Theoretical analysis of the transport critical-state ac loss in arrays of superconducting rectangular strips Enric Pardo, 1 Alvaro Sanchez, 1 Du-Xing Chen, 2 and Carles Navau 1,3 1 Grup d’Electromagnetisme, Departament de Física, Universitat Autònoma Barcelona, 08193 Bellaterra (Barcelona), Catalonia, Spain 2 ICREA and Grup d’Electromagnetisme, Departament of Física, Universitat Autònoma de Barcelona, 08193 Bellaterra (Barcelona), Catalonia, Spain 3 Escola Universitària Salesiana de Sarrià, Passeig Sant Joan Bosco 74, 08017 Barcelona, Catalonia, Spain Received 2 February 2004; revised manuscript received 29 November 2004; published 25 April 2005 We present a systematic theoretical study of current profiles, magnetic field lines, and hysteresis ac loss in linearly arranged arrays of rectangular superconducting strips subjected to a transport current. Results are obtained by means of numerical calculations assuming the critical-state model with a constant critical current density. Because finite filament thickness and magnetic coupling effects are considered, we can provide some useful hints as to how to arrange the filaments in order to reduce the ac loss in actual superconducting tapes. DOI: 10.1103/PhysRevB.71.134517 PACS numbers: 74.25.Sv, 84.71.Mn I. INTRODUCTION Some of the important applications of superconductors are based on the possibility of having a high value of super- current transported through them. For hard type II supercon- ductors, the energy loss when carrying a static transport su- percurrent is very small but it can be significant when the current is alternating. The reduction of ac loss is of funda- mental importance for the application of superconductors to actual ac electrical devices, such as power transmission cables, ac magnets, and transformers. 1–3 In recent years, a lot of effort has been made in the pro- duction of high-temperature superconducting HTS compos- ite conductors, which are made of a superconducting core with a multifilamentary structure and a normal-metal con- ducting sheath or substrate. The most common HTS compos- ite conductors are Bi-2223/ Ag tapes, coated YBCO conduc- tors, and MgB 2 tapes and wires. 3,4 The understanding of ac loss in these multifilamentary conductors is not only useful to reduce the energy dissipation but also to characterize the superconductor material properties. 5,6 Moreover, the study of the ac loss for several magnetically interacted conductors is also useful for applications. 2,7–11 The critical-state model 12,13 has been shown to be a useful tool to describe the ac loss of superconducting wires. The model assumes that the current density has a magnitude J c wherever it is nonzero. It was first applied to analytically calculate the ac loss for simple geometries, such as a cylinder, 13 a circular tube, or an infinite slab. 14 In the early 1970s, Norris paved the way for some of the modern models by calculating the ac loss produced by a transport current in a thin strip and an elliptical cross-section wire. 15 Further the- oretical advances and the discovery of HTS materials moti- vated the study of other interesting geometries, such as a cylinder with two concentric circular shells with different J c Ref. 16 and some multifilamentary geometries, like vertical and horizontal arrays of an infinite number of thin strips 17 or double thin strips. 18 Apart from the mentioned analytical studies, some nu- merical models within the critical-state model have been de- veloped to describe superconductors with transport current, like those from Norris, 19 Fukunaga et al., 20–22 Däumling, 23 and Pardo et al. 24 These numerical methods restrict the cal- culation region to the superconducting volume only. The ge- ometry investigated numerically was a rectangular strip with arbitrary thickness, 20,23,24 after which extensive work on multifilamentary tapes was done by Fukunaga et al. 21,22 An alternative approach to the critical-state model is to assume a certain EJ dependence as E  J / J c  n , where E is the elec- trical field. An interesting model considering this assumption is that developed by Brandt 25 for superconducting strips in applied magnetic fields, extended for the transport current case by Rhyner 26 and Yazawa et al. 27 Again, this model re- quires numerical calculations inside the superconductor only. Other authors applied conventional finite-element techniques to multifilamentary tapes, such as Stavrev et al. 28 In spite of all the extensive theoretical work done on ac loss, the magnetic coupling behavior between superconduct- ing filaments has not been systematically studied. To cover this lack, in this work we present accurate numerical calcu- lations and discussions of current distribution, magnetic field lines, and ac loss for matrix arrays of rectangular strips. The matrix arrangement is found in many actual tapes 22,29,30 and is a simple geometry to study for the interaction between different tapes. The calculations are performed by a numerical model based on the critical-state model with a constant critical cur- rent density, which is presented in Sec. II. In Sec. III, we first present a systematic study of the magnetic interaction among rectangular strips in both horizontal and vertical arrays, and then we use such a study as a basis to understand the mag- netic interaction in a matrix array. A comparison of the model results with both experimental data and simplified analytical models is presented in that section. Finally, we summarize our conclusions in Sec. IV. II. MODEL We consider a set of infinitely long superconducting rect- angular strips of cross-sectional dimensions 2a  2b along PHYSICAL REVIEW B 71, 134517 2005 1098-0121/2005/7113/13451712/$23.00 ©2005 The American Physical Society 134517-1