## 2 Theoretical Analysis  
### Implementation of CC Mode with ZPA Condition  
To achieve a load-independent CC output characteristic with high design freedom of the transfer gain, the DS-LCC compensation topology is proposed. Its design adopted the symmetric tuning scheme with the same compensation network parameters for the primary and secondary sides. The schematic diagram of the DS-LCC compensation wireless charging system is shown in Figure 2.  
On the primary side, the DC source \(U_{\text{DC}}\) was converted to a square wave voltage \(U_{\text{IN}}\) through the full bridge inverter (FBI) and then flowed into the DS-LCC resonance compensation topology. \(L_{1}\), \(C_{\text{PI}}\), and \(C_{\text{F1}}\) represent the primary side compensated inductor and capacitors. \(L_{2}\), \(C_{\text{P2}}\), and \(C_{\text{P2}}\) are the secondary side compensation components. \(C_{\text{F1}}\) and \(C_{\text{F2}}\) are the DC blocking capacitors to eliminate the DC current component. \(M\) is the mutual inductance between the two sides. \(Q_{1}\)-\(Q_{4}\) represent SiC metal-oxide semiconductor field-effect transistors (MOSFETs) of the FBI, while \(D_{1}\)-\(D_{4}\) represent diodes of the rectifier. \(U_{\text{O}}\) and \(I_{\text{O}}\) represent the output voltage and current of the DS-LCC topology, respectively. The \(M\)-type and \(T\)-type models are widely used to analyze WPT charging systems. When the self-conductance and the compensation capacitor should interact in the compensation network, the \(M\)-type model has more advantages than the \(T\)-type model [29]. Therefore, the \(M\)-type model is adapted to investigate the DS-LCC compensation topology, as shown in Figure 3.  
To facilitate the analysis, the equivalent \(M\)-type model in Figure 3 can be converted into the model shown in Figure 4.  
According to the [PERSON]'s voltage law (KCL), the equivalent impedance \(Z_{\text{S}}\) and the reflection impedance \(Z_{\text{R1}}\) of the secondary side can be deduced as (1):  
\[\left\{ \begin{array}{l}Z_{\text{S}}=j\omega L_{\text{S}}+\frac{1}{j\omega C_ {\text{F2}}}+\frac{1}{j\omega C_{\text{P2}}}||(j\omega L_{2}+R_{\text{AC}})\\ Z_{\text{R1}}=\frac{\omega^{2}M^{2}}{Z_{\text{S}}}\end{array} \right. \tag{1}\]  
Figure 4: Primary side circuit diagram of the DS-LCC topology with reflected impedance. Figure 3: Equivalent \(M\)-type model of the DS-LCC topology for the CC charging. Figure 2: DS-LCC compensated WPT charging systems. where the symbol \"\(||\)\" expresses the parallel connection. The input equivalent impedance \(Z_{\text{IN}}\) and the input phase angle \(Q_{\text{IN}}\) of the simplified \(M\)-type model of the DS-LCC topology can be expressed as follows:  
\[\left\{ \begin{array}{l}Z_{\text{IN}}=j\omega L_{1}+\frac{1}{j\omega C_{\text{ PI}}}||(j\omega L_{\text{P}}+\frac{1}{j\omega C_{\text{P1}}}+Z_{\text{R1}})\\ Q_{\text{IN}}=\frac{180^{\circ}}{\pi}arctan\frac{I_{\text{IN}}(Z_{\text{IN}}) }{R_{\text{e}}(Z_{\text{IN}})}.\end{array} \right. \tag{2}\]  
To obtain the ZPA for achieving CC-output operation, the input equivalent impedance \(Z_{\text{IN}}\) can be represented by Equation (3) for eliminating the imaginary part. \[Z_{\text{IN}}=j\omega L_{1}+\frac{1}{j\omega C_{\text{PI}}}+\frac{1}{\omega^{2 }C_{\text{PI}}^{2}(Z_{P}+Z_{\text{R1}})}, \tag{3}\]  
where the \(Z_{\text{P}}=j\omega L_{\text{P}}+\frac{1}{j\omega C_{\text{PI}}}+\frac{1}{j \omega C_{\text{PI}}}\), and \(Z_{\text{IN}}\) should satisfy the limited condition as follows:  
\[\left\{ \begin{array}{l}C_{\text{PI}}(\omega^{2}L_{\text{P}}C_{\text{FI}}-1) -C_{\text{FI}}=0\\ \omega^{2}L_{1}C_{\text{PI}}-1=0.\end{array} \right. \tag{4}\]  
\(Z_{\text{IN}}\) of the DS-LCC compensation topology can be rewritten as follows:  
\[Z_{\text{IN}}=\frac{L_{1}L_{2}}{C_{\text{PI}}C_{\text{P2}}\omega^{2}M^{2}R_{ \text{AC}}}. \tag{5}\]  
The input current \(I_{\text{IN}}\) and output current \(I_{\text{O}}\) can be expressed as follows:  
\[\left\{