# A Novel Secondary Side Series LCD Forward Converter with High Efficiency and Magnetic Reset
## 1 Introduction
### Analysis of the Switching Characteristics of the Switch  
According to Equation (14), the current flow through the inductor \(L_{2}\) can be derived using the following. \[i_{L2}(t)=-C_{2}\frac{d(u_{C2}(t))}{dt}=I_{L1,t0}\cdot\cos\left(\frac{t}{\sqrt {L_{2}C_{2}}}\right)+\frac{V_{C2,t0}}{\sqrt{L_{2}C_{2}}}\cdot\sin\left(\frac {t}{\sqrt{L_{2}C_{2}}}\right) \tag{18}\]  
The inductor current of \(I_{L2,t1}\) at the switch-off time can be obtained using the following. \[I_{L2,t1}=I_{L1,t0}\cdot\cos\left(\frac{t_{m}}{\sqrt{L_{2}C_{2}}}\right)+ \frac{V_{C2,t0}}{\sqrt{L_{2}C_{2}}}\cdot\sin\left(\frac{t_{m}}{\sqrt{L_{2}C_{ 2}}}\right) \tag{19}\]  
During the switch-on period, \(i_{L1}\) increases linearly from \(I_{L1,t0}\), and reaches the maximum value at the switch-off time. The maximum current of inductor \(L_{1}\) can be obtained using the following. \[I_{L1,\max}=I_{L1,t0}+\frac{V_{i}-nV_{o}}{nL_{1}}\cdot DT \tag{20}\]  
Due to the simultaneous conduction of diodes \(D_{1}\) and \(D_{3}\), the series branches of the inductor \(L_{2}\) and diode \(D_{4}\) are short-circuited, so \(i_{L2}\) remains unchanged during this process until \(i_{L1}\) decreases to be equal to \(i_{L2}\) and this process ends. In this process, the inductor \(L_{1}\) supplies energy to the load through \(D_{1}\), \(i_{L1}\) decreases linearly, and the expression of \(i_{L1}\) is shown in Equation (21). \[i_{L1}(t)=I_{L1,\max}-\frac{V_{o}}{L_{1}}\cdot t \tag{21}\]  
According to Equations (20) and (21), when \(i_{L1}\) decreases to be equal to \(i_{L2}\), the required time \(t_{1}\) can be obtained using the following equation. \[t_{1}=\frac{nL_{1}(I_{L1,t0}-I_{L2,t1})+(V_{i}-nV_{o})DT}{nV_{o}} \tag{22}\]  
When \(i_{L1}\) drops to be equal to \(i_{L2}\), \(D_{1}\) is naturally turned off. After that, \(L_{1}\) and \(L_{2}\) deliver energy to the load through diode D\({}_{3}\). The equivalent circuit is shown in Figure 5.  
As shown in Figure 5, \(i_{L1}\) and \(i_{L2}\) decrease linearly from \(I_{L2,t1}\) until the charging voltage across \(C_{2}\) increases to the maximum value, \(i_{L1}\) and \(i_{L2}\) decrease to the minimum value, and this process ends. We can obtain the expression for \(i_{L1}\) and \(i_{L2}\) using the following equation. \[i_{L1-2}(t)=I_{L2,t1}-\frac{V_{o}}{L_{1}+L_{2}}\cdot t \tag{23}\]According to Equations (7) and (22), the time \(t_{2}\) for \(L_{1}\) and \(L_{2}\) to jointly transfer energy to the load can be calculated using the following equation. \[t_{2}=t_{Lm}-t_{1}=\frac{\pi\sqrt{L_{W2}C_{2}}}{2}-\frac{nL_{1}(I_{L1,l0}-I_{L2, l1})+(V_{i}-nV_{o})DT}{nV_{o}} \tag{24}\]  
By substituting Equation (24) into Equation (23), the minimum value of \(i_{L1}\) can be obtained using the following. \[I_{L1,\text{min}}=I_{L2,l1}-\frac{V_{o}}{L_{1}+L_{2}}\cdot\left(\frac{\pi \sqrt{L_{W2}C_{2}}}{2}-\frac{nL_{1}(I_{L1,l0}-I_{L2,-2})+(V_{i}-nV_{o})DT}{nV_{ o}}\right) \tag{25}\]  
When the voltage across \(C_{2}\) reaches the maximum value, the energy stored in the capacitor C\({}_{2}\) is released to inductors \(L_{1}\), \(L_{2}\), and \(W_{2}\), and its equivalent circuit is shown in Figure 6.  
According to Figure 6, we can obtain the following equation. \[I_{L2,t1}=I_{L1,t0}\cdot\cos\left(\frac{t_{m}}{\sqrt{L_{2}C_{2}}}\right)+ \frac{V_{C2,t0}}{\sqrt{L_{2}C_{2}}}\cdot\sin\left(\frac{t_{m}}{\sqrt{L_{2}C_{ 2}}}\right) \tag{19}\]0  
The voltage across \(L_{1}\), \(L_{2}\), and \(W_{2}\) can be derived as follows. \[\left\{ \begin{array}{l}u_{L2}=-L_{2}C_{2}\frac{d^{2}u_{\text{C2}}(t)}{dt^{2}} \\ u_{L1}=-L_{1}C_{2}\frac{d^{2}u_{\text{C2}}(t)}{dt^{2}}\\ u_{W2}=-L_{W2}C_{2}\frac{d^{2}u_{\text{C2}}(t)}{dt^{2}}\end{array} \right. \tag{27}\]  
By substituting Equation (27) into Equation (26), we can obtain the following. \[-u_{\text{C2}}(t)-(L_{L2}+L_{L1}+L_{W2})\text{C2}\cdot\frac{d^{2}u_{\text{C2}} (t)}{dt^{2}}+V_{o}=0 \tag{28}\]  
Figure 5: Equivalent circuit diagram of the inductors \(L_{1}\) and \(L_{2}\) releasing energy to the load. Figure 6: Equivalent circuit of the capacitor \(C_{2}\) reverse energy storage release.