volume | PIER Journals

Abstract

Accurate prediction of magnetic forces in air-core coil systems is essential for the design and optimization. Accurate magnetic-force prediction in air-core coil systems is essential for electromagnetic actuators, wireless power transfer, inductive coupling, magnetic calibration, and coil-based positioning systems. This paper presents a software-validated computational framework for predicting axial and radial magnetic forces in practical air-core coil systems under aligned and misaligned conditions. The framework extends ideal coaxial coil-force analysis by including lateral offset, angular tilt, finite coil dimensions, discrete winding geometry, helical winding pitch, finite conductor diameter, and nonuniform conductor placement. Three complementary physics-based methods are implemented and compared: a semi-analytical elliptic-integral method for rapid coaxial and near-coaxial evaluation, a direct numerical-integration method based on the Lorentz-force interaction integral, and a filament-based discrete-winding method for practical conductor geometry. Independent Ansys Maxwell simulations are used as the finite-element software-validation reference under identical geometric and electrical conditions. This work is limited to software-based validation; physical experimental validation using real coil pairs is identified as future work. The validation results show strong agreement among the proposed methods and Ansys Maxwell simulations. The final software mesh-independence change is 0.15%, the boundary-domain sensitivity is 0.12%, and the maximum relative half-spread among the physics-based methods is 0.55%. The maximum axial-force deviation in the coaxial separation sweep is 0.62%, and the radial-force sign-symmetry error remains below 0.23%. Including practical winding details reduces the model-to-software vector error from 2.87% for the continuous-current approximation to 0.10% for the full practical winding model. A feedforward neural-network surrogate model trained using 12,000 validated samples is also introduced for rapid interpolation within the sampled design domain. The surrogate achieves R²=0.996-0.998, mean relative error below 1.20%, and maximum relative error of 5.00% on unseen test data. The proposed framework provides an accurate, efficient, and physically interpretable tool for axial and radial force prediction in practical misaligned air-core coil systems.

1. Grover, F. W., Inductance Calculations: Working Formulas and Tables, Dover Publications, New York, NY, USA, 2004.

2. Babic, Slobodan I. and Cevdet Akyel, "Calculating mutual inductance between circular coils with inclined axes in air," IEEE Transactions on Magnetics, Vol. 44, No. 7, 1743-1750, 2008.
doi:10.1109/tmag.2008.920251 Google Scholar

3. Rosa, E. B., "The self and mutual inductances of linear conductors," Bulletin of the Bureau of Standards, Vol. 4, No. 2, 301-344, 1908.
doi:10.6028/bulletin.088 Google Scholar

4. Kirsch, Andreas and Frank Hettlich, The Mathematical Theory of Time-harmonic Maxwell's Equations: Expansion, Integral, and Variational Methods, Springer, Cham, Switzerland, 2016.
doi:10.1007/978-3-319-11086-8

5. Sondhi, K., N. Garraud, D. Alabi, D. P. Arnold, A. Garraud, S. G. R. Avuthu, Z. H. Fan, and T. Nishida, "Flexible screen-printed coils for wireless power transfer using low-frequency magnetic fields," Journal of Micromechanics and Microengineering, Vol. 29, No. 8, 084006, 2019.
doi:10.1088/1361-6439/ab26ff Google Scholar

6. Xu, Xiaoyong, Zhen Huang, Wan Li, Fangliang Dong, Luning Hao, Boyang Shen, and Zhijian Jin, "Study on reducing the maximum perpendicular magnetic field of HTS coils used on synchronous generator armatures," IEEE Transactions on Applied Superconductivity, Vol. 29, No. 5, 1-5, 2019.
doi:10.1109/tasc.2019.2901585 Google Scholar

7. Pankrac, Vitezslav, "The algorithm for calculation of the self and mutual inductance of thin-walled air coils of general shape with parallel axes," IEEE Transactions on Magnetics, Vol. 48, No. 5, 1875-1889, 2012.
doi:10.1109/tmag.2011.2177854 Google Scholar

8. Lemarquand, Guy, Valérie Lemarquand, Slobodan Babic, and Cevdet Akyel, "Magnetic field created by thin wall solenoids and axially magnetized cylindrical permanent magnets," PIERS Proceedings, 614-618, Moscow, Russia, Aug. 2009.

9. Braneshi, M., O. Zavalani, and A. Pijetri, "Evaluation of axial force between two coaxial disk coils," Proceedings of the 3rd International PhD Seminar on Computational Electromagnetics, Ghent, Belgium, 2006.

10. Ravaud, R., G. Lemarquand, S. Babic, V. Lemarquand, and C. Akyel, "Cylindrical magnets and coils: Fields, forces, and inductances," IEEE Transactions on Magnetics, Vol. 46, No. 9, 3585-3590, 2010.
doi:10.1109/tmag.2010.2049026 Google Scholar

11. Maxwell, James Clerk, A Treatise on Electricity and Magnetism, Clarendon Press, Oxford, U.K., 1873.
doi:10.1017/cbo9780511709340

12. Abramowitz, M. and I. A. Stegun, Handbook of Mathematical Functions, Dover Publications, New York, NY, USA, 1965.

13. Conway, John T., "Exact solutions for the mutual inductance of circular coils and elliptic coils," IEEE Transactions on Magnetics, Vol. 48, No. 1, 81-94, 2012.
doi:10.1109/tmag.2011.2161768 Google Scholar

14. Palka, Ryszard, "Fast analytic-numerical algorithms for calculating mutual and self-inductances of air coils," Energies, Vol. 17, No. 2, 325, 2024.
doi:10.3390/en17020325 Google Scholar

15. Babic, Slobodan, Eray Guven, Kai-Hong Song, and Yao Luo, "Optimized calculation of radial and axial magnetic forces between two non-coaxial coils of rectangular cross-section with parallel axes," Computation, Vol. 12, No. 9, 180, 2024.
doi:10.3390/computation12090180 Google Scholar

16. Wang, Yiming, Xu Xie, and Hengfeng Wang, "Spatial magnetic field calculations for coreless circular coils with rectangular cross-section of arbitrary turn numbers," Progress In Electromagnetics Research M, Vol. 101, 9-23, 2021.
doi:10.2528/pierm21010802 Google Scholar

17. Giovannetti, Giulio, Nunzia Fontana, Alessandra Flori, Maria Filomena Santarelli, Mauro Tucci, Vincenzo Positano, Sami Barmada, and Francesca Frijia, "Machine learning for the design and the simulation of radiofrequency magnetic resonance coils: Literature review, challenges, and perspectives," Sensors, Vol. 24, No. 6, 1954, 2024.
doi:10.3390/s24061954 Google Scholar

18. Barmada, Sami, Nunzia Fontana, Luca Sani, Dimitri Thomopulos, and Mauro Tucci, "Deep learning and reduced models for fast optimization in electromagnetics," IEEE Transactions on Magnetics, Vol. 56, No. 3, 1-4, 2020.
doi:10.1109/tmag.2019.2957197 Google Scholar

19. Landreman, Matt, Siena Hurwitz, and Thomas M. Antonsen, "Efficient calculation of self magnetic field, self-force, and self-inductance for electromagnetic coils with rectangular cross-section," Nuclear Fusion, Vol. 65, No. 3, 036008, 2025.
doi:10.1088/1741-4326/adb04e Google Scholar

20. Yang, Yuxin, William S. P. Robertson, Azadeh Jafari, and Maziar Arjomandi, "Optimising coil design based on sensitivity analysis of magnetic force induced between misaligned coil pairs," Electrical Engineering, Vol. 107, No. 12, 15317-15328, 2025.
doi:10.1007/s00202-025-03327-w Google Scholar

21. Khan, Arbaaz, Vahid Ghorbanian, and David Lowther, "Deep learning for magnetic field estimation," IEEE Transactions on Magnetics, Vol. 55, No. 6, 1-4, 2019.
doi:10.1109/tmag.2019.2899304 Google Scholar

22. Ansys, Inc. Ansys Maxwell User's Guide, Ansys Electronics Desktop 2026 R1, Canonsburg, PA, USA, 2026.

23. Jebelli, Ali, Nafiseh Lotfi, Arezoo Mahabadi, and Mustapha C. E. Yagoub, "Unified analytical and numerical evaluation of axial magnetic force in coaxial air-core coils," Progress In Electromagnetics Research B, Vol. 117, 135-149, 2026.
doi:10.2528/pierb26031001 Google Scholar

24. Jebelli, Ali, Arezoo Mahabadi, Hicham Chaoui, and Mustapha C. E. Yagoub, "Simulation of magnetic force between two coaxial coils with air core and uniform flow in MATLAB," International Journal of Physics, Vol. 9, No. 4, 186-196, 2021.
doi:10.12691/ijp-9-4-1 Google Scholar

25. Jebelli, Ali, Arezoo Mahabadi, Mustapha C. E. Yagoub, and Hicham Chaoui, "Magnetic force calculation between magnets and coils," International Journal of Physics, Vol. 8, No. 2, 71-80, 2020.
doi:10.12691/ijp-8-2-5 Google Scholar

Prof. Ali Jebelli

Dr. Nafiseh Lotfi

Dr. Arezoo Mahabadi

Dr. Mustapha C. E. Yagoub