# Hybrid LLC Converter with Wide Range of Zero-Voltage Switching and Wide Input Voltage Operation
## 4 Circuit Characteristics and Design Example
From Equations (1), (2) and (5), the voltage transfer function Gac(f\({}_{\text{sw}}\)) is obtained in Equations (6)-(8) for high, medium and low voltage input regions. Figure 6: The equivalent resonant circuit of the proposed LLC converter with (**a**) nonlinear nonsinusoidal circuit (**b**) linear sinusoidal circuit. \[\mathrm{G_{ac}(f_{sw})}\ =\ \frac{1}{\sqrt{[1+\frac{(\frac{\mathrm{f_{sw}}}{ \mathrm{I_{r}}})^{2}-1}{\mathrm{I_{r}}}]^{2}+\frac{\mathrm{I_{r}/C_{f}(\mathrm{f_ {sw}}})^{2}-1}{\mathrm{I_{r}^{2}}(\frac{\mathrm{I_{r}/C_{f}(\mathrm{f_{sw}}}) ^{2}-1)}}}}\ =\ \frac{\mathrm{nV_{o}}}{\mathrm{V_{in-H}}} \tag{6}\] \[\mathrm{G_{ac}(f_{sw})}\ =\ \frac{1}{\sqrt{[1+\frac{(\frac{\mathrm{f_{sw}}}{ \mathrm{I_{r}}})^{2}-1}{\mathrm{I_{r}}}]^{2}+\frac{\mathrm{I_{r}/C_{f}(\mathrm{f_ {sw}}})^{2}-1}{\mathrm{I_{r}^{2}}(\frac{\mathrm{I_{r}/C_{f}(\mathrm{f_{sw}}}) ^{2}-1)}}}}\ =\ \frac{\mathrm{nV_{o}}}{2\mathrm{V_{in-M}}}\] (7) \[\mathrm{G_{ac}(f_{sw})}\ =\ \frac{1}{\sqrt{[1+\frac{(\frac{\mathrm{f_{sw}}}{ \mathrm{I_{r}}})^{2}-1}{\mathrm{I_{r}^{2}}(\frac{\mathrm{I_{r}/C_{f}(\mathrm{f_ {sw}}})^{2}-1)}}]^{2}+\frac{\mathrm{I_{r}/C_{f}(\mathrm{f_{sw}}})^{2}-1}{ \mathrm{I_{r}^{2}}(\frac{\mathrm{I_{r}/C_{f}(\mathrm{f_{sw}}})^{2}-1)}}}}\ =\ \frac{ \mathrm{nV_{o}}}{4\mathrm{V_{in-L}}} \tag{8}\]  
Therefore, the load voltage \(\mathrm{V_{o}}\) can be obtained under different input voltage conditions and expressed in Equations (9)-(11) for high, medium and low voltage input regions, respectively. \[\mathrm{V_{o}}\ =\ \frac{\mathrm{V_{in-H}}}{\mathrm{n}\sqrt{[1+\frac{( \mathrm{f_{sw}}}{\mathrm{I_{r}}})^{2}]^{2}+\frac{\mathrm{I_{r}/C_{f}(\mathrm{f_ {sw}}})^{2}-1}{\mathrm{I_{r}^{2}}(\frac{\mathrm{I_{r}/C_{f}(\mathrm{f_{sw}}} )^{2}-1)}}}}} \tag{9}\] \[\mathrm{V_{o}}\ =\ \frac{2\mathrm{V_{in-M}}}{\mathrm{n}\sqrt{[1+\frac{( \mathrm{f_{sw}}}{\mathrm{I_{r}}})^{2}-1]^{2}+\frac{2\mathrm{I_{r}/C_{f}(\mathrm{f_ {sw}}})^{2}-1}{\mathrm{I_{r}^{2}}(\frac{\mathrm{I_{r}/C_{f}(\mathrm{f_{sw}}} )^{2}-1)}}}}\] (10) \[\mathrm{V_{o}}\ =\ \frac{4\mathrm{V_{in-L}}}{\mathrm{n}\sqrt{[1+\frac{( \mathrm{f_{sw}}}{\mathrm{I_{r}}})^{2}-1]^{2}+\frac{2\mathrm{I_{r}/C_{f}(\mathrm{f_ {sw}})^{2}-1}}{\mathrm{I_{r}^{2}}(\frac{\mathrm{I_{r}/C_{f}(\mathrm{f_{sw}}} )^{2}-1)}}}} \tag{11}\]  
The studied LLC converter is operated under wide voltage input (50-400 V), the load voltage \(\mathrm{V_{o}}\) is 48 V and load power is 500 W. The resonant frequency \(\mathrm{f_{r}}\) is designed at 100 kHz. The selected magnetizing and resonant inductances are \(\mathrm{L_{m}}=6\mathrm{L_{r}}\). For high (\(\mathrm{V_{in+H}}\), 200 V-400 V), medium (\(\mathrm{V_{in-M}}\), 100 V-200 V) and low (\(\mathrm{V_{in-L}}\), 50 V-100 V) voltage input ranges, the equivalent resonant circuit models in Figure 6b are identical. Thus, the converter operated at high voltage input region is designed in the following discussion. According to Equation (6), it is clear that the proposed converter has the minimum voltage gain at \(\mathrm{V_{in}}=400\) V and has the maximum voltage gain at \(\mathrm{V_{in}}=200\) V. It is assumed that the dc gain at \(\mathrm{V_{in}}=400\) V input is designed as 0.95. From Equation (6), the primary-secondary turn \(\mathrm{n}=\mathrm{n_{p}/n_{s}}\) is obtained in Equation (12). \[\mathrm{n}\ =\ \mathrm{n_{p}/n_{s}}\ =\ \mathrm{G_{ac}(f_{sw})V_{in,max}/V_{o}}\ =\ 0.95\times 400/48\ =\ 7.916. \tag{12}\]  
The magnetic core TDK EE-55 with \(\mathrm{n}=\mathrm{n_{p}/n_{s}}=16/2=8\) is used to implement transformer T. Therefore, the voltage gains at \(\mathrm{V_{in}}=400\) V and 200 V are calculated in Equations (13) and (14). \[\mathrm{G_{ac,min}}\ =\ \mathrm{nV_{o}/V_{in,max}}\ =\ 8\times 48/400\ =\ 0.96. \tag{13}\]  
Figure 6:0  
For high voltage input range, \(\mathrm{S_{ac2}}\) is off. From Equation (4), the ac equivalent load resistance is calculated as. Figure 6:1  
The selected resonant inductance \(\mathrm{L_{r}}=7\mathrm{\mu H}\) so that the magnetizing inductance \(\mathrm{L_{m}}=6\mathrm{L_{r}}=42\ \mathrm{\mu H}\).