volume | PIER Journals

Abstract

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.

1. Trivino-Cabrera, A., J. Aguado, and J. M. Gonzalez, "Analytical characterisation of magnetic field generated by ICPT wireless charger," Electronics Letters, Vol. 53, 871-873, 2017.
doi:10.1049/el.2017.0968 Google Scholar

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.
doi:10.1109/TMTT.2017.2689747 Google Scholar

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.
doi:10.1109/TVLSI.2010.2056711 Google Scholar

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.
doi:10.1109/8.86914 Google Scholar

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.
doi:10.1109/10.135536 Google Scholar

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.
doi:10.1109/TMAG.2013.2245912 Google Scholar

7. Conway, J. T., "Inductance calculations for noncoaxial coils using Bessel functions," IEEE Transactions on Magnetics, Vol. 43, 1023-1034, 2007.
doi:10.1109/TMAG.2006.888565 Google Scholar

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.
doi:10.1109/TEMC.2018.2816030 Google Scholar

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.
doi:10.2528/PIER10101409 Google Scholar

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.
doi:10.2528/PIERB10060707 Google Scholar

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.
doi:10.1190/1.1635038 Google Scholar

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.
doi:10.1002/jnm.2120 Google Scholar

15. Wait, J. R., "Mutual electromagnetic coupling of loops over a homogeneous ground," Geophysics, Vol. 20, No. 3, 630-637, 1955.
doi:10.1190/1.1438167 Google Scholar

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.
doi:10.1190/1.1444452 Google Scholar

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.
doi:10.2528/PIERL16053003 Google Scholar

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. Google Scholar

19. Parise, M., "Transverse magnetic field of infinite line source placed on ground surface," Electronics Letters, Vol. 51, No. 19, 1478-1480, 2015.
doi:10.1049/el.2015.0636 Google Scholar

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. Google Scholar

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.
doi:10.2528/PIER07100201 Google Scholar

22. Telford, W. M., L. P. Geldart, and R. E. Sheriff, Applied Geophysics, Cambridge University Press, 1990.
doi:10.1017/CBO9781139167932

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.
doi:10.1109/LAWP.2013.2296149 Google Scholar

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.
doi:10.1109/8.481642 Google Scholar

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.
doi:10.1049/iet-map.2016.0590 Google Scholar

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.
doi:10.1109/10.503178 Google Scholar

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.
doi:10.1155/2015/187806 Google Scholar

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.
doi:10.1007/s00024-004-2586-2 Google Scholar

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. Google Scholar

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.
doi:10.1109/TEMC.2018.2790265 Google Scholar

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.
doi:10.1186/BF03351764 Google Scholar

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.
doi:10.1109/LAWP.2011.2148190 Google Scholar

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.
doi:10.1163/156939311797164945 Google Scholar

34. Parise, M. and S. Cristina, "High-order electromagnetic modeling of shortwave inductive diathermy effects," Progress In Electromagnetics Research, Vol. 92, 235-253, 2009.
doi:10.2528/PIER09022608 Google Scholar

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.
doi:10.6028/bulletin.103 Google Scholar

36. Snow, C., Formulas for Computing Capacitance and Inductance, (Circular of the Bureau of Standards No. 544), 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.
doi:10.1080/17455030.2017.1319990 Google Scholar

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.
doi:10.1016/j.aeue.2016.02.004 Google Scholar

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.
doi:10.2528/PIER12112508 Google Scholar

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.
doi:10.1109/TAP.2015.2478958 Google Scholar

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.
doi:10.1080/09205071.2014.897653 Google Scholar

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.
doi:10.1109/22.310584 Google Scholar

Dr. Marcello Salis

Dr. Marco Muzi