Although calculations on HTS are done within the magneto-quasistatic approximation to Maxwell’s Equations, magneto-quasistatics is not adequate for describing alternating currents when both inductive and capacitive effects are present. (The latter is ruled out by the magneto-quasistatic’s neglect of the displacement current.) Here we describe a more comprehensive approximation to Maxwell’s Equations, one which incorporates both inductive and capacitive effects. The approximation is the result of a straightforward revision of Ampere’s Equation, one which conserves charge but, unlike the Displacement Current, does not allow for radiation. We give the equations for scalar and vector potentials and their solutions in terms of the time dependent charge and current densities. It is shown that these solutions are not qualitatively more difficult to evaluate than the corresponding expressions in magneto-quasi statics. We also compare the magneto-quasistatic expression for energy to that in the more comprehensive approximation presented here.

Many superconductor applications contain windings with thousands of turns, such as transformers, magnets and SMES. Their design requires modelling of electromagnetic quantities, such as the AC loss. In addition, modelling of general 3D samples is necessary to interpret magnetization measurements.
This work presents a fast but accurate modelling tool for electromagnetic quantities for either cross-sectional (2D) or 3 dimensional (3D) problems. The model is based on a variational princicple in the J-q formulation, where J and q are the current and charge densities, and takes anisotropic E(J) relations from flux-cutting effects into account, as well as B-dependent critical current densities. We present, first, 2D calculations for circular coated conductor coils from few turns to 10000 (10k) and, second, 3D modelling of rectangular prisms of both 1 um or several mm thickness and 1 cm size in the other directions.
For the 2D implementation, the elapsed computing time in a table computer for n=1000 and 160000 elements in the superconductor is below 20 s and 5 hours, respectively, and it scales as n² for high n. The AC loss results agree very well with measurements for small windings made of coated conductor, supporting the validity of the model. Calculations for 10k-turn coils revealed important effects of the magnetization currents.
The 3D calculations for thin films show that the current paths under an applied magnetic field making an angle with the surface differs from that taking the perpendicular component only. In the bulk sample, the current paths are rounded close to the top and bottom surfaces, in contrast to the square current paths for infinite bars.
In conclusion, the presented 3D model describes the exotic behaviour of finite superconductors with anisotropic E(J) relation. At present, its 2D version allows full detailed modelling of large-scale applications like transformers, circular magnets, and SMES in practical short computing times.

The research leading to these results have received funding from the European Union Seventh Framework Programme [FP7/2007-2013] under grant agreement n⁰. NMP-LA-2012-280432. The authors acknowledge the use of resources provided by the SIVVP project (ERDF, ITMS 26230120002).

There is a need for reliable and scalable 3D models for superconductors. Formulations based on the magnetic field H are mostly used to model 2D or 3D non-linear problems in superconductors. In that scope, difficulties associated with the initialization of non-linear resistivity in the studied domain’s arise. While its actual initial value is zero, the typical initial value set in the numerical problem is chosen close to zero, that setting usually causes numerical errors in the solution and sometimes the numerical problem will not converge.
A combination of the typically used H formulation with the successful E formulation proposed in previous work has been implemented using the discontinuous galerkin method. A 3D model of a superconductor filament inside an air domain will be presented as a simple test case.
This new formulation allows, with the help of the E formulation, to successfully initialize the resistivity of the studied domain while taking in account the non-linearities arising in the superconductor .
Consequently, once the E formulation solves the problem at the first iteration in order to get the initial value of the resistivity, the H formulation will solve the problem in several iterations on the resistivity value. A comparison of H formulation and E-H formulation will be presented highlighting the advantages of the proposed method.

An accurate description of the thermal stability of YBCO coated conductors is of great importance for a safe design of cables and magnets manufactured with this material. The analysis of the quench phenomenon can be pushed at different levels of detail, from the description of the individual layers of the tape to the application of homogenization techniques for the study of the whole winding pack of a coil. An anisotropic 2D electro-thermal model of the YBCO tapes was developed and validated at the University of Bologna, which is able to solve the problem of the high aspect ratio of the tape in a FEM approach. This paper presents a comparison between the results of the FEM model and those of a 1D quench model based on a non-linear electric circuit coupled to a thermal model of the tape. The critical current, minimum quench energy (MQE) and normal zone propagation velocity (NZPV) are computed at different operating conditions. The results of the two models are compared to identify advantages and drawbacks of the two kinds of modeling.

In recent years, superconducting cables and multi-filamentary conductors have become attractive candidates for several power applications due to their compact design and large current capacity. However, many power applications inherently require the superconducting materials to experience hysteretic losses. This might be in the form of AC transport current, externally AC applied magnetic field, ripple currents or fields on a DC background, or a combination of the aforementioned cases. This yields a highly complex scenario that requires a careful and thorough analysis in the designing stage of the superconducting device. Consequently, there is growing interest in modeling and simulating the electromagnetic response of superconducting cables and multi-filamentary conductors, estimate their critical current and AC losses, and to optimize their design. In this work we use a 3D model based on the magnetic vector potential to simulate the current distribution in superconducting cables and multi-filamentary conductors, and to estimate their critical current and AC losses in several operating conditions. To take advantage of the 3D modeling framework, the effect of transposition in a multi-filamentary conductor is addressed. Three transposition cases are considered: full, partial and no transposition of filaments.

This work has been partly supported by the Helmholtz Association (Young Investigator Group grant VH-NG-617).

High temperature superconducting (HTS) bulks and stacks of coated conductors (CCs) can work as strong permanent magnets by trapping persistent currents. To magnetize these trapped field magnets (TFMs), the pulse field magnetization (PFM) technique is of special interest, because it can work in situ inside practical electric machines. However, the trapped field acquired by PFM is usually lower than that acquired by field cooling, because fast flux motions during PFM generate much heat and increase the temperature. In this study, a two-dimensional electromagnetic-thermal coupled model is constructed for both HTS bulks and stacks of CCs magnetized by PFM. The model uses the H formulation of Maxwell equations for the electromagnetic part. The model considers that the critical current density of HTS depends on both the temperature and the magnetic fields’ magnitude and orientation. The thermal part of the model considers the temperature-dependent heat capacity and thermal conductivity of each material. For the stacks model, coated conductor’s realistic structures of different layers are considered. Based on the model, PFM using pulses of different spatial magnetic density distributions are investigated. The penetration processes and temperature profiles are compared. The results show that high gradient fields with the peak in the centre of the sample can reduce heat generation and increase the final trapped field. Optimal designs of practical controlled magnetic density distribution coils (CMDCs) are given for bulks. The CMDCs are also applicable for increasing trapped field of stacks, given that the width of CCs could be increased to 40 mm.

This work has been partly supported by the Helmholtz Association (Young Investigator Group grant VH-NG-617).

Previous analytical models of AC losses in multifilamentary composites under a time varying external magnetic field were developed with specific strand geometries. Each of them considers the strand as a given succession of concentric layers either purely resistive or filamentary (i.e. superconducting filaments embedded in a resistive matrix). Nevertheless, as up-to-date models were built from usual strand geometries (i.e. usual layer sequences), no global expression of AC losses or screening currents distribution in transient regimes was established for any type of possible geometry (i.e. for any layer sequences).

The purpose of this paper is to present the baseline of the model developed at CEA and to describe the general analytical strategy and the assumptions which consist in using Maxwell’s equations, classic electromagnetic boundary conditions and geometric approximations. Furthermore, AC losses are generally characterized by a global time constant nτ; we here deduce effective coefficients (homogeneous to time constants) which reflect the coupling between the different screening currents. A global linear equation system linking the screening currents, the time variation of these currents and of the external magnetic field is deduced, establishing a coupling matrix built with time constant coefficients. Besides physical interpretation of the quantities defined, we numerically solve the global equation and locally quantify magnetic field, currents and AC losses in the composite. A better physical understanding of the screening currents behaviour and of the stability of the composite is tentatively achieved by the use of this analytical model and a numerical approach. In addition, applications to well-known strands (ITER, JT-60SA) and experimental confrontations are also presented together.

The ultimate scope of use for this model is to act in the framework of a physical in-depth study of the AC losses experienced by fusion cables-in-conduit, with the help of the macroscopic approaches already developed at CEA (MPAS model which is used on experimental ITER database). Analogies and possible analytical links between the different scales from strand to full cable may also be presented here.

In present scenario, efficient power transmission and distribution systems are one of the requisites to conserve energy. However, transmission and distribution of power though conventional systems involve losses due to resistance offered by the conventional conductors. Hence, power transmission through superconductors may be one of the alternative methods which can be implemented to avoid such losses. Presently, YBCO (Tc=90K) and BSCCO (Tc=110K) are the commercial superconductors used in various projects worldwide. Cooling of these superconductors is essential to retain the superconductivity of these superconductors. Even though various cooling strategies with liquid nitrogen are proposed in the past, multiphase flow arising out, while boiling of the liquid nitrogen, creates heat transfer challenges and there by degradation of current carrying capacity is reported. Hence, in the present work, Super Critical Nitrogen (SCN) is suggested as a coolant for cooling of High Temperature Superconducting (HTS) cables. The main advantage of using SCN is that the multiphase flow phenomenon is eliminated. Further, the density and viscosity of the SCN are found to be far lesser than the liquid nitrogen which contributes to lesser pumping power. Moreover, due to higher specific heat and thermal conductivity of SCN, heat transfer may be enhanced. In the present work, thermophysical properties of SCN such as viscosity, density, specific heat and thermal conductivity are considered as temperature dependent for analysis of 1m HTS cable. 3D model of HTS cable is developed in GAMBIT 2.2.30 which is exported to FLUENT TM 6.3.26 for solving governing equations of mass, momentum and energy simultaneously. For different mass flow rates (20 g/s to 24 g/s) and heat flux (0.8 W/m2 to 1.2 W/m2) thermohydraulic behavior of SCN is determined. These simulations are useful in finding friction factor, pressure drop, pumping power and heat transfer rate. It can be concluded that SCN may be used as coolant to cool the HTS cables in future. Computational Fluid Dynamics (CFD) is used as the tool in predicting the thermohydraulic behavior of HTS cable and the effect of heat flux on the performance of HTS cable is studied. The results of the simulation are validated with the experimental results available in the literature.

The Authors would like to ackknowledge the support extended by IIT Kharagpur, India.

2G Superconductor materials with its multilayer structure should have specific electrical, mechanical and thermal properties. Their design as well as their fabrication and their uses in applications still need some investigations to find their limits. The fabrication process induces residual stresses, which cause degradation in the performances of HTS conductor. It is necessary to analyse the mechanical behaviour of these 2G tapes materials in order to investigate the influence of their design on their performances when used in the applications such as cables, motors or magnets..
2G HTS materials includes extremely thin (<100 nm) and thick (>50 mm) multiple layers made out with very different materials (hard, soft metals, ceramics,...).The overall structure shows a very poor resistance against shearing stresses under bending on small diameters due to low adhesion between the different layers. The use of a conventional finite elements’ method for the analysis of this mechanical behaviour requires very fine meshes; consequently, it is extremely time consuming. In this paper, we propose a simplified mechanical model, which allows exploring easily and quickly the behaviour of the complex multilayer 2G tapes. This model is then used in parametric studies. It leads to some recommendations concerning the optimal designs (thickness of the metallic layers, selection and association of the materials,...) for industrial 2G tapes to survive to the strains and stresses from the handling winding, bending from the applications’ manufacturing processes.