IEEE TRANSACTIONS ON MAGNETICS, VOL. 48, NO. 2, FEBRUARY 2012 1015 Efficient Iterative Integral Technique for Computation of Fields in Electric Machines with Rotor Eccentricity Ioan R. Ciric , Fellow, IEEE, Florea I. Hantila , Mihai Maricaru , and Stelian Marinescu Department of Electrical and Computer Engineering, The University of Manitoba, Winnipeg, MB R3T 5V6, Canada Department of Electrical Engineering, Politehnica University of Bucharest, Spl. Independentei 313, Bucharest, 060042, Romania ICPE SA, Spl. Unirii 313, Bucharest, 030138, Romania To analyze the effects of small variations of the electric machine airgaps due to rotor eccentricity it is necessary to compute the magnetic field highly accurately. An efficient iterative integral technique is proposed, where the material nonlinearity is treated by the polariza- tion method, with the magnetic field determined at each iteration by superposing the contributions of the given electric currents and the polarizations. This computation technique has great advantages over the finite element based procedures, namely, the change in the rotor position does not require the construction of a new discretization mesh, very small airgaps can be taken into consideration without increasing the amount of computation, and the calculated magnetic field in the air is divergenceless and curlless, thus eliminating the in- troduction of spurious forces. As well, the phase voltages and the magnetic forces are easily calculated from the magnetic field quantities. Index Terms—Electric machines, iterative methods, nonlinear field computation, rotor eccentricity. I. INTRODUCTION O PERATING parameters of electric machines can strongly be influenced by the nonuniformity of the airgap, which is usually due to the rotor eccentricity. Since the variations in the airgap value are normally very small, the analysis of the corresponding magnetic field must be performed by employing highly accurate computational methods. Recently, the finite el- ement method has been used for this purpose [1]–[4], but the machine rotation and the choice of the time step, in the pres- ence of rotor eccentricity, require elaborate procedures for the construction of the discretization mesh. On the other hand, when following this approach one introduces spurious forces on the mesh element boundaries, even in the air. To take into consid- eration approximately the small modifications of the airgap, at- tempts have been made [5]–[7] to apply analytic methods, but the nonlinearity of the ferromagnetic material is not taken into account. In the procedure proposed in this paper, the nonlinearity of the magnetic field equations is dealt with by the iterative po- larization fixed point method [8], [9], the solution of the linear magnetic field problem at each iteration being obtained by using a Green function for the unbounded homogeneous space. This approach has previously been used for electric machines with uniform airgaps [10], where the pole periodicity is exploited and an associated periodic Green function is employed. Such a tech- nique cannot be applied in the case of a nonuniform airgap when the magnetic field is not periodic. In the present paper, the re- duction of the number of different entries into the coupling ma- trix, in the presence of rotor eccentricity, is achieved by using the fact that the coupling coefficient tensors are symmetric and Manuscript received June 29, 2011; accepted October 16, 2011. Date of cur- rent version January 25, 2012. Corresponding author: M. Maricaru (e-mail: mm@elth.pub.ro). Digital Object Identifier 10.1109/TMAG.2011.2173752 practically all of them have zero trace, as well as the stator tooth periodicity and the rotor pole periodicity. The coupling matrix is, in fact, constructed from three distinct submatrices, corre- sponding to the rotor-rotor, stator-stator and rotor-stator cou- plings, the first two being computed only once, whereas only the last one is corrected at each time step. II. FIELD MODELING IN NONLINEAR MEDIA The nonlinear characteristic of the ferromagnetic material is replaced by , where is a constant, and the nonlinearity is taken into account through the polarization in the form . As shown in [11], [9], the constant can be chosen such that the function is a contraction. Applying the polarization fixed point method, we start with an arbitrary polarization and then, at each iteration , the magnetic field is determined from the linear equations (1) which have a unique solution for any polarization , i.e., , where the function is nonexpansive. The polarization for the next iteration is corrected with . This yields a Picard-Banach sequence for the compo- sition of and , which is a contraction, and, thus, the method is always convergent. To increase the rate of convergence of the iterative process, an overrelaxation technique is used, the overrelaxation coefficients being between 5.5 and 16 for the field problems considered in this paper. The errors with respect to the exact solution can be monitored at each iteration step [9]. This nonlinear field problem is to be solved in a highly efficient maner at each time step within a period of revolution of the machine. III. THE LINEAR FIELD PROBLEM (1) A. Solution Procedure If an optimum local value for is chosen, then, in general, the fictitious material involved is nonhomogeneous and, usually, a 0018-9464/$31.00 © 2012 IEEE