## 3 Operation Behavior  
Before tackling this section, there are some assumptions and symbols to be given as below: (i) all the elements are viewed as ideal except the parasitic diodes and capacitors for the main switches and auxiliary switches; (ii) the input inductors can be regarded as current sources represented by \(I_{L1}\) and \(I_{L2}\); (iii) the output capacitor can be viewed as a voltage source \(V_{o}\); (iv) this converter operates in the continuous current mode (CCM); (v) the gate-driving signals for the switches \(S_{1}\), \(S_{2}\), \(S_{a}\) and \(S_{b}\) are denoted by \(v_{g1}\), \(v_{g2}\), \(v_{ga}\) and \(v_{gb}\), respectively; (vi) the voltages across the switches \(S_{1}\), \(S_{2}\), \(S_{a}\) and \(S_{b}\) are signified by \(v_{51}\), \(v_{52}\), \(v_{5a}\) and \(v_{5b}\), respectively; (vii) the currents flowing through the switches \(S_{1}\), \(S_{2}\), \(S_{a}\) and \(S_{b}\) are represented by \(i_{S1}\), \(i_{S2}\), \(i_{sa}\) and \(i_{Sb}\), respectively; (viii) the currents flowing through \(D_{1}\), \(D_{2}\) and \(D_{r}\) are indicated by \(i_{D1}\), \(i_{D2}\) and \(i_{Dr}\), respectively; (ix) the current flowing through \(L_{r}\) and the voltage across \(C_{r}\) are denoted by \(i_{Lr}\) and \(v_{Cr}\), respectively. According to the aforementioned assumptions and symbols, the converter in Figure 2 is an equivalent for that in Figure 1. In Figure 3, there are sixteen operation modes in this circuit. Since such a converter is controlled by interleave, namely, the gate-driving signal \(v_{g2}\) shifted by 180 degrees from the gate-driving signal \(v_{g1}\), the behavior of the first eight operation modes is identical to that of the last eight operation modes. Hence, only the first eight operation modes are illustrated. Figure 3: Illustrated waveforms. Figure 2: Equivalent circuit. Mode 1: \([t_{0}\leq t\leq t_{1}]\). As in Figure 4, the auxiliary switch \(S_{a}\) is switched on at the instant \(t_{0}\). Since the output diodes \(D_{1}\) and \(D_{2}\) are also switched on, thereby making the parasitic capacitor of \(S_{b}\), called \(Cs_{b}\), abruptly discharged to zero. Meanwhile, the output voltage \(V_{o}\) was imposed on the resonant inductor \(L_{r}\) and the resonant capacitor \(C_{r}\), thereby making \(L_{r}\) and \(C_{r}\) resonate. Moreover, the current \(i_{Sa}\) is equal to \(i_{Lr}\) and both are increasing, but the current \(i_{D1}\) begins to fall. As soon as \(i_{Lr}\) is equal to \(l_{L1}\), namely \(i_{Lr}(t_{1})=l_{L1}\), this mode ends. During such a mode, the accompanying equations are described as below:  
\[\left\{ \begin{array}{l}i_{Lr}(t)=\frac{(V_{o}-v_{C}(t_{0}))}{Z_{1}}\sin( \omega_{1}(t-t_{0}))+i_{Lr}(t_{0})\cos(\omega_{1}(t-t_{0}))\\ v_{Cr}(t)=V_{o}-(V_{o}-v_{Cr}(t_{0}))\cos(\omega_{1}(t-t_{0}))+i_{Lr}(t_{0})Z _{1}\sin(\omega_{1}(t-t_{0}))\end{array} \right. \tag{1}\]  
where  
\[\omega_{1}=\sqrt{\frac{1}{L_{r}C_{r}}}\text{ and }Z_{1}=\sqrt{\frac{L_{r}}{C_{ r}}} \tag{2}\]  
At the instant \(t_{0}\), \(i_{Lr}(t_{0})\) is close to zero and \(v_{Cr}(t_{0})\) is much smaller than \(V_{o}\), so the expression of the current \(i_{Lr}\) from Equation (1) can be approximately represented by  
\[i_{Lr}(t)=\frac{V_{o}}{Z_{1}}\sin(\omega_{1}(t-t_{0})) \tag{3}\]  
From Equation (3), the elapsed time for this mode can be obtained to be  
\[(t_{1}-t_{0})=\frac{1}{\omega_{1}}\sin^{-1}\!\left[\frac{l_{L1}Z_{1}}{V_{o}}\right] \tag{4}\]  
Mode 2: \([t_{1}\leq t\leq t_{2}]\). As in Figure 5, the output diode \(D_{1}\) is switched off. Meanwhile, the current \(I_{L2}\) charges the parasitic capacitor of the auxiliary switch \(S_{b}\), called \(C_{Sb}\), to the output voltage \(V_{o}\) abruptly, while the resonant inductor \(L_{r}\) resonates with the resonant capacitor \(C_{r}\) in series with the parasitic capacitor of the main switch \(S_{1}\), called \(C_{S1}\). Since the current \(i_{Sa}\) is equal to \(i_{Lr}\), the current \(I_{L1}\) is the sum of the currents of \(i_{Lr}\) and \(i_{S1}\).