* Dpto. de Conversion y Transporte de Energia ** Dpto. de Electronica y Circuitos ***Dpto. Tecnologia Industrial

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1 SQURREL CAGE NDUCTON MACHNE MODEL FOR THE ANALYSS OF SENSORLESS SPEED MEASUREMENT METHODS Jose M. Aller*, Jose A. RestrePo**, Alexander Bueno***, Maria. Gimen#ez?*'*, Gaston Pesse* Universidad Simon Bolivar - Caracas - Venezuela * Dpto. de Conversion y Transporte de Energia ** Dpto. de Electronica y Circuitos ***Dpto. Tecnologia ndustrial jaller@skynet.usb.ve Abstract n this work a squirrel cage induction machine model has been developed to include the stator current effects on the MMF space harmonics, dynamic and static eccentricity harmonics, and slot harmonics. The proposed model is intended to be used on testing the behavior of different sensorless speed measurement techniques. This dynamic model has been developed using the superposition principle and a matrix description of the machine inductance. The model has been validated using experimental data, and tested using a Time- Frequency-Distribution method for sensorless speed measurement [4]. Keywords: nduction Machine, Sensorless Speed Control, Measurements. ntroduction Modern AC machine vector control applications are widely used in today drive implementations. A growing number of researches are being done to perform the vector control technique without the use of speed sensor [,2,3]. The sensorless speed measurement can be done using direct or indirect methods [4,5]. However, the indirect methods are in general dependent on the machine model parameters. This could lead to inaccuracy of the rotor speed estimation during normal machine operation. Direct sensorless speed measurement method used in this work takes the information present on the stator voltage or current spectrum. Some of the voltage and current harmonics depend on the machine's geometry. The more important geometric factors modifying the harmonic content are: The magnetomotive force spatial distribution, the presence of rotor and stator slots, and the dynamic and static eccentricities. The flux density distribution in the airgap is given as the combination of the permeance and the MMF waves. The resulting flux density distrilbution has harmonics that are moving in relation to stator and rotor frames, therefore inducing corresponding electromotive harmonics in the stator and rotor windings. As a consequence, stator current harmonics appear when the machine is excited by a voltage source. The harmonics frequency, present in the stator current signal, is given by the following equation []: where: fi S P harmonics. nrt supply frequency. slip. number of pole pairs. Qr number of rotor slots. %S stator time nmr n'l n sn slot harmonics. rotor time harmonics. dynamic eccentricity. static eccentricity. Recent works [7,8] have lead to conclude that static eccentricity gives non balance currents in the stator windings, but does not introduces harmonic currents. Dynamic eccentricity has an important harmonic effect on the low order range. Rotor slots have the principal harmonic influence in the high order range $ EEE 243

2 [, n this work, a dynamic machine model of the induction The symmetric matrix [s,,~~] and the Cycle matrix [qkq] machine has been developed to reproduce the principal are uncoupled by the symmetric components stator current harmonics due to rotor and Stator slots, and transformation. The spatial vector transformation has the several speed sensorless techniques have been used to Same effect. Transforming to spatial vectors the prove and develop its performance with this model. Symmetric matrix and the Cycle matrix, we obtain: Additionally the model can be used to produce design specification sets steered towards the improvement of the - induction machine sensorless speed measurements. Squirrel Cage nduction Machine Model The dynamic machine's voltage equations in primitive coordinates are: [ a J][:$ 1, 2 -j=ql 2 2 a 21 (7) 2 where: Where: [Y = [V"! Vh\ VJ ;[v.] = [v. v, VJ = [O' sf k = 3n *[l a a'] [C(kB)]= 0 [i,] = [in\ 4, L]' ; [i, 1 = [L 4, ij q- k = 3n+~ -[ a a2] [~(ke)] The inductance matrices are dependent of the angular rotor position 8. Each inductance parameter in equation (2) was obtained by superposition of the permeance fluctuation due to rotor and stator slots, and rotor eccentricity. The spatial MMF distribution can be also considered in the inductance expression. To simplify. - the f = 3n + a a2] [c(ke)l -2 model we can neglected the non-sinusoida' MMF Using the spatial vector transformation [9] to simplify distribution and the rotor eccentricity. Using only the the differential equation (2), the induction machine principal harmonic due to slots' influence' the model can be written as nine different models in function inductance and resistant matrices in equation (2) have the ofits stator and rotor slots per pole pair: following structure: (3 1 Case 1: q, = 3n; n E N where: P = cosk0 cosk(0-2%) cosk(0-4%) cosk(o-'%) cosk0 cosk(0-'%) cosk(8-*%) cosk(0-4%) cos,40 n this particular case, the rotor and stator slots have not influence in the stator or rotor current performance. This case is not possible in practice, because is needed that rotor and stator slots per pole (q, and q,) have a minimum common multiplier equal one, to avoid reluctance torque problems. Most commercial induction motors have q, = 3n, but the rotor bars number is chosen to make q, # 3n7. 244

3 1.b) qr =3m+1; mczn Case3: q\ =3n+2; nen three. 1.c) 4,. =3m+2; men 3.c) 4,. =3m+2; men Case2: (, =3n+;n~N 2.a) qr = 3m; m E N n some machines the number of the stator slots per pole and phase are not an integer number [lo]. These induction motors show a more important harmonic effect in the stator currents due to the stator slot. 2.b) q,. = 3m+ 1; m E N Numerical Evaluation The principal problem in the evaluation of the previous nine differential equations is concerned with the inductance variations with the angular position 8. To reduce the numerical work, a simpler description can be performed using the flux leakage formulation in spatial vectors. Assuming linearity between fluxes and currents, the following relations can be obtained: 2.c) q, =3m+2; men [:::=[:: ;[]++ Using (1 8) in equation (2): [G = [R[F] + p[[ ~(e)][7']] = [ ~~[r(e)] [ i] + p[ i] Where: [r(e)]:= [~(e)]-' ( 9) The electric torque can be evaluated directly from the coenergy expression as: 245

4 Only the real part of the equation (20) has a physical meaning. Numerical evaluation of expression (20) can be performed quickly, because the inductance matrix is order two. The machine model's canonical form can be expressed as: $1 = [<- [R[~(B)][~] 1 pw "' =-(7:(6)-79 J P 0 = m,,, A Runge Kutta, Predictor Corrector or any other numerical integration method can be used to solve differential equation (21). n order to find the stator, currents at equally spaced points in time, a fix step $ integration method has been used. The fix time step is go5 s=-- -+ (22) QT i'.. 1 fi - ma time [S Fast Fourier Transform and in others time [S OB necessary in the speed estimation methods based on the traditional estimation techniques. Simulations Results The mathematical model of the induction machine has been implemented in MATLAB 4.2 environment. To, show the methodology performance, two different squirrel cage motors have been modeled. The first model o5 has 18 slots in the stator and 20 slots in the rotor. The - other model has 16 slots in the stator and 20 slots in the $ rotor. The following per unit machine parameters has 5 been used in the evaluation of these models: Table : nduction Machine Parameters - Models and 1 Fig. 1.- Electric Torque and angular speed - Model time [S Fig. 2.- Stator current during the steady state loaded condition - Model The stator current signal generated previously was used for measuring the machine's speed during the loaded condition. For this steady state condition the speed can 40-1''''l' 1' '1 1'11 (111'11 /1111~~~~~l~~~~~~' p 1l1' "'y - 246

5 ~ 30 ~ O 5 O Normalized Frequency Cl Normalized Frequency Fig. 4.- Stator current spectrum showing the fundamental and the harmonic components in the signal - Model Fig Zoomed spectrum of the stator current around the speed related harmonics Figure 3 and 4 show the complete stator current spectrum in each model. A zoom of the previous spectrum around the fundamental component f, is shown in figure 5 for model. Finally, figure 6 shows a zoom around the,a,, used to measure the machine s slip. n this case the L/, is approx Hz. Table 1 - Slip estimation results - Model Conclusions A model for squirrel cage AC machines has been proposed, that include the rotor and stator slot harmonic effects. A method frequently used for sensorless speed measurement has been used to validate the proposed model. The results of the simulation show that the stator current spectral content agrees with the ones found on experimental test [4]. References 20 R ll M n nlv 1 [] J. R. Cameron, W. T. Thomson, and A. B. Dow. Vibration and Current Monitoring for Detecting Air- Gar, Eccentricity in Large nduction Motor?, EE Proceedings, 133(3), May [2] M. lshida and K. wata; Steady-State Characteristics of a toraue and Speed Cointrol System of an induction motor utilizing rotor slot harmonics for slip frequency sensing. EEE Trans. Power. Electronics, Volume PE-2, NO. 3, pp July ~ Z z1.z zz Normalized Frequency Fig. 5.- Zoomed spectrum of the stator current around the fundamental component. [3] M. Riese Microcontroller mplementation of Speed Sensorless Field Oriented Control of nduction Machine 7 European Conference on Power Electronics and Applications. Volume 4, pip September Trondheim, Norway. [4] J. A. Restrepo and P. Bowler. Analysis of nduction Machine Slot Harmonics in the TF Domairi. n 247

6 Proceedings of the First nternational Caracas Electrical Machines. CEM 96. Vol. 11, pp Conference on Devices, Circuits and Systems, pp Vigo. Spain. September EEE, December [8] T. J. Sobczyk, P. Vas and C. Tassoni. Models [5] J. Cilia, G. M. Asher, K. J. Bradley and M. for nduction Motors with Air-Gap Asymmetry for Sumner. Control of a Shaft-Sensorless Position Diagnostic PurposeS. nternational Conference on nduction Motor using an Asymmetric Outer-Section Electrical Machines. CEM 96. Vol. 11, pp Vigo. Cage. 7 European Conference on Power Electronics Spain. September and Applications. Volume 4, pp September Trondheim, Norway. [9] J. M. Aller. Simple Matrix Method to ntroduce Spatial Vector Transformation and Oriented [6] Ph. L. Algert. The nature of induction Field Equations in Electric Machine Course?. machines. Znd Edition, Gordon and Breach Publishers, nternational Conference on Electrical Machines. New York, CEM 96. Vol. 111, pp Vigo. Spain. September [7] M. Johr and K. Kluszczynski. Simplified Representation of Doubly-Slotted Varying Air-Gap of [lo] M. P Kostenko and L. M. Piotrovski. Maquinas Electrical Machine. nternational Conference on Electricas -. Editorial MR. Moscu. Segunda Edicion