Extended spherical harmonic expansion using multipolar bases of equivalent magnetic sources on arbitrary surfaces

Purpose The purpose of this paper is to propose a compact model to represent the magnetic field outside the sources. This model provides the multipolar ordering of a spherical harmonic expansion far from the source while being valid in its close proximity. Design/methodology/approach The authors investigate equivalent surface sources that enable to compute the field very close to any chosen surface that encloses the source. Then the authors present a method to find an appropriate initial basis and its associated inner product that allow to construct multipolar harmonic bases for these equivalent sources, where any vector of order k produces a field that decreases at least as fast as the field produced by a multipole of order k. Finally, those bases are numerically implemented to demonstrate their performances, both far from the source and in its close proximity. Findings The charge distribution and normal dipole distribution are well-suited to construct multipolar harmonic bases of equivalent sources. These bases can be described by as few parameters as the decreasing spherical harmonic expansion. Comparison with other numerical models shows its ability to compute the field both far from the source and close to it. Originality/value A basis for normal dipole distribution has already been described in the literature. This paper presents a general method to construct a multipolar basis for equivalent sources and uses it to construct a basis for single-layer potential.