机电一体化英文论文.doc

机电一体化英文论文.doc

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AninnovativedigitalmethodforthedynamicsimulationofDCelectromechanicalsystems

ChenChaoyinga,*,P.DiBarbab,ASavinib

aDepartmentofElectricalEngineering,TianjinUniversity,300072Tianjin,People'sRepublicofChina

bDepartmentofElectricalEngineering,UniversityofPavia,27100Pavia,Italy

Abstract

Inthispaper,aninnovativedigitalsimulationmethod,named`R-K-T'method,ispresented.ThenewmethodologycombinesRunge±Kuttaandtrapezoidalmethodsandpossessestheadvantagesofbothofthem.Theerrorsfeaturingtheproposedmethodareanalysedandtheircorrectionisworkedout.Asacasestudy,thecircuitmodelofasmallDCmotor,actingastheenginestarterofaroadvehicle,isconsidered;theproposedmethodologyisappliedtocarryoutthedynamicsimulationoftheelectromechanicaldevice.Theresultsareobtainedef®cientlyandwithagooddegreeofaccuracy;inparticular,thenumericaloscillationsaresuppressed.q1998ElsevierScienceLtd.Allrightsreserved.

Keywords:

Numericalmethods;Timeintegration;Dynamicsystems;Electromechanics;DCmotor

1.Introduction

Severaldigitalmethods,suchasEuler,trapezoidal,Runge±Kuttaandlinearmultistepmethodsaregenerallyusedtocarryoutnumericalintegrationanddifferentiation.TheEulermethodissimple,butwithlowaccuracy;itscutofferrorisO（h2）,whereasthatofthetrapezoidalmethoddecreaseasO（h3）.TheRunge±Kuttamethodhasrelativelyhighaccuracybutrequireslargeamountofcomputationalwork;finally,themultistepmethodhashighaccuracy,butitcannotbeself-started[1].Therefore,thetrapezoidalmethodfindswidespreadapplicationsintransientdigitalsimulations.However,inDCsystemsimulations,thetrapezoidalmethodoftenintroducesnumericaloscillationswithequalamplitudes,sothatitsapplicationinthiscaseiscritical.SincethebackwardEulermethodcanavoidsuchoscillations,intheliterature[2],adampedtrapezoidalmethodwasproposed;thismethodintroducesadampingfactorintothetrapezoidalmethodwhicheffectivelydecreasesthenumericaloscillationsbutatthesacrificeofaccuracy.

AfteranalysingtrapezoidalandRunge±Kuttamethodscarefully,thispaperpresentsaninnovativesimulationmethod,called`R-K-T',whichcombinesRunge±Kuttaandtrapezoidalmethodsingeniously.Theadvantagesofthenewmethodare:

theRunge±Kuttamethodcanbeexpressedbythecompanionmodeljustlikethetrapezoidalmethoddoes;thenumericaloscillationscanbeattenuatedefficiently.Accordingtofrequencyspectrumanalysis,theerrorsofthemethodarecalculatedandcorrected.ItmakesitpossibletosimulateDCsystemsaccuratelyandefficiently.

2.NumericaloscillationsoftrapezoidalmethodinDCsystems

ConsideringtheinductivecircuitshowninFig.1（a）thegoverningequationis

wherecurrentiistheunknown.Usingthetrapezoidalmethodfortimeintegration,onecanget:

Wherehisthetimestepofcalculation.

Let

then

ThecompanionmodelofthatdepictedinFig.1（a）isshowninFig.1（b）.FromEq.

（1）onecanalsoget:

Fig.1.Inductiveimpedance（a）anditscompanionmodels（b）and（c）.

where

ItscompanionmodelisshowninFig.1（c）.

Suppose,whenaDCcurrent¯owsthroughtheinductiveimpedance.FromEq.（3）thevoltageresponseoftheinductivebranchcanbecalculatedas

Itcanbeseenthattheoscillationofvoltageisundepressed.

Otherwiseassume,whennˆk,thecurrentisswitchedoff,i.e.fromEq.（3）onecanget:

thatis

Thevoltageresponseisalsoanundepressedoscillation.

ItcanbeprovedthatthebackwardEulermethodcanavoidsuchanoscillation.Forinductiveimpedanceitgives:

Itcanbeseenthatun11isnotdependentonun,sothismakesitpossibletoavoidnumericaloscillationsbutgreatlyreducestheaccuracyofbackwardEulermethod.Tosolvethiscontradiction,theliterature[2]proposesatrapezoidalmethodwithdamping.Forthedifferentialequation

itgives

FortheinductiveimpedanceshowninFig.1itgives:

Whereaisthedampingfactor（0

Thismethodturnsintothetrapezoidalonewhena=0,andbecomesthebackwardEulermethodwhena=1.FromEq.（9）,itcanbeseenthatthecoeficientofunissowhenthevoltageoscillationisproduced,itcanbedampedoutquickly.Thebiggerthefactoris,themorequicklytheoscillationisreducedandtheloweraccuracycanbeobtainedbythismethod.Besides,thefactorcanbeselectedonlyaccordingtoexperience:

itsoptimumvalueisdif®culttobedetermined.

3.TheR-K-Tmethod

TheRunge±KuttamethodhashigheraccuracyandbetterstabilityinDCsystems,butitrequiresthecalculationofthevaluesofafunctionmanytimesduringasinglestep;itcannotbeexpressedbyacompanionmodellikethetrapezoidalmethod.IfonecancombinetheRunge±Kuttamethodandthetrapezoidalmethodtoformanewmethod,thenitwillpossesstheadvantagesofbothtwomethods.Takethe3rdorderRunge±Kuttamethodforexample,todeducethenewmethod.Forthedifferentialequation

bythe3rdorderRunge±Kuttamethod,onehas[3]

Fortheinductiveimpedance,onehas:

where

FromEq.（10）itfollows

Whereisthevoltageatthemidpointofthestep,whichcanbefoundbysolvingtheequationsofthesystem.Butwecalculateitbytrapezoidalmethod.Itcanbedoneintwodifferentways（A）and（B）:

（A）Taketheaveragevaluesofunandun11andlet

Substitutingun11/2fromEq.（14）intoEq.（13）Eq.（10）gives:

SubstitutingtheaboveformulaintoEq.（10）,onecanget:

where

Itisobviousthatthe3rdorderRunge±Kuttamethodwithun11/2substitutedbyEq.（14）maybeexpressedbythecompanionmodelshowninFig.1（b）,asforthetrapezoidalmethod;theparametersofthemodelare:

Thedistinguishingfeatureofthismethodisthatthecoef®-cientsofun11andunarenotequal;theirratioAmaybeusedtoattenuatethenumericaloscillationwithequalamplitudesoftrapezoidalmethod.ItturnstothetrapezoidalmethodwhenR=0,i.e.theformulaforpureinductancegivenbytrapezoidalmethod:

（B）Take

Usingthetrapezoidalmethod,onehas:

BysubstitutingEq.（19）intoeq,（13）,onecanget:

BysubstitutingtheaboveequationsintoEq.（10）,itfollows

Where

Formula（20）maybeexpressedbyacompanionmodelofinductiveimpedanceasFig.1（b）,where

Formula（20）hasalsothefunctionofattenuatingthenumericaloscillationslikeEq.（15）,anditalsoturnstothetrapezoidalmethodforpureinductancewhenR=0.

Forthe4thorderRunge±Kuttamethod,itgives:

Similarlyonecanobtainthecompanionmodelforthe4thorderRunge±Kuttamethodasfollows[4].

（A）Takingonehas:

Where

ItscompanionmodelinFig.1（b）is

（B）Takingonecanget

where

ItscompanionmodelforFig.1（b）is:

Bothofthe4thordermodelsintroducedabovealsoturntotrapezoidalmethodforpureinductancewhenRˆ0.Thus,theRunge±Kuttamethodiscombinedwithtrapezoidalmethodtoformanew`R-K-T'method,whichexhibitstheadvantagesofthesetwomethods.

4.AnalysisandcalculationoferrorfortheR-K-Tmethod

Inareal-lifesystem,voltagesandcurrents,whateverwaveformstheymayhave,canbeanalyzedbythemethodoffrequencyspectrum.Theerrorofsimulationcanbeanalyzedforeveryfrequencycomponent;thecomponentsarethenaddedtogetheraccordingtothetheoryofsuperpositiontoobtainthetotalerrors.

Letusassumethatcurrentandvoltageofacertainsystemelementare:

Wherewanyonefrequencycomponent.Letusrewritethe3rdR-K-Tmethod（20）asfollows:

SubstitutingEq.（27）intoEq.（28）,onecandeduce:

Fromtheformulaofinductiveimpedance,onehas

ThedifferenceofthetwosidesofEq.（29）representstheerrorofR-K-Tmethodforfrequencycomponent,sothattheerrorfunctioncanbede®nedas:

Iftheexcitingsourcescontainanumberoffrequencycomponents,e（v）shouldbecomputedforeveryfrequencycomponentandaddedtogether.Thesummationofallthee（v）givesthetotalerrorfunctionofthe3rdorderR-K-Tmethod.

5.CorrectionoferrorfortheR-K-Tmethod

FromEq.（29）itisclearthatthereisunbalanceintheformulaoftheR-K-Tmethodforangularfrequency;itisduetothemethoditselfandnotrelatedtotheexcitingsources.IfonewouldmatchthetwosidesofEq.（29）byaddingsomeitems,thenitcouldgivetheaccurateresultforfrequencycomponent.Letv0bethemainangularfrequencyoftheexcitingsource,inordertoconductaccuratecalculationforv0,itisnecessarytotransformEq.（29）asfollows:

ThecoeficientsofthetwosidesofEq.（32）areequalw=w0.Itmeansthatitgivesaccurateresultsforw=w0.

RestoringEq.（32）,onecandeduce:

where

Eq.（33）istheformulaoftheR-K-Tmethodaftercorrection.Iftheexcitingsourceofthesystemhasasinglefrequencyv0,thencorrectioncanbemadeforthisfrequency.Ifthesystemhasamultifrequencyexcitingsource,thencorrectionmaybemadeforoneofthedominantlowerfrequencywhichhashigheramplitude.

6.Numericalresults

Tochecktheaccuracyofthemethodpresented,thecircuitshowninFig.2hasbeenconsidered.Itsparametersare:

Theaccurateexpressionofcurrentiis:

ThetestcircuitshowninFig.2hasbeensolvedbyeachofthemodelsstatedabovewithtimesteph=0.1ms,aswellasbymeansofformula（34）givingamoreaccurateresult.Ineachcaseerrorisdefinedasthemaximumabsolute

Fig.3.Errorcurvesforeachmodel（T:

cycle）（seeTable1）.

Fig.4.Errorcurvesforeachmodel（T:

cycle）（seeTable1）.

Fig.5.Errorcurvesforeachmodel（T:

cycle）（seeTable1）.

Fig.6.Errorcurvesforeachmodel（T:

cycle）（seeTable1）.

Fig.7.Errorcurvesforeachmodel（T:

cycle）（seeTable1）.

Fi