Calculating transport AC losses in stacks of high temperature superconductor coated conductors with magnetic substrates using FEM

In this paper, the authors investigate the electromagnetic properties of stacks of high temperature superconductor (HTS) coated conductors with a particular focus on calculating the total transport AC loss. The cross-section of superconducting cables and coils is often modeled as a two-dimensional stack of coated conductors, and these stacks can be used to estimate the AC loss of a practical device. This paper uses a symmetric two dimensional (2D) finite element model based on the H formulation, and a detailed investigation into the effects of a magnetic substrate on the transport AC loss of a stack is presented. The number of coated conductors in each stack is varied from 1 to 150, and three types of substrate are compared: non-magnetic weakly magnetic and strongly magnetic. The non-magnetic substrate model is comparable with results from existing models for the limiting cases of a single tape (Norris) and an infinite stack (Clem). The presence of a magnetic substrate increases the total AC loss of the stack, due to an increased localized magnetic flux density, and the stronger the magnetic material, the further the flux penetrates into the stack overall. The AC loss is calculated for certain tapes within the stack, and the differences and similarities between the losses throughout the stack are explained using the magnetic flux penetration and current density distributions in those tapes. The ferromagnetic loss of the substrate itself is found to be negligible in most cases, except for small magnitudes of current. Applying these findings to practical applications, where AC transport current is involved, superconducting coils should be wound where possible using coated conductors with a non-magnetic substrate to reduce the total AC loss in the coil.     

Finite element analysis of AC loss in non-twisted Bi-2223 tape carrying AC transport current and/or exposed to DC or AC external magnetic field

AC losses in Bi-2223 superconducting tapes carrying AC transport current and/or exposed to DC or AC magnetic field are calculated with a numerical model based on the finite element method. Superconducting property is given by the E–J characteristic represented by a power law using equivalent conductivity. First, transport loss and magnetization loss are calculated numerically and compared with measured values. The calculated losses almost agree with the measured losses. Frequency dependencies of calculated and measured transport losses are compared with each other. Next, the influence of DC external magnetic field on the transport loss is studied. DC external magnetic field reduces n that is an exponent in the power law connecting resistivity and current density. The numerically calculated transport loss increases with increasing DC magnetic field. Finally, the total loss of superconducting tape carrying AC transport current in AC magnetic field is calculated. In the perpendicular magnetic field, the calculated total loss is lager than the sum of the transport loss and the magnetization loss, while they almost agree with each other in the parallel magnetic field.     

Reduced filament coupling in Bi(2223)/BaZrO3/Ag composite tapes

Multifilamentary Bi(2223)/Ag tapes often exhibit AC loss levels comparable to those measured in monofilamentary samples, which is partly due to the large coupling currents induced in the low resistive sheath material. Surrounding the individual filaments by electrically insulating barrier layers suppresses these currents and strongly reduces the coupling. We demonstrate this effect with various types of magnetic and self-field AC loss measurements on a series of Bi(2223)/BaZrO₃/Ag tapes. We also discuss the influence of barrier thickness, twist pitch length and filament arrangement on the measured losses in these composite conductors.     

Measuring transport current loss of BSCCO/Ag tapes exposed to external AC magnetic field

BSCCO/Ag tape superconductors are developed for electrical power applications at liquid nitrogen temperatures. In these applications, e.g., superconducting transformers and power cables, an AC transport current and an AC magnetic field are present at the same time. A set-up to measure the influence of external AC magnetic field on the transport current loss, i.e., the voltage drop across a sample supplied with an AC transport current, has been developed. The magnetic field can be applied both parallel and perpendicular to the broad side of the tape conductor. An increase of the transport current loss due to the external AC magnetic field is observed. When a DC external magnetic field is applied the increase of the self-field loss can be described well by the decrease of the critical current due to the magnetic field. In the case of an AC external magnetic field this is only a minor effect. For magnetic field amplitudes higher than a certain threshold value the transport current loss is described reasonably well by the self-field loss and a dynamic resistance contribution calculated from the DC voltage–current relation in AC magnetic field.     

The similarities in loss creation in LTS and HTS cables

The coupling current losses represent an essential contribution to AC losses in most practical superconducting conductors. The origin of this loss type is well known, being caused by induced currents in different loops consisting of superconducting and non-superconducting parts. However, the ‘current pattern' in different conductor types (strands, flat or round cables, more complicated cable structures, CICC) varies appreciably. These differences are mainly due to geometrical effects (size and shape of filaments and/or strands, their spatial distribution, conductor aspect ratio, demagnetization effects). Although the general knowledge about AC losses in low temperature cable structures is by far not complete (mainly at higher frequencies, in inhomogeneous fields and for inhomogeneous cable structure), an attempt is made to summarize those results which can be adopted to high T_c conductors and some remarks are made about new features of AC losses in these conductor types.     

Single-phase AC losses in prototype HTS conductors for superconducting power transmission lines

We report single-phase AC loss measurements on 8-, 4-, and 3-layer, multi-strand, HTS prototype conductors for power transmission lines. We use both calorimetric and electrical techniques. The agreement between the two techniques suggests that the interlayer current distribution in 1-m long conductors are representative of those in long conductors. The losses for the 8- and 4-layer conductors are in rough agreement, with the 8-layer losses being somewhat lower. The 3-layer conductor losses are substantially higher — probably due to unbalanced azimuthal currents for this configuration.     

AC losses in multifilamentary HTS-composite tapes based on BiPbSrCaCuO

Some results of AC loss measurements are presented for 19, 61, 127-filamentary Bi-2223/Ag tapes prepared by the ‘powder-in-tube' method. All measurements have been made at T=77 K under sinusoidal transport current with frequency in the range of 30–600 Hz and the current amplitude up to 30 A. The measurements have been carried out both in self field conditions and at the external magnetic field applied to the tape at the different angles. The dependencies of the AC losses on current amplitude and frequency have been obtained. It is found that for all tapes the current amplitude dependencies of the AC losses show good agreement with the Norris prediction for an elliptical or strip geometry. The AC loss dependencies on frequency were linear. The measurements of AC losses in external magnetic field show that the change of AC losses is only through the change of the critical current. So the transport AC losses in the tapes are the ‘saturation losses' that is they are different from classic hysteresis losses.     

Estimation of Jc(B) dependence from self-field alternating current (AC) losses measured on Bi-2223/Ag tapes

Transport AC losses measured in self-field conditions on multifilamentary Bi-2223 tapes are often found to be lower than those calculated within the framework of the critical state model for a bulk wire with elliptical cross section, though generally higher than predicted for a strip. This effect is sometimes ascribed to the non-ideal geometry of the tapes, which does not exactly reproduce either shape. Here we propose an alternative explanation assuming that the critical current density of superconducting material depends on magnetic field. In practice, we analyzed the AC loss curve and deduced different I_c values for the individual data points, using the standard Norris equation for elliptical conductor. This gives the relation between ‘calculated' I_c and the self-field associated to AC transport current, which can be regarded as an alternative way to qualify the dependence of J_c on magnetic field. Important is that this procedure covers the range of fields below the self-field at I_c where the measurement in background DC field can not be used to determine J_c(B).     

Computation of AC losses in high-temperature superconductors

A formulation for the computation of AC losses in technical HTS conductors by using commercial FEM packages developed for two-dimensional computation of electromagnetic problems is presented. The formulation takes into account the real current density–electric field characteristic of a conductor and the spatial dependence of the current density. Having presented the formulation, example runs comparing transport current loss behaviour between HTS and LTS conductors are given.     

Self-field AC losses of textured Bi2Sr2CaCu2O8+δ thin rods

Self-field AC losses of polycrystalline Bi-2212 thin rods textured by a Laser Floating Zone (LFZ) melting technique have been measured at 77 K. With the optimal processing parameters, these rods, of 1.6–2 mm diameter and 10 cm length, have a transport critical current density of 3 kA/cm² in the self-field which decreases to about 1.5 kA/cm² in fields of 0.02 T applied perpendicular to the rod axis. The self-field AC losses have been measured in DC magnetic fields up to 0.03 T. The measurements in zero field show that for a large current range the losses are dominated by hysteresis losses as described by the Critical State Model for a cylinder. For the measurements in DC fields the losses show an increasingly resistive-like dependence with current, while the hysteretic component expected from the CSM becomes less important. Measurements at different frequencies also indicated that the loss per cycle in fields is strongly frequency dependent.     

Numerical minimization of AC losses in coaxial coated conductor cables

Power cables are one of the most promising applications for the superconducting coated conductors. In the AC use, only small resistive loss is generated, but the removal of the dissipated heat from the cryostat is inefficient due to the large temperature difference. The aim of this work is to minimize the AC losses in a multilayer coaxial cable, in which the tapes form current carrying cylinders. The optimized parameters are the tape numbers and lay angles in these cylinders. This work shows how to cope with the mechanical constraints for the lay angles and discrete tape number in optimization. Three common types of coaxial cables are studied here to demonstrate the feasibility of optimization, in which the AC losses were computed with a circuit analysis model formulated here for arbitrary phase currents, number of phases, and layers. Because the current sharing is practically determined by the inductances of the layers, the optima were obtained much faster by neglecting the nonlinear resistances caused by the AC losses. In addition, the example calculations show that the optimal cable structure do not usually depend on the AC loss model for the individual tapes. On the other hand, depending on the cable type, the losses of the optimized cables may be sensitive to the lay angles, and therefore, we recommend to study the sensitivity for the new cable designs individually.     

Magnetic measurement of transport AC losses in Bi-2223/Ag tapes

Magnetic nature of the losses in superconducting wire carrying AC current implies that it should be possible to determine these losses in a contactless way. Ribbon-like samples are quite favorable for such an experiment, because a notable portion of magnetic flux related to losses ‘escapes' the sample volume and can be detected by an appropriate pick-up coil. In this case, a model describing the AC current penetration into the tape, based, e.g., on the critical state model, allows one to derive the losses from the pick-up coil signal. Because this signal is proportional to the number of coil turns, extension of the accessible range of measured voltages (and losses) can be achieved. We demonstrate the data obtained on a 1 cm long portion of a low-loss multifilamentary tape carrying AC current with frequency 35 Hz. The pick-up coil technique allowed us to reach loss level more than one order below the experimental limit for direct measurements.     

AC loss in a high-temperature superconducting coil

In a typical superconducting coil made of BSCCO/Ag tape, both amplitude and direction of the magnetic field determine the critical current, resistive voltage and AC loss. The distribution of the magnetic field along and across the superconducting tape in a coil is rather complex. This gives rise to the question: how accurate can one predict the critical current, V–I characteristic and AC loss of the AC coil from results of short sample measurements? To answer this question, we have measured and compared the characteristics of a short sample and a small coil employing 14 m of the same tape at 77 K. The comparison is performed as follows. First, a short sample is characterised with regard to the field dependence of the critical current, V–I characteristic and the AC loss. Second, the distribution of the magnetic field along the tape in a coil is accurately calculated. From the data, the voltage along the tape and the loss of the tape in the coil are found. Finally, the resistive voltage and the AC loss of the complete coil are calculated and compared to measured AC losses in the frequency range of 0 to 160 Hz, typical for power applications.     

AC losses and stability on large cable-in-conduit superconductors

The cable-in-conduit superconductors are preferred for applications where the AC losses and stability are a major concern, e.g., fusion magnets and SMES. A review of coupling currents loss results for both NbTi and Nb₃Sn cable-in-conduit conductors (CICC) is presented and the AC loss relevant features are listed, with special emphasis for the role of the interstrand resistance and strand coating. The transient stability approach for CICCs is discussed and the analytical models are quoted as well as the relevant experimental database. The likely spectrum of transient disturbance in CICC is reviewed and the need to account for interstrand current sharing in the design is outlined. Eventually a practical criterion for the interstrand resistance is proposed to link the stability and AC loss design.     

Transport and AC loss properties of the repaired multifilamentary REBCO superconducting tapes

For near-future applications of REBa₂Cu₃O₇ (REBCO) coated conductors to electric power cables, transformers and Superconducting Magnetic Energy Storage (SMES), the long taped wires with high performance in the transport properties have been designed and fabricated. Moreover, in order to drastically reduce AC losses in perpendicular field configuration, advanced multifilament YBCO coated conductors (MFYCCs) fabricated with technique of a laser scribing process have been also developed. In the present study, from engineering viewpoints to utilize such advanced conductors, we evaluated the transport and AC loss properties of short MFYCCs with a repaired part or a joint by a diffusion joint technique with the saddle-shaped pickup coil method.     

AC losses in multifilamentary Bi(2223) tapes with an interfilamentary resistive carbonate barrier

For the most common AC application frequencies, the main component of the AC losses in multifilamentary Bi(2223) tapes are caused by hysteresis- and coupling losses. These losses can be reduced enhancing the matrix resistivity and applying a twist to the filaments. We report on the AC loss properties of 37-filament tapes with AgAu (8 wt.%) matrix, and novel 19-filament tapes with SrCO₃ barriers between the filaments. We performed transport AC loss and magnetic AC loss measurements in parallel and perpendicular magnetic fields. Both kinds of tapes were also prepared with filament twists below a twist pitch of 20 mm. The influence of the different tape modifications on the AC loss behaviour is presented and compared with theoretical models to understand the effect of the resistive matrix. In the case of magnetic AC loss measurements, reduced AC losses due to decoupled filaments were observed for the twisted tapes with a resistive matrix in low parallel fields.     

A simple electromagnetic method of cyclic loss measurement for superconducting wires in a combined alternating transverse magnetic field and transport current

We measured cyclic losses in a superconducting wire, carrying alternating transport current, simultaneously exposed to an alternating transverse magnetic field. Samples of Bi-2223 Ag-sheathed tapes have configuration of a double-layer non-inductive coil, which itself is a pickup coil to measure the AC losses. Potential taps were attached to both terminals of the sample coil. The external field was applied along the axis of the sample coil. In this procedure, we can estimate an averaged Poynting's vector on a cylindrical surface between the two layers by means of signals from a pair of the potential taps and from pickup coils for the external magnetic field and the transport current. We can also measure a magnetization and an extended transport-current components of AC losses in addition to a total cyclic loss for a combined alternating external field and transport current. Obtained results are compared with numerical predictions of the critical state model taking into account the magnetic field dependence of critical current density.     

传输交流的Bi2223/Ag高温超导带在不同取向交流外场下的交流损耗

在高温超导的电力应用中，如超导电机、变压器等，多数情况下，高温超导带材在通以交流传输电流的同时还处于交变磁场中。此时，超导体的交流损耗不仅依赖于磁场的大小，还与磁场相对于超导带面的取向有关。本文在77K及工频50Hz情况下，实验研究了单根多芯Bi2223/Ag高温超导带及两带并联时的交流损耗随着外磁场与带面夹角的变化情况；以及交流磁场对临界电流的影响情况；并对单根带及两带并联的实验结果进行了比较与分析。     

高温超导YBCO带材的交流损耗的FEM仿真分析

对高温超导带材YBCO的交流损耗进行了仿真分析。运用有限元FEM方法,基于Maxwell方程组的数值建模,引入了基于实验数据的power-law对交流损耗进行计算。首先对自场下的交流损耗值进行了仿真计算并分析了带材的电流密度与磁场分布情况,并通过与实验测量数据做比较验证仿真方法的有效性。其次分别对不同幅值下的平行场与垂直场下的交流损耗进行了计算,将自场、平行场以及垂直场下的电流密度与磁场强度的分布以及交流损耗值进行了对比分析。     

AC loss of low temperature superconducting AC wire caused by longitudinal and azimuthal AC magnetic field components

We investigated the AC loss characteristics of a low temperature NbTi AC wire by measuring the AC transport current losses in the external AC magnetic field whose components are the longitudinal and transverse ones. The measurement results showed that the AC losses were significantly dependent on the directions and magnitudes of the external longitudinal field component. The AC losses caused by the longitudinal and azimuthal field components were estimated by our previously derived model. The theoretical results well explained the dependence of the AC losses on the longitudinal field components. It was also shown that the AC losses can be substantially reduced by the proper choice of the twisting way.