## 3 Results and Discussion  
This section will show a case study in which a DLSC, modelled as a capacitance linearly varying with the internal voltage, will be charged through a voltage source. The evolution of the electrical variables will be studied using the set of analytical expressions of Section 2, and simulations will be carried out to analyze their variation with different values of the constant k\({}_{0}\). Subsequently, the results obtained will be compared with the same DLSC, considering it as a constant capacitance with its equivalent series resistance. Table 1 shows its rated data together with the no-load voltage of the voltage source, E.  
The wiring diagram of the case study is presented in Figure 3.  
To study the time evolution of the different variables, three values will be considered for the variable k\({}_{0}\) = [0.65, 0.85, 1] and four different values will be chosen for the internal resistance of the voltage source, R\({}_{\mathrm{c}}\) (\(\Omega\)) = [0.5, 1, 3, 5]. From the obtained results, the constants C\({}_{0}\), k\({}_{c}\), k\({}_{1}\), k\({}_{2}\) and k\({}_{3}\) as defined in Equations (2), (13), (14) and (17) will be, respectively, calculated together with the value of the time constant, \(\tau\), according to (41), which, in this study, is the time taken for the internal voltage to reach 1.7067 V. All the above values are shown in Table 2. \begin{table}
\begin{tabular}{c c c c c} \hline
**E (V)** & **U\({}_{\mathrm{N}}\) (V)** & **U\({}_{\mathrm{0}}\) (V)** & **R (m\(\Omega\))** & **C\({}_{\mathrm{N}}\) (F)** \\ \hline
2.7 & 2.7 & 0 & 25 & 25 \\ \hline \end{tabular}
\end{table} 
Table 1: Rated values of the selected DLSC. Figure 4 shows the evolution of the DLSC internal voltage, u(t), for different values of the constant k\({}_{0}\) and various values of the internal resistance of the voltage source, R\({}_{\rm c}\). As can be appreciated from it, for the three models of different k\({}_{0}\), there is a coincidence in three values of the internal voltage, u(t). Two of them are obvious, since they are the initial value (they all start from the same U\({}_{0}\)) and the final value, which converge to the no-load voltage of the voltage source E = U\({}_{\rm N}\) = 2.7 V. The third one corresponds to a specific value of the g variable. This value, which will be denoted as g\({}_{\rm s}\), ensures that whatever the value of k\({}_{0}\) and the resistance, R\({}_{\rm c}\), it will be the same in all models. The value of g\({}_{\rm s}\) can be calculated as follows:  
\[\rm{g}_{s}=\left(E-\frac{U_{N}}{2}\right)\cdot W_{0}\left(\frac{U_{0}-E}{E- \frac{U_{N}}{2}}\cdot exp\left(\frac{U_{0}-E}{E-\frac{U_{N}}{2}}\right)\right). \tag{52}\]  
As shown in (52), g\({}_{\rm s}\) only depends on E, the rated voltage of the cell, U\({}_{\rm N}\) and its initial voltage, U\({}_{0}\), but is independent of all other variables, including R\({}_{\rm c}\). In the case study presented, g\({}_{\rm s}=-0.5486\) V. If this value of g\({}_{\rm s}\) is replaced in (20), where u(g) was calculated, the internal voltage at which all the models coincide is 2.1514 V, as can be appreciated from Figure 4. Below this voltage, the variable capacitance model (k\({}_{0}\) < 1) produces higher internal voltages than the conventional model, while beyond this point, the trend is reversed. By substituting this value of g\({}_{\rm s}\) in (39), the result is the time instant at which, for any value of k\({}_{0}\), all the internal voltages cross at a common point. This instant does depend on the value of R\({}_{\rm c}\).