## 2 Theoretical Analysis
### Implementation of CC Mode with ZPA Condition
\begin{array}{l}I_{\text{IN}}=\frac{U_{\text{IN}}C_{\text{PI}}C_{ \text{PI}}\omega^{2}M^{2}R_{\text{AC}}}{L_{1}L_{2}}\\ I_{\text{O}}=\frac{ULTI_{\text{IN}}}{j\omega L_{1}L_{2}}.\end{array} \right. \tag{6}\]  
As indicated by Equation (6), \(I_{\text{O}}\) only depends on \(U_{\text{IN}}\), the operating angular frequency \(\omega\), and the compensation inductance parameters \(L_{1}\) and \(L_{2}\). Therefore, the DS-LCC compensation topology can be equivalent to a current source, realizing the CC charging mode. The transconductance gain \(G_{\text{UI}}\) of the DS-LCC compensation topology can be derived as follows:  
\[G_{\text{UI}}=\left|\frac{I_{\text{O}}}{U_{\text{IN}}}\right|=\frac{M}{j \omega L_{1}L_{2}}. \tag{7}\]  
According to Equations (2) and (7), the input phase angle \(Q_{\text{IN}}\) of the input equivalent impedance and transconductance gain \(G_{\text{UI}}\) of the DS-LCC WPT charging systems are plotted in Figure 5, respectively. The diagrams of the transconductance gain \(G_{\text{UI}}\), which are regarded as the absolute value of the ratio of \(I_{\text{O}}\) to \(U_{\text{IN}}\). As shown in Figure 5a, the constant \(G_{\text{UI}}\) under the ZPA can be realized at the fixed frequency of 85 kHz, and the input phase angle of the associated input impedance against the effect of the dynamic loads. As shown in Figure 5, it can be seen that the DS-LCC compensation topology can also achieve constant current output under ZPA conditions at the frequency of 110 kHz, but the slope of the transconductance gain curve at 110 kHz was significantly higher than the slope at 85 kHz. Therefore, changes in the output current at 110 kHz were more sensitive to changes in the switching frequency than at 85 kHz. A small variation in frequency around 110 kHz could cause a significant change in the output current. In addition, the variation of the phase according to the switching frequency change at 110 kHz was larger than that at 85 kHz. Therefore, this article selected 85 kHz as the operating frequency. Additionally, it is worth noting that if the compensation parameters satisfy complex constraint conditions, the DS-LCC topology can implement CV-output by using the frequency-hopping under the conditions of ZPA [27]. However, frequency-hopping leads to the following issues:  
1. [label=(0)]
2. Iteration is needed to determine the CV-output point of attachment of the parameter settings, which leads to a high required accuracy of the compensation parameters;
3. Under the resonant condition, the WPT systems are highly sensitive to changes in the coupling parameters of the LCT. If the coupling parameters change, the ZPA conditions cannot be realized in the CV output stage;
4. The increment of the working frequency inevitably increases the loss of the WPT system and can even lead to frequency bifurcation;
5. The operating frequency of the system is limited by relevant standards; thus, it cannot be improved in a wide range. Therefore, this article analyzes the existing compensation topology that can realize the CV-output. It is found that the high-order compensation topology LCC-S can realize a load-independent CV-output. Next, a systematic theoretical analysis of the LCC-S topology is performed.