A simple and efficient explicit solution is derived for the mutual inductance of two non-coaxial coplanar circular loops, which is valid in the quasi-static as well as non-quasi-static frequency ranges. The solution is obtained by rigorously evaluating the Sommerfeld Integral describing the inductance, starting from expanding the integrand into a power series of the loop radius. As a result, a sum of simpler integrals is obtained, and term-by-term analytical integration is straightforwardly performed. The inductance is finally expressed as a series of spherical Hankel functions, with algebraic coefficients depending on the electrical size of the loops. Conducted numerical tests lead to conclude that, accuracy being equal, the proposed expression offers advantages in terms of time cost over conventional numerical integration techniques.
2. Fu, M., H. Yin, and C. Ma, "Megahertz multiple-receiver wireless power transfer systems with power flow management and maximum efficiency point tracking," IEEE Trans. Microwave Theory Techniques, Vol. 65, 4285-4293, 2017.
3. Niitsu, K., Y. Sugimori, Y. Kohama, K. Osada, N. Irie, H. Ishikuro, and T. Kuroda, "Analysis and techniques for mitigating interference from power/signal lines and to SRAM circuits in CMOS inductive-coupling link for low-power 3-d system integration," IEEE Transactions on Very Large Scale Integration (VLSI) Systems, Vol. 19, 1902-1907, 2011.
4. Angelidis, P., K. Vassiliadis, and G. D. Sergiadis, "Lowest mutual coupling between closely spaced loop antennas," IEEE Transactions on Antennas and Propagation, Vol. 39, 949-953, 1991.
5. Kwiat, D., S. Saoub, and S. Einav, "Calculation of the mutual induction between coplanar circular surface coils in magnetic resonance imaging," IEEE Transactions on Biomedical Engineering, Vol. 39, 433-436, 1992.
6. Conway, J. T., "Analytical and semi-analytical solutions for the force between circular loops in parallel planes," IEEE Transactions on Magnetics, Vol. 49, 4817-4823, 2013.
7. Conway, J. T., "Inductance calculations for noncoaxial coils using Bessel functions," IEEE Transactions on Magnetics, Vol. 43, 1023-1034, 2007.
8. Zhdanov, M. S., Geophysical Electromagnetic Theory and Methods, Elsevier, Amsterdam, 2009.
9. Parise, M., V. Tamburrelli, and G. Antonini, "Mutual impedance of thin-wire circular loops in near-surface applications," IEEE Transactions on Electromagnetic Compatibility, Vol. 61, 558-563, 2019.
10. Paul, C. R., Inductance: Loop and Partial, John Wiley & Sons, Hoboken, NJ, USA, 2010.
11. Parise, M., "Fast computation of the forward solution in controlled-source electromagnetic sounding problems," Progress In Electromagnetics Research, Vol. 111, 119-139, 2011.
12. 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.
13. 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.
14. 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.
15. Wait, J. R., "Mutual electromagnetic coupling of loops over a homogeneous ground," Geophysics, Vol. 20, No. 3, 630-637, 1955.
16. 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.
17. 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.
18. 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.
19. Parise, M., "Transverse magnetic field of infinite line source placed on ground surface," Electronics Letters, Vol. 51, No. 19, 1478-1480, 2015.
20. 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.
21. 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.
22. Telford, W. M., L. P. Geldart, and R. E. Sheriff, Applied Geophysics, Cambridge University Press, New York, 1990.
23. 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.
24. 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.
25. 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.
26. Zierhofer, C. M. and E. S. Hochmair, "Geometric approach for coupling enhancement of magnetically coupled coils," IEEE Transactions on Biomedical Engineering, Vol. 43, 708-714, 1996.
27. 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.
28. 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.
29. 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.
30. Parise, M. and G. Antonini, "On the inductive coupling between two parallel thin-wire circular loop antennas," IEEE Transactions on Electromagnetic Compatibility, Vol. 60, No. 6, 1865-1872, 2018.
31. 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.
32. 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.
33. 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.
34. Parise, M. and S. Cristina, "High-order electromagnetic modeling of shortwave inductive diathermy effects," Progress In Electromagnetics Research, Vol. 92, 235-253, 2009.
35. Rosa, E. B. and L. Cohen, "Formulae and tables for the calculation of mutual and self-inductance," Bull. Bureau Standards, Vol. 5, 1-132, 1908.
36. Snow, C., Formulas for Computing Capacitance and Inductance, (Circular of the Bureau of Standards No. 544), U. S. Govt. Printing Office, Washington DC, 1954.
37. 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.
38. Balanis, C. A., Antenna Theory: Analysis and Design, 4th Ed., John Wiley & Sons, New York, 2016.
39. Gradshteyn, I. S. and I. M. Ryzhik, Table of Integrals, Series, and Products, Academic Press, New York, 2007.
40. 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.
41. 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.
42. 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.
43. 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.
44. Kamon, M., M. J. Tsuk, and J. K. White, "FASTHENRY: A multipole accelerated 3-D inductance extraction program," IEEE Transactions on Microwave Theory and Techniques, Vol. 42, No. 9, 1750-1758, 1994.