## 2 Circuit Model and Theoretical Analysis  
### Circuit Model  
Figure 2 shows the circuit structure diagram of the proposed LC/S compensated IPT system. \(L_{\text{p}}\) and \(L_{\text{s}}\) represent the self-inductance of the transmitting and receiving coils, and \(L_{1}\) is the compensation inductance. \(R_{1}\), \(R_{\text{p}}\), and \(R_{\text{s}}\) represent the parasitic resistance of the compensation inductance and coils. Since the values of \(R_{1}\), \(R_{\text{p}}\), and \(R_{\text{s}}\) are very small, they can be ignored in the analysis in the interest of simplification. \(M\) represents the mutual inductance between the transmitting and receiving coils. The transmitting side of the system includes a constant DC power supply \(U_{\text{d}}\), a high-frequency inverter (HFI) consisting of four MOSFETs (\(Q_{1}\)-\(Q_{4}\)), a series compensation inductor \(L_{1}\), a parallel compensation capacitor \(C_{1}\), and a transmission coil \(L_{\text{p}}\); the receiving side includes a receiving coil \(L_{\text{s}}\), a series compensation capacitor \(C_{2}\), and a rectifier connected in parallel with a filter capacitor \(C_{\text{F}}\). The full-bridge rectifier provides DC power for the battery load \(R_{\text{L}}\). Figure 3 shows a simplified circuit diagram of the topology, in which the square wave voltage \(U_{\text{in}}\) driven by the HFI and the value of the equivalent load \(R_{\text{ac}}\) can be expressed as:  
\[R_{\text{ac}}=\frac{8R_{\text{L}}}{\pi^{2}},\ U_{\text{in}}=\frac{2\sqrt{2}U_{ \text{d}}}{\pi} \tag{1}\]  
According to Figure 3, the loop equation of the topology should satisfy the following:  
\[\begin{bmatrix}\mathbf{U_{\text{in}}}\\ 0\\ 0\end{bmatrix}=\begin{bmatrix}Z_{1}+Z_{\text{m}}&-Z_{\text{m}}&0\\ 0&j\omega M&Z_{\text{s}}+R_{\text{ac}}\\ -Z_{\text{m}}&Z_{\text{p}}+Z_{\text{m}}&j\omega M\end{bmatrix}\begin{bmatrix} \mathbf{I_{1}}\\ \mathbf{I_{\text{p}}}\\ \mathbf{I_{\text{s}}}\end{bmatrix} \tag{2}\]  
Figure 3: Simplified circuit diagram of the LC/S compensated IPT system.  
Figure 2: Circuit diagram of the LC/S compensated IPT system.  
where \(Z_{1}\), \(Z_{\rm m}\), \(Z_{\rm p}\), and \(Z_{\rm s}\) are equivalent impedances expressed as  
\[\left\{\begin{array}{l}Z_{1}=j\omega L_{1}\\ Z_{\rm m}=\frac{1}{j\omega C_{1}}\\ Z_{\rm p}=j\omega L_{\rm p}\\ Z_{\rm s}=j\omega L_{\rm s}+\frac{1}{j\omega C_{2}}\end{array}\right. \tag{3}\]  
\(\mathbf{I_{1}}\) is the input current phasor of the topology. \(\mathbf{I_{\rm p}}\) and \(\mathbf{I_{\rm s}}\) represent the current phasors of the transmitting and receiving loops, respectively. According to Formula (2), the expression of the current phasors can be calculated as  
\[\left\{\begin{array}{l}\mathbf{I_{1}}=\frac{(\omega^{2}M^{2}+R_{\rm ac}Z_{\rm m} +R_{\rm ac}Z_{\rm p}+Z_{\rm s}Z_{\rm m}+Z_{\rm p}Z_{\rm s})\mathbf{\mu_{\rm m}}}{AR_ {\rm ac}+B}\\ \mathbf{I_{\rm p}}=\frac{Z_{\rm m}(R_{\rm ac}+Z_{\rm s})\mathbf{\mu_{\rm m}}}{AR_{\rm ac }+B}\\ \mathbf{I_{\rm s}}=-\frac{j\omega M^{2}\mathbf{\mu_{\rm m}}}{AR_{\rm ac}+B}\end{array}\right. \tag{4}\]  
where \(A\) and \(B\) in the denominator can be expressed as  
\[\left\{\begin{array}{l}A=Z_{1}Z_{\rm m}+Z_{1}Z_{\rm p}+Z_{\rm p}Z_{\rm m}\\ B=Z_{1}Z_{\rm m}Z_{\rm s}+Z_{1}Z_{\rm p}Z_{\rm s}+Z_{\rm m}Z_{\rm p}Z_{\rm s}+ \omega^{2}M^{2}Z_{1}+\omega^{2}M^{2}Z_{\rm m}\end{array}\right. \tag{5}\]