CHAPTER 2 MATHEMATCAL MODELLING OF THREE PHASE INDUCTION MOTOR

Transcription

1 18 CHAPTER 2 MATHEMATCAL MODELLING OF THREE PHASE INDUCTION MOTOR 2.1 INTRODUCTION The convenional mehods of conrolling he speed of a hree phase inducion moor are replaced by saic power conrollers, mainly converers, which uilize hyrisors. The behavior of he inducion moor fed from saic converers is differen from he behavior of moor operaing on sinusoidal supply. A sudy of he ransien as well as he seady sae behavior of converer fed moor is useful as i provides necessary informaion regarding volage and curren of he converer which solely deermines he efficien and opimum selecion of hyrisors and communicaion circui elemens. A dynamic model of he machine which is valid for any insananeous variaion of volage and curren and adequaely describing he performance of he machine under seady sae and ransien operaing condiions can be obained by represening he machine volages and currens in erms of acual phase variables or in erms of wo axis variables or using space phasor heory. The hree phase moor is a balanced load because of he absence of zero sequence componens and herefore i is possible o represen variables of one phase in erms of he oher wo hereby reducing he oal number of Equaions describing he sysem. Thus, in such a model, a hree phase machine is replaced by an equivalen wo phase model wih boh phases in quadraure. This helps in eliminaing he muual coupling ha exiss among

2 19 he saor phases and roor phases. The ime varying self and muual inducances can be made consan by referring all he variables o a common reference frame. This common reference frame can be i) fixed o saor, ii) fixed o roor or iii) fixed o he synchronously roaing magneic field. The realizaion of a variable frequency inducion moor drive wih high performance comparable o ha of a separaely excied DC moor requires a suiable mahemaical model. The dynamic model of inducion moor will be useful in esimaing he flux of he moor, o deermine he orque angle, he phase difference beween he flux axis and saor curren vecor, o fix up he flux axis and o enable he saor curren o decompose ino wo componens, one along he flux axis and he oher perpendicular o he flux axis. Basically, inducion moor can be looked on as a ransformer wih a moving secondary, where he coupling coefficiens beween he saor and roor phases change coninuously wih he change of roor posiion, r. The machine model can be described by differenial Equaions wih ime varying muual inducances, bu such a model ends o be very complex. In order o minimize he complexiy, he machine in hree phase quaniies can be represened as an equivalen machine in wo phase quaniies. Even hough, i is made simple, he problem of ime varying quaniies sill persiss. Hence, he required quaniies are referred o a common frame of reference and by doing so; he above problem is aken care of. Three differen echniques have been proposed for referring he machine quaniies o a common reference frame. The echniques proposed by Kron deals wih referring boh he saor and roor quaniies o a common synchronously roaing reference frame. This makes he moor parameers equivalen o ha of a DC moor, which is of paramoun imporance o he FOC.

3 2 The mahemaical models of he machine used in his hesis are eiher in he synchronously roaing reference frame or in he saor reference frame depending upon he siuaion. The Volage Source Inverer (VSI) fed model is used for esimaing he flux linkages from volages and currens and for simulaing he moor for mainaining direc coupling beween he inverer and he moor. Curren Source Inverer (CSI) fed models are used for simulaions in FOC where muual coupling does no exis. Boh hese models which are used in his hesis are deal in he subsequen secions. 2.2 MODEL FOR VSI FED INDUCTION MOTOR The volage curren relaionship for a volage fed inducion moor in acual variables in he saor reference frame is useful for he analysis of he inverer fed moor. Three phase o wo phase ransformaion of such a model can be achieved if he equivalence, which is based on he principle of power invariance, beween hree and wo phase is esablished. The following assumpions are made in he hree phase o wo phase ransformaion: 1. The inducion moor is symmerical and linear. This assumpion leads o balanced windings and also neglecs sauraion. 2. Three phase source feeding he moor is balanced. 3. Uniform air gap. 4. Balanced roor and saor windings, wih sinusoidally disribued mmf and hence flux disribuion in he air gap is sinusoidal.

4 21 The moor volage equaions boh for saor and roor are given in Equaions (2.1) and (2.2). V V = V R + pl + L p(l 2) p(l 2) p(l 2) R + pl + L p(l 2) p(l 2) p(l 2) R + pl + L i i + i cos cos( + 43) cos( + 23) i pl cos( + 23) cos cos( + 43) i (2.1) cos( + 43) cos( + 23) cos i V V = pl V cos cos( + 43) cos( + 23) cos( + 23) cos cos( + 43) cos( + 43) cos( + 23) cos i i + i R + p(l + L ) p(l 2) p(l 2) i p(l 2) R + p(l + L ) p(l 2) i (2.2) p(l 2) p(l 2) R + p(l + L ) i where V a, V b, V c are he saor volages per phase, V A, V B, V C are he roor volages per phase, R s, L ls are he saor resisance and leakage reacance per phase respecively, R r, L lr are he roor resisance and leakage reacance per phase respecively, L g is he air gap inducance, i a, i b, i c are he saor currens per phase, i A, i B, i C are he roor currens per phase, p is he differenial operaor, is he flux angle. Since, hree phase inducion moors are in general symmerical, he sum of insananeous values of phase volages or currens is zero. Using his fac, he phase c curren can be expressed in erms of he oher wo phase currens as given Equaion (2.3) i c = -(i a +i b ) (2.3) ge Equaion (2.4) Subsiuing Equaion (2.3) in Equaion (2.1) and simplifying, we

5 22 V V = V V r + pl 3 L psin 3 2 r + pl 3 L psin 2 3 L psin L psin r + pl 2 3 L 2 psin 3 L psin L mpsin L mpsin r s + pl s i a i b (2.4) i A i B where L m = (2/3)L g, L s = L m +L ls and L r =L m + L lr are he muual inducance, saor inducance and he roor inducance respecively. Since, he inducances and currens are ime varying in naure, Equaion (2.4) canno be direcly applied o deermine he behavior of he moor. This difficuly is overcome by referring boh saor and roor equaions o saor reference frame. The roaing roor coils can be replaced by a se of pseudo saionary coils fixed along he axes of he saor coils. These coils carry ime varying currens i 2a, i 2b and i 2c. Currens i 2a and i 2b are relaed o i A and i B as given in Equaion (2.5). i 2a = 2 i -sin(- ) -sin 3 2b 3 sin sin(+ A )i (2.5) i B 3 where he marix 2 -sin(- ) -sin 3 3 sin sin(+ is known as ransformaion ) 3 marix T. Using his ransformaion marix and is inverse, he saor Equaions of he moor given in Equaion (2.4), afer simplificaion, can be expressed as given Equaion (2.6)

6 23 V V = R + pl R + pl i i + L p L p i i (2.6) The roor volage equaions in Equaion (2.4) can be ransformed o saor reference frame using he relaion given Equaion (2.7) V V = T V V (2.7) Equaion (2.8) Subsiuing for (V A V B ) T from Equaion (2.4), we ge he V V = TL pt i i + T R + pl R + pl T -1 i i (2.8) This can be furher simplified as given Equaion (2.9) V L p+ L = V L L L p+ L R + L p+ L L L R + L p+ L (2.9) Combining Equaions (2.6) and (2.9), we ge he Equaion (2.1) V V = V V R + pl L p R + pl L p+ L L R + L p+ L L L p+ L L L p i L i (2.1) i R + L p+ L i

7 24 Separaing he resisance drops, inducive drops, ransformer volages and speed volages, we can rewrie Equaion (2.1) as given in Equaion (2.11). (V) = (R)(I) + (L)p(I) + (Gp{})(I) (2.11) given below. Power inpu o he saor is calculaed using Equaion (2.12) as (I) (V) = (I) (R)(I) + (I) (L)p(I) + (I) (Gp{})(I) (2.12) Equaion (2.13). The orque developed, T d, by he moor is calculaed using given in T d =3PL m i 2a i b -i 2b i a (2.13) The flux linkages of he moor are calculaed using Equaion (2.14) o (2.17) as given below. = L i + L i (2.14) = L i + L i (2.15) = L i + L i (2.16) = L i + L i (2.17) The Equaion (2.1) has o be expressed as a sae space model as given in Equaion (2.18) for implemening i on a digial compuer. The sae space model is as given in Equaion (2.19) p(i) = (L) -1 (V)+ (L) -1 (R+G{})(i) (2.18)

8 25 i i p = i i V V + V V i i (2.19) i i where = 1 The oupu equaion is given by Equaion (2.2) i = 1 1 i (2.2) i i Equaion (2.19) has o be expressed in discree form as given in Equaion (2.21) and for using i along wih EKF and RTRN.

9 26 s i a (k+1) i b (k+1) = i 2a (k+1) i 2b (k+1) s s s s s s s V (k) V (k) + V (k) V (k) + ( + ) s s ( + ) s s s + ( + ) s ( + ) s s + ( ) s s ( + ) s s s ( s ) s + ( + ) s i (k) i (k) i (k) i (k) (2.21) The Equaion (2.21), which is in saionary reference frame, is used for conrolling he firing of volage source inverer. The wo phase equaion referred o any arbirary reference frame is as given in Equaion (2.22). V V V V = R + pl L L p L R + pl L L p )L R + L p )L L p )L L L p )L R + L p i i (2.22) i i

10 27 Equaion (2.22) is convered o sae space model for using i on digial compuer. Here, he saor and roor curren componens are considered as sae variables where as he saor volage componens are considered as inpus and saor curren componens are considered as oupus as given Equaions (2.23) o (2.25) as given below. x= (i i i i ) (2.23) u= (V ds V ds ) T (2.24) y= (i ds i ds ) T (2.25) The sae space represenaion of Equaion (2.22) is given in Equaions (2.26) and (2.27) px = Ax+Bu (2.26) y = Cx (2.27) where A= 1 L s L r -L m 2 -R s L r L m 2 r L m R r L m L r r 2 -L m r -R s L r -L m L 2 r L m R r -L m R s -L m L s r -L s R r -L s L r r L m L s r L m R s L s L r r -L s R r B = L s L r -L m 2 L r L r 1 -L m and = 1 -L m The model represened by Equaion (2.26) and Equaion (2.27) are convered in o discree form as given in Equaions (2.28) and (2.29) respecively.

11 28 x(k+1) = F(k)x(k) + G(k) u(k) (2.28) y(k) = Hx(k) (2.29) a a (k) a a (k) a where F(k) = (k) 1a a (k) a a a a a (k) a (k) a a a a a G(k)= a a a L s L r -L m 2 ; a = L ; a = a = L : a = L ; a = a = L ; a = L ; a = R a = L ; a = 2.3 MODEL FOR CSI FED INDUCTION MOTOR For a CSI fed inducion moor, he saor currens are known and hey are considered as inpus in sae space represenaion. Hence, he roor Equaions compleely describe he sysem. In effec, he order of he sysem represenaion becomes wo insead of four as in case of VSI fed moor. The roor flux linkages are he sae variables and erminal volages are he oupus. By suiable manipulaions of VSI fed machine Equaions, sae space Equaions of CSI fed moor can be wrien as given in Equaion (2.3).

12 29 p ( ) = ( ) ( ) ( ) + i i (2.3) The erminal volages are expressed in Equaion (2.31) V = R V i + L i p i + L i p (2.31) The discreized form of Equaion (2.3) is given in Equaion (2.32) (k + 1) (k + 1) = b b b b (k) (k) + i (k) (2.32) i (k) where b 1 =1+ r(k) - R r ; b 3 L 2 = - r r 3 ; b 3 = r 3 ; b 4=1+ r(k) 3 - R r L r Equaion (2.33) The Equaion (2.32) is convered in o d-q axes form as given in -R r p dr qr = L m L r L r r - r -R r L r dr qr + L m L r R r i ds i qs (2.33) The discreized form of Equaion (2.33) is given in Equaion (2.34) (k + 1) (k + 1) = (k) (k) + i (k) (2.34) i (k) These Equaions along wih he dynamic Equaion of moion given in Equaion (2.35) describe he complee performance of he inducion moor. = + (T T ) (2.35)

13 3 2.4 CONCLUSIONS The sae space models VSI fed and CSI fed inducion moor are necessary for simulaing he elecromagneic characerisics of he moor. These models are also useful in designing he conrollers and observers in closed loop conrol. The models may be made available in wo of he hree phase variables using Park s ransformaion. The VSI fed model is suiable for he purpose of simulaing he moor so ha here is direc coupling beween inverer and he moor. The CSI fed model is useful in he simulaions like FOC, where muual coupling beween he axes need o be eliminaed. The discreized models of he moor are necessary in making use of recursive EKF algorihm. The parameers of he mahemaical model, if opimized for an exclusive applicaion, by adaping energy efficien design, he energy losses can be reduced. The nex chaper deals wih energy efficien design of a hree phase inducion moor for opimizing he machine parameers.