# Analysis of a Three-Level Bidirectional ZVS Resonant Converter
## 3 Circuit Operation
\[t_{d}>\frac{C_{S}V_{H}}{i_{Lm,p}}=\frac{4L_{m}C_{S}V_{H}}{nV_{L}T_{sw}} \tag{3}\]  
Therefore, the maximum magnetizing inductance is derived in Equation (4). \[L_{m}\leq\frac{nV_{L}t_{d}T_{sw}}{4C_{S}V_{H}} \tag{4}\]  
**Step 4 (\(t_{3}\leq t<t_{4}\)):** At \(t_{3}\), \(v_{CS3}=v_{CS4}=0\). Since \(i_{Lr}(t_{3})\) is positive, \(D_{S3}\) and \(D_{S4}\) are conducting. Power devices \(S_{3}\) and \(S_{4}\) can turn on after \(t_{3}\) under zero voltage condition. Since \(i_{Lr}(t_{3})>0\), \(D_{a}\) is forward biased. The leg voltage \(v_{ab}=-V_{H}/2\) and \(v_{Cf}=v_{CS1}=v_{CS2}=V_{H}/2\). When \(i_{Lr}\) decreases and \(i_{Lr}\)\(<0\), \(D_{a}\) becomes off. On the secondary side, \(i_{Q2}(t_{3})<0\) and \(i_{Q3}(t_{3})<0\). Therefore, \(Q_{2}\) and \(Q_{3}\) turn on to conduct the secondary-side current, the primary-side voltage \(v_{Lm}=-nV_{L}\) and \(i_{Lm}\) decreases. **Step 5 (\(t_{4}\leq t<t_{5}\)):** The secondary-side switch currents \(i_{Q2}=i_{Q3}=0\) at \(t_{4}\). Then, \(Q_{2}\) and \(Q_{3}\) turn off. In this step, \(v_{ab}=-V_{H}/2\) and \(L_{r}\), \(L_{m}\) and \(C_{r}\) are resonant. Figure 3: The corresponding equivalent circuits related to six operating steps for forward power operation (**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 6 (\(t_{5}\leq t<\)\(T_{sw}\)+\(t_{0}\)):** At \(t_{5}\), \(S_{3}\) and \(S_{4}\) turn off. In this step, \(i_{Lr}(t_{5})<0\) and \(v_{CS1}\) and \(v_{CS2}\) decrease. The ZVS condition of \(S_{2}\) and \(S_{1}\) is the same as \(S_{4}\) and \(S_{3}\) in Equation (1). The step 6 is ended at time \(T_{sw}\)+\(t_{0}\). The _LLC_ resonant circuit is controlled to achieve ZVS operation and the bidirectional power operation. The resonant circuit is based on the fundamental frequency analysis to achieve load voltage regulation. According to the switching status of power devices \(S_{1}\)\(\sim\)\(S_{4}\) and \(Q_{1}\)\(\sim\)\(Q_{4}\), the voltage values \(V_{H}\)/2 and \(-V_{H}\)/2 are observed on \(v_{ab}\), and the other voltage values \(nV_{L}\) and \(-nV_{L}\) are generated on the magnetizing inductor voltage \(v_{Lm}\). \(L_{r}\), \(C_{r}\), \(L_{m}\) and \(R_{ac,L}\) operate as a filter to suppress the high order harmonics. The root mean square (\(rms\)) voltages at the fundamental frequency for input and output sides are \(v_{ab,rms}=\sqrt{2}V_{H}/\pi\) and \(v_{Lm,rms}=2\sqrt{2}nV_{L}/\pi\). Based on the power balance between the primary-side and the secondary-side of transformer, the primary-side load resistance is expressed as \(R_{ac,L}=8n^{2}R_{L}/\pi^{2}\). The transfer function \(G_{H}\_{L}(s)\) between the output and input sides in Figure 2a is obtained as:  
Figure 1:0  
Figure 1:1  
where \(F=\)\(f_{sw}\)/\(f_{r}\), \(f_{r}=1/(2\pi\sqrt{L_{r}C_{r}})\), \(K_{1}=L_{m}\)/\(L_{r}\) and \(Q_{1}=\sqrt{L_{r}/C_{r}}/R_{ac,L}\). From the given input voltage \(V_{H}\), the output voltage \(V_{L}\) and the circuit parameters \(L_{r}\), \(C_{r}\), \(L_{m}\) and \(R_{L}\), the switching frequency is obtained from Equation (6). For reverse power flow shown in Figure 1c, the developed converter transfers power from \(V_{L}\) terminal to \(V_{H}\) terminal. \(S_{ac}\) is turned on and \(L_{b}\), \(L_{r}\) and \(C_{r}\) are operated as a series resonant circuit to achieve voltage \(V_{H}\) regulation. Power devices \(Q_{1}\)\(\sim\)\(Q_{4}\) are controlled with PFM scheme and \(D_{S1}\)\(\sim\)\(D_{S4}\) work as a full-wave rectifier. When \(|i_{Lr}|\)\(>\)\(|i_{Lb}|\), \(D_{S1}\) and \(D_{S2}\) or \(D_{S3}\) and \(D_{S4}\) are conducting. Since the _LLC_ resonant circuit by \(L_{r}\), \(C_{r}\) and \(L_{b}\) is operated at the inductive load, power devices \(Q_{1}\)\(\sim\)\(Q_{4}\) are operated at the zero-voltage turn-on switching. Figure 4a shows the ac equivalent resonant circuit at reverse power flow operation. \(L_{b}\) and \(R_{ac,H}\) are the parallel inductance and ac equivalent resistance.