The Simulation of an Induction Motor in Transient Behavior Taking into Account the Saturation Effect

Transcription

1 The Siulation of an Inuction Motor in Transient Behavior Taking into Account the Saturation Effect Ileana Calofirescu "Politehnica" University of Bucharest, Bucharest, Roania, Abstract It is known the fact that for a igitally controlle inuction achine, taking into account the saturation effect iplies soe unknown eleents, like the nonlinearity of the agnetization curve. In this paper is stuie the inuction otor in transient behavior consiering the saturation effect (the nonlinear expression between the agnetization inuctance L an the agnetization current I ), [1]. This has been possible by creating an optiization progra evelope in Matlab, [2]. Also there ha been use ata fro laboratory tests, [3], but because of the laboratory conitions the inuction otor coul not be use at the given rate values. The Matlab progra is linke to another progra, evelope in Siulink, thus obtaining the electroagnetic torque, the stator an rotor currents in A-B-C three phase coorinates syste an the stator an rotors currents in -q orthogonal coorinates syste, the echanical velocity, the rotors current pulsation, the electrical slip. To begin, there ha been use the Clark an Park transforations,[4], because the usage of constant orthogonal voltages siplifies the progra. Then there were use the stator an rotors currents equations. For the echanical part of the inuction otor were calculate the viscosity coefficient an the inertial oent, [5] There ha been create several subsystes in which was introuce the optiization ae for the inuction otor. I. INTRODUCTION The oeling of an inuction otor which takes into account the saturation effect has encountere ifficulties because of the unknown electroagnetic paraeters. This paper presents a oel of the nonlinear inuction otor in Matlab/Siulink prograing environent, which has results that are very siilar to the ones in the ata sheet. The nonlinearity of the inuction otor is given by the agnetic saturation effect. The leakage inuctances an the ain inuctances are highlighte by the relationship between the fluxes an the currents. This iplies that the agnetization curve I ( L ) is use for an optiization for the nonlinear inuction otor. If there are ajor perturbations of the stators frequency (2-3 ties the value of the rate stator current), the leakage inuctances are influence by the saturation path of the ain fluxes. Therefore it can be ae a segregation between the two saturation aspects (the ain flux path an the leakage flux path). It is custoary that the leakage inuctance are consiere constant an equal (L s =L r ); if the saturation of the leakage paths is taken into account, only if the saturations paths are separate, the inuctances are exclusively consiere in relation with the currents winings. The agnetization of the rotor core of the inuction achine occurs in AC current at the stator an rotor, but at ifferent frequencies. Also, the hysteresis phenoenon evelops in a ifferent anner fro the rotor to the stator. The skin effect of the inuction otor is evelope in the assive conuctors an in the assive parts of the electric achine (eep rotor bars). The skin effect is influence by the agnetic saturation of the ain flux paths (assives parts) or the saturation of the leakage paths (the eep rotor bars). Incluing in the atheatical equations the two conitions, is possible to create a realistic oel, but only through nuerical ethos of fiel. To ensure equivalence between the real achine an the atheatical oel there has been use the Park transforation in orthogonal coorinate syste (eaning the inverse atrix is equal to the transpose atrix, therefore resulting the equivalence between the apparen powers). Also the equivalence between the real achine an the atheatical oel ust be satisfie fro other points of view, such as the electroagnetic torque, the agnetic energy store in the air gap, the leakage inuctances an the ain path inuctances. II. ALGORITHM AND SOFTWARE IMPLEMENTATION For the oeling of a nonlinear inuction otor are use the equations of the voltage, as in the atrix for, equation (1), [6], as it follows: [ u srq ] = [Z].[i srq ] + [L]. [ i srq ]. (1) The next equations, (2), (3) an (4) are base on the orthogonal coorinates syste -q for the stator an rotor. an us usq [ u srq ] = an ur urq ( Ls + L ) [L] = L ( Ls + Lq) Lq is isq [ i srq ] =. r (2) i irq L ( Lr + L ) L q ( Lr + Lq) (3) /12/$ IEEE

2 Rs ωs.( Ls + L ) 0 ωs. L.( ). 0 [Z] ωs Ls + L Rs ωs L =. 0 ωr. L Rr ωr.( Lr + L ) ωr. L 0 ωr.( Lr + L ) Rr ω r an ω s are the angular velocities of the stator an respectively of the rotor. These are the inuction otor equations in saturate behavior, the Park transforation being in respect of the rotating fiel, in steay-state behavior. The electroagnetic torque s for is as in equation (5): M e= p. L.( isq. ir is. ir ). (5) The electroagnetic torque is use for the valiation of the oel for the inuction otor, [7] an [8]. The coupling (agnetization) inuctance, L, between the stator an the rotor of the inuction otor has a sinusoial variation in respect to the rotors position. This inuctance has the following forula, as in equation (6), after the reucing at the stator: 3 L L0s 2 (4) =. (6) The iron losses occur only in the stator, except large variations of the rotors frequency. III. RESULTS The siulation of the atheatical oel in orthogonal -q coorinate syste of the nonlinear inuction otor is evelope in Matlab an Siulink prograing environent. For the atheatical oel of the inuction otor was use Siulink prograing environent, an in Matlab has been ae an optiization for the agnetization curve. The necessary ata for this optiization were obtaine through the no-loa test an rop spee test of the inuction otor. an outsie power engine, which has less then 10% of the inuction otors power; the inuction otor is fe fro a variable voltage transforer or a synchronous generator alternator, which has a voltage with values between 0.05 an 1.25 of the rate voltage U n of the inuction otor. The iron losses can be taken into account by shortcircuiting in a syetric anner the stator wining on the an q axes. This test also perits the knowlege of the iron loss resistance s variation R fe in respect to the agnetization current at a given frequency ω s. Generally speaking, the iron losses (fro here it can be obtaine the iron resistance in the stator wining) epen on the stators frequency. If the stators frequency, ω s, is oifie an the phase voltage is also oifie, collaborate with the fact that the synchronis spee is kept, all these lea to an estiation of the epenence of the iron losses resistance to ω s an I of ( represents the stator current on one phase for the no-loa test). In the Park transforation, the flux is copose of the agnetization flux, φ, an the leakage fluxes, φ an φ q. In the Siulink progra were foun these quantities: the stator an rotors voltages for the orthogonal -q coorinate syste, U s an U sq, the stator an rotors currents also in orthogonal -q coorinate syste, I s, I sq, I r an I rq, the stator an rotors currents in three phase A-B-C coorinate syste, the electrical slip, s, the rotor currents pulsation, ω r, an the echanical spee, Ω. Fig. 2. The stators voltage on ax. The next plot, figure 2, represents the stator voltage on axis, U s, which has a very sall value, approxiately 10-8 V, what ay be approxiate to zero. In the next graphic, figure 3, are represente the stator an rotor currents in -q orthogonal coorinate syste, which have also constants values, after the inuction otor enters in the steay-state behavior. Fig. 1. The schee of the nonlinear inuction otor in Siulink In figure 1, is presente the schee of a nonlinear inuction otor built in Siulink. For the calculation of the stator an rotor voltages in -q coorinate syste Park an Clark transforations were use in this schee. It was necessary to perfor an optiization for the agnetization curve, with the ata obtaine fro a no-loa test. The no-loa test eans that the inuction achine works at synchronous spee ω r = ωs in steay-state behavior. To obtain a synchronous spee the achine is rotate with Fig. 3. The stator an rotors current in -q coorinate syste.

3 The plots that are represente in the figures 4 an 5 represent the stator an rotor currents in three phase coorinate syste A-B-C, I sa, I sb, I sc, I ra, I rb, I rc. This was possible because of the usage of the inverse Park transforation, starting with the -q orthogonal currents, neglecting the hoopolar coponent, I s0, for the stator an also for the rotor, I r0. The rotor pulsation is integrate to fin the values of the rotor currents; the rotor pulsation has the following for: ωr = ωs ω. (7) In the plot isplaye in figure 6 shows the electroagnetic torque, M e. It can be notice the fact that the electroagnetic torque enters the steay-state behavior after 5.3 s, having a value of nearly 2 N, keeping in in that the resistant torque is of 1 N. To calculate the electroagnetic torque were use algebraic loops of the stator an rotor currents. There also have been use algebraic loops for the calculation of the rotors pulsation, the leakage inuctances of the stator an of the rotor, an last but not least the agnetization inuctance. In the graphic in figure 7, is presente the electrical slip of the inuction otor, where it can be observe that the values for the electrical slip are between 1 an 0.05, confiring that the inuction achine works as a otor. Fig. 4. The stators currents in A-B-C coorinates syste. Fig. 7. The electrical slip. Fig. 5. The rotor currents in A-B-C coorinates syste. The rotor currents have a sinusoial for, having an approxiate perio for the transient behavior of 4.5s. The stator current has an approxiate value of 12A in steaystate behavior, while the rotor currents reach saller values, aroun 8A, in the sae behavior. Fig. 8. The rotors currents pulsation. In the plot that is presente in figure 8 is plotte the rotor currents pulsation, ω r, which has a value of approxiately 17 ra/s after the transient behavior. Fig. 6. The electroagnetic torque. Fig. 9. The echanical spee.

4 The echanical spee is presente in figure 9, having a stable value at about 300 ra/s. The echanical spee has been calculate with an algebraic loop, which takes into account the rotor currents pulsation. The agnetization current, i has a atheatical expression that can be written in respect to the rotor an stator currents on -q axes, i an i q. i = is + ir, i q isq + irq =. (8) The agnetization current oule can be written as it follows in equation (9): 2 2 (9) i = ( i + i ). Each of the agnetization fluxes on the q axes, φ an φ q, can be escribe by the following atheatical forulas, (10): q ϕ = L. i, q = L. iq ϕ. (10) The agnetization inuctance is note with the sybol L, an it can be euce with the help of the otor s agnetization curve. The agnetization flux, φ, of the inuction otor without the resistant torque has the following atheatical expression, (11): ϕ = L i. (11) In the expression above L is the agnetization inuctance an i represents the agnetization current. s For us = there is the next atheatical expression: s = ( Lσ s. is + ϕ ). (12) If the leakage inuctances, L σs, are assue to be constants, the first expression of the u s becoes: s is = Lσ s +. (13) The ifferential agnetization flux on axis is in respect to the static agnetizating inuctance: = ( L. i ). (14) The variation of the agnetization inuctance L with the i oule, can be written as follows: = L i. + i L., 2 2 i i i i q q i + i q =. +.. i i (15) The ifferential agnetization flux on axis is given by the following expression: i = L. + L i i i..( i i iq + i i ). By eveloping the ters in equation (16) is obtaine the next expression: 2 L i i = ( L +. ) i i L i. iq i q +. i i (16) (17) It can be observe that the agnetization flux on the i axis epens on i q an. The sae reark it can be sai for the flux on q axis: L = L + L i q iq iq L q = L +. i (18) (19) The inuctances that appear in the above expressions, L an L q, are the agnetization inuctances on an q axes. In the linear behavior, these two inuctances are equal to L. It can be sai that calculation ethos are a siplifie oel of the inuction achine, where L q is the crossover inuctance between the an q axes, which is equal to zero if consiere the linear oel. For the optiization were use the ata provie by an oscilloscope; the oscilloscope was use for viewing the current curve at the no-loa test. These ata have been processe in a Matlab progra, being achieve a thir egree polynoial for the agnetization curve. IV. CONCLUSIONS In the present paper is esire the achieveent of a realistic oel for the inuction otor, taking onto account the nonlinearity of the agnetization curve. This was ae possible by creating an optiization oel an ientifying the agnetization inuctance L. The optiization was ae in Matlab/Siulink, using ata that was retrieve in the context of laboratory tests for an inuction otor. Soe of this test is the no-loa test to establish the agnetization curve an the DC test to eterine the stator resistance. It has to be taken into consieration the fact that the test were ae uner laboratory conitions an the inuction otor coul not be use at the rate values given by the ata sheet. Therefore the results

5 coul be copare only in ters of shape, but not in ters of nuerical values. The Matlab optiization can show the graph of the agnetization curve, after the ata processing fro the otor. The ientification of the electroagnetic paraeters of the inuction otor be as accurate as possible. ACKNOWLEDGMENT The work has been co-fune by the Sectoral Operational Prograe Huan Resources Developent of the Roanian Ministry of Labor, Faily an Social Protec-tion through the Financial Agreeent Posru/88 /1.5/s/ REFERENCES [1] Ion Bolea, Paraetrii asinilor electrice, ientificare, estiare si valiare, e. Acaeiei Roane, Bucuresti, pp , , , [2] Won Young Yang, Wenwu Cao, Tae-Sang Chung, John Morris, Applie Nuerical Methos using MATLAB, 1998 [3] I. S. Gheorghiu, Al. S. Fransua, Tratat e Masini electrice, vol al III-lea, Masini Asincrone, e. Acaeiei R. S. Roania, Bucuresti, [4] D. O. Kisch, Reglarea vectoriala a asinilor e curent alternativ, e. ICPE, [5] M. Juffer, Traité Electricité, Vol. 10, Machines électriques, Press Polytechniques et Universitaires Roanaes, Paris, 1995, pp.384. [6] C. Ghita, Moelare si paraetrii convertoarelor electroecanice, e. Printech, Bucuresti, pp 60-64,78, 89-97, [7] Mihai Iorache, Mihai Dogaru, Dragoş Niculae, Ileana Calofirescu, "Coputer Aie Analysis of Inuction Motor", Annals of the University of Craiova, Electrical Engineering, No. 34, 2010, pp , ISSN [8] M. Iorache, Ileana Calofirescu, D. Niculae, M. Dogaru, Siulation of Inuction Motor Using State Equations, Proceeing of the 7 th International Syposiu Avance Topics in Electrical Engineering ATEE 11, May 2011 Bucharest, ROMANIA, Eitura Politehnica Press, pp , ISSN: , Print ISBN: