## 2 System Topology and Circuit Theory Analysis  
_Topology Analysis of SS-Type Wireless Power Transfer System_  
The SS-type topology is a commonly used control structure in wireless power transfer systems. The SS-type controller is a state feedback controller based on the system's state-space model, enabling precise control of system state and output. In wireless power transfer systems, the SS-type controller is primarily used to control the output power and achieve optimal efficiency transmission by controlling the magnitude and waveform of the output power. Magnetic-coupling systems have four basic topologies: SS, SP, PS, and PP. In magnetic-coupling systems, the SP (Series-Parallel), PS (Parallel-Series), and PP (Parallel-Parallel) configurations complement the SS (Series-Series) topology as the four basic topologies. SS (Synchronous-Synchronous): In an SS configuration, the magnetic coupling system's input and output sides are synchronous, meaning that both the driving and driven components rotate at the same speed. SP (Synchronous-Passive): In an SP configuration, the driving side is synchronous while the driven side is passive. The driving component rotates at a constant speed and transfers motion to the passive component through magnetic coupling. PS (Passive-Synchronous): In a PS configuration, the driving side is passive while the driven side is synchronous. In this setup, the passive component receives motion from the synchronous component through magnetic coupling. PP (Passive-Passive): In a PP configuration, both the driving and driven sides are passive. There is no synchronous component involved, and both sides rely on magnetic coupling for motion transfer without active driving force. The SS-type system is widely used due to its inherent resonance frequency (\(\omega\)), which remains unchanged with variations in equivalent resistance. In this study, a series-resonant WPT system with a single transmitter and receiver was investigated. A typical SS topology is shown in Figure 1, where Vac represents the AC voltage source; Lp and Ls denote the primary and secondary resonant inductors; and Cp and Cs represent the primary and secondary resonant capacitors, together forming the resonant network of the entire system. Rp and Rs represent the internal resistance of the primary and secondary coils, respectively, while RL represents the equivalent resistance of the system. Energy transfer between the coils is based on electromagnetic induction, where M denotes the mutual inductance between the coils, and the coupling coefficient can be calculated as \(\mathrm{k}=\frac{\mathrm{M}}{\sqrt{\mathrm{L}_{\mathrm{p}}\mathrm{L}_{\mathrm{s}}}}\). The following is a schematic diagram of the SS-type wireless power transfer system topology:  
Based on the above diagram, Kirchhoff's Voltage Law (KVL) equations can be written as follows:  
\[\left\{ \begin{array}{l}\mathrm{v}_{\mathrm{ac}}=\mathrm{z}_{1}\mathrm{I}_{ \mathrm{P}}-\mathrm{j}\omega\mathrm{MI}_{\mathrm{S}}\\ 0=\mathrm{z}_{2}\mathrm{I}_{\mathrm{S}}-\mathrm{j}\omega\mathrm{MI}_{\mathrm{P} }\end{array} \right. \tag{1}\]  
Figure 1: SS-type topology. By solving Equations (1) and (2) simultaneously, the input impedance \(Z_{1}\) and output impedance \(Z_{2}\) can be obtained:  
\[ \begin{cases}\begin{aligned} & Z_{1}=R_{P}+j\omega L_{P}-\frac{1}{j \omega C_{P}}\\ & Z_{2}=R_{S}+R_{L}+j\omega L_{S}-\frac{1}{j\omega C_{S}}\end{aligned}\end{cases}  \tag{2}\]  
Rearranging the above equations, the input current \(I_{P}\) and output current \(I_{S}\) can be determined:  
\[ \begin{cases}I_{P}=\frac{Z_{2}V_{\infty}}{Z_{1}Z_{2}+\omega^{2}M^{2}}\\ & I_{S}=\frac{j\omega MV_{\infty}}{Z_{1}Z_{2}+\omega^{2}M^{2}}\end{cases}  \tag{3}\]  
From the above equations, due to resonance, the imaginary part (j) can be disregarded.