## 3 Mathematical Modelling of Multi-Output Forward Converter with Weighted Control
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Value** \\ \hline DC Input Voltage & 50–80 V \\ DC Output Voltage/Current & 15 V/1 A, 12 V/2 A, 3,3 V/2 A \\ Np, N1, N2, N3, Np, Naux & 22, 16, 13, 4, 22, 8 turn \\ IC PWM Controller & FS7M0880 \\ DR1, DR2, DR3 & MBR40250 TG Schottky Diode \\ DF1, DF2, DF3 & MBR40250 TG Schottky Diode \\ Dreset & UF4007 Diode \\ CO1, CO2, CO3 & 2200 uF/25 V \\ CP1, CP2 & 330 uF/25 V \\ LP1, LP2 & 1.2 uH/10 A \\ Fixed Operating Frequency & 67 kHz \\ Output Power under Full Load & 45.6 W \\ \hline \hline \end{tabular}
\end{table} 
Table 5: First key power stage parameters of forward converter with weighted control. Our feedback control design is made to control the PWM on the primary side by isolating the voltage information received from the secondary side. Since KRF = 0.1 is taken in our circuit, it works in CCM mode. Gvc is the control-to-output transfer function of the forward converter. \[\mathrm{G_{vc}}=\mathrm{KR_{L}}\frac{\mathrm{N_{P}}}{\mathrm{N_{S1}}}\frac{1+ \mathrm{s/w_{z}}}{1+\mathrm{s/w_{p}}} \tag{20}\]  
\[\mathrm{where~{}w_{z}=\frac{1}{R_{c1}C_{o1}},~{}w_{p}=\frac{1}{R_{L}C_{o1}}}\]  
\(\mathrm{R_{L}}\) is the effective total resistance of the controlled output, defined as \({V_{o1}}^{2}/\mathrm{Po}\), which is the voltage to current conversion ratio of the FPS, and corresponds to \(\mathrm{I_{pk}}/\mathrm{V_{FB}}\), where \(\mathrm{I_{pk}}\) is the peak drain current and VFB is the feedback voltage for a given operating condition. The output-to-control transfer function of Figure 4 is defined as follows:  
\[\frac{\mathrm{V_{FB}}}{\mathrm{V_{o1}}}=\frac{\mathrm{w_{i}}}{\mathrm{s}}\frac {1+\mathrm{s/w_{zc}}}{1+\mathrm{s/w_{pc}}} \tag{21}\]  
\[\mathrm{where~{}w_{i}=\frac{R_{B}}{R_{1}R_{C}C_{F}},~{}w_{zc}=\frac{1}{(R_{1} +R_{F})C_{F}},~{}w_{pc}=\frac{1}{R_{B}C_{B}}}\]  
Tables 6 and 7 summarize the compensation network parameters for the forward converter with weighted control. KA431 is used as the Voltage Reference IC, and PC817A is used as the optocoupler. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Value** \\ \hline Voltage divider resistor (R1,R2) & 22 k\(\Omega\) \\ Voltage divider resistor (Ro) & 2.49 k\(\Omega\) \\ Optocoupler diode resistor (RD) & 1 k\(\Omega\) \\ KA431 Bias resistor (RBias) & 0.68 k\(\Omega\) \\ Feedback pin capacitor (CB) & 39 nF \\ Feedback capacitor (CF) & 47 nF \\ Feedback resistor (RF) & 1 k\(\Omega\) \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Second key compensation network parameters for forward converter with weighted control. Figure 4: Multi-output forward converter closed-loop control system. The control parameter results of the forward converter with weighted control and the compensation network parameters for the forward converter with weighted control are designed in this context. Bode plot curves are required to gain information about the stability of the designed circuit. We examined the system stability by entering the necessary data in the MATLAB 2024a software environment. In Figure 5, the blue, pink and green lines, respectively, are the control-to-output, output-to-control and open-loop curves. The crossover frequency (fc) of the circuit with weighted control is 431 Hz. When an additional LC filter (post-filter) is applied, the crossover frequency should be set below \(1/3\) of the corner frequency of the post-filter for introducing a \(-180\) degrees phase drop [33]. This condition is met in both of our circuits. A Type-II compensator is used in our proposed design. Power stages' crossover frequency (control-to-output) is 133 Hz and power stages' phase margin \(101^{\circ}\). If a type II control was not designed or a feedback system was designed without using compensator, the phase margin of the system would be \(101^{\circ}\) and there wouldn't be fast response in the system to design.