Mathematical Model of the Three-Phase Induction Machine for the Study of Steady-State and Transient Duty Under Balanced and Unbalanced States

Transcription

1 Chapte 1 Mathematical Model of the Thee-Phae Induction Machine fo the Study of Steady-State and Tanient Duty Unde Balanced and Unbalanced State Alecandu Simion, Leonad Livadau and Adian Munteanu Additional infomation i available at the end of the chapte 1. Intoduction A pope tudy of the induction machine opeation, epecially when it come to tanient and unbalanced dutie, equie effective mathematical model above all. The mathematical model of an electic machine epeent all the equation that decibe the elationhip between electomagnetic toque and the main electical and mechanical quantitie. The theoy of electical machine, and paticulaly of induction machine, ha mathematical model with ditibuted paamete and with concentated paamete epectively. The fit mentioned model tat with the cognition of the magnetic field of the machine component. Thei mot impotant advantage conit in the high geneality degee and accuacy. Howeve, two majo diadvantage have to be mentioned. On one hand, the computing time i athe high, which omehow dicountenance thei ue fo the eal-time contol. On the othe hand, the ditibuted paamete model do not take into conideation the influence of the tempeatue vaiation o mechanical poceing upon the mateial popetie, which can vay up to 5% in compaion to the initial tate. Moeove, paticula contuctive detail (fo example lot o ai-gap dimenion), which eentially affect the paamete evaluation, cannot be alway ealized fom technological point of view. The mathematical model with concentated paamete ae the mot popula and conequently employed both in cientific liteatue and pactice. The equation tand on eitance and inductance, which can be ued futhe fo defining magnetic fluxe, electomagnetic toque, and et.al. Thee model offe eult, which ae globally acceptable but cannot detect impotant infomation concening local effect (Ahmad, 1; Chiaon, 5; Kaue et al., ; Ong, 1998; Sul, 11). 1 Livadau et al., licenee InTech. Thi i an open acce chapte ditibuted unde the tem of the Ceative Common Attibution Licene ( which pemit uneticted ue, ditibution, and epoduction in any medium, povided the oiginal wok i popely cited.

2 4 Induction Moto Modelling and Contol The family of mathematical model with concentated paamete compie diffeent appoache but two of them ae moe popula: the phae coodinate model and the othogonal (dq) model (Ahmad, 1; Boe, 6; Chiaon, 5; De Doncke et al., 11; Kaue et al., ; Maino et al., 1; Ong, 1998; Sul, 11; Wach, 11). The fit categoy wok with the eal machine. The equation include, among othe paamete, the mutual tato-oto inductance with vaiable value accoding to the oto poition. A conequence, the model become non-linea and complicate the tudy of dynamic pocee (Boe, 6; Maino et al., 1; Wach, 11). The othogonal (dq) model ha begun with Pak theoy nine decade ago. Thee model ue paamete that ae often independent to oto poition. The eult i a ignificant implification of the calculu, which became moe convenient with the defining of the pace phao concept (Boldea & Tutelea, 1; Maino et al., 1; Sul, 11). Stating with the claic theoy we deduce in thi contibution a mathematical model that exclude the peence of the cuent and angula velocity in voltage equation and ue total fluxe alone. Baed on thi appoach, we take into dicuion two contol tategie of induction moto by pinciple of contant total flux of the tato and oto, epectively. The mot conitent pat of thi wok i dedicated to the tudy of unbalanced dutie geneated by upply aymmetie. It i peented a compaative analyi, which confont a balanced duty with two unbalanced dutie of diffeent unbalance degee. The tudy ue a woking tool the Matlab-Simulink envionment and povide vaiation chaacteitic of the electic, magnetic and mechanical quantitie unde tanient opeation.. The equation of the thee-phae induction machine in phae coodinate The tuctue of the analyzed induction machine contain: 3 identical phae winding placed on the tato in an 1 electic degee angle of phae diffeence configuation; 3 identical phae winding placed on the oto with a imila diffeence of phae; a contant ai-gap (cloe lot in an ideal appoach); an unatuated (linea) magnetic cicuit that allow to each winding to be chaacteized by a main and a leakage inductance. Each phae winding ha W tun on tato and W tun on oto and a hamonic ditibution. All inductance ae conideed contant. The chematic view of the machine i peented in Fig. 1a. The voltage equation that decibe the 3+3 cicuit ae: d a db dc ua ia, ub ib, uc ic (1) dt dt dt da db d C ua ia, ub ib, uc ic () dt dt dt In a matix fom, the equation become:

3 Mathematical Model of the Thee-Phae Induction Machine fo the Study of Steady-State and Tanient Duty Unde Balanced and Unbalanced State 5 d dt abc uabc iabc (3) d dt ABC uabc iabc (4) (a) (b) Figue 1. Schematic model of thee-phae induction machine: a. eal; b. educed oto The quantitie in backet epeent the matice of voltage, cuent, eitance and total flux linkage fo the tato and oto. Obviouly, the total fluxe include both main and mutual component. Futhe, we define the elf-phae inductance, which have a leakage and a main component: Ljj=Lσ+Lh fo tato and LJJ=LΣ+LH fo oto. The mutual inductance of two phae placed on the ame pat (tato o oto) have negative value, which ae equal to half of the maximum mutual inductance and with the main elf-phae component: Mjk=Ljk=Lhj=Lhk. The expeion in matix fom ae: L Lh (1 / ) Lh (1 / ) Lh L (1 / ) Lh L Lh (1 / ) Lh (1 / ) Lh (1 / ) Lh L L h L LH (1 / ) LH (1 / ) LH L (1 / ) LH L LH (1 / ) LH (1 / ) LH (1 / ) LH L L H co co u co u u u L L L co u co co t u co co co (5-1) (5-) (5-3)

4 6 Induction Moto Modelling and Contol whee u denote the angle of 1 (o π/3 ad). The analyi of the induction machine uually educe the oto cicuit to the tato one. Thi opeation equie the alteation of the oto quantitie with the coefficient k=w/w by complying with the conevation ule. The new value ae: u ku ; k ; i 1/ k i ; abc ABC abc ABC abc ABC W W W h H h W h h k ; L k L L ; (6) W W W W WW ; h W W h L k L L L kl L whee the eluctance of the flux path have been ued. The new matice, with oto quantitie denoted with lowecae lette ae: L Lh (1 / ) Lh (1 / ) Lh L k L (1 / ) Lh L Lh (1 / ) Lh (1 / ) Lh (1 / ) Lh L L h co co u co u u u L kl L Lh co u co co t u co co co (7-1) (7-) By vitue of thee tanfomation, the voltage equation become: d abc diabc d L iabc uabc iabc iabc L dt dt dt d di dl i dt dt dt abc abc t abc uabc iabc iabc L (8) By uing the notation: L L 3Lh L L 3Lh L L L Lh L L 3Lh L L L L L L 3L L h h (9) and afte the epaation of the cuent deivative, (8) can be witten unde opeational fom a follow:

5 Mathematical Model of the Thee-Phae Induction Machine fo the Study of Steady-State and Tanient Duty Unde Balanced and Unbalanced State 7 L L hl L h ia ib ic ia co ib co uic co u L L L L L Lh 3Lh L ia in ib in uic in u L Lh L Lh 3Lh,6 L ib ic ua ub uc u a L L L L Lh ua co ub co uuc co u, L L L hl L h ib ic ia ib co ic co uia co u L L L L L Lh 3Lh L ib in ic in uia in u L Lh L Lh 3Lh,6 L ic ia ua ub uc u b L L L L Lh ub co uc co uua co u, L L L hl L h ic ia ib ic co ia co uib co u L L L L L Lh 3Lh L ic in ia in uib in u L Lh L Lh 3Lh,6 L ia ib ua ub uc u c L L L L Lh uc co ua co uub co u, L L Lh Lh ia ia co ib co uic co u,6 ib ic L L L L Lh L Lh u co co co a ub u uc u ua ub uc L L L L 3Lh L 3Lh ua L h ia in ib in uic in u L L L Lh i b i c, L L L Lh Lh ib ib co ic co uia co u,6 ic ia L L L L Lh L Lh u co co co b uc u ua u ua ub uc L L L L 3Lh L 3Lh ub L h ib in ic in uia in u L L L Lh i c i a, L L

6 8 Induction Moto Modelling and Contol L Lh Lh ic ic co ia co uib co u,6 ia ib L L L L Lh L Lh u co co co c ua u ub u ua ub uc L L L L 3L L 3L h h uc L h ic in ia in u ib in u L L L L i i L h a b L, (1) Beide (1), the equation concening mechanical quantitie mut be added. To thi end, the electomagnetic toque ha to be calculated. To thi effect, we tat fom the coenegy expeion, W m, of the 6 cicuit (3 ae placed on tato and the othe 3 on oto) and we take into conideation that the leakage fluxe, which ae independent of otation angle of the oto, do not geneate electomagnetic toque, that i: W m abc abc abc abc abc abc i L L i i L L i i L i t t t (11) The magnetic enegy of the tato and the oto doe not depend on the otation angle and conequently, fo the electomagnetic toque calculu nothing but the lat tem of (11) i ued. One obtain: 1 dl Te pi abc iabc t d 1 plhin ia ia ib ic ib ia ib ic ic ia ib ic 3 plhco iaic ibibia icicib ia (1) The equation of toque equilibium can now be witten unde opeational fom a: J kz 1 plh in ia ia ib ic ib ia ib ic p i i i i 3co i i i i i i i i i T c a b c a c b b a c c b a t (13) whee ω epeent the otational pulatance (o otational pulation). The imulation of the induction machine opeation in Matlab-Simulink envionment on the bai of the above equation ytem i athe complicated. Moeove, ince all equation depend on the angula peed than the peciion of the eult could be quetionable mainly fo the tudy of apid tanient. Conequently, the ue of othe vaiable i undetandable. Futhe, we hall ue the total fluxe of the winding (3 motionle winding on tato and othe otating 3 winding on oto).

7 Mathematical Model of the Thee-Phae Induction Machine fo the Study of Steady-State and Tanient Duty Unde Balanced and Unbalanced State 9 It i well known that the total fluxe have a elf-component and a mutual one. Taking into conideation the ule of educing the oto cicuit to the tato one, the matix of inductance can be witten a follow: 1 l 1 / 1 / co co u co u 1 / 1 l 1 / co u co co u 1 / 1 / 1 l co u co u co co co u co u 1 l 1 / 1 / u Lh u l u u o 1 / 1 / 1 Labcabc Lh co co co 1 / 1 1 / co co c l Now, the equation ytem (8) can be witten hotly a: (14) d abcabc uabcabc, i abcabc, whee : abcabc Labcabc iabcabc dt (15) By uing the multiplication with the ecipocal matix: abcabc abcabc abcabc abcabc abcabc, abcabc abcabc abcabc L L L i o i L (16) than (15) become: 1 d abcabc uabcabc, L abcabc abcabc dt (17) Thi i an expeion that connect the voltage to the total fluxe with no cuent involvement. Now, pactically the ecipocal matix mut be found. To thi effect, we uppoe that the ecipocal matix ha a imila fom with the diect matix. If we ue the 1 condition: Labcabc Labcabc 1, than though tem by tem identification i obtained: 1 1 L abcabc LD co co co L LhL LhL u u LhL L LhL co u co co u LhL LhL L co u co u co co co u co u L LhL LhL co u co co u LhL L LhL co u co u co LhL LhL L (18) whee the following notation have been ued: LD LhL LhL LL LL LhLL 3 ; 3 ( ) 3 3 ; ; L L L L L L L L L L L L L L L L h h h h (19)

8 1 Induction Moto Modelling and Contol Futhe, the matix poduct i calculated:, Labcabc abcabc, which i ued in (17). Afte a convenient gouping, the ytem become: 1 a h h d L L L L L L u dt LD LD LD a a b c a b c co 3 c b in b h h d L L L L L L u dt LD LD LD b b c a a b c co 3 a c in c h h d L L L L L L u dt LD LD LD c c a b a b c co 3 b a in a h h d L L L L L L u dt LD LD LD a a b c a b c co 3 b c in b h h d L L L L L L u dt LD LD LD b b c a a b c co 3 c a in c h h d L L L L L L u dt LD LD LD c c a b a b c co 3 a b in (-1) (-) (-3) (-4) (-5) (-6) Fo the calculation of the electomagnetic toque we can ue the pinciple of enegy conevation o the expeion of toed magnetic enegy. The expeion of the electomagnetic toque coeponding to a multipola machine (p i the numbe of pole pai) can be witten in a matix fom a follow: 1 d L p abcabc Te abcabc abcabc t d To demontate the validity of (1), one ue the expeion of the matix in ode to calculate it deivative: 1 L abcabc (1), (18),

9 Mathematical Model of the Thee-Phae Induction Machine fo the Study of Steady-State and Tanient Duty Unde Balanced and Unbalanced State 11 d 1 Labcabc 3 d u u u u u u in in in in in in in in in in in u in u in u in in u in u in u in () whee the following notation ha been ued: 1 3 (3 / )( L L ) L L / L h (3) Thi expeion define the pemeance of a thee-phae machine fo the mathematical model in total fluxe. Obevation: One can ue the geneal expeion of the electomagnetic toque whee the diect and ecipocal matice of the inductance (which link the cuent with the fluxe) hould be eplaced, that i: 1 dl 1 dl T pi i p L L e abcabc t abcabc 1 abcabc 1 abcabc abcabc abcabc d t t d 1 1 dlabcabc Te p abcabc abcabc t d (4) A moe convenient expeion that depend on inθ and coθ, lead to the electomagnetic toque equation in fluxe alone: 3 3 co Te 1/ p a a b c b b c a c c a b in a b c b c a c a b (5) Ultimately, by getting togethe the equation of the 6 electic cicuit and the movement equation we obtain an 8 equation ytem, which can be witten unde opeational fom: L LhL LhL L a ua b c LD LD LD a b c co 3 c b in L LhL LhL L b ub c a LD LD LD a b c co 3 a c in (6-1) (6-)

10 1 Induction Moto Modelling and Contol L LhL LhL L c uc a b LD LD LD a b c co 3 b a in L LhL LhL L a ua b c LD LD LD a b c co 3 b c in L LhL LhL L b ub c a LD LD LD a b c co 3 c a in L LhL LhL L c uc a b LD LD LD a b c co 3 a b in k / J p J 1/ p3 in a a b c z 3co b b c a c c a b T a b c b c a c a b t d dt (6-3) (6-4) (6-5) (6-6) (6-7) (6-8) Thi equation ytem, (6-1)-(6-8) allow the tudy of any opeation duty of the thee-phae induction machine: teady tate o tanient unde balanced o unbalanced condition, with imple o double feeding. 3. Mathematical model ued fo the tudy of teady-tate unde balanced and unbalanced condition Geneally, the ymmetical thee-phae quiel cage induction machine ha the tato winding connected to a upply ytem, which povide vaiable voltage accoding to cetain law but have the ame pulation. Pactically, thi i the cae with 4 wie connection, 3 phae and the neutal. The um of the phae cuent give the cuent along neutal and the homopola component can be immediately defined. The analyi of uch a machine can ue the ymmetic component theoy. Thi i the cae of the machine with two unbalance a concen the upply. The tudy can be done eithe uing

11 Mathematical Model of the Thee-Phae Induction Machine fo the Study of Steady-State and Tanient Duty Unde Balanced and Unbalanced State 13 the equation ytem ( ) o on the bai of ymmetic component theoy with thee ditinct mathematical model fo each component (poitive equence, negative equence and homopola). The vat majoity of electic dive ue howeve the 3 wie connection (no neutal). Conequently, thee i no homopola cuent component, the homopola fluxe ae zeo a well and the um of the 3 phae total fluxe i null. Thi i an aymmetic condition with ingle unbalance, which can be tudied by uing the diect and invee equence component when the tanfomation fom 3 to axe i mandatoy. Thi appoach pactically eplace the thee-phae machine with unbalanced upply with two ymmetic thee-phae machine. One of them poduce the poitive toque and the othe povide the negative toque. The eultant toque come out though upepoition of the effect The abc-αβ model in total fluxe The opeation of the machine with unbalance can be analyzed by conideing cetain expeion fo the intantaneou value of the tato and oto quantitie (voltage, total fluxe and cuent eventually, which can be tanfomed fom (a, b, c) to (α, β, ) efeence fame in accodance with the following pocedue : 1 1/ 1/ a 3 / 3 / b 3 / / / c (7) We define the following notation: L 3 LhL LhL L L t ; ( LD) 3LhL 3LhL L L L 3L LhL LhL 1 ; ( LD) 3LhL 3LhL L L L 6L 3 L / Lh 1 t 3 L / L3 L / Lh L L L 3 LhL LhL L L t; ( LD) 3LhL 3LhL L L L 3L LhL LhL 1 ; ( LD) 3LhL 3LhL L L L 6L 3 L / Lh 1 t 3 L / L3 L / Lh L L (8-1) (8-)

12 14 Induction Moto Modelling and Contol 3LhL L 3L 1 ; ( LD) 3L 3L LL / LhL L 3LhL L 3L 1 ; ( LD) 3L 3L LL / LhL L (8-3) (8-4) By uing thee notation in (17) and afte convenient gouping we obtain: d dt d dt d dt b c d dt d dt d dt a b c a 1 ta ua b c 3 a b c co 3c b in 1 tb ub c a 3 a b c co 3a c in 1 tc uc a b 3 a b c co 3b a in 1 ta ua b c 3 a b c co 3b c in 1 tb ub c a 3 a b c co 3c a in 1 tc uc a b 3 b c a co 3a b in (9-1) (9-) (9-3) (9-4) (9-5) (9-6) Typical fo the cage machine o even fo the wound oto afte the tating heotat i hotcicuited i the fact that the oto voltage become zeo. The equation of the ix cicuit get diffeent a a eult of cetain convenient math opeation. (9-) and (9-3) ae multiplied by (-1/) and aftewad added to (9-1); (9-3) i ubtacted fom (9-); (9-1), (9-) and (9-3) ae added togethe. We obtain thee equation that decibe the tato. Similaly, (9-4), (9-5) and (9-6) ae ued fo the oto equation. The new equation ytem i:

13 Mathematical Model of the Thee-Phae Induction Machine fo the Study of Steady-State and Tanient Duty Unde Balanced and Unbalanced State 15 d dt d dt d dt d dt d dt d dt t u co in u in co u t u co in u in co u (3-1,, 3) (3-4, 5, 6) Futhe, the movement equation ha to be attached. It i neceay to etablih the detailed expeion of the electomagnetic toque in fluxe alone tating with (5) and uing convenient tanfomation: 3 Te 3/ p in co (31) Ultimately, the 8 equation ytem unde opeational fom i: u co in (3-1) u in co (3-) u (3-3) t u co in (3-4) u in co (3-5) t u 3 (3-6) kz / J p J 3/ p in co T (3-7) t d dt (3-8)

14 16 Induction Moto Modelling and Contol Thee equation allow the tudy of thee-phae induction machine fo any duty. It ha to be mentioned that the electomagnetic toque expeion ha no homopola component of the total fluxe. 3.. The abc-dq model in total fluxe Fo the tudy of the ingle unbalance condition i neceay to conide expeion of the intantaneou value of the tato and oto quantitie (voltage, total fluxe and eventually cuent in a,b,c efeence fame) whoe um i null. The eal quantitie can be tanfomed to (d,q) efeence fame (Simion et al., 11). By uing the notation (8-1), (8-), (8-3) and (8-4) then afte convenient gouping we obtain (Simion, 1): d dt d dt d u co in u in co dt d dt co in in co (33-1, ) (33-3, 4) Futhe, the movement equation (31) mut be attached. The opeational fom of the equation ytem (4 electic cicuit and movement equation) i: u co in (34-1) u in co (34-) co in (34-3) in co (34-4) 3 k / J p J 3/ p in co T z t (34-5) d dt (34-6) The equation et ( ) and ( ) pove that a thee-phae induction machine connected to the upply ytem by 3 wie can be tudied imilaly to a two-phae machine

15 Mathematical Model of the Thee-Phae Induction Machine fo the Study of Steady-State and Tanient Duty Unde Balanced and Unbalanced State 17 (two-phae mathematical model). It paamete can be deduced by linea tanfomation of the oiginal paamete including the upply voltage (Fig. a). Figue. Induction machine chematic view: a.two-phae model; b. Simplified view of the total fluxe in tato efeence fame; c. Idem, but in oto efeence fame The winding of two-phae model ae denoted with (α, β) and (α, β) in ode to tace a coepondence with the eal two-phae machine, whoe ubcipt ae (a, b) and (a, b) epectively. We hall ue the ubcipt x and y fo the quantitie that coepond to the thee-phae machine but tanfomed in it two-phae model. Thi i a ightful aumption ince (α, β) axe ae collinea with (x, y) axe, which ae commonly ued in analytic geomety. Futhe, new notation (35) fo the flux linkage of the ight membe of the equation ( ) will be defined by following the next ule: - pojection um coeponding to oto flux linkage fom (α, β) axe along the two tato axe (denoted with x and y that i ψx, ψy) when they efe to the flux linkage fom the ight membe of the fit two equation, Fig. b. - pojection um coeponding to tato flux linkage fom (α, β) axe along the two oto axe (denoted with X and Y that i ψxs, ψys) when they efe to the flux linkage fom the lat two equation, Fig. c. x co in, y in co XS co in, YS in co (35) Some apect have to be pointed out. When the machine opeate unde motoing duty, the pulation of the tato flux linkage fom (α, β) axe i equal to ω. Since the otational pulation i ω then the pulation of the oto quantitie fom (α, β) axe i equal to ω=ω=ω ω. The pulation of the oto quantitie pojected along the tato axe with the ubcipt x and y i equal to ω. The pulation of the tato quantitie pojected along the oto axe with the ubcipt XS and YS i equal to ω. The equation ( ) become:

16 18 Induction Moto Modelling and Contol ( ) u (36-1) x ( ) u (36-) y ( ) (36-3) XS ( ) (36-4) The fit two equation join the quantitie with the pulation ω and the othe two, the quantitie with the pulation ω = ω. The expeion of the magnetic toque, in total fluxe and oto poition angle become: o a econd equivalent expeion: e YS 3/ 3 y x T p (37) 3 T (3 / ) p YS (38) XS e which how the total ymmety of the two-phae model of the thee-phae machine egading both tato and oto. The equation of the fou cicuit togethe with the movement equation (37) unde opeational fom give: u x (39-1) u y XS YS k / J p J 3/ p3 x y T (39-) (39-3) (39-4) (39-5) z t d dt (39-6) Thi lat equation ytem allow the tudy of tanient unde ingle unbalance condition. It i imila with the fequently ued equation (Pak) but contain a vaiable only total fluxe and the otation angle. Thee ae no cuent o angula peed in the voltage equation. 4. Expeion of electomagnetic toque Fo the teady tate analyi of the ymmetic thee-phae induction machine, one can define the implified pace phao of the tato flux, which i collinea to the total flux of the

17 Mathematical Model of the Thee-Phae Induction Machine fo the Study of Steady-State and Tanient Duty Unde Balanced and Unbalanced State 19 (α) axi and ha a 3 time highe modulu. In a imila way can be obtained the pace phao of the tato voltage and oto fluxe and the ytem equation ( ) that decibe the teady tate become: j 3 3 j j j 3 j 3 j3e j 3e U j j e j e 3/ in T p e When the peed egulation of the cage induction machine i employed by mean of voltage and/o fequency vaiation then the imultaneou contol of the two total flux pace vecto i difficult. A conequence, new tategie moe convenient can be choen. To thi effect, we hall deduce expeion of the electomagnetic toque that include only one of the total flux pace vecto eithe fom tato o oto. (4) 4.1. Vaiation of the toque with the tato total flux pace vecto One of the method ued fo the contol of induction machine conit in the opeation with contant tato total flux pace vecto. Fom (4), the oto total flux pace vecto i: j j 3 j e ;( );in ;co whee θ i the angle between tato and oto total flux pace vecto. Thi angle ha the meaning of an intenal angle of the machine. The expeion of the magnetic toque that depend with the tato total flux pace vecto become: Te p3e j p3e j (co jin ) (4) 3 3 in. 4 p 3 3 p 3 3 Auming the ideal hypothei of maintaining contant the tato flux, fo example equal to the no-load value, then the pull-out toque, Temax, coepond to inθ = 1 that i: (41) o T 3 inco 1, andt p ; 3 U c e max emax p3 / / tt (43)

18 Induction Moto Modelling and Contol Now an obevation can be fomulated. Let u uppoe an ideal tatic convete that opeate with a U/f=contant=k1 tategy. Fo low upply fequencie, the pull-out toque deceae in value ince the denominato inceae with the pulatance deceae, ω (Fig. 3). Within cetain limit at low fequencie, an inceae of the upply voltage i neceay in ode to maintain the pull-out toque value. In othe wod, U/f = k, and k>k1. Electomagnetic toque Te [Nm] 13 1 U 11 N f N 1.79U N 9.75f N 8.58U N.5f N 7.36U N.5f N Angula velocity Ω [ad/] Figue 3. Mechanical chaacteitic, Me=f(Ω) at Ψ3=cont Voltage U3 [V] 4 3 =.3 =.1 = ω /ω N Figue 4. eultant tato voltage v. pulatance U3=f(ω) at Ψ3=cont. (1,91Wb)

19 Mathematical Model of the Thee-Phae Induction Machine fo the Study of Steady-State and Tanient Duty Unde Balanced and Unbalanced State 1 A pope contol of the induction machine equie a tategy baed on U/f = vaiable. Moe peciely, fo low fequency value it i neceay to inceae the upply voltage with epect to the value that eult fom U/f = cont. tategy. At a pinch, when the fequency become zeo, the upply voltage mut have a value capable to compenate the voltage dop upon the equivalent eitance of the winding. Lately, the moden tatic convete can be paameteized on the bai of the catalog paamete of the induction machine o on the bai of ome laboatoy tet eult. Fom (4) we can deduce: U 3 3 ( j ) 3 j U3 tt j A BC (44) and futhe: U U F () F () G () whee F G 3 tt : ( ) ; ( ) if the tem νtt wa neglected. By inpecting the quae oot tem, which i vaiable with the lip (and load a well), we can point out the following obevation. - Contant maintaining of the tato flux fo low pulation (that i low angula velocity value including tat-up) can be obtained with a ignificant inceae of the upply voltage. The additional inceaing of the voltage depend popotionally on the load value. Analytically, thi fact i caued by the pedominance of the tem G againt F, (45). Fom the viewpoint of phyical phenomena, a highe voltage in cae of evee tat-up o low fequency opeation i neceay fo the compenation of the leakage fluxe afte which the tato flux mut keep it pecibed value. - Contant maintaining of the tato flux fo high pulation (that i angula peed cloe o even ove the ated value) equie an inignificant ie of the upply voltage. The U/f atio i cloe to it ated value (ated value of U and f) epecially fo low load toque value. Howeve, a cetain inceae of the voltage i equied popotionally with the load degee. Analytically, thi fact i now caued by the pedominance of the tem F againt G, (45). - In concluion, the eultant tato flux emain contant fo U/f =contant=k1 tategy if the load toque i mall. Fo high load (epecially if the opeation i cloe to the pullout point), the maintaining of the tato flux equie an inceae of the U/f atio, which mean a ignificant ie of the voltage and cuent. If the machine paamete ae etablihed, then a vaiation ule of the upply voltage can be ettled in ode to have a contant tato flux (equal, fo example, to it no-load value) both fo fequency and load vaiation. Fig. 4 peent (fo a machine with pedetemined paamete: upply voltage with the amplitude of 49 V (Ua=346.5V); ==; Lh=,9; Lσ= Lσ=,1; J=,5; p=; kz=,; (45)

20 Induction Moto Modelling and Contol ω1=314,1 (SI unit)) the vaiation of the eultant tato voltage with the pulatance (in pe unit deciption) fo thee contant lip value. The vaiation i a taight line fo educed load and ha a cetain inflection fo low fequency value (a few Hz). Fo unde-load opeation, a ignificant inceae of the voltage with the fequency i neceay. Thi fact i moe viible at high lip value, cloe to pull-out value (in ou example the pull-out lip i of,33). The vaiation ule baed on U=f(ω) tategy (applied to the uppe cuve fom Fig. 4) povide an opeation of the moto within a lage ange of angula peed (fom tat-up to ated point) unde a developed toque, whoe value i cloe to the pull-out one. Obviouly, the input cuent i athe high (4-5 I1N) and ha to be educed. Pactically, the opeation point mut be placed within the uppe and the lowe cuve, Fig. 4. It i alo eay to notice that the opeation with highe fequency value than the ated one doe not geneally equie an inceae of the upply voltage but the developed toque i lowe and lowe. In thi cae, the output powe keep the ated value. 4.. Vaiation of the toque with the oto total flux pace vecto Uually, the electic dive that demand high value tating toque ue contant oto total flux pace vecto tategy. The tato total flux pace vecto can be witten fom (41) a: j j ; e ; ( );in ;co (46) and the expeion of the electomagnetic toque on the bai of oto flux alone become: p T p e j e 3 3 (47) Auming the ideal hypothei of maintaining contant the oto flux, fo example equal to the no-load value, then the electomagnetic toque expeion i: 3 p3 3 p 3 U 3 3 p3 U 3 e 3 T p (48) whee the voltage and pulation i uppoed to have ated value. Taking into dicuion a machine with pedetemined paamete (upply voltage with the amplitude of 49 V (Ua=346.5V); ==; Lh=,9; Lσ= Lσ=,1; J=,5; p=; kz=,; ω1=314,1 (SI unit)) then the expeion of the mechanical chaacteitic i: T 3 3,14 96,43 U3N e 13,57 N 3,17 (49)

21 Mathematical Model of the Thee-Phae Induction Machine fo the Study of Steady-State and Tanient Duty Unde Balanced and Unbalanced State 3 which i a taight line, A1 in Fig. 5. The two inteection point with the axe coepond to ynchonim (Te=, Ω=ω/=157) and tat-up (Te=995 Nm, Ω=) epectively. The pull-out toque i extemely high and act at tat-up. Thi behavio i caued by the hypothei of maintaing contant the oto flux at a value that coepond to no-load opeation (when the oto eaction i null) no matte the load i. The compenation of the magnetic eaction of the oto unde load i hypothetical poible though an uneaonable inceae of the upply voltage. Pactically, the pull-out toque i much lowe. Anothe uneaonable poibility i the maintaining of the oto flux to a value that coepond to tat-up ( = 1) and the upply voltage ha it ated value. In thi cae the expeion of the mechanical chaacteitic i (5) and the inteection point with the axe (line A, Fig. 5) coepond to ynchonim (Te=, Ω=ω/=157) and tat-up (Te=78 Nm, Ω=) epectively. 33,14 Te k,5 (5) 96,43 The upply of the tato winding with contant voltage and ated pulation detemine a vaiation of the eultant oto flux within the hot-cicuit value (Ψk=,5Wb) and the ynchonim value (Ψ=1,78Wb). The opeation point lie between the two line, A1 and A, on a poition that depend on the load toque. When the upply pulation i two time malle (and the voltage itelf i two time malle a well) and the eultant oto flux i maintained contant to the value Ψ=1,78Wb, then the mechanical chaacteitic i decibed by the taight line B1, which i paallel to the line A1. Similaly, fo Ψk=,5Wb, the mechanical chaacteitic become the line B, which i paallel to A Ψ =1.78Wb ω =314-1 Electomagnetic toque Te[Nm] Ψ k=.5wb ω =314-1 Ψ k=.5wb ω =157-1 B Ψ =1.78Wb ω =157-1 B1 A A Angula velocity Ω [ad/] Figue 5. Mechanical chaacteitic Te=f(Ω), Ψ=cont.

22 4 Induction Moto Modelling and Contol Voltage U [V] =.3 =.1 =.1 1 Figue 6. eultant tato voltage v. pulatance, U=f(ω) at Ψ=cont. (1.3Wb) When the applied voltage and pulation ae two time malle egading the ated value then the opeation point lie between B1 and B ince the oto flux vaie within Ψk=,5Wb (hot-cicuit) and Ψ=1,78Wb (ynchonim). The contol baed on contant oto flux tategy enue paallel mechanical chaacteitic. Thi i an impotant advantage ince the induction machine behave like hunt D.C. moto. A econd apect i alo favoable in the behavio unde thi tategy. The mechanical chaacteitic ha no ecto of untable opeation a the uual induction machine ha. The modification of the flux value (geneally with deceae) lead to a diffeent lope of the chaacteitic, which mean a ignificant deceae of the toque fo a cetain angula peed. The quetion i what vaiation ule of U/ω mut be ued in ode to have contant oto flux? The expeion of the modulu of the eultant oto flux can be witten a: U 3 A B C U otational pulatance ω [ad/] A B C (51) tt tt with : A ; B ; C ; ( ) /. Fig. 6 peent the vaiation of the tato voltage with pulatance at contant eultant oto flux (1,3 Wb), which ae called the contol chaacteitic of the tatic convete connected to the induction machine. The peented chaacteitic coepond to thee contant lip value, =,1 (no-load)-cuve 1, =,1 (ated duty)-cuve and =,3 (cloe to pull-out point)-cuve 3. It can be een that the opeation with high lip value (high load) equie an inceaed tato voltage fo a cetain pulation. A a matte of fact, the atio U3/ω mut be inceaed with the load when the pulatance (pulation) and the angula peed ie a well. Such a tategy i indicated fo fan, pump o load machine with peed-dependent toque.

23 Mathematical Model of the Thee-Phae Induction Machine fo the Study of Steady-State and Tanient Duty Unde Balanced and Unbalanced State 5 When the pulation of the tato voltage i low (mall angula velocitie) then the toque that ha to be ovecame i mall too, but it will ie with the peed and the fequency along a paabolic vaiation. Since the uppe limit of the toque i given by the limited powe of the machine (themal conideation) then thi tategy equie additional pecaution a concen the afety device that potect both the tatic convete and the upply ouce itelf. The analyi of the quae oot tem fom (51) geneate imila emak a in the above dicued contol tategy. Finally i impotant to ay that a contol chaacteitic mut be pecibed fo the tatic convete. Thi chaacteitic hould be implified and geneally educed to a taight line placed between the cuve 1 and fom Fig Study of the unbalanced dutie The unbalanced dutie (geneated by upply aymmetie) ae geneally analyzed by uing the theoy of ymmetic component, accoding to which any aymmetic thee-phae ytem with ingle unbalance (the um of the applied intantaneou voltage i alway zeo) can be equated with two ymmetic ytem of oppoite equence: poitive (+) (o diect) and negative (-) (o invee) epectively. Thee ae two poible way fo the analyi of thi poblem. a. When the amplitude of the phae voltage ae diffeent and/o the angle of phae diffeence ae not equal to π/3 then the unbalanced thee-phae ytem can be eplaced with an equivalent unbalanced two-phae ytem, which futhe i taken apat in two ytem, one of diect equence with highe two-phae amplitude voltage and the othe of invee equence with lowe two-phae amplitude voltage. Uually, thi equivalence poce i obtained by uing an othogonal tanfomation. Not only voltage but alo the total fluxe and eventually the cuent mut be etablihed fo the two eulted ytem. The quantitie of the unbalanced two-phae ytem can be witten a follow: U 1 1/ 1/ Ua 3 3 U b U U Ua; U U 3 / 3 / Ub 3 3 U 1/ 1/ 1/ U c U ; Ua Ub Uc c (5) Futhe, the unbalanced quantitie ae tanfomed to balanced quantitie and we obtain: U ( ) j /6 1 1 j U U ( ) U e ju, o: U ( ) 1 j U j /6 U ( ) U e ju a b a b / ; The quantitie of the thee-phae ytem with ingle unbalance can be witten a follow: / (53) j j j a b c u U co t U Ue ; U kue ; U U(1 ke ) (54) a

24 6 Induction Moto Modelling and Contol and futhe: j/6 j( / ) j/6 j( / ) ( ) ( ) U U( e ke )/ ; U U( e ke )/ (55) Modulu of thee component can be detemined at once with: ( ) ( ) U U 1 k kin( / 6) / ; U U 1 k kin( / 6) / (56) Fo the tanfomation of the unbalanced two-phae quantitie in balanced two-phae component (53) mut be ued: U U U U ju ju ( ) ( ) ( ) ( ) (57) The matix equation of the two-phae model i witten in a convenient way heeinafte: U j U j ( j) y ( j) x (58) Uing elementay math (multiplication with contant, addition and ubtaction of diffeent equation) we can obtain the equation of the two-phae diect (MD) and invee (MI) model: (MD) U ( ) j ( ) j d x( ) (59) (MI) U ( ) j ( ) j i x( ) (6) We have defined the lip value fo the diect (+) and epectively invee (-) machine: d ; i with the inteelation expeion: i. The two machine-model ceate elf-contained toque, which act imultaneouly upon oto. The eultant toque emege fom upepoition effect pocedue (Simion et al., 9; Simion & Livadau, 1). The equation et (59), fo MD, give two equation: which give futhe U ( ) ( j ) ; j (61) ( ) x( ) ( ) x( )

25 Mathematical Model of the Thee-Phae Induction Machine fo the Study of Steady-State and Tanient Duty Unde Balanced and Unbalanced State 7 j U U j j ( ) ( ) ( ) ; x( ) ; ( ) ( ) ( ) (6) Similaly, fo MI we obtain: U( ) ( j) ( ) x( ) ; ( ) j( ) x( ) (63) j U U j j ( ) ( ) ( ) ; x( ) ; ( ) ( ) ( ) To detemine the electomagnetic toque developed unde unbalanced upply condition we ue the ymmetic component and the upepoition effect. The mean electomagnetic toque MD eult fom (5) but tanfomed in implified complex quantitie: (64) T 3p 3 e( ) 3e j( ) x( ) 3p U ( ) A B C (65) Similaly, the expeion of the mean electomagnetic toque MD i: T 3 e( ) 3e j( ) x( ) U ( ) 3p 3p A B C (66) The mean eultant toque, a a diffeence of the toque poduced by MD and MI, can be witten by uing (65) and (66): T eez 3p 3 ( ) U U ( ) A BC A( ) B( ) C (67) tt whee we have defined the notation: A; B; C; and ( ) ( ) U U 1 k k in( /6); U U 1 k k in( /6) (68) Finally, the expeion of the mean eultant toque with the lip i: 3 p 3U 1 k k in( /6) 1 k k in( T /6) eez A B C A( ) B( ) C (69) The influence of the upply unbalance upon Te=f() chaacteitic ae peented in Fig. 7. To thi effect, let u take again into dicuion the machine with the following paamete: upply voltage with the amplitude of 49 V (Ua=346.5V) and π/3 ad. hifted in phae; ==; Lh=,9; Lσ= Lσ=,1; J=,5; p=; kz=,; ω1=314,1 (SI unit). The chaacteitic coeponding to the thee-phae ymmetic machine i the cuve A (the motoing pull-out

26 8 Induction Moto Modelling and Contol toque i equal to 14 Nm and obviouly Ua(-) = ). If the voltage on phae b keep the ame amplitude a the voltage in phae a, fo example, but the angle of phae diffeence change with π/4=7,5 degee (fom π/3=16π/4 to 17π/4 ad.) then the new chaacteitic i the B cuve. The pull-out toque value deceae with appox. 1% but the pull-out lip keep it value. Othe two unbalance degee ae peented in Fig. 7 a well. Electomagnetic toque Te [Nm] 15 1 k=1, β=16π/4; u n=% 75 k=1, β=17π/4; u n=8% 5 k=1, β=18π/4; u n=16% 5 k=.71, β=π/3; u n=7% Figue 7. Te=f() chaacteitic fo diffeent unbalance degee Uually, the unbalance degee of the upply voltage i defined a the atio of invee and diect component: A B C D Slip [-] U ( ) k k U ( ) k k 1 in( /6) un 1[%] 1 in( /6) (7) The cuve A, B, C, and D fom Fig. 7 coepond to the following value of the unbalance degee: un= ; 8%; 16% and 7%. The highet unbalance degee (7% - cuve D) caue a deceae of the pull-out toque by 4%. b. The econd appoach take into conideation the following eaoning. When the amplitude of the thee-phae upply ytem and/o the angle of the phae diffeence ae not equal to π/3 then the unbalanced ytem can be eplaced by two balanced theephae ytem that act in oppoition. One of them i the diect equence ytem and ha highe voltage and the othe i the invee equence ytem and ha lowe voltage. A tanfomation of the unbalanced voltage and total fluxe into two ymmetic ytem i again neceay. In othe wod, thee i an unbalanced voltage ytem (Ua, Ub, Uc),

27 Mathematical Model of the Thee-Phae Induction Machine fo the Study of Steady-State and Tanient Duty Unde Balanced and Unbalanced State 9 which i eplaced by the diect and invee ymmetic ytem. The mean eultant toque i the diffeence between the toque developed by the two ymmetic machinemodel. Taking into conideation thei lip value (d = and i = -) we can deduce the toque expeion: 3 a1 a1 a a T 3/ p [3e j 3e j ] (71) eez T eez U 3p 3 3 3U a1 a A B C A( ) B( ) C (7) and thi i the ame with (69) a we expected. 6. Simulation tudy upon ome tanient dutie of the thee-phae induction machine 6.1. Symmetic upply ytem The mathematical model decibed by the equation ytem (6-1 8) allow a complete imulation tudy of the opeation of the thee-phae induction machine, which include tatup, any udden change of the load and baking to top eventually. To thi end, the machine paamete (eitance, main and leakage phae inductance, moment of inetia coeponding to the oto and the load, coefficient that chaacteize the vaiable peed and toque, etc.) have to be calculated o expeimentally deduced. At the ame time, the value of the load toque and the expeion of the intantaneou voltage applied to each tato phae winding ae known, a well. The oto winding i conideed hot-cicuited. Uing the above mentioned equation ytem, the tuctual diagam in the Matlab-Simulink envionment can be caied out. Additionally, fo a complete evaluation, vitual ocillogaph fo the viualization of the main phyical paamete uch a voltage, cuent, magnetic flux, toque, peed, otation angle and cuent o pecific chaacteitic (mechanical chaacteitic, angula chaacteitic o flux hodogaph) fill out the tuctual diagam. The tudy of the ymmetic thee-phae condition in the Matlab-Simulink envionment take into conideation the following paamete value: thee identical upply voltage with the amplitude of 49 V (Ua=346.5V) and π/3 ad. hifted in phae; ua=ub=uc= ince the oto winding i hot-cicuited; ==; Lh=,9; Lσ= Lσ=,1; J=,5; p=; kz=,; ω1=314,1 (SI unit). The equation ytem become: ua 135,71 3,14 3,14 co 55,67( )in a b c a b c c b ub 135,71 3,14 3,14 co 55,67( )in b c a b c a a c

28 3 Induction Moto Modelling and Contol uc 135,71 3,14 3,14 co 55,67( )in c a b c a b b a 135,71 3,14 3,14 co 55,67( )in a b c a b c b c 135,71 3,14 3,14 co 55,67( )in b c a b c a c a 135,71 3,14 3,14 co 55,67( )in c a b c a b a b,4 4 3,14 in a a b c b b c a c c a b 3 co a b c b c a c a b T t (73-1-7) 1 (73-8) 49 j(314,1 t) 49 j(314,1t,94) 49 j(314,1t4,188) ua e ; ub e ; uc e ; U U U 49 amax bmax cmax (73-9) It ha to be mentioned again that the above equation ytem allow the analyi of the theephae induction machine unde any condition, that i tanient, teady tate, ymmetic o unbalanced, with one o both winding (fom tato and oto) connected to a upply ytem. Geneally, a upplementay equiement upon the tato upply voltage i not mandatoy. The cae of hot-cicuited oto winding, when the oto upply voltage ae zeo, include the wound oto machine unde ated opeation ince the tating heotat i hot-cicuited a well. The peented imulation take into dicuion a vaying duty, which conit in a no-load tat-up (the load toque deive of fiction and ventilation and i popotional to the angula peed and have a teady tate ated value of appox. 3 Nm) followed afte,5 econd by a udden loading with a contant toque of 5 Nm. The imulation eult ae peented in Fig. 8, 1, 1, 14 and 15 and denoted by the ymbol S-5. A econd imulation iteate the peented vaying duty but with a load toque of 1 Nm, ymbol S-1, Fig. 9, 11 and 13. Finally, a thid imulation take into conideation a load toque of 15 Nm, which i a value ove the pull-out toque. Conequently, the falling out and the top of the moto in t,8 econd mak the vaying duty (ymbol S-15, Fig. 16, 17, 18 and 19). The S-5 imulation how an upwad vaiation of the angula peed to the no-load value (in t,1 econd), which ha a weak ovehoot at the end, Fig. 8. The 5 Nm toque enfocement detemine a deceae of the peed coeponding to a lip value of 6,5%. In the cae of the S-1 imulation, the tat-up i obviouly imila but the loading toque detemine a much moe ignificant deceae of the angula peed and the lip value get to 5%, Fig. 9.

29 Mathematical Model of the Thee-Phae Induction Machine fo the Study of Steady-State and Tanient Duty Unde Balanced and Unbalanced State 31 otational pulatance ω [ad/] Time t [] Figue 8. Time vaiation of otational pulatance S-5 otational pulatance ω [ad/] Time t [] Figue 9. Time vaiation of otational pulatance S-1 In the fit moment of the tat-up, the electomagnetic toque ocillate aound 1 Nm and afte the load toque enfocement, it get to appox. 53 Nm fo S-5, Fig. 1 and to appox. 1 Nm fo S-1, Fig. 11. The opeation of the moto emain table fo the both dutie. The behavio of the machine i vey inteeting decibed by the hodogaph of the eultant oto flux (the locu of the head of the eultant oto flux phao), Fig. 1 and 13. With the connecting moment, the oto fluxe tat fom (O point on the hodogaph) and tack a cokcew to the maximum value that coepond to ynchonim (ideal no-load opeation), S point on the hodogaph.

30 3 Induction Moto Modelling and Contol Electomagnetic toque Te [Nm] Time t [] Figue 1. Time vaiation of electomagnetic toque S-5 Electomagnetic toque Te [Nm] Time t [] Figue 11. Time vaiation of electomagnetic toque S-1 [Wb] - O [Wb] N - S Figue 1. Hodogaph of eultant oto flux S-5

31 Mathematical Model of the Thee-Phae Induction Machine fo the Study of Steady-State and Tanient Duty Unde Balanced and Unbalanced State 33 [Wb] - O [Wb] F - S Figue 13. Hodogaph of eultant oto flux S-1 The enfocement of the load toque detemine a deceae of the eultant oto flux, which i popotional to the load degee, and i due to the oto eaction. The locu of the head of the phao become a cicle whoe adiu i popotional to the amplitude of the eultant oto flux. The peed on thi cicle i given by the oto fequency that i by the lip value. It i inteeting to notice that the load toque of 5 Nm caue a unique otation of the oto flux whoe amplitude become equal to the egment ON (Fig. 1) wheea the 1 Nm toque caue appox. 4 otation of the oto flux and the amplitude OF i ignificantly malle (Fig. 13). If the expeion (1) and () ae alo ued in the tuctual diagam then both tato and oto phae cuent can be plotted. The tato cuent coeponding to a phae ha the fequency f1=5 Hz and get a tat-up amplitude of appox. 7 A. Thi value deceae to appox. 6 A (no-load cuent) and afte the toque enfocement (5 Nm) it ie to a table value of appox. 14 A, Fig. 14. The oto cuent on phae a, which ha a fequency value of f = f1, get a imila (appox. 7 A) tat-up vaiation but in oppoition to the tato cuent, ia. Then, it value deceae and the fequency go cloe to zeo. The loading of the machine ha a eult an inceae of the oto cuent up to 13 A and a fequency value of f 3Hz, Fig. 15. The fact that the cuent vaiation ae inuoidal and keep a contant fequency i an agument fo a table opeation unde ymmetic upply condition. Stato phae cuent ia [A] Time t [] Figue 14. Time vaiation of tato phae cuent S-5

32 34 Induction Moto Modelling and Contol oto phae cuent ia [A] Time t [] Figue 15. Time vaiation of oto phae cuent S-5 otational pulatance ω [ad/] Time t [] Figue 16. Time vaiation of otational pulatance S-15 (tat-up to locked-oto) Electomagnetic toque Te [Nm] Time t [] Figue 17. Time vaiation of electomagnetic toque S-15 The thid imulation, S-15, ha a imila tat-up but the enfocement of the load toque detemine a fat deceleation of the oto. The pull-out lip ( 33%) happen in t,5 econd afte which the machine fall out. The angula peed eache the zeo value in t,8 econd, Fig. 16, and the electomagnetic toque get a value of appox. 78 Nm. Thi value can be conideed the locked-oto (tating) toque of the machine, Fig. 17.

33 Mathematical Model of the Thee-Phae Induction Machine fo the Study of Steady-State and Tanient Duty Unde Balanced and Unbalanced State 35 The decibed citical duty that involve no-load tat-up and opeation, oveloading, falling out and top i plotted in tem of eultant oto flux and angula peed veu electomagnetic toque. The hodogaph (Fig. 18) put in view a cuai cokcew ection, coeponding to the tat-up, chaacteized by it maximum value epeented by the egment OS. The falling out tack the cokcew SP with a deceae of the amplitude, which i popotional to the deceleation of the oto. The point P coepond to the locked-oto poition (=1). Fig. 19 peent the dynamic mechanical chaacteitic, which how the vaiation of the electomagnetic toque unde vaiable opeation condition. Duing the noload tat-up, the opeation point tack ucceively the point O, M, L and S, that i fom locked-oto to ynchonim with an ocillation of the electomagnetic toque inide cetain limit ( +Nm to -5Nm). The enfocement of the oveload toque lead the opeation point along the downwad cuve SK chaacteized by an ocillation ection followed by the untable falling out ection, KP. The PKS cuve, togethe with the maked point (Fig. 19) can be conideed the natual mechanical chaacteitic unde motoing duty. + [Wb] - P + O [Wb] - S Figue 18. Hodogaph of eultant oto flux S-15 (tat-up to locked-oto) otational pulatance ω [ad/] S 1 5 O 1 P K L M Electomagnetic toque T e [Nm] Figue 19. otational pulatance v. toque S-15 (tat-up to locked-oto) 6.. Aymmetic upply ytem A imulation tudy of the thee-phae induction machine unde unbalanced upply condition and vaying duty (tat-up, udden toque enfocement and baking to top

34 36 Induction Moto Modelling and Contol eventually) i poible by uing the ame mathematical model decibed by the equation ytem (6-1 8). The value of the eitant toque and the expeion of the intantaneou phae voltage have to be tated. Since the oto winding i hot-cicuited, the upply oto voltage ae ua=ub=uc=. On thi bai, the tuctual diagam ha been put into effect in the Matlab-Simulink envionment. A egad the unbalanced thee-phae upply ytem, it ha to be mentioned that the phae voltage ae no moe equal in amplitude and the angle of phae diffeence may have othe value than π/3 ad. In any event, the um of the intantaneou value of the applied voltage mut be zeo, that i ua+ub+uc=. A an agument fo thi eemingly containt tand the fact that the vat majoity of the thee-phae induction machine ae connected to the indutial ytem via thee upply lead (no neutal). The imulation peented hee take into dicuion an induction machine with the ame paamete a above that i: ==; Lh=,9; Lσ= Lσ=,1; J=,5; p=; kz=,; ω1=314,1 (SI unit). Conequently, the equation (73-1) - (73-8) keep unchanged. The expeion (73-9) have to be modified in accodance with the aymmety degee. Two vaying dutie unde unbalanced condition have been imulated. The fit (denoted NS-1) i chaacteized by an aymmety degee, un = 16,5% and the following upply voltage: 49 j(314,1 t) 375 j(314,1t1,96) 49 j(314,1t3,97) ua e ; ub e ; uc e ; un 16,5% (74) The imulation eult ae peented in Fig.,, 4, 5 and 8. The econd tudy imulation (denoted NS-) ha an aymmety degee of un = 7% given by the following tato voltage: 49 j(314,1 t) 346,43 j(314,1t,357) 346, 43 j(314,1t3,95) ua e ; ub e ; uc e ; un 7% (75) The imulation eult ae peented in Fig. 1, 3, 6, 7 and 9. The vaying dutie ae imila to thoe dicued above and conit in a no-load tat-up (the load toque deive of fiction and ventilation and i popotional to the angula peed and have a teady tate ated value of appox. 3 Nm) followed afte,5 econd by a udden loading with a contant toque of 5 Nm. In compaion to ymmetic upply, the unbalanced voltage ytem caue a longe tat-up time with appox. % fo NS-1 (Fig. ) and with 5% fo NS- (Fig. 1). Moeove, the highe aymmety degee of NS- lead to the cancelation of the ovehoot at the end of the tat-up poce. At the ame time, ignificant peed ocillation ae noticeable duing the opeation (no matte the load degee), which ae highe with the inceae of the aymmety degee. Thee ocillation have a contant fequency, which i twice of the upply voltage fequency. They epeent the main caue that detemine the pecific noie of the machine with unbalanced upply ytem.

35 Mathematical Model of the Thee-Phae Induction Machine fo the Study of Steady-State and Tanient Duty Unde Balanced and Unbalanced State 37 otational pulatance ω [ad/] Time t [] Figue. Time vaiation of otational pulatance NS-1 (tat-up + udden load) otational pulatance ω [ad/] Time t [] Figue 1. Time vaiation of otational pulatance NS- (tat-up + udden load) The inpection of the electomagnetic toque vaiation (Fig. and 3) how the peence of a vaiable ocillating toque, whoe fequency i twice the upply voltage fequency (in ou cae 1 Hz) and ovelap the aveage toque. Thi ocillating component i demontated by the analytic expeion of the intantaneou toque, which i witten uing nothing but total flux linkage (5). The ymmetic component theoy, fo example, i not capable to povide infomation about thee ocillating toque. At the mot, thi theoy evaluate the aveage toque, pobably with inheent eo. Coming back to the toque vaiation, one can ee that the amplitude ocillation inceae with the aymmety degee, but thei fequency keep unchanged. Electomagnetic toque Te [Nm] Time t [] Figue. Time vaiation of electomagnetic toque NS-1