A new method for the calculation of magnetic fields

Abstract

Abstract Magnetic fields in systems with unsaturated yokes can be favourably calculated by means of the boundary element method. This method consists in the formulation and numerical solution of integral equations for the potentials or field sources on the material surfaces. The different formulations of boundary value problems are finally always reduced to the task of solving a Fredholm equation with logarithmically singular integral kernel. A new technique is developed for the efficient and fast solution of such problems. It combines the Galerkin method of approximation — using interpolation kernels or modified Lagrange polynomials as test functions — with a partly analytical evaluation of the integral kernel. This technique, so far demonstrated for electrostatics, can now also be applied to magnetic systems with finite permeability of the yoke material. The feasibility of the method is demonstrated for some unconventional forms of magnetic lenses and for deflection systems. A further generalization is the field calculation in configurations of several round electrodes or pole pieces which are not aligned on a common optic axis but have arbitrary positions and orientations in space. The field calculation can be performed by an iterative solution of a sequence of coupled two-dimensional problems, which is much faster than a general three-dimensional procedure. Moreover, a hybrid technique for the field calculation in systems with saturated yokes, and finally a technique for studying small deviations from symmetries are briefly sketched.