For a general multi-coil magnetic coupling wireless power transfer (MC-WPT) system with different coil properties and non-negligible cross couplings, there is no analytical solution which connects the resonant frequencies of the system to its electric parameters. Establishing such a generalized analytical solution not only permits quantitative analysis of resonant points but also provides a generic platform for the physical interpretation of resonance. For this purpose, a novel modeling and analysis method of integrating time-domain and frequency-domain is proposed in this paper, with that all possible resonant points of arbitrary coil MC-WPT systems can be quickly and accurately determined. Firstly, an accurate state space model of a general MC-WPT system with arbitrary number of transmitters, repeaters and receivers is established and the analytical current response on each circuit is derived. Then several key system concepts about resonance are elucidated, based on which the generalized analytical expressions of resonant points of MC-WPT systems, including resonant frequencies and corresponding powers and efficiencies, are derived. Further, the physical interpretation along with the mathematic relationships of several key easy-confusing concepts are deduced, which provides systematic theoretical criteria for the selection of operating frequency to achieve the resonant states for MC-WPT systems.

To reveal the general characteristic of the maximum power transfer points of MC-WPT systems, and to quickly and accurately determine all possible maximum power transfer points for arbitrary coil MC-WPT systems, a novel theoretical analysis method is proposed in this paper. Based on the modal theory, an accurate time-domain model of an n-coil MC-WPT system is established and the analytical current response on each coil is derived. According to the current response of the load, a detailed power spectrum distribution is presented, based on which the mathematic expressions of all possible maximum power transfer points, including the maximum output powers and corresponding operating frequencies, are deduced. The theoretical results are verified in experiment prototypes practically. The experimental results show that: 1) the output power reaches its maximum value at the modal frequencies rather than the designed natural frequency of the resonators in MC-WPT systems; 2) there are n modal frequencies of an n-coil MC-WPT system. The analysis results provide useful information to choose the operating frequency for practical MC-WPT systems with arbitrary number of coils.