1. Department of Electrical Engineering Southern Taiwan University Robot and Servo Drive Lab. Cogging Torque of Brushless DC Motors Due to the Interaction Between the Uneven Magnetization of a Permanent Magnet and Teeth Curvature Adviser: Ming-Shyan Wang Student: Cian-Yong Fong Student ID: MA120215 2015/5/1 S. J. Sung, S. J. Park, and G. H. Jang PREM, Department of Mechanical Engineering, Hanyang University, Seoul 133-791, Korea Samsung Electro-Mechanics Company, Suwon 443-743, Korea IEEE TRANSACTIONS ON MAGNETICS, VOL. 47, NO. 7, JULY 2011 1923-1928

2. Department of Electrical Engineering Southern Taiwan University Outline 1.Abstract 2.Introduction 3.Finite Element Analysis 4.Experimental Verification 5.Conclusions 6.References 2015/5/1 Robot and Servo Drive Lab. 2

3. Department of Electrical Engineering Southern Taiwan University Abstract 2015/5/1 Robot and Servo Drive Lab. 3 This research investigates the characteristics of cogging torque in brushless DC motors due to the interaction between the uneven magnetization of a permanent magnet and the shape of the teeth. The excitation frequencies of the cogging torque are the harmonics of the least common multiple of the poles and slots in ideal brushless DC motors. However, this research numerically and experimentally illustrates that the uneven magnetization of a permanent magnet also generates the harmonics of slot number. Magnitudes of cogging torque are affected not only by the uneven magnetization of a permanent magnet but also by the shape of the teeth.

4. Department of Electrical Engineering Southern Taiwan University Introduction 2015/5/1 Robot and Servo Drive Lab. 4 Cogging torque is one of the major sources of vibration and noise in brushless DC (BLDC) motors. Many researchers have addressed the sources and characteristics of cogging torque. The driving frequencies of cogging torque are the harmonics of the least common multiple of the poles and slots in an ideal BLDC motor. However, manufacturing errors introduce additional driving frequencies to the cogging torque, one of which is the uneven magnetization of a permanent magnet (PM), resulting in the generation of the harmonics of slot number.

5. Department of Electrical Engineering Southern Taiwan University Introduction 2015/5/1 Robot and Servo Drive Lab. 5 Fig. 1. (a) Measured and (b) enlarged surface flux density of a PM with 12 poles.

6. Department of Electrical Engineering Southern Taiwan University Introduction 2015/5/1 Robot and Servo Drive Lab. 6 TABLE I MEASURED UNEVEN MAGNETIZATION OF EIGHT PMS Table I shows the measured uneven magnetization of eight PMs in the same magnetizing fixture by increasing the magnetizing voltage. It shows that even the PMs with same magnetizing voltage have different level of uneven magnetization because the grain or particle structure of each PM is different.

7. Department of Electrical Engineering Southern Taiwan University Finite Element Analysis 2015/5/1 Robot and Servo Drive Lab. 7 TABLE II SPECIFICATION OF ANALYSIS MODEL

8. Department of Electrical Engineering Southern Taiwan University Finite Element Analysis 2015/5/1 Robot and Servo Drive Lab. 8 TABLE III RADIUS OF CURVATURE OF CHAMFERED TEETH Fig. 2. Geometry of teeth. Fig. 2 shows the geometry of the teeth curvature. Table III shows the three models with their respective radii of tooth curvature,, and. These radii are applied to the models with 8P12S and 12P9S, respectively.

9. Department of Electrical Engineering Southern Taiwan University Finite Element Analysis 2015/5/1 Robot and Servo Drive Lab. 9 Fig. 3. Uneven magnetization patterns of (a)1/4 model of 8P12S and (b)1/3 model of 12P9S.

10. Department of Electrical Engineering Southern Taiwan University Finite Element Analysis 2015/5/1 Robot and Servo Drive Lab. 10 TABLE IV FREQUENCY COMPONENTS AND THEIR AMPLITUDE OF COGGING TORQUE DUE TO AN UNEVEN MAGNETIZATION PATTERN OF 8P12S TABLE V FREQUENCY COMPONENTS AND THEIR AMPLITUDE OF COGGING TORQUE DUE TO AN UNEVEN MAGNETIZATION PATTERN OF 12P9S

11. Department of Electrical Engineering Southern Taiwan University Finite Element Analysis 2015/5/1 Robot and Servo Drive Lab. 11 Fig. 4. Cogging torques of (a) 8P12S and (b) 12P9S due to teeth shape in an ideally magnetized case.

12. Department of Electrical Engineering Southern Taiwan University Finite Element Analysis 2015/5/1 Robot and Servo Drive Lab. 12 Fig. 5. Frequency spectra of cogging torques of (a) 8P12S and (b) 12P9S due to teeth shape in an ideally magnetized case.

13. Department of Electrical Engineering Southern Taiwan University Finite Element Analysis 2015/5/1 Robot and Servo Drive Lab. 13 Fig. 6. Cogging torques of (a) 8P12S and (b) 12P9S due to teeth shape in an unevenly magnetized case.

14. Department of Electrical Engineering Southern Taiwan University Finite Element Analysis 2015/5/1 Robot and Servo Drive Lab. 14 Fig. 7. Frequency spectra of cogging torques of (a) 8P12S and (b) 12P9S due to teeth shape in an unevenly magnetized case.

15. Department of Electrical Engineering Southern Taiwan University Finite Element Analysis 2015/5/1 Robot and Servo Drive Lab. 15 where is the magnet flux crossing the air gap and is the total reluctance through which the flux passes. In case of unevenly magnetized PMs, the reduction of the radius of teeth curvature increases the variation of reluctance with respect to the rotating angle so that the amplitude of slot harmonics increases. This research suggests that the teeth with small radius of curvature increase the amplitude of the slot harmonic which is generated by uneven magnetization. If the increment of the slot harmonic is greater than the decrement of the harmonic of the least common multiple, the teeth curvature is not an effective method for reducing the cogging torque.

16. Department of Electrical Engineering Southern Taiwan University Finite Element Analysis 2015/5/1 Robot and Servo Drive Lab. 16 Fig. 8. (a) Peak value of cogging torque, (b) magnitude of 24th harmonic and (c) magnitude of 12th harmonic with an increase in the uneven magnetization of the one pole in 8P12S.

17. Department of Electrical Engineering Southern Taiwan University Finite Element Analysis 2015/5/1 Robot and Servo Drive Lab. 17 Fig. 9. (a) Peak value of cogging torque, (b) magnitude of 36th harmonic and (c) magnitude of 9th harmonic with an increase in the uneven magnetization of the one pole in 12P9S.

18. Department of Electrical Engineering Southern Taiwan University Experimental Verification 2015/5/1 Robot and Servo Drive Lab. 18 TABLE VI SPECIFICATIONS OF THE EXPERIMENTAL SAMPLES This research experimentally verified the interaction between uneven PM magnetization and teeth curvature in a BLDC motor with 8P12S. Sample A has the teeth with a radius of curvature, and sample B motor has the teeth with a radius of curvature. Table VI shows the major design parameters for Samples A and B.

19. Department of Electrical Engineering Southern Taiwan University Experimental Verification 2015/5/1 Robot and Servo Drive Lab. 19 Fig. 10. Experiment setup for measuring surface flux density.

20. Department of Electrical Engineering Southern Taiwan University Experimental Verification 2015/5/1 Robot and Servo Drive Lab. 20 Table VII shows the eight peak values of the surface magnetic flux density along the PMs, illustrating that the peak values of the PM have 3.6% variation in ample A and 4.1% in Sample B. TABLE VII PEAK VALUES OF MEASURED SURFACE FLUX DENSITY AND ESTIMATED RESIDUAL FLUX DENSITY

21. Department of Electrical Engineering Southern Taiwan University Experimental Verification 2015/5/1 Robot and Servo Drive Lab. 21 Fig. 11. Measured and simulated cogging torques of (a) Sample A (sample with chamfered teeth) and (b) Sample B (sample with no chamfered teeth).

22. Department of Electrical Engineering Southern Taiwan University Experimental Verification 2015/5/1 Robot and Servo Drive Lab. 22 Fig. 12. Frequency spectra of measured and simulated cogging torques of (a) Sample A (sample with chamfered teeth) and (b) Sample B (sample with no chamfered teeth)

23. Department of Electrical Engineering Southern Taiwan University Conclusions 2015/5/1 Robot and Servo Drive Lab. 23 This research numerically and experimentally demonstrated that the driving frequencies and their amplitudes are dependent on the teeth curvature as well as the magnetization status of the PM. Because it is very difficult to uniformly magnetize the PM of BLDC motors, the uneven magnetization of PMs inevitably introduces the harmonic of the slot number, and its amplitude may increase with the introduction of teeth curvature. In cases of uneven PM magnetization, teeth curvature is not an effective solution for reducing cogging torque. The cogging torque can be effectively reduced by determining optimal teeth curvature once the range of uneven magnetization of PM in a BLDC motor is identified.

24. Department of Electrical Engineering Southern Taiwan University References [1] G. H. Jang, J. W. Yoon, N. Y. Park, and S. M. Jang, “Torque and unbalanced magnetic force in a rotational unsymmetric brushless DC motors,” IEEE Trans. Magn., vol. 32, no. 5, pp. 5157–5159, 1996. [2] Y. D. Yao, D. R. Huang, J. C. Wang, S. H. Liou, T. F. Ying, and D. Y. Chiang, “Simulation study of the reduction of cogging torque in permanent magnet motors,” IEEE Trans. Magn., vol. 33, no. 5, pp.4095–4097, 1997. [3] T. Y. Yoon, “Magnetically induced vibration in a permanent-magnet brushless dc motor with symmetric pole-slot configuration,” IEEE Trans. Magn., vol. 41, no. 6, pp. 2173–2179, 2005. [4] A. Hartman and W. Lorimer, “Undriven vibrations in brushless DC motors,” IEEE Trans. Magn., vol. 37, no. 2, pp. 789–792, 2001. [5] D. Akihiro and Y. Shinichi, “Cogging torque investigation of PM motors resulting from asymmetry property of magnetic poles: Influence of performance variation between permanent magnets,” Elect. Eng. Jpn., vol. 163, no. 3, pp. 95–102, 1992. [6] D. Hanselman, Brushless Permanent Magnet Motor Design, 2nd ed.: The Writers’ Collective, 2003, pp. 111–114. 2015/5/1 Robot and Servo Drive Lab. 24

25. Department of Electrical Engineering Southern Taiwan University 2015/5/1 Robot and Servo Drive Lab. 25 Thanks for your attention.