# DC Converter with Wide Soft Switching Operation, Wide Input Voltage and Low Current Ripple
## 4 Circuit Analysis  
For low input voltage mode (\(V_{in}=V_{in,\ min}\)\(\sim\)\(2V_{in,\ min}\)), two boost converters and one full-bridge structure _LLC_ converter is used to reduce input ripple current and obtain soft switching turn-on operation. However, only one boost converter and a half-bridge structure _LLC_ converter are adopted for high input voltage mode (\(V_{in}=2V_{in,\ min}\)-\(4V_{in,\ min}\)). According to voltage-second balance on \(L_{1}\) and \(L_{2}\), the output voltage \(V_{C}\) of the boost converter is obtained in Equation (1). \[V_{C}=V_{in}/(1-d)=V_{in}/(1-0.5)=2V_{in} \tag{1}\]  
where \(d=0.5\) for \(S_{1}\)\(\sim\)\(S_{4}\) under low voltage mode and for \(S_{1}\) and \(S_{2}\) under high voltage mode. Since the PWM signals of \(S_{1}\) and \(S_{2}\) are phase shifted with respective to the signals of \(S_{3}\) and \(S_{4}\) by half switching cycle, the inductor current ripples \(\Delta i_{L1}\) and \(\Delta i_{L2}\) can be eliminated by each other. Therefore, the resultant input current ripple \(\Delta i_{in}=\Delta i_{L1}+\Delta i_{L2}\) is reduced to zero. The voltage rating on \(S_{1}\)\(\sim\)\(S_{4}\) is equal to the boost voltage \(V_{C}\) (=\(2V_{in}\)). The frequency control scheme [16] is adopted to analysis the circuit features and voltage gain of the adopted _LLC_ converter. To implement wide input voltage operation, two operating modes (low and high input voltage modes) are operated. The magnetizing inductor voltage at fundamental frequency is derived in (2). \[V_{Lm,rms}=\sqrt{2}nV_{o}/\pi \tag{2}\]  
Based on the full-bridge _LLC_ converter (low voltage mode) and the half-bridge _LLC_ converter (high voltage mode), the input voltage of the resonant converter at fundamental frequency \(V_{ab,\ rms}\) is derived in (3). \[V_{ab,rms}=\left\{ \begin{array}{l}2\sqrt{2}V_{C}/\pi=4\sqrt{2}V_{in,L}/\pi, S_{ac}\ on\\ \sqrt{2}V_{C}/\pi=2\sqrt{2}V_{in,H}/\pi,\ S_{ac},S_{3}:off,S_{4}:on\end{array} \right. \tag{3}\]  
The primary-side resistance of transformer \(T\) at fundamental switching frequency is derived in Equation (4). \[R_{ac}=\frac{v_{Lm,rms}}{i_{s,T}/n}=\frac{2n^{2}}{\pi^{2}}R_{o} \tag{4}\]  
Figure 3: Converter operated at high input voltage mode (**a**) main PWM signals, (**b**) step \(1\), (**c**) step \(2\), (**d**) step \(3\), (**e**) step \(4\), (**f**) step \(5\), (**g**) step \(6\). The voltage gain of the equivalent resonant tank (\(C_{r}\), \(L_{r}\), \(L_{m}\) and \(R_{ac}\)) is derived in Equation (5). \[ \begin{split}|G|&=\frac{V_{L_{m},rms}}{V_{ab,rms}}=| \frac{\frac{R_{ac}V_{ab,rms}L_{m}}{R_{ac}+V_{ab,rms}L_{m}}}{\frac{1}{V_{ab,rms}C }+\frac{R_{ac}V_{ab,rms}L_{m}}{R_{ac}+V_{ab,rms}L_{m}}}|\\ &=\frac{1}{\sqrt{\left[1+\frac{1}{L_{m}}\left(1-\frac{1}{P2} \right)\right]^{2}+Q^{2}\left(F-\frac{1}{P}\right)^{2}}}=\left\{\begin{array}[ ]{l}\frac{HV_{a}}{V_{a,L}},S_{ac}:on\\ \frac{HV_{a}}{2V_{m,H}},S_{ac},S_{3}:off,\,S_{4}:on\end{array}\right.\end{split}  \tag{5}\]  
where \(Q=\sqrt{L_{r}/C_{r}}/R_{ac}\), \(L_{n}=L_{m}/L_{r}\) and \(F=f_{sw}/f_{r}\). The gain curves of \(|G|\) related to \(F\) and \(Q\) are shown in Figure 4 for the adopted prototype circuit. From Equation (5), the output voltage \(V_{o}\) can be obtained and expressed in Equation (6). \[V_{o}=\left\{ \begin{array}{l}\frac{4V_{in,L}}{n\sqrt{\left[1+\frac{1}{L_{m}} \left(1-\frac{1}{P2}\right)\right]^{2}+Q^{2}\left(F-\frac{1}{P}\right)^{2}}}, S_{ac}:on\\ \frac{2V_{in,H}}{n\sqrt{\left[1+\frac{1}{L_{m}}\left(1-\frac{1}{P2}\right) \right]^{2}+Q^{2}\left(F-\frac{1}{P}\right)^{2}}},S_{ac},S_{3}:off,\,S_{4}:on \end{array} \right. \tag{6}\]  
It is clear that the output voltage is related to the frequency ration \(F\), quality factor \(Q\) and inductor ratio \(L_{n}\). If \(F\) equals unity, \(V_{o}\) is independent to \(l_{n}\) and \(Q\). Owing to the fact that \(V_{in,H}\) is designed as two times of \(V_{in,L}\), two output voltage equations in (5) are identical.