# Analysis and Implementation of a Hybrid DC Converter with Wide Voltage Variation
## 4 Circuit Characteristics and Design Example
\begin{array}{ll}M_{ac,F,M}(s)&=\frac{mV_{dc,f}}{V_{ac,f}}=\frac{\sqrt{2}mV_ {s}/\pi}{\sqrt{2}V_{\mathrm{inj}}/\pi}=\frac{mV_{s}}{V_{\mathrm{inj}}}\\ &=\frac{sL_{m,1}/\left(sn^{2}L_{r,2}+\frac{n^{2}}{8L_{r,2}}+R_{\omega,k}\right) }{sL_{r,1}+\frac{1}{sL_{r,1}}+\left[sL_{m,1}/\left(sn^{2}L_{r,2}+\frac{n^{2}} {8L_{r,2}}+R_{\omega,k}\right)\right]}\times\frac{R_{\omega,k}}{sn^{2}L_{r,2}+ \frac{n^{2}}{8L_{r,2}}+R_{\omega,k}}\\ \end{array} . \tag{5}\]  
\[ \begin{array}{ll}M_{ac,F,H}(s)&=\frac{mV_{dc,f}}{V_{ac,f}}=\frac{2\sqrt{2} mV_{s}/\pi}{\sqrt{2}V_{\mathrm{inj}}/\pi}=\frac{mV_{s}}{V_{\mathrm{inj}}}\\ &=\frac{sL_{m,1}/\left(sn^{2}L_{r,2}+\frac{n^{2}}{8L_{r,2}}+R_{\omega,f} \right)}{sL_{r,1}+\frac{1}{sL_{r,1}}+\left[sL_{m,1}/\left(sn^{2}L_{r,2}+\frac{ n^{2}}{8L_{r,2}}+R_{\omega,f}\right)\right]}\times\frac{R_{\omega,f}}{sn^{2}L_{r,2}+ \frac{n^{2}}{8L_{r,2}}+R_{\omega,f}}\\ \end{array} . \tag{6}\]  
\[ \begin{array}{ll}M_{ac,B}(s)&=\frac{V_{ab,f}/\pi}{V_{\omega,f}}=\frac{V_{ \mathrm{inj}}}{2 mV_{s}}\\ &=\frac{sL_{m,2}/\left(sL_{r,1}/n^{2}+\frac{1}{sn^{2}C_{r,1}}+R_{\omega,k} \right)}{sL_{r,2}+\frac{1}{sL_{r,2}}+\left[sL_{m,2}/\left(sL_{r,1}/n^{2}+\frac{ 1}{sn^{2}C_{r,1}}+R_{\omega,k}\right)\right]}\times\frac{R_{\omega,k}}{sL_{r,1} /n^{2}+\frac{1}{sn^{2}C_{r,1}}+R_{\omega,k}}\\ \end{array} . \tag{7}\]  
The following parameters are defined: \(Q_{F,h}=\sqrt{L_{r,1}/C_{r,1}}/R_{\omega,k}\), \(Q_{F,f}=\sqrt{L_{r,1}/C_{r,1}}/R_{\omega,c,f}\), \(Q_{B,h}=\sqrt{L_{r,2}/C_{r,2}}/R_{in,e,h}\), \(K_{F}=L_{m,1}/L_{r,1}\), \(K_{B}=L_{m,2}/L_{r,2}=K_{F}\), \(F_{F}=f_{sw}/f_{r,1}\) and \(F_{B}=f_{sw}/f_{r,2}=F_{F}\). The voltage gains in the four operation modes in  
Figure 5: Resonant circuits using fundamental frequency analysis: (**a**) low voltage range input with forward power flow; (**b**) medium voltage range input with forward power flow; (**c**) high voltage range input with forward power flow; (**d**) backward power flow operation. (4)-(7) can be expressed in (8)-(10) for the low, medium, and high voltage range inputs under forward power flow and in (11) for backward power flow operation. \[ \begin{array}{ll}\big{|}M_{ac,F,L}(Q_{F,h},K_{F},F_{F})\big{|}&=\frac{nV_{in}}{ 2V_{in}}\\ &=\frac{nV_{in}}{\sqrt{[1+\frac{1}{K_{F}^{2}}-\frac{1}{K_{F}f_{F}^{2}}]^{2}+Q_{ F,h}^{2}[F_{F}(2+\frac{1}{K_{F}})-\frac{1}{F_{F}}(2+\frac{2}{K_{F}}-\frac{1}{K_{F} f_{F}^{2}})]^{2}}}\ \.\end{array}  \tag{8}\]  
\[ \begin{array}{ll}\big{|}M_{ac,F,M}(Q_{F,h},K_{F},F_{F})\big{|}&=\frac{nV_{in}} {V_{in}}\\ &=\frac{n}{\sqrt{[1+\frac{1}{K_{F}^{2}}-\frac{1}{K_{F}f_{F}^{2}}]^{2}+Q_{F,h}^ {2}[F_{F}(2+\frac{1}{K_{F}})-\frac{1}{F_{F}}(2+\frac{2}{K_{F}}-\frac{1}{K_{F} f_{F}^{2}})]^{2}}}\ \.\end{array}  \tag{9}\]  
\[ \begin{array}{ll}\big{|}M_{ac,F,H}\Big{(}Q_{F,f},K_{F},F_{F})\big{|}&=\frac{2 nV_{in}}{V_{in}}\\ &=\frac{\sqrt{[1+\frac{1}{K_{F}^{2}}-\frac{1}{K_{F}f_{F}^{2}}]^{2}+Q_{F,f}^{2}[ F_{F}(2+\frac{1}{K_{F}})-\frac{1}{F_{F}}(2+\frac{2}{K_{F}^{2}}-\frac{1}{K_{F} f_{F}^{2}})]^{2}}}\ \.\end{array}  \tag{10}\]  
\[ \begin{array}{ll}\big{|}M_{ac,B}(Q_{B},K_{B},F_{B})\big{|}&=\frac{V_{in}}{2 nV_{in}}\\ &=\frac{V_{in}}{\sqrt{[1+\frac{1}{K_{B}}-\frac{1}{K_{B}f_{B}^{2}}]^{2}+Q_{B}^ {2}[F_{B}(2+\frac{1}{K_{B}})-\frac{1}{F_{B}}(2+\frac{2}{K_{B}^{2}}-\frac{1}{K _{B}f_{B}^{2}})]^{2}}}\ \.\end{array}  \tag{11}\]  
It is clear that Equations (8)-(11) have a similar voltage transfer function but different DC voltage gain. In the following sub-section, a design example of the hybrid DC converter is discussed and presented. For forward power flow, the input and output electric parameters are \(V_{in}\) = 60 V-480 V, \(V_{o}\) = 40 V-52 V, \(I_{o,max}\) = 7.7 A, and \(f_{r,1}\) = 100 kHz. For the backward power flow operation, the electric parameters are \(V_{o}\) = 42 V-48 V, \(V_{in}\) = 400 V, \(P_{in,max}\) = 400 W, and \(f_{r,2}\) = 100 kHz. The constant current/constant voltage is adopted to charge the battery under forward power flow operation.