DC motors: dynamic model and control techniques ... - LAAS

1 DCmotors:dynamicmodelandcontroltechnique sLucaZaccarianContents1 eldandconductors.. forcein a rotatingcoil.. EMFe ect..52 Thebasicequationsof .. motors.. diagramof theDCmotor..93 Statorvoltagecontrolwithconstant ..114 Statorvoltagecontrolwithconstant ..145 Armature-Current .. eldandconductorsAmagnetic eldis generatedby a owingcurrent: it is in particularexperiencedintheneighborhood of a ectof a magnetic eldmay be experimented,forinstance,by positioninga magnetcloseto a wirewherea current is owing;whatcanbeobserved is thatthemagnetexperiencesa force,which is dueto themagnetic eldgeneratedby thecurrent owingin themagnetis moved aroundthewire,theforcechangesdependingon thepositionsassumedby particular,it canbe observedthatthemagnetic elddecreasesas thedistancefromthewireincreasesandincrea sesas thecurrent experiment aimsto verifythata owingcurrent indeedgeneratesa vector eld~B. This eldmay be thought of as thesumof in nitecontributionsd~Bdueto allthein nitesimalsegments of wired~l, wherethecurrenti segment inducesamagnetic eldat any point in particular,theBiotandSavartlawstatesthat ,if~ris thevectorconnectingthewiresegmentd~lto a genericpointpin thespace,thecontributiond~Bof themagnetic eldin thepointpdueto thesegmentd~lis givenbyd~B=id~l ~rjrj3;(1)wherethesymbol~denotesthatthec onsideredquantity is a vectorof 3andthesymbol (1)providesa tool forcomputingthemagnetic eldassociatedto any conductorwherea current is particular,by suitablyintegratingit forthecaseof a solenoid,it turnsoutthat,if thecornere ectsareneglected(namely, thesolenoidis longenough),the eldinsidethewindingsis constant andparalleltotheaxisof thesolenoid,anditsmagnitudeis givenby [6, p.]

2 136]j~Bj= lN i;(2)wherelis thelengthof thesolenoid, is themagneticpermeability of thedielectricinsidethesolenoid,Nis thenumber of turnsof thewire(seeFigure1).Considernow a surfaceSlocatedin a magnetic eld~B; themagnetic ux, or ux owingthroughthesurfaceis de nedastheintegralalongthesurfaceof thenormalcomponent of themagnetic eld: :=ZS~B ~n dS;where~nis theperpendicularto particular,if a uniformmagnetic eld~Bapproachesa atsurfacewithanangleofincidence andtheareaof thesurfaceisA, then =j~BjAcos :(3)1 Figure1: Themagnetic eldin theneighborhood of a solenoidwherea currenti equation(2).Sincethecrosssectionof thesolenoidhasconstant area,the ux owingthrougha genericsurfacenormalto thesolenoidaxisis constant areaof thecrosssectionof thesolenoid,sincethemagnetic eldis parallelto thesolenoidaxis( , = 0),the uxmay be computedsubstitutingequation(2)in equation(3): =K0N i;(4)whereK0:= forcein a rotatingcoilOnceunderstood thata movingchargegeneratesa magnetic eld,whataboutif a movingchargepassesthroughan exogenousmagnetic eld?

3 TheLorentz forceequationis a goodstartingpoint to understandthemechanicalaspectsof forceis theforceexperiencedby a movingchargein an electromagnetic ~Edenotestheelectric eld,~Bdenotesthemagnetic eld,qis a chargemovingwithaspeed~vin thespace,thenthechargeexperiencesa forcegivenby~F=q(~E+~v ~B):(5)Now, considera wirewherea uniformcurrentiis eldtobe zero(thisis thecaseof a DCmotor),withreferenceto thequantity of chargedqfoundin anin nitesimalsectiond~`of thewire,theforcein equation(5)may be computedas afunctionofi:d~F=dq ~v ~B=dqd~`dt ~B=dqdtd~` ~B=i d~` ~B:2liiB FFigure2: Forceexperiencedby a current-carryingconductorlocatedin a uniformmagnetic turnsoutthat,if denotestheangleof incidencebetweenthemagnetic eldandastraight wireof lengthl, thenthemagnitudeof theforce~Fis givenbyj~Fj=i lj~Bjsin ;(6)and~Fis orientedontheperpendicularto theplanespannedby themagnetic eldandthewirefollowingtheright-hand-scre wrule(seeFigure2).Now, onthebasisof equation(6),considera rigidrectangularcoilconstitutedby a singlewirewherea currenti ows,suitablylocatedin anuniformexogenousmagnetic seenfromFigure3;1iflis thelengthof thewireperpendicularto themagnetic eld,thentwo forcesareappliedto of equation(6)is 2(dependingonwhich sideof thecoilis considered),thenit turnsoutthatthemagnitudesof thetwoforcesarethesame:j~Fj=j~Bji l:(7)Sincethecoilis square,thecurrenti owsin oppositedirectionsonthetwo sidesofthecoil;thus,thetwo forcesFgeneratea torqueTexertedat thecenterof thecoilthatis dependent ontheangularposition of thecoilwithrespectto themagnetic ,if thelengthof anedgeof thesquareisd, thenT= 2j~Fjd2sin =j~Fjdsin.

4 (8)whence,takinginto account equation(7),equation(8)yieldsT=j~Fjdsin =j~Bji l dsin :(9)1 Asusual,a dot meansthatthe owingcurrent is exitingthepage,whilea cross meansthatthecurrent is d=2 BFigure3: Torqueexperiencedby a coilin a uniformmagnetic 2 2 TFigure4: Pro leof thetorqueTexperiencedby thecoilin a magnetic eld,as therotationangle equation(9)andnoticethatthetorquehighlyd ependsontherotorcoilangularposition . Imaginethatthecoilis in therotorof a motor ;thentheresultingtorqueis highlydependent onthemotorposition;moreover,if noloadtorqueis present, themotorkeepsturningclockwiseandcounter- clockwise;as a matterof fact,if thecoilturnsof anangle , thetorqueexertedhasthesameamplitudebutop positesign(seeFigure4).Sincethegoalis to have themotorto exerta constant torqueforany position , thesolutionadoptedis to insertontherotorshafta commutatorconstitutedby two segmentsconnectedto therotorwindingsandbrushesthatslidebetwe enthesegments as therotorturns(seeFigure5).Insuch a con guration,thesignof thecurrent owingin thecoilchangesat each halfrevolutionof themotor;thus,if thesegments areproperlypositionedwithrespecttothecoi lposition,thetorquepro leof Figure4 willbe suitablyinvertedduringhalfrevolution(see Figure5).

5 Now, oncethissolutionis adopted,althoughthetorquehasalways thesamesign,stillitis highlydependent ontherotorposition;theobvioussolutionto thisproblemis to increasethenumber of coilsin therotorandthesegments of thecommutator,connectingeach pairofoppositesegments to a coilin such a way thatwhenthebrushesactivatethatcoil,thero torangleis = 2( , themaximumtorqueposition).It turnsoutthat,ifNindependent4 T 0B 0iFigure5: Torqueexertedby themotor,whena two segments commutatoris ,thecommutatorwillhave 2 Nsegments andthetorquepro lewillbe constitutedby 2 Nhalfsinusoids(relatedto thetorquesTiexperiencedby eachcoil)overlappedin onesingleperiod (seeFigure6, where cdenotesa genericcommutationangle).In such a way thetorqueripplemay be decreasedas much as neededby increasingNandtheresidualripplemay be assumedto be lteredby themechanicalsystem,so thattheresultingtorquemay be approximatedasT=i l dj~Bj:(10)Finally, consideringthat,by equation(3),the ux owingthroughtherotoris pro-portionaltoj~Bj(notethattheangle in equation(3)is equalto =2; as a matterof factthe uxis assumedto be owingstraight insidethemotor),it turnsoutT=K i;(11)whereK :=l d= EMFe ectIn Figure4, therotorconductorscarrycurrent suppliedfromanexternalsourceandarelocate din anexogenousmagnetic eld;whence,forcesareexperiencedby thecoilandatorqueis ,dueto therotationof thecoil,theconductorsthemselves cutthemagnetic ux,thus generatinganelectro-motive force( , aninducedvoltagein therotorwindings)whosemagnitudemay be computedby meansof theFaraday'slaw of induction.

6 Thislaw statesthatif thereis a variationof the ux c owingin theinternalsurfaceof a closedwire,thenanelectro-motive forceeis inducedin thewire,accordingto:e= d cdt:(12)5Ti T c c c c c c c c cFigure6: Torqueexertedby themotor,whena multiplesegments commutatoris thisparticularcase,noticethatthe ux owingin thecoilis givenby equation(3)substituting withtheangle (t) of themotorandAwiththeareaof theinternalsurfaceof ,computingthederivative in equation(12),theback electro-motiveforceor,moreeasily, theback EMFmay be computedase= ddt(j~BjAcos (t))=j~BjA_ (t) sin (t)=j~BjA !(t) sin (t);where!:=d (t)dtdenotestheangularspeedof ,duetothepresenceof thecommutator,thecoilalwaysoperatesin aneighborhood theposition = =2, thensin (t) 1 andtheback EMFmay be writtenase=j~BjA !(t).Finally, similarlyto thecasestudiedin equations(10)and(11),theback EMFmay beexpressedas a functionof the ux ,andit canbe veri edeasilythate=K !;(13)whereK is thesameconstant as theonein equation(11).2 Thebasicequationsof theDCmotorThesetof equationsherereported,constitutesa modelof theDCmotor,which may6be representedas a thismodel,withrespectto a realmotorare1.

7 Theassumptionthatthemagneticcircuitis linear(such an assumptionis approximate,sincethemetalpartsarenotperf ectlysmoothandthereis some uxdispersioninsidethemotor,moreover,duet o saturationof themetal,equation(4)does notholdforhighvaluesofi);2. theassumptionthatthemechanicalfrictionis onlylinearin themotorspeed;namely,onlyviscousfriction is assumedto be present in themotor(such anassumptionisapproximatesinceCoulomb frictionis usuallyexperiencedin motors). ,in a DCmotor,themagnetic uxisgeneratedby electricalpower is transformedin mechanicalpower istheoneexplainedin ,theactualimplementationsof thisresultarevarious,as amatterof fact,sincethemagnetic eldBarisesfromthestatorcoils,notonlyther otorcoilsmay rotatewithrespectto thestator,butalsothestatorsupplymay rotate(inanelectricalsense)by increasingthenumber of coilsandby a ,a simplemodel,which appliestotheabove cases(providedpropertransformationsarepe rformedonthesystemvariables)willbe themotorwillbe assumedto have a singlecoilcharacterizedby aninduc-tanceLedueto thewindingsanda resistanceRedueto dispersionsin theconductor(seeFigure7).

8 Theequationassociatedwithsuch anelectriccircuitis givenbyve(t) =Lediedt+Reie:(14)Sincerelation(14)is linear,by transformingin theLaplacedomainthesignals,it canbewrittenie(s)ve(s)=Ke1 + es;(15)whereKe:=1 Reis thestatorgainand e:=LeReis assumedto be a singlecoilcharacterizedby inductanceLaandresistanceRa(seeFigure7), butit hasto be takeninto account theback EMFof themotorin equation(13).Theequationassociatedwithsu ch anelectriccircuitis givenbyva(t) =Ladiadt+Raia+e:(16)Again,sincerelation( 16)is linear,by transformingin theLaplacedomainthesignals,itcanbe writtenia(s)va(s) e(s)=Ka1 + as;(17)7whereKa:=1 Rais therotorgainand a:=LaRais : Electricalequivalent schemeof a account theresultsof (inparticular,equations(11)and(13)),thef ollowingtwo equationsholdfortheback EMFeandthetorqueexertedby themotorTM:TM=K ia;(18)e=K !:(19)Asregardsto the ux owingin themotor,the ux is generatedby ,as remarkedin ,the uxis proportionalto thecurrentie,by equation(4),it canbe writtenTM=K ieia;(20)e=K ie!

9 ;(21)whereK:=K now dealwiththemechanicalrepresentationof torque,whilesuppliedby ,which is characterizedby therotorinertiaJandtheviscousfrictioncoe cientF. It hasalsoto be takeninto account thatin any operatingenvironment a loadtorqueis exertedonthemotor;then,ifTLis theloadtorque,thefollowingequationmay be written:TM TL=Jd!dt+F !:(22)Asfortheelectricalcase,alsoforthem echanicalequations,a lineartransferfunctionmay be associatedto equation(22):!(s)TM(s) TL(s)=Km1 + ms;(23)whereKm:=1 Fis themechanicalgainand m:=JFis motorsOften( , in roboticapplications),thespeedrequiredby theloadis too low as comparedto thenominalspeedof thiscases,gearsareintroducedbetweenthemo torandtheload,thus reducingby a factorntheangularvelocity of dampingandinertiadueto thepresenceof theadditionalrotatingcogwheelsof thegear,themechanicalcouplingbetweenthel oadandthemotoris alteredby correctlyunderstandthee ectsof thegear,the stthingto remarkisthatdampingandinertiaarenotthesa meif measuredat theinputor at theoutputof areinterestedin thecompletecharacterizationof themotorblock, let'sreferto theoutputquantities,anddenotebyFGtheinte rnaldampingof thegearandbyJGtheinternalinertiaof ,noticethat,sincethepower exertedby themotoris thesameat theinputandattheoutputof thegear,denotingbyT0 Mand!

10 0thetorqueandthespeedat theoutputofthegear,it canbe writtenTM!=T0M!0;and,since!0=!=n, thenT0M=n equationsin equation(22),andtakinginto account theincreaseof dampingandinertiadueto thecogwheelsof thegear,it turnsout3T0M TL= (JG+n2J)d!0dt+ (FG+n2F)!0:(24)Bya comparisonbetweenequations(22)and(24),it is stressedthatthepresenceof thegearhighlyincreasestheinertiaandtheda mpingof themotorfromthepoint of viewof addition,whena gearis insertedin anactuatorsystem,backlashis experiencedonitsoutputdueto thecouplingbetweenthecogwheelsof riseto nonlinearitiesthatmay leadto instability e thisreason,especiallyin highprecisionsystems,direct-drive motorsmay exertreasonabletorquesatlow speeds,whencetheydonotneedgearsto drive ,thesemotorsmay notbe adoptedforhighpower tasks, diagramof theDCmotorByimplementingequations(14),(1 6),(20),(21),and(22)in a nonlinearblock diagram,theresultshownin Figure8 is theblock diagram,thevariable represents therotorangularposition(whence,!=_ ).Thenonlinearmodelresultsin a two-input,one-outputmap,havinga disturbanceinputTLandwithfourstatevariab les,relatedto theenergystoredin theinductanceLe; theenergystoredin theinductanceLa;2 Thenominalspeedof themotoris thespeedcorrespondingto thee exertedby theload,whenceit shouldnotbe +ReieKKva=Ladiadt+Raia!