Section 7 - Controllers for Electric Drive Systems

Transcription

1 ELEC463 Scton 7 - Controllrs for 7. Introducton An lctrcal drv normally has a numbr of control loops rgulatng a numbr of varabls of th drv. Of cours som drvs, such as V-f drvs, may b opratd n opn loop. Howvr such drvs ar usd n low prformanc adjustabl-spd drvs whch ar usd n applcatons such as convyrs, pumps, and drlls n whch only slow adjustmnt of spd s rqurd. Many procsss n today s ndustral drvs and applancs rqur fast rspons and prcs control of poston, spd and torqu, n ordr to mt strngnt spcfcatons of th drvn procsss. In ths scton w wll dscrb what rols control loops play n a drv systm and thr basc structurs. Powr Input, Controllrs Powr Convrtr Motor Load Fgur Snsors for lctrc drv systms h hrarchcal control systm consstng of currnt, spd and poston, (n ths ordr of hghst to lowst dynamcs) as ndcatd n th top fgur of pag 9, Scton, rqurs snsors for ach stag. hs ar for currnt, spd and poston. Modrn spd snsors mploy dgtal puls countrs whch count pulss from magntc or optcal shaft mountd puls gnrators. hs combn spd and poston snsors nto on. Som drv applcatons also rqur voltag snsors, for xampl, for DC drvs wth fld control for opraton abov th bas spd. In such an applcaton, th voltag at th motor (armatur) trmnals must rman lmtd to a maxmum valu whn fld s ncrasd n ordr to rduc th spd. 7.. Currnt snsors Shunt currnt snsors Currnt snsors can b n th form of shunts whch carry th currnts to b snsd. hs shunts hav vry low rsstanc (typcally n mllohms), so as not to mpd th flow of currnt apprcably. h voltag across ths shunts (also n mllvolts, typcally) s proportonal to th currnts thy carry. hr man dsadvantags ar: () nductanc of th flamnt (for wr-wound lmnt) () tmpratur snstvty (for rsstv flamnt) () th lack of solaton from th powr lns (v) th nd for dffrntal amplfrs wth hgh common-mod rjcton rato (CM). 7. Controllrs for F. ahman (EE, UNSW) lctrc drv systms Octobr 6, 05

2 ELEC463 Shunt snsors ar good ovr a wd frquncy rang, DC to a fw hundrd khz. Wth nductanc mnmzd shunts, th cut-off frquncy (-3dB frquncy) of rsstv shunts can rach svral MHz, or vn F rang. DC or AC Powr supply lns Convrtr shunt Non solatd dffrntal amplfr Wth hgh CM Fgur 7. Hall currnt snsors Hall currnt snsors mploy a hghly prmabl cor to produc a magntc fld, whch s proportonal to th currnt bng masurd n th currnt carryng conductor, n an ar-gap. h Hall gnrator, mad of matrals lk G, S, InSb, In, As, and GaAs and stuatd n th ar-gap, ncluds a constant currnt sourc drvng a tmpratur- compnsatd constant currnt through th Hall lmnt. h voltag across th Hall gnrator s amplfd usng addtonal crcutry and s solatd from th powr crcut. S Fgur 7.3. h tmpratur drft s typcally blow 0.05%/C. h lnarty of th snsor dpnds also on th lnarty of th magntc cor and s brought about by propr szng of th ar-gap lngth. h zro currnt offst s anothr drawback whch s lmnatd by usng addtonal crcutry. Fgur 7.3 An opn-loop Hall currnt snsor. 7. Controllrs for F. ahman (EE, UNSW) lctrc drv systms Octobr 6, 05

3 ELEC463 A closd-loop Hall snsor mploys a col of a larg numbr of turns and a hgh-gan lnar pushpull amplfr and a load rsstor to produc a sgnal proporton to th currnt bng masurd as shown n Fgur 7.4. Fgur 7.4 Closd-loop Hall currnt snsor h amplfd Hall voltag drvs a currnt through th col whch cancls th B fld n th argap.., brngs th Hall voltag to zro. Bcaus th Hall snsor s subjctd to nar zro B fld, vrtually no tmpratur compnsaton s rqurd and th lnarty of th magntc cor bcoms rrlvant. Wth, say, 000 turns on th cor, th snstvty of th snsor s ma/a. h out voltag can b asly slctd by slctng th rsstanc of th load rsstor. h opn-loop Hall currnt snsor oprats wth low powr consumpton, whch s also ndpndnt of th currnt bng masurd. hs s not so wth th closd loop Hall currnt snsor, wth nablty to handl larg ovr-currnts du to th addtonal powr loss n th col. h addtonal wght, volum and powr dsspaton rqurmnts do not xst for th opnloop Hall currnt snsor. Modrn Hall currnt snsors hav -3dB frquncy of 00kHz. A vry rcntly dvlopd currnt snsor tchnology uss magnto-rsstv lmnts whch chang thr rsstanc n th prsnc of a magntc fld. hs hav bandwdths hghr than MHz and thr solaton proprty bttr than Hall snsors. Som HV drv systm manufacturrs hav alrady startd to us ths snsors to tak advantag of thr lattr qualty. Not that shunt rsstors (of hgh valus) and Hall snsors can b asly confgurd to srv as voltag snsors. 7. Controllrs for 3 F. ahman (EE, UNSW) lctrc drv systms Octobr 6, 05

4 ELEC Spd and poston snsors Analogu tacho-gnrators Shaft-mountd analogu tacho-gnrators produc an output voltag whch s proportonal to th spd of th motor shaft. DC and AC typs ar avalabl. DC tacho-gnrators ar small brushd DC gnrators whch ar smlar to a brushd DC machn wth prmannt magnt xctaton n th stator. h rotor (armatur) wndng ar n rotor slots and connctd to th outsd va commutator sgmnts and brushs. h numbr of commutator sgmnts s lmtd. h DC output voltag, whch s proportonal to th shaft spd, suffrs from th brush contact drops and th nos du to commutaton whch vars n ampltud and frquncy wth spd. Wth such spd snsors, ths nos s rsponsbl for th maxmum bandwdth lmtaton a drv can achv. AC tacho-gnrators ar small AC gnrators wth prmannt magnt xctaton n th rotor and thr-phas wndngs n th stator whch ar smlar to an AC machn. h ampltud and frquncy of th AC output voltags ar proportonal to th spd of th shaft. A thr-phas dod brdg rctfr s usd to convrt th AC voltags to DC. h DC output voltag, whch s proportonal to th shaft spd, suffrs from th dod forward voltag drops and th rppls n rctfr output vary n ampltud and frquncy wth spd. hs nos s rsponsbl for th maxm bandwdth lmtaton of a drv, wth such spd snsors. Dgtal tacho-gnrators Dgtal tacho-gnrators produc a crtan numbr pulss pr rvoluton of th shaft, whch ar countd ovr a samplng prod to obtan th spd of th shaft. hs gnrators may mploy lctromagntc (rsolvr) or optcal (ncodr) dvcs. solvrs (synchro-rsolvrs) hs shaft mountd dvcs hav thr wndngs all stuatd n th stator, as shown n fgur 7.5 (a - c). h -wndng has snusodal xctaton n th rang of khz. h rotor prsnts a varabl rluctanc couplng of th wndng wth th othr two S wndngs whch ar n quadratur (90 dgrs dsplacd). h S wndngs hav vn multpl pols of appropratly hgh numbrs. h ampltuds of th nducd voltags n th S-S and S3-S4 wndngs ar sn and cosn modulatd. Fgur 7.5(a) Cut-out vw of rsolvr 7. Controllrs for 4 F. ahman (EE, UNSW) lctrc drv systms Octobr 6, 05

5 ELEC463 Fgur 7.5(b) Fgur 7.5(c) solvr output sgnals. h followng sgnals for th rsolvr ar avalabl: V V snt m V V K sn sn t s m V V K cos sn t s m K and K ar qual and angl s zro. hs sgnals ar passd through a solvr to Dgtal Convrtr (DC) to xtract th sncosn wavs usng pak dtctors, multplcaton and dgtzaton va ADCs. ato of ach sgnal rlatv to th rfrnc V s usd, so that changs to th absolut valus do not affct th accuracy of masurmnt. h solvr and th DC ar normally suppld by th sam supplr. 7. Controllrs for 5 F. ahman (EE, UNSW) lctrc drv systms Octobr 6, 05

6 ELEC463 S S34 DC = Pak dtctor Multplr VCO, Countr Latch DAC Spd Count Clockout Drcton Fgur 7.6 Encodrs Currntly wdly usd, ths shaft mountd optcal dvcs gnrat puls tran whch ar proportonal to spd. hr ar two typs, th ncrmntal and absolut ncodrs. Fgur 7.7 Incrmntal ncodr schmatc In th ncrmntal typ, collmatd lght bams from thr sutably locatd LEDs ar ntrsctd by gratngs on a movng dsc. hr bams produc thr dgtal pulss A, B and Z at th outputs of thr rcvng optcal snsor transstors. h frquncy of ths pulss s proportonal to th spd of th shaft whch rotats th dsc. h A and B pulss ar n quadratur,.., 90 dgrs out of phas. hs pulss can b usd to obtan th drcton of rotaton and th spd. For spd snsng, th pulss ar countd n an up-down countr (not shown n Fgur blow) accordng to th drcton of rotaton, usng up to 4 dgs of th pulss n a cycl, gvng a rsoluton of up to 4 tms puls numbr pr rotaton. Dscs wth a fw hundrd to svral thousand pulss wth bandwdth hghr than 300kHz ar avalabl. 7. Controllrs for 6 F. ahman (EE, UNSW) lctrc drv systms Octobr 6, 05

7 ELEC463 Fgur 7.8 Encodr puls multplcaton and drcton snsng h Z puls occurs only on n a rvoluton and s usd to rst th countr to zro. Onc t don, th contnt of th up-down countr gvs th absolut poston of th shaft. h countr output s normally Gry codd, so that ts contnt can b rad wth rror no mor than count. h absolut ncodr hav Gry codd tracks, as many as th numbr of bts of ts rsoluton, on th dsk and t also uss as many collmatd LED lght sourcs and dtctors as th numbr of tracks (bts). h dtctors provd Gry codd absolut poston of th shaft drctly. Fgur 7.9 Dscs for a 4- and a -bt absolut ncodr. 7. Controllrs for 7 F. ahman (EE, UNSW) lctrc drv systms Octobr 6, 05

8 ELEC An lmntary poston control systm Consdr th poston control systm of fgur 7.0 n whch th drv movs th pontr at th rght-hand sd from on poston to anothr, accordng to th poston of th pontr at th lfthand sd. h rfrnc poston s st manually by a potntomtr th output of whch s a voltag proportonal to. A sparatly DC motor drvs th load pontr through a rducton gar box wth a spd rato of N: from motor to load. + a + K c G I F m GB J L o N: Fgur 7.0 An lmntary poston control systm W assum that th motor dvlopd torqu s gvn by m K (7.) a whr a s th motor armatur currnt suppld by th convrtr of gan G and K s th torquconstant of th motor. hs convrtr dlvrs th currnt a whch s drctly proportonal to th rfrnc sgnal to th convrtr (xstnc of a currnt control loop s thrfor mpld). hus, G (7.) a W assum that th load has no frctonal and fxd trms and that t only conssts of momnt of nrta J L. h total motor and load nrta, rfrrd to th motor shaft, s J,. h torqu balanc quaton s dm d m m J J (7.3) dt dt Hr s th load angular poston of th load and m motor shaft. Hnc N s th angular poston of th d dt K a KG (7.4) JN JN h rror sgnal s assumd to b proportonal to rror btwn th nput and th output shafts, producd by subtractng th sgnals from th two potntomtrs. h angular poston rror s amplfd by a factor K c n an rror amplfr. hus K K (7.5) c o c 7. Controllrs for 8 F. ahman (EE, UNSW) lctrc drv systms Octobr 6, 05

9 ELEC463 Substtutng (7.5) nto (7.4), c d KGK (7.6) dt J N d d (7.7) dt n n n dt hs quaton dscrbs an un-dampd scond-ordr systm wth an un-dampd natural frquncy gvn by KGK c n rad/sc and fn n J N Hz (7.8) hs mpls that f th nput shaft s movd by th output shaft wll contnu to oscllat about th fnal poston ndfntly, as ndcatd n fgur 7.. hs s du to th fact that no mchansm for nrgy dsspaton was ncludd n our modl of th drv. 7.. Dampng usng vscous couplng Fgur 7. Lt us now consdr som vscous frcton couplng as ndcatd n th fgur 7.(a). h dampng factor of th couplng rfrrd to th motor shaft s D n Nm/rad/sc. hus d m dm m c J D K GK dt dt = J d d KGK c D (7.9) dt dt N Equaton (7.9) s smlar to th quaton of dynamcs of a dampd scond ordr systm gvn by n d dt D d (7.0) J dt n 7. Controllrs for 9 F. ahman (EE, UNSW) lctrc drv systms Octobr 6, 05

10 ELEC463 hs quaton dscrbs a dampd scond-ordr systm. h rspons to a stp nput of s ndcatd n fgur 7.(b). Undrdampd Motor + GB Good Vscous Couplng, D L + Ovrdampd t Fgur 7.(a) Fgur 7.(b) h dampng factor whch causs th oscllatons to d down may b n th frcton wth ar, barngs or a mchancal vscous dampr; or t may b part of th load as n a strrr. A numbr of paramtrs n th abov systm can b adjustd to nflunc th systm rspons, th formost bng th gan K c whch s mbddd n an lctronc crcut. h mchancal dampr rqurd for a satsfactory dampd rspons s oftn qut larg compard to th sz of th motor. It s also wastful n trms of nrgy and xpnsv. (In spt of ths drawbacks, mchancal vscous damprs ar wdly usd n low-cost drv applcatons, mplyng hug loss of nrgy and sz of such drv systms) Constant load torqu L and vscous dampng D L Lt us now consdr that th load also has a fxd torqu L. hs load torqu rfrrd to th motor shaft s L Lm (7.) N h systm quaton thn bcoms d d Lm KGKc J D K (7.) dt dt N N whr, and K K GK N c s th systm gan. Bcaus bcoms d dt d and dt d d for a fxd rfrnc poston, quaton (7.) dt dt d d Lm J D K (7.3) dt dt N 7. Controllrs for 0 F. ahman (EE, UNSW) lctrc drv systms Octobr 6, 05

11 ELEC463 In th stady-stat, all transnts wll hav dd down, so that Lm K N Lm L L NK N KGK c / N NKGKc (7.4) hus, f a fxd load torqu s prsnt, a poston rror wll also xst for ths systm. o rduc t to zro, nfntly larg systm gan or gar rato wll b rqurd. 7.. Dampng va spd fdback As mntond n th prvous scton, adquat dampng through mchancal vscous couplng s xpnsv and wastful. Consdr th systm ndcatd n fgur 7.3 whch ncluds a spd snsor or tacho-gnrator () on th motor shaft. h spd snsor s a small (a small DC gnrator or a puls countr) whch s coupld to th motor shaft. Agan, all frcton s nglctd n th systm of fgur o K c G 3 = K 3 m I F a m, m GB N: J L +, Fgur 7.3 A poston controllr wth a spd mnor loop. h tacho-gnrator producs sgnal 3 whch s proportonal to th spd m of th motor. h systm quatons ar K c K3m (7.5) h torqu balanc quaton of th systm s d c c 3m m J K GK K GK K (7.6) dt d d J K K K (7.7) dt dt whr K KGK c and N K KGKcK3 (7.8) Equaton (7.7) s dntcal to (7.9) for th systm wth mchancal dampr, whch mpls that th vlocty dampng va spd fdback ntroducs dampng n th sam way as a mchancal 7. Controllrs for F. ahman (EE, UNSW) lctrc drv systms Octobr 6, 05

12 ELEC463 dampr dos. (Not that ths s a cas of drvatv fdback, whr vlocty s th drvatv of poston fdback). In gnral, a mnor or nnr loop wth a vlocty (or rat) fdback provds dampng to th major or outr loop Error n poston followng Poston control systms oftn hav to track a movng poston rfrnc. hs mans that th rror should b small whn th nput rfrnc s movng at a rasonabl rat. Consdr th systm wth mchancal vscous dampr and a load torqu L. h systm quaton s d d (7.9) dt dt N Lm J D K In th stady-stat, acclratons wll hav dd down so that Dd Kdt NK Lm In th stady-stat, f th targt attans th spd of th movng rfrnc,.., D Lm K NK, o (7.0) (7.) whr s th spd of th movng rfrnc (th ramp rat of ). hus th followng rror s proportonal to th dampng factor tms th rat at whch th poston rfrnc movs. If a tacho-gnrator s usd for spd fdback n addton to som mchancal dampng, th trackng or followng rror s gvn by D K 3 Lm o K NK (7.) It s clar from (7.) that tacho-gnrator (or rat) fdback ncrass th followng rror. It can b rducd by usng hghr systm gan and gar rato. Fgur Controllrs for F. ahman (EE, UNSW) lctrc drv systms Octobr 6, 05

13 ELEC Spd control In a spd control systm, th spd rfrnc may b st by a potntomtr or t may b nput from lswhr. h spd fdback sgnal s obtand from a tacho-gnrator. + K c G I F a m GB J L 3 = K 3 m N: Fgur 7.5 Usng th analyss of prvous sctons d d Lm KGKc J D K (7.3) dt dt N N whr and K K GK N c. For a fxd spd rfrnc, d d dt dt, so that d Lm J D K dt N (7.4) d Lm J D K D dt N (7.5) Assumng that n th stady-stat, spd varatons hav dd down, Lm D N D K (7.6) hus, th stady-stat spd rror s dtrmnd by load torqu, vscous frcton, rfrnc spd and systm gan as ndcatd n quaton (7.6). 7. Controllrs for 3 F. ahman (EE, UNSW) lctrc drv systms Octobr 6, 05

14 ELEC Cascad control structur It was shown abov that t s advantagous to hav a spd (.., rat or drvatv) control as th mnor loop of a poston control systm. It s also apparnt that an armatur currnt control loop, whch lads to torqu control bcaus m K a K G. For th DC motor usd n ths xampl, a s proportonal to th torqu appld to th load. hus, th nnr loop for spd was a torqu control loop. Not that for a fxd nrta systm, torqu s drctly rlatd to acclraton, so that ths nnr loop may also b sn as an acclraton control loop. hus, w hav usd, wthout bng awar, a control loop for th drvatv of spd (, acclraton) as th mnor loop of th spd control systm. Not also that th nnr currnt control loop allows th maxmum currnt through th motor (or th maxmum acclraton of th load) to b lmtd by lmtng th output of th spd controllr. hs fatur of th nnr currnt loop has th addd advantag of protctng th motor, or th powr smconductors of th convrtr drvng th motor. An nnr currnt loop s thus almost mandatory n all drvs. Othr control loops such as flux and voltag controllrs may also b mployd n ordr to mt crtan rqurmnts of spcfc drvs. h cascadd, mult-loop control structur mntond abov and ndcatd n fgur 7.6 works wll whn t s assumd that succssv nnr loops ar progrssvly fastr by a factor from 6-0 n ach stag. hs s oftn known as d-couplng of ach stag from ts followng or succdng stags, so that ach stag (or control loop) can b dsgnd ndpndntly of othr loops. rf m d/dt CONV M E Poston Controllr Spd Controllr orqu and Flux Controllr Fgur 7.6 Hrarchcal cascadd drv control structur. In ths structur, th nnrmost loop has th hghst bandwdth. h trm bandwdth normally rfrs to th ablty of a control loop to follow ts rfrnc nput. If th bandwdth of ach subsystm follows ths rul, thn ach stag can b thought of as ndpndnt of othrs and thrfor can b dsgnd and tund n solaton. Wth ths approach, th dsgn of an laborat drv systm s smplfd. Furthrmor, ndvdual sctons of th drv can b asly tstd, tund and mantand. Important physcal constrants, such spd lmt, currnt ampltud and rat lmts (, acclraton lmt), voltag lmt, flux lmt, and so on, can b ndvdually mposd on th rlvant stag, kpng all saf opratng lmts undr tght control. It can b argud, howvr, that such cascadd control systms ar nhrntly slow, bcaus control sgnals hav to pass through many control loops to tak ffct. In ordr to mtgat such dlays, fd-forward and dsturbanc obsrvr tchnqus whch bypass som of th fdback loops ar also usd n lctrcal drvs. In ths scton, w wll only consdr th cascadd, hrarchcal, mult-loop structur, comprsng smpl wll-known controllrs. 7. Controllrs for 4 F. ahman (EE, UNSW) lctrc drv systms Octobr 6, 05

15 ELEC ypcal Controllrs A. Proportonal Controllr A proportonal controllr provds for a straght gan to amplfy th rror sgnal. It has no frquncy dscrmnatory proprts. Wth th nput and fdback rsstanc valus ndcatd n th fgur 7.7, th total gan of th controllr s K p = K+ whr K /. - K p K Fgur 7.7 For ths controllr, K B. ransnt vlocty fdback controllr (7.7) It was shown n sctons that a followng rror wll xst n a control systm whn a movng (ramp) rfrnc s trackd. Vlocty (.., rat or drvatv) fdback whch s usd to damp th systm rspons ncrass ths rror. o ovrcom ths followng rror du to vlocty fdback, transnt vlocty fdback can b usd as ndcatd n th fgur 7.8. o K c K 3 C Fgur 7.8 h spd (vlocty) sgnal s passd through an C crcut at th nput of th amplfr crcut. An nput currnt occurs only whn th spd sgnal changs. In th stady-stat, th capactor s fully chargd, so that no addtonal followng rror n th stady-stat du to th vlocty fdback can xst. Proof: Lt us assum that th gan of th amplfr s unty. h vlocty sgnal s K 3. Laplac transform of th rror sgnal s gvn by 7. Controllrs for 5 F. ahman (EE, UNSW) lctrc drv systms Octobr 6, 05

16 ELEC463 E K c K3 sc (7.8) sc Assumng no frcton and fxd load torqu s c m 3 KGK KsC (7.9) J sc sc J KGK c Ns sc K NsC In th stady-stat, acclratons hav dd down, so that 3 s 0 s. hus (7.30) s 0 whr = C. (7.3) s s In th stady-stat, whn th nput and output mov at th sam vlocty, s s (7.3) hus, no followng rror wll occur n th stady-stat whn vlocty fdback s usd. Not that w hav assumd that D and L ar nglgbl. C. Phas advanc or rror rat fdback controllr hs s a lag-lad controllr whr th rror rat s fd back. C = K C K c o = K Fgur 7.9 Agan consdrng a poston control systm wth a purly nrta load, KGK c s E (7.33) J N whr E s th Laplac transform of rror. h abov ntwork can b analyzd by tratng th Laplac oprator s as th complx frquncy j. hus 7. Controllrs for 6 F. ahman (EE, UNSW) lctrc drv systms Octobr 6, 05

17 ELEC463 o E s C s C (7.34) sc E s C E E (7.35) o KGK c s sce sceo JN (7.36) In th stady-stat, whn acclratons hav dd down s C s C 0 (7.37) snc Kc and o Kc. hus Cs Cs (7.38) In th stady-stat, whn th vlocts ar qual,, s s and th rght hand sd s zro. hus, dpndng on th rlatv valus of and, may lag or lad. In othr words, n th absnc of D and L, no followng rror s ntroducd f =. D. Intgral Controllr In th transnt vlocty and rror rat fdback schms, th followng rror wll xst f vscous frcton and load torqu ar prsnt. If such loads ar prsnt, th systm gan has to b nfnty to rduc th rror to zro. Vry larg gans wll mak any physcal systms unstabl, unlss bandwdth lmtatons xst. On way to mploy nfnt gan n th stady-stat s to us and ntgrator. h ntgrator amplfs th stady-stat rror as long as t xsts and allows th controllr output to grow untl th rror s lmnatd. Normally, a proportonal-plus-ntgral (PI) acton s usd. h drvatv componnt of a PID controllr s normally not usd n th control systm of a drv systm, bcaus of th usually nosy fdback sgnals n a drv systm. Instad, drvatv sgnals ar obtand through snsors such as tacho-gnrators and currnt snsors. h structur of a PI controllr s ndcatd n fgur Controllrs for 7 F. ahman (EE, UNSW) lctrc drv systms Octobr 6, 05

18 ELEC463 C = K K c o = K Fgur 7.0 Consdr that th poston control systm has som vscous frcton. W assum that o c K (7.39) E K SC c (7.40) Kc s (7.4) As bfor, J s Lm K GKc Ds (7.4) N N s Lm KGKc sjs sds s s N N (7.43) In th stady-stat, th lft-hand sd s zro, so that s 0 and bcaus s s 0 n th stady-stat. hrfor, wth ths ntgral controllr, thr wll b no stady-stat rror vn n th prsnc of D and L. h controllr crcuts dscrbd abov ar usually mplmntd n analogu crcuts usng opratonal amplfrs. Now-a-days dgtal mplmntatons wth DSPs ar gradually rplacng such controllrs. Dgtal controllrs wll not b tratd n ths cours. 7. Controllrs for 8 F. ahman (EE, UNSW) lctrc drv systms Octobr 6, 05

19 ELEC Convrtr rprsntaton h transfr charactrstc of a powr convrtr n a drv systm s usually rprsntd as a constant gan lmnt. Whn opraton s dscontnuous, convrtr gan may fall sharply. hs s avodd by propr slcton of drv paramtrs, such as th machn nductanc and swtchng frquncy. Addtonally, th convrtr may also hav som dlay du to ts swtchng natur Convrtr voltag gan Powr Supply c Convrtr K V dc Fgur 7. For contnuous conducton, K rmans constant. For a fully-controlld thyrstor convrtr, v V cos a (7.44) dcmax Usng cosn wghtd frng angl control, c cos Ecmax (7.45) V v K (7.46) dc max a c c Ecmax v a K c Fgur Controllrs for 9 F. ahman (EE, UNSW) lctrc drv systms Octobr 6, 05

20 ELEC463 For a half-controlld convrtr V (7.47) dc max va cos V dcmax K V dc max cos cos c Fgur 7.3 V dcmax c (7.48) whr c For a PWM DC-DC convrtr, th output DC voltag of a convrtr s gvn by Va DVdc K (7.49) c 7.6. Convrtr dlays - c A. Swtchng dlay t s manly dtrmnd by th convrtr swtchng frquncy. 0msc for - phas, F.C. brdg convrtr (50 Hz supply) 3.3msc for 3-phas, brdg convrtr (50 Hz supply) 00sc for PWM choppr wth a choppng frquncy of 0kHz. c c OFF ON OFF Fgur Controllrs for 0 F. ahman (EE, UNSW) lctrc drv systms Octobr 6, 05

21 ELEC463 h dlay d s random, varyng btwn 0 and s, whch s th swtchng prod. h ffctv dlay, statstcally, s half of th maxmum dlay whch s th swtchng prod s. hus, d s.hs dlay s smlar n acton to transport lag n othr control systms. hs s rprsntd by fgur 7.5. c d K s V dc Fgur 7.5 h convrtr charactrstc wth ths dlay may b rprsntd n on of th forms gvn blow: s K d (7.50) s d K K s d If d s small compard to a or m, t can b nglctd. h powr convrtr swtchng frquncy s oftn slctd suffcntly hgh, n ordr to avod d bcomng comparabl to a or m. For a DC motor drv wth nglgbl D and L and for m crcut may thus b rprsntd as n fgur 7.6. a, th motor and th convrtr c s d V a / K K s s a E m Fgur ol of th nnr currnt loop h nnr loop nabls torqu to b controlld drctly. It smplfs th spd controllr dsgn. A contnuous currnt controllr, bng an acclraton controllr, provds dampng n th vlocty loop. Currnt lmts (both n ampltud and rat) can b xrcsd asly through th currnt controllr, thus also lmtng th maxmum currnt that flows through th motor armatur for all condtons of opraton ncludng stallng. h structur of th currnt controlld systm s shown n fgur Controllrs for F. ahman (EE, UNSW) lctrc drv systms Octobr 6, 05

22 ELEC463 * * Convrtr Spd Controllr Currnt ampltud lmt Currnt Controllr Fgur 7.7 Contnuous currnt controllr ppl n currnt fdback may load th swtchng dvcs n th convrtr unvnly. o prvnt ths, th currnt controllr always ncluds a fltr whch ffctvly rducs rppl n th currnt fdback sgnal Currnt control amplfr (CCA) dsgn h nnr currnt loop acts as th torqu loop of a DC motor. Apart from contnuously rgulatng th currnts to thr dsrd valus, currnt control loops allow rat and ampltud lmts of th motor to b appld va th currnt rfrncs. h currnt loop rgulats th armatur currnt ovrcomng changs of th opratng condtons n th drv, such as th back mf and load dsturbanc. h back mf of a drv changs rathr slowly than currnt. hs s bcaus of th larg mchancal tm constant compard to th lctrcal tm-constant. h frng angl or duty cycl of a convrtr may chang abruptly as a stp nput and th actual currnt n th armatur rsponds to ths chang fully bfor th spd, and hnc back mf, changs sgnfcantly. h maxmum gan of th currnt control loop must b dtrmnd takng ths nto account. For th currnt controlld systm, th armatur thus appars largly as a stalld rotor. In othr words, th stalld rotor s th worst (or th hghst gan) stuaton for whch th currnt controllr should b dsgnd. h stalld (or an unxctd motor) motor s ndcatd n fgur 7.8. h armatur crcut of th fr-runnng rotor s ndcatd n fgur 7.9. Powr Supply I arf I a Currnt Controllr Convrtr V a / a s a I a Fgur 7.8 Currnt loop of th stalld or unxctd motor 7. Controllrs for F. ahman (EE, UNSW) lctrc drv systms Octobr 6, 05

23 ELEC463 / a s a K J s D Fgur 7.9 Currnt loop of th motor wth rotor fr to rotat. A wll-dsgnd currnt control loop ovrcoms th dsturbanc of th back-mf and forcs th armatur currnt to follow th currnt rfrnc wthn a bandwdth gvn by /. h bandwdth of th currnt controlld systm s usually dctatd by th rqurd dynamcs of th torqu control loop. hs bandwdth s also wll blow th swtchng frquncy of th convrtr. hus Ia (7.5) I s arf whr s th bandwdth of th nnr currnt loop. Consdrng th worst cas of th stalld armatur of fgur 7.8, th purpos of th controllr s to rndr th currnt systm of fgur 7.30(a) to 7.30(b). hus s / a s s s a (7.5) 7. Controllrs for 3 F. ahman (EE, UNSW) lctrc drv systms Octobr 6, 05

24 ELEC463 I arf Currnt Controllr + Convrtr V a / a s a I a I a (a) I arf + s I a I a Fgur 7.30(b) whr s s th transfr functon of th Proportonal+Intgral (PI) currnt controllr. A s smpl dsgn rul s pol-zro cancllaton so that L a a and a a (7.53) Not that / s th zro-db cross-ovr frquncy or th bandwdth of th currnt loop. Exampl: Gvn: a = 0.5, L a = 5mH, Swtchng frquncy, f s = 300Hz h torqu loop must hav a bandwdth of 50Hz. A PI controllr whch gvs a torqu control bandwdth of 50Hz s dsgnd hr rad/sc a 7. Controllrs for 4 F. ahman (EE, UNSW) lctrc drv systms Octobr 6, 05

25 ELEC463 L 0.05 a a 0. a 0.5 Not that th valu for ncluds th convrtr gan K. must b dvdd by th convrtr gan n ordr to obtan th PI controllr gans. Currnt controllr: G c (s) = 0.s 0.055s Opn Loop ransfr Functon of th systm: A fltr, 0.s / s 0.s 0.003s s F, wth a cut off frquncy wll blow 300Hz s normally addd, so as to lmt th 300Hz swtchng frquncy rppl nto th currnt controllr va th currnt fdback. hs cut off frquncy, as a rul of thumb, s typcally two octavs blow th swtchng frquncy. hs factor s xpctd to gv suffcnt attnuaton to th 300Hz swtchng rppl whch s fd back. Gan db 0 F wthout currnt controllr Currnt controllr, G(c) 0 OLF of th currnt control loop 0 / a / 300 F rad/sc = / = 34 rad/sc log Fgur 7.3 Bod dagram of th nnr currnt (torqu) control loop. h rqurd PI controllr wth fltr s thus gvn by G c 0.s (s) = 0.055s 0.003s 7. Controllrs for 5 F. ahman (EE, UNSW) lctrc drv systms Octobr 6, 05

26 ELEC463 whr th fltr cut-off frquncy s at 300/4 = 75Hz, so that f / h abov dsgn has bn carrd out wthout consdrng th phas margn at th gan crossovr frquncy ( ). If th brak frquncs and F for th controllr ar chosn proprly, th phas margn s xpctd to b adquat. Whn th armatur s fr to rotat, th nput voltag to currnt transfr charactrstc s a scond ordr systm. Howvr, th dsgn prncpl of th currnt controllr rmans th sam as abov. h controllr zro ( + s) s mad to cancl on of th two armatur pols whch s at th hghr frquncy. h fltr consdraton also rmans th sam as bfor. h pol-zro cancllaton wll work fn f a and L a do not chang much durng opraton, so that th cancllaton rmans vald at all tms. Varaton of ths paramtrs wll lad to dtunng of th controllr n a ral applcaton. It may or may not lad to a sgnfcant problm. A mor prcs controllr must b dsgnd f d-tunng s not accptabl for an applcaton. 7.8 Spd controllr dsgn ssus h purpos of th spd controllr s to rgulat th spd of th drv clos to th spd rfrnc. to follow a spcfd spd profl wth spcfd dynamcs. Usually, t can b assumd that th nnr currnt loop s much fastr than th spd control dynamcs (or th spd controllr bandwdth). Irf W assum that Ia Irf wthn th spd control bandwdth. If th load D and Lm s ar nglgbl, th spd control systm block dagram s as shown n fgur 7.3. rf G s (s) Spd Controllr s K g I a m K J s Fgur 7.3 From Fgur 7.3, G(s) s K /Js (7.54) G (s) K K / J s rf s g whr G s (s) s th spd controllr. Wth a hgh-gan spd controllr 7. Controllrs for 6 F. ahman (EE, UNSW) lctrc drv systms Octobr 6, 05

27 ELEC463 G s K K s t g J s (7.55) wthn th spd control bandwdth. hus, rf (7.56) K g h nos n th spd snsor (tacho-gnrator) output sgnal s an mportant factor n th dsgn of th spd controllr. h nos, f passd on to th currnt controllr wll b furthr amplfd and may gv rs to huntng or lmt cycl opraton of th spd loop. hus, th spd control amplfr G s (s) should nclud a sutabl fltr to adquatly fltr out ths nos. Consdr th PI spd controllr and th rst of th systm, wth th currnt controlld systm rprsntd by Ia. I s arf rf s s s Fs s K m J s K g Fgur 7.33 h Bod dagram of fgur 7.34 shows th opn loop and closd loop systm plots. h phas plot has bn omttd, wth th assumpton that th 0dB crossovr frquncy of th OLF gvs th closd-loop systm bandwdth s. Gan db F wthout spd controllr 0 = / 0 Spd controllr, G s (s) OLF of th spd control loop 0 CLF of th spd control loop s / Fs log Fgur 7.34 Bod dagram of th spd control loop. 7. Controllrs for 7 F. ahman (EE, UNSW) lctrc drv systms Octobr 6, 05

28 ELEC Smplfd rprsntatons and control of drv systms Consdr th block dagram of fgur 7.35 n whch ndvdual lmnts (blocks) ar rprsntd n trms of thr transfr functons n th Laplac doman. + Z C (s) - Lm (s) f G A (s) (convrtr + motor) G L (s) (Load) Controllr H F (s) (Fltr) H (s) (Snsor) Fgur 7.35 Hr, G A (s), G C (s), G L (s), H (s) and H F (s) rprsnt th transfr functons of th powr convrtr plus th motor, th controllr, th load, th snsor (of spd n ths xampl) and th fltr followng th spd snsor, rspctvly. h rfrnc nput for spd and th fdback sgnal ar connctd to a summng juncton of an opratonal amplfr through rsstors and f, rspctvly. h abov systm can b smplfd to th systms of fgurs 7.36 and E (s) f / + - / f Z C (s) G A (s) G L (s) H f (s) H (s) Fgur 7.36 E (s) + f / - G C (s) G A (s) G L (s) H (s) H f (s) Fgur 7.37 Not that th output of th fltr followng a snsor s normally th sgnal wth whch th ral world dals wth. 7. Controllrs for 8 F. ahman (EE, UNSW) lctrc drv systms Octobr 6, 05

29 ELEC463 In gnral, f th ndvdual control blocks ar approxmatd as frst-ordr systms and ar mutually d-coupld, manng that ach block oprats n a frquncy band whch s far outsd th frquncy bands of all othr blocks, thn th abov systms can b rprsntd by a transfr functon of th form K G ( s ) (7.57) s s s3 s4 whr s = tc. 3 and 4 gnrally rprsnt th fltrs on currnt and spd fdback sgnals. Whn 3 and 4 ar much smallr tm constants than and, th abov may b approxmatd by G(s) K s s s s (7.58) 7.5. DC motor drv wth domnant armatur and load pols A DC motor spd control systm wth currnt and spd controllrs fall n ths catgory. and rprsnt pols assocatd wth th currnt and spd loops, and 3 and 4 ar for pols assocatd wth fltrs for armatur currnt and spd sgnals. For such a systm, a proportonal plus ntgral controllr s of th form s s s s G c ( s ) (7.59) s o F F h fltr tm constants F and F n th controllr ar small. hs ar rqurd for sak of ralzablty of G c (s), and do not affct th prformanc of th systm wthn th control bandwdth. On optmzaton crtron (Ksslr s) stpulats that, and o K s. Wth ths stpulaton, th opn and closd-loop transfr functons of th complt systm ar gvn by G(s) G(s)G c(s) s V(s) G V(s) G s s s s s s (7.60) (7.6) Not that two fltr tm constants F and F ar ncludd n G c (s) for th sak of ts ralzablty. hs can b rlgatd to frquncs far hghr than th rang of ntrst and can b gnord for furthr analyss of th systm. For a unt stp nput of v, th output v s gvn by, s v( t ) sn 4 t t s (7.6) for t Controllrs for 9 F. ahman (EE, UNSW) lctrc drv systms Octobr 6, 05

30 ELEC463 A typcal output s ndcatd n fgur *(t) (t) 4.7 s t Fgur 7.38 For a DC motor drv wth nglgbl frcton and fxd load torqu, th transfr functon G (s) s of th form G ( s ) K s s ss (7.63) whr s s th sum of a numbr of short tm-constants, assocatd wth snsors, swtchng frquncy and so on. h currnt controllr of th DC motor wth back mf has such a charactrstc. A sutabl PI controllr for ths systm s s s s s Gc( s ) (7.64) s o F F For ths systm, Ksslr s optmzaton crtron stpulats that 4s, 8K and s o (7.65) h Opn-Loop transfr functon s thn OLF 4ss 8s s s s h closd-loop transfr functon of th complt systm s thn CLF: V( s ) 4ss (7.66) V 3 3 ( s ) 4ss 8s s 8s s h pak ovrshoot of ths systm to a unt stp unt s usually unaccptabl as ndcatd by th rspons of fgur hs ovrshoot s usually rducd by nsrtng a frst ordr fltr of tm constant = 4 s n th rfrnc crcut. Wth an appropratly dsgnd fltr, as ndcatd n Fgur 7.40, th dampd rspons (n blu) of Fgur 7.39 may b obtand. 7. Controllrs for 30 F. ahman (EE, UNSW) lctrc drv systms Octobr 6, 05

31 ELEC463 Fgur 7.39 V f / + 4s s + G c (s) G (s) V Fgur Fld currnt controllr for a DC motor h transfr functon G (s) for a fld controlld systm has on domnant tm-constant, whch s th L f / f of th fld crcut, as for th fld currnt control of a DC motor. A sutabl controllr s th form G () s c s o s s F (7.67) h currnt controllr for th fld crcut s dsgnd smlarly as for th armatur currnt controllr. Hr cancls th fld crcut pol at L f / f and / F s two octavs blow th swtchng frquncy of th fld convrtr. 7. Controllrs for 3 F. ahman (EE, UNSW) lctrc drv systms Octobr 6, 05

32 ELEC Poston controllr dsgn ssus o mov th load to th rfrncd poston wth spcfd dynamcs. o follow a movng rfrnc wth spcfd followng rror. o hold th rfrnc poston wth spcfd - charactrstc. h nnr spd control loop provds for dampng of th poston control loop. I Consdr th poston control systm of fgur 7.4 n whch G s (s) and a ar th Iarf s nnr spd and currnt control systms rspctvly. Lt H(s) b th closd-loop transfr charactrstc of th nnr spd loop. M(s) s th currnt to spd transfr charactrstc. hus s Gs s M(s) H(s) (7.68) E G (s)m(s)k F(s) g s J s s Fgur 7.4(a) rf H(s) /s G(s) K p Fgur 7.4(b) H(s) (s) s rf (s) Kp H(s)G(s) s (7.69) 7. Controllrs for 3 F. ahman (EE, UNSW) lctrc drv systms Octobr 6, 05

33 ELEC463 h opn-loop transfr charactrstc s Kp H(s)G(s) OLF G p( s ) (7.70) s h pol at th orgn s du to poston snsor. hs snsor has a fxd phas-lag of 90º, whch rducs th phas-margn. Consquntly, a PI rgulator as a poston-controllr s almost mpossbl. Obvously, som phas-lad by th poston controllr s rqurd. A. G(s) Compnsaton (s) Lt Gs (s) (7.7) For >>, th controllr s lad-lag. Its paramtrs must b chosn such that th poston loop has nough phas-lad for stablty. B. H(s) Compnsaton Lt F s s (7.7) If th spd controllr has a hgh gan, s H(s) (7.73) K F(s) K g g hs may provd for th dsrd lvl of phas-lad (margn) for th poston controllr. Howvr, th addtonal lag n th spd controllr may hav stablty problm wthn th spd controlld systm. 7. Controllrs for 33 F. ahman (EE, UNSW) lctrc drv systms Octobr 6, 05