机电一体化英文论文范本模板.docx

机电一体化英文论文范本模板.docx

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机电一体化英文论文范本模板

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，when

aDCcurrent¯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〈a<1）。

Thismethodturnsintothetrapezoidalonewhena=0,andbecomesthebackwardEulermethodwhena=1.FromEq。

（9）,itcanbeseenthatthecoeficientofunis

sowhenthevoltageoscillationisproduced，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

Where

isthevoltageatthemidpointofthestep，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）Taking

onehas:

Where

ItscompanionmodelinFig.1（b）is

（B）Taking

onecanget

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:

cy