## 2 CC&OV-PT Controlled Two-Stage WPT System
### The Proposed Double-Side Feedback PT Control Strategy
#### 2.2.2 Cause and Analysis of Low-Frequency Oscillations  
With regards to the Buck converter, the variation of the output voltage can be presented in terms of the inductor current,  
\[\Delta v_{1}=\frac{1}{C_{1}}\int_{0}^{T_{0}}i_{\rm C}dt=\frac{1}{C_{1}}\int_{ 0}^{T_{0}}(i_{\rm L}-i_{1})dt \tag{8}\]  
where \(i_{\rm C}\) and \(i_{\rm L}\) are the output capacitor current and inductor current of the Buck converter, respectively. Term \(i_{1}\) is the input current of the full-bridge inverter, and \(\Delta v_{1}\) is the voltage variation of capacitor \(C_{1}\) inside a switching cycle. Assuming that the switching frequency of the Buck converter is \(f_{0}\), and the switching frequency of the inverter is \(f_{1}\), for the design convenience, the following relation is applied in this work,  
\[f_{1}=\frac{mf_{0}}{2}\ m=1,2,3\cdots N \tag{9}\]  
Therefore, their corresponding angular frequencies and periods can be described by:  
\[\left\{ \begin{array}{l}\omega_{1}=2\pi f_{1}=\frac{m}{2}\omega_{0}\\ T_{1}=\frac{1}{f_{1}}=\frac{2}{m}T_{0}\end{array} \right. \tag{10}\]  
where \(\omega_{0}\) and \(\omega_{1}\) are the equivalent angular speed of the Buck converter and the inverter, \(m\) is an integer, and \(T_{0}\) and \(T_{1}\) are their switching periods, respectively. Regarding the high power pulse and low power pulse, the waveforms of inductor current \(i_{\rm L}\), capacitor current \(i_{\rm C}\) as well as the input current \(i_{1}\) of the inverter are shown in Figure 4.  
The variation slope \(k_{\rm p}\) of the inductor current is expressed by:  
\[k_{\rm p}=\left\{ \begin{array}{l}k_{\rm r}=\frac{V_{\rm r}-V_{\rm r}}{V_{ \rm L}}\ nT_{0}<t\leq(n+D_{\rm v})T_{0}\\ k_{f}=-\frac{1}{L}\ (n+D_{\rm v})T_{0}<t\leq(n+1)T_{0}\end{array} \right. \tag{11}\]  
Figure 4: Illustrative waveforms of inductor current \(i_{\rm L}\), capacitor current \(i_{\rm C}\) and input current \(i_{\rm I}\). Inside a switching cycle, the variation of inductor current can be calculated by:  
\[\Delta i_{\rm L}=i_{\rm L}(nT_{0}+T_{0})-i_{\rm L}(nT_{0})=\frac{(V_{\rm i}D_{v}- V_{1})T_{0}}{L} \tag{12}\]  
According to the operation principle of the inverter circuit, we have:  
\[i_{1}=\frac{4(R_{2}+R_{\rm Lx})V_{1}|\sin\omega_{1}t|}{\pi\big{[}R_{1}(R_{2}+R_ {\rm Lx})+\omega_{1}^{2}M^{2}\big{]}} \tag{13}\]  
According to Figure 4, since the calculation of the shadow area corresponding to the high power pulse and the low power pulse is the same, only the high power pulse is taken as an example. The area can be calculated by:  
\[S_{\rm H1}=\int_{nT_{0}}^{(n+1)T_{0}}i_{\rm L}(t)dt=\frac{T_{0}^{2}}{2L}(-V_{ \rm i}D_{\rm H}^{2}+2V_{\rm i}D_{\rm H}-V_{1})+T_{0}i_{\rm L}(nT_{0}) \tag{14}\]  
\[S_{\rm H2}=\int_{nT_{0}}^{(n+1)T_{0}}i_{1}(t)dt=\frac{8m(R_{2}+R_{\rm Lx})V_{ 1}}{\pi\omega_{1}\big{[}R_{1}(R_{2}+R_{\rm Lx})+\omega_{1}^{2}M^{2}\big{]}} \tag{15}\]  
\[\Delta v_{1}=\frac{1}{C_{1}}S_{\rm H3}\ =X+Y+Z \tag{16}\]  
where \(X\), \(Y\) and \(Z\) are expressed by,  
\[\left\{ \begin{array}{l}X=\frac{T_{0}^{2}}{2L_{0}}\big{(}-V_{\rm i}D_{\rm H}^ {2}+2V_{\rm i}D_{\rm H}-V_{1}\big{)}\\ Y=\frac{T_{0}}{C_{1}}i_{\rm L}(nT)\\ Z=-\frac{8m(R_{2}+R_{\rm Lx})V_{1}}{\omega_{1}C_{1}\big{[}R_{1}(R_{2}+R_{\rm Lx })+\omega_{1}^{2}M^{2}\big{]}}\end{array} \right. \tag{17}\]  
It can be seen that \(\Delta v_{1}\) is decided by three terms, \(X\), \(Y\) and \(Z\). The values of \(X\) and \(Z\) are in fact constant when the system parameters are determined, and the value of \(Y\) is decided by \(i_{\rm L}(nT)\) and hence \(\Delta v_{1}\) is decided by \(i_{\rm L}(nT)\). According to (16), with regards to high and low power pulses, we have  
\[\Delta v_{1}=\left\{ \begin{array}{l}X^{\rm H}+Y+Z\ P=P_{\rm H}\\ X^{\rm L}+Y+Z\ P=P_{\rm L}\end{array} \right. \tag{18}\]  
where \(X^{\rm H}\) and \(X^{\rm L}\) are the values of \(X\) corresponding to high power pulse and low power pulse states, respectively. Assuming \(\Delta v_{\rm C}=0\), \(i_{\rm L}(nT)\) can be obtained by:  
\[i_{\rm L}(nT)=\left\{