# Modeling a Dual-Mode Controller Design
## 3 Controller Design  
To improve the frequency response of a dual-mode fly-back converter and simultaneously implement the converter performance, an adaptive compensator for obtaining the transfer function of \(G_{v_{o}v_{g}}(s)\) and \(G_{v_{o}d}(s)\) is required. \(G_{v_{o}d}(s)\) is an important keyword of the system loop gain in a DMFBC and can be used to determine the performances of voltage and current loop compensators. A peak current control integrated circuit (IC) L6561 with a transient mode control performance was used in the fly-back converter. The fly-back converter is equipped with an adaptive detection circuit and judge circuit, and can be operated under heavy load or light load conditions to improve the converter efficiency of the converter.  
When the fly-back converter with the constant duty cycle turn off control was operated under heavy load conditions, the converter entered to the CCM, the RMS current went through the power switch, and the quick diode was reduced. In contrast, the conduction loss increased. To apply the ZCD detection function of the IC L6561 control [22], the control gate drive signal was kept off until it was turned on in the next cycle of ZCD voltage level detection.  
The ZCD action function can be estimated as follows:  
\[\left\{\begin{array}{l}V_{ZCD}>V_{ZCD_{limit}}\ =\ 5.7V,\ GD\ =\ 0V\ (Low)\\ V_{ZCD}<V_{ZCD\_Tiger}\ =\ 0.7V,\ GD\ =\ 15V\ (High)\end{array}\right.. \tag{31}\]  
A DMFBC with a ZCD detection strategy was presented in this study. The voltage-adjustable controller, optocoupler, PWM comparator, and fly-back converter were included in the control system. The transfer function of the circuit presented in Figure 5 can be expressed as follows. Figure 6 displaysthe fly-back converter and voltage regulation control system. Figure 6 presents a simplified block diagram of the dual-mode control, which is represented using dotted boxes.  
\[G_{1}(s)\ =\ \frac{s+w_{z}}{R_{f4}C_{b1}s\big{(}s+w_{p}\big{)}} \tag{32}\]  
where  
\[w_{z}\ =\ \frac{1}{R_{f4}C_{b1}}\ (\mathrm{rad}/sec)w_{p}\ =\ \frac{C_{b1}+C_{b2}}{R_{f4}C_{b1}C_{b2}}\ ( \mathrm{rad}/sec)\]  
Figure 5: Block diagram of the voltage closed-loop control of the DMBFC.  
Figure 6: Simplified circuit diagram of dual-mode control.  
\[G_{2}(s)\ =\ \frac{\hat{\varrho}_{b}}{\hat{\varrho}_{a}}\ =\ \frac{CTR\cdot R_{fb2}}{R_{fb3}} \tag{33}\]  
and  
\[G_{3}(s)\ =\ \frac{\hat{\varrho}_{comp}}{\hat{\varrho}_{b}}\ =\ -\left(\frac{R_{fb3}}{R_{fb1}} \right)\ =\ -\frac{\left[\frac{\left(\frac{1}{C_{comp}}+R_{comp}\right)\cdot C_{15}}{ \left(\frac{1}{C_{comp}}+R_{comp}\right)+C_{15}}\right]}{R_{fb1}} \tag{34}\]  
and  
\[G_{4}(s)\ =\ K_{2}\ =\ \frac{\hat{d}}{\hat{\varrho}_{comp}}\ =\ \frac{1}{V_{M}} \tag{35}\]  
and  
\[G_{5}(s)\ =\ \frac{\hat{\varrho}_{o}}{\hat{d}}\ =\ nV_{g} \tag{36}\]  
and  
\[H(s)\ =\ \frac{R_{f2}}{R_{f1}+R_{f2}} \tag{37}\]  
\(G_{1}(s)\), \(G_{2}(s)\), \(G_{3}(s)\), \(G_{4}(s)\), \(G_{5}(s)\), and \(H(s)\) can be combined to obtain the open- and closed-loop voltage transfer function of a fly-back converter:  
\[T_{FLY\_OPEN}(s)\ =\ G_{1}(s)\cdot G_{2}(s)\cdot G_{3}(s)\cdot G_{4}(s)\cdot G_{5 }(s)\cdot H(s) \tag{38}\]  
Moreover, the closed-loop voltage transfer function can be expressed as follows:  
\[T_{FLY\_CLOSED}(s)\ =\ \frac{G_{1}(s)\cdot G_{2}(s)\cdot G_{3}(s)\cdot G_{4}(s) \cdot G_{5}(s)}{1+G_{1}(s)\cdot G_{2}(s)\cdot G_{3}(s)\cdot G_{4}(s)\cdot C_{ 5}(s)\cdot H(s)} \tag{39}\]  
The function characteristics of one compensator \(G_{V}(s)\) are selected and explained by the following transfer function:  
\[G_{1}(s)\ =\ K\frac{s+w_{z}}{s\cdot\left(s+w_{p}\right)} \tag{40}\]  
Adding a zero point to the compensator reduces the phase lag and adds stability to a fly-back converter. It is expected that adding one pole point (\(w_{p}\)) suppresses the switch noise to further increase the loop gain. Furthermore, it is expected that adding one pole point to the original point of the system would increase the low frequency gain of the studied fly-back converter.