Torque Ripple Minimization in PM Synchronous Motors Using Tooth Shape Optimization

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1 27 Torque Ripple Minimization in PM Synchronous Motors Using Tooth Shape Optimization Ali Jabbari* Department of Mechanical Engineering, *Corresponding author Mohsen Shakeri Department of Mechanical Engineering, Ali Nabavi Niaki Department of Electrical and Computer Engineering, Abstract: Several approaches are available to reduce pulsating torque components in permanent magnet motors. One of the known techniques is stator tooth shape modifications. This paper presents an integrated algorithm applicable for tooth shape optimization of permanent magnet motors. This method focuses on the reduced basis technique, finite element analysis and Taguchi method and is conducted on tooth shape optimization of a 3-phases radial flux PMSM motor. Keywords: PMSM Motor- Pulsating Torque- Tooth Shape Optimization Reference: Jabbari, A., Shakeri, M. and Nabavi Niaki, A., "Torque Ripple Minimization of PM Synchronous Motors Using Tooth Shape Optimization", Int J Advanced Design and Manufacturing Technology, Vol. 3, No. 2,2010, pp Biographical notes: A. Jabbari is a PhD candidate of mechanical engineering in Babol University of Technology. He received his MSc in manufacturing from the University of Mazandaran, Iran. His current research interests include mechatronic and permanent magnet brushless motor design. M. Shakeri is an associate professor in mechanical engineering department at. He received his BSc from Shiraz University, MSc from Tehran University, and his PhD from Toyohashi University of Technology, all in mechanical engineering. His research interests include fuel cell, micro machining, computer-aided machining, Feature-based CAD, and CAPP. A. Nabavi Niaki is Assistant Professor of electrical and computer engineering at the Babol University of Technology, Babol, Iran. His research interest is in unified power flow controller (UPFC) area.

2 28 1 INTRODUCTION PM motors have been accepted in industrial drives for high performance applications due to their various attractive features such as high torque density and low acoustic noise. Pulsating torque minimization is becoming necessary since the motors are finding application in torque-ripple-sensitive applications such as electric power steering, robotics, etc. Ideally, the desired pulsating torque minimization scheme should not only minimize the torque ripple due to cogging torque, but also minimize the other sources of torque ripple without sacrificing the average torque. Since the cogging torque is induced by the magnetic coupling between the rotor and the stator, it is greatly affected by the configuration of the stator. The stator configuration effects on magnetic field distribution such that the flux leakage and core saturation can be avoided. With the development of powerful computers, more efficient shape optimization approaches are employed. The huge number of design parameters needed to identify tooth shape and also high simulation time, make the application of most shape optimization algorithms impractical. Design of tooth shape was performed using several optimizations methods. In [1], [2], two rectangular slots are placed on tooth surface and the size of slots is calculated. tooth shape modelling by using non-uniform rational B-Splines with some control points as design parameters is demonstrated in [3],[4]. In [5], [6], tooth shape is divided into two regions and then, the proper angle and radii for each region calculated using optimization approach. In [7]-[9], it is proposed that the salient pole be divided into several regions. The main drawback of these optimization approaches, is sheer number of variables needed for shape optimization and also the generality of the process. The presented algorithm in this paper is based on Taguchi method in which several independent variables influence a dependent variable. In which, X represents the basis vector and, b, represents the weighting factor. If the number of boundary points is m, then by applying this method, the number of variables is decreased to n (the number of weighting coefficients). The geometrical features of the basis shapes can be defined in polar system by the r and Ө co-ordinates of their boundary points. These co-ordinates define the basis vectors. All of the basis shapes have to be defined in the same fashion, and therefore, all the resulting basis vectors will have the same dimension. Figure 1 shows an edge defined by a set of points. Fig. 1 Definition of basis vectors. The area of the resulting shape may be greater than limit area; which is the maximum allowable tooth shape area. Therefore, geometry scaling is applied as 1/Σb. Appropriate starting basis shapes are required to employ the algorithm to find optimum tooth shape design. But in some instances in which the design is completely new, a different material, or a different number of slots per poles per phases, may not be possible to begin with a reasonable set of starting basis shapes. In those cases, we may not obtain the optimum tooth shape by solving the shape optimization problem just once. The problem can be solved in multiple levels as shown in figure 2 in which the optimization procedure guides the designer progressively in selecting viable basis shapes. 2 OPTIMIZATION PROCEDURE In this section, optimization process which is based on reduced basis technique is presented. The application of this technique is verified and accepted in metal forming process [10], however, it is suggested for tooth shape optimization in this research and required to be adopted for it. The first step in this approach is to select appropriate basis shape and to build basis vectors with the boundary points of basis shapes and finally to combine them linearly using weighting coefficients as follow Select Appropriate Basis Shapes Identify Design variables Determine Design points FEM Analysis n X = i= 1 b i X i (1) Constraint satisfied? Optimum tooth Fig. 2 Design optimization process.

3 29 3 METHOD EVALUATION The shape optimization of a synchronous permanent magnet motor is presented in this research. The FEM method is used to calculate the pulsating torque and design of experiments points. A cross-section of the motor is shown in figure 3. Specifications of the motor are presented in Table 1. Figure 4 shows three curves selected as initial trial basis shapes. B 1 assumed as polyline, B 2 as arc and B 3 as spline. Finite element simulations of the basis shapes are performed to find the cogging torque for initial analysis as shown in figure 5. The peak value cogging torques of the Basis shapes are , , and N.m, respectively. From the initial analysis, it can be said that the B 2 is more successful than the other two shapes in reducing the cogging torque. Therefore, the contribution of B 2 must be more than the other basis shapes, which must be recognized by the optimizer. Fig.4 The selected basis shapes, B 1 : Arc, B 2 : Polyline, B 3 : Spline. The reduced basis technique is applied to three basis vectors and the number of design variables is decreased to three, which are the weights for each basis vector. By changing these weights, it is possible to obtain various resultant tooth shapes for the optimizer to find the best combination of these weights. Nine experiments points are generated to conduct simulation. All of the resultant tooth shapes are scaled to maintain in a limited area. Simulations are conducted at these experiments points to find the cogging torque and to build the Taguchi models for optimization. B 3 B 2 B 1 Fig.3 The investigated motor section. Table 1 Characteristics of the demonstrated motor Rotor Stator Outer Dia. 32 mm Outer Dia mm Inner Dia mm Inner Dia. 10 mm Pole No. 8 Slot No. 12 Magnet Tooth 42 Arc 1.8 mm Mag. Yoke 1.2 mm Thick. 4 mm Mag. Arc mm Slot Open 1.18 mm Magnet TDKfb9b Thickness Tip 0.92 mm Slot Skew 0 Stack Lamination 6.25 mm M19-0.5mm Fig.5 Finite element analysis of initial model. The Taguchi design of experiments approach is used to study the effects of these three variables simultaneously. Based on known variation of cogging torque with respect to different factors, each factor is considered to have three levels. Therefore, a nine levels orthogonal array has been selected to run the experiments. Table 2 shows the factors and their levels and the layout for the selected array and the value of cogging torque for all nine trials is also presented in Table 3. The experiments are carried out using FEM and the cogging torque obtained from each experiment. Table 2 Initial weighting coefficients Level b 1 b 2 b

4 30 Table 3 The values of peak cogging for all 9 levels FFM experiments Considering cogging torque as cost function, the results are studied. The main effects table, which presents the mean value of cogging torque for each factor at all levels, is shown in Table 4. Analysis this mean values, it is observed that for each weighting factor, its averages in three levels are close. In other words, parameters b 1, b 3 and especially b 2 have very low variances and it mean that the optimization is successful and the values of parameters in three levels are properly selected. This result was confirmed through FEM. If a parameter has high variances, the optimum shape is considered as a new basis shape instead of the shape with high variances. This procedure is repeated according to multi-level design process until all factors have low variances. Table 4 Mean value table of each factor Level of b 1 of b 2 of b Having performed the analysis of results, the predicted optimum result must be verified through carrying out experiments at optimum combination of factors. The optimum values for weighting factor b 1, b 2 and b 3 are 0.75, 0.1, and 0.9, respectively. The cogging torque of the resultant shape is N.m. It is clear that most of the contribution is from B 3 (b 3 =0.9). There is also a significant contribution from B 1 (b 1 =0.75). Figure 6 shows the optimum resultant tooth shape achieved by the proposed motor. The peak value of the cogging torque for basis shapes are , , and N.m, which has been reduced significantly to N.m by this method. Since the resultant cogging torque value is within the allowable limit using ANOVA, it is validated that the resultant shape is optimum. Therefore the procedure is conducted in a single level. 4 COST, FEASIBILITY AND RECOMMENDATIONS There are several cost functions can be applied in optimization procedure. Cogging torque, torque ripple and efficiency are the most important functions. Multi-objective shape optimization can also be used as the cost function. The optimum curves achieved by using the proposed method are smooth, empirical and can be manufactured by computer numerical control (CNC) machines. It is a common practice to use spline curves as basis shapes. In these cases, the number of shape variables reduces to the number of control points. If the designer has enough experience in selecting appropriate basis shapes, the second method is more recommended. Increasing the number of basis shapes also enables the designer to obtain a better tooth shape, but the computation time also increases to build an approximation model. It is important to mention that if expert knowledge is available, then practical basis shapes can be selected and the optimum tooth shape can be obtained in a single level. This design technique can also be used for 3-D tooth shape optimization, but this will also increase the number of FEM simulations. 5 CONCLUSION A tooth shape optimization method for permanent magnet motors is introduced in this paper for pulsating torque minimiztion. The reduced basis method aids in the use of the design of experiments models for optimization. The presented algorithm has been applied on a PM synchronous motor. An optimum tooth shape has been achieved by the implemented algorithms, starting from three basis shapes such as polyline, arc, and spline. The cogging torque has been reduced significantly by applying the optimization method. Optimum Initial Fig. 6 Optimum resultant shape versus initial shape. REFERENCES [1] Koh, C. S., Yoon, H. S., Nam, K. W. and Choi, H. S., "Magnetic Pole Shape Optimization of Permanent Magnet Motor for Reduction of Cogging Torque", IEEE Trans. On Magnetics, Vol. 33, No. 2, Mar 1997, pp [2] Chung, T. K., Kim, S. K. and Hahn, S. Y., "Optimal Pole Shape Design for the Reduction of Cogging Torque

5 31 of Brushless DC Motor Using Evolution Strategy", IEEE Transactions on Magnetics, Vol. 33, No. 2, Mar 1997, pp [3] Chun, J. S., Jung, H. K. and Yoon, J. S., "Shape Optimization of Closed Slot Type Permanent Magnet Motors for Cogging Torque Reduction Using Evolution Strategy", IEEE Transactions On Magnetics, Vol. 33, No. 2, Mar 1997, pp [4] Wang, S., Jeong, S. and Yoon, H., "Continuum Shape Design Sensitivity Analysis of Magnetostatic Field Using Finite Element Method", IEEE Transactions on Magnetics, Vol. 35, No. 3, May 1999, pp [5] Yao, Y. D., Huang, D. R., Wang, J. C., Liou, S. H., Wang, S. J., Ying, T. F. and Chiang, D. Y., "Simulation Study of the Reduction of Cogging Torque in Permanent Magnet Motors", IEEE Transactions on Magnetics, Vol. 33, No. 5, Sep 1997, pp [6] Huang, D. R., Ying, T. F., Wang, S. J., Zhou, C. M., Lin, Y. K., Su, K. W. and Hsu, C. I. G., "Cogging Torque Reduction of a Single-Phase Brushless DC Motor", IEEE Transactions on Magnetics, Vol. 34, No. 4, Jul 1998, pp [7] Hsu, L. Y. and Tsai, M. C., "Tooth Shape Optimization of Brushless Permanent Magnet Motors for Reducing Torque Ripples", Journal of Magnetism and Magnetic Materials, Vol. 282, November 2004, pp [8] Wang, S. and Kang, J., "Shape Optimization of BLDC Motor Using 3-D Finite Element Method", IEEE Transactions on Magnetics, Vol. 36, No. 4, Jul 2000, pp [9] Huang, D. R., Fan, C. Y., Wang, S. J., Pan, H. P., Ying, T. F., Chao, C. M. and Lean, E. G., "A New Type Single-Phase Spindle Motor for HDD and DVD", IEEE Transactions on Magnetics, Vol. 35, No. 2, Mar 1999, pp [10] Thiyagarajan, N. and Grandhi, R. V., "Multi-Level Design Process for 3-D Preform Shape Optimization in Metal Forming", Journal of Material Processing Technology, Vol. 170, 2005, pp

Effect of Cleaning Level on Topology Optimization of Permanent Magnet Synchronous Generator

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