# DC-DC Buck Converters with Quasi-Online Estimation of Filter Capacitor Equivalent Parameters
## Appendix A  
Computing formulas \(\hat{R}_{se}\) and \(\hat{C}_{e}\) obtained from system (13) have the expression (A1)  
\[\hat{R}_{se}=\frac{\hat{T}_{2}\cdot R_{ext1}-\hat{T}_{1}\cdot R_{ext2}}{\hat{T }_{1}-\hat{T}_{2}},\ \ \ \hat{C}_{e}=\frac{\hat{T}_{1}-\hat{T}_{2}}{R_{ext1}-R_{ext2}}\.\] (A1)  
They are valid in the following working assumptions:  
1 The values of \(R_{se}\), \(\hat{C}_{e}\), \(R_{ext1}\) and \(R_{ext2}\) are constant during the measurement of \(\hat{T}_{1}\) and \(\hat{T}_{2}\) from the signal \(V_{0}(t)\) and have the same value for both transients. 2 The values of \(R_{ext1}\) and \(R_{ext2}\) are measured beforehand at a temperature equal to that during the measurement of \(\hat{T}_{1}\) and \(\hat{T}_{2}\). Let x be the measured value of a variable in (A1) or a calculated value for \(R_{se}\) or \(\hat{C}_{e}\), \(\hat{x}\) the true value of \(x\) and \(\varepsilon_{x}\) the measurement or calculation error of \(x\) defined by formula (A2):  
\[\varepsilon_{x}=\frac{x-\tilde{x}}{\tilde{x}}\,\ \ \ x\in\{R_{ext1},R_{ext2},\ \hat{T}_{1e},\ \hat{T}_{2x}\}.\] (A2)By applying this formula to \(\hat{R}_{se}\) si \(\hat{C}_{e}\) in (A1), the Formulas (A3) and (A4) for error calculation \(\varepsilon_{R_{se}}\) and \(\varepsilon_{\hat{C}_{e}}\) are derived. They used the notations (A5):  
\[\varepsilon_{\hat{R}_{se}}=\frac{\rho_{R}\Big{(}1+\varepsilon_{\hat{T}_{2}} \Big{)}\big{(}1+\varepsilon_{R_{ext1}}\big{)}-\rho_{\tau}\Big{(}1+\varepsilon_ {\hat{T}_{1}}\Big{)}\big{(}1+\varepsilon_{R_{ext2}}\big{)}}{\rho_{2}[\rho_{ \tau}\Big{(}1+\varepsilon_{\hat{T}_{1}}\Big{)}-\Big{(}1+\varepsilon_{\hat{T}_ {2}}\Big{)}]}-1,\] (A3)  
\[\varepsilon_{\hat{C}_{e}}=(1+\rho_{1})\frac{1+\varepsilon_{\hat{T}_{1}}-\frac {1+\varepsilon_{\hat{T}_{2}}}{\rho_{\tau}}}{1+\varepsilon_{R_{ext1}}-\frac{1+ \varepsilon_{R_{ext2}}}{\rho_{R}}}-1,\] (A4)  
\[\rho_{R}=\frac{\widetilde{R_{ext1}}}{\widetilde{R_{ext2}}},\rho_{T_{e}}=\frac {\widetilde{\hat{T}_{1}}}{\widetilde{\hat{T}_{2}}},\rho_{1}=\frac{\widetilde{ \widetilde{R}_{se}}}{\widetilde{R_{ext1}}},\rho_{2}=\frac{\widetilde{\widetilde {R}_{se}}}{\widetilde{R_{ext2}}}.\] (A5)  
To obtain a picture of the problem, we consider the experimental data in columns 2 and 3 of Table A1, taken from [28], Table VI. They refer to five electrolytic capacitors and were obtained using a VEN who provided \(R_{ext1}\cong 980.7692\)\(\Omega\) and \(R_{ext2}\cong 0.08999984\)\(\Omega\). In columns 4 and 5, the values obtained with formulas (A1) are inserted. To roughly establish the error ranges in which the calculated values for \(\hat{R}_{se}\) and \(\hat{C}_{e}\) fall, we assume that the measurement errors of \(\hat{T}_{1e}\), \(\hat{T}_{2e}\), \(R_{ext1}\) and \(R_{ext2}\) are located in the ranges \(\varepsilon_{\hat{T}_{1}}\in[-0.01,0.01]\), \(\varepsilon_{\hat{T}_{2}}\in[-0.02,0.02]\), \(\varepsilon_{R_{ext1}}\in[-0.01,0.01]\) and \(\varepsilon_{R_{ext2}}\in[-0.02,0.02]\), that \(\widetilde{\hat{T}_{1}}\) and \(\widetilde{\hat{T}_{2}}\) take exactly the values from Table A1, and \(R_{ext1}^{*}\) and \(R_{ext2}^{*}\) have the values of \(R_{ext1}\) and \(R_{ext2}\) specified above. Consequently, using Equation (A5) yields the results in Table A2. Two observations emerge from these results for the case when \(\rho_{1}\ll 1\) and \((3\div 4)\rho_{2}\approx 1\):  
3. The estimation errors of \(C_{e}\) with Formula (A1) are within \(\pm 2\%\). 4. \(R_{se}\) estimation errors with Formula (A1) vary practically between limits of \(\pm 25\%\) to \(\pm 37\%\). The wide error limits from the last observation are due to the relatively small values of the \(\rho_{2}\) ratio. This finding is corroborated by (A3) which highlights a hyperbolic variation of \(\varepsilon_{R_{ex}}\) with respect to \(\rho_{2}\). The lower the \(\rho_{2}\), the higher the \(\varepsilon_{R_{ex}}\). In the range \(\rho_{2}\in[0.2,\,1]\) inverse \(1/\rho_{2}\) decreases from the value \(5\) to the value \(1\).