INDUCTION MOTOR MODELLING FOR VECTOR CONTROL PURPOSES

Transcription

1 Helini Univeity of echnology Depatent of Electical and Counication Engineeing Laoatoy of Electoechanic enillinen oeaoulu ähö- ja tietoliienneteniian oato ähöeaniian laoatoio Epoo 000 Rapotti 63 INDUCION MOOR MODELLING FOR VECOR CONROL PURPOSES Micea Popecu Helini Univeity of echnology Depatent of Electical and Counication Engineeing Laoatoy of Electoechanic enillinen oeaoulu Sähö- ja tietoliienneteniian oato Sähöeaniian laoatoio

2 Popecu M., Induction Moto Modelling fo Vecto Contol Pupoe, Helini Univeity of echnology, Laoatoy of Electoechanic, Repot, Epoo 000, 44 p. Keywod: Induction oto, vecto contol, d-q odel, continuo tie doain, dicete tie doain, lineaization Atact Widely ued in any indutial application, the induction oto epeent the tating point when an electical dive yte ha to e deigned. In oden contol theoy, the induction oto i decied y diffeent atheatical odel, accoding to the eployed contol ethod. In the yetical thee-phae veion o in the unyetical two-phae veion, thi electical oto type can e aociated with vecto contol tategy. hough thi contol ethod, the induction oto opeation can e analyed in a iila way to a DC oto. he goal of thi eeach i to uaize the eiting odel and to develop new odel, in ode to otain a unified appoach on odelling of the induction achine fo vecto contol pupoe. Stating fo vecto contol pinciple, the wo ugget the d-q ae unified appoach fo all type of the induction oto. Howeve, the pace vecto analyi i peented a a tong tool in odelling of the yetical induction achine. When an electical oto i viewed a a atheatical yte, with input and output, it can e analyed and decied in ultiple way, conideing diffeent efeence fae and tate-pace vaiale. All the atheatical poile odel ae illutated in thi epot. he uggetion fo what odel i uitale fo what application, ae defined a well. A the pactical ipleentation of the vecto contol tategie equie digital ignal poceo (DSP), fo the continuo tie doain odel ae deived the dicete tie doain odel. he dicete odel peit the ipleentation of the atheatical odel of the induction oto, in ode to otain high efficiency enole dive. he taility of thee vaiou odel i analyed. Ditiution: Helini Univeity of echnology Laoatoy of Electoechanic P.O. Bo 3000 FIN-005 HU el: Fa: E-ail: electoechanic@hut.fi Micea Popecu ISBN ISSN Picaet Oy Helini 000

3 3 Peface hi wo wa caied out in the Laoatoy of Electoechanic, Helini Univeity of echnology. he ai of the wo i to develop and etalih an unified appoach on the odelling of the induction oto, thee-phaed and ingle-phaed, fo vecto contol pupoe. I would lie to epe y gatitude to y upeviing Pofeo apani Joinen, head of the Laoatoy of Electoechanic, whoe help and guidance ade poile the fullfilent of thi wo. I a oliged to D. Anteo Aio and Maiu Rou, fo the enefical dicuion elated to the otacle, that I had to ovecoe in witting thi wo. A lot of than, I owe to Maianna Beecha, fo the eviion of the language. I wih to than to all the ee of the laoatoy whoe nae ae not entioned hee fo the ind atophee and help in eveyday life. Lat ut not leat, y thanful thought go towad y faily, who even faaway fo e, encouaged and undetood e. I wih to highlight that the iage of y little daughte, Miuna Ioana, inpied e all the tie, and theefoe thi wo i dedicated to he. Epoo, Octoe 000 Micea Popecu

4 4 Conten Atact.. Peface.... Lit of pincipal yol..... Vecto contol of induction oto - Oveview Intoduction. Algoith of Vecto Contol...3 Field Oientation Contol.4 Diect oque Contol. Continou-tie doain linea odel of the thee-phae induction achine... Intoduction. Voltage and flu linage equation.3 Space vecto equation fo thee-phae induction achine...4 Vectoial equation yte in a coon efeence fae..5 Induction achine equation with tato efeed oto vaiale...6 Intantaneou electoagnetic toque.7 Geneal equation of the induction achine in diffeent efeence fae..7.. Pe unit yte..7.. Stationay efeence fae equation. Bloc diaga Roto efeence fae. Bloc diaga Synchonou efeence fae. Bloc diaga..8. D-Q Ae odel of the thee-phae induction achine Model with cuent pace vecto a tate-pace vaiale Model with flue linage pace vecto a tate-pace vaiale.8.3. Model with ied cuent -flu pace vecto a tate-pace vaiale.9. Vecto contol tategie fo thee-phae induction achine..9.. Stato flu field oientation (SFO).9.. Roto flu field oientation (RFO).9.3. Ai-gap flu field oientation (AFO).9.4. Stato cuent oientation (SCO) Roto cuent oientation (RCO) Continuo-tie doain odel of the ingle-phae induction achine 3.. Intoduction 3.. Voltage and flu-cuent equation of the ingle-phae induction achine Analyi of the ingle-phae induction achine in tationay efeence fae Analyi of the teady-tate opeation fo the yetical ingle-phae induction achine Analyi of the unyetical ingle-phae induction achine 3.6. Continuo linea odel fo ingle-phae induction achine Linea Γ odel of the yetical ingle-phae induction achine

5 Linea invee Γ odel of the yetical ingle-phae induction achine Univeal odel of the yetical ingle-phae induction achine Linea Γ odel of the unyetical ingle-phae induction achine Linea invee Γ odel of the unyetical ingle-phae induction achine Univeal odel of the unyetical ingle-phae induction achine D-Q ae odel of the ingle-phae induction achine Model with cuent pace vecto a tate-pace vaiale Model with flue linage pace vecto a tate-pace vaiale Model with ied cuent-flue pace vecto a tate-pace vaiale Vecto contol tategie fo ingle-phae induction achine Stato flu field oientation (SFO) Roto flu field oientation (RFO) Ai-gap flu field oientation (AFO) Stato cuent oientation (SCO) Roto cuent oientation (RCO) 4. Matheatical dicete odel fo the thee-phae induction achine Intoduction Bilinea tanfoation ethod (utin) 4.3. Fowad-diffeence ethod (Eule) Bacwad-diffeence ethod 4.5. Z-doain tanfe function 4.6. Staility analyi. 5. Matheatical dicete odel fo the ingle-phae induction achine. 5.. Intoduction. 5.. Bilinea tanfoation ethod (utin) 5.3. Fowad-diffeence ethod (Eule) Bacwad-diffeence ethod 5.5. Z-doain tanfe function 6. Lineaiation of the induction achine atheatical odel Intoduction 6.. hee-phae induction achine Single-phae induction achine Refeence

6 6 Lit of pincipal yol: Scala vaiale ae denoted y plane lette. Vecto vaiale ae denoted y undelined lette. Boldface yol ae ued fo ati vaiale. ABCDEF,,,,, aij B H H I (a,,c) I(a,,c) Ĩ (d,q) Ĩ (a,) Ĩ (d,q) Ĩ (a,) I I i (a,,c) i(a,,c) i(a,) i(a,) i(a,b,c) i(x,y) i(d,q) i (d,q) i (d,q)r i( RI, ) i ( RI, ) i i J j Kp,i, K, L (a,,c)(,) tate-pace coefficient ati coefficient fo tate-pace vaiale vicou fiction coefficient elative inetia contant tanfe function ati intantaneou tato phae cuent fo the thee-phae induction achine intantaneou oto phae cuent fo the thee-phae induction achine cople tato phae cuent in d-q co-odinate fo teady-tate analyi cople tato phae cuent in phyical co-odinate fo teady-tate analyi cople oto phae cuent in d-q co-odinate fo teady-tate analyi cople oto phae cuent in phyical co-odinate fo teady-tate analyi tato cuent pace vecto oto cuent pace vecto intantaneou tato phae cuent fo the thee-phae induction achine intantaneou oto phae cuent fo the thee-phae induction achine intantaneou tato phae cuent fo the ingle-phae induction achine intantaneou oto phae cuent fo the ingle-phae induction achine phae cuent fo a thee-phae yte phae cuent fo an othogonal two-phae yte intantaneou tato phae cuent in d-q co-odinate intantaneou efeed oto phae cuent in d-q co-odinate Γ odel intantaneou efeed oto phae cuent in d-q co-odinate tato phae cuent in cople co-odinate and pe unit yte oto phae cuent in cople co-odinate and pe unit yte tato cuent pace vecto in pe unit yte efeed oto cuent pace vecto in pe unit yte inetia contant cople opeato popotional, epectively integative contant fo PI contolle tanfoation ati fo ac co-odinate to d-q co-odinate tun atio fo the unyetical ingle-phae achine elf-inductance fo tato phae, epectively oto

7 7 Ll(,) Ll(,a) lm l l M(a,,c),(,) N, P p R a e L t U (a,,c) U (a,,c) U( RI, ) (d,q) U U (a,) U U u (d,q) u (d,q) u u Wc X l leaage inductance fo yetical tato phae, epectively oto leaage inductance fo unyetical tato phae: ain, epectively auiliay agnetiation inductance in pe unit yte total tato inductance in pe unit yte total oto inductance in pe unit yte utual inductance fo tato phae, epectively oto tun nue fo tato phae, epectively oto nue of pole deivative opeato pe unit tato phae eitance fo the thee-phae induction achine tato phae eitance fo the yetical induction achine ain tato phae eitance fo the unyetical ingle-phae achine auiliay tato phae eitance fo the unyetical ingle-phae achine oto phae eitance fo the yetical induction achine efeed oto phae eitance fo the yetical induction achine Laplace opeato citical lip fo the induction achine apling peiod intantaneou electoagnetic toque load toque tie intantaneou tato phae voltage fo the thee-phae induction achine intantaneou oto phae voltage fo the thee-phae induction achine tato phae voltage in cople co-odinate and pe unit yte cople tato phae voltage in d-q co-odinate fo teady-tate analyi cople tato phae voltage in phyical co-odinate fo teady-tate analyi tato voltage pace vecto oto voltage pace vecto intantaneou tato phae voltage in d-q co-odinate intantaneou oto phae voltage in d-q co-odinate tato voltage pace vecto in pe unit yte efeed oto voltage pace vecto in pe unit yte agnetic coenegy ae vaiale value fo pe unit yte pace vecto vaiale agnetiation eactance tato phae leaage eactance fo the yetical induction achine

8 8 l(,a) l L L(d,q) M M(d,q) z α γ γ (d,q) RI, ψ (d,q) ψ (d,q) ψ (d,q) ψ ( RI, ) ψ ( RI, ) (d,q) (a,,c) (d,q) (d,q)r (a,,c) γ R σ θ θ, Ω n tato phae leaage eactance fo the unyetical induction achine: ain, epectively auiliay oto leaage eactance Γ odel equivalent leaage eactance fo yetical induction achine Γ odel equivalent leaage eactance fo unyetical induction achine Γ odel equivalent agnetiation eactance fo yetical induction achine Γ odel equivalent agnetiation eactance fo unyetical induction achine dicete co-odinate pace vecto opeato Γ odel tun atio fo yetical induction achine Γ odel tun atio fo unyetical induction achine eal, epectively iaginay coponent of a ati deteinant tato flu linage in d-q co-odinate and pe unit yte agnetiating flu linage in d-q co-odinate and pe unit yte oto flu linage in d-q co-odinate and pe unit yte tato flu linage in cople co-odinate and pe unit yte oto flu linage in cople co-odinate and pe unit yte tato flu linage in d-q co-odinate and flu linage unit pe econd. tato flu linage flue in a o a co-odinate efeed oto linage flue in d-q co-odinate and flu linage unit pe econd Γ odel efeed oto linage flue in d-q co-odinate and flu linage unit pe econd oto flu linage in a o a co-odinate efeed oto flu linage pace vecto fo thee-phae induction achine tato flu linage pace vecto fo thee-phae induction achine aitaily flu linage pace vecto fo thee-phae induction achine agnetic eluctance leaage facto fo tato phae, epectively oto phae peifeical diplaceent etween tato and oto pace vecto peifeical diplaceent etween tato and aitay pace vecto elative angula fequency in pe unit yte angula fequency of the upply yte ae angula fequency of the upply yte ated angula fequency of the induction achine in electical degee oto angula fequency in electical degee

9 9. VECOR CONROL OF INDUCION MOORS - OVERVIEW.. Intoduction he electical DC dive yte ae till ued in a wide ange of indutial application, although they ae le eliale than the AC dive. hei advantage conit in iple and pecie coand and contol tuctue. he AC dive, oetie oe epenive ut fa oe eliale, (Rajaheaa et al. 996) equie cople oden contol technique. he deign of a contol yte i ealied in two ipotant tep:. he dive yte ha to e conveted into a atheatical odel, in ode to accoplih the analyi and the evaluation of the yte.. he ipoed epone of the dive yte i otained though an optial egulato, when etenal petuation ae peent. he induction oto ae elatively cheap and ugged achine ecaue thei contuction i ealied without lip ing o coutato. hee advantage have deteined an ipotant developent of the electical dive, with induction achine a the eecution eleent, fo all elated apect: tating, aing, peed eveal, peed change, etc. he dynaic opeation of the induction achine dive yte ha an ipotant ole on the oveall pefoance of the yte of which it i a pat. hee ae two fundaental diection fo the induction oto contol: Analogue: diect eaueent of the achine paaete (ainly the oto peed), which ae copaed to the efeence ignal though cloed contol loop; Digital: etiation of the achine paaete in the enole contol chee (without eauing the oto peed), with the following ipleentation ethodologie: Slip fequency calculation ethod; Speed etiation uing tate equation; Etiation aed on lot pace haonic voltage; Flu etiation and flu vecto contol; Diect contol of toque and flu; Oeve-aed peed enole contol; Model efeence adaptive yte; Kalan filteing technique; Senole contol with paaete adaptation; Neual netwo aed enole contol; Fuzzy-logic aed enole contol. Anothe claification of the contol technique fo the induction achine i ade y Holtz (998) fo the point of view of the contolled ignal: a) Scala contol: a. Voltage/fequency (o v/f) contol; a. Stato cuent contol and lip fequency contol. hee technique ae ainly ipleented though diect eaueent of the achine paaete. ) Vecto contol:. Field oientation contol (FOC):... Indiect ethod;... Diect ethod;. Diect toque and tato flu vecto contol. hee technique ae ealied oth in analogue veion (diect eaueent) and digital veion (etiation technique) he developent of accuate yte odel i fundaental to each tage in the deign, analyi and contol of all electical achine. he level of peciion equied of thee odel depend entiely on the deign tage unde conideation. In paticula, the atheatical deciption ued in

10 0 achine deign equie vey fine toleance level a tated y Naae et al. (980) and Muata et al (990). Howeve, in the developent of uitale odel fo contol pupoe, it i poile to ae cetain auption that conidealy iplify the eulting achine odel. Nonethele, thee odel ut incopoate the eential eleent of oth the electoagnetic and the echanical yte fo oth teady tate and tanient opeating condition (Nowotny and Lipo - 996). Additionally, ince oden electic achine ae invaialy fed fo witching powe conveion tage, the developed oto odel hould e valid fo aitay applied voltage and cuent wavefo. hi wo peent uitale odel fo ue in digital cuent contol of the induction oto. In addition, the liit of the validity of thee odel ae uaied and, in oe cae, the odel ae etended to account fo oe non-idealitie of the achine. Uually, the following auption ae ade (Loenz et al. 994): No agnetic atuation, i.e. achine inductance i not affected y cuent level. No aliency effect i.e. achine inductance ae not function of poition. Negligile patial f haonic i.e. tato winding ae aanged to poduce inuoidal f ditiution. he effect of the tato lot ay e neglected. hee i no finging of the agnetic cicuit. he agnetic field intenity i contant and adially diected aco the ai-gap. Eddy cuent and hyteei effect ae negligile. he oden contol theoy fo an electical dive yte equie the eitence of a eal-tie, taile, and pecie atheatical odel fo each coponent of the yte. he analyi and the deign of the nueical coand fo uch yte depend on the hadwae and oftwae eouce. If in counication technique the eal-tie epone of the yte i not alway copuloy, in indutial pocee the eal-tie epone of the dive yte i eential. he oft nueical coand fo the electical dive yte i fa oe fleile to ipleent than the hadwae veion. Fo the latte, lately thee i an intene eeach effot fo ipleenting ASIC (application pecific integated cicuit). he nueical coand of the electical dive yte i a challenging ta ainly due to the DSP (digital ignal poceing) technology. Now it i poile to ealie linea and non-linea technique fo ipleenting continuo and dicete atheatical odel of the entie eleent of an electical dive yte, including the electical achine (Xu and Nowotny - 990, 99). Fo the AC dive thee ae eveal olution fo ipleenting the coand and the contol of the yte. A quic uay of the eiting technologie aleady out thee in the field i given elow: DC Dive Initially the DC dive wee ued fo vaiale peed contol ecaue they could eaily achieve a good toque and peed epone with high accuacy. Field oientation of the oto i achieved uing a echanical coutato with uhe. In DC, toque i contolled uing the aatue cuent and field cuent. he ain dawac of thi technique i the educed eliaility of the DC oto - the fact that uhe and coutato wea down and need egula evicing; that DC oto can e cotly to puchae; and that they equie encode fo poitional feedac. AC Dive he evolution of AC vaiale peed dive technology ha een patly diven y the deie to eulate the pefoance of the DC dive, uch a fat toque epone and peed accuacy, while uing out, cheap to puchae and elatively aintenance-fee AC oto (Keleen and Iec - 987). AC Dive, fequency contolled uing PWM With thi technique, oetie nown a cala contol, the field oientation of the oto i not ued. Intead, the fequency and the voltage ae the ain contol vaiale and ae applied to the tato winding. he tatu of the oto i ignoed, eaning that no peed o poition ignal i fed

11 ac. he dive i theefoe egaded a an open-loop dive. hi type of dive i uitale fo application uch a pup and fan, which do not equie high level of accuacy o peciion. AC Dive, flu vecto contol uing PWM Hee, field oientation i achieved y atheatical odelling uing icopoceo and feedac of oto peed and angula poition elative to the tato field y ean of an encode (Va - 990). hi eult in a dive with geate taility and capale of fat toque epone and accuate peed contol. But the dawac i the need fo the encode, which educe dive yte eliaility and add cot. he contolling vaiale in a DC dive fo toque ae aatue cuent and field cuent, and aatue voltage fo toque. AC dive uing the PWM pinciple; howeve, ue voltage and fequency a the contolling vaiale and thee ae contolled y a device called a odulato. A odulato add conideale delay in the eponivene of a oto to change in toque and peed. Futheoe, with flu vecto AC dive, a tacho-geneato o poition encode i invaialy needed to otain any eal degee of accuacy. Such device ae cotly and copoie the iplicity of the AC induction oto. AC Dive, enole flu vecto he flu vecto contolled dive with encode feedac doe offe vey high level of pefoance aco a wide powe ange and hould not e confued with enole vecto - o open loop vecto - dive, which offe pefoance only lightly upeio to that of a tandad invete uing cala contol (Rajaheaa et al )... Algoith of vecto contol he induction oto ae vey coon ecaue they ae inepenive and out, finding ue in eveything fo indutial application uch a pup, fan, and lowe to hoe appliance. aditionally, induction oto have een un at a ingle peed, which wa deteined y the fequency of the ain voltage and the nue of pole in the oto. Contolling the peed of an induction oto i fa oe difficult than contolling the peed of a DC oto ince thee i no linea elationhip etween the oto cuent and the eulting toque a thee i fo a DC oto. he technique called vecto contol can e ued to vay the peed of an induction oto ove a wide ange. It wa initially developed y Blache (97-973). In the vecto contol chee, a cople cuent i yntheied fo two quadatue coponent, one of which i eponile fo the flu level in the oto, and anothe which contol the toque poduction in the oto. Eentially, the contol pole i efoulated to eele the contol of a DC oto. Vecto contol offe a nue of enefit including peed contol ove a wide ange, pecie peed egulation, fat dynaic epone, and opeation aove ae peed. he vecto contol algoith i aed on two fundaental idea. he fit i the flu and toque poducing cuent. An induction oto can e odelled ot iply (and contolled ot iply) uing two quadatue cuent athe than the failia thee phae cuent actually applied to the oto. hee two cuent called diect (I d ) and quadatue (Iq) ae eponile fo poducing flu and toque epectively in the oto. By definition, the I q cuent i in phae with the tato flu, and I d i at ight angle. Of coue, the actual voltage applied to the oto and the eulting cuent ae in the failia thee-phae yte. he ove etween a tationay efeence fae and a efeence fae, which i otating ynchonou with the tato flu, ecoe then the pole. hi lead to the econd fundaental idea ehind vecto contol. he econd fundaental idea i that of efeence fae. he idea of a efeence fae i to tanfo a quantity that i inuoidal in one efeence fae, to a contant value in a efeence fae, which i otating at the ae fequency. Once a inuoidal quantity i tanfoed to a contant value y caeful choice of efeence fae, it ecoe poile to contol that quantity with taditional popotional integal (PI) contolle.

12 Vecto tanfo he Pa and Clae vecto tanfo ae one of the ey to vecto contol of induction oto. I) Clae tanfo he fowad Clae (943) tanfo doe a agnitude invaiant tanlation fo a thee phae yte into two othogonal coponent. If the neutal - gound connection i neglected, the vaiale in a thee-phae yte (A, B, and C) u i equal to zeo, and thee i a edundant infoation. heefoe, the yte can e educed to two vaiale, called X and Y. he Clae tanfo i given y: ia () t ix () t co( γ) co( γ) ib () t iy () t 3 0 in( γ) in( γ) () ic () t whee: π γ 3 Uing the elation: ia() t ib() t i C () t 0 () and the fact that: π 4π (3) co co 3 3 hu, the Clae tanfo can e iplified to: ia () t ix () t (4) ib() t ( ia() t ic () t ) 3 he Clae tanfo can alo e undetood uing a vecto diaga a hown in Fig... In the figue, A, B, and C ae the ae of a thee phae yte, each offet 0 fo the othe. X and Y ae the ae of a two vaiale yte whee X i choen to e coincident with A. o pefo the Clae tanfo of a thee vaiale yte (i A, i B, i C ), i X i equal to i A and i Y i the caled pojection of i B and i C onto the Y ai. he caling i neceay to peeve the ignal agnitude though the tanfo. B i B Y i Y i A, X i C 0 i X i A C Fig... Clae anfo Vecto Diaga he Clae tanfo peeve the agnitude, and ealie a quadatue etween the cuent coponent.

13 3 II) Pa tanfo he Pa (99) tanfo i a vecto otation, which otate a vecto (defined y it quadatue coponent) though a pecified angle. he Pa tanfo function ipleent the following et of equation: Out ( t) co( θ ) in( θ) In () t Out y( t) in( θ ) co( θ) In y () t (5) whee θ i the angle to otate the vecto though. A evee vecto otation can e accoplihed iply y changing the ign on the in (θ) input value. he vecto otation i illutated y Fig... Soe efeence (Va -990, Nowotny and Lipo - 996) decie the Pa tanfo a a coination of the Cla and Pa tanfo peented hee. Beaing into a thee-vaiale-to-two tanfo (i.e. the Clae tanfo) and a vecto otation i done fo efficiency of calculation: with epaate Pa and Clae tanfo, only two tigonoetic calculation ae equied a oppoed to 6 in the taditional Pa tanfo. Q Y i Y i D i Q θ i D X 0 i X Fig... Pa tanfo vecto diaga. 3. Field oientation contol (FOC) Vecto contol technique have ade poile the application of induction oto fo highpefoance application whee taditionally only DC dive wee applied (Holtz - 995). he vecto contol chee enale the contol of the induction oto in the ae way a epaately ecitation DC oto. A in the DC oto, toque contol of induction oto i achieved y contolling the toque cuent coponent and flu cuent coponent independently. he aic chee of indiect and diect ethod of vecto contol ae hown in Fig he diect vecto contol ethod depend on the geneation of unit vecto ignal fo the tato o ai-gap flu ignal. he ai-gap ignal can e eaued diectly o etiated fo the tato voltage and cuent ignal. he tato flu coponent can e diectly coputed fo tato quantitie. In thee yte, oto peed i not equied fo otaining oto field angle infoation. In the indiect vecto contol ethod, the oto field angle and thu the unit vecto ae indiectly otained y uation of the oto peed and lip fequency. Flu θ Stato Fig..3. Poition of the oto flu vecto

14 4 Flu coand oque coand FOC Invete IM Voltage Cuent θ Slip fequency calculation Fig..4. Indiect vecto contol ethod Speed eno Flu coand oque FOC Invete Voltage Cuent IM coand θ Flu vecto eaueent o etiation Fig..5. Diect vecto contol ethod Speed eno Fundaental equieent fo the FOC ae the nowledge of two cuent (if the induction oto i ta connected) and the oto flu poition. Knowledge of the oto flu poition i the coe of the FOC. In fact if thee i an eo in thi vaiale the oto flu i not aligned with d-ai and the cuent coponent ae incoectly etiated. In the induction achine the oto peed i not equal to the oto flu peed (thee i a lip peed; a uch, a pecial ethod to calculate the oto flu poition (angle) i needed. he aic ethod i the ue of the cuent odel. han to FOC it ecoe poile to contol, diectly and epaately, the toque and flu of the induction oto. Field oiented contolled induction achine otain evey DC achine advantage: intantaneou contol of the epaate quantitie allowing accuate tanient and teadytate anageent.. 4. Diect toque contol he ot oden technique i diect toque and tato flu vecto contol ethod (DC). It ha een ealied in an indutial way y ABB, y uing the theoetical acgound popoed y Blahe and Depenoc duing hi olution i aed oth on field oiented contol (FOC) a well a on the diect elf-contol theoy. Stating with a few aic in a vaiale peed dive the aic function i to contol the flow of enegy fo the ain to a poce via the haft of a oto. wo phyical quantitie decie the tate of the haft: toque and peed. Contolling the flow of enegy depend on contolling thee

15 5 quantitie. In pactice eithe one of the i contolled and we pea of "toque contol" o "peed contol". When a vaiale peed dive opeate in toque contol ode the peed i deteined y the load. oque i a function of the actual cuent and actual flu in the achine. Liewie when opeated in peed contol the toque i deteined y the load. Vaiale peed dive ae ued in all indutie to contol peciely the peed of electic oto diving load anging fo pup and fan to cople dive on pape achine olling ill cane and iila dive. he idea i that oto flu and toque ae ued a piay contol vaiale which i contay to the way in which taditional AC dive contol input fequency and voltage, ut i in pinciple iila to what i done with a DC dive, whee it i uch oe taightfowad to achieve. In contat, taditional PWM and flu vecto dive ue output voltage and output fequency a the piay contol vaiale ut thee need to e pule width odulated efoe eing applied to the oto. hi odulato tage add to the ignal poceing tie and theefoe liit the level of toque and peed epone tie poile fo the PWM dive. In contat, y contolling oto toque diectly, DC povide dynaic peed accuacy equivalent to cloed loop AC and DC yte and toque epone tie that ae 0 tie fate. It i alo claied that the DC doe not geneate noie lie that poduced y conventional PWM AC dive. And the wide pectu of noie ean that aplitude ae lowe which help to contol EMI and RFI eiion. he aic tuctue of diect toque and tato flu vecto contol i peented in Fig..6. Flu coand oque coand Flu contolle oque contolle Voltage vecto election Invete Voltage Cuent IM Flu vecto eaueent o etiation Speed eno Fig..6. Baic tuctue of diect toque and flu vecto contol In DC field oientation i achieved without feedac uing advanced oto theoy to calculate the oto toque diectly and tato flu without uing a odulato o a equieent fo a tachogeneato o poition encode to feed ac the peed o poition of the oto haft. Both paaete ae otained intead fo the oto itelf. DC configuation alo elie on two ey developent - the latet high-peed ignal poceing technology and a highly advanced oto odel peciely iulating the actual oto within the contolle. A DSP (digital ignal poceo) i ued togethe with ASIC hadwae to deteine the witching logic of the invete. he oto odel i pogaed with infoation aout the oto, which enale it to deteine paaete including tato eitance, utual inductance atuation coefficient and oto inetia. he odel alo encopae tepeatue copenation, which i eential fo good tatic peed accuacy without encode.

16 6 In noal opeation, eaueent of the two oto phae cuent and the dive DC lin voltage, togethe with infoation aout the witching tate of the invete ae fed into the oto odel. he oto odel then output contol ignal, which ae accuate etiate of the actual oto toque and actual tato flu. All contol ignal ae tanitted via optical lin fo high peed. In thi way, the eiconducto witching device of the invete ae upplied with an optiu witching patten fo eaching o aintaining an accuate oto toque. Alo, oth haft peed and electical fequency ae calculated within the oto odel. hee i no need to feedac any haft peed o poition with tachoete o encode to eet the deand of 95% of indutial application. Howeve, thee will alway e oe pecial application whee even geate peed accuacy will e needed and when the ue of an encode ipove the accuacy of peed contol in DC. But even then, the encode doe not need to e a cotly o a accuate a the one ued in taditional flu vecto dive, a DC only ha to now the eo in peed, not the oto poition. he dive will have a toque epone tie typically ette than 5. hi copae with epone fo oth flu vecto PWM dive and DC dive fitted with encode. he newe enole flu vecto dive now eing launched y othe dive anufactue have a toque epone eaued in hunded of illiecond. DC alo povide eceptional toque contol lineaity. Fo the fit tie with an open loop AC dive, toque contol can e otained at low fequencie, including zeo peed, whee the noinal toque tep can e inceaed in le than. he dynaic peed accuacy of DC dive i ette than any open loop AC dive and copaale to DC dive, which ue feedac. DC ing othe pecial function, not peviouly availale with AC dive, including autoatic tating in all oto electoagnetic and echanical tate. hee i no need fo additional paaete adjutent, uch a toque oot o tating ode election, uch a flying tat. DC contol autoatically adapt itelf to the equied condition. In addition, aed on eact and apid contol of the dive inteediate DC lin voltage, DC can withtand udden load tanient caued y the poce, without any ovevoltage o ovecuent tip.

17 7. CONINUOUS-IME DOMAIN LINEAR MODELS OF HE HREE-PHASE INDUCION MACHINE.. Intoduction Until the lat decade the thee-phae induction achine wa ainly ued in contant peed dive due to the contol yte pefoance, not to the opeating pinciple of the achine. Nowaday, thi ituation i copletely changed. With the technical poge in powe electonic and icoelectonic, the thee-phae induction achine contol ecoe vey fleile and highly efficient. Since 983, the yea when the digital ignal poceo (DSP) appeaed, the contol theoy fo thi type of achine wa peanently ipoved. New atheatical odel have to e ipleented fo the thee-phae induction achine in ode to analye it opeation oth dynaically and in teady-tate... Voltage and flu linage equation he fit atheatical odel fo the dynaic analyi of the induction achine wa aed on the two eal ai efeence fae, developed initially y Pa (99) fo the ynchonou achine. Uing the yetic configuation of the induction achine, Kovac and Racz (959) have elaoated the pace cople vecto theoy, and otained a odel fo the teady-tate analyi of the achine. Both theoie ae ued fo odelling the thee-phae induction achine. he following auption ae ade when a coplete equation yte i witten to decie the continuou-tie linea odel of the induction achine (Kaue et al. 995): Geoetical and electical achine configuation i yetical; Space haonic of the tato and oto agnetic flu ae negligile; Infinitely peeale ion; Stato and oto winding ae inuoidally ditiuted in pace and eplaced y an equivalent concentated winding; Saliency effect, the lotting effect ae neglected; Magnetic atuation, aniotopy effect, coe lo and in effect ae negligile; Winding eitance and eactance do not vay with the tepeatue; Cuent and voltage ae inuoidal te. End and finging effect ae neglected All thee auption do not alte in a eiou way the final eult fo a wide ange of induction achine.

18 8 a θ a U a I a U a I a U c I c c U I U I U c I c c Fig... he eal odel of the thee-phae induction achine with thee tato winding and thee oto winding Fo the achine tato in Fig.. if we chooe the tato efeence fae, the voltage equation ae a follow: d a Ua I a dt d U I (6-8) dt d c Uc I c dt whee Ua, U, U c ae the intantaneou tato voltage, Ia, I, I c ae the intantaneou tato cuent, a c i the tato winding eitance and a,, c ae the total agnetic flue fo the thee tato winding. he flu-cuent elation ae deteined afte detailing the total flu of a tato winding. Fo the othe two winding, thee ae valid iila elation: a aa a ca aa a ca (9) whee the flu coponent ae: the agnetic flu poduced y tato phae cuent a in the tato phae winding a aa the agnetic flu poduced y tato phae cuent in the tato phae winding a a the agnetic flu poduced y tato phae cuent c in the tato phae winding a ca the agnetic flu poduced y oto phae cuent a in the tato phae winding a aa the agnetic flu poduced y oto phae cuent in the tato phae winding a a ca the agnetic flu poduced y oto phae cuent c in the tato phae winding a hee coponent ae coputed with the epeion: L I M I aa a a M I a a M I ca ca c aa aa a M I a a M I ca ca c

19 9 Self-inductance L a ha two coponent, one ceated y the linage agnetic flu L a and the econd ceated y the leaage agnetic flu L la : L a La Lla (0) he utual inductance, which i conideed to e equal due to the achine yety, can alo e plit in two coponent. Howeve, the leaage flu ceated coponent in the utual inductance can e neglected. It eult that: Ma Ma Mac Mca La () he utual inductance within the tato and oto winding vaie with the elative pace poition etween the. he tato flu ceated y cuent fo oto phae a in tato phae a depend on the angle value θ: w w Maa Maa La coθ La coθ () w w t whee t epeent the tun atio ultiplied y the winding facto atio. In a iila way the elation fo the othe utual inductance can e witten: π Ma Ma La co θ t 3 (3-4π Mca Mac La co θ t 3 4) Due to the yetical configuation of the induction achine, we can deduce the total agnetic flu fo the tato phae winding a epeed a follow: a ( La Lla) Ia LaI LaIc LaIa coθ t (5) π 4π LaI co θ LaIc co θ t 3 t 3 Fo the oto winding, y uing a oto efeence fae, it can e developed a iila equation yte to the tato cae: da Ua I a dt d a U I (6-8) dt d c Uc I c dt whee: Ua, U, U c ae the intantaneou oto voltage, I a, I, I c ae the intantaneou oto cuent, a c i the oto winding eitance and a,, c ae the total agnetic flue fo the thee oto winding. he total oto agnetic flu fo the winding a i decied y: a aa a ca aa a ca (9) LaIa MaI McaIc MaaIa MaI McaIc In thi cae the utual inductance i:

20 0 M M L M M L M M L a a a ca ac a aa aa t a coθ (0-) π Ma Ma t La co θ 3 (-4) 4π Mca Mac t La co θ 3 Due to the yetical winding and oto configuation, one can wite the following elation: L a t L a L (5) t hough a iila algoith a that one fo the tato and oto phae a, epectively a, it i poile to otain anothe fou equation: two fo the tato phae, and c, and two fo the oto phae and c. All i final equation can e gouped in a ati fo a follow: d [ ] [ U] [ I] dt d [ ] [ U] [ I] (6-9) dt L I M I [ ] [ ] [ ] [ ] [ ] [ ] [ L] [ I] [ M ] [ I] whee: [ ], [ ], [ ], [ ], [ ], [ ] U U I I epeent the tanpoe ati fo the tato and oto voltage, cuent, epectively flu vecto. A an eaple ae given the flu ati: [ ] [ ] a c [ ] [ ] a c (30-3) So we get: La Lla La La σ [ L] La La Lla L a La σ (3) La La La L la σ La Lla La La σ L La La Lla L a La σ La La La L la σ [ ] (33)

21 [ M ] π 4π coθ co θ co 3 θ 3 4π π La co θ coθ co θ t 3 3 π 4π co θ co θ coθ 3 3 (34) 4π π coθ co θ co θ 3 3 π 4π [ M ] t La co θ coθ co θ 3 3 4π π co θ co θ coθ 3 3 Note: ) [ M ] [ M ] the utual tato inductance ati equal the tanpoe ati of the utual oto inductance; la a la a ) σ L L ; σ L L ae the tato, epectively oto leaage facto. L L L L a a a a 3) he ati yte deteined aove epeent the flu-cuent equation et fo the thee-phae induction achine in a efeence fae attached epaately to each aatue..3. Space vecto equation fo thee-phae induction achine Conideing the auption ade in the peviou paagaph, the pace vecto notation and concept intoduced y Racz and Kovac (959) ae paticulaly ueful. In thi appoach, all vaiale ae epeented y pola vecto indicating the agnitude and angula poition fo the otating inuoidal ditiution. A thee-phae vaiale yte can e uniquely decied though a pace vecto, which i a cople te and tie dependent (t) and a eal hoopola coponent 0 (t): ( α α ) a c t () 3 0() t ( a c) 3 whee: α e α e π j 3 4π j 3 j j 3 3 and a,, c ae the phae vaiale. he eal ai diection coincide with that one of phae a. Uually, the neutal connection fo a thee-phae yte i open, o that the hoopola coponent equal zeo. he phae vaiale can e eaily otained fo the pace vecto notation: [ a c ] () t () t () t Re α α t () (35) (36)

22 he phae voltage fo the induction achine can e epeed with the help of the pace vecto tanfoation: U ( Ua α U α Uc ) 3 (37) d ( I α I α I ) ( α α ) 3 a c 3 dt a c o in a condened fo: d U I (38) dt whee U, I, ae pace vecto fo tato voltage, cuent and flu. Siilaly, we get the oto equation: d U I (39) dt whee U, I, ae pace vecto fo oto voltage, cuent and flu, epectively. Since the achine i conideed agnetically linea, the tato and flu linage will e deteined a follow, aing the notation: [ Α ] [ α α ] 3 3 we get: [ Α] [ ] [ Α] [ L] [ I] [ Α] [ M] [ I] whee: σ 3 [ Α] [ L] α α L a σ La σ [ Α] σ π 4π coθ co θ co θ 3 3 4π π 3 j [ Α] [ M] α α L co θ coθ co θ L [ Α] e θ 3 3 π 4π co θ co θ coθ 3 3 he condened tato flu-cuent equation eult fo the lat thee equation: 3 3 j La σ I L I e θ (40) whee:, I, I ae pace vecto notation fo tato flu linage, cuent and oto cuent. Siilaly, the oto flu linage-cuent equation i deductile: [ Α] [ ] [ Α] [ L] [ I ] [ Α] [ M] [ I ] (4) whee:

23 3 σ 3 σ [ Α] [ L ] α α L σ L σ [ Α] a 4π π coθ co θ co 3 θ 3 π 4π 3 j [ Α] [ M ] α α L co θ coθ co θ L [ Α] e θ 3 3 4π π co θ co θ coθ 3 3 Finally, we get the condened fo of the oto flu linage-cuent equation: 3 3 j La σ I L I e θ (4) whee, I, I ae the pace vecto notation fo oto flu linage, cuent and tato cuent epectively. Fo an eaie anipulation of the equation we ae the notation: 3 L La σ 3 L La σ 3 3 M La t La t he geneal et of voltage and flu linage equation, witten in pace vecto notation, i: d U I dt d U I (43-46) dt jθ LI M e I jθ LI M e I Soe ipotant concluion have to e dawn egading thi ode of deciing the achine equation, though pace vecto notation: he cala equation yte witten in natual efeence fae i tanfoed in a 4 vecto equation yte. hi fo i equivalent to utituting the eal induction achine equipped with thee-phae winding on tato and oto with a fictive achine equipped with ingle phae winding on tato and oto; An inconvenience of the developed yte i that the tato equation yte i witten in tato efeence fae, and the oto equation yte i witten in oto efeence fae, aing the analyi of the achine difficult; he utual inductance depend on the elative oto poition. In Fig.. i illutated the new fictive odel of the induction achine fo the pace vecto point of view theoy.

24 4 a θ a U I U I c Fig...Model of the thee-phae induction achine with fictive one tato winding and one oto winding (pace vecto notation) c.4. Vectoial equation yte in a coon efeence fae he analyi of an induction achine dive yte ha to e ade when the tato and oto vaiale ae epeented in a coon efeence fae. When uing the ae pace vecto notation, we can define an aitay efeence fae, which otate with the angula velocity, and accoding to Fig. 3, the following elation i valid: j () t () t t () e θ (47) whee: θ (t) i the tie vaiale elative angle etween the new efeence fae and the tationay efeence fae initially conideed.; (t) epeent the pace vecto fo the new efeence fae. he evee tanfoation elation i: jθ ( t) ( t) ( t) e (48) he hoopola coponent eing a cala vaiale, i independent fo the choen efeence fae. Re θ Re 0 Fig..3. anfoation into an aitay efeence fae In an aitay coon efeence fae, the vectoial voltage and flu linage equation ecoe:

25 jθ ( ) d L I M e I U I U e I e dt jθ j ( θ θ) θ ( ) d L I e M e I jθ jθ dθ dθ dt dt dt dt jθ jθ di jθ jθ di jθ I e L e j e I M e j e I d ( LI MI ) jθ e jθ d I j ( L ) I MI e I j dt dt jθ ( ) dt j( θ θ) j( θ θ) ( ) dli M e I dli e M e I j( θ θ) j( θ θ) U I U e I e dt dt j( θ ) ( ) ( ) ( ) ( ) ( ) ( ) θ j θ θ di j θ θ d θ θ j θ di θ j θ θ d θ θ I e L e j e I M e j e I dt dt dt dt dli ( MI ) ( ) j θ θ j( θ θ) d e I j ( ) ( LI MI ) e I j ( ) dt dt (50) whee, ae the angula velocity of the aitay efeence fae, epectively the angula velocity of the induction achine. Finally, if we conide P the nue of pole fo the induction achine, the electical angula velocity of the oto i (P / ) and the vectoial equation of the induction achine witten in a coon aitay efeence fae which otate with angula velocity ae: d U I j dt d ( ) U I j dt LI MI LI MI 5 (49) (5-54) he aove equation yte epeent the atheatical odel of the induction achine with tato and oto equipped with one fictive winding each in a coon aitay efeence fae. It ha to e highlighted that the utual inductance doe not depend on the elative oto poition (Kovac - 984). R θ S θ U I Fig..4.Model of the thee-phae induction achine with fictive one tato winding and one oto winding (pace vecto notation) epeented in a coon aitay efeence fae U I

26 6.5. Induction achine equation with tato efeed oto vaiale A coplete unified equation yte fo the induction achine i otained when oth tato and oto vaiale ae epeed in a coon efeence fae and the oto vaiale ae alo tato efeed. Uing the tun atio and the winding facto atio we otain: I I t U U (55-58) t t t Now the tato and oto flu linage equation can e witten a: 3 3 La σ I La I σ La I t M ( I I ) t 3 3 L a a a t t ( ) σ I L I L I M I I t σ t If the following notation ae intoduced: L σ L the tato leaage inductance l a l t σ a L L the oto leaage inductance 3 3 LM t M La t La the ain agnetiation inductance I I I the agnetiation cuent pace vecto M M LM I M l Ll I the agnetiation flu the tato leaage flu l L l I the oto leaage flu he flu linage equation get the following fo: L I L I (6) l M M l M ( L I L IM ) ( ) (6) l M l M t t o in efeed vaiale we otain: ( L L ) I L I L I L I (63) l M M M l M M M ( L L ) I L I L I L I (64) whee L and L ae the total tato, epectively oto elf-inductance. he efeed oto voltage i given now y the elation: ( l M) d P U I j ( l M) dt (65) Finally we can wite a coplete equation yte, in a coon aitay efeence fae, with oto vaiale efeed to the tato, which define the induction achine. he eult i a atheatical odel of the thee-phae induction achine equipped with two fictive winding, otating with angula velocity. he iplified epeentation of thi yte i given in Fig..5. (59) (60)

27 7 d U I j dt d P U I j ( ) dt LI LMI L I L I M (66-69) S R θ θ U U I I Fig..5. Model of the thee-phae induction achine with fictive one tato winding and one oto winding (pace vecto notation) epeented in a coon aitay efeence fae, and efeed oto vaiale.6. Intanteou electoagnetic toque he input powe fo a thee-phae induction achine with wounded oto i: * * i() Re{ } { } Pt U I Re U I (70) * * whee i the phae nue (3), U, U ae the tato and oto voltage pace vecto, I, I ae the conjugate tato and oto cuent pace vecto. Uing a detailed elation fo the pace vecto with oto vaiale efeed to the tato, we get: Pt i() UaIa UI UcIc U a I a U I U c I c (7) Fo the coplete equation yte of the induction achine, the following epeion can e otained fo the input powe of the achine: 3 * d * * * d * P * Pi Re I I I j I I I I j I (7) dt dt 3 d * * * d P * Pi Re I I I I j I I dt dt (73) he fit two te fo the powe equation epeent the Joule effect lo, the following two te epeent the electoagnetic powe due to the tie vaiation of the agnetic enegy, and the lat te tand fo the echanical powe availale at the achine haft, if the hyteezi lo, eddy cuent lo and the tay loe ae neglected. he echanical powe will e:

28 8 * P ( ) ( ) 3 * * Re M M P j LI L I I LI L I I (74) 3 P * 3P * Re j LM I I LM I { I I } 4 Uing the flu linage elation, thee epeion ae deductile fo the induction achine echanical powe: 3P * { } { } { } * 3P 3P P * I I I I LM I I I (75) 3P L * * M 3P LM I { I } I { 4 L } 4 whee: LL LM which give fou epeion fo the intantaneou electoagnetic toque if the echanical powe i divided y the oto angula velocity: 3P * { } { } { } * 3P 3P * e I I I I LM I I I (76) 3P L * * M 3P LM I { I } I { 4 L } 4 he toque and the oto peed a elated y the echanical equation: d e J L P (77) dt whee: J i the inetia of the oto and in oe cae the connected load. he fit te on the ighthand ide of the equation i the inetial toque. he load toque L i poitive fo a toque load on the haft of the induction achine. Conideing the voltage, flu linage and echanical equation, the coplete elation yte of the induction achine can e witten: d U I j dt d U I j ( ) dt P LI LMI L I L I M { } { } * * { * } * d { I } J L (78-8) 3P 3P 3P e I I I I LM I I I P LM I 4 L P dt he aove et of equation i valid fo the following condition: All the equation ae witten in a coon efeence fae, which otate with the aitay angula velocity. he oto vaiale ae efeed to the tato; he tato and oto vaiale ae decied y the pace vecto notation; he achine paaete ae contant in a coon efeence fae fo oth aatue; he echanical equation i witten in eal doain, and i independent of the choen efeence fae.

29 9.7. Geneal equation of the induction achine in diffeent efeence fae When the atheatical odel of the induction achine i etalihed, eveal efeence fae can e eployed depending on application and the choen tategy contol. hee ae eveal ain efeence fae: tationay fied to the tato, ynchonouly fied to the oto, and evolving with an angula velocity equal to: the ai-gap flu, the oto flu, the tato voltage, the oto cuent pace vecto. he tanfoation fo one efeence fae to anothe i ade y eeping contant the value of the..f. a all the efeence fae ae enegetically equivalent. Fo the induction achine, the phaoial diaga of the ain pace vecto i illutated in Fig..6. he ignificance of the inde i a follow: tationay efeence fae lined to phae a of the tato winding otating efeence fae lined to the oto haft ai-gap flu efeence fae tato total ynchonou efeence fae oto flu ynchonou efeence fae aitay ynchonou efeence fae he angle depicted in Fig..6 ae in electical degee. d L i M L i M θ d d q θ θ θ θ d d d Fig..6. Definition of the electical angle etween diffeent efeence fae.7.. Pe unit yte It i often convenient to epe achine paaete and vaiale a pe unit quantitie. he atheatical odel in pe unit epeentation of the induction achine ha oe ain featue (Kaue et al. 995): All paaete have aiu value equal to unit; wo iila yte can e copaed eaie; he digital contol i eadily ipleented; When uing the pe unit yte fo witing the achine equation, the following oevation have to e ade: a) he ae toque i not the ated one. A the ated powe output geneally occu at a peed (ated peed) lightly le than ynchonou, the ae toque will e le than ated toque y the atio of ated peed to ynchonou peed of the achine. ) In pe unit yte, the inductance value i equal to the eactance value.

30 30 c) If the flu linage pe econd i ued in the induction achine equation, it i pe unitied y dividing y ae voltage. he following value of the achine paaete ae ued a ae vaiale: U U U n,a (),n I I I Z n.a (),n 3 S U I 3U I π f (),n ()n n n n n U I 4π f P P P U L I S 3P 4 J H S U n.a I n n,a -phae voltage, aiu value -phae cuent, aiu value -ae ipedance -ae powe -ae tato vaiale angula velocity -ae oto vaiale angula velocity -ae tato flu -ae inductance -ae toque -inetia contant he tato voltage equation in pe unit yte ecoe: U I I d j U U I dt U U dψ u Ri j Ωψ n dt (83-84) Siilaly we otain the oto voltage equation: dψ u R i j ( Ω Ω) ψ (85) n dt whee the elative angula velocity i: Ω In the peviou pe unit epeion we hould note that the ultiplying coefficient of the tie flu deivative i neceay, a the tie i not efeed. he elative tie, defined a follow, can e ued: t telative t nt π (86) n whee n epeent the tato voltage upply peiod. Finally the coplete equation et in pe unit yte fo the induction achine i:

31 3 dψ Ωψ dt u Ri j dψ u R i j ( ) ψ Ω Ω ψ dt i Mi i Mi ψ dω dt * { ψ i } 3P I H 4 L (87-9).7.. Stationay efeence fae equation. Bloc diaga If the tato voltage ae unalanced o dicontinuou and the oto-applied voltage ae alanced o zeo, the ot appopiate choice fo the efeence fae i the one fied to the tato. hi tationay efeence fae wa fit eployed y Stanley (938). In a tationay efeence fae, fied to the tato, the aitay angula velocity i zeo, ( 0) and the induction achine equation yte ecoe: u dψ dt Ri dψ u R i j ψ Ω ψ dt i i M ψ i i M * { ψ i } (9-96) dω 3P I L dt H 4 M he total leaage facto i decied y elation σ t and if we note: δ the M following elation ae deductile: M i ψ ψ δ δ (97-98) M i ψ ψ δ δ A new equation yte can e witten in the two-ai co-odinate yte:

32 3 u q dψ Ri q dt dψ d ud Ri d dt dψ u R i q q q Ω ψ d dt dψ u R i Ω ψ ψ ψ ψ d d d q dt i i q q M q i i d d M d i i q q M q ψ i i d d M d q (99-07) dω 3P ( ψdiq ψ qd i ) L dt H 4 If the induction achine i equipped with cage oto, then the oto voltage i zeo. A coplete loc diaga fo the induction achine in tationay efeence fae uing a input tato voltage and cuent and load toque, and a output the peed, i illutated in Fig..7. i a i i c /3 i d - 3 -/ i q R R - - n n / / ψ d ψ q * / Ω u a u u c /3 u d - 3 -/ u q * - - /(H) L Fig..7. Bloc diaga of the induction achine in tationay efeence fae.7.3. Roto efeence fae equation. Bloc diaga he choice of the efeence fae fo the dynaic analyi of the induction achine, epecially when the oto cicuit ae unalanced, i oe convenient to e fied to the oto fae. hi efeence fae i in fact the Pa tanfoation, initially developed fo ynchonou achine and than applied to the induction achine y Beeton (Fitzgeald et al -990). he ethod of efeing the achine vaiale to a oto efeence fae i ot ueful fo field oiented contol