample1.9)SecondquadranthysteresisloopforAlnico57/10/2024MagneticCircuitsandMagneticMaterials117/10/2024MagneticCircuitsandMagneticMaterials12ThenumericalsolutionforExample1.9(seetheloadlineshownonthepreviousslide):Bg=0.30TNoticethattheair-gapfluxdensityisabout?ofBrApplicationofpermanentmagnetsOperatingapermanent-magnetdevicewillsubjectittodemagnetizingforcesThefigureshowstheeffectsofthisasaminorhysteresiscurve,approximatedasarecoilline7/10/2024MagneticCircuitsandMagneticMaterials13SummaryThetheoryofmagneticcircuitsisdevelopedandusedforanalysisofwindingsonmagneticcoresThenonlinearmagneticpropertiesofironcorescauseshysteresisandeddycurrentlossunderACexcitationPermanent-magnetpropertieswereintroduced7/10/2024MagneticCircuitsandMagneticMaterials14Chapter2Transformers7/10/2024Transformers152.1IntroductionThetransformerconsistsoftwoormorecoils(orwindings)coupledbymutualmagneticfluxIftheprimarywindingisconnectedtoanalternatingvoltagesource,analternatingfluxwilllinkthesecondarywinding,inducinganalternatingsecondaryvoltageMosttransformersconsideredherehavehigh-permeabilityironcorestoincreasethedegreeofcouplingbetweenthewindings7/10/2024Transformers162.2No-loadconditionsWithsecondaryopen,thereisnoloadonthetransformerAlloftheprimarycurrentisexcitingcurrent,thatis,itgoestomagnetizethecoreandtoovercomecorelossesCurrentsandvoltagesareequivalentsinusoidalvalues7/10/2024Transformers17TransformerwithopensecondaryNo-loadphasordiagramNo-loadexcitationcurrentIe=Im+IcMagnetizingcurrentImlagsvoltageE1by90

Core-losscurrentIcisinphasewithvoltageE17/10/2024Transformers18No-loadphasordiagram2.3Effectofsecondarycurrent:idealtransformerIdealtransformermodel:Neglectlossesandmagnetizingcurrent,andassumeallthefluxinthemagneticcorelinksbothwindingsFaraday’sLawstatesthatinducedvoltageequalstimerateofchangeoffluxlinkingacoilAmpere’sLawstatesthatprimarycoilMMFisequaltothesecondarycoilMMF7/10/2024Transformers19IdealtransformerandloadFaraday’slaw:v1=e1=N1df/dtv2=e2=N2df/dtv1/v2=N1/N2Ampere’slaw:N1i1=N2i2i1/i2=N2/N1Idealtransformer:v1/v2=N1/N2andi1/i2=N2/N1Losslesssincepowerin=poweroutv1i1=v2i27/10/2024Transformers20IdealtransformerReflectedimpedanceAnimpedanceonthesecondaryofanidealtransformermaybereflectedorreferredtotheprimarybythesquareoftheturnsratio7/10/2024Transformers212.4TransformerreactancesandequivalentcircuitsWeuseacircuitthatisequivalentatitsterminaltoincludethemainnon-idealeffectsofatransformer:LeakagereactancesX1andX2torepresenteffectsofleakagefluxWindingresistancesR1andR2torepresentconductorlossMagnetizingreactanceXmtorepresenteffectsofMMFtomagnetizethecoreCore-lossresistanceRctorepresentcorelosses7/10/2024Transformers227/10/2024Transformers23StepsindevelopmentoftransformerequivalentcircuitPrimaryresistanceandleakagereactance.Magnetizingreactanceandcorelossresistance.Addingidealtransformerandsecondaryimpedance.Reflectingsecondaryimpedancetoprimaryofidealtransformer.ApproximateequivalentcircuitsExcitationbranchimpedancesarelargecomparedtootherbranchesApproximateequivalentcircuitscalledcantilevercircuitsUsetheformthatismostconvenient7/10/2024Transformers24ApproximateequivalentcircuitsAtnormalload,theexcitingcurrentmaybeneglectedForlargepowertransformers,thewindingresistancesaresmall7/10/2024Transformers25Example2.5Example2.5:Atransformersuppliedbyafeeder.NeglectexcitingcurrentReferallparameterstothehigh-voltagesideofthetransformer7/10/2024Transformers26AutotransformersAutotransformersareveryefficientforsmallturnsratios(fromabout1:2to2:1)Efficiencyisgainedbymetallicandmagneticcouplingbetweenprimaryandsecondary7/10/2024Transformers27Three-phasetransformersThreesingle-phasetransformerscanbeconnectedasathree-phasetransformerbankinfourwaysshowninFig.2.19(nextslide)windingsattheleftaretheprimariesthoseattherightarethesecondariesprimarywindinginonetransformercorrespondstothesecondarywindingdrawnparalleltoitSeeExamples2.8and2.9fortypicalcalculations7/10/2024Transformers28Three-phasetransformerconnections7/10/2024Transformers29InstrumenttransformersInstrumenttransformersincludevoltagetransformersandcurrenttransformersVoltagetransformersorpotentialtransformers(PT’s)areusedasalternatingvoltagetransducers,usuallysteppingdownhighvoltagetolowfeedingahighimpedance(e.g.,avoltmeter)Currenttransformers(CT’s)areusedasalternatingcurrenttransducers,usuallysteppingdownhighcurrenttolowfeedingalowimpedance(e.g.,anammeter)7/10/2024Transformers30Per-unitsystemQuantitiessuchasvoltageV,currentI,powerP,reactivepowerQ,apparentpowerS,andimpedanceZ,

canbeexpressedinper-unitformasfollows:7/10/2024Transformers31BaseapparentpowerSisusedasbaseforPandQ:VAbase=Vbase×Ibase

Baseimpedance:Zbase=Vbase/Ibase

Rulesforusingper-unitsystemSelectacommonVAbaseforthesystemSelectabasevoltageatapointinthesystemChooseallotherbasevoltagesinthesameratioastheturnsratioofanytransformerencounteredConvertallquantitiestoperunitonthechosenbasesAnalyzethecircuitinperunitConvertallquantitiestovolts,amperes,etc.bymultiplyingtheirper-unitvaluesbytheircorrespondingbasevalues7/10/2024Transformers32RemarksThisprocedurewillnormallyremovetheidealtransformersintheper-unitcircuitInmostsingletransformerproblems,thebaseVAandvoltagesarechosentobetheratedvalues,whichwillsatisfytherulesonthepreviousslidePowertransformerimpedancesareusuallygiveninpercent,whichis100%timestheper-unitvalueStudyExamples2.12and2.13forillustrationoftheuseofper-unitequations7/10/2024Transformers33SummaryThetransformerisacomponentofacsystemswhereitisusedtotransformvoltages,currents,andimpedancestoappropriatelevelsforoptimaluseTransformersserveasvaluableexamplesoftheanalysistechniquesusedlaterTheyillustratepropertiesofmagneticcircuits,suchasmmf,fluxandexcitingcurrent7/10/2024Transformers34Chapter3

Electromechanical-Energy-ConversionPrinciples7/10/2024ElectromechanicalEnergyConversion35IntroductionConversionofenergyfromelectricaltomechanicalformorfrommechanicaltoelectricalformthroughthemediumofelectricormagneticfieldsDevicesincludeTransducers(sensorsandspeakersForce-producingdevices(actuators)Continuousenergyconverters(motorsandgenerators)7/10/2024ElectromechanicalEnergyConversion363.1Forceandtorque

inmagnetic-fieldsystemsLorentzforcelawgivestheforceF[N]onachargeq[C]movingwithvelocityv[m/s]throughanelectricfieldE[V/m]andamagneticfluxdensityB[T]F=q(E+v×B)7/10/2024ElectromechanicalEnergyConversion37Right-handrulefordirectionofforceduetomagneticfield3.2EnergybalancemethodThismethodusesasimplerphysicalpicturebasedonenergyconservationDoesnotrequireddetailedfielddistributionsFocusesonthemostimportantforcesortorquesCanbeappliedeasilytopracticaldevicesEnergybalanceforamagnetic-fielddevice7/10/2024ElectromechanicalEnergyConversion38Signconventionisfor:electricalenergyinputandmechanicalenergyoutputInamotor,signswillbepositiveInagenerator,signswillbenegative7/10/2024ElectromechanicalEnergyConversion39SimpleenergyconversiondeviceEnergybalanceforadevice7/10/2024ElectromechanicalEnergyConversion403.3Energyinsinglyexcitedmagnetic-fieldsystemsAmagneticcircuitwithacoil,anairgap,andamovablesectionofthecoreThedeviceisanelectromagneticrelay7/10/2024ElectromechanicalEnergyConversion417/10/2024ElectromechanicalEnergyConversion42IntegrationpathsforWfldConservativesystemhasenergythatdoesnotdependontheparticularpathWeusepath2forevaluatingWfld7/10/2024ElectromechanicalEnergyConversion437/10/2024ElectromechanicalEnergyConversion443.4Determinationofmagneticforceandtorquefromenergy7/10/2024ElectromechanicalEnergyConversion45Forceortorque?Inatranslationalsystem,useforcefandpositionxInananalogousrotatingsystem,useTandangularpositionqTorqueisforcetimeradiusarm,tendingturnarotoraboutanaxis7/10/2024ElectromechanicalEnergyConversion463.5DeterminationofmagneticforcefromcoenergyAstatefunctionrelatedtoenergyiscalledcoenergyW’7/10/2024ElectromechanicalEnergyConversion47RemarksCoenergyisoftenconvenientsincecurrentisamoreeasilymeasuredthanfluxlinkageEitherformulation(energyorcoenergy)willgivethecorrectresults,ifusedcorrectlySeefigures3.10and3.11inthetexttoverifythattheygiveidenticalresults7/10/2024ElectromechanicalEnergyConversion483.6Multiplyexcitedmagnetic-fieldsystemsSystemswithmorethanonecoilarecalledmultiplyexcitedConsideradoublyexcitedrotatingsystem(twocoilsandtwoelectricalsources)butonerotatingshaft7/10/2024ElectromechanicalEnergyConversion497/10/2024ElectromechanicalEnergyConversion50Rememberthatpartialderivativesaretakenwithrespecttoonevariableasifallothervariableswereconstant7/10/2024ElectromechanicalEnergyConversion51IntegrationpathtoobtainfieldenergyforadoublyexcitedmagneticfieldsystemLinearmagneticcoreSpecialcaseoflinearmagnetics7/10/2024ElectromechanicalEnergyConversion52SystemswithpermanentmagnetsPermanent-magnetsystemswillneedspecialconsideration,sincetheenergyinthemagneticfieldwillbezeroonlyifthemagnetisdemagnetizedWemodelthisbyafictitiouswindingthathasacurrenttocounteractthefluxofmagnetforthepurposeoffindinganenergyexpression7/10/2024ElectromechanicalEnergyConversion537/10/2024ElectromechanicalEnergyConversion54Thefictitiouscurrentif=0atnormalconditions,butanonzerovalueonpath1atogivenomagneticforceonthatpath(theintegraliszeroonpath1a)Forsystemswithbothpermanentmagnetsandwindings:Useafictitiouscoiltoreplacethemagnet:(Ni)equiv=-H’cd7/10/2024ElectromechanicalEnergyConversion55Analysisnowproceedsasifthemagneticcircuithadtwocoils.Seeexample3.9fordetailsDynamicequations7/10/2024ElectromechanicalEnergyConversion56TheelectromagneticforceffldandtheinductanceL(x)arecalculatedbythemethodsdevelopedearlierTheequationsarenon-linearanddifficulttosolveSection3.9.1discussessimulationwithMATLAB/SimulinkSection3.9.2discusseslinearization(whichisequivalenttosmall-signalanalysis7/10/2024ElectromechanicalEnergyConversion57SummaryEnergyinanelectromechanicalsystemreliesoncouplingmagneticorelectricfieldWehavelookedatsingly-anddoubly-excitedmagneticfieldsystems,withtranslationalmotionorrotationinonedimensionEnergy-basedmethodssimplifytheanalysisandallowcalculationofforceortorquebydifferentiatingenergyorcoenergywithrespecttothedisplacement7/10/2024ElectromechanicalEnergyConversion58Chapter4IntroductiontoRotatingMachines7/10/2024IntroductiontoRotatingMachines594.1ElementaryConceptsRotatingmachines:voltagesareinducedinwindingsorgroupsofcoilsbyrotationofamagneticfieldpastawindingorrotationofawindingthroughthefield,orbydesigningthemagneticcircuitsothatthereluctancevarieswithrotationoftherotorSincethefluxlinkingacoilchangescyclically,atime-varyingvoltageisinducede=dl/dtAgroupsuchcoilscarryingACcurrentsisoftencalledanarmaturewinding7/10/2024IntroductiontoRotatingMachines607/10/2024IntroductiontoRotatingMachines61Statorofa100-MVAthree-phasesynchronousgeneratorunderconstruction.(GeneralElectricCompany.)Armatureofadcmotor.(BaldorElectric/ABB)InanACsynchronousmachine,thearmatureistypicallyonthestatorInaDCmachine,thearmatureislocatedontherotorDCandsynchronousmachinestypicallyhavefieldwindingscarryingDCtosetupthemainoperatingflux,usuallylocatedonthestatorforDCmachinestherotorofACsynchronousmachinesSomemachines,especiallymotors,usemagnetsinsteadoffieldwindingsInductionmachinesdonothavefields,butproducefluxsimilarlytotransformersManytypesofmachinesexist,butverysimilarphysicalprinciplesgoverntheirperformance7/10/2024IntroductiontoRotatingMachines624.2IntroductiontoACandDCmachinesTraditionalacmachinesareclassifiedassynchronousorinductionmachinesSynchronousmachines:rotorcurrentsaresupplieddirectlyfromthestationaryframe,througharotatingcontactforexampleInductionmachines:rotorcurrentsareinducedintherotorwindingsbymagneticinductionfromthestatorwindings7/10/2024IntroductiontoRotatingMachines63Fieldhasasinglepairofpoles,soitisatwo-polemachineArmatureherehasasinglecoilofNturnsFieldisexcitedthroughbrushescontactingsliprings,orbybrushlessexcitationsystemIftheair-gapfluxissinusoidalinspace,theinducedvoltageinthearmatureissinusoidalintime,asthemachinerotatesatconstantspeed7/10/2024IntroductiontoRotatingMachines64Schematicviewofasimple,two-pole,single-phasesynchronousgeneratorManymachineshavemorethantwopoles.Afour-polesynchronousmachine,whichwillrotateathalfthespeedofatwo-polemachineifthefrequencyisthesame7/10/2024IntroductiontoRotatingMachines65IdealizedfluxdistributionandwaveformofgeneratedvoltageFour-polesingle-phasesynchronousgeneratorForconvenienceinanalyzingmachineswithmorethantwopoles,defineelectricalangleandelectricalspeed,asfollows:7/10/2024IntroductiontoRotatingMachines66SubscripteindicateselectricalunitswhilemindicatesmechanicaloractualunitsThisisusefulsincetherearepoles/2completewavelengthsorcyclesinone(mechanical)revolutionInductionMachinesStatorwindingsareessentiallythesameasasynchronousmachineRotorwindingiselectricallyshort-circuitedandoftenhasnoexternalconnections,derivingitsexcitationbymagneticinductionAlsocalledasynchronousmachinesCommonconstructionforaninductionmotorusesthesquirrel-cagerotorwithnoexternalconnectionSquirrel-cageinductionmotorsarethemostcommontypeofmotorusedtoday7/10/2024IntroductiontoRotatingMachines67CagerotorhasbarsthatareshortedbyendringsInexpensivetoconstructandyetveryruggedRotorcurrentsareinducedastherotorslipspastthestatorfluxwave,whichrotatesatsynchronousspeedFluxwavesetupbytherotorcurrentsrotatesatsynchronousspeed,andinteractswiththestatorfluxtoproducetorqueThismachineisverysimilartoatransformer,butwithrotationofwindings7/10/2024IntroductiontoRotatingMachines68Cutawayviewofa460-V,7.5hpsquirrel-cageinductionmotor.DCMachinesAsimplifieddcgeneratorarmaturewinding(asinglecoilofNturns)isshownThecommutatorisacylindricalstructurewithtwosegmentsattachedtotherotor,servingasamechanicalrectifiertoconverttheacinthearmaturecoiltodcatthestationarybrushes7/10/2024IntroductiontoRotatingMachines69ElementarydcmachineDCinthefieldsetsupastationaryfluxThecommutatorcausesarmaturefluxtobefixedinspacebetweenthefieldpolesInteractionoffluxessetsuptorque7/10/2024IntroductiontoRotatingMachines70Air-gapfluxdistributionandvoltagewaveform4.3MMFofDistributedWindingsPracticalarmaturewindingsareusuallydistributed,orspreadoveranumberofslotsConsideronephaseofanacthree-phasetwo-polewinding(calledafull-pitchwindingsinceeachcoilspanspradians)FourieranalysisgivesthespacefundamentalcomponentoftheMMF,developedinAppendixB7/10/2024IntroductiontoRotatingMachines717/10/2024IntroductiontoRotatingMachines72Themmfofonephaseofadistributedtwo-pole,three-phasewindingwithfull-pitchcoils.7/10/2024IntroductiontoRotatingMachines73Thepeakvalueofthespacefundamentalisgiveninthefollowingequation,wherekwisthewindingfactorthataccountsforthedistributionofthewinding(seeAppendixBfordetails)ThefactorkwNphistheeffectivenumberofseriesturnsperphaseTypicalvaluesforkware0.85to0.90ConsiderthedcmachinewithanarmaturewindingdistributedovermanyslotsAnapproximationtothemmfisasawtoothwave7/10/2024IntroductiontoRotatingMachines74Crosssectionofatwo-poledcmachine7/10/2024IntroductiontoRotatingMachines75Currentandmmfwaveofidealizeddcmachine4.4MagneticFieldsinRotatingMachineryMachinewithauniformairgapandasinglefull-pitchN-turncoilonahighlypermeableironcore7/10/2024IntroductiontoRotatingMachines76Diagramofmachine7/10/2024IntroductiontoRotatingMachines77MMFandfielddistributionsSpacefundamentalfieldpeakvalue:4.5RotatingMMFWavesSingle-phasewindingproducesapulsatingMMFthatcanberesolvedintotwoequalrotatingwaves,rotatinginoppositedirectionsPolyphasewindingproducesarotatingMMFthathasconstantamplitudeandconstantspeedinsteadystateThefigureonthenextslideshowsagraphicalexplanationwhilethetextgivesamathematicalderivationforthethree-phasecase7/10/2024IntroductiontoRotatingMachines787/10/2024IntroductiontoRotatingMachines79Theproductionofarotatingmagneticfieldbymeansofthree-phasecurrentsFistheresultantofvectoradditionofFa+Fb+Fc4.6GeneratedVoltageFluxdensityisnearlysinuosoidalinspacePhase-afluxlinkage:7/10/2024IntroductiontoRotatingMachines80Generatedvoltage(constantfluxinnormalsteadystate):DCMachinesCommutatoractsasarectifier,givingaveragevoltage:7/10/2024IntroductiontoRotatingMachines814.7TorqueinNon-Salient-PoleMachinesTorquecanbefoundfromeitheracoupled-circuitpointofview,orfromamagnetic-fieldpointofviewRotorcurrentirandstatorcurrentiswithanglebetweenthemagneticaxesCoupledcircuit:T=-(poles/2)Lsris

irsinqmeMagneticfield:T

-(poles/2)Fs

FrsindsrWheredsrisanglebetweenstatorandrotorMMF’s7/10/2024IntroductiontoRotatingMachines82SummaryThephysicalprocessesproductionoftorqueandgeneratedvoltageinrotatingmachinesarequitesimilar,althoughthedetailsofmachineconstructionanddetailsofanalysisvaryTorqueisproducedbyinteractionsofthemagneticfieldsofstatorandrotorVoltagesaregeneratedbyrelativemotionofamagneticfieldwithrespecttoawinding7/10/2024IntroductiontoRotatingMachines83Chapter5SynchronousMachines7/10/2024SynchronousMachines84IntroductionTheDCfieldisnormallyontherotorandACarmaturewindingisonthestatorDCfieldcurrentsetsupmainair-gapfluxandACarmaturehandlestheelectricalpowerRotorrotatesinsynchronismwiththemagneticfieldsetupbythearmaturewindingsRotorspeedisproportionaltothefrequencyofthearmaturecurrentsinsteadystate7/10/2024SynchronousMachines85Stator-rotormutualinductanceis

Laf

cos(qme)Insteadystate

qme=wet+de0Open-circuitarmaturevoltageis 7/10/2024SynchronousMachines86Simplifiedthree-phasesynchronousmachineStatormutualinductanceis Lab=Laa0cos(120?)=-?Laa0Statorselfinductanceis La=Laa0+Lal

Balancedthree-phasecurrentsia+ib+ic=0 7/10/2024SynchronousMachines877/10/2024SynchronousMachines88Motorsignconventionsimplymeansthereferencedirectionsarechosenfornormalmotoroperation7/10/2024SynchronousMachines89Synchronous-machineequivalentcircuits:(a)motorreferencedirectionand(b)generatorreferencedirection.7/10/2024SynchronousMachines90IaVa(Ra+jXs)IaEafIaEaf(Ra+jXs)IaVaSynchronousmotorandgeneratorphasordiagramsbothhavinglaggingpowerfactorattheterminals(IalaggingVa)MotorGenerator5.3Open-andShort-CircuitCharacteristicsTheopen-circuitarmaturevoltageandshort-circuitarmaturecurrent,bothplottedversusthefieldcurrent,allowcomputationofthesynchronousreactanceXsOpen-circuitcharacteristicwillshowsignificantsaturationandisoftencalledopen-circuitsaturationcurveShort-circuitcharacteristic(plotteduptoslightlyaboveratedcurrent)willnot,sincetheexcitationislowtheseconditions7/10/2024SynchronousMachines917/10/2024SynchronousMachines92Open-andshort-circuitcharacteristicsshowingequivalentmagnetizationlineforsaturatedoperatingconditions7/10/2024SynchronousMachines93SaturatedandunsaturatedvaluesofXs:SaturatedandunsaturatedvaluesofXsinperunit:Inlargesynchronousmachines,thearmatureresistanceismuchsmallerthanthesynchronousreactance.5.4Steady-StatePower-AngleCharacteristic7/10/2024SynchronousMachines94(a)Impedanceinterconnectingtwovoltages(b)phasordiagramMaximumpowerdeliveredbyasynchronousmachineisaspecialcaseofpower-anglecharacteristicforpowertransferthroughaseriesimpedanceZ=R+jX,fromE1toE2ForsimplicityZjX7/10/2024SynchronousMachines955.5Steady-StateOperatingCharacteristics7/10/2024SynchronousMachines96Constructionusedforderivationofasynchronousgeneratorcapabilitycurve.7/10/2024SynchronousMachines97ThisequationgivesthefieldheatinglimitThearmatureheatinglimitissimplythemachinevolt-ampererating(ratedcurrentatratedterminalvoltage)7/10/2024SynchronousMachines98Phasordiagramforconstant-poweroperationatconstantterminalvoltage.NotethatlesslaggingormoreleadingcorrespondstosmallerexcitationvoltagemagnitudeEafandsmallerfieldcurrentIfIa3VajXsIa2Eaf1Ia2Ia1Eaf3Eaf2jXsIa3jXsIa17/10/2024SynchronousMachines99SynchronousGeneratorVCurves5.6EffectsofSalientPolesIntroductiontosynchronouslyrotatingdqaxistheoryResolvearmaturephasorsintodandqaxiscomponentsDirect(d)axisisalignedwithfieldpoles,quadrature(q)axisismidwaybetweenpoles7/10/2024SynchronousMachines1007/10/2024SynchronousMachines101Inductanceisfluxlinkageperunitampere:Thed-axiscurrentproducesMMFthatseesashortairgap(acrosspolefaces)sothed-axisinductanceislargeTheq-axiscurrentproducesMMFthatseesalongairgap(interpolarregion)sotheq-axisinductanceisrelativelysmallSalient-polemachinesareanalyzedwithaphasordiagramrepresentingd-andq-axisquantitiesseparately:IdseesXdandIqseesXqwhereXd>XqNosimpleequivalentcircuitisavailable7/10/2024SynchronousMachines1027