A Two-Dimensional Numerical Solution of Transient Magnetic Flux Distribution in Electric Machines Considering Magnetic Saturation and Hysteresis

Abstract

In recent electric machines, it is needed to analyse the transient behaviors of the magnetic fluxes which relate closely to the dynamic characteristics. In this paper, first, there are introduced two-dimensional fundamental equations with respect to the vector potential from the Maxwell equations in the electro-magnetic field. Here, the vector potential is used to easily obtain both the transient magnetic flux and the eddy current distributions in the electric machines. In order to solve numerically the above fundamental equations, the nodal equation at each node of the triangular meshes, by which the region to be analysed is subdivided, is obtained by a new nodal method, which is an improvement over the conventional nodal method by using any shape of the triangle. Next, the time derivative of the vector potential in the nodal equation, which is needed to calculate the eddy current, is approximated by the Crank-Nikolson finite difference method. Also, the authors show a numerical method for deriving the nonlinear reciprocal permeabilities from the magnetization and the hysteresis curves of the structural iron cores of the electric machines, in which the Frölich formula is used. Furthermore, from every nodal equation with the above approximated time derivative and the boundary conditions, the final fundamental nonlinear simultaneous nodal equations are derived. Lastly, the calculation processes to solve numerically the nodal equations are shown.