## 2 Materials and Methods
### Electrical Analysis of a DLSC Charged or Discharged with a Voltage Source
This energy depends on the constants \(\mathrm{C}_{0}\), \(\mathrm{k}_{\mathrm{c}}\) and the internal voltage, \(\mathrm{u}\), according to the following equation:  
\[i=-\frac{dq}{dt}=-\frac{dq}{du}\cdot\frac{du}{dt}. \tag{3}\]0  
The \(\mathrm{C}_{\mathrm{E}}\) value is called \"Energetic Capacitance\", as it is used to obtain the energy stored in a variable capacitance DLSC, with an equation similar to that used in a conventional capacitor. This, as with the dynamic capacitance, also depends on the internal voltage. By substituting the former (20) in (32), the instantaneous stored energy as a function of g is finally obtained:  
\[i=-\frac{dq}{dt}=-\frac{dq}{du}\cdot\frac{du}{dt}. \tag{3}\]1  
If the DLSC starts from an initial internal voltage, \(\mathrm{U}_{0}\), and after a time, \(\mathrm{t}\), reaches an internal voltage, \(\mathrm{u}\), the discharged energy, \(\mathrm{e}_{\mathrm{dch}}\), will be:  
\[i=-\frac{dq}{dt}=-\frac{dq}{du}\cdot\frac{du}{dt}. \tag{3}\]2  
On the other hand, the energy consumed by the voltage source, \(\mathrm{E}\), \(\mathrm{e}_{\mathrm{E}}\), can be also obtained by integration, bearing in mind that:  
\[i=-\frac{dq}{dt}=-\frac{dq}{du}\cdot\frac{du}{dt}. \tag{3}\]3  
Replacing in (35) the value of \(\mathrm{p}_{\mathrm{E}}\) as a function of g, obtained in (26), and the value of the derivative of g(t) with respect to time, calculated in (29), and integrating the energy consumed or generated, depending on whether it is a charge or discharge of the DLSC can be obtained:  
\[i=-\frac{dq}{dt}=-\frac{dq}{du}\cdot\frac{du}{dt}. \tag{3}\]4  
Finally, it is necessary to develop an equation that expresses time as a function of g. The variable g as a function of time was calculated earlier in (19). From the definition of the Lambert W function itself, it must be satisfied that:  
\[i=-\frac{dq}{dt}=-\frac{dq}{du}\cdot\frac{du}{dt}. \tag{3}\]5  
Taking neperian logarithms on both sides of (37), the function \(\mathrm{t}\)(g) is obtained:  
\[i=-\frac{dq}{dt}=-\frac{dq}{du}\cdot\frac{du}{dt}. \tag{3}\]6  
By replacing \(\mathrm{k}_{1}\), \(\mathrm{k}_{2}\) and \(\mathrm{k}_{3}\), previously and respectively calculated in (13), (14) and (17) in (38), the time, \(\mathrm{t}\), can be expressed as a function of g and other known variables:  
\[i=-\frac{dq}{dt}=-\frac{dq}{du}\cdot\frac{du}{dt}. \tag{3}\]7  
As can be deduced from (39), for g to reach a zero value, the time, \(\mathrm{t}\), should tend to infinity. As in first-order circuits with conventional capacitors that are charged or discharged through a voltage source, the time constant, \(\tau\), can be defined as the elapsed time for the following equality to be satisfied, valid for both charging and discharging (e being the Euler's number):  
\[i=-\frac{dq}{dt}=-\frac{dq}{du}\cdot\frac{du}{dt}. \tag{3}\]8Considering the value of \(t\) obtained in (39) and the relationship between \(g\), \(u\) and \(E\) from (20), the \"time constant\" of the circuit, \(\tau\), can be determined by the following equation:  
\[i=-\frac{dq}{dt}=-\frac{dq}{du}\cdot\frac{du}{dt}. \tag{3}\]9  
In contrast to conventional capacitors, the time constant is not only dependent on the passive elements but is also a function of the initial voltage of the DLSC, \(U_{0}\) and the no-load voltage, \(E\), of the source. This result is congruent, since for this model, the capacitance of the DLSC is a function with linear dependence on its internal voltage, whereas in a conventional capacitor, the capacitance remains invariant.