IEEE TRANSACTIONS ON MAGNETICS, VOL. 42, NO. 10, OCTOBER 2006 3431 Torque Calculation in Finite Element Solutions of Electrical Machines by Consideration of Stored Energy David G. Dorrell, Mircea Popescu, and Malcolm I. McGilp Department of Electronics and Electrical Engineering, SPEED Laboratory, University of Glasgow, Glasgow, G12 8LT, U.K. This paper studies the calculation of torque in a permanent magnet motor finite element simulation using an energy difference method. It is found that this is a simple yet powerful method of torque calculation and allows both pulsating and average torque to be calculated. The method is compared to other techniques. Index Terms—Brushless permanent magnet motor, finite element analysis, torque calculation. I. INTRODUCTION T ORQUE calculation of a rotating machine in a finite ele- ment solution (FEA) may seem straightforward, however, here can be considerable error. The most common method for calculating torque uses a Maxwell stress tensor integral. How- ever, unless the model is well meshed, the results will be poor. Even with a good mesh, there are errors and Salon [1] recom- mends that several integral paths at different radii in the air gap is used and an average taken. Several methods exist to calculate the torque with the aim of minimizing the error and producing good calculation, and these were recently addressed in [2] and [3]. These varied in their approaches and use techniques, such as the virtual work and Maxwell stress filter methods. Many methods have been implemented in commercial software (e.g., PC-FEA from the SPEED Laboratory, University of Glasgow, Glasgow, U.K.). These techniques rely on a good mesh, which is one that exhibits geometrical symmetry in terms of the air gap, poles, and slots; automatic mesh generators now address this [4]. The air gap should also be multilayer and ideally, as rotation takes place, the shape of the elements should stay constant. Alterna- tively, an early paper [5] addresses a technique which uses the change in stored energy to calculate the torque, and this method is addressed in this paper. II. TEST MACHINE An example machine is used to test the method (Fig. 1) using PC-FEA. This is a brushless permanent-magnet six-pole three- phase machine. Two half-magnet pitches are modeled giving the minimum periodicity. In the paper, some of the results are compared to other methods already implemented in PC-FEA and tests conducted under different loadings. This machine is a known servo alternating current machine and its parameters are given in Table I. The stator teeth are shown in Fig. 1 and it can be seen that the tooth tips are split to reduce the cogging torque; the actual machine has one stator slot skew; however, here we will assume zero skew to produce cogging torque. Digital Object Identifier 10.1109/TMAG.2006.879067 TABLE I MOTOR PARAMETERS Fig. 1. Machine cross section. III. STORED ENERGY AND VIRTUAL WORK METHODS While intuitively, it may seem reasonable to reduce the step angle, many methods rely on differences in stored energy or 0018-9464/$20.00 © 2006 IEEE