## 3 Control of the WPT System  
In this section, the control variables that affect the system behavior are introduced. In order to match the charging profile of batteries, the control algorithm should adjust the control variables during the charging process to provide the required current or power in different charging modes (i.e., CC, CV, and the cutoff). The typical battery charging curve is shown in Figure 4. In the CC stage, the battery voltage increases, and the peak charging power is reached at the end of the CC phase. After that is the CV stage, during which the charging current needs to be gradually reduced until the battery voltage reaches the cutoff level; then, the charging process is terminated. The charging speed or the maximal charging power can be adjusted via the charging current \(I_{\rm bat}\). Therefore, the peak charging power in the fast charging mode \(P_{\rm bat,f}\) is greater than \(P_{\rm bat,s}\) in the slow mode. The control parameters of the WPT system should, therefore, be selected to adjust the charging current, not only for adjusting the charging current for different charging speeds but also for winding down in the CV stage. In this work, the duty cycles on both sides \(D_{1,2}\), as well as the phase difference \(\varphi_{21}\), are selected as control parameters. Figure 3: The active power plane in the control parameters (primary and secondary duty cycles and the phase difference between inverter and rectifier voltages). The fundamentals of primary and secondary voltages in the form of a function of \(D_{1,2}\) can be calculated from (9) as follows:  
\[U_{1,2}=\frac{2\sqrt{2}U_{1\text{DC},2\text{DC}}}{\pi}\sin(\frac{\pi D_{1,2}}{2}), \tag{20}\]  
By substituting (20) into (15), the charging power as a function of control parameters can be obtained as follows:  
\[P_{1\text{s},2\text{s}}(D_{1},D_{2},\varphi_{21})=\mp P_{\text{max}}\sin(\frac {\pi D_{1}}{2})\sin(\frac{\pi D_{2}}{2})\sin(\varphi_{21}), \tag{21}\]  
where  
\[P_{\text{max}}=\frac{8U_{1\text{DC}}U_{2\text{DC}}(t)}{\pi^{2}}\omega_{\text{d }}^{3}MC_{1\text{f}}C_{2\text{f}}. \tag{22}\]  
As shown in (21) and (22), the charging power increases (or decreases) when the battery voltage \(U_{2\text{DC}}(t)\) rises (or falls) in the forward (or reverse) power transfer even if the control variables are kept constant. For the forward power transfer, the control variables \(D_{1}\) and \(D_{2}\) can be used mainly to match the charging profile. The maximal forward charging power \(P_{\text{max}}\) can be reached by selecting the full duty cycles \(D_{1}=D_{2}=1\) and adjusting the phase difference to \(\varphi_{21}=-90^{\circ}\). For the slow charging mode or during the CV stage, the duty cycles can be set to less than 1 to reduce the charging power. And to reduce losses in power electronics, extra reactive power should be added by adjusting \(\varphi_{21}\) to achieve ZVS. However, to reach the required power, there are multiple sets of solutions of \(D_{1}\) and \(D_{2}\), which provide the freedom to optimize the charging efficiency. With (14) and (15), and when the phase difference is set to \(\varphi_{21}=-90^{\circ}\), the efficiency of the WPT analog path (i.e., without power electronics) for the design frequency \(\eta_{\text{a}}\) is obtained as follows:  
\[\eta_{\text{a}} =\frac{-P_{2}(\varphi_{21}=-90^{\circ})}{P_{1}(\varphi_{21}=-90^ {\circ})} \tag{23}\] \[=\frac{U_{2}C_{2\text{f}}(\omega_{\text{d}}MC_{1\text{f}}U_{1}-C _{2\text{f}}R_{2}U_{2})}{U_{1}C_{1\text{f}}(\omega_{\text{d}}MC_{2\text{f}}U_ {2}+C_{1\text{f}}R_{1}U_{1})}. \tag{24}\]  
Based on the equation above, the optimized voltage ratio with maximum efficiency is derived by solving  
\[\frac{\text{d}}{\text{d}G_{21}}\eta_{\text{a}}(G_{21})=0, \tag{25}\]  
where  
\[G_{21}=\frac{U_{2}}{U_{1}}, \tag{26}\]  
Figure 4: Typical battery charging curves. Illustrated are slow and fast charging modes, each consisting of CC and CV stages.