THE advancement in the manufacturing of permanent magnets

IEEE TRANSACTIONS ON MAGNETICS, VOL. 43, NO. 8, AUGUST 2007 3435 Design Consideration to Reduce Cogging Torque in Axial Flux Permanent-Magnet Machines Delvis Anibal González, Juan Antonio Tapia, and Alvaro Letelier Bettancourt Electrical Engineering Department, University of Concepcion, Concepción Chile In designing new topologies for permanent-magnet machines based on rare earth magnets, it is necessary to diminish the undesired cogging torque. This paper presents a 3-D finite-element analysis to evaluate the effect of magnet shape and stator displacement on cogging torque reduction, for axial flux machines. It analyzes the final electromagnetic torque for the proposed configurations. Finally, it presents the resultant cogging torque waveform for a 5.0 kw prototype, based on our optimization techniques. Index Terms Axial flux machine, cogging torque, permanent-magnet machine. I. INTRODUCTION TABLE I MACHINE DIMENSIONS AND MAGNETIC CHARACTERISTICS THE advancement in the manufacturing of permanent magnets makes them available in different grades, which include a wide range of properties and application requirements. The rare earth magnets, like neodymium-iron-boron (NdFeB) and samarium-cobalt (SmCo), produced by sintering or bonding techniques allow the obtainment of some interesting magnet shapes. The introduction of rare earth magnets into the market with low prices and diverse shapes has been stimulating the design optimization of permanent-magnet machines for industrial applications. The new age of permanent-magnet machines is characterized by high flux density in the air gap. These are very attractive machines due to their high power density. However, the permanent-magnet machine presents two undesired pulsating torque components. The first is produced from the harmonic content of the voltage and current waveform. The other, called the cogging torque, is caused by the interaction between the magnet flux and the stator anisotropy, due to the slotting. Therefore, the slotless machines are not affected by cogging torques. In this paper, the cogging torque reduction is studied, particularly, by the influence of the magnet shape. The analyzed configuration was an axial flux machine (AFM) with two rotors and double central stator. These topologies allow for the introduction of a displacement between both stator sides, reducing the resulting cogging torque. The study is based on finite-element analysis (FEA), using 3-D simulation. In order to obtain a comparative result from the studied topologies, some considerations were taken into account. In all studied cases the machine s dimensions and magnetic characteristics were the same (see Table I). For the different magnet shapes analyzed the magnet/pole ratio areas were around 5/6, in order to maintain the electromagnetic torque development. For the simulations the elements numbers in the volume s region were the same. All of the presented analyses were compared with the cogging torque in the trapezoidal magnet, not a skewed machine. Its maximum point was considered the base value. II. COGGING TORQUE IN PM MACHINE The cogging torque is caused by the variation of the magnetic energy stored in the air gap, due to the permanentmagnet flux with the angular position of the rotor. In more detail, it appears that due to the interaction between the rotor magnetic flux and the variations of the stator s reluctance,by the slotting The number of periods of the cogging torque waveform in the number slots can be given in [1], by (1) (2) Digital Object Identifier 10.1109/TMAG.2007.899349 Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. where and are the stator slot and the pole number, respectively, and the denominator is the highest common dividend between and. 0018-9464/$25.00 2007 IEEE

3436 IEEE TRANSACTIONS ON MAGNETICS, VOL. 43, NO. 8, AUGUST 2007 Fig. 1. Magnet with a rounded border. Fig. 3. Magnet with a rounded face (concave). magnet moves toward the tooth, the reluctance varies slowly and the maximum cogging torque decreases. With the augment of the length of the rounded portion, the cogging torque is reduced. The maximum reduction is to 63.0%, obtained for. Notice that the waveform is enhanced in the rounded border configuration. Fig. 2. Cogging torque for a rounded border magnet. III. MAGNET SHAPE INFLUENCES As was explained in the previous section, the cogging torque depends on two parameters. The first was flux imposed in the air gap by the magnet. In order to increase the power density, it is necessary to get a high air-gap flux. Therefore, it is required to diminish the effect of the other parameter, the reluctance, to reduce the maximum cogging torque. In the AFM, the basic shape of the magnet and teeth are trapezoidal. In this case, the interaction between the complete magnet border and a tooth occurs at the same time, producing a maximum cogging torque. This effect is reduced if a whole magnet edge does not move in front of a tooth at the same time. It can be used to modify the magnet or the tooth shape. Four configurations of magnets were studied using FEA, and the results are explained below. With the purpose of minimal diminishment of the electromagnetic torque, only a portion of the magnet, equivalent to one slot pitch, was modified. For the proposed machine topology the slot pitches are 7.5. A. Magnet With a Rounded Border The first configuration analyzed was the magnet with a rounded border, depicted in Fig. 1. The parameter is the length of the rounded portion. The cogging torque waveform was presented for another two intermediate values of as a percent of the slot width, shown in Fig. 2. If the border of magnet was rounded the interaction with the teeth increased gradually, not instantaneously. While the B. Magnet With a Rounded Face (Concave) Another way to get a gradual interaction between the teeth and the magnet is with rounded magnet faces. In this case, the magnet has a concave rounded face (See Fig. 3). A suitable choice between both the borders of the magnets and its corresponding teeth is important to obtain a minimal reluctance variation. The magnet force with the coming tooth and of the salient tooth should be of the same magnitude, but opposite directions, to get the total cogging torque reduction. The parameter, in Fig. 3, is the length of the rounded portion of the magnet. The cogging torque waveform is depicted in Fig. 4. The magnet with a rounded face introduces a significant reduction of maximum cogging torque value. In the proposed topology and for, it is possible to diminish the maximum cogging torque magnitude by up to 40%. C. Magnet With a Rounded Face (Convex) The magnet with a rounded face, with a convex arc on one edge, was also analyzed. Similar to the previous case, is the length of the rounded portion of the magnet and its maximum value corresponding to one slot width. In Fig. 5, the magnet is depicted with a convex shape. The resulting cogging torque waveforms for three different values and trapezoidal magnet is presented in Fig. 6. The convex magnet introduces a displacement to the left for the maximum cogging torque value with respect to a trapezoidal magnet waveform. It is due to the displacement of magnet area with respect to the angular position. In the convex magnet, the reduction of the cogging torque capability is similar for a concave shape. The maximum cogging torque reduction, for, is 60% compared with a trapezoidal magnet. So, other factors, like an electromagnetic torque

GONZÁLEZ et al.: DESIGN CONSIDERATION TO REDUCE COGGING TORQUE IN AXIAL FLUX PERMANENT-MAGNET MACHINES 3437 Fig. 4. Cogging torque with a rounded face magnet (concave). Fig. 6. Cogging torque with a rounded face magnet (convex). Fig. 5. Magnet with a rounded face (convex). Fig. 7. Skewed magnet. or a magnet price, should be more important when choosing between both configurations. D. Skewed Magnet The skewed magnet or teeth was a technique presented in previous work. Both of them produce the same result in cogging torque reduction. This paper shows the effect of the skewed magnet in the studied machines. Fig. 7 depicts the magnet configuration and the stator teeth. The parameter is the inclination angle of both magnet edges. The minimal cogging torque is obtained when is equal to the period of the cogging torque waveform [1], but it introduces an important reduction to the electromagnetic torque as well The corresponding cogging torque waveforms, obtained for three different values of as a percent of slot pitch and for an unskewed magnet, are compared in Fig. 8. (3) The skewed magnets generate a displacement of the cogging torque waveform in the same way, referring to Fig. 8. On the other hand, the skewed magnet to one slot pitch produces a 61% cogging torque reduction, in the analyzed topology. IV. STATOR DISPLACEMENT The AFM studied is composed of two lateral rotors and a dual central stator. Therefore, the interaction of each rotor magnet with the corresponding stator teeth produces a cogging torque waveform in both sides of the machine. The sum of these is the total cogging torque development of the machine. If a displacement between both stator sides is introduced, each cogging torque keeps their waveform, but appears to phase out in time. Consequently, the total cogging torque is reduced. A displacement factor can be defined now by the relationship between the displacement angle and the slot pitch (4)

3438 IEEE TRANSACTIONS ON MAGNETICS, VOL. 43, NO. 8, AUGUST 2007 Fig. 8. Cogging torque with a skewed magnet. Fig. 10. Maximum torque value for the analyzed magnet configurations. Fig. 9. Cogging torque peaks for the variations of K. Then, the factor was varied from 0.0 to 1.0 and the peak magnitudes of cogging torque achieved is shown in Fig. 9. The smallest value obtained was 0.38 pu, from. It is important to know that the displacement of stators also reduces the maximum electromagnetic torque. This effect will be explained in the next section. V. ELECTROMAGNETIC TORQUE The torque development in the axial flux machines depends on the multiplication of the electromagnetic force and the radio, However, the vector can be calculated as (5) (6) where is the rms current in the coil, is the flux density imposed by the magnet, and the is the portion of the coil that has direct interaction with the magnet flux. Therefore, if the magnet shape is modified to reduce the total amount of contact with the teeth borders to diminish the cogging torque, at the same time the interaction between magnet flux and is also reduced. So, the maximum electromagnetic torque should be decreased. In Fig. 10, the maximum torque value for all of the analyzed magnet configurations is depicted. From the finite-element analysis, the maximal torque reduction is produced by a rounded face magnet with a convex shape. The electromagnetic torque drops about 7.6%. Nevertheless, in the concave magnet the reduction is 5.5%. The rounded magnet edge produces a torque value 4% less than the rate value, while the skew of magnet only reduces the maximum value of the electromagnetic torque to 3.3%, for an inclination angle of 7.5. Introducing a displacement between both stator sides affects the maximum electromagnetic torque also. With no displacement between the stator sides, each set of stator winding MMF and rotor magnet are interacting at the same time, generating 50% of total torque. But, if one stator side is displaced to reduce the cogging torque, both torque components appear in different moments. Consequently, the total torque produced by the machine is reduced. Fig. 11 shows the resultant torque as the displacement factor increases. As was depicted in Fig. 9, a minimum cogging torque is produced from 0.25 to 0.75 of values. For this range, the maximum torque reduction is only 2.3%. VI. PROTOTYPE Based on the obtained results, a 5.0 kw, 8-pole prototype was built. In order to reduce the cogging torque, two studied techniques were employed: skewing the magnet and displacement of the stator side. The presented prototype has the capability to extend the speed range over the rate value. The high-speed operation needs to control the air-gap flux, keeping the voltage at 1.0 pu. Using the

GONZÁLEZ et al.: DESIGN CONSIDERATION TO REDUCE COGGING TORQUE IN AXIAL FLUX PERMANENT-MAGNET MACHINES 3439 Fig. 13. Measurement and simulated cogging torque for the presented prototype. Fig. 11. Resulting torque reduction by displacement between the stator sides. Fig. 12. AFM prototype: (a) rotor with skewed magnet and (b) trapezoidal stator teeth. Fig. 14. Axial flux permanent-magnet prototype. vector control technique a negative flux is injected in -axes, reducing the final air-gap flux [4]. However, the magnet imposes a high reluctance path for the negative flux. So, the rotor pole had another modification. A portion of the magnet was substituted by an iron piece [5] (see Fig. 12). Finally, the rotor magnet was skewed to 7.5. The presented prototype had a displacement between both stator sides. The displaced angle was 3.5, corresponding to. The measurements of the cogging torque were compared with the FEA simulations results and both waveforms are shown in Fig. 13 [6] [8]. The maximum values of the cogging torque measurement and FEA results were 4.75 and 4.0 Nm, respectively. In the simulation of prototype with not skewed magnet and without stator displacement, the peak value of cogging torque was 18.2 Nm. Therefore, the final reduction of cogging torque was to 73.9%. VII. CONCLUSION In this paper, the influence of the magnet shape and displacement of the stator sides to reduce the cogging torque was investigated. The effect of the proposed techniques in electromagnetic torque, for axial flux machine, was studied also. From the obtained result, the magnet skew and the stator side displacement are excellent techniques to reduce the cogging torque, keeping the electromagnetic torque at a high value. The combination of these can produce a minimal cogging torque magnitude. The concave and convex shape modification of the magnet face generates the same cogging torque reduction, about 60%. But the concave magnets remain in the electromagnetic torque of 2.0% of the upper convex configuration. On the other hand, the magnets with rounded borders do not produce a high cogging torque reduction, but keep a fine value of the electromagnetic torque. The combination with another technique can be used to generate a good result. Finally, the results had been used to build a 5.0 kw, 8-pole axial flux machine prototype, with extended speed range, shown in Fig. 14. The design included the skewing of the magnets and the displacement between stator sides. These modifications generate an important reduction in the final cogging torque. ACKNOWLEDGMENT This work was supported by the Chilean Research Council (Fondecyt), through the Project # 1070493 and Direccion de Investigación, University of Concepción, through Project # 206. 092.047-1.0. REFERENCES [1] N. Bianchi and S. Bolognani, Design techniques for reducing the cogging torque in surface-mounted PM motors, IEEE Trans. Ind. Appl., vol. 38, no. 5, pp. 1259 1265, Sep./Oct. 2002.

3440 IEEE TRANSACTIONS ON MAGNETICS, VOL. 43, NO. 8, AUGUST 2007 [2] M. Aydin, R. Qu, and T. Lipo, Cogging torque minimization technique for multiple-rotor, axial-flux, surface-mounted-magnet-pm motor: Alternating magnet pole-arc in facing rotors, presented at the IEEE Industry Application Conf. Annu. Meeting, Oct. 2003. [3] Z. Q. Zhu and D. Howe, Influence of design parameters on cogging torque in permanent machines, IEEE Trans. Energy Convers., vol. 15, no. 4, pp. 407 412, Dec. 2000. [4] S. Morimoto, Y. Takeda, T. Hirasa, and K. Taniguchi, Expansion of operating limits for permanent magnet motor by current vector control considering inverter capacity, IEEE Trans. Ind. Appl., vol. 26, no. 5, pp. 866 871, Sep./Oct. 1990. [5] J. A. Tapia, D. A. González, R. R. Wallace, and M. A. Valenzuela, Axial flux surface mounted PM machine with field weakening capability, Recent Developments on Electrical Drives, Springer Editorial Aug. 2006. [6] C. S. Koh, B.-K. Kang, J.-S. Ryu, and J.-S. Seoul, The effects of the distribution of residual magnetization on the cogging torque and switching signals in permanent magnet (PM) motors, IEEE Trans. Magn., vol. 38, no. 2, pp. 1217 1220, Mar. 2002. [7] Y. Yang, X. Wang, R. Zhang, T. Ding, and R. Tang, The optimization of pole arc coefficient to reduce cogging torque in surface-mounted permanent magnet motors, IEEE Trans. Magn., vol. 42, no. 4, pp. 1135 1138, Apr. 2006. [8] C. Schlensok, M. H. Gracia, and K. Hameyer, Combined numerical and analytical method for geometry optimization of a PM motor, IEEE Trans. Magn., vol. 42, no. 4, pp. 1211 1214, Apr. 2006. Manuscript received April 22, 2006; revised May 1, 2007. Corresponding author: D. A. Gonzalez (e-mail: degonzale@gmail.com). Delvis Anibal González was born in Santa Clara, Cuba. He received the Electrical Engineer degree and the Master of Science degree from the Central University of Las Villas, Cuba, in 1997 and 2001, respectively. Currently, he is pursuing the Ph.D. degree from the Electrical Engineering Department, University of Concepcion, Chile. His research interests include design, optimization, and control of electrical machines. Juan Antonio Tapia (M 03) was born in Concepcion, Chile. He received the B.E.E. and M.E.E degrees in electrical engineering from the University of Concepcion, Concepcion, Chile, in 1991 and 1997, respectively and the Ph.D. degree from the University of Wisconsin, Madison, in 2002. Since 1992, he has been with the Department of Electrical Engineering, University of Concepcion, as an Associate Professor. His primary areas of interest are electromechanical analysis and electrical machines design for AC adjustable speed applications. Alvaro Letelier Bettancourt was born in Vallenar, Chile. He received the Electrical Engineer and the Master of Science degrees from the University of Concepcion, Chile, in 2004 and 2005, respectively. He was a Training Engineer at BHPBilliton, Escondida, Chile. His areas of interest are control and design of electrical machines.