机电一体化英文论文.doc

机电一体化英文论文.doc

《机电一体化英文论文.doc》由会员分享，可在线阅读，更多相关《机电一体化英文论文.doc（20页珍藏版）》请在冰豆网上搜索。

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,thenitwill