# Analysis and Implementation of a Hybrid DC Converter with Wide Voltage Variation
## 4 Circuit Characteristics and Design Example  
There are four operating modes in the proposed circuit, as shown in Figure 1. In each mode, the PFM scheme is selected to control the active switches so that the voltage gain of the resonant circuit is depended on the switching frequency. The equivalent resonant circuits in Figure 1b-e can be redrawn and is shown in Figure 4. For the forward power flow operation, the leg voltage \(V_{ab}\) is a square-wave voltage with \(\pm V_{in}\) (low voltage range input condition) or \(\pm V_{in}/2\) (medium and high voltage range input conditions). The leg voltage \(V_{de}\) on the right-hand side of the converter is a square-wave voltage with \(\pm V_{o}/2\) (low  
Figure 3: PWM signals and state circuits under backward power flow: (**a**) PWM signals; (**b**) State 1; (**c**) State 2; (**d**) State 3; (**e**) State 4; (**f**) State 5; (**g**) State 6; (**h**) State 7; (**i**) State 8.  
and medium voltage range input conditions) or \(\pm V_{o}\) (high voltage range input condition). For the backward power flow operation, the leg voltages \(V_{ab}\) = \(\pm\)\(V_{in}/2\) and \(V_{de}\) = \(\pm\)\(V_{o}\). According to the Fourier Series Analysis, the square-wave voltage can be expressed as the summation of the fundamental sinewave voltage and the higher-order harmonic components. Since the higher-order harmonic components are less than the fundamental frequency voltage, only the fundamental sinewave voltage is adopted in the following consideration. The analysis of fundamental frequency is used in [23] to analyze the resonant converter and simplify the circuit analysis. Figure 5 provides the resonant circuits with the fundamental frequency analysis method under four operating modes. If the leg voltage \(V_{ab}\) = \(\pm\)\(V_{in}/2\) (Figure 4b-d) or \(\pm\)\(V_{in}\) (Figure 4a), then its fundamental root-mean-square (\(rms\)) voltage can be obtained as \(V_{ac,f}\) = \(\sqrt{2}V_{in}/\pi\) (Figure 5b-d) or \(2\sqrt{2}V_{in}/\pi\) (Figure 5a). Similarly, the rms voltage \(V_{dc,f}\) = \(\sqrt{2}V_{o}/\pi\) (Figure 5a,b) or \(2\sqrt{2}V_{o}/\pi\) (Figure 5c,d) if \(V_{de}\) = \(\pm\)\(V_{o}/2\) (Figure 4a,b) or \(\pm\)\(V_{o}\) (Figure 4c,d). In Figure 5, \(R_{o,e,h}\) and \(R_{o,ef}\) are the fundamental load resistances under the half-wave diode rectifier (Figure 5a,b) and full-wave diode rectifier (Figure 5c) operation with forward power flow operation. \[R_{o,e,h}=\frac{2n^{2}}{\pi^{2}}R_{o}. \tag{1}\]  
\[R_{o,e,f}=\frac{8n^{2}}{\pi^{2}}R_{o}. \tag{2}\]  
Similarly, the fundamental load resistance \(R_{in,e,h}\) (Figure 5d) under backward power flow operation is given in Equation (3). \[R_{in,e,h}=\frac{2}{\pi^{2}n^{2}}R_{in}. \tag{3}\]  
Figure 4: Equivalent resonant circuits under four operating modes: (**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. The voltage transfer functions in Figure 5a-d are derived in Equations (4)-(7), where \(M_{ac,F,L}\), \(M_{ac,F,M}\), and \(M_{ac,F,H}\) are the voltage transfer functions at the low, medium, and high voltage range inputs under forward power flow operation and \(M_{ac,B}\) is the voltage transfer function under backward power flow operation. \[ \begin{array}{ll}M_{ac,F,L}(s)&=\frac{mV_{dc,f}}{V_{ac,f}}=\frac{\sqrt{2}mV_ {f}/\pi}{2\sqrt{2}V_{\mathrm{inj}}/\pi}=\frac{mV_{s}}{2V_{\mathrm{inj}}}\\ &=\frac{sL_{m,1}/\left(sn^{2}L_{r,2}+\frac{n^{2}}{8L_{r,2}}+R_{\omega,k}\right) }{sL_{r,1}+\frac{1}{sL_{r,1}}+\left[sL_{m,1}/\left(sn^{2}L_{r,2}+\frac{n^{2}}{8 L_{r,2}}+R_{\omega,k}\right)\right]}\times\frac{R_{\omega,k}}{sn^{2}L_{r,2}+ \frac{n^{2}}{8L_{r,2}}+R_{\omega,k}}\\ \end{array} . \tag{4}\]  
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