# Synchronous Rectification Method for CLLC Resonant Converters Based on State-Trajectory Models
## 3 SR Method for CLLC Resonant Converters Based on State-Trajectory Models
### SR Method in the Below-Resonance Region  
There are four modes in the below-resonant state: mode 1, mode 2, mode 4, and mode 5. As shown in Figure 6a, the key waveforms corresponding to these modes are illustrated, while Figure 6b presents the trajectory diagrams and equivalent circuits for the four modes. \[\left[\frac{i_{L_{t1N}}(t)+i_{L_{t2N}}(t)}{k_{2}}\right]^{2}+\left[u_{C_{t1N}}( t)+u_{C_{t2N}}(t)-(1+u_{C_{t2N,O}})\right]^{2}=\left(\frac{I_{1O}}{k_{2}} \right)^{2} \tag{18}\]  
\[\left[\frac{i_{L_{t1N}}(t)+i_{L_{t2N}}(t)}{k_{2}}\right]^{2}+\left[u_{C_{t1N}}( t)+u_{C_{t2N}}(t)-(1-u_{C_{t2N,O}})\right]^{2}=\left(\frac{I_{1O}}{k_{2}} \right)^{2} \tag{19}\]  
where \(k_{2}=\sqrt{k+1}\), \(I_{1O}\) are quantities related to the initial state of the circuit. Figure 4: Plane trajectory of the above-resonant state. (**a**) Key waveforms in the above-resonant region. (**b**) Equivalent circuit corresponding to the state plane trajectory. Figure 5: Plane trajectory of the above-resonant state. It is straightforward in the literature to simplify the two segments AB and CD by arguing that, in the case of \(L_{m}\gg L_{r1}\), where AB and CD are constant in the case of modes 2 and 5, these two segments can be thought of as the two segments CP and AQ. Thus, the duration of \(i_{Lr1N_{A}}+i_{Lr2N_{A}}=i_{Lr1N_{Q}}+i_{Lr2N_{Q}}\), \(i_{Lr1N_{C}}+i_{Lr2N_{C}}=i_{Lr1N_{P}}+i_{Lr2N_{P}}\) and mode 1 is half the period and is no longer related to the voltage gain and load. Figure 7 shows an enlarged view of the state trajectory of mode 2 of the CLLC resonant converter. From this figure, it can be observed that the exact trajectory of mode 2 is not linear. If the duration of mode 1 and mode 4 is \(t_{r}/2\), then BR in mode 4 becomes redundant. Therefore, based on the analysis according to Figure 8, a relatively large error occurs when performing synchronous rectification. Due to the symmetry of the operating state of the resonant converter, it is only necessary to analyze half a cycle, focusing on mode 2 and mode 4. The duty cycle of the synchronous rectified signal is derived as follows:  
\[D_{SR\_Duty}=0.5-t_{1I}f_{s} \tag{20}\]  
Equation (21) \(\Delta u_{Cr1N\_AB}\) is the sum of the resonant capacitor voltage difference between the two points AB. \(i_{Lr1N\_II}\) is the resonant inductor current because, in the period \(t_{II}\), the secondary resonant current \(i_{Lr2N\_II}\) is zero, so \(t_{II}\) is calculated only by the primary resonant inductor current and the voltage of the resonant capacitor. Because AB is a circular arc, the process is very complicated in solving for the \(u_{Cr1N\_AB}\) time. To simplify the solution process, the tangent lines of modes 2 and 4 are constructed using geometrical methods and solved within the triangle. The tangents of modes 2 and 4 intersect at a point \(B^{\prime}\); as shown in Figure 7a, the perpendiculars \(B^{\prime}H^{\prime}\) and BH from AB are drawn and almost coincident, proving that the errors of \(B^{\prime}H^{\prime}\) and BH are minimal. This \(\Delta U_{Cr1N\_AB}\) can be expressed as \(\Delta U_{Cr1N\_AB}\), which follows from the tangent principle:  
\[\Delta u_{Cr1N\_AB^{\prime}}=k_{R\_IV}\Delta u_{Cr1\_AQ}/(k_{Q\_IV}+k_{A\_II}) \tag{21}\]  
\(k_{R\_IV}\) and \(k_{R\_II}\) in (21) are the slopes of \(AB^{\prime}\) and \(RB^{\prime}\), respectively, and \(k_{R\_IV}\), \(k_{R\_II}\) can be obtained by substituting the coordinates of point A and point B into (3) and (17). \[k_{A\text{I}}=\frac{Z_{1}}{Z_{2}}\times\frac{u_{Cr1N\_A}+-1}{i_{Lr1N\_A}} \tag{22}\]  
\[k_{R\_IV}=\frac{Z_{1}}{Z_{2}}\times\frac{u_{Cr1N\_Q}+u_{Cr2N\_Q}+-nU_{outN}+1} {i_{Lr1N\_Q}+i_{Lr2N\_Q}} \tag{23}\]  
where \(u_{Cr1N\_A}\) and \(u_{Cr1N\_Q}\) are the normalized resonant capacitor voltages at points A and B. As shown in Figure 7b, as the output voltage and load increase, point A will reach the right side of the vertex of the mode 2 trajectory, \(x_{OII}=1\).