Optimal Design of a Permanent Magnet Linear ... - IEEE Xplore

IEEE TRANSACTIONS ON MAGNETICS, VOL. 48, NO. 2, FEBRUARY 2012

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Optimal Design of a Permanent Magnet Linear Synchronous Motor With Low Cogging Force Chang-Chou Hwang1 , Ping-Lun Li2 , and Cheng-Tsung Liu3 Department of Electrical Engineering, Feng Chia University, Taichung 407, Taiwan Electrical and Communications Engineering, Feng Chia University, Taichung 407, Taiwan Department of Electrical Engineering, National Sun Yat-Sen University, Kaohsiung 804, Taiwan In this paper, a technique for reducing the cogging force in slotted permanent magnet linear synchronous motors (PMLSMs) is presented. A two-step minimization process that depends on the slotting and end effects is proposed. Taguchi’s parameter method, coupled with 2-D and 3-D finite element analysis (FEA), is employed to minimize the cogging force. Index Terms—Cogging force, finite element analysis (FEA), permanent Magnet linear synchronous motor (PMLSM), Taguchi’s parameter method.

I. INTRODUCTION

P

ERMANENT magnet linear synchronous motors (PMLSMs) are becoming an alternative solution to rotational motors in many industrial applications, such as transportation and factory automation systems and the packaging industry [1]. One of the main reasons is they can provide linear motion without using any mechanical interface. However, a severe drawback in slotted PMLSMs is cogging force or detent force, which can increase the thrust ripple and may introduce a disturbance in positioning precision. The cogging force in a slotted PMLSM is caused by slotting and end effects. Slotting effect is due to the interaction between permanent magnets (PMs) and the armature slotted core with the wavelength of one slot pitch. In addition, end effect arises from the interaction between PMs and the finite length of armature core with the wavelength of one pole pitch. Several techniques have been proposed to reduce cogging force by many researchers, including the proper pole and slot combination selection [2], [3], magnet length adjustment [4], [5], the asymmetric arrangement of magnets [4], [5], magnets or slots skewing [1], [4]–[6], semi-closed slot design [4], [5], air-gap length variation [4], [5], suitable armature length selection [7], proper armature outlet edge shape design [8], proper tooth edge chamfering [9], auxiliary poles (APs) utilizing [10], and control strategies [4], [11]. This paper is concerned with only the design aspect to minimize the cogging force and maximize the thrust force. Design techniques for reducing cogging force in slot and end regions are proposed, and Taguchi’s parameter method coupled with FEA is conducted. Since the PMLSM is an inherent 3-D geometry and requires a considerable amount of engineering labor and computer resources to set up the FEA model and analyze the results,

Manuscript received June 24, 2011; accepted October 08, 2011. Date of current version January 25, 2012. Corresponding author: C.-C. Hwang (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMAG.2011.2172578

a two-step minimization process, which depends on the slot or end region, is used in order to alleviate some of the problems. In the first step, a technique for optimal design of slot region is considered only, and the Taguchi’s parameter method coupled with 2-D FEA is conducted, while in the second step, regions in both slots and two ends, which are equipped with APs and aluminum alloys (ALs), are considered, and Taguchi’s parameter method coupled with 3-D FEA is employed to minimize the cogging force [11]–[13]. The techniques have been confirmed by experimental results obtained with a laboratory prototype. II. DESIGN CONSIDERATIONS As mentioned in the previous section, the cogging force in the slotted PMLSM is caused by the slotting and end effects. Therefore, the reduction of cogging force is achieved by a two-step minimization process depending on the slotting or end effect. In the first step only the slotting effect is considered, and in the second step both the slotting and end effects are concerned. Theoretically, the use of semi-closed slots should reduce the cogging force. The price paid for using this design is the increasing difficulty in winding manufacturing and cost. Therefore, open slot is used in this investigation. However, an open slot leads to a higher resultant cogging force than a semi-closed slot. To mitigate the open slot effect, proper selection of slot and magnet pole number combination is number one of the simplest, most effective, and common techniques. and significantly affects the cogging The combination of force. To illustrate this factor, consider the number of cogging for one pole translation, as given in (1) [2], [3] periods (1) where LCM(x, y) is the least common multiple of its argument. It has been shown that the smaller the factor , the larger the cogging force [2], [3]. Therefore, increasing is the most significant method to minimize the cogging force. Table I comand pares the machine performance characteristics for four combinations of the PMSLMs, with the same design data, using 2-D FEA software FLUX2D [14]. It is seen that the configuration of 8-pole/9-slot (8P/9S) generates a lower cogging

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TABLE I PERFORMANCE COMPARISON

F

: Average thrust force; F

TABLE III L-9 OA AND MACHINE PERFORMANCE

: peak-to-peak values of cogging force.

Fig. 1. Initial design of motor. (a) Front view. (b) Chamfered tooth.

TABLE II MACHINE PARAMETERS Fig. 2. Main factor effects on (a) thrust force and (b) cogging force.

TABLE IV EFFECTS OF VARIOUS FACTORS ON MOTOR PERFORMANCE INDICES

force than any other configurations. Therefore, it is a good candidate for use in this application. Fig. 1 shows the initial design of a single-sided, 8P/9S PMLSM. The open slotted mover consists of a stack of thin laminated iron core, which carries doubled-layer non-overlapping windings. The stator is constituted by a yoke mounted with PMs. The machine parameters and simulation results are given in Table II. It is seen that the peak-to-peak value of the is 57.5 N, which is about 38.68% of the cogging force . Taguchi’s parameter method is average thrust force utilized to minimize the cogging force in the next section. III. FIRST-STEP OPTIMIZATION Taguchi’s parameter method provides designers with a systematic and efficient approach for conducting numerical experiments to determine near optimum settings of design parameters. In this technique, an orthogonal array (OA) that depends on the number of factors and levels included is used to study the parameter space [11]–[13]. Fig. 1(b) and Table III show the selected four factors A, B, C, and D affecting the cogging force due to the slotting effect. A is the width of the tooth in mm, B is the width of the chamfered tooth edge in mm, C is the height of the magnet in mm, and D is the width of the magnet in mm. Each factor has three levels, the performance of the machine in

the matrix experiments can then be calculated using 2-D FEA, as shown in Table III. The influence of each factor on the average thrust force and cogging force are as shown in Fig. 2. It is seen that factor-level combinations (A3, B1, C3, and D3) contribute to the maximization of average thrust force, and factor-level combinations (A1, B2, C1, and D1) contribute to the minimization of cogging force. There is no element selected to constitute the elements of the optimum design for minimum cogging force and maximum average thrust force, as all factors are used to regulate the values of cogging force and average thrust force. Therefore, before selecting the optimal setting parameters for robust design, it is necessary to determine the relative importance of various factors on machine performance by conducting analysis of variance (ANOV). To conduct ANOV, the sum of the squares is calculated first. For example, the factor A sum of squares can be calculated as (2) The , , and can be obtained in the same manner. These results are summarized in Table IV.

HWANG et al.: OPTIMAL DESIGN OF A PERMANENT MAGNET LINEAR SYNCHRONOUS MOTOR WITH LOW COGGING FORCE

TABLE V COMPARISON RESULTS

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TABLE VI L-9 OA AND MACHINE PERFORMANCE

Fig. 4. Main factor effects on (a) average thrust force and (b) cogging force.

TABLE VII EFFECTS OF VARIOUS FACTORS ON MOTOR PERFORMANCE INDICES

Fig. 3. PMLSM topology showing geometrical definitions. (a) Front view. (b) Top view.

It is seen in Fig. 2 and Table IV that, the best combination and maximum is of design parameters for minimum determined as (A3, B1, C1, and D1). The performance of the optimized machine was obtained by using 2-D FEA again. Table V compares the data of the machine between the initial design and the Taguchi parameter method. It is seen that the cogging force reduces from the initial design of 57.5 N to the Taguchi parameter method of 37 N, and consequently the average thrust force is slightly reduced. However, it can be seen that, the ratio between cogging force and average thrust force reduces from the initial design of 38.68% to the Taguchi parameter method of 27.52%. IV. SECOND-STEP OPTIMIZATION To minimize the component of cogging force caused by the end effect, two APs are fixed on both sides of the mover, as shown in Fig. 3 [10]. The AP contains an iron tooth, and an AL is connected between the mover and the AP. As stated in the introduction, skewing is the most effective technique to reduce cogging force. In this paper, both the two APs and the magnets are skewed in the opposite directions, as in Fig. 3(b). In the initial design, the width of one side of an AP is designed as equal , then the optimal lengths of skew on the to 1/4 pole pitch other side are determined for minimum cogging force, using the Taguchi parameter method. As before, four factors, E, F, G, and H, which correspond to the four design variables, are each chosen with three levels, as

shown in Fig. 3. To reduce the number of independent variables, the geometric parameters E, F, and H are normalized by dividing . Here E is the magnet skew all dimensions by the pole pitch length (levels 0, 0.25, and 0.5), F is width of an AL (levels 1, 0.5, and 0.25), G is the distance between the bottom of an AP and the bottom of the slot tooth in mm (levels 0, 3, and 6), and H is the width of an AP, which skews the AP (levels 1, 0.25, and 0.5). The standard OA L-9 used for matrix numerical experiments is as shown in Table VI. The values of the average thrust force and cogging force for each experiment are conducted using 3-D FEA software FLUX3D, as shown in Table VI [14]. The influence of each factor on average thrust force and cogging force are as shown in Fig. 4. As stated in the previous section, it is necessary to determine the relative importance of various factors on machine performance by conducting analysis of variance (ANOV) during the course of optimization using the Taguchi parameter method. To conduct ANOV, the sum of , , , squares is first calculated. The results of and are summarized in Table VII. It is noted in Fig. 4 that, factor-level combinations (E1, F2, G1, and H3) contribute to the maximization of average thrust force, and factor-level combinations (E3, F2, G2, and H3) contribute to minimization of cogging force. Therefore, elements

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Fig. 5. Flux density distribution within the optimized design of the PMLSM.

TABLE VIII COMPARISON RESULTS Fig. 7. Comparison of average thrust forces.

design optimization for the minimization the cogging force and the maximization of the average thrust force of a PMLSM. It has been shown that the technique presented in this paper is effective for obtaining design parameters with low levels of cogging force for PMLSMs. As shown in this paper, the selection of a suitable width of an AL and the skew length of an AP are very effective in minimizing the cogging force and maximizing of the average thrust force of PMLSMs. REFERENCES

Fig. 6. Comparison of cogging forces.

F2 and H3 are immediately selected to constitute the elements of the optimum design for maximum average thrust force and minimum cogging force. On the other side, factors E and G are used to regulate the values of average thrust force and cogging force. It is seen in Table VII that, for factor E, the magnet skew length (76.35%) to (37.4%). Factor G, has larger effect on the distance between the bottom of an AP and the bottom of the (28.06%) to (2.08%). slot tooth, has larger effect on Therefore, the best combination of design parameters for minand maximum is determined as (E1, F2, G2, imum and H3). Fig. 5 shows the flux density distribution within the optimized machine, and the performance of the optimized machine was obtained by reusing 3-D FEA. Table VIII compares the data of the machine between the initial, the first-step optimized, the second-step optimized designs, and measurement. Figs. 6 and 7 compare the cogging force and average thrust force, respectively. It can be seen that the cogging force reduces from the initial design of 57.5 N to the second-step optimized design of 17.97 N, and that the reduction of cogging force results in a small reduction of thrust force. Also, it is seen that the values of predicted and measured of cogging force and thrust force are good agreement with each other as shown in Figs. 6 and 7. V. CONCLUSION This paper applied two-step optimization using the Taguchi parameter method, coupled with 2-D and 3-D FEA, in order to

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