Finite element method analysis of the coupling effect between superconducting filaments of different aspect ratio

Using 3D finite element method (FEM) modelling, the present work investigates the coupling effect between two finite superconductors through a resistive matrix. This effect, related to the finite length of the conductors, is typically three-dimensional and cannot be analyzed by the widely utilized 2D models. Superconductors are modelled with the non linear power law E=Ec(J/Jc)^n, which has been implemented in the FEM software. The main focus is to demonstrate the feasibility of such calculations and to establish the correlation between the coupling effect and the aspect ratio of the conductor cross-section, in order to extend the existing theory, which is only precise for superconductors of infinite slabs, fully penetrated by the magnetic field. The effect of other parameters such as the conductor length and the gap between the superconducting filaments is also considered. The latter is a parameter which does not feature in the approximate theories.

Numerical modelling of high temperature superconducting tapes and cables

Francesco Grilli

This Ph.D. thesis is focused on the numerical modelling of high-Tc superconductors (HTS) at the operating temperature of 77 K (liquid nitrogen). The purpose of numerical modelling is to precisely calculate the current and field distributions inside HTS devices (tapes, cables) and the corresponding AC losses, which are one of the most important limiting factors for a large-scale application of such materials. From the electrical point of view, superconductors are characterized by a strongly non-linear voltage-current relation, which defines the transition from superconducting to normal state. In the case of HTS, the steepness of this transition is smoother than for low-Tc superconductors (LTS), so that the commonly used critical state model (CSM) gives a too simplified representation of their electromagnetic behaviour and can be used for a qualitative description only. In this thesis the finite element method (FEM) has been used for precisely computing the current and field distributions as well as for evaluating the AC losses in HTS devices. The superconducting transition is modelled with a power-law relation, E(J) = Ec(J/Jc)n, which is derived from the fit of transport measurements. Firstly, the results obtained with the software package FLUX3D on multi-filamentary tapes have been validated by means of a comparison with the ones obtained by another software package (FLUX2D). The results from FLUX2D, having already been successfully compared with experimental measurements within the framework of two previous Ph.D. theses at LANOS, have been used as reference. Secondly, two power-law models, which take into account the spatial variation of the critical current density inside HTS tapes and its strongly anisotropic dependence on the magnetic field, have been implemented in FLUX3D. In most cases, this latter dependence is extremely important, since the transport capacity of the superconductor is considerably reduced (and its power loss sensibly increased) by the presence of a magnetic field. Afterwards, FEM modelling of HTS tapes has been extended to cables. HTS cables are in general composed by different layers of several tapes and have a quite complex three-dimensional structure: in fact, the layers are wound around a central cylindrical support with different pitch lengths and relative winding orientations, in order to obtain a uniform repartition of the transport current among the layers, which minimizes the total AC losses. For overcoming the difficulties of a direct 3D FEM simulation, a simple electrical model, which allows to find the optimal pitch lengths and whose results are the input data for a 2D FEM evaluation of the AC losses, has been developed. FEM computations have also been used to investigate the influence of the non-uniformity of the tape properties (contact resistance, critical current, power index) on the global loss performance of a single-layer HTS cable. As an alternative to FEM computations, an equivalent circuit model of HTS cables has been utilized. It describes the cable from the macroscopic point of view and allows to compute the current repartition among the layers and the corresponding AC losses, without the necessity of having detailed information about individual tapes. In the framework of the European project BIG-POWA, I have collaborated to the assembling process of a HTS power-link at the Pirelli Labs, in the Milan region, Italy. Finally, 3D simulations have been performed to extensively study the coupling effect between two superconducting filaments via the resistive matrix, which is a typical case where analytical solutions exist for very peculiar geometries and physical conditions only. FEM simulations have been utilized to study the dependence of the filament coupling on the physical and geometrical parameters of the conductors.

EPFL2004

Finite Element Method Modelling of Superconductors: from 2D to 3D

Bertrand Dutoit, Francesco Grilli

A 3D numerical modelling technique for solving problems involving superconducting materials is presented. The model is implemented in finite element method software and is based on a recently developed 3D formulation for general electro-magnetic problems with solid conductors. It has been adapted for modelling of superconductors with non-linear resistivity in 3D, characterized by a power-law E-J relation. It has firstly been compared with an existing and verified 2D model: compared are the current density distribution inside the conductors and the self-field ac losses for different applied transport currents. Secondly, the model has been tested for computing the current distribution with typical 3D geometries, such as corner-shaped and twisted superconductors. Finally, it has been used with two superconducting filaments in the presence of external magnetic field for verifying the existence of coupling currents. This effect deals with the finite length of the conductors and cannot be taken into account by 2D models.

2004

3D Modeling of Coupling Between Superconducting Filaments via Resistive Matrix in AC Magnetic Field

Bertrand Dutoit, Francesco Grilli, Yong Yang

The ac loss of superconducting composite depends strongly on coupling between superconducting filaments via the resistive matrix. The established technique for loss reduction using twisted filaments relies on the decoupling of the filaments below a critical coupling field Bc, which increases with the reduction of the twist pitch and the matrix conductivity. Although the concept of Bc may be clearly demonstrated using two infinite slabs of finite length, further details on its correlation with the filament/conductor geometry are not yet available. The main obstacle is due to the fact that any accurate analysis of such a problem must be carried out in 3d. In this paper, we describe the initial results from 3d modeling using Cedrat‘s Flux3D, for which a superconductor module for handling power-law E-J characteristics was developed. Using a simple model of two rectangular superconductors connected through a normal metal, we demonstrate the feasibility for quantitative modeling of their coupling behavior over a wide range of field sweep rates for different conductor geometries. Typical examples were given for cases not addressed by the existing approximate theory, as well as for the evolution of field profile for varying field sweep rate. ample of a very low resistivity in the matrix, it is also suggested that onset of coupling is strongly dependent on the shape of the conductor and individual filaments. Qualitatively speaking, conductors with a large aspect ratio have a greater tendency to couple due to a larger demagnetising effect, as confirmed by experimental studies. While the mechanism that promotes the coupling of the filaments through the resistive matrix in ac magnetic fields is well understood conceptually, quantitative calculation of the critical coupling field is only limited to the simplified scenario of infinite slabs. The main difficulty is that we are dealing with a 3d problem with conductors of finite length and induced currents flowing both along and perpendicular to the long axis. Fortunately 3d calculations of superconductors in magnetic fields are becoming a reality thank to the rapid enhancement in computing power to desktop computers and progress in the implementation of numerical techniques for modeling superconductors.