## 3 Results and Discussion
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline
**Case** & **k\({}_{0}\)** & **C\({}_{0}\) (F)** & **k\({}_{\rm c}\)** & **k\({}_{\rm c}\)** & **k\({}_{\rm 1}\)** & **k\({}_{\rm 2}\)** & **k\({}_{\rm 3}\)** & **\(\tau\) (s)** \\ \hline
1 & & & & & 0.5 & & 0.0564 & & 11.911 \\
4 & & & & 1 & 0.0289 & & -0.3087 & & 23.255 \\
7 & 0.65 & 16.25 & 3.2407 & 3 & 0.1920 & & 0.0098 & & 68.631 \\
10 & & & & & 5 & & 0.0059 & & 114.07 \\ \hline
2 & & & & & 0.5 & & 0.0663 & & 12.605 \\
5 & 0.85 & 21.25 & 1.3889 & 1 & 0.0966 & & 0.0339 & & 24.609 \\
8 & & & & & 3 & 0.0115 & -0.2010 & & 72.627 \\
11 & & & & & 5 & & 0.0069 & & 120.64 \\ \hline
3 & & & & & 0.5 & & 0.0762 & & 13.125 \\
6 & & & & 1 & 0 & 0.0390 & & 25.625 \\
9 & & & & & 3 & 0.0132 & & 75.625 \\
12 & & & & & 5 & & 0.0080 & & 125.62 \\ \hline \end{tabular}
\end{table} 
Table 2: Constants C\({}_{0}\), k\({}_{\rm c}\), k\({}_{1}\), k\({}_{2}\), k\({}_{3}\) and \(\tau\) for different values of k\({}_{0}\) and R\({}_{\rm c}\). Figure 3: Wiring diagram of the case study. Table 3 shows the times at which, with all the models, the same internal voltage is reached as a function of the values of \(R_{c}\). The value of the current at those instants is also displayed. Figure 5 shows the difference between the calculation of the internal voltage with variable capacitance, \(k_{0}=[0.65,0.85]\), and the conventional model with constant capacitance, \(k_{0}=1\), for different values of \(R_{c}\). As can be seen, the maximum difference is independent of the value taken by \(R_{c}\), which only affects the instant at which it appears. In this example, the maximum discrepancy between the conventional DLSC model and the one with \(k_{0}=0.65\) is 0.1738 V.  
Figure 6 shows the evolution of the instantaneous current of the circuit, i(t), calculated with the three values of the constant \(k_{0}\) and for different values of \(R_{c}\). In the same instants in which the internal voltage, u, coincides, the current also coincides, whatever the value of \(k_{0}\), although in this case, the values of the current are different for each \(Rc\) value, since i(t) depends on this resistance, as was earlier indicated by means of expression (9). It can be observed how, in the calculation of the current, the difference between modelling the DLSC with variable capacitance and with constant capacitance is minimal. It can therefore be concluded that the usual RC series model is perfectly valid for i(t) calculation. Figure 4: Evolution of the internal voltage of the DLSC, u(t), for different values of \(k_{0}\) and \(R_{c}\). The points marked with a rhombus indicate the value of the internal voltage that is equal for all values of \(k_{0}\) and \(R_{c}\). \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{4}{c}{\(R_{c}\) (\(\Omega\))} \\ \cline{2-5}  & **0.5** & **1** & **3** & **5** \\ \hline t (s) & 20.92 & 40.84 & 120.52 & 200.2 \\ \hline i (A) & –1.0449 & –0.5352 & –0.1814 & –0.1092 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Time instants at which the internal voltage u = 2.1514 V is the same for different values of \(k_{0}\) and \(R_{c}\), and current values, i(t) (A), at those instants. Figure 7 shows the difference between the current obtained with variable capacitance, k\({}_{0}\) = [0.65, 0.85], and the conventional model with constant capacitance, k\({}_{0}\) = 1, for different values of R\({}_{\rm c}\). As can be appreciated, the higher the value of R\({}_{\rm c}\), the less difference there is between considering the model with variable capacitance and the RC series model with constant capacitance. Figure 8 shows the time evolution of the energy dissipated in the internal resistance of the DLSC, e\({}_{\rm d}\)(t), presented in (30) and calculated with the three values of the constant k\({}_{0}\) and different values of R\({}_{\rm c}\). The higher the value of the constant k\({}_{0}\), at any instant of time, the higher the energy dissipated.