# Analysis of a Three-Level Bidirectional ZVS Resonant Converter
## 3 Circuit Operation  
For forward power delivery, the electric power energy is transferred from \(V_{H}\) side to \(V_{L}\) side and \(S_{ac}\) is OFF. \(S_{1}\)\(\sim\)\(S_{4}\) are controlled with PFM scheme. Due to PWM signals of \(S_{1}\)\(\sim\)\(S_{4}\), there is a square wave with \(-V_{H}\)/2 or \(V_{H}\)/2 on the leg voltage \(v_{ab}\). However, \(Q_{1}\)\(\sim\)\(Q_{4}\) are operated as the synchronous switches instead of the rectifier diodes in conventional full-bridge rectifier to reduce conduction loss. The equivalent resonant circuit and PWM waveforms for forward power delivery are provided in Figure 2. To realize the ZVS operation of \(S_{1}\)\(\sim\)\(S_{4}\), the input impedance of \(LLC\) circuit must be inductive. Figure 3 gives the corresponding equivalent circuits related to six operating steps in a switching period under \(f_{r}\) (resonant frequency) \(>\)\(f_{sw}\) (switching frequency). It is assumed that the \(L_{r}\) represents the external series resonant inductance and the leakage inductance of transformer and \(C_{r}\) represents the external series resonant capacitance and the parasitic capacitance on transformer winding turns. The output capacitances \(C_{S1}\)\(-\)\(C_{S4}\) are assumed to be identical. In the same manner, \(C_{Q1}\) = \(\ldots\) = \(C_{Q4}\). Since the current \(i_{C}\)/ on \(C_{f}\) is less than \(i_{S1}\) and \(i_{S2}\) in mode 1 and \(i_{S3}\) and \(i_{S4}\) in mode 4, \(i_{Cf}\) is ignored in PWM waveforms. Therefore, \(i_{S1}\) is equal to \(i_{S2}\) in steps 1-3 and 6 and \(i_{S3}\) is equal to \(i_{S4}\) in steps 3-6. Figure 1: Proposed converter (**a**) converter structure (**b**) forward power flow operation (\(S_{ac}\) off) (**c**) backward power flow operation (\(S_{ac}\) on, \(S_{1}\)\(\sim\)\(S_{4}\) off). Figure 3: \(Cont\). Figure 2: Forward power operation (**a**) the equivalent \(LLC\) resonant circuit (**b**) pulse-width modulation (PWM) waveforms. Figure 3: _Cont._  
**Step 1 (\(t_{0}\leq t<t_{1}\)):** At \(t<t_{0}\), \(i_{Lr}<0\). Thus, \(i_{Lr}\) discharges \(C_{S1}\) and \(C_{S2}\) are discharged. At \(t_{0}\), \(v_{CS1}=v_{CS2}=0\). Thus, \(D_{S1}\) and \(D_{S2}\) are conducting due to \(i_{Lr}<0\). The ZVS operation of \(S_{1}\) and \(S_{2}\) can be achieved after time \(t_{0}\). If \(i_{Lr}<0\), \(D_{b}\) is forward biased. The leg voltage \(v_{ab}=v_{Cf}=v_{CS3}=v_{CS4}=V_{H}/2\). Since \(i_{Lr}>i_{Lm}\), \(Q_{1}\) and \(Q_{4}\) turn on to conduct the secondary-side current. When \(i_{Lr}\) increases and \(i_{Lr}\)\(>0\), \(D_{b}\) becomes off. In this step, the magnetizing voltage \(v_{Lm}\) is equal to \(nV_{L}\), where \(n=n_{p}/n_{s}\) is the transformer turns ratio, and \(i_{Lm}\) increases. The ripple current \(\Delta i_{Lm}\) in step 1 is equal to \(nV_{L}\Delta t_{01}/L_{m}\) where \(\Delta t_{01}=t_{1}-t_{0}\). The resonant frequency in step 1 is \(f_{r}=1/(2\pi\sqrt{L_{r}C_{r}})\). **Step 2 (\(t_{1}\leq t<t_{2}\)):** If \(f_{sw}<f_{r}\), \(i_{Q1}\) and \(i_{Q4}\) will decrease to zero ampere at \(t_{1}\). Thus, \(Q_{1}\) and \(Q_{4}\) can turn off after time \(t_{1}\). In step 2, the leg voltage \(v_{ab}=V_{H}/2\) and \(L_{r}\), \(L_{m}\) and \(C_{r}\) are resonant. **Step 3 (\(t_{2}\leq t<t_{3}\)):** At \(t_{2}\), \(S_{1}\) and \(S_{2}\) turn off. The positive current \(i_{Lr}(t_{2})\) will charge \(C_{S1}\) and \(C_{S2}\). On the other hand, \(C_{S3}\) and \(C_{S4}\) are discharged in this step. The ZVS operation of \(S_{3}\) and \(S_{4}\) is expressed in Equation (1). \[i_{Lm,p}\geq\frac{V_{H}}{2}\sqrt{\frac{C_{S}}{L_{r}}} \tag{1}\]  
where \(i_{Lm,p}\) is the peak current on \(L_{m}\) and \(C_{S}=C_{S1}=\ldots=C_{S4}\). The peak current \(i_{Lm,p}\) is calculated from Equation (2). \[i_{Lm,p}=\frac{\Delta i_{Lm}}{2}\approx\frac{nV_{L}T_{sw}}{4L_{m}} \tag{2}\]  
The dead time \(t_{d}\) between \(S_{3}\) and \(S_{1}\) (or \(S_{4}\) and \(S_{2}\)) is approximately expressed in Equation (3).