Evolutionary Mechanism of Frequency Splitting in Tri-Coil Dual-Load MCR–WPT Systems Considering Cross-Coupling Effects

In multi-coil, multi-load magnetically coupled resonant wireless power transfer (MCR–WPT) systems, the non-negligible cross-coupling among multiple resonators, including the transmitter (Tx), receiver 1 (Rx1), and receiver 2 (Rx2), introduces complex frequency-splitting behavior through the Tx–Rx1, Tx–Rx2, and Rx1–Rx2 coupling paths, severely constraining transmission efficiency and operational stability. In practical multi-receiver WPT applications, receiver-side cross-coupling is often unavoidable and may shift the maximum-power and maximum-efficiency points away from the designed resonant frequency. Clarifying this mechanism is therefore important for coil arrangement, impedance matching, and stable multi-load power delivery. This paper establishes an equivalent circuit model to derive analytical expressions for input impedance, load power, and efficiency. Based on this framework, the formation mechanism of frequency splitting under concurrent coupling paths is systematically investigated. The results indicate that dominant coupling paths dictate the positions and magnitudes of primary split peaks, while cross-coupling between receivers induces local modal reconfiguration and energy redistribution, leading to secondary or minor characteristic peaks. Both simulation and experimental results demonstrate that the coupling coefficient primarily governs the frequency-splitting trajectory, whereas load resistance predominantly modulates peak amplitudes. For k=0.520, the split-frequency peaks in the two-coil benchmark occur at 64.6 kHz and 103.8 kHz, showing good agreement with the calculated modal frequencies. In the tri-coil dual-load system, pronounced power peaks around 63 kHz and 112 kHz further confirm the shift of the maximum-power transfer points under asymmetric coupling and loading conditions. Furthermore, under strong-coupling conditions, the maximum power transfer point shifts from the nominal resonant frequency toward the system’s inherent modal frequencies. This study elucidates the evolution of frequency splitting in tri-coil dual-load systems, providing a theoretical foundation for parameter optimization in multi-node WPT networks.