# DC-DC Buck Converters with Quasi-Online Estimation of Filter Capacitor Equivalent Parameters
## 2 Materials and Methods
### Calculation of the Equivalent Parameters of a Capacitor  
Discharging a capacitor, modeled by an R-C series connection, across a resistor or charging it from a constant voltage source through a resistor are processes described by first-order systems (1) and (5), respectively. The time constants \(T_{1}\) and \(T_{2}\) correspond to the product of the equivalent capacitance \(C_{e}\) of the capacitor and the total resistance of the circuit according to Equation (12).  
\[T_{1}=C_{e}\cdot(R_{se}+R_{ext1}),\ \ T_{2}=C_{e}\cdot(R_{se}+R_{ext2}). \tag{12}\]  
\(R_{se}\) represents the equivalent ESR, and \(R_{ext1}\) and \(R_{ext2}\) the resistances in the discharge/charge circuit external to the capacitor.  
PO or PO2 provides the estimates \(\hat{T}_{1}\) and \(\hat{T}_{2}\) of the time constants and calculates the estimates \(\hat{C}_{e}\) and \(\hat{R}_{se}\) of the equivalent parameters as the solutions of system (13):  
\[\left\{\begin{matrix}\hat{C}_{e}\cdot\left(\hat{R}_{se}+R_{ext1}\right)=\hat{ T}_{1}\\ \hat{C}_{e}\cdot\left(\hat{R}_{se}+R_{ext2}\right)=\hat{T}_{2}\end{matrix}\right. \tag{13}\]  
The calculation formulas of \(\hat{C}_{e}\) and \(\hat{R}_{se}\) resulting from system (13) show different sensitivities in relation to the errors affecting the values of \(\hat{T}_{1}\), \(\hat{T}_{2}\), \(R_{ext1}\) and \(R_{ext2}\). The statement is substantiated by the analysis undertaken in Appendix A. While the value of \(\hat{C}_{e}\) is a little sensitive in relation to these errors, the value of \(\hat{R}_{se}\) presents a high sensitivity. From this perspective, ensuring the PO2--tuning condition of \(K\) becomes a mandatory requirement for the implementation of PO2, as it affects the value of \(\hat{T}_{2}\).  
\begin{table}
\begin{tabular}{c c c} \hline \hline \(K\) & \(\hat{T}_{2}\)[s] & \(\Delta\hat{T}_{2\%}\) \\ \hline
2.9 & 0.000930 & 6.970 \\
2.95 & 0.000965 & 3.540 \\
2.98 & 0.000988 & 1.420 \\
2.99 & 0.000993 & 0.713 \\
3.00 & 0.001000 & 0.000 \\
3.01 & 0.001007 & \(-\)0.720 \\
3.02 & 0.001014 & \(-\)1.440 \\
3.05 & 0.001036 & \(-\)3.630 \\
3.10 & 0.001074 & \(-\)7.360 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Estimated \(T_{2}\) values according to the \(K\) parameter of PO2 for the cases in Figure 6.  
Two Schemes of DC-DC Buck Converter with Estimation of Equivalent Parameters of Filter Capacitor Using PO  
The two buck converter configurations proposed in the principle schemes in Figure 7a,d are obtained by adding a variable electrical network (VEN) [28] and a microcontroller to a switching regulator circuit based on the LM2596 integrated circuit [29]. They allow the implementation of the measurement process, the mentioned observers, and the management of the VENs. In the first scheme, an auxiliary source of stabilized DC voltage is included as well. The schemes are shown in Sections 3.2 and 3.3. The part represented in black corresponds to the basic scheme in [29], and the part represented in red includes the performed adaptations. The POs are implemented in the microcontroller in the discrete-time system version (8).