Finite Element Modeling and Performance Analysis of a Permanent Magnet Synchronous Machine Saadoun 1,. Guersi,A. Ouari 3 1- Université Badji Mohtar,, Faculté des Sciences de l Ingénieur, saadoun_a@yahoo.fr - Université Badji Mohtar,, Faculté des Sciences de l Ingénieur, guersi54@yahoo.fr 3- Université Badji Mohtar,, Faculté des Sciences de l Ingénieur, ouari_a@yahoo.fr Abstract: This paper is concerned with the validation of finite element model of a three phase four pole permanent magnet synchronous machine and the development of some post processing functions needed to analyse the numerical solution of the Poisson s equation governing the magnetic field problem. Due to the lac of field control in permanent magnet synchronous machines, fewer operating characteristics are available for performance assessment in comparison with wound field synchronous machines. To mae up for this limitation, alternative approaches are required to extract the missing information from the numerical solution. In this respect, multiples simulations of different operating conditions of the devices are required to derive the operating characteristics of the device. Unfortunately some vital processing functions such as the harmonic analysis of the air gap field distribution or the computation of the time varying stator voltage are not available in the freeware pacage FEMM which has been used in the present wor. Consequently the needed post processing functions required for the prediction of the synchronous performance of the device have been developed. Key words: Poisson s equation, finite element modelling, permanent magnet synchronous machine, harmonic analysis, emf computation. OMECATURE Axy (, ) ΩΓ ( ) ν x, y J J m g l τ fdd.. magnetic potential vector solution domain bounded by the contour Γ ν reluctivity of the media in the x, y directions conductors current density magnet equivalent current density number of slots per pole and per phase effective length of a coil side pole pitch 1. Introduction flux density distribution The evaluation of the steady state performance of synchronous machine is based on the analysis of the operating characteristics which describe the relationship between the main electric quantities of the machine. Such characteristics describe not only any operating condition but also provide a mean to compute some vital parameters of the machine such as the different reactances needed to build a realistic equivalent circuit model for performance prediction. Unlie conventional synchronous machine, permanent magnet machines do not have a controlled field excitation; as a result fewer steady state operating characteristics can be obtained. either the V-curves nor the zero power factor test can be carried out on permanent magnet synchronous machine; more importantly, the slip test, commonly used to determine the direct and quadrature axis synchronous reactance, is not feasible because of the absence of the rotor field winding. Therefore, the designer has to rely only on the finite element solution of the Poisson s equation governing the magnetic field problem to overcome this lac of information. The numerical solution gives basically the flux density distribution, fdd, along the air gap. As general rule, only the fundamental of the air gap flux distribution is relevant in the calculation of synchronous performance of the machine in the first instance. However, the high order flux density harmonics, inevitably present in the air gap field waveform, affect adversely the performance because of the side effects they bring about.. Field analysis of the permanent magnet machine.1 Finite element modelling The pre-processor program of the finite element pacage PED and the FEMM v 4. have been used to implement the two dimensional finite element model of the permanent machine. The domain of
solution Ω, bounded by the contourγ, of the magneto static field equation A A ( ν y ) + ( ν x ) = ( J + Jm) (1) x x y y expressed in variational form, as an energy functional F = + A A ν y νx JA dx dy () x y ΩΓ ( ) has been restricted to a pole pitch of the machine as shown in figure 1. Figure : Flux contours from the FEM FMM pacage over one pole pitch Figure 1: Flux contours and air gap fdd from the FEM PED pacage over one pole pitch Retrieval of the solution of the field problem is made through the post-processing functions of the pacage. It consists mainly of the graphic display of the flux density contours, the flux density distribution waveform along the air gap as shown in figures from 1 to 3 and finally the calculation of the electromagnetic torque. Figure 3: Flux density distribution along the air gap: - top: FMM output - bottom: reconstructed waveform The air gap fdd is also given as a set of tabulated values at equally spaced intervals. Because the Fourier analysis of the air gap fdd is not incorporated in the pacage, this table is subsequently used to perform this vital function as far as performance evaluation is concerned.
.. Fourier analysis of the air gap fdd The performance prediction and the analysis of electric machines depend to a large extent on the accurate evaluation of the air gap field. In the case of permanent magnet machine, such an analysis becomes more complicated because the air gap flux density is often difficult to predict [3, 5]. The amplitude of the fundamental air gap field waveform is related to the woring point of the magnet and shows in the first instance whether the optimisation of the magnet dimensions have been achieved. Moreover, the directions of the direct and quadrature axis are not quite obvious since they depend not only on the armature current distribution but also on some design parameters of the machine as well, such as the pole pitch to pole arc ratio. The departure of the direction of the resultant air gap field with respect to the no load field axis, i.e. the load angle, gives a measure of the stiffness of the magnet to withstand the demagnetising effect of the stator current. For these reasons, a FORTRA program has been devised to perform the Fourier analysis of a discrete sampled function [4]. A discrete function F( θ p ), defined periodically over an interval by a set of equally spaced points θ p = p, p = 0,1,..., 1 (3) procedure is that the Fourier coefficient will be evaluated exactly because the sine and cosine functions involved in their computation remains orthogonal if the discrete samples are evenly spaced. It is obvious that if the number of sample is increased, the computed Fourier coefficient will converge to the true coefficients of the original signal; this explains the slight difference between the curves. 3. Analysis of the FEM solution From the previous figures, it can be seen that, although the flux line contours obtained using the two different fem pacages are quite similar, the air gap flux density distributions depicted in figure 1 and 3 present some differences which cannot be ignored if accurate performance prediction is sought. The differences in the air gap fdd waveform can be accounted only from the accuracy of the computed normal component of the magnetic flux in the air gap region. To single out the best solution, not only a comparison with the measured flux density distribution is required but also some post processing has to be devised. The harmonic analysis program can be modified slightly to implement a filtering function of the computed fdd. Figure 4 shows the measured fdd along the air gap. The bottom waveform has been obtained after filtering the original signal shown by the top trace. where is the number of sample, can be written as A π π F A cos B 1 0 ( θ) = + ( θ + sin θ) = 1 π + A cos θ (4) where 1 π A = F( θ )cos θ, = 0,1,..., p p p= 0 1 π B = F( θ )sin θ, = 1,.., 1 (5) p p p= 0 = + = H A B, ϕ atan B from which the amplitude and the phase of the th flux density harmonic are found. To chec the validity of the program, the fdd given by the pacage has to be regenerated according to equation (4). For the sae of comparison, the original curve obtained from the pacage and the reconstructed are displayed together in figure 3. One remarable property of such a numerical A Figure 4: Measured low pass filtered air gap flux density distribution at no load. 4. Computation of the induced voltage in the armature windings Comparison of the fdd waveforms shown in figure 4, 1 and 3 highlights the effect of the high order harmonics. Although the shape of these signals is much closer, after filtering, they are still different because the finite element solution of the magnetostatic field problem does not reveals the
effect of the time harmonics in the actual air gap fdd which prevail during the rotation of the machine. Hence, the computation of the time varying induced e.m.f in the stator windings provides the best way to chec the accuracy of the numerical solution. The computation of the phase and line voltages relies first of all on the harmonic analysis of the air gap flux density, the dimension of the machine and finally on the parameters of the windings [6]. The induced voltage within a particular conductor of the winding is derived from the flux-cutting rule as e () t = B ( θ ) lv (6) c r Finally, the time varying phase voltage can be found as γ e () t = E sin( θ + ϕ + ( g 1) ) (11) p p = 1 Figure 5 provides a direct illustration of the usefulness of the e.m.f computation post processing function since it enables a quic assessment of the validity of the modelling procedure of the machine. Once the reliability of the model has been established, the computation of the other pertinent parameters of the machine can be carried out confidently. where B ( θ) = B sin( θ + ϕ ) (7) r m = 1 represents the harmonic resolution of the normal component of the air gap field using the Fourier analysis program described in the previous section. The induced e.m.f ec () t e () t = E sin( θ + ϕ ) (8) c m = 1 E m is made up from the individual contribution of due to each spatial flux density harmonic. Written in terms of the average flux per pole and the dimensions of the machine E m [6] can be expressed as π Em = Φ f = τ lbm f1 (9) In a full pitch winding, the distribution of the coils belonging to the same phase belt σ introduces not only a phase shift γ between the induced e.m.f of two consecutive coil sides but also brings about a reduction the phase voltage. The latter is no longer equal to the arithmetic sum of the induced voltage in the coil sides as for concentrated windings but equal to the modulus of the geometric sum of the e.m.f induced in each coil side. This reduction is taen into account by the introduction of the distribution factor Figure 5: Measured and computed line voltage from the fdd shown in figure 1. 5. Results and discussion K dn nσ sin =, σ = gγ (10) nγ gsin Over the last two decades, the advent of many dedicated finite element pacages for electrical machine design has eased to a large extent the computation air gap f.d.d of the normal component of the magnetic flux density. The fundamental is needed to evaluate all relevant machine parameters
such as the phase and line voltage, electromagnetic torque, machine reactance, etc. However, for most of these pacages, the procedure design relies on the direct solution of the elliptic partial equation governing the magnetic field problem. Such a field oriented approach can hide some practical aspects of the procedure design can be misleading, for instance, in the present case, the results of both pacage seems different but in fact the finite element solution need only a further processing. By filtering the computed f.d.d shown in figures 1 and 3, the filtered f.d.d waveform is now closer in shape to the measured air gap fdd as illustrated in figures 4. 6. Conclusion In the present wor, it has been shown that the harmonic analysis the computed air gap is a vital aspect of the procedure design as far as accuracy of the finite element solution is concerned. Moreover, if the filtering post processing function can be helpful to appreciate the side effects of the high order harmonics, the e.m.f post processing function provides a simple and efficient mean to chec the reliability of the whole design. References [1] Meeer D., Finite element Method Magnetics v4.0, User s Manual, January 006. [] Biddlecombe C.S., Diserens.J., Riley C.P., Simin J., PED User s Guide, R-81-089, Rutherford Appleton aboratory Aug 1989. [3] Binns, K.J, Riley C.P., Wong, T.M., The efficient evaluation of torque and field gradient in permanent magnet with small air gap, IEEE Transactions on Magnetics, Vol. MAG. 1, 6, pp 435-438, ov. 1985. [4] Hamming R.W., Introduction to applied numerical analysis, McGraw-Hill, ew Yor, 1971. [5] Binns, K.J, Wong, T.M., Analysis and performance of high field permanent magnet synchronous machine, IEE Proc., Vol. 131, Pt B 6, pp 5-58, ov. 1984. [6] Kosteno M., Piotrosi.: Electric machines: Alternating current machines, Vol., Mir Publishers, Moscow, 1963. Acnowledgement We wish to than: David Meeer for providing indly the freeware FEMM v4. and his constant upgrade of the software, which now competes with commercial pacages in terms of quality. Helmut Michels for providing indly the Graphic Plotting ibrary Dislin v8.4 which have used in this wor. The Sybase Inc. for its free Integrated Development Environment built around the Watcom Fortran compiler.