Vol. 90

Front:[PDF file] Back:[PDF file]
Latest Volume
All Volumes
All Issues
2020-12-26

The Magnetic Field Produced from a Conical Current Sheet and from a Thin and Tightly-Wound Conical Coil

By Matthew Smith, Nikiforos Fokas, Kevin Hart, Slobodan Babic, and Jerry P. Selvaggi
Progress In Electromagnetics Research B, Vol. 90, 1-20, 2021
doi:10.2528/PIERB20091806

Abstract

Mathematical expressions for the components of the magnetic field produced by a conically-shaped current sheet and by a tightly-wound conical coil are presented. The conical current sheet forms the frustum of a cone. In the limit as the top radius of the frustum approaches the bottom radius, a cylindrical current sheet is formed. Mathematical expressions for the magnetic field produced by a cylindrical current sheet are then compared to known and published results.

Citation


Matthew Smith, Nikiforos Fokas, Kevin Hart, Slobodan Babic, and Jerry P. Selvaggi, "The Magnetic Field Produced from a Conical Current Sheet and from a Thin and Tightly-Wound Conical Coil," Progress In Electromagnetics Research B, Vol. 90, 1-20, 2021.
doi:10.2528/PIERB20091806
http://jpier.org/PIERB/pier.php?paper=20091806

References


    1. Snow, C., "Magnetic fields of cylindrical coils and annular coils," Natl. Bur. Stand., Appl. Math. Ser., Vol. 38, 1-29, Dec. 1953.

    2. Snow, C., "Formula for the inductance of a helix made with wire of any section," Sci. Pap. Bur. Stand., Vol. 21, No. 537, 431-519, Feb. 1926.
    doi:10.6028/nbsscipaper.212

    3. Jackson, J. D., Classical Electrodynamics, 3rd Ed., 180-181, John Wiley and Sons, New York, 1999.

    4. Smythe, W. R., Static and Dynamic Electricity, 3rd Ed., 282-283, McGraw-Hill, New York, 1968.

    5. Wolfram Research Inc., , Mathematica, Version 12.1, Champaign, IL, 100 Trade Center Drive, 61820-7237, USA, 2020.

    6. Hart, S., K. Hart, and J. P. Selvaggi, "Analytical expressions for the magnetic field from axially magnetized and conically shaped permanent magnets," IEEE Trans. Magn., Vol. 56, No. 7, 1-9, Jul. 2020.
    doi:10.1109/TMAG.2020.2992191

    7. Flax, L. and E. Callaghan, Magnetic field from a finite thin cone by use of Legendre polynomials, 1-41, NASA TN D-2400, Lewis Researh Center, Cleveland, Ohio, USA, Aug. 1963.

    8. Snow, C., "Formulas for computing capacitance and inductance," National Bureau of Standards Circular, Vol. 544, Sep. 1, 1954.

    9. Snow, C., "Hypergeometric and legendre functions with applications to integral equations of potential theory," Nat. Bur. Stand. Appl. Math. Ser., Vol. 19, 228-252, May 1, 1952.

    10. Cohl, H. S. and J. E. Tohline, "A compact cylindrical Green’s function expansion for the solution of potential problems," The Astrophysical Journal, Vol. 527, 86-101, Dec. 1999.
    doi:10.1086/308062

    11. Cohl, H. S., A. R. P. Rau, J. E. Tohline, D. A. Browne, J. E. Cazes, and E. I. Barnes, "Useful alternative to the multipole expansion of 1/r potentials," Phys. Rev. A, Vol. 64, 052509-5, Oct. 2001.

    12. Cohl, H. S., J. E. Tohline, and A. R. P. Rau, "Developments in determining the gravitational potential using toroidal functions," Astron. Nachr., Vol. 321, 363-372, Nov. 2000.

    13. Selvaggi, J. P., Multipole analysis of circular cylindrical magnetic systems, Ph.D. dissertation, Rensselaer Polytech. Inst., Troy, NY, USA, 2005.

    14. Selvaggi, J., S. Salon, O. Kwon, and M. V. K. Chari, "Calculating the external magnetic field from permanent magnets in permanent-magnet motors-an alternative method," IEEE Trans., Vol. 40, No. 5, 3278-3285, Sep. 2004.

    15. Selvaggi, J. P., S. Salon, and M. V. K. Chari, "An application of toroidal functions in electrostatics," Am. J. Phys., Vol. 75, No. 8, 724-727, Apr. 2007.
    doi:10.1119/1.2737473

    16. Selvaggi, J. P., S. Salon, O. Kwon, and M. V. K. Chari, "Computation of the three-dimensional magnetic field from solid permanent-magnet bipolar cylinders employing toroidal harmonics," IEEE Trans. Magn., Vol. 43, No. 10, 3833-3839, Oct. 2007.
    doi:10.1109/TMAG.2007.902995

    17. Selvaggi, J. P., S. Salon, and M. V. K. Chari, "Computing the magnetic induction field due to a radially-magnetized finite cylindrical permanent magnet by employing toroidal harmonics," PIERS Proceedings, 244-251, Cambridge, USA, Jul. 5–8, 2010.

    18. Selvaggi, J. P., S. Salon, and M. V. K. Chari, "Employing toroidal harmonics for computing the magnetic field from axially magnetized multipole cylinders," IEEE Trans. Magn., Vol. 46, No. 10, 3715-3723, Oct. 2010.
    doi:10.1109/TMAG.2010.2051558

    19. Selvaggi, J. P., S. Salon, O. Kwon, and M. V. K. Chari, "Calculating the external magnetic field from permanent magnets in permanent-magnet motors — An alternative method," IEEE Trans. Magn., Vol. 40, No. 5, 3278-3285, Sep. 2004.
    doi:10.1109/TMAG.2004.831653

    20. Selvaggi, J. P., S. Salon, O. Kwon, and M. V. K. Chari, "Computation of the three-dimensional magnetic field from solid permanent-magnet bipolar cylinders employing toroidal harmonics," IEEE Trans. Magn., Vol. 43, No. 10, 3833-3839, Oct. 2007.
    doi:10.1109/TMAG.2007.902995

    21. Whittaker, E. T. and G. N. Watson, A Course of Modern Analysis, 4th Ed., 281-301, Cambridge at University Press, 1952.

    22. Wang, Z. X. and D. R. Guo, Special Functions, 135-209, World Scientific, Singapore, 1989.
    doi:10.1142/9789812779366_0004

    23. Hanson, M. T. and I. W. Puja, "The evaluation of certain infinite integrals involving products of Bessel functions: A Correlation of formula," Quart. Appl. Math., Vol. 55, No. 3, 505-524, Sep. 1997.
    doi:10.1090/qam/1466145

    24. Colavecchia, F. D., G. Gasaneo, and J. E. Miraglia, "Numerical evaluation of Appell’s F1 hypergeometric function," Comp. Phys Comm., Vol. 138, 29-43, Mar. 2001.
    doi:10.1016/S0010-4655(01)00186-2

    25. Bailey, W. N., Appell’s Hypergeometric Functions of Two Variables, Ch. 9 in Generalised Hypergeometric Series, 73-83 and 99–101, Cambridge University Press, Cambridge, England, 1935.

    26. Byrd, P. F. and M. D. Friedman, Handbook of Elliptic Integrals for Engineers and Physicists, Springer-Verlag, Berlin Heidelberg GMBH, 1954.
    doi:10.1007/978-3-642-52803-3

    27. Callaghan, E. E. and S. H. Maslen, The magnetic field from a finite solenoid, 1-23, NASA TN D-465, Lewis Researh Center, Cleveland, Ohio, USA, Oct. 1960.

    28. Slater, L. J., Generalized Hypergeometric Functions, Cambridge Univ. Press, Cambridge, U.K., 1966.

    29. Conway, J. T., "Exact solutions for the magnetic fields of axisymmetric solenoids and current distributions," IEEE Trans. Magn., Vol. 37, No. 4, 2977-2988, Jul. 2001.
    doi:10.1109/20.947050

    30. Brown, G. V. and L. Flax, Superposition calculation of thick solenoid fields from semi-infinite solenoid tables, 1-23, NASA TN D-2494, Lewis Research Center, Cleveland, Ohio, USA, Sep. 1964.

    31. Conway, J. T., "Exact solutions for the mutual inductance of circular coils and elliptic coils," IEEE Trans. Magn., Vol. 48, No. 1, 81-94, Jan. 2012.
    doi:10.1109/TMAG.2011.2161768

    32. Conway, J. T., "Analytical solutions for the Newtonian gravitational field induced by matter within axisymmetric boundaries," Mon. Not. R. Astron. Soc., Vol. 316, 540-554, Feb. 2000.
    doi:10.1046/j.1365-8711.2000.03523.x

    33. 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, Jan. 2010.
    doi:10.1109/TMAG.2009.2026574

    34. Conway, J. T., "Exact solutions for the mutual inductance of circular coils and elliptic coils," IEEE Trans. Magn., Vol. 48, No. 1, 81-94, Jan. 2012.
    doi:10.1109/TMAG.2011.2161768

    35. 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

    36. Conway, J. T., "Noncoaxial inductance calculations without the vector potential for axisymmetric coils and planar coils," IEEE Trans. Magn., Vol. 44, No. 4, 453-462, Apr. 2008.
    doi:10.1109/TMAG.2008.917128

    37. Conway, J. T., "Non coaxial force and inductance calculations for bitter coils and coils with uniform radial current distributions," 2011 International Conference on Applied Superconductivity and Electromagnetic Devices, 61-64, Sydney, NSW, 2011.
    doi:10.1109/ASEMD.2011.6145068

    38. Babic, S., C. Akyel, J. Martinez, and B. Babic, "A new formula for calculating the magnetic force between two coaxial thick circular coils with rectangular cross-section," Journal of Electromagnetic Waves and Applications, Vol. 29, No. 9, 1181-1193, 2015.
    doi:10.1080/09205071.2015.1035807

    39. 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, Feb. 2013.
    doi:10.1109/TMAG.2012.2212909

    40. Babic, S. I. and C. Akyel, "Calculating mutual inductance between circular coils with inclined axes in air," IEEE Trans. Magn., Vol. 44, No. 7, 1743-1750, Jul. 2008.
    doi:10.1109/TMAG.2008.920251

    41. Babic, S., F. Sirois, C. Akyel, G. Lemarquand, V. Lemarquand, and R. Ravaud, "New formulas for mutual inductance and axial magnetic force between a thin wall solenoid and a thick circular coil of rectangular cross-section," IEEE Trans. Magn., Vol. 47, No. 8, 2034-2044, Aug. 2011.
    doi:10.1109/TMAG.2011.2125796

    42. Babic, S., F. Sirois, C. Akyel, G. Lemarquand, V. Lemarquand, and R. Ravaud, "Correction to “New formulas for mutual inductance and axial magnetic force between a thin wall solenoid and a thick circular coil of rectangular cross-section," IEEE Trans. Magn., Vol. 48, No. 6, 2096-2096, Jun. 2012.
    doi:10.1109/TMAG.2011.2180733

    43. 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, Sep. 2010.
    doi:10.1109/TMAG.2010.2049026

    44. Ravaud, R., G. Lemarquand, V. Lemarquand, and C. Depollier, "Analytical calculation of the magnetic field created by permanent-magnet rings," IEEE Trans. Magn., Vol. 44, No. 8, 1982-1989, Aug. 2008.
    doi:10.1109/TMAG.2008.923096

    45. Ravaud, R., G. Lemarquand, and V. Lemarquand, "Magnetic field created by tile permanent magnets," IEEE Trans. Magn., Vol. 45, No. 7, 2920-2926, Jul. 2009.
    doi:10.1109/TMAG.2009.2014752

    46. Ravaud, R., G. Lemarquand, and V. Lemarquand, "Force and stiffness of passive magnetic bearings using permanent magnets. Part 1: Axial magnetization," IEEE Trans. Magn., Vol. 45, No. 7, 2996-3002, Jul. 2009.
    doi:10.1109/TMAG.2009.2016088

    47. Ravaud, R., G. Lemarquand, and V. Lemarquand, "Force and stiffness of passive magnetic bearings using permanent magnets. Part 2: Radial magnetization," IEEE Trans. Magn., Vol. 45, No. 9, 3334-3342, Sep. 2009.
    doi:10.1109/TMAG.2009.2025315

    48. Babic, S., C. Akyel, M. M. Gavrilovic, and K. Wu, "New closed-form expressions for calculating the magnetic field of thin conductors with azimuthal current direction," 4th International Conference on Telecommunications in Modern Satellite, Cable and Broadcasting Services. TELSIKS’99 (Cat. No.99EX365), Vol. 1, 44-47, Nis, Yugoslavia, 1999.

    49. Stoll, J. C., P. L. Yohner, and J. C. Laurence, Magnetic fields due to solid and hollow conical conductors, 1-131, NASA SP-3022, Lewis Research Center, Cleveland, Ohio, USA, Aug. 1965.