volume | PIER Journals

Abstract

A toroidal field coil (TFC) is composed of several individual toroidal coils (ITCs), which are connected in a series and distributed in a toroidal and symmetrical form. Cross section of ITCs is rectangular or negligible. This paper presents analytical equations for mutual inductance of two ITCs applicable to Tokamak reactors using the filament method. These equations are based on those formulated by Neumann. The numerical analysis of the integrations resulting from these equations is solved using the extended three-point Gaussian algorithm. The finite element method (FEM) is employed to verify the mutual inductance equations of ITCs. The results obtained using the FEM, when dimensional parameters of ITCs are changed, confirm the analytical and empirical results showing an error of less than 0.2043% in the worst case. This indicates the reliability of the presented equations.

1. Alexander, R. and P. Garabedian, "Choice of coils for a fusion reactor," Proceedings of the National Academy of Sciences, Vol. 104, No. 30, 12250-12252, Jul. 24, 2007.
doi:10.1073/pnas.0705409104

2. Alizadeh Pahlavani, M. R. and A. Shoulaie, "A novel approach for calculations of helical toroidal coils inductance usable in reactor plasmas," IEEE Trans. on Plasma Science, Vol. 37, No. 8, 1593-1603, Aug. 2009.
doi:10.1109/TPS.2009.2023548

3. Alizadeh Pahlavani, M. R. and A. Shoulaie, "Elimination of compressive stress in helical toroidal coils advanced structure for reactor plasmas," IEEE Trans. on Plasma Science, to be published.
doi:10.1109/TPS.2009.2023548

4. Alizadeh Pahlavani, M. R. and A. Shoulaie, "Behavioral study of stress in toroidal, solenoidal, and helical toroidal coils with similar ring structures for reactor plasma," International Review of Electrical Engineering, Vol. 4, No. 1, 146-158, Jan.--Feb., 2009.

5. Alizadeh Pahlavani, M. R. and H. A. Mohammadpour, "Electromagnetic torque and force analysis of toroidal field coil using numerical and experimental results applicable to tokamak reactors," IEEE Trans. on Plasma Science, Vol. 38, No. 7, Jul., 2010.

6. Shiri, A., M. R. Alizadeh Pahlavani, and A. Shoulaie, "A new and fast procedure for calculation of the magnetic forces between cylindrical coils," International Review of Electrical Engineering, Vol. 4, No. 5, Sept.--Oct., 2009.

7. Shiri, A., D. Esmaeil Moghadam, M. R. Alizadeh Pahlavani, and A. Shoulaie, "Finite element based analysis of magnetic forces between planar spiral coils," Journal of Electromagnetic Waves and Applications, Vol. 2, 311-317, May, 2010.

8. Alizadeh Pahlavani, M. R., H. A. Mohammadpour, and A. Shoulaie, "Numerical and experimental analysis of modular toroidal coil inductance applicable to tokamak reactors," IEEE Trans. on Plasma Science, Vol. 38, No. 2, 113-120, Feb., 2010.
doi:10.1109/TPS.2009.2037628

9. Babic, S. I. and C. Akyel, "New mutual inductance calculation of the magnetically coupled coils: Thin disk coil-thin wall solenoid," Journal of Electromagnetic Waves and Applications, Vol. 20, No. 10, 1281-1290, 2006.
doi:10.1163/156939306779276794

10. Akyel, C., S. I. Babic, and M.-M. Mahmoudi, "Mutual inductance calculation for noncoaxial circular air coils with parallel axes," Progress In Electromagnetics Research, Vol. 91, 287-301, 2009.
doi:10.2528/PIER09021907

11. Akyel, C., S. I. Babic, S. Kincic, and J. P. Lagacé, "Magnetic force calculation between thin circular coils and thin filamentary circular coil in air," Journal of Electromagnetic Waves and Applications, Vol. 21, No. 9, 1273-1283, 2007.

12. Shiri, A. and A. Shoulaie, "A new methodology for magnetic force calculations between planar spiral coils," Progress In Electromagnetics Research, Vol. 95, 39-57, 2009.
doi:10.2528/PIER09031608

13. Koledintseva, M. Y., J. L. Drewniak, T. P. Van Doren, D. J. Pommerenke, M. Cocchini, and D. M. Hockanson, "Method of edge currents for calculating mutual external inductance in a microstrip structure," Progress In Electromagnetics Research, Vol. 80, 197-224, 2008.
doi:10.2528/PIER07101504

14. Koledintseva, M. Y., J. L. Drewniak, T. P. Van Doren, D. J. Pommerenke, M. Cocchini, and D. M. Hockanson, "Mutual external inductance in stripline structures," Progress In Electromagnetics Research, Vol. 80, 349-368, 2008.
doi:10.2528/PIER07111503

15. Ravaud, R., G. Lemarquand, V. Lemarquand, S. I. Babic, and C. Akyel, "Mutual inductance and force exerted between thick coils," Progress In Electromagnetics Research, Vol. 102, 367-380, 2010.

16. Conway, J. T., "Inductance calculations for noncoaxial coils using Bessel functions," IEEE Trans. Magn., Vol. 43, No. 3, 1023-1034, Mar., 2007.
doi:10.1109/TMAG.2006.888565

17. Zhao, P. and H.-G. Wang, "Resistances and inductances extraction using surface integral equation with the acceleration of multilevel green function interpolation method," Progress In Electromagnetics Research, Vol. 83, 43-54, 2008.
doi:10.2528/PIER08032001

18. Wang, H.-G. and P. Zhao, "Combining multilevel Green's function interpolation method eith volume loop bases for inductance extraction problems," Progress In Electromagnetics Research, Vol. 80, 225-239, 2008.
doi:10.2528/PIER07102101

19. Babic, S. I. and C. Akyel, "New analytic-numerical solutions for the mutual inductance of two coaxial circular coils with rectangular cross section in air," IEEE Trans. Magn., Vol. 42, No. 6, 1661-1669, Jun., 2006.
doi:10.1109/TMAG.2006.872626

20. Babic, S. I. and C. Akyel, "Improvement in calculation of the self- and mutual inductance of thin-wall solenoids and disk coils," IEEE Trans. Magn., Vol. 36, No. 4, 1970-1975, Jul., 2000.
doi:10.1109/TMAG.2000.875240

21. Akyel, C., S. I. Babic, and S. Kincic, "New and fast procedures for calculating the mutual inductance of coaxial circular coils (disk coil-circular coil)," IEEE Trans. Magn., Vol. 38, No. 5, 2367-2369, Sep., 2002.
doi:10.1109/TMAG.2002.803576

22. Babic, S. I., C. Akyel, and S. J. Salon, "New procedures for calculating the mutual inductance of the system: Filamentary circular coil-massive circular solenoid," IEEE Trans. Magn., Vol. 39, No. 3, 1131-1134, May, 2003.
doi:10.1109/TMAG.2003.810550

23. Babic, S. I., S. Salon, and C. Akyel, "The mutual inductance of two thin coaxial disk coils in air," IEEE Trans. Magn., Vol. 40, No. 2, 822-825, Mar., 2004.
doi:10.1109/TMAG.2004.824810

24. Akyel, C., S. I. Babic, K. Wu, and F. J. Mojca, "Improvement in the mutual inductance calculation of thin coaxial circular coils with constant current density in air," WSEAS Trans. Commun., Vol. 5, No. 6, 970-977, 1109--2742, Jun., 2006.

25. Babic, S. I. and C. Akyel, "An improvement in the calculation of the self inductance of thin disk coils with air-core," WSEAS Trans. Circ. Syst., Vol. 3, No. 8, 1621-1626, Oct., 2004.

26. Akyel, C. and S. I. Babic, "Improvement in the mutual inductance calculation between coaxial circularcoils of rectangular cross section and thin coaxial circular coils with constant current density in air," WSEAS Trans. Circuits Syst., Vol. 6, No. 5, 473-480, 1109--2734, May, 2007.

27. Tang, W., X. He, T. Pan, and Y. L. Chow, "Synthetic asymptote formulas of equivalent circuit components of square spiral inductors," Journal of Electromagnetic Waves and Applications, Vol. 20, No. 2, 215-226, 2006.
doi:10.1163/156939306775777206

28. Venkov, G., M. W. McCall, and D. Censor, "The theory of low-frequency wave physics revisited," Journal of Electromagnetic Waves and Applications, Vol. 21, No. 2, 229-249, 2007.
doi:10.1163/156939307779378763

29. Pennington, R. H., Introductory Computer Methods and Numerical Analysis, 4th Ed., Macmillan, New York, 1970.

30. Anderson, D. A., J. C. Tannehill, and R. H. Pletcher, Computational Fluid Mechanics and Heat Transfer, Hemisphere, Washington, DC, 1984.

31. Grover, F. W., Inductance Calculation Working Formulas and Tables, No. 11, 77-87, Dover Publications, Inc., New York, 1946.

32. Grover, F. W., Inductance Calculation Working Formulas and Tables, No. 16, 142-162, Dover Publications, Inc., New York, 1946.

Dr. Mohammad Reza Pahlavani

Dr. Abbas Shiri