# Parameter Identification Method of a Double-Layer Supercapacitor by Using a Real Voltage Source
## 2 Materials
### Electrical Analysis of an SC Connected to an RVS  
When an SC modeled with the main branch only is connected to a real constant voltage source (\(\mathrm{E}\), \(\mathrm{R_{c}}\)), a time function, g(t), measured in volts, can be defined, as demonstrated in [31], whose expression is as follows:  
\[\mathrm{g(t)}=\frac{1}{\mathrm{k_{m}}}\cdot\mathrm{W_{0}}\big{(}\mathrm{k_{p}} \cdot\mathrm{exp}(-\mathrm{k_{n}}\cdot\mathrm{t})\big{)}. \tag{4}\]  
The function \(\mathrm{W_{0}(x)}\) is the main or upper branch of the Lambert-W function [52; 53]. The constants \(\mathrm{k_{m}}\) [\(\mathrm{V^{-1}}\)], \(\mathrm{k_{n}}\) [\(\mathrm{s^{-1}}\)], and \(\mathrm{k_{p}}\) (dimensionless) depend on the variables of the  
Figure 2: Schematic diagram of the connection between the SC and the RVS.  
main branch (C\({}_{0}\), k\({}_{c}\), and R), those of the real voltage source (E, R\({}_{c}\)), and the initial internal voltage of the SC, U\({}_{0}\), as shown in the following expressions:  
\[\mathrm{k}_{m}=\frac{2\cdot\mathrm{k}_{c}}{C_{0}+2\cdot\mathrm{k}_{c}\cdot \mathrm{E}}. \tag{5}\]  
\[\mathrm{k}_{n}=\frac{1}{(\mathrm{R}_{c}+\mathrm{R})\cdot(C_{0}+2\cdot\mathrm{ k}_{c}\cdot\mathrm{E})}. \tag{6}\]  
\[\mathrm{k}_{p}=\mathrm{k}_{m}\cdot(U_{0}-E)\cdot\exp(\mathrm{k}_{m}\cdot(U_{0 }-E)). \tag{7}\]  
In (7), \(\exp(x)\) represents the exponential function. The variable g(t), as defined in (4), presents a positive sign when the SC is discharging on the RVS (U\({}_{0}>E\)) and a negative sign otherwise (U\({}_{0}<E\)). The instantaneous values of the internal voltage, u(t), the discharge current, i(t), and the external voltage of the cell, u\({}_{\mathrm{co}}\)(t), can be obtained from the following equations, which are derived from the function g(t) [31]:  
\[\mathrm{u}(t)=E+g(t). \tag{8}\]  
\[\mathrm{i}(t)=\frac{g(t)}{R_{c}+R}. \tag{9}\]  
\[\mathrm{u}_{\mathrm{co}}(t)=E+\left(\frac{R_{c}}{R_{c}+R}\right)\cdot g(t). \tag{10}\]  
The function g(t) can be defined as the difference between the internal voltage of the SC, u(t), and the no-load voltage of the RVS, E. It is also possible to calculate the time that must elapse for the variable g(t) to reach a specific value, given an initial value g\({}_{0}=U_{0}-E\). This time, t, presents the following value:  
\[\mathrm{t}=2\cdot\mathrm{k}_{c}\cdot(R_{c}+R)\cdot\left[\left(\frac{C_{0}}{2 \cdot\mathrm{k}_{c}}+E\right)\cdot\log\left(\frac{U_{0}-E}{g}\right)+U_{0}-E- g\right]. \tag{11}\]  
In (11), log(x) represents the Neperian or natural logarithm of x. To simplify the expressions that will be part of the final solution, the following constants are defined:  
\[\mathrm{k}_{1}=\frac{C_{0}}{2\cdot\mathrm{k}_{c}}+E. \tag{12}\]  
\[\mathrm{k}_{2}=1+\frac{R}{R_{c}}. \tag{13}\]  
\[\mathrm{k}_{3}=\frac{1}{2\cdot\mathrm{k}_{c}\cdot(R_{c}+R)}. \tag{14}\]  
The first constant, k\({}_{1}\), is measured in units of voltage [V], the second, k\({}_{2}\), is dimensionless, and the third, k\({}_{3}\), is measured in [V\(\cdot\)s\({}^{-1}\)]. The functions u(g) and u\({}_{\mathrm{co}}\)(g), derived from (8) and (10), respectively, facilitate the straightforward determination of g(u) and g(u\({}_{\mathrm{co}}\)). By composing these functions with t(g), as presented in (11), the time required for the internal and external voltage to reach a specified value can be calculated, thereby obtaining the functions t(u) and t(u\({}_{\mathrm{co}}\)). The function t(u\({}_{\mathrm{co}}\)) will be employed to compute the three parameters of interest. The final expression, as a function of the constants k\({}_{1}\), k\({}_{2}\), and k\({}_{3}\), is as follows:  
\[\mathrm{t}=\frac{1}{\mathrm{k}_{3}}\cdot\left[\mathrm{k}_{1}\cdot\log\left( \frac{U_{0}-E}{k_{2}\cdot(u_{\mathrm{co}}-E)}\right)+U_{0}+\frac{R}{R_{c}} \cdot E-\mathrm{k}_{2}\cdot u_{\mathrm{co}}\right]. \tag{15}\]