# Modeling a Dual-Mode Controller Design
## 2 QRF Converter with Dual-Mode Control
### Small-Signal AC Model  
In steady state, the output voltage of a DMFBC is constant. On the basis of the state average method [24], a small-signal equation of DMFBC was obtained. The small-signal equation can be represented as follows:  
\[L\frac{d\hat{t}(t)}{dt}\ =\ D\hat{v_{g}}(t)-D^{\prime}\frac{\hat{v_{o}}}{n}+ \left(V_{g}+\frac{V}{n}-IR_{on}\right)\hat{d}(t)-DR_{on}\hat{t}(t) \tag{20}\]  
and  
\[C\frac{d\hat{v_{o}}(t)}{dt}\ =\ \frac{D^{\prime}\hat{t}(t)}{n}-\frac{\hat{v}(t)}{ R}-\frac{\hat{t}\hat{d}(t)}{n} \tag{21}\]  
and  
\[\hat{t_{g}}(t)\ =\ D\hat{t}(t)+\hat{t}\hat{d}(t). \tag{22}\]  
Figure 3: Curve of the relationship between \(f_{sw}\) and \(V_{g}\). The hats sign represents AC change. The relationship between stability and the equivalent series resistance of the output capacitor of a dual-mode fly-back converter (DMFBC) should be considered in greater detail, because the left-plane of the system circuit has a zero point that is produced by the output capacitor ESR which affects system stability. Moreover, the analysis technique of a conventional linear circuit was used in this proposed fly-back converter and the transfer function of DMFBC was derived for verification. To discuss the relationship between input and output voltages, first, the duty cycle was set to zero (\(\hat{d}(t)~{}=~{}0\)). Then, the open-loop voltage transfer function of the output to the input \(G_{v_{g}v_{g}}\) was derived. The input voltage (\(\hat{v_{g}}(t)~{}=~{}0\)) was set and derived, and the voltage transfer function of the control to the input \(G_{v_{g}d}\) was presented. These transfer functions describe how a control change influences input voltage. The relationships between the output voltage response and the different input voltage states were discussed. The relationship between \(G_{v_{g}v_{g}}\) and \(G_{v_{g}d}\) is the most prevalent in terms of the input voltage loop gain. Figure 4 displays a small-signal AC equivalent circuit model of a DMFBC through the aforementioned derivation. Many studies have discussed the derivation of a small-signal pulse-width modulator (PWM) equivalent circuit. In addition to considering the derivative of the primary power stage of a small-stage mode that has an effective duty cycle (\(d_{eff}\)) to output voltage and the loop control transfer function, we derived the input voltage to output voltage loop control transfer function and the effective duty cycle to output voltage loop control transfer function. A series of mathematical formulas that describe the technical aspects of a QRF converter, the transfer functions \(G_{v_{0}v_{g}}(s)\) and \(G_{v_{g}d}(s)\), were obtained in the s-domain of QRF converter. Moreover, we analyzed and derived the transfer function of a DMFBC to reveal the relationship between a pole-zero point and an asymptotic gain model. The output to input voltage open-loop transfer function can be estimated as follows:  
\[G_{v_{0}v_{g}}(s)~{}=~{}\frac{\hat{v_{0}}(s)}{\hat{v_{g}}(s)}\bigg{|}_{d(s)~{} =~{}0}~{}=~{}G_{do}\cdot\frac{1}{1+\frac{s}{\left(\frac{s}{w_{0}}\right)^{2}}} \tag{23}\]  
and  
\[G_{v_{0}v_{g}}(s)~{}=~{}\frac{\hat{v_{0}}(s)}{\hat{v_{g}}(s)}\bigg{|}_{\hat{d}( s)~{}=~{}0}~{}=~{}\left(\frac{nD}{D^{\prime}}\right)\frac{1}{1+s\frac{nL}{ \left(D^{\prime}\right)^{2}}+s^{2}\frac{n^{2}RLC}{\left(D^{\prime}\right)^{2}}} \tag{24}\]  
where  
\[G_{do}~{}=~{}\left(\frac{nD}{D^{\prime}}\right)\!\!w_{o}~{}=~{}\frac{D^{\prime }}{\sqrt{nLC}}Q~{}=~{}\sqrt{\frac{LC}{n}}\cdot\frac{D^{\prime}R}{L}.\]  
\(G_{do}\), \(\omega_{o}\) and \(Q\) are the analyzed keywords of the transfer function \(G_{v_{0}v_{g}}(s)\) of the dual-mode control of a fly-back converter. In this mode, the physical parameter values \(R\), \(L\), and \(C\) serve a crucial role in designing the parameters for a QRF converter. To implement these parameters, the R, L, and C values  
Figure 4: Small-signal ac equivalent circuit of the DMBFC. used were respectively 3.61 \(\Omega\), 170 \(\Omega\), and 2000 \(\upmu\)F at the heavier load.