机电一体化英文论文.doc
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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,whennk,thecurrentisswitchedoff,i.e.fromEq.(3)onecanget:
thatis
Thevoltageresponseisalsoanundepressedoscillation.
ItcanbeprovedthatthebackwardEulermethodcanavoidsuchanoscillation.Forinductiveimpedanceitgives:
Itcanbeseenthatun11isnotdependentonun,sothismakesitpossibletoavoidnumericaloscillationsbutgreatlyreducestheaccuracyofbackwardEulermethod.Tosolvethiscontradiction,theliterature[2]proposesatrapezoidalmethodwithdamping.Forthedifferentialequation
itgives
FortheinductiveimpedanceshowninFig.1itgives:
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