This paper overviews the computation of the alternating current (AC) loss in high temperature superconductors (HTS) based on the finite-element method (FEM) using H-Formulation. A great amount of studies shows that the AC loss calculation results from H-Formulation agree well with the experiment results. As one of the most reliable numerical modeling methods, H-Formulation is able to calculate the AC loss from a wide range of HTS topologies. In this paper, we review these contributions on the AC loss calculation using H-Formulation, from small-scale HTS tapes and coils, HTS cables, to large-scale HTS applications.

This paper reviews the modeling of high-temperature superconductors (HTS) using the finiteelement method (FEM) based on the H-formulation of Maxwell's equations. This formulation has become the most popular numerical modeling method for simulating the electromagnetic behavior 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. INDEX TERMSReview, H-formulation, high temperature superconductor (HTS), finite-element method (FEM).

Superconducting flux pumps are the kind of devices which can generate direct current into superconducting circuit using external magnetic field. The key point is how to induce a DC voltage across the superconducting load by AC fields. Giaever [1] pointed out flux motion in superconductors will induce a DC voltage, and demonstrated a rectifier model which depended on breaking superconductivity. Klundert et al. [2, 3] in their review(s) described various configurations for flux pumps all of which relied on inducing the normal state in at least part of the superconductor. In this letter, following their work, we reveal that a variation in the resistivity of type II superconductors is sufficient to induce a DC voltage in flux pumps and it is not necessary to break superconductivity. This variation in resistivity is due to the fact that flux flow is influenced by current density, field intensity, and field rate of change. We propose a general circuit analogy for travelling wave flux pumps, and provide a mathematical analysis to explain the DC voltage. Several existing superconducting flux pumps which rely on the use of a travelling magnetic wave can be explained using the analysis enclosed. This work can also throw light on the design and optimization of flux pumps.

This paper presents a comprehensive AC loss study of a circular HTS coil. The AC losses from a circular double pancake coil were measured using the electrical method. A 2D axisymmetric H-formulation model using FEM package COMSOL Multiphysics has been established, which was able to make consistency with the real circular coil used in the experiment. To model a circular HTS coil, a 2D axisymmetric model provided better accuracy than a general 2D model, and was also more efficient than a 3D model. Three scenarios have been analysed: Scenario 1 AC transport current and DC magnetic field (experiment and simulation); Scenario 2 DC transport current and AC magnetic field (simulation); Scenario 3 AC transport current and AC magnetic field (simulation and experimental data support). The angular dependence analysis on the coil under the magnetic field with the different orientation angle  has been carried out for all three scenarios. For Scenario 3, we investigated the effect of relative phase difference ∆ between AC current and AC field on the total AC loss of the coil. To summarise, we have carried out a current/field/angle/phase dependent AC loss (I, B, , ∆) study of circular HTS coil, which could potentially benefit the future design and research of HTS AC systems.

High-T c Superconducting (HTS) flux pumps are capable of injecting flux into a superconducting circuit, which can achieve persistent current operation for HTS magnets. In this paper, we studied the operation of a rectifier-type HTS flux pump. The flux pump employs a transformer to generate high alternating current in its secondary winding which is connected to an HTS load shorted by an HTS bridge. A high frequency AC field is intermittently applied perpendicular to the bridge, thus generating flux flow. The dynamic resistance caused by the flux flow "rectifies" the secondary current, resulting in a direct current in the load. We have found that the final load current can be easily controlled by changing the phase difference between the secondary current and bridge field. Bridge field of frequency ranging 10Hz-40Hz, magnitude ranging 0-0.66T was tested. Flux pumping was observed for field magnitude of 50mT or above. We have found that both higher field magnitude and higher field frequency result in a faster pumping speed and a higher final load current. This can be attributed to the influence of dynamic resistance. The dynamic resistance measured in the flux pump is comparable with the theoretical calculation. The experimental results fully support a first order circuit model. The flux pump is much more controllable than travelling wave flux pumps based on permanent magnets, which makes it promising for practical use.