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When modeling superconducting devices based on REBCO tapes andworking near or above the critical current value (i.e. I> Ic), the power-law model is not always accurate. In our previous works, we proposed the overcritical current model, based on a combination of fast pulsed current measurements and finite elementan alysis. The overcritical current model was provided in the form of look-up tables and was validated experimentally. We showed that the overcritical current model could better reproduce experimental measurements than the power-law model, and that the power-law model predicts a faster quench than the overcritical current model. In this contribution, we propose the eta-beta model, a mathematical expression to model analytically the overcritical current model, based on measurements performed between 77 K and 90 K in self-field conditions. The proposed model is verified by comparing DC fault measurements with the results of numerical simulations using the eta-beta model to represent the electrical resistivity of the superconducting layer of REBCO tapes., 20 pages, 5 figures, 2 tables, journal paper

Although the H-formulation has proven to be one of the most versatile formulations used to accurately model superconductors in the finite element method, the use of vector dependent variables in non-conducting regions leads to unnecessarily long computation times. Additionally, in some applications of interest, the combination of multiple magnetic components interacting with superconducting bulks and/or tapes leads to large domains of simulation. In this work, we separate the magnetic field into a source and reaction field and use the H-$��$ formulation to efficiently simulate a superconductor surrounded by magnetic bodies. We model a superconducting cube between a pair of Helmholtz coils and a permanent magnet levitating above a superconducting pellet. In both cases, we find excellent agreement with the H-formulation, while the computation times are reduced by factors of nearly three and four in 2-D and 3-D, respectively. Finally, we show that the H-$��$ formulation is more accurate and efficient than the H-A formulation in 2-D.

The estimation of losses in high-temperature superconductors (HTS) is fundamental during the design of superconducting devices since losses can strongly influence the cooling system requirements and operating temperature. Typically, 2D finite element analysis is used to calculate AC losses in HTS, due to the lack of analytical solutions that can accurately represent complex operating conditions such as AC transport current and AC external applied magnetic field in coils. These 2D models are usually a representation of an infinitely long arrangement. Therefore, they cannot be used to analyze end effects and complex 3D configurations. In this publication, we use the T-A homogenization in 3D for the analysis of superconducting coils. This allows simulating complex geometries such as racetrack coils. We show that this approach has lower computation time than the currently available 3D homogenization of the H formulation. The approach is first validated against measurements and 2D axisymmetric solutions. Then, it is used to estimate losses and study the electromagnetic behavior of a racetrack coil.

The development of the high-temperature superconductors (HTS) conductors has allowed the development of diverse superconductor devices. Some of these devises, like the power generators and high-field magnets, are classified as large-scale HTS systems, because they are made of hundreds or thousands of turns. Mathematical models are required to address the analysis of these kind of systems. This task cannot be done by means of analytical models, because they are limited to the analysis of simple assemblies. The finite-element models using the H formulation have been extensively used during the last years. Nevertheless, the use of H formulation models to analyze large-scale systems is hindered by the excessive computational load. The recently proposed T-A formulation models have allowed building more efficient models for systems made of HTS tapes. Additionally, the homogenization and multi-scaling methods have been successfully applied in conjunction with the H and T-A formulations, these simplification methods allows reducing the required computational resources. In this article a new simplification method, called densification, is proposed. The strategies emerging from the combined use of the formulations and the simplification methods already mentioned are extensively explored, and the comprehensive validation and comparison of all the resulting strategies is presented.

The H-formulation, used abundantly for the simulation of high temperature superconductors, has shown to be a very versatile and easily implementable way of modeling electromagnetic phenomena involving superconducting materials. However, the simulation of a full vector field in current-free domains unnecessarily adds degrees of freedom to the model, thereby increasing computation times. In this contribution, we implement the well known H-$��$ formulation in COMSOL Multiphysics in order to compare the numerical performance of the H and H-$��$ formulations in the context of computing the magnetization of bulk superconductors. We show that the H-$��$ formulation can reduce the number of degrees of freedom and computation times by nearly a factor of two for a given relative error. The accuracy of the magnetic fields obtained with both formulations are demonstrated to be similar. The computational benefits of the H-$��$ formulation are shown to far outweigh the added complexity of its implementation, especially in 3-D. Finally, we identify the ideal element orders for both H and H-$��$ formulations to be quartic in 2-D and cubic in 3-D, corresponding to the highest element orders implementable in COMSOL.

The emergence of second-generation high temperature superconducting 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 real devices in real applications. The numerical models based on finite element method using the H formulation of the Maxwells 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. Then 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 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 HTS large-scale systems under slow ramping cycles of practical importance on personal computers.

The emergence of second-generation high temperature superconducting tapes has favored the development of large-scale superconductor systems. The mathematical models for superconductors have evolved from simple analytical models to complex numerical models. The available analytical models are just capable of estimating electromagnetic quantities in single wires or small arrays. The numerical models like the H formulation finite element models are useful for the analysis of more medium-size systems, but their application in large-scale systems is problematic due to the excessive cost in terms of memory and the computation time. Then it is necessary to devise new strategies to tackle this challenge. The homogenization and the multi-scale approach are methodologies that have successfully simplified the description of the systems allowing the study of large-scale systems. Recently, efficient calculations have been achieved using the T-A formulation. In this manuscript, we analyze the adequate order of the elements used in the T-A formulation. More importantly, we propose a series of adaptations to the homogenization and multi-scale methodologies in order to allow their implementation to the T-A formulation models. The proposed strategies are validated by means of comparing them with the H reference model of a racetrack coil. The computation time and memory demand are substantially reduced up to a point that makes it possible to perform real-time simulations for slow ramping cycles. At the same time the comparison demonstrates a very good agreement with respect to the reference model, both at a local and global level, is achieved., Comment: arXiv admin note: substantial text overlap with arXiv:1811.09193

In recent years, commercial HTS superconductors have gained an increasing interest for their use in applications involving large-scale superconductor systems. These systems are typically made from hundreds to thousands of turns of conductors. Due to the large number of turns, the simulations of a whole system can become prohibitive in terms of computing time and load. Therefore, an efficient strategy which does not compromise the accuracy of calculations is needed. Recently, a method, based on a multi-scale approach, showed that the computational load can be lowered by simulating, in detail, only several significant tapes from the system. The main limitation of this approach is the inaccuracy of the estimation of the background magnetic field. To address this issue, we consider the following two complementary strategies. The first strategy consists in the iterative implementation of the multi-scale method. The multi-scale method solves itself a dynamic problem, the iterative implementation proposed here is the iterative application of the multi-scale method, and a dynamic solution is obtained at each iteration. The second strategy is a new interpolation method for current distributions, based on the inverse cumulative density function interpolation technique. With respect to conventional interpolation methods, a more realistic current density distribution is then obtained, which allows for a better estimation of the background magnetic field, and consequently, a better estimation of the hysteresis losses. In contrast with previous works, here we do not focus only on the estimation of the hysteresis losses, but also the estimation of background field and the current density distribution is addressed.

The ability of stacks of superconducting tapes to trap large magnetic fields makes them ideal candidates for creating powerful permanent magnets of compact size and mass. Experimentally, several techniques are used to trap the maximum possible field in a given practical application. However, regardless of the magnetization method used, there is a physical limit to the maximum magnetic field that a given superconducting magnet can trap. This limit is given by the geometric design, the particular superconducting material used and the temperature of operation. Knowing the maximum possible trapped field is important for device design as it provides an upper limit for applications such as magnetic bearings or rotating machinery. In this work we present a collection of finite element method (FEM) models in 2D and 3D capable of estimating the maximum trapped field of stacked tape superconducting magnets. The models are computationally fast and can be used to perform parametric studies with ease. For the case of square stacks tape magnets, various sizes are considered and their estimated maximum trapped field is compared with experimental results.

The estimation of the critical current ($I_c$) and AC losses of high-temperature superconductor (HTS) devices through modeling and simulation requires the knowledge of the critical current density ($J_c$) of the superconducting material. This $J_c$ is in general not constant and depends both on the magnitude ($B_{loc}$) and the direction ($��$, relative to the tape) of the local magnetic flux density. In principle, $J_c(B_{loc},��)$ can be obtained from the experimentally measured critical current $I_c(B_a,��)$, where $B_a$ is the magnitude of the applied magnetic field. However, for applications where the superconducting materials experience a local field that is close to the self field of an isolated conductor, obtaining $J_c(B_{loc},��)$ form $I_c(B_a,��)$ is not a trivial task. It is necessary to solve an inverse problem to correct for the contribution derived from the self field. The methods presented in the literature comprise a series of approaches dealing with different degrees of mathematical regularization, such as the use of brute force or optimization methods to fit the parameters of preconceived non linear formulas. In this contribution, we present a parameter-free method that provides excellent reproduction of experimental data and requires no human interaction or preconception of the $J_c$ dependence with respect to the magnetic field. In particular, it allows going from the experimental data to a ready-to-run $J_c(B_{loc},��)$ model in a few minutes.

High temperature superconducting (HTS) bulks can be magnetized to become powerful trapped field magnets (TFMs), which are promising for high-performance electrical applications. To magnetize such TFMs, pulsed field magnetization (PFM) is supposed to substitute field cooling (FC) to provide in situ magnetization. However, the heat generation during PFM, which reduces the trapped field has always been an issue, so numerical simulation of the process is important to provide optimal magnetization strategies. In this paper, HTS bulks magnetized by PFM are simulated with an axisymmetric electromagnetic-thermal coupled model based on H-formulation. Influences of important yet difficult-to-characterize parameters of HTS bulks including n values in the E-J Power Law and B0 in the Kim's Law are investigated. Furthermore, controlled magnetic density distribution coils (CMDCs) which generate a non-uniform field are suggested to further improve the trapped field of HTS bulks compared to split coils proposed previously.

Roebel cables made of HTS coated conductors are regarded as promising cables for winding applications in virtue of their large engineering current density and low losses. The composing meander-shaped strands are assembled very tightly into the cable, which results in a strong electromagnetic interaction between them. This interaction profoundly influences the effective self-field critical current ($I_c$) of the cable, which is much lower than the sum of the $I_c$s of the composing strands. Also the AC losses are influenced by the material's properties and by the geometrical configuration of the cable. Being able to predict the effective critical current and AC losses of such cables is very important for a proper design of applications: due to the complexity of the cable's geometry and of the material's properties, this prediction can only be performed with advanced numerical tools. In this contribution we use finite-element-based models to compute the effective $I_c$ and the AC losses of Roebel cables composed of 31 strands using tapes from two manufacturers. The AC losses are analyzed in the simultaneous presence of transport current and background perpendicular field proportional to the current, which mirrors the situation occurring in a winding. Our models include the angular dependence of $J_c(B,\theta)$ at~77 K, which is very different for the two materials. By means of a successful comparison of the simulation results to experimental data obtained with a calorimetric method measuring the evaporation of liquid nitrogen, this work confirms the applicability and efficiency of our numerical techniques for simulating the electromagnetic behavior of Roebel cables and devices thereof.

Different magnet applications require compact high current cables. Among the proposed solutions, the Twisted Stacked Tape Cable (TSTC) is easy to manufacture and has very high tape length usage efficiency. In this kind of cables the tapes are closely packed, so that their electromagnetic interaction is very strong and determines the overall performance of the cable. Numerical models are necessary tools to precisely evaluate this interaction and to predict the cable's behavior, e.g. in terms of effective critical current and magnetization currents. For this purpose, we developed a fully three-dimensional model of a TSTC, which not only takes into account the twisted geometry of these cables, but is also able to account for the contact resistances of the current terminations. The latter can have profound influence on the way the current is partitioned among the tapes, especially on short laboratory prototypes. In this paper, we first use the numerical model to compute the critical current and the magnetization AC loss of a twisted tape, showing the differences with the case of a straight tape. Then, we use it to calculate the current distribution in a TSTC cable, comparing the results with those experimentally obtained on a cable composed of four straight stacked tapes. The results show the ability of the model to simulate twisted conductors and constitutes a first step toward the simulation of TSTC in high-field magnet applications. The presented modeling approach is not restricted to the TSTC geometry, but may be used for any cable configuration with periodical translational symmetry.

Numerical models for computing the effective critical current of devices made of HTS tapes require the knowledge of the Jc(B,theta) dependence, i.e. of the way the critical current density Jc depends on the magnetic flux density B and its orientation theta with respect to the tape. In this paper we present a numerical model based on the critical state with angular field dependence of Jc to extract the Jc(B,theta) relation from experimental data. The model takes into account the self-field created by the tape, which gives an important contribution when the field applied in the experiments is low. The same model can also be used to compute the effective critical current of devices composed of electromagnetically interacting tapes. Three examples are considered here: two differently current rated Roebel cables composed of REBCO coated conductors and a power cable prototype composed of Bi-2223 tapes. The critical currents computed with the numerical model show good agreement with the measured ones. The simulations reveal also that several parameter sets in the Jc(B,theta) give an equally good representation of the experimental characterization of the tapes and that the measured Ic values of cables are subjected to the influence of experimental conditions, such as Ic degradation due to the manufacturing and assembling process and non-uniformity of the tape properties. These two aspects make the determination of a very precise Jc(B,theta) expression probably unnecessary, as long as that expression is able to reproduce the main features of the angular dependence. The easiness of use of this model, which can be straightforwardly implemented in finite-element programs able to solve static electromagnetic problems, is very attractive both for researchers and devices manufactures who want to characterize superconducting tapes and calculate the effective critical current of superconducting devices.

In this paper, we present a general review of the status of numerical modelling applied to the design of high temperature superconductor (HTS) devices. The importance of this tool is emphasized at the beginning of the paper, followed by formal definitions of the notions of models, numerical methods and numerical models. The state-of-the-art models are listed, and the main limitations of existing numerical models are reported. Those limitations are shown to concern two aspects: one the one hand, the numerical performance (i.e. speed) of the methods themselves is not good enough yet; on the other hand, the availability of model file templates, material data and benchmark problems is clearly insufficient. Paths for improving those elements are provided in the paper. Besides the technical aspects of the research to be further pursued, for instance in adaptive numerical methods, most recommendations command for an increased collective effort for sharing files, data, codes and their documentation.

The current density and field distributions in polygonally arranged thin superconducting tapes carrying AC current are derived under the assumption of the critical state model. Starting from the generic Biot-Savart law for a general polygonal geometry, we derive a suitable integral equation for the calculation of the current density and magnetic field in each tape. The model works for any transport current below $I_c$, which makes it attractive for studying cases of practical interest, particularly the dependence of AC losses on parameters such as the number of tapes, their distance from the center, and their separation.

With this paper we provide an effective method to solve a large class of problems related to the electromagnetic behavior of thin superconductors. Here all the problems are reduced to finding the weight functions for the Green integrals that represent the magnetic field components; these latter must satisfy the mixed boundary value conditions that naturally arise from the critical state assumptions. The use of the Erd��lyi-Kober operators and of the Hankel transforms (and mostly the employment of their composition properties) is the keystone to unify the method toward the solution. In fact, the procedure consists always of the same steps and does not require any peculiar invention. For this reason the method, here presented in detail for the simplest cases that can be handled in analytical way (two parts boundary), can be directly extended to many other more complex geometries (three or more parts), which usually will require a numerical treatment. In this paper we use the operator technique to derive the current density and field distributions in perfectly conducting and superconducting thin discs and tapes subjected to a uniform magnetic field or carrying a transport current. Although analytical expressions for the field and current distributions have already been found by other authors in the past by using several other methods, their derivation is often cumbersome or missing key details, which makes it difficult for the reader to fully understand the derivation of the analytical formulas and, more importantly, to extend the same methods to solve similar new problems. On the contrary, the characterization of these cases as mixed boundary conditions has the advantage of referring to an immediate and na��ve translation of physics into a consistent mathematical formulation whose possible extension to other cases is self-evident., The final publication is available at http://link.springer.com/article/10.1007/s00033-011-0185-5

The AC loss in ReBCO coated conductors is large in situations when the conductors are subjected to a considerable magnetic field, like in rotating machines, transformers and high-field magnets. Roebel cables can reduce the AC loss in these cases. However, computer simulations are needed to interpret the experiments, understand the loss mechanisms, reduce the AC loss by optimising the Roebel cable and design the cryogenic system. In this article, we simulate and discuss the AC loss due to an applied magnetic field making an arbitrary angle with the cable and taking into account a realistic anisotropic field dependence of the critical current density. We study the AC loss in the superconductor parts for the limits of very high coupling currents and completely uncoupled strands. The simulations for the uncoupled case also describe a double pancake coil with no transport current. For the simulations, we use two different numerical methods with complementary strengths. This serves as a mutual check of the correctness of the simulation results, which agree to each other. Opposite than expected, we found that the AC loss does not only depend on the perpendicular component of the applied magnetic field. We also found that the AC loss for applied fields with an orientation below 7 degrees with the strands surface is reduced more than one order of magnitude comparing to an untransposed cable. Therefore, we recommend to use Roebel cables for windings with important parallel components of the magnetic field, such as transformers and high-field magnets., Comment: 28 pages, 11 figures

The high-Tc 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. 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 (DOFs), tolerance settings and the approximate time taken per cycle for each model., 22 pages, 7 figures, 2 tables

Intrinsically stable magnetic levitation between superconductors and permanent magnets can be exploited in a variety of applications of great technical interest in the field of transportation (rail transportation), energy (flywheels) and industry. In this contribution, we present a new model for the calculation of levitation forces between superconducting bulks and permanent magnet, based on the $H$-formulation of Maxwell's equations coupled with an Arbitrary Lagrangian-Eulerian formulation. The model uses a moving mesh that adapts at each time step based on the time-change of the distance between a superconductor bulk and a permanent magnet. The model is validated against a fixed mesh model (recently in turn validated against experiments) that uses an analytical approach for calculating the magnetic field generated by the moving permanent magnet. Then, it is used to analyze the magnetic field dynamics both in field-cooled and zero-field-cooled conditions and successively used to test different configurations of permanent magnets and to compare them in terms of levitation forces. The easiness of implementation of this model and its flexibility in handling different geometries, material properties, and application scenarios make the model an attractive tool for the analysis and optimization of magnetic levitation-based applications.

The design of a high-temperature superconducting (HTS) Superconducting Magnetic Energy Storage (SMES) coil with solenoidal and toroidal geometry is carried out based on a commercially available second-generation HTS conductor. An SMES system of practical interest (1 MW/5 s) is considered. The comparison between ideal toroidal and solenoidal geometry is first discussed, and the criteria used for choosing the geometrical parameters of the coils' bore are explained. The design of the real coil is then carried out, and the final amount of conductor needed is compared. A preliminary comparison of the two coils in terms of ac loss during one charge and discharge cycle is also discussed.