This paper presents an efficient method for evaluating the flux linkage between two circular loops located on the top surface of a plane multilayer soil. The method consists of a rigorous procedure, which leads to expressing the flux as a sum of products of Bessel functions. First, the integral representation for the mutual inductance is cast into a form where the integration range is continued to the negative real axis. Subsequently, the non-oscillating part of the integrand is replaced with a rational approximation, arising from using a well-known least squares-based fitting algorithm. Finally, analytical integration is performed by applying the theorem of residues. As a result of the proposed method, the flux linkage between the loops is expressed as a finite sum of products of Bessel functions. Since no assumptions are made in the mathematical derivation, the obtained explicit expression is valid regardless of the operating frequency. Numerical tests are performed to show the advantages of the proposed method with respect to standard numerical integration techniques. In particular, it is seen how the use of the derived series representation for the inductance with 50 terms permits to achieve the same accuracy as conventional Gauss-Kronrod numerical integration technique, with the advantage of reducing the computation time by at least 8 times.
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 — Volume 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. Parise, M., V. Tamburrelli, and G. Antonini, "Mutual impedance of thin-wire circular loops in near-surface applications," IEEE Trans. on Electromagnetic Compatibility, Vol. 61, No. 2, 558-563, 2019.
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. 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.
11. Wait, J. R., "Mutual electromagnetic coupling of loops over a homogeneous ground," Geophysics, Vol. 20, No. 3, 630-637, 1955.
12. 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.
13. Telford, W. M., L. P. Geldart, and R. E. Sheriff, Applied Geophysics, Cambridge University Press, Cambridge, UK, 1990.
14. 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.
15. 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.
16. 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.
17. 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.
18. 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.
19. 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.
20. 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.
21. 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.
22. 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.
23. 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.
24. Parise, M., "Fast computation of the forward solution in controlled-source electromagnetic sounding problems," Progress In Electromagnetics Research, Vol. 111, 119-139, 2011.
25. Constable, S. C., R. L. Parker, and C. G. Constable, "Occam’s inversion: A practical algorithm for generating smooth models from electromagnetic sounding data," Geophysics, Vol. 52, No. 3, 289-300, 1987.
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., "A highly accurate analytical solution for the surface fields of a short vertical wire antenna lying on a multilayer ground," Waves in Random and Complex Media, Vol. 28, 49-59, 2018.
28. 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.