VECTOR CONTROLLED-INDUCTION MOTOR DRIVE: OPERATION AND ANALYSIS

Transcription

1 VECTOR CONTROLLED-INDUCTION MOTOR DRIVE: OPERATION AND ANALYSIS Ass. Pof. Yass G. Dssouky and Eng. Mona Moussa Aab Acadmy fo Scinc and Tchnology & Maitim Tanspot Dpt. of Elctical and Comput Contol Eng. Miami, Alxandia, P.O. Box: 1024, Egypt Abstact Many industial applications qui a continuously adjustabl ang of sp. DC motos hav bn usd in such divs. Howv, DC motos a xpnsiv, pohibitiv in hazadous atmosphs and qui fqunt commutatos and bushs maintnanc. Induction Motos (IM s), on th oth hand, a chap, uggd, hav no commutatos, and a suitabl fo high-spd applications but thy xhibit highly coupld nonlina multi-vaiabl stuctus. Th availability of solid-stat contolls, although mo complx than thos usd fo DC motos, has mad it possibl to div induction motos in th sam way as a spaatly xcitd DC moto using vcto contol schm wh toqu contol is achivd by contolling th toqu and flux cunt componnts of th amatu cunt indpndntly. Vcto contol has a fast toqu spons which allows accuat toqu, spd o position contol and it can opat th induction moto in a fou-quadant mod. Also, dcoupld flux and toqu contol allows opating induction machin with a low load toqu. This pap psnts th opation and analysis of th indict-vcto contolld induction moto div. Th basic pincipl of vcto contol induction moto is psntd. Th dynamic modl is xplaind. Th ffct of toqu and flux poducing componnts of stato cunt on stady stat pfomanc is studid. Th tansint spons of vcto contolld induction moto is analyzd and thn compad with that of th DC machin to show analogy btwn both divs. Th vsal of spd diction and fild wakning opation a xplaind. A compaison btwn vcto contolld induction moto and choppd dc divs a compad. List of symbols V, V : stady stat synchonous fam d-q stato voltags, V I, I : stady stat synchonous fam d-q stato cunts, A I d, I q : stady stat synchonous fam d-q oto cunts, A v, v : instantanous synchonous fam d-q stato voltags, V i, i : instantanous synchonous fam d-q stato cunts, A i d, i q : instantanous synchonous fam d-q oto cunts, A v s, v s : instantanous stationay fam d-q stato voltags, V i s, i s : instantanous stationay fam d-q stato cunts, A

2 ψ, ψ ψ d, ψ q T, T L ω, ω sl ω, ω * P : synchonous fam d- and q- stato flux linkags, wb : synchonous fam d- and q- oto flux linkags, wb : load and lctomagntic toqu, Nm : synchonous and slip spd, ad/sc : actual and command oto spd, ad/sc : No. of pols 1. Intoduction DC and induction motos hav svd industial applications that qui vaiabl spd otating shafts. DC machins hav taditionally dominatd most of div systms. Although th machin is mo xpnsiv, th contol pincipls and convt quipmnt quid a somwhat simpl. Thi main dawback is th psnc of commutatos and bushs, which sults in a ncssity fo fqunt maintnanc. Induction machins, on th oth hand, a uggd and lss xpnsiv, and thy hav bn favod fo almost all constant spd div applications. Howv, thi us as a vaiabl spd div has bn damatically complx, sinc th quipmnts ndd a vy xpnsiv and suff fom hug complxitis in ign and implmntation. AC divs gnally qui mo complicatd contol algoithms implmntd by fast al tim signal pocssing. Th fficincy of th spd contol opation is ath low compad to th ncountd difficultis. Rcnt advancmnts in pow lctonics and micocomputs hav mad it possibl to implmnt sophisticatd contol tasks at asonabl cost. Thfo, induction machin div bcam a viabl altnativ to DC divs in many applications, such as obotics, machin tools and olling mills, wh fast dynamic spons, paamt insnsitiv contol chaactistics and apid covy fom spd dop causd by impact loa a quid. Th vcto contol mthod has sval disadvantags such as th cost of th quid two cunt snsos and th oto spd snso. 2. Spd contol of IM An induction moto is ssntially a constant-spd moto, clos to synchonous spd, whn connctd to a constant-voltag and constant-fquncy pow supply. A simpl and conomic mthod of induction moto contol is to vay its stato voltag at th fixd supply fquncy using tiacs (Mohan, Undkand and W. Robbins, 1989). This is chaactizd by a low valu of stating toqu and a poo dynamic and static pfomanc. In od to gnat th highst possibl toqu p amp of stato cunt, and hnc th bst possibl utilization of th availabl cunt capability of th div, th machin flux lvl must b kpt constant and closd to its nominal valu as th moto opating conditions vay. Indict flux gulation schms such as th "volt/htz" contol and th "slip/cunt" contol hav bn xtnsivly usd in industy (Hughs, Coda, and

3 Andad, 1993) but thy fail to povid satisfactoy tansint pfomanc. Highst possibl contol pfomanc fom an induction machin is obtaind using vcto contol o fild ointd contol (Bos, 1986), wh th stato cunt o voltag spac phasos a contolld in magnitud and position. 3. Fundamntals of vcto contol IM Th scala contol lats only to th magnitud contol of a vaiabl. In vcto contol, both magnitud and phas of a vcto a contolld, wh an indpndnt o dcoupld contol of flux and toqu of th moto is pfomd such that, ach of th flux and toqu contol loops contibut in od to gulat both of th magnitud and fquncy of th stato voltag that mans th machin is contolld lik a spaatly xcitd DC moto whos amatu and fild cunts a othogonal o d-coupld vctos wh th fild cunt is st to maintain th atd fild flux, and toqu is changd by changing th amatu cunt. Thfo th toqu snsitivity mains maximum in both tansint and stady stat opations. If th induction moto is considd in a synchonously otating fnc fam, wh th sinusoidal vaiabls appa as DC quantitis at stady stat, th dict and quadatu componnts of th stato cunt in synchonously otating fnc fam a analogous simila to th fild cunt and amatu cunts of th DC moto (Vas, 1992). With th indict mthod of vcto contol, th flux position is calculatd by fist calculating th slip spd, giving th spd of th flux lativ to th oto spd. Th slip spd is thn intgatd, giving th slip angl, o th angl of flux lativ to th oto position. Th oto position is ith masud using a shaft position ncod, o calculatd by intgating th masud oto spd. Th oto angl is addd to th slip angl to giv th position of th flux lativ to th stato. Th calculatd flux angl is usd to pfom th tansfomations btwn stationay and synchonous fnc fams. By using a twin-axis synchonously otating fnc fam alignd with th flux vcto, th stato cunt can b split up into a flux poducing componnt and a toqupoducing componnt, both of which a DC valus at stady-stat conditions which allows dcoupld contol of flux and toqu. Th flux poducing cunt componnt (d-axis) povi a slow spons mchanism to chang th flux in th machin. Th toqu poducing cunt componnt (q-axis) allows fast contolld changs of toqu (Bos, Simos, Ccluis, Rajashkaa and Matin, 1995). So, th contol dynamics bcom dcoupld sinc th activ and activ componnts of cunt a ppndicula and hav no mutual ffcts. Sval mtho hav bn poposd to implmnt fild-ointd contol. Basically, th schms can b classifid into dict (Blaschk, 1972) and indict (Hass, 1969) mtho. Th diffnc btwn ths two mtho lis in th way that oto flux magnitud and position a calculatd. In th dict mthod, th oto flux is masud using sach coils o Hall-ffct snsos built into th ai gap of th machin at th

4 tim of manufactu, o th flux is calculatd using a flux modl of th machin. Du to th difficulty in obtaining accuat stimats flux at low spd, th dict mthod of vcto contol is nomally limitd to opating at spd gat than 10% of th bas spd, unlss th flux is calculatd using oto spd masumnt and stato cunt (Vas, 1992). In th indict mthod of vcto contol, th slip angl which is dfind as th diffnc btwn th oto and th synchonous angl, is calculatd using th masud stato cunt and oto spd, giving th position of th oto flux-linkag whn addd to th oto angl. This mthod havily dpn on th oto tim constant. It avoi th quimnt of flux acquisition by using known moto paamts to comput th appopiat moto slip fquncy to obtain th id flux position (Wad, 1995). Vcto contol givs a fast toqu spons and a wid possibl spd contol ang fom na zo spd to ov th maximum atd spd without xcding th moto's atd voltag, cunt o pow, in ith diction, including motoing o gnating opation. Also, fildwakning opation, with popotionally lss than atd toqu, is availabl as th spd is incasd abov atd spd. Th disadvantags of vcto contol a th quimnts fo a shaft spd snso and cunt tansducs on th stato. A pow invt with a fast switching spd, is quid. Vcto contol is also lativly complx mathmatically, quiing a fast micopocsso o digital signal pocsso fo implmntation. Also, th machin paamts hav to b known accuatly. 4. Basic pincipal and dynamic modl Th dynamic modl of th vcto contolld IM can b concludd as follows. Th oto voltags a xpssd as (Hughs, Coda, and Andad, 1993): d d 0 Ri d ( ) q (1) dt q d 0 Ri q ( ) d (2) dt v v d Rsi (3) dt d Rsi (4) dt Th flux linkags a givn by: si m i d L L (5) si mi q L L (6) d i d mi L L (7)

5 q i q m i L L (8) By aligning th d axis with th oto flux instad of th ai-gap flux, th oto flux in th q axis must b zo. Hnc, d q q 0 (9) dt This implis that: d ( say) constant (10) Substituting (9) and (10) into (1) and (7) and aanging to gt: L R d dt L mi Taking Lapalc tansfom fo (11), th tansf function G(s) can b xpssd as follows: G s i s Lm s L s 1 R Also, th slip spd can b calculatd by subsisting (9) and (10) into (2) and (8) and aanging as follows: (11) (12) i L mr sl (13) L Substituting vcto contol condition in th toqu xpssion in (Bolda and Nassa, 1992) sults in: T 3 P Lm ( )( ) 2 2 L i Eqn. (12) shows that flux lvl contol is achivd by contolling dict componnt of stato cunt and (14) shows that toqu contol is achivd by contolling both dict and quadatu componnts of stato cunt. Fig. 1 xplains th indict voltag vcto contol implmntation. Aft bing masud, th th phas cunts a convtd into stationay fam d s -q s stato cunts (i s, i s ) and thn to synchonously otating fam d -q stato cunts (i, i ) using th absolut flux position (θ ). Th two command cunts (i *, i * ) a compad with th actual cunts (i, i ) and th os a implmntd to th cunt contolls whos outputs a th synchonously otating fam d -q stato voltags (v, v ) which in tun a convtd into th phas fnc voltags using th absolut flux position. Th invt dtmins th th phas PWM voltags applid to th moto. Also, th cunt i is inputtd to th tansf function, G(s), givn in (12) to calculat th flux ψ. Th slip spd, which is calculatd as givn in (13), is addd to th oto spd to find th synchonous spd, ω. (14)

6 5. Stady stat analysis At stady stat, at of chang of flux quals zo. Also, fo vcto contol, th flux ψ q quals zo. By substituting ths conditions into IM d-q modl givn in (1) to (4) to gt: 0 R I d (15) I q d 0 R (16) V V d q R I (17) q d R I (18) Eqn (15) dnots that th cunt I d quals zo. By applying th condition of th vcto contol in (6), it is found that: I L Iq (19) Lm Wh th tm L is th tun's atio of th cunt in th q-axis. Also by substituting (5) and (6) L m into (3) and (4) spctivly, th stady stat quations tnd b: V V R I s s d 2 L m I Ls (20) L R I I L (21) s d mi L (22) I L mr sl (23) d L Substituting (22) into (14), th lctomagntic toqu quation is givn by: T 2 m 3 P L II (24) 2 2 L Equations (15) to (24) hlp to pdict a stady opating point and to plot moto chaactistic cuvs vsus spd as follows. 5.1 Effct of load toqu on stady stat pfomanc. Stady stat pfomanc vsus oto spd is shown in fig. 2, fo two diffnt load toqus, whil kping th command cunt I constant to kp th flux constant at atd valu. It is concludd that: Roto flux, which is alignd with d -axis, is constant indpndnt on ω no load toqu, sinc th d-axis synchonous fam stato cunt is kpt constant.

7 Q-axis synchonous fam stato cunt, I incasd whn toqu is incasd indpndnt on ω as long as I is constant maning that th cunt, I is th toqu-poducing componnt of th stato cunt. Slip spd, ω sl incasd whn toqu is incasd bcaus ω sl dpn on both I, which is toqu dpndnt, and ψ, which is constant, indpndnt on ω. Synchonous spd ω incass as th toqu T L o th spd ω incass wh th spd ω is th sum of ω and ω sl which incass whn T L. Voltag V incass with th incas of spd ω and cunt I as sn fom (20). This voltag is analogous to fild voltag in DC moto but thy do not bhav in th sam way wh voltag of DC moto is only dpndnt on fild poducing cunt, whil th voltag V is dpndnt on both toqu and fild poducing cunts. Fom (21), it is concludd that th incas of cunt I and consquntly th positiv tm (R s I ) is mo dominant than th incas of spd ω and consquntly th ngativ tm L s ω I which causs th incas of V. This voltag is analogous to amatu voltag in DC moto wh both voltags dpnd on both toqu and fild poducing cunts. 5.2 Effct of d- axis command cunt on stady stat pfomanc Fig. 3 shows stady stat pfomanc whn spd vais, whil kping load toqu constant, fo two diffnt opating command cunts I. It is concludd that: Cunt I dcass whn cunt I incass, to maintain constant load toqu T L indpndnt on ω. Roto flux incass whn cunt I incass maning that I is th flux-poducing componnt of stato cunt, indpndnt on ω. Th slip spd ω sl dcass indpndnt on ω wh ω sl dpn on I and ψ d. Th incas of I, and consquntly th flux ψ, with th dcas of I, la to th dcas of ω sl As I incass, ω sl dcass and consquntly th synchonous spd ω sinc it is th sum of ω and ω sl. Fom (20), it is concludd that th dcas of ω and I, is mo dominant than th incas of I which causs th dcas of V. Fom (21), it is concludd that th incas of I is mo dominant than th dcas of I, which causs th incas of V. 6. Tansint Rspons To show th analogy btwn th pfomanc of a vcto contolld induction moto and chopp contolld spaatly xcitd DC moto having th sam pow ating, th tansint

8 spons of th two divs a compad. Fig 4 shows th connction diagam of th DC moto div wh th amatu and th fild a supplid fom a DC supply via a class-e and class-a chopp spctivly. Th dynamic modl of th spaatly xitd DC moto is illustatd in (Mohan, Undkand and W. Robbins, 1989) whil th modl divd in sction (4) is usd fo th vcto contolld induction moto. Fig. 5 and fig. 6 show th pfomanc of th DC moto (to th lft) and th vcto contolld induction moto (to th ight) fo both a stp command spd whil kping constant load toqu and fo stp command load toqu whil kping constant spd command signal, spctivly. It is concludd that th vcto contolld IM bhavs with th sam mann as th DC moto div. Th q-axis synchonous cunt I and d-axis synchonous cunt I in th IM a analogous to th amatu cunt, I a and fild cunt I f in th DC moto, spctivly such that whn th command toqu incass whil kping spd command constant, both I and I a incas bcaus thy a th toqu poducing cunts. Also, whn th command spd incass whil kping load toqu constant, both I and I f main constant whil th voltags V and V a incas to incas back EMF acoss th machins. Fig. 7 and fig. 8 show th wavfom of th IM amatu cunt fo ach cas. Fo stp spd command, it is sn that cunt magnitud dos not chang bcaus load toqu dos not chang whil cunt fquncy is incasd bcaus th synchonous spd incasd (Sn,1990). Fo stp load toqu, th pak valu of th sinusoidal phas-a-cunt, will incas, whil its fquncy main constant. Fig. 7 and 8 pov that vcto contol of induction moto contols magnitud and position of amatu cunt. 7. Baking and vsal of otation Th spd diction of th induction moto is vsd by vsing th phas squnc of th th voltags. On advantag of th vcto-contolld div is th as of vsing spd with th sam hadwa configuation by vsing spd command signal which sults in vsing toqu poducing componnt cunt and consquntly spd diction of th moto. Fig. 9 shows th tansint spons of th div whn th command spd is changd suddnly. Th spd is vsd though gnativ baking mod wh th pow is covd back to th supply that mans vcto contol is capabl to div IM in fou-quadant opations (Edwad and Sn, 1988). Thfo, vcto contolld induction moto divs can b usd fo high pfomanc application such as svo divs and stl mill contols. Dynamic baking o gnation can b implmntd. Whn th toqu command fnc is simply vsd in polaity a toqu will b poducd in th opposit diction and mchanical ngy not dissipatd in th winding sistanc is tund to th supply (Lin, Su and Yuan, 1997).

9 8. Fild wakning opation On advantag of using vcto contol of an induction machin is th capability of opation at sp abov th bas spd in th fild-wakning gion. This is possibl by diving th machin at its maximum atd voltag, but with a fquncy abov th atd supply fquncy. In vcto contol, th is achivd by contolling th fquncy supplid to th machin; such that th spd is incasd by ducing th flux poducing command fnc cunt. This sults in a wak fild in th machin ducing th back EMF tms analogous to a spaatly xcitd DC machin. Blow bas spd, th machin opats at constant flux but abov th bas spd, th flux is waknd invsly popotional to th spd duing which, th pow availabl is constant, without xcding th voltag ating of th machin. Th disadvantag with fild wakning is that th maximum lvl of toqu availabl is ducd du to th low oto flux. With fnc to fig. (1), th command cunt i * is adjustd such that, whn th command spd ω * is blow th bas spd duing constant toqu gion, th command stato cunt i * is kpt constant, whas, * whn ω is abov th bas spd, th cunt i * is automatically dcasd such that th pow is kpt constant without xcding th atd voltag. Fig. 10 shows th pfomancs of th machin using vcto contolld div in fild wakning gion. At th tim t 1 th command spd ω * is incasd fom 93% to 105% of bas spd and th command d-axis stato cunt i * is ducd fom 100% to 60% of atd valu. Again, at tim t 2 th command spd is incasd fom 105% to 110% of bas spd and th command d-axis stato cunt i * is ducd fom 60% to 30% of atd valu. By ducing th fild command cunt to an appopiat lvl dpnding on th load toqu, and incasing th toqu command cunt to compnsat fo th ducd flux, mo fficint opation of th induction machin is possibl (Rajashkaa, Kavamua and Matsus, 1996). This is only possibl wh th load toqu is small and significant pow would othwis b lost in maintaining th atd flux. 9. Compaison btwn IM and DC div Tabl (1) shows a compaison btwn vcto contolld IM div shown in fig. 1 and spaatly xcitd DC moto div shown in fig. 4. It should b notd that th pow ating of th application is an impotant facto to dcid which div is pfd. Fo low pow ating applications, although th DC moto is mo xpnsiv than IM but th ovall DC div is lss xpnsiv than ovall IM div bcaus th vcto contol should b implmntd by a high spd signal pocsso but fo high pow ating applications, th IM div is lss xpnsiv than DC moto div as th machin cost dominats th cost of th whol div (Wily and Sn, 1997).

10 IM div DC Div M/C Lss Mo Cost Pow Elctonics Mo Lss Contol Cicuit Mo Lss Masumnts Mo Lss Maintnanc Lss Mo Fild Wakning Availabl Availabl Spd vsal Availabl Availabl Tabl (1). Compaison btwn IM div and DC div. 10. Conclusion This pap has studid th indict vcto-contolld induction moto div. Th convntional spd contol mtho a viwd and th fundamntal concpt of vcto contol and th dynamic modl a psntd. Th stady stat and tansint analysis a illustatd wh th ffct of both toqu and d-axis command cunt on pfomanc is analyzd. Th vcto contolld induction moto bhavs almost lik a spaatly xitd DC moto. It is shown that th q-axis synchonous stato cunt i bhavs in a simila mann as th amatu cunt I a and th d-axis synchonous stato cunt i bhavs in a simila mann as th fild cunt I f. Vcto contol of induction moto can chang instantanously magnitud and position of amatu cunt. Rvsing spd diction of IM can b achivd using th sam hadwa configuation. Also, th induction moto can un abov bas spd at th sam atd voltag via fild wakning opation mod. Each of DC moto and vcto contolld induction moto has advantags and disadvantags. Th choic btwn thm dpn on th natu of th application. 11. Rfncs [1] Mohan N., Undkand T. and Robbins W., Pow Elctonics: Convts, Application and Dsign, John Wily & Sons, [2] Hughs A., Coda J., and Andad D. A.,"An Insid Look at Cag Motos with Vcto Contol", EMD, pp , [3] Bos B.K., "Pow Elctonics and AC Divs" Pntic-Hall, NJ, [4] Vas P., Elctical Machins and Divs: A spac vcto thoy appoach, Oxfod Univsity Pss, [5] Bos B.K., Simos M.G., Ccluis D.R., Rajashkaa K. and Matin R., Spd snsolss Hybid vcto contolld induction moto div, in IEEE-IAS Confnc, Ann. Mting, pp , 1995.

11 [6] Blaschk F., Th Pincipl of Fild Ointation as Applid to th Nw Tansvkto Closd- Loop Contol Systm fo Rotating-Machins, Simns Rviw, Vol. 39, No. 5, pp , [7] Hass K., Zu Dynamik Dlzahlglt Antib mit stomichtgspistn Asynchon- Kuzschu Laufmaschimss Damstadt,Tchn.Hochsch., Diss., 1969 [8] Wad S., Paamts idntification fo vcto contolld induction moto, Ph.D. thsis, Hiot-Watt Univsity, Sp [9] Bolda I. and S.A. Nassa, Vcto Contol of AC Divs, CRC Pss, [10] Sn P.C., "Elctic Moto Divs and Contol-Past, Psnt, and Futu", IEEE Tans. Indus. Elcton., Vol. 37, No. 6, Dc [11] Edwad Y.Y. and Sn P.C., Dcoupling contol of induction moto div, IEEE Tans on IE, Vol. 35, No. 2, May [12] Lin F.J., Su H.and Yuan C., A High Pfomanc Induction Moto Div with onlin moto tim constant stimation, IEEE Tans. on EC, Vol. 21 No. 4, Dc [13] Rajashkaa K., Kavamua A. and Matsus K., Snsolss contol of AC moto divs, IEEE pss, NJ., [14] Wily J. and Sn P. C., Pincipls of Elctic Machin and Pow Elctonics, 2 nd Edition, Lhigh Pss, 1997.

12 دراسة تحليلية إلدارة محرك الحث المغناطيسي بالنظام المتجه أدى التقدم في استخدام منظومات إلكترونيات القوى الصناعية في إدارة االالت الي امكانية تشغيل محرك الحث والتحكم في سرعته بطريقة مماثلة تماما لمحرك التيار المستمر. تقدم هذه الورةذة الثحةيذة دراسذة تحليليذة للذتحكم المتجهي لمحرك الحث المغناطيسي باستخدام طريقة مثاشرة, تم شرح المثذاد النظريذة ااساسذية التذي تو ذ فكرة التحكم المتجهذي لمحذرك الحذث وكذهلت اسذتنتاذ نمذويذ تحليلذي. تذم دراسذة تذرثير كذ مذر المركثذة ال ميذة والمركثة الفيضية لتيار عضذو االسذتنتاذ علذي االسذتجابة النهائيذة للمحذرك وكذهلت تذم االسذتنتاذ علذج االسذتجابة النهائيذة للمحذرك وكذهلت تذم اسذتنتاذ االسذتجابة االنتقاليذة لمحذرك الحذث وعمذل مقارنذة بينذه وبذير محذرك التيذار المستمر لثيان صحة ادعاء تشغيل محرك الحث بطريقة مماثلة لمحرك التيار المستمر وكهلت عقد جدول مقارنة بير إدارة المحرك لثيان ال وامل المختلفة التج ةد تؤثر في اختيار أحدهما

13 Command Spd * i * i * i * i * i * i + - * Spd i * Contoll - i G(s) d o o Cunt Contoll Cunt Contoll V V i i S Eqn. (12) 1 Gain + - sl S Synchounous to Stationay fam Stationay to Synchounous fam V V i i 2-phas to 3-phas convsion 3-phas to 2-phas convsion Snso Va Vb Vc ia ib ic DC Supply PWM Invt IM Fig. 1 Block diagam of th indict fild ointd contol. (a) (b) (c) (d) Fig. 2 Rlationship btwn ω and: (a) v, (b)v, (c) ω and (d) i

14 (a) (b) (c) (d) () (f) Fig. 3 Rlationship btwn ω and: (a) v, (b) v, (c) ω, (d) i, () ψ d, (f) ω sl Fdback Fig. 4 Closd loop spd contol of DC moto with inn cunt loop.

15 (a) (b) (c) (d) () (f) Fig. 5 Rlationship btwn tim and: (a) amatu I a, (b) q-axis synchonous cunt i, (c) DC moto spd ω, (d) induction moto spd ω, () fild cunt I f, (f) d-axis synchonous cunt i, fo stp chang in spd and constant load toqu.

16 (a) (b) (c) (d) Fig. 6 Rlationship btwn tim and: (a) amatu cunt I a, (b) q-axis synchonous cunt i, (c) fild cunt I f, (d) d-axis synchonous cunt i, fo stp chang in load toqu. (a) Fig. 7 Rlationship btwn tim and: (a) th phas-a voltag Va, (j) Th phas-a cunt Ia (b)

17 (a) (b) Fig. 8 Rlationship btwn tim and: (a) th phas-a-cunt I a, (g) Th phas-a-voltag V a, (a) (b) (c) (d) Fig. 9 Rlation of tim and (a) output spd, (b) toqu, (c) th phas voltags, (d) synchonous stato cunt. (a) (b)

18 (c) (d) Fig. 10 Rlationship btwn tim and:(a) ω, (b) ψ, (c) i q * and(d) i a-max