journal article·2025-09-25Open access

Modelling Magnetisation and Transport AC Loss of HTS Tapes near Ferromagnetic Materials Using an Integral Equation Method

Calvin C. T. Chow, K. T. Chau, Francesco Grilli

Applied Sciences· Volume 15· Issue 19

Abstract

The integral equation formulation of Maxwell’s equations proposed by Brandt provides an alternative to the H and T-A formulations for modelling high-temperature superconducting (HTS) tapes. A modified version of Brandt’s method in the literature models ferromagnetic domains near the tapes by considering the ferromagnetic domains as equivalent surface current. This paper extends this method by including the effect of external magnetic field acting on the ferromagnetic and HTS domains. The proposed method is used on a benchmark problem, which considers an HTS tape with a ferromagnetic substrate under an external time-varying magnetic field. The results agree closely (error in average ac loss less than 3%) with the widely-used T-A formulation implemented in COMSOL down to 2 mT. In addition, the proposed method is also applied to HTS tapes carrying transport ac current in a slot of a machine’s stator iron core, and HTS tapes in a stator iron slot in a machine under working conditions. It is found that ac loss calculated by the proposed method increases as the discretization size of the ferromagnetic material’s boundary decreases, and overshoots the value calculated by the T-A formulation in COMSOL when using very fine discretization.
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Research context

OpenAlex topics: Magnetic Properties and Applications · Electric Motor Design and Analysis · Superconducting Materials and Applications

SciVX discovery tags are terminology-based. OpenAlex topics are source-provided classifications.

Affiliations

Research Centre for Electric Vehicles and Department of Electrical and Electronic Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong SAR, China

Institute for Technical Physics (ITEP), Karlsruhe Institute of Technology (KIT), 76131 Karlsruhe, Germany

Funding

Hong Kong Research Grants Council · The Hong Kong Polytechnic University

References (52)

Show source-provided references
1. High-temperature superconductors and their large-scale applications10.1038/s44287-024-00112-y ↗
2. Paidpilli, M., and Selvamanickam, V. (2022). Development of RE-Ba-Cu-O superconductors in the U.S. for ultra-high field magnets. Supercond. Sci. Technol., 35.10.1088/1361-6668/ac5162 ↗
3. Yanagisawa, Y., Hamada, M., Hashi, K., and Maeda, H. (2022). Review of recent developments in ultra-high field (UHF) NMR magnets in the Asia region. Supercond. Sci. Technol., 35.10.1088/1361-6668/ac5644 ↗
4. High temperature superconducting rotating electrical machines: An overview10.1016/j.egyr.2022.11.173 ↗
5. Mitchell, N., Zheng, J., Vorpahl, C., Corato, V., Sanabria, C., Segal, M., Sorbom, B., Slade, R., Brittles, G., and Bateman, R. (2021). Superconductors for fusion: A roadmap. Supercond. Sci. Technol., 34.10.1088/1361-6668/ac0992 ↗
6. Ferracin, P., Ambrosio, G., Arbelaez, D., Brouwer, L., Barzi, E., Cooley, L., Garcia Fajardo, L., Gupta, R., Juchno, M., and Kashikhin, V. (2022). Towards 20 T hybrid accelerator dipole magnets. IEEE Trans. Appl. Supercond., 32.10.1109/TASC.2022.3152715 ↗
7. Zhang, G., Wang, H., Qiu, Q., Zhang, Z., Xiao, L., and Lin, L. (2020). Recent progress of superconducting fault current limiter in China. Supercond. Sci. Technol., 34.10.1088/1361-6668/abac1f ↗
8. Yazdani-Asrami, M., Seyyedbarzegar, S., Sadeghi, A., de Sousa, W.T.B., and Kottonau, D. (2022). High temperature superconducting cables and their performance against short circuit faults: Current development, challenges, solutions, and future trends. Supercond. Sci. Technol., 35.10.1088/1361-6668/ac7ae2 ↗
9. Diyun, W., Chau, K.T., Chunhua, L., Shuang, G., and Fuhua, L. (2012). Transient stability analysis of SMES for smart grid with vehicle-to-grid operation. IEEE Trans. Appl. Supercond., 22.10.1109/TASC.2011.2174572 ↗
10. Numerical evaluation of ac losses in HTS wires with 2D FEM formulated by self magnetic field10.1109/TASC.2003.812415 ↗
11. Pecher, R., McCulloch, M., Chapman, S.J., Prigozhin, L., and Elliott, C.M. (2003). 3D-modelling of bulk type-II superconductors using unconstrained H-formulation. Inst. Phys. Conf. Ser., 181.
12. Shen, B., Grilli, F., and Coombs, T. (2020). Review of the ac loss computation for HTS using H formulation. Supercond. Sci. Technol., 33.10.1088/1361-6668/ab66e8 ↗
13. Liang, F., Venuturumilli, S., Zhang, H., Zhang, M., Kvitkovic, J., Pamidi, S., Wang, Y., and Yuan, W. (2017). A finite element model for simulating second generation high temperature superconducting coils/stacks with large number of turns. J. Appl. Phys., 122.10.1063/1.4995802 ↗
14. Zhang, H., Zhang, M., and Yuan, W. (2017). An efficient 3D finite element method model based on the T–A formulation for superconducting coated conductors. Supercond. Sci. Technol., 30.10.1088/1361-6668/30/2/024005 ↗
15. Huber, F., Song, W., Zhang, M., and Grilli, F. (2022). The T-A formulation: An efficient approach to model the macroscopic electromagnetic behaviour of HTS coated conductor applications. Supercond. Sci. Technol., 35.10.1088/1361-6668/ac5163 ↗
16. Superconductors of finite thickness in a perpendicular magnetic field: Strips and slabs10.1103/PhysRevB.54.4246 ↗
17. Calculation of AC losses in HTSC wires with arbitrary current voltage characteristics10.1016/S0921-4534(98)00430-4 ↗
18. Numerical modeling of the critical state and calculation of ac losses and current profiles in multi-filamentary Bi-2223 tapes
19. The voltage signal on a superconducting wire in ac transport10.1088/0953-2048/18/5/019 ↗
20. Losses in Bi-2223/Ag tape at simultaneous action of ac transport and ac magnetic field shifted in phase10.1088/1742-6596/43/1/016 ↗
21. Influence of gaps in monolayer superconducting cable on ac losses10.1088/1742-6596/43/1/219 ↗
22. AC loss analysis on high-temperature superconductors with finite thickness and arbitrary magnetic field dependent voltage–current relation10.1063/1.368825 ↗
23. Nguyen, D.N., Sastry, P.V.P.S.S., Knoll, D.C., Zhang, G., and Schwartz, J. (2005). Experimental and numerical studies of the effect of phase difference between transport current and perpendicular applied magnetic field on total ac loss in Ag-sheathed (Bi,Pb)2Sr2Ca2Cu3Ox tape. J. Appl. Phys., 98.10.1063/1.2064314 ↗
24. Nguyen, D.N., Sastry, P.V.P.S.S., and Schwartz, J. (2007). Numerical calculations of the total ac loss of Cu-stabilized YBa2Cu3O7-δ coated conductor with a ferromagnetic substrate. J. Appl. Phys., 101.10.1063/1.2712180 ↗
25. Zhong, J., Zou, S., Lai, L., Chen, P., and Deng, S. (2022). Fast evaluation of the critical current of high-temperature superconducting coils based on the integral method. J. Appl. Phys., 132.10.1063/5.0112003 ↗
26. Lucchini, F., Torchio, R., Morandi, A., and Dughiero, F. (2024). A Fast Integral Equation J-φ Formulation for Superconducting Structures. IEEE Trans. Appl. Supercond., 34.10.1109/TASC.2024.3366189 ↗
27. Zhang, H., Kang, X., Song, P., Guan, M., Qu, T., and Wang, X. (2024). An advanced computational approach for evaluating screening-current-induced strain in REBCO HTS coils considering electromagnetic-mechanical coupling. IEEE Trans. Appl. Supercond., 34.10.1109/TASC.2023.3345294 ↗
28. Kang, X., Zhang, H., Tong, Y., and Wang, X. (2025). Bidirectionally coupled electromagnetic modeling of inductive behavior and screening currents in HTS no-insulation coils. Supercond. Sci. Technol., 38.10.1088/1361-6668/adedbf ↗
29. Elbaa, M., Berger, K., Douine, B., Halit, M., Ailam, E.H., and Bentridi, S.E. (2018). Analytical modeling of an inductor in a magnetic circuit for pulsed field magnetization of HTS bulks. IEEE Trans. Appl. Supercond., 28.10.1109/TASC.2018.2809438 ↗
30. Otten, S., and Grilli, F. (2019). Simple and fast method for computing induced currents in superconductors using freely available solvers for ordinary differential equations. IEEE Trans. Appl. Supercond., 29.10.1109/TASC.2019.2949240 ↗
31. Otten, S., Kario, A., Demenčik, E., Nast, R., and Grilli, F. (2020). Anisotropic monoblock model for computing ac loss in partially coupled Roebel cables. Supercond. Sci. Technol., 33.10.1088/1361-6668/ab9939 ↗
32. Musso, A., Breschi, M., Ribani, P.L., and Grilli, F. (2021). Analysis of ac loss contributions from different layers of HTS tapes using the A–V formulation model. IEEE Trans. Appl. Supercond., 31.10.1109/TASC.2020.3033587 ↗
33. Lai, L., and Gu, C. (2021). AC loss calculation in REBCO coils or stacks by solving the equation of motion for current using an integration approach. Supercond. Sci. Technol., 34.10.1088/1361-6668/abc567 ↗
34. Lai, L., and Gu, C. (2022). J model for studying ac magnetization loss in 3D cable structures. Supercond. Sci. Technol., 35.10.1088/1361-6668/ac55f4 ↗
35. Kang, X., and Wang, X. (2023). A homogenised anisotropic J-model for accelerating computations of screening current profile in large-scale HTS magnets. Supercond. Sci. Technol., 36.10.1088/1361-6668/acb66f ↗
36. Chow, C.C.T., Grilli, F., and Chau, K.T. (2023). Numerical modelling of HTS tapes under arbitrary external field and transport current via integral method: Review and application to electrical machines. Supercond. Sci. Technol., 36.10.1088/1361-6668/ace701 ↗
37. Chow, C.C.T., Zhang, M., Chau, K.T., and Grilli, F. (2024). Modeling HTS coils with different coupling scenarios via integral method. IEEE Trans. Appl. Supercond., 34.10.1109/TASC.2024.3450991 ↗
38. Gottardi, E., Krop, D.C.J., and Lomonova, E.A. (2025). A hybrid model for studying ac losses in HTS tapes of a linear machine. J. Phys. Conf. Ser., 3054.10.1088/1742-6596/3054/1/012011 ↗
39. Kalsi, S. (2021). Influence of magnetic iron teeth in motors employing superconductors for field excitation and ac stator windings. IEEE Trans. Appl. Supercond., 31.10.1109/TASC.2021.3062485 ↗
40. Patel, A., Climente-Alarcon, V., Baskys, A., Glowacki, B.A., and Reis, T. (2018, January 7–9). Design considerations for fully superconducting synchronous motors aimed at future electric aircraft. Proceedings of the 2018 IEEE International Conference on Electrical Systems for Aircraft, Railway, Ship Propulsion and Road Vehicles & International Transportation Electrification Conference (ESARS-ITEC), Nottingham, UK.10.1109/ESARS-ITEC.2018.8607734 ↗
41. Lee, C.H.T., Chau, K.T., Liu, C., Ching, T.W., and Chen, M. (2015). A new magnetless flux-reversal HTS machine for direct-drive application. IEEE Trans. Appl. Supercond., 25.10.1109/TASC.2015.2388693 ↗
42. Wang, Y., Sun, J., Zou, Z., Wang, Z., and Chau, K.T. (2013). Design and analysis of a HTS flux-switching machine for wind energy conversion. IEEE Trans. Appl. Supercond., 23.10.1109/TASC.2013.2242113 ↗
43. A novel coaxial magnetic gear using bulk HTS for industrial applications10.1109/TASC.2010.2040609 ↗
44. Zhu, H., Ma, J., Feng, B., Zhang, Z., and Mellor, P. (2025). J-Φ coupled model based on integral methods and magnetic networks for superconductor. Supercond. Sci. Technol., 38.10.1088/1361-6668/adcd04 ↗
45. Statra, Y., Menana, H., and Douine, B. (2021). Semianalytical modeling of ac Losses in HTS stacks near ferromagnetic parts. IEEE Trans. Appl. Supercond., 31.10.1109/TASC.2020.3033697 ↗
46. Statra, Y., Menana, H., and Douine, B. (2022). Integral modeling of ac losses in HTS tapes with magnetic substrates. IEEE Trans. Appl. Supercond., 32.10.1109/TASC.2021.3139383 ↗
47. (2025, August 22). Benchmark #2. Available online: https://htsmodelling.com/benchmark-2.
48. Computation of 2-D current distribution in superconductors of arbitrary shapes using a new semi-analytical method10.1109/TASC.2007.902117 ↗
49. Assessment of the computational performances of the semi-analytical method (SAM) for computing 2-D current distributions in superconductors10.1109/TASC.2009.2019582 ↗
50. Sadiku, M.N.O. (2015). Elements of Electromagnetics, Oxford University Press. [6th ed.].
51. Benkel, T., Lao, M., Liu, Y., Pardo, E., Wolfstadter, S., Reis, T., and Grilli, F. (2020). T-A-formulation to model electrical machines with HTS coated conductor coils. IEEE Trans. Appl. Supercond., 30.10.1109/TASC.2020.2968950 ↗
52. Pardo, E., Grilli, F., Liu, Y., Wolftadler, S., and Reis, T. (2019). AC loss modeling in superconducting coils and motors with parallel tapes as conductor. IEEE Trans. Appl. Supercond., 29.10.1109/TASC.2019.2899148 ↗