# Analysis and Implementation of a Hybrid DC Converter with Wide Voltage Variation
## 3 Operation Principle
### Low Voltage Range Input (\(V_{in,min}\leq V_{in}<2V_{in,min}\)) under Forward Power Flow  
If the input voltage is in the low voltage range, the full-bridge resonant circuit is used on the primary side and the half-wave diode rectifier is adopted on the secondary side to achieve the maximum voltage gain. Therefore, \(S_{ac1}\) and \(S_{5}\)-\(S_{8}\) are off and \(S_{ac2}\) is on, as shown in Figure 1b. From the pulse-width modulation signals of \(S_{1}\)-\(S_{4}\), \(D_{55}\), and \(D_{56}\), one can observe that there are eight switching states (Figure 2a) in every switching cycle if \(f_{sw}\) (switching frequency) \(<\)\(f_{r}\) (resonant frequency). The circuit components are assumed as \(C_{r,2}=n^{2}C_{r,1}\) and \(L_{r,2}=L_{r,1}/n^{2}\), where \(n=n_{p}/n_{s}\), to simplify the circuit analysis. Therefore, the series resonant frequency on the primary and secondary sides are identical, \(f_{r,1}=1/(2\pi\sqrt{L_{r,1}C_{r,1}})=f_{r,2}=1/(2\pi\sqrt{L_{r,2}C_{r,2}})\). The state circuit operations related to the eight switching states are provided in Figure 2b-i. State 1 [\(t_{0}\leq t<t_{1}\)]: When \(t=t_{0}\), \(v_{51,cc}=v_{54,cc}=0\). \(i_{L,r,1}(t_{0})<0\) will flow through diodes \(D_{51}\) and \(D_{54}\). Then, \(S_{1}\) and \(S_{4}\) can turn on at zero-voltage switching. The secondary side current \(i_{L,2}\) flows through diode \(D_{55}\) to charge capacitor \(C_{3}\). In this state, \(i_{L,1}\) increases from a negative value to zero ampere. State 2 [\(t_{1}\leq t<t_{2}\)]: When \(t>t_{1}\), \(i_{L,r,1}\) becomes a positive value. Then \(i_{L,r,1}\) flows through \(S_{1}\) and \(S_{4}\). Forward power flow is from \(V_{in}\) to \(V_{o}\) in State 2 operation. The converter leg voltages are \(V_{ab}=V_{in}\) and \(V_{de}=V_{C3}=V_{o}/2\). The resonant frequency in State 2 is given as \(f_{r,1}=1/(2\pi\sqrt{L_{r,1}C_{r,1}})\). State 3 [\(t_{2}\leq t<t_{3}\)]: When \(f_{sw}\) is less than \(f_{r,1}\), \(i_{L,r,2}\) will be decreased to zero at time \(t_{2}\). The rectifier diode \(D_{55}\) becomes off. In State 3, the resonant frequency of \(i_{L,r,1}\) is \(f_{p,1}=1/(2\pi\sqrt{C_{r,1}(L_{m,1}+L_{r,1})})\). No power is transferred to output load in this state. State 4 [\(t_{3}\leq t<t_{4}\)]: When \(t=t_{3}\), \(S_{1}\) and \(S_{4}\) are turned off. \(i_{Lr,1}\) discharges \(C_{52}\) and \(C_{53}\) where \(C_{52}\) and \(C_{53}\) are the output capacitors of \(S_{2}\) and \(S_{3}\), respectively. If the primary current \(i_{Lr,1}(t_{3})=\hat{i}_{Lm,1}>V_{in}\sqrt{2C_{S,p}/L_{r,1}}\), where \(\text{C}_{S,p}=\text{C}_{S1}=\text{C}_{S2}=\text{C}_{S3}=\text{C}_{S4}\) (\(\text{C}_{S1}\)-\(\text{C}_{S3}\) are output capacitors of \(\text{S}_{1}\)-\(\text{S}_{4}\), respectively) and \(\hat{i}_{Lm,1}\) is the peak value of \(i_{Lm,1}\), then \(C_{52}\) and \(C_{53}\) can be discharged to zero voltage at time \(t_{4}\). State 5 [\(t_{4}\leq t<t_{5}\)]: The capacitor voltages \(v_{CS2}=v_{CS3}=0\) (or \(v_{52,cc}=v_{53,cc}=0\)) at time \(t_{4}\). Then \(i_{Lr,1}(t_{4})>0\) will flow through \(D_{52}\) and \(D_{53}\). The zero-voltage turn-on operation of \(S_{2}\) and \(S_{3}\) can be implemented after time \(t_{4}\). The secondary side current \(i_{Lr,2}\) will be decreased and \(D_{56}\) becomes forward biased. The primary curet \(i_{Lr,1}\) is decreased and capacitor \(C_{4}\) is charged in State 5. State 6 [\(t_{5}\leq t<t_{6}\)]: After \(t>t_{5}\), \(i_{Lr,1}<0\). Then, \(D_{52}\) and \(D_{53}\) become reverse biased. \(i_{Lr,1}\) will flow through \(S_{2}\) and \(S_{3}\). In State 6, \(i_{Lr,1}\) is decreased and \(C_{4}\) is charged by \(i_{Lr,2}\). The resonant frequency is \(f_{r,1}=1/(2\pi\sqrt{L_{r,1}C_{r,1}})\). State 7 [\(t_{6}\leq t<t_{7}\)]: At \(t=t_{6}\), the secondary side current \(i_{Lr,2}=0\) and \(D_{56}\) becomes reverse biased. The resonant frequency in this state is \(f_{p,1}=1/(2\pi\sqrt{C_{r,1}(L_{m,1}+L_{r,1})})\). State 8 [\(t_{7}\leq t<T_{sw}+t_{0}\)]: When \(t=t_{7}\), \(S_{2}\) and \(S_{3}\) turn off. \(i_{Lr,1}(t_{7})<0\) will discharge \(C_{S1}\) and \(C_{54}\).