In the past few years magneto-optical flux imaging (MOI) has come to take an increasing role in the investigation and understanding of critical current densities in high-Tc superconductors (HTS). This has been related to the significant progress in quantitative high-resolution magneto-optical imaging of flux distributions together with the model-independent determination of the corresponding current distributions. We review in this article the magneto-optical imaging technique and experiments on thin films, single crystals, polycrystalline bulk ceramics, tapes and melt-textured HTS materials and analyse systematically the properties determining the spatial distribution and the magnitude of the supercurrents. First of all, the current distribution is determined by the sample geometry. Due to the boundary conditions at the sample borders, the current distribution in samples of arbitrary shape splits up into domains of nearly uniform parallel current flow which are separated by current domain boundaries, where the current streamlines are sharply bent. Qualitatively, the current pattern is described by the Bean model; however, changes due to a spatially dependent electric field distribution which is induced by flux creep or flux flow have to be taken into account. For small magnetic fields, the Meissner phase coexists with pinned vortex phases and the geometry-dependent Meissner screening currents contribute to the observed current patterns. The influence of additional factors on the current domain patterns are systematically analysed: local magnetic field dependence of jc(B), current anisotropy, inhomogeneities and local transport properties of grain boundaries. We then continue to an overview of the current distribution and current-limiting factors of materials, relevant to technical applications like melt-textured samples, coated conductors and tapes. Finally, a selection of magneto-optical experiments which give direct insight into vortex pinning and depinning mechanisms are reviewed.

https://iopscience.iop.org/article/10.1088/0034-4885/65/5/202/pdf

Magneto-optical studies of current distributions in high-Tc superconductors

This paper presents a new numerical model for computing the current density, field distributions and AC losses in superconductors. The model, based on the direct magnetic field H formulation without the use of vector and scalar potentials (which are used in conventional formulations), relies on first-order edge finite elements. These elements are by construction curl conforming and therefore suitable to satisfy the continuity of the tangential component of magnetic field across adjacent elements, with no need for explicitly imposing the condition . This allows the overcoming of one of the major problems of standard nodal elements with potential formulation: in the case of strong discontinuities or nonlinearities of the physical properties of the materials and/or in presence of sharp corners in the conductors' geometry, the discontinuities of the potentials' derivatives are unnatural and without smoothing artifices the convergence of the algorithm is put at risk. In this work we present in detail the model for two-dimensional geometries and we test it by comparing the numerical results with the predictions of analytical solutions for simple geometries. We use it successively for investigating cases of practical interest involving more complex configurations, where the interaction between adjacent tapes is important. In particular we discuss the results of AC losses in superconducting windings.

https://iopscience.iop.org/article/10.1088/0953-2048/20/1/004/pdf

Development of an edge-element model for AC loss computation of high-temperature superconductors

This review focuses on the superconducting properties of MgB2 that are relevant for power applications. The reversible mixed state parameters are the most important, since they define the limiting conditions for loss-free currents: the transition temperature, the upper critical field and the depairing current. They also determine the flux pinning energy, the pinning force and the elastic properties of the flux line lattice and, therefore, strongly influence the critical current densities. The magnetic properties of magnesium diboride are anisotropic and influenced by the two different energy gaps of the σ- and π-bands. Whereas the transition temperature could not be enhanced significantly during the past five years, the upper critical field was considerably increased by impurity scattering or doping. Flux pinning is very weak in MgB2 single crystals and was only improved by irradiation techniques so far. In polycrystalline samples, grain boundary pinning seems to play the dominant role. High critical currents close to the theoretical limit were found in c-axis oriented thin films. The anisotropy of the upper critical field strongly reduces the critical currents in untextured MgB2 at high magnetic fields, where the supercurrents become highly percolative, since not all grains are superconducting anymore. The performance of polycrystalline wires and tapes was significantly improved during the past few years by increasing the upper critical field and by reducing its anisotropy. Pinning seems to be nearly optimized in many forms of this material, but the connectivity between the grains might be further improved.

https://iopscience.iop.org/article/10.1088/0953-2048/20/12/R01/pdf

Magnetic properties and critical currents of MgB2

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 particularly 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, which are based on different techniques and implementations, has been proposed by researchers around the world, with the purpose of being able to estimate ac losses in HTSs quickly and accurately. This paper 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.

Computation of Losses in HTS Under the Action of Varying Magnetic Fields and Currents

A homogenization method to model a stack of second generation High Temperature Superconducting tapes under AC applied transport current or magnetic field has been obtained. The idea is to find an anisotropic bulk equivalent for the stack such that the geometrical layout of the internal alternating structures of insulating, metallic, superconducting, and substrate layers is “washed” out while keeping the overall electromagnetic behavior of the original stack. We disregard assumptions upon the shape of the critical region and use a power law E–J relationship allowing for overcritical current densities to be considered. The method presented here allows for a computational speedup factor of up to 2 orders of magnitude when compared to full 2-D simulations taking into account the actual dimensions of the stacks without compromising accuracy.

Calculation of alternating current losses in stacks and coils made of second generation high temperature superconducting tapes for large scale applications

In this paper we present a new 2D numerical model for AC loss computation in high-temperature superconductors, based on the use of edge finite elements. These elements are curl conforming by construction and involve the continuity of the tangential component of the magnetic field between adjacent elements. In this way they do not require the imposition of the zero divergence of the magnetic field at each time step, which, with standard nodal elements, increases the null space of the matrix and the risks of divergence of the algorithm. The model, implemented in the software package COMSOL, uses the two magnetic field components as state variables. It has been tested for computing the losses of YBCO coated conductor tapes in two cases of practical interest, which cannot be investigated by analytical models.

Modeling High-Temperature Superconducting Tapes by Means of Edge Finite Elements

The current density and magnetic field distributions in thin conductors are important for several applications, and they can be computed by solving integral equations This paper describes the implementation of a one-dimensional (1D) integral equation in a finite-element model This numerical method does not require the use of ad hoc assumptions to avoid logarithmic divergences of the current density at the conductor's edges and, by using a coupling with 2D electromagnetic models, it can be used to solve cases of increasing complexity With respect to commonly used 2D models, it overcomes the typical problems linked to the mesh of conductors with high aspect ratio, such as the use of large memory and long computing times

https://iopscience.iop.org/article/10.1088/0953-2048/21/10/105008/pdf

Integral equations for the current density in thin conductors and their solution by the finite-element method

In recent years many large scale demonstrators and prototypes of superconducting fault current limiters have been successfully developed and tested. Within the European Project ECCOFLOW (www.eccoflow.org), it is the first time that a resistive-type superconducting fault current limiter is developed for two different locations and that a permanent installation is foreseen. The limiter has a rating of 20 kV and 1 kA and will be tested in a busbar and transformer feeder application. The paper summarizes the conceptual design of this innovative limiter and reports in detail about the development of the super- conducting limiting elements, their integration into a cryostat and the design of the whole limiter including cooling and grid integration. As a main result it can be summarized that the ECCOFLOW limiter fulfills all requirements according to the two different specifications. Approximately 3 km of 12 mm wide YBCO tape will be used to realize a three phase system.

Conceptual Design of a 24 kV, 1 kA Resistive Superconducting Fault Current Limiter

Nickel-sheathed MgB2 tapes were fabricated by means of the powder-in-tube method with a monocore configuration of the superconductor. The metallic tubes were filled with commercial reacted powders and cold-worked, respectively, by groove rolling, drawing, and rolling to flat tapes of about 4 mm in width and 0.35 mm in thickness. Portions of the manufactured conductors were heat-treated in an argon atmosphere at temperatures up to 940 °C. Conductors were characterized by transport measurements to determine the magnetic field dependence of the critical current density, the irreversibility line, as well as to establish an initial relationship between the thermo-mechanical treatment and the current carrying capacity of the manufactured MgB2 samples. Our optimized samples for the low field operation carry a critical current density of at least 4 × 105 A cm−2 at 4.2 K, 1 T. The main outcome of this study, however, is that the optimal heat treatment condition changes if the MgB2 conductor has to be employed respectively for a low- or high-field operation. Finally, the first successful demonstration of the current carrying capability of longer MgB2 tapes is also reported in this paper.

Luciano Martini

Papers

Development of an edge-element model for AC loss computation of high-temperature superconductors

Modeling High-Temperature Superconducting Tapes by Means of Edge Finite Elements

Integral equations for the current density in thin conductors and their solution by the finite-element method

Conceptual Design of a 24 kV, 1 kA Resistive Superconducting Fault Current Limiter

Fabrication and properties of monofilamentary MgB2 superconducting tapes