Electromagnetic modelling of superconductors with a smooth current–voltage relation: variational principle and coils from a few turns to large magnets

Abstract

Many large-scale applications require electromagnetic modelling with\nextensive numerical computations, such as magnets or 3-dimensional (3D) objects\nlike transposed conductors or motors and generators. Therefore, it is necessary\nto develop computationally time-efficient but still accurate numerical methods.\nThis article develops a general variational formalism for any ${\\bf E}({\\bf\nJ})$ relation and applies it to model coated-conductor coils containing up to\nthousands of turns, taking magnetization currents fully into account. The\nvariational principle, valid for any 3D situation, restricts the computations\nto the sample volume, reducing the computation time. However, no additional\nmagnetic materials interacting with the superconductor are taken directly into\naccount. Regarding the coil modelling, we use a power law $E(J)$ relation with\nmagnetic field-dependent critical current density, $J_c$, and power law\nexponent, $n$. We test the numerical model by comparing the results to\nanalytical formulas for thin strips and experiments for stacks of pancake\ncoils, finding a very good agreement. Afterwards, we model a magnet-size coil\nof 4000 turns (stack of 20 pancake coils of 200 turns each). We found that the\nAC loss is mainly due to magnetization currents. We also found that for an $n$\nexponent of 20, the magnetization currents are greatly suppressed after 1 hour\nrelaxation. In addition, in coated conductor coils magnetization currents have\nan important impact on the generated magnetic field; which should be taken into\naccount for magnet design. In conclusion, the presented numerical method\nfulfills the requirements for electromagnetic design of coated conductor\nwindings.\n