In this paper, we present a full-wave semi-analytical solution to calculate the self and mutual impedances of two coupled spiral inductors with rectangular cross sections. In low-frequency electromagnetism, the self and mutual impedance of planar spiral inductors can be obtained based on the eddy current approximation, where the displacement current is disregarded. As the frequency increases, the size of the system can be designed to be smaller. However, the displacement current becomes more important in inductively-coupled systems. By directly deriving the Maxwell's equations without the eddy current assumption, the obtained full-wave model could be applied to both homogeneous and planarly layered media for wireless power transfer systems. Compared to the traditional methods, the newly derived impedances show a considerable discrepancy at GHz frequencies for millimeter-sized inductors, indicating the significance of the displacement current if the operating frequency of wireless power transmission reaches the GHz-range.
2. Grover, F. W., Inductance Calculations, Dover Publications, New York, 1946.
3. Hurley, W. G. and M. C. Duffy, "Calculation of self and mutual impedances in planar magnetic structures," IEEE Trans. Magn., Vol. 31, No. 4, 2416-2422, 1995.
4. Zhong, G. and C. K. Koh, "Exact closed-form formula for partial mutual inductances of rectangular conductors," IEEE Trans. Circ. Syst. Fund. Theor. Appl., Vol. 50, No. 10, 1349-1352, 2003.
5. Babic, S. I. and C. Akyel, "New analytic-numerical solutions for the mutual inductance of two coaxial circular coils with rectangular cross section in air," IEEE Trans. Magn., Vol. 42, No. 6, 1661-1669, 2006.
6. Snow, C., Formulas for Computing Capacitance and Inductance, 544, National Bureau of Standards Circular, Washington, DC , 1954.
7. Mohan, S. S., M. Hershenson, S. P. Boyd, and T. H. Lee, "Simple accurate expressions for planar spiral inductances," IEEE J. Solid-State Circuits, Vol. 34, No. 10, 1419-1424, 1999.
8. Nguyen, M. Q., Z. Hughes, P. Woods, Y.-S. Seo, S. Rao, and J.-C. Chiao, "Field distribution models of spiral coil for misalignment analysis in wireless power transfer systems," IEEE Trans. Microw. Theory Tech., Vol. 62, No. 4, 920-930, 2014.
9. Savio, A., M. Carmina, A. Richelli, L. Colalongo, and Z. M. Kovacs-Vajna, "A new lumped model for on-chip inductors including substrate currents," Proceedings of the 15th International Conference on Microelectronics, 2003. ICM 2003, 9-11, 2003.
10. Richelli, A., L. Colalongo, M. Quarantelli, M. Carmina, and Z. M. Kovacs-Vajna, "A fully integrated inductor-based 1.8-6-V step-up converter," IEEE Journal of Solid-State Circuits, Vol. 39, No. 1, 242-245, 2004.
11. Poon, A. S. Y., S. O’Driscoll, and T. H. Meng, "Optimal frequency for wireless power transmission into dispersive tissue," IEEE Trans. Antennas Propag., Vol. 58, No. 5, 1739-1750, 2010.
12. Kim, S., J. S. Ho, and A. S. Y. Poon, "Wireless power transfer to miniature implants: Transmitter optimization," IEEE Trans. Antennas Propag., Vol. 60, No. 10, 4838-4845, 2012.
13. Kim, S., J. S. Ho, and A. S. Y. Poon, "Midfield wireless powering of subwavelength autonomous devices," Phys. Rev. Lett., Vol. 110, 203905, 2013.
14. Park, S. I., "Enhancement of wireless power transmission into biological tissues using a high surface impedance ground plane," Progress In Electromagnetics Research, Vol. 135, 123-136, 2013.
15. Chew, W. C., Waves and Fields in Inhomogeneous Media, IEEE Press, 1995.
16. Chew, W. C. and B. Anderson, "Propagation of electromagnetic waves through geological beds in a geophysical probing environment," Radio Science, Vol. 20, No. 3, 611-621, 1985.
17. Chew, W. C., S. Barone, B. Anderson, and C. Hennessy, "Diffraction of axisymmetric waves in a borehole by bed boundary discontinuities," Geophysics, Vol. 49, No. 10, 1586-1595, 1984.
18. Paulus, M., P. Gay-Balmaz, and O. J. F. Martin, "Accurate and efficient computation of the Green’s tensor for stratified media," Physical Review E, Vol. 62, 5797-5807, 2000.
19. Tsang, L., C. Huang, and C. Chan, "Surface electric fields and impedance matrix elements of stratified media," IEEE Trans. Antennas Propag., Vol. 48, No. 10, 1533-1543, 2000.
20. Dural, G. and M. I. Aksun, "Closed-form Green’s functions for general sources and stratified media," IEEE Trans. Microw. Theory Tech., Vol. 43, No. 7, 1545-1552, 1995.
21. Zhao, J. S., W. C. Chew, C. C. Lu, E. Michielssen, and J. M. Song, "Thin-stratified medium fast-multipole algorithm for solving microstrip structures," IEEE Trans. Microw. Theory Tech., Vol. 46, No. 4, 395-403, 1998.
22. Gabriel, S., R. W. Lau, and C. Gabriel, "The dielectric properties of biological tissues: III. Parametric models for the dielectric spectrum of tissues," Phys. Med. Biol., Vol. 41, No. 11, 2271-2293, 1996.