## 5 Case Study
### Converter Modeling  
The basic operation of the circuit in Continuous Conduction Mode (CCM), can be described by two states: first when the transistor is ON (Figure 9) and when it is OFF (Figure 10).  
In the ON state, the inductance is charged with the energy from the source, meanwhile in the OFF stage it is discharged, enabling the capacitor to be charged with the energy flow. Hence, as explained in [6], the converter has a linear control profile, considering its voltage conversion as \(V=(D)*V_{g}\), where from this point in this paper \(D\) is the duty cycle of the PWM, which is the control signal of the converter.  
Therefore, one of the most used and first-choice methods for the mathematical modeling of converters is the small-signal approach (Small-Signal Model), developed by [PERSON] and [PERSON], where in [14] the work aims to describe the dynamics of the converter as a set of time-invariant equations, valid for the entire switching cycle.  
Figure 8: Cascade controller scheme for voltage and current in a buck converter, symbolizing the relationship of the controllers with the real converter.  
Figure 10: Buck converter equivalent model when the transistor is OFF.  
Figure 9: Buck converter equivalent model when the transistor is ON.  
Hence, the first step is to average all possible states (ON and OFF transistor states in this case) to obtain an average model, which removes variations over time but retains the non-linearities (described in [6]) of the system. As a second step, linearization is done by approximating the nonlinear relationships between the circuit variables under small-signal assumptions, which produces a dynamic model averaged over time in the presence of a small-signal excitation. Finally, the obtained model is transferred to the frequency domain, in order to obtain the transfer functions of the power stage dynamics as explained in [6].  
Consequently, after applying the small-signal analysis to the states of the circuit, the transfer functions that represent the Output voltage and Inductance current behaviors against the control signal (Duty cycle), is given by Equations (10) and (11), respectively.  
\[G_{vd}(s)=\frac{\hat{V}_{out}}{\hat{D}}=\frac{(CR_{C}V_{g}R)s+RV_{g}}{\text{Den}} \tag{10}\]  
\[G_{id}(s)=\frac{\hat{t}_{L}}{\hat{D}}=\frac{V_{g}(CR+CR_{C})s+V_{g}}{\text{Den}} \tag{11}\]  
where in the denominator in both cases (Equations (10) and (11)), is given by the expression defined by Equation (12).  
\[\text{Den}=(CLR+CLR_{C})s^{2}+(L+RCR_{C}+RCR_{L}+CR_{C}R_{L})s+(R+R_{L}) \tag{12}\]  
Henceforth, Equation (10) is defined as the objective function for the output voltage, and Equation (11) the objective function for the current in the inductor, which are the functions for the multi-objective optimization. Hence, Figure 8 can be redrawn as shown in Figure 11, where the buck converter block was substituted with the small-signal transfer functions.  
Nevertheless, it is important to highlight that the input for the converter in Figure 8 was the PWM, with the duty cycle as control parameter. Yet, the duty cycle is still the input parameter for \(G_{vd}\) and \(G_{id}\), but the expected input is the duty cycle value, without any pulsing source for the transfer functions.