DESIGN AND ANALYSIS OF SCULPTED ROTOR INTERIOR
PERMANENT MAGNET MACHINES
By
Steven Lee Hayslett
A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
Mechanical Engineering - Doctor of Philosophy
2022
ABSTRACT
DESIGN AND ANALYSIS OF SCULPTED ROTOR INTERIOR PERMANENT
MAGNET MACHINES
By
Steven Lee Hayslett
Design of interior permanent magnet electrical machines is complex. Interior permanent
magnet machines offer a good balance of cost, efficiency, and torque/power density. Max-
imum torque and power production of an interior permanent magnet machine is achieved
through balancing design choices related to the permanent magnet and salient features. The
embedded magnet within the salient structure of the rotor lamination results in an increase
in harmonic content. In addition, interaction of the armature, control angle, and rotor re-
luctance structure creates additional harmonic content. These harmonics result in increased
torque ripple, radial forces, losses, and other unwanted phenomena. Further improvements
in torque and power density, and techniques to minimize harmonics, are necessary. Typical
interior permanent magnet machine design results at the maximum torque per amp con-
dition are at neither the maximum magnet nor maximum salient torque, but at the best
combination of the two. The use of rotor surface features to align the magnet and the reluc-
tance axis allows for improvement of torque and power density. Reduction of flux and torque
harmonics is also possible through careful design of rotor sculpt features that are included
at or near the surface of the rotor.
Finite element models provide high fidelity and accurate results to machine performance,
but do not give insight into the relationship between design parameters and performance.
Winding factor models describe the machine with a set of Fourier series equations, providing
access to the harmonic information of both parameters and performance. Direct knowledge of
this information provides better insight, clear understanding of interactions, and the ability
to develop a more efficient design process. A new analytical winding function model of the
single-V IPM machine is introduced, which considers the sculpted rotor and how this model
can be used in the design approach of machines.
Rotor feature trends are established and utilized to increase design intuition and reduce
dependency upon the lengthy design of experiment optimization processes. The shape and
placement of the rotor features, derived from the optimization process, show the improvement
in torque average and torque ripple of the IPM machine.
Copyright by
STEVEN LEE HAYSLETT
2022
ACKNOWLEDGMENTS
First, I would like to express my deepest thanks to Dr. Elias Strangas for his patience,
support, and guidance in enabling me to complete my Ph.D. program. It has been a great
experience working with him over the past years. I have learned much and gained perspective
from him.
Second, I would like to pay my special regards to the members of the Electric Machines
and Power Electronics Research (EMPowER) Laboratory at Michigan State University.
Thank you, Dr. Thang Pham, Dr. Anmol Aggarwal, Matt Meier, Tiraruek Ruekamnu-
aychok, Cristian Lopez Martinez, Dr. Reemon Haddad, Dr. William Jensen, Dr. Abdullah
Alfehaid, and Dr. Shanelle Foster.
I would also like to thank my Ph.D. guidance committee members Dr. Andre Benard,
Dr. Abraham Engeda, and Dr. George Zhu.
This would have not been possible without the endless support of my wife Ratsada and
son Tyler. In addition, I am grateful to my parents Lyle and Marge for encouraging me to
pursue a lifetime of learning.
Finally, I would like to pay my special regards to my inspiring colleagues and mentors,
Dr. Sinisa Jurkovic, Dr. Khawja Rahman, Dr. Yochan Son, Dr. Tausif Husain, and Peter
Savagian.
v
TABLE OF CONTENTS
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x
KEY TO SYMBOLS AND ABBREVIATIONS . . . . . . . . . . . . . . . . . xv
Chapter 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Objective and Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3.1 Transportation Sector Trends . . . . . . . . . . . . . . . . . . . . . . 5
1.3.2 Interior Permanent Magnet Machines . . . . . . . . . . . . . . . . . . 6
1.3.3 Rotor Modifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3.4 Analytical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.3.4.1 Review of Recent Analytical Models . . . . . . . . . . . . . 14
1.4 Organization of Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
Chapter 2 Fundamentals: Magnetics . . . . . . . . . . . . . . . . . . . . . . . 17
2.1 Stator with Sinusoidally Distributed Windings . . . . . . . . . . . . . . . . . 19
2.2 Salient Rotor Geometry Analysis . . . . . . . . . . . . . . . . . . . . . . . . 20
2.3 Permanent Magnets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.3.1 Permanent Magnet Properties and Equivalent Current . . . . . . . . 24
2.3.2 Simplified Permanent Magnet Analysis . . . . . . . . . . . . . . . . . 30
2.4 Magnetic Dipole and Equivalent Magnetic Currents . . . . . . . . . . . . . . 32
2.5 Torque Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.6 Flux Linkage and Inductance . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.7 First Analysis of Machine Design . . . . . . . . . . . . . . . . . . . . . . . . 38
2.7.1 Ideal Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.7.2 Magnet Geometry Effects . . . . . . . . . . . . . . . . . . . . . . . . 42
Chapter 3 Fundamentals: Implementation and Control . . . . . . . . . . . 46
3.1 Voltage Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.1.1 Voltage Equations in ABC . . . . . . . . . . . . . . . . . . . . . . . . 48
3.1.2 The Clarke Transformation: Voltage Equations in αβ . . . . . . . . . 50
3.1.3 The Park Transformation: Voltage Equations in DQ Coordinates . . 51
3.2 Torque Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.3 First Analysis of Machine Performance and Control . . . . . . . . . . . . . . 56
3.3.1 Example Machine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.4 Relationship to Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
Chapter 4 MMF Permeance Theory . . . . . . . . . . . . . . . . . . . . . . . 64
vi
4.1 Air Gap Permeance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.2 Rotor Reluctance Path . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.3 Stator Slots by Permeance Function . . . . . . . . . . . . . . . . . . . . . . . 68
4.4 Windings, MMF, Linear Current Density . . . . . . . . . . . . . . . . . . . . 70
4.4.1 Stator MMF Modifications for the Second Reluctance Path . . . . . . 71
4.5 Sculpted and Slotted Features by MMF Function . . . . . . . . . . . . . . . 73
4.5.1 Application of Superposition for Slotted Features . . . . . . . . . . . 74
4.5.2 Magnet MMF and Sculpted Rotor . . . . . . . . . . . . . . . . . . . . 76
4.5.3 MMF of the Second Reluctance Path due to Rotor Sculpting . . . . . 79
4.6 Flux Linkage and Inductance . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.7 Example Machine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
Chapter 5 Space and Time Harmonics of Fields, Permeance, and Torque 91
5.1 Fourier Series Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.1.1 Current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.1.2 Permeance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5.1.3 Winding and Turns Functions . . . . . . . . . . . . . . . . . . . . . . 93
5.1.4 MMF and Linear Current Density . . . . . . . . . . . . . . . . . . . . 94
5.2 Rotating Harmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
5.2.1 Armature MMF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
5.2.2 Primary Reluctance Path Flux Density Harmonics . . . . . . . . . . . 96
5.2.3 Linear Current Density Harmonics . . . . . . . . . . . . . . . . . . . 97
5.2.4 Torque Harmonic Orders from Air Gap Flux Density and Linear Cur-
rent Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.2.5 Magnet Flux Density and Torque Harmonics . . . . . . . . . . . . . . 100
5.2.6 Field and Torque Harmonics Relationship to Control Angle . . . . . . 101
5.3 Example Machine Mechanical Harmonics . . . . . . . . . . . . . . . . . . . . 103
Chapter 6 Design for Minimal Torque Ripple . . . . . . . . . . . . . . . . . 106
6.1 Review of Design Methods and Features . . . . . . . . . . . . . . . . . . . . 106
6.2 Application of Analytical Model to Example Machine . . . . . . . . . . . . . 107
6.2.1 Sculpting Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
6.3 Model Validation: Radial Flux Density . . . . . . . . . . . . . . . . . . . . . 109
6.4 Model Validation: Torque Ripple . . . . . . . . . . . . . . . . . . . . . . . . 111
6.5 Torque Ripple Components . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
6.6 Investigation of Design Features . . . . . . . . . . . . . . . . . . . . . . . . . 114
6.6.1 Magnet Pole Arc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
6.6.2 Single Pair Symmetrical Rotor Sculpt Feature . . . . . . . . . . . . . 115
6.6.3 Single Asymmetrical Rotor Sculpt Feature . . . . . . . . . . . . . . . 117
6.6.4 Two Symmetrical Rotor Sculpt Features . . . . . . . . . . . . . . . . 117
6.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
Chapter 7 Optimal Design for Minimal Torque Ripple . . . . . . . . . . . . 122
7.1 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
7.2 Electric Machine Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
vii
7.3 Minimization of Torque Harmonics . . . . . . . . . . . . . . . . . . . . . . . 123
7.3.1 Optimization Methodology . . . . . . . . . . . . . . . . . . . . . . . . 124
7.4 Optimal Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
7.5 Finite Element Detailed Search of Design Space . . . . . . . . . . . . . . . . 129
7.6 Exhaustive Search . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
7.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
Chapter 8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
8.1 Closing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
8.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
viii
LIST OF TABLES
Table 3.1: Example Machine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
Table 4.1: Example Motor Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 82
Table 5.1: Armature MMF Harmonic Sequencing Summary, negative sequence (+),
positive sequence (-), not applicable (n/a). Not applicable is due to cancel-
lation of triplen harmonics of the 3 phases or even MMF components are
not possible due to convolution. . . . . . . . . . . . . . . . . . . . . . . . 97
Table 5.2: Linear Current Density Electrical Harmonic Sequence Value . . . . . . . 98
Table 5.3: Air Gap Field Spatial Electrical Harmonic Convolution, Horizontal - Flux
Density Harmonics, Vertical - Linear Current Density Harmonics . . . . . 99
Table 5.4: Air Gap Field Spatial Electrical Harmonic Convolution and Electrical Torque
Harmonic, Horizontal - Flux Density Harmonics, Vertical - Linear Current
Density Harmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
Table 5.5: Airgap flux density Fourier coefficients dependency on reference frame cur-
rents. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
Table 5.6: Mechanical Harmonics of Example Machine . . . . . . . . . . . . . . . . . 105
Table 6.1: Example motor parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . 107
Table 6.2: Two sculpt feature parameters. . . . . . . . . . . . . . . . . . . . . . . . . 118
Table 7.1: Motor Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
Table 7.2: Analytical Model Sculpt Feature Designs . . . . . . . . . . . . . . . . . . 127
ix
LIST OF FIGURES
Figure 1.1: Interior Permanent Magnet Machine . . . . . . . . . . . . . . . . . . . . 7
Figure 1.2: A Single Pole of an Interior Permanent Magnet Machine: a) slot, b) tooth,
c) stator yoke, d) winding/conductor, e) rib (secondary reluctance path),
f) bridge, g) stator OD, h) stator ID, i) rotor OD, j) barrier, k) rib (primary
reluctance path), l) magnet, n) airgap . . . . . . . . . . . . . . . . . . . 8
Figure 1.3: IPM rotor types: single V (left), delta (center), double V (right). . . . 10
Figure 1.4: Example Interior Permanent Magnet Machine with Rotor Sculpt Features 11
Figure 2.1: Radial and Tangential Flux Density in Air Gap . . . . . . . . . . . . . . 18
Figure 2.2: Sinudsoidally Distributed Three Phase Windings . . . . . . . . . . . . . 20
Figure 2.3: Doubly Sinusoidal Air Gap: The salient rotor is shown in blue and the
stator in yellow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
Figure 2.4: Integration Path for Ampere’s Law, Current Sheet (red) . . . . . . . . . 23
Figure 2.5: Magnetization Curve [1] . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
Figure 2.6: Demagnetization Curve [1] . . . . . . . . . . . . . . . . . . . . . . . . . . 26
Figure 2.7: Permanent Magnet Model: Am area of magnet, Ag area of gap, Bm mag-
net flux density, Hm magnet field strength, Bg airgap flux density, Hg air
gap field strength, lm length of magnet, g length of airgap, N1 I1 amp turns 27
Figure 2.8: Equivalent Current Permanent Magnet Model: Am area of magnet, Ag
area of gap, Bm magnet flux density, Hm ′ magnet field strength, B airgap
g
flux density, Hg air gap field strength, lm length of magnet, g length of
airgap, N1 I1 amp turns . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
Figure 2.9: Magnet Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
Figure 2.10: Flux Paths: Crossing the Air Gap (red) , Leakage (blue) . . . . . . . . . 31
Figure 2.11: Magnetic Dipole From Current Loop [2] . . . . . . . . . . . . . . . . . . 34
Figure 2.12: Inegration Path for Flux and Flux Linkage . . . . . . . . . . . . . . . . 38
x
Figure 2.13: Basic Geometry, δ = 45◦ : toroidal core (yellow), rotor (orange), north
magnets(red), south magnet(blue) . . . . . . . . . . . . . . . . . . . . . 39
Figure 2.14: Theoretical Torque Versus Current Angle IPM Machine , δ = 0◦ : Reluc-
tance Torque (blue), Magnet Torque (red), Total (black dashed) . . . . 40
Figure 2.15: Theoretical Torque Versus Current Angle IPM Machine , δ = 45◦ : Re-
luctance Torque (blue), Magnet Torque (red), Total (black dashed) . . . 41
Figure 2.16: Torque Benefit of Ideal Aligned Axis Design as Compared to Traditional
Alignment (% increase) . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
Figure 2.17: Torque Benefit as Function of Alignment Angle and Magnet Width as
Percentage of Pole Span: ideal case torque (black dashed), 100% magnet
span torque (red), 66% magnet span torque (green), 50% magnet span
torque (blue) , 50% magnet span torque (blue), magnet physical vs actual
shift angle (blue dashed) , magnet physical vs actual shift angle (blue
dashed dotted) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
Figure 3.1: Motor Structure with Voltage Eq . . . . . . . . . . . . . . . . . . . . . . 47
Figure 3.2: Single Coil Reluctance System . . . . . . . . . . . . . . . . . . . . . . . . 53
Figure 3.3: Example Machine Current Plane δ = 0◦ , current limit (blue circle), volt-
age limit (green ellipse), maximum torque per ampere (MTPA) trajectory
(orange curve), maximum speed of MTPA (orange *), maximum speed at
full current (purple x), short circuit current (purple circle). . . . . . . . 59
Figure 3.4: Example Machine Current Plane δ = 45◦ ,current limit (blue circle), volt-
age limit (green ellipse), maximum torque per ampere (MTPA) trajectory
(orange curve), maximum speed of MTPA (orange *), maximum speed at
full current (purple x), short circuit current (purple circle). . . . . . . . . 60
Figure 3.5: Example Machine Torque and Power Speed Curves δ = 0◦ . . . . . . . . 61
Figure 3.6: Example Machine Torque and Power Speed Curves δ = 45◦ . . . . . . . . 62
Figure 4.1: Permeability of IPM Rotor . . . . . . . . . . . . . . . . . . . . . . . . . . 67
Figure 4.2: Stator slots physical and air gap model: tooth width wth , slot width ws ,
slot height hs , air gap g, slot pitch τs , tooth percenatage of slot width γ,
effective slot height hs1 , stator coordinate α, effective stator air gap gs . 69
Figure 4.3: Constant air gap machine [3] . . . . . . . . . . . . . . . . . . . . . . . . 71
xi
Figure 4.4: MMF modifications for secondary reluctance paths: MMF of the arma-
ture(blue), average MMF across the 2nd reluctance path (green), modified
second reluctance path MMF (modified second reluctance path MMF =
MMF of the armature - average MMF across the 2nd reluctance path). . 72
Figure 4.5: Useage of Superposition and Equivalent Magnetization Currents (a) slot-
ted stator (b) toroidal stator (c) stator with EMC (d - e) modified EMC [4] 75
Figure 4.6: Magnet Pole Equivalent Winding Factor . . . . . . . . . . . . . . . . . . 77
Figure 4.7: Scupted Rotor Superposition (a) sculpted rotor (b) unsculpted rotor (c)
sculpted rotor equivalent magnetic currents . . . . . . . . . . . . . . . . 77
Figure 4.8: Sculpted Rotor Winding Function . . . . . . . . . . . . . . . . . . . . . 78
Figure 4.9: Sculpted Rotor Magnet Pole Flux Density . . . . . . . . . . . . . . . . . 78
Figure 4.10: Sculpted Rotor Reluctance Counter Dipole Current . . . . . . . . . . . . 79
Figure 4.11: Example Motor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
Figure 4.12: Example Motor Air Gap Permeance Functions Versus Stator Coordinate 84
Figure 4.13: Example Motor Stator Winding Functions Versus Stator Coordinate . . 85
Figure 4.14: Example Rotor Equivalent Magnet Winding Functions Versus Stator Co-
ordinate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
Figure 4.15: Example Stator Conductor Width Function Versus Stator Coordinate . 86
Figure 4.16: Analytically Calculated Stator MMF for Primary and Secondary Reluc-
tance Paths Versus Stator Coordinate . . . . . . . . . . . . . . . . . . . 87
Figure 4.17: Analytically Calculated Motor Radial Flux Density Versus Stator Coor-
dinate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
Figure 4.18: Example Motor Magnet Sculpt Feature Winding Function Versus Stator
Coordinate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
Figure 4.19: Example Motor Reluctance Sculpt Feature MMF Versus Stator Coordi-
nate, MMF due to sculpt feature based on primary reluctance path (blue),
MMF due to sculpt feature based on secondary reluctance path (red) . . 89
Figure 4.20: Example Motor Flux Density Sculpted vs Smooth Versus Stator Coordinate 89
xii
Figure 4.21: Example Motor Torque Ripple Versus Rotor Position Using MMF Perme-
ance Model and Maxwell Stress Tensor . . . . . . . . . . . . . . . . . . . 90
Figure 5.1: Permeance and Winding Functions . . . . . . . . . . . . . . . . . . . . . 93
Figure 6.1: Rotor sculpt features. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
Figure 6.2: Radial flux densities with sinusoidally distributed windings (N = 200) at
various currents and control angles. . . . . . . . . . . . . . . . . . . . . . 110
Figure 6.3: Radial flux densities with distributed windings (2 SPP) at various currents
and control angles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
Figure 6.4: Radial flux densities with single symmetrical sculpt feature located at
τ1 = 50%, W1 = 10%, and D1 = 1.2 mm. . . . . . . . . . . . . . . . . . 111
Figure 6.5: Radial flux densities (reluctance only) with single asymmetrical sculpt
feature located at τ = 50%, W = 20%, and D = 1.2 mm. . . . . . . . . 111
Figure 6.6: Smooth rotor IPM torque ripple. . . . . . . . . . . . . . . . . . . . . . . 112
Figure 6.7: Sculpt feature torque ripple. . . . . . . . . . . . . . . . . . . . . . . . . . 112
Figure 6.8: Analytical model torque ripple components at various control angles. . . 113
Figure 6.9: Effects of magnet pole arc τp loaded at Iss = 200 and β = 135◦ . . . . . . 115
Figure 6.10: Symmetrical sculpt feature effects compared to finite elements: D1 = 1.2
mm, W1 = 5%. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
Figure 6.11: Symmetrical sculpt feature effects with phasor diagram: D1 = 1.2 mm,
W1 = 9%. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
Figure 6.12: Sculpt feature effects N = 1. . . . . . . . . . . . . . . . . . . . . . . . . . 118
Figure 6.13: Single asymmetrical sculpt feature effects with phasor diagram: D1 = 1.2
mm, W1 = 5%. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
Figure 6.14: Two sculpt features of 12th order electrical torque phasor plot. . . . . . . 120
Figure 7.1: Two pole equivalent Scrupted Rotor IPM model . . . . . . . . . . . . . . 123
Figure 7.2: Rotor Sculpt Feature, bridge stress (red arrow), sculpt stress (green arrow) 125
Figure 7.3: Rotor Stresses, 5000 rpm, D1=2mm, Z1=0.1 . . . . . . . . . . . . . . . 126
xiii
Figure 7.4: Comparison of Sculpt Feature Effects to Average Torque (FE and Model),
Iss = 200A, β = 112◦ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
Figure 7.5: Comparison of Sculpt Feature Design Effects to Torque Ripple (FE and
Model), Iss = 200A, β = 112◦ . . . . . . . . . . . . . . . . . . . . . . . . 128
Figure 7.6: FE Torque Comparison of Analytical Model Optimized Sculpt Features,
Iss = 200A, β = 180◦ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
Figure 7.7: Finite Element Extended Search of Design 2, Iss = 200A, β = 112◦ ,
Constraint: Dashed Red Line, Analytical Design: Red * . . . . . . . . . 130
Figure 7.8: Finite Element Extended Search of Design 3, Iss = 200A, β = 112◦ ,
Analytical Design: Red * . . . . . . . . . . . . . . . . . . . . . . . . . . 131
Figure 7.9: Finite Element Extended Search of Design 4, Iss = 200A, β = 112◦ ,
Constraint: Dashed Red Line, Analytical Design: Red * . . . . . . . . . 131
Figure 7.10: Finite Element Exhaustive Search, Order = 6, Iss = 200A, β = 112◦ ,
Design 2 (red), Design 3 (green), Design 4 (yellow) . . . . . . . . . . . . 132
Figure 7.11: Finite Element Exhaustive Search, Order = 12, Iss = 200A, β = 112◦ ,
Design 2 (red), Design 3 (green), Design 4 (yellow) . . . . . . . . . . . . 132
Figure 7.12: Finite Element Exhaustive Search, Order = 0, Iss = 200A, β = 112◦ ,
Design 2 (red), Design 3 (green), Design 4 (yellow) . . . . . . . . . . . . 133
xiv
KEY TO SYMBOLS AND ABBREVIATIONS
Iss phase current (peak)
β control angle
p pole pairs
id d-axis current
iq q-axis current
Ia phase a current
Ib phase b current
Ic phase c current
If magnet equivalent field current
θ stator or rotor azimuthal coordinate
ϕ rotor position
δ magnet offset
l stack length
g air gap dimension
Λr1 primary reluctance path permeance
Λr2 secondary reluctance path permeance
Λm magnet path permeance
Na phase a winding factor
Nb phase b winding factor
Nc phase c winding factor
Nr equivalent magnet winding factor
na phase a turns function
nb phase b turns function
nc phase c turns function
Ca phase a conductor density function
xv
Cb phase b conductor density function
Cc phase c conductor density function
Fa phase A MMF
Fb phase B MMF
Fc phase C MMF
FP M rotor magnet MMF
Fabc 3 phase MMF
Fr2 modified Fabc for 2nd reluctance path
J volume current density
K linear current density
Kabc linear current density of 3 phase windings
Br1 (θ, ϕ) primary reluctance path radial flux density
Br2 (θ, ϕ) secondary reluctance path radial flux density
Bm (θ, ϕ) magnet path radial flux density
Tr1 primary reluctance torque
Tr2 secondary reluctance torque
Tm magnet torque
ld d- axis inductance
lq q- axis inductance
λd d- axis flux linkage
λq q- axis flux linkage
λm d- axis flux linkage of magnet
Rx radius
Hx magnetic field intensity
µ◦ permeability of free space
A area
MMF magnetomotive force
xvi
EMF electromotive force
PM permanent magnet machine
PMAC permanent magnet alternating current machine
IPM interior permanent magnet machine
IM induction machine
EMC equivalent magnetization current
MEC magnetic equivalent circuit
FE finite element
xvii
Chapter 1
Introduction
1.1 Problem Statement
The IPM machine is increasingly being utilized throughout the transportation industry as
a primary source of propulsion due to its good efficiency, torque, and power density. IPM
machines bury the magnet within the rotor to achieve a balance of cost, performance, capa-
bility, efficiency, and power and torque density. Ideally, only a constant torque is produced
from a sinusoidal distribution of airgap flux densities. The embedding of the magnet within
the salient structure of the rotor lamination, coupled with distributing the winding in dis-
crete locations results in airgap flux density harmonics. These harmonics result in torque
pulsations, radial force harmonics, losses, and other unwanted phenomena. Torque ripple is
important to minimize as it is one of the main causes of vibration, premature failure, drive-
line oscillations, and acoustic noise. The goal of this work is to minimize torque harmonics
of IPM machines.
Electric machine design choices to minimize torque ripple are often at odds with increasing
torque density or decreasing manufacturing cost. Discrete choices within the armature and
rotor must be made to improve torque ripple trade off. Basic design features are shown in
fig. 1.2. The armature choices include both number of slots and how the winding is placed
within those slots. The rotor choice is limited to the number of barriers and the strength
1
of the magnet. Interactions of the rotor and stator discrete choices are important and have
been well researched. The discrete choices in the machine alone are not enough to address
the minimization of torque pulsations, and further work is needed. The focus of this work
will be on the design of features at the surface of the rotor, sculpting features, to minimize
torque pulsations while preserving average torque.
Finite element analysis has been the tool of choice for the design of IPM machines. This is
because finite element analysis accurately predicts experimental results across a broad range
of machines and has been utilized to evaluate and compare machine types and validate
analytical solutions. Often, finite elements have been coupled with optimization methods,
requiring large sets of design of experiments. The amount of computational resources is
significant. This research seeks to develop computationally efficient methods which reduce
computational effort and yield an accurate solution.
Strategic use of symmetrical rotor sculpt features provides an additional degree of free-
dom in design to improve torque ripple in IPM machines at a cost of torque average. En-
hancements in rotor sculpt features has the potential to preserve the cost of torque average.
Analytical solutions can be applied for analysis with a reduced computational effort. How-
ever, extension to design optimization is often overlooked as analytical methods may not
capture all physical effects, such as fringing or saturation. To date, no analytical methods
have addressed the minimization of torque ripple by the design of rotor sculpt features. The
development of an accurate and low cost model for rotor sculpt features is needed to reduce
reliance on computationally expensive heuristic finite element methods.
2
1.2 Objective and Contributions
An analytical approach to minimize IPM machine torque ripple through the optimal design
of asymmetrical sculpt features is presented in this thesis. Rotor sculpting is shown to affect
both average torque and torque pulsations. Torque pulsations from a smooth IPM’s rotor
are minimized utilizing this new analytical model to design sculpt features. The model ad-
equately predicts performance. It not only minimizes specific harmonics of torque but also
finds the optimal shape of the sculpt feature taking advantage of the computational efficiency
of analytical models. An example IPM machine is modeled with this framework’s perme-
ance functions, winding functions, current sources. A novel winding function framework is
presented based upon the IPM machine’s flux sources: the first reluctance path, the second
reluctance path, and the magnet path as shown in fig. 1.1. New to this framework are the
necessary modifications to armature Magneto Motive Force (MMF) to calculate the effects
of the equipotential nature of the rotor’s salient features. The winding function method is
extended to account for rotor surface modifications utilizing an additional MMF term. The
optimization problem is defined with the objective to minimize specific orders of torque rip-
ple. The analytical model along with optimization is used to make a smooth rotor machine
better with the design of sculpting features. Starting from the FE results of the smooth ro-
tor, the torque harmonics are determined and used to set a target for the analytical method
to cancel. The analytical method is used to determine the required sculpt geometry in order
to cancel the targeted torque harmonic. The analytical model is validated through FE. This
is because it has been well established finite elements accurately predicts experimental re-
sults across a broad range of machines [5–8] and have been utilized to evaluate and compare
machine types [8–11] and validate analytical solutions [12–22]. Both smooth and sculpted
3
rotor airgap flux densities and torque are presented and compared. Three sculpted rotor
designs were developed with the analytical model. Analytical design accuracy is validated
first by comparing analytical performance to finite elements. Secondly, the geometries of the
analytical optimum are compared to the FE optimum. Significant computational savings
are demonstrated using the analytical model to minimize torque harmonics.
To achieve the above-stated objective, the contributions of this work are:
• Voltage and Torque equations considering the impacts of asymmetrical features to the
IPM machine.
• An analytical winding factor model for the IPM machine accounting for the first re-
luctance path, second reluctance path, and magnet path.
• An analytical winding factor method allowing for the accurate modeling of rotor sculpt
features.
• Demonstration of the analytical methods accuracy and computational efficiency bench-
marked with finite elements.
• A design technique to design sculpt features through vector summation to achieve
desired counter-torque amplitude and phase.
• An analytical method to minimize torque harmonics using asymmetrical rotor features
demonstrated to achieve computational efficiency better than and accuracy similar to
finite elements.
4
1.3 Background
1.3.1 Transportation Sector Trends
Recent emphasis on the impacts of the transportation industry has generated interest in
technologies that reduce energy consumption. As a result, the industry is rapidly evolving
around the electrification of the propulsion system, in particular automobiles and aeroplanes.
The automotive industry has focused on development of battery electric vehicles [23, 24],
plugin hybrid electric vehicles [25], and hybrid electric vehicles [26,27]. Similarly, aircraft are
being converted to replace propulsion and any number of types of onboard energy systems,
including hydraulic, mechanical, and pneumatic [28].
The electrified propulsion system consists of a battery pack, inverter, and electrical ma-
chine. The battery pack stores and converts energy electrochemically through direct current.
The inverter converts electrical direct current/voltage from the battery pack to an alternat-
ing current/voltage to the terminals of the electrical machine. The electrical machine, or
motor, performs electromechanical energy conversion. As it is still early in the technology life
cycle of electric cars and planes, many improvements must be made to gain wide acceptance.
Performance and financial hurdles remain for the entire system.
The choice in machine type is important with common choices being: permanent magnet
machines (PM), interior permanent magnet machines (IPM), and induction machines (IM)
each having a known possibility. Surface mount permanent magnet machines achieve high
torque density but have speed limits due to magnet retention, magnet losses, and a high
magnet cost [29, 30]. Induction machines save on material cost, with no need for rare earth
materials, but have lower torque density, lower efficiency, and require special measures for the
increased loss in the rotor. Interior permanent machine machines, bury the magnet interior
5
to the machine and achieve a balance of cost, efficiency, torque/power density [31]. The
interior permanent magnet machine has been at the center of attention for the development
of these future systems [23–26].
Highly efficient, torque and power-dense electric machines are needed for these future
applications [32] as this drives cost savings for both the system and the machine. Electrical
machine cost is fundamentally dependent upon the machine’s volume, materials, and man-
ufacturing process. Diameter and length selections may be made with T ∝ D2 L, where
T is torque, D is diameter, and L is length, based on prior designs while considering the
need for improved torque and power density [33]. Decreasing the volume of the electrical
machine requires shifting the power of the machine to higher speeds, reducing the torque of
the machine.
1.3.2 Interior Permanent Magnet Machines
A two dimensional illustration of a four-pole IPM machine is contained in fig. 1.1. The IPM is
fundamentally constructed of two components a stator and a rotor. The stator, often referred
to as the armature, is the mechanically grounded part of the machine. It is constructed of
slots, teeth, a yoke, and three phase windings. The armature teeth and yoke are constructed
of a magnetically permeable iron alloy. The teeth and yoke allow for easy flow of magnetic
flux to and from the air gap of the machine. The slots allow space for the copper windings.
The windings are distributed within the slots and produce a Magneto Motive Force (MMF)
which in turn creates the distribution of current dependant radial magnetic flux density. The
placement of the windings also creates a linear current density along the bore of the stator,
creating a tangential component of flux density. When arranged and controlled properly the
windings produce a rotating set of fields to produce torque.
6
Figure 1.1: Interior Permanent Magnet Machine
As shown in fig. 1.2 the rotor, the mechanically rotating part is constructed of an iron
core, barriers, and permanent magnets. The iron core is constructed of ribs and bridges. The
ribs control the distribution of flux density while the bridges mechanically couple all parts
together. Barriers provide air pockets, which assist the ribs in directing flux, and contain
embedded magnets. The permanent magnets are embedded within the rotor and produce
an MMF which is independent of the current.
From the perspective of the rotor, a direct axis (d-) and a quadrature axis (q-) of the
machine are electromagnetically defined. The d- axis is the primary axis of which the per-
manent magnet flux density flows. The magnet flux density flows through the magnet into
the central rib of the magnet pole, into the airgap, through the stator teeth and yoke, and
7
Figure 1.2: A Single Pole of an Interior Permanent Magnet Machine: a) slot, b) tooth, c)
stator yoke, d) winding/conductor, e) rib (secondary reluctance path), f) bridge, g) stator
OD, h) stator ID, i) rotor OD, j) barrier, k) rib (primary reluctance path), l) magnet, n)
airgap
returns into the adjacent opposite magnet pole. This permanent magnet flux density path
is shown in 1.1 as a red ellipse. The q- axis is the axis in which the armature-induced flux
flows. This flux is produced from the armature MMF. Two paths result in armature induced,
one through the primary reluctance path, and a second through the secondary reluctance
path. The primary reluctance path flux is shown as the solid blue ellipse and the second
reluctance path flux is shown as the dashed blue ellipse.
8
1.3.3 Rotor Modifications
Rotor sculpting among other design features is usually left to trial and error techniques rather
than analytical methods which could provide more insight. Researchers have investigated
rotor modifications to alter the airgap, modify airgap flux, and improve torque harmonics.
The first feature type is pole shaping, which creates a small airgap near the d -axis and an
increased airgap in the region of the q-axis. The torque ripple was reduced for the single
magnet flat magnet IPM and optimized with a differential evolution algorithm and finite
elements [34]. A surface-mount PM pole-shaped machine was studied with an analytical
solution to the field in [35]. The 2D solution was confirmed both by finite element and
testing. The pole-shaped single flat magnet IPM was optimized with a response surface
method within FE [36]. This included the use of rotor core modifications as well; both FE
and experimental results were presented. The flat magnet IPM pole shape was optimized,
along with the creation of design rules for the ratio of q-axis and d-axis airgap length in [37].
The single V magnet-shape IPM was improved with pole shaping using finite elements in [38].
Cogging torque and back emf were measured. A third harmonic was added to the pole shape
in [39], which studied the machine in finite elements. A second feature type is in the rotor
core, which creates a small hole in the rotor core near the airgap to redirect flux. Holes in
the rotor core’s second reluctance path of the single magnet IPM were shown to decrease
torque ripple using finite elements in [40]. The double V magnet IPM machine with improved
torque ripple, due to holes in rotor iron core and rotor surface sculpt features, was shown
to improve torque ripple but lower average torque in [41]. The delta magnet IPM shape
included modified internal rotor features to improve for average torque and decrease iron
loss in [42].
9
Figure 1.3: IPM rotor types: single V (left), delta (center), double V (right).
The third and final feature type is sculpting the rotor surface at the airgap to redirect flux.
The single flat magnet IPM machine cogging torque was reduced in [43] and experimentally
verified. Figure 1.4 shows a rotor pole with sculpted features. A grid on/off optimization of
the rotor surface was conducted on the single flat magnet IPM using finite elements in [44,45],
resulting in an asymmetric rotor surface with a reduction in torque ripple and maintaining
average torque. The double V-magnet IPM torque ripple was minimized with both rotor
core and surface sculpted features in [41]. The delta-magnet IPM machine torque ripple was
minimized with rotor surface sculpt features in [46]. Then, a general analytical expression for
torque harmonics was developed and utilized to optimize the solution with finite elements.
Flux density from the magnet is independent from the armature current, and the flux
density from the reluctance path is dependent upon the armature. Torque from the magnet
is proportional to the armature current while the reluctance torque is proportional to the
square of the armature current. For any given operating point there is minimum current
needed at the optimum control angle. Traditionally, the IPM machine design results in this
Maximum Torque per Amp (MTPA) at neither the magnet nor salient maximum torque,
but at the best combination of the two. Recently researchers have looked into aligning the
magnet and the reluctance axis to improve torque and power density of the machine [47–49].
This does not come without compromise as features to shift the magnet flux introduces
additional harmonics to the air gap. Alignment through the use of surface features has been
10
Figure 1.4: Example Interior Permanent Magnet Machine with Rotor Sculpt Features
shown to provide harmonic amplification of the permanent magnet flux density and overall
torque production [50].
1.3.4 Analytical Methods
Analytical expressions for the airgap and torque harmonics are developed for the IPM in [51,
52]. The synchronous reluctance of torque harmonics presented in [53] is extended to the
IPM machine in [52]. The expressions are useful in setting the stator slot and rotor barrier
counts but do not model the machine.
Analytical models build intuitive relationships between the physical geometry of the
machine to its airgap and flux density harmonics. Directly solving the Laplacian–Poisson
is difficult [54, 55]. Subdomain models break the model into pieces in which the Laplacian–
Poisson can be more readily solved [56, 57]. Magnetically Equivalent Circuits (MEC) divide
11
the geometry into smaller manageable pieces [58]. Methods depending on winding functions
allow for the geometry and harmonics to be described, but the second reluctance path can be
difficult to model. The airgap harmonics of the salient pole permanent magnet synchronous
machine are presented in [59] but does not address torque ripple or the secondary reluctance
path of the IPM machine. The rotor permeance path is approximated in [60] to determine
the torque ripple of the machine under study but does not fully describe an IPM machine.
The double V shaped IPM is presented in [61], in which flux densities are calculated through
a MEC model and described with a Fourier series. The single V IPM presented in [62]
considers the pole cap effect but does not consider torque ripple harmonics. The single
V, delta and double V IPM rotor configurations are shown in Figure 1.3. Moreover, the
airgap harmonics in permanent magnet synchronous machines were calculated in [63, 64],
but the effect of the second reluctance path on the airgap harmonics was not included in the
calculations.
The flux density in the air gap is used to compute the main proprieties of an electric ma-
chine, such as flux linkage, back-EMF, inductance, forces, and torque. Analytical, magnetic
equivalent circuit (MEC), and finite element methods can all be used to calculate the air
gap fields for both permanent magnet and armature excitations. Analytical methods [65]
are primarily based upon direct solutions of Maxwell’s Equations. Often idealization of the
geometry and material properties is utilized, leading to inaccurate results. As geometries and
material properties become more complex analytical methods become prohibitively difficult.
The magnetic equivalent circuit method subdivides the magnetic and electric fields into cir-
cuits of permeances and MMFs [66]. Carefully placed permeance or reluctance elements allow
flux to flow in predetermined directions. Geometry, time harmonics, and saturation are then
calculated at a reduced fidelity with reasonable speed. The finite element method solves the
12
partial differential equations by subdividing the complex geometry into many elements [67].
The full detail of geometry, windings, and material nonlinearity can be considered. The
finer the mesh and time step, the more accurate results can be obtained. Increased accuracy
comes at the expense of time and cost.
Recent research has been conducted in the application of MMF permeance theory to
various machines, including switched reluctance, doubly salient synchronous reluctance, sur-
face mount permanent magnet, synchronous reluctance, and the interior permanent magnet
machine. Cogging torque and acoustic noise were studied and minimized for the surface
permanent magnet machine in [68]. A modulator function was utilized in conjunction with
MMF-Permeance theory was used to develop a general formulation spanning multiple ma-
chines [69]. Air gap flux density was computed for the embedded surface mount machine
in [59]. A mixture of analytical and finite element techniques was used to compute complex
permeances and air gap pressure models to conduct a detailed NVH study [70]. Synchronous
reluctance rotors with a primary salient feature were presented in [71] and [72]. No load op-
eration of the flux switching machine was investigated in [73] utilizing MMF permeance
theory. Torque production of the doubly salient synchronous reluctance machine was stud-
ied in [74]. Han, Jahns, and Soong studied the interior permanent magnet machine, using a
simplified model MMF permanence model, to relate the torque ripple of the armature MMF
and magnet MMF interaction [51]. Dutta, Rahman, and Chong studied the inductance
properties of the interior magnet machine, focusing on the primary reluctance path [75].
The importance of considering the equal potential pole cap effect of the second reluctance
path was presented in [62], while focusing on the calculation of the inductance. Later the
same authors followed up extending the method to the calculation of magnet eddy current
losses [76]. Inductance properties of interior permanent magnet machines are studied in [77].
13
The reluctance portion of the IPM machine is studied by partitioning the design into d- and
q- axis components in [78]. Torque calculation of a five phase IPM machine is performed
in [79] using a new Lorentz force method, utilizing a combined magnetic equivalent circuit
and permeance model. A novel rotor design is presented and the open circuit back emf
properties are investigated in [15].
1.3.4.1 Review of Recent Analytical Models
The analytical MEC model is used to design a reduced magnet cost single V consequent
pole (CP) machine with the same average torque as single V IPM in [17]. The models are
developed based on zones and regions, allowing for an assumption of the open circuit flux
density distribution. Two flux sources (magnets) and six reluctance network paths are used
to create the open circuit single V IPM model. Open circuit flux density is assumed to
take a trapezoidal waveform where the flux density is determined spatially from the fluxes
in the MEC reluctance network. Similarly, the single V CP IPM network consists of one
magnet flux source and five reluctance paths. In both cases, the rotors reluctance path
reaction to the armature loading is not computed. The open circuit flux densities are used
to quickly determine an equivalent single V CP IPM fundamental to that of the single V
IPM fundamental. To guide design, the method is used to find an equivalent consequent pole
open circuit flux density fundamental to the traditional IPM flux density. Finite elements
are relied on to complete the study of the torque performance.
Multi-barrier synchronous reluctance and Permanent Magnet Assisted Synchronous Re-
luctance Machines (PMSynRM) are modeled using conformal mapping and magnetic equiva-
lent circuits in [22]. Hyperbolic shaped flux barriers are assumed. Conformal transformations
are employed to the rotors flux barrier geometry to compute the magnetic reluctance. The
14
reluctance values calculated from conformal mapping are subsequently used in the reluctance
network values of the MEC model. The MEC model considers MMF sources of both the
armature and magnet. Loaded and open circuit flux densities, average torque, and torque
ripple are compared to finite elements with reasonable accuracy.
The slotless U-type IPM machine open circuit flux density is analytically modeled with
a subdomain method solving Laplace’s and Poisson’s equations in [21]. Analytical equations
are derived and presented for each subdomain. Results are validated against finite elements.
The model is divided into four regions, which consist of the airgap and magnets. The
governing system of partial differential equations is developed, along with simplifications,
interface, and boundary conditions. Separation of variables is used to develop the general
solutions of the PDEs, and they are written as a Fourier series. The system of equations is
solved and compared to FE. Strong agreement of the radial and tangential flux density is
shown between the FE and the subdomain methods.
The analytical models discussed were developed for multiple purposes. The MEC method
is used in [17] to quickly estimate the open circuit flux density fundamental of the single-V
IPM and single-V CP IPM machines. The armature reaction of the reluctance features is
not considered by the model, and finite elements are used to finish the designs. Conformal
mapping is used in [22] to determine the reluctance of a multi-barrier PMSynRM and further
evaluated using a MEC network. Both open circuit and loaded conditions are evaluated for
airgap flux densities and torque performance and compared to finite elements. The analysis
is not extended to the design. The subdomain methodology is employed in [21] for the
analysis of the U-shape IPM machine open circuit conditions. Both tangential and radial
flux densities are shown to match finite element results. The model requires further extension
to consider the torque performance due to a loaded armature.
15
1.4 Organization of Thesis
Development of the fundamental magnetic model of IPM machine is presented in Chapter
2. This fundamental model is made up of sinusoidally distributed wingdings, a doubly sinu-
soidal salient rotor, and permanent magnets. Analysis resulting from the geometry, including
torque, flux linkage, and inductance is developed. The governing differential equations of
voltage and output equation of torque are developed for the case of the traditional align-
ment of magnets and for the case where the magnet axis is better aligned to produce torque
coincident with the reluctance path in chapter 3. MMF permeance theory is presented in
chapter 4 developing the necessary theory for IPM machines and sculpted rotor features.
An example machine is presented along with components of torque contributed from the
primary reluctance path, secondary reluctance path, and magnet path. Chapter 5 develops
the analysis of torque harmonics and the contributions of the various components. Chapter 6
presents features enabling mitigation of torque ripple. Chapter 7 develops a design method-
ology based on MMF-Permeance theory to improve intuition and reduce design iterations.
A design of a machine is presented with a comparison of MMF-Permeance to linear finite
element theory results.
16
Chapter 2
Fundamentals: Magnetics
Rotating electric machines are built from two primary parts: a stator and a rotor. The stator
is the stationary part, containing a multi-phase winding or armature, and a magnetically
permeable iron core. The rotor is the rotating component, containing permeable iron salient
features and a permanent magnet. The cylindrical rotor is surrounded by a cylindrical air
gap and further contained within the cylindrical stator.
The stator, fed by winding alternating current, produces a rotating magneto-motive force
(MMF) at its bore. The MMF interacting with the stator’s permeable iron core, the rotor’s
permeable iron salient features, and the air gap, produces, a radial air gap flux density. In
addition, the permanent magnets of the rotor interact with the same structure adding radial
air gap flux density. The stator three phase winding, located close to the inner bore of the
stator, also produces a tangential magnetic field, related to the linear current density of the
winding. The tangential magnetic field interacts with the radial flux density, as shown in
figure 2.1, to produce torque. The portion of this torque due to the air gap radial flux density
of the magnet is referred to as ’magnet torque’. The air gap radial flux density associated
with the salient portions of the rotor produce ’reluctance torque’. These torque components,
composed of the product of current, id and iq , and flux terms, can be observed within the
17
Figure 2.1: Radial and Tangential Flux Density in Air Gap
torque equation of the interior permanent magnet machine eq. (2.1).
1 3
T = (l − lq )id iq + λm iq (2.1)
22 d
The radial flux densities of the air gap pass through the three phase windings of the
stator. Each phase of the winding links the radial flux density, resulting in a magnet flux
linkage and a reluctance flux linkage (or inductance term) for each phase. The individual
phase inductance and magnet flux linkage will be dependent on the rotor’s angular location
within the stator. As the rotor spins with an angular velocity, the changing flux linkages of
the windings produce an electromotive force (EMF). The permanent magnet term of voltage
is only dependent on rotational speed. The inductive term is dependent upon both rotational
speed and the current through the winding.
These processes are fundamentals of electromechanical energy conversion for rotating
18
electric machines. In the remainder of this section, these ideas will be expanded with models
based on first principles and their results.
2.1 Stator with Sinusoidally Distributed Windings
Sinusoidally distributed windings are an ideal distribution of windings, physically not pos-
sible to achieve, but the concept allows for the first analysis of the machine. The sinusoidal
distribution neglects the harmonic content of distributed windings and focuses the analysis
on the fundamental components. Each phase of the machine can be described starting with
a conductor density function n(θ) with nˆx defining the peak density of the winding over the
stator bore and p the number of pole pairs of the electric machine, as shown in equation 2.2.
For this section, we will assume the machine has p = 1 pole pairs. This equation is directly
applicable to Phase A. Phase B and Phase C will be offset by 120◦ and 240◦ respectively.
The linear current density is a product of current ( Ia , Ib , and Ic ) , total number of turns
( n̂a , n̂b , and n̂c ), and angular location θ. Since the conductor density has a sinusoidal
form, the linear current density will be similar, with the linear current density Kx , shown in
eqs. (2.3) to (2.5). A 2-pole stator constructed with sinusoidally distributed current sheets
is shown in Figure 2.2 where each sinusoidal current sheet distributes the current over the
inner surfaces of the annulus.
nx (θ) = n̂x sin(pθ) (2.2)
n̂a
Ka = i a ( )sin(θ) (2.3)
2
n̂ 2π
Kb = ib ( b )sin(θ − ) (2.4)
2 3
n̂c 2π
Kc = ic ( )sin(θ + ) (2.5)
2 3
19
Each phase has it’s own set of current sheets. Phase A is a current sheet with the maximum
current going into the page at 270◦ and the same maximum current coming out of the page
at 90◦ . Likewise, phase B and C current sheets are sinusoidally distributed but aligned 120◦
apart from phase A.
Figure 2.2: Sinudsoidally Distributed Three Phase Windings
2.2 Salient Rotor Geometry Analysis
A doubly sinusoidal airgap is used to model the salient reluctance path. This creates a rotor
shape that has opposite ends close to the stator in one dimension, and far away from the
stator in the other dimension. This shape provides a salient feature the stator’s magnetic
field can pull on.
To define a doubly sinusoidal air gap, the rotor shape has to be modeled. The variable θ
is the angular coordinate along the stator bore, and ϕ is the position of the rotor. The rotor
is defined by these variables in eq. (2.6), where Rrot is the rotor radius, Rsta is the stator
20
radius, and gnom is the nominal air gap.
Rrot = (Rsta − gnom − A) + A cos(2θ − 2ϕ) (2.6)
The air gap due to the salient rotor geometry is the difference between the inner radius of
the stator and the radius of the salient rotor.
g = Rsta − Rrot (2.7)
The rotor geometry is shown in fig. 2.3.
Figure 2.3: Doubly Sinusoidal Air Gap: The salient rotor is shown in blue and the stator in
yellow.
The air gap function g defines the salient rotor within the armature. Ampere’s Law,
eq. (2.8), is used to calculate air gap flux. It states that for any closed loop the sum of
the magnetic field in the direction of the length elements is equal to the electric current
enclosed in the loop. In basic terms, the magnetic field in space around an electric current
21
is proportional to the electric current which serves as its source. The reluctance portion of
the air gap flux is calculated from the rotor shape using Ampere’s Law. To simplify the
calculation a few assumptions are made:
• air gap flux only flows radially
• the stator and rotor iron have infinite permeability
• the air gap has the permeability of free space
• all materials are homogeneous and linear (no saturation)
• as a result, the magnetic field in the stator and rotor is negligible.
I Z X
Hdl = JdS = I (2.8)
l S
A suitable integration path must be chosen. The integration path is chosen such that it
encloses currents and cuts through the air gap. The portions of the integration path that
extend through the air gap are radially oriented. Radial orientation in the air gap follows
the flux path which will result in direct calculation of this quantity. To consider the driving
currents to form a closed loop the integration path must cross the air gap in two places. If
the air gap is different lengths or the materials enclosed are not homogeneous calculation
of the air gap flux is complicated. In this case, we will further simplify by choosing a path
that maintains equal air gap length on both sections crossing the air gap. To meet all these
requirements the path shown in figure 2.12 is chosen. Since the stator is made of material
with infinite permeability, Hiron = 0, Ampere’s law for a phase becomes eq. (2.11). The
magnetic field for any angular position can be determined and the radial flux density can be
computed using the constitutive equation eq. (2.12).
22
Figure 2.4: Integration Path for Ampere’s Law, Current Sheet (red)
Z Z θ
2 n̂
JdS = i1 s1 sin(θ)rdθ (2.9)
S θ1 2
θ2
n̂
Hg1 (θ) · 2gair (θ) = −i1 r s1 cos(θ) (2.10)
2
θ1
θ+π
n̂
−i1 r 2s1 cos(θ)
Hg1 (θ) = θ (2.11)
2gair (θ)
B = µo H (2.12)
23
2.3 Permanent Magnets
2.3.1 Permanent Magnet Properties and Equivalent Current
Permanent magnet materials can be described in terms of three vector quantities [80]. The
flux density, B, (or magnetic induction) describes the concentration and direction of magnetic
flux at a point in space. The magnetic field vector, H, describes the field in space created
by a current through a wire. The magnetization vector, M, describes the internal state of
the magnet. The units for each property are listed below.
• B is flux density units T
• H is magnetic field m A
• M is magnetization m A
These three vectors are related through their constitutive relationship, which can also be
re-written in terms of the polarization vector J with units of T .
⃗ = µ0 H
B ⃗ + µ0 M⃗ = µ0 H⃗ + J⃗ (2.13)
Alone, this constitutive equation is not enough to describe the behavior of a permanent
magnet; the non-linear properties including hysteresis must be considered. Figure 2.5 shows
a typical magnetization curve for a permanent magnet, where the horizontal axis is the
magnetic field and the vertical axis is the magnetization. Starting at no magnetization
and no field, the origin of the graph, the magnetic field is increased and the magnetization
increases. As the field is increased further, the magnetization begins to saturate. This
point, Hs , is called the magnetic field saturation point. Continuing to increase the magnetic
24
field, the magnetization fully saturates at the point Ms . Once the magnetization is fully
saturated, the magnetic field is reduced to constant. The magnetization does not follow the
original trajectory but settles at the point Br . This point is called the residual flux density.
The residual flux density is also the point where the magnetization equals the flux density.
The process described is completely within the first quadrant and is the process used to
magnetize a permanent magnet. The second quadrant is where the magnet operates in an
electric machine. The residual flux Br and the intrinsic coercivity Hci are the key points
that define the operation of the magnet within the second quadrant.
Magnetization [T]
Br Ms
1
0.5
Hci Magnetic Field [kA/m]
-2000 -1500 -1000 -500 500 1000 1500 2000
-0.5
Hs
-1
Figure 2.5: Magnetization Curve [1]
Magnet operation within the second quadrant is shown in figure 2.6, it shows the flux
density and magnetization of the magnet eventually reduce to zero as the field becomes
increasingly negative. The second quadrant is where the magnet operates within an interior
25
permanent magnet machine, this is due to both the machines air gap and the loading from
the armature. A simple circuit consisting of coils, a magnet, infinitely permeable steel, and
Figure 2.6: Demagnetization Curve [1]
an airgap is effective in describing the operation of the magnet within an interior permanent
magnet machine [80]. This circuit is shown in figure 2.7 and is further idealized in equation
set 2.14.
µ0 Ms = Js ≥ Br
Br ≥ µ0 Hc
(2.14)
Hci ≥ Hc
(Br /2)2 ≥ µ0 (BH)max
Assuming no flux leakage in the magnet reluctance circuit, it can be described with equation
set 2.15. First among these equations are the magnet voltage drops of the magnet and the
air gap. Next continuity of total flux across both magnet and airgap must be maintained.
26
Figure 2.7: Permanent Magnet Model: Am area of magnet, Ag area of gap, Bm magnet
flux density, Hm magnet field strength, Bg airgap flux density, Hg air gap field strength, lm
length of magnet, g length of airgap, N1 I1 amp turns
27
Figure 2.8: Equivalent Current Permanent Magnet Model: Am area of magnet, Ag area of
gap, Bm magnet flux density, Hm ′ magnet field strength, B airgap flux density, H air gap
g g
field strength, lm length of magnet, g length of airgap, N1 I1 amp turns
28
Finally the constitutive equations for the magnet and the the airgap.
Hm lm + Hg g = N1 I1
Bm Am = Bg Ag
(2.15)
Bg = µ0 Hg
Bm = Br + µ0 µr Hm
The load line equation, eq. (2.16), is obtained by solving the first of set eq. (2.15). The
load line is used to solve for the magnet flux density as a function of magnetic field for each
current.
Ag
Bm = u0 (−Hm lm + N1 I1 ) (2.16)
gAm
′
The magnet constitutive equation can be further simplified by rewriting in terms of Hm .
Br ′ )
Bm = µ0 µr ( + Hm ) = µ0 µr (Hm (2.17)
µ0 µr
Hm′ l + H g = N I + Br l
m g 1 1 m
µ0 µr
Bm Am = Bg Ag
(2.18)
Bg = µ0 Hg
Bm = ′
µ0 µr Hm
Br
Nf If = lm (2.19)
µ0 µr
The system of equations are now re-written and solved to find the equivalent coil current,
equation 2.19, and the four unknowns Hm ′ , H , B , and B . This equivalent coil current
g m g
gives way to represent a system of magnets as a single phase coil wound around a material
29
with the same permeability as the magnet.
2.3.2 Simplified Permanent Magnet Analysis
The permanent magnet’s ability to hold flux density is due to its large hysteresis properties.
A demagnetized magnet will begin with properties at the origin of a magnetization curve
with no flux density and no field strength. To magnetize a magnet a large positive magnetic
field H will need to be applied and as a result, the flux density will increase. Upon relaxation
of the magnetic field H = 0 the magnetized magnet flux density will settle at its remnant
value Br . To bring the magnet back to zero flux density will require a negative magnetic field
applied equal to the magnet’s coercivity level Hc . Figure 2.9 shows the operating portion
of a B-H curve for a magnetized magnet. In most cases the second quadrant is where the
Figure 2.9: Magnet Properties
magnet will operate.
A simple framework is described first beginning with conservation of the magnet flux, Φ
and MMF F . Equation 2.20 considers the remnant flux Φr , the flux which does not cross
the airgap (leakage) Φσ , and the flux entering the magnetic circuit air gap Φg . Equation
30
Figure 2.10: Flux Paths: Crossing the Air Gap (red) , Leakage (blue)
2.21 sums the MMFs of the leakage and air gap terms, where Rg is the reluctance to cross
the airgap, and Rσ is the leakage reluctance.
X
Φ = 0 : Φr − Φσ − Φg = 0 (2.20)
X
F = 0 : Φg Rg − Φσ Rσ = 0 (2.21)
The resulting MMFs can be expressed in terms of each other and fed back into the conser-
vation of flux.
Φσ Rσ
Φg = (2.22)
Rg
Φr
Φσ = Φr (2.23)
1+ RR
σ
g
31
The remnant flux, Φr , is related to the remnant flux density , Br , multiplied by its area.
Φr = Br A = Br L(∆θ)r (2.24)
The reluctance of the leakage term, Rσ is related to the magnet thickness l, coercivity of the
magnet Hc , and remnant flux. The air gap reluctance is tied to the airgap of the machine,
g, permeability of free space, µ◦ , and area A.
Hc l Hc l
Rσ = = (2.25)
Br A Br L(∆θ)r
1 g 1 g
Rg = = (2.26)
µ◦ A µ◦ L(∆θ)r
With these few assumptions, the magnet leakage and the air gap flux density can be computed
as:
Br
Bσ = Hc l
(2.27)
1 + µ0 B g
r
Bg = Br − Bσ (2.28)
This models the magnet flux to be dependent on air gap, magnet thickness, coercivity, and
remnant flux.
2.4 Magnetic Dipole and Equivalent Magnetic Cur-
rents
The study of magnetic fields in free space begins with a subset of Maxwell’s equations
consisting of Gauss’s law 2.29 of magnetism and Ampere’s law 2.30. Both can be rewritten
32
in their respective integral form 2.31 and 2.32 made possible through the divergence theorem.
∇·B =0 (2.29)
∇ × B = µ0 J (2.30)
I
B · ds = 0 (2.31)
S
I
B · dl = µ0 I (2.32)
C
Gauss’s law states the total flux through a closed surface must be zero, therefore; a magnetic
monopole does not exist and magnetic flux is conserved. Ampere’s law states that the
magnetic field along a closed path is proportional to the current in which it encloses. Vector
calculus provides that in the case of the divergence of a vector field V is zero, a vector field
W exist such that ∇ × W = V . This allows for the magnetic field density B to be written
in terms of it’s vector magnetic potential A as shown in equation 2.33.
B =∇×A (2.33)
Using 2.30 along with 2.33 and assuming ∇ · A = 0 Poisson’s equation is formed.
∇2 A = −µ0 J (2.34)
Depending upon the complexity of the problem, a combination of these formulations is used
for the solution of the magnetic field. Solutions can be found with Poisson’s equation, sub-
domain analytical solutions [81], and finite element solutions [82].
33
The magnetic dipole in free space is needed to develop the idea of equivalent current
densities in place of magnetization states. Figure 2.11 shows a magnetic dipole in free space
formed by a loop of radius b and current of I. The solution at far fields, when R >> b, and
solved in spherical coordinates, in terms of magnetic vector potential is shown in 2.35 [4].
Where the magnetic dipole moment m is written as m = az Iπb2 .
µ m × aR
A= 0 (2.35)
4πR2
This dipole in free space can be used to explain magnetism at the atomistic level, where
Figure 2.11: Magnetic Dipole From Current Loop [2]
small circulating currents are formed by the process of magnetization. This magnetization
aligns the individual dipoles atoms and modifies the orbital spin of the electrons for each
34
atom. The macroscopic volume density of magnetization ,M , with units of A/m, is computed
through a sum of the individual microscopic dipoles.
n
P
mk
k=1
M = lim (2.36)
∆v→0 ∆v
Shown in [4], the magnetization vector M is equivalent to both a volume current density,
Jm with units of A2 , and a surface current density Jms with units of m A.
m
Jm = ∇ × M (2.37)
Jms = M × an (2.38)
Given M , the flux density B can be found first by computing both Jm and Jms then to
determine the magnetic vector potential A. Uniform M within a magnetic material will
result in no volume current density and only a surface current density Jm on its borders. If
space variations of M exist within a material a net volume current density will exist.
In summary, a magnetic dipole can be represented as the sum of equivalent volume
and surface current density (Jm and Jms), and utilized as an equivalent magnetization
current(EMC). A magnetic dipole inside a material with constant magnetization M can be
represented by a current loop formed at the exterior boundary of the dipole.
2.5 Torque Analysis
Torque can be calculated with the Maxwell stress tensor and knowledge of the magnetic
fields. It connects the electromagnetic fields and the mechanical forces they produce. It is
35
a tool to compute the air gap torque of an electric machine. Equation 2.39 shows the form
of the stress tensor applied to the computation of electric machine torque, where r is the air
gap radius, l is the length of the air gap, Bn is the radial air gap flux density, and A is the
linear current density of the armature.
Z 2π
T = r2 l · Bn (θ)K(θ)dθ (2.39)
θ=0
This equation shows that the torque generated is a product of linear current density and flux
density. This equation is used to compute the torque from all sources of flux density, including
permanent magnets and the armature. To maximize the permanent magnet contribution
to torque the linear current density needs to be simply aligned to the permanent magnet
radial flux density. The reluctance flux density will produce maximum torque when the
linear current density and the radial flux density are balanced. It can be seen that the net
sum of the product Bn and A from equation 2.39 have to be balanced and same signed to
produce maximum positive torque. This occurs when the current vector is 45◦ away from the
minimum air gap. This produces the maximum torque by creating the maximum product of
linear current density and flux density.
The air gap flux density for the reluctance path can be superimposed for all three phases.
Brel (θ) = Bg1 (θ) + Bg2 (θ) + Bg3 (θ) (2.40)
The magnet flux density is superimposed with the reluctance air gap flux densities. For
this analysis, the flux density due to the magnet is kept separate to allow for the magnet
and reluctance torque to be computed separately. In discrete form, torque equations for
36
reluctance and magnet become eq. (2.41) and eq. (2.42). The reluctance torque and magnet
torque can now be added.
n
r2 l
X
Trel = Brel (θ)K(θ)∆θ (2.41)
0
n
r2 l
X
Tmag = Bmag (θ)K(θ)∆θ (2.42)
0
T = Trel + Tmag (2.43)
2.6 Flux Linkage and Inductance
Determination of flux linkage, λ, and inductance L, are required for further analysis and the
control of the machine. They are needed for the development of voltage and torque equations.
Assuming a previously determined air gap flux density B(θ), fig. 2.12 illustrates the necessary
parameters and variables of geometry to calculate the flux linkage and inductance. The
phase windings are constructed with a sinusoidally distributed turns density n(θ) with a
phase vector quantity pointing to the right. The air gap is defined by a rotor of radius r and
the gap g. The magnetic flux over a pole, Φ(θ), requires integrating the radial flux density
over the span [83].
Z θ
Φa (θ) = µ0 Hr (θ)lrdθ (2.44)
θ− π
p
The magnetic flux over the pole span in conjunction with turns density n(θ) allows for the
calculation of flux linkage.
Z π
2p
λa = Φa (θ)na (θ)rdθ (2.45)
−π
2p
37
Figure 2.12: Inegration Path for Flux and Flux Linkage
Finally the inductance of the phase is determined by the ratio of flux linkage and current.
λa
La = (2.46)
ia
2.7 First Analysis of Machine Design
Basic components, shown in figure 2.13, include a permeable toroid, a three phase winding,
a permeable salient rotor structure, and a magnet pair. Sinusoidally distributed windings
along the inner radius of the toroidal core form the armature. Rotor saliency is achieved
through a doubly sinusoidal air gap, equation 2.47, where gm is the minimum air gap, A1 is
the amplitude of the air gap variance, ϕ is the rotor mechanical position initial offset. Half
sinusoidal magnets, defined by equation 2.48, are embedded into the rotor surface where Rm
38
Figure 2.13: Basic Geometry, δ = 45◦ : toroidal core (yellow), rotor (orange), north mag-
nets(red), south magnet(blue)
is the inner radius of the magnet, Am is the amplitude of the magnet cutout, and m defines
the magnet span. Saliency is designed through the amplitude of the doubly sinusoidal air gap
and can be further enhanced through the amplitude of the magnet cutout, the magnet span,
and the magnet offset angle. Alignment of the magnet and reluctance torque is achieved
through the magnet offset angle δ.
Rr = (Rs − gm − A1 ) + A1 cos(2θ − 2ϕ) (2.47)
Rm = Rr − Am cos m(θ − ϕ − δ) (2.48)
Resulting torques with traditional alignment, δ = 0◦ , and non traditional alignment, δ = 45◦ ,
are shown in figures 2.14 and 2.15. Magnet and reluctance properties of both cases are the
same. For the traditional IPM magnet alignment δ = 0◦ it can be seen that the reluctance
39
and magnet torque does not occur at the same current angle. When IPM magnet is aligned
at δ = 45◦ it can be seen that the reluctance and magnet torque occur at the same current
angle and maximize torque production.
1
0.8
0.6
0.4
Torque (Nm)
0.2
0
-0.2
-0.4
Reluctance
-0.6 Magnet
Total
-0.8
-1
-180 -135 -90 -45 0 45 90 135 180
Current Angle (deg)
Figure 2.14: Theoretical Torque Versus Current Angle IPM Machine , δ = 0◦ : Reluctance
Torque (blue), Magnet Torque (red), Total (black dashed)
40
1
0.8
0.6
0.4
Torque (Nm)
0.2
0
-0.2
Reluctance
-0.4 Magnet
Total
-0.6
-0.8
-1
-180 -135 -90 -45 0 45 90 135 180
Current Angle (deg)
Figure 2.15: Theoretical Torque Versus Current Angle IPM Machine , δ = 45◦ : Reluctance
Torque (blue), Magnet Torque (red), Total (black dashed)
41
2.7.1 Ideal Case
In this section, the interaction of the salient and magnet features is not considered. This
assumption is utilized to determine the maximum theoretical benefit of the aligned axis
structure and requires ignoring the effects of magnet geometry upon the geometry of the
salient rotor. Included in the ideal case, the magnet material proprieties are altered to main-
tain a constant magnet torque independent of magnet alignment or air gap. This second
assumption allows for the ratio of magnet torque to reluctance torque to be fixed as the
magnet axis alignment is varied. Under these ideal conditions, it is confirmed fig. 2.16, the
optimum magnet alignment occurs at 45◦ . A design that is overly reluctant or dominated
by the magnet yields little benefit from adjustment of the magnet alignment angle. During
design, this magnet to inductance ratio must be considered. The torque benefit of magnet
alignment, as compared to the traditional IPM magnet alignment, is computed as a per-
centage and presented in figure 2.16. The maximum torque benefit of 16% occurs when the
magnet torque is 2.5 times the reluctance torque. The most benefit can be achieved when
the magnet torque to reluctance torque ratio is between 1 and 5, centered around a magnet
alignment angle of 45◦ .
2.7.2 Magnet Geometry Effects
The embedded magnet has two effects: a primary one to provide flux that creates torque
and a secondary one, that alters the reluctance path. When the embedded magnet direction
coincides with the axis of maximum reluctance, the saliency ratio is enhanced and more
torque is realizable. As the magnet is aligned between the minimum and maximum reluc-
tance, the span of the magnet alters the reluctance path; so much so, that the current angle
42
Figure 2.16: Torque Benefit of Ideal Aligned Axis Design as Compared to Traditional Align-
ment (% increase)
43
at the torque zero crossing no longer coincides with the geometrical angle of the minimum
and maximum air gaps. The analytical magnet model and the modified air gap are both
considered during the calculation of the magnet and reluctance air gap flux densities. The
magnet span as a percentage of pole pitch is varied along with the magnet axis current angle.
A constant armature current was imposed. Figure 2.17 shows that when the geometry of the
embedded magnet is included in the reluctance torque computation, optimum results do not
occur if the span is too wide. The ideal case torque is presented alongside the torque with
different magnet spans. When the magnet spans the full pole pitch the maximum torque is
not realizable; as the magnet span shrinks, the ideal location is approached. As the magnet
alignment angle is varied, the geometrical and physical axes, of the magnet and reluctance
are no longer coincident. The shift, or difference between geometrical and physical axes, is
also presented in figure 2.17 by comparing the magnet torque and reluctance torque zero
crossing shifts. The impact of this is minimal using practical alignment angles between
0◦ and 45◦ . Clearly, magnet geometry must be considered when designing the adjustment
features for the location of the magnet flux.
44
12 30
11 50% span 25
66% span
100% span
ideal
10 magnet zero crossing 20
Torque (Nm) shift angle (deg)
reluctance zero crossing
9 15
8 10
7 5
6 0
0 10 20 30 40 50 60 70 80 90
magnet axis alignment angle (deg)
Figure 2.17: Torque Benefit as Function of Alignment Angle and Magnet Width as Per-
centage of Pole Span: ideal case torque (black dashed), 100% magnet span torque (red),
66% magnet span torque (green), 50% magnet span torque (blue) , 50% magnet span torque
(blue), magnet physical vs actual shift angle (blue dashed) , magnet physical vs actual shift
angle (blue dashed dotted)
45
Chapter 3
Fundamentals: Implementation and
Control
3.1 Voltage Equations
The synchronous PMAC machine is comprised of a permeable stator, a 3 phase winding
within the stator, and a permeable rotor with both salient and magnet features. Figure 3.1
is a simple model useful in the development of the governing voltage and torque equations.
The 3 phase windings can each be represented within the stationary frame as separate vectors
A, B, and C. The three phase vectors, fixed in direction, but not in amplitude can be vector
summed and represented in −αβ coordinates. The α coordinate is aligned with the phase A
vector and points to the right. The β coordinate is orthogonal to the α and pointed upward.
2 1 − 12 − 12
[Ts ] = √ √ (3.1)
3
3
0 2 − 23
1 0
√
[Ts ]inv = − 1
3 (3.2)
2 2
√
− 2 − 23
1
46
Transformation of ABC quantities to the αβ coordinate system can be done through the
coordinate transformation [Ts ]. [Ts ] and [Ts ]inv transform the coordinates from the 3 phase
vectors to a stationary reference frame αβ and back again. These are presented as am-
plitude invariant transformations. This transformation is often referred to as the 3 to 2
transformation and in its generic form shown below.
xa
1 1
x α 2 1 − 2 −
2
= (3.3)
3
√ √ xb
3 3
0 2 − 2
xβ
xc
xa 1 0
1 √3 xα
xb = − (3.4)
2 2
√ xβ
xc − 21 − 23
Another useful variable transformation is to look at the variables as seen by the rotating
Figure 3.1: Motor Structure with Voltage Eq
47
from −dq. The matrix Tr (ϕ) and Tr (ϕ)−1 transform the stationary frame values to the
rotating frame of reference as it rotates at angular velocity ω.
cos ϕ sin ϕ
[Tr (ϕ)] = (3.5)
− sin ϕ cos ϕ
cos ϕ − sin ϕ
[Tr (ϕ)]−1 = (3.6)
sin ϕ cos ϕ
These allow the αβ vector to be written as dq coordinates.
xd cos ϕ sin ϕ xα
= (3.7)
xq − sin ϕ cos ϕ xβ
xα cos ϕ − sin ϕ xd
= (3.8)
xβ sin ϕ cos ϕ xq
The traditional rotor frame of reference −dq has the −d axis aligned with maximum reluc-
tance and the −q axis is aligned with the minimum reluctance. Traditional placement of the
d axis would also align with the magnet. In this case, the magnet is aligned off of the -d
axis by angle δ. The rotor rotates at speed ω and a rotor position of ϕ.
3.1.1 Voltage Equations in ABC
The voltage equations consist of a voltage drop due to the resistive elements and the time
rate of change of the flux linkages. Where the voltage for each phase is written as vx , the
48
current is ix , flux linkage λx , and phase resistance Rs .
va ia λa
d
vb = Rs ib + λb (3.9)
dt
vc ic λc
Flux linkage of each phase is made up of the flux densities from both permanent magnet and
the inductive sources. Self inductance is written as Lxx and mutual inductance is written as
Mxy . Where as the permanent magnet flux linkage due to the magnet for each phase is λmx
λa ia Laa Mab Mac ia λma
λb = Lls ib + Mba Lbb Mbc ib + λmb (3.10)
λc ic Mca Mcb Lcc ic λmc
Angular position of rotor ϕ is . As the rotor changes angular position, ϕ ,the inductance of
the phases change with 2ϕ and the flux linkage of the magnets change with ϕ.
λma λm cos(ϕ − δ)
λmb = λm cos(ϕ − 2π − δ) (3.11)
3
λmc 2π
λm cos(ϕ + 3 − δ)
49
The self and mutual inductance contain a leakage term Lls , the average terms LA , and the
position dependant term LB .
Laa = Lls + LA − LB cos[2(ϕ)]
2π (3.12)
Lbb = Lls + LA − LB cos[2(ϕ − )]
3
2π
Lcc = Lls + LA − LB cos[2(ϕ + )]
3
1 π
Mab = Mba = − LA − LB cos[2(ϕ − )]
2 3
1 π (3.13)
Mac = Mca = − LA − LB cos[2(ϕ + )]
2 3
1
Mbc = Mcb = − LA − LB cos[2(ϕ + π)]
2
Ld + Lq
LA = (3.14)
2
Lq − Ld
LB = (3.15)
2
3.1.2 The Clarke Transformation: Voltage Equations in αβ
The voltage equations expressed in three phase quantities can are written in eq. (3.9). In
vector form these equations are written in eq. (3.16).
d⃗
⃗vabc = Rs⃗iabc + λ (3.16)
dt abc
Applying [Ta ] to the phase voltage equations, ⃗vabc , transforms the voltage equations to αβ
coordinates.
d d
⃗vαβ = Rs⃗iαβ + [Ta ][Lsabc ][Ta−1 ]⃗iαβ + ⃗λmαβ (3.17)
dt dt
50
With some work, the stationary frame flux linkages are simplified to contain doubly sinusoidal
and cosinusoial terns in Eq. 3.18.
3 3 3
λα Lls + 2 LA − 2 LB cos(2ϕ) − 2 LB sin(2ϕ) iα
=
λβ 3
− 2 LB sin(2ϕ) 3 3
Lls + 2 LA + 2 LB cos(2ϕ) iβ
cos(ϕ − δ)
+ λm (3.18)
sin(ϕ − δ)
The stationary frame voltage equation is simplified to the form shown in Eq. 3.19.
d
⃗vαβ = Rs⃗iαβ + ⃗λαβ (3.19)
dt
3.1.3 The Park Transformation: Voltage Equations in DQ Coor-
dinates
Transformation of variables from the stationary frame to the rotating is shown complex
vector form with eq. (3.20). Likewise, the Clarke transformation complex vector form is
shown in eq. (3.21). The complex rotation and its conjugate are shown in eq. (3.22) and
eq. (3.23).
⃗xαβ = ⃗xdq ejϕ (3.20)
⃗xdq = ⃗xαβ e−jϕ (3.21)
e−jϕ = cos(ϕ) − j sin(ϕ) (3.22)
ejϕ = cos(ϕ) + j sin(ϕ) (3.23)
51
First the Park transformation is applied to Eq. 3.19.
d ⃗ jϕ
⃗vdq ejϕ = Rs⃗idq ejϕ + λ e (3.24)
dt dq
The derivative term is expanded in Eq. 3.25.
d ⃗ jϕ d ⃗ jϕ dϕ
λdq e = λdq e + j ⃗λdq ejϕ (3.25)
dt dt dt
Finally the dq voltage equations are expressed as the following.
d
vd = Rs id + ld i − ωlq iq − ωλm sin(δ) (3.26)
dt d
d
vq = Rs iq + lq iq + ωld id + ωλm cos(δ) (3.27)
dt
3.2 Torque Equation
In this section, the torque equation is developed for the IPM machine. An energy balance
of the electro mechanical system is utilized to develop the equation. Using the single coil
armature salient rotor shown in fig. 3.2 as the starting point.
dλ
v = Ri + (3.28)
dt
Multiplying the single coil voltage equation by idt results in an expression for incremental
work, where the incremental joule losses are Ri2 dt, the infinitesimal electrical work is vidt,
52
Figure 3.2: Single Coil Reluctance System
and idλ is sum of incremental energy stored combined with increment work from torque.
vidt = Ri2 dt + idλ (3.29)
This assumption is valid when other losses, such as magnetic hysteresis, are not considered.
Considering eq. (3.29) the incremental work of the electro-mechanical system contains the
electrical energy supplied, vidt, the lost joule energy, Ri2 dt, and the magnetic energy, idλ.
The differential magnetic energy , idλ, of the system contains the balance of stored magnetic
energy and the mechanical output of the system.
idλ = dwm + τ dϕ (3.30)
53
The full differential form of stored energy dwm is shown in eq. (3.31).
∂wm (λ, ϕ) ∂wm (λ, ϕ)
dwm = dλ + dϕ (3.31)
∂λ ∂ϕ
Combining eq. (3.30) and eq. (3.31) the energy balance becomes eq. (3.32).
∂wm (λ, ϕ) ∂wm (λ, ϕ)
idλ − τ dϕ = dλ + dϕ (3.32)
∂λ ∂ϕ
Upon careful inspection of eq. (3.32) the torque can be expressed in terms of stored magnetic
energy.
∂wm (λ, ϕ)
idλ = dλ (3.33)
∂λ
∂wm (λ, ϕ)
−τ dϕ = dϕ (3.34)
∂ϕ
∂wm (λ, ϕ)
τ (λ, ϕ) = − (3.35)
∂ϕ
Due to saturation, it is useful to consider the co-energy wm ′ of the system. Co-energy
is remaining balance from the product of current and flux linkage and the stored magnetic
energy.
wm′ (i, ϕ) = iλ − w (λ, ϕ) (3.36)
m
Through similar analyis the torque can be expressed in terms of co-energy.
dwm ′ (i, ϕ) = λdi + τ dϕ (3.37)
Extending the concepts of stored magnetic energy, the torque equations for the machine
shown in fig. 3.1 is developed. Flux linkage and its energy are expressed in terms of both
54
inductive and permanent magnet sources.
dwm ′ (i, ϕ) = diλ + idλ − dw (λ, ϕ) (3.38)
m
dwm′ (i, ϕ) = λdi + τ dϕ (3.39)
∂wm ′ (i, ϕ) ∂wm ′ (i, ϕ)
λdi + τ dϕ = di + dϕ (3.40)
∂i ∂ϕ
∂wm ′ (i, ϕ)
τ= (3.41)
∂ϕ
⃗λ = [L]⃗i + λ⃗m (3.42)
1h i 1 ⃗ T⃗
wm (ia , ib , ic , λm , ϕ) = λa ia + λb ib + λc ic = λ i (3.43)
2 2
1 h⃗t ⃗ ⃗ t
⃗
i
wm = i [L]i + λm i (3.44)
2
Separating the inductive and permanent magnet terms the IPM torque equation can now be
expressed as eq. (3.45).
t
∂( 21 ⃗λT⃗i) 1 ∂(⃗it [L]⃗i) 1 ∂(λ⃗m ⃗i)
τ= = + (3.45)
∂ϕ 2 ∂ϕ 2 ∂ϕ
With much work, the torque equation for the interior permanent magnet machine considering
a magnet to reluctance angle of δ is shown in eq. (3.46).
!
13
τ= (ld − lq )id iq + λm cos(δ)iq − sin(δ)id (3.46)
22
55
3.3 First Analysis of Machine Performance and Con-
trol
The steady state torque, voltage, and power equations are shown in eq. (3.47), eq. (3.48),
eq. (3.49), and eq. (3.50). These are the fundamental equations to analyze and control of
electric machines. Resistance and loss terms do not have an effect on the torque and only
affect the voltage and electrical power. In some analysis it may be useful to assume the
phase resistance is negligible.
!
13
τ= (ld − lq )id iq + λm cos(δ)iq − sin(δ)id (3.47)
22
vd = Rs id + −ωlq iq − ωλm sin(δ) (3.48)
vq = Rs iq + ωld id + ωλm cos(δ) (3.49)
3 1
P = τω = vd id + vq iq (3.50)
22
To determine the performance of the electric machine both the current limit, Imax , and
the voltage limits Vmax must be considered. The current limit is limited by the power
devices of the inverter or the electric machine’s thermal capability and is represented by the
equation eq. (3.51). The voltage limit is the maximum phase voltage that the inverter can
apply, limited by the specific pulse width modulation technique used.
Imax ≤ i2d + i2q (3.51)
Vmax ≤ vd2 + vq2 (3.52)
56
Viewed on the current plane, where id is the horizontal and iq is the vertical axis, the current
limit equation is a circle of radius Imax with its center at the orgin. Plotting voltage limit
on the same current plane requires further analysis through the combination of the lossless
steady state voltage equations and the voltage constraint into eq. (3.53). In this form the
voltage equation becomes an ellipse on the current plane where If q and If d locate the center
of the voltage ellipse and the denominators define the minor and major axis dimensions.
iq + If q id + If d
2
+ 2
≤1 (3.53)
Vmax Vmax
(−ωlq )2 (−ωld )2
λm cos(δ) = lq If q (3.54)
λm sin(δ) = ld If d (3.55)
The maximum operable speed at specific current levels can be determined by eq. (3.56).
s
2
Vmax
ω= (3.56)
(lq iq + λm sin(δ))2 + (ld id λm cos(δ))2
The operation of the electric machine is bounded by the current circle and the voltage ellipse
but not fully defined by both. At startup the machine runs inside of the current circle and
voltage ellipse. The current to operate within these bounds is described as maximum torque
per amp (MTPA). To realize the MTPA trajectory, an optimization problem is needed, as
shown in eq. (3.57).
!
13
min τ = (ld − lq )id iq + λm cos(δ)iq − sin(δ)id
−τ 22
(3.57)
s.t. Iss = i2d + i2q
57
The machine operates on this trajectory until the critical speed or base speed of the machine
is reached. The machine begins to field weaken while operating at Imax . As the machine
speed increases, a second critical speed is reached where the machine is both operating at
Imax and Vmax . Beyond this second critical speed the machine begins to operate along the
maximum torque per volt (MTPV) trajectory. To realize the MTPV trajectory, an additional
optimization problem is used in eq. (3.58).
!
13
min τ = (ld − lq )id iq + λm cos(δ)iq − sin(δ)id
−τ 22
(3.58)
s.t. Vmax = vd2 + vq2
3.3.1 Example Machine
An example machine is explored to illustrate the principles of control and the effects of
the magnet displacement angle δ. Parameters of the machine studied are listed in table 3.1.
Analysis of two variants of the machine at δ = 0◦ and δ = 45◦ is shown in fig. 3.3 and fig. 3.4.
Machine Parameters Value
d-axis inductance ld 9 mH
q-axis inductance lq 22.5 mH
magnet flux linkage λm 0.3 Vs
maximum phase voltage Vmax 100 V
maximum phase current Imax 100 A
Table 3.1: Example Machine
Both figures illustrate the current circle, the voltage ellipse, the MTPA trajectory, critical
speed for MTPA, the critical speed for MTPV, and the MTPV trajectory. The current
trajectory of the MTPA is notably different. When the magnet angle δ = 0◦ the current
angle of the trajectory is not constant and varies with current magnitude. When the magnet
angle δ = 45◦ the current angle of the trajectory is constant. Additionally the center of the
58
voltage ellipse is different for both cases. The traditional magnet angle of δ = 0◦ is located
on the d-axis current and the δ = 45◦ alignment has shifted the center of the voltage ellipse
into the third quadrant. The torque and power curves for the two machines are shown in
fig. 3.5 and fig. 3.6. By shifting the magnet to δ = 45◦ both the torque and the peak power
are improved.
current limit
MTPA
w MTPA
Isc
voltage limit w MTPA
MTPV
w MTPV
50
-200 -150 -100 -50 50 100 150 200
-50
Figure 3.3: Example Machine Current Plane δ = 0◦ , current limit (blue circle), voltage limit
(green ellipse), maximum torque per ampere (MTPA) trajectory (orange curve), maximum
speed of MTPA (orange *), maximum speed at full current (purple x), short circuit current
(purple circle).
59
current limit
MTPA
w MTPA
Isc
voltage limit w MTPA
MTPV
w MTPV
50
-200 -150 -100 -50 50 100 150 200
-50
Figure 3.4: Example Machine Current Plane δ = 45◦ ,current limit (blue circle), voltage limit
(green ellipse), maximum torque per ampere (MTPA) trajectory (orange curve), maximum
speed of MTPA (orange *), maximum speed at full current (purple x), short circuit current
(purple circle).
60
100 5000
MTPA T
90 MTPA P 4500
FW T
80 FW P 4000
MTPV T
70 MTPV P 3500
Torque - Nm
60 3000
Power - W
50 2500
40 2000
30 1500
20 1000
10 500
0 0
0 50 100 150 200 250 300 350 400
speed - rad/s
Figure 3.5: Example Machine Torque and Power Speed Curves δ = 0◦
61
100 5000
MTPA T
90 MTPA P 4500
FW T
80 FW P 4000
MTPV T
70 MTPV P 3500
Torque - Nm
60 3000
Power - W
50 2500
40 2000
30 1500
20 1000
10 500
0 0
0 50 100 150 200 250 300 350 400
speed - rad/s
Figure 3.6: Example Machine Torque and Power Speed Curves δ = 45◦
62
3.4 Relationship to Design
The computation of torque is not enough for machine design. It is necessary to stay within
the constraints of current and voltage. As the design evolves, these properties must be
considered to meet torque, power, efficiency, torque ripple, and other requirements. To
achieve these needs flux linkage and inductive properties of the electrical machine must be
determined and related to performance and control. Asymmetrical rotor sculpt features
require consideration of non-conventional alignment of the magnet to the reluctance, δ ̸= 0,
new voltage and torque equations are needed. As a result new methods for control of torque
as needed for the magnet alignment of δ ̸= 0.
63
Chapter 4
MMF Permeance Theory
Modeling air gap permeability and its reaction MMF provides an analytical model that can
be solved quickly and provides insight into the design which is discussed in [59, 60, 84–86].
This method is used to compute the radial flux density of the machine Br through modeling
the airgap permeance, Λr , and the MMF, Fr , and a constant reflecting the pole to pole
symmetry in eq. (4.1).
Br (θ, ϕ) = 2Λr (θ, ϕ)Fr (θ, ϕ) (4.1)
Accurate descriptions of stator and rotor permeance are required, along with the MMF of
the stator windings and the rotor magnets. These models have been traditionally applied
to embedded surface mount permanent magnet machines approximating the behavior of
the interior permanent magnet machine. This chapter will establish the necessary MMF
permeance framework to model IPM machines with rotor sculpt features.
4.1 Air Gap Permeance
Magnetic circuits are analogous to electrical circuits; they replace voltage with MMF and
resistance with reluctance. Permeance, P is a property of allowing the passage of magnetic
flux and is the inverse of reluctance, R. Both are related to the permeability µ, cross sectional
area A and the length of the air gap, g. The MMF, F , is defined by the stator current and
64
winding arrangement.
1 µA
P= = (4.2)
R g
F = ΦR (4.3)
Hence the magnetic flux, Φ,is the product of MMF and permeance.
1
Φ = PF = F (4.4)
R
Considering that total magnetic flux density, Φ = BA, is the product of the flux density and
area, these concepts can be applied locally using eq. (4.1).
Stator and rotor geometry, individually and together, require the air gap permeability
function Λr to be described as functions of the angular position with respect to the station-
ary frame, α, and the angular position of the rotor, ϕ. The air gap permeance function in
its simplest form is the ratio of permeability of free space and the air gap thickness. Ac-
commodating both stator and rotor the air gap function is described as combination of the
rotor, grt , and stator, gso , air gaps [60].
µ0
Λtot (ϕ, θ) = (4.5)
grt (ϕ, θrt ) + gso (ϕ, θso )
4.2 Rotor Reluctance Path
Interior permanent magnet machines feature a rotor with embedded magnets serving to
create both a magnet driven MMF and a salient structure. A single barrier IPM rotor struc-
ture, as shown in figures 1.2, 4.1, and 4.4, shows the common features, including embedded
magnets, high reluctance barriers which the magnets fit within, a rib structure under the
65
magnet, a rib structure above the magnet, and bridges to hold the two ribs together. Highly
permeable steel rib structures along with the high reluctance of air barriers form the basis
of the salient structure. The rib behind the magnet is aligned with the q- axis at the air
gap providing two functions. Firstly it allows flux to flow in the q-axis, also referred to as
the primary reluctance path. Secondly, it provides a permeable path for the backside of
the permanent magnet. The rib above the magnet, provides a secondary reluctance path,
allowing q axis flux to loop through and allowing the magnet flux to easily flow into the air
gap of the machine.
Modeling the reluctance structure of interior permanent magnet machines has primarily
been done by assuming an inset surface mount structure [59,60,84–86]. Dajuku and Gerling
considered the primary reluctance path by calculating the two permeances, Λp1 and Λp2
[84, 85]. Koo and Nam considered a similar structure applying a Taylor series expansion for
the reluctance path [60]. Each of these describes the primary reluctance path as it allows the
magnetic flux to flow between two adjacent q-axis. Figure 4.1 shows the graphic description
(green) of the permeabilities of the primary path. At the location of the q axis the permeance,
Λp1 , is a function related to the air gap of the electric machine.
µ
Λp1 = 0 (4.6)
g
The permeance, Λp2 , is related to the flux that flows through the -d axis to yoke of the rotor.
This flow-through flux passes through both the air gap, g, and the magnet barrier, hm .
µ0
Λp2 ≈ (4.7)
g + hm
66
Figure 4.1: Permeability of IPM Rotor
67
The permeances Λp1 and Λp2 allow modeling the flux which passes through the air gap,
into the yoke of the rotor, and returns through the air gap. It does not take into account the
reluctance flux looping through the secondary reluctance path. Vagati, et al., [53] considered
the structure of multiple reluctance paths for a synchronous reluctance machine but did not
take into account the incorporation of magnets. For a single barrier IPM, as shown in fig. 4.4,
the secondary reluctance can be modeled as the permeabilities shown in blue. The region in
which the secondary path is present, Λsd2 , the air gap permeance is related to only the air
gap.
µ
Λsd2 = 0 (4.8)
g
In the regions where the secondary path is not present, Λsd1 , the permeability is zero.
Λsd1 = 0 (4.9)
This pattern effectively locates the secondary reluctance paths presence, but cannot be
directly used with eq. (4.1) to calculate flux.
4.3 Stator Slots by Permeance Function
The air gap shown so far has been a smooth stator bore with a single barrier IPM rotor.
Figure 4.2 shows the opposite, a smooth rotor bore with stator slots and the approximate
function. As illustrated, magnetic flux within airgap will cross from rotor to stator, predom-
inately in a radial fashion [60]. Magnetic flux crossing the air gap within the slots will not
travel the full height of the slot as the distance to the edge of the slot is much shorter. The
flux lines in red have been drawn to illustrate this. The air gap from the stators perspective
68
is approximated with gs (α), where at the tooth tip gs (α) = 0 and within the slot the air gap
takes on a triangular nature with height hs and width γτs . Where γ is simply the percentage
of the tooth width that occupies the slot pitch τs = wth + ws .
Figure 4.2: Stator slots physical and air gap model: tooth width wth , slot width ws , slot
height hs , air gap g, slot pitch τs , tooth percenatage of slot width γ, effective slot height
hs1 , stator coordinate α, effective stator air gap gs
Combining the permeances of the rotor air gap, g, and stator air gap is gs is a series circuit
operation and can be calculated with equation 4.5. As shown in section 4.2 it is easiest to
express the air gap of the rotor in terms of permeability as the secondary reluctance path
has permeances of zero where the material is not present. This would require modeling of an
infinite air gap; therefore, it is more convenient to model the permeance directly. In contrast,
the air gap of the stator, gs , has infinite permeance where the stator teeth exist; therefore, it
69
is best to represent the stator permeance directly with airgap. The permeance of the rotor
air gap is shown below, while the air gap of the stator is shown in figure 4.2.
µ0
Λrt (ϕ, θ) = (4.10)
grt (ϕ, θrt )
Combining the the rotor air gap permeance and stator air gap results in an effective total
air gap which can be used to calculate total air gap permeance.
gtot (θso , ϕ) = Λ−1
rt µ0 + gso (θso , ϕ) (4.11)
4.4 Windings, MMF, Linear Current Density
Rotating electrical machines typically consist of a non-moving stator and a moving rotor.
The permanences and MMFs of the stator and rotor interact within the air gap to produce a
magnetic field and flux density. The MMFs of the armature is produced by current flowing
through the armature, similarly, a rotor magnet MMF can be replicated with a current
and a winding. The modeling of turns and winding functions can be first described by
an elementary non-salient structure shown in figure 4.3. The turns function n(ϕ, θ) is the
number of the turns enclosed by the path abcda. It is dependant upon the starting point of
integration. With some work the winding function, N (ϕ, θ) is defined as follows:
N (ϕ, θ) = n(ϕ, θ)− < n(ϕ, θ) > (4.12)
70
Figure 4.3: Constant air gap machine [3]
where, < n(ϕ, θ) > is the average of the turns function.
Z 2π
1
< n(ϕ, θ) >= n(ϕ, θ)dϕ (4.13)
2π 0
When the winding function, N (ϕ, θ), is described with a Fourier series, the amplitude of
each order is referred to as a winding factor. The MMF for any winding set is the product
of its winding function and the current flowing within the winding.
F = N (ϕ, θ)I (4.14)
4.4.1 Stator MMF Modifications for the Second Reluctance Path
Solving for the loop through flux condition of the second reluctance path imposes a different
constraint, the sum of the magnetic flux across the region represented by Λsd2 must equal
zero. Modification of the MMF is required. Figure 4.4 demonstrates these MMF modifica-
71
Figure 4.4: MMF modifications for secondary reluctance paths: MMF of the armature(blue),
average MMF across the 2nd reluctance path (green), modified second reluctance path MMF
(modified second reluctance path MMF = MMF of the armature - average MMF across the
2nd reluctance path).
72
tions for the second reluctance path. The blue curve represents the armature MMF of the
armature over the surface bound by Λsd2 . The green line is the average MMF across the
second reluctance path. The red line is the difference of armature MMF and average arma-
ture MMF affecting the second reluctance path. Using the difference of the armature and
average MMFs effectively holds the condition that the sum of all magnetic flux across this
surface is equal to zero. This process needs to be repeated for each nonprimary reluctance
path where the condition of balanced magnetic flux is required.
The rotor second reluctance path, and resulting looping flux density, is a reaction to
the net change of armature MMF across the pole cap, but not the local average. This
is illustrated by the second reluctance path airgap flux, shown by the blue dashed line, in
fig. 1.1. Modification to the armature MMF by removing its mean satisfies local conservation
of armature flux condition and is made possible through (4.15). The symbols F and
< Fabc (θ, ϕ)NP M (θ) > represent the modified armature MMF and its average over the span
of the second reluctance path.
F (θ, ϕ) = Fabc (θ, ϕ)NP M (θ, ϕ)
− < Fabc (θ, ϕ)NP M (θ, ϕ) > NP M (θ, ϕ) (4.15)
1 R 2π F (θ, ϕ)N
2π 0 abc P M (θ, ϕ)dθ
< Fabc (θ, ϕ)NP M (θ, ϕ) >= (4.16)
1 R 2π | N
2π 0 P M (θ, ϕ) | dθ
4.5 Sculpted and Slotted Features by MMF Function
Notched or slotted features are a necessity of most electrical machines. These allow for
magnetic flux to flow through the iron teeth of the stator and current to flow the copper
73
windings within the slots. Features may be placed in the stator as a design feature to reduce
torque ripple. Similar features on the surface of the rotor, referred to as sculpt features, are
also used to reduce unwanted torque ripple. In its present state, MMF-permeance theory
does not adequately describe these features, as it has been developed with a constant air
gap dimension [3]. This section will present an extension of MMF-permeance theory de-
scribing both slots and sculpts based upon equivalent magnetic currents (EMC) [81] and the
equivalent dipole [4].
4.5.1 Application of Superposition for Slotted Features
Utilization of the magnetic dipole - surface current equivalence requires superposition. Bourou-
jeni et al. [81] applied EMC within the subdomain solution method and focused on the solu-
tion of the cogging torque of a surface mount permanent magnet machine. As a first analysis,
the fields were found for a first machine with a non-slotted stator 4.5 (b) simply filling the
slots with iron and maintaining the air gap dimension. A second stator, shown in 4.5 (c), is
constructed with a large airgap at a dimension equal to the slot depth and original air gap.
Lateral currents are positioned coincident with the slotted edges of the original machine.
The lateral currents are dependent on the radial flux density found in the first machine (a)
at the position of the slot edges. It is assumed the flux density within the slot is constant
therefor the radial flux density is averaged across both sides of the slot.
Br
Jms = az (4.17)
µ0
Equal but opposite currents are applied along the lateral edges 4.5(d) to counter the flux
within the slot and direct it to the tooth. Due to the fringing seen in the two-dimensional
74
Figure 4.5: Useage of Superposition and Equivalent Magnetization Currents (a) slotted
stator (b) toroidal stator (c) stator with EMC (d - e) modified EMC [4]
75
solution, the lateral currents (d) are converted to arc currents on the ID of the stator(e).
Some iteration is required by the author to insure the fields at the center of the slot (d)
are equal to the fields in (e). The solution of the arc currents with no magnets (e) and the
magnet-only fields (b) are superimposed to obtain the slotted stator result. The arc currents
enable the usage of Maxwell’s stress tensor to determine cogging torque.
4.5.2 Magnet MMF and Sculpted Rotor
The prediction of airgap flux density for buried magnet IPM machines with sculpted rotor
features requires a multi-part analysis. Equation 2.19 establishes the equivalent current for
the magnet. For the case of the buried magnet, these currents cannot be directly applied in
MMF permeance theory, as the span of the magnet pole is much wider than the length of
the magnet. In this case, the following relationship based on conservation of flux is utilized.
′ τmag
Imagnet = If (4.18)
τmagpole
To apply this, conductors are placed at the edges of the second reluctance path surface and
the ensuing turns function n, winding function N , and MMF can be formed for the embedded
magnet rotor. The air gap permeance function must take into account both the thickness of
the magnet and the physical air gap dimension. Figure 4.6 illustrates the conductor, turns,
and winding function for the 4 pole rotor.
Superposition and MMF-Permeance theory is used, to adjust for the redistribution of
flux density due to the added feature of the sculpted rotor. The sclupted rotor concept is
shown in figure 4.7 (a). The unsculpted rotor, figure 4.7 (b), analyzed above with the flux
density measured along the lateral features of the sculpted EMCs is calculated. The EMC
76
Figure 4.6: Magnet Pole Equivalent Winding Factor
is applied to figure 4.7 (c), and in this case, the winding functions for each magnet pole
are considered separately. Figure 4.8 illustrates the EMC current conductor direction and
Figure 4.7: Scupted Rotor Superposition (a) sculpted rotor (b) unsculpted rotor (c) sculpted
rotor equivalent magnetic currents
the resulting winding function taken over the pole. Figure 4.9 demonstrates the usage of
superposition and the effect the sculpt feature has on redistribution of flux density. First, in
the process, a smooth magnet pole rotor is solved for the flux density (a south pole is shown).
77
Next, the air gap flux density along the lateral edges of the sculpt feature is measured and
used to determine the equivalent magnetic currents. The effective winding factor across the
pole is computed and applied, determining the flux density countering flux within the sculpt
feature and increasing flux outside of the sculpt feature.
Figure 4.8: Sculpted Rotor Winding Function
Figure 4.9: Sculpted Rotor Magnet Pole Flux Density
78
4.5.3 MMF of the Second Reluctance Path due to Rotor Sculpting
Rotor sculpting reluctance flux distribution. The redistribution of flux density and MMF is
possible with the use of the equivalent dipole concept. Due to the changing reluctance flux
density over the sculpt feature, the process of redistribution requires breaking the geometry
into smaller discrete dipoles, allowing for the assumption of homogeneous flux density to
hold.
Figure 4.10: Sculpted Rotor Reluctance Counter Dipole Current
Figure 4.10 illustrates the process of analyzing the effects sculpt feature to the reluctance
path. They can be broken down into the following steps, across the airgap of the second
reluctance feature:
1. Define the dipole counter-current with a sufficient number of points in the sculpt feature
to achieve reasonable accuracy. Utilization of the turns function from the magnet sculpt
feature can simplify the determination of sculpting locations, assuming a constant
sculpt depth.
2. determine the opposite second dipole counter current,
79
3. sum both the dipole counter current and the opposite dipole counter current, effectively
creating the current-turns function for the sculpted feature,
4. create the MMF function over the pole path by removing the mean from the current-
turns function,
5. calculate the air gap flux density impacts due to the sculpt feature using the increased
air gap of equivalent dipole current,
6. determine the total air gap flux density by superimposing the sculpted feature air gap
flux density upon the smooth rotor air gap flux density.
This process enables MMF to be redistributed by the sculpt feature and allows for the
separation of sculpt and smooth rotor flux densities.
4.6 Flux Linkage and Inductance
Flux linkage is a key parameter of the machine. Flux is the area integral of flux density but
does not alone determine the flux linking the coil.
Zb
Φ = rl B(ϕ, θ)dθ (4.19)
a
Flux linkage includes how much flux density links the winding. Its calculation requires the
use of the winding function. The winding function contains key information, such as: where
the coils are, what direction they are wound, and the number of turns. Integrating the
product of flux density and the winding function over the area provides the flux linkage, λ,
80
for each phase x.
Z2π
λx = rl B(ϕ, θ)Nx (ϕ, θ)dθ (4.20)
0
The inductance of each phase is simply the ratio of flux linkage to the current applied.
λx (ϕ, θ)
Lx = (4.21)
Ix (ϕ, θ)
Knowledge of the flux linkage and rotational speed provides the necessary information to
determine the phase voltage.
dλ dϕ
Vx = (4.22)
dϕ dt
4.7 Example Machine
The MMF Permeance model of an example electrical machine is presented in this section.
The example machine is shown in fig. 4.11 with the parameters in table 4.1. A four pole,
three phase, single flux barrier, 24 slot, interior permanent magnet machine is modeled. The
machine is for development purposes and not tied to any specific performance requirement.
The iron components of the stator and rotor are colored gray and are assumed to have
infinite permeability. The permanent magnets are embedded in the rotor are colored blue
and have typical proprieties associated with NdFeB magnets. The stator is designed to have
negligibly small slots, such that the bore of the stator can be considered smooth. Windings
are placed around the interior of the stator, within slots, and are colored red, green, and
blue to represent the three phases. Rotation of the rotor is counter clockwise, it is shown in
figure 4.11 at a position of zero mechanical degrees.
The MMF-Permeance model requires a description of the air gap permeance of the rotor,
81
Figure 4.11: Example Motor
Parameter Value Unit
Number of pole pairs 2
Rotor diameter 100 mm
Number of phases 3
Slots per pole per phase 2
Number of slots 24
Magnet pole width 66 % of pole pitch
Primary reluctance path width 20 % of pole pitch
Sculpted rotor feature width 25 % of pole pitch
Number of magnets per pole 2
Magnet thickness 5 mm
Magnet Length (per magnet) 20 mm
Relative permeability of magnet 1.099
Rotor airgap 1 mm
Primary reluctance path second air gap 6 mm
Sculpt feature depth 5 mm
Table 4.1: Example Motor Parameters
82
MMF functions for each of the three phases, the equivalent MMF function of the rotor
magnets, and linear current density functions represent the windings. For each magnet pole,
the modification of the stator MMF must be considered to describe the second reluctance
path flux density. These alone are sufficient to describe smooth rotors. They are described
with a Fourier series, allowing modification for each rotor position. Equivalent magnetic
dipole currents are used to describe the sculpted feature, redistributing the MMF and flux
densities.
The base permeance functions of the smooth rotor inside a smooth bore stator include
a description of the primary reluctance path, the second reluctance path, and the magnet
path. The primary reluctance path has high permeance proportional to the inverse of airgap,
aligned with the q axis of the rotor, and minimal permeance in the region of the magnet
pole. The high permeance is associated with the main dimension of the rotor air gap, while
the minimal permeance includes the thickness of the air gap and the magnet. In the region
of the q- axis the secondary reluctance path and the magnet pole have no permeance but
differ in the region of the magnet pole. The magnet path must consider both the main air
gap dimension and the magnet thickness, the second reluctance path only needs to consider
the air gap between the stator and rotor. For the example motor, the permeance functions
are shown in fig. 4.12.
Windings are distributed among the stator slots sequenced to produce the desired MMF,
linear current density, and resulting flux densities. Each turn of the winding is defined
by both a position along the bore and a direction. The position of the turn is related to
the available slot positions. The vector direction of the turn must either be positive (out
of the page) or negative (into the page). These positive and negative directions of turns
are used to create a magnetized dipole when current flows. Specially arranged these turns
83
Air Gap Peremance
1600
Primary Reluctance Path
1400 Secondary Reluctance Path
Magnet Path
1200
inverse air gap [1/m]
1000
800
600
400
200
0
-200
0 45 90 135 180 225 270 315 360
theta [deg]
Figure 4.12: Example Motor Air Gap Permeance Functions Versus Stator Coordinate
create an MMF which approximates a sinusoidal distribution. Figure 4.13 illustrates the
stator conductor direction and location, the turns function, and the winding function for the
example motor. The winding function multiplied by the current for each phase produces the
MMF of the armature.
In a similar fashion to the stator winding functions, a winding function can be created to
replicate the MMF of the rotor magnets. Each pole has conductors with a positive direction
and a negative direction, forming alternating north and south poles. The conductors are
distributed at the edges of the magnet pole span. To produce the equivalent MMF of the
magnets, the current is applied through the windings. Figure 4.14 shows the conductor
location and direction, turns function, and the winding function of the equivalent rotor
magnet winding.
Stator windings produce an MMF, which creates a radial flux density in the airgap, but
also creates a tangential component of field intensity. In the analysis of torque, employing the
84
phase A
Stator Winding
conductor direction
phase B
2
phase C
0
-2
0 45 90 135 180 225 270 315 360
theta [deg]
2
turns function
0
-2
0 45 90 135 180 225 270 315 360
theta [deg]
winding fucntion
2
0
-2
0 45 90 135 180 225 270 315 360
theta [deg]
Figure 4.13: Example Motor Stator Winding Functions Versus Stator Coordinate
Equivalent Magnet Winding
conductor direction
2
0
-2
0 45 90 135 180 225 270 315 360
theta [deg]
2
turns function
0
-2
0 45 90 135 180 225 270 315 360
theta [deg]
winding fucntion
2
0
-2
0 45 90 135 180 225 270 315 360
theta [deg]
Figure 4.14: Example Rotor Equivalent Magnet Winding Functions Versus Stator Coordinate
85
Maxwell stress tensor, this tangential component is required. Conductor location, direction,
and an assumed width are required. The current applied through the phase is used along
with the assumed width to compute the linear current density. The conductor location
function presented in fig. 4.13 is expanded to include the assumed width in fig. 4.15. The
linear current density, or tangential field intensity, is calculated by the multiplication of the
current density and the conductor width and location function.
Conductor Location and Width Function
2
exactA
exactB
1.5
exactC
fourier series A
1 fourier series B
fourier series C
0.5
Turns 0
-0.5
-1
-1.5
-2
0 45 90 135 180 225 270 315 360
theta [deg]
Figure 4.15: Example Stator Conductor Width Function Versus Stator Coordinate
Stator MMF is directly computed from the phase current and the winding function of
the individual phases. The total MMF is the sum of each respective phase MMF. This stator
MMF applies to the primary reluctance path but does not apply to the equipotential nature
of the secondary reluctance path. Modification of the MMF from the stator must be made
to account for this. Figure 4.16 provides an example of how this primary MMF is modified
to the equipotential secondary MMF across the magnet pole region of the rotor. Radial
airgap flux density can now be solved for each component of flux density using the MMF
86
permeance equation as shown in figure 4.17.
Example Motor Stator MMF Iss=400A Beta=135deg Position=0deg
1000
Primary Path
800 Secondary Path
600
400
MMF [A-turns]
200
0
-200
-400
-600
-800
-1000
0 45 90 135 180 225 270 315 360
theta [deg]
Figure 4.16: Analytically Calculated Stator MMF for Primary and Secondary Reluctance
Paths Versus Stator Coordinate
Example Motor Flux Density Iss=400A Beta=135deg Position=0deg
2
1.5
1
Flux Density [T]
0.5
0
-0.5
-1
primary reluctance
secondary reluctance
-1.5
magnet
total
-2
0 45 90 135 180 225 270 315 360
theta [deg]
Figure 4.17: Analytically Calculated Motor Radial Flux Density Versus Stator Coordinate
Superposition of the magnetic dipole, representing the sculpted feature, requires the
calculation of the smooth rotor radial air gap flux density. The radial air gap flux density of
87
the machine without sculpting features is used the compute the current density. Combining
the current density with the sculpt feature dimensions, the equivalent current is calculated.
The resulting winding functions of the sculpted magnet features, figure 4.14, and MMF of
the sculpted reluctance features, figure 4.19 are used for a modification of the air gap flux
density.
Example Motor Magnet Sculpt Widing Function
1
0.8
0.6
Winding Function [turns]
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
0 45 90 135 180 225 270 315 360
theta [deg]
Figure 4.18: Example Motor Magnet Sculpt Feature Winding Function Versus Stator Coor-
dinate
Finally, torque can be analyzed for each rotor position providing details of the torque
ripple. Using the MMF Permeance method allows for the torque ripple components for
the primary reluctance path, secondary reluctance path, magnet, and sculpt features to
be analyzed separately and combined. Within this example, torques are calculated using
Maxwell’s stress tensor as discussed in section 2.5.
88
Example Motor Reluctance Sclupt MMF Iss=400A Beta=135deg Position=0deg
1000
primary
800 secondary
600
400
MMF [A-turns]
200
0
-200
-400
-600
-800
-1000
0 45 90 135 180 225 270 315 360
theta [deg]
Figure 4.19: Example Motor Reluctance Sculpt Feature MMF Versus Stator Coordinate,
MMF due to sculpt feature based on primary reluctance path (blue), MMF due to sculpt
feature based on secondary reluctance path (red)
Example Motor Radial Flux Density Iss=400A Beta=135deg Position=0deg
2
1.5
1
Flux Denstiy [T]
0.5
0
-0.5
Smooth
-1 Sculpted
-1.5
-2
0 45 90 135 180 225 270 315 360
theta [deg]
Figure 4.20: Example Motor Flux Density Sculpted vs Smooth Versus Stator Coordinate
89
Example Motor, Iss=400 A, Beta=135 deg
25
Total Smooth
Total Sculpted
Rel Tot
20 Magnet Smooth
Primary Reluctance Smooth
Secondary Reluctance Smooth
Torque [Nm]
15
10
5
0
0 10 20 30 40 50 60 70 80 90
mechanical rotor position [degrees]
Figure 4.21: Example Motor Torque Ripple Versus Rotor Position Using MMF Permeance
Model and Maxwell Stress Tensor
90
Chapter 5
Space and Time Harmonics of Fields,
Permeance, and Torque
5.1 Fourier Series Description
Permeance, winding functions, turns densities, MMF, and linear current density may be
described with the Fourier series. Airgap flux density and torque are derived from these base
equations. While average torque is maintained during rotation, harmonic content also results.
This section develops the Fourier series description of the three phase IPM machine, described
in table 4.1. Equations are expressed in terms of electrical frequency of the machine, with
the rotor spatial coordinate as θ, and rotor position of ϕ = ωt. Simplified Fourier series are
developed in this section to illustrate harmonic contributions and interactions.
5.1.1 Current
Phase current amplitude, Iss , and control angle, β, are controlled in order to develop torque.
Transformed to the rotor reference frame, current are described with the direct axis current,
Id , and quadrature axis current, Iq . The phase currents, Ia , Ib , and Ic , are determined from
91
the rotating reference frame currents and rotor position ϕ.
Id = Iss cos(β) (5.1)
Iq = Iss sin(β) (5.2)
Ia (ϕ) = Id cos(ϕ) − Iq sin(ϕ) (5.3)
2π 2π
Ib (ϕ) = Id cos(ϕ − ) − Iq sin(ϕ − ) (5.4)
3 3
2π 2π
Ic (ϕ) = Id cos(ϕ + ) − Iq sin(ϕ + ) (5.5)
3 3
5.1.2 Permeance
Permeance functions result from the geometry of the IPM rotor as shown in Fig. 5.1. Three
functions are needed to describe the single-V IPM machine: the primary reluctance path,Λr1 ,
shown as a solid red line, the second reluctance path Λr2 , shown as a dashed gray line, and
the permeance of the magnet path, Λm , shown as a solid gray line. Each function contains
a constant and even order harmonics due to pole pair geometry.
X∞
Λr1 (ϕ, θ) = Ar1◦ + Ar1n cos(nθ − nϕ) (5.6)
n=2,4,6..
X∞
Λr2 (ϕ, θ) = Ar2◦ + Ar2n sin(nθ − nϕ) (5.7)
n=2,4,6..
X∞
ΛP M (ϕ, θ) = AP M◦ + AP Mn sin(nθ − nϕ) (5.8)
n=2,4,6..
92
Permeance functions for the second reluctance path, Λr2 , and magnet path, ΛP M , are based
off the winding function, NP M , and related to the their respective airgaps, g.
Figure 5.1: Permeance and Winding Functions
5.1.3 Winding and Turns Functions
Winding, Nx (θ), and conductor density, Cx (θ), functions are dependant upon the spatial
coordinate θ, Each function contains the Fourier coefficients for amplitude and phase. Con-
ductor density functions for each phase are Ca , Cb , and Cc represent the location and width
of each conductor. The conductor density function is related to the winding function by
93
C = dNdθ . These contain odd harmonics due to the alternating polarities between poles.
X∞
Ca (θ) = −an sin n(θ) (5.9)
n=1,3,5,7,9..
∞
X 2π
Cb (θ) = −an sin n(θ − ) (5.10)
3
n=1,3,5,7,9..
∞
X 2π
Cb (θ) = −an sin n(θ + ) (5.11)
3
n=1,3,5,7,9..
Where, winding functions for each phase are Na (θ), Nb (θ), and Nc (θ) are written as follows,
also contain all odd harmonics.
X∞
Na (θ) = av cos v(θ) (5.12)
v=1,3,5..
∞
X 2π
Nb (θ) = av cos v(θ − ) (5.13)
3
v=1,3,5..
∞
X 2π
Nc (θ) = av cos v(θ + ) (5.14)
3
v=1,3,5..
Finally, the equivalent magnet winding function for the rotor is NP M (θ).
X∞
NP M (θ, ϕ) = bv cos(vθ − vϕ) (5.15)
v=1,3,5,7..
5.1.4 MMF and Linear Current Density
The MMF is the product of the winding function, Nx , and current, Ix . The MMF functions
of the motor include phase A, Fa , phase B, Fb , phase C, Fc , rotor magnet, FP M . The total
94
MMF of the armature (stator), Fabc , is the sum of all the phases.
∞ ∞
X 2π X 2π
Fabc = Iss cos(ϕr ) av cos v(θ) + Iss cos(ϕr − ) av cos v(θ − )+
3 3
v=1,3,5.. v=1,3,5..
∞
4π X 4π
Iss cos(ϕr − ) av cos v(θ − ) (5.16)
3 3
v=1,3,5..
Where the permanent magnet MMF is the product of equivalent current IP M , the winding
magnets function, and ratio of pole span, τm and magnet width wm .
τm
FP M (θ, ϕ) = NP M (θ, ϕ)IP M (5.17)
wm
Br
IP M = lm (5.18)
µ0 µr
Only a subset of the armature MMF reacts with the second reluctance path permeance.
Λr2 (θ)
Fr2 (θ, ϕ) ⊆ Fabc (θ, ϕ) (5.19)
max Λr2 (θ)
Linear current density is the sum of the product of conductor densities and current for
each phase.
Kabc (θ, ϕ) = Ca (θ, ϕ)Ia (pϕ) + Cb (θ, ϕ)Ib (pϕ) + Cc (θ, ϕ)Ic (pϕ) (5.20)
95
5.2 Rotating Harmonics
5.2.1 Armature MMF
MMF is the product of winding functions and phase currents. A rotating MMF results from
the rotor position dependant phase currents and spatially dependant winding functions.
With much work and using the trigonometric identity, eq. (5.21) , the rotating system can
be broken down into a positively and negatively rotating harmonics.
1
cos A cos B = [cos(A − B) + cos(A + B)] (5.21)
2
This sequence involves grouping like terms where ϕ − vθ as the positively rotating sequence
ϕ + vθ as the negatively rotating sequence
∞ ∞
X 3 X 3
Fabc = Iss av cos(ϕ − vθ) + Iss av cos(ϕ + vθ) (5.22)
2 2
v=1,7,13,19.. v=5,11,17..
Even harmonics will not naturally show up, as the winding functions are derived from square
waves. The resulting MMF components are centered around harmonic multiples of 6. A
summary of the sequence effects is shown in the Table 5.1.
5.2.2 Primary Reluctance Path Flux Density Harmonics
The permeance function of the primary reluctance path has only a constant and even elec-
trical harmonics. The armature MMF harmonics only contain odd orders and no triplen
harmonics. The convolution of the two results in only the odd order harmonics with triplen
harmonics.
96
positive sequence negative sequence
harmonic ϕr − vθ ϕr + vθ
1 +
2 n/a - even n/a - even
3 n/a - triplen n/a - triplen
4 n/a - even n/a - even
5 -
6 n/a - triplen n/a - triplen
7 +
8 n/a - even n/a - even
9 n/a - triplen n/a - triplen
10 n/a - even n/a - even
11 -
12 n/a - triplen n/a - triplen
13 +
14 n/a - even n/a - even
15 n/a - triplen n/a - triplen
16 n/a - even n/a - even
17 -
18 n/a - triplen n/a - triplen
19 +
20 n/a - even n/a - even
21 n/a - triplen n/a - triplen
Table 5.1: Armature MMF Harmonic Sequencing Summary, negative sequence (+), positive
sequence (-), not applicable (n/a). Not applicable is due to cancellation of triplen harmonics
of the 3 phases or even MMF components are not possible due to convolution.
X∞
Brel1 = [Ar1◦ + Ar1n cos(nθ − nϕ)]
n=2,4,6..
∞ ∞
X 3 X 3
[Iss av cos(ϕr − vθ) + Iss av cos(ϕr + vθ)] (5.23)
2 2
v=1,7,13,19.. v=5,11,17..
5.2.3 Linear Current Density Harmonics
The linear current density is a product of the phase current and the conductor density
functions. Equation 5.20 can be simplified and re-written as equation 5.24. The electrical
97
harmonic description is considered.
∞ ∞
X 2π X 2π
Kabc = Iss cos(ωt) an cos(nθ) + Iss cos(ωt − ) an cos(nθ − n )
n
3 n 3
∞
4π X 4π
+ Iss cos(ωt − ) an cos(nθ − n ) (5.24)
3 n 3
Each harmonic can be simplified by considering sequence effects of the phases.
∞
X
Kabc (n) = Iss an cos(nθ ± ωt) (5.25)
n
The value of the ± becomes dependant on the order of the harmonic, which is summarized
in table 5.2. A pattern emerges where the fundamental rotates clockwise, triplen harmonics
equate to zero, harmonics immediately before triplen harmonics rotate counter clockwise,
and harmonics immediately after triplen harmonics rotate counter clockwise.
Order n Sign Rotation Direction
1 - clockwise
5 + counter clockwise
6 n/a
7 - clockwise
11 + counter clockwise
12 n/a
13 - clockwise
17 + counter clockwise
18 n/a
Table 5.2: Linear Current Density Electrical Harmonic Sequence Value
98
5.2.4 Torque Harmonic Orders from Air Gap Flux Density and
Linear Current Density
Torque is calculated from the spatial integral of the Maxwell stress tensor. Integration filters
out all but combinations that produce a constant component; periodic functions without
an offset do not survive. Each combination of the radial and tangential field harmonics
gives rise to two possible harmonics, resulting from both addition and subtraction of the
harmonic orders. For each position, integration of the fields results in a torque. As rotor
position changes like orders of B and K contribute to torque to their nearest multiple of 6,
resulting in the torque of the machine changing with rotor position.
Concentrating on the convolution of the flux density with linear current density orders
two possible harmonics, shown in table 5.3. Many harmonic combinations are possible due
to the convolution of radial and tangential flux density orders, most do not produce torque.
Only convolution of like orders results in a constant value, which is necessary to produce
torque.
K\B 1 3 5 7 9 11 13 15 17 19
1 0 2 -2 4 -4 6 -6 8 -8 10 -10 12 -12 14 -14 16 -16 18 -18 20
5 4 6 2 8 0 10 -2 12 -4 14 -6 16 -8 18 -10 20 -12 22 -14 24
7 6 8 4 10 2 12 0 14 -2 16 -4 18 -6 20 -8 22 -10 24 -12 26
11 10 12 8 14 6 16 4 18 2 20 0 22 -2 24 -4 26 -6 28 -8 30
13 12 14 10 16 8 18 6 20 4 22 2 24 0 26 -2 28 -4 30 -6 32
17 16 18 14 20 12 22 10 24 8 26 6 28 4 30 2 32 0 34 -2 36
19 18 20 16 22 14 24 12 26 10 28 8 30 6 32 4 34 2 36 0 38
Table 5.3: Air Gap Field Spatial Electrical Harmonic Convolution, Horizontal - Flux Density
Harmonics, Vertical - Linear Current Density Harmonics
Maxwell stress tensor is the spatial integral of the convolution of air gap radial flux density
and armature linear current density. This integration filters out all but combinations that
produce a zeroth order; periodic functions without an offset do not survive. Torque harmonics
99
which result from the convolution flux density harmonics are shown in the following table.
K\B 1 5 7 11 13 17 19
1 0
5 6
7 6
11 12
13 12
17 18
19 18
Table 5.4: Air Gap Field Spatial Electrical Harmonic Convolution and Electrical Torque
Harmonic, Horizontal - Flux Density Harmonics, Vertical - Linear Current Density Harmon-
ics
5.2.5 Magnet Flux Density and Torque Harmonics
The flux density of the rotor magnet expressed as a Fourier series is shown in equation
5.26. In this description, the rotor position, ωt, is multiplied by the order of the harmonic.
The linear current density of the armature, as shown in section 5.2.3, is dependent on rotor
position through the phase current.
∞
X
Bm (θ, ωt) = 2Λm If bv cos(vθ − vωt) (5.26)
v=1,3,5,7..
The Maxwell stress tensor combines both the magnet radial flux density and armature linear
current density.
∞ ∞
Z 2π ! !
Tm = r2 l2Λm If Iss ·
X X
bv cos(vθ − vωt) an cos(nθ ± ωt) dθ (5.27)
θ=0 v n
The trigonometric identity 5.21, along with the Maxwell stress tensor, is used to illustrate
the 5th torque harmonic dependency on rotor position. With some work, it is shown the
100
fundamental of the phase current interacts with both the fifth and seventh spatial harmonics
of the flux density to produce sixth harmonics in torque. The sixth torque harmonic is only
dependant upon rotor position, and the component which is dependant upon the spatial
information is null.
Z 2π !
Tm = r2 lΛ m If Iss · b5 a5 [cos(−6ωt) + cos((10θ − 4ωt)] dθ (5.28)
θ=0
5.2.6 Field and Torque Harmonics Relationship to Control Angle
Consideration of control angle is necessary in the design and analysis of optimal fields,
torques, and harmonics. For the average values this importance is highlighted by the torque
equation 5.29, and voltage equations 5.30 and 5.31. Written with the traditional -d and
-q axis, parameters and states include inductances, ld and lq , the permanent magnet flux
linkage λm , armature resistance Rs , magnet alignment to reluctance axis ,δ, currents, id
and iq , and rotor speed ω. These governing equations, along with current, voltage, and
speed constraints, determine the operating currents related to maximum torque per ampere
(MTPA) and field weakening. The d- and q- axis currents influence the average performance
and also contribute to the airgap flux density and torque harmonics.
!
13
τ= (ld − lq )id iq + λm cos(δ)iq − sin(δ)id (5.29)
22
vd = Rs id + −ωlq iq − ωλm sin(δ) (5.30)
vq = Rs iq + ωld id + ωλm cos(δ) (5.31)
The Fourier series description of the airgap flux density for the permanent magnet and
101
armature induced reluctance flux densities reveals symmetries within the rotors frame of
reference. The permanent magnet airgap radial flux density has even symmetry and is not
dependent upon armature current, hence the spatial Fourier series expansion is based upon a
cosine. The current dependant flux density, due to the interaction of armature MMF and the
rotors salient structure, results in radial airgap flux densities with even symmetry (cosine)
due to a d- axis current and odd symmetry (sine) with a q- axis current. The windings
tangential flux density creates odd symmetry with a d- axis current and even symmetry
with a q- axis current. The resulting radial and tangential flux density space harmonics, due
to n current angle command, vector sum from its d- and q- axis constituents.
B(θ, Iss , β) = Λ(θ)F (θ, i⃗d ) + Λ(θ)F (θ, i⃗q ) (5.32)
id iq O.C.
magnet (radial) n/a n/a even
rotor reluctance (radial) even odd n/a
winding (tangential) odd even n/a
Table 5.5: Airgap flux density Fourier coefficients dependency on reference frame currents.
Airgap flux density symmetry extends to the discussion of torque and its harmonics.
Average torque (and torque harmonics) is generated from like symmetries of radial and
tangential flux densities. Opposite symmetries produce zero average torque along with har-
monics. The permanent magnet flux density interaction with the -q axis currents produces
average torque and harmonics. Likewise, reluctance flux densities similarly generate average
torque.
Z 2π
Tmag (ϕ, i⃗d , i⃗q ) = r2 l Bmag (θ, ϕ)K(θ, ϕ, i⃗d ) + Bmag (θ, ϕ)K(θ, ϕ, i⃗q ) dθ (5.33)
θ=0
102
Freezing of the reluctance airgap flux density due to reference frame current is required to
obtain the interaction with its conjugate tangential flux density. Knowledge of the torque
harmonics and flux density harmonics due to the -d and -q axis current, provides complete
knowledge of harmonics over the entire current plane.
Z 2π
Trel (ϕ, i⃗d , i⃗q ) = r2 l Brel (θ, ϕ, i⃗d )K(θ, ϕ, i⃗d ) + Brel (θ, ϕ, i⃗d )K(θ, ϕ, i⃗q )+
θ=0
Brel (θ, ϕ, i⃗q )K(θ, ϕ, i⃗d ) + Brel (θ, ϕ, i⃗q )K(θ, ϕ, i⃗q ) dθ (5.34)
T⃗mag(n) (⃗id + ⃗iq ) = T⃗(n) (⃗id ) + T⃗(n) (⃗iq ) (5.35)
T⃗rel(n) (⃗id + ⃗iq ) = T⃗rel(n) (⃗id ,⃗id ) + T⃗rel(n) (⃗id ,⃗iq ) + T⃗rel(n) (⃗iq ,⃗id ) + T⃗rel(n) (⃗iq ,⃗iq ) (5.36)
Torque and its harmonics are produced from interactions of the same harmonic of tangen-
tial and radial flux density. For the IPM machine, average torque is produced by components
of fundamentals that are in phase with each other. Furthermore, torque is produced from
the orthogonality of d- and q- axis winding tangential flux densities and radial flux densities.
Field harmonics and non-orthogonal interactions produce the undesired torque harmonics.
5.3 Example Machine Mechanical Harmonics
Mechanical harmonics of the example machine are listed in table 4.1, without the rotor
sculpt feature is presented in section 5.3. Mechanical harmonics are a product of electrical
harmonics and pole pairs, with the fundamental mechanical order defined as one mechanical
rotation. Since this machine is a two pole pair machine, p = 2, sinusoidal currents are applied
with two electrical cycles per mechanical rotation. Permeance functions are doubly sinusoidal
103
for each pole pair and, as a result, have a fourth mechanical order. Conductor density and
winding functions contain both the triplen harmonics and typical 5th/7th, 11th/13th, and
additional orders multiplied by the number of poles, p. Phase MMFs, conductor densities,
and linear current densities retain these harmonics. The summation of all phase quantities
leads to the dropping of the triplen harmonics from the total MMF and total linear current
densities. Triplen harmonics are not removed from the 2nd reluctance path MMF Fr2 . Flux
densities contain both triplen and winding harmonics. Calculation of torque from Maxwell
stress tensor and convolution air gap variables only even order torque harmonics remain.
104
Quantity Mechanical Order
Id 0
Iq 0
Ia 2
Ib (pϕ) 2
Ic (pϕ) 2
Λr1 (θ) 0,4,8,12,16,24,28,32,36,44...
Λr2 (θ) 0,4,8,12,16,20,24,28,32,36,40...
Λm (θ) 0,4,8,16,20,28,32,36,40...
Na (θ, ϕ) 2,6,10,14,18,22,26,30,34,38,42...
Nb (θ, ϕ) 2,6,10,14,18,22,26,30,34,38,42...
Nc (θ, ϕ) 2,6,10,14,18,22,26,30,34,38,42...
Nr (θ) 2,6,10,14,18,22,26,30,34,38,42...
Ca (θ, ϕ) 2,6,10,14,18,22,26,30,34,38,42...
Cb (θ, ϕ) 2,6,10,14,18,22,26,30,34,38,42...
Cc (θ, ϕ) 2,6,10,14,18,22,26,30,34,38,42...
Fa (θ, ϕ) 2,6,10,14,18,22,26
Fb (θ, ϕ) 2,6,10,14,18,22,26
Fc (θ, ϕ) 2,6,10,14,18,22,26
Fm (θ, ϕ) 2,6,10,14,18,22,26
Fabc (θ, ϕ) 2,10,14,22,26,34,38,46,50
Fr2 (θ, ϕ) 2,6,10,14,18,22,26,30,34,38,42
Kabc (θ, ϕ) 2,10,14,22,26,34,38
Br1 (θ, ϕ) 2,6,10,14,18,22,26,30
Br2 (θ, ϕ) 2,6,10,14,18,22,26,30
Bm (θ, ϕ) 2,6,10,14,18,22,26,30
Tr1 0,12,24,36,48
Tr2 0,12,24,36,48
Tm 0,12,24,36,48
Table 5.6: Mechanical Harmonics of Example Machine
105
Chapter 6
Design for Minimal Torque Ripple
6.1 Review of Design Methods and Features
The design of IPM machines is multidisciplinary and complex. Many methods have been
developed to optimize and design the machine, which relies on computationally expensive
processes. Each of these methods requires multiple runs to explore and optimize the re-
sult. Taguchi’s design of experiment techniques was used to optimize torque production
of IPM machines and minimize torque ripple in [87, 88]. Im et al., used the response sur-
face methodology to minimize torque ripple considering both the flux barrier geometry and
residual flux of the permanent magnet [89]. Response surface methodology was applied to a
surface mount PMAC machine focusing on magnet pole shaping in [35]. Multi-objective and
multi-load point optimization of using the response surface method to minimize torque ripple
and maximize average torque is presented in [90]. Lebensztajn and Marretto presented the
use of Kriging in the optimization of electromagnetic devices, reducing the sample size nec-
essary while maintaining accuracy [91]. Kriging along with genetic algorithms are presented
in [92, 93]. A Multi-objective MOGA technique was utilized in [52]. Each of these optimiza-
tion techniques effectively minimizes torque ripple but leaves little guidance to design.
Features on or near the rotor surface have been utilized to reduce torque harmonics. The
use of notches at the rotor surface placed on the primary reluctance path and/or secondary
106
Parameter Value Unit
Pole Pairs 4
Stator Slots 48
Number of Phases 3
Stack Length 83 mm
Rotor Diameter 161.15 mm
Airgap Length 0.75 mm
Magnet Pole Arc % of Pole Pitch 63.8 %
Barrier Type Single V
Magnet Thickness 6.48 mm
Magnet Width 16 × 2 mm
Permanent Magnet Remnant 1.19 T
Permeability of Iron ∞ H/m
Permeability of Bridge Features 4π · 10−7 H/m
Table 6.1: Example motor parameters.
reluctance path is explored in [94]. Surface dimples on the flat magnet IPM design are
studied in [43, 95], where the first utilized finite elements with some analytical method to
guide design, and the former is reliant purely on finite elements for design. IPM machines
with dimples at the surface and holes near the surface are discussed in [36,41]. Bread loafing
of the secondary reluctance path is considered in [38]. Rotor surface features are staggered
in place of skew features in [34]. Barrier design is altered in [96] to minimize torque ripple.
Each of these methods has merits that are shown to be effective.
6.2 Application of Analytical Model to Example Ma-
chine
The method developed in chapter 4 is validated with a well-known industrialized IPM ma-
chine. Details of a well-known industrialized traction motor are included in table 7.1, and the
geometry is modeled analytically within Matlab and finite elements within Ansys Maxwell.
107
Focusing on the effects of the rotor, the stator geometry has been idealized with no slots.
Both sinusoidally distributed stator windings and the production configuration of discretely
placed windings are modeled. Only the stator winding harmonics interactions with rotor
geometry harmonics are considered. With the assumption of infinite permeability, the bridge
features are omitted. Airgap fringing in the region of the magnet barrier is not considered.
Design parameters are studied within this section using the analytical winding function
model previously validated. Rotor sculpt features are included along with their additional
MMF term.
6.2.1 Sculpting Geometry
The sculpted rotor IPM machine geometry design space to be explored is shown on a single
pole of the example machine in Figure fig. 6.1. Rotor primary and secondary reluctance
paths are shown in green with no bridge features. The stator, shown in gray, continues to
have omitted its slot features, and distributed windings, orange, are placed within the airgap.
The primary design parameters are centered on the rotor effects, which include the ratio
of primary and secondary reluctance path and sculpt features. The magnet pole arc width
is varied. Up to two sets of symmetrically placed sculpt features are placed on the second
reluctance path. The symmetrical feature span locations τ1 and τ2 define the symmetrical
location of the feature in terms of its percentage of the magnet pole arc span. Single asym-
metrical features are described with a similar parameter but with only one sculpt feature on
the pole. In this single asymmetrical case, the feature location, τ , is set to be positive for
right-hand side placement and negative for the left-hand side placement. The depths D1 and
D2 are measured from the outer surface of the rotor to the root of the sculpt feature. The
widths W1 and W2 are measured in terms of a single feature percentage of the pole span.
108
6.3 Model Validation: Radial Flux Density
This section compares the radial flux density results of the analytical model and finite ele-
ments while varying: (1) winding type, (2) current, (3) control angle and (4) rotor sculpt
features. Figures 6.2–6.5 plot the flux density along the rotors spatial coordinate, θ, over a
single pole pair. Both sinusoidal and distributed windings are compared. The q-axis, which
is aligned to the rotor minimum reluctance, occurs at θelec = 90◦ and θelec = 270◦ . The
d -axis is aligned, which is aligned to the smooth rotors permanent magnet maximum flux
linkage, occurs at θelec = 0◦ , θelec = 180◦ , and θelec = 360◦ . In all results, the finite element
and the analytical model result in comparable flux densities.
Flux densities shown for sinusoidal windings, fig. 6.2, illustrate the changing airgap flux
density harmonics with current and control angle. When the phase current is set to zero,
only the permanent magnet field is present. As current and current angle increases, the flux
density becomes more jagged, with the case of a fully negative d -axis current displaying the
most harmonic content. It is clear that as the negative d -axis current becomes dominant, so
do the reluctance path harmonics. Harmonic effects of the discretely distributed windings
are shown in fig. 6.3. As current increases, so do the airgap reluctance harmonics. In all
cases, the analytical model and finite element results agree with reasonable accuracy.
The effects of rotor sculpt features on the second reluctance path are shown in
Figures 6.4 and 6.5. A symmetrical pair of sculpting features are shown with distributed
windings in fig. 6.4. The flux densities of the smooth rotor and sculpted rotor are plotted.
Reduced flux density in the region of the sculpt features is observed, and sculpt features
are located approximately at θelec = 30◦ , 150◦ , 210◦ , 270◦ . This flux density from the sculpt
features is conserved and redistributed across the regions of the second reluctance path. The
109
Figure 6.1: Rotor sculpt features.
Iss=0 Iss=200 =90 Iss=200 =135
Model
Flux Density [T] Flux Density [T] Flux Density [T]
FE
elec
[deg] elec
[deg] elec
[deg]
Iss=400 =135 Iss=400 =180
Flux Density [T] Flux Density [T]
elec
[deg] elec
[deg]
Figure 6.2: Radial flux densities with sinusoidally distributed windings (N = 200) at various
currents and control angles.
Iss=100 =90 Iss=400 =135
model
Flux Density [T] Flux Density [T]
FE
elec
[deg] elec
[deg]
Figure 6.3: Radial flux densities with distributed windings (2 SPP) at various currents and
control angles.
110
Iss=400 =130°
Smooth FE
Sculpted FE
Flux Density [T]
Sculpted Model
elec
[deg]
Figure 6.4: Radial flux densities with single symmetrical sculpt feature located at τ1 = 50%,
W1 = 10%, and D1 = 1.2 mm.
reluctance flux density single sculpt feature is plotted in fig. 6.5. Similar to the symmetrical
sculpt features, the flux density is reduced in the region of the sculpt feature. In all cases,
the analytical model and finite element results agree with reasonable accuracy.
Sinusoidal Windings Iss=400 =130° Distributed Windings Iss=400 =130°
Smooth FE
Flux Density [T] Flux Density [T]
Sculpted FE
Sculpted Model
elec
[deg] elec
[deg]
Figure 6.5: Radial flux densities (reluctance only) with single asymmetrical sculpt feature
located at τ = 50%, W = 20%, and D = 1.2 mm.
6.4 Model Validation: Torque Ripple
Torque ripple of the smooth rotor IPM, table 7.1, is compared between finite element and
the analytical model in fig. 6.6. Good agreement between the finite element and analyti-
cal models is observed. The torque ripple effects of two symmetrical rotors sculpt features
are demonstrated in fig. 6.7, directly calculated by the analytical model, whereas the finite
111
element model requires two runs, once with and once without sculpting features, to deter-
mine the sculpt feature effects.A good correlation between the model and finite elements is
demonstrated and shown.
Torque Ripple Iss=400 =130 °
100
FE
90 Model
80
70
Torque [Nm]
60
50
40
30
20
10
0
0 90 180 270 360
elec
[deg]
Figure 6.6: Smooth rotor IPM torque ripple.
Sculpt Feature Torque Ripple, Iss =200 = 135 o
analytical
FE
Torque [Nm]
Rotor Electrical Postion [deg]
Figure 6.7: Sculpt feature torque ripple.
6.5 Torque Ripple Components
In this section, the torque ripple results of the analytical model are studied while varying
current and control angle. Figure 6.8 plots the torque ripple for a complete electrical cycle
112
of the example machine. The torque components for the first reluctance path, second reluc-
tance path, total reluctance torque, magnet torque, and total machine torque are plotted.
Magnet torque and its harmonics are dominant at lower currents, but the reluctance paths
cannot be ignored. As current is increased, the reluctance torque increases relative to the
magnet torque. The stronger field weakening currents cause the contribution of the reluc-
tance features torque to increase. The dominant torque harmonic orders are the 6th and
12th electrical orders. In the design, both the torque harmonic amplitudes and phases of
each of the components need to be considered as the sculpt feature design will provide the
counter torque at the counter phase.
Example Motor, Iss=200, =90 Example Motor, Iss=200, =135 Example Motor, Iss=200, =180
Total
Rel Tot
Magnet
1st Rel
Torque [Nm] Torque [Nm] Torque [Nm]
2nd Rel
elec
[degrees] elec
[degrees] elec
[degrees]
Example Motor, Iss=400, =90 Example Motor, Iss=400, =135 Example Motor, Iss=400, =180
Torque [Nm] Torque [Nm] Torque [Nm]
elec
[degrees] elec
[degrees] elec
[degrees]
Figure 6.8: Analytical model torque ripple components at various control angles.
113
6.6 Investigation of Design Features
In this section, the effects of design features are demonstrated to influence both torque
harmonic amplitudes and phases. Carefully applied, these effects are used to design counter
torque harmonics. Second reluctance path pole arc, sculpt feature type (symmetrical/asymmetrical),
sculpt feature location, sculpt feature depth, and sculpt feature width can all be used to de-
sign an appropriate counter torque to reduce the machine’s torque harmonics. While mildly
affecting average torque, the second reluctance path pole arc, τp , strongly affects the phase
of the 12th electrical order torque harmonic. A single pair of symmetrical sculpt features
placed upon the second reluctance path pole arc reduces the average torque. Feature po-
sition provides 12th electrical order torque harmonic phasing, and the feature width, and
depth directly affect the torque harmonics amplitude. The single asymmetrical feature is
shown to increase average torque when placed on a specific side of the second reluctance
path pole arc. The asymmetrical feature placement can also be used to modify the phase of
both 6th and 12th electrical order torque harmonics. Finally, feature phasor summation is
shown to be effective in combining the effects of multiple design features, further providing
the ability to design both the amplitude and phase of these minimizing torque harmon-
ics. These relationships provide the necessary intuition to reduce computationally intensive
design steps.
6.6.1 Magnet Pole Arc
Magnet pole arc span, τm , effects upon the torque harmonics, without rotor sculpt features,
are explored. Figure 6.9 shows the average, 6th, and 12th harmonics of torque as a function
of τm . For this case, magnet torque is dominant. Although not always the case, it is just
114
Total First Reluctance Path Second Reluctance Path Magnet
0
6
Torque [Nm] Torque [Nm] Torque [Nm] Torque [Nm]
12
Magnet Arc Span of Pole [%] Magnet Arc Span of Pole [%] Magnet Arc Span of Pole [%] Magnet Arc Span of Pole [%]
Phase [deg] Phase [deg] Phase [deg] Phase [deg]
Magnet Arc Span of Pole [%] Magnet Arc Span of Pole [%] Magnet Arc Span of Pole [%] Magnet Arc Span of Pole [%]
Figure 6.9: Effects of magnet pole arc τp loaded at Iss = 200 and β = 135◦ .
as important to follow the trends of individual torque components. Total, first reluctance,
and magnet average machine torque are reduced as the pole arc is increased, and only the
second reluctance torque increases the average torque. The 12th order torque harmonic
is dominant, with primary contributions from the magnet and the second reluctance path,
whereas the 6th order torque harmonic is mostly contributed to by the primary reluctance
and magnet. Rotor geometry has a strong influence on the phase of the 12th order torque
harmonics, whereas the 6th harmonic is less affected by rotor geometry.
6.6.2 Single Pair Symmetrical Rotor Sculpt Feature
A single symmetrical rotor sculpt feature torque is studied in Figures 6.10–6.12. Only the
effects of the sculpt feature torque are plotted. In Figures 6.10 and 6.11, a single sculpt
feature position is varied, with fixed width, W1 , and fixed depth, D1 , along the magnet pole
arc. Rotor sculpt features hurt average torque, as the phase is 180◦ out of phase with the
smooth rotor average torque. Figure 6.10 compares the analytical model to finite elements
and shows precise agreement with the phase and matching trends for torque amplitude.
115
Using a wider feature width, D1 , fig. 6.11 translates amplitude and phase plots to a phasor
representation. The 12th harmonic is the dominant torque in both amplitudes and choice
of phase.
I ss = 200 A = 135° I ss = 200 A = 135° I ss = 200 A = 135°
Torque Amplitude 6th [Nm] Torque Amplitude 12th [Nm]
Average Torque [Nm]
1
span location [%] 1
span location [%] 1
span location [%]
I ss = 200 A = 135° I ss = 200 A = 135°
phase 6 th [deg] phase 12 th [deg]
analytical
FE
1
span location [%] 1
span location [%]
Figure 6.10: Symmetrical sculpt feature effects compared to finite elements: D1 = 1.2 mm,
W1 = 5%.
Amplitude, I ss=200, =135 Phase, Iss=200, =135 Complex Phasor, I ss=200, =135
0
Torque [Nm] Torque [Nm] Torque [Nm]
6
12
6
12
18
Span % of Magnet Arc [%] Span % of Magnet Arc [%] Torque [Nm]
Figure 6.11: Symmetrical sculpt feature effects with phasor diagram: D1 = 1.2 mm, W1 =
9%.
Figure 6.12 includes the sculpt features width and depth effects. Increasing sculpt fea-
ture width, W1 , and/or the sculpt features depth, D1 , increases the amplitude of the sculpt
116
features torque harmonic. The primary influence on the torque harmonic amplitude is the
width of the sculpt feature. Sculpt feature width and depth do not affect the torque har-
monics phase.
6.6.3 Single Asymmetrical Rotor Sculpt Feature
A single sculpt feature resulting in asymmetrical placement upon the rotor surface is studied
in this section. Similar parameters D1 , W1 , and τ1 are used to describe the features width,
depth, and location. In the asymmetrical case, the location, τ1 , is described with the same
location parameter, wherein this case, a positive τ1 results on the right side of fig. 6.1 and a
negative τ1 results in sculpt feature placement on the left hand side. Figure 6.13 compares
the model to finite element and shows precise agreement with phase and matching trends for
torque amplitude. In the asymmetric sculpt feature case, a torque improvement is possible
due to the aligned axis effect from τ1 > 0. The placement of the sculpt feature allows for
the placement of the torque harmonic phase angle across a broad range of phases. Negative
values of τ1 result in the largest amplitudes of the 12th electrical torque harmonic.
6.6.4 Two Symmetrical Rotor Sculpt Features
More than one symmetrical rotor sculpt feature can be used. In this section, it is shown
that the components of a first symmetrical feature can be combined with that of the second
symmetrical rotor sculpt feature. The MMF-permeance model is validated by comparing to
finite element results in fig. 6.7.
To illustrate this concept, the parameters of the two sculpt feature sets of fig. 6.1 are
shown in table 6.2. Through vector summation, the two vectors were used to create a 12th
117
Torque Harmonic (6th) Torque Harmonic (12th)
W 9%, D 1.5mm
W 9%, D 3mm
Torque [Nm] Torque [Nm]
W 9%, D 6mm
W 4%, D 1.5mm
W 4%, D 3mm
W 4%, D 6mm
W2%, D 1.5mm
W 2%, D 3mm
W 2%, D 6mm
Span % of Magnet Arc [%] Span % of Magnet Arc [%]
Torque Harmonic (6th) Torque Harmonic (12th)
Phase [deg] Phase [deg]
Span % of Magnet Arc [%] Span % of Magnet Arc [%]
Figure 6.12: Sculpt feature effects N = 1.
order counter torque with a phase of −116◦ .
Feature Value Unit
τ1 82 % of magnet pole arc
τ2 46.5 % of magnet pole arc span
W1 5.5 % of pole span
W2 5.5 % of pole span
D1 1.2 mm
D2 1.2 mm
Table 6.2: Two sculpt feature parameters.
Figure 6.14 illustrates the first (red) and second 12th order electrical torque (green)
phasors. The two phasors combine to create the effective total phasor (blue). This phasor
summation is plotted along with the torque complex mapping of the previous single feature
design sweep. These symmetrical rotors sculpt features are designed to mitigate the 12th
order electrical torque harmonics to near zero. A single feature or multiple features can be
118
I ss = 400 A = 135° I ss = 400 A = 135° I ss = 400 A = 135°
Torque Amplitude 6th [Nm] Torque Amplitude 12th [Nm]
Average Torque [Nm]
1
span location [%] 1
span location [%] 1
span location [%]
I ss = 400 A = 135° I ss = 200 A = 135°
analytical
phase 12 th [deg]
FE
phase 6 th [deg]
1
span location [%] 1
span location [%]
Figure 6.13: Single asymmetrical sculpt feature effects with phasor diagram: D1 = 1.2 mm,
W1 = 5%.
119
designed to minimize the torque ripple. The sculpt features are not without consequence,
as the average torque is negatively affected.
Sculpt Feature Torque Phasor (12th Order Electrical), Iss =200 = 135 o
map
first
second
first + second
Torque [Nm]
Torque [Nm]
Figure 6.14: Two sculpt features of 12th order electrical torque phasor plot.
6.7 Conclusions
This chapter has presented an analytical modeling and design approach to reduce torque
ripple with rotor sculpt features. By carefully placing rotor sculpt features and rotor barrier
features, average torque can be maintained while minimizing torque harmonics. Sculpt
features torque amplitude and phase are shown to be in close agreement in 6.10 and 6.13. The
analytical model accurately predicts the phase in the case of symmetrical and asymmetrical
features providing an opportunity to guide design. Contributions of this section include:
• A new analytical winding factor modeling approach for the single V IPM machine re-
lating the rotor’s first reluctance feature, second reluctance feature, permanent magnet
features, sculpt features, and stator windings to the resulting torque harmonics;
• An analytical modeling approach accounting for both symmetrical and torque aligning
120
asymmetrical rotor sculpt features;
• Results from the analytical model providing valuable insights for identifying rotor
feature design improvements;
• Design approach for placement of rotor sculpt features to minimize torque ripple while
maintaining average torque;
• Demonstration of close agreement of radial flux density and torque harmonics results
between the analytical model and that of finite element results.
These results enable better design insight and an efficient design process through the use
of an analytical model.
121
Chapter 7
Optimal Design for Minimal Torque
Ripple
7.1 Methods
This section describes the electric machine parameters, sculpt feature geometry, analytical
modeling, and optimization method utilized to minimize torque harmonics.
7.2 Electric Machine Parameters
To evaluate the analytical MMF permanence method, a single barrier IPM machine has been
modeled using both FE and analytical methods. The model is constructed and evaluated
based on a well-known smooth rotor industrial IPM machine. Parameters of this machine
are included in section 7.2.
A schematic of a two-pole rotor describing the sculpting features to this machine is
given in fig. 7.1. Variables Y1 and Y2 , define sculpt feature location in terms of half of the
magnet pole arc span, αm , where Y1 is a positive percentage and Y2 as a negative percentage
(counter-clockwise position). Feature width Z1 and Z2 , define the sculpt feature width in
terms of percentage of the magnet pole arc span, αm . A single sculpt feature depth, D1 ,
is utilized for both features. The asymmetrical features will be used to take advantage of
122
Parameter Value Unit
Pole Pairs 4
Stator Slots 48
Number of Phases 3
Stack Length 83 mm
Rotor Diameter 161.15 mm
Air Gap Length 0.75 mm
Magnet Pole Arc % of Pole Pitch 63.8 %
Barrier Type Single V
Magnet Thickness 6.48 mm
Magnet Width 16x2 mm
Permanent Magnet Remnant 1.19 T
mkg
Permeability of Iron ∞
s2 A2
mkg
Permeability of Bridge Features 4π · 10−7
s2 A2
Table 7.1: Motor Parameters
average the possible torque increase but also to target specific torque harmonics.
Figure 7.1: Two pole equivalent Scrupted Rotor IPM model
7.3 Minimization of Torque Harmonics
The developed analytical model is used to design sculpting features upon the smooth rotor
machine defined in Table section 7.2. This computationally efficient model is integrated with
a genetic optimization tool, GOSET [97], to determine the geometry to minimize the torque
123
ripple via targeted torque harmonics. All optimization is done with a fixed current Iss and
angle β, in the constant torque region.
7.3.1 Optimization Methodology
The design optimization of sculpt features to minimize the 6th or 12th order torque compo-
nent amplitude. In previous work, it was shown that the analytical model predicts sculpt
feature effects of torque harmonic phase precisely and the amplitude with modest accu-
racy [98]. Starting with a smooth rotor, optimizing with sculpting features is done using
the smooth rotor FE result and the sculpted rotor analytical model. Addressing both the
amplitude and phase, the torque harmonic design objective is expressed as a complex num-
ber. The targeted smooth rotor FE torque harmonic is rotated 180◦ , T ∗ , to determine the
desired torque harmonic produced by the sculpt features.
F E (n)e⃗iπ
T⃗ ∗ (n) = T⃗sm (7.1)
To minimize the torque harmonic, the difference between the finite element smooth rotor
torque harmonic conjugate, T⃗ ∗ (n), and the analytical models sculpt feature torque harmonic
effect, T⃗ScEf
M DL , further defines the optimization objectives in (7.3) and (7.4). Subtracted in
f
complex form, the amplitude of the difference is later used in the objective function.
M DL (n) = T
T⃗ScEf ⃗ M DL (n) − T⃗ M DL (n) (7.2)
f sc sm
The first, (7.3), utilizes the sculpt feature location, Y 2, width, Z2, and depth D1 . The
second, (7.4), utilizes the sculpt feature location, Y1 , width, Z1 , and depth D1 . The opti-
124
mization problems utilize Y1 are Y2 to respectively minimize the 6th order and 12th order
torque harmonics. Feature locations Y1 and Y2 set the phase of the torque harmonics intro-
duced by the sculpt feature [98]. Feature Z1 and Z2 are designed to set the amplitude of
injected torque harmonic.
The main design constraint of the rotor is mechanical stress. The centrifugal forces
coupled with the stress intensification due to the barrier features. Thinner bridges increase
magnetic performance but increase stress. For this analysis, the bridge thickness is left
constant, and the sculpt feature placement is varied. Stress concentrations occur in either
the bridge or sculpt feature as shown in fig. 7.2. At 5000rpm the rotor stresses, shown in
fig. 7.3, near yield as the sculpt feature approaches the barrier. A further constraint is applied
on feature location in combination with feature size to avoid manufacturing and mechanical
limitations of feature placement near the barriers.
Figure 7.2: Rotor Sculpt Feature, bridge stress (red arrow), sculpt stress (green arrow)
125
Bridge and Sculpt Feature Stress, D1=2mm Z2=0.1
450
400 yield stress
bridge
sculpt
350
VonMises Stress [MPa]
300
250
200
150
100
0.5 0.55 0.6 0.65 0.7 0.75 0.8
Y2
Figure 7.3: Rotor Stresses, 5000 rpm, D1=2mm, Z1=0.1
min T⃗ ∗ (12) − T⃗ScEf
M DL (12)
f
T,id ,iq
s.t. − 90% < Y2 < 0%
(7.3)
2% < Z2 < 50%
|Y2 | + |Z2 | ≤ 91%
min T⃗ ∗ (6) − T⃗ScEf
M DL (6)
f
T,id ,iq
s.t. 30% < Y1 < 90%
(7.4)
2% < Z1 < 50%
|Y1 | + |Z1 | ≤ 91%
Each problem was set up with a population of 200 and 10 generations. To confirm the
genetic algorithm’s accuracy, optimizations were repeated multiple times to confirm stable
results.
126
7.4 Optimal Designs
The optimization process minimized the 12th harmonic or 6th harmonic torque, three designs
are of interest. Table 7.2, highlights these designs. Design 1 designates the starting point of
the smooth rotor design. Design 2 and Design 3 seek to minimize the 12th order harmonic
of torque. Design 4 seeks to minimize the 6th order torque harmonic.
Y1 Z1 Y2 Z2 D1
Design 1 smooth rotor
Design 2 - - -86.6% 4.3% 2mm
Design 3 - - -32.1% 20.3% 2mm
Design 4 75.9% 14.1% - - 2mm
Table 7.2: Analytical Model Sculpt Feature Designs
Sculpt feature effects, based on (7.2), are shown in Figs. 7.4 and 7.5, examining which
shows the torque resulting from the analytical design relative to the objective and FE results.
Sculpt feature effects to average torque, fig. 7.4, shows reduced average torque for both
Design 2 and Design 3. Increased average torque is achieved with Design 4. Both the radial
flux density fundamental amplitude and phase account for these changes. Torque amplitude
trends are the same for all designs but with limited accuracy. Sculpt feature effects on torque
harmonics of the designs are shown in Fig. fig. 7.5. Designs 2 and 3 are shown analytically
to achieve the target T ∗ , when checked in FE Design 2 underachieves the desired torque
amplitude, and Design 3 overachieves it. A strong correlation holds between the analytical
model sculpt feature torque harmonic effect phase and that found with the finite element
model. Design 4 shows a good correlation in torque harmonic phase between the analytical
model and finite element model, and the amplitude shows that the effect to the 6th harmonic
is limited.
The three designs proposed to minimize torque ripple, table 7.2, are confirmed in FE and
127
Average Torque Sculpt Feature Effects
0.5
0
-0.5 FE
Model
Torque [Nm]
-1
-1.5
-2
-2.5
2 3 4
Design
Figure 7.4: Comparison of Sculpt Feature Effects to Average Torque (FE and Model), Iss =
200A, β = 112◦
Design 2, n=12 Design 3, n=12
Torque (imag) [Nm] Torque (imag) [Nm]
2 2
0 0
-2 -2
-2 0 2 -2 0 2
Torque (real) [Nm] Torque (real) [Nm]
Design 4, n=6
Torque (imag) [Nm]
2
T * (n)
T MDL
ScEff
(n)
0
T FE
ScEff
(n)
-2
-2 0 2
Torque (real) [Nm]
Figure 7.5: Comparison of Sculpt Feature Design Effects to Torque Ripple (FE and Model),
Iss = 200A, β = 112◦
128
shown in fig. 7.6. Design 2 is shown to reduce average torque 1%, reduce the 6th torque
harmonic 3%, and reduce the 12th harmonic of torque ripple by 50%. Design 3 reduces
average torque by 8%, increases the 6th harmonic amplitude by 6%, and the 12th harmonic
by 75%. Design 4 improves average torque by 1%, reduces the 6th torque harmonic by 17%,
and reduces the 12th harmonic by 23%.
Average 6th Order
30 3
Torque Amplitude [Nm]
Torque [Nm]
20 2
10 1
0 0
1 2 3 4 1 2 3 4
Design Design
12th Order
3 Design 1: smooth rotor
Torque Amplitude [Nm]
Design 2: Y2=-86.6% Y2=4.3% D1=2mm
2 Design 3: Y2=-32.1% Y2=20.3% D1=2mm
Desgin 4: Y1=75.9% Z2=14.1% D1=2mm
1
0
1 2 3 4
Design
Figure 7.6: FE Torque Comparison of Analytical Model Optimized Sculpt Features, Iss =
200A, β = 180◦
7.5 Finite Element Detailed Search of Design Space
To validate the optimized analytical design, a detailed search, in the region of the designs in
table 7.2, is performed in FE. Figures 7.7, 7.8, and 7.9 show the amplitude of the machine’s
torque harmonic as a function of the design variables. The constraint, |Yx | + |Zx | ≤ 91%, is
represented as a dashed red line in Figs. 7.7 and 7.9. For all extended searches, the analytical
designs are designated with a red asterisk. Comparing Design 2 to extended search results
in fig. 7.7 shows the genes identified in table 7.2 are in proximity of the minimum and design
optimum within 3%. Full cancellation of the 12th harmonic is not possible with design 2
since the geometry achieving the torque harmonic minimum is beyond the allowable design
129
space. The comparison between design 3 and extended search in fig. 7.8 shows the analytical
design is within the optimum 2% and with a small modification able to fully cancel the torque
harmonic. Figure 7.9, shows that Design 4 cannot fully cancel the intended torque harmonic
but is at the minimum along with the constraint. For each design, gradients around the
optimum are approximately 0.25N %
m further showing the analytical design sufficiently finds
the region of optimum for each design. The new analytical model coupled with GOSET
optimization algorithms yields similar geometry and performance as the FE extended search
method.
Figure 7.7: Finite Element Extended Search of Design 2, Iss = 200A, β = 112◦ , Constraint:
Dashed Red Line, Analytical Design: Red *
7.6 Exhaustive Search
The effectiveness of the analytical model with optimization is compared to finite elements
and an exhaustive search. Variables Y1 , Y2 , Z1 , and Z2 are swept, with an increment of
1%, over the same range as the optimization problem. Figures 7.12 7.10 and 7.11 plot the
amplitude of the average torque and 6th and 12th order torque ripple. The optimized designs
are plotted as stars. The optimization of each design using the analytical model required
130
Figure 7.8: Finite Element Extended Search of Design 3, Iss = 200A, β = 112◦ , Analytical
Design: Red *
Figure 7.9: Finite Element Extended Search of Design 4, Iss = 200A, β = 112◦ , Constraint:
Dashed Red Line, Analytical Design: Red *
131
approximately 45 minutes, whereas an exhaustive search within FE required 72 hours.
Figure 7.10: Finite Element Exhaustive Search, Order = 6, Iss = 200A, β = 112◦ , Design 2
(red), Design 3 (green), Design 4 (yellow)
Figure 7.11: Finite Element Exhaustive Search, Order = 12, Iss = 200A, β = 112◦ , Design
2 (red), Design 3 (green), Design 4 (yellow)
132
Figure 7.12: Finite Element Exhaustive Search, Order = 0, Iss = 200A, β = 112◦ , Design 2
(red), Design 3 (green), Design 4 (yellow)
7.7 Conclusion
This chapter has presented an analytical modeling and design optimization approach to
reduce torque ripple with rotor sculpt features. By carefully placing rotor sculpt features
and rotor barrier features, average torque can be maintained while torque harmonics are
minimized. An average torque increase is presented in fig. 7.4, while reduction of harmonics
is presented in fig. 7.5. Contributions of this chapter include:
1. Asymmetrical sculpting features onto the rotor can address the torque harmonic con-
tents, specifically the 6th and the 12th orders.
2. An analytical MMF model is developed and is used to guide the design process in place
of FEA.
3. The analytical model with GOSET optimization yields similar geometry as the bench-
marked FE method with an extended search.
133
4. The analytical model with GOSET optimization yields similar torque pulsation reduc-
tion as the benchmarked FE method with an extended search.
5. A single use of the analytical model to design sculpt features, has saved significant
computational time.
6. Non-standard IPM design can result in the reduction of torque harmonics without a
sacrifice of average torque, however a cost of additional control complexity.
7. Given a predefined torque harmonic target, an analytical model was developed and
used with optimization to design sculpt features to minimize ripple.
134
Chapter 8
Conclusion
8.1 Closing
The purpose of this dissertation is to investigate the design of interior permanent magnet
machines with the look to minimize torque pulsations and improve torque density through the
design of the rotor. The development of an MMF permeance model for the traditional IPM
machine is presented. Treatment of surface features is at the center of this work. Analysis
of the airgap surface features is made possible through the use of an additional MMF term,
based upon the equivalent magnetic dipole. Evaluation of the winding, permeance, and
surface harmonics will is presented alongside their interactions. Further improvement of
torque density and torque ripple is developed through the use of these surface harmonics. A
design process allowing for vector summation is presented. Torque ripple is minimized with
the analytical model and optimization enabling quick and efficient design.
8.2 Future Work
Future work includes an extension of the winding factor model to a broader set of problems.
The following questions are posed:
• How can the MMF Permeance method be extended to include the effects of fringing?
135
• What are the impacts of saturation to torque harmonics?
• How can the MMF Permeance method be extended to treat the saturated case?
• Can the rotor equivalent magnetic current be extended to problems solved in finite
elements?
• Can the torque harmonics be improved across multiple operating points, or the entire
operating range?
• How do the sculpt feature affect the windage losses and convection coefficients?
• Can the optimization speed be further improved using a gradient based optimization
method?
• How can the torque ripple minimization process be developed to handle different op-
erating temperatures?
• How does the proposed design methods fit in with the larger design process?
136
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