# Parameter Observer Useable for the Condition Monitoring of a Capacitor
## 2 Materials and Methods
### Method for Estimating the Parameters of a Capacitor  
The process we refer to occurs in the circuit in Figure 9 and consists of discharging a real capacitor through an ideal resistor \(R_{ext}\). In the R-C model used for the capacitor, R represents the ESR. The voltage at the terminals of the capacitor is \(y\), and the discharging current \(i\).  
As a working hypothesis, we consider that both C and R vary in time, resulting in the so-called \"circuit time constant \(T\)\" being the function of time (29):  
\[T(t)=C(t)\cdot[R(t)+R_{ext}]. \tag{29}\]  
We also consider that \(y(0)=y_{0}>0\). Under these circumstances, the discharging process is described by the model (1). Assuming the measurement of \(y(t)\) by sampling, the situation is identical to that described in Section 2.2. Consequently, we can determine the signal \(\left\{\hat{T}[k]\right\}_{k\in N}\) using the tracking subsystem (22)-(24), the linking Formulas (25) and (26), and the variation \(\left\{\hat{T}_{m}[k]\right\}_{k\in N}\) by Formula (27).  
Given that for the sampling instants (29) becomes \(T[k]=C[k]\cdot(R[k]+R_{ext})\), we can consider that beginning at the end of the transient regime, Equation (30) is valid:  
\[\hat{T}[k]=C[k]\cdot(R[k]+R_{ext}). \tag{30}\]  
Since, as will be seen below, \(\left\{\hat{T}[k]\right\}_{k\in N}\
eq const.\), this result highlights that \(T\) varies during the discharging process. Naturally, we are interested in the variations \(\left\{C[k]\right\}_{k\in N}\) and \(\left\{R[k]\right\}_{k\in N}\). Their determination from a single Equation (30) is impossible because at each moment \(k\cdot h\), Equation (30) presents two unknown values. Hence, a second equation is needed, and that means a new capacitor's discharge from the same initial condition \(y_{0}\) through a different external resistance. However, this implies a process for which the dynamics at time \(k\cdot h\) are characterized by a different pair \((C[k],R[k])\).  
Figure 8: Estimation of the parameter \(\left\{T[k]\right\}_{k\in N}\) for the sampled signal corresponding to (28): (**a**) the signal (28) and its estimate; (**b**) the estimated time constant.  
Figure 9: The analysed circuit.  
For this reason, the only calculation alternative for the presented situation is the operation with equivalent values \(\hat{T}_{e}\), \(\hat{C}_{e}\), \(\hat{R}_{e}\) of the variations of \(T\), \(C\), \(R\) linked by Equation (31):  
\[\hat{T}_{e}=\hat{C}_{e^{\prime}}(\hat{R}_{e}+R_{ext})\,, \tag{31}\]  
and splitting the discharging process into at least two stages during which \(\hat{C}_{e}\) and \(\hat{R}_{e}\) can be considered constant. Let us admit in this context the possibility of performing the discharge of the same capacitor in two stages through two different external resistors, \(R_{ext1}\) and \(R_{ext2}\), assigned for each stage. As a consequence, the value of \(\hat{T}_{e}\), changes for the first and second stage to \(\hat{T}_{e1}\) and \(\hat{T}_{e2}\), respectively, and we obtain the system of Equation (32) with respect to \(\hat{C}_{e}\) and \(\hat{R}_{e}\):  
\[\left\{\begin{array}{l}\hat{T}_{e1}=\hat{C}_{e^{\prime}}(\hat{R}_{e}+R_{ext1 })\\ \hat{T}_{e2}=\hat{C}_{e^{\prime}}(\hat{R}_{e}+R_{ext2})\end{array}\right.. \tag{32}\]  
The above-described situation can be implemented with a good approximation using the discharging scheme in two stages, as shown in Figure 10a. At \(t_{s}\), at the end of the first stage, the switch closes, determining the voltage variation \(y(t)\) presented in Figure 10b. This is due to the variation in the equivalent resistance \(R_{ext}\) with its different values, \(R_{ext1}\) and \(R_{ext2}\), according to Equation (33):  
\[R_{ext1}=R_{L}\,\ \ R_{ext2}=\frac{R_{L}\cdot R_{a}}{R_{L}+R_{a}}. \tag{33}\]  
The idea of modifying the value of the discharging resistor during the process appears in [6], which also shows the modification of the discharging resistor by connecting additional resistors in parallel.