1460 IEEE TRANSACTIONS ON MAGNETICS, VOL. 41, NO. 5, MAY2005 Saturation and Space Harmonics in the Complex Finite Element Computation of Induction Motors S. Mezani , B. Laporte , and N. Takorabet Department of Electronic and Electrical Engineering, University of Sheffield, Sheffield S1 3JD, U.K. GREEN-ENSEM-INPL, 54516 Vandoeuvre, France The authors present a method to obtain steady state performances of induction motors including space harmonics and saturation. The principle of the method consists on a spectral decomposition of the source field in the air gap, to compute each elementary problem corresponding to a harmonic order and then to use superposition to determine the total field. The saturation is included in an average sense using effective reluctivities. Computations are compared to experiments. Index Terms—Eddy currents, finite element methods, induction motors, moving bodies. I. INTRODUCTION T HE computation of induction machines using time step- ping techniques has achieved a good level of accuracy [1], [2]. These methods are unfortunately time consuming, espe- cially when only the steady state performances, under sinusoidal excitation, are needed. To compute the steady state operation, the use of the com- plex representation is very attractive. However, this method is strictly valid only in the linear case, so its adaptation in the sat- urated case is a difficult task. Since the torque of induction mo- tors depends essentially on the velocity, the movement modeling represents another difficulty [3]. In this paper, saturation and movement are considered in the complex finite element modeling of induction motors. Spatial harmonics are accounted for using superposition principle that leads to Alger’s chain equivalent circuit [4]. II. ELECTROMAGNETIC MODELING In a two-dimensional (2-D) approximation, we can use a mag- netic vector potential (mvp) having only one component in the axial direction, which depends on the plan coordinates. Two do- mains and are considered (Fig. 1). The radii of the circular boundaries and are respec- tively and . Thus, the mvp has two determinations, noted in and in . Introducing the rotor angular velocity , the coordinates of the points and are related by (1) In and , and satisfy in (2) in (3) In the above equations, represents the magnetic reluctivity, the electric conductivity and the slot’s current density. Digital Object Identifier 10.1109/TMAG.2005.844554 Fig. 1. Stator and rotor domains. In addition, and must verify on (4) on (5) The air gap is included both in and ; the mvp is then calculated twice in where it satisfies Laplace’s equa- tion. If the relations (4) and (5) were strictly verified, these two determinations would be coincident. We propose in this paper to ensure the coincidence, until a certain order, of the Fourier series expansion of these relations. We assume that the machine is supplied from a balanced three-phase sinusoidal system of currents and only one time pul- sation is present in the source currents. These currents, when flowing through a three-phase balanced winding, produce in the air gap rotating fields. If we assume the linearity of the ferro- magnetic materials, superposition can be used. The following expansion for the mvp can be written: on (6) on (7) where, according to (1), . and are phasors and stands for “real part of.” 0018-9464/$20.00 © 2005 IEEE