# Parameter Observer Useable for the Condition Monitoring of a Capacitor
## 4 Discussion  
The purpose of this article is first of all theoretical. The proposed POs were used in analytical examples and for experimental data regarding the electrolytic capacitor COUT in Figures 12 and 13. For both capacitors tested (100 uF/25 V and 470 uF/25 V), the experimental data were processed using the discrete-time PO. The main aspects that we retain from the study are summarized below in points 1-7. The figures and/or the equations that support these aspects are specified in brackets. 1. During the capacitor's discharging process, the parameter \(T\) of the circuit, currently called the time constant, varies (Figures 15a,c and 16a,c). According to [19], the variation in the estimated value \(\hat{T}(t)\) reflects that both \(C\) and \(ESR\) vary in time. From a practical perspective, this conclusion implies the defining of equivalent values: \(\hat{T}_{e}\), \(\hat{C}_{e}\), \(\hat{R}_{e}\). To determine the equivalent value \(\hat{T}_{e}\) of \(T(t)\), the variation in the average value \(\left\{\hat{T}_{m}[k]\right\}\) was calculated using the discrete-time PO, and \(\hat{T}_{e}\) was extracted by the \"bisector method\" (Figures 11, 15b,d and 16b,d). 2. The working hypothesis of simplifying the scheme in Figure 12 as in Figure 13 means that the discharge current on the left side of COUT is negligible with respect to the discharge current on the right side. The hypothesis is confirmed by simulation with a Simscape Electrical model. 3. Obtaining the values \(\hat{C}_{e}\), \(\hat{R}_{e}\) requires two discharging experiments. To maintain, as far as possible, the same discharging regime, according to the method presented in [6] a two-stage discharge was used (Figure 10). For each capacitor, the two stages were treated separately (Figures 15 and 16). As the time required for the two discharge steps is much longer than the ripple's discharge time, the method can be applied as in [6], only when the converter is stopped. Unlike [6], where the capacitor discharge process exceeds 10 s, in our paper the discharge time is less than 0.02 s. As a result,the necessary stop time of the converter is quite short. The difference is due to the inertia of the discharging electrical circuits to which the capacitors belong. 4. The estimation of the \(T\) parameter with both proposed POs is affected at the beginning by a short transient regime caused by the inconsistency between the initial state variables of the PO with the observer's input. The duration of the transient may be shortened by appropriately adopting the parameters of the observer. As the transient cannot be eliminated, it was avoided in the calculation of \(\hat{T}_{m}(t)\) by choosing the moment \(t_{0}\). The attenuation time of the estimation error of the PO used for the experimental data is a maximum of \(10^{-3}\) s (Figure 14 et seq. ), and so is smaller than the attenuation of approx. \(5\cdot 10^{-3}\) s that occurred in [3], and much smaller than that of approx. \(50\cdot 10^{-3}\) s that occurred in [27; 28]. 5. The study of the influence of noise filtering on \(\hat{T}(t)\), applied for 100 uF, does not show favorable noticeable consequences on the equivalent value \(\hat{T}_{e}\) (Figure 16b,d). 6. In the theoretical examples, for which the values of \(T(t)\) can be verified analytically, the PO ensures the correct estimation of \(T\) after the attenuation of the transient regime (Figures 3, 4, 7 and 8). Since a theoretical verification of \(T(t)\) is impossible in the practical case of the two capacitors, we analysed the equivalent values (40) and (41). Both values of \(\hat{C}_{e}\) are in the range of 20 Hz-25 kHz of the measured frequency characteristics \(C(f)\), while the values of \(\hat{R}_{e}\) are below 20 Hz. It should be also noted that the calculated values of \(\hat{R}_{e}\) with Formula (36) are very sensitive in relation to the values of \(\hat{T}_{e1}-\hat{T}_{e2}\), \(R_{L}\) and \(R_{a}\), while the values of \(\hat{C}_{e}\) are less sensitive. 7.