In an attempt to unveil the impact of the material law selection on the numerical modelling and analysis of the electromagnetic properties of superconducting coils, in this paper we compare the four most common approaches to the E-J power laws that serve as a modelling tool for the conductivity properties of the second generation of high-temperature superconducting (2G-HTS) tapes. The material laws considered are: (i) the celebrated E-J critical-state like-model, with constant critical current density and no dependence with the magnetic field; (ii) the classical Kim’s model which introduces an isotropic dependence with the environment magnetic field; (iii) a semi-empirical Kim-like model with an orthonormal field dependence, J c ( B ) , widely used for the modelling of HTS thin films; and (iv) the experimentally measured E–J material law for SuperPower Inc. 2G-HTS tapes, which account for the magneto-angular anisotropy of the in-field critical current density J c ( B ; θ ) , with a derived function similar to Kim’s model but taking into account some microstructural parameters, such as the electron mass anisotropy ratio ( γ ) of the superconducting layer. Particular attention has been given to those physical quantities which within a macroscopic approach can be measured by well-established experimental setups, such as the measurement of the critical current density for each of the turns of the superconducting coil, the resulting distribution of magnetic field, and the curve of hysteretic losses for different amplitudes of an applied alternating transport current at self-field conditions. We demonstrate that although all these superconducting material laws are equally valid from a purely qualitative perspective, the critical state-like model is incapable of predicting the local variation of the critical current density across each of the turns of the superconducting coil, or its non-homogeneous distribution along the width of the superconducting tape. However, depending on the physical quantity of interest and the error tolerance allowed between the numerical predictions and the experimental measurements, in this paper decision criteria are established for different regimes of the applied current, where the suitability of one or another model could be ensured, regardless of whether the actual magneto angular anisotropy properties of the superconducting tape are known.

The macroscopic electromagnetic behaviour of a type-II superconducting wire for alternating current power transmission under constant magnetic field conditions is captured by the numerical solution of the Maxwell equations under the framework of the critical state principle and the so-called integral formulation, also known as J-formulation. Time-dependent distributions for the flux front profiles of local current density, magnetic flux, and cycles of magnetic moment are presented. We have found that, regardless of the intensity of the applied magnetic field, the first cycle of Itr(t) defines the period of magnetic stabilization of the SC wire where two plateaus with constant magnetic moment can be measured. Then, a cyclic monotonic behaviour with well-defined flux-front boundaries has been identified, with clear signatures of current-like and field-like flux front profiles. This observation has allowed us to establish semi-analytical approaches of flux-tracking for the local dynamics of current density of AC SC wires immersed in a constant transverse magnetic field, from which all the macroscopical electromagnetic quantities of interest such as the magnetic field, magnetic moment, and power density of energy losses can be calculated. The observations reported for a rounded SC wire, define an adequate benchmark for the implementation of flux-tracking approaches in other 2D symmetries, such as SC strips, where the flux-front profile for isolated excitations can be formulated by exact geometrical expressions.

The AC losses induced by an alternating transport current in type-II superconductors is a well known phenomena which still attracts much attention due to its intrinsic relevance for the proper development of practical applications. In the case of single core superconducting cables of cylindrical crosssection, it is possible to find exact analytical solutions at self field conditions, and it has been believed for nearly two decades that the use of an ideal soft ferromagnetic sheath with negligible magnetization losses will not affect the electromagnetic properties of the superconducting wire, and on the contrary due to the shielding magnetic properties of the ferromagnet, the total AC losses of the SC wire have to be reduced or as maximum they must be equal to the one for the bare superconductor at self-field conditions, what contraries the experimental evidences that show a non-negligible increase on the AC losses. In this paper, we explain the physical nature of this mysterious increase on the AC losses for rounded superconducting/ferromagnetic heterostructures, which for the sake of generality, it has been solved within the critical state theory and, a magnetic multipolar expansion which enables the direct coupling of the magnetostatic properties of the superconductor and an ideal soft ferromagnet. A significant increase on the transient electric field during the excitation period has been observed, which might have utter implications on the adequate choosing of insulation materials for superconducting/ferromagnetic heterostructures.

Remarkable features on the magnetic moment of type-II superconducting wires of cylindrical shape, subjected to direct current conditions (DC) and transverse oscillating (AC) magnetic fields, are reported. We show how for relatively low amplitudes of the applied magnetic field, Ba, the superconducting wire rapidly develops a saturation state, |Mp|, characterizing the limits of magnetization loops that exhibit a Boolean-like behaviour. Regardless of the premagnetization state of the superconducting wire, we show how after two cycles of magnetic relaxation, boolean-like ±Mp states can be measured during the entire period of time from which the external magnetic field B0 ranges from 0 to ±Ba, with the signs rule defined by the sign of the slope ∆B0y(t). In addition, for the practical implementation of superconducting DC wires sharing the right of way with AC lines, we report that for relatively low values of magnetic field, Ba ≤ BP /2, being BP the analytical value for the full penetration field in absence of transport current, Itr, the use of semi-analytical approaches for the calculation of AC-losses leads to a significant underestimation of the actual contribution of the induction losses. This phenomena is particularly relevant at dimensionless fields ba < 1 − i 2/3 a , being ba = Ba/BP and, ia = Ia/Ic the amplitude of an AC or DC transport current, due to the local motion of flux front profiles being dominated by the occurrence of transport current. On the other hand, we have found that regardless of the nature of the transport current, either be DC or AC, when a transverse oscillating magnetic field greater than the classical limit ba = (1 − ia) is applied to the SC wire, the difference between the obtained AC losses in both situations results to be negligible indistinctly of the approach used, semi-analytical or numerical. Thus, the actual limits from which the estimation of the AC-losses can be used as an asset for the deployment of DC SC wires sharing the right of way with AC lines, against the sole use of SC wires for the transmission of AC transport current, are established.

Type-II superconductors are expected to be extensively used in the designing of DC power grids due to their reduced use of space, high transport current capability, and nearly-zero resistive losses. Nevertheless, these systems will have to share the right of way of the currently installed AC network, reducing the costs of development associated to the superconducting cables without the need of increasing the right of way. Under the theoretical framework of the critical state model and the numerical solution of Maxwell equations in the magneto quasi-steady approach, in this paper we present a comprehensive study of the effects of applying an AC transverse magnetic field to a type-II SC wire of rounded cross section it utilized for direct current power transmission. Our numerical results can be used as a practical benchmark for determining the minimal losses and magnetic impact of DC SC lines subjected to external oscillating magnetic fields. The local dynamics of the flux front profile of current density, the resulting density of magnetic flux, the total magnetic moment of the wire, and the curve of AC losses for different conditions of DC current and transverse AC magnetic field are presented. Our results are compared with simplified analytical approaches, demonstrating the importance of considering the concomitant action of DC current and AC magnetic field by the use of numerical methods, mainly at low values of applied magnetic field relative to the intensity of transport current and the full penetration profile of current density.