# DC-DC Buck Converters with Quasi-Online Estimation of Filter Capacitor Equivalent Parameters
## 2 Materials and Methods
### Scheme Based on the Implementation of the Principle from Figure 7d  
The proposed scheme has the configuration of Figure 12. The converter supplies a load that is not purely resistive. The circuit was made for \(R_{1}=0\) and \(R_{2}=0\). In Figure 7d, the switches swij were implemented with T-swij transistors. Each of the capacitors C21 and C22 of the filter capacitor bank independently discharges across the resistors of the VEN, initially across the resistance \(R_{ext1}\), then across the resistance \(R_{ext2}\
eq R_{ext1}\). Any discharge of a capacitor is immediately followed by a recharge of it from the converter, triggered when the PO completes the determination of the discharge time constant. Equivalent parameters \(\hat{C}_{e,i}\) and \(\hat{R}_{se,i}\) of the two capacitors C21, \(i=1,2\) are calculated as solutions of the systems of Equation (15). The load impedance does not appear in the equations. Unlike the scheme in Figure 9, the scheme in Figure 12 contains two temperature sensors \(S_{\theta 1}\) and \(S_{\theta 2}\) that measure the operational temperature of each capacitor. \begin{table}
\begin{tabular}{c c c c} \hline \(\hat{T}_{1}\)**[ms]** & \(\hat{T}_{2}\)**[ms]** & \(\hat{C}_{e}\)**[uF]** & \(\hat{R}_{se}\)**[\(\Omega\)]** \\ \(\sigma_{\hat{T}_{1}}\)**[\%]** & \(\sigma_{\hat{T}_{2}}\)**[\%]** & \(\sigma_{\hat{C}_{e}}\)**[\%]** & \(\sigma_{R_{se}}\)**[\%]** \\ \hline
[18.4718 \(\div\) 18.4822] & [0.71102 \(\div\) 0.71285] & [202.088 \(\div\) 202.213] & [0.20516 \(\div\) 0.21592] \\
[0.13774 \(\div\) 0.19571] & [0.27062 \(\div\) 0.46067] & [0.14702 \(\div\) 0.20012] & [4.90394 \(\div\) 8.2696] \\ \hline \end{tabular}
\end{table} 
Table 4: Ranges for the parameters of C2 in Figure 9 determined by PO2 for \(K=3.291\) V and their standard deviations. Figure 12: Buck converter scheme made based on the principle scheme in Figure 7d. in portions 3 and 4 where filtering is performed with C21 and \(\overline{V_{L}}=3.2974\) V in normal operation when filtering is completed with both capacitors. In Table 5, the results obtained for six sets of 25 experiments are systematized. The average values of the parameters on the set of 150 experiments have the values in (16). Each set of experiments was repeated again with a period 0.3 \(s\) (3.33 Hz). Also, the duration of an experiment did not exceed 40 ms.  
\[\hat{C}_{e\_1}=460.4057\ \mu\text{F},\ \hat{R}_{se\_1}=0.345\ \Omega.\ \hat{C}_{e\_2}=210.8587\ \mu\text{F},\ \hat{R}_{se\_2}=0.86953\ \Omega. \tag{16}\]  
The values \(\hat{C}_{e\_1}\) and \(\hat{R}_{se\_1}\) roughly correspond to the frequency range 65 Hz \(\div\) 80 Hz from the frequency characteristics in Figure 13a,b. The different values of the frequencies are due to the non-sinusoidal nature of the experimental regime and from the measurement errors mentioned in Appendix A. Stage 2: Context: The converter feeds a resistive-inductive load. In series with the load resistance \(R_{L}=20\ \Omega\) were connected separately, in series, then in parallel, two coils with \(R^{\prime}=1.775\ \Omega/\text{L}\
u=0.463\ \text{mH}\) and \(R^{\prime\prime}=1.8\ \Omega/\text{L}^{\prime\prime}=0.426\ \text{mH}\). For each of the 4 cases, six series of 25 experiments were performed. On the set of 600 experiments, the average values were obtained: \(\overline{\hat{C}_{e\_1}}=465.8374\ \mu\text{F},\ \overline{\hat{R}_{se\_1}}=0.3123\ \Omega\) for C21 and \(\overline{\hat{C}_{e\_2}}=211.34\ \mu\text{F},\ \overline{\hat{R}_{se\_2}}=0.87595\ \Omega\) for C22. The equivalent capacitance values are practically within \(\pm 1\%\) of those in Table 5, while the ESR values are within much higher limits.