机电一体化英文论文Word格式.doc
- 文档编号:7781972
- 上传时间:2023-05-09
- 格式:DOC
- 页数:20
- 大小:1.18MB
机电一体化英文论文Word格式.doc
《机电一体化英文论文Word格式.doc》由会员分享,可在线阅读,更多相关《机电一体化英文论文Word格式.doc(20页珍藏版)》请在冰点文库上搜索。
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,thenitwillpossesstheadvantagesofbothtwomethods.Takethe3rdorderRunge±
Kuttamethodforexample,todeducethenewmethod.Forthedifferentialequation
bythe3rdorderRunge±
Kuttamethod,onehas[3]
Fortheinductiveimpedance,onehas:
FromEq.(10)itfollows
Whereisthevoltageatthemidpointofthestep,whichcanbefoundbysolvingtheequationsofthesystem.Butwecalculateitbytrapezoidalmethod.Itcanbedoneintwodifferentways(A)and(B):
(A)Taketheaveragevaluesofunandun11andlet
Substitutingun11/2fromEq.(14)intoEq.(13)Eq.(10)gives:
SubstitutingtheaboveformulaintoEq.(10),onecanget:
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:
ItscompanionmodelinFig.1(b)is
(B)Takingonecanget
ItscompanionmodelforFig.1(b)is:
Bothofthe4thordermodelsintroducedabovealsoturntotrapezoidalmethodforpureinductancewhenR0.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:
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:
Fig.5.Errorcurvesforeachmodel(T:
Fig.6.Errorcurvesforeachmodel(T:
Fig.7.Errorcurvesforeachmodel(T:
Fi
- 配套讲稿:
如PPT文件的首页显示word图标,表示该PPT已包含配套word讲稿。双击word图标可打开word文档。
- 特殊限制:
部分文档作品中含有的国旗、国徽等图片,仅作为作品整体效果示例展示,禁止商用。设计者仅对作品中独创性部分享有著作权。
- 关 键 词:
- 机电 一体化 英文 论文