This article presents a review of the finite-element method (FEM) model based on the H formulation of Maxwell’s equations used to calculate AC losses in high-temperature superconductor (HTS) tapes, cables and windings for different applications. This model, which uses the components of the magnetic field as state variables, has been gaining a great popularity and has been in use in tens of research groups around the world. This contribution first reviews the equations on which the model is based and their implementation in FEM programs for different cases, such as 2D longitudinal and axis-symmetric geometries and 3D geometries. Modeling strategies to tackle large number of HTS tapes, such as multi-scale and homogenization methods, are also introduced. Then, the second part of the article reviews the applications for which the H formulation has been used to calculate AC losses, ranging from individual tapes to complex cables and large magnet windings. Afterwards, a section is dedicated to the discussion of the H formulation in terms of computational efficiency and easiness of implementation. Its pros and cons are listed. Finally, the last section draws the main conclusions.

This paper reviews the modelling of high-temperature superconductors (HTS) using the finite-element method (FEM) based on the H-formulation of Maxwell’s equations. This formulation has become the most popular numerical modelling method for simulating the electromagnetic behaviour of HTS, especially thanks to the easiness of implementation in the commercial finite-element program COMSOL Multiphysics. Numerous studies prove that the H-formulation is able to simulate a wide scope of HTS topologies, from simple geometries such as HTS tapes and coils, to more complex HTS devices, up to large superconducting magnets. In this paper, we review the basics of the H-formulation, its evolution from 2D to 3D, its application for calculating critical currents and AC losses as well as magnetization of HTS bulks and tape stacks. We also review the use of the H-formulation for large-scale HTS applications, its use to solve multi-physics problems involving electromagnetic-thermal and electromagnetic-mechanical couplings, and its application to study the dynamic resistance of superconductors and flux pumps.

High-T_c Superconducting (HTS) magnets operating in persistent current mode suffer a current decay due to flux creep of superconductor and joint resistance. Flux pumps are able to inject direct current into superconducting circuit to compensate the current decay, without the thermal loss caused by current leads. In this work, we proposed a flux pumping mechanism for HTS coils, with an experimental verification and an analytical model. The basic principle we have used is that flux flow can be triggered when the superconductor carrying a direct current is subjected to a perpendicular AC magnetic field. Low frequency alternating current is induced in a loop of YBCO tape using an AC field. A portion of the tape which we refer to as the “bridge” shorts a superconducting coil. A high frequency AC field is applied perpendicular to the bridge tape when alternating current in the tape reaches one polarity. This triggers a net flux flow and results in a current increase in the coil. The proposed flux pump has clear physics and is easily controllable, which may make it promising in practical use.

This paper presents the modeling of second generation (2 G) high-temperature superconducting (HTS) pancake coils using finite element method. The axial symmetric model can be used to calculate current and magnetic field distribution inside the coil. The anisotropic characteristics of 2 G tapes are included in the model by direct interpolation. The model is validated by comparing to experimental results. We use the model to study critical currents of 2 G coils and find that 100 μV/m is too high a criterion to determine long-term operating current of the coils, because the innermost turns of a coil will, due to the effect of local magnetic field, reach their critical current much earlier than outer turns. Our modeling shows that an average voltage criterion of 20 μV/m over the coil corresponds to the point at which the innermost turns’ electric field exceeds 100 μV/m. So 20 μV/m is suggested to be the critical current criterion of the HTS coil. The influence of background field on the coil critical current is also studied in the paper.