Mika Lyly

Comparison of three eddy current formulations for superconductor hysteresis loss modelling

As is well known, the superconductor hysteresis loss modelling problem may be formulated as an eddy current (EC) problem in which the resistivity of the superconducting region is modelled with a power law. We compare three EC formulations suitable for the modelling of superconductor hysteresis losses. Namely, the a?v?j-, T??- and h-formulations are discussed. We review these formulations, and through simulation results the properties of these formulations are discussed and their suitabilities for different modelling situations are compared. Special attention is paid to the h-formulation: we investigate the effects of the modelling decisions related to resistivity of the air region in an h-formulation based EC solver. According to the results, these decisions affect the energy distribution of the field solution and may even lead to seemingly contradictory behaviour.

An H-formulation-based three-dimensional hysteresis loss modelling tool in a simulation including time varying applied field and transport current: the fundamental problem and its solution

When analytic solutions are not available, finite-element-based tools can be used to simulate hysteresis losses in superconductors with various shapes. A widely used tool for the corresponding magnetoquasistatic problem is based on the H-formulation, where H is the magnetic field intensity, eddy current model. In this paper, we study this type of tool in a three-dimensional simulation problem. We consider a case where we simultaneously apply both a time-varying external magnetic field and a transport current to a twisted wire. We show how the modelling decisions (air has high finite resistivity and applied field determines the boundary condition) affect the current density distribution along the wire. According to the results, the wire carries the imposed net current only on the boundary of the modelling domain, but not inside it. The current diffuses to the air and back to the boundary. To fix this problem, we present another formulation where air is treated as a region with 0 conductivity. Correspondingly, we express H in the air with a scalar potential and a cohomology basis function which considers the net current condition. As shown in this paper, this formulation does not fail in these so-called AC-AC (time varying transport current and applied magnetic field) simulations.

Finite Element Simulations of Twisted NbTi Conductors

Low-temperature superconducting wires, NbTi and Nb3Sn, designed for ac applications, such as CERN and ITER magnets, are composed of twisted multifilament structures. Under time-varying applied magnetic field, twisting decreases the induced electromotive forces between the filaments and is therefore an effective method to reduce interfilamentary coupling. In order to study coupling losses computationally with high precision, 3-D numerical models are needed. In this work, we use 3-D finite-element method simulations to study hysteresis and coupling losses in NbTi superconductors. We investigate the effect of twist pitch on ac losses. In practice, NbTi wires cannot be studied at the filament level due to the extremely complex geometries. However, the manufacturing of these wires is done by using filament bundles. Therefore, we consider wires that consist of homogenized filament bundles embedded in normal conducting matrix. In particular, we consider the effect of barriers around filament bundles and how filaments should be arranged in bundles to minimize the losses. The simulations show that the qualitative behavior of the model is consistent with the analytical results and it can be used, e.g., in optimization processes, where a comparison of wire geometries is needed. Additionally, when considering the coupling of filaments, the barrier plays a very important role in minimizing the losses.

Validation of Homogenized Filament Bundle Model in AC Loss Computations

In high current ac applications, such as CERN and ITER magnets, superconducting cables are used. These cables typically consist of several individual strands and therefore have very complex geometries. Even a single NbTi strand designed for ac applications may consist of tens of thousands of filaments embedded in a normal conducting matrix. It is not practical to numerically model the cable or the strand at the filament level in the ac loss analysis due to extremely long computation times. However, an approximation for the ac losses can be computed if the filament zones or bundles are considered as a homogeneous mixture of superconducting filaments and matrix metal. This means, for example, substantial savings in the computation times of optimization procedures where hundreds of geometries are analysed. In this paper we compare the ac losses computed with both a homogenized bundle model and a detailed model (all or some of the bundles are modeled at the filament level) for different NbTi strand geometries to benchmark the accuracy of the bundle approximation. The results show that the bundle model is a suitable approximation, especially for certain bundle geometries.

Magnesium Diboride Wires With Nonmagnetic Matrices—AC Loss Measurements and Numerical Calculations

In the superconducting applications, the wires are exposed to time-varying magnetic field when the current changes. This generates losses which can be minimized by reducing filament size, twisting the wire, and increasing the transverse resistivity. However, the high losses of magnesium diboride wires often arise from magnetic sheath materials, and therefore, this work presents new type of wires with nonmagnetic matrix and multi-filamentary structure. The results of AC loss measurements, in external sinusoidal magnetic field, are presented. Two MgB2 samples were measured both in two temperature ranges, as two different set-ups were used, one with fixed LHe bath temperature 4.2 K. Second one enabled operation temperatures from 23 K up to the critical temperature of 39 K. Amplitude of magnetic field of the former set-up was up to 0.8 T and frequency range was from 0.1 to 1.4 Hz. In the latter one, the maximum amplitude was 28 mT, and the frequencies were 72 and 144 Hz. The results evidenced that the superconducting filaments were uncoupled and the measurements agreed with theoretical models based on this assumption. In practice, the uncoupling was modeled so that the net current in each filament was set to zero.

Current-Penetration Patterns in Twisted Superconductors in Self-Field

In a straight and untwisted superconducting wire, the current penetrates from the edges and flows along the wires axis. The generated self-field is in the wires cross-section plane and always perpendicular to the current. Beans critical state model (CSM) describes the mesoscopic current and magnetic field penetration in such a configuration. Also, by using power-law resistivity, an eddy current problem can be formulated to simulate such a case. However, when a conductor is twisted, the self-field is no longer perpendicular to the helicoidal trajectories of the filaments. When this non-perpendicularity of the magnetic field occurs, the current does not flow along the helicoidal lines of the filaments. If that was the case, the central regions of the filaments would not be shielded from the magnetic field. In this paper we investigate current penetration in a twisted configuration by means of numerical simulations. We used the finite element method to solve eddy current formulations where the resistivity in the superconducting regions was described with an isotropic power-law model.

An Optimization of a NbTi Wire Cross Section to Attain Low AC-Losses and High Critical Current

Removing the heat generated by AC-losses in large-scale superconducting applications has high costs. This is also the case in semi-DC magnets such as the ones of large particle accelerators where magnets are ramped synchronously with the beam energy. Also, these magnets require large cables made of several individual strands. This makes even the AC-loss analysis of such magnets very difficult. However, to attain low-loss full-size magnets, it becomes necessary to use wire designs that provide low AC-losses and adequate stability. The design of individual wires includes at least the determination of filament sizes and locations and the selection of matrix material. At the same time a high critical current and good workability must be ensured. With numerical analysis it is possible to consider these factors and design new strands and cables having low AC-losses for specific applications. In this paper we consider how to employ a numerical tool for the optimization of the NbTi wire cross-section to achieve low AC-loss and high Ic.

Measured Performance of Different Solders in Bi2223/Ag Current Leads

In superconducting magnet systems the current leads are usually divided into two parts. Normal metal like brass or copper is often used in the first part from the room temperature to the temperature of the radiation shield. The second part down to the magnet is made of high temperature superconductors (HTS). HTS leads can reduce the conductive heat load because they have poor thermal conductivity. Since HTS wires are lossless with direct current and have fair tolerance to the magnetic field only Ohmic losses are generated in the contact resistances at the current terminals. Thus, efficient current leads require that appropriate solders are used to reduce the contact resistances. In this paper, the thermal and contact resistances as well as thermal conduction losses of Bi-2223/Ag current leads are experimentally investigated using indium- and tin-based solders at operation temperatures between 20 and 77 K. Finally, the measured data is utilized to design an efficient HTS current lead to the current range of 0-1000 A.

Design Process for a NbTi Wire With New Specification Objectives: Technical Design Constraints and Optimization of a Wire Layout Considering Critical Current and AC Losses

New ac applications set high requirements for the performance of superconductors. In particular, in particle accelerators, where large magnets are frequently charged and discharged, low ac losses and high stability must be provided. Otherwise, the cooling costs become intolerable, or the reliability cannot be guaranteed. Although the NbTi conductors have been available for a long time, new wire configurations are needed to fulfill new high performance requirements. Development work of a superconducting wire requires adjusting many parameters, such as filament size and zones and the composition of matrix metal. In addition, there are some technical limits for different design parameters due to the manufacturing process. In this paper, we consider the design process and the design parameters and their limitations from the manufacturers point of view. As a part of this NbTi wire design, the use of numerical ac loss computation and an optimization method is demonstrated in designing of a cross-section layout for minimizing the ac losses and to achieve a high critical current. This optimization method is among the first steps of numerical optimization procedures, which considers the geometrical optimization, and more studies are needed before it could become a method of practical use. However, in such a case, substantial savings in time and costs in a design process of a new wire can be achieved.

A Time-Harmonic Approach to Numerically Model Losses in the Metal Matrix in Twisted Superconductors in External Magnetic Field

The first NbTi superconductor was developed in 1962 at Westinghouse. During 50 years, the manufacturing process of NbTi wires became highly optimized, and complex wire structures, which are needed in reducing ac losses, can be produced these days. Twisted multifilamentary structures generate many challenges from the modelers point of view. Considering numerical modeling, NbTi wires are too complicated to model with filament-level details. This is due to the high number and very nonlinear resistivity of filaments. Consequently, simplified approaches are needed for wire modeling. In this paper, a time-harmonic approach to model losses in the metal matrix is introduced. It is based on the linear approximation of filaments. The method to determine the linear resistivity of filaments is based on the definition of the skin depth and the radius of filaments. This can be done in advance without solving a nonlinear problem in 3-D. The suitability of the time-harmonic approach is benchmarked against the H-formulated 3-D eddy current model (ECM) with power-law resistivity. According to simulations, the losses in the metal matrix predicted by the time-harmonic approach agreed well with the nonlinear ECM when the filaments were uncoupled. The results achieved with the proposed approach similarly followed the effects of geometrical changes in the wire structure on losses as the nonlinear ECM and predicted the circumstances where the filaments became partially coupled. The simulation times were considerably lower with the new approach. From the manufacturers perspective, it is important to design conductors where the filaments stay uncoupled and have low ac losses in such a situation. Thus, the new approach can provide an effective design tool in developing new superconductors.