Analytical and numerical computation of the no-load induction motors Dan M. Ionel University of Glasgow, Glasgow, Scotland, UK and Mihai V. Cistelecan Research Institute for Electrical Machines, Bucharest 225 Keywords Electromagnet, Induction motor, Abstract In the paper a comparison between the analytical and numerical method of no-load computation will be made taking into account the initial hypothesis and computational effort face to the concrete obtainable results. It is concluded that the reduced fixed-mesh FEM analysis of the is very suitable for no-load computation because it relates directly to the well known equivalent circuit parameters. The symmetry of the machine allows for the computation only on a small part of the magnetic circuit so that the time of computation can be reduced even when a relatively fine mesh is used. However, for the optimal design of the motor it appears that analytical methods are very suitable owing to the simplicity and accuracy of the design. 1. Introduction It is well known that a sinusoidal mmf in the air gap of the induction motor leads to a non-sinusoidal air gap flux density. Associated harmonics are called saturation harmonics. These harmonics are rotating at the synchronous speed and their amplitude depends on the degree of the iron saturation, that means the iron mmf of the teeth and the yokes. For a long time the air gap flux density shape was characterised by the flattening factor α defined as the ratio of the pole average flux density to the maximum one and some additional curves were used in the magnetic circuit computation to determine the flattening factor as function of the teeth saturation factor[1]. The increase of the flattening factor α from the unsaturated value 2/π to values up to 0.8-0.85 is related to the high degree of the teeth saturation. Later it was observed that the saturation of the yokes implies the decrease of the flattening factor[2], so that there may exist with high degree of saturation having the flattening factor close to the unsaturated value[3]. Another problem is related to the computation of the yoke mmf taking into account the non-uniformity of the field on both radial and tangential directions. In the classical method some additional curves for correction coefficients are used depending on the maximum flux density in the neutral axis of the rotating field. As it was demonstrated[3], this coefficient is depending on the shape of the flux density itself and also on the quality of the electrical sheet. Having COMPEL The International Journal for Computation and Mathematics in Electrical and Electronic Engineering, Vol. 17 No. 1/2/3, 1998, pp. 225-231. MCB University Press, 0332-1649
COMPEL 17,1/2/3 226 these in mind the analytical method should offer the possibility to compute the shape of the air-gap flux density in its interdependence with the teeth and yokes mmfs without above mentioned additional curves. 2. Analytical method for no-load computation At no-load there are no currents in the rotor and, because of the symmetry, only a half pole of the machine will be considered. In the developed analytical method only one additional curve is necessary, the magnetising curve of the magnetic material expressed in a.c. maximum values at rated frequency. The space harmonics of the air gap mmf are neglected and also the tooth pulsation due to the air gap reluctance variation. As it is stated in the literature[2] the saturation affects mainly the fundamental space harmonic because it has the highest amplitude. The teeth mmf is simply computed taking into account the actual shape and cross section of the teeth and the value of the air gap flux density in the front of the actual tooth. For high values of the teeth flux density the longitudinal flux in the slot should be considered[3]. For the computation of the yoke mmf one can have in mind that the flux in the yoke is not constant and the flux density at the point θ depends on the shape of the flux density from 0 to θ, the reference frame being the synchronous one (the air gap flux density is maximum at θ = 0 and vanishes at θ = π/2p). If the yoke flux density is considered only by its tangential component the computation can be simplified without a significant decrease of accuracy of the results. If the variable is the angular co-ordinate θ the functional equation of the air-gap flux density can be derived from the magnetic circuit law applied to a curve which passes across the air gap in the point characterised by the angle θ: where the iron mmf F i (θ) represents the sum of the stator and rotor teeth and yokes mmfs corresponding to the co-ordinate θ. The maximum value of the air gap flux density is noted B g (θ = 0). The air gap flux density equation can be solved iteratively by turning the angular continuous variable into a discrete one and dividing the interval (0, π/2p) into n intervals. In this case the yoke mmf computed at the average diameter D yav of the yoke can be written for the stator or the rotor as: (1) (2) where h y is the height of the yoke, D a is the diameter of the armature at the air gap side, p is the number of the pole pairs and k i is the lamination filling factor. The function f(*) represents the magnetisation curve expressed by H = f(b). The λ m is given by additional notation:
The tooth mmf depends only on the value of the air gap flux density in front of the respective tooth. The yoke mmf depends on all of the values B gk since the distribution of the field in the yokes depends entirely on the air gap flux density. Hence the necessity that the method of determining the yoke mmf should be applicable to any shape of the air gap flux density. The equation (2) represents a simplified Simpson approximation of the integral of the yoke mmf. Whenever the yokes are unsaturated the system of equation derived from (1) turns into n independent equations having B gk as unknowns. If the discrete variable is θ k = kπ/(2pn) the new system of equation can be expressed by: Using the mentioned discretization the functional equation of the air gap flux density becomes a non-linear system of equation, the unknown being the values B gk = B g (θ k ). The system can be solved iteratively starting from a supposed shape of the air gap flux density. The resultant shape, after the convergence is obtained, leads to the value of the no-load induced voltage and to the no-load current. Two successive computational cycles are needed, one for determining of the air gap flux density shape at the given maximum value and the second to determine the maximum value of the air gap flux density at given phase voltage. The resulting shape of the gap flux density can be analysed from the point of view of the saturation harmonics. 3. FEM analysis of no-load induction motor The numerical analysis starts from the stator in-slot currents supposed to be time sinusoidal. The rotor currents are supposed to be negligible. For no-load computation usually a reduced fixed mesh model is used covering one pole of the machine. The assumption is made that all currents and field variables vary sinusoidally in time[5]. This assumption allows for using the complex variables in the two dimensional field equation: in which the over-bars indicate complex quantities. The magnetic vector potential A and the current density J are supposed to have only components normal to the cross section. The local reluctivities ν are used in the same manner as in the analytical method as ratio between the time maximum values of the field quantities. The current density J is given from the supposed phase current, the three phase winding arrangement and the moment of time in which the computation is performed. In the particular case the computation may be repeated for (3) (4) (5) 227
COMPEL 17,1/2/3 228 different values of the currents (time variation) and/or different relative position of the rotor. The mesh is automatically performed (Figure 1) and should be viewed in detail when necessary (Figure 2). Of course the density of the finite elements should be appropriate to local conditions and the mesh should have more elements in the air gap and also in the places where there are big variations in the electric or magnetic properties. After the field problem is solved by known methods the field pattern may be analysed in order to obtain the useful values of the flux densities in the teeth, yokes and air gap and also the corresponding no-load current. If the end winding inductance is supposed or computed by classical methods the phase Figure 1. One-pole mesh Figure 2. Enlarged mesh (detailed)
leakage reactance may be obtained. Because there is not relative motion between stator and rotor meshes the model will lead to the neglecting of the effects produced by the movement of the stator and rotor teeth past each other. This does not necessarily mean that such effects must be altogether neglected in the performance computation. Fourier analysis of the air gap flux distribution will enable the magnitudes of slot order fields to be determined. These may be used in standard ripple torque and high frequency loss expressions in place of field amplitudes obtained in the analytical (classical) manner by multiplying mmf and air gap permeance harmonics[4]. However, it may be claimed that this approach has the advantage face to classical approach in that the flux density waves are determined taking into account the tooth tip saturation. 4. Computation results Both analytical and numerical method for no-load computation were applied to a 3kW, four-poles squirrel cage induction motor in the frame 112. The electrical silicon sheet had 3.6W/kg at 1 T and 50Hz and the magnetisation characteristic had B 25 = 1.58 T[6]. The shaft was considered to be made from non-magnetic material. The rotor and the stator teeth have a constant cross section, that means they are with parallel sides. The number of stator/rotor slots was 36/28. In Figure 3 is presented the field pattern corresponding to a quarter of the cross section of the motor. There is a relation between the shaded colours and the field intensity in the magnetic circuit. One can observe the curvature of the field lines in the neutral axis of the yokes, where the flux density has maximum values and there exists some discharge of the yokes in the tooth basis. Starting from the field computation the values of flux densities in the teeth and yokes were determined and compared with those resulting from analytical method. The comparison is presented in Table I and one can observe that the values are in a close relation. Of course the advantage of the numerical method consists in the fact that it can emphasise the variation of the flux density on the height of the yokes or on the width of the teeth. In the analytical method only the average values of these flux densities could be obtained. 229 Flux density [T] Rotor yoke Stator yoke Rotor tooth Stator tooth Analytical method 1.44 1.75 1.71 1.66 Numerical (FE) method 1.40 1.81 1.73 1.69 Table I. Analytical and numerical (FEM) computation results (comparison) In Figure 4 is represented the radial component of the air gap flux density as it was computed by both methods on a half pole for the relative position of the rotor as in Figure 3. Because of the teeth saturation one can see a strong flattening of the resulting curve (the dotted line). The pulsation of the air gap flux density owing to relative stator/rotor teeth position is pointed out from FEM results. The no-load current computed by the two methods was also in a
COMPEL 17,1/2/3 230 Figure 3. Field pattern Air-gap flux density [T] 1.4 1.2 Analytical results FEA results 1 0.8 0.6 0.4 Figure 4. Radial air gap flux density 0.2 0 0 10 20 30 40 50 60 70 80 90 Angular coordinate [elect. deg] close relation (4.34A for FEM, 4.54A analytical and 4.48A measured on the actual motor). 5. Conclusions In the paper a comparison has been made between the analytical and numerical (FEM) method for investigation of the no-load running of. The no-load current was the main object of comparison but some remarks were made on the iron loss and the flux densities in the air gap and in the main parts of the magnetic circuit. The main conclusion is that the fixed mesh FEM computation offers much the same information about the induction motor no-
load running as analytical method. It is difficult to say that one or another of the methods gives more accurate results for a large range of knowing the widespread magnetic properties of the cold rolled electrical sheets. From the researchers point of view, of course, the FEM method offers some interesting results related to slot and tooth to tooth leakage flux, pulsation of the air gap flux density and other local properties of the main or leakage field. Applying the FEM method for different time varying currents in the slots one can obtain useful information about differential reactance. As the optimisation of the design is concerned, where the computation of the induction motor must be performed for different loads and voltages for a thousand times, the analytical method should be chosen because of its simplicity and speed. References 1. Liwschitz, M., Calcul des machines eletriques, Spes Lausane-Dunod, Paris, 1967, pp. 1-53. 2. Weh, H., Analitische Behandl.des mangetische Kreise von Asynchronmaschinen, AfE, 1961, H1, pp. 27-40. 3. Cistelecan, M. and Onica, P., Computation of the magnetic circuit and establishing of the electrical sheet quality influence on the induction motor performance, ICEM 88, Pisa, Italy, Vol. 1, pp. 237-42. 4. Williamson, S., Induction motor modelling using finite elements, ICEM 95, Paris, Vol. 1, pp. 1-8. 5. IEC Publication 404-8-4, Magnetic materials. Spec. for colled rolled N/O magnetic steel sheet and strip, 1986. Further reading Eastham, J.F., Lai, H.C., Demeter, E., Ionel, D.M. and Postnikov, V.I., Advanced finite element analysis of rotating machines, International Conference OPTIM, Vol. 4, Brasov, Romania, May 1996, pp. 1081-90. 231