734 IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 39, NO. 3, MAY/JUNE 2003 Modeling of Iron Losses of Permanent-Magnet Synchronous Motors Chunting Mi, Member, IEEE, Gordon R. Slemon, Life Fellow, IEEE, and Richard Bonert, Member, IEEE Abstract—Permanent-magnet (PM) motors offer potential en- ergy savings as compared with induction motors because of the vir- tual elimination of rotor loss and the reduction of stator loss from operation near unity power factor. In PM machines, iron losses form a significant fraction of the total loss partly due to the non- sinusoidal flux density distribution. Design optimization therefore requires good means of predicting these iron losses. Finite-element analysis can be employed but this approach is cumbersome and costly when used in the many iterations needed in optimizing the design. This paper presents a set of improved approximate models for the prediction of iron loss. They can be used in design optimiza- tion programs and, since they are directly related to machine di- mensions and material properties, they also provide quick insight into the effects of design changes. A time-stepped finite-element method is employed to evaluate the iron losses in a range of typ- ical PM machines and the results are used to evaluate the adequacy of the models. The predictions of overall iron losses are then com- pared with measurements made on two PM motors. Index Terms—Core losses, eddy currents, hysteresis, iron losses, permanent magnets (PMs), permanent-magnet (PM) machines, synchronous motors. I. INTRODUCTION P ERMANENT-MAGNET (PM) motors are challenging the monopoly of induction machines in many applications such as pumps, fans, and compressor drives where the higher initial cost can be rapidly paid back by energy savings [1]. In PM motors, iron losses form a larger proportion of the total losses than is usual in induction machines. This is partly due to the elimination of significant rotor slip loss and is partly due to the reduction of stator loss from operation at near-unity power factor. Optimum design of PM motors therefore requires good means for predicting these iron losses [2]. It is accepted that finite-element analysis can produce a good estimate of iron losses but this approach is cumbersome and costly when used in the many iterations needed in the optimizing design. This paper presents a set of improved approximate models for the prediction of iron losses of surfaced-mounted PM motors. An earlier paper [3] developed expressions for iron losses of surface-mounted PM motors based on a number of approxima- Paper IPCSD 03–009, presented at the 2001 Industry Applications Society Annual Meeting, Chicago, IL, September 30–October 5, and approved for pub- lication in the IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS by the Electric Machines Committee of the IEEE Industry Applications Society. Manuscript submitted for review June 19, 2002 and released for publication January 23, 2003. C. Mi is with the Department of Electrical and Computer Engineering, Uni- versity of Michigan, Dearborn, MI 48128-1491 USA (e-mail: mi@ieee.org). G. R. Slemon and R. Bonert are with the Department of Electrical and Computer Engineering, University of Toronto, Toronto, ON M5S 3G4, Canada (e-mail: g.slemon@utoronto.ca; r.bonert@utoronto.ca). Digital Object Identifier 10.1109/TIA.2003.810635 tions. This simple analytical iron-loss model has been used by a few authors to determine the iron losses of PM synchronous motors [4]–[8]. While the overall application of these approxi- mations produced an acceptable prediction of the total measured losses in an experimental machine, the validity of each approx- imation remained in doubt. In this paper, these approximations are examined in turn and their results are compared with those obtained by finite-element analysis. II. IRON LOSS DENSITY Measurements of iron losses in magnetic material are tradi- tionally made with sinusoidal flux density of varying frequency and magnitude. The total iron-loss density is commonly expressed in the following form for sinusoidally varying mag- netic flux density with angular frequency : W/m (1) where and are the hysteresis and the eddy-current loss density, respectively, and are hysteresis and eddy cur- rent constants, and is the Steinmetz constant, all of which depend on the lamination material. These constants can be ob- tained by curve fitting from manufacturer’s data. Typical values for grades of silicon iron laminations used in small and medium induction motors, with the stator frequency given in radians per second, are in the ranges – , – , and – . An expression for the classical eddy-current loss can also be developed based on the resistivity of the core material [14]. The result is generally found less than that obtained from the second term of (1). The iron-loss expression in (1) is only valid for sinusoidal flux density. In most PM motors, the variation in flux density in the stator core is far from sinusoidal. In this situation, while the hys- teresis loss is still easy to evaluate as it depends only on the peak value of the flux density assuming that there are no minor hys- teresis loops, the eddy-current losses evaluated using only the fundamental component of flux density may be much lower than the measured values [9]. An alternative approach which includes the harmonics of the flux density can be employed [10]–[13], but this approach involves the complex evaluation of these har- monics in each finite element of the machine. For the eddy current it is convenient to represent the average loss density as a function of the time rate of change of the vector flux density [3], [14] W/m (2) 0093-9994/03$17.00 © 2003 IEEE