## 2 Theoretical Analysis
### Implementation of CV Mode with ZPA Condition  
The LCC-S topology has received considerable attention for its ability to provide a stable CV-output characteristic under dynamic loads. The circuit diagram of the LCC-S compensation topology is presented in Figure 6, where the \(C_{\text{FF}}\) represents the \(C_{\text{F3}}\) and \(L_{2}\) in parallel, and in series with \(C_{\text{F2}}\), which is described in detail in Section 3. The composition of the primary side of the LCC-S topology is the same as that of the DS-LCC compensation topology, which can generate a constant current in the transmitting coil. The difference between the two topologies is that the secondary side is a resonant network composed of \(L_{\text{S}}\) and \(C_{\text{FF}}\) in series. Figure 5: (**a**) Transconductance gain of the DS-LCC topology; (**b**) input phase angle of the input equivalent impedance. Figure 6: LCC-S compensated WPT charging systems. The \(M\)-type model analysis method is also applicable to the analysis of the LCC-S compensation topology. According to the foregoing analysis, the \(M\)-type model is applied to investigate the LCC-S compensation topology, as shown in Figure 7.  
To intuitively study the LCC-S compensation topology, the equivalent \(M\)-type model in Figure 7 can be converted into the model, as shown in Figure 8.  
The equivalent impedance \(Z_{\text{S}}\) and the reflection impedance \(Z_{\text{R2}}\) of the secondary side can be derived as follows:  
\[\left\{ \begin{array}{l}Z_{\text{S}}=j\omega L_{\text{S}}+\frac{1}{j\omega \text{C}_{\text{FF}}}+R_{\text{AC}}\\ Z_{\text{R2}}=\frac{w^{2}M^{2}}{Z_{\text{S}}}.\end{array} \right. \tag{8}\]  
The input equivalent impedance \(Z_{\text{IN}}\) and the input phase angle \(Q_{\text{IN}}\) can be represented as follows:  
\[\left\{ \begin{array}{l}Z_{\text{IN}}=j\omega L_{1}+\frac{1}{j\omega\text{C}_ {\text{PI}}}||(j\omega L_{\text{P}}+\frac{1}{j\omega\text{C}_{\text{PI}}}+Z_{ \text{R2}})\\ Q_{\text{IN}}=\frac{180^{\circ}}{\pi}arctan\frac{\text{Im}(Z_{\text{IN}})}{R_{ \text{c}}(Z_{\text{IN}})}.\end{array} \right. \tag{9}\]  
According to [27], when the operating angular frequency is operated at \(\omega_{\text{CV}}=1/\sqrt{L_{1}\text{C}_{\text{PI}}}=1/\sqrt{\text{C}_{\text {FI}}(L_{\text{P}}-L_{1})}=1/\sqrt{L_{\text{S}}\text{C}_{\text{FF}}}\), the input equivalent impedance \(Z_{\text{IN}}\) can be rewritten as (10):  
\[Z_{\text{IN}}=\frac{\omega^{2}\text{C}_{\text{FI}}L_{1}R_{\text{AC}}}{j\omega (L_{\text{P}}-L_{1}-\frac{1}{\omega^{2}\text{C}_{\text{FI}}})R_{\text{AC}}+ \omega^{4}M^{2}\text{C}_{\text{PI}}\text{C}_{\text{FI}}}. \tag{10}\]  
To achieve soft switching in the ZPA condition, it is required that the imaginary part of the Equation (10) must be equal to zero, which can be expressed as (11):  
\[L_{1}=\frac{\text{C}_{\text{FI}}}{\text{C}_{\text{FI}}+\text{C}_{\text{PI}}} L_{\text{P}}. \tag{11}\]  
As indicated from the Equation (10), \(Z_{\text{IN}}\) can be rewritten as (12):  
\[Z_{\text{IN}}=\frac{L_{1}^{2}}{M^{2}}R_{\text{AC}}. \tag{12}\]  
Figure 8: Primary side circuit diagram of the LCC-S topology with reflected impedance. Figure 7: Equivalent \(M\)-type model of the LCC-S compensation topology for the CV-output. The input and output currents of the LCC-S topology can be represented as (13):  
\[\left\{ \begin{array}{l}I_{\text{IN}}=\frac{U_{\text{IN}}}{Z_{\text{IN}}}=\frac{U_ {\text{IN}}M^{2}}{L_{1}^{2}R_{\text{AC}}}\\ I_{\text{P}}=\frac{U_{\text{IN}}}{j\omega L_{1}}\\ I_{\text{O}}=\frac{l\omega M_{\text{IP}}}{Z_{\text{S}}}=\frac{U_{\text{IN}}M}{L _{1}R_{\text{AC}}}.\end{array} \right. \tag{13}\]  
The input voltage \(U_{\text{O}}\) of the rectifier and the voltage transfer gain \(G_{\text{UU}}\) can be represented as:  
\[\left\{ \begin{array}{l}U_{\text{O}}=I_{\text{O}}R_{\text{AC}}=\frac{MU_{ \text{IN}}}{L_{1}}\\ G_{\text{UU}}=\left|\frac{U_{\text{LO}}}{U_{\text{AB}}}\right|=\frac{M}{L_{1}}.\end{array} \right. \tag{14}\]  
Figure 9a,b is based on Equations (9) and (14), respectively.