## 2 Modified Boost Converter (MBC)
### Design of Closed-Loop Controller  
By considering that the output voltage of a boost converter should be kept constant by the load or input voltage changes, the key point for controller design is finding an equation between the output voltage and one of the inductor currents or capacitor voltage derivatives and it can be obtained from Equation (9):  
\[C_{0}\frac{dV_{0}}{dt}+\frac{1}{R}V_{0}=(1-d)i_{LY}=u;\;C_{2}5V_{0}+\frac{1}{R }V_{0}=V_{0}\Big{(}C_{0}S+\frac{1}{R}\Big{)}=(1-d)i_{LY}=u \tag{18}\]  
Other obtained equations cannot present a direct mathematical relation for the output voltage and Equation (9) is more suitable. Here, the final destination is generating a PWM signal for the power switch that will change according to these amendments to guarantee a fixed output voltage. In Equation (18), u is the PI controller output signal and d is the duty cycle of PWM, which will be implemented to power MOSFET. Equation (18) can be rewritten in a simpler way as:  
\[d=1-\frac{u}{i_{LY}} \tag{19}\]  
Figure 5 illustrates the closed-loop form of the PI controller based on Equation (18). Since the goal is receiving a fixed DC voltage at the endpoints of the converter on the load side, a sampling of this voltage was done and compared with a reference voltage in order to be applicable by a microcontroller. Therefore, the sampled voltage should be at a comparable level of the reference voltage. For example, for a 120 V as the fixed output voltage, if the reference voltage chosen is 4 V, the sampled voltage should be decreased through high resistors and be equal to 4 V. For other amounts of the output voltage above or under the 120 V, this sampled voltage will change between 3 to 5 V to be comparable with the reference voltage. After passing through the controller, the control signal u will be compared with the current of the second inductor \(i_{LY}\), and as Equation (19) shows, the desired duty cycle is supplied for the power switch. Figure 6 presents the controller structure for the proposed converter. Figure 5: Closed-loop form of the Proportional-Integral (PI) controlled cascade boost structure. The general form of a PI controller comes from:  
\[G(s)=k_{p}+\frac{k_{i}}{s}=\frac{k_{p}s+k_{i}}{s}\quad\text{ \ \ \ or \ \ \ \ }G(j\omega)=k_{p}+\frac{k_{i}}{j\omega} \tag{20}\]  
where \(K_{P}\) and \(K_{l}\) are the proportional and integral gains of the PI controller. We can put Equation (20) in a feedback loop by a PI controller as presented in Figures 5 and 6 by considering Equation (19). The transfer function of closed-loop form of this feedback is equal with:  
\[G_{F}=\frac{G_{o}}{1+G_{o}}=\frac{\frac{1}{C_{2}}\Big{(}k_{p}+k_{i}\Big{)}}{s^{ 2}+\Big{(}\frac{1+R_{P}}{RC_{2}}\Big{)}+\frac{k_{i}}{C_{2}}}\Rightarrow G_{F}= \frac{\frac{1}{C_{2}}\Big{(}k_{p}s+k_{i}\Big{)}}{s^{2}+2\xi\omega_{0}s+\omega _{o}^{2}} \tag{21}\]  
\(K_{P}\) and \(K_{l}\) are the proportional and integral coefficients of the PI controller. It is easy to gain:  
\[\left\{ \begin{array}{c}\frac{1+R_{P}}{RC_{2}}=2\xi\omega_{0}\\ \frac{k_{i}}{C_{2}}=\omega_{0}^{2}\end{array} \right.\Rightarrow\left\{ \begin{array}{c}k_{p}=2\xi\omega_{0}C_{2}-\frac{1}{R}\\ k_{i}=\omega_{0}^{2}C_{2}\end{array} \right. \tag{22}\]  
The representation of the frequency domain in the kp and ki plane can be introduced by:  
\[\left[ \begin{array}{cc}X_{RP}&X_{Ri}\\ X_{lp}&X_{li}\end{array} \right]=\left[ \begin{array}{c}k_{p}\\ k_{i}\end{array} \right]=\left[ \begin{array}{c}0\\ -\omega_{0}\end{array} \right] \tag{23}\]  
By solving this equation for \(\omega\
eq 0\);  
\[K_{p}(\omega,\partial_{A},\xi) =\frac{-Re(\omega)-\frac{1}{\xi}(A_{A}cos\theta_{A}-B_{A}sin \theta_{A})}{X(\omega)} \tag{24}\] \[K_{l}(\omega,\partial_{A},\xi) =\frac{\omega(Im(\omega)+\frac{1}{\xi}(A_{A}sin\theta_{A}+B_{A} cos\theta_{A}))}{X(\omega)}\]  
\[X(\omega)=\left(