New compact and singularity free formulations for the magnetic field produced by a finite cylinder considering linearly varying current density

Abstract

This paper presents new compact analytical expressions for the magnetic field calculation produced by a finite cylindrical sheet. A linearly varying surface current density between the ends of the cylindrical sheet has been assumed as current source. The expressions presented in this work are substantially more compact if compared with the ones currently available in the literature. Since the solutions are given in terms of complete elliptic integrals of the first, second and third kind, and the last two ones diverge at certain arguments, the field expressions also diverge at these critical points, even though the field solution is finite. These singular points are located at the ends of the current sheet cylinder. New analytical expressions are also presented for these critical points avoiding the singularities in an elegant way. The radial component of the magnetic field strength is separately discussed, and it has been proven that its solution at singular points diverges. To improve the computational performance on the evaluation of magnetic field at critical points, an alternative formulation is presented, where complete elliptic integrals of the second and third kind are replaced with alternative functions. Numerical algorithms for the computation of these functions are also presented.