This paper presents an exact expression for the mutual impedance of two coaxial loops located on the surface of a conductive ground. The semi-infinite complete integral representation for the impedance is first converted into a finite integral. Then the spherical Hankel function contained in the integrand is expanded according to Gegenbauer addition theorem. This makes it possible to perform analytical integration and express the mutual impedance as a sum of products of spherical Bessel functions. Since no simplifying assumption is introduced in the mathematical derivation, the obtained formula is valid in quasi-static as well as non-quasi-static frequency ranges. Numerical examples show how, accuracy being equal, the proposed expression is less computationally expensive than standard Gauss-Kronrod numerical integration technique.
2. Parise, M., "Exact electromagnetic field excited by a vertical magnetic dipole on the surface of a lossy half-space," Progress In Electromagnetics Research B, Vol. 23, 69-82, 2010.
3. Kong, F. N., S. E. Johnstad, and J. Park, "Wavenumber of the guided wave supported by a thin resistive layer in marine controlled-source electromagnetics," Geophysical Prospecting, Vol. 29, No. 10, 1301-1307, 2003.
4. Shastri, N. L. and H. P. Patra, "Multifrequency sounding results of laboratory simulated homogeneous and two-Layer earth models," IEEE Trans. Geosci. Remote Sensing, Vol. 26, No. 6, 749-752, 1988.
5. Farquharson, C. G., D. W. Oldenburg, and P. S. Routh, "Simultaneous 1D inversion of loop-loop electromagnetic data for magnetic susceptibility and electrical conductivity," Geophysics, Vol. 68, No. 6, 1857-1869, 2003.
6. Ward, S. H. and G. W. Hohmann, "Electromagnetic theory for geophysical applications," Electromagnetic Methods in Applied Geophysics, Theory, Vol. 1, 131-308, edited by M. N. Nabighian, SEG, Tulsa, Oklahoma, 1988.
7. Zhdanov, M. S., Geophysical Electromagnetic Theory and Methods, Elsevier, Amsterdam, 2009.
8. Beard, L. P. and J. E. Nyquist, "Simultaneous inversion of airborne electromagnetic data for resistivity and magnetic permeability," Geophysics, Vol. 63, No. 5, 1556-1564, 1998.
9. Parise, M., "Quasi-static vertical magnetic field of a large horizontal circular loop located at the earth's surface," Progress In Electromagnetics Research Letters, Vol. 62, 29-34, 2016.
10. Wait, J. R., "Mutual electromagnetic coupling of loops over a homogeneous ground," Geophysics, Vol. 20, No. 3, 630-637, 1955.
11. Spies, B. R. and F. C. Frischknecht, "Electromagnetic sounding," Electromagnetic Methods in Applied Geophysics, Vol. 2, 285-426, edited by M. N. Nabighian, SEG, Tulsa, Oklahoma, 1988.
12. Parise, M., "Second-order formulation for the quasi-static field from a vertical electric dipole on a lossy half-space," Progress In Electromagnetics Research, Vol. 136, 509-521, 2013.
13. Tiwari, K. C., D. Singh, and M. K. Arora, "Development of a model for detection and estimation of depth of shallow buried non-metallic landmine at microwave x-band frequency," Progress In Electromagnetics Research, Vol. 79, 225-250, 2008.
14. Telford, W. M., L. P. Geldart, and R. E. Sheriff, Applied Geophysics, Cambridge University Press, Cambridge, UK, 1990.
15. Parise, M., "An exact series representation for the EM field from a circular loop antenna on a lossy half-space," IEEE Antennas and Wireless Propagation Letters, Vol. 13, 23-26, 2014.
16. Parise, M., "Full-wave analytical explicit expressions for the surface fields of an electrically large horizontal circular loop antenna placed on a layered ground," IET Microwaves, Antennas & Propagation, Vol. 11, 929-934, 2017.
17. Palacky, G. J., "Resistivity characteristics of geologic targets," Electromagnetic Methods in Applied Geophysics, Vol. 1, 52-129, Nabighian, M. N., Ed., SEG, Tulsa, Oklahoma, 1988.
18. Parise, M., "On the surface fields of a small circular loop antenna placed on plane stratified earth," International Journal of Antennas and Propagation, Vol. 2015, 1-8, 2015.
19. Singh, N. P. and T. Mogi, "Electromagnetic response of a large circular loop source on a layered earth: A new computation method," Pure and Applied Geophysics, Vol. 162, 181-200, 2005.
20. Parise, M., "Efficient computation of the surface fields of a horizontal magnetic dipole located at the air-ground interface," International Journal of Numerical Modelling: Electronic Networks, Devices and Fields, Vol. 29, 653-664, 2016.
21. Wait, J. R., "Fields of a horizontal loop antenna over a layered half-space," Journal of Electromagnetic Waves and Applications, Vol. 9, 1301-1311, 1995.
22. Parise, M. and G. Antonini, "On the inductive coupling between two parallel thin-wire circular loop antennas," IEEE Trans. on Electromagnetic Compatibility, Vol. 60, No. 6, 1865-1872, 2018.
23. Kong, J. A., L. Tsang, and G. Simmons, "Geophysical subsurface probing with radio-frequency interferometry," IEEE Transactions on Antennas and Propagation, Vol. 22, No. 4, 616-620, 1974.
24. Singh, N. P. and T. Mogi, "Effective skin depth of EM fields due to large circular loop and electric dipole sources," Earth Planets Space, Vol. 55, 301-313, 2003.
25. Parise, M., "On the use of cloverleaf coils to induce therapeutic heating in tissues," Journal of Electromagnetic Waves and Applications, Vol. 25, 1667-1677, 2011.
26. Ryu, J., H. F. Morrison, and S. H. Ward, "Electromagnetic fields about a loop source of current," Geophysics, Vol. 35, No. 5, 862-896, 1970.
27. Parise, M., "Fast computation of the forward solution in controlled-source electromagnetic sounding problems," Progress In Electromagnetics Research, Vol. 111, 119-139, 2011.
28. Watson, G. N., A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, UK, 1994.
29. Parise, M., "Improved Babylonian square root algorithm-based analytical expressions for the surface-to-surface solution to the Sommerfeld half-space problem," IEEE Transactions on Antennas and Propagation, Vol. 63, 5832-5837, 2015.
30. Parise, M., "An exact series representation for the EM field from a vertical electric dipole on an imperfectly conducting half-space," Journal of Electromagnetic Waves and Applications, Vol. 28, No. 8, 932-942, 2014.