## 3 Operation Behavior
Therefore, \(I_{Lr}\) will force \(C_{S1}\) to discharge, thereby causing the voltage across \(S_{1}\), called \(v_{S1}\), to be dropped. As soon as \(v_{S1}\) is zero, this mode ends. During such a mode, the accompanying equations are described as below:  
\[\left\{\begin{array}{l}i_{Lr}(t)=\frac{V_{1}}{Z_{1}}\sin(\omega_{2}(t-t_{1} ))+I_{1}\cos(\omega_{2}(t-t_{1}))-I_{1}+i_{Lr}(t_{1})\\ v_{S1}(t)=\frac{C}{C_{S1}}(V_{1}\cos(\omega_{2}(t-t_{1}))-I_{1}Z_{2}\sin( \omega_{2}(t-t_{1}))-V_{1})+\\ \frac{I_{L1}}{C_{r}+C_{S1}}(t-twhere  
\[ \begin{array}{l}C=\frac{C_{r}C_{S1}}{C_{r}+C_{S1}}\,\ \omega_{2}=\sqrt{\frac{1}{L \mathcal{C}}}=\sqrt{\frac{C_{r}+C_{S1}}{L_{r}C_{S1}}}\\ Z_{2}=\sqrt{\frac{L}{\mathcal{C}}}=\sqrt{\frac{L_{r}(C_{S1}+C_{r})}{C_{r}C_{S1 }}}\\ V_{1}=V_{o}-v_{Cr}(t_{1})\,\ I_{1}=i_{Lr}(t_{1})-\frac{C}{C_{S1}}I_{L1}\end{array}  \tag{6}\]  
As the voltage \(v_{S1}\) is identical to the voltage \(v_{Cr}\), the resonant current \(i_{Lr}\), which flows through the auxiliary switch \(S_{a}\), will be a maximum value, called \(I_{Lr\text{-peak}}\), which can be expressed as  
\[I_{Lr\text{-peak}}=\frac{\sqrt{V_{1}^{2}+(I_{1}Z_{2})^{2}}}{Z_{2}}+\frac{C}{C_{ S1}}I_{L1} \tag{7}\]  
Mode 3: \([t_{2}\leq t\leq t_{3}]\). As in Figure 6, the voltage across \(S_{1}\), called \(v_{S1}\), is zero due to the parasitic diode of the switch \(S_{1}\), called \(D_{S1}\), conducting the current. Therefore, as \(S_{1}\) is switched on during this mode, \(S_{1}\) has ZVT turn-on without conduction. Since \(L_{r}\) resonates with \(C_{r}\), the voltage on \(L_{r}\) is identical to the minus voltage across \(C_{r}\), thereby causing \(i_{Lr}\) to begin to drop. Moreover, the current \(I_{L1}\) is the sum of the currents of \(i_{Lr}\), \(i_{S1}\) and \(i_{Dr}\). As soon as \(i_{Lr}\) drops to \(I_{L1}\), this mode ends. During such a mode, the accompanying equations are described as below:  
\[\left\{ \begin{array}{l}i_{Lr}(t)=i_{Lr}(t_{2})\cos(\omega_{1}(t-t_{2}))- \frac{v_{C}(t_{2})}{Z_{1}}\sin(\omega_{1}(t-t_{2}))\\ v_{Cr}(t)=v_{Cr}(t_{2})\cos(\omega_{1}(t-t_{2}))+i_{Lr}(t_{2})Z_{1}\sin( \omega_{1}(t-t_{2}))\end{array} \right. \tag{8}\]  
Figure 5: Current path in mode 2. Figure 6: Current path in mode 3. **Lemma 4**: \([t_{3}\leq t\leq t_{4}]\)_. As in Figure 7, the current \(i_{L,r}\) is equal to the current \(I_{L1}\) at the instant \(t_{3}\), and hence both the currents \(i_{S1}\) and \(i_{Dr}\) are zero, thereby causing the diodes \(D_{S1}\) and \(D_{r}\) to be switched off. During this mode, the main switch \(S_{1}\) is switched on with conduction. Meanwhile, the resonant inductor \(L_{r}\) still resonates with the resonant capacitor \(C_{r}\), and \(i_{Lr}\) is still reduced, but \(i_{S1}\) increases from zero gradually. Moreover, the current \(I_{L1}\) is the sum of the currents \(i_{S1}\) and \(i_{Sa}\), and \(i_{Sa}\) is identical to \(i_{Lr}\). As \(i_{Lr}\) drops to zero, the auxiliary switch \(S_{a}\) is switched off, thereby causing \(S_{a}\) to operate under ZCS turn-off. As soon as \(i_{Lr}\) is identical to zero, this mode ends. **Lemma 5**: \([t_{4}\leq t\leq t_{5}]\)_. As in Figure 8, the main switch \(S_{1}\) remains switched on. During this mode, the resonant inductor \(L_{r}\) still keeps resonating with the resonant capacitor \(C_{r}\), and the current \(i_{Lr}\) increases in the negative direction and flows through \(S_{1}\), thereby making the parasitic diode of the auxiliary switch \(S_{a}\), called \(D_{Sa}\), switched on. Moreover, as soon as \(i_{Lr}\) goes from the minimum value to zero, this mode ends. **Lemma 6**: \([t_{5}\leq t\leq t_{6}]\)_. As in Figure 9, the main switch \(S_{1}\) still remains switched on. During this mode, the resonant inductor \(L_{r}\), with its current in the positive direction, resonates with the resonant capacitor \(C_{r}\), thereby making the resonant diode \(D_{r}\) switched on and hence a resonant loop is generated. Moreover, the current in the main switch \(S_{1}\), called \(i_{S1}\), is identical to the current \(I_{L1}\).