## 2 The Proposed Converter
### Modes of Operation  
Throughout the process of a completed switching cycle, from \(t_{0}\)-\(t_{8}\), the converter operates in eight modes, as in the following exposition. The key waveforms are exhibited in Figure 4.  
Figure 4: The key waveforms in a complete switching cycle. Mode I [\(t_{0}\), \(t_{1}\)]: The equivalent circuit of mode I is depicted in Figure 5. The switch \(Q_{2}\) turns off at the instant of \(t_{0}\), the capacitance \(C_{Q2}\) starts to charge, and the voltage increases. On the other hand, \(C_{Q1}\) is discharging simultaneously. Furthermore, the charging capacity of \(C_{Q2}\) is equal to the discharging capacity of \(C_{Q1}\) due to the coupling effect and the balance of the transformer. There is no energy exchange in the transformer. The leakage inductance \(L_{p}\) resonates with \(C_{Q1}\) and \(C_{Q2}\). The winding capacitance of transformer \(C_{p1}\) starts charging and \(C_{p2}\) discharges. In addition, \(I_{Cp1}=I_{Cp2}\). When the voltage of \(C_{p1}\) reaches \(-V_{b}\), \(V_{Cp2}\) reaches \(V_{b}\) and this mode comes to the end. Since there is no energy exchange between the primary side and the secondary side in mode I, the voltage across \(C_{b}\) can be considered constant, and the equivalent circuit of mode I can be depicted in Figure 6.  
Therefore:  
\[\left\{ \begin{array}{l}V-L_{m1}\frac{d_{1}}{dt}+M\frac{dt_{2}}{dt}-\frac{1}{ C_{01}}\int i_{1}dt=0\\ V-L_{m2}\frac{d_{2}}{dt}-M\frac{d_{1}}{dt}-\frac{1}{C_{Q2}}\int i_{2}dt=0\end{array} \right. \tag{5}\]  
As \(L_{m1}=L_{m2}=L_{m}\), \(C_{Q1}=C_{Q2}=C_{Q}\), neglecting the effect of the leakage inductance, it is \(M=L_{m}\), so that yields:  
\[i_{1}=-i_{2} \tag{6}\]  
Combining (5) and (6):  
\[V-2L_{m}\frac{d_{1}}{dt}-\frac{1}{C_{Q}}\int i_{1}dt=0 \tag{7}\]  
The solution of (7) is:  
\[ \begin{array}{l}i_{1}(t)=\frac{l_{m}}{\sqrt{2\cos\psi}}cos(\omega t+\psi)\\ \omega=\frac{1}{\sqrt{2L_{m}C_{Q}}}\\ I_{m}=\frac{1}{\sqrt{2L_{m}C_{Q}}}\end{array}  \tag{8}\]  
Figure 5: The equivalent circuit of mode I.  
Figure 6: Approximated equivalent circuit for no energy exchanging between the primary side and secondary side. Mode II [\(t_{1}\),\(t_{2}\)]: The equivalent circuit of mode II is depicted in Figure 7. In this mode, the voltage of winding capacitance \(C_{p1}\) remains \(-V_{b}\), and \(V_{Cp2}\) remains \(V_{b}\). \(C_{Q2}\) keeps charging, and the voltage increases. Since \(C_{Q1}\) is discharging, the voltage decreases as well. A small level of high-frequency oscillated current flows through the second-side rectifier diode \(D_{O2}\). When the voltage of \(V_{Q1}\) drops to 0 and \(V_{Q2}\) rises to 2\(V_{b}\), \(C_{Q1}\) and \(C_{Q2}\) complete discharging and charging and mode II finishes. Mode III [\(t_{2}\),\(t_{3}\)]: The equivalent circuit of mode III is depicted in Figure 8.When the resonance on the primary side of the transformer finishes, the current through \(C_{Q1}\), \(C_{Q2}\), \(C_{p1}\), and \(C_{p2}\) returns to 0, but the voltage is still the same, respectively. \(I_{Q1}\) and \(I_{LP}\) go to 0 gradually so that the MOSFET \(Q_{1}\) operates in ZCS when it turns on. From the moment of \(t_{2}\), the current through \(D_{O2}\) returns to 0, and \(D_{O2}\) operates in ZCS in the stage of energy exchange between the primary and secondary sides. Mode IV [\(t_{3}\),\(t_{4}\)]: As \(V_{Q1}\) had already been 0 at \(t_{3}\), the MOSFET \(Q_{1}\) turns on in ZVS. Mode IV is the major process that transfers energy from the primary side to the secondary side. No more current flows in \(C_{Q1}\), \(C_{Q2}\), \(C_{p1}\), or \(C_{p2}\), but the voltage on them is invariable. Energy transmits from the push-pull side to the high-voltage side through a high-frequency transformer on the coupling of magnetizing inductance \(L_{m1}\). The current in \(Q_{1}\) increases gradually in sinusoidal form from 0 to the maximum (about 10 A), then decreases to 0 later again, the current in \(D_{O2}\) charges the voltage-doubling capacitance in the same form and synchronously offers energy for the load.