In this paper the equivalent impedance of resonator arrays for wireless power transfer systems is obtained in closed-form from a continued fraction expression. Using the theory of difference equations, the continued fraction is described as the general term of a complex sequence defined by recurrence, and its convergence is analyzed. It is shown that the equivalent impedance can be easily found in closed-form in terms of the system parameters. In this way, the obtained closed-form expressions may help electrical engineers to quickly predict the behaviour of a system with the changes of its parameters. Some numerical examples of the theoretical results are given and discussed. Finally, the analytical formulae obtained in this work are validated with measurements and a good agreement is observed.
2. 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. on Microwave Theory and Techniques, Vol. 62, No. 4, 920-930, Apr. 2014.
3. Li, S. and C. C. Mi, "Wireless power transfer for electric vehicle applications," IEEE Journal of Emerging and Selected Topics in Power Electronics, Vol. 3, No. 1, 4-17, Mar. 2015.
4. Kamineni, A., G. A. Covic, and J. T. Boys, "Analysis of coplanar intermediate coil structures in inductive power transfer systems," IEEE Trans. on Power Electronics, Vol. 30, No. 11, 6141-6154, Nov. 2015.
5. Zhong, W., C. K. Lee, and S. Y. R. Hui, "General analysis on the use of Tesla’s resonators in domino forms for wireless power transfer," IEEE Trans. on Industrial Electronics, Vol. 60, No. 1, 261-270, Jan. 2013.
6. Wang, B., W. Yerazunis, and K. H. Teo, "Wireless power transfer: Metamaterials and array of coupled resonators," Proceedings of the IEEE, Vol. 101, No. 6, 1359-1368, Jun. 2013.
7. Hui, S. Y. R., W. Zhong, and C. K. Lee, "A critical review of recent progress in mid-range wireless power transfer," IEEE Trans. on Power Electronics, Vol. 29, No. 9, 4500-4511, Sept. 2014.
8. Alberto, J., G. Puccetti, G. Grandi, U. Reggiani, and L. Sandrolini, "Experimental study on the termination impedance effects of a resonator array for inductive power transfer in the hundred kHz range," Proc. 2015 IEEE Wireless Power Transfer Conference (WPTC 2015), 1-4, Boulder, CO, USA, May 2015.
9. Monti, G., L. Corchia, L. Tarricone, and M. Mongiardo, "A network approach for wireless resonant energy links using relay resonators," IEEE Trans. on Microwave Theory and Techniques, Vol. 64, No. 10, 3271-3279, Oct. 2016.
10. Moon, S. and G. W. Moon, "Wireless power transfer system with an asymmetric four-coil resonator for electric vehicle battery chargers," IEEE Trans. on Power Electronics, Vol. 31, No. 10, 6844-6854, Oct. 2016.
11. Stevens, C. J., "Magnetoinductive waves and wireless power transfer," IEEE Trans. on Power Electronics, Vol. 30, No. 11, 6182-6190, Nov. 2015.
12. Syms, R. R. A., I. R. Young, and L. Solymar, "Low-loss magneto-inductive waveguides," Journal of Physics D: Applied Physics, Vol. 39, No. 18, 3945, 2006.
13. Puccetti, G., C. J. Stevens, U. Reggiani, and L. Sandrolini, "Experimental and numerical investigation of termination impedance effects in wireless power transfer via metamaterial," Energies, Vol. 8, No. 3, 1882-1895, 2015.
14. Beato-Lopez, J., C. de la Cruz Blas, A. Mitra, and C. Gomez-Polo, "Electrical model of giant magnetoimpedance sensors based on continued fractions," Sensors and Actuators A: Physical, Vol. 242, 73-78, 2016.
15. Gong, Z., Z. Tang, S. Mukamel, J. Cao, and J. Wu, "A continued fraction resummation form of bath relaxation effect in the spin-boson model," The Journal of Chemical Physics, Vol. 142, No. 8, 084103, 2015.
16. Puccetti, G., U. Reggiani, and L. Sandrolini, "Experimental analysis of wireless power transmission with spiral resonators," Energies, Vol. 6, No. 11, 5887-5896, 2013.
17. Elaydi, S., An Introduction to Difference Equations, Springer Science & Business Media, 2005.
18. Alberto, J., U. Reggiani, and L. Sandrolini, "Circuit model of a resonator array for a WPT system by means of a continued fraction," Proc. 2016 IEEE 2nd Int. Forum on Research and Technologies for Society and Industry Leveraging a Better Tomorrow (RTSI), 1-6, Bologna, Italy, Sept. 2016.
19. Alberto, J., U. Reggiani, and L. Sandrolini, "Study of the conducted emissions of an IPT system composed of an array of magnetically coupled resonators," Proc. 2017 IEEE Int. Symposium on Electromagn. Compat. Signal/Power Integrity (EMCSI), 623-628, Washington, DC, USA, Aug. 2017.
20. Kazimierczuk, M. K., "Class D voltage-switching MOSFET power amplifier," IEE Proceedings B-Electric Power Applications, Vol. 138, No. 6, 285-296, IET, 1991.