## 3 Proposed CC and CV Charging Modes and Parameter Modification
### The Parameter Modification  
On the primary side of the WPT charging systems, the input voltage can be modulated by the FBI into a square wave voltage with a 0.5-T duty cycle. Assuming the output voltage \(U_{\text{IN}}\) is expanded by the Fourier series, it can be expressed as:  
\[U_{\text{IN}}=\frac{4l_{\text{DC}}}{\pi}\sum_{n=1,3,5...}\frac{\sin(n\varphi)}{ n}. \tag{15}\]  
where \(n\) is the number of harmonics and \(\varphi\) is the phase angle. Because the quality factor of the resonance compensation topology is quite high in the WPT system, the \(l_{\text{IN}}\) current component corresponding to the super-harmonics are very small. Therefore, fundamental harmonic analysis can be used without affecting accuracy. If only the fundamental component is considered, \(n=1\), the relationships between the input DC source and root-mean-square (RMS) values of the output voltage \(U_{\text{IN}}\) and current \(I_{\text{IN}}\) of FBI can be expressed as the following equation according to [30]:  
\[U_{\text{IN}}=\frac{2\sqrt{2}}{\pi}U_{\text{DC}},I_{\text{IN}}=\frac{2\sqrt{2} }{\pi}I_{\text{DC}}. \tag{16}\]  
The secondary side, due to the output low-pass filter, is only composed of capacitance. Thus, the RMS input voltage \(U_{\text{O}}\) of the rectifier, and the connection between the charging current \(I_{\text{BAT}}\) of the battery charger and the RMS input current \(I_{\text{O}}\) of the rectifier can be derived according to [26], which is represented as (17):  
\[U_{\text{O}}=\frac{2\sqrt{2}}{\pi}U_{\text{BAT}},I_{\text{BAT}}=\frac{2\sqrt{ 2}}{\pi}I_{\text{O}}. \tag{17}\]  
According to (17), the rectifier output resistance \(R_{\text{L}}\) can be expressed as (18):  
\[R_{\text{L}}=\frac{\pi^{2}}{8}R_{\text{AC}}. \tag{18}\]  
The parameter adjustment process of the proposed WPT systems typically includes two steps: First, the compensation inductors \(L_{1}\) and \(L_{2}\) are designed according to the CC-output \(I_{\text{BAT}}\) and the rated output power \(P_{\text{O}}\) required by the DS-LCC compensation topology. \(L_{1}\) and \(L_{2}\) symmetric design by using traditional tuning method [20] can be expressed as (19):  
\[L_{1}=L_{2}=\frac{\pi\sqrt{2}MU_{\text{IN}}}{4U_{\text{BAT}}}. \tag{19}\]  
Here, \(M=k\sqrt{L_{\text{P}}L_{\text{S}}}\), by substituting Equation (16) into (19), \(L_{1}\) and \(L_{2}\) can be rewritten as follows:  
\[L_{1}=L_{2}=\frac{k\sqrt{L_{\text{P}}L_{\text{S}}}U_{\text{DC}}}{U_{\text{BAT }}}. \tag{20}\]  
Figure 12: (a) Control logic for \(S_{1}\) and \(S_{2}\) of the battery charger and (b) circuit diagram for performing the switching action.  
Second, all the compensation capacitors can be determined according to the given resonance conditions.  
\[\begin{array}{c}C_{\rm PI}=\frac{1}{\omega_{0}^{2}L_{1}},C_{\rm P2}=\frac{1}{ \omega_{0}^{2}L_{2}},C_{\rm FI}=\frac{1}{\omega_{0}^{2}(L_{\rm P}-L_{1})},\\ C_{\rm F2}=\frac{1}{\omega_{0}^{2}(L_{\rm S}-L_{1})},C_{\rm FF}=\frac{1}{ \omega_{0}^{2}L_{\rm S}}.\end{array} \tag{21}\]  
On the secondary side of the LCC-S compensation topology, the sum of the impedances of \(C_{\rm F2}\), \(C_{\rm F3}\), and \(L_{2}\) should be the same as the impedance value \(C_{\rm FF}\), as shown in Figure 10c.  
\[\frac{1}{j\omega C_{\rm FF}}=\frac{1}{j\omega C_{\rm F2}}+\frac{1}{j\omega C_{ \rm F3}}||j\omega L_{2}. \tag{22}\]  
According to Equations (21) and (22), the capacitance of \(C_{\rm F3}\) can be expressed as:  
\[C_{\rm F3}=\frac{1}{L_{2}\omega_{0}^{2}}-\frac{C_{\rm FF}C_{\rm F2}}{C_{\rm FF }-C_{\rm F2}}. \tag{23}\]  
As indicated by Equations (15)-(23), the compensation capacitance and inductance values of the WPT charging systems can be obtained. In order to more clearly present the parameters design procedure of the proposed hybrid compensation topology is shown in Figure 13.  
Moreover, the proposed high-order hybrid compensation topology has high degree of freedom, and by designing compensation parameters of the WPT charging systems can yield different levels of charging current and charging voltage within a certain range.