## 3 Mathematical Modelling of Multi-Output Forward Converter with Weighted Control
However, thanks to the type II compensator, the open loop crossover frequency is increased to 431 Hz and open-loop PM is reduced to \(68.9^{\circ}\), as shown in Table 8. Briefly, the system's stability has been made faster by allowing slightly overshoot. As can be seen from Figure 5, the type-II compensator is used boosting \(53.6^{\circ}\) at 431 Hz which is also the crossover frequency designed for this system. (see Appendix A) \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Value** \\ \hline Control-to-output DC gain & 9 \\ Control-to-output DC zero & 1809 Hz \\ Control-to-output pole & 15 Hz \\ Feedback integrator gain (fi) & 462 Hz \\ Feedback zero (fz) & 147.3 Hz \\ Feedback pole (fpo) & 1361 Hz \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Third key control parameter results of forward converter with weighted control. Figure 5: CCM forward converter control-to-output and output-to-control transfer functions with weighted control. Thanks to weighted averages, the design works by obtaining suitable voltage references, and thus, the desired cross-regulation is obtained. To briefly mention the weighted control method, Figure 6 shows in detail the voltage information received from two outputs of a three-output forward converter. To create our optimal design, we first needed to obtain information about the output voltages by using the classical control format. A random selection of weights may not benefit the experiment in cross-regulation. For example, if the converter to be tested has four winding outputs, classical control is provided from only one winding. Then, the windings which have the most positive voltage change and the most negative voltage change are selected as references. Then, the design is made. The multi-output forward converter is designed with resistors at the desired rates. Thanks to the advantages provided by our design, we applied our design by taking, to a certain extent, the weighted voltage supply from each of the 15 V and 12 V outputs. Thanks to Equations (22) and (23), the output weights of our converter can be adjusted by using W1, W2 and W3 = 0, in other words, i3 = 0. \[\mathrm{i_{o}=i_{1}+i_{2}+i_{3}=w_{1}i_{0}+w_{2}i_{0}+w_{3}i_{0}} \tag{22}\]  
\[\mathrm{w_{1}+w_{2}+w_{3}=1} \tag{23}\]  
KA431 is a 2.5 V reference amplifier. If we use R\({}_{\mathrm{o}}\) (2.49 k\(\Omega\)), i0 is approximately 1 mA. We did not receive voltage information from the third output. The expected voltages for V1 and V2 are 15 V and 12 V, respectively. By choosing R1 (22 k\(\Omega\)) and R2 (22 k\(\Omega\)) in Equation (24), we have determined that i0 is approximately 1 mA. As in this design, the desired R1 and R2 can be selected to satisfy Equation (24). That is, another R1 and R2 could have been chosen that satisfied Equation (24):  
\[\mathrm{i_{o}=i_{1}+i_{2}=\frac{V_{1}-2.5}{R_{1}}+\frac{V_{2}-2.5}{R_{2}}} \tag{24}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameters** & **Crossover Frequency (fc)** & **PM** & **GM** \\ \hline Control-to-Output & 133 Hz & 101\({}^{\circ}\) & Inf \\ Open-Loop & 431 Hz & 68.9\({}^{\circ}\) & Inf \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Fourth key system control parameter results of forward converter with weighted control. Figure 6: Proposed weighted control method. \(\rm R_{bias}\) and \(\rm R_{D}\) must be designed to satisfy the following conditions:  
\[\frac{\rm V_{O}-V_{\rm OP}-2.5}{\rm R_{D}}>\rm I_{FB} \tag{25}\]  
\[\frac{\rm V_{OP}}{\rm R_{bias}}>1\rm mA \tag{26}\]  
where \(\rm V_{OP}\) is optodiode forward voltage drop, which is typically 1 V for PC817A, and \(\rm I_{FB}\) is the feedback current of the FPS, which is typically 0.9 mA for FS7M0880.