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.

The emergence of second-generation high temperature superconducting (HTS) tapes has favored the development of large-scale superconductor systems. The mathematical models capable of estimating electromagnetic quantities in superconductors have evolved from simple analytical models to complex numerical models. The available analytical models are limited to the analysis of single wires or infinite arrays that, in general, do not represent actual devices in real applications. The numerical models based on the finite element method using the H formulation of Maxwell’s equations are useful for the analysis of medium-size systems, but their application in large-scale systems is problematic due to the excessive computational cost in terms of memory and computation time. Therefore it is necessary to devise new strategies to make the computation more efficient. The homogenization and the multi-scale methods have successfully simplified the description of the systems allowing the study of large-scale systems. Also, efficient calculations have been recently achieved using the T–A formulation. In the present work, we propose a series of adaptations to the multi-scale and homogenization methods so that they can be efficiently used in conjunction with the T–A formulation to compute the distribution of current density and hysteresis losses in the superconducting layer of superconducting tapes. The computation time and the amount of memory are substantially reduced up to a point that it is possible to achieve real-time simulations of large-scale HTS systems under slow ramping cycles of practical importance on personal computers.

A Conductor on Round Core (CORC ^® ) cable wound with a high temperature superconductor is an important cable concept for high current density applications. The design of a CORC cable makes understanding its electromagnetic performance—for example its AC losses—challenging. This paper presents a thorough study of CORC cables by combining experimental and numerical methods. In particular, it focuses on understanding how the cable structure influences the magnetization losses and on how these can be reduced. A novelty of this paper lies in the use of a new T-A formulation, which, for the first time, is employed for three-dimensional modelling of a CORC cable with real geometry. The use of the new T-A formulation in finite element software enables the study of how the winding direction and multiple-layer structure affect the magnetization losses of CORC cables. Moreover, influence of striation in CORC cables is studied as an effective way to reduce their losses. A CORC cable with striated tapes shows a significant magnetization loss reduction at high magnetic fields, in comparison to its counterpart without striated tapes. At low magnetic fields, tape striation leads to an increase in loss when the number of filaments is low, then the loss drops with a further increase in the number of filaments, but this loss reduction is much weaker than that at high fields. This paper provides an efficient tool for investigating the electromagnetic behaviour of CORC cables, which can provide valuable guidance in designing CORC cables with minimized losses for high energy physics and energy conversion applications.

In recent years, the T-A formulation has emerged as an efficient approach for modelling the electromagnetic behaviour of high-temperature superconductor (HTS) tapes in the form of coated conductors (CCs). HTS CCs are characterized by an extremely large width-to-thickness ratio of the superconducting layer, normally up to 1000 ∼ 6000, which in general leads to a very large number of degrees of freedom. The T-A formulation considers the superconducting layer to be infinitely thin. The magnetic vector potential A is used to calculate the magnetic field distribution in all simulated domains. The current vector potential T is used to calculate the current density in the superconducting layer, which is a material simulated with a highly nonlinear power-law resistivity. This article presents a review of the T-A formulation. First, the governing equations are described in detail for different cases (2D and 3D, cartesian and cylindrical coordinates). Then, the literature on the implementation of T-A formulation for simulating applications ranging from simple tape assemblies to high field magnets is reviewed. Advantages and disadvantages of this approach are also discussed.

Numerical models are powerful tools to predict the electromagnetic behavior of superconductors. In recent years, a variety of models have been successfully developed to simulate high-temperature-superconducting (HTS) coated conductor tapes. While the models work well for the simulation of individual tapes or relatively small assemblies, their direct applicability to devices involving hundreds or thousands of tapes, e.g., coils used in electrical machines, is questionable. Indeed, the simulation time and memory requirement can quickly become prohibitive. In this paper, we develop and compare two different models for simulating realistic HTS devices composed of a large number of tapes: (1) the homogenized model simulates the coil using an equivalent anisotropic homogeneous bulk with specifically developed current constraints to account for the fact that each turn carries the same current; (2) the multi-scale model parallelizes and reduces the computational problem by simulating only several individual tapes at significant positions of the coil’s cross-section using appropriate boundary conditions to account for the field generated by the neighboring turns. Both methods are used to simulate a coil made of 2000 tapes, and compared against the widely used H-formulation finite-element model that includes all the tapes. Both approaches allow faster simulations of large number of HTS tapes by 1–3 orders of magnitudes, while maintaining good accuracy of the results. Both models can therefore be used to design and optimize large-scale HTS devices. This study provides key advancement with respect to previous versions of both models. The homogenized model is extended from simple stacks to large arrays of tapes. For the multi-scale model, the importance of the choice of the current distribution used to generate the background field is underlined; the error in ac loss estimation resulting from the most obvious choice of starting from a uniform current distribution is revealed.