Rotor loss in permanent-magnet brushless AC machines

1612 IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 36, NO. 6, NOVEMBER/DECEMBER 2000 Rotor Loss in Permanent-Magnet Brushless AC Machines Kais Atallah, Member, IEEE, David Howe, Philip H. Mellor, and David A. Stone, Member, IEEE Abstract—The eddy-current loss in the permanent magnets of brushless ac machines is usually neglected, since the fundamental air-gap field usually rotates in synchronism with the rotor, and time harmonics in the current waveform and space harmonics in the winding distribution are generally small. However, an important category of brushless ac machine design is emerging in which the fundamental component of the stator MMF has fewer poles than the rotor, the torque being developed by a higher order MMF harmonic. The fundamental and lower order MMF harmonics can then give rise to significant rotor eddy currents. An analytical model is developed to predict rotor-induced eddy currents in such machines, and to quantify the effectiveness of circumferentially segmenting the permanent magnets in reducing the rotor loss. Index Terms—Brushless machines, eddy currents, losses. I. INTRODUCTION N PERMANENT-MAGNET brushless dc machines, the stator magnetic field rotates incrementally as the winding currents are sequentially commutated. This can lead to a significant eddy-current loss in the rotor magnets and core, particularly in forced-cooled, high electric loading machines, such as traction motors [1]. There has, therefore, been considerable research into the prediction of the rotor loss and its minimization, for which analytical techniques [2], [3] have been developed which can account for skin effect. However, these cannot model the effect of circumferentially segmenting the permanent magnets, which can be an effective method for reducing the eddy-current loss [4], which is not the case with axial segmentation [5]. In contrast, the rotor loss in brushless ac machines is usually considered to be negligible, since high-order time harmonics in the stator current waveform and space harmonics in the winding distribution are generally small. However, machine designs are emerging for which the fundamental component of the stator MMF has fewer poles than the permanent-magnet rotor, and in which the torque is developed by the interaction of a higher order stator MMF harmonic with the permanent magnets. By way of example, Fig. 1(a) shows a three-phase 36-slot machine in which the seventh harmonic of the six-pole fundamental Fig. 1. Permanent-magnet brushless ac machines with rotor pole number > fundamental stator MMF pole number. (a) Three-phase external rotor machine. (b) Six-phase fault-tolerant machine. Paper IPCSD 00–040, presented at the 1999 IEEE International Electric Machines and Drives Conference, Seattle, WA, May 9–12, and approved for publication in the IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS by the Electric Machines Committee of the IEEE Industry Applications Society. Manuscript submitted for review July 2, 1999 and released for publication June 19, 2000. The authors are with the Department of Electronic and Electrical Engineering, University of Sheffield, Sheffield, S1 3JD, U.K. (e-mail: [email protected]; [email protected]; p.h.mellor @sheffield.ac.uk; [email protected]). Publisher Item Identifier S 0093-9994(00)10433-5. MMF distribution interacts with the 42-pole rotor to generate the output torque. Such a design facilitates modularity and ease of manufacture, important considerations for cost-sensitive applications, such as traction drives [6]. Fig. 1(b) shows another example, a six-phase 12-slot machine, which has a two-pole fundamental stator MMF, the torque being developed by the interaction of the fifth space harmonic MMF with the ten-pole rotor. Since the phase windings are physically, magnetically, I (a) (b) 0093–9994/00$10.00 © 2000 IEEE ATALLAH et al.: ROTOR LOSS IN PERMANENT-MAGNET BRUSHLESS AC MACHINES 1613 TABLE I PARAMETERS OF BRUSHLESS AC MACHINES Fig. 2. Analytical model. (a) External rotor motor. (b) Internal rotor motor. and thermally isolated, its fault tolerance is significantly higher than that of conventional brushless machine designs, a major consideration for safety-critical applications [7], [8]. In both machines shown in Fig. 1, the torque-creating stator MMF harmonic results from the short chording of the stator winding. However, lower order MMF harmonics, including the fundamental, also exist which can be comparable in magnitude, and can, therefore, induce significant eddy currents in the surface-mounted rotor permanent magnets, which may result in excessive heating. An analytical method for predicting the eddycurrent loss in the magnets is, therefore, presented, and used to quantify the effectiveness of circumferentially segmenting the magnets as a means of reducing the loss. Fig. 3. Variation of eddy-current loss with rotor speed at rated current (one magnet segment per pole arc). (a) Three-phase machine. (b) Six-phase machine. II. ANALYTICAL MODEL The model is formulated in two-dimensional polar coordinates, and is applicable to both internal and external rotor motors. The stator and rotor cores are assumed to be infinitely permeable, while the permanent magnets have an arbitrary pole arc , Fig. 2. The eddy currents which are induced in the magnets by the low-order space harmonic MMFs are assumed to be resistance limited, since the magnets in the machines under consideration have a relatively high electrical resistivity and a low recoil permeability. Hence, the skin depth, at the inducing frequencies of interest, is greater than both the pole arc and the radial thickness of the magnets. The stator winding is represented by an equivalent current sheet, which for a -phase machine is given by for for for (1a) 1614 IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 36, NO. 6, NOVEMBER/DECEMBER 2000 Fig. 4. Variation of eddy-current loss with rotor speed at rated current (two magnet segments per pole arc). (a) Three-phase machine. (b) Six-phase machine. Fig. 5. Variation of eddy-current loss with rotor speed at rated current (four magnet segments per pole arc). (a) Three-phase machine. (b) Six-phase machine. or, with respect to the rotor reference system, by for for for (1b) , is the space harmonic order, and where are the fundamental number of pole pairs associated with the stator winding and rotor, respectively, is the rotor angular ve, where and are locity, and Fig. 6. Influence of number of magnet segments per pole arc on eddy-current loss. ATALLAH et al.: ROTOR LOSS IN PERMANENT-MAGNET BRUSHLESS AC MACHINES 1615 the number of series turns per phase and the peak phase curis the winding factor. The induced rent, respectively, and eddy-current density in the permanent-magnet arc segments is given by (2) where is the electrical resistivity of the permanent magnets, and is the magnetic vector potential distribution, given by the solution of Laplace’s equation (3) (7c) in which subject to the boundary conditions (4) and are the circumferential components of the flux where density and magnetic field, respectively, and is an integration constant which ensures that zero net total current flows in each permanent-magnet arc segment at any instant, i.e., (5) for (7d) for Similarly, for an internal rotor machine The induced eddy-current loss in one permanent-magnet segment of an external rotor machine is derived as (8) (6) where where (9a) (7a) in which in which for for for (7b) for (9b) 1616 IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 36, NO. 6, NOVEMBER/DECEMBER 2000 and TABLE II EDDY-CURRENT LOSS IN SINTERED NdFeB ROTOR MAGNETS THREE-PHASE MACHINE OF (9c) TABLE III IN SINTERED Sm Co OF SIX-PHASE MACHINE EDDY-CURRENT LOSS ROTOR MAGNETS in which for (9d) for , and “ ” to . It where “ ” applies to will be noted that for a rotor comprising a multipole magnetized , the term in (6) and (8) is cylindrical magnet, i.e., zero. III. EFFECT OF CIRCUMFERENTIALLY SEGMENTING MAGNETS The analysis has been applied to the two machine designs shown in Fig. 1, viz: a 42-pole 36-slot three-phase machine [Fig. 1(a)], and a ten-pole 12-slot six-phase machine [Fig. 1(b)]. Table I gives the parameters of both machines. Loss predictions from the analytical model have been compared to losses deduced from a series of two-dimensional time-stepped moving-boundary finite-element analyses, assuming the permanent magnets to be electrically conducting but to have zero remanence. Thus, they do not account for the eddy-current loss component caused by the variation of the magnet working point which results from stator slotting. Figs. 3–5 show the influence of the rotor speed on the eddycurrent loss at rated stator current, for one, two, and four insulated magnet segments per pole arc, respectively. It can be seen that, although the analytical model assumes resistance-limited eddy currents, good agreement is obtained. Fig. 6 shows the influence of the number of magnet segments per pole arc on the per-unit eddy-current loss. It can be seen that a significant reduction in loss can be achieved by increasing the number of segments per pole arc, albeit, for these particular machine designs, segmentation is more effective for the threephase machine. The use of from two to four segments per pole arc is practical, as well as very effective. Tables II and III show the effect of circumferentially segmenting the rotor magnets on the eddy-current loss associated with the different stator MMF space harmonics, for the three- Fig. 7. Variation of eddy-current loss with rotor speed at rated current (multipole magnetized ring magnet). (a) Three-phase machine. (b) Six-phase machine. ATALLAH et al.: ROTOR LOSS IN PERMANENT-MAGNET BRUSHLESS AC MACHINES 1617 been shown that circumferentially segmenting the magnets can be a very effective means of reducing the eddy-current loss when the rotor pole number fundamental stator MMF pole number. REFERENCES Fig. 8. Variation of ratio of eddy-current loss in a ring magnet to the eddy-current loss in 42 insulated segments for the three-phase machine (finite-element prediction) and six-phase machines, respectively. It can be seen that the effect of circumferentially segmenting the magnets on the eddycurrent loss is greatest for the lower order space harmonics, viz, first and fifth for the three-phase machine and first and seventh for the six-phase machine. Fig. 7 shows the variation of the rotor loss, at rated stator current, with rotor speed assuming the rotors to have a single multipole magnetized ring magnet. It can be seen that, in this case, the accuracy of the analytical model reduces significantly as the rotor speed is increased, since for the inducing frequencies under consideration, the skin depth of the permanent-magnet material is only a fraction of the first space harmonic pole pitch. The eddy currents, therefore, are not resistance limited, as, for example, in the one-magnet-segment-per-pole-arc three-phase machine, for which the pole arc is 12 mm, while the pole pitch of the first space harmonic is 111 mm. At 2000 r/min, for example, the inducing frequency of the first space harmonic is 600 Hz, at which the skin depth of the permanent-magnet material is 22 mm. Therefore, the developed analytical model is applicable only when the rotor has at least one magnet segment per pole arc, which is the case for the majority of brushless ac machines. Fig. 8 shows the variation of the ratio of the eddy-current loss in the rotor of the three-phase machine when the magnet is a single multipole magnetized ring to that in 42 insulated magnet segments, i.e., one magnet segment per pole arc. Clearly, for the design of brushless machines under consideration, multipole magnetized ring magnets are inappropriate, since the eddy-current loss is likely to be excessive. IV. CONCLUSIONS An analytical method for predicting the eddy-current loss induced in the permanent magnets of brushless ac machines by low-order stator MMF harmonics has been developed. It has [1] N. Schofield, K. Ng, Z. Q. Zhu, and D. Howe, “Parasitic rotor losses in a permanent magnet brushless traction machine,” in Proc. IEMDC’97, 1997, pp. 200–204. [2] A. K. Nakargatti, O. A. Mohammed, and N. A. Demerdash, “Special losses in rotors of electronically commutated brushless dc motors induced by nonuniformly rotating armature mmfs,” IEEE Trans. Power App. Syst., vol. PAS-101, pp. 4502–4506, Dec. 1982. [3] F. Deng, “Commutation-caused eddy-current losses in permanent magnet brushless dc motors,” IEEE Trans. Magn., vol. 33, pp. 4310–4318, Sept. 1997. [4] K. Yoshida, K. Kesamaru, and Y. Hita, “Eddy currents analysis of surface-mounted-PMSM by finite element method,” in Proc. ICEM’98, 1998, pp. 1821–1825. [5] J. L. Kirtley Jr., M. Tolikas, J. H. Lang, C. W. Ng, and R. Roche, “Rotor loss models for high speed PM motor-generators,” in Proc. ICEM’98, 1998, pp. 1832–1837. [6] D. Bauch-Panetzky, R. Sonnenburg, R. Zoller, and R. Stepputat, “Electric drives excited by sintered NdFeB magnets, especially designed for traction applications in passenger vehicles,” in Proc. ISATA’98, 1998, pp. 189–206. [7] B. C. Mecrow, A. G. Jack, J. A. Haylock, and J. Coles, “Fault-tolerant permanent magnet machine drives,” Proc. IEE—Elect. Power Applicat., vol. 143, no. 6, pp. 437–442, 1996. [8] G. M. Raimondi, R. D. McFarlane, C. M. Bingham, K. Atallah, D. Howe, P. H. Mellor, R. Capewell, and C. Whitley, “Large electromechanical actuation systems for flight control surfaces,” presented at the IEE Colloq. All-Electric Aircraft, London, U.K., 1998, Paper 7/1–7/6. Kais Atallah (M’98) was born in Constantine, Algeria, in 1964. He received the Ingenieur d’Etat degree in electrical power engineering from the Ecole Nationale Polytechnique, Algiers, Algeria, and the Ph.D. degree from the University of Sheffield, Sheffield, U.K. From 1993 to 2000, he was a Research Associate in the Department of Electronic and Electrical Engineering, University of Sheffield, where he is currently a Lecturer. His research interests include the modeling of iron losses in electrical machines, multipole impulse magnetization, and the design of novel topologies of permanent-magnet machines for automotive and aerospace applications. David Howe received the B.Tech. and M.Sc. degrees from the University of Bradford, Bradford, U.K., and the Ph.D. degree from the University of Southampton, Southampton, U.K., in 1966, 1967, and 1974, respectively, all in power engineering. He has held academic posts at Brunel and Southampton Universities, and spent a period in industry with NEI Parsons Ltd., working on electromagnetic problems related to turbogenerators. He is currently Lucas Professor of Electrical Engineering at the University of Sheffield, Sheffield, U.K., where he heads the Electrical Machines and Drives Research Group. His research activities span all facets of controlled electrical drive systems, with particular emphasis on permanent-magnet excited machines. He is the author of more than 200 publications in the fields of machines, drives, and motion control systems. Prof. Howe is a Chartered Engineer in the U.K. and a Fellow of the Institution of Electrical Engineers, U.K. 1618 IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 36, NO. 6, NOVEMBER/DECEMBER 2000 Philip H. Mellor was born in Rangoon, Burma, in 1957. He received the B.Eng. and Ph.D. degrees from the Department of Electrical Engineering, Liverpool University, Liverpool, U.K., in 1978 and 1981, respectively. From 1981 to 1984, he was with the GEC’s Hirst Research Center, Wembley, U.K., as a Project Leader in power electronic drives. In 1984, he joined Liverpool University as a Lecturer, where he researched the thermal modeling and power electronic control of induction and permanent-magnet machines. In 1990, he was appointed to the Electrical Machines and Drives Group, University of Sheffield, Sheffield,, U.K., where he is a Reader in electrical engineering. His research interests lie in electrically controlled rotary and linear motion for applications such as aircraft electrical actuation and traction systems for electric and hybrid vehicles. David A. Stone (M’90) received the B.Eng. degree in electronic engineering from the University of Sheffield, Sheffield, U.K., and the Ph.D. degree from Liverpool University, Liverpool, U.K., in 1984 and 1989, respectively. He is currently a Member of Academic Staff, University of Sheffield, specializing in power electronics and machine drive systems. His current research interests are hybrid electric vehicles, battery charging, EMC, and novel lamp ballasts for low-pressure fluorescent lamps.