Electromagnetic Modeling Of Permanent Magnets Machine Coupled To The Circuit Equation By Time Stepping Technique M. Ounnadi, R. Saraoui, N. Benamrouche Laboratoire des Technologies Avancées de Génie Electrique (LATAGE), Université Mouloud Mammeri de Tizi-Ouzou, Département délectrotechnique, BP 17 RP DZ 15000, Tizi Ouzou, Algéria Email : m.ounnadi@yahoo.fr Y. Boutora Université Mouloud Mammeri de Tizi-Ouzou, Faculté de Génie Electrique et Informatique; Département délectrotechnique, BP17 RP, Tizi Ouzou 15000, Algeria Abstract—In this paper, a method of modeling and numerical simulation of a permanent magnet motor using time-stepping finite-element technique is presented. In the proposed model, the electromagnetic field equations, the stator circuit equation are solved simultaneously at each ti me step. The motion equation is not considering in this paper, the step of displacement is constant. The movement is taken into account by the moving band method; and one of the recent efficient renumbering method which is suitable for cylindrical structure is applied for renumber the mesh induce by the finite element method. The method is adapted for taking into account the movement by the moving band method. The renumbering is done in the aim of minimizing the profile of the stiffness matrix then, CPU time. The resolution is made by block elimination schemes, thus the direct method is used and we benefice to the profile reduction. Keywords-component; Electrical Circuit; Finite element method; permanent magnet machine; profile reduction; moving band. I. INTRODUCTION Accurate modeling of elect rical machines requires simultaneous solution of the finite element method (FEM) equations and circuit equations of the windings [1, 2]. When electrical machines are connected to external circuits, additional circuit equations are also coupled into the system. There are two basic approaches to coupling the FEM with circuit equations. One is direct coupling where the field and circuit equations are coupled directly together and solved simultaneously. The other one is indirect coupling. In this case, the FEM and circuit simulator are treated as separate systems. They communicate with each other by means of coupling coefficients. This approach simplifies the resolution of the problem but is not always applicable in practice case. When the interaction between magnetic circuit and electric circuit become important, it is necessary to use the direct coupling. For this, this latest strategies is used in our work. In this paper a direct coupling between magnetic field equation and circuit equation is presented. The resolution of the global system is carried out with Euler method (backward difference type). An application on the permanent magnet machine is made for a constant speed. The results of the current in the stator phases are shown. A finite element program is elaborated in Fortran 90 language, the program is efficacious because it include a recent nodes renumbering method [3] and then we have used direct met hod; then the results was accurate and the execution time is reduced. II. COUPLING MAGNETIC AND ELECTRIC CIRCUIT A. Electromagnetic model The governing equation of the magnetic field in the permanent magnet machines is represented by Maxwell's equation in the form of a magnetic vector potential by [4]: ry rx B x B y S is t A y A y x A x ) / ( . . (1) Where is the magnetic reluctivity, A is the magnetic vector potential, J is the exciting current density, is the electric conductivity; S is / is the density of stator current; rx B et ry B are the x and y component of the remanence flux density r B of the permanent magnet. The approximation function of magnetic vector potentiel in each finite element is given by: i i i A y x A 3 1 , (2) Where, i is the element shape functions and i A the ith element of A. The Galerkine formulation of the equation (1) leeds finaly to the matrix system (3): 0 ). / ( . . . . G Q S is A t T A K (3) Where : K . is the stiffness matrix, A is the unknown vector, T . the conductance matrix, (i s/S), [Q] : the source vector due to the currents and G the source matrix due to the permanent magnets. International Conference on Information Processing and Electrical Engineering ICIPEE12 April 14-16, 2012 p66 University of Tebessa Science and Technologies Faculty Electrical Engineering Department Route de Constantine, 12002 Tebessa, Algerie www.univ-tebessa.dz genieelectrique@mail.univ-tebessa.dz