COGGING torque is one of the major sources of vibration

Transcription

1 IEEE TRANSACTIONS ON MAGNETICS, VOL. 47, NO. 7, JULY Cogging Torque of Brushless DC Motors Due to the Interaction Between the Uneven Magnetization of a Permanent Magnet and Teeth Curvature S. J. Sung 1, S. J. Park 2, and G. H. Jang 1 PREM, Department of Mechanical Engineering, Hanyang University, Seoul , Korea Samsung Electro-Mechanics Company, Suwon , Korea 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. The cogging torque can be effectively reduced by determining optimal teeth curvature once the range of uneven magnetization of permanent magnet in a brushless DC motor is identified. Index Terms Brushless DC motor, cogging torque, teeth curvature, uneven magnetization. I. INTRODUCTION 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 [1]. 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 [2], [3]. Fig. 1 shows the measured magnetic flux density along the inner surface of a PM in a 12-pole spindle motor of a computer hard disk drive, illustrating that the PM of a BLDC motor is not magnetized uniformly. 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. Many researchers have tried to reduce the cogging torque, and teeth curvature has been considered as a solution [4]. However, the uneven magnetization of PMs in addition to teeth curvature should be considered in order to effectively reduce the cogging torque of a BLDC motor. This research investigates cogging torque due to the interaction between teeth curvature and uneven PM magnetization. The magnetic field is calculated using a finite element method, and the cogging torque is calculated using the Maxwell stress tensor. The characteristics of cogging torque are investigated using spectral analysis, and the analysis results are experimentally verified. Manuscript received December 02, 2010; revised February 17, 2011; accepted February 17, Date of current version June 24, Corresponding author: G. H. Jang ( ghjang@hanyang.ac.kr). Color versions of one or more of the figures in this paper are available online at Digital Object Identifier /TMAG Fig. 1. (a) Measured and (b) enlarged surface flux density of a PM with 12 poles. II. FINITE ELEMENT ANALYSIS This research developed finite element models of two BLDC motors of computer hard disk drives, one with eight poles and twelve slots (8P12S) and another with twelve poles and nine slots (12P9S). Table II shows the specifications of these motors. The developed finite element models have 9,948 and 10,098 triangular elements with three nodes, and the cogging torque is calculated at every degree for 360 degrees by using the Maxwell stress tensor. In this research, teeth are modeled by gradually reducing the radius of curvature of the teeth and using the smaller radius of tooth curvature,, instead of the outer radius of a stator, R. Fig. 2 shows the geometry of the teeth curvature. Table III shows the three models with their respective radii of /$ IEEE

2 1924 IEEE TRANSACTIONS ON MAGNETICS, VOL. 47, NO. 7, JULY 2011 TABLE I MEASURED UNEVEN MAGNETIZATION OF EIGHT PMS TABLE II SPECIFICATION OF ANALYSIS MODEL Fig. 3. Uneven magnetization patterns of (a) 1/4 model of 8P12S and (b) 1/3 model of 12P9S. 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 Fig. 2. Geometry of teeth. TABLE III RADIUS OF CURVATURE OF CHAMFERED TEETH tooth curvature,, and. These radii are applied to the models with 8P12S and 12P9S, respectively. Since there are many different cases of uneven PM magnetization, a reasonable numerical model which represents all cases of uneven magnetization is required to generalize the effect of the interaction between a PM and teeth curvature. Akihiro et al. showed that the overall uneven magnetization pattern of a model can be presented by investigating one of the partial periodic models [5]. Fig. 3 shows the uneven magnetization patterns of a 1/4 partial model with 2P3S from the full model with 8P12S and a 1/3 partial model with 4P3S from the full model with 12P9S. Tables IV and V show the frequency components of cogging torque and their amplitudes due to the uneven magnetization patterns of 8P12S and 12P9S, respectively. The amplitudes

3 SUNG et al.: COGGING TORQUE OF BRUSHLESS DC MOTORS 1925 Fig. 4. Cogging torques of (a) 8P12S and (b) 12P9S due to teeth shape in an ideally magnetized case. Fig. 6. Cogging torques of (a) 8P12S and (b) 12P9S due to teeth shape in an unevenly magnetized case. Fig. 5. Frequency spectra of cogging torques of (a) 8P12S and (b) 12P9S due to teeth shape in an ideally magnetized case. of the frequency components are normalized to the amplitude of the least common multiple harmonic of the poles and slots in the ideally magnetized case. The data in the tables show that uneven magnetization of one pole generates all of the possible harmonic frequencies. Therefore, the uneven magnetization of PMs in which one pole is more magnetized than the others can be used in this research. Fig. 7. Frequency spectra of cogging torques of (a) 8P12S and (b) 12P9S due to teeth shape in an unevenly magnetized case. Fig. 4 shows the simulated cogging torques of the analysis models with 8P9S and 12P9S in the case of the ideally magnetized PM, illustrating that the teeth with the small radius of curvature decrease the peak value of the cogging torque. Fig. 5 shows the frequency spectra of the simulated cogging torques of the analysis models of 8P9S and 12P9S in the case of the ideally magnetized PM. It shows that the excitation frequencies of the cogging torques are the harmonics of the least common multiple

4 1926 IEEE TRANSACTIONS ON MAGNETICS, VOL. 47, NO. 7, JULY 2011 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. 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. of the poles and slots in ideal BLDC motors and that the teeth with small radius of curvature decrease their amplitude. Figs. 6 and 7 show the simulated cogging torques and their frequency spectra for an unevenly magnetized PM in which one pole is 5% more highly magnetized than the others. The figures show that the uneven magnetization of PMs introduces the harmonics of the slot number to the cogging torque in addition to the harmonics of the least common multiple of the poles and slots. The cogging torque can be represented in the following equation [6]: 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 (1) 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. Figs. 8 and 9 show how uneven magnetization of the PM interacts with teeth curvature in the production of cogging torque. The uneven magnetization of a pole increases from 0% to 10% in increments of 1%. The amplitude of the ideal harmonic of the cogging torque is hardly affected by the increase in the uneven magnetization of the PM, but the amplitude of the slot harmonic increases not only due to the teeth with small radius of curvature but also due to the uneven magnetization. In both the 8P12S and 12P9S models, teeth curvature is effective for reducing the cogging torque when the uneven magnetization is less than 5%. However, teeth curvature adversely increases the cogging torque when the uneven magnetization is greater than 5%.

5 SUNG et al.: COGGING TORQUE OF BRUSHLESS DC MOTORS 1927 TABLE VI SPECIFICATIONS OF THE EXPERIMENTAL SAMPLES Fig. 11. Measured and simulated cogging torques of (a) Sample A (sample with chamfered teeth) and (b) Sample B (sample with no chamfered teeth). Fig. 10. Experiment setup for measuring surface flux density. TABLE VII PEAK VALUES OF MEASURED SURFACE FLUX DENSITY AND ESTIMATED RESIDUAL FLUX DENSITY III. EXPERIMENTAL VERIFICATION 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. This research measured the magnetic flux density along the inner surfaces of the PMs using a Gauss-meter. Fig. 10 shows the experiment setup for measuring the surface flux density. Table VII shows the eight peak values of the surface magnetic flux density along the PMs, illustrating that the peak values of 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) the PM have 3.6% variation in Sample A and 4.1% in Sample B. Figs. 11 and 12 show the measured and simulated cogging torques and their frequency spectra for Samples A and B, respectively. The cogging torque was measured using a commercial torque meter, and the simulated cogging torque was calculated using finite element models with measured magnetization patterns. To simulate the uneven magnetization of the PMs, the

6 1928 IEEE TRANSACTIONS ON MAGNETICS, VOL. 47, NO. 7, JULY 2011 BH characteristics were estimated from the measured surface flux density. Since the residual magnetic flux density was between 0.67 and 0.73 T, the smallest peak value was assumed to be 0.67 T, and the other values were linearly interpolated using the BH characteristics of the finite element model. The simulated results matched well with the measured results. As predicted in the simulation, the amplitude of the 24th harmonic, the least common multiple of the poles and slots, was reduced due to the use of the teeth with small radius of curvature. However, the teeth in Sample A introduced the high amplitude of the 12th harmonic originating from the slot number, even though the uneven magnetization of Sample A was smaller than that of Sample B. These experimental results verified the predicted simulation results. IV. CONCLUSIONS 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. 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 , [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 , [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 , [4] A. Hartman and W. Lorimer, Undriven vibrations in brushless DC motors, IEEE Trans. Magn., vol. 37, no. 2, pp , [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 , [6] D. Hanselman, Brushless Permanent Magnet Motor Design, 2nd ed.: The Writers Collective, 2003, pp