Efficient calculation of the self-inductance, self-force, and internal magnetic field for thin electromagnetic coils

There exist several quantities of conducting coils that are difficult to numerically evaluate yet are relevant to the design and optimization of magnetic confinement fusion reactors. Lorentz forces are a limiting factor in reactor design due to coil stresses. A conductor's internal magnetic field also describes stress and strain as well as a superconducting coil's proximity to its quench limit. Its self-inductance measures magnetic energy. When computed between coils (e.g., mutual forces), these quantities are simple to evaluate, though when computed on a single coil (e.g., self-force), evaluation is difficult due to a source point singularity. Importantly, attempts to treat coils as infinitesimally thin fail as the self-inductance, internal magnetic field, and self-force all diverge. Instead, we present novel models for these quantities using non-singular integral formulae of reduced dimensions. These formulae were determined rigorously by dividing the domain of integration of the magnetic vector potential into two regions, exploiting the unique assumptions of each region, and expanding in high aspect ratio. Our formulae show good agreement to the full calculations under the high aspect ratio limit, both analytically for a torus and numerically for HSX stellarator coils.