# DC-DC Buck Converters with Quasi-Online Estimation of Filter Capacitor Equivalent Parameters
## Appendix A
The objective pursued in the estimation methods of the equivalent parameters of a capacitor modeled by a serial connection \(\mathbb{R}-\mathbb{C}\) is to achieve the approximations \(\hat{R}_{se}\cong\widetilde{\hat{R}}_{se}\) and \(\hat{\mathcal{C}}_{e}\cong\widetilde{\hat{\mathcal{C}}}_{e}\), i.e., \(\varepsilon_{\hat{R}_{se}}\cong 0\) and \(\varepsilon_{\hat{\mathcal{C}}_{e}}\cong 0\). Assuming that the measurement errors can be minimized from a technical point of view, the means of achieving the objective consists in the appropriate manipulation of the values of the ratios \(\rho_{R}\cong\rho_{T_{e}}\) and \(\rho_{1}\) or \(\rho_{2}\), practically of the values of the resistances \(R_{ext1}\) and \(R_{ext2}\). To obtain \(\varepsilon_{\hat{R}_{ex}}\) as close as possible to \(0\) the value of \(R_{ext2}\) must be adopted as low as possible, so that \(\rho_{2}\) is as high as possible. Ensuring high values of \(\rho_{2}\) is in principle a critical problem from the electrical point of view because discharging the capacitor over a resistance of value \(2\times R_{se}\), from an initial voltage \(V_{0}(0)\) suitable for measurement, is equivalent to short-circuiting it. Moreover, the value of \(T_{2}\) becoming very small, the time available for measuring \(T_{2}\) becomes very short, which could also become a critical problem for the online implementation of the measurement method in order to monitor the capacitor parameters. Also, a small value of \(T_{2}\) is likely to cause an increase in the absolute value of the measurement error \(\left|\varepsilon_{\hat{T}_{2}}\right|\). Hence, \(R_{ext2}\) should be chosen so that the initial current \(V_{0}(0)/\left(R_{ext2}+R_{se}\right)\) is, on the one hand, as high as possible, and on the other hand, it does not endanger the capacitor and the resistor. To obtain a \(\varepsilon_{\hat{\mathcal{C}}_{e}}\) as close as possible to \(0\) the ratio \(\rho_{R}\) must have a value as high as possible. Once \(R_{ext2}\) is adopted, there are no problems adopting the value of \(R_{ext1}\). In conclusion, for designing the monitoring circuit, in order to provide the best estimates for both Formula (A1), it is necessary to assure:  
* \(\rho_{R}\gg 1\), i.e., \(R_{ext1}\gg R_{ext2}\);
* \(\rho_{2}\), i.e., \(\widetilde{\hat{R}}_{se}/\widetilde{R}_{ext2}\), as close as possible to \(1\). If these requirements are not met, the \(\hat{R}_{se}\) formula leads to erroneous estimates, strongly influenced by the errors in the measurements of \(\hat{T}_{1e}\), \(\hat{T}_{2e}\), \(R_{ext1}\) and \(R_{ext2}\), even when they are only a few percent. In this case, the formula of \(\hat{\mathcal{C}}_{e}\) will also produce errors outside the range of \(\pm 2\%\). For DC-DC buck converters in this paper, the following values were used:  
* Converter in Figure 9: \(R_{ext1}=R_{L}=91.19\)\(\Omega\), \(R_{ext2}=3.311\)\(\Omega\), \(\hat{R}_{se}=0.21054\)\(\Omega\), \(\rho_{R}=27.54\) and \(\rho_{2}=0.06358\);
* Converter in Figure 12: \(R_{ext1}=3.311\)\(\Omega\), \(R_{ext2}=89.16\)\(\Omega\), \(\hat{R}_{se,1}=0.345\)\(\Omega\), \(\hat{R}_{se,2}=0.86953\)\(\Omega\), \(\rho_{R}=26.92\), \(\rho_{2}\_{1}=0.10419\) for C21 and \(\rho_{2}\_{2}=0.2626\) for C22.