## 3 Mathematical Modelling of Multi-Output Forward Converter with Weighted Control  
In this section, we will additionally mention the mathematical calculations of the design stages using Equation (1) to Equation (26). The circuit diagram of a multi-output forward converter with weighted control is shown in Figure 3[25,33]. We also recommend that readers look at the datasheets of the FS7M0880, PC817 and KA431 commercial integrated circuits used in the design. Figure 2: Idealized steady-state waveforms of forward converter for continuous mode operation. First and foremost, the system properties of the forward converter must be determined. In our design, we have taken the outputs as follows: First Output: 15 V-1 A; Second Output: 12 V-2 A; and Third Output: 3.3 V-2 A. Moreover, the input range is 50 V-80 V. However, the minimum input voltage of the circuit is designed to be 48 V. The expected efficiency is thought to be around 75%. This is calculated by taking the lowest efficiency predicted for the load changes at the outputs when designing the relevant circuit. In this way, a suitable circuit that will work stably is designed. In this case, the expected maximum power consumption is 60.8 W. Another step is to determine the auxiliary winding reset of the transformer. One criterion of the forward converter is that the transformer needs to be reset during the switching-off period. Therefore, an additional reset auxiliary must be recruited. The efficiency of the circuit will be increased with the auxiliary winding reset method. However, it is a disadvantage that this method brings an additional winding to the transformer. One of the design stages should be selected as 50% below the duty ratio to prevent saturation in the transformer. Generally, a duty max ratio of 45% is selected [33]. The maximum voltage for the MOSFET and the maximum duty ratio are given by the following equations:  
\[\mathrm{V_{ds}}^{\mathrm{max}}=\mathrm{V_{DC}}^{\mathrm{max}}\bigg{(}1+\frac{ \mathrm{N_{P}}}{\mathrm{N_{r}}}\bigg{)} \tag{1}\]  
\[\mathrm{D_{max}}\leq\frac{\mathrm{N_{P}}}{\mathrm{N_{P}}+\mathrm{N_{r}}} \tag{2}\]  
One of the most important issues in a forward converter is that the PWM control (IC) operates up to a certain current. One of the factors that determines this current is the output current ripple factor. The current ripple factor (\(\mathrm{K_{RF}}\)) is selected to be as low as possible. Thus, drain current is reduced. It is reasonable to set \(\mathrm{K_{RF}}\) = 0.1- 0.2. We chose \(\mathrm{K_{RF}}\) = 0.1 in our experiment. \[\mathrm{I_{ds}}^{\mathrm{peak}}=\mathrm{I_{EDC}}(1+\mathrm{K_{RF}}) \tag{3}\]  
Figure 3: Multi-output forward converter with weighted control. \[{\rm I_{ds}}^{\rm rms}=\sqrt{{\rm I_{EDC}}\Big{(}3+{K_{RF}}^{2}\Big{)}\frac{D_{max} }{3}} \tag{4}\]  
\[{\rm where~{}I_{EDC}=\frac{P_{in}}{V_{DC}^{min}D_{max}}} \tag{5}\]  
Another issue is that the winding must be wound on the primary side, to prevent the transformer from saturating. Where \({\rm Ae}\) = 173 mm\({}^{2}\), \({\rm\Delta B}\) was taken as 0.3 Tesla for ETD 44. \[{\rm N_{P}}^{\rm min}=\frac{V_{DC}^{min}D_{max}}{A_{e}f_{s}{\rm\Delta B}}\;10^{6} \tag{6}\]  
The next step is to determine the number of turns for each winding of the transformer and the auxiliary winding reset. \[{\rm n}=\frac{N_{P}}{N_{S1}}=\frac{V_{DC}^{min}D_{max}}{V_{0}+V_{F1}} \tag{7}\]  
\[{\rm N_{s(n)}}=\frac{V_{o(n)}+V_{F(n)}}{V_{o1}+V_{F1}}{N_{S1}} \tag{8}\]  
\[\mathrm{V_{ds}}^{\mathrm{max}}=\mathrm{V_{DC}}^{\mathrm{max}}\bigg{(}1+\frac{ \mathrm{N_{P}}}{\mathrm{N_{r}}}\bigg{)} \tag{1}\]0  
We will now explain the stages in order; the points to be taken into consideration are stated in the previous steps.