Calculation of mutual inductance of two coaxial thick coils with rectangular cross section by using cylindrical multipole expansion

A triple power series expansion for mutual inductance of two coaxial thick coils with rectangular cross section is obtained. Previously developed calculation methods consist of either semi-analytical methods containing numerical calculations or exact expressions containing integrals that cannot be analytically evaluated. In contrast, the triple series expansion obtained in this paper is a complete and purely analytical expression - a triple power series. The triple sum contains one free parameter that is restricted by necessary and sufficient convergence conditions. To apply the method in practice the sums are truncated, with the choice of parameter significantly impacting convergence properties and obtained precision. A theoretical analysis and empirical tests were carried out to devise a heuristic algorithm for finding a good value for the free parameter. The method's effectiveness and very high performance was verified by comparing it to the relevant literature. Especially good results are achieved in the far field case with sub-microsecond execution times and more than 10 significant digits of precision.