Calculation of the multipole coefficients of 2-dimensional magnetic fields from the spectrum analysis computed by FLUX2D

Two-dimensional fields in regions with constant material properties (magnetic permeability, dielectric permittivity, heat conductivity, etc.) can be derived from a complex potential function W, where the real and imaginary parts correspond to the scalar potential function F and the stream function Y respectively. In source-free regions (no currents, electric charges, heat sinks, etc.), the field, is conservative and its complex potential W is a harmonic function where F and Y satisfy the Laplace equation. In such regions W can be expressed as a power series, the coefficients of which being the, multipole components of the field with respect to the origin of the series expansion. This note describes how these multipole coefficients are related to the amplitudes and phase angles as computed by, FLUX2D in the spectrum analysis of the magnetic vector potential. In the case of electrical or thermal problems, these same considerations apply by analogy.