# Synchronous Rectification Method for CLLC Resonant Converters Based on State-Trajectory Models
## 3 SR Method for CLLC Resonant Converters Based on State-Trajectory Models
### Above-Resonant Synchronous Rectification Method  
Through the analysis in the Section 2, it can be seen that the resonant converter has a total of six modes. There are four modes in the over-resonant state--mode 1, mode 3, mode 4, and mode 6--as shown in Figure 4a for the modes corresponding to the middle of the key waveforms and Figure 4b for the trajectory diagrams and equivalent circuits corresponding to the four modes. The following expressions for the trajectories corresponding to the four modes are obtained from the KVL analysis circuit:  
\[\left[\left(i_{L\text{r}1N}(t)+i_{L\text{r}2N}(t)\right)\right]^{2}+\left[u_{ C\text{r}1N}(t)+u_{C\text{r}2N}(t)-(1-N)\right]^{2}=4P_{1}^{2} \tag{1}\]  
\[\left[\left(i_{L\text{r}1N}(t)+i_{L\text{r}2N}(t)\right)\right]^{2}+\left[u_{ C\text{r}1N}(t)+u_{C\text{r}2N}(t)-(1+N)\right]^{2}=4P_{1}^{2} \tag{2}\]  
\[\left[\left(i_{L\text{r}1N}(t)+i_{L\text{r}2N}(t)\right)\right]^{2}+\left[u_{ C\text{r}1N}(t)+u_{C\text{r}2N}(t)-(-1+N)\right]^{2}=4P_{1}^{2} \tag{3}\]  
\[\left[\left(i_{L\text{r}1N}(t)+i_{L\text{r}2N}(t)\right)\right]^{2}+\left[u_{ C\text{r}1N}(t)+u_{C\text{r}2N}(t)-(-1-N)\right]^{2}=4P_{1}^{2} \tag{4}\]  
\(i_{L\text{r}1N}(t)+i_{L\text{r}2N}(t)\) and \(u_{C\text{r}1N}(t)+u_{C\text{r}2N}(t)\) are the sums of the resonant inductor currents and resonant capacitor voltages on the primary and secondary sides normalized to different modes, where the at-harmonic voltage gain is \(N=\frac{nL_{\text{r}1N}}{U_{\text{r}1N}}\). \[D_{SR\_Delay}=\theta_{IV}/\omega_{r} \tag{5}\]  
\[\theta_{IV}=\theta_{1}-\theta_{2} \tag{6}\]  
\[ \begin{cases}\theta_{1}=\arccos\frac{(V_{C\text{r}1N_{E}}+V_{C\text{r}2N_{E}} )-x_{O_{IV}}}{R_{IV}}\\ \theta_{2}=\arccos\frac{(V_{C\text{r}1N_{E}}+V_{C\text{r}2N_{E}})-x_{O_{IV}}} {R_{IV}}\end{cases}  \tag{7}\]  
where \(\theta_{IV}\) is the angle of the center of the circle corresponding to the state IV, \(\theta_{IV}\) is as shown in Equation (6), and, according to the expression in Equation (7), it can be obtained that the final expression of \(\theta_{IV}\) is as follows:  
\[\theta_{IV}=\arccos\frac{(V_{C\text{r}1N_{E}}+V_{C\text{r}2N_{E}})-x_{O_{IV} }}{R_{IV}}-\arccos\frac{(V_{C\text{r}1N_{E}}+V_{C\text{r}2N_{E}})-x_{O_{IV}}} {R_{IV}} \tag{8}\]  
where \(V_{C_{\text{r}1N_{E}}}+V_{C_{\text{r}2N_{E}}}\) and \(V_{C_{\text{r}1N_{E}}}+V_{C_{\text{r}2N_{E}}}\) are the sums of normalized resonant capacitance voltages \(C_{\text{r}1},\ C_{\text{r}2}\) at the points E and F, respectively. Points E and F are the start and end points of mode 4, \(R_{IV}\) is the radius corresponding to the pair of modes 4, \(O_{IV}\) is the center of the circle of mode 4, and \(x_{O_  
In Equation (9), in which \(u_{C\text{r1N}_{-E}}+u_{C\text{r2N}_{-E}},u_{C\text{r1N}_{-H}}+u_{C\text{r2N}_{-H}}\), and \(u_{C\text{r1N}_{-F}}+u_{C\text{r2N}_{-F}}\) correspond to the sum of the resonant capacitor \(u_{C\text{r1N}_{-F}}+u_{C\text{r2N}_{-F}}\) voltages for modes E, F, and H, and \(i_{L\text{r1N}_{-E}}+i_{L\text{r2N}_{-E}}i_{L\text{r1N}_{-H}}+i_{L\text{r2N}_{ -H}}\), and \(i_{L\text{r1N}_{-F}}+i_{L\text{r2N}_{-F}}\) correspond to the sum of the resonant capacitor \(i_{L\text{r}}\),\(i_{L\text{r2}}\) currents for modes E, F, and H, \(R_{{}_{I}}\) is the radius of the mode 1 trajectory, and \(R_{{}_{I}V}\) is the radius of the trajectory of mode 4. \[nU_{outN}=(u_{C\text{r1N}_{E}}+u_{C\text{r2N}_{E}})/(u_{C\text{r1N}_{F}}+u_{C \text{r2N}_{F}}) \tag{10}\]  
Next, the output current \(I_{outN}\) is determined by solving, and the output current can be solved by the difference between the primary resonant inductor current \(i_{L\text{r1N}}\) and the excitation inductor current \(i_{L\text{mN}}\):  
\[I_{\text{outN}}=\frac{f_{\text{N}}}{\pi}\int^{t_{\text{s}}}\lvert i_{L\text{rN }}-i_{L\text{mN}}\rvert dt \tag{11}\]  
Half of the cycle consists of mode 1 and mode 4, where \(L_{m}\) is clamped in a voltage of magnitude \(nU_{out}\). The other half of the state consists of mode 3 and mode 5, where \(L_{m}\) is clamped in a voltage of magnitude \(-nU_{out}\).