# Interleaved Boost Converter with ZVT-ZCT for the Main Switches and ZCS for the Auxiliary Switch
## 3 Basic Operating Principles  
Prior to the circuit analysis, there are some assumptions as below: (i) all the main switches and diodes are viewed as ideal; (ii) no parasitic resistances exist in the inductor and capacitors; (iii) the ideal input inductor can be considered as current source such that \(L_{1}\), \(L_{2}\), and \(V_{in}\) can be removed. ; (iv) the ideal output capacitor can be regarded as a voltage source such that \(C_{o}\) and \(R_{o}\) can be removed. Based on the above, the circuit in Figure 2 is an equivalent circuit for Figure 1. In Figure 3, there are twenty-two operating states. Since this converter is controlled by interleave, the behavior of the first eleven states is the same as that of the last eleven states. Therefore, only the first eleven states are described. Figure 1: Proposed soft-switching interleaved boost converter. Figure 2: Equivalent circuit of the circuit shown in Figure 1.  
State 1: \([t_{0}\leq t\leq t_{1}]\). As shown in Figure 4, the main switches \(S_{1}\) and \(S_{2}\) are turned off but the auxiliary switch \(S_{a}\) is turned on, whereas the freewheeling diodes \(D_{1}\) and \(D_{2}\) as well as the resonant diodes \(D_{r1}\) and \(D_{r2}\) are turned on. During this state, the resonant inductor \(L_{r}\) is magnetized, and hence the resonant inductor current \(i_{L,r}\) is linearly increased. In addition, the energy required by the load is provided by the current \(l_{L}\), which is equal to the current \(l_{L1}\) plus the current \(l_{L2}\). As soon as \(i_{L}\) is equal to \(l_{L}\), \(D_{1}\) and \(D_{2}\) are turned off, and hence this state comes to the end. During this interval, the corresponding state equation can be expressed as follows:  
\[i_{Lr}(t)=\frac{V_{o}}{L_{r}}(t-t_{0})+i_{Lr}(t_{0}) \tag{1}\]  
Figure 3: Illustrated waveforms for the proposed converter. As \(t=t_{1}\), \(i_{L}(t_{1})=I_{L}\), and hence the corresponding time elapsed is as follows:  
\[(t_{1}-t_{0})=\frac{[I_{L}-i_{Lr}(t_{0})]L_{r}}{V_{o}} \tag{2}\]  
State 2: \([t_{1}\leq t\leq t_{2}]\). As shown in Figure 5, the main switches \(S_{1}\) and \(S_{2}\) are still turned off but the auxiliary switch \(S_{a}\) is still turned on, whereas the freewheeling diodes \(D_{1}\) and \(D_{2}\) are turned off but the resonant diodes \(D_{r1}\) and \(D_{r2}\) are still turned on. During this state, the parasitic capacitors of \(S_{1}\) and \(S_{2}\), called \(C_{S1}\) and \(C_{S2}\), resonate with the resonant inductor \(L_{r}\), thereby making the resonant inductor current \(i_{Lr}\) keep increasing. In addition, the output capacitor \(C_{o}\) provides energy to the load. The moment \(C_{S1}\) and \(C_{S2}\) are discharged to zero, the parasitic diodes of the main switches \(S_{1}\) and \(S_{2}\), called \(D_{S1}\) and \(D_{S2}\), are turned on, and hence this state comes to the end. During this interval, the corresponding state equation can be represented as follows:  
\[\left\{ \begin{array}{l}i_{Lr}(t)=I_{L}+\frac{v_{S}(t_{1})}{Z_{2}}\sin\omega_ {2}(t-t_{1})-[I_{L}-i_{Lr}(t_{1})]\cos\omega_{2}(t-t_{1})\\ v_{S}(t)=Z_{2}[I_{L}-i_{Lr}(t_{1})]\sin\omega_{2}(t-t_{1})+v_{S}(t_{1})\cos \omega_{2}(t-t_{1})\end{array} \right. \tag{3}\]  
where  
\[\omega_{2}=\sqrt{\frac{1}{L_{r}C_{S}}}\,,\,Z_{2}=\sqrt{\frac{L_{r}}{C_{S}}}\,, \,C_{S}=C_{S1}+C_{S2}\,\,\text{and}\,\,v_{S1}=v_{S2}=v_{S} \tag{4}\]  
Figure 4: Current flow in state 1. Figure 5: Current flow in state 2. And hence, the corresponding time elapsed is as follows:  
\[(t_{2}-t_{1})=\frac{\pi}{2\omega_{2}} \tag{5}\]  
State 3: \([t_{2}\leq t\leq t_{3}]\). As shown in Figure 6, the main switches \(S_{1}\) and \(S_{2}\) are still turned off but the auxiliary switch \(S_{a}\) is still turned on, whereas the freewheeling diodes \(D_{1}\) and \(D_{2}\) are still turned off but the resonant diodes \(D_{r1}\) and \(D_{r2}\) are still turned on.