# Analysis of a Three-Level Bidirectional ZVS Resonant Converter
## 3 Circuit Operation
Figure 4b gives the main PWM waveforms and Figure 5 demonstrates the corresponding equivalent circuits at the reverse power flow operation. Figure 1:2 \(Cont\). Figure 1:3 The proposed circuit operated at reverse power flow (**a**) the equivalent _LLC_ resonant circuit (**b**) PWM waveforms. Figure 1:4 _Cont._  
Figure 1:5 The corresponding equivalent circuits related to six operating steps under backward power transfer (**a**) step 1 circuit (**b**) step 2 circuit (**c**) step 3 circuit (**d**) step 4 circuit (**e**) step 5 circuit (**f**) step 6 circuit. **Step 1 (\(t_{0}\leq t<t_{1}\)):** This step starts at \(t_{0}\) when \(v_{\text{CQ4}}=v_{\text{CQ1}}=0\). Then, the \(D_{\text{Q4}}\) and \(D_{\text{Q1}}\) conduct and \(v_{Q2,ds}=v_{Q3,ds}=V_{L}\). Due to \(D_{\text{Q1}}\) and \(D_{\text{Q4}}\) are conducting, \(v_{\text{Q4},ds}\) and \(v_{\text{Q1},ds}=0\) and \(Q_{\text{1}}\) and \(Q_{\text{4}}\) can turn on under zero voltage. Due to \(i_{Lr}(t_{0})+i_{\text{Lb}}(t_{0})\textless 0,\)\(D_{\text{S1}}\) and \(D_{\text{S2}}\) are forward biased, \(C_{H1}\) is charged, \(v_{Lm}=nV_{L}\), \(v_{ab}=V_{H}/2\) and \(i_{Lm}\) and \(i_{Lb}\) both increase. Before switches \(Q_{1}\) and \(Q_{4}\) turn off, \(i_{\text{D51}}\) and \(i_{\text{D52}}\) will decrease to zero if \(f_{sw}<f_{r}=1/2\pi\sqrt{C_{r}L_{r}}\). **Step 2 (\(t_{1}\leq t<t_{2}\)):** At time \(t_{1}\), \(i_{\text{D52}}=i_{\text{D51}}=0\) and \(D_{\text{S2}}\) and \(D_{\text{S1}}\) are off. \(L_{r}\), \(L_{b}\), and \(C_{r}\) are series resonant at frequency \(f_{p}=1/2\pi\sqrt{C_{r}(L_{b}+L_{r})}\). **Step 3 (\(t_{2}\leq t<t_{3}\)):**\(Q_{4}\) and \(Q_{1}\) turn off at \(t_{2}\). \(C_{Q2}\) and \(C_{Q3}\) are discharged in step 3. The ZVS condition of \(Q_{3}\) and \(Q_{2}\) are obtained in Equation (7). Figure 1:6  
where \(C_{Q}=C_{\text{Q1}}=\)\(\ldots=C_{\text{Q4}}\), \(i_{Lm,p}\approx nV_{L}/(4L_{m}f_{sw})\) and \(i_{Lb,p}\approx V_{H}/(8L_{b}f_{sw})\). At \(t_{3}\), \(v_{\text{CQ3}}(t_{3})=v_{\text{CQ2}}(t_{3})=0\). The time interval \(\Delta t_{23}\) is expressed in Equation (8). Figure 1:7  
where \(t_{d}\) is dead time between \(Q_{4}\) and \(Q_{3}\) or \(Q_{2}\) and \(Q_{1}\). **Step 4 (\(t_{3}\leq t<t_{4}\)):** Step 4 starts at \(t_{3}\) when \(v_{\text{CQ2}}=v_{\text{CQ3}}=0\). Therefore, \(D_{\text{Q3}}\) and \(D_{\text{Q2}}\) conduct and \(Q_{3}\) and \(Q_{2}\) can turn on under zero voltage. In step 4, \(D_{\text{S3}}\) and \(D_{\text{S4}}\) conduct, \(v_{ab}=-V_{H}/2\), \(v_{Lm}=-nV_{L}\), and \(i_{Lm}\) and \(i_{Lb}\) both decrease. **Step 5 (\(t_{4}\leq t<t_{5}\)):**\(i_{\text{D53}}=i_{\text{D54}}=0\) at \(t_{4}\). In this step, \(Q_{2}\) and \(Q_{3}\) are still in the on state so that \(v_{Lm}=-nV_{L}\). \(L_{b}\), \(C_{r}\) and \(L_{r}\) are series resonant. **Step 6 (\(t_{5}\leq t<T_{sw}\)+\(t_{0}\)):**\(Q_{2}\) and \(Q_{3}\) turn off at \(t_{5}\). Then, \(C_{\text{Q1}}\) and \(C_{\text{Q4}}\) are discharged and \(v_{\text{CQ1}}=v_{\text{CQ4}}=0\) at \(t_{sw}+t_{0}\). The proposed converter has the similar operation principle for both forward and reverse power operation. For the reverse power operation, \(Q_{1}\text{\text{\textminus}}\text{\textminus}\text{\textminus}\text{\textminus} \textinput and \(V_{L,max}\) = 52 V output is designed to be unity. The transformer turns ratio is calculated in Equation (11). Figure 1:8  
In the prototype circuit, the selected primary and secondary turns are \(n_{H}\) = 48 and \(n_{L}\) = 6. Thus, the actual transformer turns ratio is \(n\) = \(n_{H}\)/\(n_{L}\) = 8. With the adopted turns ratio, the actual maximum and minimum voltage gains at \(V_{L,nom}\) = 48 V condition are given in Equations (12) and (13). Figure 1:9  
Figure 3:0  
The control parameters \(K_{1}\) and \(Q_{1}\) can be selected at full load \(P_{L,full}\) and minimum input voltage \(V_{H,min}\) conditions. To reduce circulating current, the inductor ratio \(K_{1}\)=10 is used in this prototype circuit. For \(Q_{1}\) = 0.38 and \(K_{1}\) = 10, it can obtain the peak gain of \(\left|G_{H,L}\right(\)s\()\) is 1.13.