HANDBOOK OF SMALL ELECTRIC MOTORS William H Yeadon, P.E Editor in Chief Alan W Yeadon, P.E Associate Editor Yeadon Energy Systems, Inc Yeadon Engineering Services, P.C McGraw-Hill New York Chicago San Francisco Lisbon London Madrid Mexico City Milan New Delhi San Juan Seoul Singapore Sydney Toronto Library of Congress Cataloging-in-Publication Data Handbook of small electric motors / William H Yeadon, editor in chief, Alan W Yeadon, associate editor p cm ISBN 0-07-072332-X Electric motors, Fractional horsepower I Yeadon, William H II Yeadon, Alan W TK2537 H34 621.46—dc21 2001 00-048974 Copyright © 2001 by The McGraw-Hill Companies, Inc All rights reserved Printed in the United States of America Except as permitted under the United States Copyright Act of 1976, no part of this publication may be reproduced or distributed in any form or by any means, or stored in a data base or retrieval system, without the prior written permission of the publisher DOC/DOC ISBN 0-07-072332-X The sponsoring editor for this book was Scott Grillo and the production supervisor was Sherri Souffrance It was set in Times Roman by North Market Street Graphics Printed and bound by R R Donnelley & Sons Company McGraw-Hill books are available at special quantity discounts to use as premiums and sales promotions, or for use in corporate training programs For more information, please write to the Director of Special Sales, Professional Publishing, McGraw-Hill, Two Penn Plaza, New York, NY 10121-2298 Or contact your local bookstore This handbook is intended to be used as a reference for information regarding the design and manufacture of electric motors It is not intended to encourage or discourage any motor type, design, or process Some of the configurations or processes described herein may be patented It is the responsibility of the user of this information to determine if any infringement may occur as a result thereof Information contained in this work has been obtained by The McGraw-Hill Companies, Inc (“McGraw-Hill”) from sources believed to be reliable However, neither McGraw-Hill nor its authors guarantee the accuracy or completeness of any information published herein, and neither McGraw-Hill nor its authors shall be responsible for any errors, omissions, or damages arising out of use of this information This work is published with the understanding that McGraw-Hill and its authors are supplying information but are not attempting to render engineering or other professional services If such services are required, the assistance of an appropriate professional should be sought This handbook is dedicated to my wife, Luci Yeadon, who took most of the photographs for it William H Yeadon Editor in Chief CONTRIBUTORS Larry C Anderson John S Bank American Hoffman Corporation (Sec 3.14) Phoenix Electric Manufacturing Company (Sec 3.15) Warren C Brown Link Engineering Company (Sec 3.10.6) Joseph H Bularzik Peter Caine Magnetics International, Inc (Sec 2.5) Oven Systems, Inc (Sec 3.16) David Carpenter Vector Fields, Ltd (contributed the finite-element plots in Chap 4) John Cocco Loctite Corporation (Sec 3.17) Philip Dolan Oberg Industries (Sec 3.10.5) Birch L DeVault Cutler-Hammer (Sec 10.6) Brad Frustaglio Yeadon Energy Systems, Inc (Sec 6.4) Francis Hanejko Hoeganaes Corporation (Sec 2.6) Duane C Hanselman Daniel P Heckenkamp Leon Jackson Dan Jones University of Maine (Secs 5.1.4 and 10.11) Cutler-Hammer (Sec 10.6) LDJ Electronics (Sec 3.19) Incremotion Associates (Secs 5.1.3, 10.12, and 10.13) Douglas W Jones University of Iowa (Secs 5.2.10 and 10.8 to 10.10) Mark A Juds Eaton Corporation (Secs 1.1 to 1.12) Robert R Judd Judd Consulting Associates (Secs 2.2 and 2.3) Ramani Kalpathi Dana Corporation (Sec 10.7) John Kauffman Phelps Dodge Magnet Wire Company (Sec 2.10) Todd L King H R Kokal Eaton Corporation (Sec 10.6) Magnetics International, Inc (Sec 2.5) Robert F Krause Magnetics International, Inc (Sec 2.5) Barry Landers Electro-Craft Motion Control (Chap 9) Roger O LaValley Magnetic Instrumentation, Inc (Sec 3.18) Bill Lawrence Oven Systems, Inc (Sec 3.16) Andrew E Miller Software and motor designer (Secs 4.5, 6.4.3, and 6.4.4) Stanley D Payne Windamatics Systems, Inc (Sec 3.10.4) Derrick Peterman LDJ Electronics (Sec 3.19) xi xii CONTRIBUTORS Curtis Rebizant Integrated Engineering Software (contributed the boundary element plots in Figs 5.58 to 5.61) Earl F Richards University of Missouri (Secs 1.14, 4.1, 6.2, 6.3, and 8.4 to 8.8) Robert M Setbacken Renco Encoders, Inc (Secs 10.1 to 10.5) Karl H Schultz Schultz Associates (Secs 3.1 to 3.9) Joseph J Stupak Jr Oersted Technology Corporation (Sec 2.8) Chris A Swenski Yeadon Energy Systems, Inc (Secs 3.6.5, 6.4, and 7.4) Harry J Walters Oberg Industries (Sec 3.10.5) Alan W Yeadon Yeadon Engineering Services, PC (Secs 3.10, 3.11, and 4.2 to 4.5) Luci Yeadon handbook) Luci’s Photography (contributed most of the photographs in this William H Yeadon Yeadon Engineering Services, PC (Secs 1.13, 2.1, 2.9, 2.11, 3.10 to 3.12, 4.3, 4.6 to 4.8, 5.1 to 5.3, 6.1, 6.4, 7.1 to 7.4, and 8.1 to 8.3) ACKNOWLEDGMENTS Over the course of my career I have had the privilege to meet many of the giants of this industry Many I have met through my association with the Small Motors and Motion Association (SMMA) and others through business relationships Included among them are Dr Cyril G Veinott, Professor Philip H Trickey, Dr Ben Kuo, Dr Duane Hanselman, and those authors who have contributed to this handbook There is, however, one person of whom I must make special mention He is Dr Earl Richards, Professor Emeritus of the University of Missouri at Rolla This man never ceases to amaze me He is always willing to help out selflessly with projects of this type I have taught many motor design courses with him When a student asks questions of him, he can start at the lowest level of understanding necessary and develop in a very understandable way a logical and reasonable answer to the question His ability to communicate and teach is truly amazing He has been very helpful in the preparation of this book I also need to acknowledge the dedication of my secretary, Kristina Wodzinski Without her tireless effort this work would not have been completed xv ABOUT THE CONTRIBUTORS LARRY C ANDERSON (Sec 3.14) is an applications consultant with American Hofmann Corporation, one of the world’s leading manufacturers of precision balancing machines He has been with the company since 1990 and performs unbalance analysis on rotating assemblies for manufacturers worldwide He holds a BS degree in electrical engineering technology and has over 20 years experience, with the past focused on the electric motor industry JOHN S BANK (Sec 3.15) is the executive vice president of Phoenix Electric Manufacturing Company and is responsible for coordinating new product development and developing advanced strategies He received his bachelor’s degree in business administration (magna cum laude) from the University of Michigan in 1981 and his JD from UCLA in 1984 He is also a Certified Public Accountant in the state of Illinois (1981) and a licensed real estate broker in the state of Illinois (1981) Mr Bank currently serves on the board of directors of SMMA (1995–present) and EMERF (1997–present) He is the Company Representative and Voting Member of NEMA (1992–present), EMCWA (1992–present) and NAM (1992–present) WARREN C BROWN (Sec 3.10.6) graduated with a BSME from Michigan State University in 1966 and with an MBA from Michigan State University in 1968 He was the Manager/Director MIS of Burroughs Corporation in Detroit, Michigan, from 1968 to 1982 He directed sales and marketing at Link Engineering Company from 1982 to 1990 Since 1990, he has been vice president for motor products of Link Engineering Company He has been a member of SAE, ESD, and SMMA JOSEPH H BULARZIK (Sec 2.5) is a staff engineer He received a BS in chemistry from Arizona State University, Tempe, in 1982 He received a PhD in chemistry from the University of California, Berkeley, in 1987 He conducted postdoctoral research in the field of superconducting oxides at Princeton University, Princeton, New Jersey, in 1989 He was an assistant professor of chemistry at Lycoming College, Williamsport, Pennsylvania, from 1987 to 1989 He has seven years of experience in magnetic materials research He is a member of ASM He has worked in research for Magnets International, Inc., East Chicago, Indiana, since 1994, and he worked in research at Inland Steel Company, East Chicago, Indiana, from 1990 to 1994 PETER CAINE (Sec 3.16) graduated from the University of Wisconsin in Platteville with a BS in industrial engineering His career at Oven Systems, Inc., has included applications engineering, custom product sales, and management For the past three years, he has managed the electric motor equipment division DAVID CARPENTER (Chap finite-element plots) received a first-class honors BSc in electrical engineering from the University of Southampton, England, in 1979 and joined GEC Ltd as an induction motor design engineer In 1986 to 1987 he was appointed as visiting professor at Lakehead University, Canada, and in the following year he received an MSc from Coventry University, England After joining Vector Fields Ltd as an application engineer in 1991, he received a PhD from the University of Bath, England, in 1993 He was appointed to the position of vice president of Vector Fields, Inc., United States, in 1995 He is a Charter Engineer and a member of the IEEE JOHN COCCO (Sec 3.17) is the director of Loctite Corporation’s North American Application Engineering Center For the past 10 years, he has been working with Loctite Corporation’s cusC.1 C.2 ABOUT THE CONTRIBUTORS tomer base, developing adhesive and sealant applications for use in small motors In the past two years, he has conducted several design seminars at original equipment manufacturers focusing on this topic He holds a bachelor’s degree in chemical engineering and is a licensed Professional Engineer PHILIP DOLAN (Sec 3.10.5) graduated from Marquette University with a BA He was vice president of Marketing for Oberg Industries and had previous experience in plant management and strategic planning BIRCH L DEVAULT (Sec 10.6) was born in Pittsburgh, Pennsylvania, in 1946 He received a BS in electrical engineering from the University of Pittsburgh in 1967 He joined the Westinghouse Electrical Graduate Student Course in 1967 In 1968, he joined the Westinghouse Standard Control Division, Beaver, Pennsylvania, as an associate design engineer In 1981, he joined the Control Division in Asheville, North Carolina Since February of 1994, he has been a senior development engineer with Cutler-Hammer, Milwaukee, Wisconsin, responsible for the design and application of magnetic motor control He is a Registered Professional Engineer in the state of Pennsylvania He has eight patents in the area of motor control He is a member of IEEE He has published papers related to motor control in TAPPI and IEEE publications BRAD FRUSTAGLIO (Sec 6.4) has a BSME from Michigan Technological University and is a design engineer for Yeadon Energy Systems, Inc FRANCIS HANEJKO (Sec 2.6) is a metallurgical engineer and received his BS and MS degrees from Drexel University He has been employed by the Hoeganaes Corporation for 22 of the last 25 years During that time, he has held numerous positions in the sales and marketing and research and development departments His current position is manager of electromagnetics and customer applications in the research and development department, with responsibilities for customer service and product development He is a past chairman of the Philadelphia Section of the APMI DUANE C HANSELMAN (Secs 5.1.4 and 10.11) is an associate professor in electrical engineering at the University of Maine He holds PhD and MS degrees from the University of Illinois He is a senior member of IEEE and an associate editor of the IEEE Transactions of Industrial Electronics He is the author of numerous articles on motors and motion control He has published several textbooks, including Brushless Permanent-Magnet Motor Design and MATLAB Tools for Control System Analysis and Design (McGraw-Hill, 1994) DANIEL P HECKENKAMP (Sec 10.6) received his BS in mechanical engineering from the University of Wisconsin, Milwaukee, in 1983 In 1981, he joined the Square D Company in Milwaukee, where he was responsible for the design of industrial lifting magnets and their applications In 1983, he transferred to the Square D Controls Division, where he was responsible for contactor development He joined Cutler-Hammer’s controls division in 1988 as a product development engineer, where he has been responsible for the design and maintenance of contactors and overload relays His current position is principal engineer LEON JACKSON (Sec 3.19) received an AS from Port Huron Junior College in 1957 He also received a BS in electrical engineering from Wayne State University in 1960 He also attended the University of Loyola for business administration He received honors from the Tau Betta Pi educational honor society and the Etta Kappa Nu engineering honor society for academic achievement He has worked for General Magnetic Corporation and LDJ Electronics, Inc., where he is currently president He is a member of the IEEE Magnetics Society DAN JONES (Secs 5.1.3, 10.12, and 10.13) has a BSEE from Hofstra University and a MS in mathematics from Adelphi University He is a member of ASME, IEEE, ISA, and AIME He has 38 years experience in the motor business He founded Incremotion Associates in 1982 and has previously worked for such companies as Vernitron, Printed Motors, Inc., Singer-Kearfott, Electro-Craft Corporation, Data Products Corporation and IMC Magnets Corporation ABOUT THE CONTRIBUTORS C.3 DOUGLAS W JONES (Secs 5.2.10 and 10.8 to 10.10) is an associate professor of computer science at the University of Iowa He received his PhD in computer science from the University of Illinois, Urbana, in 1980 He completed his BS in physics at Carnegie-Mellon University in 1973 His research interests are discrete event simulation, resource protection in architecture, operating systems, system Programming Languages, and the history of computing MARK A JUDS (Secs 1.1 to 1.12) has BS and MS degrees in mechanical engineering from the University of Wisconsin He is currently a senior principal engineer for Eaton Corporation’s Innovations Center, where he designs electromagnetic devices He also has expertise in heat transfer and mechanical dynamics ROBERT R JUDD (Secs 2.2 and 2.3) is currently president of Judd Consulting Associates, Inc., a general ferrous metallurgy and electrical-sheet consulting firm He acquired his doctorate in materials science from Carnegie-Mellon University and holds a bachelor’s degree in mechanical engineering from the University of Rochester He spent 30 years in principal research positions for U.S Steel and Ispat-Inland For three years he served as director of research and development for Johnstown Corporation, a large ferrous foundry and fabrication firm He has also taught general metallurgy at Carnegie-Mellon University His professional activities include ASM, AIME, MPIF and the ASTM A-6 subcommittee on magnetics He is also the treasurer and organizing committee member of the annual Conference on the Properties and Application of Magnetic Materials He holds patents in the powder metallurgy and soft magnetic material fields RAMANI KALPATHI (Sec 10.7) was a senior project engineer with Dana Corporation He completed his PhD in electrical engineering at Texas A&M University in 1994 and has been with Dana for the past five years Recently he has returned home to start his own consulting firm in Madras, India His interests are in the areas of power electronics and control of switchedreluctance motors JOHN KAUFFMAN (Sec 2.10) graduated from Purdue University with a BA in industrial economics in 1963 He has worked for Phelps Dodge Magnet Wire Company for 35 years He holds four patents for magnet wire and cable products and equipment TODD L KING (Sec 10.6) received BS and MS degrees in electrical engineering from the University of Wisconsin-Madison in 1978 and 1980, respectively He joined Borg Warner Corporate Research Center, Des Plaines, Illinois, in 1980, where he worked in analysis of motors and actuators and the design of automotive controls, actuators, and sensors He joined Eaton Corporate Research and Development Center, Milwaukee, Wisconsin, in 1988 as a senior engineer specialist, where he worked in the design of actuators for appliance, automotive, aerospace, hydraulic, and truck products He also worked in the design and analysis of commercial and industrial motor controls He became the engineering manager for the Design Analysis Technology Group in 1990 and added systems technology in the Eaton Innovation Center, where he has responsibility for defining the strategic direction of systems technology for the corporation HAROLD R KOKAL (Sec 2.5) is a senior staff engineer He received his BS and MS degrees in metallurgical engineering from the University of Minnesota, Minneapolis, in 1964 and 1970, respectively He has 30 years experience in process and product research He is a member of APMI and AIME He has worked in research at Magnetics International, Inc., East Chicago, Indiana, since 1992 He worked in research at Inland Steel Company, East Chicago, Indiana, from 1985 to 1992, and at U.S Steel Corporation, Coleraine, Minnesota, and Monroeville, Pennsylvania, from 1968 to 1985 He was an MRRC Research Fellow at the University of Minnesota, Minneapolis, from 1965 to 1966 ROBERT F KRAUSE (Sec 2.5) is a technical director He received his BS and PhD degrees in material science from Notre Dame University, South Bend, Indiana, in 1962 and 1966, respectively He has 31 years experience in metallurgy and magnetic materials He is a member of the ASM and IEEE He has worked in research at Magnetics International, Inc., Burns Harbor, Indiana, since 1991 He worked in research at Inland Steel Company, East Chicago, Indiana, C.4 ABOUT THE CONTRIBUTORS from 1987 to 1991; at Crucible Steel Company, Pittsburgh, Pennsylvania, from 1986 to 1987; at Westinghouse Electric Corporation, Churchill, Pennsylvania, from 1972 to 1986; and at U.S Steel Corporation, Monroeville, Pennsylvania, from 1966 to 1972 BARRY LANDERS (Chap 9) has 24 years of experience in the design and testing of ac and dc motors, including writing electrical and mechanical design and testing software for fractionalhorsepower ac, brush DC, and brushless dc motors, as well as for fine-pitch custom gearing In addition, he has 17 years of experience in spectral analysis of sound, vibration, and current on these motor types and on ball bearings as received, as well as in failure analysis of field problems As a senior project engineer and registered Professional Engineer, he currently has responsibility for an engineering development, analysis, and test group for ac and dc products at Electro-Craft Motion Control, Gallipolis, Ohio (a Rockwell Automation business) ROGER O LAVALLEY (Sec 3.18) is a senior application engineer with Magnetic Instrumentation, Inc He has 25 years experience in the area of magnetic applications In his present position he is responsible for reviewing customer requirements for the magnetizing, demagnetizing, and measuring of permanent magnets and magnet assemblies and for proposing the appropriate equipment and complete systems BILL LAWRENCE (Sec 3.16) has a BSME and an MBA from Marquette University He has worked in sales of servo electric motors at Moog, Inc., and in sales of specialty motors at Doerr Electric He is currently the vice president of Oven Systems, Inc ANDREW E MILLER (Secs 4.5, 6.4.3, and 6.4.4) has a BS in chemical engineering from Michigan Technological University He has several years of experience in software design and three years of experience in the motor design industry STANELY D PAYNE (Sec 3.10.4) is the vice president engineer at Windamatics Systems, Inc., Fort Wayne, Indiana DERRICK PETERMAN (Sec 3.19) has over eight years experience with magnetics research and instrumentation He completed a BA in physics at Washington University, St Louis, Missouri, in 1989 and a PhD in physics at Ohio State University in 1996 He currently holds the position of magnetic measurement specialist at LDJ Electronics CURTIS REBIZANT (Figs 5.58 to 5.61 boundary element plots) is an engineer at Integrated Engineering Software, which produces and markets software for electromagnetic, thermal, and structural system simulation He has a BS in electrical engineering from the University of Manitoba and has extensive experience with electromagnetic CAE software EARL F RICHARDS (Secs 1.14, 4.1, 6.2, 6.3, and 8.4 to 8.8) is Professor Emeritus of Electrical Engineering in the School of Engineering, University of Missouri, Rolla He received his PhD from the University of Missouri He has 16 years of field experience in motor design and over 36 years of experience in the instruction of motor technology His professional emphasis is on electromechanical, power, and control systems He currently instructs graduate-level engineering courses and is frequently sought as an industrial and legal consultant ROBERT M SETBACKEN (Secs 10.1 to 10.5) is vice president of engineering at Renco Encoders, Inc He received his MSME degree in 1979 He has developed and tested analog and digital electromechanical and hydraulic servosystems for the military and commercial interests Since joining Renco in 1990, he has been involved with the design and manufacture of incremental rotary optical encoders for the industrial and office automation industries KARL H SCHULTZ (Secs 3.1 to 3.9) holds a BSME from Western Michigan University He is a senior member of SME and a member of SMMA He has 25 years experience in manufacturing and management with such companies as General Signal, General Electric, Emerson Electric, Clark Equipment, Chrysler, and Cincinnati Milacron, and his own consulting firm 10.126 CHAPTER TEN (a) (b) FIGURE 10.98 Sinusoidal torque and current waveforms: (a) principles of the electrical circuit, and (b) principles of torque generation DRIVES AND CONTROLS 10.127 TABLE 10.21 Ratios of Torque and Voltage Constants for Brushless Motors in Three-Phase Sine-Wave Systems, Motor Speed in rad/s Kt [N ⋅ m/ARMS] = 1.732 Ke Kt [lb ⋅ ft/ARMS] = 1.2775 Ke Kt [lb ⋅ in/ARMS] = 15.33 Ke Kt [oz ⋅ in/ARMS] = 245.28 Ke Ke[VRMS,LL/rad ⋅ s−1] = 0.5774 Kt Ke[VRMS,LL/rad ⋅ s−1] = 0.7828 Kt Ke[VRMS,LL/rad ⋅ s−1] = 6.523 × 10−2 Kt Ke[VRMS,LL/rad ⋅ s−1] = 4.007 × 10−3 Kt [VRMS,LL/rad ⋅ s−1] [VRMS,LL/rad ⋅ s−1] [VRMS,LL/rad ⋅ s−1] [VRMS,LL/rad ⋅ s−1] [N ⋅ m/ARMS] [lb ⋅ ft/ARMS] [lb ⋅ in/ARMS] [oz ⋅ in/ARMS] and 54 mm (2.126 in) OD for the 36-slot The magnet arc widths were varied from 70 to 85° mechanical, and the actual measured torque versus position waveforms were recorded by using a plotter and mounted torque transducer The stator stack axial lengths (SALs) were all 3.0 in long Persson also tested stators with no slot skew and with either a 3⁄4-slot skew or a 1-slot skew Table 10.21 shows Persson’s data summary Persson was interested in the torque ripple measured for the 18 stator slot and skew combinations The impact of skew on reducing torque ripple is also demonstrated The impact of more teeth per pole in reducing torque ripple was not demonstrated by the experimental data There was no attempt to alter tooth shapes, air gaps, or magnet material, just to supply the reader with a reference point in terms of solid experimental data Magnet material used was a low grade of ferrite magnet material, so there was no magnetic saturation in any of the motor soft iron (steel) members What is more astounding is the shape of the motor T versus θ waveform The FIGURE 10.99 Static torque function: 12 slots, 80° arc, no skew 10.128 CHAPTER TEN FIGURE 10.100 skew Static torque function: 12 slots, 80° arc, 3⁄4-slot commutation interval (angle) is 60° electrical as in the previous theoretical examples Figures 10.99 and 10.100 show the static torque function for the 12-slot stator in both unskewed and 3⁄4-skewed stator configurations, respectively The overall shape is flat for trapezoidal drive conditions with a maximum drop of percent in torque over the 60° commutation interval Most of the torque drop in Fig 10.100 was caused FIGURE 10.101 skew Static torque function: 12 slots, 70° arc, no 10.129 DRIVES AND CONTROLS FIGURE 10.102 skew Static torque function: 12 slots, 70° arc, 3⁄4-slot by the motor’s cogging torque The 3⁄4-slot skewed version (Fig 10.100) reduces the torque drop to percent of total (peak) value over the 60° electrical interval The overall torque versus position waveform shape for both skewed and unskewed stators (with an 80° arc width) is trapezoidal in shape Figures 10.101 and 10.102 show the torque versus position plot for a 12-slot stator with a 70° magnet arc width Figure 10.101 displays the torque vs position plot with no stator slot skew The cogging torque signature is the primary cause of the percent torque variation over the 60° commutation interval in this setup Skewing the stator stack by 3⁄4 slot changes the torque profile into a quasi-sinusoidal torque profile with lower instantaneous torque variation (see Table 10.22) Figure 10.103 illustrates the torque profile of the 24-slot BLDC stator, no skew, with an 85° rotor magnet arc width There is a percent torque drop-off, but the waveform has a near-trapezoidal torque profile Adding a 1-slot skew (Fig 10.104) smoothes the torque profile, and the waveform over the 60° electrical interval is closer to a trapezoidal waveform Figures 10.105 and 10.106 show the effects on waveform of a 70° arc with a 24-slot stator, both skewed and unskewed, respectively The waveform becomes more sinusoidal The 36-slot lamination performance is shown in Figures 10.107 and 10.108 Figure 10.107 displays the 80° rotor magnet arc width with no stator skew The overall torque profile is decidedly sinusoidal There is TABLE 10.22 Subjective Evaluation of Expected Torque Ripple 12 slot 24 slot 36 slot Magnet arc Straight Slot skew Straight Slot skew Straight Slot skew 85° 80° 70° 16 16 7 14 14 13 26 10 20 10.130 CHAPTER TEN FIGURE 10.103 skew Static torque function: 24 slots, 85° arc, no a 13 percent drop-off in torque magnitude over the 60° electrical commutation interval, and the torque waveform shows the cogging torque “bumps” quite prominently The 1-slot skew version of the 80° rotor magnet and 36-slot lamination displays a quasi-trapezoidal waveform with a 10 percent torque drop-off over the 60° electrical commutation interval The final torque-position waveform shows the last torque profile, which describes a 36-slot lamination with a 70° rotor magnet arc width The overall shape is definitely sinusoidal in both unskewed and skewed examples The torque drop-off has increased to 26 percent (Fig 10.109) and 20 percent (Fig 10.110) Figures 10.111 and 10.112 display copies of the actual 12-slot and 24-slot laminations used by Persson FIGURE 10.104 arc, 1-slot skew Static torque function: 24 slots, 85° DRIVES AND CONTROLS FIGURE 10.105 skew 10.131 Static torque function: 24 slots, 70° arc, no Summarizing Persson’s results: q q q Decreasing the magnet arc width will lead to sinusoidal torque waveforms Skewing the rotor will reduce cogging and increase the tendency toward quasisinusoidal torque waveforms Increasing the number of stator slots per phase per pole (n ≥ 3) will increase the tendency toward sinusoidal torque waveforms FIGURE 10.106 1-slot skew Static torque function: 24 slots, 70° arc, 10.132 CHAPTER TEN FIGURE 10.107 skew q Static torque function: 36 slots, 80° arc, no Use a wide pole arc magnet and a minimum number of stator slots per phase per pole (n = 1) to achieve a trapezoidal torque profile 10.13.2 Phase Resistance This section develops a basic comparison chart for the theoretically anticipated values for line-to-line resistance for the four major line-winding configurations Using the phase resistance X values, the line-to-line resistance for a given configuration is as shown in Table 10.23 FIGURE 10.108 skew Static torque function: 36 slots, 80° arc, 1-slot DRIVES AND CONTROLS FIGURE 10.109 skew 10.133 Static torque function: 36 slots, 70° arc, no 10.13.3 Establish Speed-Torque Curve—No-Load, Load, and Peak Torque Values The brushless dc motor has one major difference from its brush dc counterpart All testing including the “simple” back-emf Ke measurement requires that the brushless dc motor be tested with its electronic drive package The brushless dc motor that uses rare earth magnets can usually outperform its companion drive in the region of maximum torque capability, and there are commutation limits based on motor speed FIGURE 10.110 1-slot skew Static torque function: 36 slots, 70° arc, 10.134 CHAPTER TEN FIGURE 10.111 12-slot lamination times the number of commutation switching events per revolution These two limits shape the torque-speed profile Figure 10.113 shows a representative torque-speed limit curve for a sinusoidal drive showing these limitations The calculations for establishing the various torque-speed performance points does proceed in the same manner used for a PM brush dc motor, except that the torque and speed limits established for the specific motor-drive combination must be used to limit the calculated values at both ends of the motor torque-speed curve The BLDC motor’s torque and speed performance is linear over a major portion of the torque-speed curve as described FIGURE 10.112 24-slot lamination 10.135 DRIVES AND CONTROLS TABLE 10.23 Basic Comparison Chart Phase resistance Hookup configuration Line-to-line resistance X X X X Y (half) Y (full) Delta Independent X 2X ⁄3X X Once the torque constant Kt and the phase resistance Rph have been established for a representative design, one can use simple algebraic formulas to compute the motor’s basic torque versus speed performance The basic performance parameters from the design example are shown in the Fig 10.114 computer simulation.The computed values for Kt = 21.046 (oz и in)/A or 0.149 (N и m)/A and RLL = 0.275 Ω for a full-wave Y-connected BLDC motor are used to establish the load torque Tload, peak torque Tpk, and other operating parameters Tload = Iload Kt FIGURE 10.113 Typical BLDC motor torque-speed limit curve (10.10) 10.136 FIGURE 10.114 CHAPTER TEN Example simulation program results DRIVES AND CONTROLS 10.137 where Iload is the value established by the motor’s thermal constants Et Tpk = ᎏ Kt RLL (10.11) ET = terminal voltage (10.12) ET Ipk = ᎏ RLL (10.13) where The load torque can be computed by using Eq (10.10) if the load current has been established.The inner line on the Fig 10.113 representative torque-speed curve is the motor’s safe operating ambient conditions (SOAC) curve All load points (torque and speed) to the left of this boundary line are rated and continuous values The Iload is located somewhere along this curve The actual motor terminal voltage would create a straight-line torque-speed curve based on constant voltage which would intersect the SOAC curve at a specific point For purposes of illustration, let us assume that this BLDC motor can dissipate 74 W at a specific point on the SOAC line Using the estimated hot resistance value for RLL which is assumed to be at 100°C above ambient, the actual load current can be computed using the formula in Eq (10.14): ෆ ͙Pdiss ILD = ᎏᎏ = 13.8 A RLL,hot (10.14) where RH,hot = (0.275)(1.4) = 0.385 Ω Therefore, using Eqs (10.10), (10.11), and (10.12), the load torque Tload is computed at 290 oz и in The Tpk is calculated as (160/0.385) (21.046) = 8746 oz и in or 61.94 N и m The peak current is calculated as 415.8 A The peak values are theoretical, because one can expect the drive limits to engage well before these calculated peak values would be reached The establishment of the theoretical no-load speed NNL,th can be done by using the Ke value computed in equation 10.15 ET NNL,th = ᎏᎏ Ke (10.15) where: ET = motor terminal voltage Ke = motor’s Y-configuration back-emf 160 NNL,th = ᎏᎏ = 10.28 krpm 15.562 where NNL,th is the motor’s no-load speed Therefore, 10,280 rpm is the peak theoretical no-load speed One can estimate or calculate the actual no-load speed if no load current is available In any case, since the BLDC no-load rotating losses (windage, bearing system, friction, etc.) are very low, conservatively in the region of to percent, an adjusted no-load speed value of 9972 rpm (97 percent of theoretical value) can be established Load speed NL can be computed as displayed in Eq (10.16) NL = NNL (1 − α) (10.16) 10.138 CHAPTER TEN where α is the ratio of Tload/TPK,th ΂ ΃ 290 NL = (9972) − ᎏᎏ = (9972)(0.967) = 9641 rpm 8746 Any speed point along the constant-voltage derived torque-speed curve can be computed in a similar manner until the boundary for peak performance (intermittent duty limit line in Fig 10.113) is reached The values used in the Fig 10.114 simulation are the original cold (20°C) value for Ru = 0.275 Ω, so the calculated values in the simulation in Fig 10.114 are 40 percent higher than the ones computed in this section Still, the procedures are the same 10.13.4 Optimizing Motor Constant Km The thousands of applications that use a BLDC motor make it impossible to determine a simple criteria in selecting the best BLDC motor for a given application For motor designers, there is a single figure of merit that can determine if the motor design engineer has optimized the design in terms of magnetics and stator winding conditions The figure of merit (motor constant Km), along with the motor’s dimensions does provide a relatively simple but effective method of evaluating a BLDC motor design This also assumes that the proper magnet material has been selected, based on cost as well as dimensional requirements T Kt Km = ᎏᎏ = ᎏᎏ ͙Win ͙ෆ ෆ RLL (10.17) where T = any torque value Win = input power at the chosen torque value Kt = motor torque constant RLL = motor line-to-line resistance This constant, when maximized for a given motor volume, ensures the design engineer of the best possible design Many motor mechanical, magnetic, and electrical parameters are integrated into motor constant Km Maximizing Km is maximizing the motor’s copper to iron ratio for that specific motor size If one reviews the design process, the Kt value is primarily tied to magnet flux, number of turns, pole count, unit path reluctance, and so forth RLL is a function of available slot space, winding pattern, pole count, and so forth Optimizing both performance parameters optimizes the motor design in terms of the motor’s magnetic circuit and winding selection Remember that other user requirements, such as the lowest possible winding inductance, would bias the motor constant Km to very low numbers of turns (conductors) Minimizing rotor inertia for certain incremental motion applications would also bias the Km optimization process as described The Km value listed in Fig 10.114 is 40.17 (oz и in)/͙W One could significantly ෆ increase Km by the following means: q q Increasing the number of poles Increasing the rotor OD up to a certain OD dimension; then the Km value would decrease beyond this dimensional limit DRIVES AND CONTROLS q q 10.139 Changing to high-energy magnets Increasing motor OD or AL Usually the motor’s available input voltage and current levels will act as a limit to optimizing Km beyond a certain limit The challenge for the motor design engineer is to maximize Km within specified electric inputs or friction and inertia requirements which will yield the best overall magnetic design 10.13.5 Power, Losses, Efficiency, and Others There are two groups of motor performance parameters, the conventional and the special performance parameters A list of the conventional motor performance parameters includes those parameters or figures of merit that relate to motor power For example, Eq (10.18) determines dissipated power I2R, Eq (10.19) computes input power, Eq (10.20) calculates output power, and Eq (10.21) establishes the power efficiency at that specific load point PLD These parameters are designated as the conventional performance parameters and are computed as follows: Pdiss = ILD,RMS RLL (10.18) Pin = ET ILD,RMS (10.19) TLDNLD Pout = ᎏᎏ K (10.20) where K = 1352 for oz и in and rpm values K = for N и m and rad/s values P Efficiency = ᎏout ᎏ Pin (10.21) The measuring of motor voltage and currents has become much more difficult with the inverter and the power amplifier unable to be separated from the motor The VRMS and IRMS current values are the most important ones to measure and require special equipment In most design computations, a 100°C temperature rise value of RLL is used Computations of these values are shown in Fig 10.114, where a more optimistic room ambient RLL value is employed The more specialized figures of merit include motor characteristics such as theoretical acceleration T/J, peak power rate (PPR), rated power rate (RPR), peak power density (PPD), and rated power density (RPD) The first three figures of merit are used in incremental motion applications The other two are more often used in automotive, aerospace, and portable instrument applications Theoretical acceleration can be determined by Eq (10.22) Tacc = Jm α where α = acceleration, rad/s2 Tacc = Tpk Jm = the motor rotor inertia or T ᎏacc = α ᎏ JT (10.22) 10.140 CHAPTER TEN This equation can be modified in the inertia term (Jm) to include the load inertia JL for load matching conditions Power rate is defined as the rate of change of power with respect to time and is important in incremental motion applications Equations (10.23) and (10.24) detail the method of computation of peak and rated power rates (PPR and RPR) T2 I2 RLL pk PPR = ᎏpk = ᎏᎏ ᎏ Jm Γm (10.23) where Γm is the machine’s mechanical time constant, T2 I2 RLL R R RPR = ᎏᎏ = ᎏᎏ Jm Γm (10.24) Power rate can be expressed in kilowatts per second The BLDC motor with the highest power rate will produce the shortest time to move from one point to another Power density or power per unit volume has become popular in recent years with the advent of battery operated motors and actuators These figures of merit relate output power to the unit volume and, indirectly, unit weight Equations (10.25) and (10.26) show the simple formulas for computing these figures of merit Peak output power PPD = ᎏᎏᎏ unit volume (10.25) Rated output power RPD = ᎏᎏᎏ unit volume (10.26) The units are watts per cubic inch or watts per cubic meter The higher the value of power density, the more power per unit volume available ... 10.51n Kirchhoff’s law, 1.16, 1.91 in analysis of dc series motors, 4.14 in analysis of PMDC motors, 4.42 in analysis of single-phase motors, 6.13, 6.74 in analysis of three-phase motors, 10.107...Library of Congress Cataloging-in-Publication Data Handbook of small electric motors / William H Yeadon, editor in chief, Alan W Yeadon, associate editor p cm ISBN 0-07-072332-X Electric motors, ... E motors in PMDC motors, 4.45–4.46, 4.115 in polyphase motors, 6.100 in series dc and ac motors, 4.128–4.130 in single-phase motors, 6.56 in switched-reluctance motors, 5.102 in synchronous motors,