Roberto Brambilla

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

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 conductors 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.

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

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.

AC losses in thin superconductors: the integral equation method applied to stacks and windings

In this paper we present a method for computing transport current ac losses in interacting thin superconductors. The method solves the integral equations for the sheet current density distribution and is specifically developed for those configurations where the symmetry of the current density distributions allows writing the equation in a self-consistent form, without the need for using an auxiliary 2D model to describe the interaction between superconducting tapes. This results in very short computation times and therefore the model can be very useful for optimizing the design of superconducting devices. The method has been tested for different cases of practical applications and the ac loss results have been compared with those obtained with analytical models and with experiments.

Integral Equations for Computing AC Losses of Radially and Polygonally Arranged HTS Thin Tapes

In this paper, we derive integral equations for radially and polygonally arranged high-temperature superconductor thin tapes, and we solve them by finite-element method. The superconductor is modeled with a nonlinear power law, which allows the possibility of considering the dependence of the parameters on the magnetic field or the position. The ac losses are computed for a variety of geometrical configurations and for various values of the transport current. Differences with respect to existing analytical models, which are developed in the framework of the critical state model and only for certain values of the transport current, are pointed out.

Design and Development of 15 MVA Class Fault Current Limiter for Distribution Systems

As a consequence of electric power systems growth and further interconnection, the fault current levels increase. Therefore there is the interest in devices which are able to limit fault currents. The superconducting fault current limiter (SFCL) is an innovative device able to limit the short-circuit currents to lower levels, so that no component or underrated switching equipment in the system becomes overloaded but can be operated safety. In this work, we report on design and development of a 9 kV/15 MVA-class SFCL for MV distribution system, based on 1G and 2G HTS. The project final goal is to develop a SFCL for a MV distribution system, including design, fabrication, test at a high power test facility, installation and operation in the Milan area distribution network. AC losses for different HTS winding arrangements and SFCL short-circuit simulation results for two selected network case studies are reported and discussed.

Edge and top/bottom losses in non-inductive coated conductor coils with small separation between tapes

In this paper we consider two different finite-element models for computing ac losses in coils composed of coated conductors: a 2D model based on solving Maxwells equations by means of edge elements and a 1D model based on solving the integral equations for the current density in the tapes. The models are tested for a configuration of practical interest, a non-inductive solenoidal coil for fault current limiter applications. We focused our attention on the conditions when differences between the two models are expected to emerge, for example when the tapes are closely packed or when the dependence of the critical current density on the local magnetic field is taken into account. We present and discuss several cases, offering possible explanations for the observed differences of ac loss values.

Current Density Distribution in Multiple YBCO Coated Conductors by Coupled Integral Equations

In this paper we present a new model to compute the current density distribution and the AC losses in multiple YBCO coated conductors by means of coupled integral equations implemented in a finite-element (FE) environment. The model considers the superconducting tapes as 1D objects and allows overcoming the problem of the mesh of conductors of large aspect ratio, typical of 2D implementations; as a result, it is also much faster. The interaction between the different conductors is taken into account by means of a coupling of the electromagnetic variables. The magnetic field distribution in the space between the conductors is computed by coupling the 1D model with a standard 2D magnetic model, which uses the computed current density distributions as sources for the magnetic field. Results are compared with those obtained with standard 2D implementations and with experimental results.

Critical state solution and alternating current loss computation of polygonally arranged thin superconducting tapes

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 Ic, 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.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 Ic, 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.

The critical state in thin superconductors as a mixed boundary value problem: analysis and solution by means of the Erdélyi–Kober operators

With this paper, we provide an effective method to solve a large class of problems related to the electromagnetic behavior of thin superconductors. Here, all the problems are reduced to finding the weight functions for the Green integrals that represent the magnetic field components; these latter must satisfy the mixed boundary value conditions that naturally arise from the critical state assumptions. The use of the Erdélyi–Kober operators and of the Hankel transforms (and mostly the employment of their composition properties) is the keystone to unify the method toward the solution. In fact, the procedure consists always of the same steps and does not require any peculiar invention. For this reason, the method, here presented in detail for the simplest cases that can be handled in analytical way (two-part boundary), can be directly extended to many other more complex geometries (three or more parts), which usually will require a numerical treatment. In this paper, we use the operator technique to derive the current density and field distributions in perfectly conducting and superconducting thin discs and tapes subjected to a uniform magnetic field or carrying a transport current. Although analytical expressions for the field and current distributions have already been found by other authors in the past by using several other methods, their derivation is often cumbersome or missing key details, which makes it difficult for the reader to fully understand the derivation of the analytical formulas and, more importantly, to extend the same methods to solve similar new problems. On the contrary, the characterization of these cases as mixed boundary conditions has the advantage of referring to an immediate and naïve translation of physics into a consistent mathematical formulation whose possible extension to other cases is self-evident.

Analysis of magnetic field and geometry effects for the design of HTS devices for AC power applications

The performance of HTS devices is strongly influenced by local values of the current and field distributions. In this paper, we investigate the influence of the magnetic field and the geometrical configuration on the loss behavior of a 200 kVA FCL prototype, composed by Bi-2223/Ag tapes wound around a cylindrical support. The investigation is performed by means of finite element computations, with the use of an axisymmetric 2D A-V formulation for taking into account the cylindrical geometry. The electrical behavior of the superconductor is described by means of a B-dependent E-J power-law relation, derived from experimental measurements with a field of different orientation. Several geometrical configurations are analyzed and compared, in order to find the ones with the lowest AC loss.