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ReferenceFrameTheory

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ReferenceFrameTheory DevelopmentandApplications

PaulC.Krause

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LibraryofCongressCataloging-in-PublicationData

Names:Krause,PaulC.,author.

Title:Referenceframetheory:developmentandapplications/PaulC. Krause,PCKrauseandAssociates.

Description:Firstedition.|Hoboken,NewJersey:Wiley-IEEEPress, [2020]|Includesbibliographicalreferenceandindex.

Subjects:LCSH:Relativity(Physics)

Classification:LCCQC173.5.K732020(print)|LCCQC173.5(ebook)|DDC 530.11–dc23

LCrecordavailableathttps://lccn.loc.gov/2020029315

LCebookrecordavailableathttps://lccn.loc.gov/2020029316

Setin9.5/12.5ptSTIXTwoTextbySPiGlobal,Chennai,India

PrintedintheUnitedStatesofAmerica. 10987654321

Contents

AbouttheAuthor xv

Preface xvii

1ABriefHistoryofReferenceFrameTheory 1 References 3

2Tesla’sRotatingMagneticField 5

2.1Introduction 5

2.2RotatingMagneticFieldforSymmetricalTwo-PhaseStator Windings 5

2.3RotatingMagneticFieldforSymmetricalThree-PhaseStator Windings 11

2.4RotatingMagneticFieldforSymmetricalTwo-PhaseRotor Windings 13

2.5RotatingMagneticFieldforSymmetricalThree-PhaseRotor Windings 15

2.6ClosingComments 17 References 17

3Tesla’sRotatingMagneticFieldandReferenceFrame Theory 19

3.1Introduction 19

3.2TransformationofTwo-PhaseSymmetricalStatorVariablestothe ArbitraryReferenceFrame 20

3.3TransformationofTwo-PhaseSymmetricalRotorVariablestothe ArbitraryReferenceFrame 24

3.4TransformationofThree-PhaseStatorandRotorVariablestothe ArbitraryReferenceFrame 26

3.5BalancedSteady-StateStatorVariablesViewedfromAnyReference Frame 31

3.6ClosingComments 35 References 35

4EquivalentCircuitsfortheSymmetricalMachine 37

4.1Introduction 37

4.2Flux-LinkageEquationsforaMagneticallyLinearTwo-Phase SymmetricalMachine 37

4.3Flux-LinkageEquationsintheArbitraryReferenceFrame 39

4.4TorqueExpressioninArbitraryReferenceFrame 41

4.5InstantaneousandSteady-StatePhasors 42

4.6Flux-LinkageEquationsforaMagneticallyLinearThree-Phase SymmetricalMachineandEquivalentCircuit 45

4.7ClosingComments 49 References 50

5SynchronousMachines 51

5.1Introduction 51

5.2SynchronousMachine 51

5.3EquivalentCircuitForThree-PhaseSynchronous Generator 53

5.4ClosingComment 57 Reference 57

6BrushlessdcDrivewithFieldOrientation 59

6.1Introduction 59

6.2ThePermanent-MagnetacMachine 59

6.3InstantaneousandSteady-StatePhasors 62

6.4FieldOrientationofaBrushlessdcDrive 65

6.5TorqueControlofaBrushlessdcDrive 75

6.6ClosingComments 78 References 79

7FieldOrientationofInductionMachineDrives 81

7.1Introduction 81

7.2FieldOrientationofaSymmetricalMachine 81

7.3TorqueControlofField-OrientatedSymmetrical Machine 86

7.4ClosingComments 89 References 89

8AdditionalApplicationsofReferenceFrameTheory 91

8.1Introduction 91

8.2NeglectingStatorTransients 91

8.3SymmetricalComponentsDerivedbyReferenceFrameTheory 93

8.4MultipleReferenceFrames 97

8.5ClosingComments 97 References 97

Index 99

AbouttheAuthor

PaulC.Krause receivedaBSEEin1956,aBSMEin1957,andaMSEEin1958 fromtheUniversityofNebraska.HereceivedaPhDinEEin1961fromKansas University.Hetaughtatseveralcollegesfor52yearsretiringin2009,after39 yearsasafullprofessor,withPurdueUniversity.Heisthe2010recipientofthe IEEENikolaTeslaAward.Hehaspublishedover100technicalpapersandfour textbooksinElectricMachinesandDriveswithtwointheirthirdedition.

Preface

ThechangeofvariableswasintroducedtomeatthebeginningofthePhD program.Atthattime,itwasamysteryastowherethetransformationoriginated, anditremainedamysteryfornearly60years.AlthoughtherehasbeenconsiderableworkintheareasinceParkwrotehis1929paper,noonetomyknowledge hassetforththebasisofthetransformation.Withtheadventofthecomputerand powerelectronics,referenceframetheoryhasbecomenecessaryintheteaching ofmachinesanddrives.However,theconceptofreferenceframetheorywas difficulttoteachsincethetransformationseemedtoappearfromoutoftheblue. Onehadtomoreorlessacceptthatitworked.

In2016,duringthewritingof“IntroductiontoPowerandDriveSystems,”the connectionbetweenTesla’srotatingmagneticfieldandreferenceframetheorywas uncovered,anditthenbecameclearthatthechangeofvariablesmadethesubstitutevariablesportraythecorrectviewofTesla’srotatingmagneticfieldfrom agivenreferenceframe.Thisgavemeaningtothetransformation,anditisnow easiertounderstand.Thisinformationwaslaterpublishedinthethirdeditionof “ElectromechanicalMotionDevices”andanIEEEPaper.

Thisbookcoverssomeoftheaspectsofreferenceframetheorythattheauthor hasbeeninvolvedwithduringthepast60years,fromthearbitraryreferenceframe tothebasisofreferenceframetheoryandfieldorientation.Otherworkincludingneglectingstatortransients,multiplereferenceframes,andtherelationofthe transformationtosymmetricalcomponentsarealsodiscussed.

ItisinterestingthatreferenceframetheoryiscontainedinTesla’srotatingmagneticfield,whichisthebasisofallknownrealtransformations.Moreover,since thearbitraryreferenceframecanbeusedtoderivesymmetricalcomponents, Tesla’srotatingmagneticfield,althoughnotasdirect,canalsobeconsidered thebasisforcomplextransformationssuchasthesymmetricalcomponent transformation.Thus,itappearsthatalltransformations,realandcomplex,used inthepoweranddrivesareascanbetracedbacktoTesla’srotatingmagnetic field.

Preface

Thisbookiswrittenfortheengineersworkinginthepoweranddrivesareas asareferenceandforthegraduatestudentswhowanttoknowmoreaboutthe historyandbasisofreferenceframetheory.Itcanbeusedasatextbook,andsome instructorsmaywanttheirclasstobecomemoreinformedregardingreference frametheoryandrequireitasareferencetext.

PaulKrause

ABriefHistoryofReferenceFrameTheory

Inthelate1920s,R.H.Park,ayoungMITgraduateworkingforGE,wroteapaper onanewmethodofanalyzingasynchronousmachine[1].Heformulatedachange ofvariablesthat,ineffect,replacedthevariables(voltages,currents,andfluxlinkages)associatedwiththestatorwindingsofasynchronousmachinewithvariables associatedwithfictitious,sinusoidallydistributedwindingsrotatingattheelectricalangularvelocityoftherotor.Thischangeofvariables,whicheliminated theposition-varyinginductancesfromthevoltageequations,isoftendescribed astransformingorreferringthestatorvariablestotherotorreferenceframe. Althoughhedidnotrefertothisas“referenceframetheory,”itwasitsbeginning.

Thisnewapproachtomachineanalysisfoundlimiteduseuntiltheadventof thecomputer.PerhapsadiscussionwrittenbyC.H.Thomasinthelate1950s wasthefirsttosetforthamethodofusingParksequationstoestablishastable computersimulationofasynchronousmachinewhichisstillbeingusedtoday [2].Thisopenedthedoortotheuseofachangeofvariablestoanalyzeproblems involvingelectricmachinessincethissamemethodofsimulationisusedtodayfor thesimulationofallsynchronousandinduction-typemachines.

Therehavebeennumerouschangesofvariablesthathavebeensetforthafter Park’swork.Inthelate1930s,C.H.Stanley[3]employedachangeofvariablesin theanalysisofinductionmachines.Heshowedthattherotor-position-dependent inductancesinthevoltageequationsofaninductionmachine,whicharedue toelectriccircuitsinrelativemotion,couldbeeliminatedbyreplacingthe rotorvariableswithsubstitutevariablesassociatedwithsinusoidallydistributed stationarywindings.Thisisoftendescribedastransformingorreferringtherotor variablestoaframeofreferencefixedinthestatororthestationaryreference frame.Aboutthesametime,E.Clarke[4]setforthanalgebraictransformation forthree-phasestationarycircuitstofacilitatetheirsteady-stateandtransient analysesofthree-phaseacpowersystems.Shereferredtothesesubstitute variablesasalpha,beta,andzerocomponents.

In[5],G.Kronintroducedachangeofvariablesthateliminatedthe rotor-position-dependentinductancesofasymmetricalinductionmachine bytransformingboththestatorandtherotorvariablestoareferenceframerotatinginsynchronismwiththefundamentalelectricalangularvelocityofthestator variables.Thisreferenceframeiscommonlyreferredtoasthesynchronously rotatingreferenceframe.

D.S.Breretonetal.[6]employedachangeofvariablesthatalsoeliminatedthe rotor-position-varyinginductancesofasymmetricalinductionmachine.Thiswas accomplishedbytransformingthestatorvariablestoareferenceframerotatingat theelectricalangularvelocityoftherotor.

Park,Stanley,Kron,andBreretonetal.developedchangesofvariableseachof whichappearedtobeunique.Consequently,eachtransformationwasderivedand treatedseparatelyintheliteratureuntilitwasnotedin1965[7]thatallknownreal transformationsusedinmachineanalysiswerecontainedinonetransformation. TheArbitraryReferenceFramewasintroducedin[7]asageneralreferenceframe thatcontainedallknowntransformationssimplybyassigningthespeedofthereferenceframe.Forexample,when ��,thespeedofthe q and d axes,issetequalto zero,wehaveStanley’sandClarke’stransformations;with �� = ��r ,wehavePark’s andBrereton’stransformations;andwhen �� = ��e ,wehaveKron’s.Althoughthis wasaninterestingobservation,theconnectiontoTesla’srotatingmagneticfield wasnotmade.Althoughitshouldhavebeen,sincemovingfromonereference frametoanotherchangesonlythefrequencythatweobserveTesla’srotatingmagneticfield.

Inarecentpaper[8],theconnectionbetweenTesla’srotatingmagneticfield andthearbitraryreferenceframewassetforth.ItwasshownthatthetransformationtothearbitraryreferenceframewascontainedinTesla’sexpressionforthe rotatingmagneticfield.Moreover,oncethesymmetricalstatorandrotoraretransformedtothearbitraryreferenceframe,wehavethe q and d voltageequations forallmachines.Theonlythingthatmustbetransformedaretheflux-linkage equationsforthemachinebeingconsidered[9].

Upuntilthewritingof[10],thetransformationsweregivenwithoutanyexplanationastothebasisofthetransformation.Itwasacceptedwithoutquestion. Althoughitwaspossibletoobtainthetransformationbyreferringthe abc axes toa qd-axis,therewasnotananalyticalbasisforthetransformation.Thisplagued machineanalystsfornearlyahundredyears.

Duringthewritingof[10]itwasfoundthattheequationforTesla’srotating magneticfieldcontainedthebasiswehadallbeentryingtofindsincePark’swork. Thisformsthemachineanalysisin[10]andwasexplainedin[9].Thisapproach tomachineanalysisisthesubjectofthenexttwochapters.InChapter2,werefer Tesla’srotatingmagneticfieldtoarotatingaxis.InChapter3,weestablishthe connectionbetweenTesla’srotatingmagneticfieldandreferenceframetheory.

References

1 Park,R.H.(1929).Two-reactiontheoryofsynchronousmachines–generalized methodofanalysis–partI. AIEETrans. 48:716–727.

2 Riaz,M.(1956).Analoguecomputerrepresentationsofsynchronousgenerators involtage-regulationstudies. Trans.AIEEPowerApp.Syst. 75:1178–1184.See discussionbyC.H.Thomas.

3 Stanley,C.H.(1938).Ananalysisoftheinductionmotor. AIEETrans. 57(Supplement):751–755.

4 Clarke,E.(1943). CircuitAnalysisofA-CPowerSystems,Vol.1–Symmetrical andRelatedComponents.NewYork:Wiley.

5 Kron,G.(1951). EquivalentCircuitsofElectricMachinery.NewYork:Wiley.

6 Brereton,D.S.,Lewis,D.G.,andYoung,C.G.(1957).Representationofinductionmotorloadsduringpowersystemstabilitystudies. AIEETrans. 76: 451–461.

7 Krause,P.C.andThomas,C.H.(1965).Simulationofsymmetricalinduction machinery. IEEETrans.PowerApp.Syst. 84:1038–1053.

8 Krause,P.C.,Wasynczuk,O.,O’Connell,T.C.,andHasan,M.(2018).Tesla’s contributiontoelectricmachineanalysis.Presentedatthe2018Summer MeetingofIEEE(5-9August2018),Portland,OR.

9 Krause,P.C.,Wasynczuk,O.,Pekarek,S.D.,andO’Connell,T.C.(2020). ElectromechanicalMotionDevices,3e.Wiley,IEEEPress.

10 Krause,P.C.,Wasynczuk,O.,O’Connell,T.C.,andHasan,M.(2017). IntroductiontoPowerandDriveSystems.NewYork:Wiley,IEEEPress.

Tesla’sRotatingMagneticField

2.1Introduction

ItistoldthatwhenTeslawasastudenthearguedwithhisprofessorthatthere mustbeabetterwaytodesignanelectricmachinethanthedcmachinebeing demonstratedinclass.Nearlyadecadelater,Teslainventedtheinductionmachine whereheemployedtherotatingmagneticfield[1].Therotatingmagneticfield hasbeenreferredtoastherotatingair-gapmmf,rotatingmagneticpoles,andthe rotatingfield.NotonlyisTesla’srotatingmagneticfieldthebackboneoftheac machineoperation,itisalsothekeytotheanalysisofacmachines.Theimportance ofviewingTesla’srotatingmagneticfieldfromanyframeofreferencecannotbe overemphasized.Weshowthatthetransformtoareferenceframeestablishesthe substitutevariablesassociatedwiththefictitiouscircuitstoportrayTesla’srotatingmagneticfieldasviewedfromthatreferenceframe[2,3].Understandingthis connectionisveryhelpfultomachineandelectricdriveanalysis.Thischapteris devotedtoestablishingtheconceptofviewingTesla’srotatingmagneticfieldin thearbitraryreferenceframe.

2.2RotatingMagneticFieldforSymmetrical Two-PhaseStatorWindings

AnidealizedsinusoidallydistributedwindingisshowninFigure2.2-1.Eachcircle, ⊗ or ⊙,has ncs coils.Forthepositivecurrent ias thewindingdistributionfrom 0 <��s < 2�� maybeapproximatedas

Sinusoidal approximate

Figure2.2-1 Sinusoidallydistributedstatorwinding.(a)Approximatesinusoidal distribution.(b)Developeddiagramof(a).(c)Sinusoidalapproximationof(b).

where Np isthepeakturnsdensityinturns/radian.If Ns representsthenumberof turnsoftheequivalentsinusoidallydistributedwinding,then,using �� asadummy variableofintegration

= ∫ �� 0 Np sin �� d�� = 2Np (2.2-3)

ThedevelopeddiagramisthelinearversionofFigure2.2-1a,and ��s ispositive fromrighttoleft.Thisdiagramisvalidsincetheair-gaplengthisverysmallcomparedtotheradiusoftherotor.Also,the ⊗ and ⊙ arethenotationthatweuse fromhereonforasinusoidallydistributedwinding.

WecanobtainaplotofthemmfbyapplyingAmperesLaw, ∮ H ⋅ dL = in ,as showninFigure2.2-2,whereonehalfofthemmfisdroppedacrosseachequal

2.2RotatingMagneticFieldforSymmetricalTwo-PhaseStatorWindings

Figure2.2-2 Closedpathsofintegrationandtheplotof mmf as . airgap.Figure2.2-3isaplotofthesinusoidalapproximationofmmf as shownin Figure2.2-2g.

Asetofsymmetrical,two-polestatorwindingsisshowninFigure2.2-4.The windingsareidenticalandsinusoidallydistributed,andtheairgapisuniform. Currententersthepaperat ⊗ andoutofthepaperat ⊙.Therotorwindingsare notshown.

Themmfforeachwindingmaybeexpressedas mmf as = Ns 2 ias cos ��s (2.2-4)

as-axis

as=ias cos ϕs

Figure2.2-3 Plotofsinusoidalapproximationof mmf as showninFigure2.2-2g.

bs-axis

Figure2.2-4 Elementarysymmetrical, two-poletwo-phasesinusoidallydistributed statorwindings. mmf bs = Ns 2 ibs sin ��s

axis

Thetotalair-gapmmfduetothetwo-polestatorwindings,whichisTesla’srotatingmagneticfieldwouldbe

mmf s = mmf as + mmf bs = Ns 2 (ias cos ��s + ibs sin ��s )

Thisisaveryimportantrelationship.Letthesteady-statecurrentsbeabalanced two-phaseset

Ibs = √2Is sin[��e t + ��esi (0)]

where Is isthermsvalueofthephasecurrent, ��e istheelectricalangularvelocityin rad/s,and ��esi (0) isthephaseangleofthecurrents.Forthisbalancedset, Ibs =−jIas . Ifwesubstitute(2.2-7)and(2.2-8)into(2.2-6),weseethatforthisbalancedset therotatingmagneticfieldtravelscounterclockwiseat ��e .Letuslookattherotatingmagneticfieldfromarotatingaxis.Todothisweintroducethe q-axis,which rotatesatanangularvelocity ��.Ifwelet �� betheangularpositionfromthe as-axis,

2.2RotatingMagneticFieldforSymmetricalTwo-PhaseStatorWindings 9

Figure2.2-5 Symmetrical,two-pole two-phasesinusoidallydistributedstator windingswithathirdmagneticaxis(q-axis).

Thedisplacementorcoordinate �� showninFigure2.2-5isfromthe q-axisjust as ��s isthedisplacementorcoordinatefromthe as-axis.Letusrelatethisdisplacementtoanadjacentdisplacementonthestator.FromFigure2.2-5

Ifwesubstitute(2.2-10)into(2.2-6),weobtain

TheconceptofTesla’srotatingfieldcanbeexplainedbyassumingthatthe steady-statephasecurrentsare(2.2-7)and(2.2-8)with �� esi (0) = 0.Attimezero, I as ismaximumand I bs iszero.Theair-gapmmfisorientedalongthe as-axis. Whenthetimehasadvancedtowhere ��e t = �� 2 , I as = 0,and I bs ismaximum,the air-gapmmfhasrotatedfromthe as-axistothe bs-axiswithanangularvelocityof ��e .Thatis,themmfhasrotatedcounterclockwiseat ��e relativetoanobserveron thestator.

Equation(2.2-11)iscornerstonetoreferenceframetheorywhichweconsiderin Chapter3;however,letusfirstdealwithsteady-stateconditions.Ifwesubstitute balancesteady-statecurrentsgivenby(2.2-7)and(2.2-8)into(2.2-11)andafter somework

Now, ��e istheangularfrequencyoftheelectricalsystem,and �� istheangular velocityofthe q-axis,where �� isthedisplacementfromthe q-axis,and �� (0)isthe timezeropositionofthe q-axis(2.2-9)whichwehavelettobezeroin(2.2-12).We continuetoset �� (0)equaltozerounlessotherwisespecified.

Ifnowwelettheangularvelocityofthe q-axistobezero(�� = 0)and �� (0) = 0, then �� = ��s ,andwewouldbeviewingTesla’srotatingmagneticfieldasastationaryobserver.Inthiscase,(2.2-12)becomes

wherewehaveaddedasuperscript s toemphasizethatweareobservingTesla’s rotatingmagneticfieldfromastationaryframeofreference.If ��esi (0) and ��s are constants,mmf s s isviewedasasinusoidalvariationoffrequency ��e .If,forexample, ��s iszero,wewouldbepositioned(fixed)atthe as-axis(��s = 0inFigure2.2-5), andthemmf s s wouldbepulsatingat ��e andthemmfisrotatingcounterclockwise. Letusthinkaboutthisforaminute.WeareobservingTesla’srotatingmagnetic fieldaswestandonthestator.Weseetherotatingmagneticfieldpulsatingat ��e .It seemslogicalforthistwo-poledevicethatthevariablesassociatedwiththisrotatingmagneticfieldinthisframeofreferencewouldvaryat ��e .Thisissomethingwe alreadyknew.Thatis,thebalancedsetofcurrentsthatproducesthisviewofthe rotatingmagneticfieldisthebalancedsteady-statesetgivenby(2.2-7)and(2.2-8). Ifwenowlet �� = ��e ,(2.2-12)becomes

where ��e isthedisplacementfromthe q-axiswhichisrotatingat ��e .Since ��esi (0) isconstantforbalancedsteady-stateconditions,(2.2-14)isasinusoidalfunction ofthespatialvariable ��e (thedisplacementfromthe q-axis).Itsmaximumvalue occursat ��e = ��esi (0) , whichisthephaseangleofthebalancedsteady-statestator currents.Inotherwords,weareobservingTesla’srotatingmagneticfieldaswerun at ��e inthecounterclockwisedirectionaroundtheairgap,andthisfieldappears constanttousforbalancedsteady-stateconditionsandthevariablesmustbedc. Since ��e istheangularfrequencyof Ias and Ibs ,andthe q-axisisalsorotatingat ��e orinsynchronismwiththeelectricalsystem,thisisreferredtoasthe synchronously rotatingframeofreference.Therein,sincewehave360∘ vision,Tesla’srotating magneticfieldappearstousasasinusoidalfunctionof ��e (aspacesinusoid)with themaximumvalueat ��e = ��esi (0) fromthe q-axisandstationaryrelativetous sincewearerotatingat ��e counterclockwise.ThisisdepictedinFigure2.2-6, where ��esi (0) isselectednegativetocorrespondtoinductivecircuits.Apositive mmf e s isasouthpoleduetothestatorcurrents (Ss ) withthemaximumintensityat ��e = ��esi (0).Anegativemmf e s representsastatornorthpole (Ns ). WecanviewTesla’srotatingmagneticfieldfromframesofreferenceotherthan astationaryobserver.InFigure2.2-6,thereferenceframeisrotatingat ��e inthe counterclockwisedirection.TheimportanceofFigure2.2-6cannotbeoveremphasized.Itistheconnectionbetweendisplacementandtime.Wefindthatthis conceptenablesaclearvisualizationoftheoperationofelectricmachinesand