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Conductors made of high-temperature (HTS) wires helically wound in one or more layers on round tubes (CORT) are compact, flexible, and can carry a large amount of current. Although these conductors were initially developed for DC applications, e.g. in magnets, it is worth considering their use for AC, e.g. in underground cables for medium voltage grids and with currents in the kA-range. In these cases, the major challenge is reducing AC losses. In contrast to a straight superconducting wire, in a helical arrangement, due to superconducting shielding, the current does not follow the direction of the wires, but takes a non-trivial zig-zag path within the individual HTS wires (Garber pattern). This includes current components across the thickness of the superconducting layers, so that the often used thin-shell approximation does not hold. In this contribution, we studied a one-layer three-wire CORT by means of fully three-dimensional simulations, based on the H-formulation of Maxwell's equations implemented in the commercial software package COMSOL Multiphysics. As a result of our simulations, the peculiar current profiles were confirmed. In addition, the influence of current, pitch angle, and frequency on the AC losses was studied. We found an optimum for the pitch angle and that the current profiles strongly depend on frequency.

The modeling and analysis of superconducting coils is an essential task in the design stage of most devices based on high-temperature superconductors (HTS). These calculations allow verifying basic estimations and assumptions, proposing improvements, and computing quantities that are not easy to calculate with an analytical approach. For instance, the estimation of losses in HTS is fundamental during the design stage 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 homogenization of the T-A formulation in 3D for the analysis of superconducting coils with complex geometries where a 2D approach can not provide accurate analyses and verification of assumptions. The modeling methodology allows an easier implementation in commercial software (COMSOL Multiphysics) in comparison with the currently available 3D H homogenization, despite the complexity of the geometry. This methodology is first validated with a racetrack coil (benchmark case) by comparing the results with the well-established H formulation. Then, the electromagnetic behavior of coils with more complex geometries is analyzed., Comment: arXiv admin note: text overlap with arXiv:2103.06590

Numerical models based on the finite-element method (FEM) are popular tools for investigating the macroscopic electromagnetic behavior of high-temperature superconductor (HTS) applications. This article explains how to use the $T$-$A$ formulation for modeling HTS coils in 2D with different coupling scenarios between the turns. First we consider a racetrack coil wound from one piece of superconducting tape. Then we consider a coil obtained by winding a cable composed of different HTS tapes. In the latter case, the tape turns are either electrically connected along their entire length or just at the two ends of the coil: in the model, these two different types of electrical connection are implemented with the help of the electrical circuit module. The current density distributions and the AC losses of the coils in the different coupling scenarios are compared and discussed. The limits of applicability of the presented approach are pointed out. The model is developed for the straight section of racetrack coils, but can be easily adapted to axisymmetric geometries.

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., Comment: 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-$\phi$ 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-$\phi$ 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.

One of the main tasks during the design of a superconducting electrical machine is the estimation of losses in the superconducting coils. These losses can be decisive in such applications since they influence the cooling power requirements and the overall efficiency of the machine. In this publication, we focus on the dissipation in the stator superconducting coils of a synchronous machine for a wind turbine application. The T-A formulation of Maxwell's equations is used in a 2D finite element model to analyse the behaviour of the magnetic field around the coils and calculate losses. Particular attention is given to the position of the coils inside a slot and several coil configurations are presented. It is shown that certain coil arrangements lead to a significantly lower total loss, a more uniform loss distribution, which ultimately leads to the possibility of increasing the operating temperature.

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-$\phi$ formulation in COMSOL Multiphysics in order to compare the numerical performance of the H and H-$\phi$ formulations in the context of computing the magnetization of bulk superconductors. We show that the H-$\phi$ 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-$\phi$ 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-$\phi$ formulations to be quartic in 2-D and cubic in 3-D, corresponding to the highest element orders implementable in COMSOL.

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., Comment: 22 pages, 7 figures, 2 tables

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.

In recent years, the $H$ formulation of Maxwell's equation has become the de facto standard for simulating the time-dependent electromagnetic behavior of superconducting applications with commercial software. However, there are cases where other formulations are desirable, for example for modeling superconducting turns in electrical machines or situations where the superconductor is better described by the critical state than by a power-law resistivity. In order to accurately and efficiently handle those situations, here we consider two published approaches based on the magnetic vector potential: the $T$-$A$ formulation of Maxwell's equations (with power-law resistivity) and Campbell's implementation of the critical state model. In this contribution, we extend the $T$-$A$ formulation to thick conductors so that large coils with different coupling scenarios between the turns can be considered. We also revise Campbell's model and discuss it in terms of its ability to calculate AC losses: in particular, we investigate the dependence of the calculated AC losses on the frequency of the AC excitation and the possibility of using quick one-step (instead of full cycle) simulations to calculate the AC losses.

In the last 10-15 years, the most common strategy in the development of High Temperature Superconducting (HTS) cable for magnets has been to imitate Low Temperature Superconducting (LTS) cable designs. However, requirements for LTS materials are not valid for HTS materials, which are extremely stable. For example, non-twisted multifilamentary Bi-2223 tapes have been successfully used in several magnets. This paper review stability and analyse inductance and AC losses in non-twisted stack of HTS tapes. Numerical calculations show that twisting has negligible effect on inductance variations in a stack of tapes. Regarding AC losses, any magnet built with coated conductors have larger losses than LTS ones, because of the aspect ratio and large width of the tape. If a wide tape is replaced by a non-twisted stack of narrow tapes, losses and residual magnetisation could be reduced. In contrast with multifilamentary wires, twisting a stack of tapes reduces losses only marginally. Therefore, cables composed of non-twisted stack could be designed to have losses comparable to the one of twisted stack concepts. Few examples of large cables for fusion applications are discussed. Designs based on non-twisted stacks can be simpler, more robust and cost effective than twisted ones., Comment: 20 pages, 10 figures, submitted to Cryogenics

Stacks of superconducting tapes can trap much higher magnetic fields than conventional magnets. This makes them very promising for motors and generators. However, ripple magnetic fields in these machines present a cross-field component that demagnetizes the stacks. At present, there is no quantitative agreement between measurements and modeling of cross-field demagnetization, mainly due to the need of a 3D model that takes the end effects and real micron-thick superconducting layer into account. This article presents 3D modeling and measurements of cross-field demagnetization in stacks of up to 5 tapes and initial magnetization modeling of stacks of up to 15 tapes. 3D modeling of the cross-field demagnetization explicitly shows that the critical current density, $J_c$, in the direction perpendicular to the tape surface does not play a role in cross-field demagnetization. When taking the measured anisotropic magnetic field dependence of $J_c$ into account, 3D calculations agree with measurements with less than 4 % deviation, while the error of 2D modeling is much higher. Then, our 3D numerical methods can realistically predict cross-field demagnetization. Due to the force-free configuration of part of the current density, J, in the stack, better agreement with experiments will probably require measuring the Jc anisotropy for the whole solid angle range, including $J$ parallel to the magnetic field., Comment: 36 pages, 20 figures

The emergence of second generation (2G) high-temperature superconductor (HTS) tapes has favored the development of HTS magnets for their applications in areas such as NMR, MRI and high field magnets. The screening current-induced field and the field drift are two major problems hindering the use of HTS tapes in the mentioned areas. Both problems are caused by the screening current, then it is necessary to have a modeling strategy capable to estimate such phenomena. Thus far, the H formulation has been the most successful and used approach to model medium-size systems (hundreds of tapes). However, its application to large-scale systems is still impaired by excessive computation times and memory requirements. Homogenization and multi-scaling strategies have been successfully implemented to increase the computational efficiency. In this contribution, we show that using the homogenization technique with the recently developed T-A formulation allows reducing the computation time and the amount of memory up to the point that real-time simulations of slow ramping cycles of large-scale systems are possible. The T-A homogeneous model also allows systematically investigating the screening current using numerical simulations.

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 finite element method programs for different cases, such 2D longitudinal and axis-symmetric geometries, 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 $H$ formulations 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.

Modelling high temperature superconductor (HTS) motors remains challenging mainly due to the high aspect ratio of these conductors but also because of the properties of the magnetic materials. This paper presents a 2D time dependent model to assess the AC losses of superconducting motors based on the new T-A formulation, which by using Finite Element Methods (FEM), allows its implementation in commercial software. The T-A formulation computes the magnetic flux density with different Maxwell's equations depending on the areas of the motor and makes it possible to use the thin strip approximation i.e. the HTS tapes are modelled as infinitely thin lines. The model is then expected to tackle the high aspect ratio of the HTS as well as decreasing both the mesh complexity and the computing time. The first objective of the paper is to validate the method in 2D by evaluating the AC losses of a specific synchronous motor called SUTOR; the computed results are compared with good agreements to those assessed with the MEMEP method, already validated. In a second part, the same losses are computed, taking into account the anisotropy of Jc with the implementation of a data set based on experimentally measured Ic at 65 K and 77 K.

Many superconducting power applications contain REBCO coils. Superconducting motors are very interesting for mobile applications like aviation, thanks to their high power per unit weight. In order to reduce the inductance, multi-tape conductors are often used in windings but parallel tapes present coupling AC loss. In this article, we analyze the AC loss of single coils and a full stator of a motor made of windings of parallel tapes, taking several coupling situations into account. Our computer modeling method combines self-programmed and commercial finite-element methods to calculate the current density and AC loss. We found that isolating the tapes along the length provides almost the same AC loss reduction as that obtained by full transposition. This reveals that there are still many unexploited strategies to reduce the AC loss in multi-tape conductors., Comment: 5 pages, 6 figures

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

Superconductors have been being applied to a variety of large-scale power applications, including magnets, electric machines, and fault current limiters, because they can enable a compact, lightweight and high efficiency design. In applications such those mentioned above, superconducting coils are always a key component. For example, in a superconducting electric machine, the superconducting coils are used to generate the main flux density in the air gap, which is significantly important for the energy conversion. It is the performance of the superconducting coils that plays an essential role in determining the performance of the device. However, the performance of a superconducting coil is limited by its critical current, which is determined by temperature and the magnitude and orientation of the magnetic field inside the superconductors. Hence, in-depth investigations to estimate the critical current of the superconducting coils are necessary before manufacturing. Available transient simulation models to estimate the critical current are through the H- and T-A formulations of Maxwell's equations. Both methods consider the same current ramp-up process occurring in experiments. Besides these transient models, static simulations can also be used: a modified load-line method and the so-called P-model, which is based on the asymptotic limit of Faraday's equation when time approaches infinity. To find the best way to calculate the critical current, the four methods are used to estimate the critical current of a double pancake superconducting coils and results are compared with experiments. As a conclusion, T-A formulation, P-model, and the modified load-line methods are recommended for estimating the critical current of the superconducting coils.

This paper presents the alternating current (AC) loss analysis on high-temperature superconductor (HTS) Cross-Conductor (CroCo) cables, in order to evaluate whether they could be utilized for electrical power transmission. The modeling of HTS CroCo cables was based on a cable assembled at the Karlsruhe Institute of Technology (KIT) and the AC loss calculation was based on the H-formulation model implemented in the finite-element method (FEM) software package COMSOL Multiphysics. The AC loss calculations have been carried out for isolated single-phase CroCo cable and three-phase CroCo cables. The AC loss angular dependence of a particular phase of CroCo cables during three phase operation has been studied. The current distributions of individual tapes within CroCo cables have been investigated.

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.

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.

Fluorescent Microthermographic Imaging, a method using rare-earth fluorescent coatings with temperature-dependent light emission, was used for quench investigation in high temperature superconductors (HTS). A fluorophore was embedded in a polymer matrix and used as a coating on top of an HTS tape, while being excited with UV light and recorded with a high-speed camera. Simultaneously, the tape was pulsed with high amplitude, short duration DC current, and brought to quench with the help of a localized defect. The joule heating during a quench influences the fluorescent light intensity emitted from the coating, and by recording the local variations in this intensity over time, the heating of the tape can be visualized and the developed temperatures can be calculated. In this paper, the fluorophore Europium tris[3-(trifluoromethylhydroxymethylene)- (+)-camphorate] (EuTFC) provided sufficient temperature sensitivity and a usable temperature range from 77 K to 260 K. With the help of high-speed recordings, the normal zone development was imaged in a 20 \mu m copper stabilized HTS tape held in a liquid nitrogen bath, and using a calibration curve, the temperatures reached during the quench have been calculated.

Electrical machines employing superconductors are attractive solutions in a variety of application domains. Numerical models are powerful and necessary tools to optimize their design and predict their performance. The electromagnetic modeling of superconductors by finite-element method (FEM) is usually based on a power-law resistivity for their electrical behavior. The implementation of such constitutive law in conventional models of electrical machines is quite problematic: the magnetic vector potential directly gives the electric field and requires using a power-law depending on it. This power-law is a non-bounded function that can generate enormous uneven values in low electric field regions that can destroy the reliability of solutions. The method proposed here consists in separating the model of an electrical machine in two parts, where the magnetic field is calculated with the most appropriate formulation: the H-formulation in the part containing the superconductors and the A-formulation in the part containing conventional conductors (and possibly permanent magnets). The main goal of this work is to determine and to correctly apply the continuity conditions on the boundary separating the two regions. Depending on the location of such boundary -- in the fixed or rotating part of the machine -- the conditions that one needs to apply are different. In addition, the application of those conditions requires the use of Lagrange multipliers satisfying the field transforms of the electromagnetic quantities in the two reference systems, the fixed and the rotating one. In this article, several exemplary cases for the possible configurations are presented. In order to emphasize and capture the essential point of this modeling strategy, the discussed examples are rather simple. Nevertheless, they constitute a solid starting point for modeling more complex and realistic devices.

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 ($\theta$, relative to the tape) of the local magnetic flux density. In principle, $J_c(B_{loc},\theta)$ can be obtained from the experimentally measured critical current $I_c(B_a,\theta)$, 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 $J_c(B_{loc},\theta)$ form $I_c(B_a,\theta)$ 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 $J_c(B_{loc},\theta)$ model in a few minutes.

High temperature superconducting (HTS) bulks or stacks of coated conductors (CCs) can be magnetized to become trapped field magnets (TFMs). The magnetic fields of such TFMs can break the limitation of conventional magnets (<2 T), so they show potential for improving the performance of many electrical applications that use permanent magnets like rotating machines. Towards practical or commercial use of TFMs, effective in situ magnetization is one of the key issues. The pulsed field magnetization (PFM) is among the most promising magnetization methods in virtue of its compactness, mobility and low cost. However, due to the heat generation during the magnetization, the trapped field and flux acquired by PFM usually cannot achieve the full potential of a sample (acquired by the field cooling or zero field cooling method). The multi-pulse technique was found to effectively improve the trapped field by PFM in practice. In this work, a systematic study on the PFM with successive pulses is presented. A 2D electromagnetic-thermal coupled model with comprehensive temperature dependent parameters is used to simulate a stack of CCs magnetized by successive magnetic pulses. An overall picture is built to show how the trapped field and flux evolve with different pulse sequences and the evolution patterns are analyzed. Based on the discussion, an operable magnetization strategy of PFM with successive pulses is suggested to provide more trapped field and flux. Finally, experimental results of a stack of CCs magnetized by typical pulse sequences are presented for demonstration.

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 current-voltage characteristics of superconductors are commonly determined by means of a standard 4-probe DC measurement technique. In coated conductors, however, one often encounters problems due to weak or no stabilization and in such case, when the risk of burning the tape is high, the 4-probe approach can fail. In other cases such as in HTS cable structures, besides vulnerability and high risk of damage, it may also be difficult to achieve a high critical current level with a power supply that is normally used to characterize single strands. We therefore investigated an alternative method to determine the DC I-V curve, based on measuring the AC transport loss as a function of frequency in overcritical current range. In the AC regime, both the hysteretic loss component and the joule dissipation are present for applied transport currents exceeding the critical current Ic. In this work, AC loss measurements at different frequencies are related to different electric fields in DC and a curve similar to the DC I-V is obtained. From this curve, the power index N and Ic values are estimated. AC regime operation can be more beneficial for two reasons: firstly, because very high currents can be achieved easily by using a transformer; secondly, because the effective joule heat dissipation with overcritical AC currents is much lower than that with DC of the same amplitudes. In the experiment, we used a 4 mm wide YBCO reference tape with 20 {\mu}m of Cu-stabilization. For the comparison with AC transport loss, the sample was measured with the standard 4-probe technique and remained stable even in overload condition. To support our study, numerical simulations, reproducing both transient and stationary operation conditions, were performed. A similar relation between the AC current transport loss and the I-V curves was also observed in the simulations., Comment: EUCAS 2015

The design of a HTS SMES coil with solenoidal and toroidal geometry is carried out based on a commercially available 2G HTS conductor. A 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 discharge cycle is also discussed.

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.

High temperature superconducting (HTS) stacks of coated conductors (CCs) can work as strong trapped field magnets (TFMs) and show potential in electrical applications. Pulsed field magnetization (PFM) is a practical method to magnetize such TFMs, but due to heat generation during the dynamic process, it cannot achieve a trapped field as high as field cooling can. In this work, we construct a 2D electromagnetic-thermal coupled model to simulate stacks of HTS CCs with realistic laminated structures magnetized by PFM. The model considers temperature and anisotropic magnetic field dependent Jc of HTS and other temperature dependent thermal and electrical material properties. Based on the model, a configuration of controlled magnetic density distribution coils is suggested to improve the trapped field compared to that obtained by ordinary solenoids.

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, as for example coils used in electrical machines, is questionable. Indeed the simulation time and memory requirement can quickly become prohibitive. In this article, 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 speeding-up simulations of large number of HTS tapes by 1-3 orders of magnitudes, while keeping a good accuracy of the results. Such models could be used to design and optimize large-scale HTS devices., Comment: 17 pages, 11 figures

In order to transport sufficiently high current, high-temperature superconductor (HTS) tapes are assembled in cable structures of different forms. In such cables, the tapes are tightly packed and have a strong electromagnetic interaction. In particular, the generated self-field is quite substantial and can give an important contribution in reducing the maximum current the cable can effectively carry. In order to be able to predict the critical current of said cable structures, a static numerical model has been recently proposed. In this contribution, we present in detail the implementation of such models in different programming environments, including finite-element-based and general numerical analysis programs, both commercial an open-source. A comparison of the accuracy and calculation speed of the different implementations of the model is carried out for the case of a Roebel cable. The model is also used to evaluate the importance of choosing a very accurate description of the angular Jc(B) dependence of the superconductor as input for the material's property. The numerical codes, which are open-source, are made freely available to interested users., Comment: The accompanying models descried in this work are available for free download at the HTS modeling workgroup under http://www.htsmodelling.com/?p=917

Hysteretic losses in MgB2 wound superconducting coils of a 500 kW synchronous hybrid generator were estimated as part of the European project SUPRAPOWER led by the Spanish company Tecnalia Research and Innovation. Particular interest was given to the losses found in tapes in the superconducting rotor caused by the magnetic flux ripples originating from the conventional stator during nominal operation. To compute the losses, a 2D Finite Element Method was applied to solve the H-formulation of Maxwell's equations considering the nonlinear properties of both the superconducting material and its surrounding Ni matrix. To be able to model all the different turns composing the winding of the superconducting rotor coils, three geometrical models of single tape cross section of decreasing complexity were studied: 1) the first model reproduced closely the actual cross section obtained from micrographs, 2) the second model was obtained from the computed elastoplastic deformation of a round Ni wire, 3) the last model was based on a simplified elliptic cross section. The last geometry allowed validating the modeling technique by comparing numerical losses with results from well-established analytical expressions. Additionally, the following cases of filament transpositions were studied: no, partial and full transposition. Finally, choosing the right level of geometrical details to predict the expected behavior of individual superconducting tapes in the rotor, the following operational regimes were studied: Bias-DC current, ramping current under ramping background field, and magnetic flux ripples under DC background current and field.

Nowadays, there is growing interest in using superconducting wires or tapes for the design and manufacture of devices such as cables, coils, rotating machinery, transformers and fault current limiters among others. Their high current capacity has made them the candidates of choice for manufacturing compact and light cables and coils that can be used in the large scale power applications described above. However, the performance of these cables and coils is limited by their critical current, which is determined by several factors, including the conductor's material properties and the geometric layout of the device itself. In this work we present a self-consistent model for estimating the critical current of superconducting devices. This is of large importance when the operating conditions are such that the self-field produced by the current is comparable to the overall background field. The model is based on the asymptotic limit when time approaches infinity of Faraday's equation written in terms of the magnetic vector potential. It uses a continuous E-J relationship and takes the angular dependence of the critical current density on the magnetic flux density into account. The proposed model is used to estimate the critical current of superconducting devices such as cables, coils, and coils made of transposed cables with very high accuracy. The high computing speed of this model makes it an ideal candidate for design optimization.

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.

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.

The optimization of a HTS cable design with respect to AC-losses is of crucial importance for the economic viability of the respective concept. However the experimental determination of AC-losses is not straightforward since for short cable samples the distribution of current among the super-conducting tapes is mainly determined by the contact resistances of the individual tapes. The resulting inhomogeneous current distribution definitely falsifies the results. To solve this experimental problem we present a new experimental technique. The setup is a 2m-long three phase concentric cable model for which, within each phase, the superconducting tapes (up to 30) are connected in series. The Cu-braid backwards conductors were assembled in a rotational symmetric cage type arrangement, such that their self fields at the cable cancel. If experimental peculiarities of this setup, as the strong inductive coupling between the phases and the suitable positioning of the voltage contact leads, are correctly taken into account, the currents can be controlled independently and the electrical properties of the cable can be measured unambiguously. In this paper preliminary results are presented. The work is part of the German government funded cable project AMPACITY (1 km / 20 kV/ 2 kA)

In this paper we develop a method based on the critical state for calculating the current and field distributions and the AC losses in a cable made of curved thin superconducting tapes. The method also includes the possibility of considering spatial variation of the critical current density, which may be the result of the manufacturing process. For example, rare-earth based coated conductors are known to have a decrease of the transport properties in near the edges of the tape: this influences the way current and field penetrate in the sample and, consequently, the AC losses. We demonstrate that curved tapes arranged on a cylindrical former behave as an infinite horizontal stack of straight tapes, and we compare the AC losses in a variety of working conditions, both without and with the lateral dependence of the critical current density. This model and subsequent similar approaches can be of interest for various applications of coated conductors, including power cables and conductor-on-round-core (CORC) cables.

We address the problem of modeling termination resistances which are largely responsible for the uneven distribution of currents in superconducting cables. For such purpose we present three DC models. In a first model a 0D circuit-like approach considering a continuous E-J relationship is presented. A second model uses the 2D H-Formulation of Maxwell's equations, with a new contribution to the electric field term that takes into account the voltage drop due to termination resistances. A third model, based on the 3D H-Formulation of Maxwell's equations, uses a novel technique to simulate both the termination resistances and the superconducting cable within a compact framework that calculates both contributions using two non-connected domains. Advantages and disadvantages of each model are discussed. Particular applications for which a given model is best fitted are also considered. The models' predictions are in good agreement with experimental results for a stacked-tape cable composed of 4 HTS tapes. Overall, this work presents a palette of three different numerical tools for calculating the current distribution in cables composed of multiples tapes, where the termination resistance is also taken into account. The choice of one model over another depends on the particular application and on the degree of precision needed

Roebel cables made of HTS coated conductors can carry high currents with a compact design and reduced AC losses. They are therefore good candidates for manufacturing coils for HTS applications such as motors and generators. In this paper we present the experimental DC and AC characterization of several coils assembled from a 5 meter long Roebel cable built at KIT, which differ in the number of turns and turn-to-turn spacing. Our experiments, supported by finite-element method (FEM) calculations, show that a more tightly wound Roebel coil, despite having a lower critical (and therefore operating) current, can produce a higher magnetic field than a loosely wound one. For a given magnetic field produced at the coil's center, all the coils have similar AC losses, with the exception of the most loosely wound one, which has much higher losses due to the relatively large current needed to produce the desired field. The experiments presented in this paper are carried out on the geometry of pancake coils made of Roebel cables, but they are exemplary of a more general strategy that, coupling experiments and numerical simulations, can be used to optimize the coil design with respect to different parameters, such as tape quantity, size, or AC loss, the relative importance of which is dictated by the specific application.

Striation of HTS coated conductors (CCs) as a way to reduce their magnetization AC losses has been the subject of intense research in the past years by several groups. While the principle of this approach is well understood, its practical application on commercial material to be used in power application is still far to be implemented due to manufacturing and technological constraints. Recent advances in tape quality and striation technology are now enabling systematic investigations of the influence of the number of filaments on AC loss reduction with a consistency that was not available in the past. In this work we demonstrate the technological feasibility of reducing the magnetization losses of commercially available CC by striating them into a high number of filaments (up to 120). The loss reduction exceeds one order of magnitude and does not come at the expense of current-carrying capability: samples with 10 and 20 filaments are unaffected by the striation process, while samples with 80 and 120 filaments still retain 80 and 70% of the current-carrying potential, respectively. We also investigate the transverse resistivity in order to understand the paths followed by the coupling currents: we found that the coupling current prevalently flows in the metallic substrate, rather than in and out of the filaments. Finally, we use oxidation as a method to reduce the coupling currents and losses. The contribution of this work is three-fold: 1) It describes the know-how to produce a large number of high quality striations in commercially available CCs, greatly reducing their losses without extensively degrading their transport properties; 2) It provides a comprehensive characterization of said samples (e.g. measurements in a wide frequency range, transverse resistance profiles, influence of oxidation on DC and AC behavior); 3) It provides new insight on the patterns of the coupling currents.

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.

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.

Use of 2G HTS coated conductors in several power applications has become popular in recent years. Their large current density under high magnetic fields makes them suitable candidates for high power capacity applications such as stacks, coils, magnets, cables and current leads. For this reason, modeling and simulation of their electromagnetic properties is very desirable in the design and optimization processes. For many applications, when symmetries allow it, simple models consisting of 1D or 2D representations are well suited for providing a satisfying description of the problem at hand. However, certain designs such as racetrack coils and finite-length or non-straight stacks, do pose a 3D problem that cannot be easily reduced to a 2D configuration. Full 3-D models have been developed, but their use for simulating superconducting devices is a very challenging task involving a large-scale computational problem. In this work, we present a new method to simulate the electromagnetic transient behavior of 2G HTS stacks and coils. The method, originally used to model stacks of straight superconducting tapes or circular coils in 2D, is now extended to 3D. The main idea is to construct an anisotropic bulklike equivalent for the stack or coil, such that the geometrical layout of the internal alternating structures of insulating, metallic, superconducting and substrate layers is reduced while keeping the overall electromagnetic behavior of the original device. Besides the aforementioned interest in modeling and simulating 2G HTS coated conductors, this work gives a further step towards efficient 3D modeling and simulation of superconducting devices for large scale applications.

In this contribution we develop a refined numerical model of pancake coils assembled from a coated conductor Roebel cable, which includes the angular dependence of the critical current density $J_c$ on the magnetic field and the actual (three-dimensional) shape of the current lead used to inject the current. Previous works of ours indicate that this latter has an important influence on the measured value of the AC losses. For the simulation of the superconductor, we used two alternative models based on different descriptions of the superconductor's properties and implemented in different mathematical schemes. For the simulation of the current lead we use a full three-dimensional finite-element model. The results of the simulation are compared with measurements and the main issues related to the modeling and the measurement of Roebel coils are discussed in detail.

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.

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., Comment: IEEE Transactions on Applied Superconductivity, 2013