Vol. 103
Latest Volume
All Volumes
PIERM 114 [2022] PIERM 113 [2022] PIERM 112 [2022] PIERM 111 [2022] PIERM 110 [2022] PIERM 109 [2022] PIERM 108 [2022] PIERM 107 [2022] PIERM 106 [2021] PIERM 105 [2021] PIERM 104 [2021] PIERM 103 [2021] PIERM 102 [2021] PIERM 101 [2021] PIERM 100 [2021] PIERM 99 [2021] PIERM 98 [2020] PIERM 97 [2020] PIERM 96 [2020] PIERM 95 [2020] PIERM 94 [2020] PIERM 93 [2020] PIERM 92 [2020] PIERM 91 [2020] PIERM 90 [2020] PIERM 89 [2020] PIERM 88 [2020] PIERM 87 [2019] PIERM 86 [2019] PIERM 85 [2019] PIERM 84 [2019] PIERM 83 [2019] PIERM 82 [2019] PIERM 81 [2019] PIERM 80 [2019] PIERM 79 [2019] PIERM 78 [2019] PIERM 77 [2019] PIERM 76 [2018] PIERM 75 [2018] PIERM 74 [2018] PIERM 73 [2018] PIERM 72 [2018] PIERM 71 [2018] PIERM 70 [2018] PIERM 69 [2018] PIERM 68 [2018] PIERM 67 [2018] PIERM 66 [2018] PIERM 65 [2018] PIERM 64 [2018] PIERM 63 [2018] PIERM 62 [2017] PIERM 61 [2017] PIERM 60 [2017] PIERM 59 [2017] PIERM 58 [2017] PIERM 57 [2017] PIERM 56 [2017] PIERM 55 [2017] PIERM 54 [2017] PIERM 53 [2017] PIERM 52 [2016] PIERM 51 [2016] PIERM 50 [2016] PIERM 49 [2016] PIERM 48 [2016] PIERM 47 [2016] PIERM 46 [2016] PIERM 45 [2016] PIERM 44 [2015] PIERM 43 [2015] PIERM 42 [2015] PIERM 41 [2015] PIERM 40 [2014] PIERM 39 [2014] PIERM 38 [2014] PIERM 37 [2014] PIERM 36 [2014] PIERM 35 [2014] PIERM 34 [2014] PIERM 33 [2013] PIERM 32 [2013] PIERM 31 [2013] PIERM 30 [2013] PIERM 29 [2013] PIERM 28 [2013] PIERM 27 [2012] PIERM 26 [2012] PIERM 25 [2012] PIERM 24 [2012] PIERM 23 [2012] PIERM 22 [2012] PIERM 21 [2011] PIERM 20 [2011] PIERM 19 [2011] PIERM 18 [2011] PIERM 17 [2011] PIERM 16 [2011] PIERM 14 [2010] PIERM 13 [2010] PIERM 12 [2010] PIERM 11 [2010] PIERM 10 [2009] PIERM 9 [2009] PIERM 8 [2009] PIERM 7 [2009] PIERM 6 [2009] PIERM 5 [2008] PIERM 4 [2008] PIERM 3 [2008] PIERM 2 [2008] PIERM 1 [2008]
2021-06-11
The Analytical Formula for Calculating the Self-Inductance for the Circular Coil of the Rectangular Cross-Section with a Non-Uniform Current Density
By
Progress In Electromagnetics Research M, Vol. 103, 15-26, 2021
Abstract
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.
Citation
Slobodan Babic Matthew Smith Nikiforos Fokas Yuriy Langer Jerry P. Selvaggi , "The Analytical Formula for Calculating the Self-Inductance for the Circular Coil of the Rectangular Cross-Section with a Non-Uniform Current Density," Progress In Electromagnetics Research M, Vol. 103, 15-26, 2021.
doi:10.2528/PIERM21040905
http://www.jpier.org/PIERM/pier.php?paper=21040905
References

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