测控专业 毕业论文 外文翻译 中英文对照.docx

测控专业 毕业论文 外文翻译 中英文对照.docx

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测控专业毕业论文外文翻译中英文对照

Oscillation,InstabilityandControlofStepperMotors

LIYUCAOandHOWARDM.SCHWARTZ

DepartmentofSystemsandComputerEngineering,CarletonUniversity,1125ColonelByDrive,

Ottawa,ONK1S5B6,Canada

（Received:

18February1998;accepted:

1December1998）

Abstract

Anovelapproachtoanalyzinginstabilityinpermanent-magnetsteppermotorsispresented.Itisshownthattherearetwokindsofunstablephenomenainthiskindofmotor:

mid-frequencyoscillationandhigh-frequencyinstability.Nonlinearbifurcationtheoryisusedtoillustratetherelationshipbetweenlocalinstabilityandmidfrequencyoscillatorymotion.Anovelanalysisispresentedtoanalyzethelossofsynchronismphenomenon,whichisidentifiedashigh-frequencyinstability.Theconceptsofseparatorsandattractorsinphase-spaceareusedtoderiveaquantitytoevaluatethehigh-frequencyinstability.Byusingthisquantityonecaneasilyestimatethestabilityforhighsupplyfrequencies.Furthermore,astabilizationmethodispresented.Ageneralizedapproachtoanalyzethestabilizationproblembasedonfeedbacktheoryisgiven.Itisshownthatthemid-frequencystabilityandthehigh-frequencystabilitycanbeimprovedbystatefeedback.

Keywords:

Steppermotors,instability,nonlinearity,statefeedback.

1.Introduction

Steppermotorsareelectromagneticincremental-motiondeviceswhichconvertdigitalpulseinputstoanalogangleoutputs.Theirinherentsteppingabilityallowsforaccuratepositioncontrolwithoutfeedback.Thatis,theycantrackanysteppositioninopen-loopmode,consequentlynofeedbackisneededtoimplementpositioncontrol.SteppermotorsdeliverhigherpeaktorqueperunitweightthanDCmotors;inaddition,theyarebrushlessmachinesandthereforerequirelessmaintenance.Allofthesepropertieshavemadesteppermotorsaveryattractiveselectioninmanypositionandspeedcontrolsystems,suchasincomputerharddiskdriversandprinters,XY-tables,robotmanipulators,etc.

Althoughsteppermotorshavemanysalientproperties,theysufferfromanoscillationorunstablephenomenon.Thisphenomenonseverelyrestrictstheiropen-loopdynamicperformanceandapplicableareawherehighspeedoperationisneeded.Theoscillationusuallyoccursatsteppingrateslowerthan1000pulse/s,andhasbeenrecognizedasamid-frequencyinstabilityorlocalinstability,oradynamicinstability.Inaddition,thereisanotherkindofunstablephenomenoninsteppermotors,thatis,themotorsusuallylosesynchronismathighersteppingrates,eventhoughloadtorqueislessthantheirpull-outtorque.Thisphenomenonisidentifiedashigh-frequencyinstabilityinthispaper,becauseitappearsatmuchhigherfrequenciesthanthefrequenciesatwhichthemid-frequencyoscillationoccurs.Thehigh-frequencyinstabilityhasnotbeenrecognizedaswidelyasmid-frequencyinstability,andthereisnotyetamethodtoevaluateit.

Mid-frequencyoscillationhasbeenrecognizedwidelyforaverylongtime,however,acompleteunderstandingofithasnotbeenwellestablished.Thiscanbeattributedtothenonlinearitythatdominatestheoscillationphenomenonandisquitedifficulttodealwith.

Mostresearchershaveanalyzeditbasedonalinearizedmodel.Althoughinmanycases,thiskindoftreatmentsisvalidoruseful,atreatmentbasedonnonlineartheoryisneededinordertogiveabetterdescriptiononthiscomplexphenomenon.Forexample,basedonalinearizedmodelonecanonlyseethatthemotorsturntobelocallyunstableatsomesupplyfrequencies,whichdoesnotgivemuchinsightintotheobservedoscillatoryphenomenon.Infact,theoscillationcannotbeassessedunlessoneusesnonlineartheory.

Therefore,itissignificanttousedevelopedmathematicaltheoryonnonlineardynamicstohandletheoscillationorinstability.ItisworthnotingthatTaftandGauthier,andTaftandHarnedusedmathematicalconceptssuchaslimitcyclesandseparatricesintheanalysisofoscillatoryandunstablephenomena,andobtainedsomeveryinstructiveinsightsintotheso-calledlossofsynchronousphenomenon.Nevertheless,thereisstillalackofacomprehensivemathematicalanalysisinthiskindofstudies.Inthispaperanovelmathematicalanalysisisdevelopedtoanalyzetheoscillationsandinstabilityinsteppermotors.

Thefirstpartofthispaperdiscussesthestabilityanalysisofsteppermotors.Itisshownthatthemid-frequencyoscillationcanbecharacterizedasabifurcationphenomenon（Hopfbifurcation）ofnonlinearsystems.Oneofcontributionsofthispaperistorelatethemid-frequencyoscillationtoHopfbifurcation,thereby,theexistenceoftheoscillationisprovedtheoreticallybyHopftheory.High-frequencyinstabilityisalsodiscussedindetail,andanovelquantityisintroducedtoevaluatehigh-frequencystability.Thisquantityisveryeasytocalculate,andcanbeusedasacriteriatopredicttheonsetofthehigh-frequencyinstability.Experimentalresultsonarealmotorshowtheefficiencyofthisanalyticaltool.

Thesecondpartofthispaperdiscussesstabilizingcontrolofsteppermotorsthroughfeedback.Severalauthorshaveshownthatbymodulatingthesupplyfrequency,themid-frequencyinstabilitycanbeimproved.Inparticular,PickupandRussellhavepresentedadetailedanalysisonthefrequencymodulationmethod.Intheiranalysis,Jacobiserieswasusedtosolveaordinarydifferentialequation,andasetofnonlinearalgebraicequationshadtobesolvednumerically.Inaddition,theiranalysisisundertakenforatwo-phasemotor,andtherefore,theirconclusionscannotapplieddirectlytooursituation,whereathree-phasemotorwillbeconsidered.Here,wegiveamoreelegantanalysisforstabilizingsteppermotors,wherenocomplexmathematicalmanipulationisneeded.Inthisanalysis,ad–qmodelofsteppermotorsisused.Becausetwo-phasemotorsandthree-phasemotorshavethesameq–dmodelandtherefore,theanalysisisvalidforbothtwo-phaseandthree-phasemotors.Uptodate,itisonlyrecognizedthatthemodulationmethodisneededtosuppressthemid-frequencyoscillation.Inthispaper,itisshownthatthismethodisnotonlyvalidtoimprovemid-frequencystability,butalsoeffectivetoimprovehigh-frequencystability.

2.DynamicModelofStepperMotors

Thesteppermotorconsideredinthispaperconsistsofasalientstatorwithtwo-phaseorthree-phasewindings,andapermanent-magnetrotor.Asimplifiedschematicofathree-phasemotorwithonepole-pairisshowninFigure1.Thesteppermotorisusuallyfedbyavoltage-sourceinverter,whichiscontrolledbyasequenceofpulsesandproducessquare-wavevoltages.Thismotoroperatesessentiallyonthesameprincipleasthatofsynchronousmotors.Oneofmajoroperatingmannerforsteppermotorsisthatsupplyingvoltageiskeptconstantandfrequencyofpulsesischangedataverywiderange.Underthisoperatingcondition,oscillationandinstabilityproblemsusuallyarise.

Figure1.Schematicmodelofathree-phasesteppermotor.

Amathematicalmodelforathree-phasesteppermotorisestablishedusingq–dframe-referencetransformation.Thevoltageequationsforthree-phasewindingsaregivenby

va=Ria+L*dia/dt−M*dib/dt−M*dic/dt+dλpma/dt,

vb=Rib+L*dib/dt−M*dia/dt−M*dic/dt+dλpmb/dt,

vc=Ric+L*dic/dt−M*dia/dt−M*dib/dt+dλpmc/dt,

whereRandLaretheresistanceandinductanceofthephasewindings,andMisthemutualinductancebetweenthephasewindings._pma,_pmband_pmcaretheflux-linkagesofthephasesduetothepermanentmagnet,andcanbeassumedtobesinusoidfunctionsofrotorposition_asfollow

λpma=λ1sin（Nθ）,

λpmb=λ1sin（Nθ−2/3）,

λpmc=λ1sin（Nθ-2/3）,

whereNisnumberofrotorteeth.Thenonlinearityemphasizedinthispaperisrepresentedbytheaboveequations,thatis,theflux-linkagesarenonlinearfunctionsoftherotorposition.

Byusingtheq;dtransformation,theframeofreferenceischangedfromthefixedphaseaxestotheaxesmovingwiththerotor（refertoFigure2）.Transformationmatrixfromthea;b;cframetotheq;dframeisgivenby

Forexample,voltagesintheq;dreferencearegivenby

Inthea;b;creference,onlytwovariablesareindependent（iaCibCicD0）;therefore,theabovetransformationfromthreevariablestotwovariablesisallowable.Applyingtheabovetransformationtothevoltageequations

（1）,thetransferredvoltageequationintheq;dframecanbeobtainedas

vq=Riq+L1*diq/dt+NL1idω+Nλ1ω,

vd=Rid+L1*did/dt−NL1iqω,

Figure2.a,b,candd,qreferenceframe.

whereL1DLCM,and!

isthespeedoftherotor.Itcanbeshownthatthemotor’storquehasthefollowingform

T=3/2Nλ1iq

Theequationofmotionoftherotoriswrittenas

J*dω/dt=3/2*Nλ1iq−Bfω–Tl,

whereBfisthecoefficientofviscousfriction,andTlrepresentsloadtorque,whichisassumedtobeaconstantinthispaper.

Inordertoconstitutethecompletestateequationofthemotor,weneedanotherstatevariablethatrepresentsthepositionoftherotor.Forthispurposethesocalledloadangle_[8]isusuallyused,whichsatisfiesthefollowingequation

Dδ/dt=ω−ω0,

where!

0issteady-statespeedofthemotor.Equations（5）,（7）,and（8）constitutethestatespacemodelofthemotor,forwhichtheinputvariablesarethevoltagesvqandvd.Asmentionedbefore,steppermotorsarefedbyaninverter,whoseoutputvoltagesarenotsinusoidalbutinsteadaresquarewaves.However,becausethenon-sinusoidalvoltagesdonotchangetheoscillationfeatureandinstabilityverymuchifcomparedtothesinusoidalcase（aswillbeshowninSection3,theoscillationisduetothenonlinearityofthemotor）,forthepurpo