## 5 Design Procedure and Application Example
Therefore, \(\delta v_{cb}\) is selected as 10% of the steady-state value of \(v_{cb}\), which, according to expression (19), is equal to 48 V. In addition, the switching frequency of the NEC boost converter is selected to be below 100 kHz, which ensures the correct operation of commercial MOSFETs as observed in the implementation of the boost converter reported in [25]. Finally, Table 2 summarizes the design characteristics for the NEC boost converter for this example. Applying Equation (24) leads to a minimum \(L_{2}\) inductance of 147 \(\mu\)H; so, the commercial values \(L_{1}=L_{2}=150\)\(\mu\)H are selected. Similarly, applying Equation (25) leads to a minimum \(C_{cb}\) capacitance of 1.13 \(\mu\)F, selecting a commercial value \(C_{cb}=1.2\)\(\mu\)F for this application. The \(C_{pv}\) capacitance is calculated from Equation (26) using the values of \(\delta i_{1}=\delta i_{2}\), which results in a minimum capacitance of 108.33 \(\mu\)F; thus, the commercial value \(C_{pv}=110\)\(\mu\)F is adopted. The parameters of the P&O algorithm are calculated following the procedure proposed by [PERSON] et al. in [34], obtaining a perturbation period \(T_{a}=500\)\(\mu\)s and a perturbation magnitude \(\Delta v_{po}=200\) mV. Since [PERSON] also demonstrated that the settling time of the PV \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline \(L_{2}\) current ripple & \(\delta i_{2}\leq 0.39\) A at \(S=250\) W/m\({}^{2}\) \\ \(L_{1}\) inductor & \(L_{1}=L_{2}\) \\ \(C_{cb}\) voltage ripple & \(\delta v_{cb}\leq 4.8\) V \\ PV voltage ripple & \(\delta v_{pv}\leq 9\) mV \\ Switching frequency & \(F_{sw}\leq 100\) kHz \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Design characteristics for the NEC boost converter. Figure 12: Electrical characteristics of the PV panel. voltage must be shorter than \(T_{a}\) to ensure the P&O stability, such a settling time is defined as \(t_{s}=400\)\(\upmu\)s \(<T_{a}\). Concerning the SMC, Equation (54) is used to calculate the hysteresis width needed to limit the switching frequency under 100 kHz, obtaining \(H=0.667\) A. Then, the voltage controller parameters are calculated from Equations (60) and (64) to ensure the desired \(t_{s}\) value, obtaining \(k_{p}=2.96\) A/V and \(k_{i}=19.98\) kA/V. It is important to remark that the global stability of the SMC is ensured when the dynamic restrictions (47) and (48) are fulfilled. Those limits depend on the derivative of the PV current, which can be analyzed using the ideal single diode model, whose parameters are calculated following the procedure reported in [35]: \(i_{pv}=i_{sc}-A\cdot e^{B\cdot z_{pv}}\), where \(A=896.8\)\(nA\), \(B=0.7029\) V\({}^{-1}\), and the short-circuit current is considered proportional to the irradiance condition \(S\) as \(i_{sc}=(5/1000)\cdot S\). Then, the PV current derivative is \(\frac{d{i_{pv}}}{dt}=\frac{d{i_{sc}}}{dt}-A\cdot B\cdot e^{B\cdot y_{pv}}\), where \(\frac{d{i_{sc}}}{dt}\) is proportional to the derivative of the irradiance \(\frac{d{S}}{dt}\). In order to test the proposed solution under strong perturbations, a very fast change on the irradiance condition \(\frac{d{i_{sc}}}{dt}=1000\) (W/m\({}^{2}\))/ms is considered, which corresponds to a change of one sun in a single millisecond, i.e., from fully irradiated to completely shaded in one millisecond. Evaluating the limits given in (47) and (48) results in \(-0.35\) A/\(\upmu\)s \(<\frac{d{i_{s}}}{dt}<0.21\) A/\(\upmu\)s, but with the aim of imposing a single dynamic limit, the most restrictive condition is selected as \(\left|\frac{d{i_{r}}}{dt}\right|<0.21\) A/\(\upmu\)s. The current reference \(i_{r}\) is provided by the voltage controller as \(i_{r}=k_{p}\cdot(v_{pv}-v_{r})+k_{i}\cdot\int\left(v_{pv}-v_{r}\right)\ dt\); thus, the dynamic limitation must be described in terms of the voltage reference \(v_{r}\).