CHAPTER 1. INDUCTION MOTOR MODEL. GENERALITIES. 1.1 - Equations of the induction motor model. 1.1.1 – Introduction. A dynamic model of the machine subjected to control must be known in order to understand and design vector controlled drives. Due to the fact that every good control has to face any possible change of the plant, it could be said that the dynamic model of the machine could be just a good approximation of the real plant. Nevertheless, the model should incorporate all the important dynamic effects occurring during both steady-state and transient operations. Furthermore, it should be valid for any changes in the inverter’s supply such as voltages or currents [ROM 1]. Such a model can be obtained by means of either the space vector phasor theory or two-axis theory of electrical machines. Despite the compactness and the simplicity of the space phasor theory, both methods are actually close and both methods will be explained. 1.1 Induction motor model. Generalities. For simplicity, the induction motor considered will have the following assumptions: § Symmetrical two-pole, three phases windings. § The slotting effects are neglected. § The permeability of the iron parts is infinite. § The flux density is radial in the air gap. § Iron losses are neglected. § The stator and the rotor windings are simplified as a single, multi-turn full pitch coil situated on the two sides of the air gap. sA' sB sC ra' rb rc rc' rb' ra sC' sB' sA Figure 1.1. Cross-section of an elementary symmetrical three-phase machine. 1.1.2 – Voltage equations. The stator voltages will be formulated in this section from the motor natural frame, which is the stationary reference frame fixed to the stator. In a similar way, the rotor voltages will be formulated to the rotating frame fixed to the rotor. In the stationary reference frame, the equations can be expressed as follows: dy (t) (1.1) u (t) = R i (t) + sA sA s sA dt dy (t) (1.2) u (t) = R i (t) + sB sB s sB dt dy (t) (1.3) u (t) = R i (t) + sC sC s sC dt 1.2 Induction motor model. Generalities. Similar expressions can be obtained for the rotor: dy (t) (1.4) u (t) = R i (t) + ra ra r ra dt dy (t) (1.5) u (t) = R i (t) + rb rb r rb dt dy (t) (1.6) u (t) = R i (t) + rc rc r rc dt The instantaneous stator flux linkage values per phase can be expressed as: 2p 4p (1.7) ysA = LsisA + MsisB + MsisC + Msrcosqmira + Msrcos(qm + 3)irb + Msrcos(qm + 3)irc 4p 2p y sB = M sisA + L si sB + M sisC + Msr cos(qm + 3)i ra + M srcosqm i rb + M srcos(qm + 3)irc (1.8) 2p 4p y sC = MsisA + MsisB + LsisC + Msr cos(qm + 3)i ra + Msr cos(qm + 3)i rb + M srcosqmi rc (1.9) In a similar way, the rotor flux linkages can be expressed as follows: 2p 4p yra = M srcos(-qm )isA + Msrcos(- qm + 3)isB + Msrcos(- qm + 3)isC + Lrira + Mrirb + Mrirc (1.10) 4p 2p y rb = M srcos(- q m + 3)isA + M sr cos(- qm )isB + M srcos(-q m + 3)i sC + M r ira + L ri rb + M r i rc (1.11) 2p 4p y rc = M srcos(- q m + 3)isA + M sr cos(- q m + 3)isB + M srcos(- qm )isC + M r ira + Lr irb + M ri rc (1.12) Taking into account all the previous equations, and using the matrix notation in order to compact all the expressions, the following expression is obtained: éu sA ù é R s + pLs pM s pM s pM srcosqm pM srcosq m1 pM srcosq m2 ù éisA ù ê ú êu ú êi ú ê sBú ê pM s R s + pLs pM s pM srcosqm2 pM srcosq m pM srcosqm1 ú ê sB ú êusC ú ê pM s pM s R + pLs pM srcosq pM srcosq pM srcosq ú êisC ú ê ú = ê s m1 m2 m ú × ê ú u i ê ra ú ê pM srcosqm pM srcosq m1 pM srcosq m2 R r + pLr pM r pM r ú ê ra ú ê ú ê ú ê ú u pM srcosq pM srcosq pM srcosq pM r R + pLr pM r i ê rb ú ê m2 m m1 r ú ê rb ú u i ëê rc ûú ëêpM srcosq m1 pM srcosq m2 pM srcosqm pM r pM r R r + pLr ûú ëê rc ûú (1.13) 1.3 Induction motor model. Generalities. 1.1.3 – Applying Park’s transform. In order to reduce the expressions of the induction motor equation voltages given in equation 1.1 to equation 1.6 and obtain constant coefficients in the differential equations, the Park’s transform will be applied. Physically, it can be understood as transforming the three windings of the induction motor to just two windings, as it is shown in figure 1.2 [VAS 1]. sQ sB rb Wm ra Wm ra rb sA sD rc sC Figure 1.2 Schema of the equivalence physics transformation. In the symmetrical three-phase machine, the direct- and the quadrature-axis stator magnitudes are fictitious. The equivalencies for these direct (D) and quadrature (Q) magnitudes with the magnitudes per phase are as follows: 1 1 1 éu s0 ù é ù éusA ù ê ú 2 2 2 u = c × ê cos q cos q - 2p cos q + 2p ú × êu ú ê sD ú ê ( 3) ( 3) ú ê sB ú (1.14) êu ú ê sin sin 2p sin 2p ú êu ú ë sQ û ë- q - (q - 3) - (q + 3)û ë sC û 1 éusA ù é cos q - sin q ù éus0 ù ê 2 ú ê ú êu ú = c × 1 cos q - 2p - sin q - 2p × u ê sB ú ê 2 ( 3) ( 3)ú ê sD ú (1.15) êu ú ê 1 cos q + 2p - sin q + 2p ú êu ú ë sC û ë 2 ( 3) ( 3)û ë sQ û Where "c" is a constant that can take either the values 2/3 or 1 for the so-called non-power 2 invariant form or the value 3 for the power-invariant form as it is explained in section 1.3.3. These previous equations can be applied as well for any other magnitudes such as currents and fluxes. Notice how the expression 1.13 can be simplified into a much smaller expression in 1.16 by means of applying the mentioned Park's transform. 1.4 Induction motor model. Generalities. u i é sD ù é Rs + pLs - Lspqs pLm - Lm (P × w m + pqr )ù é sD ù ê ú ê ú ê ú u L pq R + pL L P × w + pq pL i ê sQ ú = ê s s s s m ( m r ) m ú × ê sQ ú ê ú ê ú ê ú ura pL - L (pq - P × w ) R + pL - L pqr ira ê ú ê m m s m r r r ú ê ú u i (1.16) ëê rb ûú ëêLm (pqs - P × w m ) pLm L rpqr R r + pLr ûú ëê rb ûú 3 Where Ls = Ls - M s , Lr = Lr - M r and Lm = 2 M sr . 1.1.4 – Voltage matrix equations. If the matrix expression 1.16 is simplified, new matrixes are obtained as shown in equations 1.17, 1.18 and 1.19 [VAS 1]. 1.1.4.1 – Fixed to the stator. It means that ws = 0 and consequently wr = -wm. éusD ù é R s + pLs 0 pLm 0 ù éisD ù ê ú ê ú ê ú usQ 0 R + pL 0 pL isQ ê ú = ê s s m ú × ê ú êu rd ú ê pLm P × w m Lm R r + pLr P × w m Lr ú êird ú ê ú ê ú ê ú u i (1.17) ëê rq ûú ë- P × w m Lm pLm - P × w m Lr R r + pLr û ëê rq ûú 1.1.4.2 – Fixed to the rotor. It means that wr = 0 and consequently ws = wm. éusD ù éRs + pLs - LsP wm pLm - Lm P wm ù éisD ù ê ú ê ú ê ú usQ L P w R + pL L P w pL isQ ê ú = ê s m s s m m m ú × ê ú êu rd ú ê pLm 0 R r + pLr 0 ú ê ird ú ê ú ê ú ê ú u i (1.18) ëê rq ûú ë 0 pLm 0 Rr + pLr û ëê rq ûú 1.1.4.3 – Fixed to the synchronism. It means that wr = sws. éusD ù éRs + pLs - Lsw s pLm - Lm w s ù éisDù ê ú ê ú ê ú usQ L w R + pL L w pL isQ ê ú = ê s s s s m s m ú × ê ú êu rd ú ê pLm - Lm sw s R r + pLr - Lrsw s ú êird ú ê ú ê ú ê ú u i (1.19) ëê rq ûú ë L msw s pLm Lrsw s R r + pLr û ëê rq ûú 1.5 Induction motor model. Generalities. 1.2 – Space phasor notation. 1.2.1 – Introduction. Space phasor notation allows the transformation of the natural instantaneous values of a three- phase system onto a complex plane located in the cross section of the motor.