Reduced models of self-force, stored energy, and critical current for design optimization of electromagnets

To design magnetic fusion facilities, the Lorentz force on coils and magnetic energy must be computed to ensure a support structure is feasible. Also, the magnetic field inside the conductor is needed to compute the superconducting critical current. For these force, energy, and internal field calculations, the coils cannot be simply approximated as infinitesimally thin filaments due to divergences when the source and evaluation points coincide, so more computationally demanding calculations are usually required, resolving the finite cross-section of the conductors. Here, a new alternative method enables these self-field quantities to be computed rapidly and accurately within a 1D filament model. The method exploits an expansion in small conductor width compared to other macroscopic length-scales associated with the coil such as radii of curvature. We also assume a rectangular conductor cross-section, building on work by Hurwitz for coils with circular cross-section. In our method, the reduced integrals are regularized such that they rigorously match the true high-dimensional integrals. The new filament model exactly recovers analytic results for a circular coil, and accurately reproduces high-fidelity finite-cross-section calculations for a stellarator coil. Due to the efficiency of the model here, it is well suited for use inside design optimization.