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1、TheSteppermotorcontrolcircuitbebasedonSinglechipmicrocomputerTheAT89C51isalow-power,high-performanceCMOS8-bitmicrocomputerwith4KbytesofFlashprogrammableanderasablereadonlymemory(PEROM).ThedeviceismanufacturedusingAtmeshigh-densitynonvolatilememorytechnologyandiscompatiblewiththeindustrystandardMCS-5
2、1instructionsetandpinout.Theon-chipFlashallowstheprogrammemorytobereprogrammedin-systemorbyaconventionalnonvolatilememoryprogrammer.Bycombiningaversatile8-bitCPUwithFlashonamonolithicchip,theAtmelAT89C51isapowerfulmicrocomputerwhichprovidesahighly-flexibleandcost-effectivesolutiontomanyembeddedcontr
3、olapplications.FunctioncharacteristicTheAT89C51providesthefollowingstandardfeatures:4KbytesofFlash,128bytesofRAM,32I/Olines,two16-bittimer/counters,afivevectortwo-levelinterruptarchitecture,afullduplexserialport,on-chiposcillatorandclockcircuitry.Inaddition,theAT89C51isdesignedwithstaticlogicforoper
4、ationdowntozerofrequencyandsupportstwosoftwareselectablepowersavingmodes.TheIdleModestopstheCPUwhileallowingtheRAM,timer/counters,serialportandinterruptsystemtocontinuefunctioning.ThePower-downModesavestheRAMcontentsbutfreezestheoscillatordisablingallotherchipfunctionsuntilthenexthardwarereset.PinDe
5、scriptionVCC：Supplyvoltage.GND：Ground.Port0：PortOisan8-bitopen-drainbi-directionalI/Oport.Asanoutputport,eachpincansinkeightTTLinputs.WhenIsarewrittentoportOpins,thepinscanbeusedasHighimpedanceinuts.PortOmayalsobeconfiguredtobethemultiplexedIoworderaddress/databusduringaccessestoexternalprogramandda
6、tamemory.InthismodePOhasinternalpullups.PortOalsoreceivesthecodebytesduringFlashprogramming,andoutputsthecodebytesduringprogramverification.Externalpullupsarerequiredduringprogramverification.PortlPort1isan8-bitbi-directionalI/Oportwithinternalpullups.ThePOrt1outputbufferscansink/sourcefourTTLinputs
7、.WhenIsarewrittentoPort1pinstheyarepulledhighbytheinternalpullupsandcanbeusedasinputs.Asinputs,Port1pinsthatareexternallybeingpulledlowwillsourcecurrent(IIL)becauseoftheinternalpullups.Port1alsoreceivesthelow-orderaddressbytesduringFlashprogrammingandverification.Port2Port2isan8-bitbi-directionalI/O
8、portwithinternalpullups.ThePort2outputbufferscansink/sourcefourTTLinuts.When1sarewrittentoPort2pinstheyarepulledhighbytheinternalpullupsandcanbeusedasinputsAsinputs,Port2pinsthatareexternallybeingpulledlowwillsourcecurrent,becauseoftheinternalpullups.Port2emitsthehigh-orderaddressbyteduringfetchesfr
9、omexternalprogrammemoryandduringaccessestoexternaldatamemorythatuse16-bitaddresses.Inthisapplication,itusesstronginternalpullupswhenemittingIs.Duringaccessestoexternaldatamemorythatuse8-bitaddresses,Port2emitsthecontentsoftheP2SpecialFunctionRegistenPort2alsoreceivesthehigh-orderaddressbitsandsomeco
10、ntrolsignalsduringFlashprogrammingandverification.Port3Port3isan8-bitbi-directionalI/Oportwithinternalpullups.ThePort3outputbufferscansink/sourcefourTTLinputs.WhenIsarewrittentoPort3pinstheyarepulledhighbytheinternalpullupsandcanbeusedasinputs.Asinputs,Port3pinsthatareexternallybeingpulledlowwillsou
11、rcecurrent(IIL)becauseofthepullups.Port3alsoservesthefunctionsofvariousspecialfeaturesoftheAT89C51aslistedbelow:Port3alsoreceivessomecontrolsignalsforFlashprogrammingandverification.RSTResetinputAhighonthispinfortwomachinecycleswhiletheoscillatorisrunningresetsthedevice.ALE/PROGAddressLatchEnableout
12、putpulseforlatchingthelowbyteoftheaddressduringaccessestoexternalmemory.Thispinisalsotheprogrampulseinput(PROG)duringFlashprogrammingJnnormaloperationALEisemittedataconstantrateof1/6theoscillatorfrequency,andmaybeusedforexternaltimingorclockingpurposes.Note,however,thatoneALEpulseisskippedduringeach
13、accesstoexternalDataMemory.Ifdesired,ALEoperationcanbedisabledbysettingbitOofSFRlocation8EH.Withthebitset,ALEisactiveonlyduringaMOVXorMOVCinstruction.Otherwise,thepinisweaklypulledhigh.SettingtheALE-disablebithasnoeffectifthemicrocontrollerisinexternalexecutionmode.PSENProgramStoreEnableisthereadstr
14、obetoexternalprogrammemory.WhentheAT89C51isexecutingcodefromexternalprogrammemory,PSENisactivatedtwiceeachmachinecycle,exceptthattwoPSENactivationsareskippedduringeachaccesstoexternaldatamemory.EA/VPPExternalAccessEnable.EAmustbestrappedtoGNDinordertoenablethedevicetofetchcodefromexternalprogrammemo
15、rylocationsstartingatOOOOHuptoFFFFH.Note,however,thatiflockbit1isprogrammed,EAwillbeinternallylatchedonreset.EAshouldbestrappedtoVCCforinternalprogramexecutions.Thispinalsoreceivesthe12-voltprogrammingenablevoltage(VPP)duringFlashprogramming,forpartsthatrequire12-voltVPP.XTALlInputtotheinvertingosci
16、llatoramplifierandinputtotheinternalclockoperatingcircuit.XTAL2OutputfromtheinvertingoscillatoramplifierOscillatorCharacteristicsXTALlandXTAL2aretheinputandoutput,respectively,ofaninvertingamplifierwhichcanbeconfiguredforuseasanon-chiposcillator,asshowninFigureLEitheraquartzcrystalorceramicresonator
17、maybeused.Todrivethedevicefromanexternalclocksource,XTAL2shouldbeleftunconnectedwhileXTALlisdrivenasshowninFigure2.Therearenorequirementsonthedutycycleoftheexternalclocksignal,sincetheinputtotheinternalclockingcircuitryisthroughadivide-by-twoflip-flop,butminimumandmaximumvoltagehighandlowtimespecifi
18、cationsmustbeobserved.Figure1.OscillatorConnectionsFigure2.ExternalClockDriveConfigurationIdleModeInidlemode,theCPUputsitselftosleepwhilealltheonchipperipheralsremainactive.Themodeisinvokedbysoftware.Thecontentoftheon-chipRAMandallthespecialfunctionsregistersremainunchangedduringthismode.Theidlemode
19、canbeterminatedbyanyenabledinterruptorbyahardwarereset.ltshouldbenotedthatwhenidleisterminatedbyahardwarereset,thedevicenormallyresumesprogramexecution,fromwhereitleftoff,uptotwomachinecyclesbeforetheinternalresetalgorithmtakescontrol.On-chiphardwareinhibitsaccesstointernalRAMinthisevent,butaccessto
20、theportpinsisnotinhibited.ToeliminatethepossibilityofanunexpectedwritetoaportpinwhenIdleisterminatedbyreset,theinstructionfollowingtheonethatinvokesIdleshouldnotbeonethatwritestoaportpinortoexternalmemory.Power-downModeInthepower-downmode,theoscillatorisstopped,andtheinstructionthatinvokespowerdowni
21、sthelastinstructionexecuted.Theon-chipRAMandSpecialFunctionRegistersretaintheirvaluesuntilthepower-downmodeisterminated.Theonlyexitfrompower-downisahardwarereset.ResetredefinestheSFRsbutdoesnotchangetheon-chipRAM.TheresetshouldnotbeactivatedbeforeVCCisrestoredtoitsnormaloperatinglevelandmustbeheldac
22、tivelongenoughtoallowtheoscillatortorestartandstabilize.ProgramMemoryLockBitsOnthechiparethreelockbitswhichcanbeleftunprogrammed(U)orcanbeprogrammed(P)toobtaintheadditionalfeatureslistedinthetablebelow.Whenlockbit1isprogrammed,thelogiclevelattheEApinissampledandlatchedduringreset.Ifthedeviceispowere
23、dupwithoutareset,thelatchinitializestoarandomvalue,andholdsthatvalueuntilresetisactivated.ItisnecessarythatthelatchedvalueofEAbeinagreementwiththecurrentlogiclevelatthatpininorderforthedevicetofunctionproperly.IntroductionSteppermotorsareelectromagneticincremental-motiondeviceswhichconvertdigitalpul
24、seinputstoanalogangleoutputs.Theirinherentsteppingabilityallowsforaccuratepositioncontrolwithoutfeedback.Thatis,theycantrackanysteppositioninopen-loopmode,ConsequentlynofeedbackisneededtoimplementpositioncontrolSteppermotorsdeliverhighereaktorqueperunitweightthanDCmotors;inaddition,theyarebrushlessm
25、achinesandtherefbrerequirelessmaintenance.AllofthesepropertieshavemadesteppermotorsaVeryattractiveselectioninmanypositionandspeedcontrolsystems,suchasincomputerharddiskdriversandprinters,XY-tables,robotmanipulators,etc.Althoughsteppermotorshavemanysalientproperties,theysufferfromanoscillationOrunsta
26、blephenomenon.Thisphenomenonseverelyrestrictstheiropen-loopdynamicerformanceandapplicableareawherehighspeedoperationisneeded.TheoscillationUsuallyoccursatsteppingrateslowerthan100Opulse/s,andhasbeenrecognizedasamid-frequencyinstabilityorlocalinstability1,oradynamicinstability2.Inaddition,thereisanot
27、herkindofunstablephenomenoninsteppermotors,thatis,themotorsusuallyloseSynchronismathighersteppingrates,eventhoughloadtorqueislessthantheirpull-outtorque.Thisphenomenonisidentifiedashigh-frequencyinstabilityinthispaper,becauseitappearsatmuchhigherfrequenciesthanthefrequenciesatwhichthemid-frequencyos
28、cillationoccurs.Thehigh-frequencyinstabilityhasnotbeenrecognizedaswidelyasmid-frequencyinstability,andthereisnotyetamethodtoevaluateit.Mid-frequencyoscillationhasbeenrecognizedwidelyforaverylongtime,however,acompleteunderstandingofithasnotbeenwellestablished.ThiscanbeattributedtoIhenonlinearitythatd
29、ominatestheoscillationphenomenonandisquitedifficulttodealwith.384L.CaoandH.M.SchwartzMostresearchershaveanalyzeditbasedonalinearizedmodel1.Althoughinmanycases,thiskindoftreatmentsisvalidoruseful,atreatmentbasedonnonlineartheoryisneededinordertogiveabetterdescriptiononthiscomplexphenomenon.Forexample
30、,basedonalinearizedmodelonecanonlyseethatthemotorsturntobelocallyunstableatsomesupplyfrequencies,whichdoesnotgivemuchinsightintotheobservedoscillatoryphenomenon.Infact,theoscillationcannotbeassessedunlessoneusesnonlineartheory.Therefore,itissignificanttousedevelopedmathematicaltheoryonnonlineardynam
31、icsIohandletheoscillationorinstability.ItisworthnotingthatTaftandGauthier3,andTaftandHamed4usedmathematicalconceptssuchaslimitcyclesandSeparatricesintheanalysisOfoscillatoryandunstablephenomena,andobtainedsomeveryinstructiveinsightsintotheSocalledlossofsynchronousphenomenon.Nevertheless,thereisstill
32、alackofaComprehensivemathematicalanalysisinthiskindofstudies.Inthispaperanovelmathematicalanalysisisdeveloedtoanalyzetheoscillationsandinstabilityinsteppermotors.Thefirstpartofthispaperdiscussesthestabilityanalysisofsteppermotors.ItisShownthatthemid-frequencyoscillationcanbecharacterizedasabifurcati
33、onphenomenon(Hopfbifurcation)ofnonlinearsystems.OneofcontributionsofthispaperistorelatethemidfrequencyoscillationtoHopfbifurcation,thereby,theexistenceoftheoscillationisprovedtheoreticallybyHopftheory.High-frequencyinstabilityisalsodiscussedindetail,andanovelquantityisintroducedtoevaluatehigh-freque
34、ncystability.Thisquantityisveryeasytocalculate,andcanbeusedasacriteriatopredicttheonsetofthehigh-frequencyinstability.Experimentalresultsonarealmotorshowtheefficiencyofthisanalyticaltool.ThesecondpartofthispaperdiscussesstabilizingcontrolofsteppermotorsIhroughfeedback.Severalauthorshaveshownthatbymo
35、dulatingthesupplyfrequency5,themidfrequencyinstabilitycanbeimproved.Inparticular,PickupandRussell6,7havepresentedadetailedanalysisonthefrequencymodulationmethod.Intheiranalysis,Jacobiserieswasusedtosolveaordinarydifferentialequation,andasetofnonlinearalgebraicequationshadtobesolvednumerically.Inaddi
36、tion,theiranalysisisundertakenforatwo-phasemotor,andtherefore,theirconclusionscannotapplieddirectlytooursituation,whereathree-phasemotorwillbeconsidered.Here,wegiveamoreelegantanalysisforstabilizingsteppermotors,wherenocomplexmathematicalmanipulationisneeded.Inthisanalysis,ad-qmodelofsteppermotorsis
37、used.Becausetwo-phasemotorsandthree-phasemotorshavethesameq-dmodelandtherefore,theanalysisisvalidforbothtwo-phaseandthree-phasemotors.Uptodate,itisonlyrecognizedthatthemodulationmethodisneededtosuppressthemidfrequencyoscillation.Inthispaper,itisshownthatthismethodisnotonlyvalidtoimprovemid-frequency
38、stability,butalsoeffectivetoimprovehigh-frequencystability.2. DynamicModelofStepperMotorsThesteppermotorconsideredinthispaperconsistsofasalientstatorwithtwo-phaseorIhreephasewindings,andapermanent-magnetrotor.Asimplifiedschematicofathree-phasemotorwithonepole-pairisshowninFigure1.Thesteppermotorisus
39、uallyfedbyavoltagesourceinverter,whichiscontrolledbyasequenceofpulsesandproducessquare-wavevoltages.Thismotoroperatesessentiallyonthesameprincipleasthatofsynchronousmotors.Oneofmajoroperatingmannerforsteppermotorsisthatsupplyingvoltageiskeptconstantandfrequencyofpulsesischangedataverywiderange.Under
40、thisoperatingcondition,oscillationandinstabilityproblemsusuallyarise.Figure 1. Schematicmodelofathree-phasesteppermotorAmathematicalmodelforathree-phasesteppermotorisestablishedusingq-dframereferencetransformation.Thevoltageequationsforthree-phasewindingsaregivenbyVa=Ria+L*diadt-M*dibdt-M*dicdt+dpma
41、dt,Vb=Rih+L*dibdt-M*diadt-M*dicdt+dpmbdt,Vc=RicL*dicdLM*diadt-M*dibdt+dpmcdt,whereRandLaretheresistanceandinductanceofthephasewindings,andMisthemutualinductancebetweenthephasewindings,_pm,_pmhand_pmoaretheflux-linkagesofthephasesduetothepermanentmagnet,andcanbeassumedtobesinusoidfunctionsofrotorposi
42、tion_asfollowpma-lsi(N),pmb=sin(N-23),pr11c-si(N-23),whereNisnumberofrotorteeth.Thenonlinearityemphasizedinthispaperisrepresentedbytheaboveequations,thatis,theflux-linkagesarenonlinearfunctionsoftherotorposition.Byusingtheq;dtransformation,theframeofreferenceischangedfromthefixedphaseaxestotheaxesmo
43、vingwiththerotor(refertoFigure2).Transformationmatrixfromthea;b;Cframetotheq;dframeisgivenby8Forexample,voltagesintheq;JreferencearegivenbyInthea;b;Creference,onlytwovariablesareindependent(iaC历CicD0);therefore,theabovetransformationfromthreevariablestotwovariablesisallowable.Applyingtheabovetransfo
44、rmationtothevoltageequations(1),thetransferredvoltageequationintheq;dframecanbeobtainedasVq=RiqL*diqdt+NLid+N,Vd=Rid+L*diddt-NLiq,(5)Figure 2. a,b,candd,qreferenceframewhereLIDLCMand/isthespeedoftherotor.Itcanbeshownthatthemotor,storquehasthefollowingform2T=32NiqTheequationofmotionoftherotoriswritte
45、nasJ*ddt=32*Niq-Bko-Tl,where旦/isthecoefficientofviscousfriction,andTlrepresentsloadtorque,whichisassumedtobeaconstantinthispaper.Inordertoconstitutethecompletestateequationofthemotor,weneedanotherstatevariablethatrepresentsthepositionoftherotor.Forthispurposethesocalledloadangle_8isusuallyused,which
46、satisfiesthefollowingequationDdt=o,where/Oissteady-statespeedofthemotor.Equations(5),(7),and(8)constitutethestatespacemodelofthemotor,forwhichtheinputvariablesarethevoltagesIqandvd.Asmentionedbefore,steppermotorsarefedbyaninverter,whoseoutputvoltagesarenotsinusoidalbutinsteadaresquarewaves.However,b
47、ecausethenon-sinusoidalvoltagesdonotchangetheoscillationfeatureandinstabilityverymuchifcomparedtothesinusoidalcase(aswillbeshowninSection3,theoscillationisduetothenonlinearityofthemotor),forthepurposesofthispaperwecanassumethesupplyvoltagesaresinusoidal.Underthisassumption,wecanget叫andIdasfollowsVq=
48、VmCOS(N),Vd=VmSin(N3),whereVmisthemaximumofthesinewave.Withtheaboveequation,wehavechangedtheinputvoltagesfromafunctionoftimetoafunctionofstate,andinthiswaywecanrepresentthedynamicsofthemotorbyaautonomoussystem,asshownbelow.Thiswillsimplifythemathematicalanalysis.FromEquations(5),(7),and(8),thestate-
49、spacemodelofthemotorcanbewritteninamatrixformasfollowsX=F(X,u)=AX+Fn(X)+Bu,(10)whereXDTigid/_UT,UDT/l77UTisdefinedastheinput,and/1DMOisthesupplyfrequency.TheinputmatrixisdefinedbyThematrixAisthelinearpartofF.J.andisgivenbyFn.XrepresentsthenonlinearpartofEJandisgivenbyTheinputtermuisindependentoftime,andthereforeEquation(10)isau
