# Parameter Identification Method of a Double-Layer Supercapacitor by Using a Real Voltage Source
## 4 Results and Discussion
To ensure that the end user can work without encountering any issues, it is most common to indicate the rated value as the minimum obtained in the quality control process or to combine it with the typical and maximum values. When a manufacturer presents a single rated value, it signifies that the actual capacitance of the cell will be at least that value, but may be up to 20% higher, within the usual tolerances. The specifications may also appear, as in other electronic components, such as the rated value, C\({}_{\text{N}}\), together with the tolerances, e.g., (-5%, +10%). Similarly, the series resistance is typically presented by its maximum value, although the average or typical value may also be provided. Table 2 presents the tolerances for capacitance and series resistance. The value designated as the rated capacitance is highlighted in bold. The temperature at which these values were measured is typically provided by the manufacturer: 20 \({}^{\circ}\)C or 25 \({}^{\circ}\)C. Figure 6 depicts a photograph of the cells tested, some of which have their connection cables already soldered. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Manufacturer** & **C\({}_{\text{min}}\) [F]** & **C\({}_{\text{typical}}\) [F]** & **C\({}_{\text{max}}\) [F]** & **R\({}_{\text{typical}}\) [m\(\Omega\)]** & **R\({}_{\text{max}}\) [m\(\Omega\)]** & **U\({}_{\text{N}}\) [V]** \\ \hline Maxwell Technologies, San Diego, CA, USA. & **150** & 165 & 180 & 4.0 & 6.5 & 3.0 \\ Maxwell Technologies, San Diego, CA, USA. & **310** & - & - & - & 2.2 & 2.7 \\ Eaton, Cleveland, OH, USA & 380 & **400** & 440 & - & 3.2 & 2.7 \\ DGH, Knowles Corporation, Itasca, IL, USA & 540 & **600** & 780 & - & 3.5 & 2.7 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Rated values of the tested SCs. Figure 5: Full instrumentation used in the experimental tests. Figure 7 illustrates the u\({}_{\rm co}\)(t) voltage evolution, in the four cases, where the cells were connected to the RVS with E = 2.7 V. The SCs started with a zero initial voltage, except for the 600 F cell, which exhibited a residual voltage of 5 mV derived from previous tests. The source was activated 20 s after the beginning of data acquisition (19.2 s for the 400 F cell). The initial external voltage, u\({}_{\rm co1}\), is dependent on the values of the resistances of the wires connecting the cell (R\({}_{\rm w1}\), R\({}_{\rm w2}\), R\({}_{\rm w3}\), and R\({}_{\rm w4}\)). This is demonstrated in (17) and (38). Figure 8 shows the charge current, i(t), in the four cases. The resistor R\({}_{\rm lim}\) has been selected with such a value that ensures the maximum current does not exceed 4.5 A, to provide a safety margin, given that the source is not capable of supplying more than 5 A. The lower the capacitance value, the faster the cell will charge and the faster the current will decrease. Figure 6: Tested SCs (capacitances of 150 F, 310 F, 400 F, and 600 F). For the 400 F and 310 F cells, the resistance value of the connection cables (R\({}_{\rm w3}\), R\({}_{\rm w4}\)) has been considered. Figure 7: Measurement of the external voltage, u\({}_{\rm co}\)(t), for the 150 F (red), 310 F (black), 400 F (magenta), and 600 F (blue) cells. To calculate the parameters of the SCs under test, only the first 80 s of the data acquisition, measured from the moment of connection of the RVS, were taken, to avoid this way the secondary branch having time to charge. With these samples and the methods described above, the results shown in Table 3 were obtained. For the application of the 3 PM, the time instants t\({}_{1}\) = 0, t\({}_{2}\) = 40, and t\({}_{3}\) = 80 s were considered. These measurements, taken just at those instants, were not averaged with the samples close to each other, so the values obtained with the LSM method will be more accurate. Even so, the results produced by both methods are very similar.