Title ANALYSIS AND DETERMINATION OF COGGING TORQUE AND UNBALANCED MAGNETIC FORCES IN PERMANENT MAGNET SPINDLE MOTORS FOR HARD DISK DRIVES LI JIANGTAO (M. Eng., Xi’an Jiaotong Univ., P. R. China) A DISSERTATION SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2006 Acknowledgements Acknowledgements I would like to express my most sincere and heartfelt gratitude to Dr. Z. J. Liu with Data Storage Institute, Singapore and Prof. M. A. Jabbar with Electrical and Computer Engineering, National University of Singapore, for their guidance, patience and supports during the entire course of my Ph. D project. Without their judicious advices and supports, my completion of the research work would not have been possible. It is my utmost honor to be under their supervision. I would like to extend my gratitude to Dr. C. Bi, Dr. Q. Jiang, and Dr. X. K. Gao, who have kindly shared their knowledge and research experiences with me. My appreciation also goes to all the staffs and students of Data Storage Institute, who have helped me in one way or another. I also wish to thank all of my friends for their encouragements and assistance to my studies in Singapore. On a personal note, I am truly grateful to my parents and my wife, whose solid supports have accompanied me all the time. National University of Singapore I Table of Contents Table of Contents Title I Acknowledgements I Table of Contents . II Summary VI List of Tables IX List of Figures XI List of Symbols . XIX Introduction 1.1 Hard Disk Drives and Permanent Magnet Brushless DC Motors 1.2 Technical Problems Related to Spindle Motors in Hard Disk Drives 1.3 Performance Evaluations and Design Optimization of Spindle Motors 12 1.4 Organization of the Dissertation 14 Review on Computer Modeling and Analysis of PM Brushless DC Motors 17 2.1 Electromagnetic Field in PM Brushless DC Motors 17 2.2 Electromagnetic Forces in PM Brushless DC Motors . 27 2.3 Effect of Calculation Error in Tangential Field Component on Calculation National University of Singapore II Table of Contents of Electromagnetic Forces . 30 2.4 Minimization of Cogging Torque and Unbalanced Magnetic Pull 34 Effect of Pole Transition over Slot Opening 38 3.1 Introduction 38 3.2 Pole Transition over Slot Model 40 3.2.1 Mathematical Model . 41 3.2.2 Scalar Potential Distribution along Slot Opening . 54 3.2.3 Flux Density Distribution in the Air Gap 56 3.2.4 Effect of Curvature 59 3.2.5 Tangential Force Waveform 62 3.3 Cogging Torque Calculation by Superposition Approach . 64 3.3.1 General Procedure . 64 3.3.2 Case Studies 67 3.4 Conclusion . 74 Closed Form Solution of the Magnetic Field in PM Machines 75 4.1 Introduction 75 4.2 Scalar Potential Distribution on the Stator Surface in Slotted PM Motor76 4.3 Air Gap Field in Spindle Motors without LoadingError! Bookmark not defined. 4.4 Armature Reaction Field 97 4.5 Back Electrical Motive Force 103 4.6 Magnetic Field in PM Motors with Rotor Eccentricity . 104 National University of Singapore III Table of Contents 4.7 Closed Form Solution of Electromagnetic Forces in PM Machines 113 5.1 Introduction 113 5.2 Maxwell Stress Tensor Method 114 5.3 Cogging Torque 119 5.4 Conclusion . 112 5.3.1 Influence of Pole-Arc to Pole-Pitch Ratio 125 5.3.2 Influence of Slot-Opening to Slot-Pitch Ratio 128 5.3.3 Effect of Rotor Eccentricity 130 Unbalanced Magnetic Pull . 135 5.4.1 Influence of Pole-Arc to Pole-Pitch Ratio 138 5.4.2 Influence of Slot-Opening to Slot-Pitch Ratio 139 5.4.3 Effect of Rotor Eccentricity 139 5.5 Running Torque 145 5.6 Comparisons with Experimental Results . 151 5.7 Conclusion . 155 Minimization of Cogging Torque and UMP in PM Spindle Motors . 156 6.1 Introduction 156 6.2 Constrained Optimization Problem 157 6.3 Powell’s Methods . 159 6.4 Objective Functions and Design Variables 162 6.5 A Case Study 165 6.6 Cogging Torque Minimization . 169 National University of Singapore IV Table of Contents 6.7 6.8 6.6.1 Peak Cogging Torque 169 6.6.2 Peak Cogging Torque to Running Torque Ratio . 178 UMP Minimization 180 6.7.1 Average Radial Force 180 6.7.2 UMP Ripple to Average Radial Force Ratio . 182 Conclusion . 183 Combined Analytical and Numerical Approach for Design Optimization . 184 7.1 Introduction 184 7.2 Response Surface Methodology 185 7.2.1 Concept of RSM . 185 7.2.2 Limitation of RSM 191 7.3 Combined Analytical and Numerical Approach 197 7.4 Conclusion . 205 Conclusions and Future Work 206 References 212 Appendix 228 List of Publications . 231 National University of Singapore V Summary Summary With the rapid development of data storage technology, the permanent magnet spindle motors in hard disk drives (HDDs) become smaller and smaller in size. In recent years, following the successful commercial applications of inch format micro-drives, the smaller HDDs of 0.85 inch and 0.5~0.75 inch format begin to attract the attention due to the fast development of wearable electronic devices such as pocket drives, mobile phones and digital cameras with high storage capacities. In such high precision applications, the design requirements for the electromagnetic forces are extremely stringent. Electromagnetic forces such as the cogging torque and unbalanced magnetic pull (UMP) generated in the permanent magnet spindles are of great concerns as they may cause undesirable speed pulsation, mechanical vibration and acoustic noise which in turn limit the recording density. A lot of efforts have been made to derive an effective technique to suppress the cogging torque and UMP. However, to obtain the accurate performance predictions speedily remains a challenging task in engineering practice. Numerical methods such as the finite element method (FEM) are widely adopted in the National University of Singapore VI Summary study of field distributions and evaluation of the cogging torque and UMP. The accuracy achieved is usually very high, but they are not suitable for design optimization purposes if a large number of design parameters need to be investigated for the special kind of PM motors used in hard disk drives. Analytical modeling as an alternative has its own merits as it normally demands for less computer time and therefore allows for fast analysis of PM motors of various magnetic structures. It also provides deep insight into the underlying physical processes and is helpful in establishing a relationship between the performance and the dominant design parameters. Though, it is usually very hard to obtain the analytical solution of instantaneous magnetic fields in electromagnetic devices, which in turn forms the basis for accurate evaluation of electromagnetic forces. Several attempts have been made previously by researchers to obtain analytical solutions of instantaneous magnetic fields in permanent magnet motors. However, the existing analytical approaches, such as those based on magnetic field energy or the estimation of the net tangential forces acting on the slot walls, generally suffer lack of accuracy when predicting the instantaneous magnetic fields and the cogging torque over a wide range of design parameters. This is mainly due to the fact that the previous analytical approaches rely on the radial component of the magnetic flux density in the air gap for PM motors with slotted stator. However, detailed investigations show that the cogging torque prediction relies heavily on the accuracy of the air gap flux density distribution, especially in proximity of slot openings. Based on a comprehensive study of the mechanism of the cogging torque in this dissertation, a new exact analytical solution of the boundary National University of Singapore VII Summary value problem associated with the instantaneous fields in PM motors is developed for the detailed analysis of cogging torques. The electromagnetic forces in the radial direction developed in PM motors are also derived and calculated based on the model. The results are in good agreement with the numerical simulations using the finite element method as shown by the comparisons in which a wide range of design parameters, including the choices of pole number and slot number combinations, were investigated. Due to the applications of the aerodynamic and fluid dynamic bearing system, there exists the rotor eccentricity, and it affects both the amplitude and the waveforms of the electromagnetic forces. Using the new analytical solution, both the amplitude and the frequency spectrum can be calculated accurately when the rotor eccentricity is present, which is not possible using the previously existing analytical methods. In order to obtain a global optimum, a large design space needs to be explored and a large number of computations are required, which are usually not affordable if numerical simulation tools are used to evaluate the target function. In this dissertation, an optimization approach based on combined analytical solutions and numerical search is devised to cope with the problem. The target functions are evaluated by analytical solutions first to locate a sub-region where the global optimum may exist. Then, the numerical simulation based Response Surface Methodology (RSM) is applied over the sub-region to obtain the final optimum. The effectiveness of the approach is demonstrated using design optimization case studies. National University of Singapore VIII List of Tables List of Tables Table 1.1 The dimension of hard disk drives and their applications . Table 1.2 Dimensions of PM motors used for different formats of HDDs (in-hub design) . Table 1.3 Rotation speeds of spindles and their applications . Table 3.1 Relationship between radial field motor and slot model . 60 Table 3.2 Slot phase shift in an 8-pole 9-slot PM motor . 66 Table 3.3 Parameters of an 8-pole 9-slot PM motor . 68 Table 3.4 Parameters of an 8-pole 6-slot PM motor . 71 Table 3.5 Parameters of a 12-pole 9-slot PM motor . 72 Table 5.1 Coil arrangement in 8-pole 6-slot PM motors 146 Table 5.2 Coil arrangement in 8-pole 9-slot PM motors 147 Table 5.3 Coil arrangement in 8-pole 12-slot PM motors 148 Table 6.1 List of optimization variables 163 Table 6.2 Structure parameters of the studied PM motor . 166 Table 6.3 One step in Powell’s method . 168 Table 6.4 Comparison of the orginal design and the optimal design 169 Table 6.5 Determination of cogging torque period (p = 3) . 170 National University of Singapore IX References electrical machines,” IEEE Transactions on Magnetics, vol. 33, no. 2, pp. 20182021, Mar. 1997. [82] J. Mizia, K. Adamiak, A. R. Eastham, and G. E. 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National University of Singapore 227 Appendix Appendix Table A. 1: Specifications of the 8-pole/6-slot motor Operation speed 11,000 rpm Running current 0.64 A Rotor outer diameter 23 mm Rotor inner diameter 18.6 mm Radial magnet thickness 1.485 mm Pole-arc to pole-pitch ratio Air gap length 0.28 mm Shaft diameter 7.0 mm Axial lamination thickness 7.7 mm Tooth tip width 1.25 mm Tooth body height 2.78 mm Slot opening 1.26 mm Number of turns per phase 66 Diameter of the conductor 0.3 mm Resistance per phase 0.514Ω Magnet Polymer NdFeB Stator iron M-17 Rotor iron Mild steel National University of Singapore 228 Appendix Table A. 2: Specifications of the 8-pole/9-slot motor Operation speed 11,000 rpm Running current 0.42 A Rotor outer diameter 23 mm Rotor inner diameter 18.6 mm Radial magnet thickness 1.485 mm Pole-arc to pole-pitch ratio Air gap length 0.28 mm Shaft diameter 7.0 mm Axial lamination thickness 4.1 mm Tooth tip width 1.1 mm Tooth body height 2.71 mm Slot opening 1.26 mm Number of turns per phase 66 Diameter of the conductor 0.285 mm Resistance per phase 0.56Ω Magnet Polymer NdFeB Stator iron M-17 Rotor iron Mild steel National University of Singapore 229 Appendix Table A. 3: Specifications of the 8-pole/12-slot motor Operation speed 7200 rpm Running current 0.19 A Rotor outer diameter 28 mm Rotor inner diameter 26 mm Radial magnet thickness mm Pole-arc to pole-pitch ratio Air gap length 0.2 mm Shaft diameter 11 mm Axial lamination thickness 3.5 mm Tooth tip width mm Tooth body height 2.78 mm Slot opening 1.5 mm Number of turns per phase 160 Diameter of the conductor 0.25 mm Resistance per phase 1.35 Ω Magnet Polymer NdFeB Stator iron M-17 Rotor iron Mild steel National University of Singapore 230 List of Publications List of Publications Journals [1] J. T. Li, Z. J. Liu, M. A. Jabbar, and X. K. Gao, “Design optimization for cogging torque minimization using response surface methodology,” IEEE Transactions on Magnetics, vol.40, no.2, pp. 1176-1179, March, 2004. [2] Z. J. Liu, J. T. Li and H. H. Long, “Sensitivity analysis for write field with respect to design parameters for perpendicular recording heads,” Journal of Applied Physics, vol. 97, pp.515-517, May, 2005. [3] Z. J. Liu, J. T. Li and M. A. Jabbar, “Prediction of cogging torque by superposition of torque due to pole transition over slot,” submitted to IEEE Transactions on Energy Conversion. [4] Z. J. Liu, J. T. Li, H. T. Wang and H. L. Li, “Distribution of slanted write field for perpendicular recording heads with shielded pole,” IEEE Transactions on Magnetics, vol. 41, no. 10, pp. 2908-2910, Oct. 2005. [5] Z. J. Liu, J. T. Li and M. A. Jabbar, “Magnetic field and forces calculation in permanent magnet machines part I: effect of magnet pole transition over slot opening,” submitted to IEEE Transactions on Magnetics. [6] Z. J. Liu, J. T. Li and M. A. Jabbar, “Magnetic field and forces calculation in permanent magnet machines part II: field distribution in air gap and magnetic National University of Singapore 231 List of Publications forces prediction,” submitted to IEEE Transactions on Magnetics. [7] Z. J. Liu, H. H. Long, E. T. Ong, E. P. Li, and J. T. Li, “Dynamic simulation of high density perpendicular recording head and media combination,” IEEE Transactions on Magnetics, vol. 42, no. 4, pp. 943-946, April,2006. [8] Z. J. Liu, J. T. Li and M. A. Jabbar, “Prediction and analysis of magnetic forces in permanent magnet brushless dc motor with rotor eccentricity”, accepted for publication in Journal of Applied Physics, 2006 Jun Issue. [9] Z. J. Liu, J. T. Li and M. A. Jabbar, “Prediction of Cogging Torque by Superposition of Torque due to Pole Transition over Slot,” submitted to IEEE Transactions on Industrial Application [10] Z. J. Liu, J. T. Li, C. Bi, and Q. Jiang, “A Numerical Approach for Accurate Prediction of Magnetic Field in Permanent Magnet Motors,” to be published on IEEE Transactions on Magnetics, vol. 24, no. 10, Oct. 2006 [11] Z. J. Liu, J. T. Li, K. S. Chai, and L. Wang, “Direct Solution of Medium Field and Cross Track Characteristics of Read Sensor,” to be published on IEEE Transactions on Magnetics, Oct. 2006 [12] J. T. Li and Z. J. Liu, “Unbalanced Magnetic Forces in Permanent Magnet Motors with Rotor Eccentricity,” to be published on International Journal of Applied Electromagnetics and Mechanics [13] Z. J. Liu and J. T. Li, “Predicting playback waveform of giant magneto-resistive read sensor,” to be published on International Journal of Applied Electromagnetics and Mechanics National University of Singapore 232 List of Publications Conferences [1] X. K. Gao, J. T. Li, Z. Xie, and Z. J. Liu, “Application of robust design techniques to electromagnetic devices design optimization,” Proceedings of the International Conference on Scientific & Engineering Computation, pp.304-307, Dec. 2002. [2] J. T. Li, Z. J. Liu, M. A. Jabbar, and X. K. Gao, “Design optimization for cogging torque minimization using response surface methodology,” 14th Conference on the Computation of Electromagnetic Fields, Saratoga Springs, New York, USA, vol. II, pp. 130-134, July, 2003. [3] Z. J. Liu, M. A. Jabbar, J. T. Li, “Optimal and robust design for HDD PM spindle motor assisted by an analytical field solution,” 6th International Symposium on Electric and Magnetic Fields, Aachen, Germany, Symposium Reports, pp. 147-150, Oct., 2003. [4] X. K. Gao, Z. Q. Shen, Z. J. Liu, T. S. Low, C. H. Lau, and J. T. Li, “Robust design of electromagnetic devices using web-based RSM tool,” 6th International Symposium on Electrical and Magnetic Field, Aachen, Germany, Symposium Reports, pp. 123-127, Oct., 2003 [5] J. T. Li, Z. J. Liu, M. A. Jabbar, “Optimal design for permanent magnet spindle motor using combined analytical and numerical methods,” 9th Joint MMM/Intermag Conference, Anaheim, California, USA, Jan., 2004 [6] J. T. Li, H. H. Long and Z. J. Liu, “Analytical write field prediction for perpendicular recording Heads,” Asia-Pacific Magnetic Recording Conference 2004, FT-02, Aug, 2004 National University of Singapore 233 List of Publications [7] Z. J. Liu, J. T. Li and H. H. Long, “Sensitivity analysis for write field with respect to design parameters for perpendicular recording heads,” 49th Annual Conference on Magnetism & Magnetic Materials, HB-11,Florida, USA, Nov. 2004. [8] H. H. Long, J. T. Li, Z. J. Liu, "Parallelization in 3d finite element micromagnetic simulation of magnetization processes in perpendicular magnetic recording," Proceedings of the International Conference on Scientific & Engineering Computation, Parallel Session (PS-8), Computational Electronics and Electromagnetics no. 3, June. 2004. [9] Z. J. Liu, J. T. Li and M. A. Jabbar, “Calculation of cogging torque in permanent magnet motor with improved accuracy,” 15th Conference on the Computation of Electromagnetic Fields, Shen Yang,China, June, 2005. [10] Z. J. Liu, H. H. Long, E. P. Li and J. T. Li, "Dynamic simulation of high density perpendicular recording head and media combination," 15th Conference on the Computation of Electromagnetic Fields, Shen Yang, China, 2005 [11] H. H. Long, T. Liu, J. T. Li, H. L. Li and Z. J. Liu, "Finite element micromagnetic simulation of switch dynamic of composite media," 3rd International Conference on Materials for Advanced Technologies (ICMAT), 2005. [12] Z. J. Liu, J. T. Li and M. A. Jabbar, “Cogging torque prediction by superposition of torque due to pole transition over slot,” IEEE International Conference on Electric Machines and Drives, pp. 1219-1224, May 15, 2005. [13] Z. J. Liu, J. T. Li, H. T. Wang and J. P. Wang, “Distribution of slanted write field for perpendicular recording heads with shielded pole,” CP-08, Intermag conference, Japan, 2005. National University of Singapore 234 List of Publications [14] Z. J. Liu, H. H. Long, J. T. Li and J. P. Wang, “Finite element micromagnetic simulation of switch dynamic of tilted high density perpendicular recording,” Intermag conference, Japan, 2005. [15] Z. J. Liu, J. T. Li and M. A. Jabbar, “Magnetic analysis of magnetic forces in permanent magnet motors for HDDs with various types of rotor eccentricity,” 50th Magnetism and Magnetic Materials Conference, California, USA, 2005. [16] Z. J. Liu, H. H. Long, E. T. Ong, E. P. Li, and J. T. Li, “Dynamic simulation of high density perpendicular recording head and media combination”, PD3-14, Compumag Conference, Shenyang, China, 2005. [17] Z. J. Liu, J. T. Li, H. H. Long, E. P. Li, E. T. Ong, and K. S. Chai, “Calculation of dynamic write field for perpendicular recording head”, EMC Zurich 2006 Conference, Singapore. [18] Z. J. Liu, J. T. Li, Q. Jiang, C. Bi, “A numerical approach for accurate prediction of magnetic field in permanent magnet motors,” IEEE International Magnetics Conference, San Diego, California, May 8-12, 2006. [19] Z. J. Liu, J. T. Li, K. S. Kao, L. Wang, “Direct solution of medium field and cross track characteristics of read sensor”, IEEE International Magnetics Conference, San Diego, California, May 8-12, 2006. [20] J. T. Li and Z. J. Liu, “Unbalanced Magnetic Forces in Permanent Magnet Motors with Rotor Eccentricity,” Asia-Pacific Symposium on Applied Electromagnetics and Mechanics (APSAEM2006), 20-21 July 2006, University of Technology Sydney, NSW, Australia [21] Z. J. Liu and J. T. Li, “Predicting playback waveform of giant magneto-resistive read sensor,” Asia-Pacific Symposium on Applied Electromagnetics and Mechanics National University of Singapore 235 List of Publications (APSAEM2006), 20-21 July 2006, University of Technology Sydney, NSW, Australia National University of Singapore 236 [...]... to allow the hard disk to function properly The size of spindle motors used in hard disk drives continues to be reduced and the speed increased throughout past years As the size of hard disk drives is decreasing, the dimensions of the spindle motors also decrease in order to fit in the hard disk drives Table 1.2 shows the dimensions of the spindles used in hard disk drives of different formats Table... on magnetic forces are especially concerned in permanent magnet spindle motors used in the high precision and high capacity hard disk drives, which refer to the cogging torque and unbalanced magnetic pull (UMP) These factors may cause undesirable effects on the precision of the rotation [8]-[12] if not properly addressed Therefore, the design requirements on the magnetic forces for HDD spindle motors. .. used in hard disk drives 2) To build a model for predicting the electromagnetic forces developed in PM motors with sufficient accuracy required for design and analysis of PM spindle motors for high precision HDD applications National University of Singapore 14 Chapter 1 Introduction 3) To develop a combined analytical and numerical optimization methodology to minimize cogging torque and UMP in permanent. .. considerations for miniaturized spindles and undesirable axial forces Unbalanced magnetic pull is generated by the combination of asymmetrical magnetic National University of Singapore 10 Chapter 1 Introduction structure and symmetrical slot structure or the combination of symmetrical magnet structure and asymmetrical slot structure In symmetrical motor structure, the radial magnetic forces are canceled out in. .. Dimensions of PM motors used for different formats of HDDs (in- hub design) Form Factor (inch) Diameter (mm) 5.25 56.0 3.50 25.0 2.50 20.0 1.8 12.0 Since the increasing of the spindle speed could improve both random-access and sequential performance of the hard disk drive, the spindle speed has kept increasing At one time, all PC hard disks spun at 3,600 RPM In the early 1990s, manufacturers started to investigate... represents the working region of the magnet It can be observed that such type of magnet shows very good linearity in the working region Figure 1.4 A typical demagnetization curve of NdFeB permanent magnet There are two major types of PM motors used in hard disk drives: the interior-rotor PM motors and exterior-rotor PM motors The exterior-rotor motor is commonly used for modern hard disk drives In the exterior-rotor... electromagnetic induced vibrations are very critical to the design of PM spindle motors used in ultra-high density HDDs However, the previous analytical medels suffer from inaccuracy in predicting the air gap field and magnetic forces due to oversimplification of the physical problems, and in many circumstances are unsuitable for performing the required design and analysis of PM motors The motivation of. .. presented in this theis is to develop an analytical model that is accurate and helpful to identify and understand the influences of the leading design parameters on the magnetic field and forces developed in PM motors The scope of the research work described in the thesis are: 1) To develop a new accurate and fast analytical model to investigate the electromagnetic fields and forces in permanent magnet motors. .. starting torque is required to overcome the initial stiction force between the head and media and the cogging torque as well Thus, the cogging will also pose a problem, although not very severe, in making the starting difficult In design of PM brushless motors, skewing of stator slots can be an effective way to reduce the cogging torque This technique is not commonly used in HDD spindle motors because of. .. applications of 1 inch format micro -drives (as shown in Figure 1.3), the smaller format hard disk drives of 0.85 inch and 0.5~0.75 inch are emerging National University of Singapore 3 Chapter 1 Introduction Figure 1.3 One-inch micro-drive Table 1.1 The dimension of hard disk drives and their applications Platter Diameter Typical Form Factor Application Oldest PCs, used in servers through the 5.12 mid-1990s and . Title ANALYSIS AND DETERMINATION OF COGGING TORQUE AND UNBALANCED MAGNETIC FORCES IN PERMANENT MAGNET SPINDLE MOTORS FOR HARD DISK DRIVES LI JIANGTAO. XIX 1 Introduction 1 1.1 Hard Disk Drives and Permanent Magnet Brushless DC Motors 1 1.2 Technical Problems Related to Spindle Motors in Hard Disk Drives 9 1.3 Performance Evaluations and Design. are extremely stringent. Electromagnetic forces such as the cogging torque and unbalanced magnetic pull (UMP) generated in the permanent magnet spindles are of great concerns as they may cause