# Hybrid Resonant Converter with Three Half-Bridge Legs for Wide Voltage Operation
## 3 Circuit Characteristics  
The presented circuit is controlled by frequency modulation. The input voltage \(V_{in}\) has a variation between 50 and 400 V and the output voltage \(V_{o}\) is regulated at 48 V. The fundamental harmonic analysis (FHA) proposed by [22] is selected to calculate the voltage transfer function in the proposed converter by frequency modulation. When 50 V \(\leq V_{in}\) \(<\) 100 V (low voltage range), only \(Q_{1}\)-\(Q_{4}\) are operated (Figure 2a) to achieve high voltage gain. The square voltage waveform with voltage values of \(\pm V_{in}\) is obtained on the leg voltage \(v_{ab}\). If 100 V \(\leq V_{in}\) \(<\) 200 V (medium voltage range), then only power switches \(Q_{1}\), \(Q_{2}\), \(Q_{5}\), and \(Q_{6}\) are controlled to control load voltage, and AC switch \(S\) is always on (Figure 2b). The square waveform with voltage values of \(\pm V_{in}\) is obtained on the leg voltage \(v_{ac}\). If 200 V \(\leq V_{in}\) \(\leq\) 400 V (high voltage range), then only \(Q_{1}\) and \(Q_{2}\) are operated to control the load voltage, and switches \(S\) and \(Q_{6}\) are always on (Figure 2c). The square waveform with voltage values of 0 and \(V_{in}\) is obtained on the leg voltage \(v_{ac}\). The fundamental root mean square (_rms_) values of the leg voltages \(v_{ab,rms}\) (in the low voltage range) and \(v_{ac,rms}\) (in the medium and high voltage ranges) are calculated as:  
\[v_{ab,rms}=2\sqrt{2}V_{in},\ 50\ V\ \leq V_{in}<100\ V, \tag{1}\]  
\[v_{ac,rms}=\left\{ \begin{array}{cc}2\sqrt{2}V_{in},\ 100\ V\ \leq V_{in}<200\ V\\ \sqrt{2}V_{in},\ 200\ V\ \leq V_{in}<400\ V\end{array} \right.. \tag{2}\]  
Figure 5: Proposed DC converter under high input voltage range: (**a**) PWM waveforms; (**b**) state 1; (**c**) state 2; (**d**) state 3; (**e**) state 4; (**f**) state 5; (**g**) state 6. The turns ratio of transformer \(T\) under low input voltage range is \(n_{1}=n_{p}/n_{s}\). However, the turn-ratio of transformer \(T\) under medium and high input voltage ranges is \(n_{2}=2n_{p}/n_{s}=2n_{1}\). The fundamental _rms_ magnetizing voltage \(v_{Lm,rms}\) under different input voltage ranges is expressed as  
\[v_{Lm,rms}=\left\{ \begin{array}{ll}2\sqrt{2}n_{1}V_{o},&50~{}V~{}\leq V_{in} <100~{}V\\ 2\sqrt{2}n_{2}V_{o}=4\sqrt{2}n_{1}V_{o},&100~{}V~{}\leq V_{in}<400~{}V.\end{array} \right. \tag{3}\]  
The relationship between the AC equivalent resistor \(R_{eq}\) on the input-side and the DC load resistor \(R_{o}\) is obtained in Equation (4):  
\[R_{eq}=\left\{ \begin{array}{ll}\frac{8n_{1}^{2}R_{o}}{\pi^{2}},&50~{}V~{} \leq V_{in}<100~{}V\\ \frac{8n_{2}^{2}R_{o}}{\pi^{2}}=\frac{32n_{1}^{2}R_{o}}{\pi^{2}},&100~{}V~{} \leq V_{in}<400~{}V.\end{array} \right. \tag{4}\]  
Figure 6 shows the corresponding resonant tank of the presented circuit. The voltage transfer function of the corresponding resonant tank is derived and expressed in Equation (5):  
\[|G_{ac}|=\frac{v_{out,LLC}}{v_{in,LLC}}=1/\sqrt{\left[1+\frac{f_{n}^{2}-1}{l_{ n}f_{n}^{2}}\right]^{2}+x^{2}\left(f_{n}-\frac{1}{f_{n}}\right)^{2}}=\left\{ \begin{array}{ll}\frac{n_{1}V_{o}}{V_{o}},&50~{}V~{}\leq V_{in}<100~{}V\\ \frac{2n_{1}V_{o}}{V_{o}},&100~{}V~{}\leq V_{in}<200~{}V\\ \frac{4n_{1}V_{o}}{V_{o}},&200~{}V~{}\leq V_{in}<400~{}V,\end{array} \right. \tag{5}\]  
where \(x=\sqrt{\left(L_{r,eq}/C_{r,eq}\right)}/R_{eq}\) is the quality factor where \(L_{r,eq}=L_{r1}\) and \(C_{r,eq}=C_{r1}\) for low input voltage range and \(L_{r,eq}=L_{r1}+L_{r2}=2L_{r1}\) and \(C_{r,eq}=C_{r1}\times C_{r2}/\left(C_{r1}+C_{r2}\right)=C_{r1}/2\) for medium and high input voltage ranges; \(f_{n}=f_{sn}/f_{r}\) is the frequency ratio, and \(l_{n}=L_{m,eq}/L_{r,eq}\) is the inductor ratio.