Vector Control of Unsymmetrical Two-Phase Induction Machines

Transcription

1 Vcto Contol of Unsytical TwoPhas Induction Machins Mica Popscu, Dan M. Ionl *), Daid G. Doll SPEED aboatoy, Dpatnt of Elctical Engining, Unisity of Glasgow, Glasgow G T, Scotland, UK *) Innsys Boo Copton, Cntal Rsach and Dlopnt, Huddsfild HD J, England, UK Abstact A gnal odl suitabl fo th cto contol of th unsytical twophas induction achins is psntd in th pap. Both th oto flux ointd and th stato flux ointd contol statgis a analysd and thi suitability fo low and highspd opation is discussd. In th athatical odl th achin ain and auxiliay winding a psntd in th stationay fnc fa and th ffct of co losss and th ain flux path satuation a includd. A spcial tchniqu is usd to odl diffnt lls of satuation on th two axs of agntisation. Th odl can b ployd fo th accuat siulation of both th stadystat and th tansint opation of th unsytical twophas induction achins. I. INTRODUCTION Th unsytical twophas induction achins a widly usd in high olu cocial applications du to thi lati low cost and high liability. Conntionally, ths achins a fd fo a singlphas ac ains supply. To achi aiabl spd opation a pow lctonics int can b usd. Although in this cas a configuation with a singl phas input / th phas output int diing a th phas induction oto sas to b conoically pfabl, it is blid that fo ctain applications th tu potntial of th unsytical twophas induction oto di is not yt xhaustd. Th pfoanc of a twophas induction oto contolld by fild ointd thods has bn analysd in [], basd on a siplifid lina nonfd odl. It is nown that ion losss and agntic satuation a aong th ain causs of oto paat aiation, which unlss suitabl copnsatoy asus a ployd can lad to significant pfoanc dtioation in an lctonically contolld di. In this spct sal odn contol tchniqus ha bn poposd, but only fo th thphas induction oto dis [][]. A gnal thoy fo th unifid tatnt of co losss and satuation in th dq odls of lctical achins has bn dscibd in [7]. Th ain objcti of this pap is to psnt an oiginal thod dlopd fo dtuning th ffcts of th ion losss and ain flux path satuation in cto contolld unsytical two phas induction oto dis, wh th dq oto paats a diffnt along th two axs. It will b shown that th thod nsus satisfactoy pfoanc copnsation and is suitabl both fo oto flux and stato flux ointd contol. II. MATHEMATICA MODE Th poposd odl is basd on sal assuptions, gnally accptabl []: ) only th fundantal spachaonic coponnt of th aigap flux distibution is considd, ) th is a uniqu distinct nonlina agntisation cu along ach of th two othogonal axs of th achin (th agnticflux satuation ffcts a odlld and supposition is not usd), ) th co losss a odlld though a nonlina sisto associatd with th total stato flux linag, ) th stay losss, th agntic diffusion (i.. dpba ffct), th tpatu ffcts and th otational losss a nglctd, but th odl can b futh dlopd to includ th. Sinc th stato windings of th unsytical twophas induction achin a not idntical, th oto paats a constant only in th stationay fnc fa. In dq coodinats, th achin odl, psntd in Fig., is athatically dscibd though th following (th list of sybols is gin in Appndix I): stato oltag quations = R i p ( i i ) p () M lm q f q = R A i la p ( i id f ) p () d oto oltag quations = R R i q R lr p i q R d R p q ω () = R i p i ω p () R d R lr d R q R d flux linag quations = ( i i ) (5) lm q f q = ( i i ) () = i (7) la df d lr q = i () lr d = ( i i i ) = ( i ) i (9) q qf q q q = ( i i i ) = ( i ) i () d df d d d cunt quations i i = i i () qf q i i = i i () df d i i i = q d () Bcaus th achin is unsytical, th ti aiation of th agntisation fluxlinag in dq axs is gin by:

2 d d dd i d d i q did = dt di i i i dt dd i d diq diq di i i dt d q d q i q q i d diq = dt di i i i dt d q q idiq did di i i dt By intoducing th following notations: d d = dynaic inductanc in axis d d d di d d i d i q = dynaic inductanc in axis q ds q d = static inductanc in axis d () (5) q i = q s static inductanc in axis q, th inducd oltag in th agntising banch bcos: d d di di d q d = = d d q dt dt dt () d q diq did q = = q qd dt dt dt (7) wh th inductanc ts a dscibd by th lations: = ( ) cos ρ () d ds dd ds ( d d d s ) sin ρ d q = (9) = ( ) cos ρ () q qs qd qs ( qd qs ) sin ρ qd = () iq ρ = a cta n () id Th lctoagntic toqu quation and th chanical ont quation a: P T = ( qsi ( i i iqf ) () P dsi ( i i idf ) = ( i i ) T T = J pω B ω () Th achin paats can b coputd, pfably by FEA, o asud though classical xpintal thods. Th alu of th sistanc usd to odl th ion loss is dtind fo a standad noload tst, at fist with sinusoidal supply at atd fquncy, and thn with a PWM supply at atd fquncy. Th sa tst is usd to asu disct points on th agntising cu of th achin. Th thods fo th asunt of th oto paats a dscibd in o dtail in fncs [][]. Und tansint opation, th syst of quations ()() dscibs th pfoanc of th twophas induction Fig. Th dq axs quialnt cicuit of th unsytical twophas induction achin including th ion loss and th ain flux path satuation achin. Th oltag atix quation in stataiabl fo is [9]: = A p x B x (5) T x = i, i, i, id R T =,,, R M lm lr (/) ω i i qf R qf R A la lr ω i df i i R R R df wh: R q f i i i = = q f R M R q f a n d i = i id f = R d f R A R d f lm q qd q qd dq la d dq d A = (7) q qd lr q qd dq d dq lr d RM Rqf RM Rqf RA Rdf B= RA Rdf ω ω ω ω ( ) R ds R ( lr ds ) qs lr qs R () Th odl dscibd abo was ifid by copaing coputd and asud sults. III. ROTOR FUX VECTOR CONTRO OF AN UNSYMMETRICA TWOPHASE INDUCTION MACHINE As th stato windings of th unsytical twophas induction oto a unbalancd, th achin paats a diffnt in btwn th d and th q axis and th cto contol pincipls ha to b iplntd in a spcial way. Th wafo of th lctoagntic toqu donstats that th syst is unbalancd. En if th othogonal stato q d i R R

3 cunts i and i a of qual aplitud th toqu contains an AC t as it can b sn fo quation (). In od to liinat th AC t fo th toqu xpssion so lations btwn th stato cunts w poposd in []. How, ths lations a alid only und lina assuptions (satuation and ion co losss a nglctd). Futho, th odl poposd in [] was iplntd using a nonfd quialnt cicuit, which quis so coplicatd asunts of th oto and stato agntising and utual inductanc. A nw lation fo coputing th toqu can b dtind by substituting (7)(), in (): i qs i ds P (9) T = qs ds wh: i = i i qf i = i i df = lr qs = lr ds () ds q s As =, a toqu xpssion without a stadystat oscillating t can b obtaind by iposing th following conditions: i = i () i = i = i () ds qs By substituting th aiabls i and i, into (9), th toqu xpssion bcos: P d s T = ( i i ) () d R Basd on this nw xpssion, which is quialnt to that of a sytic achin, i.. no oscillating ts a psnt, a cto contol odl can b iplntd fo th twophasd unsytical achin. Fo ()() and (7)(), th dynaic lation btwn th oto flux and th stato cunts is obtaind as: p = ω qsi () p = ω i (5) wh: ds = ; = RR RR In an abitay fnc fa, using ()(5) and th conditions ()(), th oto flux of a twophas induction achin can now b did as: p = ( ω ω ) i () a a a ds a a p = ( ω ω ) i (7) a a a ds a a wh ω a is th angula spd of th abitay fnc fa. Roto flux ointd contol syst. Basd on th did oto flux odl, th oto fild ointd contol syst can b adaptd to a twophas induction achin. Th achin quations in th oto flux fnc fa can b obtaind fo () and (7) and thn using th constaint condition fo oto flux fild ointation contol. Th sulting quations a as follows: ω qs = i () p = i (9) d s wh is th oto flux aplitud, ω = ω ω and ω is th angula spd of th oto flux as sn fo th stato. Th toqu xpssion in th oto flux ointd contol syst bcos: P d s T = i () d R Fig. shows an indict oto flux ointd contol syst, with th fncs dnotd by th * supscipt. Th oltag souc int and PWM odul is fd by signals poid by cunt contolls (PI, fuzzy, o nual ntwos typ). Th input of ths contolls is th diffnc btwn th stiatd and asud alus of th stato cunt ducd by th quialnt ionloss banch cunt. Th influnc of satuation is iplntd in th spd o oto flux stiato. Bsid th agntisation cus in th d q axs, th stiato quis th nowldg of th R q,d f (f) function. Th atix quations (5)() can b iplntd ith fo siulation, o on a DSP boad fo stiating th oto paats, whn th inputs a th tinal quantitis (oltags, cunts) of th oto. IV. STATOR FUX VECTOR CONTRO OF AN UNSYMMETRICA TWOPHASE INDUCTION MACHINE Th lctoagntic toqu as a function of th stato fluxlinag and cunt is gin by: lm i ds P qs qs ds T = la lm la i i i qs ds ds qs ()

4 ω Spd contoll Fild waning T * DR Calculat i* Eq. Calculat i* Eq. i* i* Tansfo Rotational To Stationay Eqs., (iidf) R id R iq V V (iiqf) θ VSI & PW M Calculat ω Eq. 9 ω ω ω dt ain aux Fig. Bloc diaga of indict oto flux ointd cto contol syst with twophas induction achin IM Th pious quation can b futh siplifid: P T = i i lm la i i () Usually, th stato ain and auxiliay windings ha an idntical configuation, and diff only though th nub of tuns and th diat of th copp wi. In this cas, th atio of th ain and auxiliay laag quals th tuns atio squad ( ). Th conditions ()() fo th stato cunts bco: i = i ; i = i Equation () can b wittn as: P T ( i i ) () To xpss th dynaic lations as a function of th stato cunts and fluxs, (5)() can b substitutd in ()(5), sulting in: p ω = σ pi () ωσ i i p ω = σ pi (5) ω σ i i wh σ = /( ). d(q)s d(q) s d(q )R d(q )S Th dynaic quations fo an abitay fnc fa can b did by substituting th ω t with ω ω a. Stato flux ointd contol syst. flux of th twophas induction achin. Th following quations a obtaind: = () i = (7) s T = () P s p = σ pi s s ω σ s i i Dcoupl fo dict stato flux ointd achin. (9) Th daxis coponnt of th stato cunt fo th stato flux ointd contol syst is not only dpndnt on th associatd daxis coponnt of th stato flux, but also on th qaxis coponnt of th stato cunt. Th coupling btwn th toqu poducing cunt coponnt i and th flux poducing cunt coponnt i is shown in (9). As a consqunc, any chang in on cunt coponnt without changing th oth accodingly will caus a tansint in th stato flux. This disadantag is oco by using a dcoupl in od to copnsat th coupling ffct [], []. Th coplt bloc diaga of th dict stato flux ointd syst of twophas induction achins is dpictd in Fig.. Th dsciption of th dcoupl is obtaind as follows: K i s i = idq F f ( p) s = idq K p s p (5) Basd on th cto odl gin by ()(5) th fildointd pincipls [] can b applid to contol th stato

5 ω Spd contoll Fild waning T * s Calculat i* Eq. R Dcoupl Eqs.5 idq i* Tansfo Rotational To Stationay Eqs., i* (iiqf) Rid Riq V V (iidf) θ =ag( s) V s () VSI & PWM ω s = s s Stato flux stiato Spd stiato Eqs: 555 ain aux i s, IM Fig. Bloc diaga of dict stato flux ointd cto contol syst with twophas induction achin wh F f (p) is th tansf function of a PI typ flux contoll (soft coputing tchniqus such as fuzzy o nualntwo can also b ployd), K p and K i a th cofficints of th PI typ contoll, and s = s * s s is th diffnc btwn th fnc and stiatd alu of stato flux. In od to dcoupl i fo i, th following has to b fulfilld: i ( σ p / ) = ω σ i (5) d q d R s i σ pi (5) ωs = ( σ i ) wh ω s = ω ω psnts th slip angula fquncy. Whn poply tund, th dcoupl dscibd by th abo lations gnats coct alus of stato stiatd cunts, so that th stato flux is not altd by any changs in th fnc lctoagntic toqu T*, n duing tansints. Fo a coplt snsolss cto contol of an unsytical twophas induction oto, th stiatd stato flux is dtind fo th infoation poidd by th tinal asud quantitis (oltags and cunts), as dscibd in []. Th lctical fquncy is obtaind though th lation: p p ω = (5) S Th ffcts of th agntic satuation and co losss can b includd in th achin odl in such a ann that solly stato cunt gons th toqu and flux poduction. It should b ntiond that in stato flux ointd contol opation, th achin psnts a stadystat pullout toqu that liits th stabl opating gion. V. EXPERIMENTA AND COMPUTED RESUTS An xpintal alidation of th poposd odl was pfod on a capacito un twophas induction oto, with th paats gin in Appndix II. Th coputd and th xpintal toqu slip chaactistic of th oto with both stato windings ngisd and no unning capacito a shown in Fig.. On should not that th atd and pullout toqu a lss than xpctd fo th cas of th un capacito connctd, as th oto is dsignd to opat fo such conditions. Th dict connction of th two stato windings to th ains supply, was chosn in od to ify th odl poposd in Fig. without th capacito influnc. To illustat th copaati adantags of th poposd cto contol statgy th sa twophas induction oto was siulatd opating fo ains supply with a unning capacito and opating in a cto contol di without a unning capacito, spctily. Du to spac liitations, only a fw siulation sults a psntd h. Th toqu against ti and th toqu against spd cus of th oto in linstat capacito un opation a shown in Figs. 5, and th pfoanc in cto contol opation is displayd in Figs. 7 and. Two ipotant iponts a achid though cto contol opation: a significant duction of th pulsating coponnt of th oto toqu and dcayd spd oscillations aound th stadystat opation point. 5

6 . 5 Toqu (N) Toqu (N) s lip Fig. Toquslip chaactistic of th xapl twophas oto (calculatd, o asud) ti (sc) Fig. 5 Siulatd toqu s. ti cu of unsytical twophas induction oto opating fo ains supply with unning capacito Toqu (N) spd (ot/in) Fig. Siulatd toqu s. spd cu of unsytical twophas induction oto opating fo ains supply with unning capacito Toqu (N) ti (sc) Fig. 7 Siulatd toqu s. ti cu of cto contolld unsytical twophas induction oto Toqu (N) spd (ot/in) Fig. Siulatd toqu s. spd cu of cto contolld unsytical twophas induction oto VI. CONCUSIONS Th quialnt cicuit odl poposd in th pap fo th unsytical twophas achins includs both th ffct of th agntic satuation and th ion losss and thfo lads to a pfoanc ipont of cto contolld achins. Futh wo will b undtan in od to includ into th odl also th paat aiation with th tpatu, th stay losss and th dpba ffct. Th nuical sults fo th stadystat pfoanc of an xapl capacitoun twophas induction oto ag satisfactoily with th xpintal alus. Th siulatd tansint pfoanc illustats th adantags of th cto contol di in ts of ducd toqu and spd ippl. Th cto contol fo this typ of induction achin can b achid ith though a oto o stato flux ointd statgy and th iplntation can b don on th sa DSP boad bcaus th sa oto paats a usd in both sions. It is anticipatd that th stato flux contol is o suitabl fo low spd applications, whil th oto flux contol syst is applicabl fo highspd applications. APPENDIX I ist of pincipal sybols ω synchonous spd ω oto spd p diffntial opato R M, R A, ain, auxiliay stato windings sistanc R R oto winding sistanc R q,df quialnt ionloss sistanc (d and q axis) lm, la ain, auxiliay stato laag inductanc ds,qs agntising inductanc (d and q axis) lr oto laag inductanc tuns atio auxiliay/ain windings T lctoagntic toqu J oto intia,,, stato / oto flux (d and q axis), stato oltags (d and q axis) i,,, stato / oto cunt (d and q axis) i df,qf ionloss quialnt stato cunt

7 APPENDIX II Twophas induction oto data Ratd output pow: 75 W Ratd fquncy: 5 Hz Ratd spd: p Ratd oltag: V Nub of pols: Intia:. g Running capacito: µf/ V. Stato ain winding sistanc: 5.5 Ω Stato auxiliay winding sistanc:. Ω Roto sistanc:.95 Ω Ionloss sistanc (in d axis): 59 Ω Ionloss sistanc (in q axis): 7 Ω Stato ain winding laag actanc:.5 Ω Stato auxiliay winding actanc:.5 Ω Roto laag actanc: 5.5 Ω Mutual actanc (in d axis):.7 Ω Mutual actanc (in q axis):. Ω REFERENCES [] M.B.R. Coa, C.B. Jacobina, A.M.N. ia and E.R.C. DaSila: Fild ointd contol of a twophas induction oto di, in Pocdings of IEEEPESC 9, Fuuoa, Japan, 99, pp [] P. Vas, Elctical Machins and Dis, Clandon Pss, Oxfod, 99. [] P.C. Kaus, O. Waszynchu an.d. Sudhoff, Analysis of Elctical Machiny, IEEE Pss, Nw Yo, 995. [] P. Vas, Paat Estiation, Condition Monitoing, and Diagnosis of Elctical Machins, Clandon Pss, Oxfod, 99 [5] P. Vas, Snsolss Vcto and Dict Toqu Contol, Clandon Pss, Oxfod 99 [] E. i, A unifid appoach to ain flux satuation odlling in DQ axis odls of induction achins, IEEE Tans. Engy Con., ol., no., 995, pp.55. [7] I. Bolda an.a. Nasa, Unifid tatnt of co losss and satuation in th othogonal axis odls of lctic achins, IEE Poc., Pt. B, ol., no., 97, pp. 5. [] S. D. Uans, Stadystat, lupdpaat odl fo capacitoun, twophas induction otos, IEEE Tans. Ind. Appl., ol., no., 99, pp.979. [9] M. Popscu, Analysis and Modlling of SinglPhas Induction Moto with Extnal Roto fo Dostic Applications, in Conf. Rc. IEEE IAS Annual Mting,, Ro, Italy. [] X. Xu, R. DDonc and D.W. Nootny, A stato flux ointd induction achin di, in Conf. Rc. PESC, Apil 9, USA, pp. 77. [] R. DDonc, F. Pofuo, M. Pastolli, P. Fais, Copaison of unisal fild ointd (UFO) contolls in diffnt fnc fas, Tans. Pow Elct., ol., no., 995, pp. 5. 7