# DC-DC Buck Converters with Quasi-Online Estimation of Filter Capacitor Equivalent Parameters
## 2 Materials and Methods
### Parameter K of PO2  
The parameter \(K\) in Equation (8) represents the limit value of the asymptotic increasing exponential variation of the observed signal \(v(t)\) of Equation (4), i.e., of the signal that the designer supposes to apply to the PO2 input. In practical applications, the correct use of PO2 requires an accurate estimation of the value of \(K\) related to the applied signal \(v(t)\). As a motivation of this statement, Figure 5 is considered. The exponential in the figure has the equation \(K\cdot e^{-t/T}\) with \(K=K_{0}\) The parameter \(T\) being constant, its value is equal to the length of the subtangent corresponding to the current point \(A\) on the exponential curve, i.e., to the segment \(B_{0}C_{0}\) measured on the asymptote \(v(t)=K_{0}\). When \(A\) changes its position, successively passing through the positions \(A^{\prime}\) and \(A^{\prime\prime}\), the length of the subtangent does not change. If we wrongly consider for the same exponential that the asymptote is in another position, i.e., that the parameter \(K\) has a value different from \(K_{0}\), the property is no longer preserved. Thus, if we consider \(K=K_{1}>K_{0}\) the length of the subtangent, \(B_{1}C_{1}\) is greater than the subtangent \(B_{0}C_{0}\); moreover, as \(A\) passes through the positions \(A^{\prime}\), \(A^{\prime\prime}\), \(\ldots\), it increases tending to infinity. If we consider that \(K=K_{2}<K_{0}\), the length of the subtangent \(B_{2}C_{2}\) is smaller than the subtangent \(B_{0}C_{0}\). When the exponential reaches the value \(K_{2}\) the length of the subtangent takes the value \(0\), and then disappears. Figure 4: Parameter observer on two edges: (**a**) the structure of PO2; (**b**) the block diagram of the connection OS-PO2 for discrete time processing. Analytically, the above discussion translates into the analysis of Equation (9), where \(v_{K_{0}}(t)\) is the signal interpreted by PO2 when \(K\) is tuned correctly, i.e., \(K=K_{0}\), and \(v_{K}(t)\) is the signal interpreted by PO2 when \(K\) is not tuned correctly, i.e., \(K\
eq K_{0}\). \[v_{K}(t)=v_{K_{0}}(t)+(K-K_{0})\Big{(}1-e^{-\frac{t}{T}}\Big{)},\ \ t\geq 0. \tag{9}\]  
The effect of the additional term in (9) is illustrated in Figure 6. In Figure 6a are displayed more characteristics of the type such as that in Figure 3b, of parameter \(K\), generated by PO2 when the signals of Equation (7) were applied to its input with \(K\in\{2.9,2.95,2.98,2.99,3,3.01,3.02,3.05,3.1\}\). The group includes the \(K=3\) case. With \(\hat{T}_{1,K}(t)\) we denote the segments of \(\hat{T}(t)\) on the time interval on which \(\sigma(t)\) decreases and with \(\hat{T}_{2,K}(t)\) the segments corresponding to the time interval on which \(\sigma(t)\) increases. The latter are detailed in Figure 6b. All estimates \(\hat{T}_{1,K}(t)\) overlap and stabilize at the value \(\hat{T}_{1}=2\cdot 10^{-3}\) s. This is explained by the fact that in processing the decreasing variations of \(v(t)\) PO2 does not use the value of \(K\). Among the curves \(\hat{T}_{2,K}(t)\), only the curve \(\hat{T}_{2,3}(t)\) stabilizes at the correct value \(\hat{T}_{2}=10^{-3}\)\(s\). All other curves fail to stabilize. After the damping of the transients that appears starting from the moment \(t=0.8\ \cdot\ 10^{-3}\) s they have either a monotonically increasing or a monotonically decreasing shape, as \(K\) takes values lower or higher than the value 3. The practical conclusion from the above presentation is:  
To apply PO2 to a physical system, it is necessary to tune the value of the parameter \(K\) in Equation (8) as precisely as possible to match the value that the same-named parameter assumes in the observed signal (4) generated by the physical system. We call this condition \"PO2--tuning condition of \(K\)\". Figure 5: The influence of the \(K\) parameter of PO2 on \(T\) value. The PO2--tuning condition of \(K\) is more experimental. Tuning can be completed in several ways. The simplest way is to consider \(K\) equal to \(v(t)\) final value after stabilization.