This work presents an efficient method that allows to accurately calculate the time-harmonic vertical magnetic field generated at the center of a large current-carrying coil of wire positioned above a layered ground. The method consists of evaluating the integral representation for the vertical magnetic field by using a hybrid procedure. At first, the direct and ideal reflected fields are extracted from the total magnetic field and expressed in explicit form. Then, the non-analytic part of the integrand of the remaining contribution is replaced with a sum of partial fractions, obtained by using a rational function fitting algorithm. Finally, the resulting sum of integrals is analytically evaluated and turned into a sum of modified Bessel functions of the second kind. The obtained expression for the magnetic field is then used to evaluate the voltage induced in a small receiving loop co-axial with the transmitting loop.
2. Constable, S. C., R. L. Parker, and C. G. Constable, "Occam's inversion: A practical algorithm for generatlng smooth models from electromagnetic sounding data," Geophysics, Vol. 52, No. 3, 289-300, 1987.
3. 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.
4. 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.
5. Parise, M., "Exact EM field excited by a short horizontal wire antenna lying on a conducting soil," AEU — International Journal of Electronics and Communications, Vol. 70, No. 5, 676-680, 2016.
6. 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.
7. Parise, M., "On the use of cloverleaf coils to induce therapeutic heating in tissues," Journal of Electromagnetic Waves and Applications, Vol. 25, No. 11–12, 1667-1677, 2011.
8. Romano, D., I. Kovacevic-Badstubner, M. Parise, U. Grossner, J. Ekman, and G. Antonini, "Rigorous dc solution of partial element equivalent circuit models including conductive, dielectric,and magnetic materials," IEEE Transactions on Electromagnetic Compatibility, Vol. 62, No. 3, 870-879, 2020.
9. 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.
10. Ward, S. H. and G. W. Hohmann, "Electromagnetic theory for geophysical applications," Electromagnetic Methods in Applied Geophysics, Theory — Volume 1, 131-308, edited by M. N. Nabighian, SEG, Tulsa, Oklahoma, 1988.
11. 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.
12. Spies, B. R. and F. C. Frischknecht, "Electromagnetic sounding," Electromagnetic Methods in Applied Geophysics, Volume 2, 285-426, edited by M. N. Nabighian, SEG, Tulsa, Oklahoma, 1988.
13. Parise, M., L. Lombardi, F. Ferranti, and G. Antonini, "Magnetic coupling between coplanar filamentary coil antennas with uniform current," IEEE Transactions on Electromagnetic Compatibility, Vol. 62, 622-626, 2020.
14. 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.
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 Prop. Letters, Vol. 13, 23-26, 2014.
16. Werner, D. H., "An exact integration procedure for vector potentials of thin circular loop antennas," IEEE Transactions on Antennas and Propagation, Vol. 44, 157-165, 1996.
17. 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.
18. 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.
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, No. 10, 1301-1311, 1995.
22. Parise, M. and G. Antonini, "On the inductive coupling between two parallel thin-wire circular loop antennas," IEEE Transactions on Electromagnetic Compatibility, Vol. 60, 1865-1872, 2018.
23. 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.
24. 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.
25. Watson, G. N., A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge (UK), 1944.
26. Gustavsen, B. and A. Semlyen, "Rational approximation of frequency domain responses by vector fitting," IEEE Transactions on Power Delivery, Vol. 14, 1052-1061, 1999.
27. Parise, M. and S. Cristina, "High-order electromagnetic modeling of shortwave inductive diathermy effects," Progress In Electromagnetics Research, Vol. 92, 235-253, 2009.
28. Parise, M., "A study on energetic efficiency of coil antennas used for RF diathermy," IEEE Antennas and Wireless Propagation Letters, Vol. 10, 385-388, 2011.