Modern Permanent
Magnet Electric Machines
The late 1980s saw the beginning of the PM brushless machine era, with the
invention of high-energy density permanent magnets (PM) and the development
of power electronics. Although induction motors are now the most popular
electric motors, the impact of PM brushless machines on electromechanical
drives is significant. Today, PM machines come second to induction machines.
Replacement of electromagnetic field excitation systems by PMs brings the
following benefits:
•• No electrical energy is absorbed by the field excitation system and thus
there are no excitation losses, causing substantial increase in efficiency
•• Higher power density (kW/kg) and/or torque density (Nm/kg) than
electromagnetic excitation
•• Better dynamic performance than motors with electromagnetic excitation (higher magnetic flux density in the air gap)
•• Simplification of construction and maintenance
•• Less expensive for some types of machines
Modern Permanent Magnet Electric Machines: Theory and Control serves as a
textbook for undergraduate power engineering students who want to supplement
and expand their knowledge in the fundamentals of magnetism, soft magnetic
materials, permanent magnets (PMs), calculation of magnetic circuits with PMs,
modern PM brushed DC machines and their controls, modern PM brushless
DC motors and drive control, and modern PM generators. The book can help
students learn more about electrical machines and can serve as a prescribed text
for teaching elective undergraduate courses such as modern permanent magnet
electrical machines. Since the book is written in a simple scientific language and
without redundant mathematics, it can also be used by practicing engineers and
managers employed in electrical machinery or electromagnetic device industries.
Modern Permanent
Magnet Electric Machines
Theory and Control
Jacek F. Gieras
PBS University of Science and Technology, Bydgoszcz, Poland
Jian-Xin Shen
Zhejiang University, Hangzhou, China
First edition published 2023
by CRC Press
6000 Broken Sound Parkway NW, Suite 300, Boca Raton, FL 33487-2742
and by CRC Press
4 Park Square, Milton Park, Abingdon, Oxon, OX14 4RN
CRC Press is an imprint of Taylor & Francis Group, LLC
© 2023 Jacek F. Gieras and Jian-Xin Shen
Reasonable efforts have been made to publish reliable data and information, but the author and
publisher cannot assume responsibility for the validity of all materials or the consequences of their use.
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Library of Congress Cataloging‑in‑Publication Data
Names: Gieras, Jacek F., author. | Shen, Jian-Xin, author.
Title: Modern permanent magnet electric machines : theory and control /
Jacek F. Gieras, Jian-Xin Shen.
Description: First edition. | Boca Raton : CRC Press, 2023. | Includes
bibliographical references and index.
Identifiers: LCCN 2022022229 (print) | LCCN 2022022230 (ebook) | ISBN
9780367610586 (hardback) | ISBN 9780367610616 (paperback) | ISBN
9781003103073 (ebook)
Subjects: LCSH: Permanent magnet motors. | Electric motors,
Brushless--Design and construction.
Classification: LCC TK2537 .S4544 2023 (print) | LCC TK2537 (ebook) | DDC
621.46--dc23/eng/20220916
LC record available at https://lccn.loc.gov/2022022229
LC ebook record available at https://lccn.loc.gov/2022022230
ISBN: 978-0-367-61058-6 (hbk)
ISBN: 978-0-367-61061-6 (pbk)
ISBN: 978-1-003-10307-3 (ebk)
DOI: 10.1201/9781003103073
Typeset in CMR10 font
by KnowledgeWorks Global Ltd.
Publisher’s note: This book has been prepared from camera-ready copy provided by the authors.
Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
1
FUNDAMENTALS OF MAGNETISM . . . . . . . . . . . . . . . . . . . .
1.1
Atom, spin, magnetic dipole moment . . . . . . . . . . . . . . . . . . . . .
1.2
Magnetic permeability, magnetization vector, magnetic
susceptibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3
Classification of materials according to magnetic permeability
1.4
Hysteresis loop of ferromagnetic materials . . . . . . . . . . . . . . . . .
1.5
Comparison of soft and hard magnetic materials . . . . . . . . . . . .
1.6
Analogies in electric and magnetic circuits . . . . . . . . . . . . . . . . .
1.7
Effect of ferromagnetic core inside a coil . . . . . . . . . . . . . . . . . . .
1.8
Applications of magnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.8.1
Electric motors and generators . . . . . . . . . . . . . . . . . . . .
1.8.2
Magnetic storage of data . . . . . . . . . . . . . . . . . . . . . . . . .
1.8.3
Loudspeakers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.8.4
Lift electromagnet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.8.5
Magnetic core memory . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.8.6
Magnetoresistive random-access memory (MRAM) . .
1.8.7
Cathode ray tube (CRT) . . . . . . . . . . . . . . . . . . . . . . . . .
1.8.8
Nuclear magnetic resonance (NMR) spectroscopy . . . .
1.8.9
Magnetic resonance imaging (MRI) . . . . . . . . . . . . . . . .
1.8.10 Magnetic levitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.8.11 Cyclotrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.8.12 Tokamak . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.8.13 MHD generators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.9
Biot-Savart law, Faraday’s law and Gauss’s law . . . . . . . . . . . .
1.9.1
Biot–Savart law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.9.2
Faraday’s law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.9.3
Gauss’s law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.10 Maxwell’s equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.10.1 Maxwell’s first equation . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1
3
4
8
9
11
12
13
13
13
14
14
16
16
17
18
19
20
21
22
22
23
23
24
25
26
26
vi
2
3
Contents
1.10.2 Maxwell’s second equation . . . . . . . . . . . . . . . . . . . . . . .
1.10.3 Maxwell’s third equation . . . . . . . . . . . . . . . . . . . . . . . . .
1.10.4 Maxwell’s fourth equation . . . . . . . . . . . . . . . . . . . . . . . .
1.11 Magnetic vector potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.12 Speed of electromagnetic wave and theory of relativity . . . . . .
1.13 Maxwell’s equations in application to electrical machines . . . .
1.14 Force in electromagnetic field . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
27
28
29
29
30
33
34
35
SOFT MAGNETIC MATERIALS . . . . . . . . . . . . . . . . . . . . . . . . .
2.1
Classification of soft ferromagnetic materials . . . . . . . . . . . . . . .
2.1.1
Laminated silicon steels . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.2
High-saturation cobalt alloys . . . . . . . . . . . . . . . . . . . . .
2.1.3
Amorphous ferromagnetic alloys . . . . . . . . . . . . . . . . . .
2.1.4
Soft magnetic composites (SMC) . . . . . . . . . . . . . . . . . .
2.1.5
Permalloys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.6
Nanocrystalline composites . . . . . . . . . . . . . . . . . . . . . . .
2.1.7
Solid ferromagnetic steels . . . . . . . . . . . . . . . . . . . . . . . .
2.2
Losses in ferromagnetic materials . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1
Hysteresis losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.2
Eddy-current losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.3
Excess eddy-current losses . . . . . . . . . . . . . . . . . . . . . . . .
2.2.4
Additional losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3
Engineering approach to calculation of core losses . . . . . . . . . .
2.4
Ferromagnetic cores . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.1
Transformers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.2
Electronic devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.3
DC machines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.4
Switched reluctance machines (SRM) . . . . . . . . . . . . . .
2.4.5
Induction machines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.6
Synchronous turbogenerators . . . . . . . . . . . . . . . . . . . . .
2.4.7
Synchronous hydrogenerators . . . . . . . . . . . . . . . . . . . . .
2.4.8
Permanent magnet (PM) brushless motors . . . . . . . . .
2.4.9
Segmented stator and rotor cores . . . . . . . . . . . . . . . . . .
2.4.10 3D cores made of soft magnetic composites (SMC)
for special electric machines . . . . . . . . . . . . . . . . . . . . . .
2.4.11 Solid ferromagnetic rotors . . . . . . . . . . . . . . . . . . . . . . . .
2.5
Magnetic circuits of electrical machines for recycling . . . . . . . .
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
39
39
40
42
44
45
47
47
49
51
51
53
53
54
54
54
55
55
56
57
57
58
59
60
60
PERMANENT MAGNETS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1
Early history of permanent magnets (PM) . . . . . . . . . . . . . . . . .
3.2
Earth’s magnetic field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3
What is a permanent magnet (PM)? . . . . . . . . . . . . . . . . . . . . . .
65
65
67
69
60
61
61
63
Contents
Hysteresis loop, demagnetization curve, recoil line, magnetic
energy density and intrinsic magnetization . . . . . . . . . . . . . . . . .
3.5
Temperature coefficients and Curie temperature . . . . . . . . . . . .
3.6
PM materials used in construction of electrical machines . . . .
3.6.1
Alnico . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.6.2
Ferrites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.6.3
Rare-earth magnets SmCo and NdFeB . . . . . . . . . . . . .
3.7
Nanocomposite magnets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.8
Shape of demagnetization curves of ferrite and rare
earth PMs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.9
Simplified method of finding the operating point of a PM . . . .
3.10 Main flux and leakage flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.11 B–H and Φ–F coordinate systems . . . . . . . . . . . . . . . . . . . . . . . .
3.12 Operating point for PM magnetized outside the machine . . . .
3.12.1 PM without pole shoes in open space . . . . . . . . . . . . . .
3.12.2 PM with pole shoes in open space . . . . . . . . . . . . . . . . .
3.12.3 PM inside an external magnetic circuit . . . . . . . . . . . . .
3.12.4 PM with a complete external armature system . . . . . .
3.13 Operating point for magnetization without armature . . . . . . . .
3.14 Mallinson–Halbach array . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
vii
3.4
69
72
74
74
76
77
81
82
84
85
86
87
87
88
88
89
91
93
95
4
CALCULATION OF MAGNETIC CIRCUITS
WITH PMs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
4.1
Methods of calculation of magnetic circuits with PMs . . . . . . . 97
4.2
Permeance evaluation by dividing the magnetic field into
simple solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
4.3
Graphical methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
4.4
Analytical approach to calculation of magnetic circuits with
PMs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
4.5
Calculation of magnetic circuits with PMs using an
equivalent reluctance network . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
4.6
Calculation of magnetic circuits with PMs using the FEM . . . 107
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
5
PM BRUSH DC MACHINES AND THEIR CONTROL . . 113
5.1
Why PM machines? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
5.2
Construction of a brush-type PM DC machine . . . . . . . . . . . . . 114
5.3
Principle of operation of a PM brush DC machine . . . . . . . . . . 115
5.4
Windings of a slotted rotor (armature) of a brush-type DC
machine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
5.5
Construction of a coreless rotor winding with an inner PM . . 120
5.6
Coreless rotor windings: Maxon versus Faulhaber winding . . . 122
5.7
PM brush DC motor with cylindrical rotor and foil winding . 124
5.8
Disk-type PM brush DC motors with printed rotor winding . . 126
viii
Contents
5.9
5.10
Fundamentals of transient analysis of PM brush DC motors . 128
Speed control of a brush-type PM DC motor . . . . . . . . . . . . . . . 130
5.10.1 Three-phase fully controlled rectifier . . . . . . . . . . . . . . . 131
5.10.2 Chopper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
5.10.3 H-bridge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
5.11 PM brush DC servomotors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
5.12 Applications of brush-type PM DC motors . . . . . . . . . . . . . . . . 136
5.12.1 Toys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
5.12.2 Auxiliary motors for automobiles . . . . . . . . . . . . . . . . . . 139
5.12.3 Vibration motors for mobile phones . . . . . . . . . . . . . . . 141
5.12.4 Robotic vehicles for Mars missions . . . . . . . . . . . . . . . . 144
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
6
PM BRUSHLESS DC MOTORS AND DRIVE
CONTROL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
6.1
From PM DC brushed to PM DC brushless motors . . . . . . . . . 149
6.2
Construction of rotors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
6.3
Sinusoidally excited and square wave motors . . . . . . . . . . . . . . . 152
6.4
Method of changing DC bus voltage and speed control . . . . . . 156
6.5
Unipolar and bipolar operating mode . . . . . . . . . . . . . . . . . . . . . 158
6.6
Six-step commutation: two phases on . . . . . . . . . . . . . . . . . . . . . 159
6.7
Three phases on: 180-degree conduction . . . . . . . . . . . . . . . . . . . 161
6.8
Rotor position sensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
6.8.1
Hall sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
6.8.2
Encoders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
6.8.3
Resolvers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
6.8.4
Sensorless control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
6.9
Mathematical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
6.10 Cogging torque . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
6.11 The smallest and the biggest PM brushless motors in the
world . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
6.12 Wiring diagram for a solid-state converter-fed PM brushless
motor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
6.13 Integrated circuits (IC) for control of PM brushless motors . . 175
6.14 Practical electromechanical drive system . . . . . . . . . . . . . . . . . . 177
6.15 Selected applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
6.15.1 Computer hard disk drives (HDD) . . . . . . . . . . . . . . . . 177
6.15.2 Two-phase PM brushless motors for computer
cooling fans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
6.15.3 PM brushless motors integrated with an electronic
control circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
6.15.4 Hybrid electric vehicles . . . . . . . . . . . . . . . . . . . . . . . . . . 185
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
Contents
ix
7
PM SYNCHRONOUS MOTORS AND DRIVE
CONTROL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
7.1
Fundamental equations for synchronous machines . . . . . . . . . . 193
7.1.1
Speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
7.1.2
Air gap magnetic flux density . . . . . . . . . . . . . . . . . . . . . 193
7.1.3
Electromotive force (EMF) . . . . . . . . . . . . . . . . . . . . . . . 194
7.1.4
Armature line current density and current density . . . 195
7.1.5
Electromagnetic power . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
7.1.6
Synchronous reactance . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
7.2
Location of the armature current in the d-q coordinate
system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
7.3
Armature reaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
7.4
Phasor diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
7.5
Input and electromagnetic power . . . . . . . . . . . . . . . . . . . . . . . . . 204
7.6
How to obtain zero d-axis current Iad = 0 . . . . . . . . . . . . . . . . . 206
7.7
Influence of d-axis current on the power factor . . . . . . . . . . . . . 206
7.8
Vector control of PM synchronous motors . . . . . . . . . . . . . . . . . 208
7.9
Starting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
7.9.1
Asynchronous starting . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
7.9.2
Starting by means of an auxiliary motor . . . . . . . . . . . 212
7.9.3
Frequency-change starting . . . . . . . . . . . . . . . . . . . . . . . . 213
7.10 Comparison of PM synchronous motors with induction
motors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
8
AXIAL AND TRANSVERSE FLUX MOTORS . . . . . . . . . . . 217
8.1
Axial flux disk motors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
8.1.1
Force and torque of axial flux motors . . . . . . . . . . . . . . 218
8.1.2
Double-sided motor with internal PM disk rotor . . . . 220
8.1.3
Stator cores of axial flux motors . . . . . . . . . . . . . . . . . . 220
8.1.4
Main dimensions of axial flux motors . . . . . . . . . . . . . . 221
8.1.5
Double-sided axial-flux motors with a single stator . . 223
8.1.6
Single-sided motors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
8.1.7
Ironless double-sided motors . . . . . . . . . . . . . . . . . . . . . . 229
8.1.8
Multidisk motors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232
8.2
Transverse flux motors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236
8.2.1
Principle of operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236
8.2.2
EMF and electromagnetic torque . . . . . . . . . . . . . . . . . . 239
8.2.3
Armature winding resistance . . . . . . . . . . . . . . . . . . . . . 241
8.2.4
Armature reaction and leakage reactance . . . . . . . . . . . 241
8.2.5
Magnetic circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
8.2.6
Advantages and disadvantages . . . . . . . . . . . . . . . . . . . . 244
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
x
9
Contents
HIGH-SPEED PM BRUSHLESS MACHINES . . . . . . . . . . . . 247
9.1
Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
9.2
Main dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248
9.3
Mechanical requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
9.4
Fundamental problems in design . . . . . . . . . . . . . . . . . . . . . . . . . 252
9.5
Stator design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
9.6
Rotor design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256
9.7
Mechanical design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
9.8
Thermal issues and cooling technologies . . . . . . . . . . . . . . . . . . . 260
9.9
Directed energy weapon (DEW) . . . . . . . . . . . . . . . . . . . . . . . . . . 263
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
Appendix A Conversion of units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
A.1 Conversion of units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
A.1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
A.1.2 Conversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268
A.1.3 Some physical constants . . . . . . . . . . . . . . . . . . . . . . . . . 268
Appendix B Lenz’s law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
Appendix C Right-handed cork-screw rule . . . . . . . . . . . . . . . . . . . . 271
Appendix D The right-hand grip rule . . . . . . . . . . . . . . . . . . . . . . . . . 273
Appendix E Left-hand and right-hand rules . . . . . . . . . . . . . . . . . . . 275
Symbols and Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
Preface
Permanent magnet (PM) machines use permanent magnets in place of wound
field excitation systems thus enabling a lighter, more compact, simpler and
more efficient machine with a wide speed range and linear characteristics.
All of the earliest inventors of electrical rotating machines used PMs in
their designs. But these first PM machines had severe limitations as far as their
practical application was concerned. Up until the 20th century, PM materials
were limited to naturally occurring magnetite, commonly called lodestone.
These PM materials had very low magnetic energy. In 1901 the so-called
Heusler alloys (CuMnAl) were reported, which had outstanding properties
compared to magnetite. In 1917 cobalt steel alloys were introduced with the
maximum energy product (BH)max not exceeding 10 kJ/m3 . Discovery of
Alnico in the 1950s allowed for increasing the maximum energy product up to
40 kJ/m3 . In the 1950s, ceramic (ferrite) PMs appeared and were used in small
motors and electromagnetic devices. A breakthrough came with invention of
samarium-cobalt PMs in the 1960s and the announcement of neodymium-ironboron PMs in the 1980s. The maximum energy product of samarium-cobalt
PMs is up to 240 kJ/m3 and of neodymium-iron-boron PMs over 400 kJ/m3 .
Future PM materials will probably be based on nanocomposite materials.
High energy density PMs and development of power electronics started the
PM brushless machine era in the late 1980s. Although, induction motors are
now the most popular electric motors, the impact of PM brushless machines
on electromechanical drives is significant. PM brush machines are also used,
but as small motors or motors for special applications.
This book is intended to serve as a textbook for undergraduate Power
Engineering students in order to supplement and enlarge their knowledge in
the fundamentals of magnetism, soft magnetic materials, permanent magnets
(PMs), calculation of magnetic circuits with PMs, modern PM brushed DC
machines and their control, modern PM brushless DC motors and drive control and modern PM generators. It can supplement knowledge of Electrical
xii
Preface
Machines or serve as a prescribed textbook for teaching an elective graduate
course in Modern Permanent Magnet Electrical Machines. Since the book is
written in simple scientific language and without redundant mathematics, it
can also be used by practicing engineers and managers employed in electrical
machinery or electromagnetic device industries.
The authors have produced this textbook without any support from funding agencies and/or industry in European Union countries, the United States,
or China.
Any suggestions for improvement, constructive criticism and corrections
from students, engineers and professors are most welcome.
Prof. Jacek F. Gieras, PhD, DSc, IEEE Life Fellow
Glastonbury, CT, U.S.A., E-mail: jgieras@ieee.org
Prof. Jian-Xin Shen, PhD, IET Fellow
Hangzhou, China, Email: j_x_shen@zju.edu.cn
1
FUNDAMENTALS OF MAGNETISM
1.1 Atom, spin, magnetic dipole moment
Fig. 1.1 shows the model of the atom and structure within the atom. The
proton and electron in a hydrogen atom both have spin (Fig. 1.2). They can
be spinning in the same or opposite direction. About once every 10 million
years, the electron flips its spin and emits a radio photon of wavelength 0.21
m.
Spin: electron acts like a spinning charge and contributes to magnetic
dipole moment m.
Fig. 1.1. How the atom is built.
The magnetic field of a bar magnet and magnetic field of a current loop
look the same [71]. Fig. 1.3 shows the magnetic flux line about a hypothetical
dipole and about a current loop (coil). Magnetic dipole moment m is a vector
pointing out of the plane of the current loop and with a magnitude equal to
the product of the current and loop area, i.e.,
m = nIS Am2
(1.1)
2
Modern Permanent Magnet Electric Machines
Fig. 1.2. The proton and electron in a hydrogen atom both have spin.
where n is the vector normal to the surface S of the orbit, i.e., area enclosed by
the circulating current, and I is the circulating electric current. It means that
the elementary magnetic source is the dipole formed by the loop of current.
Fig. 1.3. Magnetic flux lines: (a) due to a hypothetical dipole; (b) magnetic dipole
modeled as a current loop. m is the magnetic dipole moment.
Fig. 1.3 shows alignment of atomic dipole moments. Particles with spin can
possess a magnetic dipole moment, just like a rotating electrically charged body
in classical electrodynamics.
The electric current
I=
e
ev
=
A
T
2πR
(1.2)
where the charge of electron (elementary charge) e = 1.60217662 × 10−19 C,
T is the time of one revolution, the surface of the orbit S = πR2 and the
linear speed v = RΩ, and the angular speed Ω = 2πn, the rotational speed is
n. The magnetic dipole moment of one atom expressed as a scalar
m = IS =
ev
1
1
πR2 = evR = eΩR2 Am2
2πR
2
2
(1.3)
Fundamentals of Magnetism
3
Fig. 1.4. Alignment of magnetic dipole moments in: (a) ferromagnetic materials
such as iron, nickel, cobalt; (b) most materials.
Fig. 1.5. The dipole is formed by the loop of current: (a) orbital magnetic dipole;
(b) magnetic field lines above electron.
The mass of the electron is me = 9.10938356 × 10−31 kg. An orbiting electron
is equivalent to the magnetic dipole moment. For N dipoles
m = nN IS Am2
(1.4)
1.2 Magnetic permeability, magnetization vector,
magnetic susceptibility
Magnetic permeability is the measure of the ability of a material to support
the formation of a magnetic field within itself. The magnetic permeability of
free space, also known as the magnetic constant is
µ0 = 0.4π × 10−6 H/m
(1.5)
On 20 May 2019, a revision to the SI system went into effect, making the
vacuum permeability no longer a constant but rather a value that needs to
be determined experimentally. For comparison, the electric permittivity, also
4
Modern Permanent Magnet Electric Machines
known as the electric constant is
ϵ0 =
1
× 10−9 F/m
36π
(1.6)
The volume magnetic susceptibility χ is a dimensionless quantity defined by
the following relationship
M = χH A/m
(1.7)
where M is the magnetization of the material (the magnetic dipole moment
per unit volume) [A/m] and H is the magnetic field strength vector [A/m].
The magnetic flux density B is related to the magnetic field intensity H as
B = µ0 (H + M) = µ0 H + µ0 χH = µ0 (1 + χ)H = µH
(1.8)
Because vectors B, H and M are parallel, eqn (1.8) can be expressed in scalar
form, i.e.,
B = µ0 (H + M ) = µ0 H + µ0 χH = µ0 (1 + χ)H = µH
(1.9)
The magnetic permeability as a function of magnetic susceptibility
µ = µ0 (1 + χ)
(1.10)
µ = µ0 µr
(1.11)
because
the relative magnetic permeability is
µr = 1 + χ
(1.12)
1.3 Classification of materials according to magnetic
permeability
Depending on the relative magnetic permeability (1.12), all materials can be
divided into the following groups:
ˆ
ˆ
ˆ
ˆ
ferromagnetic materials with χ >> 1 and µr >> 1, strongly attracted by
magnetic field, e.g., Fe, Co, Ni, Cd and their alloys;
ferrimagnetic materials, displaying a weak form of ferromagnetism, e.g.,
magnetite Fe3O4, yttrium-iron garnet YIG;
antiferromagneti c materials, similar to ferromagnetic and ferrimagnetic
materials, where spins of electrons align in a regular pattern with neighboring spins pointing in opposite directions, e.g., Cr, FeMn, NiO;
paramagnetic materials with 0 < χ < 1 and µr > 1, weakly attracted by
magnetic field, e.g., Al, Na, Cl, U, antiferromagnetics; and
Fundamentals of Magnetism
ˆ
5
diamagnetic materials with χ < 0 and µr < 1, weakly repelled by magnetic
field, e.g., Cu, Ag, Au, Sn, superconductors.
Types of magnetic behavior of materials are shown in Table 1.1 and in Fig.
1.6.
Table 1.1. Types of magnetic behavior of materials (in order of decreasing strength)
Material
Spin alignment
Examples
All spins align parallel
Fe, Co, Ni, Gd, Dy
to one another; spontaneous
SmCo5 , Sm2 Co1 7
magnetization M = a + b
Nd2 Fe1 4B
Most spins parallel to one another,
magnetite Fe3 O4 ,
Ferrimagnetic
some spins antiparallel; spontaneous yttrium iron garnet YIG,
magnetization M = a − b > 0
GdCo5
Periodic parallel-antiparallel;
Antiferromagnetic
spin distribution
Cr, FeMn, Ni
M =a−b=0
Spins tend to align parallel
oxygen, sodium,
Paramagnetic
to an external magnetic field
aluminum, calcium,
M = 0 at H = 0; M > 0 at H > 0
uranium
Spins tend to align antiparallel
superconductors, N, Cu,
Diamagnetic
to an external magnetic field
Ag, Au, water,
M = 0 at H = 0; M < 0 at H < 0
organic compounds
Ferromagnetic
Fig. 1.6. Simplified plots of spins for different materials: (a) ferromagnetic; (b)
ferrimagnetic; (c) antiferromagnetic; (d) paramagnetic; (e) diamagnetic.
Ferromagnetic materials have a large, positive susceptibility χ to an external magnetic field. They exhibit a strong attraction to magnetic fields and
are able to retain their magnetic properties after the external field has been
removed.
6
Modern Permanent Magnet Electric Machines
Fig. 1.7. Domains in ferromagnetic material: (a) unmagnetized; (b) magnetized.
A magnetic domain is a region within a magnetic material in which the
magnetization M is in a uniform direction. They must be separated by domain
walls. This means that the individual magnetic moments of the atoms are
aligned with one another and they point in the same direction (Fig. 1.7).
In nonferromagnetic materials, these domains are randomly aligned. As a
result, the total magnetic field of that ferromagnetic material is zero. When
all the magnetic domains are aligned in the same direction, then their magnetic
moments are added.
Fig. 1.8. Magnetic dipole moment per unit volume as a function of magnetic field
intensity and magnetic susceptibility as a function of temperature for: (a) paramagnetic materials; (b) diamagnetic materials.
Paramagnetic materials have a small, positive susceptibility χ to magnetic
fields (Fig. 1.8a). These materials are slightly attracted by a magnetic field
and the material does not retain the magnetic properties when the external
field is removed.
Fundamentals of Magnetism
7
Fig. 1.9. Pyrolytic graphite block levitated by four cube NdFeB PMs. For sale by
Apex Magnets www.apexmagnets.com/magnets/pyrolytic-graphite-block.
Diamagnetic materials have a weak, negative susceptibility χ to magnetic
fields (Fig. 1.8b). Diamagnetic materials are slightly repelled by a magnetic
field and the material does not retain the magnetic properties when the external field is removed. For example, pyrolytic carbon (PyC) has one of the
largest negative susceptibilities at room temperature of any diamagnetic material and is repelled by and external magnetic field. It has χ ≈ −4.5 × 10−4
in one direction and χ ≈ −0.85 × 10−4 in the two remaining directions. A
pyrolytic carbon sheet can be levitated above PMs (Fig. 1.9).
In diamagnetic materials the magnetic moment opposes the field. For
the same applied field, progressively stronger moments are present in paramagnetic, ferrimagnetic and ferromagnetic materials. Fig. 1.10 shows the
Fig. 1.10. Magnetization curves B versus H for ferromagnetic, ferrimagnetic, paramagnetic, diamagnetic materials and vacuum.
8
Modern Permanent Magnet Electric Machines
magnetization curves, i.e., magnetic flux density B versus magnetic field intensity H curves for different materials.
1.4 Hysteresis loop of ferromagnetic materials
A hysteresis loop shows the relationship between the induced magnetic flux
density B and the magnetic field strength H (Fig. 1.11a). The hysteresis loop
of ferromagnetic materials can be measured using toroidal samples and using
a connection diagram as in Fig. 1.11b.
Fig. 1.11. Measurement of hysteresis loops: (a) hysteresis loop of ferromagnetic
materials; (b) connection diagram for measurement of hysteresis loops using toroidal
samples.
Starting with an unmagnetized sample both B and H are at zero. If the
magnetization current is increased in a positive direction to some value, the
magnetic field strength H increases linearly with the current and the flux
density B also increases as shown by the curve from point 0 to the saturation
point. Domain configuration during several stages of magnetization is shown
in Fig. 1.11. In the saturation region, all domains are fully aligned under the
external magnetic field.
Now, if the magnetizing current in the coil is reduced to zero, the magnetic
field H circulating around the core also reduces to zero. However, the magnetic
flux density does not reach zero due to the residual magnetism present within
the core and it is equal to remanent magnetic flux density Br also called
retentivity.
To reduce the flux density at point Br to zero, it is necessary to reverse
the current in the coil. The magnetic field intensity, which must be applied to
null the residual flux density, is called a coercive force Hc . This coercive force
reverses the magnetic field re-arranging the magnetic domains until the core
becomes unmagnetized at point Hc .
Fundamentals of Magnetism
9
If the magnetizing current is reduced again to zero, the residual magnetism
Br present in the core will be equal to the previous value but in reverse.
Again reversing the magnetizing current in the coil to a positive direction
will cause the magnetic flux density to reach zero. As before, increasing the
magnetizing current further in a positive direction causes the core to reach
saturation.
Fig. 1.12. Domain configuration during several stages of magnetization.
1.5 Comparison of soft and hard magnetic materials
The ferromagnetic materials can be categorized into soft magnetic materials
and hard magnetic materials. Hysteresis loops for soft and hard magnetic
materials are plotted in Fig. 1.13.
Soft magnetic materials can be easily magnetized and demagnetized at low
magnetic field. Thus, their coercivity Hc is low and permeability is high. Soft
magnetic materials are suitable for applications of magnetic cores or recording
heads.
Hard magnetic materials are difficult to magnetize, but once magnetized,
they are difficult to demagnetize. Since large magnetic field intensity is required to demagnetize, their coercivity Hc is high and highly sensitive to the
10
Modern Permanent Magnet Electric Machines
Fig. 1.13. Hysteresis loops for: (a) soft magnetic materials; (b) hard magnetic
materials.
microstructure. In hard magnetic materials (permanent magnets) domains remain aligned even when the external magnetic field is removed. Hard magnetic
materials are suitable for applications such as permanent magnets (PMs) and
magnetic recording media.
Comparison of soft and hard magnetic materials is given in Table 1.2.
Table 1.2. Comparison of soft and hard magnetic materials
Soft magnetic material
Hard magnetic material
Hysteresis loop area is small
Hysteresis loop area is large
Low hysteresis losses because
High hysteresis losses because
of small hysteresis loop
of large hysteresis loop
Can be easily magnetized
Cannot be easily magnetized
and demagnetized
and demagnetized
Require small value of H
Require large value of H
for magnetization.
for magnetization.
Domain wall moves easily
Domain wall does not move easily
High susceptibility and relative
Low susceptibility and relative
magnetic permeability
magnetic permeability
Low remanence and coercivity
High remanence and coercivity
Eddy-currents can be limited
Eddy-currents cannot be limited
by making laminations
in solid cubes
Examples: silicon steel, cobalt
Examples: Alnico, Barium Ferrite,
steels, amorphous alloys,
SmCo, NdFeB
ferrites, garnets
Applications: ferromagnetic cores
Applications: permanent magnets (PMs)
of electrical machines and transformers,
electromagnets, computer data storage
Fundamentals of Magnetism
11
1.6 Analogies in electric and magnetic circuits
Table 1.3 shows basic analogies in electric and magnetic circuits. Table 1.4
shows a comparison of Ohm’s and Kirchhoff’s laws for electric and magnetic
circuits.
Table 1.3. Basic analogies in electric and magnetic circuit
Quantity
Electric circuit
Magnetic circuit
Voltage
Electric voltage U [V]
Magnetic voltage Vµ [A]
Voltage
Electromotive
Magnetomotive
source
force EMF E [V]
force MMF F [A]
Current/Magnetic flux Electric current I [A]
Magnetic flux Φ [Wb]
Resistance/Reluctance Resistance R [Ω = 1/S] Reluctance Rµ [1/H = A/(Vs)]
Conductance/
Conductance G [S = 1/Ω] Permeance Gµ [H = Vs/A]
Permeance
Electric
Magnetic
Constant
conductivity σ [S/m]
permeability µ [H/m]
Table 1.4. Ohm’s and Kirchhoff’s laws for electric and magnetic circuits
Law
Electric circuit
Magnetic circuit
R = UI
Rµ = Φµ
G = UI
Gµ = VΦµ
l
R = σs
l
Rµ = µs
G = σs
l
Gµ = µs
l
P
I=0
P
U−
P
V
Ohm’s law
2nd Ohm’s law
Kirchhoff’s current law
Kirchhoff’s voltage law
P
RI = 0
P
Vµ −
Φ=0
P
Rµ Φ = 0
12
Modern Permanent Magnet Electric Machines
1.7 Effect of ferromagnetic core inside a coil
The presence of a ferromagnetic core made of soft magnetic material increases
the magnetic flux density inside the coil (Fig. 1.14). If the axial length of the
coil L is much longer than the diameter of the coil, the Ampere’s circuital law
can be written as
HL ≈ N I
(1.13)
where N is the number of turns of the coil and I is the electric current in the
coil. The magnetic flux density for the coil without a ferromagnetic core is
B = µ0 H ≈ µ0
NI
L
(1.14)
Fig. 1.14. Long round coil (solenoid): (a) without ferromagnetic core; (b) with
ferromagnetic core.
For the coil with a ferromagnetic core
NI
(1.15)
L
where µr is the relative magnetic permeability of the core. Thus, the ferromagnetic core increases the magnetic flux density inside a coil (multiplier effect),
i.e.,
B = µ0 µr H ≈ µ0
Bnet = Bcoil + Bcore
Bnet > Bcoil
(1.16)
Fundamentals of Magnetism
13
1.8 Applications of magnetism
1.8.1 Electric motors and generators
Electric motors and generators are electromechanical energy conversion devices. An electric motor (Fig. 1.15) converts electrical energy into mechanical
energy and a generator (Fig. 1.16) converts mechanical energy into electrical
energy. Motors and generators have their ferromagnetic cores made of soft
magnetic materials. The role of ferromagnetic cores is to increase the magnetic flux density in the air gap and to direct the magnetic flux in the desired
direction. The magnetic flux penetrates through the path with the lowest reluctance. A ferromagnetic core has much lower reluctance than the air gap.
To reduce the eddy current losses, ferromagnetic cores are laminated.
Fig. 1.15. Small-power induction motors.
1.8.2 Magnetic storage of data
Magnetic storage and retrieval devices include tape, flexible disk, and rigid
disk drives used for audio, video, and data processing applications. The magnetic recording process involves relative motion between a magnetic medium
(tape or disk) and a stationary or rotating read/write magnetic head.
Magnetic media is made up of a thin layer that can record a magnetic
signal supported by a thicker film backing. The top coat consists of a magnetic
pigment. The binder holds the magnetic particles together. The magnetic layer
(top coat) records and stores the magnetic signals that are written to it. The
backing film supports the magnetic top coat and reduces tape friction and
distortion.
14
Modern Permanent Magnet Electric Machines
Fig. 1.16. Large turbogenerator in a power plant.
A traditional hard disk drive (HDD) is made up of very thin platters
of material, coated with a magnetic medium, which stores data in magnetic
patterns. Each platter in a hard drive can store billions of bits of data, and
three or more platters are stacked on top of each other. A spindle, which runs
through the center of each platter, spins the platters at speeds from 5400 to
15,000 rpm while a read/write head on each side of the platter reads data.
The head must completely avoid touching the platters or the disks will crash,
causing a loss of data. The heads are driven by a PM linear motor of voice-coil
type. The platters are driven by a rotary PM brushless motor.
Magnetic storages of data, i.e., computer HDD, magnetic tapes, floppy
disks and magnetic cassette are shown in Fig. 1.17.
1.8.3 Loudspeakers
A loudspeaker (or loud-speaker or speaker) is an electroacoustic transducer,
i.e., a device which converts an electrical signal into an acoustic sound providing the most faithful reproduction. The moving coil type of loudspeaker is
the type that is most commonly seen. It consists of a cone attached to a coil
that is held within a magnetic field (Fig. 1.18).
1.8.4 Lift electromagnet
Lift electromagnets are usually DC electromagnets with an axial symmetry
magnetic circuit and ring-shape coil (Fig. 1.19). They are used for lifting heavy
ferromagnetic objects like scrap metal, steel sheets, steel pipes, car bodies, etc.
Fundamentals of Magnetism
15
Fig. 1.17. Magnetic storages of data: (a) computer HDD; (b) computer magnetic
tape and floppy disks; (c) magnetic cassette.
Fig. 1.18. Moving-coil loudspeaker: (a) general view; (b) construction. 1 – PM, 2 –
moving voice coil, 3 – cone, 4 – cone suspension, 5 – support chassis, 6 – electrical
leads, 7 – input voltage signal, 8 – air movement.
Attraction force of an electromagnet is given by the following equation
Fz = µ0
(iN )2
Sg
4g 2
(1.17)
where µ0 is the magnetic permeability of free space, i is the current in the
coil, N is the number of turns, g is the nonferromagnetic air gap between the
core and ferromagnetic body being attracted, and Sg is the area of the air gap
per two poles.
16
Modern Permanent Magnet Electric Machines
Fig. 1.19. Lift electromagnet: (a) general view; (b) construction. 1 – pot-type core
made of mild steel, 2 – ring-shaped coil, 3 – terminals.
1.8.5 Magnetic core memory
Magnetic-core memory (MCM) was the predominant form of random-access
computer memory between about 1955 and 1975. Core memory uses toroids
(rings) of a hard magnetic material (usually a semi-hard ferrite) as transformer
cores, where each wire threaded through the core serves as a transformer
winding (Fig. 1.20). Three or four wires pass through each core. Each core
stores one bit of information. Distance between rings is about 1 mm.
Fig. 1.20. Magnetic-core memory: (a) memory chip; (b) principle of operation.
1.8.6 Magnetoresistive random-access memory (MRAM)
Magnetoresistive random-access memory (MRAM), also called a core memory,
is a type of non-volatile random-access memory which stores data in magnetic
domains (Fig. 1.21). MRAM uses magnetic storage elements instead of electric
Fundamentals of Magnetism
17
used in conventional RAM. Tunnel junctions are used to read the information
stored in MRAM, typically a “0” for zero point magnetization state and “1”
for antiparallel state.
Fig. 1.21. Magnetoresistive random-access memory (MRAM): (a) memory chip;
(b) principle of operation.
1.8.7 Cathode ray tube (CRT)
The cathode ray tube (CRT) was used in old TV sets and computer monitors
(Fig. 1.22). The “cathode rays” are in fact beams of electrons, and magnets
can be used to bend their path. The CRT is filled with gas, which glows
when electrons hit it. The ideal CRT is enclosed by Helmholtz coils to allow
a varying magnetic field to be applied. In the absence of Helmholtz coils, a
strong NdFeB PM should suffice to bend the electron beam.
Fig. 1.22. Cathode ray tube (CRT): (a) general view; (b) construction.
18
Modern Permanent Magnet Electric Machines
In the CRT, electrons are ejected from the cathode and accelerated through
a voltage, gaining some velocity of 600 km/s for every volt they are accelerated
through. Some of these fast-moving electrons crash into the gas inside the tube,
causing it to glow. Helmholtz coils can then be used to apply a quantifiable
magnetic field by passing a known current through them.
A magnetic field will cause a force to act on the electrons which is perpendicular to both their direction of travel and the magnetic field. This causes a
charged particle in a magnetic field to follow a circular path. The faster the
motion of the particle, the larger the circle traced out for a given field or,
conversely, the larger the field needed for a given radius of curvature of the
beam.
1.8.8 Nuclear magnetic resonance (NMR) spectroscopy
Nuclear magnetic resonance spectroscopy, commonly referred to as NMR, is
a technique, which exploits the magnetic properties of certain nuclei to study
physical, chemical, and biological properties of matter (Fig. 1.23).
Fig. 1.23. Nuclear magnetic resonance (NMR) spectroscopy: (a) spectrometer; (b)
principle of operation.
Many nuclei have spin and all nuclei are electrically charged. If an external
magnetic field is applied, an energy transfer is possible between the base
energy to a higher energy level (generally a single energy gap). The energy
transfer takes place at a wavelength that corresponds to radio frequencies (60
to 1000 MHz) and when the spin returns to its base level, energy is emitted
at the same frequency. The signal that matches this transfer is detected with
sensitive radio receivers and processed in order to yield an NMR spectrum for
the nucleus concerned. NMR spectra are unique, well-resolved, analytically
tractable and often highly predictable for small molecules.
Typical high-resolution NMR spectrometers have a superconducting magnet to generate high magnetic fields. Modern NMR spectrometers use magnetic flux density of 1.0 to 20 T.
Fundamentals of Magnetism
19
Isidor I. Rabi (Nobel Prize in Physics, 1944) demonstrated the phenomenon of NMR in 1937, and Felix Bloch and Edward Mills Purcell, working
independently, demonstrated in December 1945 and January 1946 the use of
RF waves to detect NMR signals (joint Nobel Prize in Physics, 1952). With
their discovery, nuclear magnetic spectroscopy was born.
1.8.9 Magnetic resonance imaging (MRI)
Magnetic resonance imaging (MRI) is a medical imaging technique used in
radiology to form pictures of the anatomy and the physiological processes of
the body (Fig. 1.24).
Fig. 1.24. Magnetic resonance imaging (MRI): (a) Examination of a patient; (b)
MRI scanner cutaway.
MRI scanners use strong magnetic fields, magnetic field gradients, and radio waves to generate images of the organs in the body. The strong magnetic
field is produced by superconducting electromagnets. Today, hospitals routinely use machines with magnetic flux density of 1.5 T to 3.0 T. There are
MRI scanners in research laboratories around the world with magnetic flux
density over 10 T.
Every MRI patient has an RF coil placed near the part of the body being
scanned. This coil is a radio transceiver that can communicate with your
hydrogen atoms via RF waves. The technologist uses that coil to send RF
pulses at the body part under examination. The pulses are precisely timed to
achieve the resonance.
MRI does not involve X-rays or the use of ionizing radiation, which distinguishes it from computed tomography (CT) or computerized axial tomography (CAT) scans and positron-emission tomography (PET) scans. Magnetic
resonance imaging is a medical application of NMR.
20
Modern Permanent Magnet Electric Machines
1.8.10 Magnetic levitation
The term levitation comes from the Latin word levitas-atis = lightness
and refers to raising an object against the force of gravity in such a way
that it remains suspended without any physical contact.
Levitation was the supernatural belief of being able to raise any object
and hold it in midair by the use of spiritual energy. It was a pseudoscience
because objects cannot defy gravity without a proven scientific method that
allows it to, such as quantum levitation or magnetic levitation.
Electromagnetic levitation (EML) uses attraction forces between an electromagnet with controlled air gap and a ferromagnetic plate or rail. The attraction force between electromagnet and a ferromagnetic plate with controlled air gap g between the magnet poles and plate is expressed by eqn
(1.17).
Electrodynamic levitation (EDL) uses repulsive forces between currents
induced in a nonferromagnetic conductive body and source magnetic field.
Fig. 1.25. Transrapid maglev train: (a) at Pudong International Airport in Shanghai; (b) principle of operation.
The Transrapid Shanghai is the only commercial maglev high-speed train
in the world (Fig. 1.25). There are also other maglev trains in China, Japan
and South Korea, but these are low-speed maglev trains with maximum speed
of 110 km/h. Construction of the Maglev Line in Shanghai began in March
2001 and public commercial services commenced on January 1, 2004. The
Shanghai maglev uses German Transrapid technology, i.e., attraction forces
between vehicle-mounted electromagnets and a track-mounted reaction rail
(EML), which also serves as a ferromagnetic core for linear synchronous motor
(LSM) armature windings mounted in the track. Attraction forces are also
used for lateral stabilization (guidance).
Fundamentals of Magnetism
21
The length of the double-track Maglev Line between Pudong International
Airport and Longyang Road Station (outskirts of Shanghai) is 30.5 km. The
travel time at maximum approved speed 431 km/h is 7 min 26 s. At this speed
and travel time, the Maglev Train consumes 1600 kWh electrical energy. There
are 115 trains per day in both directions. The train accelerates from standstill
to 350 km/h in 2 min. The speed is controlled by the input frequency of the
LSM from 0 to 300 Hz. The current of LSMs ranges from 1200 to 2000 A
during acceleration and decreases to one-third full current when the vehicle
cruises at a constant speed.
1.8.11 Cyclotrons
Linear accelerators (also called linacs), cyclotrons, and synchrotrons are designed for acceleration of charged particles, usually electrons, protons, and
isotopes, as well as subatomic particles, to incredibly high speeds (Fig. 1.26).
Fig. 1.26. Cyclotron: (a) 520 MeV TRIUMF cyclotron at University of British
Columbia, Canada; (b) construction of a cyclotron.
Cyclotrons accelerate particles along an outward spiral path and are held
in that path by a static electromagnetic field perpendicular to the spiral path.
Charged particles get injected into the center of the cyclotron into a vacuum
chamber between two hollow D-shaped metal electrodes (called dees). An
alternating RF voltage of several thousand volts is applied to one dee and
then the other. The output energy of particles is
1
q 2 B 2 R2
mv 2 =
(1.18)
2
2m
where m is the mass of the particle, v is the velocity of the particle, q is the
charge of the particle, B is the magnetic flux density limited to about 2 T for
electromagnets with ferromagnetic cores and R is the radius of the dees.
E=
22
Modern Permanent Magnet Electric Machines
The largest cyclotron in the world is the 17.1-m TRIUMF cyclotron at
University of British Columbia, Vancouver, Canada, with an outer orbit radius
of 7.9 m, extracting protons at up to 510 MeV, which is 3/4 of the speed
of light. In Fig. 1.26a the top half of the cyclotron is raised. TRIUMF can
accelerate 1,000 trillion particles per second to 224,000 km/s.
The cyclotron was invented by Ernest O. Lawrence in 1929–1930 at the
University of California, Berkeley, and patented in 1932 (Nobel prize in
physics, 1939).
1.8.12 Tokamak
A tokamak is a device which uses a powerful magnetic field to produce controlled thermonuclear fusion power and confine hot plasma in the shape of a
torus (Fig. 1.27). In order for fusion to occur in the very hot gas (plasma),
the plasma must be heated to temperatures in excess of 150 million degrees
Celsius. To achieve this, the plasma is actively held away from the walls of
the tokamak container by using powerful magnetic fields.
The toroidal field coils in the ITER tokomak (France) can produce a total
magnetic energy of 41 GJ and a maximum magnetic flux density of 11.8 T.
Fig. 1.27. Tokamak: (a) cutaway; (b) construction.
Borrowed from Russian (to(roidalnaya) ka(mera) (s) ma(gnitnymi)
k(atushkami), which means toroidal chamber with magnetic coils. Tokamaks
were initially conceptualized in the 1950s by Soviet physicists Igor Tamm and
Andrei Sakharov, inspired by a letter by Oleg Lavrentiev.
1.8.13 MHD generators
A magnetohydrodynamic generator (MHD generator) is a converter that converts the kinetic energy of an electrically conductive fluid, in motion with
Fundamentals of Magnetism
23
respect to a steady magnetic field, into electricity (Fig. 1.28). The MHD
generator uses hot conductive plasma as the moving conductor. Plasma is
a fourth state of matter. It is a gas in which atoms have been broken up into
free-floating negative electrons and positive ions.
Fig. 1.28. MHD generator: (a) prototype; (b) principle of operation.
1.9 Biot-Savart law, Faraday’s law and Gauss’s law
Maxwell’s equations were derived in 1864–1865 from the earlier Biot–Savart
law (1820), Faraday’s law (1831), and Gauss’s law (1840).
1.9.1 Biot–Savart law
The Biot-Savart law gives the differential magnetic flux density dB at a point
P2 , produced by a current element Idl at point P1 , which is filamentary and
differential in length, as shown in Fig. 1.29a. This law can best be stated in
vector form as
Z
Idl × 1r
1
(1.19)
H=
4π l
r2
where the subscripts indicate the point to which the quantities refer, I is
the filamentary current at P1 , dl is the vector length of current path (vector
direction same as conventional current) at P1 , 1r is the unit vector directed
from the current element Idl to the location of dH, from P1 to P2 , r is the
scalar distance between the current element Idl to the location of dB, the
distance between P1 and P2 , and dH is the vector magnetostatic field intensity
at P2 .
24
Modern Permanent Magnet Electric Machines
Fig. 1.29. Graphical display of: (a) the vector magnetostatic flux density dB at P2
produced by a current element Idl at P1 (Biot-Savart law); (b) charge Q is enclosed
by the closed surface S (Gauss’s law).
The Biot-Savart law is similar to Coulomb’s law of magnetostatics.
Jean-Baptiste Biot (21 April 1774–3 February 1862) was a French physicist,
astronomer, and mathematician who co-discovered the Biot-Savart law of magnetostatics, established the reality of meteorites, made an early balloon flight,
and studied the polarization of light. The mineral biotite and Cape Biot in
Greenland were named in his honor.
Félix Savart (30 June 1791–16 March 1841), a French physicist and mathematician who is primarily known for the Biot–Savart law of magnetostatics,
which he discovered together with his colleague Jean-Baptiste Biot. His main
interest was in acoustics and the study of vibrating bodies. A particular interest
in the violin led him to create an experimental trapezoidal model. He gave his
name to the savart, a unit of measurement for musical intervals, and to Savart’s
wheel—a device he used while investigating the range of human hearing.
1.9.2 Faraday’s law
Faraday’s law says that a time-varying or space-varying magnetic field induces
an EMF in a closed loop linked by that field:
dΦ(x, t)
∂Φ ∂Φ ∂x
e = −N
= −N
+
(1.20)
dt
∂t
∂x ∂t
where e is the instantaneous EMF induced in a coil with N turns and Φ is
the magnetic flux (the same in each turn).
Fundamentals of Magnetism
25
Michael Faraday (22 September 1791–25 August 1867) was an English scientist who contributed to the study of electromagnetism and electrochemistry.
His main discoveries include the principles underlying electromagnetic induction, diamagnetism and electrolysis. An enormously important discovery for the
future of both science and technology was the electromagnetic induction law
(1831). Faraday discovered that a varying magnetic field causes electricity to
flow in an electric circuit. Previously, people had only been able to produce electric current with a battery. Faraday was one of the major players in the founding
of the new science of electrochemistry, which studies events at the interfaces of
electrodes with ionic substances. In 1834 he discovered Faraday’s laws of electrolysis. In 1836 Faraday discovered that when any electric conductor becomes
charged, all the extra charge sits on the outside of the conductor. This means
that the extra charge does not appear on the inside of a room or cage made of
metal (invention of the Faraday Cage).
1.9.3 Gauss’s law
The total electric flux passing through any closed imaginary surface enclosing
the charge Q is equal to Q (in SI units). The charge Q is enclosed by the
closed surface and is called Q enclosed, or Qen (Fig. 1.29b). The total flux ΨE
is thus equal to
I
I
ΨE =
dΨE =
DS · dS = Qen
S
S
H
where S indicates a double integral over the closed surface S and Ds is the
electric flux density through the surface S. The mathematical formulation
obtained from the above equation
I
DS · dS = Qen
(1.21)
S
is called Gauss’s law after K. F. Gauss. The Qen enclosed by surface S, due
to a volume charge density ρV distribution, becomes
Z
Qen =
ρV dV
(1.22)
V
where V is the volume.
Gauss’s law cannot be mistaken for Gauss’s theorem, also called the divergence theorem. It relates a closed surface integral of DS · dS to a volume
integral of ∇ · DdV involving the same vector, i.e.,
I
Z
DS · dS =
∇ · DdV
(1.23)
S
V
It should be noted that the closed surface S encloses the volume V .
26
Modern Permanent Magnet Electric Machines
Johann Carl Friedrich Gauss (30 April 1777—23 February 1855) was a German mathematician and physicist who made significant contributions to many
fields in mathematics and science. In physics, Gauss’s law, is a law relating the
distribution of electric charge to the resulting electric field. The surface under
consideration may be a closed one enclosing a volume such as a spherical surface.
In vector calculus, the divergence theorem, also known as Gauss’s theorem, is a
result that relates the flux of a vector field through a closed surface to the divergence of the field in the volume enclosed. His discoveries in mathematics and
astronomy led to Gauss’ appointment as professor of mathematics and director
of the observatory at Gottingen, where he remained in his official position until
his death.
1.10 Maxwell’s equations
James Clerk Maxwell was born in Edinburgh on the 13th June 1831. Although the family moved to their estate at Glenlair, near Dumfries, shortly
afterwards, James returned to Edinburgh to attend school at The Edinburgh
Academy. He continued his education at the Universities of Edinburgh and
Cambridge. In 1856, at the early age of 25, he became Professor of Physics
at Marischal College, Aberdeen. From there he moved first to King’s College, London, and then, in 1871, to become the first Professor of Experimental
Physics at Cambridge, where he directed the newly created Cavendish Laboratory. Maxwell died at Cambridge on November 5, 1879 of abdominal cancer.
1.10.1 Maxwell’s first equation
Maxwell introduced so-called displacement current, the density of which is
∂D/∂t, where D is the electric flux density (displacement) vector (Fig. 1.30).
There is a continuity of the displacement current and electric current J, e.g.,
in a circuit with a capacitor. The differential form of Maxwell’s first equation
is
curlH = J +
∂D
+ curl(D × v) + v divD
∂t
∇×H=J+
∂D
+ ∇ × (D × v) + v ∇ · D
∂t
(1.24)
or
where J is the density of the electric current, ∂D/∂t is the density of the displacement current, curl(D × v) is the density of the current due to the motion
of a polarized dielectric material, and vdivD is the density of the convection
Fundamentals of Magnetism
current. For v = 0
curlH = J +
∂D
∂t
27
(1.25)
Fig. 1.30. AC circuit with capacitor. Maxwell has proved that the displacement
current and electric current, under certain conditions, are extensions of one another.
The last equation in the Cartesian coordinate system has the following scalar
form:
∂Hy
∂Dx
∂Hz
−
= Jx +
∂y
∂z
∂t
∂Hz
∂Dy
∂Hx
−
= Jy +
∂z
∂x
∂t
(1.26)
∂Hy
∂Hx
∂Dz
−
= Jz +
∂x
∂y
∂t
1.10.2 Maxwell’s second equation
Maxwell’s second equation in the differential form is
curlE = −
∂B
− curl(B × v)
∂t
∇×E=−
∂B
− ∇ × (B × v)
∂t
(1.27)
or
For v = 0
curlE = −
∂B
∂t
(1.28)
28
Modern Permanent Magnet Electric Machines
The scalar form of the last equation in Cartesian coordinate system is
∂Ey
∂Bx
∂Ez
−
=−
∂y
∂z
∂t
∂Ex
∂Ez
∂By
−
=−
∂z
∂x
∂t
(1.29)
∂Ey
∂Ex
∂Bz
−
=−
∂x
∂y
∂t
For magnetically isotropic bodies
B = µ0 µr H
(1.30)
−6
where µ0 = 0.4π × 10 H/m is the magnetic permeability of free space, and
µr is the relative magnetic permeability.
For magnetically anisotropic materials, e.g., cold-rolled electrotechnical
steel sheets



 
Hx
Bx
µ11 µ12 µ13
 By  =  µ21 µ22 µ23   Hy 
µ31 µ32 µ33
Bz
Hz
If the coordinate system 0, x, y, z is the same as the axes of anisotropy


 

Bx
Hx
µ11 0 0
 By  =  0 µ22 0   Hy 
0 0 µ33
Bz
Hz
Since µ11 = µ0 µrx , µ22 = µ0 µry , and µ33 = µ0 µrz
Bx = µ0 µrx Hx ,
By = µ0 µry Hy ,
Bz = µ0 µrz Hz
(1.31)
1.10.3 Maxwell’s third equation
From Gauss’s law (1.21) for the volume charge density ρV and through the
use of Gauss’s theorem (1.23), Maxwell’s third equation in differential form is
divD = ρV
(1.32)
or
∇ · D = ρV
In scalar form
∂Dy
∂Dz
∂Dx
+
+
= ρV
∂x
∂y
∂z
(1.33)
Fundamentals of Magnetism
29
1.10.4 Maxwell’s fourth equation
The physical meaning of the equation
I
B · dS = 0
S
is that there are no magnetic charges. By means of the use of Gauss’s theorem
(1.23) the Maxwell fourth equation in differential form is
divB = 0
(1.34)
or
∇·B=0
In scalar form
∂By
∂Bz
∂Bx
+
+
=0
∂x
∂y
∂z
(1.35)
1.11 Magnetic vector potential
Through the use of the identity
div curlA = 0
(1.36)
and Maxwell’s fourth equation (1.34), which states that the divergence of B
is always zero everywhere, the following equation can be written
curlA = B
or
∇×A=B
(1.37)
The vector A defined according to eqn (1.37) is called the magnetic vector
potential . On the assumptions that µ = const, ϵ = const, σ = const, v = 0,
and divD = 0, Maxwell’s first equation (1.24) can be written in the form
∂E
∂t
Putting the magnetic vector potential (1.37) and using the identity
curlB = µσE + µϵ
curl curlA = grad divA − ∇2 A
the eqn (1.38) takes the form
grad divA − ∇2 A = µσE + µϵ
∂E
∂t
(1.38)
(1.39)
30
Modern Permanent Magnet Electric Machines
Since divA = 0 and for power frequencies 50 or 60 Hz σE >> jωϵE, the
magnetic vector potential A for sinusoidal fields can be expressed with the
aid of Poisson’s equation
∇2 A = −µJ
(1.40)
In scalar form
∇2 Ax = −µJx
∇2 Ay = −µJy
∇2 Az = −µJz
(1.41)
1.12 Speed of electromagnetic wave and theory of
relativity
The theory of relativity assumes that the speed of light is the same for all
observers.
1
= constant
c= √
µ0 ϵ0
(1.42)
The speed of light in a vacuum is c = (2.997930 ± 0.000003) × 108 m/s. Let
us make a clock whose timing is based on a pulse of light bouncing between
two mirrors as shown in Fig. 1.31.
Fig. 1.31. Stationary clock and astronaut’s clock.
If the distance between mirrors is 0.3 m, the light travels this distance
within 1 ns (1 ns = 10−9 s).
3 × 108
m
m
m
= 3 × 108 −9 = 0.3
s
10 ns
ns
According to the Pythagorean theorem
(ct′ )2 = (vt′ )2 + (ct)2
Fundamentals of Magnetism
31
Fig. 1.32. The astronaut’s time t′ at speed v > 0 is longer than time t at speed
v = 0.
Thus, the astronaut’s time t′ at speed v > 0 is longer than time t at speed
v = 0 (Fig. 1.32), i.e.,
1
1
t′ = q
=
2
γ
1 − vc2
(1.43)
in which the Lorentz’s coefficient
r
γ=
1−
v2
c2
(1.44)
Fig. 1.33. Lorentz coefficient γ according to eqn (1.44) and its reciprocal 1/γ as
functions of speed v.
The Lorentz’s coefficient γ and its reciprocal are plotted as a function of
speed v in Fig. 1.33. For example, at speed v = 0.99999c the astronaut’s time
t′ = 223.6t.
32
Modern Permanent Magnet Electric Machines
Albert Einstein was born on March 14th 1879 at Ulm, Württemberg, Germany. He attended elementary school at the Luitpold Gymnasium in Munich.
He enjoyed classical music and played the violin.
In 1889 a Polish medical student, Max Talmud, frequently visited the family
and became a tutor to Einstein, introducing him to higher mathematics and
philosophy.
In 1894, the family moved to Milan, Italy. Einstein subsequently renounced
his German citizenship to avoid military service and enrolled at the Swiss Federal
Polytechnic School in Zurich.
Finding it difficult to get employment, Einstein tutored children, until 1902,
when the father of his friend, Marcel Grossman, recommended him as a clerk in
the Swiss patent office in Bern. Einstein married Maleva Maric in 1903. In May,
1904 they had their first son, Hans Albert, and then a second son, Eduard in
1910. He divorced in 1919 and then married Elsa Löwenthal in the same year.
While studying James Maxwell’s description of the nature of light, Einstein
discovered that the speed of light was constant, which conflicted with Isaac Newton‘s laws of motion, and it was this realization that led Einstein to formulate
the principle of relativity.
In 1905 he submitted a paper for his doctorate at the Polytechnic Academy
in Zurich and in the same year published four important papers in the physics
journal, Annalen der Physik .
Einstein subsequently went to the University of Berlin, as director of the
Kaiser Wilhelm Institute for Physics from 1913 to 1933.
In November 1915, Einstein completed the General Theory of Relativity,
which he considered correct because it accurately predicted the perihelion of
Mercury’s orbit around the sun, which fell short in Newton’s theory. This theory also predicted a measurable deflection of light around the sun when a planet
orbited nearby, and it was confirmed by observations made by Sir Arthur Eddington during the solar eclipse of 1919. In 1921, A. Einstein received the Nobel
Prize in Physics for his explanation of the photoelectric effect.
In December, 1932 Einstein decided to leave Germany because of the Nazis.
He took a position at the Institute for Advanced Study at Princeton, New Jersey,
becoming a U.S. citizen in 1940.
In the summer of 1939, Einstein became aware of Germany’s success with the
fission of the Uranium atom and wrote a letter to President Roosevelt to alert
him of the possibility of a Nazi bomb. Roosevelt invited Einstein to meet with
him and this led to the Manhattan Project.
He spent the rest of his career trying to develop a unified field theory for the
forces of the universe, refuting the accepted interpretation of quantum physics.
However, in his later years, he stopped opposing quantum theory and tried to
incorporate it, along with light and gravity, into the larger unified field theory
he was trying to develop.
On April 17, 1955 he died of an abdominal aortic aneurysm.
Fundamentals of Magnetism
33
1.13 Maxwell’s equations in application to electrical
machines
Electrical machines and transformers most often operate at power frequencies,
i.e., 50 or 60 Hz. It means that
ˆ
ˆ
at power frequencies, displacement currents are much lower than electric
conduction currents, so that displacement currents can be neglected;
it can also be assumed that there are no convection currents and no currents due to the motion of a polarized dielectric material, i.e.,
curl(D × v) = 0
v divD = 0
(1.45)
In DC brush machines and in synchronous machines, only the voltage due to
rotation is induced in the armature winding. Electromagnetic phenomena in
these machines are described by the following Maxwell’s equations
curlH = J
curlE = −curl(B × v)
divD = 0
(1.46)
divB = 0
It is necessary to say that a synchronous machine is a particular case of a DC
brush machine, because a synchronous machine does not have a rectifier (operation as a generator) or electromechanical inverter (operation as a motor).
An electromechanical rectifier-inverter consists of a commutator and brushes.
In induction machines, in addition to the voltage induced due to rotation,
there is also transformer action voltage in the rotor (or the secondary of a
linear motor), so that Maxwell’s second equation in application to electrical
machines contains both rotation and transformer action voltage (variation of
magnetic flux density with time), i.e.,
curlH = J
∂B
− curl(B × v)
∂t
divD = 0
curlE = −
(1.47)
divB = 0
In the case of a transformer, which is a static converter of electrical energy,
Maxwell’s second equation has the form
curlE = −
∂B
∂t
(1.48)
34
Modern Permanent Magnet Electric Machines
The remaining equations are the same as in the case of rotating electrical
machines.
Classical theory of electrical machines assumes that the magnetic field
along the axial length of the core (direction perpendicular to the direction
of motion and parallel to the shaft of a rotating machine) does not change.
Performance characteristics of electrical machines are calculated analytically
on the basis of the 1D or at most the 2D distribution of electromagnetic field.
1.14 Force in electromagnetic field
In electromagnetism, the Lorentz force (or electromagnetic force) is the combination of electric and magnetic force on a point charge q due to electromagnetic fields (Fig. 1.34), i.e.,
F = q(E + v × B)
(1.49)
where qE is the electric force and q(v × B) is the magnetic force.
Fig. 1.34. Visualization of Lorentz’s force.
The equation of motion of a free particle of charge q and mass m moving
in electric and magnetic fields is
dv
= qE + qv × B
(1.50)
dt
This equation of motion (1.50) was first verified in a famous experiment carried
out by J.J. Thompson, the physicist from Cambridge University, in 1897.
Thomson was investigating cathode rays.
If a particle is subject to a force F and moves a distance ∆r in a time
interval ∆t, then the work done on the particle by the force is
m
∆W = F · ∆r
(1.51)
Fundamentals of Magnetism
35
The power input to the particle from the force field is
∆W
=F·v
∆t→0 ∆t
P = lim
where v is the velocity of particle. From the Lorentz force law, eqn (1.49), the
power input to the particle moving in electric and magnetic fields is
P = qv · E
(1.52)
A charged particle can gain or lose energy only from an electric field, but not
from a magnetic field. This is because the magnetic force is always perpendicular to the direction of motion of the particle (Fig. 1.34) and does not do
any work on the particle. This explains why in particle accelerators, magnetic
fields are often used to guide particle motion, e.g., in a circle, while the actual
acceleration is performed by electric fields.
Hendrik A. Lorentz, Dutch physicist, was born at Arnhem, the Netherlands,
on July 18, 1853. He introduced the force law in 1892. During the 19th century
he clarified important connections between electricity, magnetism and light. In
1892 he presented his electron theory that in matter there are charged particles,
electrons, that conduct electric current and whose oscillations give rise to light.
H. Lorentz’s electron theory could explain Pieter Zeeman’s discovery in 1896
that the spectral lines corresponding to different wavelengths split up into several
lines under the influence of a magnetic field. He shared the 1902 Nobel Prize
in Physics with P. Zeeman for the discovery and theoretical explanation of the
Zeeman effect. The so-called Lorentz transformation (1904) was based on the
fact that electromagnetic forces between charges are subject to slight alterations
due to their motion, resulting in a minute contraction in the size of moving
bodies. It not only adequately explains the apparent absence of the relative
motion of the Earth with respect to the ether, as indicated by the experiments
of Michelson and Morley, but also paved the way for Einstein’s special theory
of relativity. Until his death he was Chairman of all Solvay Congresses, and
in 1923 he was elected to the membership of the International Committee of
Intellectual Cooperation of the League of Nations. Of this committee, consisting
of only seven of the world’s most eminent scholars, he became the president in
1925. Lorentz died at Haarlem on February 4, 1928.
Summary
Particles with spin can possess a magnetic dipole moment, just like a rotating
electrically charged body in classical electrodynamics.
A current loop behaves as a magnetic dipole and has a magnetic momentum. Magnetic dipole moment is a vector pointing out of the plane of the
current loop and with magnitude equal to the product of the current and loop
area m = nIS.
36
Modern Permanent Magnet Electric Machines
Magnetic permeability is the measure of the ability of a material to support
the formation of a magnetic field within itself. The magnetic permeability of
free space, also known as the magnetic constant, is µ0 = 0.4π × 10−6 H/m.
The volume magnetic susceptibility χ is a dimensionless quantity defined
as M = χH, where M is the magnetization vector and H is the magnetic field
strength vector. The magnetic permeability µ = µ0 (1 + χ).
Depending on relative magnetic permeability µr = 1 + χ, all materials can
be divided into ferromagnetic with µr >> 1, paramagnetic with µr > 1 and
diamagnetic with µr < 1. Also there are ferrimagnetic materials displaying
a weak form of ferromagnetism and antiferromagnetic materials, similar to
ferromagnetic and ferrimagnetic materials.
Ferromagnetic materials exhibit a strong attraction to magnetic fields and
are able to retain their magnetic properties after the external field has been
removed.
A magnetic domain is a region within a magnetic material in which the
magnetization is in a uniform direction.
A hysteresis loop shows the relationship between the induced magnetic
flux density B and the magnetic field strength H. The wider the hysteresis
loop, the harder the material is magnetically. In the saturation region, all
domains are fully aligned under the external magnetic field.
The presence of a ferromagnetic core inside a coil (solenoid) increases the
net magnetic flux density, i.e., Bnet = Bcoil + Bcore .
Magnetism has found broad applications, e.g., electric motors and generators, magnetic storages of data, loudspeakers, lift electromagnets, magnetic
core memory, MRAM, CRT, NMR, MRI, magnetic levitation, cyclotrons,
tokamak, MHD generators.
The special theory of relativity owes its origins to Maxwell’s equations of
the electromagnetic field (A. Einstein).
Maxwell’s equations are fundamental equations of electromagnetic fields,
which were published in 1873 on the basis of earlier laws: Biot-Savart law
(1820), Faraday’s law (1831) and Gauss’s law (1840).
Maxwell’s first equation says that the curls of vector H are due to the
electric current with density J, displacement current with density ∂D/∂t,
currents due to the motion of a polarized dielectric material with density
curl(D × v) and convection current with density vdivD = vρ.
Maxwell’s second equation says that the curls of vector E are due to variation of the magnetic flux density with time ∂B/∂t and motion of the magnetic
field curl(B × v) relative to the electric circuit.
Maxwell’s third equation is a generalization of Gauss’s law and its extension on alternating quantities divD = ρ.
Maxwell’s fourth equation divB = 0 says that the lines of the vector B
always penetrate through the closed surface because there are no magnetic
charges (monopoles).
Fundamentals of Magnetism
37
The magnetic vector potential A is defined as ∇ × A = B. The magnetic
vector potential for sinusoidal fields can be expressed with the aid of Poisson’s
equation ∇2 A = −µJ.
The theory of relativity assumes that the speed of light c = 2.99793 × 108
m/s is the same for all observers.
Electrical machines and transformers most often operate at power frequencies of 50 or 60 Hz, and at these frequencies, the displacement currents
∂D/∂t are much lower than electric conduction current J, so displacement
currents can be neglected. Also, there are no currents due to motion of a polarized dielectric material curl curl(D × v) = 0 and no convection currents
vdivD = vρ = 0.
In electromagnetism, the Lorentz force (or electromagnetic force) is the
combination of electric force qE and magnetic force q(v × B) on a point
charge q due to electromagnetic fields, i.e., F = q(E + v × B).
2
SOFT MAGNETIC MATERIALS
2.1 Classification of soft ferromagnetic materials
Soft ferromagnetic materials have a small area of hysteresis loop, hysteresis
losses are low, retentivity and coercivity are low, they can be easily magnetized
and demagnetized, require a small value of H for magnetization, their domain
walls move easily, and susceptibility and permeability values are high.
Soft ferromagnetic materials used in construction of ferromagnetic cores
for electrical machines and electromagnetic devices can be classified as
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
laminated silicon steels;
high saturation cobalt alloys;
amorphous ferromagnetic alloys;
amorphous alloys;
soft magnetic composites (SMC);
permalloys;
nanocrystalline composite;
solid ferromagnetic steels.
The hysteresis loop shows the “history-dependent” nature of magnetization of a ferromagnetic material. Once the material has been magnetized to
the saturation level, the magnetic field intensity can then be dropped to zero
and the material will retain some remanent magnetic flux density Br . It remembers its history (Fig. 2.1).
Soft magnetic materials are characterized by the magnetization curve, i.e.,
magnetic flux density versus magnetic field intensity B = f (H) and specific
core loss curve ∆pF e = f (B) at constant frequency.
Magnetization curve B = f (H) is the locus of the tips of a family of
hysteresis loops being measured using DC currents (Fig. 2.2a).
The specific core losses curve is the plot of specific core losses (W/kg)
versus magnetic flux density ∆pF e = f (B) at constant frequency f = const
(Fig. 2.2b).
40
Modern Permanent Magnet Electric Machines
Fig. 2.1. Hysteresis loop and orientation of magnetic domains in a ferromagnetic
material.
Fig. 2.2. Basic characteristics of soft magnetic materials: (a) magnetization curve
B = f (H); (b) ∆pF e = f (B).
2.1.1 Laminated silicon steels
Addition of 0.5% to 3.25% of silicon (Si) increases the resistivity (reduces
eddy current losses) and improves magnetization curves B-H of low-carbon
steels. Silicon content, however, increases hardness of laminations and, as a
consequence, shortens the life of stamping tooling.
Nonoriented electrical steels are Fe-Si alloys with random orientation of
crystal cubes and practically the same properties in any direction in the plane
of the sheet or ribbon. Nonoriented electrical steels are available as both fully
processed and semi-processed products. Fully processed steels are annealed
to optimum properties by the manufacturer and ready for use without any
additional processing.
Soft Magnetic Materials
41
Annealing is a heat treatment that alters the physical and sometimes chemical properties of a material to increase its ductility and reduce its hardness,
making it more workable.
Silicon steels are generally specified and selected on the basis of allowable
specific core losses (W/kg or W/lb). The most universally accepted grading of
electrical steels by core losses is the American Iron and Steel Industry (AISI)
system, the so-called M-grading. The M number, e.g., M19, M27, M36, etc.,
indicates the maximum specific core losses in W/lb at 1.5 T and 50 or 60 Hz,
e.g., M19 grade specifies that losses shall be below 1.9 W/lb = 4.2 W/ kg at
1.5 T and 60 Hz.
The magnetization curve B = f (H) of M19 silicon steel is plotted in Fig.
2.3 and the specific loss curve ∆p = f (B) for f = 50 Hz of M19 silicon steel
is plotted in Fig. 2.4. Core losses at 60 Hz are approximately 1.27 times the
core losses at 50 Hz.
Fig. 2.3. Magnetization curve B = f (H) of cold-rolled isotropic silicon steel sheet
Armco-DI-MAX M19.
To reduce eddy current losses, steel sheets are covered on both sides with
insulating material. The stacking factor is the ratio of the thickness d of a
single sheet without insulating layers to the thickness of the sheet with doublesided insulation d + 2∆, i.e.,
d
d + 2∆
where ∆ is the thickness of single-sided insulation.
ki =
(2.1)
42
Modern Permanent Magnet Electric Machines
Fig. 2.4. Specific loss curve ∆p = f (B) for f = 50 Hz of cold-rolled isotropic silicon
steel sheet Armco-DI-MAX M19.
For modern high-efficiency, high-performance applications, there is a need
for operating AC devices at higher frequencies, i.e., 400 Hz to 10 kHz. Because of the thickness of the standard silicon ferromagnetic steels of 0.25 mm
(0.010”) or more, core loss due to eddy currents is excessive. Nonoriented
electrical steels with thin gauges (down to 0.025 mm thick) for ferromagnetic
cores of high-frequency rotating machinery and other power devices are manufactured, e.g., by Arnold Magnetic Technologies Corporation, Rochester, NY,
USA.
2.1.2 High-saturation cobalt alloys
Iron–cobalt (Fe-Co) alloys with Co contents ranging from 15 to 50% have
the highest known saturation magnetic flux density, up to 2.39 T at room
temperature. They are the natural choice for applications such as aerospace
(motors, generators, transformers, magnetic bearings) where mass and space
savings are of prime importance. Additionally, the iron-cobalt alloys have the
highest Curie temperatures of any alloy family and have found use in elevated
temperature applications. The nominal composition, e.g., for Hiperco 50 from
Carpenter , PA, U.S.A. is 49% Fe, 48.75% Co, 1.9% V, 0.05% Mn, 0.05% Nb
and 0.05% Si. Hiperco 50 has the same nominal composition as Vanadium
Permendur and Permendur V .
The specific mass density of Hiperco 50 is 8120 kg/m3 , modulus of elasticity 207 GPa, electric conductivity 2.5 × 106 S/m, thermal conductivity 29.8
W/(m K), Curie temperature 940◦ C, specific core loss about 76 W/kg at 2
T, 400 Hz and thickness from 0.15 to 0.36 mm.
Similar to Hyperco 50 is Vacoflux 50 (50% Co) cobalt-iron alloy from
Vacuumschmelze, Hanau, Germany (Fig. 2.5 and 2.6).
Soft Magnetic Materials
43
Fig. 2.5. Magnetization curves B = f (H) of Vacoflux 50 and Vacoflux 17 .
Fig. 2.6. Specific loss curves ∆p = f (B)of Vacoflux 50 at 50, 60, 400 and 1000 Hz.
44
Modern Permanent Magnet Electric Machines
Fe-Co alloy laminations have found main applications in aerospace motors,
generators and transformers and also in magnetic bearings.
2.1.3 Amorphous ferromagnetic alloys
Core losses can be substantially reduced by replacing standard electrical laminated steels with amorphous ferromagnetic alloys. Metglass amorphous alloys
(Honeywell (Allied-Signal)) have specific core losses at 1 T and 50 Hz from
0.125 to 0.28 W/kg.
Amorphous alloy ribbons based on alloys of iron, nickel and cobalt are
produced by rapid solidification of molten metals at cooling rates of about
1060 C/s. The alloys solidify before the atoms have a chance to segregate or
crystallize. The result is a metal alloy with a glass-like structure, i.e., a noncrystalline frozen liquid.
The efficiency of a standard small 550 W induction motor is 74%. It means
that power losses dissipated in this motor are 137 W. Replacing the standard
core with amorphous alloy core, the losses are reduced to 88 W, so that the
efficiency increases to 84%. Application of amorphous ribbons to the mass
production of electric motors is limited by their hardness, i.e., 800 to 1100 in
Vicker’s scale which requires alternative cutting methods as a liquid jet.
Fig. 2.7. Technology of amorphous ribbon production [84].
The material in liquid state is poured on a rotating copper drum (Fig.
2.7). The speed of cooling should be fast enough not to allow the formation
of a crystal structure. To help in formation of the amorphous state, a small
addition of metalloid (mostly boron) is made in order to improve viscosity
of the molten metal. Since the ribbon should be cooled very quickly, it is
thin with the thickness not exceeding 50 µm. Also the width of the ribbon is
limited, usually not exceeding 20 cm. The magnetic parameters of amorphous
ribbon can be improved by annealing, especially by annealing in a longitudinal
magnetic field.
Soft Magnetic Materials
45
Fig. 2.8. Magnetization curve B = f (H) of Metglas 2705M (Metglas, Morristown,
NJ, USA). DC – magnetization under DC current, 50 Hz – magnetization at 60 Hz.
Fig. 2.9. Specific loss curve ∆p = f (B) of Metglas 6025 SA1 alloy (Metglas, Morristown, NJ, USA).
Magnetization curve B = f (H) of Metglas 2705M is plotted in Fig. 2.8,
while specific loss curves ∆p = f (B) of Metglas 6025 SA1 at different frequencies are plotted in Fig. 2.9.
2.1.4 Soft magnetic composites (SMC)
New soft magnetic powder materials, which are competitive with traditional
steel laminations, have recently been developed in the U.S.A. and Sweden.
Powder metallurgy is used in the production of ferromagnetic cores of small
46
Modern Permanent Magnet Electric Machines
Fig. 2.10. Soft magnetic powder composite: (a) high-purity iron powder; (b) electrically insulated surface; (c) “nano-coated” ferromagnetic particles.
electrical machines or ferromagnetic cores with complicated 3D shapes. Powder materials are recommended for 3D magnetic circuits such as claw-pole,
transverse flux (TFMs), disc-type and recyclable machines. Specific core losses
at 1 T and 100 Hz are 9 W/kg for Accucore (U.S.A.) and 12.5 W for Somaloy
500 (Sweden). At 10 kA/m, the magnetic flux density is 1.72 T for Accucore
and 1.54 T for Somaloy 500 .
The components of soft magnetic powder composites are iron powder, dielectric (epoxy resin) and filler (glass or carbon fibers) for mechanical strengthening (Fig. 2.10). Powder composites for ferromagnetic cores of electrical machines and apparatus can be divided into
(a) dielectromagnetics and magnetodielectrics,
(b) magnetic sinters.
Magnetization curve B = f (H) of Somaloy—500 is plotted in Fig. 2.11,
while specific loss curves ∆p = f (B) of at different frequencies are plotted in
Fig. 2.12.
Fig. 2.11. Comparison of magnetization curves B = f (H) of Accucore and Somaloy—500 .
Soft Magnetic Materials
47
Fig. 2.12. Specific losses ∆p = f (B) at frequencies from 50 to 1000 Hz and temperature 5000 C for Somaloy—500 with specific density 7300 kg/m3 and 0.5% Kenolube.
2.1.5 Permalloys
Permalloy is the term for a Nickel Iron (Ni-Fe) ferromagnetic alloy. Generically, it refers to an alloy with about 20% Fe and 80% Ni content. Permalloy
has a high magnetic permeability, low coercivity, near zero magnetostriction,
and significant anisotropic magnetoresistance. The low magnetostriction is
critical for industrial applications, where variable stresses in thin films would
otherwise cause a ruinously large variation in magnetic properties.
Permalloy 80 is a highly ferromagnetic Nickel-Iron-Molybdenum alloy,
with roughly 80% Ni and 15% Fe and 5% Mo content. It is useful as a magnetic
core material in electrical and electronic equipment. Commercial Permalloy
alloys typically have relative permeability of around µr ≈ 100, 000, saturation
magnetic flux density Bsat ≈ 0.75 T, specific mass density ρ = 8740 kg/m3 ,
electric conductivity σ = 1.72 × 106 S/m. Magnetization curves B = f (H)
and core loss curves ∆P = f (B) are plotted in Fig. 2.13.
2.1.6 Nanocrystalline composites
In 1988 Y. Yoshizawa, S. Oguma and K. Yamauchi from Hitachi Metals Company proved that after appropriate annealing of a Fe-based amorphous ribbon, it is possible to create very small grains of α-FeSi (average diameter
around 10 nm) embedded in an amorphous matrix. They developed the first
nanocrystalline soft magnetic material in the world, named FINEMET . This
nanocrystalline material has high saturation flux density, high permeability
®
48
Modern Permanent Magnet Electric Machines
Fig. 2.13. Characteristics of Permalloy 80 and Superpermalloy: (a) magnetization
curves B = f (H) of Permalloy 80 and Superpermalloy; (b) specific loss curves
∆P = f (B) at 5.0 to 100 kHz for Permalloy 80.
and low core losses (1/5th the core loss of Fe based amorphous metal and
approximately the same core loss as Co-based amorphous metal). It also has
stable temperature characteristics, low magnetostriction, and provides excellent performance in electromagnetic noise suppression. It will allow reduction
in size and mass of electric and electronics devices.
There are four types of FINEMET nanocrystalline material [47]:
®
ˆ
ˆ
ˆ
ˆ
H type: a magnetic field is applied in a circumferential direction during
annealing;
M type: no magnetic field is applied during annealing;
L type: a magnetic field is applied vertically to the core plane during
annealing;
S type: a magnetic field is controlled, annealing process is improved, the
highest magnetic permeability of FINEMET is obtained.
®
®
Magnetization curves B = f (H) and core loss per volume curves ∆pV =
f (B) of FINEMET are plotted in Figs 2.14 and 2.15 [47].
Nanocrystalline materials fill the gap between amorphous materials and
conventional (coarse-grained) materials. Nanocrystalline alloys are materials
based on Fe, Si, and B, with additions of Nb and Cu. Typically, they are
Soft Magnetic Materials
49
®
Fig. 2.14. Magnetization curves B = f (H) of FINEMET nanocrystalline soft
magnetic materials: (a) H-type (FT-3H); (b) M-type (FT-3M); (c) L-type (FT-3L).
Fig. 2.15. Core loss per volume curves ∆pV = f (B) of FINEMET
soft magnetic materials FT-3H, FT-3M and FT-3L at 20 kHz.
® nanocrystalline
produced through a rapid solidification process as a thin, ductile ribbon.
Initially the ribbon is in the amorphous state, then crystallized in a subsequent heat treatment to promote nano-crystallization (approx. 10-20 nm).
Once nano-crystallized, they exhibit low core loss and magnetostriction, while
maintaining high saturation induction and permeability. The properties of a
nano-crystalline material are similar to the best grades of permalloy. However, the NiFe alloys can be used in frequencies only up to 100 kHz, but the
nanocrystalline materials can work correctly at frequencies similar to the best
grades of ferrites (1–500 MHz).
2.1.7 Solid ferromagnetic steels
Solid ferromagnetic materials, such as cast steel and cast iron, are used for
salient poles, pole shoes, solid rotors of special induction motors, and reaction
50
Modern Permanent Magnet Electric Machines
rails (platens) of linear motors. Electrical conductivities of carbon steels are
from 4.5 × 106 to 7.0 × 106 S/m at 20◦ C.
Magnetization curves B = f (H) of carbon solid steels 35 and 4340 and
ferromagnetic alloy FeNiCoMoTiAl are plotted in Fig. 2.16. Magnetization
curves B = f (H) of a mild carbon steel (0.27% C) and cast iron are given in
Table 2.1.
Fig. 2.16. Magnetization curves B = f (H) of carbon solid steels 35 and 4340 and
ferromagnetic alloy FeNiCoMoTiAl.
Table 2.1. Magnetization curves B = f (H) of a mild carbon steel (0.27% C) and
cast iron
Magnetic
Magnetic field
flux density
intensity, H
B
Mild carbon steel 0.27% C Cast iron
T
A/m
A/m
0.2
190
900
0.4
280
1600
0.6
320
3000
0.8
450
5150
1.0
900
9500
1.2
1500
18,000
1.4
3000
28,000
1.5
4500
1.6
6600
1.7
11,000
Soft Magnetic Materials
51
Power losses in a solid ferromagnetic halfspace can be calculated using the
Poynting vector, i.e.,
ˆ
ˆ
active power losses per unit of surface
r
ωµ0 µrs |Hms |2
∆P = aR
2σ
2
reactive power losses per unit of surface
r
ωµ0 µrs |Hms |2
∆Q = aX
2σ
2
W/m2
(2.2)
VAr/m2
(2.3)
where ω = 2πf , µ0 = 0.4π × 10−6 H/m, µrs is the relative magnetic permeability at the surface of the halfspace, Hms is the peak value of the magnetic
field intensity at the surface of the halfspace, σ is the electric conductivity,
aR ≈ 1.45 is the coefficient taking into account variation of magnetic permeability inside the ferromagnetic body and hysteresis losses for active power
losses and aX ≈ 0.85 is a similar coefficient for reactive losses.
2.2 Losses in ferromagnetic materials
2.2.1 Hysteresis losses
As the alternating magnetic flux magnetizes the core, the energy is lost in the
core due to the hysteresis effect. The energy loss, called the hysteresis loss,
is proportional to the area of the hysteresis loop (Fig. 1.13). The hysteresis
loss depends on the ferromagnetic material of the core. The first empirical
formula for hysteresis losses was proposed by C.P. Steinmetz (Fig. 2.17) and
published in the 1892 [78], i.e.,
n
∆Ph = kh Bm
(2.4)
where kh and n are curve-fitted coefficients of actual experimental data and
Bm is the peak value of the magnetic flux density. A more accurate empirical
formula for the hysteresis loss contains the frequency f of the magnetic flux
density, i.e.,
n
∆Ph = ch f Bm
(2.5)
where Bm is the peak value of the magnetic flux density, f is the frequency and
the hysteresis constants ch and n vary with the core material. The constant
n is often assumed to be 1.6 . . . 2.0.
52
Modern Permanent Magnet Electric Machines
Charles Proteus Steinmetz (9 April 1865—26 October 1923), original name
Karl August Rudolf Steinmetz, German-American electrical engineer, born in
Breslau, now Wroclaw (Poland). As a result of his involvement in socialist movements, Steinmetz had to flee Breslau (1888). After a short stay in Zürich he immigrated to the United States in 1889, traveling by steerage. He soon obtained a
job with a small electrical firm owned by Rudolf Eickemeyer in Yonkers, NY. He
established a small laboratory at the factory, where he did much of his scientific
research.
The Steinmetz experiments on power losses in the magnetic materials used
in electrical machinery led to his first important work, the law of hysteresis.
In 1892 Steinmetz gave two papers before the American Institute of Electrical
Engineers (AIEE) on his new law concerning hysteresis loss, which brought him
a worldwide reputation. The Steinmetz method of calculation of AC circuits
was presented to an uncomprehending audience at the International Electrical
Congress in 1893. His book Theory and Calculation of Alternating Current Phenomena (coauthored with Ernst J. Berg in 1897) was read and understood by
only a very few.
In 1893 the newly formed General Electric Company (GEC) purchased Eickemeyer’s company, primarily for his patents, but Steinmetz was considered one
of its major assets. In 1894 the GEC transferred its operations to Schenectady, NY, and Steinmetz was made head of the calculating department. The
Steinmetz third major scientific achievement was in the study and theory of
electrical transients. He served as president of the AIEE in 1901–02. In his later
years Steinmetz also engaged in public affairs to a considerable degree, serving
as president of the Board of Education of Schenectady, NY, and as president of
the city council.
Fig. 2.17. Group being given a tour of the Marconi Wireless Station in Somerset,
NJ, in 1921, C.P. Steinmetz (center, in white suit), Albert Einstein (to his right).
Soft Magnetic Materials
53
Fig. 2.18. Eddy currents in: (a) solid ferromagnetic core; (b) laminated ferromagnetic core.
2.2.2 Eddy-current losses
Another source of the power loss in the ferromagnetic core is the eddy currents
induced by the alternating magnetic flux. If the magnetic flux is perpendicular,
directed toward the plane of this page and increasing with time, it induces
voltages in conductive material of the core (Fig. 2.18). Under action of these
voltages, eddy currents flow in closed loops (paths) producing power losses
i2 R, which are converted into heat. The eddy-current losses can be reduced
by decreasing the current i or increasing the resistance R. This can be done by
replacing a solid ferromagnetic core with laminated ferromagnetic core. The
eddy-current losses are proportional to the frequency f squared and the peak
magnetic flux density Bm square, i.e.,
2
∆Pe = ce f 2 Bm
(2.6)
The eddy-current constant ce depends on the electric conductivity of the material of the core and the thickness squared of laminations. An addition of
silicon reduces the electric conductivity of steel, i.e., reduces the eddy-current
losses and increases the saturation magnetic flux density.
To reduce eddy-current losses in sheet steels, all electrical steels are coated
double-sided with a thin layer of insulation, usually oxide insulation. The
stacking factor is the ratio of the thickness of the bare sheet to the thickness
of the sheet with insulation, as defined by eqn (2.1).
2.2.3 Excess eddy-current losses
There are also the so-called excess eddy-current losses, which can be estimated
as [11]
p
∆Pex = 8 σF e GSF e V0 f 1.5 B 1.5
(2.7)
where σF e is the electric conductivity of steel sheet, G = 0.1356 is a unitless
constant, V0 is the curve-fitted coefficient and SF e is a cross-sectional area of
the core.
54
Modern Permanent Magnet Electric Machines
2.2.4 Additional losses
The additional losses mainly result from the deterioration of the core materials
during the manufacturing process of the machine.
The core losses of electrical machines and electromagnetic devices may
be twice as large as comprehensive loss models predict. The electrical steel
sheets are punched, welded together and shrunk fit to the frame. This causes
residual strains in the core sheets, degrading their magnetic characteristics.
The cutting burrs make galvanic contacts between the sheets and form paths
for inter-lamination currents. Another potential source of additional losses
are the circulating currents between the parallel strands of random-wound
windings.
2.3 Engineering approach to calculation of core losses
The total power losses, neglecting the additional losses, are:
n
2
1.5
∆PF e = ∆Ph + ∆Pe + ∆Pex = ch f Bm
+ ce f 2 Bm
+ cex f 1.5 Bm
(2.8)
where
p
cex = 8 σF e GSF e V0
(2.9)
The constants ch , ce and cex are not always available. In practical calculations
of AC magnetic circuits, the core losses ∆PF e can be estimated on the basis
of the specific core losses ∆p1/50 , i.e., losses at Bm = 1.0 T and f = 50 Hz
and masses. In the case of, say, legs and yokes of a transformer, i.e.,
∆PF e = ∆p1/50
f
50
4/3
2
2
kadl Bml
ml + kady Bmy
my
(2.10)
where kadl > 1 and kady > 1 are the factors accounting for the increase in
losses due to metallurgical and manufacturing processes, ∆p1/50 is the specific
core loss in W/kg at 1 T and 50 Hz, Bl is the magnetic flux density in the
leg, By is the magnetic flux density in the core (yoke), mt is the mass of legs,
and my is the mass of yokes.
2.4 Ferromagnetic cores
Ferromagnetic cores in electromagnetic devices and machines are necessary
ˆ
ˆ
to guide the magnetic flux in the desired direction and
reduce the reluctance for the magnetic flux.
Soft Magnetic Materials
55
2.4.1 Transformers
A transformer is a static electromagnetic device for transforming electrical
energy in an AC system from one (primary) circuit into another (secondary)
circuit at the same frequency but with different values of voltages and currents. The ferromagnetic core magnetically couples the primary and secondary
circuits. Ferromagnetic cores of small single-phase transformers are shown in
Figs 2.19 and 2.20. Ferromagnetic cores of three-phase transformers are shown
in Figs 2.21.
Ferromagnetic cores of transformers can be made of anisotropic silicon
laminations provided that the direction of the magnetic flux is always in the
direction of the highest permeance (magnetic permeability) of the strip.
Fig. 2.19. Laminations for cores of small single-phase transformers: (a) shell-type;
(b) core-type.
Fig. 2.20. Wound cores of single-phase transformers.
2.4.2 Electronic devices
Ferrites have an advantage over other types of magnetic materials due to their
high electrical resistivity and low eddy-current losses over a wide frequency
range. Ferrite cores are dense, homogeneous ceramic structures made by mixing iron oxide (Fe2 O3 ) with oxides or carbonates of one or more metals such
56
Modern Permanent Magnet Electric Machines
Fig. 2.21. Laminated core of a three-phase transformer: (a) core with pressing
beams; (b) assembly of laminated core with mitered joints.
as manganese, zinc, nickel, or magnesium. They are pressed, then fired in a
kiln (furnace) to 1300◦ C, and machined as needed to meet various operational
requirements.
Different types of ferrite cores for electronics devices are shown in Fig.
2.22.
Fig. 2.22. Ferrite cores for electronics devices: (a) E-type cores for inductors and
transformers; (b) pot cores for energy storing chokes; (c) U-type core for inductors
and transformers; (d) round core for EMI suppression filter; (e) choke core clips for
anti-interference noise filters.
2.4.3 DC machines
DC machines convert DC electrical power into mechanical power (motors) or
mechanical power into DC electrical power. In a typical design the rotor consists of an armature system and commutator, which is a mechanical inverter
(motors) or mechanical rectifier (generator). The field excitation system is
Soft Magnetic Materials
57
stationary (stator) and can be designed as a field excitation winding or a set
of PMs.
Laminated ferromagnetic cores of DC machines are shown in Fig. 2.23.
Fig. 2.23. Laminated cores for DC machines: (a) rotor (armature) cores of small
DC machines; (b) main pole core with pole shoe.
2.4.4 Switched reluctance machines (SRM)
The switched reluctance machine (SRM) is a doubly-salient, singly-excited
electrical machine. The electromagnetic torque is produced by the magnetic
attraction of a steel salient-pole rotor to stator electromagnets. No rotor PMs
are needed, and the rotor carries no windings. An SRM requires a controllable solid-state converter and cannot be operated directly from a utility grid.
Laminated cores for SRMs are shown in Fig. 2.24.
2.4.5 Induction machines
Induction machines are widely used as motors in industrial, traction and
consumer device drives and as generators in wind energy conversion systems.
Their advantages include simple construction, low cost, and high reliability.
The stator (usually three-phase) rotating magnetic field, together with current
induced in the rotor, produces electromagnetic torque. The rotor can be made
as a cage rotor, wound rotor (slip-ring rotor) or solid steel rotor.
58
Modern Permanent Magnet Electric Machines
Fig. 2.24. Laminated cores for SRMs: (a) rotor cores; (b) stator cores.
Laminated magnetic circuits of induction motors are shown in Fig. 2.25.
Fig. 2.25. Laminated cores for induction motors: (a) stator cores; (b) rotor cores.
2.4.6 Synchronous turbogenerators
In synchronous machines, the rotor runs in synchronism with the stator rotating magnetic field. Synchronous machines can be designed with non-salientpole rotor (cylindrical rotor) or salient-pole rotor. Synchronous turbogenerators (turboalternators) have non-salient-pole rotors and are used for generation of electrical energy in thermal power plants.
The three-phase stator has a laminated core (Fig. 2.26a) and a rotor is
made as a solid steel forging (Fig. 2.26b).
Soft Magnetic Materials
59
Fig. 2.26. Ferromagnetic cores for large turbogenerators: (a) stator laminated cores;
(b) two-pole solid steel rotor cores.
2.4.7 Synchronous hydrogenerators
Synchronous hydrogenerators are used for generation of electrical energy in
hydro-power plants. They have large diameters and many poles. The rotor is
designed as a salient-pole rotor. Pole cores and pole shoes of the rotor are
laminated. Stator and rotor cores of a large hydrogenerator are shown in Fig.
2.27.
Fig. 2.27. Large 358-kW hydrogenator, Ingula Pumped Storage, South Africa: (a)
stator; (b) rotor. Photo courtesy of Eskom, Megawatt Park, Sandton, South Africa.
60
Modern Permanent Magnet Electric Machines
2.4.8 Permanent magnet (PM) brushless motors
PM brushless motors have a three-phase stator and PM field excitation system
in the rotor. They are classified into sine-wave motors (synchronous motors)
and square-wave motors (DC brushless motors). Most of these motors are
inverter-operated motors, although, if the rotor is equipped with a cage winding, they are self-starting constant-speed motors. Magnetic circuits of PM
brushless motors are shown in Fig. 2.28.
Fig. 2.28. Laminated cores for PM brushless motors: (a) stator and rotor cores for
surface PMs; (b) stator and rotor cores for embedded PMs.
2.4.9 Segmented stator and rotor cores
Segmentation of the stator and rotor cores offers the possibility to simplify the
coil-winding process, to increase the slot fill factor or to minimize the amount
of laminated steel. With this method, the core is split into one-tooth pitch,
two-tooth pitch, or more tooth-pitch segments. Segmented stators are used
for concentrated, non-overlapping windings. Each segment is wound and then
the segments are joined together, e.g., by laser welding. Segmented laminated
cores are shown in Fig. 2.29.
2.4.10 3D cores made of soft magnetic composites (SMC) for
special electric machines
Axial-flux, transverse-flux machines (TFM) and special machines have 3D
magnetic circuit, which are difficult to manufacture using laminated steel.
Application of ferromagnetic powder materials (SMCs) can significantly simplify the manufacturing process of 3D cores (Fig. 2.30).
Soft Magnetic Materials
61
Fig. 2.29. Segmented laminated cores for: (a) stator; (b) rotor.
Fig. 2.30. 3D cores made of soft magnetic composites (SMC) for axial-flux, TFMs
and special electric machines.
2.4.11 Solid ferromagnetic rotors
Induction motors with solid ferromagnetic rotors can be used as high-speed
machines, e.g., motors for compressors, motors for pumps, motors for drills
and generators for microturbines. A solid ferromagnetic rotor is both the
conductor for the magnetic flux and for the electric current. To improve the
performance of the induction machine, the impedance of the rotor can be
reduced by making axial slots (Fig. 2.31a), coating the external cylindrical
surface with copper layer (Fig. 2.31b), or furnishing the rotor with a cage
winding.
2.5 Magnetic circuits of electrical machines for recycling
After failure or longtime use, when repairing is not economically justified or
an electric machine is not repairable, it can be handled, generally, within the
following categories [61]:
62
Modern Permanent Magnet Electric Machines
Fig. 2.31. Solid ferromagnetic rotors for high speed induction machines: (a) rotor
with axial slots; (b) rotor coated with copper layer.
(a) discarded into the environment;
(b) placed in a permitted landfill;
(c) put to a high-value use, breaking it down into its components, melting
steel, copper and aluminum;
(d) rebuilt (totally or partially), some components discarded or reused;
(e) reused.
Since categories (a) and (b) have a negative effect on the environment (Fig.
2.32), they are not recycling. Categories (c), (d) and (e) can be classified as
recycling, because they create value at the end of life of an electric machine.
Fig. 2.32. What to do with electrical machines after failure or longtime use, when
repairing is not economically justified or the machine is not repairable?
Design and construction guidelines for recyclable electric machines include,
but are not limited to
ˆ
ˆ
the number of parts should be reduced;
all parts should be made simple;
Soft Magnetic Materials
63
Fig. 2.33. An ideal machine for recycling is a machine with sintered powder (SMC)
magnetic circuit. After crushing powder materials and conductors can be easily
separated and reused.
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
in mechanical design, both assembly and disassembly should be considered;
the number of materials should be limited;
toxic materials, e.g., beryllium copper, lead based alloys, etc., should be
avoided;
usage of recyclable materials should be maximized;
ferromagnetic, current conductive and insulating materials should not age
and, when possible, their performance should improve with time;
as many dimensions and shapes as possible should be standardized;
opportunities for using old or rebuilt parts in new machines should be
considered.
An ideal machine for recycling is a machine with ferrite PMs and sintered
powder (SMC) magnetic circuit and slotless winding. After crushing powder
materials, conductors can be separated and reused. Design and manufacture
of a recyclable electric machine is economically justified if costs of the final
product do not significantly exceed the costs of a similar non-recyclable machine.
Summary
A hysteresis loop shows the relationship between the induced magnetic flux
density B and the magnetic field strength H. In the saturation region all
domains are fully aligned under the external magnetic field. The magnetization
curve is the locus of the tips for a family of hysteresis loops measured for
64
Modern Permanent Magnet Electric Machines
DC currents. Soft magnetic materials have a narrow hysteresis loop. Hard
magnetic materials have a large area hysteresis loop.
Soft magnetic materials used in construction of ferromagnetic cores for
electrical machines and electromagnetic devices can be classified as laminated
silicon steels, high-saturation cobalt alloys, amorphous ferromagnetic alloys,
soft magnetic composites (SMC), permalloys, nanocrystalline composite, or
solid ferromagnetic steels.
Core losses consist of hysteresis losses, eddy-current losses and excess
losses. Additional losses are due to metallurgical and manufacturing processes.
The use of a ferromagnetic core can increase the strength of the magnetic
field in a coil by a factor of several hundred times what it would be without
the core. Ferromagnetic cores in electromagnetic devices and machines are
necessary to guide the magnetic flux in the desired direction and to reduce
the reluctance for the magnetic flux.
In most electrical machines, laminated silicon cores are used. In special machines like high-speed machines, laminated cobalt alloy cores are used with saturation magnetic flux density 2.39 T. 3D cores of such machines, like axial flux
machines, transfer flux machines (TFM), claw pole machines are fabricated of
soft magnetic composites (SMC) or magnetic powder materials. Amorphous
materials with structure similar to glass allow for substantial reduction of core
losses. Permalloys (Ni-Fe) have a hysteresis loop close to rectangular and can
operate at frequencies at least up to 100 kHz. Nanocrystalline materials and
ferrites can work correctly at frequencies of 1 to 500 MHz.
Ferromagnetic cores for electrical machines and electromagnetic devices
are shown in Figs 2.19 to 2.31.
Future electrical machines should be manufactured as recyclable machines.
An ideal machine for recycling is a machine with an SMC magnetic circuit.
3
PERMANENT MAGNETS
3.1 Early history of permanent magnets (PM)
Approximately 2,600 years ago (600 BC), according to a legend, a shepherd
named Magnes who lived in Magnesia near Mount Ida in Greece found that
the nails and buckle of his sandals were attracted to the rock he was standing on. Greek philosopher Thales of Miletus, ca. 600 years BC, named these
rocks lodestones. Lodestones contain magnetite, the natural magnetic material
Fe3 O4 .
An early compass was invented in China probably 400 years BC (spoon of
magnetic lodestone on a plate of bronze). The spoon or ladle is of magnetic
lodestone, and the plate is of bronze, i.e., non-ferromagnetic metal (Fig. 3.1a).
The circular center represents Heaven, and the square plate represents Earth.
The handle of the spoon representing the Great Bear, points South. This
compass was invented as a divination tool by Chinese fortune-tellers.
Fig. 3.1b shows a simple mariner’s compass. A magnetized needle pointing
North and South floats in a bowl of water with edge markings. By the time of
the Tang dynasty (7th and 8th centuries AD), Chinese scholars had devised
a way to magnetize iron needles, by rubbing them with magnetite, and then
suspending them in water.
The Chinese provide the first documented use of suspended lodestones
used as a compass. In 1088 AD, Shen Kuo described the magnetic needle
compass. The first recorded use was documented by Zheng He of the Yunnan
province between the years 1405 and 1433 AD. Zheng He led the largest ships
in the world on seven voyages of exploration to the lands around the Indian
Ocean.
In approximately 1180 AD, Englishman Alexander Neckam records the
earliest European understanding of the magnet as a guide to seamen, the
early compass.
Around 1200 AD there are references in a French poem written by Guyot
de Provins to a touched needle of iron supported by a floating straw.
66
Modern Permanent Magnet Electric Machines
Fig. 3.1. Early compasses invented in China: (a) the spoon of magnetic lodestone
on the plate of bronze; (b) magnetized needle floating in the bowl of water.
Fig. 3.2. Book De Magnete (On the Magnet) by W. Gilbert [35]: (a) cover of original
book in Latin published in 1600; (b) cover of English translation published in 1893.
In 1600, English scientist William Gilbert described how to arm loadstone
with soft iron pole tips to increase attractive force and concluded that the
Earth was a magnet. He published the book De Magnete (On the Magnet) in
1600 in Latin (Fig. 3.2) [35].
Hans Christian Oersted, a Danish physicist and chemist, discovered and
demonstrated experimentally in 1820 that electricity and magnetism are
linked. In the experiment he passed electric current through a wire, which
caused a nearby magnetic compass needle to deflect.
Permanent Magnets
67
Extending Oersted’s experimental work, André-Marie Ampère in 1820
made the revolutionary discovery that a wire carrying electric current can
attract or repel another wire next to it, which also carries electric current.
No PMs are necessary for the effect to be seen. The so-called Ampère’s force
law was discovered in 1823. Ampère’s force law for parallel currents can be
regarded as an analog of Coulomb’s law for charges. The force per unit length
L between two current elements I1 and I2 separated by a distance r is given
by the equation
µ0 I1 I2
dF
= −2
(3.1)
dL
4π r
Ampère’s force law is a consequence of the Lorentz force on the moving charge,
as given by eqn (1.49). Another important contribution to electromagnetism
was Ampère’s circuital law
I
H · dl = Ienc
(3.2)
l
where Ienc is the current enclosed by the closed loop l.
Hans Christian Oersted was born in Rudkoebing, Denmark, on 14
August 1777. Both Hans and his brother were largely self-educated and in
1793 they went to Copenhagen to study at the University of Copenhagen.
In 1796 Oersted received honors for his papers in both aesthetics and
physics. He conducted research studies and wrote a thesis entitled “The
Architectonics of Natural Metaphysics,” thus receiving his Doctor degree.
On April 21, 1820, during a lecture, he noticed a deflection in the compass
needle when he switched on and off an electric current from a battery.
Initially, he interpreted that magnetic effects radiate from all sides of a wire
carrying an electric current, just like the case of light and heat. After indepth research he found that electric current produces a circular magnetic
field while flowing through a wire. His contributions were not confined just
to physics, but to other related domains like chemistry as well. In 1825,
Oersted extracted aluminum from aluminum chloride, thus becoming the
first person to isolate the metal. Oersted died on 9 March 1851 and was
buried in the Assistens Cemetery, Copenhagen.
3.2 Earth’s magnetic field
A compass works the way it does because Earth has a magnetic field. Earth’s
magnetic field extends from the Earth’s interior to where it meets the solar
wind, i.e., a stream of charged particles emanating from the Sun. Its magnetic
flux density at the Earth’s surface ranges from 25 to 65 µT (0.25 to 0.65 Gs).
It is the field of a magnetic dipole currently tilted at an angle of 11.5◦ with
respect to Earth’s rotational axis (Fig. 3.3). Earth’s magnetic field changes
68
Modern Permanent Magnet Electric Machines
over time because it is generated by a geodynamo (in Earth’s case, the motion
of molten iron alloys in its outer core).
Fig. 3.3. Magnetic field of the Earth.
André-Marie Ampère, born January 20, 1775 was a French physicist,
natural philosopher, and mathematician who is best known for his important contributions to the study of electrodynamics. He invented the astatic
needle, a critical component of the modern astatic galvanometer, and was
the first to demonstrate that a magnetic field is generated when two parallel wires are charged with electricity. He is generally credited as one of
the first to discover electromagnetism. Even without any formal education
Ampère began a career as a science teacher. After teaching for a while in
Lyon, he accepted positions at institutions of higher learning including
the College of France and the Polytechnic School at Paris, where he was a
professor of mathematics. In the early 1820s, after learning about the electromagnetism experiments of H. C. Oersted, Ampère began to formulate
a combined theory of electricity and magnetism. His work confirmed and
validated the discoveries of Oersted. Ampère’s most significant scholarly
paper on the subject of electricity and magnetism, entitled Memoir on
the Mathematical Theory of Electrodynamic Phenomena, was published
in 1826.
Ampère was elected to the prestigious National Institute of Sciences in
1814, and was awarded a chair at the University of France in 1826. There
he taught electrodynamics and remained a member of the faculty until his
death. Ampère died June 10, 1836 in Marseilles, France, and was buried
in the Montmartre Cemetery in Paris. The ampere, the unit for measuring
electric current, was named in honor of Ampère.
Permanent Magnets
69
3.3 What is a permanent magnet (PM)?
Popular scientific literature defines a permanent magnet (PM) as an object
that produces a magnetic field and has the property of attracting any ferromagnetic material.
In electrical machine textbooks, a PM (hard magnetic material) is defined
as an object that can produce a magnetic field in the air gap of the magnetic
circuit with no field excitation winding and no dissipation of electric power.
The energy of a PM in the external space only exists if the reluctance of
the external magnetic circuit is greater than zero. If a previously magnetized
PM is placed inside a closed ideal ferromagnetic circuit, i.e., toroidal, this PM
does not show any magnetic properties in the external space.
No external energy is necessary to maintain the magnetic field. External
energy must be involved only in changing the energy of the magnetic field.
Because change in the energy of the magnetic field requires the delivery of
external energy, it is not possible to build a free-energy generator or free
energy motor (perpetuum mobile) using PMs.
3.4 Hysteresis loop, demagnetization curve, recoil line,
magnetic energy density and intrinsic magnetization
The hysteresis loop of ferromagnetic materials can be measured using toroidal
samples in a circuit as shown in Fig. 1.12. The demagnetization curve is a
part of the hysteresis loop in the second quadrant (Fig. 3.4). There are two
characteristic points: remanent magnet flux density Br or remanence and
coercive field strength Hc or coercivity.
Fig. 3.4. Characteristics of a PM (a) hysteresis loop; (b) demagnetization curve
and recoil line.
70
Modern Permanent Magnet Electric Machines
Remanent magnetic flux density Br , is the magnetic flux density corresponding to zero magnetic field intensity. High remanence means the magnet can
support higher magnetic flux density in the air gap of the magnetic circuit.
Coercive field strength Hc , is the value of demagnetizing field intensity necessary to bring the magnetic flux density to zero in a material previously magnetized (in a symmetrically cyclically magnetized condition). High coercivity
means that a thinner magnet can be used to withstand the demagnetization
field.
Fig. 3.5. Magnetization curves for: (a) soft magnetic materials; (b) hard magnetic
materials (permanent magnets).
PMs can be described by the following equation (Fig. 3.5)
B = µ0 (H + M) + Br = µ0 µr H + Br
(3.3)
while soft magnetic materials are described by eqn (1.8).
The intrinsic demagnetization curve is the portion of the Bi = f (H) hysteresis loop located in the upper left-hand quadrant, where Bi = B − µ0 H.
For H = 0 the intrinsic magnetic flux density Bi = Br .
The general relationship between the magnetic flux density B, intrinsic
magnetization (polarization) Bi due to presence of ferromagnetic material,
and magnetic field intensity H may be expressed as
B = µ0 H+Bi = µ0 (H+M )+Br = µ0 H+µ0 χH+Br = µ0 (1+χ)H+Br (3.4)
where M = χH is the magnetization vector, χ is the magnetic susceptibility,
µ0 is the magnetic permeability of free space, and µr = 1 + χ is the relative
magnetic permeability. From eqn (3.4)
Permanent Magnets
Bi = µ0 M + Br = B − µ0 H
71
(3.5)
Intrinsic coercivity Hci is the magnetic field strength required to bring to
zero the intrinsic magnetic flux density Bi of a magnetic material described
by the Bi = f (H) curve. For PM materials Hci > Hc as shown in Fig. 3.6.
Fig. 3.6. Normal demagnetization curve B(H) and intrinsic demagnetization curve
Bi (H).
If a reverse magnetic field intensity is applied to a previously magnetized,
say, toroidal specimen, the magnetic flux density drops down to the magnitude
determined by the point K (Fig. 3.7). When the reversal magnetic flux density
is removed, the flux density returns to the point L according to a minor
hysteresis loop, the so-called recoil loop. Because the recoil loop is narrow, it
can be approximated with a straight line called the recoil line (Fig. 3.4b and
3.7).
Recoil magnetic permeability µrec is the ratio of the magnetic flux density
to magnetic field intensity at any point on the demagnetization curve (Fig.
3.7), i.e.,
∆B
(3.6)
∆H
where the relative recoil permeability µrrec = 1 . . . 3.5.
Maximum magnetic energy per unit produced by a PM in the external
space is equal to the maximum magnetic energy density per volume (Fig.
3.7), i.e.,
µrec = µ0 µrrec =
wmax =
(BH)max
2
J/m
3
(3.7)
72
Modern Permanent Magnet Electric Machines
Fig. 3.7. Demagnetization curve, recoil loop, recoil line, recoil magnetic permeability and energy of a PM.
where the product (BH)max corresponds to the maximum energy density
point on the demagnetization curve with coordinates Bmax and Hmax (Fig.
3.7).
3.5 Temperature coefficients and Curie temperature
Demagnetization curves are sensitive to the temperature. Both Br and Hc
decrease as the magnet temperature increases (Fig. 3.8), i.e.,
h
i
αB
Br = Br20 1 +
(ϑP M − 20)
(3.8)
100
i
h
αH
Hc = Hc20 1 +
(ϑP M − 20)
(3.9)
100
where ϑP M is the temperature of the PM, Br20 and Hc20 are the remanent
magnetic flux density and coercive force at 20◦ C, and αB < 0 and αH < 0
are temperature coefficients for Br and Hc in %/◦ C, respectively (Table 3.1).
For example, for sintered NdFeB magnets αB = −0.09 to −0.15 %/◦ C and
αH = −0.40 to –0.80 %/◦ C. Variation of demagnetization curves B(H) and
Bi (H) with temperature for sintered NdFeB PMs is plotted in Fig. 3.8.
Curie temperature (TC ), or Curie point, is the temperature above which
certain materials lose their permanent magnetic properties, which can (in most
cases) be replaced by induced magnetism. Curie temperature is named after
Pierre Curie, who showed that magnetism was lost at a critical temperature.
Alignment of spins below and above Curie temperature is shown schematically
in Fig. 3.9.
Permanent Magnets
73
Table 3.1. Temperature coefficients and Curie temperature for common PM materials according to Arnold Magnetic Technologies, Rochester, NY, U.S.A.
Material
Alnico 5
Alnico 8
Ferrite 8
Plastiform 2401
Ferrite-Neo hybrid
Sm2 Co17
SmCo5
Bonded NdFeB
MQP-C (15% Co)
Sintered NdFeB
318 kJ/m3 (0% Co)
Reversible temperature Reversible temperature
Curie
coefficient for Br
coefficient for Hci
temperature
◦
%/◦ C
%/◦ C
C
−0.02
−0.01
900
−0.02
−0.01
860
−0.20
+0.27
450
−0.14
−0.03
−0.045
−0.04
−0.20
−0.40
−
800
700
−0.07
−0.40
470
−0.10
−0.60
310
Fig. 3.8. Comparison of B(H) and Bi (H) demagnetization curves and their variation with temperature for sintered N48M NdFeB PM. Courtesy of ShinEtsu, Japan.
Fig. 3.9. Alignment of spins below and above Curie temperature: (a) below Curie
temperature, neighboring spins align parallel to each other in the absence of an
applied magnetic field; (b) above Curie temperature, spins are randomly aligned
unless a magnetic field is applied.
74
Modern Permanent Magnet Electric Machines
3.6 PM materials used in construction of electrical
machines
There are three classes of PMs currently used for electrical machines:
ˆ
ˆ
ˆ
alnicos (Al, Ni, Co, Fe);
ceramics (ferrites), e.g., barium ferrite BaO×6Fe2 O3 and strontium ferrite
SrO×6Fe2 O3 ;
rare-earth materials, i.e., samarium-cobalt SmCo and neodymium-ironboron NdFeB.
Comparison of demagnetization curves of the above PM materials are given
in Fig. 3.10.
Fig. 3.10. Demagnetization curves for different permanent magnet materials.
Maximum energy product (BH)max according to eqn (3.7) for different PMs is shown in Fig. 3.11. The greater the maximum energy product
(BH)max , the less the PM material (Fig. 3.12). Properties of typical PM
materials used in electrical machines are given in Table 3.2.
3.6.1 Alnico
Alnico magnets are made primarily from Al, Ni, Co, Cu, Fe and sometimes Ti
(Titanium). They can be either cast or sintered . The development of Alnico
began in 1931, when T. Mishima in Japan discovered that an alloy of Fe, Ni,
and Al had a coercivity of 32 kA/m, double that of the best magnet steels at
that time.
Permanent Magnets
75
Fig. 3.11. Development of PMs 1920-2000: maximum energy product (BH)max .
Fig. 3.12. The volume of various PM materials necessary to obtain approximately
the same magnetic flux density in the air gap.
Table 3.2. Properties of typical PM materials used in electrical machines
Property of
Ferrite Alnico
SmCo
NdFeB
material
Ceramic 8 Alloy Sm2 Co1 7
Sintered
Bonded
Br , T
0.4
1.25 1.0 to 1.1 0.55 to 0.70 1.25 to 1.35
Hc , kA/m
270
55
600 to 800 180 to 450 950 to 1040
Hci , kA/m
260
55 720 to 2000 210 to 1100 1200 to 1400
(BH)max , kJ/m3 25 to 32 < 40 190 to 240 32 to 88
290 to 400
αB , %/◦ C
−0.20
−0.02
−0.03
−0.105
−0.11
αH , %/◦ C
−0.27 −0.015
−0.15
−0.4
−0.65
Tc , ◦ C
460
890
800
360
330
76
Modern Permanent Magnet Electric Machines
Cast Alnico is melted and poured into a mold. Once solidified, the material
is rough ground, then heat-treated and cooled, sometimes within a magnetic
field. When treated in the presence of a magnetic field, the magnet is called
anisotropic (oriented). It means that anisotropic Alnico magnets show better
magnetic performance in one direction than in the other.
Sintered Alnico is made from a powdered mixture of ingredients that are
pressed into a die under high pressure (in the order of tons), sintered in a
hydrogen atmosphere and then cooled either within (anisotropic) or without
(isotropic) a magnetic field.
Alnico has high remanent magnetic flux density Br and low coercive force
(Fig. 3.10). Its demagnetization curve is very nonlinear. On the other hand
Alnico magnets can operate at high temperatures (Tc = 890◦ C) and its
temperature coefficients are low both for Br (αB = −0.02 %/◦ C) and Hc
(αH = −0.015 %/◦ C). Modern cast Alnico PMs manufactured by Arnold
Magnetic Technologies, Rochester, NY, USA are presented in Table 3.3.
Table 3.3. Cast Alnico permanent magnets. Arnold Magnetic Technologies
Property
Alnico 8B Alnico 8HE Alnico 8H Alnico 9
Remanent magnetic
flux density Br , T
0.83
0.93
0.74
1.12
Coercive force Hc , kA/m
131
123
151
109
Maximum energy product
(BH)max , kJ/m3
43.8
47.7
43.8
83.6
Recoil permeability µrrec
1989
1989
1989
1273
Magnetic flux density
at (BH)max , T
0.50
0.575
0.44
0.89
Electric conductivity
at 25◦ C, ×106 S/m
2.0
2.0
2.0
2.0
Specific mass
density, kg/m3
7250
7250
7250
7250
Tensile strength, MPa
205
205
205
55
3.6.2 Ferrites
Ferrite magnets or ceramic magnets are produced by calcining1 (between 1000
to 1350◦ C) a mixture of iron oxide (Fe2 O3 ) and strontium carbonate (SrCO3 )
or barium carbonate (BaCO3 ) to form a metallic oxide. In some grades, other
chemicals such as cobalt (Co) and lanthanum (La) are added to improve the
magnetic performance. This metallic oxide is then milled to a small particle
size (less than a 1.0 mm in size, usually a few microns). Then the process has
two main production options depending on the type of magnet required.
1
heating to high temperatures in air or oxygen
Permanent Magnets
77
The first process is to press the dry fine powder in a die which results
in an isotropic magnet (e.g., ferrite C1 grade), which has better dimensional
tolerances. The hexagonal crystal structure is random allowing the magnet
to be magnetized in any direction afterwards. An external magnetic field can
also be applied to produce anisotropic magnets, e.g., ferrite C5 grade.
The second method involves mixing the fine powder with water to produce
a slurry which is then compacted in a die in the presence of an externally
applied magnetic field. The external magnetic field helps the hexagonal crystal
structure align more perfectly with the magnetic field, improving the magnetic
performance, e.g., ferrite C8 grade because the water in the slurry acts like a
lubricant. This results in an anisotropic ferrite magnet with stronger magnetic
properties, but it will possibly require additional machining stages to give the
final dimensions. Sometimes a wet extrusion is performed instead of wet die
pressing (to make arcs for example). The magnet is then cut to required size
after sintering.
There are two chemical varieties of ferrite magnet. Barium ferrite is
known by two chemical symbols: BaFe12 O19 or BaO.6Fe2 O3 (barium hexaferrite). Strontium ferrite is known also by two chemical symbols: SrFe12 O19
or SrO.6Fe2 O3 (strontium hexaferrite).
Yogoro Kato and Takeshi Takei of the Tokyo Institute of Technology synthesized the first ferrite compounds in 1930. This led to the founding of
TDK Corporation in 1935 to manufacture the material. Barium hexaferrite
(BaO.6Fe2 O3 ) was discovered in 1950 at the Philips Physics Laboratory. From
1952 it was marketed under the trade name Ferroxdure. In the 1960s, Philips
developed strontium hexaferrite (SrO.6Fe2 O3 ), with better properties than
barium hexaferrite. Barium and strontium hexaferrite dominate the market
due to their low costs. However, other materials have been found with improved properties: BaO.2(FeO).8(Fe2 O3 ) came in 1980 and Ba2 ZnFe18 O23
came in 1991.
Characteristics of high-performance FB series PMs manufactured by TDK
(formerly Tokyo Denki Kagaku Kogyo), Tokyo, Japan are shown in Table 3.4.
PM FB13B and FB14H has the addition of La and Co, while FB6N and FB6B
have no contents of La and Co.
3.6.3 Rare-earth magnets SmCo and NdFeB
Invented in the 1960s and introduced in the 1970s, SmCo magnets were the
first commercially available rare-earth PMs (Table 3.5). They offer excellent
temperature stability and a high resistance to demagnetization and corrosion.
The temperature coefficient of Br is −0.02 to −0.045%/◦ C and the temperature coefficient of Hc is −0.14 to −0.40%/◦ C . The maximum service
temperature is 350◦ C and the Curie temperature is up to 890◦ C.
The cost is the only drawback. Both Sm and Co are relatively expensive
elements due to their supply restrictions. SmCo magnets held their place as
78
Modern Permanent Magnet Electric Machines
Table 3.4. Barium ferrite permanent magnets. TDK Corporation, Tokyo, Japan
Property
FB13B
FB14H
FB6N
FB6B
Remanent magnetic
flux density Br , T
0.475 ± 0.01 0.47 ± 0.01 0.44 ± 0.01 0.42 ± 0.01
Coercive force Hc , kA/m
340 ± 20
355 ± 20 258.6 ± 12 302.4 ± 12
Intrinsic coercive force Hci , kA/m 380 ± 20
430 ± 20 262.6 ± 12 318.3 ± 12
Maximum energy product
(BH)max , kJ/m3
44.0 ± 1.6 43.1 ± 1.6 36.7 ± 1.6 33.4 ± 1.6
Recoil permeability µrrec
1.05 to 1.10
αB , %/◦ C
-0.18
αHci , %/◦ C
0.11 to 0.18
0.3 to 0.6
Specific mass
density, kg/m3
4900 to 5100
Tensile strength, MPa
35
Young’s modulus, GPa
200
Tc , ◦ C
733
the strongest magnets until their increasing production costs led engineers to
search for a cheaper alternative.
Remarkable progress with regard to lowering raw material costs has been
achieved with the discovery of a second generation of rare-earth magnets on
the basis of inexpensive neodymium (Nd). This new generation of rare-earth
PMs was announced by Sumitomo Special Metals, Japan, in 1983 at the 29th
Annual Conference of Magnetism and Magnetic Materials held in Pittsburgh,
PA, U.S.A. The Nd is a much more abundant rare-earth element than Sm.
NdFeB magnets, which are now produced in increasing quantities, have better
magnetic properties than those of SmCo, but unfortunately only at room
temperature. The demagnetization curves, especially the coercive force, are
strongly temperature dependent. The temperature coefficient of Br is −0.09
to −0.15%/◦ C and the temperature coefficient of Hc is −0.40 to −0.80%/◦ C.
The maximum service temperature is 2500 C and the Curie temperature is
3500 C. The NdFeB is also susceptible to corrosion. NdFeB magnets have great
potential for considerably improving the performance–to–cost ratio for many
applications. For this reason they have a major impact on the development
and application of PM apparatuses.
The latest grades of NdFeB have a higher Br and better thermal stability
(Table 3.6). Metallic or resin coatings are employed to improve resistance to
corrosion.
Nowadays, for the industrial production of rare-earth PMs the powder
metallurgical route is mainly used [71]. Neglecting some material specific parameters, this processing technology is, in general, the same for all rare-earth
magnet materials. The alloys are produced by vacuum induction melting or
by a calciothermic reduction of the oxides. The material is then size-reduced
Permanent Magnets
79
Table 3.5. Physical properties of Vacomax sintered Sm2 Co17 PM materials at room
temperature 200 C manufactured by Vacuumschmelze GmbH, Hanau, Germany
Vacomax
Vacomax
Vacomax
Property
240 HR
225 HR
240
Remanent flux density, Br , T
1.05 to 1.12 1.03 to 1.10 0.98 to 1.05
Coercivity, Hc , kA/m
600 to 730 720 to 820 580 to 720
Intrinsic coercivity, Hci , kA/m
640 to 800 1590 to 2070 640 to 800
(BH)max , kJ/m3
200 to 240 190 to 225 180 to 210
Relative recoil
magnetic permeability
1.22 to 1.39 1.06 to 1.34 1.16 to 1.34
Temperature coefficient αB
of Br at 20 to 1000 C, %/0 C
−0.030
Temperature coefficient αiH
of Hci at 20 to 1000 C, %/0 C
−0.15
−0.18
−0.15
Temperature coefficient αB
of Br at 20 to 1500 C, %/0 C
−0.035
Temperature coefficient αiH
of Hci at 20 to 1500 C, %/0 C
−0.16
−0.19
−0.16
Curie temperature, 0 C
approximately 800
Maximum continuous
service temperature, 0 C
300
350
300
Thermal conductivity, W/(m 0 C)
approximately 12
Specific mass density, ρP M , kg/m3
8400
Electric conductivity, ×106 S/m
1.18 to 1.33
Coefficient of thermal expansion
at 20 to 1000 C, ×10−6 /0 C
10
Young’s modulus, ×106 MPa
0.150
Bending stress, MPa
90 to 150
Vicker’s hardness
approximately 640
by crushing and milling to a single crystalline powder with particle sizes less
than 10 µm.
In order to obtain anisotropic PMs with the highest possible (BH)max
value, the powders are then aligned in an external magnetic field, pressed
and densified to nearly theoretical density by sintering. The most economical
method for mass production of simple shaped parts like blocks, rings or arc
segments of the mass in the range of a few grams up to about 100 g is a die
pressing of the powders in an approximate final shape. Larger parts or smaller
quantities can be produced from isostatically pressed bigger blocks by cutting
and slicing.
Sintering and the heat treatment that follows are done under vacuum or
under an inert gas atmosphere. Sintering temperatures are in the range of
1000 to 1200◦ C depending on the PM material with sintering times ranging
from 30 to 60 min. During annealing after sintering, the microstructure of
80
Modern Permanent Magnet Electric Machines
Table 3.6. Physical properties of Vacodym sintered NdFeB PM materials at room
temperature 200 C manufactured by Vacuumschmelze GmbH, Hanau, Germany
Vacodym
Vacodym
Vacodym
Property
633 HR
362 TP
633 AP
Remanent flux density, Br , T
1.29 to 1.35 1.25 to 1.30 1.22 to 1.26
Coercivity, Hc , kA/m
980 to 1040 950 to 1005 915 to 965
Intrinsic coercivity, Hci , kA/m
1275 to 1430 1195 to 1355 1355 to 1510
(BH)max , kJ/m3
315 to 350 295 to 325 280 to 305
Relative recoil
magnetic permeability
1.03 to 1.05
1.04 to 1.06
Temperature coefficient αB
of Br at 20 to 1000 C, %/0 C
−0.095
−0.115
−0.095
Temperature coefficient αiH
of Hci at 20 to 1000 C, %/0 C
−0.65
−0.72
−0.64
Temperature coefficient αB
of Br at 20 to 1500 C, %/0 C
−0.105
−0.130
−0.105
Temperature coefficient αiH
of Hci at 20 to 1500 C, %/0 C
−0.55
−0.61
−0.54
Curie temperature, 0 C
approximately 330
Maximum continuous
service temperature, 0 C
110
100
120
Thermal conductivity, W/(m 0 C)
approximately 9
Specific mass density, ρP M , kg/m3
7700
7600
7700
Electric conductivity, ×106 S/m
0.62 to 0.83
Coefficient of thermal expansion
at 20 to 1000 C, ×10−6 /0 C
5
Young’s modulus, ×106 MPa
0.150
Bending stress, MPa
270
Vicker’s hardness
approximately 570
the material is optimized, which increases the intrinsic coercivity Hci of the
magnets considerably. After machining to get dimensional tolerances, the last
step in the manufacturing process is magnetizing. The magnetization fields to
reach complete saturation are in the range of 1000 to 4000 kA/m, depending
on material composition.
Researchers at General Motors, MI, U.S.A., have developed a fabrication
method based on the melt-spinning casting system originally invented for
production of amorphous metal alloys. In this technology, a molten stream
of NdFeCoB material is first formed into ribbons 30 to 50-µm thick by rapid
quenching, then cold pressed, extruded and hot pressed into bulk. Hot pressing
and hot working are carried out while maintaining the fine grain to provide
high density close to 100% which eliminates the possibility of internal corrosion. The standard electro-deposited epoxy resin coating provides excellent
corrosion resistance.
Permanent Magnets
81
NdFeB magnets are mechanically very strong, not as brittle as SmCo. Surface treatment is required (nickel, aluminum chromate or polymer coatings).
Most PMs are assembled using adhesives. Adhesives with acid content
must not be used since they lead to rapid decomposition of the PM material.
NdFeB magnets cannot be used under the following conditions:
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
in an acidic, alkaline, or organic solvent (unless hermetically sealed inside
a can)
in water or oil (unless hermetically sealed)
in an electrically conductive liquid, such as electrolyte containing water
in a hydrogen-containing atmosphere, especially at elevated temperatures
since hydrogenation causes the magnet material to disintegrate
in corrosive gasses, such as Cl2 , NH3 , NOx , etc.
in the presence of radioactive rays (NdFeB magnets can be damaged by
radiation, mainly gamma ray)
Table 3.7. Super high energy density sintered NdFeB PM materials
Grade
Br , T
Hc , kA/m Hci , kA/m (BH)max , kJ/m3
HS-50AH 1.38 to 1.45 1042 to 1130 ≥ 1035
358 to 406
Hitachi
HS-47AH 1.35 to 1.42 1019 to 1106 ≥ 1115
342 to 390
HS-46CH 1.33 to 1.40 1003 to 1090 ≥ 1353
334 to 374
N50
1.38 to 1.43
≥ 820
≥ 875
366 to 405
ShinEtsu
N48M 1.35 to 1.40
≥ 995
≥ 1114
950 to 390
Vacodym
Vacuumschmelze 722 HR 1.42 to 1.47 835 to 915
> 875
380 to 415
Manufacturer
The best available sintered NdFeB magnet has Br = 1.45 T and Hc = 1100
kA/m (Table 3.7). Unfortunately, high energy and high remanent magnetic
flux density of sintered NdFeB magnets means low service temperature. Theoretical limit for NdFeB magnets is 508 kJ/m3 . Neomax Materials Co., Japan
has already produced NdFeB magnets with (BH)max = 467 kJ/m3 .
3.7 Nanocomposite magnets
Nanocomposite magnets, like many other composite materials, combine two
substances with complementary properties. Around 1990, Soviet researchers
led by Nikolay Manakov (Orenburg State University) and independently, German researchers Eckart Kneller and Reinhard Hawig (Ruhr University) proposed to place side by side two different magnetic materials: one having high
saturation magnetization and the other having a high coercive field (Fig. 3.13).
82
Modern Permanent Magnet Electric Machines
Fig. 3.13. Two different magnetic materials, one with high Br and the other with
high Hc can serve much better than either one can alone: (a) make particles; (b)
combine them; (c) consolidate [36].
This is the synergy law, which can be simply explained as 1 + 1 > 2. Two
different components can serve much better than either one can alone.
For a composite magnet to work, the high-saturation material can’t be any
more than 10 or 15 nanometers thick [36]. Otherwise, exchange forces will not
reach far enough into its interior.
In 1994 researchers calculated that using an iron-cobalt mixture for
the high saturation magnetization and stabilizing it with a samarium-ironnitrogen alloy could produce magnets with maximum energy products as
large as 1090 kJ/m3 (137 MGsOe)–more than twice the current record. In
addition, such magnets would require much smaller amounts of rare earth
elements and would better resist corrosion, a notable problem with today’s
high-performance PMs.
3.8 Shape of demagnetization curves of ferrite and rare
earth PMs
Demagnetization curve B–H of Alnico and ferrite magnets is strongly nonlinear (Fig. 3.14a), while demagnetization curve of NdFeB magnets at ambient
temperature is almost a straight line (Fig. 3.14b).
The most widely used approximation of demagnetization curves of Alnico
and ferrite PMs is the approximation by hyperbola, i.e.,
B = Br
Hc − H
Hc − a0 H
(3.10)
where
√
2 γ−1
Br
a0 =
=
Bsat
γ
γ=
(BH)max
Br Hc
(3.11)
The parameter γ is called the form factor of the demagnetization curve. Bsat is
the saturation magnetic flux density. For rare-earth PMs at room temperature
Permanent Magnets
83
Fig. 3.14. Demagnetization curves B–H of PMs at room temperature: (a) ferrites
(b) NdFeB.
20◦ C the approximation is simple(a0 = 0), i.e.,
H
B = Br 1 −
Hc
(3.12)
The position of the recoil line depends on the shape of the demagnetization
curve. For Alnico and ferrite PMs, the demagnetization curve and recoil line
are different (Fig. 3.15a). For rare-earth PMs the demagnetization curve is
almost a straight line and the recoil line coincides with the demagnetization
line (Fig. 3.15b).
Fig. 3.15. Position of the recoil line and demagnetization curves for: (a) Alnico and
ferrite PMs (nonlinear demagnetization curve); (b) rare earth PMs (almost linear
demagnetization curve). K is the point of intersection of the demagnetization curve
and recoil line corresponding to the operating point of the rare-earth PM.
84
Modern Permanent Magnet Electric Machines
3.9 Simplified method of finding the operating point of a
PM
The operating point on the demagnetization curve for rare-earth PMs can be
found using a simple graphical method (Fig. 3.16). The following assumptions
are to be made:
(a) demagnetization curve is a straight line;
(b) permeance Gg of the air gap g is linear;
(c) there is no leakage flux;
(d) there is no fringing flux;
(e) magnetic voltage drop in mild steel core is neglected.
Fig. 3.16. Simplified graphical method of finding the operating point K on the
demagnetization curve for rare-earth PMs.
The PM with its height 2hM is placed in a mild steel magnetic circuit with
the air gap g. The magnetic field strength Hg in the air gap, in PM HM and
coercivity Hc are, respectively,
Hg =
Bg
µ0
HM =
Bg
µ0 µrrec
Hc =
Br
µ0 µrrec
(3.13)
The magnetic voltage drop balance equation is
Hc hM ≈ HM hM + Hg g
(3.14)
Permanent Magnets
85
Thus, the air gap magnetic flux density
Bg ≈
Br
1 + µrrec g/hM
(3.15)
Eqn (3.15) gives quite good results for rare-earth PMs. It cannot be used for
Alnico and ferrite PMs. To obtain more accurate results, the finite element
method (FEM) should be used (Fig. 3.17).
Fig. 3.17. 2D FEM analysis of the magnetic circuit shown in Fig. 3.16: magnetic
flux lines; (b) magnetic flux density distribution.
3.10 Main flux and leakage flux
The total magnetic flux ΦM excited by a PM consists of the main (useful)
flux Φg , i.e., flux passing through the air gap, and the leakage flux ΦlM , i.e.,
the flux that omits the air gap (Fig. 3.17), according to the following equation
ΦM = Φg + ΦlM = σlM Φg
(3.16)
The coefficient of leakage flux is defined as
σlM =
ΦM
Φg + ΦlM
ΦlM
=
=1+
>1
Φg
Φg
Φg
(3.17)
Typical values of the coefficient of leakage flux for electrical machines are
σlM = 1.05 . . . 1.15 and depend on the construction of the magnetic circuit.
86
Modern Permanent Magnet Electric Machines
The resultant permeance Gt consists of the useful permeance Gg of the air
gap and the leakage permeance GlM of the PM, i.e.,
Gt = Gg + GlM = σlM Gg
(3.18)
The coefficient of leakage flux can be also defined with the aid of permeances,
i.e.,
σlM =
Gt
Gg + GlM
GlM
=
=1+
>1
Gg
Gg
Gg
(3.19)
3.11 B–H and Φ–F coordinate systems
The demagnetization curve of PMs can be drawn in two rectangular coordinate systems:
ˆ
ˆ
magnetic flux density B versus magnetic field intensity H (Fig. 3.18a);
magnetic flux Φ versus MMF F (Fig. 3.18b).
Fig. 3.18. Demagnetization curve in two coordinate systems: (a) B–H; (B) Φ–F .
The following equations show the transition from the B–H to the Φ–F coordinate system
Φ = BlM wM
F = HlM
Φr = Br lM wM
Fc = Hc lM
(3.20)
where lM is the length, wM is the width and hM is the height per pole of a
cubic PM.
Permanent Magnets
87
3.12 Operating point for PM magnetized outside the
machine
3.12.1 PM without pole shoes in open space
A PM previously magnetized is placed alone in an open space. The state of
the PM is characterized by the point K on the demagnetization curve (Fig.
3.19). The location of the point K is at the intersection of the demagnetization
curve with a straight line representing the permeance of the external magnetic
circuit (open space):
Gext =
ΦK
,
FK
tan αext =
ΦK /Φr
Fc
= Gext
FK /Fc
Φr
(3.21)
Fig. 3.19. Operating point for PM magnetized outside the machine: PM without
pole shoes in open space.
The permeance Gext corresponds to the Φ–F coordinate system and is referred
to as MMF at the ends of the PM. The magnetic energy per unit produced by
the PM in the external space is wK = BK HK /2. This energy is proportional
to the rectangle limited by the coordinate system and lines perpendicular to
the Φ and F coordinates projected from the point K.
The recoil line KGM is expressed by the internal permeance GM of the PM,
i.e.,
SM
wM lM
GM = µrec
= µrec
(3.22)
hM
hM
88
Modern Permanent Magnet Electric Machines
3.12.2 PM with pole shoes in open space
If the PM poles are furnished with mild steel pole shoes, the permeance of
the external space Gext increases to
GA =
ΦA
FA
tan αA = GA
Fc
Φr
(3.23)
The point which characterizes a new state of the PM in Fig. 3.20 moves along
the recoil line from the point K to the point A. The recoil line KGM is the
same as the internal permeance of the PM as given by eqn (3.22).
Fig. 3.20. Operating point for PM magnetized outside the machine: PM with mild
steel pole shoes in open space.
The point A is the intersection of the recoil line KGM and the straight
line OGA representing the permeance of the PM with pole shoes, given by
eqn (3.23). The energy produced by the PM in the external space decreases
as compared with the previous case, i.e., wA = BA HA /2.
3.12.3 PM inside an external magnetic circuit
The next stage is to place the PM in an external ferromagnetic circuit. The
resultant permeance of this system is
GP =
ΦP
,
FP
tan αP = GP
Fc
Φr
(3.24)
which meets the condition GP > GA > Gext .
For an external magnetic circuit without any electric circuit carrying the
armature current, the magnetic state of the PM is characterized by the point
P (Fig. 3.21), i.e., the intersection of the recoil line KGM and the permeance
line OGP .
Permanent Magnets
89
Fig. 3.21. Operating point for PM magnetized outside the machine: PM with mild
steel pole shoes placed in external magnetic circuit.
3.12.4 PM with a complete external armature system
The external armature system is complete, when the armature magnetic circuit is furnished with the winding. Demagnetizing action of the stator (armature) winding is when the excitation current is in such a direction that the
external armature magnetic field weakens the field of the PM. For this case
′
it is necessary to lay off the distance OFad
from the origin of the coordinate
′
system to the left (Fig. 3.22). The line GP drawn from the point Fad
with the
slope αP intersects the demagnetization curve at the point K ′ . This point
can be above or below the point K (for the PM alone in the open space). The
point K ′ is the origin of a new recoil line K ′ G′M . Now if the armature exciting current decreases, the operating point will move along the new recoil line
K ′ G′M to the right. If the armature current drops down to zero, the operating
point takes the position P ′ (intersection of the new recoil line K ′ G′M with the
permeance line GP drawn from the origin of the coordinate system).
Fig. 3.22. Operating point for PM magnetized outside the machine: PM with mild
steel pole shoes, external magnetic circuit and stator (armature) winding. Demagnetizing action of armature winding.
90
Modern Permanent Magnet Electric Machines
On the basis of Fig. 3.22, the energies wP ′ = BP ′ HP ′ /2, wP = BP HP /2,
and wP ′ < wP . The location of the origin of the recoil line, as well as the location of the operating point, determines the level of utilization of the energy
produced by the PM. A PM behaves in a different way than a DC electromagnet; the energy of a PM is not constant if the permeance and exciting current
of the external armature change.
The location of the origin of the recoil line is determined by the minimum
value of the permeance of the external magnetic circuit or the demagnetization
action of the external field. The MMF of the armature field acting directly
′
on the PM (in the d-axis) is −Fad
and can be determined on the basis of
the armature current, number of turns, number of poles and magnetic flux
leakage coefficient. In the general case, the maximum d-axis MMF of armature
reaction
Iamax N
(3.25)
Fadmax =
2p
where Iamax is the maximum armature current (at reversal or locked rotor),
N is the number of turns of the armature winding and 2p is the number of
poles. The MMF and magnetic field intensity acting directly on the PM
′
Fad
=
Fadmax
σlM
′
Had
=
′
Fad
hM
(3.26)
where σlM is according to eqn (3.17) or (3.19) and hM is the height of the
PM per pole.
When the armature winding is fed with a current that produces an MMF
magnetizing the PM, the magnetic flux in the PM increases to the value ΦN .
′
The d-axis MMF Fad
of the external (armature) field acting directly on the
PM corresponds to ΦN . The magnetic state of the PM is described by the
point N located on the recoil line on the right-hand side of the origin of the
coordinate system (Fig. 3.23). To obtain this point it is necessary to lay off
Fig. 3.23. Operating point for a PM magnetized outside the machine: PM with mild
steel pole shoes, external magnetic circuit and stator (armature) winding. Magnetizing action of armature winding.
Permanent Magnets
91
′
′
the distance OFad
and to draw a line GP from the point Fad
inclined by the
angle αP to the F -axis. The intersection of the recoil line and the permeance
line GP gives the point N . If the exciting current in the external armature
winding is increased further, the point N will move further along the recoil
line to the right, up to the saturation of the PM.
In analytical calculations, permeances for the main (useful) magnetic flux
and leakage magnetic fluxes are usually found by dividing the magnetic field
into simple solids (rectangular prisms, cylinders, one-half of a sphere, onequarter of a sphere, etc.).
To improve the properties of PMs independent of the external fields, PMs
are stabilized. Stabilization means the PM is demagnetized up to a value that
is slightly higher than the most dangerous demagnetization field during the
operation of a system where the PM is installed. In magnetic circuits with
stabilized PMs the operating point describing the state of the PM is located
on the recoil line.
3.13 Operating point for magnetization without
armature
The PM has been magnetized outside the armature system and has then been
placed in the armature system, e.g., the same as that for an electrical machine
with an air gap. The beginning of the recoil line is determined by the leakage
permeance Gext of the PM alone located in open space (Fig. 3.24). In order
to obtain the point K, the set of eqns (3.10), (3.21) in flux Φ–F coordinate
system is to be solved. This results in the following second-order equation:
2
a0 Gext FK
− (Gext Fc + Φr )FK + Φr Fc = 0
If a0 > 0, the MMF corresponding to the point K is
Φr
Fc
+
±
FK =
2a0
2a0 Gext
s
= b0 ±
Fc
Φr
+
2ao
2a0 Gext
2
−
Φr Fc
a0 Gext
q
b20 − c0
(3.27)
where
b0 =
Fc
Φr
+
2a0
2a0 Gext
and
c0 =
Φr Fc
a0 Gext
92
Modern Permanent Magnet Electric Machines
Fig. 3.24. Location of the operating point for the magnetization without the armature.
If a0 = 0 (for rare-earth PMs), the MMF FK is
FK =
Φr
Gext + Φr /Fc
(3.28)
The magnetic flux ΦK can be found on the basis of eqn (3.21).
The equation of the recoil line for a0 > 0 is
(i) in the B–H coordinate system
B = BK + (HK − H)µrec
(3.29)
(ii) in the flux Φ–F coordinate system
Φ = ΦK + (FK − F )µrec
SM
hM
(3.30)
For rare-earth PMs with a0 = 0, the recoil permeability µrec = (hM /SM )(Φr −
ΦK )/FK = Φr hM /(Fc SM ) and the equation of the recoil line is the same as
that for the demagnetization line, i.e.,
F
Φ = Φr 1 −
(3.31)
Fc
′
The d-axis armature MMF Fad
acting directly on the magnet usually demagnetizes the PM, so that the line of the resultant magnetic permeance,
Gt =
ΦM
′
FM − Fad
(3.32)
Permanent Magnets
93
intersects the recoil line between the point K and the magnetic flux axis.
Solving eqn (3.30) in which Φ = ΦM and F = FM , and eqn (3.32) the MMF
of the PM, is given by the equation
FM =
′
ΦK + FK µrec (SM /hM ) + Gt Fad
Gt + µrec (SM /hM )
(3.33)
For rare-earth PMs eqns (3.31) and (3.32) should be solved to obtain
FM =
′
Φr + Gt Fad
Gt + Φr /Fc
(3.34)
′
The magnetic flux ΦM = Gt (FM − Fad
) in the PM is according to eqn (3.32).
The useful flux density in the air gap can be found using the coefficient of
leakage flux (eqn 3.17) or (eqn 3.19), i.e.,
Bg =
=
Gt
Sg σlM
′
ΦM
Gt (FM − Fad
)
=
Sg σlM
Sg σlM
′
ΦK + FK µrec (SM /hM ) + Gt Fad
′
− Fad
Gt + µrec (SM /hM )
(3.35)
where Sg = lM wM is the surface of the air gap. With the fringing effect being
neglected, the corresponding magnetic field intensity is
Hg = HM =
′
FM
ΦK + FK µrec (SM /hM ) + Gt Fad
=
hM
hM [Gt + µrec (SM /hM )]
(3.36)
Similar graphical construction as in Fig. 3.24 is given in [32] for PM magnetized inside the armature system.
3.14 Mallinson–Halbach array
In 1972 J.C. Mallinson [64] of Ampex Corporation, Redwood City, CA, USA
discovered a magnetic curiosity of a PM configuration that concentrates magnetic flux on one side of the array and cancels it to near zero on the other (Fig.
3.25). Another interesting and unique quality of this configuration is that the
array of PMs is stronger than its individual components, i.e., a single PM,
because of superposition of field lines. The fundamental field is stronger by a
factor of 1.4 than in a conventional PM array.
This effect was rediscovered in the late 1970s by K. Halbach of Lawrence
Berkeley National Laboratory, Berkeley Hills, CA, USA, applied to particle
accelerators and expanded upon cylindrical configurations [38, 39, 40]. The
polarities of individual PMs in the array are arranged such that the magnetization vector rotates as a function of distance along the array.
A Halbach cylinder is a cylinder composed of rare earth PMs producing an
intense magnetic field confined entirely inside or outside the cylinder with zero
94
Modern Permanent Magnet Electric Machines
Fig. 3.25. 90-degree Mallinson–Halbach array. Magnetic curiosity of a PM configuration that concentrates magnetic flux on one side of the array and cancels it to
near zero on the other.
Fig. 3.26. Halbach cylinders: (a) λ = 2π, 2p = 2; (b) λ = π, 2p = 4; λ = 2π/3,
2p = 6.
field on the other cylindrical surface (Fig. 3.26). The magnetization vector can
be described as [38, 90]
M = Mr cos(pθ)1r + Mr sin(±pθ)1θ
(3.37)
where Mr is the remanent magnetization, 1r , 1θ are unit vectors in radial
and tangential directions, respectively, r, θ are cylindrical coordinates, p is
the number of pole pairs (wave number), the + sign is for internal field and
the − sign is for external field. The number of poles expressed with the aid of
wavelength (spatial period of the array) λ is
2π
(3.38)
λ
The uniform magnetic flux density inside the cylinder is described by the
following equation [38]
Dout
sin π/nM
B = Br ln
(3.39)
Din
π/nM
p=
Permanent Magnets
95
where Br is the remanent magnetic flux density, Dout is the outer diameter,
Din is the inner diameter and nM is the number of PM pieces per wavelength
λ. For λ = 2π the number of pole pairs is p = 2π/2π = 1, for λ = π the
number of pole pairs p = 2, and for λ = 2π/3 the number of poles p = 3
(Fig. 3.26). If the ratio of outer to inner diameters is greater than the base
of the natural logarithm e, the magnetic flux density inside the bore exceeds
the remanence flux density of the PM. Magnetic fields of over 5 T in a 2 mm
gap in a Mallinson–Halbach type PM dipole at room temperature has been
achieved [60]. The Mallinson–Halbach array has the following advantages:
ˆ
ˆ
ˆ
ˆ
the fundamental field is stronger by a factor of 1.4 than in a conventional
PM array, and thus the power efficiency of the machine is doubled;
the array of PMs does not require any backing steel magnetic circuit and
PMs can be bonded directly to a non-ferromagnetic supporting structure
(aluminum, plastics);
the magnetic field is more sinusoidal than that of a conventional PM array;
the Mallinson–Halbach array has very low back-side fields.
Summary
Magnetite, a natural magnetic material Fe3 O4 , was discovered approximately
2,600 years ago (600 BC), according to a legend, in Magnesia near Mount Ida
in Greece.
An early compass was invented in China probably 400 years BC (spoon of
magnetic lodestone on the plate of bronze).
Dutch scientist Hans Christian Oersted discovered the relationship between electricity and magnetism in 1820. French physicist Andre Ampere further expanded the discovery of Oersted in 1823.
The magnitude of the Earth’s magnetic field at the surface ranges from 25
to 65 µT (0.25 to 0.65 Gs). It is the field of a magnetic dipole currently tilted
at an angle of 11.5◦ with respect to Earth’s rotational axis.
A PM is an object that can produce a magnetic field in the air gap of the
magnetic circuit with no field excitation winding and no dissipation of electric
power. The energy of a PM in the external space only exists if the reluctance
of the external magnetic circuit is greater than zero.
The basis for the evaluation of a PM is the portion of its hysteresis loop
located in the upper left-hand quadrant, called the demagnetization curve.
The recoil loop (minor hysteresis loop) may usually be replaced with little
error by a straight line called the recoil line. This line has a slope called the
recoil magnetic permeability µrec = ∆B/∆H. The demagnetization curve and
recoil line are only the same for rare earth PMs.
The maximum magnetic energy per unit produced by a PM in the external
space is equal to the maximum energy density per volume. The maximum
96
Modern Permanent Magnet Electric Machines
magnetic energy density per volume produced by a PM is a good parameter
for comparison of different types and grades of PMs.
Demagnetization curves, especially in ferrite and NdFeB magnets, are sensitive to temperature. Both Br and Hc decrease as the magnet temperature
increases. Temperature coefficients for Br and Hc are expressed in %/◦ C.
There are three classes of PMs: Alnicos (Al, Ni, Co, Fe); ceramics (ferrites), e.g., barium ferrite BaO×6Fe2 O3 and strontium ferrite SrO×6Fe2 O3 ;
and rare-earth materials, i.e., samarium-cobalt SmCo and neodymium iron
boron NdFeB. The best available sintered NdFeB magnet has Br = 1.45 T
and Hc = 1100 kA/m. Nanocomposite magnets may replace rare earth magnets in the future. NdFeB magnets are very sensitive to hydrogen atmosphere.
In 1966 the first rare-earth magnets were developed from Samarium-Cobalt
(SmCo5 ) producing a high-energy product of 143 kJ/m3 . In 1972 further developments were made using Sm-Co (Sm2 Co17 ) to produce a higher-energy
magnet product of 238 kJ/m3 .
General Motors, Sumitomo Special Metals and the Chinese Academy of
Sciences developed Neodymium-Iron-Boron Nd2 Fe14 B (278 kJ/m3 ) in 1983.
The operating point of a PM can be found using graphical methods. The
position of the recoil line depends on the shape of the demagnetization curve.
For Alnico and ferrite PMs, the demagnetization curve and recoil line are
different. For rare-earth PMs, the demagnetization curve is almost a straight
line and the recoil line coincides with the demagnetization line.
The magnetic flux of a PM consists of the main flux and leakage flux. Only
the main flux is useful.
In analytical calculations, permeances for the main (useful) magnetic flux
and leakage magnetic fluxes are usually found by dividing the magnetic field
into simple solids (rectangular prisms, cylinders, one-half of a sphere, onequarter of a sphere, etc.).
Operating point of a PM can be determined analytically only if permeances
for air gap and leakage fluxes are accurately estimated. To obtain reliable
results, analytical calculations of PM circuits should be verified by using the
FEM.
The polarities of individual PMs in the Mallinson–Halbach array are arranged such that the magnetization vector rotates as a function of distance
along the array. The fundamental field is stronger by a factor of 1.4 than in a
conventional PM array.
4
CALCULATION OF MAGNETIC CIRCUITS
WITH PMs
4.1 Methods of calculation of magnetic circuits with
PMs
Magnetic circuits, which contain ferromagnetic materials, are nonlinear circuits. The following methods are used in engineering practice to analyze the
magnetic circuits with PMs (Fig. 4.1):
(a) graphical methods;
(b) analytical methods, i.e., analogy between electric and magnetic circuits;
(c) equivalent reluctance method (ERN);
(d) finite element method (FEM).
Graphical, analytical and ERM methods require calculation of permeances or
reluctances for the main and leakage magnetic fluxes.
Graphical methods have been presented in Sections 3.9, 3.12 and 3.13.
Fig. 4.1. Methods of analysis of magnetic circuits with PMs.
98
Modern Permanent Magnet Electric Machines
4.2 Permeance evaluation by dividing the magnetic field
into simple solids
The permeances of the simple solids shown in Fig. 4.2 can be found using the
following formulae:
(a)
(f)
lM
wM
c
lM
g
c
(b)
g
(g)
dM
g
g
(h)
(c)
lM
gav
g
g
2
2
g
c
(i)
(d)
g
c
lM
g
(j)
(e)
lM
c
wM
g
wM
g
g
2
2
c
Fig. 4.2. Simple solids: (a) rectangular prism, (b) cylinder, (c) half-cylinder, (d)
one-quarter of cylinder, (e) half-ring, (f) one-quarter of ring, (g) one-quarter of a
sphere, (h) one-eighth of a sphere, (i) one-quarter of a shell, (j) one-eighth of a shell.
Calculation of Magnetic Circuits with PMs
99
(a) Rectangular prism (Fig. 4.2a)
wM lM
g
(4.1)
πd2M
4g
(4.2)
G = 0.26µ0 lM
(4.3)
G = µ0
(b) Cylinder (Fig. 4.2b)
G = µ0
(c) Half-cylinder (Fig. 4.2c)
where gav = 1.22g and Sav = 0.322glM
(d) One-quarter of a cylinder (Fig. 4.2d)
G = 0.52µ0 lM
(4.4)
(e) Half-ring (Fig. 4.2e)
2lM
π(g/wM + 1)
(4.5)
2wM
lM
ln 1 +
π
g
(4.6)
G = µ0
For g < 3wM ,
G = µ0
(f) One-quarter of a ring (Fig. 4.2f)
2lM
π(g/c + 0.5)
(4.7)
c
2lM
ln 1 +
π
g
(4.8)
G = µ0
For g < 3c,
G = µ0
(g) One-quarter of a sphere (Fig. 4.2g)
G = 0.077µ0 g
(4.9)
(h) One-eighth of a sphere (Fig. 4.2h)
G = 0.308µ0 g
(4.10)
(i) One-quarter of a shell (Fig. 4.2i)
G = µ0
c
4
(4.11)
100
Modern Permanent Magnet Electric Machines
(j) One-eighth of a shell (Fig. 4.2j)
G = µ0
c
2
(4.12)
Fig. 4.3 shows a model of a flat electrical machine with a smooth armature
core (without slots) and a salient-pole PM excitation system. The armature
is in the form of a bar made of steel laminations. The PMs are fixed to the
mild steel rail (yoke).
The pole pitch is τ , the width of each PM is wM , and its length is lM . The
space between the pole face and the armature core is divided into a prism
eqn (4.1), four quarters of a cylinder eqn (4.4), four quarters of a ring eqn
(4.7), four pieces of 1/8 of a sphere eqn (4.10), and four pieces of 1/8 of a
shell eqn (4.12). Formulae for the permeance calculations are found on the
assumption that the permeance of a solid is equal to its average cross-section
area divided by the average length of the flux line. Neglecting the fringing
flux, the permeance of a rectangular air gap per pole (prism 1 in Fig. 4.2) is
Gg1 = µ0
wM lM
g′
(4.13)
Fig. 4.3. Flat model of a simple PM machine and division of the space occupied
by the magnetic field into simple solids: (a) longitudinal section, (b) air gap field,
(c) leakage field between PM and rotor core.
The equivalent air gap g ′ is only equal to the nonferromagnetic gap (mechanical clearance) g for a slotless and unsaturated armature. To take into account
slots (if they exist) and magnetic saturation, the air gap g is increased to
g ′ = gkC ksat , where kC > 1 is Carter’s coefficient taking into account slots,
and ksat > 1 is the saturation factor of the magnetic circuit defined as the
ratio of the MMF per pole pair to the air gap magnetic voltage drop taken
twice, i.e.,
ksat = 1 +
2(V1t + V2t ) + V1c + V2c
2Vg
(4.14)
Calculation of Magnetic Circuits with PMs
101
where Vg is the magnetic voltage drop (MVD) across the air gap, V1t is the
MVD along the armature teeth (if they exist), V2t is the MVD along the PM
pole shoe teeth (if there is a pole shoe and cage winding), V1c is the MVD
along the armature core (yoke), and V2c is the MVD along the excitation
system core (yoke).
To take into account the fringing flux, it is necessary to include all paths
for the magnetic flux coming from the excitation system through the air gap
to the armature system (Fig. 4.3), i.e.,
Gg = Gg1 + 4(Gg2 + Gg3 + Gg4 + Gg5 )
(4.15)
where Gg1 is the air gap permeance according to eqn (4.1) and Gg2 to Gg5
are the air gap permeances for fringing fluxes. The permeances Gg2 to Gg5
can be found using eqns (4.4), (4.7), (4.10), and (4.12).
In a similar way, the resultant permeance for the leakage flux of the PM
can be found, i.e.,
GlM = 4(Gl6 + Gl7 )
(4.16)
where Gl6 (one-quarter of a cylinder) and Gl7 (one-eighth of a sphere) are the
permeances for leakage fluxes between the PM and rotor yoke according to
Fig. 4.3c, and eqns (4.4) and (4.10).
In the case of simple-shaped PMs, the permeance for leakage fluxes of a
PM alone (in open space) can be found as:
Gext = µ0
2π SM
Mb hM
(4.17)
where Mb is the ballistic coefficient of demagnetization. This coefficient can
be estimated with the aid of graphs as shown in Fig. 4.4 [9]. The cross-section
area is SM = πd2M /4 for a cylindrical PM, and SM = wM lM for a rectangular
PM. In the case of hollow cylinders (rings), the coefficient Mb is practically
the same as that for solid cylinders. For cylindrical PMs with small height hM
and large cross sections πd2M /4 (button-shaped PMs), the leakage permeance
can be calculated using the following equation [9]:
Gext ≈ 0.716µo
d2M
hM
(4.18)
Eqns (4.17) and (4.18) can be used for finding the origin K of the recoil line
for PMs magnetized without an armature (Fig. 3.24).
102
Modern Permanent Magnet Electric Machines
Fig. 4.4. Ballistic coefficient of demagnetization.
4.3 Graphical methods
Graphical methods are described in Sections 3.9, 3.12 and 3.13. These methods
are based on the demagnetization curve, recoil line and graphical construction
of the operating point of the PM.
4.4 Analytical approach to calculation of magnetic
circuits with PMs
Fig. 4.5 shows the equivalent magnetic circuit of a PM system with armature.
The reluctances of pole shoes (mild steel) and the armature stack (electrotechnical laminated steel) are much smaller than those of the air gap and PM and
have been neglected. The “open circuit” MMF acting along the internal magnet permeance GM = 1/RµM is FM 0 = HM 0 hM (Fig. 3.22). For a linear
demagnetization curve HM 0 = Hc . The d-axis armature reaction MMF is
Fad , the total magnetic flux of the permanent magnet is ΦM , the leakage flux
of the PM is ΦlM , the useful air gap magnetic flux is Φg , the leakage flux of
the external armature system is Φla , the flux produced by the armature is
Φad (demagnetizing or magnetizing), the reluctance for the PM leakage flux
is RµlM = 1/GlM , the air gap reluctance is Rµg = 1/Gg , and the external armature leakage reactance is Rµla = 1/Ggla . The following Kirchhoff equations
can be written on the basis of the equivalent circuit shown in Fig. 4.5:
ΦM = ΦlM + Φg
Calculation of Magnetic Circuits with PMs
Φla =
103
±Fad
Rµla
FM 0 − ΦM RµM − ΦlM RµlM = 0
ΦlM RlM − Φg Rµg ∓ Fad = 0
Fig. 4.5. Equivalent circuit of a PM system with armature.
The solution to the above equation system gives the air gap magnetic flux:
Gg
(Gg + GlM )(GM + GlM )
Gg GM
Φg = FM o ∓ Fad
Gg + GlM
Gg GM
Gg + GlM + GM
or
Gg GM
′ Gt (GM + GlM )
Φg = FM 0 ∓ Fad
Gg GM
Gt + GM
(4.19)
where the total resultant permeance Gt for the flux of the PM is according to
eqn (3.32) and the direct-axis armature MMF acting directly on the PM is
′
Fad
= Fad
−1
Gg
Fad
GlM
= Fad 1 +
=
Gg + GlM
Gg
σlM
(4.20)
The upper sign in eqn (4.19) is for the demagnetizing armature flux and the
lower sign is for the magnetizing armature flux.
The general expressions for the coefficient of the PM leakage flux σlM are
given by eqns (3.17) and (3.19).
104
Modern Permanent Magnet Electric Machines
4.5 Calculation of magnetic circuits with PMs using an
equivalent reluctance network
The equivalent reluctance network (ERN) shown in Fig. 4.6 has been created
on the basis of the following assumptions:
(a) A symmetry axis exists every 180◦ electrical degrees (one pole pitch).
(b) The magnetic flux density, magnetic field intensity and relative magnetic
permeability in every point of each ferromagnetic portion of the magnetic
circuit (PMs, cores, teeth) is constant.
(c) The air gap leakage flux is only between the heads of teeth.
(d) The magnetic flux of the armature (primary unit) penetrates only through
the teeth and core (yoke).
(e) The equivalent reluctance of teeth per pole pitch is ℜt /Q1 , where ℜt is
the reluctance of a single tooth and Q1 is the number of teeth (slots) per
pole.
Fig. 4.6. Equivalent reluctance network of a portion of a PM brushless motor with
surface PMs. Symbols are described in the text.
Calculation of Magnetic Circuits with PMs
105
Each portion of the magnetic circuit is replaced by equivalent reluctances:
ˆ
reluctance of the PM
hM
µ0 µrrec wM LM
(4.21)
gkC
gkC
=
µ0 wM lM
µ0 αi τ LM
(4.22)
ht
µ0 µrt ct Li ki
(4.23)
ℜM =
ˆ
reluctance of the air gap
ℜg =
ˆ
reluctance of a single tooth
ℜt =
ˆ
reluctance per pole pitch of the armature core (yoke)
ℜ1c ≈
ˆ
(4.24)
reluctance per pole pitch of the yoke with the PMs
ℜ2c ≈
ˆ
τ + h1c
µ0 µr1c h1c Li ki
τ + h2c
µ0 µr2c h2c LM
(4.25)
reluctance for the PM leakage flux
ℜlM =
1
GlM
(4.26)
in which
GlM ≈ 2µ0 (0.52lM + 0.26wM + 0.308hM )
GlM ≈ 2µ0
ˆ
hM lM
+ 0.26wM + 0.308hM
xM
if
hM ≤ xM
(4.27)
if
hM > xM
(4.28)
reluctance for the air gap leakage flux
ℜlg ≈
1 5 + 4gkC /b14 1
µ0 5gkC /b14 Li
(4.29)
In the foregoing equations (4.21) to (4.29), µrrec is the relative recoil magnetic
permeability of the PM, µrt is the relative magnetic permeability of the armature tooth, µr1c is the relative magnetic permeability of the armature core
(yoke), µr2c is the relative magnetic permeability of the reaction rail (core),
hM is the height of the PM per pole, wM is the width of the PM, lM is the
106
Modern Permanent Magnet Electric Machines
length of the PM in the direction perpendicular to the plane of the traveling
field, h1c is the height of the armature core (yoke), h2c is the height of the
reaction rail core, b14 is the armature slot opening, kC is Carter’s coefficient,
τ is the pole pitch, Li is the effective length of the armature stack (in the
direction perpendicular to laminations), and xM is the distance between adjacent PMs. The magnetic flux Φf is excited by the MMF FM = Hc hM of
the PMs, the armature reaction MMF Fad is the EMF acting directly on the
PM, Φf is the PM excitation flux (flux at no load), Φg is the air gap magnetic
flux, and Φ is the magnetic flux linked with the primary winding (air gap flux
Φg reduced by the air gap leakage flux Φlg , if included).
Reluctances for leakage fluxes ℜlM , according to eqns (4.26), (4.27), and
(4.28), have been calculated by dividing the magnetic field into simple solids.
The reluctance (4.27) is a parallel connection of the reluctances of two onequarters of a cylinder (4.4), two half-cylinders (4.3), and four one-quarters of
a sphere (4.9). The reluctance (4.28) is a parallel connection of the reluctances
of two prisms (4.1), two half-cylinders (4.3), and four one-quarters of a sphere
(4.9).
The following Kirchhoff equations can be written for the magnetic circuit
presented in Fig. 4.6:
1
1
ℜt
+ Φℜ1c − Φlg ℜlg = −2Fad
Q1
2
2
(4.30)
Φg = Φ + Φlg
(4.31)
Φf = Φg + ΦlM
(4.32)
1
1
2Φf ℜM + ΦlM ℜlM + Φf ℜ2c = 2FM
2
2
(4.33)
2Φ
2(FM − Fad ) = 2Φf ℜM + 2Φg ℜg + 2Φ
1
1
ℜt
+ Φℜ1c + Φf ℜ2c
Q1
2
2
(4.34)
The solution to these equations gives magnetic fluxes in the following form,
ˆ
magnetic flux excited by PMs
ˆ
2(FM − Fad − 4Fad ℜg /ℜlg )A + 2(FM + Fad ℜlM /ℜlg )(2/ℜlM )B
AC + BD
(4.35)
magnetic flux linked with the armature winding
Φf =
Φ=
2(FM − Fad − 4Fad ℜg /ℜlg )D − 2(FM + Fad ℜlM /ℜlg )(2/ℜlM )C
AC + BD
(4.36)
Calculation of Magnetic Circuits with PMs
ˆ
107
air gap magnetic flux
Φg = ΦA +
4Fad
ℜlg
(4.37)
The leakage fluxes result from eqns (4.31) and (4.32), i.e.,
Φlg = Φg − Φ
(4.38)
ΦlM = Φf − Φg
(4.39)
In the above eqns (4.35) to (4.37),
1
A=1+
ℜlg
ℜt
4
+ ℜ1c
Q1
B = 2Aℜg + 2
ℜt
1
+ ℜ1c
Q1
2
1
C = 2ℜM + ℜ2c
2
D =1+4
(4.40)
(4.41)
(4.42)
ℜM
ℜ2c
+
ℜlM
ℜlM
(4.43)
ΦlM
Φf
(4.44)
ΦlM + Φlg
Φf
(4.45)
The coefficient of PMs leakage flux
σlM = 1 +
The coefficient of total leakage flux
σl ≈ 1 +
If ℜlg → ∞ and ℜlM → ∞, then A → 1. It means that leakage fluxes Φlg → 0
and ΦlM → 0. Thus, the magnetic flux
Φf = Φg = Φ =
2(FM − Fad )
ℜt
2 ℜM + ℜg + Q
+ 0.5(ℜ1c + ℜ2c )
1
(4.46)
4.6 Calculation of magnetic circuits with PMs using the
FEM
The finite element method (FEM) is a numerical method for solving problems
of engineering and mathematical physics. The procedure involves
108
Modern Permanent Magnet Electric Machines
1. functional minimization that satisfies the original differential equation, or
a weighted residual approach;
2. volume discretization of the geometry;
3. interpolation of the unknown using specific functions;
4. solving a set of linear equations.
The finite element method (FEM) has proved to be particularly flexible, reliable and effective in the analysis and synthesis of power-frequency electromagnetic and electromechanical devices. Even in the hands of non-specialists,
modern FEM packages are user friendly and allow for calculating the electromagnetic field distribution and integral parameters without detailed knowledge of applied mathematics.
The FEM can analyze PM circuits of any shape and material. There is no
need to calculate reluctances, leakage factors or the operating point on the
recoil line. The PM demagnetization curve is input into the finite element program, which can calculate the variation of the magnetic flux density throughout the PM system. An important advantage of finite element analysis over
the analytical approach to PM motors is the inherent ability to accurately calculate armature reaction effects, inductances and the electromagnetic torque
variation with rotor position (cogging torque).
A simple magnetic circuit with PMs will be solved with the aid of the
Ansoft Maxwell SV FEM package for magnetostatic solver. The 2D model is
shown in Fig. 4.7. It consists of a U-shaped core of mild steel and an I-shaped
core of mild steel with two PMs. The problem will be solved in seven steps.
1. Step 1: Define the model (Fig. 4.7).
2. Step 2: Assign materials (Fig. 4.8a).
3. Step 3: Set up boundaries or sources (Fig. 4.8b).
4. Step 4: Set up executive parameters (Fig. 4.8c). In this case the traction
force between two parts of core is to be calculated.
5. Step 5: Set up solution options; automatic or manual generation of mesh
(Fig. 4.9).
6. Step 6: Solve. Attraction forces and their components are calculated (Fig.
4.10).
7. Step 7: Postprocessing. Calculate the magnetic field distribution (Fig.
4.11).
Then, a coil with N I = 3000 ampturns is added (Fig. 4.12). The magnetic
flux of the coil is in the direction opposite to the flux produced by PMs.
This system can show demagnetizing action of armature reaction. The PMs
constitute the field excitation system and the coil constitutes the armature
system. Magnetic flux and magnetic flux density distributions are shown in
Fig. 4.13.
Calculation of Magnetic Circuits with PMs
109
Fig. 4.7. Step 1: Define the 2D model.
Fig. 4.8. Steps 2, 3, and 4: (a) assign materials; (b) set up boundaries or sources;
(c) set up executive parameters.
Fig. 4.9. Step 5: Set up solution options. Automatic or manual generation of mesh.
110
Modern Permanent Magnet Electric Machines
Fig. 4.10. Step 6: Solve. Rectangular coordinate system forces are in N/m. To
obtain forces in Newtons, it is necessary to multiply these forces by the dimension
perpendicular to the plane of modeling (2D field).
Fig. 4.11. Step 7: Postprocessing. Calculation of magnetic field distribution: (a)
magnetic flux; (b) magnetic flux density.
Fig. 4.12. Coil with MMF N I = 3000 A is added to simulate the demagnetizating
action of the armature reaction: (a) 2D model; (b) sources; (c) mesh generation.
Calculation of Magnetic Circuits with PMs
111
Fig. 4.13. Magnetic field distribution in a simple magnetic circuit with PM and a
coil: (a) magnetic flux; (b) magnetic flux density.
Summary
Magnetic circuits with PMs can be calculated using the following methods:
(a) graphical methods;
(b) analytical methods, i.e., analogy between electric and magnetic circuits;
(c) equivalent reluctance method (ERM);
(d) finite element method (FEM).
In the analytical approach, the magnetic circuit is calculated in a similar way
as a linear electric circuit for steady state conditions, i.e., using, for example,
Kirchhoff equations. The magnetic circuit is divided into simple solids such
as rectangular prisms, cylinders, half-cylinders, etc., as shown in Fig. 4.2.
An equivalent reluctance network (ERN) method is an improved magnetic
circuit method. The magnetic circuit of an electrical machine is divided into a
large number of sections, which are then replaced by equivalent concentratedparameter reluctances. The reluctances Rµ = 1/Gµ are estimated similar to
permeances for simple solids.
The finite element method (FEM) is a numerical method for solving problems of engineering and mathematical physics. The FEM can analyze PM
circuits of any shape and material. There is no need to calculate reluctances,
leakage factors or the operating point on the recoil line. The PM demagnetization curve is input into the finite element program, which can calculate
the variation of the magnetic flux density throughout the PM system. An
important advantage of finite element analysis over the analytical approach
to PM motors is the inherent ability to accurately calculate armature reaction effects, inductances and the electromagnetic torque variation with rotor
position (cogging torque).
5
PM BRUSH DC MACHINES AND THEIR
CONTROL
5.1 Why PM machines?
PM machines, in comparison with machines with electromagnetic excitation,
show the following advantages:
ˆ
ˆ
ˆ
ˆ
ˆ
No electrical energy is absorbed by the field excitation system, and thus
there are no excitation losses, which means a substantial increase in efficiency.
Higher power density (kW/kg) and/or torque density (Nm/kg) than when
using electromagnetic excitation.
Better dynamic performance than motors with electromagnetic excitation
(higher magnetic flux density in the air gap).
Simplification of construction and maintenance.
Reduction of prices for some types of machines.
Fig. 5.1. Comparison of PM brush DC motor with PM brushless motor: (a) PM
brush motor. 1 – rotor coils, 2 – stator PM, 3 – brushes, 4 – shaft, 5 – commutator;
(b) PM brushless motor. 1 – stator winding, 2 – stator ferromagnetic core, 3 – PM
rotor with retaining sleeve, 4 – shaft.
114
Modern Permanent Magnet Electric Machines
Fig. 5.1 shows the construction of a PM brush DC motor and PM brushless
motor. In a PM brush DC motor, the armature winding is placed on the rotor,
and in a PM brushless motor, the armature winding is placed on the stator.
5.2 Construction of a brush-type PM DC machine
Cutaway of a small PM brush DC machine is drawn in Fig. 5.2. Details of the
stator and rotor construction are shown in Fig. 5.3.
Fig. 5.2. Cutaway of a small PM brush DC servomotor: 1 – laminated ferromagnetic
rotor core, 2 – rotor (armature) winding, 3 – rotor winding connections, 4 – PM,
5 – ferromagnetic housing, 6 – brushes and brush holder, 7 – commutator, 8 – drive
shaft, 9 – bearing.
The active surface of the commutator is formed by a ring of segments
separated from each other by insulating spacers, usually mica spacers. Fig.
5.4 shows cylindrical commutators of DC machines, i.e., commutators with
segments pressed in a plastic cylinder with a mica insulation (Fig. 5.4a) and
a mica-free extruded commutator with sections pressed in a plastic cylinder
(Fig. 5.4b).
Carbon-graphite brushes of small DC machines are shown in Fig. 5.5.
Brushes are held in a brush holder and pressed to the commutator with the
aid of spiral or flat springs. For small PM DC servo-motors, brushes made
of precious-metal wires, e.g., silver—palladium are frequently used (Fig. 5.6).
The advantages of precious-metal brushes are
ˆ
ˆ
low friction;
increased reliability;
PM Brush DC Machines and Their Control
115
Fig. 5.3. Stator and rotor construction of a PM DC brush motor: (a) 2-pole stator
with segmental PMs; (b) rotor. 1 – segmental PM, 2 – steel housing, 3 – laminated
ferromagnetic rotor core, 4 – rotor (armature) winding, 5 – rotor winding connections, 6 – commutator, 7 – drive shaft.
Fig. 5.4. Cylindrical commutators: (a) commutator with segments pressed in a plastic cylinder with mica insulation; (b) mica-free extruded commutator with sections
pressed in a plastic cylinder.
ˆ
ˆ
low audible noise;
low electromagnetic interference (EMI).
To extend the operating life of DC motors that have precious metal brushes,
a capacitor long life (CLL) circuit (filter) is used. The CLL suppresses brush
sparking.
5.3 Principle of operation of a PM brush DC machine
A brush-type DC machine is a reversible machine and can operate either as
a generator or a motor. The EMF induced in the armature winding is
E=
Na p
Φn = cE Φn = kE n
a
(5.1)
116
Modern Permanent Magnet Electric Machines
Table 5.1. Comparison of graphite brushes with precious metal brushes according
to Maxon, Sachseln, Switzerland, https://www.maxongroup.com/
Graphite brushes
Precious metal brushes
◦ graphite with approx.
◦ carrier material: spring bronze
50% Cu added to reduce brush
◦ contact material: Ag alloy
resistance
(Au alloy)
◦ max. operating life with
◦ very low electrical resistance
pure electrographite brushes
◦ more complex construction
◦ few parts
Construction ◦ lead wire for brushes
◦ simple construction
◦ brushes with springs
◦ brushes preloaded
in brush holders
◦ for larger motors
◦ for small motors
Use for long ◦ for higher currents
◦ for very small currents and
operating life ◦ for frequent peak currents in
voltages in continuous operation
star-stop and reversing operation
◦ higher costs
◦ lower costs
Benefits and ◦ higher audible noise
◦ lower audible noise
drawbacks
◦ greater losses
◦ lower losses
◦ higher no-load current
◦ lower no-load current
◦ depends on the surface of the ◦ low constant contact resistance
commutator
◦ small contact surface
Contact
◦ good electrical contact only
◦ very sensitive to arcing,
response
at higher currents
therefore capacitive damping by
◦ less sensitive to moderate brush CLL is required
fire (no CLL circuit required)
◦ no lubrication
◦ lubrication with special
Lubrication ◦ graphite and the moisture
commutator lubricant
in the air acts as lubricant
◦ servo drives
Applications ◦ feeder systems, robots
◦ fans
◦ drills
◦ simple pumps
◦ screw drives
◦ Cu alloy
Commutator ◦ surface turned on the lathe
◦ Ag alloy
for perfect concentricity, cleanless ◦ surface polished or coated
and surface texture
Brush
material
Brush
resistance
PM Brush DC Machines and Their Control
117
Fig. 5.5. Carbon-graphite brushes for small DC machines: (a) pair of brushes with
spiral springs; (b) single brush with spiral spring; (c) pair of brushes with flat springs.
Fig. 5.6. Brushes made of precious metal wires.
where Na is the number of armature conductors (bars), p is the number of
pole pairs, a is the number of pairs of current parallel paths, Φ is the magnetic
flux, n is the rotational speed and cE is the EMF (armature) constant, i.e.,
Na p
(5.2)
a
Neglecting the armature reaction, for PM machines, the magnetic flux Φ =
const and the new EMF constant is
cE =
Na p
Φ
(5.3)
a
The electromagnetic torque is proportional to the armature current Ia
kE = cE Φ =
EIa
Na p 1
=
ΦIa = cT ΦIa = kT Ia
2πn
a 2π
where the torque constant
Telm =
(5.4)
Na p 1
cE
=
a 2π
2π
(5.5)
Na p 1
Φ = cT Φ
a 2π
(5.6)
cT =
Neglecting the armature reaction
kT =
118
Modern Permanent Magnet Electric Machines
From Kirchhoff’s voltage law, the terminal (input) voltage of a DC machine
is
X
U = E ± Ia
Ra ± ∆Vbr
(5.7)
where E is the voltage induced in the armature winding (called the EMF), Ia
is the armature current, and ∆Vbr is the brush voltage drop. The brush voltage
drop is approximately constant, and for the majority of typical DC motors is
practically independent of the armature current. For carbon brushes, ∆Vbr ≈ 2
V. The “+” sign is for the motor and the “−” sign is for the generator. The
armature circuit resistance is for PM brush DC machines, in general:
X
Ra = Ra + Rint
(5.8)
where Ra is the resistance of the armature winding and Rint is the resistance
of the commutation winding located on the interpoles (if exists).
5.4 Windings of a slotted rotor (armature) of a
brush-type DC machine
Cylindrical rotor DC machines are also called radial flux DC machines. The
armature winding located on the rotor can be distributed in slots (Fig. 5.7a) or
distributed uniformly on the external smooth surface of the rotor (Fig. 5.7b).
In the first case the armature winding is called a slotted winding and in the
second case the winding is called a slotless winding. In both cases, the rotor
core is usually made of silicon steel laminations. The machine with the slotted
armature winding has a small nonferromagnetic air gap (mechanical clearance)
and requires a small amount of PMs. On the other hand, the slotted rotor
generates cogging torque, i.e., torque ripple due to interaction of a PM on slot
openings in the zero-current state. A slotless machine does not produce any
cogging torque, but the large nonferromagnetic air gap (mechanical clearance
plus radial thickness of the winding) requires larger amount of PM material.
Fig. 5.7. Cylindrical construction of radial flux PM DC machines with (a) slotted
rotor; (b) slotless rotor.
PM Brush DC Machines and Their Control
119
Fig. 4.13 shows two basic types of armature windings of brush DC machines: (a) lap winding and (b) wave winding. There are the following definitions of the winding span and commutator span:
ˆ
ˆ
Span of armature winding
y = y1 ± y2
(5.9)
yc = y
(5.10)
Commutator span
where y1 is the partial first span, y2 is the partial second span and the “+”
sign is for the wave winding and the “−” sign is for the lap winding. Each coil
is connected to the adequate segment of the commutator. The diagram of the
armature winding is shown in Fig. 5.8. The armature slot is usually divided
into two layers: an upper layer and lower layer separated by an insulation. Each
coil has its left side located in the upper layer of a slot and its right side located
in the lower layer of the slots. The main feature of the armature winding of
brush DC machines is that the armature winding is a closed winding (without
beginning and end).
Fig. 5.8. Two basic types of winding for brush DC machines: (a) lap winding; (b)
wave winding.
The brushes divide the armature winding into parallel paths (Fig. 5.9). The
number of pairs of parallel paths a is equal to the number of pairs of brushes.
If the armature current is Ia , the current of the parallel path is Ia /(2a). The
number of parallel paths should be adjusted to the nominal armature current,
which is divided uniformly among all parallel paths.
The following relationship exists between the number of armature conductors Na and the number of commutator segments C:
Na = 2CNc
where Nc is the number of turns per one armature coil.
(5.11)
120
Modern Permanent Magnet Electric Machines
Fig. 5.9. Division of the lap winding into parallel paths for p = 1 and a = 1.
The absolute symmetry of the armature winding is when at any position
of the rotor and any waveform of the EMF, both the EMFs and resistances
of all parallel paths are the same. In practice, it is enough to obtain relative
symmetry, when the following conditions are to be met:
s
p
C
= integer;
= integer
= integer
(5.12)
a
a
a
If the first and second conditions are even numbers, the symmetry becomes
absolute symmetry.
5.5 Construction of a coreless rotor winding with an
inner PM
Coreless rotor winding with an inner PM of a brush DC machine is shown in
Fig. 5.10. Only the winding spins around the stationary PM. The cylindrical
PM is inside the winding. This type of winding is also called a “moving-coil”
(Fig. 5.11). The external housing made of ferromagnetic material is a return
path for the magnetic flux. Lightweight winding of copper wire spins around
PM Brush DC Machines and Their Control
121
the stationary PM instead of a heavy laminated rotor core with winding distributed in slots. This type of winding has the following advantages:
ˆ
ˆ
ˆ
ˆ
ˆ
no end turns (less copper, lower resistance, lower inductance and electromagnetic time constant);
low rotor moment of inertia (lower mechanical time constant);
mechanical symmetry of windings (good balancing);
high mechanical stiffness of the rotor (transverse layers of winding);
good heat dissipation (possible high current density).
Low inductance of the armature winding means that the electromagnetic time
constant
Te =
La
Ra
(5.13)
is low. In eqn (5.13), La is the self-inductance of the armature winding and
Ra is the armature winding resistance.
If the moment of inertia J of the rotor is low, the mechanical time constant
Tmech =
2πn0 J
JΩ0
=
Tst
Tst
(5.14)
is also low. In eqn (5.14), Ω0 = 2πn0 is the angular speed of the rotor at
no load, n0 is the rotational speed at no load and Tst is the starting torque
(locked-rotor torque).
Good heat transfer from the rotor to surrounding air is achieved because
the winding has direct contact with the outer and inner air gaps.
Fig. 5.10. Coreless rotor winding with inner PM: 1 – cylindrical 2-pole PM, 2 –
coreless moving coil-type rotor winding, 3 – support for rotor winding made of
insulating material, 4 – ferromagnetic housing; 5 – commutator, 6 – brushes made
of precious metal wire, 7 – terminals, 8 – drive shaft, 9 – bearing, 10 — bearing
cover (end bell); 11 – ring-shaped support for the PM.
122
Modern Permanent Magnet Electric Machines
Fig. 5.11. Types of coreless rotor windings (moving coil) for PM DC motors: (a),
(b) according to Maxon, Sachseln, Switzerland; (c) Faulhaber windings according
to US Patent 3360668, Portescap.
5.6 Coreless rotor windings: Maxon versus Faulhaber
winding
Construction of a moving-coil coreless armature (rotor) winding of rhombic
type (Maxon, Switzerland) is explained in Fig. 5.12. This winding is sometimes
called knitted winding. Current flow through the rhombic winding is shown in
Fig. 5.13.
Fig. 5.12. Moving-coil coreless armature (rotor) winding of rhombic type according to Maxon, Sachseln, Switzerland: (a) winding loop shape; (b) arrangement of layers; (c) complete cylindrical rhombic winding; (d) complete rotor.
www.maxonmotor.com
Cutaway of modern brush-type PM DC motor with coreless moving-coil
armature (rotor) winding manufactured by Maxon, Switzerland, is shown in
Fig. 5.14.
Moving-coil honeycomb winding according to F. Faulhaber’s invention (US
Patent 3360668) has advantages similar to rhombic winding. Construction
of this winding is explained in Fig. 5.15 (Minimotor SA, Faulhaber Group,
Croglio, Switzerland).
PM Brush DC Machines and Their Control
123
Fig. 5.13. Current flow through the rhombic winding. www.maxonmotor.com
Fig. 5.14. Brush-type PM DC motor with coreless moving-coil armature (rotor)
winding according to Maxon, Sachseln, Switzerland: 1 – PM, 2 – ferromagnetic
housing, 3 – shaft, 4 – moving-coil armature winding; 5 – commutator plate, 6 –
graphite brushes, 7 – terminals, 8 – end cover (end bell), 9 – commutator, 10 — ball
bearing.
Steady-state performance characteristics of brush-type PM DC motors
with moving-coil coreless armature (rotor) honeycomb winding according to
F. Faulhaber’s invention are plotted in Fig. 5.16. The maximum efficiency can
be found from the following simplified formula
r
ηmax =
1−
T0
Tst
!2
r
=
1−
Ia0
Iash
!2
(5.15)
124
Modern Permanent Magnet Electric Machines
Fig. 5.15. Moving-coil coreless armature (rotor) honeycomb winding according to F.
Faulhaber’s invention (US Patent 3360668): (a) winding loop shape; (b) arrangement
of honeycomb layers; (c) complete cylindrical honeycomb rotor.
where Ia0 is the no-load armature current, Iash is the locked-rotor armature
current, T0 is the no-load torque (losses) and Tst is the starting torque (lockedrotor torque).
Fig. 5.16. Steady-state performance characteristics of brush-type PM DC motors
with F. Faulhaber’s honeycomb winding: (a) output power Pout and efficiency η
versus torque T ; (b) speed n and armature current Ia versus torque T .
Cutaway of a complete brush-type PM DC micromachine with honeycomb
armature winding (Faulhaber’s winding) is shown in Fig. 5.17.
5.7 PM brush DC motor with cylindrical rotor and foil
winding
A brush-type PM DC motor with moving-coil cylindrical rotor and foil winding is shown in Fig. 5.17. Instead of armature (rotor) winding made of round
copper wire, this winding is cut from thin copper coil. Construction is similar
to other PM brush DC motors with moving-coil armature windings.
PM Brush DC Machines and Their Control
125
Fig. 5.17. Brush-type PM DC micromachine with honeycomb armature winding
(Minimotor SA, Faulhaber Group, Croglio, Switzerland: 1 – PM, 2 – moving-coil
armature (rotor) winding with transverse layers of conductors, 3 – commutator,
4 – wire brushes; 5 – terminals, 6 – ferromagnetic housing, 7 – cover (end bell),
8 – pinion.
Fig. 5.18. Brush-type PM DC motor with moving-coil cylindrical rotor and foil
winding Embest, Seoul, South Korea: 1 – winding cut from copper foil, 2 – disk
commutator, 3 – brush, 4 – cover, 5 – terminals, 6 – PM, 7 – bearing, 8 – washer,
9 – shaft, 10 – ferromagnetic housing, 11 – magnet bracket, 12 – shrink ring.
126
Modern Permanent Magnet Electric Machines
Embest, Seoul, South Korea, has developed a technology for creating motor
“coils” on flexible printed circuits as an alternative to conventional windings.
The process involves etching patterns in copper films on both sides of an
insulating layer. By connecting the two sides via holes through this layer,
dense coil patterns can be created. Among the attractions that Embest claims
for this process are low manufacturing costs, light weight, simple structures,
low inductances, high efficiencies, and easy maintenance.
The company suggests that its “film coil” technology could be applied
to a wide range of cylindrical and disk motors from small machines used in
applications such as computer peripherals, robots, office automation, through
to motors used to power golf carts, utility vehicles, electric tools and industrial
or agricultural equipment.
Embest points out that conventionally wound coreless motors are not
widely used because of their high price.
5.8 Disk-type PM brush DC motors with printed rotor
winding
Brush PM DC motors with moving-coil printed rotor winding can be designed
as disk-type (axial flux) motors (Fig. 5.19). The coils are stamped from pieces
of sheet copper, placed on both sides of a disk made of insulating material
(ceramic material, textolite, epoxy-glass laminate) and then welded, forming a
wave winding. When this motor was invented by J. H. Baudot [10], the rotor
was made using a similar method to that by which printed circuit boards
(PCB) are fabricated. Hence, this is called the printed winding motor. Wave
winding makes it possible to use fewer brushes with a large number of pole
pairs. The winding is made to obtain a closed circuit without crossing the
wires (Fig. 5.20). The active bars of the rotor are arranged radially or almost
radially. The end turns are on the upper and lower sides of the active bars.
Insulation gaps between the bars are of equal width along the entire bar and
end turns. Active bars have a trapezoidal shape. The end turns are made in
such a way that they have a constant width along their entire length and
therefore are shaped according to an involute. Lower end turns serve as a
commutator. The brushes directly touch the active parts of the bars between
the poles. Two to four pairs of poles are most commonly used (p = 2 or
p = 4). The magnetic flux of the disk-type printed motor is usually produced
by Alnico magnets.
As the rotor circuit is not insulated, the heat dissipation from the copper
into the cooling air is very good. Therefore, very high current densities can
be used, much higher than in machines with a wound rotor. High current
densities are also possible due to the lack of insulation of the rotor conductors
so that much higher temperatures of the printed winding can be assumed than
in the case of classical machines. The operating temperature of the winding
is limited by the properties of the adhesive, which sticks the electric circuit
PM Brush DC Machines and Their Control
127
Fig. 5.19. PM brush DC motor with disk-type moving-coil rotor and foil winding:
1 – moving-coil disk-type printed winding, 2 – PM, 3 – brush, 4 – cylindrical ferromagnetic housing, 5 – ferromagnetic cover.
to the insulating plate of the rotor. The thermal expansion of copper may be
an important factor in determining the limit on the load capacity.
Fig. 5.20. Printed rotor wave winding of disk-type PM brush DC motor: (a) coil
paths on both sides of the disk made of insulating material; (b) complete winding.
For the radial arrangement of the active bars, assuming that their length
is equal to the radial dimension of the poles and ignoring the insulation width
128
Modern Permanent Magnet Electric Machines
between the bars, the electromagnetic torque can be expressed as
Z rout
2
2
Telm = 2αi N
Bg Ib rdr = αi N Bg Ib (rout
− rin
)
(5.16)
rin
where αi = bp /τ is the pole width bp –to–pole pitch τ ratio, N is the number
of bars at one side of the disk, Bg is the magnetic flux density in the air gap,
Ib is the current in a single bar, rout is the outer radius of the active bar and
rin is the inner radius of the active bar. The current in the active bar can be
expressed by the current density jb in the narrowest point of the bar (radius
rin ), i.e.,
2π
jb hrin
(5.17)
N
where sbin is the cross section of the bar in the narrowest point and h is the
thickness of the layer of the conductor. To obtain the maximum electromagnetic torque, the ratio of rout to rin should be [8]
Ib = jb sbin =
√
rout
3<
< 2.59
rin
(5.18)
5.9 Fundamentals of transient analysis of PM brush DC
motors
The voltage balance equations (voltage-current equations) for DC brush machines with number of pole pairs p = 1 and number of pairs of parallel paths
a = 1 in matrix form for instantaneous values of voltage and current are
uf
Rf + pLf
0
if
=
(5.19)
ua
ΩM
Ra + pLa
ia
where ua is the armature voltage, uf is the field voltage (for machines with
electromagnetic excitation), if is the field current (for machines with electromagnetic excitation), Ω = 2πn is the angular speed, Ra is the armature
winding resistance, Rf is the field winding resistance, La is the armature winding self-inductance and Lf is the field winding self-inductance. The mutual
inductance between the armature and field winding can be expressed as
M = Na Nf
1
Rµ
(5.20)
In the above equation p = d/dt, Na is the number of armature conductors
(bars), Nf is the number of turns of the field winding, and Rµ is the reluctance
for the field winding flux Φf linked with the armature winding, i.e.,
Rµ =
Ff
Φf
(5.21)
PM Brush DC Machines and Their Control
129
Multiplying through eqn (5.21) by the field winding current if
M if = Na Nf if
Ff
1
= Na
= Na Φf
Rµ
Rµ
(5.22)
The above equation (5.22) can be applied to the analysis of PM brush DC
machines, because M if = Na Φf is a convenient notation for this type of
machine. Thus, the voltage balance equation for the armature circuit is
ua = ΩM if +(Ra +pLa )ia = ΩNa Φf +(Ra +pLa )ia = e+(Ra +pLa )ia (5.23)
The instantaneous value of the EMF induced in the armature winding
e = ΩNa Φf = 2πnNa Φf = cE Φf n
(5.24)
For PM machines (Φf = const) and
e = kE n
(5.25)
while
cE = 2πNa
kE = cE Φf
(5.26)
In classical theory of brush DC machines, the constant cE is given by eqn (5.2).
The difference (cE = Na P/a) is because, in the above transient analysis, the
angular speed Ω = 2πn is used instead of rotational speed n and it has been
assumed that the number of pole pairs p = 1 and the number of pairs of
parallel current paths a = 1.
The instantaneous electromagnetic power and instantaneous electromagnetic torque are, respectively,
pelm = eia = ΩNa Φf ia
pelm
= Na Φf ia = cT Φf ia
Ω
The electromagnetic torque for PM brush DC machines
Telm =
Telm = kT ia
(5.27)
(5.28)
(5.29)
where kT is given by eqn (5.6). Assuming infinitely high stiffness of shaft, the
torque balance equations have the following forms:
ˆ
for motor operation
J
dΩ
+ DΩ = Telm − T
dt
(5.30)
130
ˆ
Modern Permanent Magnet Electric Machines
for generator operation
J
dΩ
+ DΩ = T − Telm
dt
(5.31)
where J is the moment inertia of the rotor and other rotating masses, D is
the torsional damping constant, and T is the external torque. For modeling
and simulation, the torque balance equations (5.30) and (5.31) should be
supplemented with the voltage balance equation (5.23).
5.10 Speed control of a brush-type PM DC motor
The speed of a PM brush-type DC motor can be controlled only from the
armature circuit side by changing (Fig. 5.21):
ˆ
ˆ
the supply voltage U ;
the armature current Ia by changing the armature circuit resistance, e.g.,
with the aid of variable armature rheostat with resistance Rrhe .
Fig. 5.21. Armature circuit of a PM DC brush motor with series variable-resistance
rheostat Rrhe .
From eqns (5.1), (5.7) and (5.8) the rotational speed of a PM brush DC
motor is
n=
X
i
1 h
U − Ia
Ra + Rrhe − ∆Ubr
kE
(5.32)
where Rrhe is the resistance of a variable rheostat being in series with
the armature circuit. Since Ia = Telm /kT , the speed as a function of the
PM Brush DC Machines and Their Control
131
electromagnetic torque is
P
1
Ra + Rrhe
n=
(U − ∆Ubr ) −
Telm
(5.33)
kE
kE kT
Theoretical speed control characteristics can be obtained on the basis of eqns
(5.32) and (5.33). Variable armature terminal-voltage speed control, i.e., speed
versus electromagnetic torque n = f (Telm ) at U = const is plotted in Fig.
5.22a and variable armature rheostat speed control, i.e., speed versus armature
current n = f (Ia ) at Rrhe = const, is plotted in Fig. 5.22b.
Fig. 5.22. Theoretical speed control characteristics: (a) variable armature terminalvoltage speed control n = f (Telm ) at U = const; (b) variable armature rheostat
speed control n = f (Ia ) at Rrhe = const. Telmn = nominal (rated) electromagnetic
torque. Ian = nominal (rated) armature current.
5.10.1 Three-phase fully controlled rectifier
Typical diagram of speed-controlled DC motor drive is shown in Fig. 5.23.
For motors up to a few kilowatts, the armature converter can be supplied
from either single-phase or three-phase mains, but for larger motors threephase is always used. The main power circuit consists of a six-SCR (silicon
controlled rectifier) bridge circuit, which rectifies the incoming AC supply
to produce a DC supply to the motor armature (Fig. 5.24a). The controlled
rectifier produces a crude form of DC with a pronounced ripple in the output
voltage (Fig. 5.24b). This ripple component gives rise to pulsating currents and
fluxes in the motor, excessive eddy-current losses and commutation problems.
The firing angle is the phase angle of the voltage at which the SCR turns
on (conducts). In other words, it is the angle after which the SCR fires. For
example, if the firing angle is α = 45◦ , then up to 45◦ of the input sine wave
the SCR will not conduct. Starting from 45◦ , the SCR will conduct till the
cycle completes (through the angle of 180◦ −α = 135◦ ). This is repeated again
and again.
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Modern Permanent Magnet Electric Machines
Fig. 5.23. Schematic diagram of a speed-controlled DC motor drive with current
feedback and speed feedback.
Fig. 5.24. Three-phase fully controlled rectifier: (a) power circuit diagram; (b)
three-phase thyristor bridge waveforms in rectification mode (firing angle α = 40◦ ).
For a three-phase fully controlled rectifier, the DC rectified voltage is
√
3 2
UL cos α
(5.34)
Ud =
π
where UL is the line-to-line AC supply voltage.
5.10.2 Chopper
A chopper is a static power electronics device which converts fixed DC input
voltage to a variable DC output voltage (Fig. 5.25). It can be a step-up or
step-down chopper. When a DC voltage is supplied to the motor, current is
fed to the armature winding through brushes and a commutator. Choppers
are widely used in regulated switching power supplies and DC motor drive
applications. There are two types of choppers:
PM Brush DC Machines and Their Control
ˆ
ˆ
133
step-up chopper (Fig. 5.26a);
step-down chopper (Fig. 5.26b)
In a step-up chopper the output voltage is higher than the input voltage.
In a step-down chopper the output voltage is lower than the input voltage.
Fig. 5.25. How the chopper works.
Fig. 5.26. Choppers: (a) step-up chopper; (b) step-down chopper. CH – chopper
switch, D – diode, FD – freewheeling diode, L – inductance, Ub – battery voltage,
U – voltage across load terminals.
In a chopper-fed PM brush DC motor, the average value of the DC terminal
voltage can be varied either by pulse-width modulation (PWM) or pulsefrequency modulation (PFM), as shown in Fig. 5.27. The average value of the
DC terminal voltage is
1
U=
T
Z T
u(t)dt
(5.35)
0
5.10.3 H-bridge
An H-bridge is an electronic circuit that switches the polarity of a voltage
applied to a load, e.g., a brush DC motor (Fig. 5.28). It is called that because it looks like the capital letter H when viewed on a circuit diagram. The
great ability of an H-bridge circuit is that the motor can be driven forward
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Modern Permanent Magnet Electric Machines
Fig. 5.27. Chopper-fed PM brush DC motor: (a) simplified circuit diagram; (b)
PWM; (c) PFM.
or backward at any speed, optionally using a completely independent power
source.
The switching elements (Q1. . .Q4) are usually bipolar or field effect transistors (FETs), and in some high-voltage applications, integrated gate bipolar
transistors (IGBTs). The diodes (D1. . .D4) are called catch diodes and are
usually of a Schottky type to protect against overvoltage or undervoltage
from the motor.
Fig. 5.28. H-bridge: (a) simplified circuit diagram; (b) Q1 and Q4 are turned on,
and the motor starts spinning in the forward direction; (c) Q2 and Q3 are turned
on, and the motor starts spinning the backward direction.
PM Brush DC Machines and Their Control
135
If Q1 and Q4 are turned on, the motor starts spinning in the forward
direction (Fig. 5.28b). If Q2 and Q3 are turned on, the reverse will happen
and the motor will start spinning backwards (Fig. 5.28c).
5.11 PM brush DC servomotors
A PM brush DC servomotor is controlled from the armature winding terminals (Fig. 5.29a). Modern DC servomotors have coreless rotors, e.g., with
Faulhauber’s cylindrical windings (Fig. 5.29b) or disk-type windings (Fig.
5.29c).
Fig. 5.29. PM brush DC servomotor: (a) circuit diagram; (b) rotor with Faulhauber’s cylindrical winding; (c) rotor with disk-type winding. Ω – angular speed,
Uc – control voltage, Ia – armature (rotor) current, T – load torque, J – moment of
inertia, and D – torsional damping.
From a control ability point of view, PM brush DC servomotors must meet
the following requirements:
(a) linear mechanical characteristic n = f (T );
(b) linear regulation characteristic n = f (Uc );
(c) small time constants: mechanical Tmech and electromagnetic Te (fastacting motors);
(d) high starting torque Tst ;
(e) small control power Pc = Uc Ia at high shaft power Pout = 2πnT ;
(f) small volume envelope and small mass;
(g) self-braking.
The mechanical time constant Tmech is expressed by eqn (5.14) and the electromagnetic time constant Te is expressed by eqn (5.13). Moving coil rotors
(Fig. 5.29b and Fig. 5.29c) provide low moment of intertia J of the rotor and
consequently low mechanical time constant Tmech . The higher the starting
torque Tst , the lower the mechanical time constant Tmech .
The self-braking property of a servomotor is that the rotor should stop
immediately after the control voltage Uc is removed.
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Modern Permanent Magnet Electric Machines
A block diagram of a microcontroller for a PM brush DC servomotor is
shown in Fig. 5.30. The microcontroller unit (MCU) produces PWM signal
and controls IGBT via buffer-amplifier.
Fig. 5.30. Example of a microcontroller for a PM brush DC servomotor.
The basic parameters of a brush DC servomotor in control systems are
ˆ
relative torque and relative rotational speed
t=
ˆ
T
Tst
ν=
(5.36)
coefficient of signal
α=
ˆ
n
n0
Uc
Un
(5.37)
mechanical time constant Tmech expressed by eqn (5.14) and electromagnetic time constant Te expressed by eqn (5.13).
In the above equations the following symbols have been used: T – shaft
torque, Tst – starting torque, n – rotational speed, n0 – no-load rotational
speed, Uc – control voltage, Un – nominal voltage. Characteristics of an ideal
PM brush DC servomotor are shown in Fig. 5.31.
5.12 Applications of brush-type PM DC motors
The PM brush DC motor has many applications, for example:
ˆ
ˆ
ˆ
ˆ
ˆ
motors for toys
auxiliary motors for automobiles
domestic equipment
public life equipment
medical and healthcare equipment
PM Brush DC Machines and Their Control
137
Fig. 5.31. Characteristics of an ideal PM brush DC servomotor: (a) mechanical
characteristics; (b) regulation characteristics.
ˆ
ˆ
ˆ
ˆ
ˆ
sports equipment (fitness clubs)
linear actuators with ball screws
cordless power tools
vibration motors for mobile phones
robotic vehicles for Mars missions.
In this section, only motors for toys, auxiliary motors for automobiles, vibration motors for mobile phones and motors for Mars exploration rovers will be
discussed.
5.12.1 Toys
A PM brush DC motor for toys usually has a three-slot laminated rotor core,
three coils connected to a three-segment commutator and two-pole ferrite
magnets mounted in a steel housing (Fig. 5.32). The three-slot rotor and twopole PMs provide self-starting at any position of the rotor. Owing to ferrite
magnets, the price of very small PM brush DC motors is very low.
Adding a gear train to the mechanical output of any motor will reduce the
speed, while simultaneously increasing the torque. Gear train construction
ranges from simple plastic drive trains for toys to robust metal gear trains for
extra-high-torque applications (Fig. 5.33).
An example of a home-made toy car driven by a PM brush DC motor is
shown in Fig. 5.34.
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Modern Permanent Magnet Electric Machines
Fig. 5.32. Small PM brush DC motor for toys: (a) electromagnetic system; (b) end
plastic cover. 1 – PM, 2 – rotor laminated core with three slots and three teeth,
3 – rotor (armature) coil, 4 – steel cylindrical housing, 5 – shaft, 6 – plastic cover,
7 – carbon-graphite brushes, 8 – brush spring.
Fig. 5.33. Examples of small gears for PM brush DC motors.
Fig. 5.34. Example of drive train for toy. 1 – PM brush DC motor, 2 – gear
transmission, 3 – belt transmission, 4 – battery.
PM Brush DC Machines and Their Control
139
5.12.2 Auxiliary motors for automobiles
Most electric motors in today’s cars run from the standard 12 V automotive
system, with a belt-driven alternator to generate voltage and a lead-acid battery for storage. This arrangement has worked fine for decades, but the latest
vehicles need more and more current for comfort, entertainment, navigation,
driver assistance and safety features.
A dual-voltage 12 V and 48 V system could move some of the highercurrent loads off the 12 V battery. The advantages of using a 48 V supply are
a four times reduction in current for the same power, and an accompanying
reduction in weight in terms of cables and motor windings. Examples of highcurrent loads that may migrate to a 48 V supply include the starter motor,
turbocharger, fuel pump, water pump and cooling fans. Implementing a 48
V electrical system for these components could result in fuel-consumption
savings of around 10%.
Brush DC motors are the traditional solution for driving most electric
convenience features in an automotive body. Since the brushes provide the
commutation, these motors are simple to drive and are relatively inexpensive.
The simplicity and cost-effectiveness of brush motors still hold an advantage.
In some applications, PM brushless DC (BLDC) motors can provide significant benefits in terms of power density, thus reducing weight and providing
better fuel economy and lower emissions.
Typical applications of auxiliary electric motors in cars include
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
windshield wipers;
steerable headlights;
radiator fan;
radiator shutters;
door locks;
oil pump;
fuel pump;
water pump;
windshield water pump;
power steering;
compressor;
mirror XY;
folding side mirrors;
power sliding windows;
ride stabilization;
trailer hitch retract;
trunk/hatch lift;
sunroof;
power seats;
ventilated seats;
140
ˆ
ˆ
Modern Permanent Magnet Electric Machines
HVAC blower;
HVAC dampers;
and more. Under the hood, electric motors are becoming more common in several places. In most cases, electric motors are replacing belt-driven mechanical
components. Examples include radiator fans, fuel pumps, water pumps and
compressors. Moving these functions from a belt drive to an electric drive has
several advantages. One is that driving electric motors with modern electronics can be much more power-efficient than using belts and pulleys, leading to
benefits like higher fuel efficiency, reduced weight and lower emissions. Another advantage is that using electric motors rather than belts allows freedom
in mechanical design, as the mounting position of pumps and fans need not
be constrained by having to run a serpentine belt to each pulley.
To change seat positions, there are several common requirements. Seat
position adjustments are typically bidirectional, meaning that the drive electronics must have a way to apply voltage across the motor with either polarity
so that the seat is adjustable in both directions. The simplest and universal
solution is to use a PM brush DC motor fed from an H-bridge.
Fig. 5.35. Windshield wiper PM brush DC motor. 1 – rotor (armature), 2 – commutator, 3 – polythene vent pipe, 4 – terminals, 5 – warm gear, 6 – adjuster for
rotor end-float, 7 – plastic warm wheel, 8 – brush, 9 – steel housing, 10 – PM.
The windshield wiper motor is a small DC motor (usually PM brush-type)
that controls the movement of the windshield wipers (Fig. 5.35). To accelerate
the wiper blades back and forth across the windshield, a worm gear is used
on the output of an electric motor. The worm gear reduction can multiply the
torque of the motor by about 50 times, while slowing the output speed of the
electric motor by 50 times as well. A short cam is attached to the output shaft
of the gear reduction. This cam spins around as the wiper motor turns. There
is an electronic circuit inside the motor/gear assembly that senses when the
PM Brush DC Machines and Their Control
141
wipers are in their down position. The circuit maintains power to the wipers
until they are parked at the bottom of the windshield, then cuts the power
to the motor. This circuit also parks the wipers between wipes when they are
on their intermittent setting.
Fig. 5.36. Starter PM brush DC motor: (a) motor with solenoids and pinion; (b) set
of stator PMs. Photo Courtesy of Hangseng (Ningbo) Magnetech Co., Ltd., Zhejiang,
China.
Sometimes PM brush DC motors are used as starter motors (Fig. 5.36).
Starter motors are equipped with solenoids and a pinion. The solenoid contains
two coils that are wrapped around a moveable core. The solenoid acts as a
switch to close the electrical connection and connects the starter motor to
the battery. The pinion is a unique combination of a gear and springs. Once
the starter is engaged, the gear is extended into the gearbox housing and it
is engaged with the flywheel. This spins the engine to begin the combustion
process.
5.12.3 Vibration motors for mobile phones
The current world population is 8,021 billion as of September 2022 according
to the UN. Cell phone subscriptions exceed the worldwide population (Fig.
5.37). Now almost every cell phone device has the ability to produce vibration
alerts.
PM brush DC motors are the most popular vibration motor for cellular
phones. Vibration is generated by a small unbalanced mass fixed to the shaft
end (Fig. 5.38). When the motor spins, the unbalanced mass creates a centrifugal force that translates to vibrations. The vibration motor is one of the
smallest PM brush-type DC micro motors. Construction of a PM brush DC
vibration motor for a cellular phone is shown in Fig. 5.39.
When considering how to power a vibration motor within a mobile application, there are two significant obstacles.
First, it is likely that the battery voltage is higher than the maximum operating voltage of the motor. Running the motor above this maximum value
142
Modern Permanent Magnet Electric Machines
Fig. 5.37. Number of users of mobile phones in the world.
Fig. 5.38. Vibration motor for a cellular phone is one of the smallest PM brush
DC micro motors.
can damage the motor and cause it to fail prematurely. Therefore, a solution is required that reduces the motor’s supply voltage to an adequate level,
preferably with high efficiency.
Secondly, the output voltage from a battery will vary depending on its
charge. For example, a Li-ion battery which operates at 4.2 V fully charged,
may only produce 3.2 V when nearly depleted. If a motor was using the battery
as its supply voltage, this would lead to a reduction in performance as the
battery drained. It is preferred to provide the motor with a constant supply
voltage so that the performance and effects of the vibration are constant,
irrespective of the level of charge.
PM Brush DC Machines and Their Control
143
Both problems can be solved using one of three fairly simple solutions: (1)
a linear voltage regulator with a Zener diode in parallel with the load, (2) a
low-dropout (LDO) voltage-regulator IC, or (3) a motor drive IC.
As a result of a motor’s electromechanical operation, they can be sources
of EMI. Given that most mobile or cell phones are extremely limited in terms
of space, the vibration motor can be in close proximity to very noise-sensitive
RF circuitry. Motors can generate RF wideband noise through radiated and
conduction emissions, and they can also generate high current and voltage
spikes, which in extreme cases can damage the motor drive circuitry. Therefore
electromagnetic compatibility (EC) is a very important consideration.
A PM brush DC vibration motor can also be designed as an axial flux
motor, called a coin motor . The coin-type vibration motor is quite popular in
the mobile phone field. Because the coin motor is one of the thinnest motors
in the world (about 1.8 mm), it is suitable for thin and light smart phones.
Fig. 5.39. Construction of PM brush DC vibration motor for cellular phone. 1 —
stationary parts, 2 — rotor (armature winding) with shaft, 3 — unbalanced mass,
4 — brushes and terminals.
The iPhone 5 was the last smart phone manufactured by Apple with a PM
brush DC vibration motor (Fig. 5.40a). The next iPhones since the iPhone5
have been equipped with a linear vibration motor called a taptic motor (Fig.
5.40b,c and Fig. 5.41). A combination of “tap” and “haptic feedback,” the
taptic engine is a name Apple created for its technology that provides tactile
sensations in the form of vibrations to users of Apple devices (“haptic” =
relating to the sense of touch). Construction of a taptic motor is shown in
Fig. 5.42. This is not the Apple taptic motor .
144
Modern Permanent Magnet Electric Machines
Fig. 5.40. Vibration motors for iPhones: (a) rotary PM brush DC vibration motor
for iPhone 5; (b) linear vibration motor iPhone 6s; (c) linear vibration motor called
the “taptic motor” for iPhone 7.
Fig. 5.41. Taptic motors in iPhones 8 Plus and 13 Max Pro: (a) iPhone 8 Plus; (b)
iPhone 13 Max Pro.
For iPhones 11, 12, and 13 it is claimed that Apple introduced an updated
taptic motor, allegedly under the codename “leap haptics.” It is more useful
for a haptic touch.
5.12.4 Robotic vehicles for Mars missions
Currently, Maxon motors are involved in several projects destined for Mars.
As of 2022, 21 lander missions and 8 sub-landers (Rovers and Penetrators)
attempted to land on Mars. Of 21 landers, the Curiosity Rover, InSight Mars
Lander, Perseverance Rover, and Tianwen-1 are currently in operation on
Mars.
For example, in NASA’s InSight1 Lander a Maxon DC motor powers the
mole that hammers the measuring sensor into the ground.
Both NASA and the European Space Agency (ESA) send rovers to Mars
(Fig. 5.43). The mission is looking for former or current life. A drill will take
soil samples from a depth of 2 m, which the rover will then analyze on-site
1
Interior Exploration using Seismic Investigations, Geodesy and Heat Transport
PM Brush DC Machines and Their Control
145
Fig. 5.42. Construction of a taptic motor . 1 — voice coil, 2 — NdFeB PM, 3 — flex
circuit, 4 — moving mass, 5 — current-conducting spring, 6 — flying terminal leads,
7 — chassis, 8 — case, 9 — self-adhesive backing. Courtesy of Precision Microdrives,
London, UK, https://www.precisionmicrodrives.com/
Fig. 5.43. ESA’s ExoMars rover.
146
Modern Permanent Magnet Electric Machines
with measuring instruments. More than 50 actuators, from the wheel drive
to sample distribution to camera movement, with 17 different configurations
of brushed or brushless DC motors (such as DCX 10, DCX 22 or EC 40) in
combination with gearheads (such as GP 22 HD), brakes and encoders are
installed.
Maxon motors themselves are standard products with diameters of 20 and
25 mm that deliver an efficiency level of more than 90%. Minor modifications
were required to enable them to cope with the harsh conditions. The equipment has to be able to withstand temperature changes on the surface of Mars,
which can range from around −120◦ C to +25◦ C, vibrations, and the special
atmosphere.
Fig. 5.44. Artists concept of one of the two NASA Mars Exploration Rovers, Spirit
and Opportunity. Spirit landed on Mars at Gusev Crater on Meridiani Planum
January 25, 2004 and Opportunity landed on the opposite side of the planet at
Eagle Crater on Meridiani Planum January 25, 2004. 1 – rocker-bogie mobility
system, 2 – alpha-particle X-ray spectrometer, 3 – Mössbauer spectrometer, 4 –
rock abrasion tool, 5 – microscopic imager, 6 – magnet array (forward), 7 – solar
panels, 8 – panoramic cameras, 9 – navigation cameras, 10 – mini-thermal emission
spectrometer (at rear), 11 – UHF antenna, 12 – low gain antenna, 13 – calibration
target, 14 – high-gain antenna.
In the past, Maxon PM brush DC motors were used in Sojourner , the first
Mars rover, which landed on July 14, 1997, Spirit and Opportunity, which
landed on Mars in January 2004 (Fig. 5.44), Phoenix , which landed on May
PM Brush DC Machines and Their Control
147
Table 5.2. Comparison of Martian rovers Sojourner and Spirit and Opportunity.
Sojourner Spirit and Opportunity
Mass, kg
11
185
Height, m
0.32
1.57
Height above ground, m
0.25
1.54
Communication
8 Bit CPU
32 Bit CPU
Cameras
3 (768 × 484)
9 (1024 × 1024)
Spectrometers
1
3
Speed, m/h
3.6
36 to 100
Maxon motors
11 RE16
17 RE20
22 RE25
25, 2008 and Curiosity, which landed in August 2012. Comparison of Mars
rovers Sojourner and Spirit and Opportunity is given in Table 5.2.
Summary
Replacement of electromagnetic excitation systems with PMs brings many
advantages, amongst others, increase in efficiency, increase in power density
and improvement of dynamic performance. In a typical design of a PM brush
DC machine, the slotted rotor is equipped with the armature winding and
commutator while the stator contains PMs, which excite the magnetic flux.
Ferromagnetic housing serves as a return path for the magnetic flux.
The PM brush DC machine is a reversible machine and can operate both
as a motor (Telm = kT Ia ) and generator (E = kE n).
PM brush DC motors, due to their commutator and brushes, are less
reliable than AC cage induction motors, brushless PM motors (BLDC), and
switched reluctance motors (SRM). To improve reliability, reduce friction,
reduce acoustic noise and reduce EMI, brushes made of precious metals are
used in small PM brush DC motors.
Torque ripple can be reduced, if the rotor (armature) is made with a slotless
core, while the armature winding is uniformly distributed on the external
cylindrical surface of the rotor core. On the other hand, a slotless rotor (larger
air gap) requires more PM material.
Heat dissipation conditions in PM brush DC machines (armature winding
on the rotor) are much worse than in brushless PM motors (armature winding
on the stator).
To reduce the moment of inertia of the rotor, reduce the torque ripple,
improve dynamic performance and improve heat dissipation conditions, a
small PM brush DC motor has an inner PM, the spinning coreless winding (moving-coil winding) in the form of a cup and outer housing made of
ferromagnetic material to create a return path for the magnetic flux, e.g.,
rhombic (Maxon) winding and honeycomb (Faulhaber’s) winding. Rhombic
148
Modern Permanent Magnet Electric Machines
and honeycomb winding wound with round copper wire can be replaced with
foil winding cut from a copper tape.
PM brush DC machines can be made as disk-type (axial flux) machines.
The coreless rotor (armature) can be made of either copper wire or made as
a printed winding by etching or cutting from copper foil.
For dynamic simulation of a brush-type PM DC machine, the voltage
balance equation for the electric circuit and torque balance equations for the
mechanical system are used.
The speed of PM brush DC motors can only be controlled from the armature terminals by changing the input terminal voltage or the armature current.
The input terminal voltage is usually controlled by three-phase fully controlled
rectifiers, choppers or H-bridges. The armature current can be controlled with
the aid of variable armature rheostats.
PM brush DC servomotors with linear mechanical and regulation characteristics are used in control systems. Two-phase induction servo-motors are
not in use anymore and their production has been abandoned.
Typical applications of PM brush DC motors include toys, auxiliary motors for automobiles, domestic equipment, public life equipment, medical and
healthcare equipment, sports equipment, linear actuators with ball screws,
cordless power tools, vibration motors for mobile phones, and robotic vehicles
for Mars missions.
Very small PM brush DC motors are the most popular vibration motors in
mobile phones. In iPhones 6, 7, 8, X, 11, 12, and 13 there are linear vibration
motors, the so-called “taptic motors”.
6
PM BRUSHLESS DC MOTORS AND DRIVE
CONTROL
6.1 From PM DC brushed to PM DC brushless motors
Although the speed of PM DC brush motors can be easily controlled (5.32)
and (5.33), the fundamental disadvantage of these machines is their commutator and brushes. About 90% of maintenance relates to the commutator
and brushes. To avoid these disadvantages, the mechanical commutator with
brushes can be replaced with an electronic commutator, as shown in Fig. 6.1.
The armature winding is relocated from the rotor to the stator and PMs are
placed on the rotor. In this way, the PM brushless motor is created.
Fig. 6.1. Replacement of an electromechanical commutator with electronic commutator. 1 – armature laminated core, 2 – armature winding (end turns), 3 – electromechanical commutator, 4 – armature winding placed on the stator, 5 – PM placed on
the rotor, 6 – solid state converter.
The difference between the construction of PM DC brush motors and PM
brushless motors is explained in Fig. 6.2. The PM DC brush motor (Fig. 6.2a)
is fed from a DC source, e.g., a battery or DC power supply. For constant
speed, no power electronics is needed. The PM brushless motor (Fig. 6.2b) in
standard applications, must be fed from a solid state inverter. If the rotor is
150
Modern Permanent Magnet Electric Machines
Fig. 6.2. Differences in construction of PM machines: (a) PM DC brushed machine;
(b) PM brushless machine.
equipped with a starting cage winding, brushless motors can become a line
start synchronous motor, without the possibility of speed control.
In a PM DC brush motor the power losses occur mainly in the internal
rotor with the armature winding, which limits the heat transfer through the
air gap to the stator and consequently the armature winding current density.
In PM brushless motors, all power losses are practically dissipated in the
stator where heat can be easily transferred through the ribbed frame or, in
larger machines, liquid cooling systems, e.g., water or oil jackets can be used.
Comparison of PM DC brush and brushless motors is given in Table 6.1.
Table 6.1. Comparison of PM DC brush (commutator) and brushless motors.
Commutator (inverter)
Maintenance
Reliability
Moment of inertia
Power density
Heat
dissipation
Speed control
Brush DC motor
Mechanical commutator
Commutator and brushes
need periodical maintenance
Low
High
Medium
Poor
(rotor armature winding)
Simple
(armature rheostat
or chopper)
Brushless DC motor
Electronic commutator
Minimal
maintenance
High
Can be minimized
High
Good
(stator armature winding)
Solid state
converter
required
The armature (stator) winding made of concentrated non-overlapping coils
is simple to manufacture and provides short end turns. The concentrated-coil
PM Brushless DC Motors and Drive Control
151
winding is feasible when
Nc
= km1
GCD(Nc , 2p)
(6.1)
where Nc is the total number of armature coils, GCD is the greatest common
divisor of Nc and the number of poles 2p, m1 is the number of phases, and
k = 1, 2, 3, . . ..
6.2 Construction of rotors
Construction examples of rotors of PM brushless motors are shown in Fig.
6.3. There are the following basic constructions:
(a) surface magnets (Fig. 6.3a);
(b) spoke-type magnets (Fig. 6.3b);
(c) interior magnets (Fig. 6.3c);
(d) inset magnets (Fig. 6.3d);
(e) double-layer interior magnets (Fig. 6.3e);
(f) buried magnets asymmetrically distributed (Fig. 6.3f) according to German patent 1173178.
The surface magnet rotor has magnets magnetized radially (as in Fig. 6.3a).
An external non-ferromagnetic cylinder (sleeve) is sometimes used. It protects
the PMs against damage due to centrifugal stresses and the demagnetizing
action of the armature reaction, and provides an asynchronous starting torque
and acts as a damper.
The spoke-type magnet rotor has circumferentially magnetized PMs (Fig.
6.3b). An asynchronous starting torque can be produced with the aid of either a cage winding (if the core of the motor is laminated) or salient poles
(in the case of a motor with a solid steel core). If the shaft is ferromagnetic,
a large amount of useless magnetic flux will be directed through it. To increase the linkage flux, therefore, a spoke-type magnet rotor should always be
equipped with a nonferromagnetic shaft or nonferromagnetic bushing between
the ferromagnetic shaft and the rotor core.
The interior-magnet rotor of Fig. 6.3c has radially magnetized and alternately poled magnets. Because the magnet pole area is smaller than the pole
area at the rotor surface, the no-load air gap flux density is less than the flux
density in the magnet.
The inset-type PM rotor shown in Fig. 6.3d is very similar to the surfacemagnet rotor. The magnets are placed in slots made in the rotor core.
The rotor with double-layer interior magnets (Fig. 6.3e) magnetized radially can create strong magnetic flux density in the air gap, when rare-earth
PMs are used. Very often, rare-earth PMs are replaced with cost-effective ferrite magnets, usually distributed in three or more layers, which can provide
152
Modern Permanent Magnet Electric Machines
Fig. 6.3. Constructions of rotors of PM brushless motors: (a) surface magnets; (b)
spoke-type magnets; (c) interior magnets; (d) inset magnets; (e) double-layer interior
magnets; (f) buried magnets asymmetrically distributed.
a cost-effective alternative solution (Fig. 6.4a). If no PMs are in the axial
slot, the machine will operate as a synchronous reluctance machine with flux
barriers in the rotor core (Fig. 6.4b). Single-layer V-shaped or double-layer
magnets, especially in high-count pole rotors are also common.
An alternative construction involves a rotor with asymmetrically distributed buried magnets and cage winding (Fig. 6.3f) according to German
Patent 1173178 assigned to Siemens, also called Siemosyn. The magnets are
magnetized radially. Owing to the cage winding, the motor is self-starting and
can be directly plugged in to a 50 or 60 Hz utility grid, without any solid state
converter (line start PM synchronous motor).
6.3 Sinusoidally excited and square wave motors
PM brushless motor drives fall into the two principal classes of sinusoidally
excited and square wave (trapezoidally excited) motors. Sinusoidally excited
motors are fed with three-phase sinusoidal waveforms (Fig. 6.5a) and operate
PM Brushless DC Motors and Drive Control
153
Fig. 6.4. Brushless motors with rotors having flux barriers: (a) four-layer interiormagnet rotor; (b) four-flux barrier reluctance motor also called a synchronousreluctance motor 1 – stator core, 2 – rotor core, 3 – shaft, 4 – PMs.
on the principle of a rotating magnetic field. The speed of the rotor is equal
to the synchronous speed of the stator magnetic rotating field, i.e.,
ns =
f
p
(6.2)
where f is the input frequency and p is the number of pole pairs. They are
simply called sinewave motors or PM synchronous motors. All phase windings
conduct current at a time.
Square wave motors are also fed with three-phase waveforms shifted by
120◦ one from another, but these wave shapes are rectangular or trapezoidal
(Fig. 6.5b). Such a shape is produced when the armature current (MMF) is
precisely synchronized with the rotor instantaneous position and frequency
(speed). Surface PMs with a large effective pole arc coefficient
bp
(6.3)
τ
are usually used. In eqn (6.3) bp is the width of the PM pole and τ is the
pole pitch, i.e., the inner circumference of the stator core πD1in divided by
the number of poles 2p.
The most direct and popular method of providing the required rotor position information is to use an absolute angular position sensor mounted on
the rotor shaft. Usually, only two phase windings out of three conduct the
current simultaneously. Such a control scheme or electronic commutation is
αi =
154
Modern Permanent Magnet Electric Machines
functionally equivalent to the mechanical commutation in DC brush motors.
This explains why motors with square wave excitation are called DC brushless
motors.
A comparison of sinusoidal excitation with square wave excitation is given
in Table 6.2.
Fig. 6.5. Basic armature waveforms for three-phase PM brushless motors: (a) sinusoidally excited, (b) square wave.
Table 6.2. Sinusoidally excited versus square wave PM brushless motors.
Feeding
Sinusoidally excited
Square wave
(synchronous) motors (trapezoidally excited) motors
Three-phase sinusoidal Three-phase rectangular or
waveforms shifted
trapezoidal waveforms shifted
by 120◦
by 120◦
Number of phases
conducting at any All the three phases
given point of time
On the principle of
Operation
the rotating magnetic
field
Only two phases
Armature current
synchronized with the rotor
instantaneous position
PM Brushless DC Motors and Drive Control
155
The electromagnetic torque developed by a synchronous motor is usually
expressed as a function of the angle ψ between the q-axis (EMF Ef axis) and
the armature current Ia , i.e.,
Telm = cT Φf Ia cos ψ
(6.4)
where cT is the torque constant. For a PM motor
Telm = kT Ia cos ψ
(6.5)
where kT = cT Φf is a new torque constant. The magnetic flux Φf = const if
the armature reaction is negligible.
The maximum torque is when cos ψ = 1 or ψ = 0◦ . It means that the
armature current Ia = Iaq is in phase with the EMF Ef .
For the DC brushless motor, the torque equation is similar to eqn (5.4)
for a DC brush (commutator) motor, i.e.,
Telm = cT dc Φf Ia = kT dc Ia
(6.6)
where cT dc and kT dc = cT dc Φf are torque constants.
Similar to DC brush motors, eqn (5.1), the EMF of a PM brushless motor
can simply be expressed as a function of the rotor speed n, i.e.,
ˆ
Phase-to-neutral EMF (e.g., unipolar operation)
Ef = cE Φf n = kE n
ˆ
(6.7)
Line-to-line EMF (e.g., bipolar operation)
Ef L−L = cEL−L Φf n = kEL−L n
(6.8)
where cE , cEL−L or kE = cE Φf and kEL−L = cEL−L Φf are the EMF constants, also called the armature constants. For PM field excitation and negligible armature reaction, Φf ≈ const.
The PM brushless motor shows more advantages than its induction or
synchronous reluctance counterparts in motor sizes up to 10–15 kW. The
largest commercially available motors are rated at least at 750 kW (1000 hp).
There have also been successful attempts to build rare-earth PM brushless
motors rated above 1 MW in Germany, and a 36.5 MW PM brushless motor
by DRS Technologies, Parsippany, NJ, USA.
The armature winding of PM brushless motors is usually distributed in
slots. When cogging (detent) torque needs to be eliminated, slotless windings are used. In comparison with slotted windings, the slotless windings provide higher efficiency at high speeds, lower torque ripple and lower acoustic
noise. On the other hand, slotted motors provide higher torque density, higher
efficiency in a lower speed range, lower armature current and use less PM
material.
156
Modern Permanent Magnet Electric Machines
An auxiliary DC field winding located in the rotor or magnetic flux diverters with additional DC winding located in the stator can help to increase
the speed range over constant power region or control the output voltage of
variable speed generators. These machines are called PM brushless machines
with a hybrid field excitation system.
6.4 Method of changing DC bus voltage and speed
control
A brushless motor can be fed from a DC source. In the case of a three-phase
system (Fig. 6.6), the DC bus voltage follows:
ˆ
For a three-phase fully controlled rectifier (six controlled switches),
ˆ
UmL
cos α
(6.9)
π
where UmL is the peak value of the line voltage and α is the so-called firing
angle. The firing angle is the phase angle of the voltage at which the SCR
turns on (conducts).
For a three-phase half-controlled rectifier (3 controlled switches, 3 diodes),
ˆ
UmL
(1 + cos α)
2π
For α = 0, both eqns (6.9) and (6.10) give the same results.
For a three-phase uncontrolled rectifier (6-diode rectifier),
Ud = 3
Ud = 3
Ud = 3
UmL
π
(6.10)
(6.11)
Fig. 6.6. Example of electromechanical drive with a PM DC brushless motor, uncontrolled rectifier and three-phase inverter.
PM Brushless DC Motors and Drive Control
157
Speed can be controlled by changing the input AC voltage UAC , e.g., with
the aid of an autotransformer with diode rectifier to obtain a change in DC
bus voltage Ud . Instead of changing the AC voltage, variable-speed operation
can be achieved by changing DC bus voltage Ud .
To control the DC bus voltage Ud , a thyristor rectifier or gate turn-off
thyristor (GTO) rectifier is used. The DC bus voltage Ud in the above power
circuit shown in Fig. 6.7 is a function of firing angle α of the rectifier bridge.
Fig. 6.7. Example of an electromechanical drive with a PM DC brushless motor,
controlled rectifier and three-phase inverter.
Fig. 6.8. PWM speed control at constant DC bus voltage Ud = const: (a) PWM
technique; (b) influence of pulse width on the average voltage; (c) PWM control of
PM brushless motor drive.
Controlled solid switches are used not only for commutation, but also
for voltage control at the motor input terminals with the aid of pulse width
modulation (PWM). The principle of operation is shown in Fig. 6.8.
158
Modern Permanent Magnet Electric Machines
In the PWM technique, a desired sinusoidal waveform, the modulating
wave, is compared to a much higher-frequency triangular waveform, called
the carrier wave (Fig. 6.8a). The resultant waveform is a train of rectangular pulses. The average voltage depends on the width of pulses (Fig. 6.8b).
Because stator windings of PM brushless motors have large inductance, the
stator current obtained from switched voltage is almost identical with the current obtained from the DC voltage. Appropriate control signals are delivered
to the solid state devices (SSDs) (Fig. 6.8c).
The frequency modulation index is defined as
mf =
∆fd
fm
(6.12)
where ∆fd is the frequency deviation and fm is the modulating frequency
(sine wave).
The amplitude modulation index is the ratio of the peak value Um of modulating sinusoidal signal voltage to the peak value Uc of carrier sawtooth signal
voltage, i.e.,
Um
(6.13)
Uc
In the case of space vector modulation, 0 ≤ ma ≤ 1. The line-to-line inverter
output voltage is
ma =
ˆ
ˆ
in the case of 6-pulse commutation,
√
6
Ud ≈ 0.78Ud
(6.14)
U1L =
π
in the case of a 3-phase PWM inverter and sinusoidal output voltage,
r
3
Ud
U1L =
ma
≈ 0.61ma Ud
(6.15)
2
2
Although PWM speed control is nowadays quite common, variable DC
bus voltage speed control (Fig. 6.7) is still in use in systems, where dynamic
performance is not important.
6.5 Unipolar and bipolar operating mode
In unipolar operating mode, currents in the phase winding flows in one direction during commutation (Fig. 6.9a). Each phase is controlled by only one
SSD. In bipolar operating mode, the current in the phase winding changes
direction during commutation (Fig. 6.9b). Each phase winding is controlled
by two SSDs.
The speed of the PM brushless motor during unipolar operation is higher
than in bipolar operation, e.g., application to high-speed ventilation systems.
PM Brushless DC Motors and Drive Control
159
Fig. 6.9. Operating modes of three-phase PM brushless motors: (a) unipolar operation; (b) bipolar operation.
On the other hand, the starting torque is lower. The main advantage of unipolar operation is that it requires two times fewer SSDs than for bipolar operation.
Bipolar operation causes low starting current because the torque constant
and EMF constants are high. Current ripple is lower than in the case of
unipolar operation.
6.6 Six-step commutation: two phases on
In bipolar operation mode, the motor phase current can be of either positive
or negative polarity. A PM DC brushless motor is driven by a three-phase
inverter bridge and all six solid state switches are used. Since the conduction
period (one step) for line currents is 60◦ , this is a six-step commutation with
only two phases on. In Fig. 6.10 the DC voltage Udc is switched between phase
terminals and for the Y connection, two windings belonging to different phases
are series connected during each conduction period. Neglecting the winding
inductance, the current is
Udc − ef L
(6.16)
2R1
where ef L is line-to-line EMF. The current sequence is iaAB , iaAC , iaBC ,
iaBA , iaCA , iaCB , . . .. For this current sequence, the MMFs FAB , FAC , FBC ,
FBA , FCA , FCB ,. . . rotate counterclockwise (Fig. 6.10). Conduction occurs for
both the positive and negative half of the EMF waveform (bipolar or full-wave
operation). For sinusoidal EMF waveforms, the currents can be regulated in
such a way as to obtain approximately square waves. The electromagnetic
power and torque are always positive because negative EMF times negative
current gives a positive product. Each conduction period (one step) for line
currents is 60◦ (six-step commutation) with two phases on at any time leaving
the remaining phase floating. Current conduction period for two phases on is
120◦ . As a result, the torque ripple is substantially reduced.
At non-zero speed, the maximum torque–to–current ratio is achieved at
the peak of EMF waveforms. The current is in phase with the EMF. The
ia =
160
Modern Permanent Magnet Electric Machines
T1
V dc
D1
T3
D3
T5
D5
A
C
C
B
T2
D2
T4
D4
T6
D6
FAC
A
counter-clockwise
rotation
FAB
A
-C
4
2
1
5
1
2
FBC
-B
6
FCB
3
3
B
C
B
4
6
5
C
-A
FBA
FCA
Fig. 6.10. Switching sequence and MMF phasors for six-step commutation of a
Y-connected DC PM brushless motor. Commutation sequence is AB, AC, BC, BA,
CA, CB, etc.
commutation timing is determined by the rotor position sensors or estimated
on the basis of the motor parameters, e.g., EMF.
The average torque can be maximized and torque ripple can be minimized
if the EMF waveform has a trapezoidal shape (Fig. 6.11). For trapezoidal
operation, the peak line-to-line EMF occurs during the whole conduction period, i.e., 60◦ for line current as given in Figs 6.11 and 6.12. The EMF, i.e.,
ef AC = ef A − ef C = −ef CA = ef C − ef A and the current, e.g., iaAC = −iaCA .
The trapezoidal shape of the line-to-line EMFs is obtained by proper shaping
and magnetizing the PMs and proper designing of the stator winding. Theoretically, the flat top EMF waveforms at DC voltage Ud = const produce
square current waveforms and a constant torque independent of the rotor position (Fig. 6.12). Owing to the armature reaction and other parasitic effects,
the EMF waveform is never ideally flat. However, the torque ripple below 10%
can be achieved. Torque ripple can further be reduced by applying more than
three phases.
PM Brushless DC Motors and Drive Control
efA
30
60
el. degrees
90
120
150
180
210
phase EMFs
0
161
240
270
300
330
360
390
420
450
efB
line-to-line EMFs
efC
efAB
= efA- efB
efBC
= efB- efC
currents
iaA
iaAB
efCA
= efC - efA
iaC
iaB
iaAC =
- iaCA
iaBC
iaBA =
- iaAB
iaCA
iaCB =
- iaBC
Fig. 6.11. Phase and line-to-line trapezoidal EMF and square current waveforms
of a bipolar-driven PM brushless motor with 120◦ current conduction.
6.7 Three phases on: 180-degree conduction
Although six-step commutation is the most economical and popular method
of control of PM brushless motors, it has the following disadvantages:
ˆ
ˆ
ˆ
high torque ripple during commutation;
the efficiency of the whole system is poor;
acoustic noise can be important in the case of larger motors.
In six-step mode operation, only one upper and one lower solid state switch
are turned on at a time (120◦ conduction). With more than two switches on
at a time, a 180◦ current conduction can be achieved, as shown in Fig. 6.13.
If the full current flows, say, through one upper leg, two lower legs conduct
half of the current. All three phases always conduct the current.
Operation under stepped waveforms and a conduction angle of 180◦ is a
basis for sinusoidal operation and PWM of the space vector. The waveform
162
Modern Permanent Magnet Electric Machines
ef
efAB
efAC
efBC
(a)
el. degrees
0
12
0
60
18
0
24
0
30
0
36
0
(b)
ia
(c)
td
Fig. 6.12. Ideal three-phase six-step operation of a Y-connected DC PM brushless
motor: (a) trapezoidal line-to-line EMF waveforms, (b) current waveforms, (c) electromagnetic torque waveforms. Switching points are marked with arrows.
(a)
(b)
T1
T3
T5
A
1
ia
2
1
i
2a
C
1
T2
T4
T6
T1
T3
T5
B
A
ia
180 o
60 o
A
ia
0
1
i
2 a
ia
B
C
2
T2
T4
B
T6
el. degrees
0
1
ia
2
C
T1
T3
T5
1
ia
2
A
C
ia
3
1
ia
2
T2
T4
T6
0
1
2
3
4
5
6
B
Fig. 6.13. Three-phase bipolar-driven Y-connected DC PMBM with three phases
on at a time: (a) commutation, (b) current waveforms.
PM Brushless DC Motors and Drive Control
163
of sinusoidal voltage can be generated when a rectangular waveform is PWM
modulated by a sinusoid with the same phase and frequency.
Advantages of electromechanical drives with 180◦ -conduction sinewave
brushless motors include:
ˆ
ˆ
ˆ
better utilization of SSDs;
very small torque pulsations (EMF is very close to sinusoidal waveform);
low noise of electromagnetic origin.
6.8 Rotor position sensing
Rotor position sensing in PM DC brushless motors is done by position sensors,
i.e.,
(a) Hall elements (Fig. 6.14a);
(b) encoders (Fig. 6.14b);
(c) resolvers (Fig. 6.14c).
In rotary machines, position sensors provide feedback signals proportional to
the rotor angular position.
Fig. 6.14. Rotor position sensors: (a) Hall element; (b) encoder; (c) resolver.
6.8.1 Hall sensors
The Hall element is a magnetic field sensor that takes advantage of the phenomenon known as the Hall effect. When placed in a stationary magnetic field
and fed with a DC current, it generates an output voltage (Fig. 6.15a).
1
VH = kH Ic B sin β
δ
(6.17)
where kH is the Hall constant in m3 /C, δ is the semiconductor thickness,
Ic is the applied current, B is the magnetic flux density and β is the angle
164
Modern Permanent Magnet Electric Machines
between the vector of B and the Hall element surface. The polarity depends
on whether the pellet is passing a North or a South pole. Thus, it can be used
as a magnetic flux detector.
Fig. 6.15. Hall element: (a) principle of operation; (b) block diagram of the inner
IC.
Hall sensors come in small IC packages and usually have three pins. The
simplified IC is shown in Fig. 6.15b. Rotor position sensing of three-phase DC
brushless motors requires three Hall elements. All the necessary components
are often fabricated in an IC. In most cases, satisfactory operation requires
the mechanical separation of the Hall elements to be given by
360◦
(6.18)
m1 p
For example, in the case of a two-pole (p = 1), three-phase (m1 = 3) DC
brushless motor, a mechanical displacement of 120◦ between individual Halleffect devices is required. The sensors should be placed 120◦ apart as in Fig.
6.16a. However, they can also be placed at 60◦ intervals as shown in Fig.
6.16b. Hall sensors generate a square wave with 1200 phase difference, over
one electrical cycle of the motor. The inverter or servo amplifier drives two of
the three motor phases with DC current during each specific Hall sensor state
(Fig. 6.16c).
αH =
6.8.2 Encoders
In optical encoders, a light passes through the transparent areas of a rotating
disk (grating) and is sensed by a photodetector (Fig. 6.17). To increase the
resolution, sometimes a collimated light source is used and a mask is placed
between the grating and detector. The light is allowed to pass to the detector
only when the transparent sections of the grating and mask are in alignment.
In an incremental encoder , a pulse is generated for a given increment of
shaft angular position, which is determined by counting the encoder output
PM Brushless DC Motors and Drive Control
165
Fig. 6.16. Hall element-based three-phase position sensor: (a) Hall element spacing
for 120 electrical degrees; (b) Hall element spacing for 60 electrical degrees; (c) sensor
signals and phase current waveforms.
pulses from a reference. The grating has a single track. In the case of power
failure an incremental encoder loses position information and must be reset
to a known zero point.
An absolute encoder is a position verification device that provides unique
position information for each shaft angular location. Owing to a certain number of output channels, every shaft angular position is described by its own
unique code. The number of channels increases as the required resolution
increases. An absolute encoder is not a counting device like an incremental
encoder and does not lose position information in the case of loss of power.
To understand the importance of absolute encoders, it is good to first understand the limitations of incremental encoders. Fig. 6.19 shows how an incremental encoder uses quadrature output signals to convey position information.
166
Modern Permanent Magnet Electric Machines
Fig. 6.17. Principle of operation of an optical encoder. 1 – LED, 2 – rotating disk,
3 – photo sensor, 4 – squaring IC.
Fig. 6.18. Rotating disks of optical encoders: (a) incremental encoder; (b) absolute
encoder.
In incremental encoders, there are 4 distinct states, and those 4 states are repeated over the full rotation of the encoder. Since there are only 4 states, the
host cannot determine the encoder exact radial position without a reference.
Many incremental encoders include an index signal which occurs once per
rotation and can be used as a home location to count from.
This output is useful for obtaining speed information, direction of travel,
and can be used to count up or down from the index position. However, this
type of encoder is not useful when the host system must know the current
PM Brushless DC Motors and Drive Control
167
Fig. 6.19. Output signals of optical encoders: (a) incremental encoder; (b) absolute
encoder.
position immediately after power on. An incremental encoder can give a precise radial position, but only after physically rotating to the index location.
Unlike incremental quadrature encoders that repeat the same 4 states over
a revolution, an absolute encoder generates a unique digital word for each
position in its stated resolution. Because many absolute encoders are digital
devices, resolution is expressed as an exponent of 2, otherwise known as binary.
The numbers on the right of the absolute output (Fig. 6.19b) represent the
numeric value of the bit when it is “on” or “high.” A 6-bit (26 ) absolute
encoder can generate 64 unique digital words that represent 64 positions over
one revolution. Five positions are illustrated in Fig. 6.19b. At the blue line,
only the 20 bit is “high,” so the output is 1. At the red line, the 20 , 21 22 and
23 bits are “high,” so that 1 + 2 + 4 + 8 = 15.
6.8.3 Resolvers
A resolver is a rotary electromechanical transformer that provides outputs in
forms of trigonometric functions sin(ϑ) and cos(ϑ) of its inputs. For detecting the rotor position of brushless motors, the excitation or primary winding
is mounted on the resolver rotor and the output or secondary windings are
wound at right angles to each other on the stator core. As a result, the output signals are sinusoidal waves in quadrature; i.e., one wave is a sinusoidal
function of the angular displacement ϑ and the second wave is a cosinusoidal
function of ϑ (Fig. 6.20). Instead of delivering the excitation voltage to the rotor winding by brushes and slip rings, a rotary transformer (inductive coupling
system) is frequently used (Fig. 6.21). The rotary transformer is a transformer
168
Modern Permanent Magnet Electric Machines
with an air gap, the rotor of which is mounted on the same shaft as the rotor
of the resolver.
Fig. 6.20. Principle of operation of a rotary resolver.
Fig. 6.21. Construction of a resolver with rotary transformer: (a) longitudinal section; (b) disassembled resolver with removed rotor. 1 – stator assembly, 2 – stator of
rotary transformer, 3 – rotor of rotary transformer, 4 – rotor assembly, 5 – housing,
6 – shaft.
6.8.4 Sensorless control
There are several reasons to eliminate electromechanical position sensors:
ˆ
ˆ
ˆ
Cost reduction of electromechanical drives
Reliability improvement of the system
Temperature limits on Hall sensors
PM Brushless DC Motors and Drive Control
ˆ
ˆ
169
In motors rated below 1 W, the power consumption by position sensors
can substantially reduce the motor efficiency
In compact applications, e.g., computer hard disk drives, it may not be
possible to accommodate position sensors
In general, the position information of the shaft of PMBMs can be obtained
using one of the following techniques:
(a) Detection of back EMF (zero crossing approach, phase-locked loop technique, EMF integration approach)
(b) Detection of the stator third harmonic voltage
(c) Detection of the conducting interval of freewheeling diodes connected in
antiparallel with the solid state switches
(d) Sensing the inductance variation (in the d and q-axis), terminal voltages
and currents
6.9 Mathematical model
Assuming no rotor currents (no damper, no retaining sleeve, very high resistivity of magnets and pole faces) and the same stator phase resistances,
Kirchhoff voltage equation for a three-phase machine can be expressed in the
following matrix form (Fig. 6.22):

 


u1A
R1 0 0
iaA
 u1B  =  0 R1 0   iaB 
u1C
0 0 R1
iaC


 

L L
L
i
e
d  A BA CA   aA   f A 
LBA LB LCB
iaB + ef B
+
dt
LCA LCB LC
iaC
ef C
Fig. 6.22. Circuit diagram of a three-phase PM brushless motor.
(6.19)
170
Modern Permanent Magnet Electric Machines
For inductances independent of the rotor angular position, the self-inductances
LA = LB = LC = L and mutual inductances between phases LAB = LCA =
LCB = M are equal. For no neutral wire iaA + iaB + iaC = 0 and M iaA =
−M iaB − M iaC . Hence

 


u1A
R1 0 0
iaA
 u1B  =  0 R1 0   iaB 
u1C
0 0 R1
iaC
 

 
iaA
ef A
L−M
0
0
d
L−M
0   iaB  +  ef B 
+ 0
dt
iaC
ef C
0
0
L−M

(6.20)
The electromagnetic instantaneous power per phase at a given time instant is
pelm = ia ef and the electromagnetic instantaneous torque is
1
(ef A iaA + ef B iaB + ef C iaC )
(6.21)
2πn
For a bipolar commutation and 120◦ conduction, only two phases conduct
(tr)
(tr)
at any time instant. For example, if ef A = Ef , ef B = −Ef , ef C = 0,
Telm =
(sq)
(sq)
and iaC = 0, the instantaneous electromagnetic
iaA = Ia , iaB = −Ia
torque according to eqn (6.21) is
Telm =
(tr)
(tr) (sq)
Ia
2Ef
2πn
(6.22)
(sq)
where Ef and Ia are flat-topped values of trapezoidal EMF and square
wave current. For constant values of EMF and currents, the torque (6.22) does
not contain any pulsation.
Since ef = ωψf = (2πn/p)ψf where ψf is the flux linkage per phase
produced by the excitation system, the instantaneous torque (6.21) becomes
Telm = p(ψf A iaA + ψf B iaB + ψf C iaAC )
(6.23)
For computer simulation of PM brushless motors, eqns (6.21), (6.20) and
(6.23) must be supplemented by the torque balance equation, i.e.,
d2 ϑ
dϑ
+ Kϑ ϑ = Telm ± T
(6.24)
+ Dϑ
dt2
dt
where J is the rotor moment of inertia, D is the torsional damping constant
(friction), K is the torsional compliance constant and T is the external torque.
J
PM Brushless DC Motors and Drive Control
171
6.10 Cogging torque
The cogging effect (detent effect) is defined as the interaction between the
rotor magnetic flux and variable permeance of the air gap due to the stator
slot geometry, i.e., slot openings. The cogging effect produces torque pulsations
(Fig. 6.23), the so-called cogging torque.
Neglecting the armature reaction and magnetic saturation, the cogging
torque is independent of the stator current. The fundamental frequency of the
cogging torque is a function of the number of slots s1 , number of pole pairs p
and input frequency f . One of the cogging frequencies (usually fundamental)
can be estimated as
fc = 2ncog f ;
ncog =
LCM (s1 , 2p)
2p
if
Ncog =
2p
≥1
GCD(s1 , 2p)
(6.25)
where LCM (s1 , 2p) is the least common multiple of the number of slots s1
and number of poles 2p, GCD(s1 , 2p) is the greatest common divisor of s1
and 2p and ncog is sometimes called the fundamental cogging torque index
[43].
Fig. 6.23. Cogging torque waveforms versus rotor position angle: (a) without no
skew of stator slots; (b) with skewed stator slots.
For example, for s1 = 36 and 2p = 2, the fundamental cogging torque
index ncog = 18 (LCM = 36, GCD = 2, Ncog = 1), for s1 = 36 and 2p = 6
the index ncog = 6 (LCM = 36, GCD = 6, Ncog = 1), for s1 = 36 and
2p = 8 the index ncog = 9 (LCM = 72, GCD = 4, Ncog = 2), for s1 = 36 and
2p = 10 the index ncog = 18 (LCM = 180, GCD = 2, Ncog = 5), for s1 = 36
and 2p = 12 the index ncog = 3 (LCM = 36, GCD = 12, Ncog = 1), etc. The
larger the LCM (s1 , 2p), the smaller the amplitude of the cogging torque.
The torque ripple can be minimized both by the proper motor design
and motor control. Measures taken to minimize the cogging torque by motor
design include [32]
172
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
Modern Permanent Magnet Electric Machines
elimination of slots (slotless winding);
skewed slots (Fig. 6.23);
special shape slots and stator laminations;
selection of the number of stator slots with respect to the number of poles;
decentered magnets;
skewed magnets;
shifted magnet segments;
selection of magnet width;
direction-dependent magnetization of PMs.
Control techniques use modulation of the stator current or EMF waveforms.
6.11 The smallest and the biggest PM brushless motors
in the world
Electric ship propulsion requires large electric motors. For example, a 90,000
gt cruise ship employs two 19.5 MW synchronous motors. Low-speed PM
brushless motors offer significant savings in mass (up to 50%) and efficiency
(2% to 4% at full load and 15% to 30% at partial load) as compared to
high-speed synchronous motors with electromagnetic excitation and reduction
gears. Fig. 6.24 shows the most powerful PM brushless motor in the world for
advanced ship propulsion rated at 36.5 MW and 127 rpm.
An optimum undisturbed water inflow to the propeller and consequently
reduced propeller pressure pulses (causing vibration and noise) and increased
propulsion efficiency can be achieved with the aid of a pod propulsor . Reduction of vibration and noise considerably enhances passenger comfort. The
propeller acts as a tractor unit located in front of the pod. The pod can be
rotated through 3600 to provide the required thrust in any direction. This
eliminates the requirement for stern tunnel thrusters and ensures that ships
can maneuver into ports without tug assistance.
The smallest high-speed PM brushless motor in the world for clinical engineering applications is shown in Fig. 6.25. The stator of this PM brushless
motor is a coreless type with skewed winding. The outer diameter of the motor
is 1.9 mm, the length of the motor alone is 5.5 mm and together with gearhead is 9.6 mm (Figs 6.25). The rotor has a 2-pole NdFeB PM on a continuous
spindle. The maximum output power is 0.13 W, no-load speed 100, 000 rpm,
maximum current 0.2 A (thermal limit), and maximum torque 0.012 mNm
[32]. The high-precision rotary speed setting allows analysis of the received
ultrasound echoes to create a complex ultrasound image.
Brushless motors with planetary gearhead and outer diameter below 2
mm have many potential applications such as motorized catheters,1 minimally
invasive surgical devices, implantable drug-delivery systems and artificial organs. An ultrasound catheter consists of a catheter head with an ultrasound
1
A catheter is a tube that can be inserted into a body cavity, duct or vessel.
PM Brushless DC Motors and Drive Control
173
Fig. 6.24. PM brushless motor rated at 36.5 MW and 127 rpm for advanced ship
propulsion. Photo courtesy of DRS Technologies, Parsippany, NJ, U.S.A.
Fig. 6.25. Expanded view of the smallest electromechanical drive system in the
world with (a) PM brushless micromotor and (b) microplanetary gearhead. 1 –
housing (enclosure) of micromotor, 2 – end cap, 3 – bearing support, 4 – bearing of
micromotor, 5 – PM, 6 – shaft, 7 – armature winding, 8 – washer, 9 – end cover,
10 – ring gear, 11 – planet gear, 12 – sun gear, 13 – planetary stage, 14 – output
shaft, 15 – housing of microplanetary gearhead, 16 – bearing cover, 17 – retaining
ring. Source: Faulhaber Micro Drive Systems and Technologies - Technical Library,
Croglio, Switzerland.
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Modern Permanent Magnet Electric Machines
transducer on the motor/gearhead unit and a catheter tube for the power
supply and data wires. The site to be examined can be reached via cavities
like arteries or the urethra.2 The supply of power and data to and from the
transmit/receive head is provided via slip rings.
6.12 Wiring diagram for a solid-state converter-fed PM
brushless motor
Most PM brushless motors are fed from voltage-source, PWM solid state
converters. A power electronics converter consists of a rectifier bridge, intermediate circuit (filter) and inverter (DC to AC conversion).
Fig. 6.26. Wiring diagram for a solid state converter-fed PM brushless motor.
A wiring diagram for a converter-fed motor is shown in Fig. 6.26. To obtain
proper operation, minimize radiated noise and prevent shock hazard, proper
interconnection wiring, grounding and shielding are important. Many solid
state converters require a minimum of 1% to 3% line impedance calculated as
U10L−L − U1rL−L
× 100%
(6.26)
U1rL−L
where V10L−L is the line-to-line voltage measured at no load and V1rL−L is
the line-to-line voltage measured at full rated load. The minimum required
inductance of the line reactor is
z% =
1 U1L−L z%
H
(6.27)
2πf Ia 100
where f is the power supply frequency (50 or 60 Hz), U1L−L is the input
voltage measured line to line and Ia is the input current rating of control.
L=
2
The urethra is a tube which connects the urinary bladder to the outside of the
body.
PM Brushless DC Motors and Drive Control
175
6.13 Integrated circuits (IC) for control of PM brushless
motors
The most common configuration for sequentially applying current to a threephase PM brushless motor is to use three pairs of power MOSFETs arranged
in a bridge structure, as shown in Fig. 6.27. Each pair governs the switching of
one phase of the motor. In a typical arrangement, the high-side MOSFETs are
controlled using pulse-width modulation (PWM), which converts the input
DC voltage into a modulated driving voltage. The use of PWM allows the
start-up current to be limited and offers precise control over speed and torque
range. The PWM frequency is a trade-off between the switching losses that
occur at high frequencies and the ripple currents that occur at low frequencies.
Typically, the PWM frequency is at least an order of magnitude higher than
the frequency for maximum motor rotational speed.
Fig. 6.27. Three-phase PM brushless motor powered by three pairs of MOSFETs
arranged in a bridge structure and controlled by PWM.
There are plenty of proven integrated products on the market that can be
used as the building blocks for the circuitry.
Allegro Microsystems’ A4915 three-phase MOSFET3 driver operates as a
pre-driver for a six-power MOSFET bridge for a brushless DC motor. This
device is designed for battery-powered products. One notable feature for saving power is a low-power sleep mode which ensures the device draws minimal
current when not turning the motor. The device also features synchronous
rectification, a technique borrowed from switching voltage regulators to lower
power consumption and eliminate the need for external Schottky diodes.
3
Metal–oxide–semiconductor field-effect transistor.
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Modern Permanent Magnet Electric Machines
Microchip also offers a pre-driver for a six-power MOSFET bridge for a
brushless DC motor, but this time for small sensorless units used in automotive, home appliances and hobby products. The MCP8025 device integrates
a step-down (“buck”) switching regulator to power an external controller in
addition to two low-drop-out (LDO) linear regulators and a charge pump to
power the MOSFET bridge.
This chip keeps things simple by measuring the back EMF of the floating
winding, which is then compared to the motor’s neutral point. When the back
EMF crosses the zero point, the zero-crossing detector sends a signal to the
host controller to indicate the commutation reference point.
Fig. 6.28. Closed-loop control system for a sensored three-phase PM brushless DC
motor. Courtesy of Texas Instruments [21].
Texas Instruments’ (TI) DRV8313 takes things a step further by integrating three individually controllable half-H bridge drivers [21]. The advantage
of this arrangement is that as well as being used for three-phase PM brushless
DC motor control, the chip can be used to drive a mechanically commutated
motor (using two of the half-H bridges) or three independent solenoids. The
chip can supply up to 3.5 A from an 8 to 60 V supply.
The DRV8313 does not include sensor inputs. TI suggests that for either
sensored or sensorless operation, the chip should be teamed with a microcontroller such as the popular MSP430. Such an arrangement, as illustrated
in Fig. 6.28, provides a complete closed-loop control system for a sensored,
three-phase brushless DC motor. The circuit comprises an analog speed input,
MSP430 microcontroller supervising the PWM outputs for the power MOSFETs, a six-MOSFET bridge driver, MOSFET bridge and PM brushless DC
motor. Motor stator and rotor positions are determined by three Hall-effect
sensors which feed signals to the microcontroller.
PM Brushless DC Motors and Drive Control
177
6.14 Practical electromechanical drive system
A block diagram of a practical electromechanical drive system is shown in Fig.
6.29. The power circuit consists of a solid state converter (rectifier), intermediate circuit (capacitor for VSI) and inverter . The control circuit consists of a
controller area network (CAN), microcontroller and gate driver . A gate driver
is a power amplifier that accepts a low-power input from a controller IC and
produces a high-current drive input for the gate of a high-power transistor
such as an IGBT or power MOSFET.
An optocoupler is an electronic component that interconnects two separate
electrical circuits by means of a light-sensitive optical interface.
Fig. 6.29. Block diagram of a practical electromechanical drive system. CAN –
controller area network, HVIC – high-voltage integrated circuit, LVIC – low-voltage
integrated circuit, OPTO – optocoupler.
6.15 Selected applications
6.15.1 Computer hard disk drives (HDD)
The data storage capacity of a hard disk drive (HDD) is determined by the
aerial recording density and number of disks. The aerial density is now 155
Gbit/cm2 = 1000 Gbit/in2 (2020). The mass of the rotor, moment of inertia
and vibration increase with the number of disks. Circumferential vibration of
mode r = 0 and r = 1 causes deviations of the rotor from the geometric axis of
rotation. Disk drive spindle motors are brushless DC motors with outer rotor
designs. Drives with a large number of disks have the upper end of the spindle
fixed to the top cover with a screw (Fig. 6.30a). This “tied” construction
reduces vibration and deviations of the rotor from the center axis of rotation.
For a smaller number of disks, the so-called “untied” construction with fixed
shaft (Fig. 6.30b) or rotary shaft (Fig. 6.30c) has been adopted.
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Modern Permanent Magnet Electric Machines
Fig. 6.30. Construction of spindle motors for HDDs: (a) tied type, (b) untied type
with fixed shaft, (c) untied type with rotary shaft. 1 — stator, 2 — PM, 3 — shaft,
4 — ball bearing, 5 — base plate, 6 — disk, 7 — disk clamp, 8 — top cover, 9 —
thrust bearing, 10 — radial bearing, 11 — screw.
Heads of the HDD are driven by the so-called voice coil actuator . It is
a PM motor with limited rotary movement. The PM system consists of two
magnetized plates (top and bottom). The coreless coil moves between the
poles of the PMs. Depending on the direction of the current, the coil moves
left or right, thus moving the read/write heads.
Special design features of spindle motors are their high starting torque, limited current supply, reduced vibration and noise, and physical constraints on
volume and shape, contamination and scaling problems. High starting torque,
10 to 20 times the running torque, is required, since the read/write head
tends to stick to the disk when not moving. The starting current is limited
by the computer power supply, which severely limits the starting torque. For
a 2.5-inch, 20,000-rpm, 12-V HDD, the starting current is less than 2 A at
a starting torque of 6.2 mNm. The acoustic noise is usually below 30 dB(A)
and nonrepeatable run out maximum 2.5 × 10−5 µmm.
The choice of the number of poles determines the frequency of torque ripple
and switching frequency. Although larger numbers of poles reduce the torque
ripple, it increases switching and hysteresis losses and complicates commutation tuning and installation of rotor position sensors. Most commonly used
are four-pole and eight-pole motors. The pole-slot combination is important
in reducing the torque ripple. Pole-to-slot ratios with high least common multiple LCM (s1 , 2p) such as 8-pole/9-slot (LCM (9, 8) = 72) and 8-pole/15-slot
(LCM (15, 8) = 120) produce very small cogging torque.
Drawbacks of ball bearings include noise, low damping, limited bearing
life and nonrepeatable run out. The HDD spindle motor is now changing from
ball bearing to a fluid dynamic bearing (FDB) motor. Contact-free FDBs (Fig.
6.33) produce less noise and are serviceable for an extended period of time.
6.15.2 Two-phase PM brushless motors for computer cooling fans
Small two-phase permanent magnet (PM) brushless motors for computers
and other electronics equipment cooling fans are one of the most popular
PM Brushless DC Motors and Drive Control
179
Fig. 6.31. Computer hard disk drive (HDD). 1 — base casting, 2 — spindle, 3 —
slider and heads, 4 — actuator arm, 5 — actuator shaft, 6 — voice coil actuator,
7 — small computer systems interface (SCSI) connector, 8 — jumper pins, 9 —
jumper, 10 — power connector, 11 — tape seal, 12 — ribbon cable (attaches heads
to logic board), 13 — platters, 14 — case mounting holes, 15 — cover mounting
holes (cover not shown).
electric motors. The central processing unit (CPU) generates the most heat
in a typical personal computer (PC). This heat needs to be removed quietly
and efficiently. It is estimated that there were more than 2 billion PCs in use
in 2015. So the number of fan motors nowadays well exceeds 2 billion. In spite
of a large number of single-phase PC brushless motors installed in computers,
very few research papers have been devoted to these motors [5, 49, 62, 82, 83].
Computer cooling fans are typically based on two-phase PM brushless motors with an inner stator and outer PM rotor drawing between 1 and 50 W of
electric power. An integrated circuit (IC) on the printed circuit board (PCB)
controls the stator windings, energizes the coils, and changes the magnetic field
that interacts with PMs located in the outer rotor to keep the motor spinning. Many PC motherboards feature hardware and software that regulates
the speed of fans based on the processor and computer case temperatures.
Solutions have been proposed to provide variable speed control for two-phase
180
Modern Permanent Magnet Electric Machines
Fig. 6.32. Spindle motor of HDD. 1 — base casting, 2 — spacer ring, 3 — platter,
4 — spindle motor, 5 — motor shaft, 6 — top cover.
Fig. 6.33. Construction of FDB spindle motors for HDDs: (a) fixed-shaft spindle
motor, (b) rotating-shaft spindle motor. 1 — stator, 2 — PM, 3 — shaft, 4 — radial
bearing, 5 — thrust bearing, 6 — disk, 7 — stopper/seal, 8 — hub, 9 — spacer,
10 — clamp, 11 — base plate, 12 — attractive magnet.
brushless motor assemblies, while limiting the number of wires connecting to
such assemblies to three, a desirable cost saving objective [62, 83].
The speed control of two-phase brushless motor assemblies can be done
by adjusting the DC voltage to the motor, applied between the supply and
ground wires. The third wire is then used for the tachometer’s feedback signal.
A control IC includes a speed monitor, which receives a tachometer signal from
the fan. Control signals generated by the system PCB and provided to the
fan assembly can use the same wire as tachometer signals generated in the
fan assembly. From the user point of view, there are PC fan motors with a
two-pin, three-pin and four-pin connector.
With respect to acoustic noise, reliability, and power efficiency, the most
preferable method of fan control is the use of a high-frequency (≤ 20 kHz)
PM Brushless DC Motors and Drive Control
181
pulse width modulation (PWM) drive. The latest technology (Yen Sun Technology, Soeul, Korea) in computer cooling fans is the tip-driven fan that moves
the motor out of the hub of the fan, and puts it around the edge [80]. The
impeller blades are surrounded by a ring studded with 12 magnets, which are
acted upon by four coils that are located at the corners of the housing of the
fan. The tips of the blades can also be made of a hard magnetic material and
magnetized in a radial direction.
The cost-effective two-phase brushless motors for computer fans have a
salient-pole inner stator and ring-shaped outer PM rotor. The outer PM rotor
is integrated with the fan blades facilitating air flow. The housing is mechanically connected with the inner stator of the motor with the aid of a spider
structure. The details of construction of a PC fan motor are shown in Fig.
6.34. A Hall sensor detects the polarity of PMs, and via solid state devices,
switches the DC voltage from one stator coil to another. The speed of the
fan motor is controlled by adjusting either the DC voltage or pulse width in
low-frequency PWM.
In spite of the fact that the PM brushless motor has four dead spots per
revolution, it has good self-starting capability. Since the rotor rests between
the poles of PMs at zero-current state (Fig. 6.35), and instantly rotates 45◦
when first switched on, it will not stop on one of its dead spots.
Fig. 6.34. Construction of a PM BLDC motor drive for computer fans: (a) disassembled motor; (b) inner stator with four salient poles; (c) PCB; (d) external rotor
with 4-pole ring-shaped PM rotor.
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Modern Permanent Magnet Electric Machines
Fig. 6.35. Stator coils, PM polarity and Hall sensor position: (a) opposite coils
connected in pairs; (b) neighboring coils connected in pairs. 1 – PM, 2 – stator pole,
3 – coil of one phase, 4 – coil of the second phase, 5 – Hall sensor located between
two stator poles.
Two-phase stator winding consists of four coils wrapped around the stator
pole cores. There are four coils in the inner stator (Fig. 6.35b), while two
neighboring coils have different magnetic polarity. The coils are connected in
pairs, either each one with its opposite coils (Fig. 6.35a), or with its neighboring coils (Fig. 6.35b). Around the perimeter of the outer rotor, there are
four PMs in an N-S-N-S pattern (Fig. 6.34b, Fig. 6.35). Typically, a 12-V DC
cooling fan motor consists of a rotor-blade assembly containing a 4-pole PM,
and a 4-pole stator. A Hall sensor detects the rotating magnetic field and
switches 12 V DC from one stator coil set to another (Fig. 6.35). Varying the
supplied DC voltage can vary the speed of most fans. A 12-V DC fan might
start rotating with 3.5 to 5.0 V DC voltage applied, and increase its speed
when increasing voltage is supplied.
Typical electronic circuits for feeding and controlling PC fan motors are
shown in Fig. 6.36. The common cooling fans used in computers use standardized connectors with two to four pins. The first two pins are always used to
deliver power to the fan motor, while the rest can be optional, depending on
fan design and type:
ˆ
ˆ
ˆ
ˆ
ground;
power (+12 V);
sense: provides a tachometer signal that measures the actual speed of the
fan as a pulse train, frequency being proportional to speed (with each fan
rotation, there are two pulses sent through this pin);
control: provides a PWM signal, which gives the ability to adjust the
rotation speed without changing the input voltage delivered to the cooling
fan.
PM Brushless DC Motors and Drive Control
183
Fig. 6.36. Simplified diagrams of a built-in electronic circuit (PCB): (a) stator coils
and outer PM rotor; (b) circuit with two-pin connector (no provision to control the
fan by an external signal); (b) circuit with three-pin connector; (d) circuit with fourpin connector [10]. The “+” and “-” are power supply terminals, C is the control
pin, T is the tachometer pin (speed sensing) and HS is the Hall sensor.
The PWM is a common method of controlling computer fans. A PWM-capable
fan is usually connected to a 4-pin connector (Fig. 6.36d). The sense (tachometer) pin is used to relay the rotation speed of the fan. The control pin is an
open-drain or open-collector output, which requires a pull-up to 5.0 V or 3.3
V in the fan. Unlike linear voltage regulation, where the fan voltage is proportional to the speed, the fan is driven with a constant supply voltage; the
speed control is performed by the fan based on the control signal.
The control signal is a square wave operating at 25 kHz, with the duty cycle
determining the fan speed. Typically, a fan can be driven between about 30%
and 100% of the rated fan speed, using a signal with up to 100% duty cycle.
The exact speed behavior (linear, off until a threshold value, or a minimum
speed until a threshold) at low control levels is manufacturer dependent [7].
Speed regulators are used by many manufacturers to keep the fans quieter.
Control is performed on a temperature basis. Measurement sensors constantly
monitor temperatures (such as on cooling elements). If the temperature is too
high, then the control unit increases the operating voltage for the fan and
hence the rotor speed and air flow.
184
Modern Permanent Magnet Electric Machines
Bearings are a critical component in a cooling fan because bearings make
the fan rotate smoothly. Bearings reduce friction, allow the fan to operate at
high speeds, and are partly responsible for the life expectancy of a cooling fan
in a computer and the noise level of fans. Three types of bearings can be used
in a cooling fan: (a) sleeve bearings, (b) ball bearings, and (c) fluid dynamic
bearings.
6.15.3 PM brushless motors integrated with an electronic control
circuit
The integrated electromechanical drive also called a smart motor combines
the electromechanical, electrical and electronic components, i.e., motor, power
electronics, position, speed and current sensors, controller and protection circuit together in one package (Fig. 6.37).
Fig. 6.37. PM brushless motor integrated with an electronic control circuit and
gears. Courtesy of AVL, Graz, Austria.
The traditional concept of an electrical drive is to separate the mechanical
functions from the electronic functions, which in turn requires a network of
cables. In the smart motor or integrated drive, the electronic control, position
sensors and power electronics are mounted inside the motor against the casing,
thus reducing the number of input wires to the motor and forming a structurally sound design. The cables connected to a smart motor are generally
the power supply and a single speed signal. In addition, traditional compatibility problems are solved, the standing voltage wave between the motor and
PM Brushless DC Motors and Drive Control
185
converter (increase in the voltage at the motor terminals) is reduced, and
installation of a smart motor is simple. To obtain an even more compacted
design, sensorless microprocessor control is used. Careful attention must be
given to thermal compatibility of components, i.e., excessive heat generated
by the motor winding or power electronics module can damage other components.
Table 6.3 shows specifications of smart PM DC brushless servo motors
(3400 Series) manufactured by Animatics, Santa Clara, CA, USA. These compact units consist of a high power density PM DC brushless servo motor, encoder, PWM amplifier, controller and removable 8 kB memory module which
holds the application program for stand-alone operation, PC or PLC control.
Table 6.3. Smart PM DC brushless motors (3400 Series) manufactured by Animatics, Santa Clara, CA, U.S.A.
Specifications
3410 3420 3430 3440 3450
Rated continuous power, W
120 180 220 260 270
Continuous torque, Nm
0.32 0.706 1.09 1.48 1.77
Peak torque, Nm
1.27 3.81 4.06 4.41 5.30
No load speed, rpm
5060 4310 3850 3609 3398
Number of poles
4
Number of slots
24
EMF constant, V/krpm
9.2 10.8 12.1 12.9 13.7
Torque constant, Nm/A
0.0883 0.103 0.116 0.123 0.131
Rotor moment of inertia, kgm2 × 10−5 4.2
9.2 13.0 18.0 21.0
Length, mm
88.6 105 122 138 155
Width, mm
82.6
Mass, kg
1.1
1.6 2.0 2.5 2.9
A smart motor should be able to encompass the best of all materials and
optimal electromagnetic, thermal and mechanical design, combined with all
aspects of noise, vibration, and harshness (NVH) and component noise optimization. Therefore, the smart motor might be a brushless rare-earth PM
motor, which admits use at steady-state temperatures exceeding 180◦ C, with
liquid cooling, iron-cobalt laminations, insulating material, resins and pottings, able to withstand temperatures exceeding 200◦ C. Such a PM motor
has a built-in power electronics converter, sensors, controller, protection circuit, sometimes reduction gears and a brake.
6.15.4 Hybrid electric vehicles
Combustion engines of automobiles are one of the major oil consumers and
sources of air pollution. Oil conservation and road traffic congestion call for
186
Modern Permanent Magnet Electric Machines
new energy sources for propulsion of motor vehicles and protection of the
natural environment.
An electric vehicle (EV) is driven solely by an electric motor fed from an
on-board rechargeable energy storage system (RESS), e.g., a battery.
A hybrid electric vehicle (HEV) has a conventional combustion engine
(gasoline or diesel), electric motor and RESS, so the wheels of the vehicle are
driven by both a combustion engine and electric motor. All the energy wasted
during braking and idling in conventional vehicles is collected, stored in the
RESS and utilized in HEVs. The electric motor assists in acceleration (energy
saved by the RESS), which allows for a smaller and more efficient combustion
engine.
In most contemporary HEVs, called “charge-sustaining,” the energy for
battery charging is produced by the internal combustion engine. Some HEVs,
called “plug-in” or “charge-depleting,” can charge the battery from the utility
grid.
HEVs have many advantages over classical vehicles with gasoline or diesel
engines: most important follow:
(a) Smaller combustion engine, lower fuel consumption since part of the energy
is derived from the RESS, and improved efficiency (about 40% better fuel
efficiency than that for conventional vehicles of similar ratings).
(b) High electric motor torque at low speed with high combustion engine
torque in higher speed ranges make the torque-speed characteristic suitable
for traction requirements (Fig. 6.38).
(c) Utilization of wasted energy at braking (regenerative braking), idling and
low speeds.
(d) The use of an electric motor reduces air pollution and acoustic noise.
(e) Wear and tear on the combustion engine components decrease, so they can
work for a longer period of time.
(f) Lower maintenance costs due to reduced fuel consumption.
(g) Although the initial cost of HEVs is higher than conventional cars, their
operating costs are lower over time.
EVs and HEVs use brushless electric motors, i.e., PM brushless motors,
switched reluctance motors (SRMs) and induction motors (IMs). Simulations
indicate that a 15% longer driving range is possible for an EV with PM brushless motor drive systems compared with induction types. PM brushless motor
drives show the best efficiency, output power to mass, output power to volume
(compactness) and overload capacity factors.
In series HEVs an electric motor drives the wheels, while the combustion
engine drives the electric generator to produce electricity. In parallel HEVs
the combustion engine is the main way of driving the wheels and the electric
motor assists only for acceleration. A series–parallel HEV (similar to Toyota
Prius) is equipped with a so-called power split device (PSD), which delivers
a continuously variable ratio of combustion engine-to-electric motor power to
the wheels. It can run in “stealth mode” on its stored electrical energy alone.
PM Brushless DC Motors and Drive Control
187
Fig. 6.38. Torque-speed characteristics of a combustion engine and traction electric
motor.
The PSD (Fig. 6.39) is a planetary gear set that removes the need for
a traditional stepped gearbox and transmission components in an ordinary
gasoline-powered car. It acts as a continuously variable transmission (CVT)
but with a fixed gear ratio.
Toyota Prius NHW20 is equipped with a 1.5 l, 57 kW (5000 rpm), fourcylinder gasoline engine, 50 kW (1200 to 1540 rpm), 500 V (maximum)
PM brushless motor and nickel-metal hydride (NiMh) battery pack as a
RESS (Fig. 6.40). To simplify construction, improve transmission and achieve
smoother acceleration, the gearbox is replaced by a single reduction gear (Fig.
6.39). This is because the engine and electric motor have different torquespeed characteristics, so they can act with each other to meet the driving
performance requirements. Fig. 6.39b shows integration of a combustion engine with a generator/starter, electric motor and PSD of Toyota Prius. In a
PM brushless motor, the rotor with interior PMs has been selected because
it provides a wider torque-speed range under the size and weight restrictions
than other rotor configurations. To utilize the reluctance torque in addition
to synchronous torque, the q-axis permeance is maximized while keeping low
d-axis permeance. A double-layer PM arrangement (Fig. 6.3e) seems to be
impractical in mass production due to the high cost of manufacturing, so
single-layer V-shaped PMs have been used in the 8-pole rotor.
188
Modern Permanent Magnet Electric Machines
Fig. 6.39. Toyota Prius hybrid-electric drive: (a) block diagram, (b) engine cutaway.
Electric motors for passenger hybrid cars are typically rated from 30 to
75 kW. Water cooling offers superior cooling performance, compactness and
lightweight design over forced-air motor cooling. The water jacket cooling
permits weight reductions of 20% and size reductions of 30% as compared to
forced-air designs, while the power consumption for cooling system drops by
Fig. 6.40. How the Toyota Prius HEV is built.
PM Brushless DC Motors and Drive Control
189
75%. The use of a single water cooling system for the motor and solid state
converter permits further size reductions.
Summary
The development of PM brushless motors started in the 1980s due to progress
in rare-earth PM technology and progress in power electronics. PM brushless
motors are the highest power density and highest-efficiency motors with the
best dynamic performance.
PM brushless motor drives fall into the two principal classes of sinusoidally
excited and square wave (trapezoidally excited) motors.
Sinusoidally excited motors are fed with three-phase sinusoidal waveforms
shifted by 120◦ (Fig. 6.5a) and operate on the principle of a rotating magnetic
field. The speed of the rotor is equal to the synchronous speed ns of the
stator magnetic rotating field, given by eqn (6.2). They are simply called PM
sinewave motors or PM synchronous motors.
Square wave motors are also fed with three-phase waveforms shifted by
120◦ one from another, but these waveforms are rectangular or trapezoidal
(Fig. 6.5b). Such a shape is produced when the armature current (MMF) is
precisely synchronized with the rotor instantaneous position and frequency
(speed). The most direct and popular method of providing the required rotor
position information is to use an absolute angular position sensor mounted
on the rotor shaft. Such electronic commutation is functionally equivalent to
the mechanical commutation in DC brush motors. This explains why motors
with square wave excitation are called DC brushless motors.
Comparison between sinusoidally excited and square wave motors is given
in Table 6.2.
The DC PM brushless motor is a reversed DC brush motor in which PMs
are placed on the rotor and armature winding is placed on the stator (Fig.
6.1). The commutator (mechanical inverter) is replaced by a stationary solid
state inverter. In the case of a DC generator, the commutator (mechanical
rectifier) is replaced by a stationary solid state rectifier, either controlled or
uncontrolled. In a PM DC brushless machine with natural air cooling system,
the heat transfer and cooling conditions are much better than in a PM DC
brush machine, because nearly all losses (stator winding losses and stator
core losses) are dissipated in the stator and transferred via the housing to the
surrounding air.
A comparison of PM DC brush (commutator) and brushless motors is
given in Table 6.1.
Typically, the armature winding of PM brushless motors is distributed in
slots. When cogging (detent) torque needs to be eliminated, slotless windings
are used.
An auxiliary DC field winding located in the rotor or magnetic flux diverters located in the stator can help to increase the speed range over a constant
190
Modern Permanent Magnet Electric Machines
power region or control the output voltage of variable speed generators. These
machines are called PM brushless motors with a hybrid field excitation system.
PM synchronous motors are usually built with one of the following rotor
configurations:
(a) Surface-magnet rotor (Fig. 6.3a)
(b) Spoke-type magnet rotor (Fig. 6.3b)
(c) Interior-magnet rotor with flat PMs (Fig. 6.3c)
(d) Inset-magnet rotor (Fig. 6.3d)
(e) Rotor with double-layer interior-magnets (Fig. 6.3e)
(f) Rotor with buried magnets asymmetrically distributed (Fig. 6.3f)
There are three modes of commutation of PM DC brushless motors:
ˆ
ˆ
ˆ
Unipolar commutation with phase sequence A, B, C, A, B, . . . (Fig 6.9a)
where the current is always conducted by only one phase (the neutral point
of the stator winding must be available)
Bipolar six-step commutation with phase sequence AB, AC, BC, BA, CA,
CB, . . . (Figs 6.9, 6.10, 6.11 and 6.12) where the current is always conducted by two phases (120◦ current conduction, one step equivalent to
60◦ )
Bipolar commutation with three phases on and a conduction interval of
180◦ (Fig. 6.13)
The EMF induced in a phase winding of a DC PM brushless motor neglecting saturation and armature reaction can be simply expressed as a function
of speed n (6.7)
Ef = kE n
where kE is the EMF constant provided by manufacturers of PM brushless
machines.
The inverter output voltage is given by eqns (6.14) and (6.15) and the DC
bus voltage of a rectifier is given by eqns (6.9) to (6.11).
Rotor position sensing in a PM DC motor is done by position sensors, i.e.,
(a) Hall elements (Fig. 6.14a);
(b) encoders (Fig. 6.14b);
(c) resolvers (Fig. 6.14c).
The Hall element is a magnetic field sensor.
In optical encoders, a light passes through the transparent areas of a grating and is sensed by a photodetector.
In an incremental encoder a pulse is generated for a given increment of
shaft angular position which is determined by counting the encoder output
pulses from a reference. The rotating disk (grating) has a single track. In the
case of a power failure, an incremental encoder loses position information and
must be reset to a known zero point.
PM Brushless DC Motors and Drive Control
191
An absolute encoder is a position verification device with multichannel
output that provides unique position information for each shaft angular location. An absolute encoder is not a counting device like an incremental encoder
and does not lose position information in the case of loss of power.
A resolver is a rotary electromechanical transformer that provides output
in the form of trigonometric functions sin(ϑ) and cos(ϑ) of its inputs. As a
result the output signals are sinusoidal waves in quadrature; i.e., one wave is
a sinusoidal function of the angular displacement ϑ and the second wave is a
cosinusoidal function of ϑ. Instead of delivering the excitation voltage to the
rotor winding by brushes and slip rings, an inductive coupling system (rotary
transformer) is frequently used.
Sensorless methods of detection of the shaft position of PM brushless motors usually use one of the following techniques:
(a) Detection of back EMF (zero crossing approach, phase-locked loop technique, EMF integration approach)
(b) Detection of the stator third harmonic voltage
(c) Detection of the conducting interval of freewheeling diodes connected in
antiparallel with the solid state switches
(d) Sensing the inductance variation (in the d and q axis), terminal voltages
and currents
The mathematical model of a PM brushless motor (electrical circuit diagram)
is shown in Fig. 6.22 and is expressed by eqn (6.20), i.e.,

 


u1A
R1 0 0
iaA
 u1B  =  0 R1 0   iaB 
u1C
0 0 R1
iaC
 

 
iaA
ef A
L−M
0
0
d
L−M
0   iaB  +  ef B 
+ 0
dt
iaC
ef C
0
0
L−M

where u1A , u1B , u1C are instantaneous values of input voltages, iaA , iaB , iaC
are the instantaneous values of phase currents, ef A , ef B , ef C are instantaneous
values of EMFs, R1 is the stator winding resistance per phase, L is the stator
winding self-inductance, and M is the mutual inductance between phases.
A symmetrical stator winding has been assumed. For computer simulation
of PM brushless motors this equation must be supplemented by the torque
balance equation, i.e.,
J
dϑ
d2 ϑ
+ Dϑ
+ Kϑ ϑ = Telm ± T
dt2
dt
where J is the moment of inertia, Dϑ is the torsional damping, Kϑ is the spring
constant, Telm is the electromagnetic torque and T is the external torque.
192
Modern Permanent Magnet Electric Machines
Cogging torque is produced due to interaction of the PM rotor magnetic
flux and variable permeance of the air gap due to the stator slot geometry,
i.e., slot openings. The larger the least common multiple of the number of
slots s1 and number of poles 2p, i.e., LCM (s1 , 2p), the smaller the amplitude
of the cogging torque.
In armature winding made of concentrated non-overlapping coils, the coil
span is equal to one tooth pitch. The concentrated coil winding is feasible,
when (6.1)
Nc
= km1
GCD(Nc , 2p)
where Nc is the total number of armature coils, 2p is the number of poles,
m1 is the number of phases, k = 1, 2, 3, . . . and GCD is the greatest common
divisor of Nc and 2p. Sometimes such a winding is called the winding with
fractional number of slots per pole per phase q1 = s1 /(2pm1 ). Owing to very
short end turns, the winding losses are reduced. Very small mutual inductance
between phases causes fault tolerance.
Electromechanical drive systems with PM brushless motors and solid state
converters are shown in Figs 6.26 and 6.29. The PM brushless motor and solid
state converter should be connected in such a way as to obtain
ˆ
ˆ
ˆ
proper operation,
minimize electromagnetic interference (EMI), and
prevent shock hazard.
A typical hybrid electric vehicle (HEV) uses a PM brushless motor in addition
to a combustion engine. HEVs have many advantages over classical vehicles
with gasoline or diesel engines. The most important are
(a) smaller size of combustion engine, lower fuel consumption since part of
the energy is derived from the rechargeable energy storage system RESS,
and improved efficiency (about 40% better fuel efficiency than that for
conventional vehicles of similar ratings);
(b) high electric motor torque at a low speed with high combustion engine
torque in higher speed ranges make the torque speed characteristic suitable
for traction requirements (Fig. 6.23);
(c) utilization of wasted energy at braking (regenerative braking), idling and
low speeds;
(d) the use of electric motor reduces air pollution and acoustic noise;
(e) wear and tear on the combustion engine components decrease, so they can
work for a longer period of time;
(f) lower maintenance costs due to reduced fuel consumption;
(g) although the initial cost of HEVs is higher than conventional cars, their
operating costs are lower over time.
7
PM SYNCHRONOUS MOTORS AND
DRIVE CONTROL
A synchronous machine operates at a constant speed in absolute synchronism
with the line frequency. This means that the rotor speed is the same as that of
the rotating magnetic field excited by the stator (or armature) AC winding.
The magnetic flux of the armature rotates synchronously with the field excitation flux. The magnetic flux of the rotor can be excited either with the aid
of a field excitation winding (wound-field machine) or PMs (PM machine). In
a typical synchronous machine the armature system is stationary (stator) and
the field excitation system rotates (rotor). In a reversed design, e.g., in brushless synchronous exciters, the field excitation system is stationary (stator) and
the armature winding spins (rotor).
There is no essential difference between the stators of polyphase synchronous and induction machines of comparable rating. The stator (armature) is made of stacked-up electrotechnical steel laminations, and the stator
slots accommodate a three-phase distributed winding that sets up a rotating
magnetic field.
7.1 Fundamental equations for synchronous machines
7.1.1 Speed
In the steady-state range, the rotor speed is given by the input frequency–to–
number of pole pairs ratio ns = f /p, as given by eqn (6.2), and is equal to the
synchronous speed of the rotating magnetic field produced by the stator.
7.1.2 Air gap magnetic flux density
The first harmonic of the air gap magnetic flux density is
2
Bmg1 =
π
Z 0.5αi π
Bmg cos αdα =
−0.5αi π
4
αi π
Bmg sin
π
2
(7.1)
194
Modern Permanent Magnet Electric Machines
where, neglecting the saturation of the magnetic circuit, the peak flat-topped
value of the magnetic flux density in the air gap
Bmg =
µ0 Fexc
g ′ kC )
(7.2)
under the pole shoe can be found on the basis of the excitation MMF Fexc ,
equivalent air gap g ′ which includes the PM height hM and Carter’s coefficient
kC .
For αi = 1 the fundamental harmonic component Bmg1 is 4/π times the
Bmg peak flat-topped value.
The coefficient αi is defined as the ratio of the average-to-maximum value
of the normal component of the air gap magnetic flux density, i.e.,
Bavg
Bmg
αi =
(7.3)
If the magnetic field distribution in the air gap is sinusoidal, αi = 2/π. For
zero magnetic voltage drop in the ferromagnetic core and a uniform air gap,
the coefficient αi is expressed by eqn (6.3). The coefficient αi is also called
the pole-shoe arc bp -to-pole pitch τ ratio. The pole pitch is
τ=
πD1in
2p
(7.4)
Carter’s coefficient, kC , takes into account the slotted surface of the stator
(armature) core and can be calculated according to the following equation:
kC =
t1
t 1 − γ1 g
(7.5)
t1 =
πD1in
s1
(7.6)
where the slot pitch
and D1in is the inner diameter of the external stator, s1 is the number of
stator slots with the opening b14 and


s
2
b14
b14 
4  b14
arctan
− ln 1 +
(7.7)
γ1 =
π g
g
g
7.1.3 Electromotive force (EMF)
The no-load rms voltage induced in one phase of the stator winding (EMF)
by the DC magnetic excitation flux Φf of the rotor is
√
Ef = π 2f N1 kw1 Φf
(7.8)
PM Synchronous Motors and Drive Control
195
where N1 is the number of stator turns per phase, and kw1 is the stator
winding coefficient. The fundamental harmonic Φf 1 of the excitation magnetic
flux density Φf without armature reaction is
Z τ
π 2
Φf 1 = Li
Bmg1 sin
x dx = τ Li Bmg1
(7.9)
τ
π
0
where Li is the effective (ideal) length of the stator stack in the axial direction.
The EMF Ei per phase with the armature reaction taken into account is
√
Ei = π 2f N1 kw1 Φg
(7.10)
where Φg is the air gap magnetic flux under load (excitation flux Φf reduced
by the armature reaction flux). At no-load (very small armature current)
Φg ≈ Φf . Including the saturation of the magnetic circuit
Ei = 4σf f N1 kw1 Φg
(7.11)
The form factor σf depends on the magnetic saturation of armature teeth,
i.e., the sum of the air gap magnetic voltage drop MVD and the teeth MVD
divided by the air gap MVD, as given by eqn (7.37).
7.1.4 Armature line current density and current density
The peak value of the stator (armature) line current density (A/m) or specific
electric loading is defined as the √
number of conductors in all phases 2m1 N1
times the peak armature current 2Ia divided by the armature circumference
πD1in , i.e.,
√
√
√
m1 2N1 Ja sa
2m1 2N1 Ia
m1 2N1 Ia
=
(7.12)
Am =
=
πD1in
pτ
pτ
where Ja is the current density (A/m2 ) in the stator (armature) conductors
and sa is the cross section of armature conductors including parallel wires.
For air cooling systems Ja ≤ 7.5 A/mm2 (sometimes up to 10 A/mm2 ) and
for liquid cooling systems 10 ≤ Ja ≤ 28 A/mm2 . The top value is for very
intensive oil spray cooling systems.
7.1.5 Electromagnetic power
For an m1 -phase salient-pole synchronous motor with negligible stator winding resistance R1 = 0, the electromagnetic power is expressed as
U1 Ef
U12
1
1
Pelm = m1
sin δ +
−
sin 2δ
(7.13)
Xsd
2
Xsq
Xsd
where U1 is the input (terminal) phase voltage, Ef is the EMF induced by the
rotor excitation flux (without armature reaction), δ is the power angle, i.e.,
196
Modern Permanent Magnet Electric Machines
the angle between U1 and Ef , Xsd is the synchronous reactance in the direct
axis (d-axis synchronous reactance), and Xsq is the synchronous reactance
in the quadrature axis (q-axis synchronous reactance). The electromagnetic
torque
m1 U1 Ef
U12
1
1
Pelm
=
sin δ +
−
sin 2δ
Telm =
2πns
2πns Xsd
2
Xsq
Xsd
(7.14)
7.1.6 Synchronous reactance
For a salient-pole synchronous motor, the d-axis and q-axis synchronous reactances are
Xsd = X1 + Xad
Xsq = X1 + Xaq
(7.15)
where X1 = 2πf L1 is the stator leakage reactance, Xad is the d-axis armature
reaction reactance, also called the d-axis mutual reactance, and Xaq is the
q-axis armature reaction reactance, also called the q-axis mutual reactance.
The reactance Xad is sensitive to the saturation of the magnetic circuit whilst
the influence of the magnetic saturation on the reactance Xaq depends on the
rotor construction. In salient-pole synchronous machines with electromagnetic
excitation, Xaq is practically independent of the magnetic saturation. Usually,
Xsd > Xsq except for some PM synchronous machines.
The leakage reactance X1 consists of the slot, end-connection differential
and tooth-top leakage reactances [32]. Only the slot and differential leakage
reactances depend on the magnetic saturation due to leakage fields.
7.2 Location of the armature current in the d-q
coordinate system
Fig. 7.1 shows the position of the phasor of the armature current in the d-q
coordinate system. Depending on which quadrant is the armature current, the
synchronous machine can operate as an overexcited and underexcited generator
or overexcited and underexcited motor .
7.3 Armature reaction
The form factor of the field excitation is
kf =
α π
4
Bmg1
i
= sin
Bmg
π
2
(7.16)
where the pole arc-to-pole pitch ratio αi < 1. If the magnetic field distribution
is sinusoidal, αi = 2/π. For zero magnetic voltage drop in the ferromagnetic
PM Synchronous Motors and Drive Control
197
Fig. 7.1. Location of the armature current Ia in the d-q coordinate system and four
modes of operation of a synchronous machine.
core and uniform air gap the coefficient αi is expressed by eqn (6.3) The first
harmonic of the air gap magnetic flux density is expressed by eqn (7.1).
The form factors of the armature reaction are defined as the ratios of
the first harmonic amplitudes-to-maximum values of normal components of
armature reaction magnetic flux densities in the d-axis and q-axis, respectively,
i.e.,
kf d =
Bad1
Bad
kf q =
Baq1
Baq
(7.17)
The peak values of the first harmonics Bad1 and Baq1 of the armature magnetic
flux density can be calculated as coefficients of Fourier series for ν = 1, i.e.,
4
π
Z 0.5π
4
Baq1 =
π
Z 0.5π
Bad1 =
B(x) cos xdx
(7.18)
B(x) sin xdx
(7.19)
0
0
For a salient-pole motor with electromagnetic excitation and the air gap g ≈ 0
(fringing effects neglected), the d-axis and q-axis form factors of the armature
reaction are
kf d =
αi π + sin αi π
π
kf q =
αi π − sin αi π
π
(7.20)
198
Modern Permanent Magnet Electric Machines
Table 7.1. Factors kf , kf d , kf q , kad , and kaq for salient-pole synchronous machines
according to eqns (7.16), (7.17), (7.20) and (7.21).
Factor
kf
kf d
kf q
kad
kaq
0.4
0.5
αi = bp /τ
0.6 2/π 0.7
0.8
1.0
0.748 0.900 1.030 1.071 1.134 1.211 1.273
0.703 0.818 0.913 0.943 0.958 0.987 1.00
0.097 0.182 0.287 0.391 0.442 0.613 1.00
0.939 0.909 0.886 0.880 0.845 0.815 0.785
0.129 0.202 0.279 0.365 0.389 0.505 0.785
The reaction factors in the d- and q-axis are defined as
kad =
kf d
kf
kaq =
kf d
kf
(7.21)
The form factors kf , kf d and kf q of the excitation field and armature reaction
and reaction factors kad and kaq for synchronous machines according to eqns
(7.16), (7.17), (7.20) and (7.21) are given in Tables 7.1 and 7.2.
Table 7.2. Form factors of the armature reaction for PM synchronous machines
[32]
Rotor configuration
Inset type
PM rotor
d-axis
q-axis
kf d = π1 [αi π + sin αi π
+cg (π − αi π − sin αi π)]
kf q = π1 [ c1g (αi π − sin αi π)
+π(1 − αi ) + sin αi π]
cg ≈ 1 + h/g
Surface PM rotor
Buried PMs
kf d = kf q = 1
1
kf d = π4 αi 1−α
2 cos(0.5αi π)
kf q = π1 (αi π − sin αi π)
kf d = π1 (αi π + sin αi π)
kf q = π1 (αi π − sin αi π)
i
Salient-pole
rotor with
excitation winding
h = slot depth
PM Synchronous Motors and Drive Control
199
The equivalent MMFs kad Fad and kaq Faq excite their own magnetic fluxes
Φad =
2
2
2 kad Fad
= Λkf kad Fad = Λkf d Fad
π Rµ
π
π
(7.22)
Φaq =
2 kaq Faq
2
2
= Λkf kaq Faq = Λkf q Faq
π Rµ
π
π
(7.23)
The reaction factors kad and kaq are expressed by eqns (7.21). The form factor
kf of the field excitation and form factors of the armature reaction kf d and
kf q are given by eqns (7.16) and (7.20), respectively.
The permeance for the armature reaction fluxes Λ = const because it has
been assumed that the equivalent air gap in the d-axis g = const. Neglecting
the saturation of the magnetic circuit, this permeance is
Λ = µ0
τ Li
gkC
(7.24)
Each of the magnetic fluxes (7.22) and (7.23) excites in the stator (armature)
windings its own EMF of the armature reaction, i.e.,
√
Ead = π 2f N1 kw1 Φad
(7.25)
and
√
Eaq = π 2f N1 kw1 Φaq
(7.26)
For a synchronous machine with a non-salient-pole rotor, the permeances of
the magnetic circuit in the d- and q-axis are the same.
Assuming g = 0, the equivalent d-axis field MMF, which produces the
same fundamental wave flux as the armature-reaction MMF, is
√
m1 2 N1 kw1
kad Ia sin Ψ
(7.27)
Fexcd = kad Fad =
π
p
where Ia is the armature current and Ψ is the angle between the resultant
armature MMF Fa and its q-axis component Faq = Fa cos Ψ . Similarly, the
equivalent q-axis MMF is
√
m1 2 N1 kw1
Fexcq = kaq Faq =
kaq Ia cos Ψ
(7.28)
π
p
Putting magnetic fluxes (7.22) and (7.23) to eqns (7.25) and (7.26), the EMFs
of armature reaction in complex form are
√
√
2
m1 2 N1 kw1
Ead = jπ 2f N1 kw1 Λkf d
Ia sin ψ
π
π
p
4
(N1 kw1 )2
= j m1 f
Λkf d Ia sin ψ = jXad Iad
π
p
(7.29)
200
Modern Permanent Magnet Electric Machines
√
2
m1 2 N1 kw1
Eaq = jπ 2f N1 kw1 Λkf q
Ia cos ψ
π
π
p
√
(N1 kw1 )2
4
Λkf q Ia cos ψ = jXaq Iaq
= j m1 f
π
p
(7.30)
Finally, there are the following simple relationships between the EMFs and
armature reaction reactances in the d- and q-axis
Ead = jXad Iad
Eaq = jXaq Iaq
(7.31)
The d-axis armature reaction reactance with the magnetic saturation
being included is
Xad = kf d Xa = 4m1 µ0 f
(N1 kw1 )2 τ Li
kf d
πp
g′
(7.32)
where µ0 is the magnetic permeability of free space, Li is the effective length
of the stator core and the inductive reactance of the armature of a non-salientpole (cylindrical rotor) synchronous machine
Xa =
Xad
(N1 kw1 )2 τ Li
= 4m1 µ0 f
kf d
πp
g′
(7.33)
Similarly, for the q-axis
Xaq ≈ kf q Xa = 4m1 µ0 f
(N1 kw1 )2 τ Li
kf q
πp
gq′
(7.34)
For most PM configurations, the equivalent d-axis air gap g ′ in eqn (7.32)
should be replaced by
g ′ = gkC ksat +
hM
µrrec
(7.35)
and gq′ in eqn (7.34) by
gq′ = gq kC ksatq
(7.36)
where gq is the mechanical clearance in the q-axis, kC is Carter’s coefficient
for the air gap according to eqn (7.5), and ksat ≥ 1 is the saturation factor of
the magnetic circuit.
The saturation factor of the magnetic circuitP
is defined as the ratio of the
total magnetic voltage drop MVD per pole pair
Vµ to the MVD across the
air gaps 2Vµg , i.e.,
P
Vµ
ksat =
≥ 1.0
(7.37)
2Vµg
PM Synchronous Motors and Drive Control
201
The total MVD across the pole pair for a typical PM brushless machine with
stator and rotor cores and surface PMs is
X
Vµ = 2(Vµg + V1t + VP M ) + V1y + V2y
(7.38)
where V1t is the MVD along the stator teeth, VP M is the MVD along PMs,
V1y is the MVD along the stator yoke and V2y is the MVD along the rotor
yoke.
The saturation factor in the q-axis ksatq ≈ 1, since the q-axis armature
reaction reactance is practically independent of magnetic saturation.
The sum of the armature-reaction reactance Xad or Xaq and armature
leakage reactance X1 is called synchronous reactance and is given by eqn
(7.15).
Similar to induction machines, the leakage reactance X1 = 2πf L1 is due to
the stator (armature) leakage fluxes: slot leakage reactance, end-turn leakage
reactance and differential leakage reactance. The differential leakage reactance
is caused by higher space harmonics, i.e., the armature current multiplied by
the differential leakage reactance gives a voltage drop due to higher space
harmonics of the MMF.
The armature reaction reactances Xad and Xaq correspond to the mutual
(air gap) reactance Xm of an induction motor. Usually, Xsd > Xsq , except in
the case of some PM synchronous machines.
7.4 Phasor diagram
When drawing phasor diagrams of synchronous machines, two arrow systems
are used:
(a) Generator arrow system, i.e.,
Ef = U1 + Ia R1 + jIad Xsd + jIaq Xsq
= U1 + Iad (R1 + jXsd ) + Iaq (R1 + jXsq )
(7.39)
(b) Consumer (motor) arrow system, i.e.,
U1 = Ef + Ia R1 + jIad Xsd + jIaq Xsq
= Ef + Iad (R1 + jXsd ) + Iaq (R1 + jXsq )
(7.40)
Ia = Iad + Iaq
(7.41)
where
202
Modern Permanent Magnet Electric Machines
and on the basis of Fig. 7.1
Iad = Ia sin Ψ
Iaq = Ia cos Ψ
(7.42)
Fig. 7.2. Equivalent circuit of a PM synchronous machine for: (a) generator arrow
system; (b) consumer (motor) arrow system.
When the current arrows are in the opposite direction, the phasors Ia , Iad ,
and Iaq , are reversed by 180◦ . The same applies to the voltage drops. The
location of the armature current Ia with respect to the d- and q-axis for
generator and motor mode is shown in Fig. 7.1.
Phasor diagrams for synchronous generators are constructed using the generator arrow system. An overexcited generator (Fig. 7.3a) delivers an inductive
current and a corresponding reactive power to the line. The same system can
be used for motors; however, the consumer arrow system is more convenient.
To draw the phasor diagram for the underexcited motor shown in Fig. 7.3a,
the d-q coordinate system shown in Fig. 7.1 has been rotated 180◦ to obtain
the 3rd quadrant in the position of the 1st quadrant. Fig. 7.3b shows the phasor diagram using the consumer arrow system for a load current Ia lagging the
vector U1 by the angle ϕ. At this angle the motor is, conversely, underexcited
and induces, with respect to the input voltage U1 , a capacitive current component Ia sin Ψ . An overexcited motor, consequently, draws a leading current
from the circuit and delivers reactive power to it.
In the phasor diagrams shown in Fig. 7.3 the stator core losses have been
neglected. This assumption is justified only for power frequency synchronous
motors with unsaturated armature cores.
The input voltage U1 projections on the d- and q-axis follow:
ˆ
For an overexcited motor (Fig. 7.3a)
U1 sin δ = Iaq Xsq + Iad R1
U1 cos δ = Ef − Iad Xsd + Iaq R1
(7.43)
PM Synchronous Motors and Drive Control
203
Fig. 7.3. Phasor diagrams of salient-pole synchronous machines: (a) overexcited
generator; (b) underexcited motor.
ˆ
for an underexcited motor(Fig. 7.3b)
U1 sin δ = Iaq Xsq − Iad R1
U1 cos δ = Ef + Iad Xsd + Iaq R1
(7.44)
The currents of an overexcited motor
Iad =
U1 (R1 sin δ − Xsq cos δ) + Ef Xsq
Xsd Xsq + R12
(7.45)
Iaq =
U1 (R1 cos δ + Xsd sin δ) − Ef R1
Xsd Xsq + R12
(7.46)
are obtained by solving the set of eqns (7.43). Similarly, the currents of an
underexcited motor are found by solving the set of eqns (7.44). The d-axis
current of an underexcited motor is
Iad =
U1 (Xsq cos δ − R1 sin δ) − Ef Xsq
Xsd Xsq + R12
(7.47)
204
Modern Permanent Magnet Electric Machines
and the q-axis current is expressed by eqn (7.46). The rms armature current
of an underexcited motor as a function of U1 , Ef , Xsd , Xsq , δ, and R1 is
Ia =
s
×
q
2 + I2 =
Iad
aq
Ef Xsq
(Xsq cos δ − R1 sin δ) −
U1
2
U1
Xsd Xsq + R12
2
Ef R1
+ (R1 cos δ + Xsd sin δ) −
U1
(7.48)
The angle between the phasor Ia and q-axis is ψ = ϕ ∓ δ, where the “−” sign
is for an underexcited motor and the “+” sign is for an overexcited motor.
7.5 Input and electromagnetic power
Fig. 7.4. Phasor diagrams for finding the input power Pin as a function of Iad , Iaq
and load angle δ: (a) overexcited motor, (b) underexcited motor.
The phasor diagrams in Figs 7.3 and 7.4 can also be used to find the input
power. Since for an underexcited motor (Fig. 7.3b)
Ia cos ϕ = Iaq cos δ − Iad sin δ
(7.49)
Pin = m1 U1 Ia cos ϕ = m1 U1 (Iaq cos δ − Iad sin δ)
(7.50)
the input power is [32]
PM Synchronous Motors and Drive Control
205
Putting U1 sin δ and U1 cos δ according to eqns (7.44) into eqn (7.50),
2
2
Pin = m1 [Iaq Ef + Iad Iaq Xsd + Iaq
R1 − Iad Iaq Xsq + Iad
R1 ]
= m1 [Iaq Ef + R1 Ia2 + Iad Iaq (Xsd − Xsq )]
Because the stator core loss has been neglected, the electromagnetic power
is the motor input power minus the stator winding loss ∆P1w = m1 Ia2 R1 =
2
2
m1 (Iad
+ Iaq
)R1 . Thus
Pelm = Pin − ∆P1w = m1 [Iaq Ef + Iad Iaq (Xsd − Xsq )]
=
m1 [U1 (R1 cos δ + Xsd sin δ) − Ef R1 )]
(Xsd Xsq + R12 )2
(7.51)
×[U1 (Xsq cos δ −R1 sin δ)(Xsd −Xsq )+Ef (Xsd Xsq +R12 )−Ef Xsq (Xsd −Xsq )]
The electromagnetic torque developed by a salient-pole synchronous motor is
Telm =
Pelm
m1
1
=
2πns
2πns (Xsd Xsq + R12 )2
×{U1 Ef (R1 cos δ + Xsd sin δ)[(Xsd Xsq + R12 ) − Xsq (Xsd − Xsq )]
−U1 Ef R1 (Xsq cos δ − R1 sin δ)(Xsd − Xsq )
+U12 (R1 cos δ + Xsd sin δ)(Xsq cos δ − R1 sin δ)(Xsd − Xsq )
−Ef2 R1 [(Xsd Xsq + R12 ) − Xsq (Xsd − Xsq )]}
(7.52)
The last term is the constant component of the electromagnetic torque independent of the load angle δ. Putting R1 = 0, eqn (7.52) becomes the same
as eqn (7.14). Small synchronous motors have a rather high stator winding
resistance R1 that is comparable with Xsd and Xsq . That is why eqn (7.52)
is recommended for calculating the performance of small motors.
206
Modern Permanent Magnet Electric Machines
7.6 How to obtain zero d-axis current Iad = 0
To check if there is a real value of the load angle δ at which Iad = 0, the
numerator of eqn (7.47) is equated to zero, i.e.,
U1 (Xsq cos δ − R1 sin δ) − Ef Xsq = 0
Putting A = U1 Xsq , B = U1 R1 and C = Ef Xsq the following trigonometric
equation is obtained:
A cos δ − B sin δ − C = 0
or
(−B sin δ)2 = (C − A cos δ)2
After substituting sin2 δ = 1 − cos δ 2 , the following second-order linear equation is obtained:
(A2 + B 2 ) cos2 δ + 2AC cos δ + (C 2 − B 2 ) = 0
The discriminant of the quadratic equation is
∆ = b2 − 4ac
Roots of the second-order equation
√
−b − ∆
x1 =
2a
x2 =
√
−b + ∆
2a
There are two solutions
δ1 = arccos(x1 )
δ2 = arccos(x2 )
(7.53)
Sometimes both roots x1 and x2 are complex numbers. This means that probably for motoring operation, the EMF Ef is greater than the terminal phase
voltage V1 .
7.7 Influence of d-axis current on the power factor
Phasor diagrams for an underexcited PM synchronous motor are plotted in
Fig. 7.5:
ˆ
ˆ
ˆ
If the d-axis armature current Iad is large, the angle ϕ between the current
and the voltage is large, and the power factor cos ϕ is low (Fig. 7.5a).
If the d-axis armature current Iad is low, the angle ϕ between the current
and the voltage is low, and the power factor cos ϕ is high (Fig. 7.5b).
For zero d-axis armature current Iad = 0, the angle ϕ = δ and the total
armature current Iaq = Ia is torque producing (Fig. 7.5c).
PM Synchronous Motors and Drive Control
207
Fig. 7.5. Underexcited PM synchronous motor: (a) large angle ϕ; (b) small angle
ϕ; angle ϕ = 0.
Fig. 7.6. Overexcited PM synchronous motor: (a) large angle ϕ; (b) small angle ϕ;
angle ϕ = 0.
A similar effect of the d-axis current on power factor cos ϕ is for an overexcited
motor, as shown in Fig. 7.6.
For Iad = 0 the angle Ψ = 0 (between the armature current Ia = Iaq and
EMF Ef ). Therefore, the angle ϕ between the current and voltage is equal to
the load angle δ between the voltage V1 and EMF Ef , i.e.,
cos ϕ =
Ef + Ia R1
U1
(7.54)
208
Modern Permanent Magnet Electric Machines
and
2
U12 = (Ef + Ia R1 )2 + (Ia Xsq )2 ≈ Ef2 + Ia2 Xsq
(7.55)
Thus
s
cos ϕ ≈
1−
Ia Xsq
U1
2
+
Ia R1
U1
(7.56)
At constant voltage U1 and frequency (speed), the power factor cos ϕ decreases
with the load torque (proportional to the armature current Ia ). The power
factor can be kept constant by increasing the voltage in proportion to the
current increase, i.e., keeping Ia Xsq /U1 = const.
7.8 Vector control of PM synchronous motors
The basis of vector control is separation of the iad and iaq current components
and separate control of the iad and iaq current components. Separation of the
iad and iaq current components and separate control of the iad and iaq can
also be used for power factor cos ϕ correction, as shown in phasor diagrams
in Figs 7.5 and 7.6. To perform vector control, the following actions must be
taken:
1. Measure the motor phase currents.
2. Transform them into the two-phase system α, β using Clarke transformation.
3. Calculate the rotor position angle.
4. Transform stator currents into the d,q-coordinate system using BlondelPark transformation.
5. The stator (armature) current torque iaq and flux iad producing components are controlled separately by the controllers.
6. The output stator voltage space vector is transformed back from the d,qcoordinate system into the two-phase α, β system fixed with the stator
by inverse Blondel-Park transformation.
7. Using the space vector modulation (SVM), the output three-phase voltage
is generated.
PM Synchronous Motors and Drive Control
209
Edith Clarke (1883-1959) was born in Howard County, Maryland, USA.
Edith’s father died when she was 7 and her mother when she turned 12. She graduated from Vassar College, Poughkeepsie, NY in 1908 with honors. She earned
her master’s degree in Electrical Engineering (EE) from the Massachusetts Institute of Technology (MIT) in 1919, becoming the first woman to earn a degree
in that field from MIT. She joined General Electric (GE) to work as a “human
computer.” She invented a graphical calculator to be used in the solution of
electric power transmission problems. After traveling throughout Europe and
Turkey, Ms. Clarke finally achieved her life-long goal to become a salaried electrical engineer for the Central Station Engineering Department of GE in 1922.
This made her the first professionally employed female electrical engineer in the
United States.
In 1947 Ms. Clarke left GE after 26 years to teach Electrical Engineering
(EE) at the University of Texas, Austin, where she became the first female EE
full professor in the US and worked there until retirement in 1956. She became
the first woman to be elected as a Fellow of the American Institute of Electrical
Engineers (which became the Institute of Electrical and Electronic Engineers,
IEEE in 1963). She was the first woman to present an AIEE paper. In 1954, she
received a lifetime Achievement Award from the Society of Women Engineers.
The Award cited her contributions to the field in the form of her simplifying
charts and her work in system instability.
Ms. Clarke authored or co-authored nineteen technical papers between 1923
and 1951 and a two-volume reference textbook entitled Circuit Analysis of AC
Systems. Edith Clarke died in 1959 in Olney, Maryland.
Fig. 7.7. Principle of vector control of PM synchronous motors.
AC electrical machines can be analyzed in the following reference frame
(coordinate systems):
ˆ
ˆ
ˆ
stationary reference frame ABC linked to the stator, where A is phase A,
B is phase B and C is phase C;
stationary orthogonal reference frame α, β linked to the stator;
orthogonal reference frame d q linked to the rotor and rotating with the
speed of the rotor.
210
Modern Permanent Magnet Electric Machines
Fig. 7.8. Block diagram of vector control of PM synchronous motors.
The reference frame ABC and α, β are called natural reference frames.
Transformation ABC =⇒ α, β is called Clarke’s transformation. Transformation ABC =⇒ d q is called Park’s [70] transformation or the BlondelPark transformation, which originated from Blondel’s two-reaction theory for
salient-pole synchronous machines.
Robert H. Park (1902-1994) was born in Strasbourg, Germany while his father, the sociologist Robert Ezra Park, was studying and teaching at Heidelberg
University. He graduated from Massachusetts Institute of Technology (MIT) in
1923 with a degree in electrical engineering. He did post-graduate work at the
Royal Technical Institute in Stockholm, Sweden.
Park started working for General Electric, where he created his 1929 Park’s
transformation paper entitled Two-reaction Theory of Synchronous Machines,
which made him world famous.
Although his main field was electrical engineering, Park’s work covered several disciplines. At one time, he worked as a chemical engineer in charge of
physics research. In the 1950s and 1960s he owned a company in Brewster, Massachusetts, that made plastic bottles. He invented the machinery to automate
the process, and at the same time was an independent consultant.
R.H. Park was elected as an IEEE Fellow in 1965 and elected as member of
the National Academy of Engineering in 1986. In 1972, the IEEE honored him
with the Lamme Medal. He died in Providence in 1994.
PM Synchronous Motors and Drive Control
211
André-Eugène Blondel (1863–1938) was born in Chaumont, Haute-Marne,
France. In 1893 Blondel sought to solve the problem of integral synchronization,
using the theory proposed by Marie A. Cornu. This led him to invent the bifilar
and soft iron oscillographs in 1893. They remained the best way to record highspeed electrical phenomena for more than 40 years, when they were replaced by
the cathode ray oscilloscope. He built a theory of rectification with asymmetrical
electrodes for AC electric arcs (1891–1901). On the basis of earlier work of
Paul Boucherot, in 1892 he published a study on the coupling of synchronous
generators on a large AC electric grid. In 1894 he proposed the lumen and other
new measurement units for use in photometry. In 1899, he published Empirical
Theory of Synchronous Generators, which contained the basic theory of twoarmature reactions (direct and transverse). It was used extensively to explain
the properties of salient-pole AC machines. In 1909, assisted by M. Mähl, he
worked on one of the first long-distance schemes for the transmission of AC
power.
Blondel became a member of the French Academy of Sciences in 1913. He
was appointed commander of the Légion d’honneur in 1927, and was awarded
the Faraday Medal in 1937. He also received the medal of the Franklin Institute,
the Montefiore award (Belgium) and the triennial Medal of Lord Kelvin in 1929.
A. E. Blonded died in 1938 in Paris.
7.9 Starting
7.9.1 Asynchronous starting
A synchronous motor is not self-starting. To produce an asynchronous starting
torque, its rotor must be furnished with a cage winding or mild steel pole
shoes. The starting torque is produced as a result of the interaction between
the stator rotating magnetic field and the rotor currents induced in the cage
winding or mild steel pole shoes [50].
PM synchronous motors that can produce asynchronous starting torque
are commonly called line-start PM synchronous motors. These motors can
operate without solid state converters. After starting, the rotor is pulled into
synchronism and rotates with the speed imposed by the line input frequency.
The efficiency of line-start PM motors is higher than that of equivalent induction motors and the power factor can be equal to unity.
The rotor bars in line-start PM motors are unskewed, because PMs are
embedded axially in the rotor core. In comparison with induction motors,
line-start PM motors produce much higher content of higher space harmonics
in the air gap magnetic flux density distribution, current and electromagnetic
torque. Further, the line-start PM synchronous motor has a major drawback
during the starting period as the magnets generate a brake torque which decreases the starting torque and reduces the ability of the rotor to synchronize
a load. Starting characteristics of a line-start PM brushless motor are plotted
in Fig. 7.9.
212
Modern Permanent Magnet Electric Machines
torque
(a)
effect of higher
harmonics
1
3
1
0
0.8
0.6
2
0.4
0.2
0
slip
speed
(b)
n0
0
time
torque
(c)
Tload
0
time
Fig. 7.9. Characteristics of a line-start PM brushless motor: (a) steady-state torqueslip characteristic; (b) speed-time characteristic; (c) torque-time characteristic. 1 –
asynchronous torque, 2 – braking torque produced by PMs, 3 – resultant torque,
n0 – steady-state speed, Tload – load torque.
There are many constructions of line-start PM brushless motors, e.g., according to US patent 2543639, German patent 1173178 (Fig. 6.3f) or international patent publication WO 2001/06624. Fig. 7.10 shows two rotors for
line-start PM synchronous motors [87]: a rotor with conventional cage winding and a rotor with slots of different shapes in the d- and q-axis. The rotor
shown in Fig. 7.9b allows for significant reduction of the 5th, 11th, 13th, 17th
and higher odd harmonics [87].
7.9.2 Starting by means of an auxiliary motor
Auxiliary induction motors are frequently used for starting large synchronous
motors with electromagnetic excitation. The synchronous motor has an auxiliary smaller starting motor on its shaft, capable of bringing it up to the
synchronous speed, at which time, synchronizing with the power circuit is possible. The unexcited synchronous motor is accelerated to almost synchronous
PM Synchronous Motors and Drive Control
213
Fig. 7.10. Rotors of line-start PM synchronous motors with: (a) constant width
slots; (b) variable width slots. Photo courtesy of Technical University of Wroclaw,
Poland [87].
speed using a smaller induction motor. When the speed is close to the synchronous speed, first the armature voltage and then the excitation voltage is
switched on, and the synchronous motor is pulled into synchronism.
The disadvantage of this method is that it’s impossible to start the motor
under load. It would be impractical to use an auxiliary motor of the same
rating as that of the synchronous motor and installation would be expensive.
7.9.3 Frequency-change starting
Frequency-change starting is a common method of starting synchronous motors both with electromagnetic excitation and PMs. The frequency of the
voltage applied to the motor is smoothly changed from a value close to zero
to the rated value. The motor runs synchronously during the entire starting
214
Modern Permanent Magnet Electric Machines
period being fed from a variable voltage variable frequency (VVVF) solid state
inverter.
7.10 Comparison of PM synchronous motors with
induction motors
PM synchronous motors, as compared with their induction counterparts, do
not have rotor winding losses and require simple line commutated inverters
which are more efficient than forced commutated inverters. Table 7.3 contains
a comparison of the speed, power factor cos ϕ, air gap, torque-voltage characteristics, and price of synchronous and induction motors. A larger air gap in
synchronous motors makes them more reliable than induction motors. The increased air gap is required to minimize the effect of the armature reaction, to
reduce the synchronous reactance (if necessary), and to improve the stability.
Table 7.3. Comparison between PM synchronous and induction motors
Quantity
Synchronous motor
Induction motor
Constant, independent
As the load increases,
of the load
the speed decreases
Adjustable power factor
Depends on
Power factor
(controlled by solid
the air gap
cos ϕ
state converter).
pf ≈ 0.8...0.9 at rated load
Operation at pf = 1
pf ≈ 0.1 at no load
is possible
Nonferromagnetic
From a fraction
As small
air
of mm to a few
as
gap
millimeters
possible
Torque directly
Torque directly
Torque-voltage
proportional to the
proportional to the
characteristic
input voltage
input voltage squared
More expensive
Cost
Cost
than
effective
induction motor
motor
Speed
Summary
The open-circuit EMF Ef induced by the rotor PM excitation system and the
rotor magnetic flux Φf are given by the same equations as for synchronous
PM Synchronous Motors and Drive Control
215
machines with electromagnetic excitation, i.e., eqns (7.8) and (7.9), i.e.,
√
Ef = π 2f N1 kw1 Φf
Z τ
Φf 1 = Li
Bmg1 sin
0
π 2
x dx = τ Li Bmg1
τ
π
where N1 is the number of the stator turns per phase, f is the frequency
of the magnetic flux in the stator, kw1 is the stator winding factor for the
fundamental space harmonic, and αi = bp /τ . If the magnetic field distribution
in the air gap is sinusoidal, the ratio of pole-shoe arc bp to pole pitch τ is
αi = 2/π.
The armature reaction magnetic fluxes Φad (7.22) and Φaq (7.23) induce
their own EMFs, Ead (7.25) and Eaq (7.26), in the d−axis and q-axis, respectively.
The electromagnetic power Pelm and electromagnetic torque Telm at zero
stator resistance R1 = 0 of PM brushless motors are given by eqn (7.51)
and (7.14), respectively. Equations are exactly the same as for salient-pole
synchronous machines with electromagnetic field excitation.
The synchronous reactances Xsd , Xsq in the d-axis and q-axis are given
by eqns (7.15).
The form factors of armature reaction kf d , kf q are defined as the ratios of
the first harmonic amplitudes to the maximum values of normal components of
armature reaction magnetic flux densities in the d-axis and q-axis, respectively,
eqns (7.20).
The reaction factors kad , kaq in the d-axis and q-axis are defined as the
ratios of form factors of armature reaction-to-the form factor of the excitation
field kf , as given by eqns (7.21).
The values of the form factor of the excitation field kf , form factors of
the armature reaction kf d , kf q , and the reaction factors kad , kaq are given in
Tables 7.1 and 7.2.
The d-axis and q-axis armature reaction reactances for PM brushless machines are expressed by eqns (7.32) and (7.34), i.e.,
Xad = 4m1 µ0 f
(N1 kw1 )2 τ Li
kf d
πp
g′
Xaq = 4m1 µ0 f
(N1 kw1 )2 τ Li
kf q
πp
gq′
where m1 is the number of phases, µ0 = 0.4π × 10−6 H/m is the magnetic
permeability of free space, f is the stator current frequency, N1 is the number
of series turns per phase, kw1 is the winding factor for the fundamental space
harmonic, p is the number of pole pairs, τ is the pole pitch, Li is the effective
216
Modern Permanent Magnet Electric Machines
length of the stator stack, and kf d and kf q are the reaction factors. For surface
PM configurations, the equivalent air gaps are
g ′ = gkC ksat +
hM
µ0
gq′ ≈ gkC
In the above equations for the equivalent air gap: g is the mechanical clearance, kC is Carter’s coefficient, ksat is the saturation factor of the magnetic
circuit, hM is the height of PM per pole. If the rotor is equipped with a nonferromagnetic retaining sleeve, the thickness of the sleeve should be added to
the mechanical clearance g.
Equivalent circuits for a PM synchronous machine are shown in Fig. 7.2,
while phasor diagrams are shown in Fig. 7.3. Armature currents derived on
the basis of a phasor diagram for an underexcited PM synchronous motor
with the stator winding resistance R1 taken into account are given by eqns
(7.47), (7.46) and (7.48), i.e.,
Iad =
U1 (Xsq cos δ − R1 sin δ) − Ef Xsq
Xsd Xsq + R12
Iaq =
U1 (R1 cos δ + Xsd sin δ) − Ef R1
Xsd Xsq + R12
Ia =
q
2 + I2
Iad
aq
The electromagnetic torque of a PM synchronous motor with the stator winding resistance R1 taken into account is expressed by eqn (7.14).
The d-axis current Iad is the magnetizing current. The q-axis current Iaq
is the torque producing current. For Iad = 0 the angle ψ = 0 and the total
armature current Ia = Iaq is the torque producing current. The angle ψ is the
angle between the armature current Ia = Iaq and the EMF Ef . Therefore,
the angle φ between the current Iaq and voltage U1 is equal to the load angle
δ between the voltage U1 and EMF Ef , and the power factor is (7.56)
cos φ =
Ef + Ia R1
V1
A PM synchronous motor is not self-starting. The following methods are used
for starting PM synchronous motors:
ˆ
ˆ
ˆ
Asynchronous starting with the aid of additional cage winding in the rotor
Starting by means of auxiliary motor
Frequency-change starting using a solid state converter, e.g., a VVVF inverter
PM synchronous motors are more expensive motors than their induction counterparts but, on the other hand, they are more compact motors, have higher
efficiency, better dynamic performance and other advantages (Table 7.3).
8
AXIAL AND TRANSVERSE FLUX
MOTORS
The axial flux PM motor is an attractive alternative to the cylindrical radial
flux motor due to its pancake shape, compact construction and easy integration with other electromechanical components. These motors are particularly
suitable for electrical vehicles, pumps, valve control, centrifuges, fans, machine tools, robots and industrial equipment. They have become widely used
for low-torque servo and speed control applications. Axial flux PM motors,
also called disk-type motors, can be designed as double-sided or single-sided
machines, with or without armature slots, with internal or external PM rotors and with surface-mounted or interior-type PMs. Low-power axial flux PM
machines are usually machines with slotless windings and surface PMs.
As the output power of the axial flux motor increases, the contact surface
between the rotor and shaft becomes smaller. Careful attention must be given
to the design of the rotor-shaft mechanical joint as this is the principal cause
of failures of disk-type motors.
In some cases, rotors are embedded in power-transmission components
to optimize the volume, mass, power transfer and assembly time. For EVs
with built-in wheel motors the payoff is a simpler electromechanical drive
system, higher efficiency and lower cost. Dual-function rotors may also appear
in pumps, blowers, elevators and other types of machinery, bringing new levels
of performance to these products.
Transfer flux motors (TFMs) are compact motors with a 3D magnetic circuit. The key can be designed using soft magnetic composite (SMC) material,
enabling a compact and cost-efficient system with superior efficiency.
The TFM employs high energy density PMs, simple toroidal windings, and
a modular stator core, which guides the main flux through a path transverse
to the direction of rotation. Because the armature MMF is applied to every
pole, the TFM is capable of producing high torque per unit volume provided
that the pole number is high.
A conventional drive system for electric vehicles consists of the motor,
a reduction gear and a differential gear to transmit the torque to the two
driving wheels. The gears increase the costs and mass, and the axle required
218
Modern Permanent Magnet Electric Machines
between the wheels obstructs the free layout of the drive components. These
shortcomings can be avoided, if the motors are gearless and integrated with the
wheels. This requires high-output torque. Compared to conventional designs,
the TFM concept has favorable characteristics concerning specific torque and
efficiency. For this reason it seems to be especially suited for direct drive
applications.
8.1 Axial flux disk motors
8.1.1 Force and torque of axial flux motors
In the design and analysis of axial flux motors the topology is complicated by
the presence of two air gaps, high axial attractive forces, changing dimensions
with radius and the fact that torque is produced over a continuum of radii,
not just at a constant radius as in cylindrical motors.
The tangential force acting on the disk can be calculated on the basis of
Ampere’s circuital law
dFx = Ia (dr × Bg ) = A(r)(dS × Bg )
(8.1)
√
where Ia dr = A(r)dS, A(r) = Am (r)/ 2 according to eqn (7.12) for D1in =
2r, dr is the radius element, dS is the surface element and Bg is the vector
of the normal component (perpendicular to the disk surface) of the magnetic
flux density in the air gap at given radius r.
Assuming the magnetic flux density in the air gap Bmg is independent of
the radius r, the electromagnetic torque on the basis of eqn (8.1) is
dTelm = rdFx = r[kw1 A(r)Bavg dS] = 2παi kw1 A(r)Bmg r2 dr
(8.2)
where Bavg = αi Bmg according to eqn (7.3) and dS = 2πrdr. The line current
density A(r) is the electric loading per one stator active surface in the case of
a typical stator winding with distributed parameters (double-sided stator and
inner rotor) or electric loading of the whole stator in the case of an internal
toroidal-type or coreless stator.
A three-dimensional FEM analysis is required to calculate the magnetic
field, winding inductances, induced EMF and torque. The model can be simplified to a two-dimensional model by introducing a cylindrical cutting plane
at the mean radius of the magnets [24]. This axial section is unfolded into a
two-dimensional surface on which the FEM analysis can be done.
The performance characteristics can also be calculated analytically, using
simplifications and adjusting the equations derived for cylindrical motors to
disk-type motors.
Table 8.1 shows specifications of axial flux PM brushless servo motors rated
up to 2.7 kW, manufactured by E. Bautz GmbH, Weiterstadt, Germany.
Axial and Transverse Flux Motors
219
Table 8.1. Specifications of PM disk brushless servo motors manufactured by E.
Bautz GmbH, Weiterstadt, Germany
Quantity
S632D S634D S712F S714F S802F S804F
Rated power, W
680
940
910
Rated torque, Nm
1.3
1.8
2.9
Maximum torque, Nm
7
9
14
Standstill torque, Nm
1.7
2.3
3.5
Rated current, A
4.0
4.9
4.9
Maximum current, A
21
25
24
Standstill current, A
5.3
6.3
5.9
Rated speed, rpm
5000 5000 3000
Maximum speed, rpm
6000 6000 6000
Armature constant, V/1000 rpm 23
25
42
Torque constant, Nm/A
0.35 0.39 0.64
Resistance, Ω
2.5
1.8
2.4
Inductance, mH
3.2
2.8
5.4
Moment of inertia, kgm2 × 10−3 0.08 0.12 0.21
Mass, kg
4.5
5.0
6.2
Diameter of frame, mm
150
150
174
Length of frame, mm
82
82
89
Power density, W/kg
151.1 188.0 146.8
Torque density, Nm/kg
0.289 0.36 0.468
1260
4.0
18
4.7
6.6
30
7.8
3000
6000
42
0.64
1.5
4.2
0.3
6.6
174
89
190.9
0.606
1850
5.9
28
7.0
9.9
47
11.7
3000
6000
42
0.64
0.76
3.0
0.6
9.7
210
103
190.7
0.608
2670
8.5
40
10.0
11.9
56
14.0
3000
6000
50
0.77
0.62
3.0
1.0
10.5
210
103
254.3
0.809
Table 8.2. Specifications of PM disk brushless motors for medium duty electrical
vehicles according to Premag, Cohoes, NY, U.S.A.
Quantity
HV2002 HV3202 HV4020 HV5020
Continuous output power, kW
20
Short duration
Output power, kW
30
Input voltage, V
200
Torque, Nm
93.8
“Base” speed, rpm
2037
Maximum speed, rpm
6725
Efficiency
0.902
Diameter of frame, mm
238.0
Length of frame, m
71.4
Mass, kg
9
Power density, kW/kg
2.22
Torque density, Nm/kg
10.42
32
40
50
48
182
150.0
2037
6725
0.868
286.0
85.6
12
2.67
12.5
60
350
191.0
2000
6600
0.906
329.2
68.1
14
2.86
13.64
75
350
238.7
2000
6600
0.901
284.2
70.1
14
3.57
17.05
220
Modern Permanent Magnet Electric Machines
Table 8.2 shows specifications of axial flux PM brushless motors rated from
20 to 50 kW for medium capacity (1300 to 4500 kg) electrical vehicles. Their
pancake shapes make them ideal for direct wheel attachment.
8.1.2 Double-sided motor with internal PM disk rotor
In the double-sided motor with internal PM disk rotor , the armature winding
is located on two stator cores. The disk with PMs rotates between two stators.
Fig. 8.1. Axial flux double-sided brushless motor with internal PM disk rotor: 1 –
rotor, 2 – PM, 3 – three-phase stator, 4 – housing, 5 – end cover, 6 – terminal box.
Courtesy of Omni Powertrain Technologies Houston, TX, USA.
An eight-pole configuration is shown in Fig. 8.1. PMs are embedded or
glued in a nonferromagnetic rotor skeleton. The nonferromagnetic air gap is
large, i.e., the total air gap is equal to two mechanical clearances plus the
thickness of a PM with its relative magnetic permeability close to unity. A
double-sided motor with parallel connected stators can operate even if one
stator winding is broken. On the other hand, a series connection can provide
equal but opposing axial attractive forces.
8.1.3 Stator cores of axial flux motors
Normally, the stator cores are wound from electrotechnical steel strips and
the slots are machined by shaping or planing. An alternative method is first
to punch the slots with variable distances between them and then to wind the
steel strip into the form of the slotted toroidal core (Research and Development Institute of Electrical Machines VÚES in Brno, Republic of Czech). In
Axial and Transverse Flux Motors
221
Fig. 8.2. Double-sided axial flux PM brushless motor with internal PM disk rotor
and built-in brake: 1 – stator winding, 2 – stator core, 3 – disk rotor with PMs,
4 – shaft, 5 – left frame, 6 – right frame, 7 – flange, 8 – brake cover, 9 – brake flange,
10 – electromagnetic brake, 11 – encoder or resolver. Courtesy of Slovak University
of Technology STU, Bratislava and Electrical Research and Testing Institute, Nová
Dubnica, Slovakia.
addition, this manufacturing process allows for making skewed slots to minimize the cogging torque and effect of slot harmonics. Each stator core has
skewed slots in opposite directions. It is recommended to make a wave stator
winding to obtain shorter end connections and more space for the shaft. An
odd number of slots, e.g., 25 instead of 24 can also reduce the cogging torque
(VÚES Brno).
Another technique is to form the stator core segments [79]. Each segment
corresponds to one slot pitch (Fig. 8.3). The lamination strip of constant
width is folded at distances proportional to the radius. To make folding easy,
the strip has transverse grooves on opposite sides of the alternative steps.
The zigzag laminated segment is finally compressed and fixed using a tape or
thermosetting, as shown in Fig. 8.3 [79].
8.1.4 Main dimensions of axial flux motors
The main dimensions of a double-sided PM brushless motor with internal disk
rotor can be determined using the following assumptions: (a) the electric and
magnetic loadings are calculated on an average diameter of the stator core;
(b) the number of turns per phase per one stator is N1 ; (c) the phase armature
222
Modern Permanent Magnet Electric Machines
1
3
5
4
2
Fig. 8.3. Stator core segment formed from lamination strip: 1 – lamination strip,
2 – groove, 3 – folding, 4 – compressed segment, 5 – finished segment.
current in one stator winding is Ia ; (d) the back EMF per phase per one stator
winding is Ef .
The line current density per one stator is expressed by eqn (7.12) in which
the inner stator diameter should be replaced by an average diameter
Dav = 0.5(Dext + Din )
(8.3)
where Dext is the outer diameter and Din is the inner diameter of the stator
core. The pole pitch and the effective length of the stator core in a radial
direction are
τ=
πDav
2p
Li = 0.5(Dext − Din )
(8.4)
The EMF induced in the stator winding by the rotor excitation system, according to eqn (7.8), for the disk rotor synchronous motor has the following
form:
√
√
Ef = π 2ns pN1 kw1 Φf = π 2ns N1 kw1 Dav Li Bmg
(8.5)
where the magnetic flux can approximately be expressed as
Φf ≈
2
Dav
τ Li Bmg =
Li Bmg
π
p
(8.6)
The electromagnetic apparent power in two stators
2
Selm = m1 (2Ef )Ia = m1 Ef (2Ia ) = π 2 kw1 Dav
Li ns Bmg Am
(8.7)
Axial and Transverse Flux Motors
223
For series connection the EMF is equal to 2Ef and for parallel connection
the current is equal to 2Ia . For a multidisk motor the number “2” should be
replaced by the number of stators. On the other hand
Selm = m1 (2Ef )Ia =
ϵPout
η cos ϕ
(8.8)
where
ϵ=
Ef
U1
(8.9)
It is convenient to use the ratio of inner-to-outer stator diameter
kd =
Din
Dext
(8.10)
Theoretically, a PM √axial flux motor develops maximum electromagnetic
2
Li proportional to the volume
torque when kd = 1/ 3 [1]. The product Dav
of one stator is
2
Dav
Li =
1
3
(1 + kd )(1 − kd2 )Dext
8
Putting
kD =
1
(1 + kd )(1 − kd2 )
8
(8.11)
3
2
. In connection
Li = kD Dext
the volume of one stator is proportional to Dav
with eqns (8.7) and (8.8) the stator outer diameter is
s
ϵPout
(8.12)
Dext = 3 2
π kw1 kD ns Bmg Am η cos ϕ
The outer diameter of the stator
√ is the most important dimension of disk rotor
PM motors. Since Dext ∝ 3 Pout the outer diameter increases rather slowly
with the increase of the output power (Fig. 8.4). This is why small power
disk motors have a relatively large diameter. The disk rotor is preferred for
medium and large power motors. Motors with output power over 10 kW have
reasonable diameters. Also, disk construction is recommended for AC servo
motors fed with high-frequency voltage.
8.1.5 Double-sided axial-flux motors with a single stator
A double-sided motor with internal stator is more compact than the previous construction with an internal PM rotor [25, 77, 88]. In this machine the
toroidal stator core is also formed from a continuous steel tape, as in the motor
224
Modern Permanent Magnet Electric Machines
1
Dext
m
0.2
0.5
0.15
0
0
20000
0.1
40000
Pout
W
kD
0.05
60000
80000
100000
Fig. 8.4. Outer diameter Dext as a function of the output power Pout and parameter
kD for ϵ = 0.9, kw1 η cos ϕ = 0.84, ns = 1000 rpm = 16.67 rev/s and Bmg Am =
26, 000 TA/m.
with an internal PM disk. The polyphase slotless armature winding (toroidal
type) is located on the surface of the stator core. The total air gap is equal to
the sum of the thickness of the armature winding, mechanical clearance and
the thickness of the PM in the axial direction. The double-sided rotor with
PMs is located at two sides of the stator. The configurations with internal and
external rotors are shown in Fig. 8.5. The three-phase winding arrangement,
magnet polarities and flux paths in the magnetic circuit are shown in Figs 8.6
and 8.7.
The average electromagnetic torque developed by the motor according to
eqn (8.2) is
dTelm = 2αi m1 Ia N1 kw1 Bmg rdr
Integrating the above equation from Din /2 to Dext /2 with respect to r
Telm =
1
2
2
αi m1 Ia N1 kw1 Bmg (Dext
− Din
)
4
1
2
αi m1 N1 kw1 Bmg Dext
(1 − kd2 )Ia
(8.13)
4
where kd is according to eqn (8.10). The magnetic flux per pole pitch is
=
Φf = αi Bmg
2π
2p
Z 0.5Dext
rdr =
0.5Din
1 π
2
αi Bmg Dext
(1 − kd2 )
8 p
(8.14)
Axial and Transverse Flux Motors
225
The above eqn (8.14) is more accurate than eqn (7.9). Putting eqn (8.14) into
eqn (8.13) the average torque is
p
Telm = 2 m1 N1 kw1 Φf Ia
(8.15)
π
To obtain the rms torque for sinusoidal current and sinusoidal
√ magnetic flux
density, eqn (8.15) should be multiplied by the coefficient π 2/4 ≈ 1.11, i.e.,
m1
Telm = √ pN1 kw1 Φf Ia = kT Ia
2
(8.16)
where the torque constant
m1
kT = √ pN1 kw1 Φf
2
(8.17)
The EMF at no-load can be found by differentiating the first harmonic of the
magnetic flux waveform ϕf 1 = Φf sin ωt and multiplying by N1 kw1 , i.e.,
ef = N1 kw1
dϕf 1
= 2πf N1 kw1 Φf cos ωt
dt
(a)
(b)
1
1
2
2
3
3
7
7
6
5
4
4
5
4
6
Fig. 8.5. Double-sided motors with one slotless stator: (a) internal rotor, (b) external rotor. 1 – stator core, 2 – stator winding, 3 – steel rotor, 4 – PMs, 5 – resin,
6 – frame, 7 – shaft.
226
Modern Permanent Magnet Electric Machines
4
S
N
2
S
-B
A
-C
B
-A
C
-B
A
-C
B
-A
C
-B
A
-C
B
-A
C
S
N
S
1
3
1
2
4
Fig. 8.6. Three-phase winding, PM polarities and magnetic flux paths of a doublesided disk motor with one internal slotless stator. 1 — winding, 2 — PM, 3 — stator
yoke, 4 — rotor yoke.
The rms√value is obtained by dividing the peak value 2πf N1 kw1 Φf of the
EMF by 2, i.e.,
√
√
Ef = π 2f N1 kw1 Φf = π 2pN1 kw1 Φf ns = kE ns
(8.18)
where the EMF constant (armature constant)
√
kE = π 2pN1 kw1 Φf
(8.19)
The same form of eqn (8.18) can be obtained on the basis of the electromagnetic developed torque Telm = m1 Ef Ia /(2πns ) in which Telm is according to
eqn (8.16). For the toroidal-type winding, the winding factor kw1 = 1.
A motor with an external rotor, according to Fig. 8.5b, has been designed for hoist applications. A similar motor can be used as an electric car
wheel propulsion machine. Additional magnets on cylindrical parts of the rotor are sometimes added, or U-shaped magnets can be designed. Such magnets
embrace the armature winding from three sides and only the internal portion
of the winding does not produce any electromagnetic torque.
Owing to the large air gap, the maximum magnetic flux density does not
exceed 0.65 T. To produce this flux density, sometimes a large volume of PMs
is required. As the permeance component of the flux ripple associated with
the slots is eliminated, the cogging torque is practically absent. The magnetic
circuit is unsaturated (slotless stator core). On the other hand, the machine
structure lacks the necessary robustness [77].
The stator can also be made with slots (Fig. 8.7). For this type of motor,
slots are progressively notched into the steel tape as it is passed from one
mandrel to another and the polyphase winding is inserted [88]. In the case of
the slotted stator, the air gap is small (g ≈ 0.5 mm) and the air gap magnetic
flux density can increase to 0.85 T [25]. The magnet thickness is less than
50% that of the previous design, shown in Figs 8.5 and 8.6.
Axial and Transverse Flux Motors
227
Fig. 8.7. Double-sided motor with one internal slotted stator and buried PMs. 1 –
stator core with slots, 2 – PM, 3 – mild steel core (pole), 4 – nonferromagnetic rotor
disk.
There are a number of applications for medium and large power axial
flux motors with external PM rotors, especially in electrical vehicles [25, 88].
Disk-type motors with external rotors have a particular advantage in traction
applications, such as buses and shuttles, due to their large radius for torque
production. For small electric cars, the possibility of mounting the electric
motor directly into the wheel has many advantages; it simplifies the drive
system and the constant velocity joints are no longer needed [25].
8.1.6 Single-sided motors
Single-sided construction of an axial flux motor is simpler than double-sided,
but the torque produced is lower. Fig. 8.8 shows typical constructions with
surface PM rotors and laminated stators wound from electromechanical steel
strips. A single-sided motor according to Fig. 8.8a has a standard frame and
shaft. It can be used in industrial, traction and servo electromechanical drives.
The motor for hoist applications shown in Fig. 8.8b is integrated with a sheave
(drum for ropes) and brakes (not shown). It is used in gearless elevators [37].
Specifications of single-sided disk-type PM motors for gearless passenger
elevators are given in Table 8.3 [37]. Stators have from 96 to 120 slots with
three-phase short-pitch winding, insulation class F. For example, the MX05
motor rated at 2.8 kW, 280 V, 18.7 Hz has the stator winding resistance
R1 = 3.5 Ω, stator winding reactance X1 = 10 Ω, 2p = 20, sheave diameter
340 mm and weighs 180 kg.
228
Modern Permanent Magnet Electric Machines
(a)
1
(b)
2
2
1
3
3
5
5
4
6
4
Fig. 8.8. Single-sided disk motors: (a) for industrial and traction electromechanical
drives, (b) for hoist applications. 1 – stator, 2 – PM, 3 – rotor, 4 – frame, 5 – shaft,
6 – sheave.
Table 8.3. Specifications of single-sided PM disk brushless motors for gearless elevators manufactured by Kone, Hyvinkää, Finland
Quantity
MX05
MX06
MX10
MX18
Rated output power, kW
2.8
3.7
6.7
46.0
Rated torque, Nm
240
360
800
1800
Rated speed, rpm
113
96
80
235
Rated current, A
7.7
10
18
138
Efficiency
0.83
0.85
0.86
0.92
Power factor
0.9
0.9
0.91
0.92
Cooling
natural natural natural
forced
Diameter of sheave, m
0.34
0.40
0.48
0.65
Elevator load, kg
480
630
1000
1800
Elevator speed, m/s
1
1
1
4
Location
hoistway hoistway hoistway machine room
Axial and Transverse Flux Motors
229
8.1.7 Ironless double-sided motors
The ironless disk-type PM brushless motor has neither armature nor excitation ferromagnetic core. The stator winding consists of full-pitch or short-pitch
coils wound from insulated wires. Coils can be arranged in overlapping layers
like petals around the center of a flower and embedded in a plastic of very
high mechanical integrity, e.g., U.S. Patent No. 5744896 [56]. The winding can
be fixed to the cylindrical part of the frame.
The twin nonferromagnetic rotor disks have cavities of the same shape as
PMs. Magnets are inserted in these cavities and glued to the rotor disks. The
PMs of opposite polarity fixed to two parts of the rotor produce magnetic
flux, the lines of which crisscross the stator winding. The motor construction
is shown in Fig. 8.9.
Fig. 8.9. Ironless double-sided PM brushless motor of disk type: (a) expanded view;
(b) assembled motor; (c) 45◦ Malinson–Halbach array. 1 – 3-phase Litz wire stator
winding, 2 – Malinson–Halbach array of PMs, 3 – carbon fiber backing plate, 4 –
carbon fiber spoke. Courtesy of LaunchPoint Technologies, Goleta, CA, USA.
A strong magnetic flux density in the air gap is produced by PMs arranged
in a Mallinson–Halbach array. The Mallinson–Halbach array does not require
any ferromagnetic cores and excites magnetic flux density closer to the sinusoid than a conventional PM array. The key concept of the Mallinson–Halbach
array is that the magnetization vector should rotate as a function of distance
along the array (Figs 8.10 and 8.11). The magnetic flux density distribution
plotted in Fig. 8.10 has been produced with the aid of a two-dimensional FEM
analysis of an ironless motor with magnet-to-magnet air gap of 10 mm (8 mm
winding thickness, two 1 mm air gaps). The thickness of each PM is hM = 6
mm. The remanent magnetic flux density is Br = 1.23 T and the coercivity
is Hc = 979 kA/m. The peak value of the magnetic flux density in the air
230
Modern Permanent Magnet Electric Machines
(a)
900
(b)
60 0
(c)
45 0
Fig. 8.10. Magnetic flux distribution in an ironless double-sided brushless motor
excited by Mallinson–Halbach arrays of PMs: (a) 900 , (b) 600 , and (c) 450 PM array.
Axial and Transverse Flux Motors
231
(a)
(b)
Fig. 8.11. Waveforms of the normal and tangent components of the magnetic
flux density in the center of an ironless double-sided brushless motor excited by
Mallinson–Halbach arrays of PMs: (a) 900 , (b) 450 . The magnetic flux density waveforms are functions of the circumferential distance at the mean radius of the magnets.
gap exceeds 0.6 T. Three Mallinson–Halbach arrays have been simulated, i.e.,
900 , 600 and 450 . As the angle between the magnetic flux density vectors of
neighboring magnets decreases, the peak value of the normal component of
the magnetic flux density increases slightly.
Ironless motors do not produce any torque pulsations at zero current state
and can reach very high efficiency impossible for standard motors with ferromagnetic cores. Elimination of core losses is extremely important for highspeed motors operating at high frequencies. Another advantage is a very small
mass of the ironless motor and consequently high power density and torque
density. These motors are excellent for propulsion of solar-powered electric
cars [72]. The drawbacks include mechanical integrity problems, high axial
232
Modern Permanent Magnet Electric Machines
Fig. 8.12. Exploded view of the axial flux PM brushless motor with film coil ironless
stator winding. Courtesy of Embest, Soeul, South Korea.
forces between PMs on the opposite disks, heat transfer from the stator winding and its low inductance.
Small ironless motors may have printed circuit stator windings or film coil
windings. The film coil winding is stamped from a thin copper foil ribbon. The
film coil stator winding has many coil layers while the printed circuit winding
has one or two coil layers. Fig. 8.12 shows an ironless brushless motor with film
coil stator winding. Small film coil motors are used in computer peripherals,
pagers, mobile phones, flight recorders, card readers, copiers, printers, plotters,
micrometers, labeling machines, video recorders and medical equipment.
8.1.8 Multidisk motors
There is a limit on the increase of motor torque that can be achieved by
enlarging the motor diameter. Factors limiting the single-disk design are (a)
axial force taken by bearings, (b) integrity of the mechanical joint between
the disk and shaft and (c) disk stiffness. A more reasonable solution for large
torques are double- or triple-disk motors.
There are several constructions of multidisk motors [2, 3, 4, 16]. Large
multidisk motors rated at least 300-kW have a water cooling system with
radiators around the winding end connections [16]. To minimize the winding
losses the cross section of conductors is bigger in the slot area (skin effect)
than in the end connection region. Using a variable cross section means a gain
of 40% in the rated power [16]. Owing to high mechanical stresses, titanium
alloy is recommended for disk rotors.
A double-disk motor for gearless elevators is shown in Fig. 8.13 [37]. Table
8.4 lists specification data of double-disk PM brushless motors rated from 58
to 315 kW [37].
Axial and Transverse Flux Motors
233
Fig. 8.13. Double-disk PM brushless motor for gearless elevators. Courtesy of Kone,
Hyvinkää, Finland.
Table 8.4. Specifications of double-disk PM brushless motors manufactured by
Kone, Hyvinkää, Finland
Quantity
MX32 MX40 MX100
Rated output power, kW 58
Rated torque, Nm
3600
Rated speed, rpm
153
Rated current, A
122
Efficiency
0.92
Power factor
0.93
Elevator load, kg
1600
Elevator speed, m/s
6
92
315
5700 14,000
153
214
262
1060
0.93
0.95
0.93
0.96
2000 4500
8
13.5
234
Modern Permanent Magnet Electric Machines
Ironless disk motors provide a high level of flexibility to manufacture multidisk motors composed of the same segments (modules). Fractional horsepower
motors can be assembled “on-site” from modules (Fig. 8.14) by simply removing one of the bearing covers and connecting terminal leads to the common
terminal board. The number of modules depends on the requested shaft power
or torque. One of the disadvantages of this type of multidisk motor is that a
large number of bearings equal to double the number of modules are required.
(a)
(b)
Fig. 8.14. Fractional horsepower ironless multidisk PM brushless motor: (a) singlemodule, (b) four-module motor.
Motors rated at kWs or tens of kWs must be assembled using separate
stator and rotor units (Fig. 8.15). Multidisk motors have the same end bells
with cylindrical frames inserted between them. The number of rotors is K2 =
K1 + 1 where K1 is the number of stators, while the number of cylindrical
frames is K1 − 1. The shaft must be tailored to the number of modules. Like
a standard motor, this kind of motor has only two bearings.
Table 8.5 shows the specifications of single disk and multidisk PM brushless
motors manufactured by Lynx Motion Technology Corporation, New Albany,
IN, U.S.A. The multidisk motor M468 consists of T468 single disk motors.
LaunchPoint Technologies, Goleta, CA, USA, has been developing high
specific power, high-efficiency electric machines for the demanding, highreliability applications associated with hybrid electric UAV flight. The machines are an axial flux design based on dual Mallinson–Halbach array magnet rotors and coreless stators (Fig. 8.16). This combination of design features
allows for an extremely high-efficiency UAV motor or generator with good specific power. The power density of coreless axial flux motors is from 4.1 to 8.2
kW/kg. The UAV motors and generators are air cooled and create their own
air flow, so no additional cooling system or fan is required. LaunchPoint’s
hybrid electric UAV motors, generators, and alternators enhance payload,
mission time, survivability and efficiency.
Axial and Transverse Flux Motors
(a)
235
(b)
Fig. 8.15. Ironless multidisk PM brushless motors assembled using the same stator
and rotor units: (a) single-stator motor, (b) three-stator motor.
Table 8.5. Specifications of ironless single-disk and multidisk PM brushless motors
manufactured by Lynx Motion Technology Corporation, New Albany, IN, U.S.A.
Quantity
Output power, kW
Speed, rpm
Torque, Nm
Efficiency
Voltage line-to-line, V
Current, A
Armature constant line-to-line, V/rpm
Torque constant, Nm/A
Resistance d.c., phase-to-phase, Ω
Inductance line-to-line, mH
Rotor inertia, kgm2
Outer diameter, m
Mass, kg
Power density, kW/kg
Torque density, Nm/kg
T468
M468
single-disk motor multidisk motor
32.5
230
1355
0.94
432 (216)
80 (160)
1.43
17.1 (8.55)
7.2 (1.8)
4.5 (1.125)
0.48
0.468
58.1
0.56
23.3
156
1100
1355
0.94
400
243
0.8
5.58
0.00375
–
1.3
0.468
131.0
1.19
10.34
236
Modern Permanent Magnet Electric Machines
Fig. 8.16. Axial flux coreless PM motor for hybrid-electric aircraft propulsion: (a)
expanded view; (b) assembled motor. 1 – coreless stator, 2 – encapsulated stator
winding, 3 – Mallinson–Halbach array, 4 – integrated impeller for cooling. LaunchPoint Technologies, Goleta, CA, USA.
8.2 Transverse flux motors
8.2.1 Principle of operation
In a transverse flux motor (TFM) the electromagnetic force vector is perpendicular to the magnetic flux lines. In all standard or longitudinal flux motors
the electromagnetic force vector is parallel to the magnetic flux lines. The
TFM can be designed as a single-sided (Fig. 8.17a) or double-sided machine
(Fig. 8.17b). Single-sided machines are easier to manufacture and have better
prospects in practical applications.
The stator consists of a toroidal single-phase winding embraced by Ushaped cores. The magnetic flux in U-shaped cores is perpendicular to the
stator conductors and direction of rotation. The rotor consists of surface or
buried PMs and a laminated or solid core. A three-phase machine can be built
of three of the same single-phase units as shown in Fig. 8.18. The magnetic
circuits of either stator or rotor of each single-phase unit should be shifted by
3600 /(pm1 ) mechanical degrees where p is the number of the rotor pole pairs
and m1 is the number of phases. A TFM with an internal stator (Fig. 8.18a)
has a smaller external diameter. It is also easier to assemble the winding and
internal stator cores. On the other hand, the heat transfer conditions are worse
for internal than for the external stator.
If the number of the rotor PM poles is 2p, the number of the stator Ushaped cores is equal to p, i.e., the number of the stator U-shaped cores
is equal to the number of the rotor pole pairs p. Each of the U-shaped cores
creates one pole pair with two poles in the axial direction. The more poles, the
better utilization and smoother operation of the machine. The power factor
also increases with the number of poles. TFMs have usually from 2p = 24
Axial and Transverse Flux Motors
237
(a)
2
7
3
4
i
5
N
S
N
(b)
S
1
`
N
3
2
7
4
1
N
S
N
4
i
N
`
s
3
2
6
1
Fig. 8.17. PM transverse flux motor: (a) single-sided, (b) double-sided. 1 — PM,
2 — stator core, 3 – stator winding, 4 – stator current, 5 – rotor yoke, 6 – mild steel
poles shoes, 7 – magnetic flux.
(a)
(b)
Fig. 8.18. Three-phase TFM consisting of three single-phase units with: (a) internal
stator, (b) external stator.
238
Modern Permanent Magnet Electric Machines
to 72 poles. The input frequency is higher than the power frequency of 50
or 60 Hz and the speed at an increased frequency is low. For example, a
TFM with 2p = 36 fed with 180 Hz input frequency operates at the speed
ns = f /p = 180/18 = 10 rev/s = 600 rpm.
Specifications of small two-phase and three-phase TFMs manufactured by
Landert-Motoren AG, Bülach, Switzerland are shown in Table 8.6.
Table 8.6. TFMs manufactured by Landert-Motoren AG, Bülach, Switzerland.
Number of phases
Continuous torque
(no active cooling)
• at standstill, Nm
• at 300 rpm, Nm
• at 600 rpm, Nm
Efficiency
• at 300 rpm
• at 600 rpm
EMF constant, V/rpm
Torque constant, Nm/A
Rotor
Outer diameter, mm
Protection
Class of insulation
Cooling
SERVAX SERVAX
SERVAX
SERVAX
MDD1-91-2 MDD1-91-3 MDD1-133-2 MDD1-133-3
2
3
2
3
3.5
2.5
1.5
4.5
3.3
2
0.60
0.65
0.07
1.8
0.65
0.68
0.07
2.7
91
12
8
5
16
10
7
0.68
0.70
0.16
2.8
external
91
133
IP54
F
IC410
0.76
0.80
0.15
4
133
The peak value of the line current density of a single phase is [44] (see
also eqn (7.12)
√
√
p 2Ia N1
2Ia N1
=
Am =
(8.20)
2τ
πDg
where Ia is the stator (armature) rms current, N1 is the number of turns
per phase, τ is the stator pole pitch and Dg is the average air gap diameter.
At constant ampere turns-to-diameter ratio the line current density can be
increased by increasing the number of pole pairs. Since the force density (shear
stress) is proportional to the product Am Bmg , the electromagnetic torque of
the TFM is proportional to the number of pole pairs. The higher the number
of poles, the higher the torque density of a TFM.
Since at a large number of poles and increased frequency the speed is
low and the electromagnetic torque is high, TFMs are inherently well-suited
propulsion machines to gearless electromechanical drives. Possible designs of
magnetic circuits of single-sided TFMs are shown in Fig. 8.19. In both designs
Axial and Transverse Flux Motors
(a)
239
(b)
fl
netic
mag
magnetic flux
ux
S
N
N
S
Fig. 8.19. Practical single-sided TFMs: (a) with magnetic shunts and surface PMs,
(b) with twisted stator cores and surface PMs.
the air gap magnetic flux density is almost the same. However, in the TFM
with magnetic shunts the rotor can be laminated radially [44].
8.2.2 EMF and electromagnetic torque
According to eqn (7.2) the first harmonic of the magnetic flux per pole pair
per phase excited by the PM rotor of a TFM is
2
τ lp Bmg1
(8.21)
π
where τ = πDg /(2p) is the pole pitch (in the direction of rotation), lp is the
axial length of the stator pole shoe (Fig. 8.20) and Bmg1 is the first harmonic
of the air gap peak magnetic flux density. With the rotor spinning at constant
speed ns = f /p, the fundamental harmonic of the magnetic flux is
Φf 1 =
2
2
τ lp Bmg1 sin(ωt) = τ lp kf Bmg sin(ωt)
(8.22)
π
π
where the form factor kf = Bmg1 /Bmg of the excitation field is given by eqns
(7.16) in which bp is the width of the stator pole shoe (salient-pole stator).
The approximate air gap magnetic flux density Bmg can be found using eqn
(7.2) both for the magnetic circuit shown in Fig. 8.19a and Fig. 8.19b. The
instantaneous value of the sinusoidal EMF at no load induced in N1 armature
turns by the rotor excitation flux Φf 1 is
ϕf 1 = Φf 1 sin(ωt) =
dΦf 1
= ωN1 pΦf 1 cos(ωt) = 2πf N1 pΦf 1 cos(ωt)
dt
where p is the number of the stator pole pairs (U-shaped cores). The peak
value of EMF is 2πf n1 pΦf 1 . Thus, the rms value of EMF is
ef = N1 p
Ef =
√
2πf N1 pΦf 1
√
= π 2 N1 p2 Φf 1 ns
2
(8.23)
240
Modern Permanent Magnet Electric Machines
lM
lM
N
hry
N
hM
N
S
wM
S
S
N
S
wM
g
bp
bp
h0
au
wu
hw
hu
Dg
au
lp
bu
lp
wu
Fig. 8.20. Dimensions of U-shaped stator core and coil.
or
√
Ef = 2 2f N1 pτ lp kf Bmg
(8.24)
The electromagnetic power
√
Pelm = m1 Ef Ia cos Ψ = 2 2 m1 f N1 pτ lp kf Bmg Ia cos Ψ
(8.25)
where Ψ is the angle between the current Ia and EMF Ef . The electromagnetic
torque developed by the TFM is
Telm =
Pelm
m1
m1
=
Ef Ia cos Ψ = √ N1 p2 Φf 1 Ia cos Ψ
2πns
2πns
2
(8.26)
As in the case of other motors, the EMF and electromagnetic torque can be
brought to simpler forms
Ef = kE ns
and
Td = kT Ia
(8.27)
Assuming Φf 1 = const, the EMF constant and torque constant are, respectively,
√
kE = π 2 N1 p2 Φf 1
(8.28)
m1
m1
kT =
kE cos Ψ = √ N1 p2 Φf 1 cos Ψ
(8.29)
2π
2
For Iad = 0 the total current Ia = Iaq is torque-producing and cos Ψ = 1.
Axial and Transverse Flux Motors
241
8.2.3 Armature winding resistance
The armature winding resistance can be calculated approximately as
R1 ≈ k1R π[Dg ± g ± (hw + ho )]
N1
aw σ1 sa
(8.30)
where k1R is the skin-effect coefficient for resistance [32], hw is the coil height,
ho = hu − hw − au is the top portion of the “slot” not filled with conductors,
aw is the number of parallel wires, σ1 is the conductivity of the armature
conductor at a given temperature and sa is the cross section of the armature
single conductor. The “+” sign is for the external stator and the “−” sign is
for the internal stator.
8.2.4 Armature reaction and leakage reactance
The mutual reactance corresponding to the armature reaction reactance in a
synchronous machine can analytically be calculated in an approximate way.
One U-shaped core (pole pair) of the stator can be regarded as an AC electromagnet with N1 turn coil which,
when fed with the sinusoidal current Ia ,
√
produces peak MMF equal to 2Ia N1 . The equivalent d-axis field MMF per
pole pair per phase, which produces the same magnetic flux density as the
armature reaction MMF, is
√
Bad ′
Bad1 ′
2Iad N1 =
g =
g
µ0
µ0 kf d
where g ′ is the equivalent air gap and kf d = Bad1 /Bad is the d-axis form factor
of the armature reaction according to eqn (7.17). Thus, the d-axis armature
current as a function of Bad1 is
Iad =
Bad1 g ′
√
kf d µ0 2N1
(8.31)
At constant magnetic permeability, the d-axis armature EMF
√
Ead = 2 2f N1 pτ lp Bad1
(8.32)
is proportional to the armature current Iad . Thus, the d-axis armature reaction
reactance is
Xad =
Ead
τ lp
= 4µ0 f N12 p ′ kf d
Iad
g
(8.33)
Similarly, the q-axis armature reactance
Xaq =
Eaq
τ lp
= 4µ0 f N12 p ′ kf q
Iaq
g
(8.34)
242
Modern Permanent Magnet Electric Machines
The d-axis and q-axis form factors of the armature reaction can be found
in a similar way as in Section 7.3. Most TFMs are designed with surface
configuration of PMs and kf d = kf q = 1, i.e., Xad = Xaq .
Neglecting the saturation of the magnetic circuit, the equivalent air gap is
calculated
ˆ
ˆ
for the TFM with magnetic shunts (Fig. 8.19a)
hM
′
g =4 g+
µrrec
for the TFM with twisted U-shaped cores (Fig. 8.19b)
hM
g′ = 2 g +
µrrec
(8.35)
(8.36)
where g is the mechanical clearance in the d-axis, hM is the radial height of
the PM (one pole) and µrrec is the relative recoil magnetic permeability of
the PM. To take into account the magnetic saturation, the equivalent air gap
g ′ should be multiplied by the saturation factor ksat in the d-axis and ksatq
in the q-axis.
The armature reaction inductances
Lad =
2
τ lp
Xad
= µ0 N12 p ′ kf d
2πf
π
g
(8.37)
Laq =
Xaq
2
τ lp
= µ0 N12 p ′ kf q
2πf
π
g
(8.38)
The leakage inductance of the stator winding is approximately equal to the
sum of the “slot” leakage inductance and pole-top leakage reactance. The
approximate equation is
L1 ≈ µ0 π[Dg ± g ± (hw + ho )]N12 (λ1s + λ1p )
(8.39)
where hw is the height of the coil, ho = hu − hw − au is the top portion of the
“slot” not filled with conductors, the “+” sign is for the external stator and
the “−” sign is for the internal stator. The coefficients of leakage permeances
are
ˆ
coefficient of “slot” leakage permeance
λ1s =
ˆ
hw
ho
+
3bu
bu
(8.40)
coefficient of pole-top leakage permeance
λ1p ≈
where bu = wu − 2lp (Fig. 8.20).
5g/bu
5 + 4g/bu
(8.41)
Axial and Transverse Flux Motors
243
The leakage inductance according to eqn (8.39) is much smaller than that
obtained from measurements and the FEM. Good results are obtained if the
eqn (8.39) is multiplied by 3 [45]. For most TFMs, L1 > Lad and L1 > Laq .
The leakage inductance can also be estimated as a sum of three inductances, i.e., due to lateral leakage flux, “slot” leakage flux and leakage flux
about the portion of the coil not embraced by the ferromagnetic core [7].
The synchronous reactances in the d and q axes according to eqn (7.15)
are the sums of the armature reaction reactances (8.33), (8.34) and leakage
reactance X1 = 2πf L1 .
8.2.5 Magnetic circuit
Kirchhoff equations for the MVD per pole pair are
ˆ
for the TFM with magnetic shunts (Fig. 8.19a)
4
ˆ
X
Bmg
Bmg
Br
hM = 4
hM + 4
g+
HF ei lF ei
µ0 µrrec
µ0 µrrec
µ0
i
for the TFM with twisted U-shaped cores (Fig. 8.19b)
2
X
Bmg
Bmg
Br
hM = 2
hM + 2
g+
HF ei lF ei
µ0 µrrec
µ0 µrrec
µ0
i
P
where Hc = Br /(µ0 µrrec ) and i HF ei lF ei is the magnetic voltage drop in
ferromagnetic parts of the magnetic circuit (stator and rotor cores). The above
equations can be expressed with the aid of the saturation factor ksat of the
magnetic circuit
ˆ
for the TFM with magnetic shunts (Fig. 8.19a)
Br
Bmg
hM
4
hM = 4
+ gksat
µ0 µrrec
µ0
µrrec
(8.42)
where
P
ksat = 1 +
ˆ
i HF ei lF ei
4Bmg g/µ0
for the TFM with twisted U-shaped cores (Fig.8.19b)
Bmg
hM
Br
hM = 2
+ gksat
2
µ0 µrrec
µ0
µrrec
(8.43)
(8.44)
where
P
ksat = 1 +
i HF ei lF ei
2Bmg g/µ0
(8.45)
244
Modern Permanent Magnet Electric Machines
Both eqns (8.42) and (8.44) give almost the same value of the air gap magnetic
flux density, i.e.,
Bmg =
Br
1 + (µrrec g/hM )ksat
(8.46)
Please note that the saturation factors ksat expressed by eqns (8.43) and (8.45)
are different.
8.2.6 Advantages and disadvantages
The TFM has several advantages over a standard PM brushless motor, i.e.,
(a) at low rotor speed the frequency in the stator (armature) winding is high
(large number of poles), i.e., a low speed machine behaves as a high-speed
machine, which is the cause of better utilization of active materials than
in standard (longitudinal flux) PM brushless motors for the same cooling
system, i.e., higher torque density or higher power density;
(b) less winding and ferromagnetic core materials for the same torque;
(c) simple stator winding consisting of a single ring-shaped coil (cost-effective
stator winding, no end connection);
(d) unity winding factor (kw1 = 1);
(e) the more poles, the higher the torque density, higher power factor and less
torque ripple;
(f) a three-phase motor can be made of three (or multiples of three) identical
single-phase units;
(g) a three-phase TFM can be fed from a standard three-phase inverter for
PM brushless motors using a standard encoder;
(h) the machine can operate as a low-speed generator with high-frequency
output current.
Although the stator winding is simple, the motor consists of a large number
of poles (2p ≥ 24). There is a double saliency (the stator and rotor) and
each salient pole has a separate “transverse flux” magnetic circuit. Careful
attention must be given to the following problems:
(a) To avoid a large number of components, it is necessary to use radial laminations (perpendicular to the magnetic flux paths in some portions of the
magnetic circuit), sintered powders or hybrid magnetic circuits (laminations and sintered powders).
(b) The motor external diameter is smaller in the so-called “reversed design,”
i.e., with external PM rotor and internal stator.
(c) The TFM uses more PM material than an equivalent standard PM brushless motor.
(d) The power factor decreases as the load increases and special measures must
be taken to improve the power factor.
Axial and Transverse Flux Motors
245
(e) As each stator pole faces the rotor pole and the number of stator and rotor
pole pairs is the same, special measures must be taken to minimize the
cogging torque.
Summary
Disk-type motors and TFMs belong to the group of electric motors with axial flux. They are sometimes called high power density electric motors. This
name is not totally correct, because a radial flux motor can achieve the same
power density. All electromagnetic motors operate on the principle of Faraday’s induction law. Higher power density of axial flux motors can sometimes
be achieved only due to the better packaging of components.
The axial flux disk-type PM machine is an attractive alternative to the
cylindrical radial flux PM machine due to its pancake shape, compact construction and easy integration with electromechanical drive systems. Axial
flux PM motors are particularly suitable for electrical vehicles, pumps, fans,
valve control, centrifuges, machine tools, robots and industrial equipment.
The large diameter rotor with its high moment of inertia can be utilized as a
flywheel. Axial flux PM machines can also operate as small to medium power
generators. Since a large number of poles can be accommodated, these machines are ideal for low-speed applications, as for example, electromechanical
traction drives, hoists or wind generators.
The unique disk-type profile of the rotor and stator of axial flux PM machines makes it possible to generate diverse and interchangeable designs. Axial
flux PM machines can be designed as single air gap or multiple air gap machines, with slotted, slotless or even totally ironless armature. Low-power axial
flux PM machines are frequently designed as machines with slotless windings
and surface PMs.
In the design and analysis of axial flux motors the topology is complicated
by the presence of two air gaps, high axial attractive forces, changing dimensions with radius and the fact that torque is produced over a continuum of
radii, not just at a constant radius as in cylindrical motors.
As the output power of the axial flux PM machines increases, the contact
surface between the rotor and the shaft in proportion to the power becomes
smaller. Careful attention must be given to the design of the rotor-shaft mechanical joint as this is usually the cause of failures of disk-type machines.
In some cases, rotors are embedded in power-transmission components to
optimize the number of parts, volume, mass, power transfer and assembly
time. For electric vehicles (EVs) with built-in wheel motors the payoff is a
simpler electromechanical drive system, higher efficiency and lower cost. Dualfunction rotors may also appear in pumps, elevators, fans and other types of
machinery, bringing new levels of performance to these products.
In a transverse flux motor (TFM) the electromagnetic force vector is perpendicular (transverse) to the magnetic flux lines. The stator consists of a
246
Modern Permanent Magnet Electric Machines
toroidal single-phase winding embraced by U-shaped cores. The rotor consists
of surface or buried PMs and a laminated or solid core. A three-phase machine
can be built of three of the same single-phase units.
The TFM has several advantages over a standard PM brushless machine,
i.e.,
(a) better utilization of laminations and conductors than in standard (radial
flux) PM brushless machines for the same cooling system.
(b) lower winding and core losses for the same rating.
(c) simple stator winding consisting of a single ring-shaped coil (cost effective
stator winding, no end connections).
(d) unity winding factor.
(e) the more poles the higher the power density, higher power factor and less
torque ripple.
(f) a three-phase machine can be made of three (or multiple of three) identical
single-phase units.
On the other hand, special attention must be given to the cogging torque and
power factor, which drops sharply with the load. The low power factor can
be corrected by injection of a negative d-axis current component from the
power electronics converter. TFM uses more PM materials than its radial flux
counterpart.
9
HIGH-SPEED PM BRUSHLESS MACHINES
The actual trend in high-speed electromechanical drive technology and energy
generation is to use PM brushless machines. The following issues, which are
essential in electromagnetic, mechanical and thermal design of high-speed PM
brushless machines are discussed: (1) main dimensions and sizing procedure,
(2) mechanical requirements, (3) stator design guidelines including bad practices and corrective actions, (4) rotor design guidelines, (5) retaining sleeves
(cans), (6) losses in the rotor, (7) thermal and cooling issues.
9.1 Requirements
High-speed PM machines that develop rotational speeds in excess of 5000
rpm are necessary for centrifugal and screw compressors, grinding machines,
mixers, pumps, machine tools, textile machines, drills, handpieces, aerospace,
microturbines, flywheel energy storages, turbochargers, etc. [6, 12, 51, 57].
The actual trend in high-speed electromechanical drives and energy generation
technology is to use PM brushless machines, solid rotor induction machines
or switched reluctance machines (SRMs). The highest efficiency and highest
power density is achieved with PM brushless machines. Requirements for highspeed PM brushless machines include, but are not limited to:
ˆ
ˆ
ˆ
ˆ
ˆ
compact design, high power density and minimum number of components;
high efficiency and power factor close to unity over the whole range of
variable speed and variable load;
ability of the PM rotor to withstand high temperature due to losses in
retaining sleeve and PMs;
active and passive materials used for the rotor should be thermally compatible, i.e., with similar coefficient of thermal expansion;
SmCo PMs rather than NdFeB PMs should be used if the PM rotor is
integrated with a turbine rotor;
248
ˆ
ˆ
ˆ
ˆ
Modern Permanent Magnet Electric Machines
optimal cost- to-efficiency ratio to minimize the cost-to-output power ratio
of the system;
high reliability (failure rate < 5% within 80,000 h);
low cogging torque and vibration level;
low total harmonics distortion (THD).
9.2 Main dimensions
The classical sizing procedure of electrical machines uses the so-called output
equation [19, 33, 90]. The output equation requires estimation of the magnetic
flux density in the air gap (T) and stator line current density (A/m). Even
for an experienced designer, it is difficult to estimate the line current density
for a given type of high-speed machine. It is much more convenient to use the
current density in stator conductors (A/mm2 ) because the current density in
conductors allows for estimation of the Joule losses and selection of a cooling
system at the early stage of the design procedure. Given below is the alternative method of estimation of the main dimensions of electrical machines,
which is designer-friendly, especially for high-speed PM machines.
The electromagnetic apparent power Selm also called the internal apparent
power of an AC machine is
Selm = m1 Ef Ia
(9.1)
and the phase EMF Ef is expressed by eqn (7.8) and rotor magnetic flux Φf
of an AC electrical machine is expressed by eqn (7.9). The pole pitch τ is
given by eqn (7.4). Thus the electromagnetic apparent power gets the form
√
√
1
Selm = m1 π 2f N1 kw1 Bmg D1in LIa = m1 π 2ns N1 kw1 Bmg D1in LIa
p
√
= m1 π 2ns N1 kw1 Bmg D1in LJa sa
(9.2)
where the synchronous speed ns = f /p, the stator (armature) current density
Ja = Ia /sa , and sa is the cross section of bare conductors including parallel
wires. Using the slot fill factor kf ill defined as the ratio of pure copper area
(2m1 N1 sa )–to–slot cross-section area Aslot , the slot area and slot-tooth zone
area Asl are, respectively
Aslot =
2m1 N1 sa
kf ill
(9.3)
2
2
− D1in
D1y
π 2
= D1in
(ky2 − 1)
(9.4)
4
4
where D1y is the inner diameter of the stator yoke (bottoms of slots) and the
coefficient ky = D1y /D1in depends on the number of poles. Since the radial
Asl = π
High-Speed PM Brushless Machines
249
height of the stator yoke is inversely proportional to the number of poles, for
most AC electrical machines
ky =
D1y
1
≈ 1.05 +
D1in
1.5p
(9.5)
Thus
π2
3
Selm = √ ns kw1 kf ill Bmg LJa D1in
(ky2 − 1)
6 2
(9.6)
and, on the other hand
Selm = m1 Ef Ia = m1 ϵU1 Ia =
ϵPout
η cos φ
(9.7)
where η is the efficiency, cos φ is the power factor, ϵ = Ef /U1 , and the output
power is Pout = m1 U1 Ia η cos φ. Comparing right-hand sides of the above eqns
(9.6) and (9.7)
√
π
ϵPout
π 2
1
2
=
ns kw1 kf ill Bmg Ja
D1out
L D1out 3 (ky2 − 1)
η cos φ
3
4
kD
(9.8)
where the ratio of the outer–to–inner diameter of the stator
kD =
D1out
D1in
(9.9)
is equal to
ˆ
ˆ
kD ≈ 1.75 if p = 1
kD ≈ 1.05 + (1/p) if 1 < p ≤ 20.
2
The volume of the active parts of the machine is V = 0.25πD1out
L and the
V D1out product
V D1out =
3
3ϵkD
Pout
√
π 2ns kw1 kf ill (ky2 − 1)Bmg Ja η cos φ
(9.10)
Alternatively, the inner diameter D1in of the machine can be found as
s
√
6 2ϵPout
3
D1in =
(9.11)
π 2 ns kw1 kf ill (ky2 − 1)LBmg Ja η cos φ
Examples of calculations of the main dimensions on the basis of eqns (9.10)
and (9.11) and their comparison with prototypes or calculations using another
approach are shown in Table 9.1. This comparison confirms the accuracy of
eqns (9.10) and (9.11). The machine rated at 1.5 MW and 15 krpm is a
generator designed for directed energy weapon (DEW) systems (Fig. 9.1).
250
Modern Permanent Magnet Electric Machines
Fig. 9.1. High-speed, 8-pole, 72-slot PM brushless generator rated at 1.5 MW,
15 krpm, 620 V: (a) cross section with 2D magnetic flux distribution; (b) main
dimensions. The air gap including the thickness of retaining sleeve is 2.8 mm, radial
thickness of SmCo SS3218 PMs is 13.6 mm (Br = 1.155 T, Hc = 1259 kA/m), slot
fill factor kf ill = 0.5, armature current Ia = 1491 A, efficiency η = 98% and power
factor cosφ = 0.9465.
Table 9.1. Comparison of dimensions of prototypes with dimensions obtained from
eqns (9.10) and (9.11).
How the
Mass of active
results have D1in /D1out L
components
been obtained
mm
mm
kg
92 kW, 41 krpm,
prototype
65/125.5 102.9
8.2
Ja = 10.3 A/mm2
eqns (9.10),
4-pole motor
(9.11)
78/125.5 93.5
7.4
48.5 kW, 43 krpm,
prototype
45/98.5 128.0
8.7
Ja = 6.62 A/mm2
eqns (9.10),
4-pole motor
(9.11)
46/98.0 126.0
8.2
1.5 MW, 15 krpm, SPEED Adapco 205.6/280 200
88.0
Ja = 16.8 A/mm2
eqns (9.10),
8-pole generator
(9.11)
210/273
200
88.7
Machine
9.3 Mechanical requirements
The rotor diameter is limited by the bursting stress at the design speed. The
rotor axial length is limited by its stiffness and the first critical (whirling)
speed. Since the centrifugal force acting on a rotating mass is proportional to
the linear velocity squared and inversely proportional to its radius of rotation,
High-Speed PM Brushless Machines
251
the rotor must be designed with a small diameter and must have very high
mechanical integrity. The surface linear speed (tip speed) of the rotor
v = π(D1in − 2g)ns = π(D1in − 2g)
f
p
(9.12)
where g is the air gap (mechanical clearance). The maximum permissible
surface linear speed depends on the rotor construction and materials.
Fig. 9.2. Single mass flexible rotor with residual unbalance and possible modes of
oscillations: (a) 1st mode; (b) 2nd mode; (c) 3rd mode. O — center of rotation, G
— center of gravity, P — geometric center.
When the shaft rotates, centrifugal force will cause it to bend out. For a single
rotating mass m, the first critical (whirling) rotational speed and first critical
angular speed are, respectively [28]
r
r
1
K
K
ncr =
Ωcr =
(9.13)
2π m
m
The static deflection takes the form
σ=
mgL3
mg
=
48EI
K
(9.14)
EI
L3i
(9.15)
where the stiffness is
K = 48
I = πD4 /64 is the area moment of inertia, EI is the bending stiffness and L
is the bearing span (Fig. 9.2a). Distinguishing between the rotor stack (E, I,
Li ) and shaft (Esh , Ish , L), the stiffness of the shaft with rotor stack is [28]
K = 48
EI
Esh Ish
+ 48
L3i
L3
(9.16)
252
Modern Permanent Magnet Electric Machines
The modulus of elasticity of the laminated stack E is from 1% to 20% of
the modulus of elasticity Esh of the steel shaft. The stronger the clamping of
laminations, the higher the modulus E. Neglecting damping, the centrifugal
force is mΩ 2 (σ + e) and the restoring force (deflection force) is Kσ, in which
σ is the shaft deflection, e is imbalance distance (eccentricity) and σ + e is the
distance from the center of rotation to the center of gravity. From the force
balance equation
Kσ = mΩ 2 (σ + e)
(9.17)
the deflection of the shaft can be found as
e
mΩ 2 e
=
(9.18)
K(1 − mΩ 2 /K)
(Ωcr /Ω)2 − 1
The shaft deflection σ → ∞ if Ω = Ωcr . No matter how small the imbalance
distance e is, the shaft will whirl at the natural frequency. The mass rotates
about the center of rotation O if Ω < Ωcr . Points O and G are opposite each
other. The mass rotates about the center of gravity G if Ω > Ωcr . Point O
approaches point G.
It is recommended that the synchronous (rated) speed ns of the machine
should meet the following conditions [19]:
σ=
ˆ
If ns < ncr , when ncr is the first critical speed of the rotor, then
ˆ
ncr
2p
If ns > ncr , then
ns > 0.75
or
ns > 1.3ncr
ns < 1.33
ncr
2p
(9.19)
(9.20)
In the case of asymmetry of the magnetic field in the gap between the stator
and the rotor, a one-sided radial magnetic pull appears in the machine. The
nature of the magnetic pull depends on the type of magnetic asymmetry. The
value and direction of radial magnetic pull may vary. With the radial magnetic
pull being included, the first critical speed is [19]
r
1
K − Ke
ncr =
(9.21)
2π
m
where Ke is the negative spring coefficient (stiffness) induced by the electromagnetic field (magnetic pull). This coefficient is given, e.g., in [19].
9.4 Fundamental problems in design
Given below are the fundamental issues, which are essential in electromagnetic, mechanical and thermal design of high-speed PM brushless machines
[18, 32, 89]:
High-Speed PM Brushless Machines
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
253
Volume and mass: the higher the speed, the higher the power density.
Power losses and efficiency: special attention must be given to windage
and core losses.
Laminations: cobalt alloy, non-oriented silicon steel or amorphous alloy
laminations.
Stator conductors: small diameter stranded conductors or Litz wires.
Higher harmonics generated by the solid-state converters: their influence
on losses, vibration and noise.
Cooling system: intensive air or oil cooling system.
Rotor tensile hoop stresses: properly selected rotor diameter, rotor diameter–to–length ratio and rotor retaining sleeve.
Thermal compatibility of rotor materials to avoid compressing stresses on
PMs that fluctuates with the temperature.
Rotor dynamics: the first critical speed of the rotor should be much higher
or much lower than the rated speed; eqns (9.19) and (9.20).
Fig. 9.3. Duplex winding. Unavoidable small phase shift between two systems of
windings can create high currents that can thermally damage the stator winding.
9.5 Stator design
The stator core is a stack of slotted or slotless laminations. For input frequencies 400 Hz and lower, 0.2-mm thick laminations are used. For higher frequencies, 0.1-mm laminations are necessary. Vacuum impregnated coils made of
stranded conductors are inserted into slots. To minimize the space harmonics, the stator winding is made as a double-layer winding with shorted coils.
For very high speeds and low voltages, when the EMF induced in single turn
stator coils is too high, a small number of coils, single-layer winding or parallel paths (not recommended) must be used. Hollow conductors and direct
254
Modern Permanent Magnet Electric Machines
Fig. 9.4. Parallel paths can cause circulating currents in the auto-wound stator due
to random positions of conductors in coils.
water cooling are too expensive for machines rated below 200 kW. The stator
volume is affected by winding losses and heat dissipation.
In order to avoid circulating currents, excessive winding losses and hot
spots in the stator winding, it is necessary to avoid the following:
ˆ
ˆ
ˆ
ˆ
Duplex windings (Fig. 9.3). Duplex winding works well for induction machines, but it is not acceptable for high-speed PM machines.
Parallel paths (Fig. 9.4). There are circulating currents in parallel paths of
the auto-wound stator due to the random position of conductors in coils.
Concentric winding (Fig. 9.5). Concentric double-layer winding with coil
groups containing different numbers of coils is not recommended. It is much
better to use double-layer lap winding instead. Double-layer lap winding
can be auto wound.
Deep slots (Fig. 9.6). Deep slots for auto wound double-layer windings
with parallel paths are not recommended. Winding asymmetries due to coil
side location in the slot, lead to unequal impedances and unequal induced
EMFs. This causes circulating currents and more importantly, very uneven
distribution of the currents within the strand conductors (parallel wires)
of the same phase.
To minimize the losses in the retaining sleeve and PMs, torque ripple and
vibration, the stator slots should have very narrow slot openings or be closed.
In the case of closed stator slots, the slot closing bridge should be highly
saturated under normal operating conditions.
High-Speed PM Brushless Machines
255
Fig. 9.5. Concentric double-layer winding with coil groups containing different number of coils is a wrong solution. Double-layer lap winding is recommended. It can be
auto wound.
Fig. 9.6. Deep slots for auto wound double-layer windings with parallel paths are
not recommended. Asymmetries due to coil side locations in the slot lead to unequal
impedances and unequal EMFs.
256
Modern Permanent Magnet Electric Machines
Fig. 9.7. High-speed PM rotor designs: (a) surface PM rotor provides minimal
leakage flux; (b) bread-loaf surface-type PM rotor provides the highest magnetic
flux density in the air gap (large volume of PM material); (c) interior-type PM
rotor does not need any retaining sleeve, but the ferromagnetic bridge in the rotor
core between neighboring PMs must be very carefully sized.
9.6 Rotor design
PM rotor designs (Fig. 6.3) include surface-type, inset-type bread loaf or
interior-type PMs [32]. All surface-type PM rotors are characterized by minimal leakage flux. Bread loaf surface-type PM rotors provide, in addition, the
highest magnetic flux density in the air gap (large volume of PM material).
All surface-type, including bread loaf and inset-type PM rotors, can be used
only with an external rotor retaining sleeve (can). In the case of interior-type
PM rotors, the retaining sleeve is not necessary, but the ferromagnetic bridge
in the rotor core between neighboring PMs must be very carefully sized. From
an electromagnetic point of view, this bridge should be very narrow to obtain
full saturation, preventing the circulation of leakage flux between neighboring rotor poles. From a mechanical point of view, this bridge cannot be too
narrow to withstand high mechanical stresses. In practice, interior-type PM
rotors without retaining sleeves can be used at speeds not exceeding 6000
rpm.
High-Speed PM Brushless Machines
257
Fig. 9.8. Retaining sleeves for high-speed PM rotors: (a) metal sleeve; (b) carbongraphite sleeve.
Good materials for retaining sleeves are nonferromagnetic and have high
permissible stresses, low electric conductivity, low specific mass density and
good thermal conductivity. Typical materials for nonferromagnetic sleeves
are Inconel 718 (NiCoCr based alloy) with electric conductivity of 0.8 × 106
S/m (1.38% ICACS), titanium alloys, stainless steels, carbon graphite, carbon
fiber, glass fiber and reinforced plastics. The maximum temperature for metal
sleeves (Fig. 9.8a) is 290◦ C and for fiber sleeves (Fig. 9.8b) is 180◦ C. A PM
rotor with a metal retaining sleeve for a 110 kW, 70,000 rpm brushless motor
is shown in Fig. 9.9. The maximum surface linear speed for metal sleeves is
240 m/s and for fiber sleeves is 320 m/s. There are practically no eddy-current
losses in fiber sleeves; however, it is more difficult to assemble the rotors with
fiber sleeves than rotors with metal sleeves. If the magnetic saturation effect
is used effectively, a thin steel sleeve in low-power machines can sometimes be
better than a sleeve made of nonferromagnetic material.
The use of fiber material in high-speed PM machines provides some key
performance advantages. Compared to a metallic sleeve, fiber material, with
its higher strength-to-mass ratio, is much thinner, thus resulting in higher
magnetic flux density in the air gap. The disadvantages of fiber material
sleeves include lower temperature rating compared to metal sleeves, negligible
Fig. 9.9. Rotor of a 110 kW, 70 krpm PM brushless motor for an oil-free compressor.
1 – PM rotor with retaining sleeve, 2 – foil bearing journal sleeve. Photo courtesy
of Mohawk Innovative Technology, Albany, NY, USA.
258
Modern Permanent Magnet Electric Machines
Fig. 9.10. Laminated retaining sleeve: (a) single non-ferromagnetic lamination; (b)
stacked retaining sleeve.
bending stiffness as fibers are wound in hoop direction, and very low thermal
conductivity.
The allowable stress on the PMs is 80 N/mm2 . To prevent the magnets
from exfoliating, initially, a nonferromagnetic stainless steel sleeve is shrunk
on the PMs to retain them. Although the stainless steel has low electric conductivity, the losses occurred in a relatively thick sleeve can be still quite large
at speeds over 100,000 rpm. Nonconductive fiber-reinforced plastic is better
at higher speeds.
Recently, laminated sleeves stacked from non-ferromagnetic materials have
been investigated (Fig. 9.10) [42, 76]. They provide
ˆ
ˆ
ˆ
significant reduction of eddy currents;
simple manufacture using punching dies;
and can withstand high radial stresses.
The disadvantage is the limit on the radial thickness of the laminated sleeve:
the sleeve cannot be too thin.
To increase the electromagnetic coupling between the magnets and the
stator winding, the air gap should be made as small as mechanically possible.
However, the use of a small air gap increases the tooth ripple losses in the
retaining sleeve, if the sleeve is made of current-conducting material.
Active radial and axial magnetic bearings or air bearings are frequently
used. High-speed PM brushless motors integrated with magnetic bearings and
solid-state devices are used in gas compressors providing a true oil-free system, reduced maintenance and high efficiency. No auxiliary lubrication supply
system is needed, eliminating hazardous waste disposal issues.
A good manufacturing practice is to use segmented construction of rotors,
which allows use of the same segment (module) for different ratings of machines and leads to reduction of cost of fabrication. A six-pole rotor segment
is shown in Fig. 9.11. The PMs are of bread loaf type and are contained with
High-Speed PM Brushless Machines
259
Fig. 9.11. PM rotor of segmented construction of a six-pole high-speed PM brushless motor with metal retaining sleeve: (a) single segment; (b) rotor stacked with
20 segments. 1 – PM, 2 – rotor core, 3 – retaining sleeve (can). Photo courtesy of
Electron Energy Corporation, Landisville, PA, USA.
the rotor hub in a nonferromagnetic metal can. The number of rotor segments
depends on the machine rating.
9.7 Mechanical design
The objective function is generally the maximum power density at given speed
and cooling system. The power is limited by the thermal and mechanical
constraints. In the design of high-speed PM brushless motors, the following
aspects should be considered [12, 32]:
(a) Mechanical design constraints are important due to the high cyclic stress
placed on the rotor components. Materials with high fatigue life are favored. Materials with low melting points, such as aluminum, should be
avoided or restricted.
260
Modern Permanent Magnet Electric Machines
(b) Capital and operational costs are generally directly linked. The use of magnetic bearings over traditional rolling element bearings or oil lubricated
bearings is a very important consideration. The capital cost of magnetic
bearings is high, but the operational costs are less since the rotational loss
and power consumption are reduced and there is no maintenance.
(c) Dynamic analysis of the rotor assembly, including shaft, core stack and
bearing sleeves should be carried out with great detail using the 3D FEM
simulation.
(d) Static and dynamic unbalance. Even a very small unbalance can produce
high vibration. For example, a static unbalance of 0.05 N at a speed of
100,000 rpm produces an additional centrifugal force of more than 600 N.
Unbalance occurs when the center of gravity of a rotating object is not aligned
with its center of rotation. Static unbalance is where the rotor mass center
(principal inertia axis) is displaced parallel to the rotor geometric spin axis.
Dynamic unbalance is where the rotor mass center is not coincidental with
the rotational axis.
It is generally not difficult to design a high-speed PM brushless motor
rated at a few kWs and speed of 7000 to 20,000 rpm with efficiency of about
93% to 95%. The efficiency of high-speed PM brushless motors rated above
80 kW and 70,000 to 90,000 rpm should be over 96%. Core losses, windage
losses and metal sleeve losses are high. Slotless stators, amorphous cores and
foil bearings can increase the efficiency up to 98%. High-speed machines in
the multimegawatt range with slotted stators should also have efficiency up
to 98%.
9.8 Thermal issues and cooling technologies
Although the main attention in high-speed machines must be given to the
windage and core losses, the stator winding losses can also be high. For this
reason, an adequate cooling system must be selected according to the current
density in the stator winding.
Table 9.2 contains typical current densities for high-speed electrical machines with different cooling systems.
The stator yoke (back iron) with a serrated external surface behaves similar
to a surface with fins. Small fins are stamped in each lamination (Fig. 9.12a).
The stator stack has neighboring laminations shifted one from each other by
half of the fin pitch (Fig. 9.12b). In this simple way, using natural cooling
system, the current density in the stator winding can be increased from about
6 A/mm2 (smooth external surface) to 10 A/mm2 . Liquid cooling jackets
may have circumferential tubing or an axial tubing system (Fig. 9.13). A
circumferential tubing system (Fig. 9.13a) is more difficult to manufacture,
but there is more uniform cooling of the external surface of the stack and
hot spots can be avoided. An axial tubing system (Fig. 9.13b) is easier to
High-Speed PM Brushless Machines
261
Table 9.2. Typical current densities for high-speed electrical machines with different
cooling systems
Cooling system
Current density
A/mm2
Totally enclosed machine, natural ventilation
4.5 to 6.0
Fins or heat sinks, natural ventilation
6.0 to 10.0
Totally enclosed machine, external blower
7.0 to 11.0
Through-cooled machine, external blower
14.0 to 15.0
Water or oil jacket
12.0 to 15.5
Spray oil-cooled end turns of stator and/or rotor 23.0 to 28.00
Direct cooling and hollow conductors
up to 30.0
Fig. 9.12. Stator yoke (back iron) with serrated external surface: (a) single lamination; (b) stator stack. Neighboring laminations are shifted one from each other
by half of the fin pitch.
Fig. 9.13. Liquid cooling jackets: (a) circumferential tubing; (b) axial tubing.
262
Modern Permanent Magnet Electric Machines
manufacture, but there is less uniform cooling of the external surface of the
stack and probability of hot spots.
Instead of tubes with a round cross section, a cooling jacket with flat rectangular cross-section tubes is directly built in to the housing. Typical material
for a cooling jacket is aluminum, aluminum alloy or sometimes copper. Additive manufacturing (3D printing) technology can be used to make portions of
cooling jackets.
Fig. 9.14. Wet machine with spray-oil cooled end windings. Spray nozzles are installed in oil inlets.
An oil spray cooling system can be as effective as direct liquid cooling with
hollow conductors (Fig. 9.14). The current density can achieve 28 A/mm2 . The
oil is injected through the nozzles to the interior of the machine and cools the
end turns directly. A liquid spraying process can be described as consisting of
two phases: (a) breaking of the liquid into separate droplets, and (b) directing
the liquid drops onto a surface of an object. Nozzles are usually made of brass
and provide a conical or flat spray distribution pattern.
The cooling can be even more intensive if the oil flow passages are created
between conductors in slots. Of course, such cooling is limited only to the
stator windings. In the case of round conductors, oil passages are naturally
created between conductors with cylindrical cross section (Fig. 9.15a). In the
case of rectangular conductors, oil flow passages are formed using a removable
High-Speed PM Brushless Machines
263
Fig. 9.15. Cross section of stator slots with double-layer windings: (a) coils wound
with round conductors; (b) stiff coils made of rectangular conductors.
“spacer” casting process during impregnation (Fig. 9.15b). A void (about
0.5 mm gap) is left by melting out wax or pulling out a Teflon strip after
resistance heat auto dispense (RHAD) gels/cures. The reminder of the slot is
impregnated.
The temperature distribution in the cross-section area of a rectangular slot
with round conductors and oil flow through passages between conductors is
shown in (Fig. 9.16).
Normally, conductors with insulation class H (180◦ C), 220◦ C or 240◦ C are
used for stator windings. Nickel-clad copper conductors with ceramic insulation can withstand higher temperatures, up to 600◦ C. DuPont—Kapton—HN
general purpose insulation films can be applied at temperatures up to 400◦ C.
9.9 Directed energy weapon (DEW)
Directed energy weapons (DEW) take the form of lasers, high-powered microwaves, and particle beams [46, 22, 23]. They can be adopted for ground,
air, sea, and space warfare. DEWs irradiate the target with electromagnetic
energy. The so-called fluence is the energy density, i.e.,
Pdout ∆tS
J/m2
(9.22)
A
where Pdout is the DEW output power, ∆t is the duration of the DEW pulse,
0 ≤ S ≤ 1.0 is the dimensionless transmission number, also called the Strehl1
ratio, and A is the spot area on the target. To destroy soft targets, i.e., fabrics,
E=
1
Named after German physicist and mathematician Karl Strehl (1864-1940).
264
Modern Permanent Magnet Electric Machines
Fig. 9.16. Temperature distribution near stack edge at downstream end in the
cross-section area of a rectangular slot with round conductors and oil flow through
passages between conductors.
plastics, etc., approximately 1000×104 J/m2 are required, but extremely hard
targets, i.e., tanks, mine-resistant vehicles, armored trucks, etc., might require
100 000 × 104 J/m2 . Once the target has absorbed this energy, it will begin
to heat up and even burn out.
The only difference between lasers and high-energy microwaves, which are
both made up of photons, is their energy level. The photon energy
E = hf = h
c
λ
(9.23)
is a function of the frequency f , where h = 6.626 × 10−34 Js is Planck’s
constant, c = 299 792 458 m/s is the speed of light, and λ is the length of
wave.
The power generation capabilities of electron microwave tubes (MTs), i.e.,
klystrons, magnetrons, gyratrons, gridded tubes and cross-field tubes, range
from watts to megawatts at frequencies from 300 MHz to 300 GHz. Klystrons
High-Speed PM Brushless Machines
265
are the most efficient MTs and are capable of the highest peak and average
powers. A klystron is a specialized vacuum tube called a linear-beam tube.
DEWs require high-speed synchronous generators in the range of megawatts.
PM synchronous generators cannot be applied because of serious problems in
the failure modes as, for example, an inter-turn short circuit. It is impossible to reduce the field excitation to zero in PM generators. From this point
of view, wound-field synchronous generators are used so far [41, 63, 74, 75],
which in the future, can be replaced by synchronous generators with high
temperature superconducting (HTS) field excitation winding [81].
There are two difficult challenges in construction of high-speed multimegawatt generators [59, 68, 86]:
ˆ
ˆ
high power density, low envelope volume and low mass;
thermal management and heat dissipation.
Summary
The actual trend in high-speed electromechanical drives and energy generation
technology is to use PM brushless machines, solid rotor induction machines
or switched reluctance machines (SRMs). The highest efficiency and highest
power density is achieved with PM brushless machines. Requirements for highspeed PM brushless machines include
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
compact design, high power density and minimum number of components;
high efficiency and power factor close to unity over the whole range of
variable speed and variable load;
ability of the PM rotor to withstand high temperature (losses in retaining
sleeve and PMs);
active and passive materials used for the rotor should be thermally compatible, i.e., with similar coefficient of thermal expansion;
SmCo PMs rather than NdFeB PMs should be used if the PM rotor is
integrated with turbine rotor;
optimal cost-to-efficiency ratio to minimize the cost-to-output power ratio
of the system;
high reliability (failure rate < 5% within 80,000 h);
low cogging torque and vibration level;
low total harmonics distortion (THD).
The proposed new sizing equations (9.10) and (9.11) allow for easy estimation of main dimensions including selection of cooling system at the early
stage of design of a high-speed PM machine.
The rotor diameter is limited by the bursting stress at the design speed.
The rotor axial length is limited by its stiffness and the first critical (whirling)
speed.
266
Modern Permanent Magnet Electric Machines
When the shaft rotates, centrifugal force will cause it to bend out. For a
single rotating mass m, the first critical (whirling) rotational speed and first
critical angular speed are given by eqn (9.13).
The synchronous (rated) speed ns of a high-speed machine should be much
lower or much higher than the first critical speed, as given by eqns (9.19) and
(9.20).
When designing the stator winding of a high-speed PM machine it is necessary to avoid: (a) duplex winding; (b) parallel paths; (c) concentric winding
with different coil groups; (d) deep slots.
All surface-type, including bread-loaf and inset-type PM rotors, can be
used only with an external rotor retaining sleeve [32].
High-speed PM brushless machines can reach power density up to 7.0
kW/kg (with liquid cooling) and 98% efficiency [32].
High-speed PM machines can be cooled by the following methods:
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
natural ventilation;
fins or heat sinks;
external blower;
water or oil jacket;
spray oil-cooled end turns of stator and/or rotor;
hollow conductors with direct cooling.
The stator yoke (back iron) with serrated external surface behaves similar to
a surface with fins. Small fins are stamped in each lamination.
Liquid cooling jackets may have a circumferential tubing or axial tubing
system. An axial tubing system is easier to manufacture, but there is less
uniform cooling of the external surface of the stack and probability of hot
spots.
An oil-spray cooling system can be as effective as direct liquid cooling with
hollow conductors. The current density can achieve 28 A/mm2 .
Oil cooling can be intensified if the oil flow passages are created between
conductors in slots. In the case of round conductors, oil passages are naturally
created between conductors with a cylindrical cross section.
Normally, conductors with insulation class H (180◦ C), 220◦ C or 240◦ C are
used for stator windings. Nickel-clad copper conductors with ceramic insulation can withstand higher temperatures, up to 600◦ C.
Directed energy weapons (DEW) take the form of lasers, high-powered
microwaves, and particle beams. They can be adopted for ground, air, sea,
and space warfare.
DEWs require high-speed synchronous generators in the range of megawatts
[22, 23]. PM synchronous generators cannot be applied because of serious
problems in the failure modes as, for example, an inter-turn short circuit. It is
impossible to reduce the field excitation to zero in PM generators. From this
point of view, wound-field synchronous generators are used so far, which in
the future, can be replaced by synchronous generators with high-temperature
superconducting (HTS) field excitation winding.
Appendix A
Conversion of units
A.1 Conversion of units
A.1.1 Definitions
The unit of the magnetic flux density in the International System of Units
(SI) is the “tesla” (T). One tesla is equal to one Vs per square meter or
one weber per square meter. The tesla is named after the Serbian-American
inventor Nikola Tesla (1856–1943).
The unit of the magnetic flux in the SI is the “weber” (Wb). One weber is
equal to one Tm2 . The “weber” is named after the German physicist Wilhelm
Eduard Weber (1804–1891).
The unit of the magnetic field intensity or magnetic field strength
in the SI is “ampere per meter” (A/m). Ampere is a unit of electric current
equal to a flow of one coulomb per second (1 C/s). It is named after French
mathematician and physicist André-Marie Ampère (1775–1836).
The unit of the magnetic field energy density in the SI is “joule per
meter cubic” (J/m3 ). One joule is equal to the energy transferred to an object
when a force of one newton acts on that object in the direction of the motion
through a distance of one meter (1 Nm). It is also the energy dissipated as
heat when an electric current of one ampere passes through a resistance of one
ohm for one second. It is named after the English physicist James Prescott
Joule (1818–1889).
The unit of the inductance in the SI is the “henry” (H). One henry is
equal to Ωs = Vs/A. The inductance of an electric circuit is one henry when
an electric current that is changing at one A/s results in an electromotive
force (EMF) of one volt across the inductor. The henry is named after Joseph
Henry, American scientist who served as the first secretary of the Smithsonian
Institution (1797–1878).
The unit of the capacitance in the SI is the “farad”(F). One farad is the
capacitance of a capacitor that has a charge of 1 C when there is an applied
268
Modern Permanent Magnet Electric Machines
voltage drop of 1 V (1F = 1C / 1V). It is named after English scientist Michael
Faraday (1791–1867).
A.1.2 Conversion
Table A.1. Conversion of units
Quantity
SI unit
Conversion
Magnetic flux
1 T = 1 Vs/m2
density B
T (tesla)
= 10,000 Gs
Intrinsic magnetization
1 T = 1 Vs/m2
(polarization) Bi
T (tesla)
= 10,000 Gs
Magnetic
1 Wb = 1 Vs
flux Φ
Wb (weber) = 108 Mx (maxwell)
Magnetic field
1 A/cm = 0.4 π Oe
intensity
A/m
= 1.257 Oe (oersted)
(strength) H
1 Oe = 79.55 A/m
Energy density
1 J/m3 = 126 GsOe
0.5BH
J/m3
1 MGsOe = 7.958 kJ/m3
1H = Ωs =
1F =
Wb
Tm2
J
kgm2
kgm2
V
s=
=
= 2 =
= 2 2
2
A
A
A
A
C
A s
C
As
J
Ws
Nm
C2
s
s2
=
= 2 = 2 = 2 =
=
=
V
V
V
V
V
Nm
Ω
H
A.1.3 Some physical constants
Charge of electron (elementary charge) e = 1.60217662 × 10−19 C
Mass of electron me = 9.10938356 × 10−31 kg
Magnetic permeability of free space µ0 = 0.4π × 10−6 H/m
1
Electric permittivity (electric constant) ϵ0 = 36π
× 10−9 F/m
Speed of light in the vacuum c = (2.997930 ± 0.000003) × 108 m/s
Planck’s constant 6.62607004 × 10−34 m2 kg/s
Stefan Boltzmann constant 5.670367 × 10−8 kgs−3 K−4
Appendix B
Lenz’s law
Lenz’s law states that the induced B field in a loop of wire will oppose the
change in magnetic flux through the loop. This is the principle of defiance:
an induced current is always in such a direction as to oppose the motion or
change causing it. If the flux through the loop is increased, the induced field
will oppose that increase (Fig. B.1a). If the flux through the loop is decreased,
the induced field will replace that decrease (Fig. B.1b).
Fig. B.1. Lenz’s law or principle of defiance: (a) magnetic flux through the loop is
increased; (b) magnetic flux through the loop is decreased.
270
Modern Permanent Magnet Electric Machines
Heinrich Friedrich Emil Lenz, in Russian, Emil Khristianovich Lenz, was
born 12 February 1804 in Dorpat (nowadays Tartu, Estonia), at that time in
the Governorate of Livonia, the Russian Empire. After completing his secondary
education in 1820, Lenz studied chemistry and physics at the University of Dorpat. He traveled with the navigator Otto von Kotzebue on his third expedition
around the world from 1823 to 1826.
After the voyage, Lenz began working at the University of St. Petersburg,
Russia, until his death in 1865. Lenz also taught at the Petrischule in 1830 and
1831, and at the Mikhailovskaya Artillery Academy.
Lenz began studying electromagnetism in 1831. Besides the law named in
his honor, Lenz also independently discovered Joule’s law in 1842. To honor his
efforts on the problem, it is also given the name the ”Joule–Lenz law.”
Lenz eagerly participated in development of electroplating technology, invented by his friend and colleague Moritz Hermann von Jacobi, in Russian
Boris Semyonovich Jacobi (1801–1874). Jacobi was a German and Russian
engineer and physicist who worked mainly in Russia. Jacobi made substantial
contributions to galvanoplastics, electric motors, and wire telegraphy.
Lenz died on 10 February 1865 in Rome, after suffering from a stroke. A small
lunar crater on the far side of the Moon is named after him.
Appendix C
Right-handed cork-screw rule
The nature of the magnetic field around a current carrying a straight conductor is like concentric circles having their center at the axis of the conductor.
The direction of these circular magnetic lines is dependent upon the direction
of current.
If a right-handed cork screw is assumed to be held along the conductor,
and the screw is rotated such that it moves in the direction of the current,
the direction of the magnetic field is the same as that of the rotation of the
screw (Fig. C.1). This is called the right-handed cork-screw rule or Maxwell’s
right-handed cork-screw rule.
Fig. C.1. Right-handed cork screw rule.
Appendix D
The right-hand grip rule
The right-hand grip rule is used to determine the relationship between the
current and the magnetic field based upon the rotational direction. The wire
needs to be held in the right hand and the thumb should point in the direction
of the flow of current, then curl your fingers around the wire. Now, the curled
fingers show the direction of the magnetic flux lines around the wire (Fig.
D.1).
Fig. D.1. Right-hand grip rule.
The right-hand grip rule is also used to determine the direction of magnetic
polarity. When you wrap your right hand around the solenoid with your fingers
in the direction of the current, your thumb will point out the direction of the
magnetic North pole (Fig. D.2).
274
Modern Permanent Magnet Electric Machines
Fig. D.2. Right-hand grip rule is used to determine the direction of magnetic
polarity.
Appendix E
Left-hand and right-hand rules
The left-hand rule (motor) determines the direction of electrodynamic force
(Fig. E.1a). The electrodynamic force F is expressed in vector form as
dF = Idl × B
(E.1)
where I is the electric current, dl is the elementary length of the conductor,
and B is the magnetic flux density. In scalar form
F = BIl
(E.2)
Fig. E.1. Left-hand rule (a) and right-hand rule (b).
The right-hand rule (generator) determines the direction of the electromotive force (EMF) (Fig. E.1b). The EMF dE is
dE = v × B · dl
(E.3)
276
Modern Permanent Magnet Electric Machines
where v is the linear velocity of the conductor dl with respect to the magnetic
field B or linear velocity of the magnetic field B with respect to the conductor
dl. In scalar form
dE = Blv
(E.4)
Symbols and Abbreviations
A
a
line current density
number of parallel current paths of the armature winding of AC motors;
number of pairs of parallel current paths of the armature winding of
DC brush (commutator) motors
B
vector magnetic flux density
B
magnetic flux density
bp
pole shoe width
C
number of commutator segments; capacitance
cE
armature constant (EMF constant)
cT
torque constant
D
diameter; damping constant
E
EMF, rms value
Ef
EMF per phase induced by the rotor of a synchronous machine
Ei
internal EMF per phase
e
instantaneous EMF
F
force; MMF
Fexc MMF of the rotor excitation system
Fa
armature reaction MMF
f
frequency
fc
frequency of cogging torque
GCD(Nc , 2p) greatest common divisor of Nc and 2p
g
air gap (mechanical clearance)
′
g
equivalent air gap
H
magnetic field intensity
h
height
hM
height of the PM
I
electric current
Ia
armature DC or rms current
i
instantaneous value of current
J
moment of inertia
Ja
current density in the armature winding
278
K
k
k1R
kC
kad
Symbols and Abbreviations
lumped stiffness
coefficient, general symbol
skin effect coefficient for armature conductors
Carter’s coefficient
reaction factor in d-axis; coefficient of additional losses in armature
core
kaq
reaction factor in q-axis
kd1
distribution factor for the fundamental space harmonic ν = 1
kE
EMF constant kE = cE Φf
kf
form factor of the field excitation kf = Bmg1 /Bmg
ki
stacking factor of laminations
kp1
pitch factor for the fundamental space harmonic ν = 1
ksat
saturation factor of the magnetic circuit due to the main (linkage)
magnetic flux
kT
torque constant kT = cT Φf
kw1
winding factor kw1 = kd1 kp1 for the fundamental space harmonic ν = 1
L
inductance; length
LCM (s1 , 2p) least common multiple of s1 and 2p
Li
armature stack effective length
lM
axial length of PM
M
mutual inductance
m
number of phases; mass
ma
amplitude modulation index
N
number of turns
Ncog number of poles–to–GCD(s1 , 2p) ratio
n
rotational speed in rpm
n0
no-load speed
P
active power
Pelm electromagnetic power
∆P
active power losses
∆p1/50 specific core loss in W/kg at 1T and 50 Hz
p
number of pole pairs; sound pressure
Q
reactive power
R
resistance
Ra
armature winding resistance of DC commutator motors
R1
armature winding resistance of AC motors
Rµg
air gap reluctance
S
apparent power; surface
s
slip; cross-section area
s1
number of stator teeth or slots;
s2
number of rotor teeth or slots;
T
torque
Telm electromagnetic torque
Telmsyn electromagnetic synchronous or synchronizing torque
Telmrel electromagnetic reluctance torque
Symbols and Abbreviations
Tsh
Tm
t
U
V
Vµ
v
W
Wm
w
wM
X
Xad
Xaq
Xsd
Xsq
Z
α
αi
β
∆Vbr
δ
η
θ
ϑ
Λ
λ
µ
µ0
µr
ν
σ
τ
Φ
Φad
Φsq
Φf
Φl
φ
χ
Ψ
Ω
ω
shaft torque (output or load torque)
mechanical time constant
time; slot pitch
electric voltage
volume
magnetic voltage
instantaneous value of electric voltage; linear velocity
energy, J
stored magnetic energy
energy per volume, J/m3
width of PM
reactance
d-axis armature reaction (mutual) reactance
q-axis armature reaction (mutual) reactance
d-axis synchronous reactance
q-axis synchronous reactance
√
impedance Z = R + jX; | Z |= Z = R2 + X 2
electrical angle
effective pole arc coefficient αi = bp /τ
overlap angle of pole
voltage drop across commutation brushes
power (load) angle
efficiency
rotor angular position
temperature
permeance, H
specific permeance, H/m2
magnetic permeability
magnetic permeability of free space µ0 = 0.4π × 10−6 H/m
relative magnetic permeability
number of the stator νth harmonic
electric conductivity, leakage factor
pole pitch
magnetic flux
d-axis armature reaction flux
q-axis armature reaction flux
excitation magnetic flux
leakage flux
power factor angle
magnetic susceptibility
flux linkage Ψ = N Φ; angle between Ia and Ef
angular speed Ω = 2πn
angular frequency ω = 2πf
279
280
Symbols and Abbreviations
Subscripts
a
armature
av
average
b
braking
br
brush
c
commutation
cog
cogging
Cu
copper
d
direct axis; differential
elm
electromagnetic
eq
equivalent
Fe
ferromagnetic
f
field
fr
friction
g
air gap
h
hysteresis
in
inner
k
kinetic
L
load
l
leakage
M
magnet
m
peak value (amplitude)
max maximum
min minimum
n
nominal
n, t
normal and tangential components
out
output, outer
q
quadrature axis
r
relative
rhe
rheostat
rot
rotational
s
synchronous
sat
saturation
sh
shaft
st
starting
str
stray, additional
syn
synchronous or synchronizing
vent ventilation
w
winding
wind windage
x, y, z cartesian coordinate system
1
primary; stator; fundamental harmonic
2
secondary; rotor
Symbols and Abbreviations
Superscripts
inc
(sq)
incremental
square or trapezoidal wave
Abbreviations
AC
APU
AWG
CAD
CAN
CD
CLL
CPU
CRT
CSCF
CSD
CSI
CVT
DC
DSP
EDL
EE
EMI
EML
EV
FDB
FEM
GCD
GPU
GTO
HEV
HV
HVIC
IC
ICACS
IDG
IGBT
ISG
IT
LCM
LDO
LV
LVIC
alternating current
auxiliary power unit
American wire gauge
computer-aided design
controller area network
compact disk
capacitor long life
central processor unit
cathode ray tube
constant speed constant frequency
constant speed drive
current source inverter
continuously variable transmission
direct current
digital signal processor
electrodynamic levitation
electrical engineering
electromagnetic interference
electromagnetic levitation
electric vehicle
fluid dynamic bearing
finite element method
greatest common divisor
ground power unit
gate turn-off (thyristor)
hybrid electric vehicle
high voltage
high voltage integrated circuit
integrated circuit
International Annealed Copper Standard
integrated drive generator
insulated-gate bipolar transistor
integrated starter-generator
information technology
least common multiple
low drop out
low voltage
low voltage integrated circuit
281
282
Symbols and Abbreviations
MMCM
MEA
MLT
MMF
MRAM
MRI
MOSFET
MT
MVD
NiMh
NMR
NVH
PC
PCB
PM
PMBM
PSD
PWM
RAT
RESS
RF
RHAD
SCSI
SRM
SSD
SVM
TI
VF
VSCF
VSI
VSD
VVVF
magnetic-core memory
more electric aircraft
mean length of turn
magnetomotive force
magnetoresistive random-access memory
magnetic resonance imaging
metal–oxide–semiconductor field-effect transistor
microwave tube
magnetic voltage drop
nickel-metal hydride
nuclear magnetic resonance
noise, vibration, and harshness
personal computer
printed circuit board
permanent magnet
permanent magnet brushless motor
power split device
pulse width modulation
ram air turbine
rechargeable energy storage system
radio frequency
resistance heat auto dispense
small computer systems interface
switched reluctance machine; switched reluctance motor
solid state device
space vector modulation
Texas Instruments
variable frequency
variable speed constant frequency
voltage source inverter
variable-speed drive
variable voltage variable frequency
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Index
accelerator, 21, 35
actuator
voice coil, 178
air gap, 199, 200, 242
alloy
amorphous, 44
NiFe, 49
NiFeMo, 47
angle
between current and voltage, 207
firing, 131, 156
load, 207
annealing, 41
armature
current, 202, 203, 241
reaction, 200, 241
winding resistance, 241
atom
model, 1
spin, 1
biographical sketch
Maxwell J.C., 26
Ampère A.M., 68
Biot J.B., 24
Blondel A. E., 211
Clarke E., 209
Einstein A., 32
Faraday M., 25
Gauss J.C.F., 26
Lentz H.F.E., 270
Lorentz H.A., 35
Oersted H.C., 67
Park R.H., 210
Savar F., 24
Steinmetz C.P., 52
bridge
H-bridge, 134
MOSFET, 175
rectifier, 174
brush
carbon-graphite, 114
holder, 114
precious-metal wire, 115
voltage drop, 118
catheter, 172
cathode ray tube (CRT), 17
characteristic
line start, 211
mechanical, 136
performance, 218
regulation, 136
speed control, 131
chopper, 132
step-down, 132
step-up, 132
circuit
equivalent, 202
cobalt alloys
Hiperco 50, 42
Permendur, 42
Vacoflux 50, 42
coefficient
ballistic of demagnetization, 101
Carter, 100, 200, 216
290
Index
curve-fitted, 51
leakage flux, 85, 103, 107
Lorentz’s, 31
temperature, 72
coercivity, 71
cogging
frequency, 171
torque, 171
coil
Helmholtz, 17
RF, 19
commutation
bipolar, 159, 161, 170
six-step, 159
three phases on, 161
two phases on, 159, 170
commutator, 149
compass, 65
conductor
nickel clad copper, 263
constant
EMF, 155, 226, 240
Hall, 163
torque, 155, 225, 240
construction
axial flux motor, 220
line start-motors, 212
PM brushless motors, 149
PM DC brush machines, 114
continuously variable transmission
(CVT), 187
control
motion, 174
sensorless, 169
speed, 157
vector, 208
converter
voltage-source, 174
wiring diagram, 174
cooling
nozzles, 262
oil flow passages, 263
oil spray, 262
serrated surface, 262
technologies, 260
coordinate system, 86
core
3D, 60
segmented, 60
SMC, 60
transformer, 55
current
d-axis, 203
q-axis, 203
armature, 202, 203, 241
density, 128, 260
displacement, 26
square wave, 170
current density
line, 195, 238
vector, 26
cyclotron, 21
damping
torsional, 191
demagnetization curve, 74
approximation, 82
design
fundamental problems, 253
mechanical, 259
rotor, 256
stator, 254
DEW, 263
diode
catch, 134
freewheeling, 133, 169
Schottky, 134
Zener, 142
directed energy weapon (DEW), 263
drive
integrated, 184
PM brushless motor, 153
effect
cogging, 171
fringing, 93, 197
efficiency, 95
electromagnet
lift, 15
superconducting, 19
elevator, 227, 233
EMF, 24, 155, 169, 170, 194, 195, 222,
225, 239, 241
EMI, 143
encoder
absolute, 165
incremental, 164
optical, 164
Index
end turns
involute, 126
short, 151
energy
maximum magnetic, 71
of photon, 264
productBH, 72, 74
equation
Kirchhoff, 102, 106, 169, 243
matrix form, 169
Maxwell first, 26
Maxwell fourth, 29
Maxwell second, 27
Maxwell third, 28
output, 248
Poisson, 30
recoil line, 92
torque balance, 191
equivalent reluctance network (ERN),
104
EV, 186
factor
form (armature reaction), 198
form of armature reaction, 197
reaction, 198
saturation, 100, 200, 243
stacking, 53
winding, 244
FEM, 85
Ansoft Maxwell, 108
calculations, 107
field excitation system
hybrid, 156
fluence, 263
fluid dynamic bearing (FDB), 178
flux
alternating, 53
barriers, 152
leakage, 104
force
acting on disk, 218
attraction, 15
coercive, 8
electric, 34
Lorentz, 34
magnetic, 34
tangential, 218
form factor
291
demagnetization curve, 83
frequency
cogging torque, 171
gear, 137
generator
MHD, 23
overexcited, 196, 202
synchronous, 58
underexcited, 196
greatest common divisor, 151, 171
H-bridge, 134, 140
Halbach cylinder, 93
hard disk drive (HDD), 14, 177, 179,
180
harmonic
higher space, 211
reduction, 212
third, 169
HEV, 186
parallel, 186
series, 186
series–parallel, 186
Toyota Prius, 188
hydrogenation of PMs, 81
hysteresis loop, 8, 39
impedance
line, 174
index
cogging, 171
inductance
armature reaction, 242
leakage, 242
line reactor, 174
mutual, 170
self-inductance, 170
variation, 169
intrinsic
coercivity, 71
demagnetization curve, 70
magnetization, 70
inverter
VVVF, 214
klystron, 265
law
292
Index
Ampère, 218
Ampere’s circuital, 67
Ampère’s force, 67
Biot-Savart, 23
Faraday, 24
Gauss, 25
Lenz’s, 269
Maxwell, 26
least common multiple, 171, 178
levitation
electrodynamic (ELM), 20
electromagnetic (ELM), 20
line current density
PM transverse flux motor(TFM), 238
line current density
PM axial flux motor, 218
PM synchronous motor, 195
TFM, 238
loadstone, 65
losses
active, 51
additional, 54
core, 54
eddy-currents, 53
excess, 53
hysteresis, 51
reactive, 51
loudspeaker, 14
machine
DC brush, 57
high speed, 247
HTS, 265
induction, 57
PM brushless, 150
recyclable, 62
switched reluctance (SRM), 57, 247
synchronous, 193
magnet
superconducting, 18
magnetic
circuit, 102, 243
coercivity, 8
dipole moment, 1
domain, 6, 9
flux, 102, 103, 106, 199, 222
flux density, 197
flux per pole, 239
hysteresis, 8
levitation, 20
nuclear resonance spectroscopy NMR,
18
permeability, 3, 4
permeability of free space, 200
permeability recoil, 71, 92
remanent flux density, 8
resonance imaging (MRI), 19
retentivity, 8
saturation, 8, 100, 242
shunt, 243
storages, 14
susceptibility, 3, 4
vector potential, 29
magnetic field
around conductor, 271
Earth’s, 68
in a loop, 269
magnetic flux
armature reaction, 199
density, 197
fringing, 101
leakage, 85, 107
main, 85
total, 85
magnetic flux density
air gap, 93, 193, 218, 244
Hall sensor, 164
inside Halbach cylinder, 94
remanent, 70, 94
magnetization, 94
vector, 3
magnetostatic
solver, 108
magnetostriction, 47
main
dimensions, 249
Mallinson–Halbach array, 93, 95, 229,
230
materials
Accucore, 46
amorphous, 44
antiferromagnetic, 5
comparison of magnetic, 10
diamagnetic, 5, 7
ferrimagnetic, 5
ferromagnetic, 5
hard magnetic, 9
laminated silicon steel, 40
Index
nanocrystalline, 48
paramagnetic, 5, 6
permalloy, 47
soft magnetic, 9
soft magnetic composites, 46
solid ferromagnetic, 50
Somaloy, 46
memory
magnetic-core (MCM), 16
magnetoresistive random-access
(MRAM), 17
microcontroller
for brush DC motor, 136
MMF, 91, 93, 103, 199
modulation
PFM, 133
PWM, 133
space vector, 208
modulation index
amplitude, 158
frequency, 158
modulus
of elasticity, 252
moment
of inertia, 251
MOSFET, 175
motor
1.9 mm, 172
axial flux, 217
comparison, 214
converter fed, 174
coreless, 172
disk type, 126, 217
double-sided, 220, 223
film coil, 232
for changing seat position, 140
for elevator, 260
for mobile phone, 141
for toys, 137
for windshield wipers, 141
ironless, 229, 232, 235
multidisk, 232
overexcited, 196, 202
PM brushless, 60, 149
PM DC brushed, 149
PM DC brushless, 174
printed winding, 126
self-starting, 211
single-sided, 227
293
smart, 184
starter, 141
taptic, 143
transverse flux (TFM), 236, 238, 244
underexcited, 196
vibration, 141
with gearhead, 172
with gears, 137
MVD, 243
operating diagram of PM
construction, 88
without armature, 91
operating mode
bipolar, 158
unipolar, 158
parallel paths, 119
permanent magnet, 69
Alnico, 74
classes, 74
ferrite, 76
hexaferrite, 77
nanocomposite, 82
NdFeB, 78
operating point, 89, 91
properties, 74
rare-earth, 77
sintered NdFeB, 80
SmCo, 77, 79
super high energy, 81
permeance
air gap, 100
external magnetic circuit, 87
leakage, 101, 242
of simple solids, 98
pole-top, 242
resultant, 88, 103
slot, 242
perpetuum mobile, 69
phasor diagram, 201, 203, 204, 207, 209,
210
plasma, 23
PM
Alnico, 74
button-shaped, 101
ferrite, 76
NdFeB, 78
SmCo, 77
294
Index
PM brushless motor
axial flux, 218
for cooling fan, 179
for EV and HEV, 186
for HDD, 177
TFM, 236
PM rotor
inset-type, 151
interior, 151
Siemosyn, 152
spoke-type, 151
surface type, 151
PM rotor configuration
buried asymmetrical, 151
double-layer interior, 151
inset, 151
interior, 151
spoke-type, 151
surface, 151
with cage winding, 211
pod propulsor, 172
Pointing vector, 51
pole pitch, 222, 239
position sensor
encoder, 164
Hall, 163
resolver, 168
power
apparent, 222, 248
electromagnetic, 195, 205, 240, 248
factor, 208, 244
input, 204
split device PSD, 186
principle
of defiance, 269
of relativity, 30, 32
PWM, 157, 175
pyrolytic carbon, 7
ratio
average–to–maximum value, 194
cost–to–efficiency, 248
diameter, 223
outer–to–inner diameter, 249
pole-shoe arc–to–pole pitch, 215
rotor diameter—to—length, 253
torque–to–current, 159
reactance
armature reaction, 200, 241
mutual, 241
synchronous, 196, 201, 243
recoil
line, 71, 87
loop, 71
magnetic permeability, 71
rectifier
fully controlled, 131
three-phase, 131
recycling, 63
relativity theory, 30
reluctance
air gap, 105
core (yoke), 105
equivalent, 105
for leakage flux, 105, 106
tooth, 105
remanence, 70
resistance
armature winding, 241
of armature circuit, 118
resolver, 167
rotor
disk type, 217
of PM brushless machines, 149
segmented construction, 258
with inner PM, 121
rotor-shaft mechanical joint, 232
rule
cork-screw, 271
left hand, 275
right hand, 275
right-hand cork screw, 271
right-hand grip, 273
self-braking, 135
sensor
Hall, 163, 181
position, 163
servomotor
PM brush DC, 135
requirements, 135
shaft
deflection, 252
ship propulsion, 172
silicon, 40
sintering, 79
sizing procedure
high-speed machines, 249
Index
PM axial flux motor, 221
sleeve
fiberglass, 257
laminated, 258
metal, 257
retaining, 257
slot, 100
solid state switch
FET, 134
IGBT, 134, 136
solid steel, 50
span
commutator, 119
of armature winding, 119
speed
control, 130
critical, 251
surface linear, 251
synchronous, 193, 252
starting
asynchronous, 211
auxiliary motor, 212
frequency-change, 213
synchronous motor, 211
stiffness
bending, 251
system
consumer arrow, 201
generator arrow, 201
permanent magnet, 103
temperature
Curie, 72
distribution, 263
theorem
Gauss, 25
time constant
electromagnetic, 121
mechanical, 121
tokamak, 22
torque
295
cogging, 171, 245
electromagnetic, 128, 155, 170, 205,
218, 224
ripple minimization, 171
rms, 225
starting, 211
transformer
rotary, 168
Transrapid maglev, 20
units
conversion, 267
SI, 267
voltage
brush drop, 118
induced, 194
input, 202
terminal od DC machine, 118
third harmonic, 169
winding
armature, 119
concentrated, 151
construction, 119
duplex, 254
Faulhaber’s, 122
film coil, 232
honeycomb, 122
knitted, 122
lap, 119
moving coil, 121
non-overlapping, 150
printed, 232
rhombic, 122
single-phase, 236
skewed, 172
slotless, 118
slotted, 118
symmetry, 120
wave, 119