Numerical modeling of superconductors is widely recognized as a powerful tool for interpreting experimental results, understanding physical mechanisms and predicting the performance of high-temperature superconductor (HTS) tapes, wires and devices. This is especially true for ac loss calculation, since a sufficiently low ac loss value is imperative to make these materials attractive for commercialization. In recent years, a large variety of numerical models, based on different techniques and implementations, have been proposed by researchers around the world, with the purpose of being able to estimate ac losses in HTSs quickly and accurately. This article presents a literature review of the methods for computing ac losses in HTS tapes, wires and devices. Technical superconductors have a relatively complex geometry (filaments, which might be twisted or transposed, or layers) and consist of different materials. As a result, different loss contributions exist. In this paper, we describe the ways of computing such loss contributions, which include hysteresis losses, eddy current losses, coupling losses, and losses in ferromagnetic materials. We also provide an estimation of the losses occurring in a variety of power applications.

Energy applications employing high-temperature superconductors (HTS), such as motors/generators, transformers, transmission lines and fault current limiters, are usually operated in the alternate current (AC) regime. In order to be efficient, the HTS devices need to have a sufficiently low value of AC loss, in addition to the necessary current-carrying capacity. Most applications are operated with currents beyond the current capacity of single conductors and consequently require cabled conductor solutions with much higher current carrying capacity, from a few kA to up to 20-30 kA for large hydro-generators.A century ago, in 1914, Ludwig Roebel invented a low-loss cable design for copper cables, which was successively named after him. The main idea behind Roebel cables is to separate the current in different strands and to provide a full transposition of the strands along the cable direction. Nowadays, these cables are commonly used in the stator of large generators. Based on the same design concept of their conventional material counterparts, HTS Roebel cables from REBCO coated conductors were first manufactured at the Karlsruhe Institute of Technology (KIT) and have been successively developed in a number of varieties that provide all the required technical features such as fully transposed strands, high transport currents and low AC losses, yet retaining enough flexibility for a specific cable design. In the past few years a large number of scientific papers have been published on the concept, manufacturing and characterization of such cables. Times are therefore mature for a review of those results. The goal is to provide an overview and a succinct and easy-to-consult guide for users, developers, and manufacturers of this kind of HTS cables.

Many large-scale applications require electromagnetic modelling with extensive numerical computations, such as magnets or 3-dimensional (3D) objects like transposed conductors or motors and generators. Therefore, it is necessary to develop computationally time-efficient but still accurate numerical methods. This article develops a general variational formalism for any E(J) relation and applies it to model coated-conductor coils containing up to thousands of turns, taking magnetization currents fully into account. The variational principle, valid for any 3D situation, restricts the computations to the sample volume, reducing the computation time. However, no additional magnetic materials interacting with the superconductor are taken directly into account. Regarding the coil modelling, we use a power law E(J) relation with magnetic field-dependent critical current density, Jc, and power law exponent, n. We test the numerical model by comparing the results to analytical formulas for thin strips and experiments for stacks of pancake coils, finding a very good agreement. Afterwards, we model a magnet-size coil of 4000 turns (stack of 20 pancake coils of 200 turns each). We found that the AC loss is mainly due to magnetization currents. We also found that for an n exponent of 20, the magnetization currents are greatly suppressed after 1 hour relaxation. In addition, in coated conductor coils magnetization currents have an important impact on the generated magnetic field; which should be taken into account for magnet design. In conclusion, the presented numerical method fulfills the requirements for electromagnetic design of coated conductor windings. * Final version published as E Pardo, JŠouc and L Frolek 2015 Supercond. Sci. Technol. 28 044003, doi:10.1088/0953-2048/28/4/044003. Several minor typos have been corrected in the published version. 1 ReBCO stands for ReBa 2 Cu 3 O 7−x , where Re is a rare earth, typically Y, Gd or Sm. the flux creep exponent and the AC loss. Critial current density and flux-creep exponentIn this article, we use a ReBCO coated conductor tape from SuperPower [49] for all experiments. This tape is 4 mm wide, with a total of 40 µm copper stabilizer layers, a 1 µm thick superconducting layer, and a self-field critical current at 77 K of 128 A.We measured the dependence of the critical current density J c on the magnetic field magnitude |B| ≡ B and its orientation θ (see sketch in figure 3) at 77 K, as detailed in [50]. In order to extract J c from measurements of the tape critical current, I c , we corrected the spurious effects of the self-field, following the method in [50]. The reader can find the I c measurements and extracted J c for the tape used in this article in [30]. For completeness, we include the extracted J c (B, θ) relation, being J c (B, θ, J) = [J c,ab (B, θ, J) m + J c,c (B) m ] 1/m (60) with J c,ab (B, θ, J) = J 0,ab 1 + Bf (θ,J) B 0,ab β ab ,J c,c (B) = J 0,c

Many superconductor applications require cables with a high current capacity. This is not feasible with single-piece coated conductors because their ac losses are too large. Therefore, it is necessary to develop superconducting cables with a high current capacity and low ac losses. One promising solution is given by ROEBEL cables. We assembled three ROEBEL cables from commercial YBCO coated conductors. The cables have the same width but a different number of strands, which results in different aspect ratios and current capacities. We experimentally studied their ac losses under a transport current or a perpendicular magnetic field. In addition, we performed numerical calculations, which agree with the experiments, especially for the transport case. We found that in the cables there is good current sharing between the strands. We also found that stacking the strands reduces the magnetization losses. For a given critical current, thicker cables have lower magnetization ac losses. In addition, a conducting matrix is not required for a good current sharing between strands.

The current distribution, the magnetic field and the AC loss are numerically studied for pancake coils with up to 200 turns, taking into account the real thickness of the conductor and the magnetic interaction between the turns. All calculations assume the critical state model with a constant critical current density. The results show a slab-like current distribution, with the absence of magnetization currents due to the radial magnetic field. Consequently, longitudinal striation, with or without twisting, cannot reduce the self-field AC loss of this system. Finally, an analytical formula has been deduced for the AC loss at current amplitude: I m = I c , where I c is the critical current, for large mean coil radius and any number of turns.

Computing the electric eddy currents in non-linear materials, such as superconductors, is not straightforward. The design of superconducting magnets and power applications needs electromagnetic computer modeling, being in many cases a three-dimensional (3D) problem. Since 3D problems require high computing times, novel time-efficient modeling tools are highly desirable. This article presents a novel computing modeling method based on a variational principle. The self-programmed implementation uses an original minimization method, which divides the sample into sectors. This speeds-up the computations with no loss of accuracy, while enabling efficient parallelization. This method could also be applied to model transients in linear materials or networks of non-linear electrical elements. As example, we analyze the magnetization currents of a cubic superconductor. This 3D situation remains unknown, in spite of the fact that it is often met in material characterization and bulk applications. We found that below the penetration field and in part of the sample, current flux lines are not rectangular and significantly bend in the direction parallel to the applied field. In conclusion, the presented numerical method is able to time-efficiently solve fully 3D situations without loss of accuracy.

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The high-T c superconducting (HTS) dynamo is a promising device that can inject large DC supercurrents into a closed superconducting circuit. This is particularly attractive to energise HTS coils in NMR/MRI magnets and superconducting rotating machines without the need for connection to a power supply via current leads. It is only very recently that quantitatively accurate, predictive models have been developed which are capable of analysing HTS dynamos and explain their underlying physical mechanism. In this work, we propose to use the HTS dynamo as a new benchmark problem for the HTS modelling community. The benchmark geometry consists of a permanent magnet rotating past a stationary HTS coated-conductor wire in the open-circuit configuration, assuming for simplicity the 2D (infinitely long) case. Despite this geometric simplicity the solution is complex, comprising time-varying spatially-inhomogeneous currents and fields throughout the superconducting volume. In this work, this benchmark problem has been implemented using several different methods, including H-formulation-based methods, coupled H-A and T-A formulations, the Minimum Electromagnetic Entropy Production method, and integral equation and volume integral equation-based equivalent circuit methods. Each of these approaches show excellent qualitative and quantitative agreement for the open-circuit equivalent instantaneous voltage and the cumulative time-averaged equivalent voltage, as well as the current density and electric field distributions within the HTS wire at key positions during the magnet transit. Finally, a critical analysis and comparison of each of the modelling frameworks is presented, based on the following key metrics: number of mesh elements in the HTS wire, total number of mesh elements in the model, number of degrees of freedom, tolerance settings and the approximate time taken per cycle for each model. This benchmark and the results contained herein provide researchers with a suitable framework to validate, compare and optimise their own methods for modelling the HTS dynamo.