Efficient Calculation of the Self-Magnetic Field, Self-Force, and Self-Inductance for Electromagnetic Coils

The design of electromagnetic coils may require evaluation of several quantities that are challenging to compute numerically. These quantities include Lorentz forces, which may be a limiting factor due to stresses; the internal magnetic field, which is relevant for determining stress as well as a superconducting coil’s proximity to its quench limit; and the inductance, which determines stored magnetic energy and dynamics. When computing the effect on one coil due to the current in another, these quantities can often be approximated quickly by treating the coils as infinitesimally thin. When computing the effect on a coil due to its own current (e.g., self-force or self-inductance), evaluation is difficult due to the presence of a singularity; coils cannot be treated as infinitesimally thin as each quantity diverges at zero conductor width. Here, we present novel and well-behaved methods for evaluating these quantities using non-singular integral formulas of reduced dimensions. These formulas are determined rigorously by dividing the domain of integration of the magnetic vector potential into two regions, exploiting appropriate approximations in each region and expanding in a high aspect ratio. Our formulas show good agreement to full finite-thickness calculations even at low aspect ratio, both analytically for a torus and numerically for a non-planar coil of a stellarator fusion device, the helically symmetric experiment (HSX). Because the integrands of these formulas develop fine structure as the minor radius becomes infinitely thin, we also develop a method of evaluating the self-force and self-inductance with even greater efficiency by integrating this sharp feature analytically. We demonstrate with this method that the self-force can be accurately computed for the HSX coil with as few as 12 grid points. Additionally, we demonstrate that this method offers a significant speed-up compared to industry standard finite-element analysis (FEA) software: for the HSX coil, it is possible to compute the magnetic field in a fraction of a millisecond per point with this method, while FEAs require an order of an hour to evaluate across an entire mesh to a similar accuracy.