## 2 Theoretical Analysis and Proposed Hybrid IPT System
### Analysis and Implementation of the CC Charging Mode with the ZPA Operation
According to (6) and (7), for the \(\Pi-\)style network to achieve a CV output, two conditions must be satisfied: \(Z_{\Pi 3}\) should tend to infinity and \(Z_{\Pi 1}+Z_{\Pi 2}=0\). Therefore, we present a new design that uses an ACS to turn off \(C_{p2}\) such that \(Z_{\Pi 3}\) tends to infinity. Moreover, another ACS is added in \(C_{\
ot 2}\) to achieve the condition of \(Z_{\Pi 1}+Z_{\Pi 2}=0\). The proposed system's equivalent analytical circuit is presented in Figure 5:0  
As shown in Figure 9, the analysis method for the primary-side resonant network (4 and 5) is identical to that of (1 and 2) in Figure 5:1 \(L_{1}\) and \(C_{p1}\) form a \(T\)-style resonant network, which converts the voltage source \(U_{AB}\) into a current source. After adjustment by ZPA, it provides energy for the \(\Gamma\)-style resonant network composed of 4 and 2 on the secondary side. Therefore, the conversion circuit is shown in Figure 10a,b. For the compensated network on the secondary side to operate in a resonant state, \(C_{\
ot kv}\) must resonate \(L_{s}\) and \(L_{2}\) simultaneously. Thus, \(C_{\
ot kv}\) can be expressed as follows:  
Figure 5:2  
To achieve ZVS, it is necessary to realize the ZPA input at the resonance frequency point. In Figure 9, the equivalent input impedance and ZPA of the proposed IPT system after switching topologies can be expressed as:  
\[\left\{ \begin{array}{l}Z_{IN}=j\omega L_{1}+\frac{1}{j\omega C_{p1}}||(j \omega L_{p}+\frac{1}{j\omega C_{f1}}+Z_{R})\\ Q_{IN}=\frac{180^{3}}{\pi}\arctan\frac{\text{Im}(Z_{IN})}{\text{Re}(Z_{IN})} \end{array} \right. \tag{14}\]  
where \(Z_{r}\) represents the reflection impedance, which can be deduced as:  
\[Z_{R}=\frac{\omega^{2}M^{2}}{j\omega L_{s}+1/(j\omega C_{\
ot cv})+j\omega L_{ 2}+R_{ac}} \tag{15}\]  
Figure 10: The proposed IPT charger’s circuit conversion process for the CV mode: (**a**) Circuit before conversion; (**b**) simplified circuit. Figure 9: The proposed IPT charger’s equivalent analytical circuit for the CV mode. If the conditions of (8) and (13) are satisfied, the equivalent input impedance \(Z_{IN}\) can be given as follows:  
\[Z_{IN}=\frac{L^{2}R_{ac}}{M^{2}} \tag{16}\]  
To obtain the inductive input impedance with the resonant current lagging behind the resonant voltage under the ZVS condition, the resonant frequency can be slightly reduced, or \(L_{1}\) can be set to a value slightly larger than the theoretical value, reducing the inverter's switching loss. Meanwhile, the MOSFETs should have enough dead time to maintain ZVS [33]. Therefore, the rectifier's input voltage, \(U_{ab}\), can be seen as a product of the HFI output voltage, \(U_{AB}\), multiplied by the ratio of \(M\) to \(L_{1}\), which can be expressed as follows:  
\[U_{ab}=\frac{U_{AB}M}{L_{1}} \tag{17}\]  
Moreover, the proposed IPT charger's voltage transfer gain, \(G_{UU}\), in the CV mode is expressed as:  
\[G_{UU}=\frac{U_{ab}}{U_{AB}}=\frac{M}{L_{1}} \tag{18}\]  
As indicated by (18), the proposed IPT charger can realize a CV output and satisfy the charging-voltage requirements of the charging device by adjusting the inductor \(L_{1}\).