Analysis and determination of cogging torque and unbalanced magnetic forces in permanent magnet spindle motors for hard disk drives

ANALYSIS AND DETERMINATION OF COGGING TORQUE AND UNBALANCED MAGNETIC FORCES IN PERMANENT MAGNET SPINDLE MOTORS FOR HARD DISK DRIVES LI JIANGTAO M.. Summary With the rapid development o

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

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

2 Review on Computer Modeling and Analysis of PM Brushless DC Motors 17

2.1 Electromagnetic Field in PM Brushless DC Motors 172.2 Electromagnetic Forces in PM Brushless DC Motors 272.3 Effect of Calculation Error in Tangential Field Component on Calculation

of Electromagnetic Forces 30

2.4 Minimization of Cogging Torque and Unbalanced Magnetic Pull 34

3 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

4 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

4.7 Conclusion 112

5 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

5.3.1 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

5.4 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

6 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

6.6.1 Peak Cogging Torque 169

6.6.2 Peak Cogging Torque to Running Torque Ratio 178

6.7 UMP Minimization 180

6.7.1 Average Radial Force 180

6.7.2 UMP Ripple to Average Radial Force Ratio 182

6.8 Conclusion 183

7 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

8 Conclusions and Future Work 206

References 212

Appendix 228

List of Publications 231

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 1 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

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

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

List of Tables

Table 1.1 The dimension of hard disk drives and their applications 4

Table 1.2 Dimensions of PM motors used for different formats of HDDs (in-hub design) 5

Table 1.3 Rotation speeds of spindles and their applications 6

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

Table 6.6 Determination of cogging torque period (p = 4) 170

Table 7.1 Analysis of Variance 189

Table 7.2 Structure parameters of the studied PM motor 192

Table 7.3 Design parameters for testing RSM 193

Table 7.4 ANOVA 193

Table 7.5 Design Parameters 195

Table 7.6 ANOVA 195

Table 7.7 Control factors 199

Table 7.8 Levels for control factors 200

Table 7.9 Design setting matrix 201

Table 7.10 Runs of simulations and results 201

Table 7.12 ANOVA 202

Table 7.11 Coefficient of fitted response 203

Table 7.13 Comparison of the original and optimal design 204

List of Figures

Figure 1.1 Structure of a hard disk drive 2

Figure 1.2 Evolution of hard disk areal density [4] 3

Figure 1.3 One-inch micro-drive 4

Figure 1.4 A typical demagnetization curve of NdFeB permanent magnet 7

Figure 1.5 Cut-plane plot along the shaft axis of an exterior-rotor PM motor 8

Figure 1.6 Cross section view of an exterior-rotor PM motor 8

Figure 1.7 Schematic of run-outs 11

Figure 2.1 The slot model considering flux reduction 21

Figure 2.2 Schwarz-Christoffel transform 22

Figure 2.3 Simulated relative permeance function 23

Figure 2.4 Assumptions when summing up the net forces on slot walls 28

Figure 2.5 Effect of randomly distributed errors in flux density on cogging torque calculation 32

Figure 2.6 Effect of randomly distributed errors in flux density on UMP calculation 32 Figure 2.7 Comparison of flux density results from FEM and exisiting analytical technique 34

Figure 3.1 Studied regions 41

Figure 3.2 Magnetization distribution along the x- direction 43

Figure 3.3 Schematic scalar potential distribution along the stator surface 46

Figure 3.4 Comparison of scalar potential along slot opening when N-S transition over slot opening is present 55

Figure 3.5 Comparison of scalar potential along slot opening when S-N transition over slot opening is present 56

Figure 3.6 Comparison of air gap flux density (s = 0) 57

Figure 3.7 Comparison of air gap flux density (s = 0.2) 57

Figure 3.8 Comparison of air gap flux density (s = 0.5) 58

Figure 3.9 Comparison of air gap flux density (s = 1) 58

Figure 3.10 Structure of radial field PM motors 59

Figure 3.11 Comparison of air gap flux density (wt = 0o) 60

Figure 3.12 Comparison of air gap flux density (wt = 1o) 61

Figure 3.13 Comparison of air gap flux density (wt = 2o) 61

Figure 3.14 Comparison of air gap flux density (wt = 4o) 62

Figure 3.15 Force waveform comparison (b o = 1.7 mm g = 0.25 mm h m = 2 mm) 63

Figure 3.16 Force waveform comparison (b o = 1 mm g = 0.25 mm h m = 1 mm) 63

Figure 3.17 Pole-slot relationship in 8-pole 9-slot PM dc motors 65

Figure 3.18 Superposition of force waveforms in 8-pole 9-slot PM motors 66

Figure 3.19 Superposition of force waveforms in 8-pole 6-slot PM motors 67

Figure 3.20 Superposition of cogging torque profiles in an 8-pole 9-slot motor 68

Figure 3.21 Mesh in the air gap of the studied motor 69

Figure 3.22 Cogging torque simulated by FEM using MST along different air gap paths 69 Figure 3.23 Comparison of cogging torque results of virtual work and Maxwell stress

tensor methods 70

Figure 3.24 Comparison of the results from FEM, superposition method and existing analytical technique 70

Figure 3.25 Superposition of torque profiles in an 8-pole 6-slot PM motor 71

Figure 3.26 Comparisons of the results from FEM, superposition method and existing analytical technique 72

Figure 3.27 Superposition of torque profiles in a 12-pole 9-slot PM motor 73

Figure 3.28 Comparisons of the results from FEM, superposition method and existing analytical technique 73

Figure 4.1 Scalar potential along stator surface with rotor rotation of 5 degrees in an 8-pole 9-slot PM motor 82

Figure 4.2 PM motor topology 83

Figure 4.3 Solution regions 83

Figure 4.4 Magnetization distribution in the tangential direction 84

Figure 4.5 Surface flux density distribution of a radially magnetized magnet sample NP-8L ID-12P with back yoke [126] 85

Figure 4.6 Approximation of the real magnetization pattern in magnets 85

Figure 4.7 Flux lines in an 8-pole 6-slot PM motor 91

Figure 4.8 Comparison of air gap flux density in an 8-pole 6-slot PM motor (αp = 1) 91 Figure 4.9 Error of (B n×B t) comparisons in an 8-pole 6-slot PM motor (αp = 1, rotor position = 1 degree) 92

Figure 4.10 Flux lines in an 8-pole 9-slot PM motor 93

Figure 4.11 Comparison of air gap flux density in an 8-pole 9-slot PM motor (αp = 1) 93

Figure 4.12 Error of (B n ×B t) comparisons in an 8-pole 9-slot PM motor (αp = 1, rotor position = 1 degree) 94

Figure 4.13 Flux lines in an 8-pole 12-slot PM motor 95

Figure 4.14 Comparison of air gap flux density in 8-pole 12-slot PM motor (αp = 1) 95 Figure 4.15 Error of (B n ×B t) comparisons in 8-pole 12-slot PM motor (αp = 1, rotor position = 1 degree) 96

Figure 4.16 Distribution of winding current in two adjacent slots and its equivalent current distribution 97

Figure 4.17 Equivalent current distribution along the stator surface for a single coil 98

Figure 4.18 Solution region for armature reaction field 99

Figure 4.19 Connections of phase windings 101

Figure 4.20 Comparison of armature reaction field of the winding on a single tooth 102 Figure 4.21 Comparison of armature reaction field of the winding on adjacent teeth 102 Figure 4.22 Comparison of back EMF waveforms 104

Figure 4.23 Schematics of rotor eccentricity 105

Figure 4.24 The coordinates to study the effece of rotor eccentricity 106

Figure 4.25 New orthogonal coordinates 108

Figure 4.26 Comparison of radial flux density in the air gap in an 8-pole 9-slot PM motor with 40% rotor eccentricity 111

Figure 4.27 Comparison of tangential flux density in the air gap in an 8-pole 9-slot PM motor with 40% rotor eccentricity 112

Figure 5.1 Cogging torque comparisons in an 8-pole 6-slot PM motor (αp = 1) 122

Figure 5.2 Cogging torque comparisons in an 8-pole 6-slot PM motor (αp = 0.9) 122

Figure 5.3 Cogging torque comparisons in an 8-pole 9-slot PM motor (αp = 1) 123

Figure 5.4 Cogging torque comparisons in an 8-pole 9-slot PM motor (αp = 0.9) 124

Figure 5.5 Cogging torque comparisons in an 8-pole 12-slot PM motor (αp = 1) 124

Figure 5.6 Cogging torque comparisons in an 8-pole 12-slot PM motor (αp = 0.9) 125

Figure 5.7 Peak cogging torque value vs pole-arc to pole-pitch ratio in an 8-pole 6-slot

PM motor with slot opening of 12 degrees 126

Figure 5.8 Peak cogging torque value vs pole-arc to pole-pitch ratio in an 8-pole 9-slot

PM motor with slot opening of 10 degrees 127

Figure 5.9 Peak cogging torque value vs pole-arc to pole-pitch ratio in an 8-pole slot PM motor when slot opening angle is 4 degrees 127

12-Figure 5.10 Peak cogging torque value vs slot-opening to slot-pitch ratio in an 8-pole 6-slot PM motor when the pole-arc to pole-pitch ratio equals to 1 128

Figure 5.11 Peak cogging torque value vs slot-opening to slot-pitch ratio in an 8-pole 9-slot PM motor when the pole-arc to pole-pitch ratio equals to 1 129

Figure 5.12 Peak cogging torque value vs slot-opening to slot-pitch ratio in an 8-pole 12-slot PM motor when the pole-arc to pole-pitch ratio equals to 1 129

Figure 5.13 Cogging torque profile in an 8-pole 9-slot PM motor with static rotor eccentricity of 10% 131

Figure 5.14 Frequency spectrum of cogging torque in an 8-pole 9-slot PM motor in absence of rotor eccentricity 132

Figure 5.15 Frequency spectrum of cogging torque in an 8-pole 9-slot PM motor with static rotor eccentricity of 10% 132

Figure 5.16 Cogging torque profiles in an 8-pole 9-slot PM motor with rotor eccentricity of 10%, with and without whirling speed of 100 % 133

Figure 5.17 Frequency spectrum of cogging torque in an 8-pole 9-slot PM motor with rotor eccentricity of 10% and whirling speed of 100% 133

Figure 5.18 Cogging torque in an 8-pole 12-slot PM motor with and without rotor eccentricity 135Figure 5.19 Maxwell stress tensors in different coordinates 136

Figure 5.20 The x- and y- components of radial force in an 8-pole 9-slot PM motor

when the pole-arc to pole-pitch ratio is 1 137Figure 5.21 Locus of the UMP in an 8-pole 9-slot PM motor when the pole-arc to pole-

pitch ratio is 1 137

Figure 5.22 Radial force vs pole-arc to pole-pitch ratio in an 8-pole 9-slot PM motor when slot opening angle is 10 degrees 138

Figure 5.23 Radial force vs slot-opening to slot-pitch ratio in an 8-pole 9-slot PM motor when the pole-arc to pole-pitch ratio is 1 139

Figure 5.24 UMP in an 8-pole 9-slot PM motor with 10% static rotor eccentricity 140

Figure 5.25 Frequency spectrum of the radial force in an 8-pole 9-slot PM motor without rotor eccentricity 141

Figure 5.26 Frequency spectrum of the radial force in an 8-pole 9-slot PM motor with 10% static rotor eccentricity 141

Figure 5.27 UMP profiles in an 8-pole 9-slot PM motor with rotor eccentricity of 10% and whirling speed of 100% 142

Figure 5.28 Frequency spectrum of the radial force in an 8-pole 9-slot PM motor with rotor eccentricity of 10% and whirling speed of 100% 142

Figure 5.29 UMP in an 8-pole 12-slot PM motor with 10% static rotor eccentricity 143 Figure 5.30 Frequency spectrum of the radial force in an 8-pole 12-slot PM motor with static rotor eccentricity of 10% 144

Figure 5.31 UMP in an 8-pole 12-slot PM motor with 10% rotor eccentricity and 100% whirling speed 144

Figure 5.32 Structure and winding layout of 8-pole 6-slot PM motors 146

Figure 5.33 Structure and winding layout of 8-pole 9-slot PM motors 147

Figure 5.34 Structure and winding layout of 8-pole 12-slot PM motors 148

Figure 5.35 Current flowing in the phase windings 149

Figure 5.36 Ideal current scheme 149

Figure 5.37 Measurement of phase current waveforms 150

Figure 5.38 Eexperimental setup for PM brushless dc motors 151

Figure 5.39 Measured back EMF waveform 152

Figure 5.40 Calculated back EMF waveform 152

Figure 5.41 Measured running torque 153

Figure 5.42 Calculated running torque using ideal current waveform 154

Figure 5.43 Measured phase current waveform 154

Figure 5.44 Calculated running torque using real current waveform 155

Figure 6.1 Flow chart of the Powell’s method 161

Figure 6.2 Cogging torque vs pole-slot combinations (p = 3) 171

Figure 6.3 Cogging torque vs pole-slot combinations (p = 4) 171

Figure 6.4 Three-dimensional plot of peak value of cogging torque vs arc to pole-pitch ratio and slot-opening to slot-pole-pitch ratio in an 8-pole 6-slot PM motor (analytical) 173

Figure 6.5 Three-dimensional plot of peak value of cogging torque vs arc to pole-pitch ratio and slot-opening to slot-pole-pitch ratio in an 8-pole 6-slot PM motor (numerical) 173

Figure 6.6 Contour plot of peak value of cogging torque vs pole-arc to pole-pitch ratio and slot-opening to slot-pitch ratio in an 8-pole 6-slot PM motor 174

Figure 6.7 Three-dimensional plot of peak value of cogging torque vs arc to pole-pitch ratio and slot-opening to slot-pole-pitch ratio in an 8-pole 9-slot PM motor 175

Figure 6.8 Three-dimensional plot of peak value of cogging torque vs arc to pole-pitch ratio and slot-opening to slot-pole-pitch ratio in an 8-pole 12-slot PM motor (analytical) 176

Figure 6.9 Three-dimensional plot of peak value of cogging torque vs arc to pole-pitch ratio and slot-opening to slot-pole-pitch ratio in an 8-pole 12-slot PM motor (numerical) 176

Figure 6.10 Three-dimensional plot of peak cogging torque vs air gap length and magnet thickness in an 8-pole 9-slot PM motor (αp = 1, αo = 10o) 177 Figure 6.11 Three-dimensional plot of peak cogging torque vs air gap length and

magnet thickness in an 8-pole 12-slot PM motor (αp = 1, αo = 8o) 178

Figure 6.12 Three-dimensional plot of running torque vs pole-arc topole-pitch ratio and slot-opening to slot-pitch ratio in an 8-pole 12-slot PM motor 179

Figure 6.13 Three-dimensional plot of the cogging torque to running torque ratio vs pole-arc to pole-pitch ratio and slot-opening to slot-pitch ratio in an 8-pole 6-slot PM motor 179

Figure 6.14 Three-dimensional plot of average radial force vs pole-arc to pole-pitch ratio and slot-opening to slot-pitch ratio in an 8-pole 9-slot PM motor 181

Figure 6.15 Three-dimensional plot of UMP vs air gap length and magnet thickness in

an 8-pole 9-slot PM motor (αp = 1, αo = 10o) 181

Figure 6.16 Three-dimensional plot of radial force ripple vs pole-arc to pole-pitch ratio and slot-opening to slot-pitch ratio in an 8-pole 9-slot PM motor 182Figure 7.1 CCD for 2 variables 188Figure 7.2 CCD for 3 variables 189

Figure 7.3 Response surface of cogging torque over {αt ∈ [32 deg, 35 deg] αp ∈ [0.5, 1]} 192Figure 7.4 Fitted response surface over {αt ∈ [32 deg, 35 deg] αp ∈ [0.5, 1]} 194Figure 7.5 True response surface over {αt ∈ [32 deg, 35 deg] αp ∈ [0.75, 0.95]} 196Figure 7.6 Fitted Response Surface over {αt ∈ [32 deg, 35 deg] αp ∈ [0.75, 0.95]} 196Figure 7.7 Flow chart of the combined analytical and numerical approach 198Figure 7.8 Parameters of tooth structure 198

Figure 7.9 Contour plot of cogging torque to running torque ratio vs arc to pitch ratio and slot-opening to slot-pitch ratio in the 8-pole 6-slot PM motor 200

pole-Figure 7.10 The fitted response vs x1 and x2 203Figure 7.11 Cogging torque comparison of the original and optimal design 204

List of Symbols

e Back-EMF

on the spindle motor which rotates the platters at a constant speed The magnetic head

is installed at the end of the suspension The suspension is controlled by an actuator to locate the magnetic head at different data tracks and to write data along a certain track

Figure 1.1 Structure of a hard disk drive

The most noticeable historic trend in high capacity and high performance hard disk drives has been size reduction, capacity increase, and cost reduction In terms of the storage capacity, reliability and other characteristics, hard disk drives have been improved steadily along with other key components in PCs, such as the central processing unit (CPU) The storage capacity of hard disk drives is measured by the areal density [1], which is defined as the number of bits that can be stored in a unit area

It is usually expressed in bits per square inch (BPSI) The areal density of hard disk platters continues to increase at an amazing pace in recent years In 2005, Seagate has achieved a milestone with an areal density of 100 Gbits/in2 [2] Modern disks are now able to pack as much as 400 Gigabits of data onto a single 3.5" platter [3]

Figure 1.2 Evolution of hard disk areal density [4]

Figure 1.2 shows the progress of areal density over the past several decades The key HDD head technology developments are also indicated It is observed that the areal density increased by almost ten folds every ten years

On the other hand, the form factors of hard disk drives are taking a downsizing trend The 5.25" drives have now all but disappeared from the mainstream PC market, with 3.5" drives dominating the desktop and server segments In the mobile world, 2.5" drives are the standard with smaller sizes becoming more prevalent So far, following the successful commercial 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

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

5.12 5.25"

Oldest PCs, used in servers through the mid-1990s and some retail drives in the mid-to-late 1990s Now obsolete 3.74 3.5"

Standard platter size for the most common hard disk drives used in PCs

The spindle motor is responsible for turning the hard disk platters, allowing the hard drive to operate It must provide stable, reliable and consistent turning power 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 1.2 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 on how the performance could be improved by increasing the spindle speeds The next step up from 3,600 RPM was 4,500 RPM Thereafter, speeds are steadily increasing Spindles running at 7200 RPM are now used in mainstream IDE/ATA drives A SCSI drive with a running speed at 15,000 RPM was announced by Seagate in early 2000 [5] Table 1.3 shows the most common PC spindle speeds, the associated average rotational latency, and their typical applications

Table 1.3 Rotation speeds of spindles and their applications Spindle Speed (RPM) Average Latency♣ (ms) Typical Current Applications

SCSI

♣ Average latency is always equal to one half of the rotational period

The spindle motors are required to meet number of important design factors First, the

motor must be of high quality, such that it can run for thousands of hours, and tolerate

thousands of start and stop cycles without failing Second, it must be able to run

smoothly with minimum vibration, due to the tight tolerances of the platters and the

heads inside the drive Thirdly, it should not generate excessive amounts of heat or

noise Fourth, it should operate with low power losses Finally, its speed must be well

controlled so that the speed variation can be kept at an extremely low level

Permanent magnet (PM) brushless dc motors are adopted in hard disk drives Modern

permanent magnets possess high residual magnetization and high coercivity Thus,

strong flux density can be achieved using a relatively small volume of permanent

magnets In other words, PM brushless motors possess high power density and are

suitable for use in HDDs Polymer-bonded NdFeB magnets are widely used in HDD

spindles The typical demagnetization curve of NdFeB [6] is shown in Figure 1.4 The

second quadrant 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 PM motors, the permanent magnets are bounded to the inner surface of the rotor Figure 1.5 shows the cut-plane view along the shaft axis of an exterior-rotors PM motor, and the cross section view is shown in Figure 1.6

Figure 1.5 Cut-plane plot along the shaft axis of an exterior-rotor PM motor

Figure 1.6 Cross section view of an exterior-rotor PM motor

1.2 Technical Problems Related to Spindle Motors in

Hard Disk Drives

As discussed earlier, the capacity and areal density of hard disk drives are increasing while the size of the platters is decreasing This trend will continue in the future There are two basic techniques to increase the areal density, i.e., to increase the track density

or increase the linear density The track density is a measure of how tightly the concentric tracks on the disk are packed together and is expressed in track per inch (TPI) Linear density is a measure of how closely the bits are packed within a certain track length and is defined as bits per inch (BPI) [1] For an areal density of 100Gbits/in2, the track density is 143 KTPI and the linear density is 700KBPI [7] For the proposed 1Tbits/in2, the target for industry is thriving to achieve in the next 3~5 years, the track density and linear density will be much higher The requirements for the spindles are becoming more and more stringent The narrower track width and the smaller bit length require the platter to rotate much more smoothly and precisely To achieve this target, it is necessary for the magnetic forces generated in spindle motors

be well studied and designed

Two critical design factors 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 are extremely stringent when compared to PM motors for other applications

Cogging torque is the interaction between the permanent magnets and the slot geometry on the stator [13] Compared with slotless spindle motor, the slot geometry changes the uniform permeance distribution along the circumference in air gap When the magnets rotate with the rotor yoke, the net force in the circumferential direction due to the magnetic attractions on the slot walls will vary with the rotor position Therefore, the cogging torque is an inherent phenomenon in PM motors as long as the slots are present The cogging torque will act to accelerate and decelerate the motor from time to time It is one of the sources of undesirable torque pulsation and acoustic noise Usually, the running torque of the PM motors in HDDs is very small (around 2~4 mN⋅m) Hence, the cogging effect can be much more pronounced than that in other type of applications if the cogging torque is not well suppressed

Due to the smaller size and more efficient designs, newer hard disk drives require relatively small amount of power to keep their platters spinning continuously However, when the hard disk is first started up, a higher 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 manufacturability considerations for miniaturized spindles and undesirable axial forces

Unbalanced magnetic pull is generated by the combination of asymmetrical magnetic

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 all directions However, it is not the case in asymmetrical motor structures The magnitude and the phase angle of the net radial force vary with the rotor positions It is a major source of vibration and acoustic noise

in HDDs [14][15] The radial forces will cause the platter depart form a perfect circle, i.e the repeatable and non-repeatable run-outs (RRO and NRRO) [1] Figure 1.7 illustrates schematically the run-outs The two dotted circles represent the perfect tracks and the solid circle is the real track containing RRO and NRRO It should be noted that the design specifications for HDD spindle motors are featured with a very stringent requirement for RRO and NRRO, when compared to other applications Typically, the NRRO of spindle motors used in modern HDDs must be controlled within tens of nanometers

Figure 1.7 Schematic of run-outs

A critical component of the HDD spindle motor that has received much attention is the set of bearings Bearings are precision components that are placed around the shaft of the motor to support it to ensure that the spindle turns smoothly with minimum wobbling or vibration There are several kinds of bearing systems Ball bearings were widely used in early years For such type of bearing, the radial stiffness is relatively high due to the rigidity of the ball bearing However, it generates high level of noise when the rotation speed increases With the size reduction of spindle motors, the aerodynamic bearing and fluid-dynamic bearing (FDB) system were developed In such systems, the balls around the shaft are replaced by a layer of compressed air or oil The noise generated is reduced significantly, but the system becomes more sensitive to radial magnetic forces [16]-[18] In the fluid dynamic bearing system or aerodynamic bearing system, the unbalanced magnetic pull also causes rotor eccentricity, which may result in dynamic problems with additional vibration, noise, and torque pulsation [19]

As discussed above, the cogging torque and UMP of the PM spindles shall be the significant design issues In this research work presented in this dissertation, the motor performances associated with these aspects are to be studied in detail

1.3 Performance Evaluations and Design

Optimization of Spindle Motors

To investigate the spindle motor performances, the magnetic field distribution in the motor is required Normally, there are two ways to achieve such purpose One would

be the widely used numerical simulations, for example, the finite element method The other way would be the analytical methods Numerical simulation tools began to

emerge about 50 years ago and they are very mature nowadays and are able to provide very accurate simulation results It is very useful for failure analysis and design validations Generally, very fine mesh is required to obtain accurate evaluations of product performances We know that the target of investigating motor performance is

to discover the best design scheme that satisfies the desired requirements Therefore, in the design phase, a lot of simulations are necessary In order to obtain a full understanding of the relationship between system performance and the choice of design parameters, a large design space should be explored Hence, the computations can be very costly in the design phase The computing cost is sometimes not affordable

if the time of the product design cycle is of concern To the contrary, analytical method provides a much faster evaluation of the performances at a cost of loosing precision sometimes since the analytical methods are generally suitable for dealing with linear boundary value problems with relatively simple geometry Nevertheless, analytical method can provide good physical insight and is inexpensive to explore very large design space

Since the numerical tools are expensive in the optimization phase where large amount

of computations are required, many optimization schemes were devised to reduce the computations Response Surface Methodology (RSM) in combination with numerical simulations appears to be one of the promising optimization approaches It is based on statistics theories and is able to achieve satisfactory optimization results using minimum sets of system evaluations The limitation of this method is that it is only effective when the parameters vary within a relatively small region and therefore the response surface can be closely modeled by multivariable polynomials However, such region cannot be easily determined in practice In this dissertation, we will explore the

possibility of using analytical solutions to identify realistic operation region in which the RSM can be effectively employed In this way, the effectiveness of the optimization can be highly increased

1.4 Organization of the Dissertation

As discussed in this chapter, it is clearly that the 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 the research work 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 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

3) To develop a combined analytical and numerical optimization methodology to minimize cogging torque and UMP in permanent magnet motors used in ultra-high density magnetic recording systems

In this dissertation, the contents will be arranged as follows:

Chapter 2 provides a review of the previous works on calculations of the magnetic fields and magnetic forces in permanent magnet motors

Chapter 3 will introduce a new mathematical model to account for the effect of magnet pole transition passing over slot opening This newly developed model can provide more accurate results of tangential flux density in the air gap An accurate prediction of the electromagnetic force waveforms can be obtained using the new analytical model

A superposition approach is introduced to estimate the cogging torque waveform

Chapter 4 will derive the analytical solutions for the air gap fields in PM spindle motors based on the slot model in Chapter 3 The scalar magnetic potential distribution along the stator surface derived from the new slot model will be used as the boundary condition for the stator surface Then, the no-load magnetic field, the armature reaction field and back EMF expressions in slotted PM motors will be derived The magnetic field in the PM motor with rotor eccentricity will also be presented It is demonstrated that the new analytical method can provide significantly more accurate prediction of the air gap field than the previously existing analytical techniques

Chapter 5 will derive the analytical expressions of the magnetic forces in PM spindle

motors including the cogging torque, unbalanced magnetic pull and running torque The relationship between the cogging torque/UMP and design parameters such as pole-arc to pole-pitch ratio, slot-opening to slot-pitch ratio, air gap length and magnet thickness will be investigated and the effect of rotor eccentricity on the cogging torque and UMP in different types of PM motors will also be studied With accurate prediction of air gap fields, the electromagnetic forces can be computed with high accuracy in such special conditions of motor operation

In Chapter 6, efforts will be made to minimize the cogging torque and unbalanced magnetic pull using analytical methods The objective functions are determined and a suitable method is selected to perform the minimization Taking advantage of the analytical solution, the relationship between the response and design parameters is clearly illustrated, which is useful for the determination of the optimal setting of parameters This is the base of the optimization method discussed in Chapter 7

Chapter 8 draws the conclusions from the research presented in this dissertation and contains some suggestions for future work

2 Review on Computer Modeling and

Analysis of PM Brushless DC Motors

2.1 Electromagnetic Field in PM Brushless DC

As a result, the magnetic field problem is translated into solving a large number of

simultaneous linear equations Accurate predictions of magnetic fields can be obtained using FEM with complex geometry and non-linear material properties taken into consideration Therefore, it is widely adopted for calculating the magnetic fields and forces in PM spindle motors [24]-[37] However, FEM is more suitable for validation purposes in product design procedure When it is utilized to perform calculations for optimization purpose, where a large amount of computations is required to explore a wide design space, the computation load can be too heavy to afford [38] Nevertheless, FEM is still an accurate method and it is frequently used as a means for verifying computation results obtained from other methods such as analytical methods

Analytical method is an alternative technique for calculating the magnetic field distributions Algebraic expressions of the field quantities are sought for within the studied domain Usually, the analytical methods are much faster than numerical methods in terms of computation time, which results in convenience in design optimization [39] Another advantage of analytical methods is that they can be helpful

to reveal the physical insight of the problem under investigation and it is easier to study the relationship between the system response and the design parameters Therefore, development of effective analytical approaches for PM motor design syntheses continues to attract great attention from researchers [40]-[47]

A number of works have been done to present the analytical solutions for magnetic field distribution in PM machines An analytical solution of the air gap field for slotless PM motors was reported by Boules [48] In his work, the magnets were represented by equivalent surface current sheets, and the corresponding two-dimensional field distributions were described Zhu [49] also provided analytical

solutions for the magnetic fields in slotless PM motors In his approach, the permanent magnets were expressed as magnetization distribution along the circumference direction Analytical field solutions were also reported in other types of PM slotless motors such as tubular PM motors [50]

In the slotless model, the scalar magnetic potential distribution in the air gap can be expressed in polar coordinates as

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