In this article we give an analytical formula for calculating the self-inductance for circular coils of rectangular cross-section which has a non-uniform current density. Recently, the formula for calculating this important electromagnetic quantity was published in the form of the single integral whose kernel function was asum of elementary functions. However, a new formula is obtained in the form of elementary functions, single integrals, and the complete elliptic integral of the first, second and third kind. Although its development looks tedious, we obtain a rather user-friendly expression for the calculation. From the general case, the self-inductance of the thin disk coil (pancake coil) with the nonuniform current is obtained in a remarkably simple form. The results of this work are compared with different known methods, and all results are in the excellent agreement. Our approach has not been found in the literature.
1. Grover, F. W., Inductance Calculations, Chs. 2 and 13, Dover, New York, 1964.
2. Dwight, H. B., "Electrical Coils and Conductors," McGraw-Hill Book Company, New York, 1945.
3. Snow, C., "Formulas for computing capacitance and inductance," 544, National Bureau of Standards Circular Washington DC, December 1954.
4. Kalantarov, P. L., Inductance Calculations, National Power Press, Moscow, USSR/Russia, 1955.
5. Conway, J. T., "Exact solutions for the magnetic fields of axisymmetric solenoids and current distributions," IEEE Trans. Magn., Vol. 37, No. 4, 2977-2988, 2001. doi:10.1109/20.947050
6. Conway, J. T., "Trigonometric integrals for the magnetic field of the coil of rectangular cross section," IEEE Trans. Magn., Vol. 42, No. 5, 1538-1548, 2006. doi:10.1109/TMAG.2006.871084
7. Babic, S. and C. Akyel, "New formulas for mutual inductance and axial magnetic force between magnetically coupled coils: Thick circular coil of the rectangular cross-section-thin disk coil (Pancake)," IEEE Trans. Magn., Vol. 49, No. 2, 860-868, 2013. doi:10.1109/TMAG.2012.2212909
8. Ravaud, R., G. Lemarquand, S. Babic, V. Lemarquand, and C. Akyel, "Cylindrical magnets and coils: Fields, forces and inductances," IEEE Trans. Magn., Vol. 46, No. 9, 3585-3590, Sept. 2010. doi:10.1109/TMAG.2010.2049026
9. Conway, J. T., "Inductance calculations for circular coils of rectangular cross section and parallel axes using bessel and struve functions," IEEE Trans. Magn., Vol. 46, No. 1, 75-81, 2010. doi:10.1109/TMAG.2009.2026574
10. Yu, D. and K. S. Han, "Self-inductance of air-core circular coils with rectangular cross section," IEEE Trans. Magn., Vol. 23, No. 6, 3916-3921, Nov. 1987.
11. Kajikawa, K. and K. Kaiho, "Usable ranges of some expressions for calculation of the self-inductance of a circular coil of rectangular cross section," Journal of Cryogenics and Superconductivity Society of Japan, Vol. 30, No. 7, 324-332, 1995 (in Japanese). doi:10.2221/jcsj.30.324
12. Luo, J. and B. Chan, "Improvement of self-inductance calculations for circular coils of rectangular cross section," IEEE Trans. Magn., Vol. 49, No. 3, 1249-1255, Mar. 2013. doi:10.1109/TMAG.2012.2228499
13. Luo, Y., X. Wang, and X. Zhou, "Inductance calculations for circular coils with rectangular cross section and parallel axes using inverse mellin transform and generalized hypergeometric functions," IEEE Trans. on Power Electronics, Vol. 32, No. 2, 1367-1374, Feb. 2017. doi:10.1109/TPEL.2016.2541180
14. Pankrac, V., "Generalization of relations for calculating the mutual inductance of coaxial coils in terms of their applicability to non-coaxial coils," IEEE Trans. Magn., Vol. 47, No. 11, 4552-4563, Nov. 2011. doi:10.1109/TMAG.2011.2148175
15. Liang, S. and Y. Fang, "Analysis of inductance calculation of coaxial circular coils with rectangular cross section using inverse hyperbolic functions," IEEE Transactions on Applied Superconductivity, Vol. 25, No. 4, Aug. 2015.
16. Zupan, T., Z. Stih, and B. Trkulja, "Fast and precise method for inductance calculation of coaxial circular coils with rectangular cross section using the one-dimensional integration of elementary functions applicable to superconducting magnets," IEEE Transactions on Applied Superconductivity, Vol. 24, No. 2, Apr. 2014. doi:10.1109/TASC.2014.2301765
17. Bitter, F., "The design of powerful electromagnets Part II. The magnetizing coil," Rev. Sci. Instrum., Vol. 7, No. 12, 482-489, 1936. doi:10.1063/1.1752068
18. Conway, J. T., "Non coaxial force and inductancecalculations for bitter coils and coils with uniform radialcurrent distributions," 2011 International Conference on Applied Superconductivity and Electromagnetic Devices (ASEMD), 61-64, Sidney, Australia, Dec. 2011.
19. Ren, Y., F. Wang, G. Kuong, W. Chen, Y. Tan, J. Zhu, and P. He, "Mutual inductance and force calculations between coaxial bitter coils and superconducting coils with rectangular cross section," Journal of Superconductivity and Novel Magnetism, 2010.
20. Ren, Y., G. Kaung, and W. Chen, "Inductance of bitter coil with rectangular cross section," Journal of Superconductivity and Novel Magnetism, Vol. 26, 2159-2163, 2013. doi:10.1007/s10948-012-1816-6
21. Ren, Y., et al., "Electromagnetic, mechanical and thermal performance analysis of the CFETR magnet system," Nuclear Fusion, Vol. 55, 093002, 2015. doi:10.1088/0029-5515/55/9/093002
22. Babic, S. and C. Akyel, "Mutual inductance and magnetic force calculations for bitter disk coils (pancakes)," IET Science, Measurement & Technology, Vol. 10, No. 8, 972-976, 2016. doi:10.1049/iet-smt.2016.0221
23. Babic, S. and C. Akyel, "Mutual inductance and magnetic force calculations between two thick coaxial bitter coils of rectangular cross section," IET Electric Power Applications, Vol. 11, No. 3, 441-446, 2017. doi:10.1049/iet-epa.2016.0628
24. Babic, S. and C. Akyel, "Mutual inductance and magnetic force calculations between thick coaxial bitter coil of rectangular cross section with inverse radial current and filamentary circular coil with constant azimuthal current," IET Electric Power Applications, Vol. 11, No. 9, 1596-1600, 2017. doi:10.1049/iet-epa.2017.0244
25. Babic, S. and C. Akyel, "Calculation of some electromagnetic quantities for circular thick coil of rectangular cross section and pancake with inverse radial currents," IET Electric Power Applications, 2018. doi:10.1049/iet-epa.2017.0244
26. Babic, S. and C. Akyel, "Mutual inductance and magnetic force calculations for bitter disk coil (pancake) with nonlinear radial current and filamentary circular coil with azimuthal current," Journal Advances in Electrical Engineering, Hindawi, 2016.
27. Filanovsky, I. M., "On design of 60 GHz solid-state transformers modeled as coupled bitter coils," 2019 IEEE 62nd International Midwest Symposium on Circuits and Systems (MWSCAS), Dallas, TX, USA, Aug. 2019.
28. Babic, S. and C. Akyel, "Self-inductance of the circular coils of the rectangular cross-section with the radial and azimuthal current densities," Applied Physics, Open Access, Vol. 2, 352-367, 2020.
29. Yu, Y. and Y. Luo, "Inductance calculations for non-coaxial Bitter coils with rectangular cross-section using inverse Mellin transform," IET Electric Power Applications, Vol. 13, No. 1, 119-125, 2019. doi:10.1049/iet-epa.2018.5386
30. Chen, J. W., "Modeling and decoupling control of a linear permanent magnet actuator considering fringing effect for precision engineering," IEEE Trans. Magn., Vol. 57, No. 3, Mar. 2021. doi:10.1109/TMAG.2021.3050835
31. Gradshteyn, S. and I. M. Ryzhik, Table of Integrals, Seriesand Products, Academic Press Inc., New York and London, 1965.
32. Abramowitz, M. and I. S. Stegun, Handbook of Mathematical Functions, Dover, New York, NY, USA, 1972.
33. Brychkov, Y. A., Handbook of Special Functions: Derivatives, Integrals, Series and Other Formulas, CRC Press, Boca Raton, FL, USA, 2008. doi:10.1201/9781584889571