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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:
Whereaisthedampingfactor(0<
a<
1).
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