Modern Permanent Magnet Electric Machines 9780367610586, 9780367610616, 9781003103073, 0367610582

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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. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, access www.copyright.com or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. For works that are not available on CCC please contact mpkbookspermissions@tandf. co.uk Trademark notice: Product or corporate names may be trademarks or registered trademarks and are used only for identification and explanation without intent to infringe. 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

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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

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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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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69 72 74 74 76 77 81 82 84 85 86 87 87 88 88 89 91 93 95

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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

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PM 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8

BRUSH DC MACHINES AND THEIR CONTROL . . 113 Why PM machines? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 Construction of a brush-type PM DC machine . . . . . . . . . . . . . 114 Principle of operation of a PM brush DC machine . . . . . . . . . . 115 Windings of a slotted rotor (armature) of a brush-type DC machine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 Construction of a coreless rotor winding with an inner PM . . 120 Coreless rotor windings: Maxon versus Faulhaber winding . . . 122 PM brush DC motor with cylindrical rotor and foil winding . 124 Disk-type PM brush DC motors with printed rotor winding . . 126

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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

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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

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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

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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

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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: [email protected] Prof. Jian-Xin Shen, PhD, IET Fellow Hangzhou, China, Email: [email protected]

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)

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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

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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.

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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.

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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

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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 Low hysteresis losses because of small hysteresis loop Can be easily magnetized and demagnetized Require small value of H for magnetization. Domain wall moves easily High susceptibility and relative magnetic permeability Low remanence and coercivity Eddy-currents can be limited by making laminations Examples: silicon steel, cobalt steels, amorphous alloys, ferrites, garnets Applications: ferromagnetic cores of electrical machines and transformers, electromagnets, computer data storage

Hysteresis loop area is large High hysteresis losses because of large hysteresis loop Cannot be easily magnetized and demagnetized Require large value of H for magnetization. Domain wall does not move easily Low susceptibility and relative magnetic permeability High remanence and coercivity Eddy-currents cannot be limited in solid cubes Examples: Alnico, Barium Ferrite, SmCo, NdFeB Applications: permanent magnets (PMs)

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=

U I

Rµ =

Vµ Φ

G=

I U

Gµ =

Φ Vµ

R=

l σs

Rµ =

l µs

G=

σs l

Gµ =

µs l

Ohm’s law

2nd Ohm’s law

P

Kirchhoff’s current law

Kirchhoff’s voltage law

P

U−

P

I=0

P

RI = 0

P

Vµ −

Φ=0

P

Rµ Φ = 0

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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.

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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.

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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.

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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.

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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. F´ elix 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 .

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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)

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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)

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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 t′ = q 1−

= v2 c2

1 γ

(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.

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Modern Permanent Magnet Electric Machines

Albert Einstein was born on March 14th 1879 at Ulm, W¨ urttemberg, 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 L¨ owenthal 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)

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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)

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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).

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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.

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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.

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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

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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 .

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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

®

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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

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®

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

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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

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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.

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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 Z¨ urich 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).

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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.

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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.

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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

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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

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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.

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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).

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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.

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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).

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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]:

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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;

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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.

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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.

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Extending Oersted’s experimental work, Andr´e-Marie Amp`ere 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 Amp`ere’s force law was discovered in 1823. Amp`ere’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 Amp`ere’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 Amp`ere’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

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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.

Andr´ e-Marie Amp` ere, 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 Amp`ere 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, Amp`ere began to formulate a combined theory of electricity and magnetism. His work confirmed and validated the discoveries of Oersted. Amp`ere’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. Amp`ere 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. Amp`ere 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 Amp`ere.

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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.

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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)

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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)

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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.

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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.

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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.

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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 material Ceramic 8 Br , T 0.4 Hc , kA/m 270 Hci , kA/m 260 (BH)max , kJ/m3 25 to 32 αB , %/◦ C −0.20 αH , %/◦ C −0.27 Tc , ◦ C 460

Alnico Alloy 1.25 55 55 < 40 −0.02 −0.015 890

SmCo NdFeB Sm2 Co1 7 Sintered Bonded 1.0 to 1.1 0.55 to 0.70 1.25 to 1.35 600 to 800 180 to 450 950 to 1040 720 to 2000 210 to 1100 1200 to 1400 190 to 240 32 to 88 290 to 400 −0.03 −0.105 −0.11 −0.15 −0.4 −0.65 800 360 330

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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

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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

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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

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Table 3.5. Physical properties of Vacomax sintered Sm2 Co17 PM materials at room temperature 200 C manufactured by Vacuumschmelze GmbH, Hanau, Germany

Property Remanent flux density, Br , T Coercivity, Hc , kA/m Intrinsic coercivity, Hci , kA/m (BH)max , kJ/m3 Relative recoil magnetic permeability Temperature coefficient αB of Br at 20 to 1000 C, %/0 C Temperature coefficient αiH of Hci at 20 to 1000 C, %/0 C Temperature coefficient αB of Br at 20 to 1500 C, %/0 C Temperature coefficient αiH of Hci at 20 to 1500 C, %/0 C Curie temperature, 0 C Maximum continuous service temperature, 0 C Thermal conductivity, W/(m 0 C) Specific mass density, ρP M , kg/m3 Electric conductivity, ×106 S/m Coefficient of thermal expansion at 20 to 1000 C, ×10−6 /0 C Young’s modulus, ×106 MPa Bending stress, MPa Vicker’s hardness

Vacomax 240 HR 1.05 to 1.12 600 to 730 640 to 800 200 to 240

Vacomax 225 HR 1.03 to 1.10 720 to 820 1590 to 2070 190 to 225

Vacomax 240 0.98 to 1.05 580 to 720 640 to 800 180 to 210

1.22 to 1.39 1.06 to 1.34 1.16 to 1.34 −0.030 −0.15

−0.18

−0.15

−0.035 −0.16

300

−0.19 −0.16 approximately 800 350 approximately 12 8400 1.18 to 1.33

300

10 0.150 90 to 150 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

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Table 3.6. Physical properties of Vacodym sintered NdFeB PM materials at room temperature 200 C manufactured by Vacuumschmelze GmbH, Hanau, Germany

Property Remanent flux density, Br , T Coercivity, Hc , kA/m Intrinsic coercivity, Hci , kA/m (BH)max , kJ/m3 Relative recoil magnetic permeability Temperature coefficient αB of Br at 20 to 1000 C, %/0 C Temperature coefficient αiH of Hci at 20 to 1000 C, %/0 C Temperature coefficient αB of Br at 20 to 1500 C, %/0 C Temperature coefficient αiH of Hci at 20 to 1500 C, %/0 C Curie temperature, 0 C Maximum continuous service temperature, 0 C Thermal conductivity, W/(m 0 C) Specific mass density, ρP M , kg/m3 Electric conductivity, ×106 S/m Coefficient of thermal expansion at 20 to 1000 C, ×10−6 /0 C Young’s modulus, ×106 MPa Bending stress, MPa Vicker’s hardness

Vacodym 633 HR 1.29 to 1.35 980 to 1040 1275 to 1430 315 to 350

Vacodym 362 TP 1.25 to 1.30 950 to 1005 1195 to 1355 295 to 325

1.03 to 1.05

Vacodym 633 AP 1.22 to 1.26 915 to 965 1355 to 1510 280 to 305 1.04 to 1.06

−0.095

−0.115

−0.095

−0.65

−0.72

−0.64

−0.105

−0.130

−0.105

−0.55

110 7700

−0.61 −0.54 approximately 330 100 approximately 9 7600 0.62 to 0.83

120 7700

5 0.150 270 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.

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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 Manufacturer

Grade HS-50AH Hitachi HS-47AH HS-46CH N50 ShinEtsu N48M Vacodym Vacuumschmelze 722 HR

Br , T 1.38 to 1.45 1.35 to 1.42 1.33 to 1.40 1.38 to 1.43 1.35 to 1.40

Hc , kA/m Hci , kA/m (BH)max , kJ/m3 1042 to 1130 ≥ 1035 358 to 406 1019 to 1106 ≥ 1115 342 to 390 1003 to 1090 ≥ 1353 334 to 374 ≥ 820 ≥ 875 366 to 405 ≥ 995 ≥ 1114 950 to 390

1.42 to 1.47 835 to 915

> 875

380 to 415

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).

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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

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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.

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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)

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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.

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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

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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.

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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

FK

Φr Fc + ± = 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

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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.

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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) G = µ0

wM lM g

(4.1)

πd2M 4g

(4.2)

(b) Cylinder (Fig. 4.2b) G = µ0 (c) Half-cylinder (Fig. 4.2c) G = 0.26µ0 lM

(4.3)

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)

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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).

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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

−1  Gg Fad GlM = Fad = 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).

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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

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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: 2Φ

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)

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)

The coefficient of PMs leakage flux σlM = 1 + The coefficient of total leakage flux σl ≈ 1 +

ΦlM + Φlg Φf

(4.45)

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 ℜM

2(FM − Fad )  ℜt + 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

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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. 2. 3. 4.

Step 1: Define the model (Fig. 4.7). Step 2: Assign materials (Fig. 4.8a). Step 3: Set up boundaries or sources (Fig. 4.8b). 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.

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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.

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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;

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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)

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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/

Brush material Brush resistance

Construction

Use for long operating life

Benefits and drawbacks

Contact response

Lubrication

Applications

Commutator

Graphite brushes ◦ graphite with approx. 50% Cu added to reduce brush resistance ◦ max. operating life with pure electrographite brushes ◦ more complex construction ◦ lead wire for brushes ◦ brushes with springs in brush holders ◦ for larger motors ◦ for higher currents ◦ for frequent peak currents in star-stop and reversing operation ◦ higher costs ◦ higher audible noise ◦ greater losses ◦ higher no-load current ◦ depends on the surface of the commutator ◦ good electrical contact only at higher currents ◦ less sensitive to moderate brush fire (no CLL circuit required) ◦ no lubrication ◦ graphite and the moisture in the air acts as lubricant ◦ servo drives ◦ feeder systems, robots ◦ drills ◦ screw drives ◦ Cu alloy ◦ surface turned on the lathe for perfect concentricity, cleanless and surface texture

Precious metal brushes ◦ carrier material: spring bronze ◦ contact material: Ag alloy (Au alloy) ◦ very low electrical resistance ◦ few parts ◦ simple construction ◦ brushes preloaded ◦ for small motors ◦ for very small currents and voltages in continuous operation ◦ lower costs ◦ lower audible noise ◦ lower losses ◦ lower no-load current ◦ low constant contact resistance ◦ small contact surface ◦ very sensitive to arcing, therefore capacitive damping by CLL is required ◦ lubrication with special commutator lubricant

◦ fans ◦ simple pumps

◦ Ag alloy ◦ surface polished or coated

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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 =

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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)

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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.

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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).

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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)

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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.

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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.

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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

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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

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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)

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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)

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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

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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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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:

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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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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.

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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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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

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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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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.

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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

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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

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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.

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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 .

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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

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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.

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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 – M¨ ossbauer 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

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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

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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

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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

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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

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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

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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 =

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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

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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.

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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.

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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 =

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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

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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.

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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

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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

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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.

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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

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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

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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)

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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]

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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.

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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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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.

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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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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.

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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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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

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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

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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)

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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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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.

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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.

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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

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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 Rated continuous power, W Continuous torque, Nm Peak torque, Nm No load speed, rpm Number of poles Number of slots EMF constant, V/krpm Torque constant, Nm/A Rotor moment of inertia, kgm2 × 10−5 Length, mm Width, mm Mass, kg

3410 120 0.32 1.27 5060

3420 180 0.706 3.81 4310

9.2 0.0883 4.2 88.6

10.8 0.103 9.2 105

1.1

1.6

3430 220 1.09 4.06 3850 4 24 12.1 0.116 13.0 122 82.6 2.0

3440 260 1.48 4.41 3609

3450 270 1.77 5.30 3398

12.9 0.123 18.0 138

13.7 0.131 21.0 155

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

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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.

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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.

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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.

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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

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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.

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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 Bmg1

2 = π

Z

0.5αi π

Bmg cos αdα = −0.5αi π

4 αi π Bmg sin π 2

(7.1)

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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)

πD1in s1

(7.6)

where the slot pitch t1 =

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)

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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

Telm

    m1 U1 Ef U12 1 1 Pelm = sin δ + − sin 2δ = 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

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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.,

Baq1

0.5π

4 π

Z

4 = π

Z

Bad1 =

B(x) cos xdx

(7.18)

B(x) sin xdx

(7.19)

0 0.5π

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)

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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

0.748 0.703 0.097 0.939 0.129

0.900 0.818 0.182 0.909 0.202

αi = bp /τ 0.6 2/π 0.7 1.030 0.913 0.287 0.886 0.279

1.071 0.943 0.391 0.880 0.365

1.134 0.958 0.442 0.845 0.389

0.8

1.0

1.211 0.987 0.613 0.815 0.505

1.273 1.00 1.00 0.785 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

h = slot depth

kf d = kf q = 1

kf d =

4 α 1 π i 1−α2

cos(0.5αi π)

kf q =

1 (αi π π

− sin αi π)

kf q =

1 (αi π π

− sin αi π)

i

Salient-pole rotor with excitation winding

kf d =

1 (αi π π

+ sin αi π)

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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)

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Modern Permanent Magnet Electric Machines

Eaq

√ 2 m1 2 N1 kw1 = 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

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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)

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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.

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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

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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)

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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.

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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.

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211

Andr´ e-Eug` ene 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. M¨ ahl, 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 L´egion 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 Φf 1 = Li

τ

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 Rated torque, Nm Maximum torque, Nm Standstill torque, Nm Rated current, A Maximum current, A Standstill current, A Rated speed, rpm Maximum speed, rpm Armature constant, V/1000 rpm Torque constant, Nm/A Resistance, Ω Inductance, mH Moment of inertia, kgm2 × 10−3 Mass, kg Diameter of frame, mm Length of frame, mm Power density, W/kg Torque density, Nm/kg

680 940 910 1.3 1.8 2.9 7 9 14 1.7 2.3 3.5 4.0 4.9 4.9 21 25 24 5.3 6.3 5.9 5000 5000 3000 6000 6000 6000 23 25 42 0.35 0.39 0.64 2.5 1.8 2.4 3.2 2.8 5.4 0.08 0.12 0.21 4.5 5.0 6.2 150 150 174 82 82 89 151.1 188.0 146.8 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

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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 Develop´ ment Institute of Electrical Machines VUES 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, Nov´ a 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 ´ (VUES 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 1 2 2 αi m1 Ia N1 kw1 Bmg (Dext − Din ) 4

Telm =

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, Hyvink¨ aa ¨, 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, Hyvink¨ aa ¨, Finland.

Table 8.4. Specifications of double-disk PM brushless motors manufactured by Kone, Hyvink¨ aa ¨, 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, B¨ ulach, Switzerland are shown in Table 8.6. Table 8.6. TFMs manufactured by Landert-Motoren AG, B¨ ulach, 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

0.68 0.70 0.16 2.8 external 91 133 IP54 F IC410

16 10 7 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

HF ei lF ei 4Bmg g/µ0 i

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.

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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

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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;

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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

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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).

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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 results have been obtained 92 kW, 41 krpm, prototype Ja = 10.3 A/mm2 eqns (9.10), 4-pole motor (9.11) 48.5 kW, 43 krpm, prototype Ja = 6.62 A/mm2 eqns (9.10), 4-pole motor (9.11) 1.5 MW, 15 krpm, SPEED Adapco Ja = 16.8 A/mm2 eqns (9.10), 8-pole generator (9.11) Machine

Mass of active D1in /D1out L components mm mm kg 65/125.5 102.9 8.2 78/125.5 93.5 45/98.5 128.0

7.4 8.7

46/98.0 126.0 205.6/280 200

8.2 88.0

210/273

200

88.7

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,

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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)

where the stiffness is K = 48

EI L3i

(9.15)

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)

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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

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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.

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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.

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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.

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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.

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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

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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.

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(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

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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.

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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).

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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.

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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 Andr´e-Marie Amp`ere (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

Magnetic flux density B T (tesla) Intrinsic magnetization (polarization) Bi T (tesla) Magnetic flux Φ Wb (weber) Magnetic field intensity A/m (strength) H Energy density 0.5BH J/m3 1

1H = Ωs =

1F =

Conversion 1 T = 1 Vs/m2 = 10,000 Gs 1 T = 1 Vs/m2 = 10,000 Gs 1 Wb = 1 Vs = 108 Mx (maxwell) 1 A/cm = 0.4 π Oe = 1.257 Oe (oersted) 1 Oe = 79.55 A/m 1 J/m3 = 126 GsOe MGsOe = 7.958 kJ/m3

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.

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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).

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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 av b br c cog Cu d elm eq Fe f fr g h in k L l M m max min n n, t out q r rhe rot s sat sh st str syn vent w wind x, y, z 1 2

armature average braking brush commutation cogging copper direct axis; differential electromagnetic equivalent ferromagnetic field friction air gap hysteresis inner kinetic load leakage magnet peak value (amplitude) maximum minimum nominal normal and tangential components output, outer quadrature axis relative rheostat rotational synchronous saturation shaft starting stray, additional synchronous or synchronizing ventilation winding windage cartesian coordinate system primary; stator; fundamental harmonic 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 Amp`ere 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

Amp`ere, 218 Ampere’s circuital, 67 Amp`ere’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