Comparison of Alternative Equivalent Circuits of Induction Motor with Real Machine Data

Transcription

1 Comparison of Alternative Equivalent Ciruits of Indution Motor with Real Mahine Data J. radna, J. auer, S. Fligl and V. Hlinovsky Abstrat The algorithms based on separated ontrol of the motor flux and torque is used in order to gain the maximum performane from the indution mahine. To push the effiieny and dynamis limits of the IM to its limits mostly FOC or DTC ontrol strategies are used. oth are based on the knowledge of the hardly measurable variable-mahine flux. To obtain the information about inner mahine flux models based on the mahine equivalent iruit are mostly used. Therefore the auray of the equivalent iruits has diret influene on the auray of the mahine ontrol. To redue the omplexity of the mathematial model the resistanes and indutanes are onentrated to one omponent and three phase winding is assumed to be symmetrial. In order to design ontrol strategy for the indution motor, system equations and equivalent iruit must be established at first. This paper examines and ompares some of the issues of adequate mahine modeling and attempts to provide a firmer basis for seletion of an appropriate model and to onfirm or disprove the equivalene of different approahes. The results of the IM model run up are then ompared to the results obtained from the measurements on the real mahine and the equivaleny is disussed.. Keywords Indution motor Equivalent iruit IM model J. radna J. auer (&) S. Fligl V. Hlinovsky Czeh Tehnial University in Prague, Faulty of Eletrial Engineering, Tehnika 2, Prague 6, 66 27, Prague, Czeh Republi bauerja2@fel.vut.z J. radna bradnaj@fel.vut.z S. Fligl xfligl@fel.vut.z V. Hlinovsky hlinovsk@fel.vut.z J. eran et al. (eds.), Advanes in Mehanisms Design, Mehanisms and Mahine Siene 8, DOI:.7/ _2, Ó Springer Siene+usiness Media Dordreht 22 3

2 4 J. radna et al. i R L L σ σ2 i 2 is R s ir L σ Ψ σ e () Ψ L m Ψ2 (2) R 2 e s Ψ μ L μ Ψ r R r jω Ψ j( ω ω ) Ψ 2 jωψ μ j( ω ω ) Ψ r Fig. Equivalent T-iruit of an IM C-iruit of the IM Introdution Tee indution mahines (IM) are widely used as eletro-mehanis hangers in variable speed drives []. For desribing the eletromagneti proesses in indution mahines different equivalent iruits an be used. To redue the omplexity of the mathematial model, resistanes and indutanes are represented as onentrated omponents and the 3-phase winding system is assumed to be symmetrial [2]. These equivalent iruits are also used as a starting point for the design of a drive ontroller []. Moreover for the funtion of a regulation algorithm of the drive or model of a mahine, designed on the basis of the mahine s equivalent iruit, the parameters of the equivalent iruit must be known with suffiient auray In IM drive ontrollers the representations based on an equivalent T-formed iruit are used, beause they serve as a solid basis for an advaned motor ontrol. However, other authors tend to use gamma models instead [3 5]. 2 Equivalent Ciruit in T-form In the equivalent IM iruit in T-form (Fig. ) an be established two kirhhoff loops and (), (2) for the derivation of the fluxes an be written _W ðtþ e þ R i ðtþþx j W ðtþ ¼ _W 2 ðtþþr 2 i 2 ðtþþj W 2 ðtþðx x Þ¼; ðþ ð2þ where: W stator flux omplex spae vetor, W 2 rotor flux omplex spae vetor, e supplying voltage omplex spae vetor, x angular speed of the referene frame. Then () and (2) an be rewritten in a standard vetor form (3) _W ðþ¼a t T WðtÞþgðtÞ ð3þ

3 Comparison of Alternative Equivalent Ciruits 5 where A T is the state spae matrix (4) and g respets the influene of the supplying voltage. R ðl m þ2 Þ L x m R e a ðtþ A T R ðl m þ2 Þ x L m R 2 R 2ðL m þ Þ where ¼ þ 2 þ L m þ L m 2 L m R x x L m R 2 x x R 2ðL m þ Þ C A g e b ðtþ C A ð4þ 3 Equivalent Ciruit in C-form The C-equivalent iruit is seond equivalent iruit used in eletri drives. The advantage of this iruit is a simplifiation onsisting in fusion of the rotor and stator indutanes into one indutane on the rotor side without loss of information. The two loops for the stator resp. rotor flux (5) and (6) an be established too _W l ðtþ e s ðtþþr s i s ðtþþx j W l ðtþ ¼ _W r ðtþþr r i r ðtþþj W r ðtþðx x Þ¼ ð5þ ð6þ where: i s stator urrent omplex spae vetor, i r rotor urrent omplex spae vetor, leakage indutane, L l magnetizing indutane. When we use similar labeling to that for a T-iruit (5), (6) an be rewritten into standard vetor form (7) again. _WðtÞ ¼A C WðtÞþbðtÞ where A C is the system state spae matrix. R s L l R s R x 2 x R s L A C ¼ l R s R r R R 3 R r x x R r x x C A ; b e sa ðtþ e sb ðtþ C A ð7þ ð8þ

4 6 J. radna et al. Lσ L σ 2 i r L σ Lσ L σ 2 i r L σ L m L μ L m L μ Fig. 2 Equivaleny of the impedanes 4 Realulation Formulas etween T and G Ciruit We must express the realulation relations between the variables in the iruits in order to ompare them. Let us to follow the most widespread approah in text books impedane equivaleny and time onstant omparison [2 3]. Firstly, let us suppose the input voltage and the stator resistane to be the same for both iruits. Seondly we an substitute the fluxes under the assumption that in a similar way as the fluxes the rotor urrents have a ertain reounting as follows: e s ¼ e ; R s ¼ R ; i s ¼ i ; i r ¼ L mi 2 ð9þ L m þ Next we assume that indutane iruits have the same behaviour when the right-hand sides of the iruits are opened or short-iruited. From the priniple of the superposition it then follows that for the opened right-hand side the same urrent will flow through the primary iruit (left-hand side) (). The same behaviour shall be guaranteed in the ase of the short-iruit on the right-hand side () Fig. 2. For the impedane an be then written Z T ¼ x j(l m þ Þ; Z C ¼ x j L l ) L l ¼ L m þ ðþ Z T ¼ x j þ L m 2 ; Z C ¼ x j L l ) L m þ 2 L l þ L r ¼ L r þ þ L 2 r þ 2 ðþ L m L m Finally, the very last assumption is the equality of the rotor and stator time onstants: T = T s and T 2 = T r The stator time onstants an be expressed as (2). eause of (9) the equivaleny is obvious. T ¼ L m þ ; T s ¼ L l ð2þ R R s In the ase of the rotor time onstant the situation is depited in Fig 3. This an be desribed by (3).

5 Comparison of Alternative Equivalent Ciruits 7 i R L σ i s R s L σ2 i 2 i r L σ Ψ σ e () (2) Ψ e Ψ μ L s m L L μ m Ψ2 R2 L μ Ψ r R r jω Ψ jω Ψ μ j( ω ω ) Ψ 2 j( ω ω ) Ψ r Fig. 3 Equivaleny of the time onstants Fig. 4 System OMEGA 2 T 2 ¼ L m þ 2 ; T r ¼ L l þ R 2 R r ) R r ¼ R 2ðL m þ Þ 2 L 2 m ð3þ 5 Indution Motor Parameter Test ed Measuring system is based on omponents from National Instruments PXI devie. Whole system is managed by interative programme Omega 2, whih was reated in program Lab Windows CVI in language C++. Whole system is plaed in hassis with eight positions, whih protets system from disturbane and outer damage. The effiient power supply and fast data bus PCI are plaed in the base unit for diret proessing measured data in the memory of ontrol omputer (Fig. 4). Nominal parameters of the IM are summarized in (Table ), the parameters for the T-equivalent iruit alulated from the measurements are in (Table 2). These parameters were realulated aording to ( 3) to obtain the parameters for the

6 8 J. radna et al. Table Nominal values of the IM Nominal values of the IM R623? 4? Db? H2 P n (kw) 5,5 I n (A),8 f n (Hz) 5 n n (min - ) 43 U n (V) 38 J (kg m 2 ),4 Table 2 Equivalent iruit parameters T-Ciruit C-Ciruit R (X) 3,3 R S (X) 3,3 (H),26 R R (X) 2,69 R 2 (X) 2,3 (H),52 2 (H),2 L l (H),3499 L m (H),3235 Fig. 5 Simulation results IM reversal measured reversal of IM C-iruit. ased on these parameters were established two models in Matlab/ Simulink. The Fig. 5 (left) show the simulation of reversal of the IM from negative nominal speed to positive. Figure 5 (right) shows measured results by system Omega 2. 6 Conlusions Simulation results show that both equivalent iruits are equal. Also the realulation equations for transformation from T-iruit to C-iruit are orret. This is proof for the full equivalene of both iruits. However, size adjustment for rotor state variables must be undertaken if those are to be exhanged diretly between

7 Comparison of Alternative Equivalent Ciruits 9 the Gamma and the T model running in parallel. Also measured and simulated reversal of the IM show same behavior of IM and same variable sales. Aknowledgments the researh was supported by the Grant Ageny of the Czeh Tehnial University, Prague, grant No. SGS/9/OHK3/2T/3. Referenes. D.W. Novotny and T.A. Lipo, Vetor Control and Dynamis of AC Drive,.edn. Oxford, Oxford University Press, Fitzgerald, A E., Kingsley, Ch, Jr., Umans, S.D.: Eletri mahinery, MGraw-Hill ook Company, Japan, ISN X (983) 3. arnes, M.L., Gross, C.A., Comparison of indution mahine equivalent iruit models, in system theory, Proeedings of the twenty-seventh southeastern symposium on, 995, pp. 4 7 (995) 4. oglietti, A., Cavagnino, A., Ferraris, L., Lazzari, M.: Indution motor equivalent iruit inluding the stray load losses in the mahine power balane, in Industrial eletronis soiety. IECON rd annual onferene of the IEEE, 27, pp (27) 5. Ergene, L.T., Salon, S.J.: Determining the equivalent iruit parameters of anned solid-rotor indution motors, Magnetis, IEEE Trans, 4(7), (25)

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