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 apotti 63 INDUCION MOO MODELLING FO VECO CONOL PUPOSES 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, epot, 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: [email protected] Micea Popecu ISBN ISSN Picaet Oy Helini 000

3 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 oto 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.. oto flu field oientation (FO).9.3. Ai-gap flu field oientation (AFO).9.4. Stato cuent oientation (SCO) oto cuent oientation (CO) 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

4 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) oto flu field oientation (FO) Ai-gap flu field oientation (AFO) Stato cuent oientation (SCO) oto cuent oientation (CO) 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 efeence

5 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. A, B, C, D, E, F aij B H H I (a,,c) I(a,,c) Iɶ (d,q) Iɶ (a,) Iɶ (d,q) Iɶ (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) i(, I ) i (, I ) i i J j K p,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

6 7 Ll(,) Ll(,a) lm l l M(a,,c),(,) N, P p a e L t U (a,,c) U (a,,c) U (, I ) (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

7 8 l(,a) l L L(d,q) M M(d,q) z α γ γ (d,q), I ψ (d,q) ψ (d,q) ψ (d,q) ψ (, I ) ψ (, I ) (d,q) (a,,c) (d,q) (d,q) (a,,c) γ 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

8 9. VECO CONOL OF INDUCION MOOS - OVEVIEW..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, (ajaheaa 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

9 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

10 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 (ajaheaa 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.

11 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) i B ( t ) i C ( t ) 0 () and the fact that: π 4π (3) co c o 3 3 hu, the Clae tanfo can e iplified to: ia ( t) ix ( t ) (4) ib ( t) ( ia ( t) i C ( 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.

12 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

13 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

14 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 FI 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.

15 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.

16 7. CONINUOUS-IME DOMAIN LINEA MODELS OF HE HEE-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 acz (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.

17 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 Ia dt d U I (6-8) dt d c Uc Ic dt whee U a, U, U c ae the intantaneou tato voltage, I a, 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 ca aa the agnetic flu poduced y tato phae cuent c in the tato phae winding a the agnetic flu poduced y oto phae cuent a in the tato phae winding a a the agnetic flu poduced y oto phae cuent in the tato phae winding 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

18 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: M a M a M ac M ca 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: ww M aa M aa 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 M a M a La co θ t 3 (3-4π M ca M ac 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 La I La Ic La Ia coθ t (5) π 4π LaI co θ La Ic 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 Ia dt d a U I (6-8) dt d c Uc Ic dt whee: U a, U, U c ae the intantaneou oto voltage, Ia, 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) La Ia M ai M ca Ic M aaia M a I M ca Ic In thi cae the utual inductance i:

19 0 M M L M M L M M L a a a ca ac a aa aa t a coθ (0-) π M a M a t La co θ 3 (-4) 4π M ca 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)

20 [ 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; Lla La Lla La ) σ ; σ 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 acz 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) e α α ( t) (35) (36)

21 he phae voltage fo the induction achine can e epeed with the help of the pace vecto tanfoation: U ( U a α U α U c ) 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:

22 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 L I M e I jθ L I 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.

23 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. e θ e 0 Fig..3. anfoation into an aitay efeence fae In an aitay coon efeence fae, the vectoial voltage and flu linage equation ecoe:

24 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 d I jθ jθ d I jθ I e L e j e I M e j e I d ( L I M I ) jθ e jθ d I j ( L ) I M I e I j dt dt jθ ( ) dt j( θ θ ) j( θ θ ) ( ) d L I M e I d L I 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 ( ) d L I M I ( ) j θ θ j( θ θ ) d e I j ( ) ( L I 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 L I M I L I M I 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). 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

25 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 I M ) ( ) (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)

26 7 d U I j dt d P U I j ( ) dt L I LM I L I L I M (66-69) S 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 ( ) e{ } { } P t U I e 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: Pi ( t) UaIa U I 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 e I I I j I I I I j I (7) dt dt 3 d * * * d P * Pi e 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:

27 8 * P ( ) ( ) 3 * * e M M P j L I L I I L I L I I (74) 3 P * 3P * e 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: L L 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 L I LM I 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.

28 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.

29 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 i j Ω ψ n dt (83-84) Siilaly we otain the oto voltage equation: dψ u 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 n t 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:

30 3 dψ Ω ψ dt u i j dψ u i j ( ) ψ Ω Ω ψ dt i M i i M i ψ * { ψ i } d Ω 3P I dt 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 i dψ u 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:

31 3 u u q d dψ iq dt dψ id dt dψ u i q q q Ω ψ d dt dψ u 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 d (99-07) dω 3P ( ψ diq ψ q id ) 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 - - 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. oto 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

32 33 yte. anfoation fo the aitay fae to the oto fae i ade y utituting Ω Ω, o fo the tationay fae with the ati elation: d coθ inθ d inθ coθ q q coθ in θ d d q inθ coθ q (08-09) whee θ i the electical angle etween the agnetic ai of the tato flu and the agnetic ai of the oto flu. he coplete equation et eult a follow: dψ u i j Ω ψ dt dψ u i dt ψ i M i i M i ψ * { ψ i } d Ω 3P I L dt H 4 O epeed in othogonal two-ai co-odinate yte: dψ q uq iq Ωψ d dt u dψ i Ωψ d d d q dt dψ u q i q dt dψ u d i d dt ψ ψ i i q q M q i i d d M d ψ i i q q M q ψ i i d d M d q d (0-4) (5-3) dω 3P ( ψ q i d ψ di q ) L dt H 4 he loc diaga of the induction achine odel in oto efeence fae i peented in Fig..8. he input vaiale ae tato cuent and load toque, and a output vaiale i choen the angula velocity of the oto. Alo, the oto voltage ae conideed zeo (cae of quiel-cage oto). It ha to e oeved that two algeaic loop appea in thi diaga. hi inconvenient lead to a liited ue of thi efeence fae to the wound oto induction achine, when the loc diaga i iila to that fo Fig..8, whee input will e oto voltage and cuent.

33 34 i a i i c 3 ---> e -j θ /3-3 -/ i d i q - M - M / / i d i q * / Ω ψ q ψ d / / - n - n * - - /(H) L Fig..8. Bloc diaga of the induction achine in oto efeence fae (cage oto cae) i a i i c /3-3 -/ i q i d - - n n / / ψ d ψ q * / Ω u a u u c /3 u d - 3 -/ u q - * - /(H) L Fig..9. Bloc diaga of the induction achine in oto efeence fae (wound oto cae).7.4. Synchonou efeence fae equation. Bloc diaga he ynchonouly otating efeence fae, with angula velocity equal to that one of the powe upply yte, i paticulaly convenient when incopoating the dynaic chaacteitic of an induction achine into a digital copute poga ued to tudy the tanient and dynaic taility of powe yte. he ynchonouly otating efeence fae ay alo e ueful in vaiale fequency application if we ay aue that the tato voltage ae a inuoidal alanced et. It wa yteatically developed y Kovac (984) and Kaue et al (995), o Loenz et al. (994).

34 35 When a two-ai co-odinate efeence fae i eployed, it ha to e fied to diffeent vaiale of the achine. he ain configuation ae the ynchonouly efeence fae fied to: tato flu, ai-gap flu and oto flu. I) Stato flu fied ynchonou efeence fae. In ode to lin the d-ai of the ynchonou efeence fae to the tato flu pace vecto, the q- coponent of thi flu vecto i defined equal to zeo: ψ ψ ψ q 0 d he following equation et i otainale: u q i q Ω ψ d u d i d d q q q Ω Ω ψ d dt d d d Ω Ω ψ q dt q M dψ u i ( ) dψ u i ( ) 0 i i ψ q i i d d M d ψ i i ψ q q M q i i d d M d dψ dt (4) (5-33) dω 3P ψ d i q L dt H 4 he loc diaga fo Fig..0 peent the induction achine odel in ynchonou efeence fae lined to the tato flu, with input tato voltage and cuent, and load toque. he output ae ynchonou and oto angula velocity. i a i i c u a u u c > /3 e -j θ > /3 -j θ e - 3 -/ - 3 -/ i q i d u d u q - n - / D ψ d * N / - Ω Ω / /(H) L Fig..0. Bloc diaga of the induction achine in ynchonou efeence fae (lined to tato flu)

35 36 II) oto flu fied ynchonou efeence fae. In ode to lin the d-ai of the ynchonou efeence fae to the oto flu pace vecto, the q- coponent of thi flu vecto i defined equal to zeo: ψ ψ d (34) ψ 0 q he induction achine chaacteitic can e analyed with the equation yte: dψ q u q i q Ω ψ d dt u d d i d Ω ψ q dt u 0 i ( Ω Ω ) ψ q q d dψ u d 0 i d dt ψ ψ i i q q M q i i d d M d 0 i q Mi q ψ i i dψ d d M d d (35-43) dω 3P 3P LM d i q L d i q L dt H ψ ψ 4 H 4 L he loc diaga fo Fig.. peent the induction achine odel in ynchonou efeence fae lined to the oto flu, with input tato cuent, and load toque. he output ae ynchonou and oto angula velocity. he oto voltage ae conideed equal to zeo (cae of the cage oto). i a i i c 3 ---> /3 e -j θ - 3 -/ i q i d M / / M n / / D ψ d * N / M / Ω - Ω Ω /(H) - L Fig... Bloc diaga of the induction achine in ynchonou efeence fae (lined to oto flu)

36 37 III) Ai-gap flu fied ynchonou efeence fae. In ode to lin the d-ai of the ynchonou efeence fae to the ai-gap flu pace vecto, the q- coponent of thi flu vecto i defined equal to zeo: ψ ψ d (44) ψ q 0 he yte i decied y the elation: dψ q u q i q Ω ψ d dt u d d i d Ω ψ q dt dψ u 0 i ( ) q q q Ω Ω ψ d dt dψ u 0 i ( ) d d d Ω Ω ψ q dt 0 M ( i q i q ) ψ ψ ψ dψ ( i i ) d M d d i i ψ i d d M d d l d i i i q q M q l q ψ i i i q q M q l q ψ i i ψ i d d M d d l d (45-55) d Ω 3P ψ di q L dt H 4 Fig.. illutate the induction achine odel in ynchonou efeence fae lined to the aigap flu, with input tato voltage and cuent, ynchonou angula velocity and load toque. he output i oto angula velocity. i a i i c > /3 e -j θ - 3 -/ i d i q - - l n / - * ψ d / Ω u a u u c Ω l * > /3 e -j θ - 3 -/ u d u q - /(H) L Fig... Bloc diaga of the induction achine in ynchonou efeence fae (lined to ai-gap flu)

37 38 IV) Stato cuent pace vecto fied ynchonou efeence fae. In ode to lin the d-ai of the ynchonou efeence fae to the tato cuent pace vecto, the q-coponent of thi pace vecto i defined equal to zeo: c c i id (56) c iq 0 hi efeence fae ealie a pecial contol of the achine toque and peed, y uing only one cuent coponent and the tato voltage. De-coupling cicuit ae needed fo an independent contol of the peed and toque. he coplete et of equation which decie the dynaic and teady-tate opeation of the induction achine i. c c dψ q c u q Ω ψ d dt u c c c d c d i d Ω ψ q dt c c c q c q q Ω Ω ψ d dt c c c d c d d Ω Ω ψ q dt c c c d d M d c q c q M c q dψ u 0 i ( ) dψ u 0 i ( ) ψ ψ ψ i i i i c q ψ i i dψ c c c d d M d (57-65) dω 3P c c ψ q i d L dt H 4 he loc diaga fo Fig..3, ue a input the tato voltage pace vecto, the d-ai tato cuent coponent, the load toque, and the ynchonou angula velocity. A output i conideed the oto angula velocity (peed). i a i i c Ω 3 ---> /3 e -j θ c - 3 -/ i q i d - n * / ψ d * / Ω u a u u c > /3e -j θ c - 3 -/ u d u q - n / ψ q * - - /(H) L Fig..3. Bloc diaga of the induction achine in ynchonou efeence fae (lined to tato cuent pace vecto)

38 39 V) oto cuent pace vecto fied ynchonou efeence fae. In ode to lin the d-ai of the ynchonou efeence fae to the oto cuent pace vecto, the q-coponent of thi pace vecto i defined equal to zeo: c c i i d (66) c i q 0 Applying the aove condition, a coplete equation yte fo the induction achine epeed in thi paticulaly efeence fae can e deduced: c c dψ q c u q Ω ψ d dt u u c c c d c d i d Ω ψ q dt c c q c q 0 ( Ω Ω ) ψ d dt c c c d c d d Ω Ω ψ q dt c c c d d M d c q c q c q c M q dψ dψ u 0 i ( ) ψ ψ ψ ψ i i i i i i dψ c c c d d M d (67-75) dω 3P c c 3P c c ψ q i d L M i q i d L dt H 4 H 4 In Fig..4, the loc diaga that decie the induction achine odel ha a input the d-ai tato voltage coponent and cuent pace vecto, the load toque, and the ynchonou angula velocity. A output i conideed the oto angula velocity (peed). i a i i c Ω 3 ---> /3 e -j c θ - 3 -/ i q i d - ψ q * n / ψ d - / Ω u a u u c > /3 e -j c θ - 3 -/ u q u d 3P/4 * - /(H) L Fig..4. Bloc diaga of the induction achine in ynchonou efeence fae (lined to oto cuent pace vecto)

39 40 VI) Magnetiing cuent pace vecto fied ynchonou efeence fae. In ode to lin the d-ai of the ynchonou efeence fae to the agnetiing cuent pace vecto, the q-coponent of thi pace vecto i defined equal to zeo: c c c c i id i i (76) c iq 0 Conideing the aove condition fo a ynchonou efeence fae, the coplete et of equation fo the thee-phae induction achine can e otained: c c c dψ q c u q i q Ω ψ d dt u c c c d c d i d Ω ψ q dt c c c q c q q Ω Ω ψ d dt c c c d c d d Ω Ω ψ q dt c c c q q q c d c i d dψ u 0 i ( ) dψ u 0 i ( ) i 0 i i ψ ψ M i dψ c d i i i c c c c q q M q l q ψ i i i c c c c q M q q l q ψ i i c c c d d M d dω 3P c c M i d i q L dt H 4 (77-85) A loc diaga fo the atheatical odel of the induction achine in thi efeence fae i peented in Fig..5. A input ae choen the tato cuent pace vecto, the d-ai coponent of the tato voltage pace vecto, the ynchonou angula velocity and the load toque. he output i the achine (oto) angula velocity. he iilaity to the ynchonou efeence fae lined to the ai-gap flu pace vecto can e oeved, a the agnetiing cuent i diectly popotional with the ai-gap flu, though the agnetiing inductance. i a i i c 3 --> j θ e /3-3 -/ Ω * i d i q l - ψ q n l / ψ d * 3P/4 / Ω u a u u c 3 --> /3 e j θ - u d 3 -/ u d u q - /(H) L Fig..5. Bloc diaga of the induction achine in ynchonou efeence fae (lined to agnetiing cuent pace vecto)

40 4.8. D-Q Ae odel of the thee-phae induction achine he thee-phae induction achine can e odelled y uing diffeent tate-pace vaiale and eeping a input the tato voltage and the load toque, and a output the electoagnetic toque and the oto angula velocity. he poile et of cuent and flu linage pe econd pace vecto ae defined a follow (Nowotny and Lipo - 996): i i j i d q i i j i d q i i j i d q j d q j d q j d q [ ] [ i i i ] (86-9) (9) he d-q ae ae othogonal and fied to the tato, d ai coincide to the agnetic ai of the a winding. A thee ae fou voltage equation, it i neceay to conide two of the pace vecto a tatevaiale in ode to otain a olution fo the equation yte. Let the elected pai of tate-pace vaiale e denoted a,. he et of i tate-pace vaiale will e epeed in te of the two elected tate-pace vaiale: a a a a a3 a3 [ ] [ i i i ] [ ] (93) a4 a4 a 5 a 5 a6 a6 Only fouteen out of the fifteen tate-pace poiilitie epeent valid atheatical odel fo the thee-phae induction achine (the pai of tate-pace vaiale that copie aigap flu linage pe econd pace vecto and the agnetiing cuent pace vecto cannot e elected a thee vecto have the ae diection). hee ae thee type of odel:. cuent tate-pace vaiale odel;. flu linage tate-pace vaiale odel; 3. ied cuent-flu linage tate-pace vaiale odel. In conjunction with the echanical equation: P p ( e L ) (94) J we otain a coplete veion of the thee-phae induction achine odel, viewed a the ey fo a otion contol yte. he tating point fo the tate-vaiale odel i given y the voltage equation yte witten in tationay efeence fae: u L p (95) whee: i the elected et of tate-vaiale and epeent alo the output of the odel, u i the input vecto (tato voltage), L i the coefficient ati (it can e foed y eactance value, o non-dienional eleent) fo ultiplying the tie deivative of the tate-vaiale, i the coefficient ati (it can e foed y eitance and eactance value o non-dienional eleent) fo ultiplying the tate-vaiale and p tand fo the diffeential opeato (d/dt).

41 4 It eult the geneal fo of the tate-vaiale yte: p A B u (96) whee: A L B L All the following atheatical odel peit a dicetiation fo the ipleentation of contolle in the dive yte with thee-phae induction achine. Claically, the cuent tatepace and the flu tate-pace odel ae the choen option fo the anufactue of dive yte. he oiginal vecto oientation chee wa aed on the alignent of the ynchonou efeence fae to the oto flu linage pace vecto. hen the vecto contol tategy wa etended y conideing a well the tato and ai-gap flu pace vecto, a alignent of the efeence fae. Howeve, fo a coplete analyi of the vecto contol tategie, each of the i pace vecto that can e elected a tate-pace vaiale epeent a poile ai fo a efeence fae. If the agnetiing cuent pace vecto election a alignent give iila eult to the ai-gap flu pace vecto cae, the tato and oto cuent pace vecto ae till to e futhe analyed a new option fo vecto contol. An eay to follow tep algoith fo ipleenting vecto oiented contol yte i otained a follow:. A coplete atheatical odel of the thee-phae induction achine i developed in tationay efeence fae, accoding to the choen et of tate-pace vaiale;. he oto aed vaiale ae copletely epeed in the new tate-vaiale yte; 3. he oto angula velocity te i utituted with ( - ) whee i the angula velocity of the ynchonou fae; 4. he ynchonou efeence fae i elected lined to one of the pace vecto, which ean that the q-ai coponent of the efeence pace vecto i null; 5. he toque equation i coputed accoding to the elected flu o cuent pace vecto in the ynchonou efeence fae. he loc diaga of odelling an induction oto fo vecto contol pupoe i detailed in Fig..6. ynchonou fae u p ABu [a ij ] e -j θ e (cloed loop) e (open loop) l - P J tationay fae Fig..6. Bloc diaga fo of the induction achine atheatical odel Fig..7 how the geneal two-ai equivalent cicuit fo the induction achine, in an aitay efeence fae. hi equivalent cicuit epeent the tating point fo deteining the ati equation, which will e deived futhe.

42 43 d L L ( ) d l l - - u q i q L M i q u q - - q - L l L l ( ) q - u d i d L M i d u d - - Fig..7. Aitay efeence-fae equivalent cicuit fo a thee-phae, yetical induction achine Note: Fo copactne, the eleent fo Fig..7. have to e tanfoed in te of eactance, while the flu linage ha to e epeed in flu linage unit pe econd, o volt..8.. Model with cuent pace vecto tate-pace vaiale I) A cople tate vaiale, the cuent pace vecto iq, id, i q, i d ae uually aued. he tato cuent pace vecto i conideed geneally a the ight choice, ecaue it coepond to diectly eauale quantitie. hi odel i eadily availale fo voltage and flu linage pe econd equation, and it can e epeed in a ati fo a follow: D D D D 0 0 D D i q i q q u p i d D i 0 0 D D D d u D D d i q i q u q 0 0 i d D D D D i d D D u d 0 0 D D D D D D (97) whee: D If the tato, oto and total leaage facto definition ae ued: ( σ ) l ( σ ) l σ ( σ )( ) σ we otain: σ D ( σ σ σ σ ) σ

43 44 he electoagnetic toque can e coputed a: 3P e ( iq i d id i q ) (98) 4 II) he tato and the agnetiing pace vecto cuent a tate-pace vaiale iq, id, q i, id epeent anothe atheatical d-q ai odel aed on cuent pace vecto. By electing the agnetiing cuent pace vecto aong the tate-pace vaiale, it i poile to include the atuation effect in odelling the induction oto. Alo, the tato cuent pace vecto i a eauale quantity, and deteine a pecie and accuate option fo ipleenting contolle. he tate ati eleent ae all non-zeo, which iplie a coputational effot iila to the peviouly analyed odel. l iq iq l i p d i d iq D i l l l l l l q id id (99) l l l l l l 0 0 uq 0 0 u d D l 0 l 0 u q 0 l 0 l u d he intantaneou electoagnetic toque ay e epeed in te of the tate-pace vaiale: 3P e ( iq id id iq ) (00) 4 III) he thid poile coination of cuent pace vecto a tate-pace vaiale i otained y electing agnetiing and oto cuent i q, i d, iq, id. When copaed to the othe cuent pace vecto odel, the iila coputational uden fo otaining the output of the yte i oviou. he ain diffeence etween the i the peence of the gloal paaete (elfeactance) in the tato and oto cuent tate-pace vaiale odel, while in the othe two odel an accuate deteination of the leaage eactance o leaage facto i neceay. A coplete deciption of the tate-vaiale in ati notation i given elow:

44 45 l i q i q l i p d i d iq D i l l l l l l q i d id l l l l l l (0) 0 0 uq 0 0 u d D l 0 l 0 u q 0 l 0 l u d he intantaneou electoagnetic toque i epeed a follow: 3P e ( i d iq i q id ) (0) Model with flu linage a tate-pace vaaile I) When flu linage pe econd ae elected a tate-pace vaiale, the odel ae le coputationally deanding than in the cuent tate-pace vaiale veion. A each flu contain infoation aout two cuent pace vecto coponent, the tate ati contain zeo eleent. One option i to elect the tato and oto flu linage pace vecto q, d, q, d fo deciing the atheatical odel of the achine. Fo thi yte the ati equation ae a follow: 0 0 D D q q q 0 0 u p d D D d u d (03) q 0 q u q D D d d u d 0 D D he electoagnetic toque i deteined with the equivalent elation: 3P e ( q d d q ) 4 D (04) By copaion with the cuent odel, it can e oeved that the coputational uden i eentially lowe. Due to thi ipotant featue, thi odel i the ot uitale fo dicetiation in otion contol tategie. II) An altenative to odel the thee-phae induction achine with flu pace vecto a tatevaiale yte, i the election of ai-gap flu pace vecto aong the et of independent vaiale. hi choice peent the advantage of an eaie atuation effect odelling, ut the diadvantage of an inceaed coputational uden. he ai-gap flu i a eauale quantity, and thi advantage ipoe it in any pactical olution fo vecto contol chee.

45 46 A fit appoach i given y the tato and ai-gap flu pace vecto elected a tate-pace vaiale q, d, q, d. 0 0 l l q 0 0 q d l l p d q l l l l l q d ld D D l D D D l d l l l l D l D D D l uq u d l l 0 0 D D u q l l u d 0 0 D D he electoagnetic toque epeion depend on tato leaage facto: 3P e q d d q 4 l l D D l (05) (06) III) he thid option of electing flu pace vecto a tate-pace vaiale i iila to the peviou odel. It copie the ai-gap (agnetiing) flu and oto flu pace vecto q, d, q, d in the et of independent vaiale. he ae coputational effot i equied, and the electoagnetic toque i deteined in a uitale fo fo vecto contol if the oto leaage facto i nown: l l l 0 D l D D l D D q l l l q 0 p d D l D D D l D d q q 0 d l l d (07) 0 l l l l 0 0 D D uq l l u d 0 0 D D u q u d he eulting elation fo coputing the intantaneou electoagnetic toque i a follow:

46 47 3P (08) ( ) e q d d q 4 l.8.3. Model with ied cuent flu pace vecto tate-pace vaiale I) Fo otaining an acceptale coputational effot a well a eauale output quantitie, the geneal accepted olution ae the ied cuent-flu tate-pace vaiale odel. If the tato vaiale ae choen fo odelling the thee-phae induction achine yte, then a ied flu linage-cuent tate-pace vaiale odel q, d, iq, id i developed. hi atheatical odel i elected when tato flu oiented tategy i ipleented. he ati equation and the electoagnetic toque elation ae: q q p d d i q D D D i q id id D D D (09) uq u 0 0 d D u q u d 0 0 D he intantaneou electoagnetic toque elation depend only on the agnitude of the output vecto coponent. 3P e ( diq qid ) (0) 4 II) Anothe ipotant ied flu linage-cuent tate-pace vaiale odel q, d, i q, i d i that epeed in oto quantitie. hi odel epeent the optiu olution fo oto flu oiented contol tategy in a dive yte with an induction achine. Howeve, a it i ipoile to eaue the oto cuent if the achine i equipped with cage oto, thee ae liitation in uing thi odel fo vecto contol tategie. he tate-pace vaiale yte i detailed elow: q q d d q i q D uq p u d i u q 0 i D D d D i d u d D 0 D D D ()

47 48 he electoagnetic toque epeion depend only on the agnitude of the output vecto coponent: 3P e ( q i d d i q ) () 4 III) he ot ued altenative odel fo oto flu oiented contol tategie i that which copie the tato cuent and oto flu linage q, d, iq, id a tate-pace vaiale. It contain the advantage of eauale output quantitie (tato cuent) and acceptale coputational uden. he ati equation and the electoagnetic toque elation ae peented elow: 0 D D D iq i q 0 p id D D i D d q q 0 d d D D uq u d 0 0 D D u q u d and 3P e ( iq d id q ) (4) 4 (3) IV) A theoetical atheatical odel i that with ied oto cuent pace vecto and tato flu linage pace vecto a tate-pace vaiale q, d, i q, i d. It hould e noted that the advantage of thi odel i that only the tato winding paaete i neceay and a uch the influence of oto paaete i iniied. hi odel can e ued fo an unconventional tato flu oiented contol with oto cuent coponent poducing the toque and the flu. he epeion fo ipleenting thi odel ae:

48 49 D D D 0 0 i q i q D D uq i p d i D D D d u 0 0 d (5) D D q q u q d d 0 u d and 3P e ( q i d di q ) (6) 4 V) he ied tato cuent pace vecto and ai-gap flu pace vecto q, d, iq, id tate-pace vaiale elong to one of the ot cople type of odel. It peeve infoation egading oth tato and oto paaete. hi atheatical odel i the uitale choice fo the ai-gap flu oientation contol tategy. he geatet advantage of thi odel i that y uing Hall eno o tapped tato winding, all the output vecto coponent ae eauale. Due to it veatility, thi odel i widely ued in contolle ipleentation, epecially fo ediu peed application. It peit alo the iulation o odelling of the atuation effect. By copaion to the peviou ied odel, the tate-pace ati contain only non-zeo eleent, which lead to geate coputational effot. a iq l ( / ) ( / ) iq p id l ( / ) ( / ) id q D ( l ) l l ( / ) l l ( / ) q d l l ( / ) ( l ) l ( / ) l d 0 0 uq 0 0 u d D l 0 l 0 u q 0 l 0 l u d (7) he electoagnetic toque i coputed a: 3P e ( iqd idq ) (8) 4 VI) Anothe theoetical odel, iila in fo to the pecedent one, i the ied oto cuent pace vecto and ai-gap flu pace vecto a tate-pace vaiale q, d, i q, i d It ain hotcoing i the peence of the uneauale oto cuent aong the tate-pace vaiale. he ati equation of the yte i given elow:

49 50 i q l ( / ) ( / ) i q i p d l ( / ) ( / ) i d q D l l l l ( / ) l l ( / ) q d l l ( / ) l l l ( / ) l d 0 0 uq 0 0 u d D l 0 l 0 u q 0 l 0 l u d (9) he electoagnetic toque elation ecoe: 3P e ( id q iqd ) (0) 4 VII) If the agnetiing cuent pace vecto i elected a tate-pace vaiale togethe with one of the flu linage pace vecto (i.e. tato flu linage) iq, id, q, d, the coputation of the tate ati eleent give eveal null eult. hi choice fo a et of tate-pace vaiale peent only theoetical ipotance, a the output vecto coponent ae uneauale. Alo the coputational deand of the odel doe not ae it a pactical option fo vecto contol ipleentation. l l l l l ld l D D iq l l l l l iq i p d ld D l D i d q q 0 0 d l l d () 0 0 l l l l 0 0 D D uq l l u d 0 0 D D u q u d he electoagnetic toque elation: 3P e ( id q iq d ) () 4 l VIII) One othe odel analyed fo theoetical eaon, i the ied cuent-flu linage pace vecto odel which ealie a the connection etween the agnetiing cuent pace-vecto and the oto flu linage pace vecto, elected a tate-pace vaiale iq, id, q, d A the output vecto coponent cannot e eaued, thi odel, lie the peviou one, i pohiitive fo ipleentation in vecto contol tategie. Howeve, fo coputational effot, the tate ati contain the ae nue of zeo eleent (fou) and the intantaneou electoagnetic toque i deteined in a uitale fo copaale to the claical vecto contol ipleentation.

50 5 l l l l l 0 l D l D D iq l l l l l iq 0 i p d l D D l D i d q q 0 d l l d 0 l l l l 0 0 D D uq l l u d 0 0 D D u q u d (3) he electoagnetic toque i deteined a follow: 3P e ( iq d id q ) (4) 4 l.9. Vecto contol tategie fo thee-phae induction achine he ai of vecto contol i uually to decouple the tato cuent i into it flu poducing and toque poducing coponent (i d, i q epectively) in ode to otain a decoupled contol of the flu and the electoagnetic toque. Fo thi eaon a pecial efeence fae i elected fied to diffeent pace vecto vaiale. he efeence fae ha to e ynchonou, a all the pace vecto have the ae angula velocity given y the upply voltage fequency (Va - 990, 99), (Sleon - 994), (Keleen and Iec - 987). Geneally, the te of vecto contol i aociated with field oientation contol. hi ean that the pecial ynchonou efeence fae i lined to one of the flu linage pace vecto. he oiginal field oientation hee, developed oe than twenty five yea ago, wa aed on the aligneent of the efeence fae to the oto flu. Afte 985, (DeDonce and Nowotny - 988), (Edan and Hoft - 990) thi contol tategy wa etented to the ai-gap flu and to the tato flu aligneent of the ynchonuou efeence fae. Howeve, fo vecto contol chee thee ae two oe poiilitie, elated to the cuent pace vecto: tato and oto cuent. All of thee contol tategie ae invetigated in thi chapte, accoding to the odelling point of view of the induction achine. An eay to follow tep algoith fo ipleenting vecto oiented contol yte i otained a follow (Lai - 999):. A coplete atheatical odel of the thee-phae induction achine i developed in tationay efeence fae, accoding to the choen et of tate-pace vaiale;. he oto aed vaiale ae copletely epeed in the new tate-vaiale yte; 3. he oto angula velocity te i utituted with ( - ) whee i the angula velocity of the ynchonou fae; 4. he ynchonou efeence fae i elected lined to one of the pace vecto, which ean that the q-ai coponent of the efeence pace vecto i null; 5. he toque equation i coputed accoding to the elected flu o cuent pace vecto in the ynchonou efeence fae. he tanfoation of efeence fae fo the induction oto vecto contol can e uaied a hown in Fig..8.

51 5 Synchonuou efeence fae wo-ai co-odinate yte Stationay efeence fae hee-phae yte Dive Contolle Invee /3 Cla/ Pa anfoation Induction oto 3/ Cla/ Pa anfoation Fig..8. Bloc diaga of tanfoation of fae and co-odinate yte fo induction oto vecto contol.9.. Stato flu field oientation (SFO) Fo thi vecto contol tategy, the et of tate-pace vaiale foed y tato flu linage and cuent pace vecto q, d, iq, id i elected. he atheatical odel i given in the chapte dedicated to d-q odel of the thee-phae induction achine. he oto aed vaiale epeed in tate-pace vaiale te ae: i q ( q iq ) i ( i ) d d d ( D i ) q q q ( D i ) d d d (5-8) o otain the achine equation in the ynchonuou tato flu efeence fae, one ha to eliinate the oto flu fo the oto voltage equation and then foce the q coponent of the tato flu to e zeo. he tato voltage equation eain unchanged. If a cage oto i conideed, the eultant equation ae a follow: p p D iq ( d D id ) q 0 (9-30) p p D id D iq d q whee the definition ued fo the lip peed i: e. If the pecial efeence fae i fied to the tato flu linage vecto, the q-coponent of thi flu vecto i defined equal to zeo: q 0 d Fo tato flu linage equation, the q-cuent coponent ae given y: (3)

52 53 iq iq (3) iq iq he electoagnetic toque elation and lip peed can e deived in tato field oientation contol a: 3P e d iq 4 (33-34) ( D p ) iq D i d d he econd dynaic equation of the achine, how that thee i a coupling etween the tato cuent coponent. Conequently, any change in the toque poducing coponent i without changing i d accodingly will caue a tanient in the tato flu. A decouple i neceay to ovecoe thi diadvantage. heefoe the coand cuent of the d-ai coponent of the tato cuent can e calculated a follow: Ki id Kp d idq p (35-36) D iq idq D p whee K p and K i ae popotional, epectively integal coefficient of the flu contolle. hi contolle can e PI type o oft coputing technique type (fuzzy, neual-netwo) a deontated y Xu and Nowotny (988). Steady-tate pefoance of a tato flu oiented yte Letting the deivative opeato p 0, one can otain the teady-tate voltage equation of the induction achine. Afte eveal anipulation of the yte equation, the d cuent coponent ae given y: id i q iq (37) id iq it yield the lip peed equation: iq D ( ) iq 0 (38) he olution of the aove equation have to e eal, fo a given tato flu linage. hi ean that the deteinant of the econd ode equation atify the condition: ( ) D ( ) ( iq ) 4 0 (39) he aiu value fo the q coponent of the tato cuent, the lip peed and the electoagnetic toque (pull-out toque) ae: q

53 54 ( iq ) ( ) ( ) e a a a D D ( ) 3P 4 D If the angula lip velocity i lage than ( ) a (40-4) tatic intaility will aie. hi aiu value depend only on achine paaete and not on tato flu level. Neveethele, the pull-out toque depend on the quae of the flu agnitude. he pull-out toque deteine the liit fo the yte taility opeation zone, when the tato flu oiented contol i eployed. It i poile to liit the toque coand not to eceed the pull out toque fo a given tato flu. Soe ipotant concluion cand e dawn fo thi vecto contol tategy: he electoagnetic toque and tato flu poducing coponent of the tato cuent ae not decoupled; A paaete dependent decoupling netwo ha to e included; oque and flu contol doe not equie peed feed ac; Fo opeation at low peed it i difficult to etiate the tato flu; It i a good altenative fo ediu pefoance dive..9.. oto flu field oientation (FO) Fo thi vecto contol tategy, the et of tate-pace vaiale foed y oto flu linage and tato cuent pace vecto q, d, iq, id i elected. If the induction oto i equipped with wound oto, a the oto cuent ae alo eauale, the oto cuent pace vecto can e alo elected a tate-pace vaiale togethe with the oto flu linage pace vecto q, d, i q, i d. he atheatical odel i given in the chapte dedicated to d-q odel of the thee-phae induction achine. he fit cae will e analyed, fo the econd one the algoit i iila. he tato flu linage and oto cuent pace vecto coponent epeed a function in te of tate-pace vaiale ae: i i ( D i ) q q q ( D i ) d d d ( i ) q q q ( i ) d d d he voltage equation e-witten in te of the tate-pace vaiale ecoe: (43-46)

54 55 u D p i D i p q q d q d u D p i D i p d d q d q u 0 i p q q q d (47-50) p u d 0 id d q If the pecial efeence fae i fied to the oto flu linage vecto, the q-coponent of thi flu vecto i defined equal to zeo: q 0 d Fo tato flu linage equation, the q-cuent coponent ae given y: i i q q i q iq he flu poducing coponent of the tato cuent i deteined a follow: (5) (5) p d id (53) he aove elation how that thee i no need of a cuent decouple in oto field oientation chee. Both tato cuent coponent (toque and flu poducing) can e contolled independently. Steady-tate pefoance of a oto flu oiented yte Letting the deivative opeato p 0, one can otain the teady-tate voltage equation of the induction achine. Afte eveal anipulation of the yte equation, the d cuent coponent ae given y: d i q id (54) id 0 It yield the lip peed equation: d iq 0 (55) which fo a given oto flu ha alway eal olution. hu the eulting cuent contolled lip peed and the electoagnetic toque ae: iq d (56-57) 3P 3P e d iq i q 4 4 ( ) ( ) he ot ipotant featue of oto field oiented (FO) vecto contol ae:

55 56 A it i the oiginal vecto contol tategy developed y Blahe (97), the FO epeent the ot popula appoach in vecto contol tategie; It povide coplete decoupling of the toque and flu poducing coponent of the tato cuent; hee ae poile diect vecto contol tategy (the uage of eno o odel to povide feedac of the flu agnitude and oientation) and indiect vecto contol tategy (the uage of aued lip fequency elationhip to achieve field oientation); he diect vecto contol tategy i the optiu choice fo ediu and high-peed application; he indiect vecto contol tategy i the optiu choice fo low-peed application;.9.3. Ai-gap flu field oientation (AFO) Fo thi vecto contol tategy, the et of tate-pace vaiale foed y ai-gap flu linage and tato cuent pace vecto q, d, iq, id i elected. If the induction oto i equipped with wound oto, a the oto cuent ae alo eauale, the oto cuent pace vecto can e alo elected a tate-pace vaiale togethe with the ai-gap flu linage pace vecto q, d, i q, i d. he atheatical odel i given in the chapte dedicated to d-q odel of the thee-phae induction achine. A oth cae ae iila to analye, only the fit cae will e detailed. he tato and oto flu linage and oto cuent pace vecto coponent epeed a function in te of tate-pace vaiale ae: ( ) ( ) q q iq d d d i i i i ( ) q q q i ( ) q q d i q q q i d d d (58-6) (6-63) he voltage equation of the achine epeed in tate-vaiale te, ae eadily deductile now: p p uq ( ) iq q ( d ( ) id ) u p i p i ( ) d ( ) d d q ( ) q u 0 ( ) p i p ( ) i q q q d d p p u d 0 ( ) id d q ( ) iq (64-67)

56 57 If the pecial efeence fae i fied to the ai-gap flu linage vecto, the q-coponent of thi flu vecto i defined equal to zeo: q 0 d Fo ai-gap flu linage equation, the q-cuent coponent ae given y: i i i q q q i q he flu poducing coponent of the tato cuent i deteined a follow: (68) (69) p p ( ) id d ( ) iq (70) One can note that the d-ai coponent of tato cuent fo AFO i not only contollled y the d- ai ai-gap flu coponennt, ut alo y the q-ai (toque poducing) coponent of the tato cuent. It i neceay to decouple the tato cuent coponent, in ode to achieve a linea contol. Fo thi eaon, the coand of cuent of the d-ai coponent i coputed a follow: Ki id K p d idq p (7-7) ( ) iq idq ( ) p he electoagnetic toque elation and lip peed can e deived in ai-gap field oientation contol a: 3P e diq 4 [ ( ) p] i (73-74) q d ( ) id Steady-tate pefoance of an ai-gap flu oiented yte Letting the deivative opeato p 0, the teady-tate voltage equation of the induction achine ae eadily deteined. Afte eveal anipulation of the yte equation, the d cuent coponent ae given y: id iq ( ) iq (75) ( ) iq id It yield the lip peed equation: ( ) iq d iq 0 (76) he olution of the aove equation have to e eal, fo a given tato flu linage. hi ean that the deteinant of the econd ode equation atify the condition: ( d ) ( ) ( iq ) 4 0 (77) he aiu value fo the q coponent of the tato cuent, the lip peed and the electoagnetic toque (pull-out toque) ae:

57 58 ( iq ) ( ) ( ) e a a a d ( ) 3P 8 ( d ) When the angula lip velocity i lage than ( ) a (78-80) intaility will occu. One hould note that the aiu (pull-out) angula lip velocity depend only on the oto paaete and doe not depend on the ai-gap flu. Howeve, the aiu (pull-out) toque i popotional to the quae of the ai-gap flu agnitude and thu a all inceent of ai-gap flu will detetine a ignificant inceent of the electoagnetic toque. Soe ipotant concluion cand e dawn fo the ai-gap field oientation (AFO) contol tategy: he electoagnetic toque and ai-gap flu poducing coponent of the tato cuent ae not decoupled; A paaete dependent decoupling netwo ha to e included; oque and flu contol doe not equie peed feed ac; No ophiticated paaete etiation ethod o odel aed oeve i equied; It i a good altenative fo low and ediu pefoance dive a the ai-gap flu can e eaued diectly;.9.4. Stato cuent oientation (SCO) If the field oientation contol i well etalihed with ultiple pactical olution fo diffeent indutial application, an unconventional ethod of contolling the peed and toque fo the induction achine, i given y the election of cuent pace vecto a lining ai fo the ynchonuou efeence fae. he et of tate-pace vaiale i identical with the tato flu field oientation contol (SFO): the tato flu and cuent pace vecto atheatical odel i identical to that ued in SFO cae. he oto aed vaiale epeed in tate-pace vaiale te ae: c c c i q ( q iq ) i ( i ) c c c d d d ( D i ) c c c q q q ( D i ) c c c d d d q, d, iq, id. he (8-84) o otain the achine equation in the ynchonuou tato cuent efeence fae, one ha to eliinate the oto flu fo the oto voltage equation and then foce the q coponent of the tato cuent to e zeo. he tato voltage equation eain unchanged. If a cage oto i conideed, the eultant equation ae a follow:

58 59 p c c c p c D iq ( d D id ) q 0 (85-86) p c c p c c D id D iq d q 0 If the pecial efeence fae i fied to the tato cuent pace vecto, the q-coponent of thi cuent vecto i defined equal to zeo: c iq 0 (87) c i id Diffeent fo the SFO cae, we have to epe the q-flu coponent, y conideing the flu linage equation: c c q q c (88) c q q he electoagnetic toque elation and lip peed can e deived in tato cuent oientation contol a: 3P c c e q id 4 (89-90) c ( p ) q c D id d Fo the econd dynaic equation of the achine, a elation etween the d-ai coponent of the tato cuent and tato flu linage pace vecto coponent can e deduced: p c p c c D id d q 0 (9) he peviou dynaic equation of the achine, how that thee i a coupling etween the tato c flu linage coponent. Conequently, any change in the toque poducing coponent c without changing q accodingly will caue a tenient in the tato flu. A decouple i neceay to ovecoe thi diadvantage. heefoe the coand cuent of the d-ai coponent of the tato cuent can e calculated a follow: c K i c c d Kp id dq p c (9-9) q c dq p whee K p and K i ae popotional, epectively integal coefficient of the cuent contolle. hi contolle can e PI type o oft coputing technique type (fuzzy, neual-netwo). Steady-tate pefoance of a tato cuent oiented yte Letting the deivative opeato p 0, one can otain the teady-tate voltage equation of the induction achine. Afte eveal anipulation of the yte equation, the d cuent coponent ae given y: d

59 60 D c c c d q q c c d q it yield the lip peed equation: (93) c c c q i d q 0 (94) he olution of the aove equation have to e eal, fo a given tato cuent. hi ean that the deteinant of the econd ode equation atify the condition: c c d q i 4 0 (95) he aiu value fo the q coponent of the tato flu linage, the lip peed and the electoagnetic toque (pull-out toque) ae: c c id ( q ) a ( ) ( ) e a a ( id ) c 3P 4 If the angula lip velocity i lage than ( ) a (96-99) tatic intaility will aie. hi aiu value depend only on achine paaete and not on tato cuent o flu level. Nevethele, the pullout toque depend on the quae of the tato cuent agnitude. he pull-out toque deteine the liit fo the yte taility opeation zone, when the tato cuent pace vecto oiented contol i eployed. It i poile to liit the toque coand not to eceed the pull out toque fo a given tato cuent. Soe ipotant concluion can e dawn fo the tato cuent oientation (SCO) vecto contol tategy: he electoagnetic toque and tato cuent elated coponent of the tato flu linage ae not decoupled; A paaete dependent decoupling netwo ha to e included; oque and flu contol doe not equie peed feed ac; An opeation at low peed i eay to ealie a thee i no need to etiate the tato flu; It i a good altenative fo wide ange peed dive oto cuent oientation (CO) Anothe unconventional vecto contol tategy i the one wich copie a ynchonuou efeence fae lined to the oto cuent pace vecto. he oto cuent oientation contol tategy (CO) i analyed fo a cage oto induction oto. he elected et of tate-pace vaiale i given y two option if we conide the citeia of diect eauale quantitie: I) he tato flu linage and oto cuent pace vecto: q, d, i q, i d ; When the tato flu and and the oto cuent ae elected a tate-pace vaiale, one can deive the oto flu and tato cuent function in te of tate vaiale a follow:

60 6 i ( i ) c c c q q q i ( i ) c c c d d d D i c c c q q q D i c c c d d d ( ) o otain the achine equation in the ynchonuou oto cuent efeence fae, one ha to eliinate the oto flu fo the oto voltage equation and then foce the q coponent of the oto cuent to e zeo. he tato voltage equation ae e-witten alo. he eultant equation have the following fo: u u i p c c c c q q q d p c c c c d i d d q D p D p 0 i i c c c c q d q d ( ) D p c D c p c c 0 i d i q d q If the pecial efeence fae i fied to the oto cuent pace vecto, the q-coponent of thi cuent vecto i defined equal to zeo: c i q 0 (309) c c i i d Diffeent fo the FO cae, we have to epe the q-flu coponent, y conideing the flu linage equation: c c q q c (30) c q q he electoagnetic toque elation and lip peed can e deived in tato cuent oientation contol a: 3P c c e q i d 4 (3-3) c pq c c D i d d Fo the econd dynaic equation of the achine, a elation etween the d-ai coponent of the oto cuent and tato flu linage pace vecto coponent can e deduced: D p c p c c 0 i d d q (33) A coupling etween the tato flu linage coponent appea, a can e deduced fo the peviou dynaic equation of the achine. Conequently, any change in the toque poducing

61 6 c c coponent d without changing q accodingly, will caue a tanient in the tato flu. he coand cuent of the d-ai coponent of the tato cuent can e calculated in ode to ovecoe thi diadvantage: c Ki c c d Kp i d dq p (34-35) c c dq q p whee K p and K i ae popotional, epectively integal coefficient of the cuent contolle. hi contolle can e PI type o oft coputing technique type (fuzzy, neual-netwo). Steady-tate pefoance of a oto cuent oiented yte Afte eveal anipulation of the yte equation if we let the deivative opeato p 0, the d cuent coponent ae otained fo the teady-tate voltage equation of the induction achine: c D c d q (36) c d 0 It yield the lip angula velocity equation: c D D c id q 0 (37) which fo a given oto flu ha alway eal olution. hu the lip peed and the electoagnetic toque ae: c i d c q (38-39) 3P c c e q i d 4 he ot ipotant featue of oto cuent oiented (CO) vecto contol with ied flu and cuent tate-pace vaiale ae: he electoagnetic toque and oto cuent elated coponent of the tato flu linage ae not decoupled; he decoupling netwo that ha to e included i paaete independent and eay to ipleent; oque and flu contol equie peed feed ac; hee i no taility liit a thee i no pull-out lip peed o electoagnetic toque, An opeation at low peed i eay to ealie a thee i no need to etiate the tato flu; It i a good altenative fo wide ange peed dive. II) he tato and oto cuent iq, id, i q, i d. When the tato and the oto cuent ae elected a tate-pace vaiale, one can deive the tato and oto flu a function in te of tate vaiale fo the claical flu linage equation. o otain the achine equation in the ynchonuou oto cuent efeence fae, one ha to eliinate the oto flu fo the oto voltage equation and then foce the q coponent of the oto cuent to e zeo. he tato voltage equation ae e-witten alo. he eultant equation have the following fo:

62 63 u p i p i i i c c c c c q q q d d u i i p i p i c c c c c d q q d d 0 p i p i i i c c c c q q d d (30-33) c c p c p c 0 iq i q id i d If the pecial efeence fae i fied to the oto cuent pace vecto, the q-coponent of thi cuent vecto i defined equal to zeo: c i q 0 (34) c c i i d Diffeent fo the (I) cae of CO contol tategy, fo the (II) cae we have to epe the tato flu linage q-coponent, y conideing the flu linage equation: c c q iq c c (35) c q iq q Fo the econd dynaic equation of the achine, a elation etween the d-ai coponent of the oto cuent and tato cuent pace vecto coponent can e deduced: p c c p c id i q i d (36) he electoagnetic toque elation and lip peed can e deived in tato cuent oientation contol a: 3P c c e iqi d 4 (37-38) c piq c c id i d A coupling etween the d-ai and q-ai tato cuent coponent appea, a can e deduced fo the peviou dynaic equation of the achine. Conequently, any change in the toque poducing coponent c c i d without changing i q accodingly, will caue a tanient in the tato flu. he coand cuent of the d-ai coponent of the tato cuent can e calculated in ode to ovecoe thi diadvantage: c Ki c c id K p i d idq p (39-330) c c idq iq p whee K p and K i ae popotional, epectively integal coefficient of the cuent contolle. hi contolle can e PI type o oft coputing technique type (fuzzy, neual-netwo). Steady-tate pefoance of a oto cuent oiented yte Afte eveal anipulation of the yte equation if we let the deivative opeato p 0, the d flu coponent ae otained fo the teady-tate voltage equation of the induction achine:

63 64 c D d i c q (33) c d 0 It yield the lip angula velocity equation: c D D c id iq 0 (33) which fo a given oto flu ha alway eal olution. hu the lip peed and the electoagnetic toque ae: c i d c iq ( ) 3P c c e iq i d 4 he oto cuent oiented (CO) vecto contol with cuent tate-pace vaiale peent the ae featue a the ied flu-cuent tate vaiale odel. Howeve the cuent odel i eaie to e ipleented a it povide diectly the etiated value fo the two-ai co-odinate tato cuent. hu the coand fo a cuent PWM invete i eadily otainale. Alo, a feedac eaueent, the tato cuent eno ae oe eliale and cheape than the flu eno.

64 65 3. CONINUOS-IME DOMAIN LINEA MODELS OF HE SINGLE- PHASE INDUCION MACHINE 3.. Intoduction he induction achine i ued in a wide ange of application a an electical to echanical enegy convete. he ingle-phae induction achine i the ot ued convete in hoe appliance. he analyi of the induction achine i eentially the ae fo a thee-phae, two-phae o ingle-phae achine. An accuate atheatical odel fo the induction achine i neceay to e deteined in vecto contol opeation. hi odel ha to e uitale fo the analyi of oth the teady-tate opeation and the dynaic opeation of the yte. If the doule-evolving field (Veinott - 959), o the yetical coponent theoie (Fotecue - 98) peit a detailed iulation and odelling of the ingle-phae induction achine in teady-tate opeation, fo the dynaic analyi of the achine a diffeent atheatical appoach ha to e found.. Stating fo the efeence fae theoy, with voltage, cuent and flue efeed to a two-ai quadatue coodinate yte, a geneal odel fo the ingle-phae induction achine i developed accoding to Kaue (965) and Kaue et al (995). Diffeent fo the thee-phae induction achine, whee two appoache ae valid (d-q ae and pace vecto theoy), the ingle-phae induction achine i copletely decied only y a two-ai quadatue ai. 3.. Voltage and flu-cuent equation of the ingle-phae induction achine In Fig. 3. i illutated a ingle-phae induction achine. he following auption have een ade: a) electically othogonal tato winding with inuoidal ditiution; ) only the fundaental-pace-haonic-coponent of the ai-gap flu ditiution will e conideed; c) agnetic-atuation effect, coe lo and tay load loe ae negligile; d) agnetic-diffuion (i.e. deep-a) effect in the oto will e ignoed. hi auption i typically valid in all induction achine. It i futhe jutified y the fact that unde ot opeating condition, the ingle-phae induction oto will e opeating at low lip and hence the oto cuent will e at fequencie ufficiently low that agnetic-diffuion effect ae inignificant. e) tepeatue effect on winding eitance and eactance value i negligile; f) the laination agnetic peeaility i conideed infinite; g) in teady-tate opeation, the voltage and cuent ae inuoidal.

65 66 ai ai a Φ L a ai a Φ e θ a a ai a - tato oto u a i a L L i a L L i i - u Fig.3.. he eal odel of the yetical ingle-phae induction achine with quiel-cage oto epeented y two identical winding he achine oto i decied y two identical agnetically othogonal winding. Conideing an aitaily efeence fae, the patial poition of the tato winding i chaacteied y the electical angle Φ and the patial poition of the oto winding i chaacteied y the electical angle Φ. he angula peed of the oto i and the diplaceent etween tato and oto winding i θ. hee angle ae lined though the elation: Φ Φ θ (335) he voltage equation elated to the achine fo Fig.3., can e epeed a follow: ua V co t ( ) u V co( t ϕ) d a ua ia dt d u i dt (338-34) d a 0 ia dt d 0 i dt A the achine i agnetically linea, the flue ae eaily deteined fo the cuent and inductance value. Paticulaly, it can e witten:

66 67 L i M i M i M i a aa a a aa a a M i L i M i M i a a a a M i M i L i M i a aa a a aa a a ( ) M a ia M i M a ia L i he utual inductance given in the aove elation ae defined y the ucipt. Applying the ecipocity pinciple, the following identitie ae valid: M a M a M aa Maa Fo futhe developent of the odel, the te fo the peviouly equation ae gouped in a ati fo: a Li a Li a ( ) a ( L ) ia Li a he indee denote tato flue (a, ), epectively oto flue (a, ). he elf-inductance can e witten y including a leaage inductance caued y the leaage flu and a agnetiation (utual) inductance caued y the agnetic flu which lin the tato and oto coe: L Ll L ( ) L Ll L whee the fit te tand fo the leaage inductance, and the econd one fo the utual inductance etween tato and oto winding. he utual inductance can e deteined: N () L () (35) whee: N () i the tun nue and the agnetic eluctance depending on the ai-gap value and on the agnetic coe dienion. A the agnetic ai of the tato, epectively oto winding ae othogonal, it eult in null utual inductance etween the tato and oto winding. hi i one of the ain diffeence fo the thee-phae induction achine whee the tato (oto) winding ae placed at 0 0 electical pace angle, which deteine utual linage aong the ae aatue winding. Due to the elative oveent etween tato and oto winding a agnetic linage will appea. he following defining elation fo the utual inductance can e epeed: M L coθ M M L L aa L a L a L N N inθ inθ coθ (35) he electoagnetic toque of the ingle-phae induction achine can e deteined fo the genealied foce law: P Wc ( i, θ ) e ( i, θ ) (353) θ whee W c epeent the conenegy which i equal to the agnetic enegy of the linage field a follow: Wf ( Li a L i L i a L i ) Li a i a co θ (354) L i i in θ L i i in θ L i i coθ a a

67 68 In the aove elation, the upecipt ( ) tand fo efeed oto winding to the tato accoding to the epeion: N i a ia N a N N a ( ) N L L N he electoagnetic toque elation can e iplified and peented in the following fo: P e L ( iai a ii ) in θ ( iai ii a ) coθ (358) hi elation how that thee ae two coponent of the intantaneou electoagnetic toque: an aveage coponent with contant value fo a given value of the oto peed and a pulating coponent with a fequency doule of the cuent fequency ( e ). he pulating toque coponent deteine an ipotant agnetic noie fo the ingle-phae induction achine copaed to the thee-phae induction achine. he echanical equation that lin the toque and the oto peed i: d e J B L (359) P dt P whee J i the oto inetia, B i the vicou fiction coefficient aociated to the otational yte of the achine and with the echanical load, and P epeent the nue of pole fo the analyed achine Analyi of the ingle-phae induction achine in tationay efeence fae In ode to eliinate the tie dependence of the voltage and flu equation te, a vaiale tanfoation into a new efeence fae i neceay. hi tanfoation i given y the following elation (Kaue et al - 995): f K f (360) qd a f qd K f a (36) co θ inθ K inθ coθ (36) co( θ θ ) in( θ θ ) K in( θ θ ) co( θ θ ) (363) whee f can e flue, voltage, cuent in new co-odinate (d-q) o in claical co-odinate (a-). Inde () tand fo tatoic te and inde () tand fo the otoic te. One hould note that the two tanfoation atie depend on the genealied co-odinate θ which epee the peifeical diplaceent of the choen efeence fae. efeence fae lined to tato, oto o aitaily can e choen fo the polyphae induction achine. A fo the ingle-phae induction achine, the tato winding ae not identical, the only tanfoation that aintain the winding paaete (eitance, inductance) unchanged i the tationay efeence fae tanfoation (θ 0). he voltage and linage flu equation ae epeed fo the peviou auption ade at the eginning of thi chapte.

68 69 Note: Fo copactne, the flu linage will e futhe decied y flu linage unit pe econd o volt, and the inductance equivalent cicuit eleent will e tanfoed in eactance eleent. p uq iq q u i dq d d p 0 i p q q d p 0 i d d q i ( i i ) q q q q i ( i i ) d d d d i ( i i ) q q q q ( ) (368-37) d i d ( id i d) P e ( iqi d idi q ) (37) In the efeence fae yte theoy we have to note that the paaete do not depend on the elative poition etween the tato and the oto. he tie deivative (d/dt) in the aove equation i denoted a p, and and epeent the angula ae peed given y the upply fequency, epectively the electical oto peed. he q te ae efeed to the q winding and the d te ae efeed to the d winding. It i ipotant to oeve that tato winding of the ingle-phae induction achine phyically epeent the d-q co-odinate winding, diffeent fo the theephae induction achine whee the two-ai co-odinate equivalent winding ae fictive. he dynaic analyi of the yetical ingle-phae induction achine can e accoplihed y uing the equivalent cicuit fo Fig. 3.. he only odification that ha to e done in ode to otain the eal value of the teinal quantitie (voltage, cuent) i the ultiplication y (-) facto fo the auiliay (d) winding paaete. ( / ) d - u q _ i q i q ( / ) Q - u d _ i d i d Fig. 3.. Equivalent cicuit of the yetical ingle-phae induction achine confo to efeence fae yte theoy

69 Analyi of the teady-tate opeation fo the yetical ingle-phae induction achine In a wide ange of application the ingle-phae induction achine i equipped with a cage oto. Duing the teady-tate opeation, the tato paaete ae vaiale with the tato voltage fequency, and the oto paaete ae vaiale with the lip fequency -. Fo the tationay efeence fae equation, y letting the diffeential opeato p e eplaced y the cople opeato j, the following ati elation i etalihed (Kaue and hoa - 965): j 0 j 0 U q Iɶ q 0 0 j j U d Iɶ d (373) 0 Iɶ q j j 0 Iɶ d j j whee ( ) A yetical two-phae yte i defined y the identitie: Fɶ QS jfɶ DS ( ) Fɶ jfɶ Q D whee F ~ epeent a cople vaiale with cuent o voltage ignificance. he fou equation fo the aove ati elation ae intedependent. If it i ued, the vaiale lip i defined a: (378) and alo though the invee tanfoation fo the tationay efeence fae (d-q) to the phyical one (a-), the elation fo the teady-tate opeation analyi of the yetical inglephae induction achine can e witten: U a ( j ) Iɶ j ( Iɶ Iɶ ) a a a ( ) 0 j Iɶ a j( Iɶ a Iɶ a ) he aove equation ugget the equivalent cicuit fo Fig j j / U a I a j I a - Fig Equivalent cicuit of the yetical ingle-phae induction achine fo the teady-tate opeation analyi

70 7 With light odification ( ecoe 3/ ) it can e oeved that the equivalent cicuit fo Fig. 3.3 can e ued alo fo the thee-phae induction achine pefoance in teady-tate opeation. he electoagnetic toque equation i: ɶ ɶ ɶ (38) P ( P /)( * / )( / ) Ia e e ji ai a ( / ) ( P /)( / )( ) U a e ( ( )) ( ) (38) It i ipotant to highlight that the poitive value of the toque ae otainale when the lip i poitive (oto opeation) and the negative value when the lip i negative (geneato opeation). By etting the toque/lip deivative equal to zeo, the elation fo the citical lip can e otained: G G ± ( ) ( ) he poitive value coepond to oto opeation, and the negative one to geneato opeation. If at tat-up ( ) the toque i diectly dependent to the vaiation of oto eitance, a the agnetiation eactance value i conideed to e uch highe than the tato o oto eitance value, the aiu toque value i not dependent to the oto eitance value: ( P /)( / ) G U a e,a G( ) ( G ) 3.5. Analyi of the unyetical ingle-phae induction achine (385) A new equation et and a new equivalent cicuit fo the unyetical ingle-phae induction achine can e otained y eliinating fo the initial auption the one efeing to identical tato winding (Kaue - 965): Voltage equation: p uq iq q u i p d a d d p 0 i q q q p 0 i d d d Flu equation: i ( i i ) q l q q q i ( i i ) d la d d d i ( i i ) q q q q d i d ( id i d) he electoagnetic toque equation i: ( ) ( )

71 7 P ( i i i i ) (394) e q d d q he atheatical odel peented in Fig. 3.4 peit the analyi of teady tate a well a dynaic opeation of the unyetical ingle-phae induction achine. hi odel i alo uitale fo ipleenting non-lineaitie effect uch a: coe lo o atuation of the ain o leaage inductance. l (/) ( / ) d - i q i q u q _ a la ( / ) q - i d i d u d _ Fig Equivalent cicuit of the unyetical ingle-phae induction achine 3.6. Linea odel fo ingle-phae induction oto he analyi of the induction achine-dive i geneally ade uing a conventional linea atheatical achine odel, eithe in the fo of elf and utual inductance, o the failia - fo of equivalent cicuit. hi type of odel ha een developed in the peviou paagaph, dedicated to yetical and ingle-phae induction achine. When the vecto contol of thee achine i ipleented, uch ind of odel i adequate fo any ituation, epecially fo ai-gap flu oientation contol tategy. Howeve, they ae oe cople than neceay fo the analyi of ot linea achine. If fo the polyphaed induction achine two appoache ae valid elated to vecto contol analyi, i.e. the pace vecto notation and the two-ai co-odinate efeence fae, the inglephae induction achine can e copletely decied only y the latte appoach in tationay efeence fae. A fo the polyphaed induction achine, the ingle-phae veion can e contolled with a chee which aintain coect angula elationhip etween the tato cuent vecto and one flu vecto (tato, ai-gap o oto flu), y eithe diect o indiect ethod. All the field-oiented ethod uffe fo pecific theoetical and pactical pole. he indiect ethod ae highly dependent on the achine oto paaete (vaying with load and tepeatue), and have good peed contol pefoance only if pecie hat encode ae ued to calculate the electical fequency. he diect ethod peent ino dependency on oto paaete, ut ae not ale to eaue the elected flu vecto at low peed, i.e. zeo fequency. hi i why only etiation of the elected flu vecto can ae poile the total contol of the peed in diect ethod. Fo etiation, uually the teinal quantitie ae eaued and a atheatical odel with elated paaete i ued. hu, the coect etiation of the achine paaete i eential fo all type of field oiented contol chee.

72 73 he oiginal field oientation chee, developed fo the thee-phae induction achine, wa aed on the alignent of the oto flu linage ecaue the toque and the oto flu ae elated to each othe in a taightfowad anne, without any de-coupling cicuit. o calculate the oto flu, the tato and oto leaage inductance and the ain inductance ae neceay. If oe othe equivalent cicuit ae ipleented, it can e otained a Γ odel lie that one illutated y Sleon (989), whee the tato and oto leaage inductance ae viewed a a total eauale inductance Linea Γ odel of the yetical ingle-phae induction achine he claical -fo cicuit odel can e tanfoed into iple odel with no lo of infoation o accuacy. Since oto vaiale can e een fo the tato efeence fae a efeed vaiale depending on the tanfoation tun atio, we can chooe a value uch that the agnetiation inductance i equal to the total tato inductance. hi would give the following et of oto vaiale elated to thoe fo the -fo equivalent cicuit: γ d γ q d q i d i q (395) i d γ i q γ whee γ /( ). he aove elation coined with the voltage equation give the equivalent d-q odified odel fo the yetical ingle-phae induction achine peented in Fig hi configuation ha een denoted a the Γ fo odel due to it inductance tuctue. he paaete of thi equivalent cicuit ae elated to thoe of the -fo d-q atheatical odel though the elation: γ M ( ) γ ( ) γ γ L γ (396) L ( / ) d - u q _ i q M i q L ( / ) q - u d _ i d M i d Fig Equivalent linea d-q Γ fo cicuit fo the yetical ingle-phae induction achine

73 74 hee paaete can e deived diectly fo the uual no load and tandtill eaueent on the achine. hi Γ odel epeent an appopiate olution fo the analyi of cala contol and vecto contol (tato flu oiented) ingle-phae induction achine (Sleon - 994). he following equation decie the coplete opeation of the yte: Stato and oto voltage: p uq iq q u i d d d p p 0 i q q d p 0 i d d q Stato and oto flu linage: i ( i i ) q M qm M q q i ( i i ) d M dm M d d i i i ( ) i q L q M qm M q L M q ( ) (40-404) d Li d MidM Mid ( L M ) i d Electoagnetic toque: P M P e ( iq i d idi q ) ( iqd id q ) (405) In the teady-tate (p 0) the toque epeion ecoe: ( ) P q d ( ) e L whee the lip i given y: ( ) / Linea invee Γ odel of the yetical ingle-phae induction achine (406) If the aitaily tun atio of efeing the oto paaete to the tato i choen uch that the agnetiation inductance i equal to the total oto inductance, a new et of vaiale i otainale: γ d γ q d q i d i q (407) i d γ i q γ whee: γ /( ). Uing the voltage equation given fo -fo equivalent cicuit, it will eult a new configuation denoted a invee Γ fo odel hown in Fig.3.6 in which:

74 75 γ L γ γ ( ) M ( )( ) ( ) (408) γ he invee Γ fo atheatical odel of the ingle-phae induction achine i paticulaly appopiate to analye the vecto-contolled achine in oto field oiented yte. L ( / ) d - u q _ i q M i q L ( / ) q - u d _ i d M i d Fig Equivalent linea d-q invee Γ fo cicuit fo the yetical ingle-phae induction achine he following equation can e witten to decie the opeation of the achine: Stato and oto voltage: p uq iq q u i d d d p p 0 i q q d p 0 i d d q Stato and oto flu linage: i ( i i ) ( ) i i q L q M q q M L q M q i ( i i ) ( ) i i d L d M d d M L d M d i ( i i ) q M qm M q q d M i dm M ( id i d ) Electoagnetic toque: P M P P i i i i i i i i In teady-tate opeation the toque ecoe: ( ) ( ) ( ) e q d d q q d d q d q q d (409-4) (43-46) (47)

75 76 ( ) P d q e whee denote the lip fo the analyed achine. (48) Univeal odel of the yetical ingle-phae induction achine he oden theoy of the vecto contolled induction achine deontate that eveal efeence flu vecto can e choen fo an independent contol of the toque and flu in the achine (DeDonce et al - 995). A unified theoy iplifie and unifie the calculation of the achine paaete and ae a change of the efeence vecto flu oe fleile. he idea i to lin the tationay efeence fae (the only valid option fo ingle-phae induction achine) to an aitay flu vecto γ which can e deived fo the flu linage equation given at the peviou analyed odel (-fo, Γ-fo, invee Γ-fo) y ean of a tun atio tanfoation. An aitay tun atio γ i ued to ultiply the eal oto vecto flu d j q and defining the flu linage equation a: d ( ) id id γ id γ id (49-40) i i γ i γ i ( ) q q q q q ( ) ( ) γ γ i γ i i i d d d d d d γ γ i γ i i i q q q q q q he tanfoed oto cuent i equal: d(q) i i i γ γ he flu linage equation educe to: γ i γ i i γ i (4-43) d q d ; i (43) q ( ( ) ) ( ) ( ( ) ) γ ( ( ) ) i ( i i ) ( ( ) ) i γ ( ( ) ) ( ) γ ( ) ( ) i i i ( ) d d d d d d γ γ γ q q q q q q ( ) ( ) ( ) ( ) γ γ γ i γ i i γ γ γ i d d d d d d γ γ γ γ γ γ γ γ i q q q q q q hi flu linage can e epeented y the equivalent cicuit given in Fig (44-47)

76 77 (- γ ) γ(γ (γ ) ) i d γ i d i d i d ( γ) γ(γ (γ ) ) i q γ i q i q i q Fig Flu linage equivalent cicuit with aitay tun atio he oto quantitie ae now calculated with the equation: p 0 i d d q p 0 i q q d (48-430) γ hough vaiation of the aitay tun atio γ the appopiate efeence vecto flu fo contolling the achine can e elected. ale 3.I uaie the choice that have to e ade: ABLE 3.I. Specific choice of the tun atio γ un atio Matheatical odel d(q)γ Flu elected Invee Γ-fo γ d(q) oto flu γ -fo d(q) Ai-gap flu Γ-fo d(q) Stato flu γ he univeal atheatical odel given in Fig. 3.7 peit a choice etween diffeent flu vecto y electing only one paaete γ, denoted the aitay tun atio. All othe aic achine paaete eain unaffected Linea Γ odel of the unyetical ingle-phae induction achine Uually the ingle-phae induction achine i equipped with non-identical inuoidally ditiuted winding aanged in pace quadatue. Fo the -fo equivalent cicuit developed

77 78 fo thi type of achine in peviou paagaph, it can e developed a linea Γ-fo equivalent cicuit a hown in Fig Since oto vaiale can e een fo the tato efeence fae a efeed vaiale depending on the tanfoation tun atio, we can chooe a value uch that the agnetiation inductance i equal to the total tato inductance on each of the two-ai co-odinate. hi would give the following et of oto vaiale elated to thoe fo the -fo equivalent cicuit: γ d d γ q q d q i d i q (43) i d γ d i q γ q whee γ q /( l ) (43-433) γ /( ) d la Fo an identical ditiution of the winding, we have with a good appoiation la, and l thi give equal tun atio fo each ai ( γ d γ q ). he aove elation coined with the voltage equation give the equivalent d-q odified odel fo the unyetical ingle-phae induction achine. he paaete of the Γ-fo odel ae elated to thoe of the -fo d-q atheatical odel though the elation: γ Mq q l Lq ( ) γ q ( l ) l γ q γ q γ q γ Md d a Mq Ld ( ) γ d ( la ) la γ d γ q Lq ( ) Lq ( / ) d - u q _ i q M q i q a Ld ( / ) q - u d _ i d M d i d It eult an equation yte a follow: Stato and oto voltage: Fig Equivalent linea d-q Γ fo cicuit fo the unyetical ingle-phae induction achine

78 79 u u i p q q q i p d a d d p 0 i q q d p 0 i d d q Flu linage: i i ( ) q Mq q q ( ) ( ) i i i i d Md d d Mq d d ( ) ( ) ( ) i i q Mq q Mq Lq q (439-44) ( ) d Md id Md Ld i d Mq id Mq Lq i d Electoagnetic toque: P Mq P e ( iq i d idi q ) iqd id q (448) hi Γ-fo odel epeent an appopiate olution fo the analyi of cala contol and vecto contol (tato flu oiented) of the unyetical ingle-phae induction achine dive Linea invee Γ odel of the unyetical ingle-phae induction achine Anothe poile appoach on analying the unyetical ingle-phae induction achine i the one when an aitay tun atio i ued uch that the agnetiation inductance in each ai i equal to the coepondent total oto inductance. he following et of vaiale i eadily otainale: γ d d γ q q d q i d i q (449) i d γ d i q γ q whee: γ q d q γ ( ) γ (450) he linea invee Γ-fo odel of the unyetical ingle-phae induction achine i peented in Fig hi configuation i deductile fo the -fo odel uing the aove epeion and new equivalent achine paaete defined a follow:

79 80 γ Mq q γ Md d Mq γ Lq l q γ Ld la d Lq ( γ ) q(d) (45-455) Lq ( / ) d - u q --- i q Mq i q a Ld ( / ) q - u d --- i d Md i d Fig.3.9. Equivalent linea d-q invee Γ fo cicuit fo the unyetical ingle-phae induction achine he coplete equation yte i detailed elow: Stato and oto voltage: p uq iq q u i p d a d d p 0 i q q d p 0 i d d q Flu linage: i ( i i ) ( ) i i q Lq q Mq q q Mq Lq q Mq q i ( i i ) ( ) i i d Ld d Md d d Md Ld d Md d (( Mq Lq ) d Mq d ) i i i ( i i ) q Mq qm Mq q q d Md i dm Md ( id i d ) Mq ( id i d ) Electoagnetic toque: ( ) ( )

80 8 P P ( i i i i ) i i Mq e q d d q q d d q (464) P i d q i q d he cicuit odel fo Fig. 3.9, i paticulaly uitale fo analyi and undetanding the vecto contol yte with ingle-phae induction achine when the oto flu vecto i choen a efeence Univeal odel of the unyetical ingle-phae induction achine All the developed atheatical odel fo the unyetical induction achine can e geneically peented with a univeal odel that peeve the chaacteitic fo each of the equivalent fo cicuit. he aic idea i to calculate in the tationay efeence fae (the only valid option fo inglephae induction achine that aintain contant paaete) an aitay flu vecto. he flu vecto i deteined fo the flu linage equation given in vaiou fo of equivalent cicuit (-fo, Γ-fo, invee Γ-fo) y uing an aitay tun atio tanfoation. hi tun atio, that chaacteie the two-ai efeence fae, ha two value: γ q and γ d. By ultiplying the eal oto vecto flu d j q with the tun atio we can define the flu linage equation a: q ( l ) iq i q γq iq γ q iq ( ) i i γ i γ i ( ) d la d d d d d d ( ) ( ) γ γ i γ i i i d d d d d d d d d γ γ i γ i i i q q q q q q q q q he tanfoed oto cuent i d ( q) equal: ( ) i i d q i d ; i q γ d γ (469) q he flu linage equation educe to: γ i γ i i ( ( ) ) ( ) ( γ ) i d la d d d d d ( la d ) d dγ d ( ( ) ) i ( i i ) ( l ( γ q ) ) iq qγ q ( ( ) ) i ( i i ) dγ γ ( ( ) ) d d γ d γ d i d γ q ( γ q ( γ q ) ) i q γq ( iq i q ) qγ γ ( ( ) q q γ q γ q ) i q γ γ q l q q q q q γ γ γ γ d d d d d d d d q ( ) hi flu linage can e epeented y the equivalent cicuit given in Fig he oto quantitie ae now calculated with the equation:

81 8 p 0 i d d q p 0 i q q d γ q(d) ( ) hough vaiation of the aitay tun atio γ q and γ d thee ae eveal option to elect the appopiate efeence vecto flu fo contolling the ingle-phae induction achine. ale 3.II uaie the availale choice: (- γ ) q γ (γ (γ ) q q q ) i d γ q i d i d i d a ( γ ) d γ (γ (γ ) ) d d d i q γ d i q i q i q Fig Flu linage equivalent cicuit with aitay tun atio fo unyetical ingle-phae induction achine ABLE 3.II Specific choice of the tun atio γ q and γ d γ γ γ γ q d q un atio q γ d γ q ( ) γ Matheatical d(q)γ Flu elected odel Invee Γ-fo d(q) oto flu -fo d(q) Ai-gap flu l la d γ q Γ-fo d(q) Stato flu

82 83 he univeal atheatical odel given in Fig. 3.0 peit a choice etween diffeent flu vecto y electing only two paaete γ q and γ d that have pactically equal value, and denote the aitay tun atio fo each ai of the yte. A in the yetical ingle-phae induction achine cae, all othe aic achine paaete eain unaffected D-Q ae odel of the ingle-phae induction achine he ingle-phae induction achine can e odelled y uing diffeent tate-pace vaiale, eeping a input the tato voltage and the load toque, and a output the electoagnetic toque and oto angula velocity. Let u define the poile et of cuent and flu linage pe econd pace vecto a follow: i i j i d q i i j i d q i i j i d q j d q j d q j d q [ ] [ i i i ] (477-48) (483) wo-pace vecto have to e elected fo the i availale vecto. Let the elected pai of tate-pace vaiale e denoted a,. It i poile to epe the et of i tate-pace vaiale in te of the two elected tate-pace vaiale a follow: a a a a a3 a 3 [ ] [ i i i ] [ ] (484) a4 a4 a5 a 5 a6 a6 whee the coefficient a ij, depend on the choen et of tate-pace vaiale. Out of fifteen poile tate-pace veion otained fo coining two pace vecto fo the total of i, only fouteen epeent valid atheatical odel fo the ingle-phae induction achine. Pleae note that the pai of tate-pace vaiale that copie the aigap flu linage pace vecto and the agnetiing cuent pace vecto cannot e elected a thee vecto have the ae diection. hee odel can e claified in thee type: cuent pace vecto, flu linage pace vecto and ied cuent-flu linage pace vecto odel. In conjunction with the echanical equation: P p ( e L ) (485) J we otain a coplete veion of the ingle-phae induction achine odel, viewed a the ey fo a otion contol yte. Fo an unitay appoach on all the poile odel, and conideing that the agnetiing eactance ha geneally uch geate value than the leaage eactance, the following appoiation i ade without alteing the final eult: ( ) (486) la l

83 84 he tating point fo the tate-vaiale odel i given y voltage equation yte witten in tationay efeence fae: u L p (487) whee: i the elected et of tate-vaiale and epeent alo the output of the odel, u i the input vecto (tato voltage), L i the coefficient ati (it can e foed y eactance value, o non-dienional eleent) fo ultiplying the tie deivative of the tate-vaiale, i the coefficient ati (it can e foed y eitance and eactance value o non-dienional eleent) fo ultiplying the tate-vaiale and p tand fo the diffeential opeato (d/dt). It eult the geneal fo of the tate-vaiale yte: p A B u (488) whee: A L ( ) B L All the following atheatical odel can theoetically e dicetied and ued fo ipleentation of contolle in the dive yte with ingle-phae induction achine. Howeve, pactically only few of the ae uitale fo thi pupoe a the coputational effot, and iplicitly the cot of the hadwae, i a deciive facto to e conideed. When the vecto contol tategy ha to e choen, each of the i pace vecto that can e elected a tate-pace vaiale epeent a poile ai fo a efeence fae. he algoith fo ipleenting field oiented contol yte i otained a follow:. he oto flu pace vecto i epeed a function of tate-vaiale;. he oto angula velocity te i utituted with ( - ) whee i the angula velocity of the ynchonou fae; 3. he dynaic equation that elate the oto angula velocity to oto flu linage and voltage ae deduced in the new tate-vaiale yte; 4. he ynchonou efeence fae i elected lined to one of the pace vecto, which ean that the q-ai coponent of the efeence pace vecto i null; 5. he toque equation i coputed accoding to the elected flu o cuent pace vecto in the ynchonou efeence fae Model with cuent pace vecto a tate-pace vaiale I) he ot ued odel of the ingle-phae induction achine i the cuent pace vecto tatepace vaiale iq, id, i q, i d. An ipotant featue of thi odel i that the tato and leaage eactance, aitaily conideed equal in any ipleentation, ae now included in the tato and elf-eactance value. A the tato cuent ae eaily eauale quantitie, a contolle aed on thee tate vaiale give good pefoance accuacy. he odel i eadily availale fo the equation of voltage and flu linage pe econd, and it can e epeed in a ati fo a follow:

84 85 D D D D iq a i q i p d D D D D i d i q i q i d D D D D i d a D D D D 0 0 D D uq 0 0 D D u d u q 0 0 D D u d 0 0 D D (49) whee: D he electoagnetic toque can e coputed a: P e ( iq i d idi q ) (49) A it can e oeved, a hotcoing fo thi odel i the full yte ati, with all 6 eleent non-zeo value. II) Anothe atheatical d-q ai odel aed on cuent pace vecto i the one which copie the tato and the agnetiing pace vecto cuent a tate-pace vaiale iq, id, q i, id. he tate ati eleent ae all non-zeo, which iplie a coputational effot iila to the peviouly analyed odel.

85 86 l A A A A iq l a iq i p d A A A A i d iq l l l l l l i q id A A A A id la l la a l la la A A A A 0 0 A A u q 0 0 A A u d l l u q 0 0 A A u d l la 0 0 A A (493) whee: A D l l A l la D he intantaneou electoagnetic toque ay e epeed in te of the tate-pace vaiale: P e ( iq id id iq ) (494) III) he thid poile coination of cuent pace vecto a tate-pace vaiale i otained y electing agnetiing and oto cuent i q, i d, iq, id. When copaed to the othe cuent pace vecto odel, the iila coputing uden fo otaining the output of the yte i oviou. he ain diffeence etween the i the peence of the gloal paaete (elfeactance) in the fit odel, while in the othe two odel an accuate deteination of the leaage eactance i neceay. A coplete deciption of the tate-vaiale in ati notation i given elow:

86 87 whee: A l A A A A i q l a a i q p i d A A A A i d iq l l l l l l i q id A A A A i d la l a l la la a l A 0 0 A A u q 0 0 A A u d l l u q 0 0 A A u d l la 0 0 A A l l D A A A (495) A la l D he intantaneou electoagnetic toque i epeed a follow: P e ( i d iq i q id ) (496) Model with flu linage a tate-pace vaiale I) In thi cae the flu linage pace vecto, q, d, q, d ae ued fo deciing the atheatical odel of the achine. he ati equation fo thi yte ae a follow: 0 0 D D q a a q q 0 0 u p d D D d u d (497) q 0 q u q D D d d u d 0 D D whee D ha the ae ignificance a in the peviou cae. he electoagnetic toque i deteined with the equivalent elation: P e ( q d d q ) D (498) A oppoed to the cuent odel, it can e oeved that the coputational uden i utantially lowe. Due to thi ipotant featue, thi odel i the ot uitale fo dicetiation in otion contol tategie.

87 88 II) An altenative to odel the ingle-phae induction achine with flu pace vecto a tatevaiale yte, i the election of ai-gap flu pace vecto aong the et of independent vaiale. A fit appoach i given y the tato and ai-gap flu pace vecto elected a tate-pace vaiale q, d, q, d. 0 0 l l q a a 0 0 q la la d d q l l l l l l q p d la A laa la A A la d la l a l la l a l l A la A A A l la A A uq l l u d 0 0 A A u q u l la d 0 0 A A (499) whee: A D l l A la l D he electoagnetic toque epeion how that thi odel i pohiitive fo ipleentation fo odelling the ingle-phae induction achine in a vecto contol yte and ecoe: P e q d q d dq (500) l la la l which can e iplified only if the appoiation i eaonale to e ade: e q d d q l la P (50) III) he thid option of electing flu pace vecto a tate-pace vaiale i the one which copie the ai-gap (agnetiing) flu and oto flu pace vecto l q, d, q, d copaion to the peviouly analyed odel, thi one peue the ae coputational effot, ut the electoagnetic toque i deteined in a uitale fo fo vecto contol, without any uppleentay appoiation.. By

88 89 l l l 0 A l A A l A A q a la la a la q 0 p d A l A A A l A d q q 0 d l l d 0 l l l l 0 0 A A u q l la ud 0 0 A A u q u d (50) whee: A l l D A l l D he eulting elation fo coputing intantaneou electoagnetic toque i a follow: P e ( q d d q ) (503) l Model with ied cuent flu pace vecto tate-pace vaiale I) If the tato vaiale ae choen fo odelling the ingle-phae induction achine yte, then a ied flu linage-cuent tate-pace vaiale odel q, d, iq, id i developed. he ati equation and the electoagnetic toque elation ae: q a q uq p d d D D ud i q D i q D u q i d i d u d D D D D (504) P e diq qid (505) II) Anothe ipotant ied flu linage-cuent tate-pace vaiale odel q, d, i q, i d i that epeed in oto quantitie. hi odel can e eaily ued fo the oto flu oiented contol of the achine. We can decie the yte a follow:

89 90 p q q uq 0 0 d d u i q d i q 0 D u q i D D d D i d u d a a D 0 4 D D D he electoagnetic toque elation i given elow: P e q i d d i q (506) (507) III) A iila odel to the peviou one i that with tato cuent and oto flu linage q, d, iq, id a tate-pace vaiale. It epeent an altenative odel fo oto flu oiented contol tategie. he ati equation and electoagnetic toque elation ae peented elow: 0 D D D iq a i q 0 i d i p D D D d q q 0 d d D D u q u d 0 0 D D u q u d (508) and toque epeion i: P i i e q d d q (509) IV) A aely ued atheatical odel i that with ied oto cuent pace vecto and tato flu linage pace vecto a tate-pace vaiale q, d, i q, i d. Howeve, the advantage of thi odel i that eely only tato winding paaete ae neceay, o the influence of oto paaete i iniied. It can e ued fo an unconventional tato flu oiented contol with oto cuent coponent poducing the toque and the flu. he epeion fo ipleenting thi odel ae:

90 9 D D D 0 0 i q a i a q D D uq 4 i d i D d u d 0 0 q D D q u q u d p D D 0 0 d d a a 0 0 whee the ignificance of D wa aleady tated. he ain yte output, the electoagnetic toque, can e coputed a: P e q i d di q (50) (5) V) he ied tato cuent pace vecto and ai-gap flu pace vecto a tate-pace vaiale q, d, iq, id elong to one of the oe cople odel type. It peeve infoation egading oth tato and oto paaete. Diffeent fo the peviou ied odel, the tatepace ati contain only non-zeo eleent, which lead to geate coputational effot. l A A A A iq l a iq p id A A A A id q ( l ) l l l l q A d A A A d la l ( l a la ) la la A A A A 0 0 A A uq 0 0 A A ud l l u q 0 0 A A u d l la 0 0 A A (5) whee: A D l l A l la D he electoagnetic toque i coputed a:

91 9 P i i e q d d q (53) VI) A iila odel to the pecedent one, i the ied oto cuent pace vecto and ai-gap flu pace vecto a tate-pace vaiale given elow: q, d, i q, i d. he ati equation of the yte i l A A A A i q l a a iq i p d A A A A i d q ( l l ) l l l l q d A A A A d la l ( l a la ) la la a A A A A 0 0 A A uq 0 0 A A u d l l u q 0 0 A A u d l la 0 0 A A (54) whee: A D l l A D la l he electoagnetic toque elation ecoe: P e i d q i q d (55) VII) If the agnetiing cuent pace vecto i elected a tate-pace vaiale togethe with one of flu linage pace vecto iq, id, q, d, the tate ati coputation give eveal null eleent. Nevethele, the toque epeion i oe coplicated and the final elation can e witten in the ae fo a in peviou cae only though oe new auption,.

92 93 l l la l l l l l A la A l A la A iq la A ( a l la ) la l la a l iq i p d l A la A l A la A i d q q 0 0 d l l d a a 0 0 la la l l 0 0 A A u q l la u d 0 0 A A u q u d (56) whee: A l l D A D la l he electoagnetic toque elation, with the appoiation: la can e witten a: l 3 P e id q iq d q d la l l la (57) P id q iq d l he peence of ocillating te given y the poduct etween flu linage, ae thi odel pohiitive fo ipleenting in vecto contol tategie. VIII) he lat ied cuent-flu linage pace vecto odel ealie the connection etween the agnetiing cuent pace-vecto and the oto flu linage pace vecto iq, id, q, d, elected a tate-pace vaiale. he tate ati contain the ae nue of zeo eleent (fou) and the intantaneou electoagnetic toque i deteined in a uitale fo fo vecto contol ipleentation.

93 94 iq i p q d d l l l l l 0 l A l A A la a l la a l la iq 0 l A A l A i d q 0 l l d l l 0 0 A A u q l la u d 0 0 A A u q u d whee: A D l l A D l la he electoagnetic toque i deteined a follow: P e iq d id q l l l (58) (59) 3.8. Vecto contol tategie fo ingle-phae induction achine he ai of vecto contol i uually to decouple the tato cuent i into it flu poducing and toque poducing coponent (i d, i q epectively) in ode to otain a decoupled contol of the flu and the electoagnetic toque. Fo thi eaon a pecial efeence fae i elected fied to diffeent pace vecto vaiale. he efeence fae ha to e ynchonou, a all the pace vecto have the ae angula velocity given y the upply voltage fequency. A the tato winding of the ingle-phae induction oto ae uually unyetical, the vecto contol pinciple have to e ipleented in a pecial way. he achine paaete diffe fo ai d to ai q. he wavefo of the electoagnetic toque deontate the unalance of the yte. Even fo equal aplitude, othogonal tato cuent i qs and i ds, the toque contain AC te. A new toque elation ha to e developed fo each cae of vecto contol tategy of inglephae induction achine (Coea et al , 999). Geneally, the conideation ade fo the thee-phae induction achine when the vecto contol hee ha to e developed, ae valid. An eay to follow tep algoith fo ipleenting vecto oiented contol yte i otained a follow:. A coplete atheatical odel of the ingle-phae induction achine i developed in tationay efeence fae, the only efeence fae that aintain contant paaete, accoding to the choen et of tate-pace vaiale;. A uitale toque epeion i deteined, in ode to eliinate the influence of AC te; 3. he oto aed vaiale ae copletely epeed in the new tate-vaiale yte; 4. he oto angula velocity te i utituted with ( - ) whee and ae the angula velocity of the ynchonou fae, epectively the lip angula velocity;

94 95 5. he ynchonou efeence fae i elected lined to one of the pace vecto, which ean that the q-ai coponent of the efeence pace vecto i null; 6. he toque equation i coputed accoding to the elected flu o cuent pace vecto in the ynchonou efeence fae. he tanfoation of efeence fae fo the induction oto vecto contol can e uaied a hown in Fig. 3.. Synchonuou efeence fae wo-ai co-odinate yte Stationay efeence fae Single-phae yte Dive Contolle Invee / Cla/ Pa anfoation Induction oto / Cla/ Pa anfoation Fig. 3.. Bloc diaga of tanfoation of fae and co-odinate yte fo ingle-phae induction oto vecto contol Stato field oientation (SFO) Fo thi vecto contol tategy, the et of tate-pace vaiale foed y tato flu linage and cuent pace vecto q, d, iq, id i elected. he atheatical odel i given in the chapte dedicated to d-q odel of the ingle-phae induction achine. he oto aed vaiale epeed in tate-pace vaiale te ae: i q ( q iq ) i d id d ( D i ) q q q ( D i ) d d d (50-53) o otain the achine equation in the ynchonuou tato flu efeence fae, one ha to eliinate the oto flu fo the oto voltage equation and then foce the q coponent of the tato flu to e zeo. he tato voltage equation eain unchanged. If a cage oto i conideed, the eultant equation ae a follow: p p D iq d D id q 0 D p i D i p d q d q (54-55)

95 96 whee the definition ued fo the lip peed i: e. he electoagnetic toque epeion ha to e e-witten: P e diq qid (56) If we ae the notation: i i the aove toque elation ay e epeed a fo the q q yetical induction achine: P e ( d iq q id ) (57) If the pecial efeence fae i fied to the tato flu linage vecto, the q-coponent of thi flu vecto i defined equal to zeo: q 0 (58) d Fo tato flu linage equation, the q-cuent coponent ae given y: iq iq (59) i q iq he electoagnetic toque elation and lip peed can e deived in tato field oientation contol a: P e d iq ( D p ) i (530-53) q d D id he econd dynaic equation of the achine, how that thee i a coupling etween the tato cuent coponent. Conequently, any change in the toque poducing coponent i without changing i d accodingly will caue a tanient in the tato flu. A decouple i neceay to ovecoe thi diadvantage. heefoe the coand cuent of the d-ai coponent of the tato cuent can e calculated a follow: Ki id Kp d idq p (53-533) D iq idq D p whee K p and K i ae popotional, epectively integal coefficient of the flu contolle. hi contolle can e PI type o oft coputing technique type (fuzzy, neual-netwo). Steady-tate pefoance of a tato flu oiented yte Letting the deivative opeato p 0, one can otain the teady-tate voltage equation of the induction achine. Afte eveal anipulation of the yte equation, the d cuent coponent ae given y: q

96 97 D d d iq i d i d i d it yield the lip peed equation: D iq i q ( ) 0 (536) he olution of the aove equation have to e eal, fo a given tato flu linage. hi ean that the deteinant of the econd ode equation atify the condition: 4 4 D ( iq ) 0 he aiu value fo the q coponent of the tato cuent, the lip peed and the electoagnetic toque (pull-out toque) ae: ( iq ) a D ( ) a D ( ) ( ) P ( e ) a D he pole elated to tatic taility of the ingle-phae induction achine when thi vecto contol tategy i applied, ae iila to the polyphae achine cae. It ha to pointed out that the citical value fo cuent q-coponent, lip angula velocity and electoagnetic toque depend alo on the tun atio of the unyetical tato winding. If we detail thi conideation, it can e oeved that having a oto equipped with a ain tato winding with fied paaete (eitance, eactance) the pull out toque and citical oto peed ae uch that (Popecu -000): A aiu toque value and iniu oto peed ae otained if the tun atio <, which i the cae of the uual plit-phae oto configuation; he ediu value ae otained if the tun atio, which i the cae of the yetical oto configuation; A iniu toque value and aiu oto peed ae otained if the tun atio >, which i the cae of the capacito un oto configuation; Soe ipotant concluion can e dawn fo thi vecto contol tategy, eide thoe one valid fo the polyphae oto: he etiated equivalent value fo the toque coponent of the tato cuent ha to e coected y dividing to the quae of the tun atio value; hee i no need fo changing the nue of co-odinate fo the eal achine yte to the contol yte oto flu oientation (FO) Fo thi vecto contol tategy, the et of tate-pace vaiale foed y oto flu linage and tato cuent pace vecto q, d, iq, id i elected. he atheatical odel i given in the chapte dedicated to d-q odel of the ingle-phae induction achine. he tato flu linage and oto cuent pace vecto coponent epeed a function in te of tate-pace vaiale ae:

97 98 i ( D i ) q q q ( D i ) d d d ( i ) q q q d i d id he oto voltage equation e-witten in te of the tate-pace vaiale ecoe: p u q 0 iq q d p u d 0 id d q he electoagnetic toque epeion ha to e e-witten: P e d iq q id ( ) ( ) (546) If we ae the notation: iq iq the aove toque elation ay e epeed a fo the yetical induction achine: P e ( d iq q id ) (547) If the pecial efeence fae i fied to the oto flu linage vecto, the q-coponent of thi flu vecto i defined equal to zeo: q 0 (548) d Fo oto flu linage equation, the q-cuent coponent ae given y: iq iq (549) i q iq he flu poducing coponent of the tato cuent i deteined a follow: p d d i (550) he aove elation how that thee i no need of a cuent decouple in oto field oientation chee. Both tato cuent coponent (toque and flu poducing) can e contolled independently. Steady-tate pefoance of a oto flu oiented yte Letting the deivative opeato p 0, one can otain the teady-tate voltage equation of the induction achine. Afte eveal anipulation of the yte equation, the d cuent coponent ae given y:

98 99 i d d i q i d 0 It yield the lip peed equation: (55) d i 0 (55) q which fo a given oto flu ha alway eal olution. hu the eulting cuent contolled lip peed and the electoagnetic toque ae: 3 iq iq d d ( ) 4 P P P e diq ( iq ) ( iq ) By copaion with the thee-phae induction oto, fo the oto field oientation contol tategy, thee diffeence have een highlighted y Popecu and Navapecu (000): he tady-tate value fo the lip angula velocity and toque ae popotional with the tun atio value, epectively the quae of tun atio which deteine the option fo >, a the toque epone i the deteinant facto in an electical dive yte; he etiated equivalent value fo the toque coponent of the tato cuent ha to e coected y dividing to the quae of the tun atio value; hee i no need fo changing the nue of co-odinate fo the eal achine yte to the contol yte; he pole of the tato winding ayetical configuation i ovecoe y uing the value of tun atio in coputing the etiated value fo the toque and flu poducing coponent of the tato cuent Ai-gap flu field oientation (AFO) Fo thi vecto contol tategy, the et of tate-pace vaiale foed y ai-gap flu linage and tato cuent pace vecto q, d, iq, id i elected. he atheatical odel i given in the chapte dedicated to d-q odel of the ingle-phae induction achine. he tato and oto flu linage and oto cuent pace vecto coponent epeed a function in te of tate-pace vaiale ae: ( ) i ( ) q q q d d d i i i ( ) q q q i ( ) d q d i q q q i i d d d ( ) ( )

99 00 he voltage equation of the achine epeed in tate-vaiale te, ae eadily deductile now: p p u q 0 ( ) iq q d ( ) id p p u d 0 ( ) id d q ( ) iq (56-56) he electoagnetic toque epeion ha to e e-witten: P e diq qid (563) If we ae the notation: iq iq the aove toque elation ay e epeed a fo the yetical induction achine: P e ( d iq qid ) (564) If the pecial efeence fae i fied to the ai-gap flu linage vecto, the q-coponent of thi flu vecto i defined equal to zeo: q 0 d Fo the ai-gap flu linage equation, the q-cuent coponent ae given y: i i q q i i q q (565) (566) he flu poducing coponent of the tato cuent i deteined a follow: p p 3 ( ) id d ( ) iq (567) When an ai-gap field oientation contol i eployed, it i neceay to decouple the tato cuent coponent, in ode to achieve a linea contol. Fo thi eaon, the coand of cuent of the d-ai coponent i coputed a follow: Ki id K p d idq p ( ) ( ) iq idq ( ) p he electoagnetic toque elation and lip peed can e deived in ai-gap field oientation contol a: P e diq [ ( ) p ] i (570-57) q d ( ) id

100 0 Steady-tate pefoance of a ai-gap flu oiented yte Letting the deivative opeato p 0, the teady-tate voltage equation of the induction achine ae eadily deteined. Afte eveal anipulation of the yte equation, the d cuent coponent ae given y: id iq ( ) iq (57) ( ) iq i d It yield the lip peed equation: ( ) d iq iq 0 (573) he olution of the aove equation have to e eal, fo a given tato flu linage. hi ean that the deteinant of the econd ode equation atify the condition: d 4 i 0 ( ) ( q ) (574) he aiu value fo the q coponent of the tato cuent, the lip peed and the electoagnetic toque (pull-out toque) ae: d ( iq ) a 3 ( ) ( ) ( ) a P ( d ) ( ) e a ( ) Fo the ai-gap field oientation contol tategy, thee have to e pointed out eveal ipotant featue: he tun atio deteine the effect on the citical value of the electoagnetic toque of the ingle-phae achine: lowe toque fo >, and highe toque fo <, when the ain tato winding paaete ae ept contant; he pull-out value of the lip peed doe not depend on the unyetical configuation of the tato winding, and it i epeed with an identical elation to that deduced fo the thee-phae induction achine; he etiated equivalent value fo the toque coponent of the tato cuent ha to e coected y dividing to the quae of the tun atio value; hee i no need fo changing the nue of co-odinate fo the eal achine yte to the contol yte; he pole of the tato winding ayetical configuation i ovecoe y uing the value of tun atio in coputing the etiated value fo the toque and flu poducing coponent of the tato cuent Stato cuent oientation contol (SCO) Fo a coplete copaion to the vecto contol tategie applied to thee-phae induction oto, the cuent oientation contol hee have to e analyed a well. One option i if the et of tate-pace vaiale i identical with the tato flu field oientation contol (SFO): the tato flu

101 0 and cuent pace vecto q, d, iq, id. he atheatical odel i identical to that ued in SFO cae. he oto aed vaiale epeed in tate-pace vaiale te ae: c c c i q ( q iq ) i ( i ) c c c d d d ( D i ) c c c q q q ( D i ) c c c d d d (578-58) o otain the achine equation in the ynchonuou tato cuent efeence fae, one ha to eliinate the oto flu fo the oto voltage equation and then foce the q coponent of the tato cuent to e zeo. he tato voltage equation eain unchanged. If a cage oto i conideed, the eultant equation ae a follow: p c c c p c D iq d D id q 0 (58-583) p c c p c c D id D iq d q he electoagnetic toque epeion ha to e e-witten: P c c c c e d iq qid (584) c c If we ae the notation: q q the aove toque elation ay e epeed a fo the yetical induction achine: P c c c c e ( d iq qd i ) (585) If the pecial efeence fae i fied to the tato cuent pace vecto, the q-coponent of thi cuent vecto i defined equal to zeo: c q i 0 (586) c i id Diffeent fo the SFO cae, we have to epe the q-flu coponent, y conideing the flu linage equation: c c q q c (587) c q q he electoagnetic toque elation and lip peed can e deived in tato cuent oientation contol a: P c c e qid ( ) c ( p ) q c D i ( d d )

102 03 he elation etween the d-ai coponent of the tato cuent and tato flu linage pace vecto coponent can e deduced a follow: p c p c c D id d q 0 (590) Siilaly to the SFO cae, thee i a coupling, ut etween the tato flu linage coponent c ued a contol vaiale. Conequently, any change in the toque poducing coponent c without changing q accodingly, will caue a tanient in the tato flu. A decouple i neceay to ovecoe thi diadvantage: c K i c c d Kp id dq p c (59-59) q c dq p whee K p and K i ae popotional, epectively integal coefficient of the cuent contolle. hi contolle can e PI type o oft coputing technique type (fuzzy, neual-netwo). Steady-tate pefoance of a tato cuent oiented yte Letting the deivative opeato p 0, one can otain the teady-tate voltage equation of the induction achine: c c D c d q q (593) c c d q it yield the lip peed equation: c c c q id q 0 (594) he olution of the aove equation have to e eal, fo a given tato cuent. hi ean that the deteinant of the econd ode equation atify the condition: c c q ( id ) 4 0 (595) he aiu value fo the q coponent of the tato flu linage, the lip peed and the electoagnetic toque (pull-out toque) ae: c c id ( q ) a ( ) a ( d ) d ( ) c P i ( e ) a he analyi of thi vecto contol tategy lead to the ae concluion a fo the thee-phae induction achine cae, ut with the diffeence ipoed y the ayetical oto configuation and the nue of phae fo the upply voltage:

103 04 he tun atio deteine diffeent effect on the citical value of the oto peed and electoagnetic toque epone of the ingle-phae achine: the electoagnetic toque citical value eache a aiu fo >, while the oto peed eache a aiu fo <, when the ain tato winding paaete ae ept contant; he etiated equivalent value fo the toque coponent of the tato flu ha to e coected y ultypling with the quae of the tun atio value ; hee i no need fo changing the nue of co-odinate fo the eal achine yte to the contol yte; oto cuent oientation contol (CO) If the ynchonuou efeence fae i lined to the oto cuent pace vecto it eult anothe unconventional vecto contol tategy. he oto cuent oientation contol tategy (CO) i analyed fo a cage oto induction oto. hee ae two option in electing the et of tate-pace vaiale, if we conide the citeia of diect eauale quantitie: I. he tate-pace vaiale ae the tato flu linage and oto cuent pace vecto: q, d, i q, i d. When the tato flu and and the oto cuent ae elected a tate-pace vaiale, one can deive the oto flu and tato cuent function in te of tate vaiale a follow: c c c iq ( q i q ) c c d c d i d i D i c c c q q q D i c c c d d d (599-60) o otain the achine equation in the ynchonuou oto cuent efeence fae, one ha to eliinate the oto flu fo the oto voltage equation and then foce the q coponent of the oto cuent to e zeo: D p c D c p c c 0 i q i d q d ( ) D p c D c p c c 0 i d i q d q he electoagnetic toque epeion ha to e e-witten: P c c c c e q i d d i q (605) c c If we ae the notation: q q the aove toque elation ay e epeed a fo the yetical induction achine: P c c c c e ( qi d d i q ) (606) If the pecial efeence fae i fied to the oto cuent pace vecto, the q-coponent of thi cuent vecto i defined equal to zeo:

104 05 c q i 0 (607) c c i d i Diffeent fo the FO cae, we have to epe the q-flu coponent, y conideing the flu linage equation: c c q q c (608) c q q he electoagnetic toque elation and lip peed can e deived in tato cuent oientation contol a: P c c e qi d c (609-60) p q c c D i d d A elation etween the d-ai coponent of the oto cuent and tato flu linage pace vecto coponent can e deduced: D p c p c c 0 i d d q (6) Due to the coupling etween the tato flu linage coponent, any change in the toque poducing coponent without changing accodingly, will caue a tenient in the tato c d c q flu. o ovecoe thi diadvantage, the coand cuent of the d-ai coponent of the tato cuent i epeed a follow: K K i p c i c c d p d dq (6-63) c c q dq p whee K p and K i ae popotional, epectively integal coefficient of the cuent contolle. hi contolle can e PI type o oft coputing technique type (fuzzy, neual-netwo). Steady-tate pefoance Afte eveal anipulation of the yte equation if we let the deivative opeato p 0, the d cuent coponent ae otained fo the teady-tate voltage equation of the induction achine: c D c d q (64) c d 0 It yield the lip angula velocity equation: c D D c id q 0 (65) which fo a given oto flu ha alway eal olution. hu the lip peed and the electoagnetic toque ae:

105 06 i i 3 c c d d c c q q P P i i c c c c e q d q d (66-67) he oto cuent oiented (CO) vecto contol with ied flu and cuent tate-pace vaiale diffe fo the vaiant applied to the thee-phae induction achine y the following: he tanfoation etween the efeence fae (tationay to ynchonou and vice-vea) i eaie a thee i no need of changing the nue of vaiale; he ayety of the tato winding i odeled only y uing a uppleentay paaete, the tun atio ; In teady-tate opeation, the toque and lip peed epone ae influenced y the unyetical configuation of the oto: highe toque and lip peed fo <. II. he tate-pace vaiale ae tato and oto cuent iq, id, i q, i d. When the tato and the oto cuent ae elected a tate-pace vaiale, one can deive the tato and oto flu a function in te of tate vaiale fo the claical flu linage equation. o otain the achine equation in the ynchonuou oto cuent efeence fae, one ha to eliinate the oto flu fo the oto voltage equation and then foce the q coponent of the oto cuent to e zeo. he tato voltage equation ae e-witten alo. he eultant equation have the following fo: p c p c c c 0 iq i q id i d (68-69) c c p c p c 0 iq i q id i d he electoagnetic toque epeion ha the epeion: P c c c c e ( iqi d id i q ) (60) It can e oeved that the aove elation i eadily availale fo ipleenting a vecto contol tategy. When lining the pecial ynchonuou efeence fae to the oto cuent pace vecto, the q- coponent of thi cuent vecto i defined equal to zeo: c i q 0 (6) c c i d i Diffeent fo the (I ) cae of CO contol tategy, fo the (II ) cae we have to epe the tato flu linage q-coponent, y conideing the flu linage equation: c c q iq c c (6) c q iq q he electoagnetic toque elation and lip peed can e deived in thi type of oto cuent oientation contol a:

106 07 P c c e iqi d c piq c c ( d d ) i i (63-64) Fo the econd dynaic equation of the achine, a elation etween the d-ai coponent of the oto cuent and tato cuent pace vecto coponent can e epeed a: p c c p c id iq i d (65) A coupling etween the d-ai and q-ai tato cuent coponent appea, and conequently, any change in the toque poducing coponent i without changing i accodingly, will caue a c d tanient in the tato flu. o ovecoe thi diadvantage the coand cuent of the d-ai coponent of the tato cuent ha to e epeed in the following fo: K i K i i p c i c c d p d dq c q (66-67) c c iq idq p whee K p and K i ae popotional, epectively integal coefficient of the cuent contolle. hi contolle can e PI type o oft coputing technique type (fuzzy, neual-netwo). Steady-tate pefoance By letting the deivative opeato p 0, the d flu coponent ae otained fo the teady-tate voltage equation of the induction achine: c D c d iq (68) c d 0 It yield the lip angula velocity equation: c D D c i d iq 0 (69) which fo a given oto flu ha alway eal olution. hu the lip peed and the electoagnetic toque ae: c i d c iq (630-63) P c c e iqi d A copaion with the oto cuent oientation contol detailed fo the thee-phae induction achine peit to highlight the ae concluion. Howeve, thee ha to e pointed out the neceay coection: he tanfoation etween ynchonou efeence fae and the tationay efeence fae i ade without changing the nue of vaiale; he influence of the unyetical tato winding configuation can e evidenced y uing the tun atio, and it deteine diffeent effect ove the oto peed and toque epone of the oto: highe electoagnetic toque and lowe oto peed fo >, epectively lowe toque and highe oto peed fo <.

107 08 4. MAHEMAICAL DISCEE MODELS FO HE HEE-PHASE INDUCION MACHINE 4.. Intoduction Advance in vey-lage-cale integation (VLSI) technology ade poile the eal-tie odelling fo any indutial application. eal-tie iulation i ued inceaingly in the autoatic contol field. One ipotant application i the advanced AC oto contol, i.e. vecto contol, whee the ieauale quantitie, lie the cage oto paaete (flu, cuent), can e etiated y a iulato opeating in paallel with the eal oto. he iulation ecae an ipotant altenative againt the eaueent, a the latte i cople, noie enitive, and epenive. It ha to e alo entioned that the appoiate iulation alway intoduce oe eo etween the tue dynaic ehaviou and the odelled ehaviou (Vainio et al - 99). he continuou and dicete odel of the induction achine ae equivalent if thei tie epone i iila fo typical input (tep o inuoidal input). Howeve, thee ae eveal dicete atheatical odel equivalent to the continuou odel. Ion lo and atuation can e oitted if the tato and oto flue ae liited to tay elow the wide atuation ode of the ion coe, which i the cae in ot pactical application. Fo thi equivalence, the appoiation ode of the epone fo two conecutive apling ate i deteinant. Uually, when a tie continuou yte i dicetized, the deigne ha to choe the popetie of the yte that will e aintained: the zeo and pole nue, the epone chaacteitic fo pule, tep and linea input, the DC aplify, the fequency epone. In ale 4.I thee ae peented eveal poile dicetization ethod fo the tie continuou yte. ABLE 4.I No Dicetization ethod Continuou dicete equivalence equation Fowad-diffeence ethod (Eule) z Bacwad-diffeence ethod 3 Bilinea tanfoation ethod (utin) 4 Fequency pewaping ethod z z z z z, z ( z) Z L 5 Pule invaiance ethod G D [ ] 6 Step invaiance ethod ( z ) ( z ) Z L D A tan G( ) { } G ( ) G D 7 Matched zeo-pole ethod Zeo/pole fo -a i placed in dicete to ze -a. Zeo/pole fo ± i placed in dicete to z- All of the aove dicetization ethod ae copaed fo the induction achine een a a dicete yte. he auption valid fo the continuou odel ae alo valid fo the dicete yte. he eult of the dicetization ae analyed fo the point of view of coputation copleity and the epone taility fo a tep input ignal. he analyi of the diffeent odel i eential fo the deign of electical dive yte with nueical coand. A pecial attention i given to toque etiation fo the data acquiition poce (voltage o cuent quantitie).

108 09 he ynchonou efeence fae i ued fo deteining the dicete odel of the induction achine. he following paaete can e defined: Fo an eay to follow analyi, the inde will e oitted in the achine equation. he geneal fo equation ued to deteine the dicete atheatical induction achine in aitay ynchonuo efeence fae and pe unit yte ae: dψ ( t) U ( t) i ( t) j ψ ( t) n dt (63) dψ ( t) 0 i ( t) j ψ ( t) n dt (633) ψ ( ( ( M (634) ψ ( M (635) i i ; ψ γ ψ ; γ γ (636) d dt M ( te tl ) (637) Depending on the ynchonuou efeence fae the following elation ae valid: a) Stato efeence fae: l γ lm 3 * te P I { ψ i} (638) ) oto efeence fae: lm γ l 3 l M * te P I{ ψ i} l (639) c) Aigap efeence fae: γ * { ψ } 3 te P I i ψ l ( i i ) M (640) If the deteinant of the flu yte equation i noted with δ and with d it invee, the voltage equation ae e-witten a follow: dψ ( t) U ( t) d ( γ l ψ l M ψ ) j ψ ( t) (64) n dt dψ ( t) 0 d ( l ψ γ lm ψ ) j ψ ( t) (64) n dt Fo zeo initial condition, if the Laplace tanfoation i applied, it eult the ati yte: ψ γ l d j lm d ψ U n ψ γ lm d l d j ψ 0 (643)

109 0 o in ynthetiized fo: Y ( ) A Y ( ) B U ( ) (644) One can note that y copaing with the tate vaiale yte decied y the equation: X ( ) A X ( ) B U ( ) (645) Y ( ) C X ( ) D U ( ) the tate vaiale vecto i identical to the output vecto (flue vecto in thi cae). he input vecto i the voltage vecto. hough identification we otain: γ l d j lm d A n γ lm d ld j (646) B n 4.. Bilinea tanfoation ethod (utin) he elation (3) fo the ale I i nown a ilinea tanfoation o utin and it ae the connection etween continuou to dicete doain. hi ethod give ette eult than the tapezoidal appoiation, o othe dicetization ethod illutated in ale 4.II, y conideing fit fou ode of integation opeato. he deciing yte elation ecoe: z Y ( z) A Y ( z ) B U ( z) (647) z whee i the ignal apling peiod fo continuou doain, and f i the apling fequency. ABLE 4.II Method utin z z Boe- z hale z Madwed z z ( z ) 4 ( z ) z 0z ( z ) z 4z 6 ( z ) 3 3 ( z ) 8 ( z ) 3 z z 3 ( z ) 3 3 z z z 3 4 ( z ) ( z ) 6 ( z ) 4 3 z 4z z 4 6 ( z ) z 6z 66z 4 0 ( z ) 4 4 ( z ) ( z ) ( z ) ( z ) ( z ) ( z) Y A Y B U zy ( z ) Y ( z) z A Y ( z) A Y ( z ) ( z ) B U ( z ) I A z Y ( z) I A Y ( z) ( z ) B U ( z) z Y ( z) I A I A Y ( z) I A B ( z ) U ( z) z Y ( z) I A I A Y ( z) I A B ( z ) U ( z) Y ( z) C z Y ( z ) D ( z ) U ( z ) whee i the ati deteinant: I A : (648)

110 n n n M j ( f γ l d) ( f l d) ( γ l d ) I [ ( ) ( )] j f γ l d f l d n n n γ n n γ M c ( f l d ) ( f l d ) ( l d ) [ ( ) ( )] j f γ l d f l d n n n c 4 f γ l d n M c 4 f l d n M c ( f γ l d) ( f l d) ( γ l d ) n n n M j ( f γ l d ) ( f l d ) n [ ] d ( f l d ) j n n n M n d l d n n (649) (650) Oevation: d, and d ae not calculated a the induction achine i conideed to have cage oto. ψ ( n) c ψ ( n ) c ψ ( n ) d [ U ( n ) U ( n) ] (65) ψ ( n) c ψ ( n ) c ψ ( n ) d [ U ( n ) U ( n) ] whee n and n ae two conecutive apling peiod in dicete doain. he induction achine dicete odel i peented in Fig. 4.. Fo thi odel the tato voltage epeent the input vecto and the flue ae the epeed a output vecto. c z - d / z - Ψ U c c d / z - Ψ c Fig. 4.. he induction achine dicete odel he loc with thice line denote cople ultiplie. he following ati elation i otained fo the epaation of the eal and the iaginay pat of the flu equation. ψ ( n) ψ ( n ) U ( n ) U ( n) ψ I ( n) ψ I ( n ) U I ( n ) UI ( n) E F (65) ψ ( n) ψ ( n ) 0 I I ψ I ( n) ψ I ( n ) 0 whee:

111 e e e e e e e e f f f e e e e e e e e f 4 f f c c 3 4 c 43 c c c c d d c d c I I I c I I I c d I I c I I d I I I (653) f 3 f 4 d I he coplete dicete odel of the induction achine, aed on the ilinea ethod i given in Fig. 4.. he electoagnetic toque epeed in elative unit i conideing diffeent efeence fae fo the induction achine: a) Stato efeence fae 3 * 3 3 te PI{ ψ i} PI {( ψ } ( ) jψ I ) ( i j i I ) P ψ i I ψ I i 3 3 P( ψ ( γ l d ψ I lmd ψ I ) ψ I ( γ l d ψ lmd ψ )) P lm d ( ψ I ψ ψ ψ I ) (654) ) oto efeence fae: 3 * 3 3 te P γ I{ ψ i} P γ I {( ψ } ( ) j ψ I) ( i j i I ) P γ ψ i I ψ I i 3 3 P γ ( ψ ( γ l d ψ I lmd ψ I ) ψ I ( γ l d ψ lmd ψ ) ) P γ l d ( ψ I ψ ψ ψ I ) 3 P lm d ( ψ I ψ ψ ψ I ) (655) c) Aigap efeence fae: 3 * 3 3 te P I{ ψ i} PI {( ψ } ( ) j ψ I ) ( i j i I ) P ψ i I ψ I i 3 P( ψ ( γ l d ψ I lmd ψ I ) ψ I ( γ l d ψ lmd ψ ) ) 3 lmd ( ψ γ ( l γ lm) ψ ( γ l lm )) ( γ l d ψ I lmd ψ I ) P l M d ( ψ I γ ( l γ l M) ψ I ( γ l lm) ) ( γ l d ψ lmd ψ ) ( ψ ( l lm) ψ ( l lm )) ( l d ψ I lmd ψ I ) ( ψ I ( l lm) ψ I ( l lm )) ( l d ψ lmd ψ ) 3 3 P lmd P lm d ( ψ I ψ ψ ψ I ) (656) Fo the toque equation, witten in tato, oto and ai-gap efeence fae, one can note that the ae geneal elation can e ipleented, though the equivalent oto flu tand fo diffeent ignificance, i.e. only in ai-gap efeence fae it ha the eal phyical oto flu.

112 3 A it can e oeved y tudying the loc diaga of the dicete odel of the induction achine, fo a coplete ipleentation thee ae neceay 3 ultiplie loc, 3 uing loc and 4 delay loc. Howeve, thi odel can e iplified futhe. he total nue of ultiplie loc can vay fo one ipleentation to anothe, due to the place of thi opeation in the loc diaga. Alo the fequency epone i vaiale accoding to the ipleentation veion (Vainio et al - 99). e f e z - U (n) z - U I(n) f f f f3 f3 / e3 e4 e e / e3 e4 e3 e3 / e33 e34 z - z - z - Ψ (n) * Ψ I(n) Ψ I(n) * Ψ (n) - l M d oque t f4 f4 e4 e4 / e43 e44 z - Fig. 4.. Coplete dicete induction achine odel aed on ilinea tanfoation ethod (utin) 4.3. Fowad-diffeence ethod (Eule) In thi ethod the continuou-tie deivative i appoiated y the elation () fo ale 4.I o y a caled diffeence of two ucceive aple: dx X ( n ) X ( n) (657) dt whee i the apling peiod. he dicete-tie odel i in thi cae defined y the following equation: ψ ( n ) ψ ( n ) U ( n) ( γ l d ψ ( n) l Md ψ ( n) ) j ψ ( n) (658) ( ) M n n ψ ( n ) ψ ( n ) 0 l d ψ ( n) γ l d ψ ( n) j ψ ( n) (659) In ati fo it eult:

113 4 γ l d j l d ψ ( n ) ψ ( n) ( ) M n U n n n ψ ( n ) ψ ( n) 0 γ lmd ld j n Afte epaating the eal and the iaginay coponent, it will eult: ψ ( n ) j ψ I ( n ) n ( γ l d j ) [ ψ ( n) j ψ I ( n) ] [ ( ) I ( )] [ ( ) I ( )] [ ] γ l d ψ n j ψ n U n j U n n M n ψ ( n ) j ψ ( n ) γ l d ψ ( n) j ψ ( n) I n M I ( l d j ) ψ ( n) j ψ ( n) [ ] n I n n (660) (66) (66) ψ ( n ) 0 ( ) γ l d l d n n n n M ψ U ( n) ψ ( n ) γ l d 0 l d ψ ( n) U ( n) I n n n M I I n ψ ( n ) n γ lmd 0 n ld n ψ ( n) 0 ψ ( n ) 0 n γ lmd n n I ld ψ ( n) 0 I (663) o: ψ ( n ) A ψ ( n) n U ( n) (664) Fo the cuent vecto i valid the following elation: i( n) B ψ ( n) (665) o the ati fo equation: i ( n) γ l d 0 l Md 0 ψ ( n) ii ( n) 0 γ l d 0 lmd ψi ( n) i ( n) γ lmd 0 l d 0 ψ ( n) (666) i I ( n) 0 γ lmd 0 ld ψ I ( n) Fo thi cae the coplete dicete odel of the induction achine i decied in Fig. 4.3.

114 5 a U (n) n z - z - a U I(n) n z - a3 a a a4 a3 z- ψ (n) * ψ I(n) - l M d oque t a33 ψ I(n) a34 a4 z - * ψ (n) a43 z - a44 Fig Coplete dicete induction achine odel aed on fowad-diffeence ethod (Eule) 4.4. Bacwad-diffeence ethod hi iple ethod poduce a taile dicete yte fo a taile tie continuou yte. Even oe untale continuou yte can e tanfoed to taile veion though thi dicetization ethod. Howeve, a highe apling fequency ha to e adopted in ode to avoid the fequency epone ditotion of the yte. With elation () fo ale 4.I the following equation can e ipleented: z Y A Y B U (667) ( I A ) Y z Y B U nγ l d jn nl Md I A nγ l Md n ld jn (668) n n n n n M I A j I ( γ l d j ) ( l d j ) γ l d ( γ l d) ( l d) ( γ l d ) n n n M [ ( ) ( )] jn nγ l d n ld (669) n ld j n nl M d ( I A ) nγ lmd nγ l d j n (670) I A By utituting vecto Y and U with the flu vecto, epectively tato voltage, it follow that:

115 6 ψ ( n) ψ ( n ) U ( n) ( I A ) ( I A ) B ψ 0 ( n) ψ ( n ) (67) ψ ( n ) U ( n) G H ψ 0 ( n ) I I When the eal and the iaginay pat ae epaated, it eult that: ψ ( n) ψ ( n ) U ( n) ψi ( n) ψ I ( n ) UI ( n) L M (67) ψ ( n) ψ ( n ) 0 I I ψ I ( n) ψ I ( n ) 0 l l g g I I l l g g I I l l g 3 4 l l g 4 3 l l g 3 4 l l g 3 4 l l g g I I l l g g I I h h I I h h I I h 3 4 I I (673) 3 4 h I he coplete dicete atheatical odel of the achine, otained though the acwaddiffeence ethod i peented in Fig A iila analyi wa ade fo the tep and pule invaiance ethod y Vainio et al (99). he eult peit oe ipotant concluion to e dawn: he pule invaiance ethod and acwad-diffeence ethod deteine iila and vey eeling dicete atheatical odel fo the achine; he tep invaiance ethod and the fowad-diffeence ethod deteine iila and vey eeling atheatical dicete odel fo the achine. Nevethele, although the ethod ae vey iila in eult, the fequency chaacteitic of the odel diffe fo one ethod to anothe. he coputational uden fo the thee decied ethod i uaied in ale 4.III. A the DSP-ASIC ipleentation i an ipotant cot iue, the deigne ha to chooe a copoie etween the copleity and the accuacy of the odel. When the copaion i ade, one can note the advantage of the fowad-diffeence ethod (Eule) fo le coputing tie. When accuacy i the deteining facto, the ilinea tanfoation ethod (utin) ha to e choen fo the ipleentation of the atheatical dicete odel of the achine. It i alo poile to apply a hyid appoach whee the tato equation i dicetized uing the fowad-diffeence (Eule) ethod, wheea the oto equation i conveted uing the ilinea tanfoation (utin) o vicevea. A one ight epect, the eulting epone and coputational copleity ae etween thoe of the coplete ethod. hi could e eploited when ipleenting the dicetized odel with a pogaale ignal poceo.

116 7 ABLE 4.III Method \ eal opeation Addition Multiplication Delay Fowad-diffeence ethod (Eule) 7 6 Bacwad-diffeence ethod 7 4 Bilinea tanfoation ethod (utin) l U (n) l l3 z - Ψ (n) l4 3 4 l l Ψ I(n) l3 z - * l4 _ l M d l3 l3 oque t U I(n) 3 4 l33 l34 l4 l4 z - Ψ I(n) * l43 z - Ψ (n) l44 Fig Coplete dicete induction achine odel aed on acwad-diffeence ethod 4.5. Z-doain tanfe function he geneal fo of a tate-vaiale yte uing ati notation in z-doain i: ( n ) A ( n) B u ( n) y ( n) C ( n) D u ( n) [ ] ( n) ( n),..., ( n) (674) whee (n) i the vecto fo the tate vaiale, u (n) and y (n) ae the input, epectively the output vecto, and A i the tate ati. he ipule epone equence in te of the tate-vaiale deciption i given y the elation: D, fo 0 h( ) (675) C A, fo > 0

117 8 and the tanfe function ati: H ( z) h( ) z D C ( zi A) B (676) ψ 0 Fo the induction achine odel cae, thee ae fou eal output (the tato and oto flue) and two eal input (tato voltage). Alo, the tate vaiale ae the output diectly. We can etalih the notation: ( n) ( ) ψ ( n) ψ ( n) I y n ψ ( n) ψ I ( n) he tate ati A, the input vecto u and the coefficient ati B, C, and D have diffeent eleent accoding to the tanfoation ethod ued fo ipleenting the dicete atheatical achine odel. I) Bilinea tanfoation ethod (utin): he tate ati A i calculated a follow: c c I I c I c I c I c I c c I I c I c I A (677) I c c I c ci I c I c I c I c c I c I c c I I whee: n n n M j ( f γ l d) ( f l d ) ( γ l d ) I [ ( ) ( )] j f γ l d f l d n n n γ n n γ M c ( f l d ) ( f l d ) ( l d ) [ ( ) ( )] j f γ l d f l d n n n c 4 f γ l d n M c 4 f l d n M n γ n n γ M c ( f l d ) ( f l d ) ( l d ) [ ] (678) (679) j n ( f nγ l d ) ( f n ld) he input vecto i deteined with the elation: U ( n ) U ( n) u UI ( n ) UI ( n) (680) he coefficient ati ae: d di I d I d I d I di d d I I B (68) d I I d I d I d I whee: d ( f l d ) j d n n n l Mdn

118 9 By identification, we etalih the othe coefficient ati: C D (68) Fo the tanfe function ati elation, we otain a 4 X ati giving the tanfe function fo oth input to the fou output: c c I I c I c I c I z I I I I I c I c c I I c I c I z I I I I Hψ ( z) c I c I c ci I c I c I z I I I I c I c c I ci c c I I z I I I I d di I d I di d I d I d d I I d I d I d I d I (683) II) Fowad-diffeence ethod (Eule) he tate ati A i decied y the elation: n γ l d n n lmd 0 n n γ l d 0 n lm d A (684) n γ lmd 0 n l d n 0 n γ l M d n n ld he input vecto and the coepondent ati coefficient ae: n 0 U ( n) 0 n u ( n) U ( n) B ( ) I he othe ati C and D have the ae value a fo the ilinea tanfoation cae. It eult the tanfe function ati a follow: z n γ l d n n l Md 0 n 0 n z n γ l d 0 n l Md 0 n Hψ ( z) n γ lmd 0 z n ld n n γ lmd n z n l d 0 0 (687)

119 0 III) he acwad-diffeence ethod he tate ati A and coefficient ati B can e calculated with the epeion: g g I I g I g I g g I g I g I g g I I g I g A (688) g I g I g g I I g I g I g I g g I gi g g I I whee: g jg I g n ld j n nl Md G ( j I ) g g g I n γ l Md n γ l d j n ( n ld ) I n j( n I ( n ld)) nl Md j I nl Md nγ lmd j I nγ lmd ( n γ ld ) I n j( n I ( n γ l d)) (689) and: h h I I h I h I h I h I h h I I B (690) h I h I h I h whee: h jh h l d j l d I n n n M n h h jh I n γ lmd n γ l d j n H (69) U ( n) A the input vecto i u UI ( n) and the coefficient ati C and D ae calculated in a iila way to the peviou cae, it eult the following tanfe function ati fo two input (voltage) and fou output (flue): g gi I g I gi g g I z I I I I g I g I g g I I g I g z I I I I Hψ ( z) g g I g g I I g I g I z I I I I g I g g I g I g gi I z I I I I h h I I h I h I h I h I h h I I h I h I h I h (69)

120 4.6. Staility analyi If the eulting dicete-tie yte i untale, it can e tanfoed in a tale one y deceaing the apling ate. It i poile to elect the the apling ate uch that the dicete-tie yte i alway tale, auing that the pole location in the continuo doain ae nown. Howeve, thi peue an inceaed coputing uden, a thee ae poceed oe apling ate in a tie unit. heefoe it i ipotant to analye the value fo the iniu apling ate and to deteine an optiu value a tated y Fanlin et al (997). he z-doain tanfe function i otained fo an analog pototype tanfe function y uing the following utitution accoding to the appoiation ethod: z fowad-diffeence ethod (Eule) z acwad-diffeence ethod z z z ilinia tanfoation ethod (utin) hu the intedependence etween an -doain pole and the coeponding z-doain pole i: zpole pole fowad-diffeence ethod (Eule) zpole acwad-diffeence ethod pole pole zpole ilinia tanfoation ethod (utin) pole Fo a tale -doain pole (i.e., α < 0): pole α jβ A z-doain pole i tale if it i located inide the unit cicle, i.e., it odulu i le than the unit (one in elative unit): zpole < ( α ) ( β ) < α fowad-diffeence ethod (Eule) (693) α α β < < α β zpole < < ( α ) ( β ) acwad-diffeence ethod (694) α α α β > > α β ( α ) ( β ) zpole < < ( α ) ( β ) > 0 ilinia tanfoation ethod (utin) (695) Epeed in te of the apling fequency (f aple /), the aove condition ae: α β f aple > fowad-diffeence ethod (Eule) α (696) α β f aple < acwad-diffeence ethod α (697) f ilinia tanfoation ethod (utin) (698) aple > 0

121 hee epeion pove that the apling ate can alway e elected uch that the dicete-tie yte i tale if the oiginal continuo-tie yte i tale a well. he pole poition ay e vaiale due to thee ipotant facto: he iplified auption fo lineaity of the continuou-tie odel; he paaete vaiation due to envionental effect: tepeatue, huidity; he ipleentation of the yte uing fied-point digital ignal poceo. he fied point deteine ound eo and caling o quantification eo. So, even fo pole placed inide the unit cicle, thee i a poiility of untale opeation of the yte. A pole location vey nea to the unit location can e poleatic, ainly in envionent with hot wod length of the DSP. heefoe it ay e deiale to aiie the ditance etween the citical pole and the unit cicle and et the apling ate accodingly. he deivative of the odulu of the pole give the elation: d z pole α α β fowad-diffeence ethod (Eule) (699) d α α β d z d pole α α β acwad-diffeence ethod (700) 3 ( α α β ) [ α ( α β ) 4 ( α β ) 4α ] d z pole 4 ilinia tanfoation ethod (utin) (70) d ( α ) ( β ) ( α ) ( β ) By etting the deivative equal to zeo it i otained an epeion fo the optiu apling fequency: α β f aple, opt fowad-diffeence ethod (Eule) (70) β α β f aple, opt acwad-diffeence ethod (703) β f 0 ilinia tanfoation ethod (utin) a the deivative i alway poitive. (704) aple, opt > One hould notice that the aove apling ate value do not optiie the eelance etween tie-doain o fequency-doain epone of the continuo-tie and the dicete-tie yte. he tale ethod (fowad-diffeence, acwad-diffeence and ilinia tanfoation) ap an -doain point α j β into the z doain a follow: a) Fowad-diffeence ethod (Eule) ( ) α jβ z (705) α jβ whee fo the optiu apling ate; fo the iniu apling ate ) Bacwad-diffeence ethod β jα z (706) β j( ) α whee fo the optiu apling ate; fo the iniu apling ate c) Bilinia tanfoation ethod (utin) z the unit cicle, (707) i.e. the yte i tale fo any tale -doain pole (α < 0)

122 3 5. MAHEMAICAL DISCEE MODELS FO HE SINGLE-PHASE INDUCION MACHINE 5.. Intoduction Dicete-tie coputational odel have to e deived fo an advanced oto contol of quiel-cage type ingle-phae induction achine. he eal-tie vecto contol analyi fo the ingle-phae induction achine can e ealied uing atheatical dicete odel, in a iila way to the thee-phae induction achine. Soe diffeence will appea due to the ayety of the ingle-phae induction achine tato configuation. he ae geneal conideation egading the dicetiation poce, valid fo the thee-phae achine, apply. he tating point fo conveting the continuou-tie odel of the ingle-phae achine into a dicete one i the yte of voltage and flu linage equation. Howeve, y copaion with the thee-phae induction achine dicetiation two ain diffeence have to e highlighted: he efeence yte i tationay, fied to the tato; he ayetical configuation of the tato winding peit only the analyi in two-ai coodinate yte. he pace vecto notation cannot e ued. he coplete et of equation fo an unyetical ingle-phae induction achine uing the univeal atheatical odel, with flu linage pe econd unit and eactance eleent, i a follow: Stato voltage equation: d uq iq q dt ( ) d ud a id d dt Flu linage equation: γ i γ i i ( ( ) ) ( ) ( γ ) i d la d d d d d ( la d ) d dγ d ( ( ) ) i ( i i ) ( l ( γ q ) ) iq qγq ( ( ) ) i ( i i ) dγ γ ( ( ) ) d d γ d γ d i d γ ( γ ( γ ) ) i γ ( i i ) qγ γ ( ( ) q q γ q γ q ) i q γ γ q l q q q q q γ γ γ γ d d d d d d d d q q q q q q q q oto voltage equation (cage oto cae): p 0 i d d q p 0 i q q d γ q(d) (70-73) (74-76)

123 4 he tanfoed oto cuent i d d ; i q γ d q i d(q) equal: i i q γ (77) he electoagnetic toque equation i: P P P e ( γd iq i d γq idi q ) q i d d i q d iq qid P q d d q ( ) γ d γ q (78) If the flu linage ae choen a independent vaiale, the cuent can e deduced fo the epeion: q q γ q iq D d d γ d id D (79-7) q q γ q i q γ D i d d γ d d γ d D whee: l l q D ecapping fo the continuou linea atheatical odel fo the unyetical ingle-phae induction achine, thee ae thee pecific choice of the tun atio γ q and γ d : a) he oto flu i elected a efeence (invee Γ-fo odel): γ q (73) γ d γ q ( ) ) he ai-gap flu i elected a efeence (-fo odel): γ γ (74) q d c) he tato flu i elected a efeence (Γ-fo odel): l γ q γ la d γ q If the invee of D i noted with d, the voltage equation will e e-witten in a new fo: (75)

124 5 d u d d q q q q γ dt d u d d a d d d d γ dt 0 d d d q q q d γ γ dt (76-79)) d 0 d d d d d q γ γ dt Fo null initial condition, if the Laplace tanfoation i applied, it eult a ati yte: q d 0 d / γ 0 q uq d 0 a d / 0 a d /( γ ) d u d q d / γ 0 d / γ /( ) q 0 (730) d 0 d / γ ( / ) d / γ d 0 o ynthetically: Y ( ) A Y ( ) B U ( ) (73) By copaing to the geneal fo of a tate vaiale yte: X ( ) A X ( ) B U ( ) (73) Y ( ) C X ( ) D U ( ) It can e oeved that the tate vaiale vecto i identical to the output vecto (in thi cae the flu linage pe econd vecto). he input vecto i the voltage vecto (in thi cae only the tato voltage, a the oto i hot-cicuited). By identification, we otain: d 0 d / γ 0 0 a d / 0 a d /( γ ) A d / γ 0 d / γ /( ) ( ) 0 d / γ ( / ) d / γ B 5.. Bilinea tanfoation ethod (utin) ecapping fo the atheatical dicete odel of thee-phae induction achine analyi, the following elation i the ilinea tanfoation (utin) fo continuou to dicete doain: z z he ati equation fo continuou doain ecoe in dicete: z Y ( z) A Y ( z) B U ( z) (735) z which give the elation fo the input vecto Y(z): Y( z) f I A f I A z Y( z) f I A B z U ( z) (736) whee: ( ) ( ) ( ) ( )

125 6 f ± d 0 d / γ 0 a d a d 0 f ± 0 γ f I A d d 0 f (737) ± γ γ d d 0 ± f ± γ γ he epeion of the flu linage pe econd in the dicete tie doain i otained when uing the ilinea tanfoation: q ( n) q ( n ) uq ( n ) uq ( n) d ( n) d ( n ) ud ( n ) ud ( n) C D (738) q ( n) q ( n ) 0 d ( n) d ( n ) 0 ( f ) ( f ) C A A ( f A) (739) D he aove elation lead to the atheatical dicete odel of the ingle-phae induction achine fo Fig. 5.. he electoagnetic toque i coputed uing the tato and oto flu linage a independent vaiale: P P d e q d d q ( q d d q ) (740) ( ) γ d γ q γ Conideing the continuou tie doain and uing the Laplace tanfoation, the oto angula velocity value i eadily availale fo the toque epeion: P ( e L ) J (74) he cuent vecto can e deteined accoding to the following ati equation: iq d 0 d / γ 0 q i d 0 d / 0 d /( γ ) d (74) i q d / γ 0 d / γ 0 q i d 0 d /( γ ) 0 d /( γ ) d o in a condened fo: i( n) F ( n) (743)

126 7 c d c z - U d(n) z - U q(n) d d d d3 d3 c3 c4 c c c3 c4 c3 c3 c33 c34 z - z - z - d(n) * q(n) q(n) * d(n) - P_ d /( γ M ) oque e d4 c4 c4 d4 c43 c44 z - Fig.5.. Coplete dicete ingle-phae induction achine odel aed on ilinea tanfoation ethod (utin) he analyi of the loc diaga given in Fig. how that fo a coplete ipleentation of the odel 7 ultiplie, 3 addition opeation and 4 delay loc ae neceay. Howeve, thi odel can e iplified depending on the type of ipleentation. he geneal tuctue of the phyical ipleentation depend on the placeent of the loc and the yte fequency epone will e odified accodingly Fowad-diffeence ethod (Eule) he yte will ecoe untale if the apling peiod fo a dicete odel i incoectly choen. he tanfoation fo the continuou-tie to the dicete-tie doain can e ade uing the Eule ethod, which y definition i: d ( n ) ( n) (744) dt whee i the apling peiod. he voltage equation fo the ingle-phae induction achine with cage oto ae tanfoed a follow:

127 8 uq ( n) d q ( n) d q ( n) γ q q q q d γ γ q ( n ) ( n) a d ( n ) d ( n) ud ( n) d d ( n) d d ( n) γ 0 d ( n) d ( n) ( n ) ( n) d ( n ) d ( n) 0 d d ( n) d d ( n) q ( n) γ γ which give the ati elation fo the flu linage pe econd in dicete epeentation: q ( n) ( ) q ( n ) d 0 d / γ 0 q ( n) uq ( n) d ( n ) 0 a d / 0 a d /( γ) d ( n) u d ( n) q( n ) d / γ 0 d / γ / q( n) 0 d( n ) 0 d / γ d / γ d( n) 0 (749) o in a condened fo: ( n ) E ( n) u( n) (750) A the flu linage pe econd vecto i the independent vaiale vecto, the cuent ae deteined accoding to the following ati equation: iq d 0 d / γ 0 q i d 0 d / 0 d /( γ ) d (75) i q d / γ 0 d / γ 0 q i d 0 d /( γ ) 0 d /( γ ) d o in a condened fo: i( n) F ( n) (75) Siila to the ilinea tanfoation cae, the electoagnetic toque i coputed with the elation: P P d e q d d q ( q d d q ) ( ) γd γ (753) q γ Fo the fowad-diffeence ethod (Eule), the atheatical dicete odel of the ingle-phae induction achine i decied y the loc diaga in Fig. 5.. hi odel i chaacteied y the following nue of loc equeted fo ipleentation: 9 addition, 5 ultiplie and 6 delay loc. By copaion with the peviou dicetiation ethod, it i oviouly the iple ipleentation tuctue fo the fowad-diffeence ethod. he coputational uden i.8 tie geate with the ilinea tanfoation than it i with the fowad-diffeence ethod.

128 9 e U q(n) z - z - e3 U d(n) z - e e4 e3 e33 z- q(n) * d(n) d(n) - (P/) M d/( γ ) oque e e34 e4 z - * q(n) e43 z - e44 Fig.5.. Coplete dicete ingle-phae induction achine odel aed on fowad-diffeence ethod (Eule) 5.4. Bacwad-diffeence ethod hi iply to apply dicetiation ethod allow the ipleentation of a tale dicete yte if the analogue veion in continuou tie doain i alo tale. Uing thi ethod, even untale continuou odel can e tanfoed in dicete tale yte. Nevethele, it ut e tated that due to the ditotion in the fequency epone of the yte, a lowe apling peiod ha to e ued. he flu linage pe econd vecto can e otained y applying the elation ued fo the ae tanfoation type fo the thee-phae induction achine to the ingle-phae induction achine: z Y ( z) A Y ( z ) B U ( z) ( I A ) Y ( z) z Y ( z) B U ( z) (754) o in a ati fo: q ( n) q ( n ) uq ( n) d ( n) d ( n ) ud ( n) ( I A ) ( I A ) (755) q ( n) q ( n ) 0 d ( n) d ( n ) 0 o in a condened fo: ( n) G ( n ) H u( n) (756) whee:

129 30 d d 0 0 γ a d a d g g g3 g4 0 0 γ g g g3 g 4 G ( I A ) d d g3 g3 g33 g34 0 γ γ g4 g4 g43 g44 d d 0 γ γ (757) and: H G (758) he ati eleent g ij, i,j,, 3, 4 epeion ae not detailed hee, due to the pace liitation. Lie in the peviou cae, the electoagnetic toque epeent the output of the dicete yte: P P d e q d d q ( q d d q ) (759) ( ) γ d γ q γ he dicete atheatical odel otained though the acwad tanfoation ethod i illutated in Fig A copaative coputational uden fo diffeent ethod of dicetiation i given in ale 5.I. ABLE 5.I Method \ eal opeation Addition Multiplication Delay Fowad-diffeence ethod (Eule) Bacwad-diffeence ethod 7 4 Bilinea tanfoation ethod (utin) he following concluion can e dawn egading diffeent ipleentation option of a dicete atheatical odel fo the ingle-phae induction achine: the pule invaiance ethod and acwad-diffeence ethod deteine iila dicete atheatical odel fo the achine; the tep invaiance ethod and the fowad-diffeence ethod deteine iila atheatical dicete odel fo the achine. the fequency chaacteitic of the odel diffe fo one ethod to anothe. the fowad-diffeence ethod (Eule) peent the iniu coputing tie. when accuacy i the deteining facto, the ilinea tanfoation ethod (utin) ha to e choen fo the ipleentation of the atheatical dicete odel of the achine. it i alo poile to apply a hyid appoach whee the tato and oto equation ae dicetied uing diffeent tanfoation ethod.

130 3 g U q(n) h g g3 z - q(n) h3 g4 h h4 g g q(n) g3 g4 z - * (P/) d / ( γ ) M U d(n) h h3 h h4 g3 g3 g33 g34 g4 g4 z - d(n) * _ oque e g43 z - d(n) g44 Fig Coplete dicete ingle-phae induction achine odel aed on acwad-diffeence ethod 5.5. Z-doain tanfe function ecapping oe geneal conideation of linea algea, a tate-vaiale yte uing ati notation in z-doain i decied a: ( n ) A ( n) B u ( n) y ( n) C ( n ) D u ( n ) (760) [ ] ( n) ( n),..., ( n) whee (n) i the vecto fo the tate vaiale, u (n) and y (n) ae the input and the output vecto epectively, and A i the tate ati. he ipule epone equence in te of the tate-vaiale deciption i given y the elation: D, fo 0 h( ) C A, fo > 0 and the tanfe function ati: 0 (76) H ( z ) h( ) z D C ( zi A) B (76) Fo the analyed yte, i.e. ingle-phae induction achine, thee ae fou eal output (the tato and oto flu linage pe econd) and two eal input (tato voltage). Alo, the tate vaiale ae the output diectly. We can etalih the notation:

131 3 q ( n) ( n) ( n) y ( ) (763) d n q ( n) d ( n) he tate ati A, the input vecto u and the coefficient ati B, C, and D have diffeent eleent accoding to the tanfoation ethod ued fo ipleenting the dicete atheatical odel fo the ingle-phae induction achine. I) Bilinea tanfoation ethod (utin): By copaing the aleady etalihed elation with the geneal fo of a tate-vaiale deciption, fo thi tanfoation ethod we have the notation: he input vecto: uq ( n) uq ( n ) ud ( n) ud ( n ) u ( n) (764) 0 0 he tate ati: f d 0 d / γ 0 a d a d 0 f 0 γ A d d 0 f γ γ d d 0 f γ γ (765) f d 0 d / γ 0 a d a d 0 f 0 a a a3 a4 γ a a a 3 a 4 d d 0 f a3 a3 a33 a 34 γ γ a4 a4 a 43 a44 d d 0 f γ γ he coefficient ati i: f d 0 d / γ 0 a d a d 0 f γ B 3 4 d d 0 f γ γ d d 0 f γ he eleent of the ati A and B equie too uch pace to e detailed hee. γ (766)

132 C D (767) We otain the tanfe function decied y the 4 ow, 4 colun non-zeo eleent ati: z a a a3 a4 3 4 h ( z) h ( z) h3 ( z) h4 ( z) a z a a 3 a h ( z) h ( z) h3 ( z) h4 ( ( z) z) H (768) a3 a3 z a33 a h3 ( z) h3 ( z) h33 ( z) h34 ( z) a4 a 4 a 43 z a h4 ( z) h4 ( z) h43 ( z) h44 ( z) whee the eleent h ij of the tanfe function ati ay e coputed though linea algeaic coputation. he tanfe function fo the cuent vecto conideed a output in elation to the voltage vecto a input can e deteined a well: h ( z) h3 ( z) h ( z) h3 ( z) γ γ h ( z) h 4( z) h ( z) h 4 ( z) γ γ H i ( z) d (769) h 3( z) h ( z) h 3 ( z) h ( z) γ γ γ γ h 4( z) h ( z) h 4 ( z) h ( z) ( γ ) γ ( γ ) γ he coeponding input vecto will e: u u uq ( z ) u u whee: u ud ( z ) II) Fowad-diffeence ethod (Eule) he tate ati A i decied y the elation: d 0 d / γ 0 0 d / 0 d /( γ ) a a A (770) d / γ 0 d / γ / 0 d / γ d / γ he input vecto and the coepondent ati coefficient ae: 0 uq ( n) 0 u ( n) B (77-77) ud ( n) he othe ati C and D have the ae value a fo the ilinea tanfoation cae. he tanfe function ati eult a follow:

133 34 z d 0 d / γ z a d / 0 ad /( γ) 0 H ( z) d / γ 0 z d / γ / d / γ z d / γ 0 0 (773) A detailed epeion fo each eleent of the ati H (z) with 4 ow, colun of non-zeo eleent, i eyond the cope of thi wo. he tanfe function valid fo conideing cuent vecto a output, when voltage vecto epeent the input, i identical in yolic fo with the ilinea tanfoation ethod cae. III) Bacwad-diffeence ethod Siila to the cae peviouly analyed, the tate ati A conit the ain coputational uden when deteining the tanfe function ati H. By copaing the etalihed elation fo the dicete doain odel though thi ethod, we get the following notation: d d 0 0 γ d d a a a a 0 0 a a 3 4 γ a a a3 a 4 A d d a3 a3 a33 a34 0 γ γ a4 a4 a43 a44 d d 0 γ γ (774) and the input vecto u and the coefficient ati B ae: uq ( n) ud ( n) u ( n) B A ( ) 0 0 he ati C and D ae identical to that one peviouly deteined. We otain the tanfe function ati H a 4 ow, 4 colun ati with non zeo eleent: z a a a a a a a a a z a a a a a a a H ( z) (777) a3 a 3 z a33 a 34 a3 a3 a33 a 34 a4 a4 a43 a44 a4 a4 a43 a44 he coplete epeion fo coputing the tanfe function ati eleent equie intene linea algea coputation, and theefoe it i ecoended to deteine thee value accoding to the concete achine paaete. All the conideation ade fo the cuent vecto input cae ae alo valid.

134 35 6. LINEAISAION OF HE INDUCION MACHINE MAHEMAICAL MODEL 6.. Intoduction An ipotant pole elated to the odelling of the induction achine i the non-lineaity of the equation that decie it opeation. hi phenoenon appea in the voltage equation and the electoagnetic toque elation a well, due to the poduct etween the tate vaiale. When a contol yte with induction achine i deigned, it i vey ueful to lineaie the achine equation. Baically, the lineaied equation ae otainale in two way (Kaue et al - 995), (oge - 965). Fitly, the ot ued ethod i the aylo eie epanion of a paticula vaiale (fo eaple voltage, cuent, flu linage, o toque) aound the teady-tate opeating point and then y neglecting all econd-ode te. Altenatively, it i poile to otain the lineaied equation y epeing all the vaiale a the u of thei value in the opeating point and thei inceental value, y neglecting the te which copie poduct of inceental value and y eliinating the teady-tate te. hi i called the all-ignal fo of the achine equation. he eult i a diffeential lineaied equation et which decie the dynaic ehaviou of the achine when all diplaceent fo the opeation point ae peent. he induction achine can e analyed in thi way a a linea yte, and it i poile to apply the aic linea yte theoy in ode to copute the eigenvalue and to etalih tanfe function fo ue in the deign of contol fo thee achine. By definition, the initial diplaceent of the tate-vaiale ae conideed to e zeo. hat i, any achine vaiale can e witten a: (778) 0 whee 0 i the value fo the vaiale in the fied-opeation point, and i an inceental value fo thi value. he equation et fo the tate-vaiale ae: p( t) A ( t ) B u( t ) E z( t ) ( ) y( t) C ( t ) D u( t) whee i the tate-vaiale vecto, u i the input vecto, z i the petuation vecto, y i the output vecto, A the tate yte ati, B the input ati, C the output ati, D the inputoutput ati, E the petuation ati. If u i et to zeo the geneal olution of the hoogenou o foce-fee linea diffeential equation ecoe: t e A K (78) t wheek i a vecto foed y an aitay et of initial condition. he eponential e A epeent the unfoced epone of the yte. It i defined a the tate tanition ati. Sall-ignal taility i aued if all eleent of the tanition ati appoach zeo ayptotically a tie appoache infinity. Fo the taility analyi, the chaacteitic equation of A, defined a follow, i ued: det ( A I ) 0 (78) In the peviou equation I i the identity ati and ae the oot of the chaacteitic equation, efeed a eigenvalue, chaacteitic oot o latent oot. he eigenvalue povide a iple ean of pedicting the ehaviou of an induction achine at any alanced opeating condition. Fo a eal eigenvalue, the induction achine ha an eponential epone, and ignifie a oveent away fo the opeating point. A eal eigenvalue i poitive

135 36 ove the poitive-lope egion of the toque-peed cuve, and ecoe negative afte aiu teady-tate toque. When the eigenvalue ae cople they occu a conjugate pai and ignify a ode of ocillation of the tate vaiale. Negative eal pat coepond to ocillation, which deceae eponentially with tie, eaning a tale condition, while poitive eal pat coepond to an eponential inceae with tie, an untale condition. Uually, the tating point fo the achine yte analyi i fo the voltage equation, coined with the echanical equation. So, if the input vecto i foed y the tato and oto voltage plu the load toque, the ati equation of the yte i witten a follow: u L p Lp (783) [ ] whee [ Lp ] i denoted a the otional ipedance ati of the achine. It i poile to otain the eigenvalue y utituting the diffeential opeato p with the oot yol, thu the eigen-otional ipedance ati Z can e foed, and the chaacteitic equation will e: det [ Z ( )] 0 (784) he deign and analyi of contol aociated with achine (i.e. vecto contol) equie the tanfe function of the actual electical achine, viewed a a yte. Uing the peviou tatevaiale et of equation, and utituting the diffeential opeato p with the Laplace opeato, the input-output tanfe function can e epeed a follow, conideing no petuation (z 0): y ( ) H ( ) C [ I A] B D u( ) If the input vecto i zeo, the output-petuation tanfe function can e coputed a: y ( ) H ( ) [ ] z C I A E z( ) 6.. hee-phae induction achine lineaiation (785) (786) In the concete cae of the yetical thee-phae induction achine, the pe unit veion i elected fo copactne, and the input vecto u i choen to e the voltage vecto. Pactically, it contain two te fo the tato voltage, and two te fo the oto voltage epeed in twoai coodinate yte. Howeve, a ingle input vaiale o a linea coination of eveal input vaiale can alo e elected. In thi foulation, we can epe u a: u G u i (787) wheeg i a colun ati and u i i an input vaiale uch a a linea coination of eveal input vaiale i.e. the aplitude of the teinal voltage. he petuation vecto i uually given y the vaiation of the load toque: z L (788) ecapping fo the d-q odelling of the thee-phae induction achine, thee ae fouteen poile et of tate-vaiale: cuent, flu linage o ied flu linage and cuent pai. Let the elected pai of inceental tate-pace vaiale e denoted a, and detailed a: d j q (789) j d q he all-ignal epeion fo electoagnetic toque can e deduced fo the lage-ignal toque epeion: K (790) ( ) e q d d q

136 37 whee K denote a contant coeponding to the elected et of tate-vaiale (i.e. K fo flu D linage a tate-vaiale). If we apply the all-diplaceent appoach to the tate-vaiale, it eult: K (79) ( ) e q0 d d0 q d0 q q0 d Fo a coplete deciption of the lineaied induction achine odel, we have to include alo the echanical equation, epeed in all-ignal fo: p ( e L ) (79) H he tating point fo the lineaied tate-vaiale odel i given y voltage equation yte witten in tationay efeence fae: u L p (79) whee: i the elected et of tate-vaiale and epeent alo the output of the odel, u i the peviouly detailed input vecto, L i the coefficient ati (it can e foed y eactance value, o non-dienional eleent) fo ultiplying the tie deivative of the tate-vaiale, i the coefficient ati (it can e foed y eitance and eactance value o non-dienional eleent) fo ultiplying the tate-vaiale. Finally, the eleent fo the thee-phae induction achine atheatical odel in lineaied fo will e: p( t) A ( t ) B u( t ) E z( t ) ( ) y( t) C ( t ) D u( t) whee: ( t) q d q d (795) u ( t) uq ud uq ud L (796) and: A M S B M L 0(4,) M ; S l 0(,4) 0 H l K d0 q0 d0 q0 0 0 (, ) (, ; d0 d0 d0 q0 q0 q0 ( ) C I5 D ( ) he petuation vecto can e etacted fo the input vecto a: z ( t) L (80) and:

137 38 E H (80) he eigenvalue can e deteined fo the chaacteitic equation. he othe way of deteining the oot of thi equation i y uing the otional ati ipedance: det [ Z ( ) ] det[ M S ] 0 (803) In Fig. 6. i illutated the all-ignal equivalent cicuit of the thee-phae induction achine in tationay efeence fae: u q i q L l L M L l 0 d - i q d0 u q - L l L l 0 q - q0 - u d i d L M i d u d - - Fig. 6.. Sall-ignal tationay efeence-fae equivalent cicuit fo a thee-phae, yetical induction achine 6.3. Single-phae induction achine lineaiation In the concete cae of the unyetical ingle-phae induction achine, the flu linage pe econd veion i elected fo copactne, and the input vecto u i choen to e the voltage vecto. Copaing with the yetical thee-phae induction achine cae, the ae geneal conideation ae valid. he ayetical configuation of thi induction achine type can e included eadily in the tate-vaiale equation fo y uing the tun atio value. he all-ignal epeion fo electoagnetic toque can e deduced fo the lage-ignal toque epeion in two fo, depending on the elected et of tate-vaiale: I) Flu linage odel o cuent odel: K (804) ( ) e q d d q II) Mied flu linage and cuent odel ( cuent pace vecto, flu linage pace vecto): e K q d d q (805) whee K denote a contant coeponding to the elected et of tate-vaiale. Fo eaple: P K D if the tato and oto flu linage ae elected a tate-vaiale. If we apply the all-diplaceent appoach to the tate-vaiale, it eult: I) Flu linage odel o cuent odel:

138 39 ( ) K (806) e q0 d d0 q d0 q q0 d II) Mied flu linage and cuent odel ( cuent pace vecto, flu linage pace vecto): e K q0 d d0 q d0 q q0 d (807) Fo a coplete deciption of the lineaied induction achine odel, we have to include alo the echanical equation, epeed in all-ignal fo: P p ( e L ) (808) J Following ae the final eleent, which decie the lineaied odel of the ingle-phae induction achine: p( t) A ( t ) B u( t ) E z( t ) (809-80) y( t) C ( t ) D u( t) whee: ( t) q d q d (8) u ( t) uq ud uq ud L (8) A M S B M L 0(4,) J M ; S l (,4) 0 0 P l K d0 q0 d0 q0 o l K d0 qo d0 q0 0 0 (, ) (, ) ; d0 d0 d0 q0 q0 q0 (83-84) C I5 D (85-86) he petuation vecto can e etacted fo the input vecto a: P z ( t) L and E J (87-88) he eigenvalue can e coputed eithe fo the chaacteitic equation o y uing the otional ipedance ati of the achine: det [ Z ( ) ] det[ M S ] 0 (89) In Fig. 6. i illutated the all-ignal equivalent cicuit of the ingle-phae induction achine in tationay efeence fae:

139 40 u q i q L l L M L l 0 d (/) - i q d0 (/) u q - u d a i d L la L M - L l 0 q q0 - i d u d - - Fig. 6.. Sall-ignal tationay efeence-fae equivalent cicuit fo a ingle-phae, unyetical induction achine

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