## 3 Design of Analog Control Circuit  
The design procedure of the simplified analog control circuit is introduced in this subsection. The proposed control circuit is depicted in Figure 2, and the parameters of the dynamic buck converter are given in Table 1[32]. An input filter capacitor \(C_{i}\) of 470 uF and 0.2 \(\Omega\) ESR is added to the power converter input to accommodate the discontinuous input current. The feedback network gain, \(\beta\), is assumed to be 5/14. Based on the existence and stability conditions, the proportional and integral gains \(K_{p}\) and \(K_{i}\) are set to 27.6 and 13.8 \(\times\) 10\({}^{4}\), respectively. In addition, a scaling factor, \(\gamma\), of 0.4 is selected to scale down the control parameters values and fit the practical range. General-purpose op-amps such as LF356 [4] and LM318 [3] can be used to construct the analog control circuit. \begin{table}
\begin{tabular}{l c c} \hline \hline
**Description** & **Parameter** & **Value** \\ \hline \hline \multicolumn{1}{l}{Inductor} & \(L\) & 56 uH \\ \multicolumn{1}{l}{Output capacitor} & \(C_{O}\) & 2.2 uF \\ \multicolumn{1}{l}{Nominal load resistance} & \(R\) & 75.00 \(\Omega\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The parameters of the dynamic buck converter. The design steps of the simplified control circuit components can be summarized as follows:  
* Output voltage sensor: The feedback network gain, \(\beta\), is set to \(5/14\), and \(\beta=R_{B}/(R_{A}+R_{B})\), where \(R_{A}\) and \(R_{B}\) are the resistors of the output voltage sensor. If the value of \(R_{A}\) is assumed to be \(9.1\) k\(\Omega\), then \(R_{B}\) is \(5.1\) k\(\Omega\). * Summing amplifier: The resistors \(R_{S1}\), \(R_{S2}\), and \(R_{S3}\) for the summing op-amp can be set to \(5.1\) k\(\Omega\). * PWM generator: Since the input voltage, \(V_{I}\), is \(28\) V, the ramp voltage of the pulse-width modulator \(V_{T}=\beta V_{I}=10\) V, and the switching frequency \(f_{s}\) is set to \(1\) MHz. * Scaling factor: If a scaling factor, \(\gamma\), is required to scale down \(V_{T}\) to \(4\) V, then \(\gamma\) can be set to \(0.4=R_{I1}/R_{I2}\). Hence, if \(R_{I2}\) is chosen as \(5.1\) k\(\Omega\), then \(R_{I1}\) is \(2\) k\(\Omega\). * Proportional gain \(K_{p}\): According to [33], the proportional gain is defined as \(K_{p}=R_{2}/R_{1}\). Thus, if the value of \(K_{p}\) is \(27.6\) and \(R_{1}\) is selected as \(33\) k\(\Omega\), then \(R_{2}\) becomes \(910\) k\(\Omega\). * Integral gain \(K_{i}\): In [33], the integral gain is defined as \(K_{i}=1/(R_{1}\) C). If the value of \(K_{i}\) is set to \(13.8\times 10^{4}\) and \(R_{1}\) is \(33\) k\(\Omega\), then \(C\) can be chosen as \(220\) pF. It should be noted that the sum of the feedback resistors \(R_{A}\) and \(R_{B}\) must be greater than the maximum load resistor to reduce the loading effect of the feedback network [33]. High-precision feedback resistors should be selected to obtain an accurate output voltage measurement. It is also worth noting that the values of the gains \(K_{p}\) and \(K_{i}\) given in this research are not unique. Different values can be chosen within the practical range if they satisfy the existence and stability conditions [31]. The proposed control equation in (13), along with the selected control parameters, can be written as  
\[\left\{ \begin{array}{l}\bar{u}_{e}=0.4\big{[}27.6(V_{r}-\beta\overline{v}_{ O})+13.8\times 10^{4}\int(V_{r}-\beta\overline{v}_{O})dt+\beta \overline{v}_{O}\big{]}\\ V_{T}=4\ V,\end{array} \right. \tag{20}\]  
and the corresponding analog control circuit is given in Figure 2.