# Parameter Identification Method of a Double-Layer Supercapacitor by Using a Real Voltage Source
## 4 Results and Discussion  
For the experimental tests, a Keithley(tm) 2440 SourceMeter (Keithley Instruments, Inc., Cleveland, OH, USA), configured as an ideal voltage source, was used, with an output voltage of 2.7 V set [54]. This piece of equipment can establish voltages at its external terminals between 5 uV and 42 V, with a measurement resolution of 1 uV. The connection cables between the source and the SC have an electrical resistance whose values are R\({}_{\text{w1}}\) (positive) and R\({}_{\text{w2}}\) (negative). The internal resistance of the SC will be denoted by R\({}_{0}\) (Figure 4). As the source can supply a maximum current of 5 A, a series resistor, R\({}_{\rm lim}\), of 496.7 m\(\Omega\) was connected to limit the current set when the source is turned on. The resistances of the wires leading from the R\({}_{\rm lim}\) resistor to the SC were also measured; their values are R\({}_{\rm w3}\) (positive) and R\({}_{\rm w4}\) (negative). Given that the wires and connectors are not identical, the resistances R\({}_{\rm wj}\) (j = 1, 2, 3, 4) are different. To achieve the highest precision in measuring these low resistances, a 4-wire connection was employed. The values obtained are presented in Table 1, where the initial internal voltage of each cell, U\({}_{0}\), is also shown. In consideration of the diagram depicted in Figure 2, the resistance R\({}_{\rm c}\) used in the theoretical formulas described in Section 3 is equal to:  
\[\rm R_{c}=R_{w1}+R_{w2}+R_{lim}. \tag{38}\]  
When the estimation methods described in Section 3 are applied, using the external voltage u\({}_{\rm co}\)(t) (Figure 4), the value of the obtained resistance, R, can be expressed as the sum of the internal resistance of the SC, R\({}_{0}\), and the resistances of the cables, R\({}_{\rm w3}\) and R\({}_{\rm w4}\). Subtracting those two latter from the value of R allows R\({}_{0}\) to be obtained. The voltage u\({}_{\rm co}\)(t) is measured by connecting its positive terminal to the analog input AII of a DA-NI-USB 6343\({}^{\rm TM}\) data acquisition board, which has a resolution of 16 bits, and the negative terminal to its GND port. The resolution of the voltage measurement with the analog inputs is 0.152 mV and its accuracy is 1.52 mV [55]. The voltage u\({}_{\rm a}\)(t) was measured at the analog input AII0 (Figure 4). The two analog inputs were configured in single-ended mode with an input range from \(-\)5 V to +5 V. The charging current, i(t), was obtained with the difference u\({}_{\rm a}\)(t) \(-\) u\({}_{\rm co}\)(t) divided by the resistance R\({}_{\rm lim}\). The chosen sampling period was 1 ms. The full instrumentation setup is shown in Figure 5. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**C\({}_{\rm N}\) [F]** & **R\({}_{w1}\) [m\(\Omega\)]** & **R\({}_{w2}\) [m\(\Omega\)]** & **R\({}_{w3}\) [m\(\Omega\)]** & **R\({}_{w4}\) [m\(\Omega\)]** & **U\({}_{0}\) [mV]** \\ \hline
150 & 59.30 & 59.90 & 0 & 0 & 0 \\
310 & 45.78 & 66.22 & 2.23 & 2.45 & 0 \\
400 & 54.25 & 57.08 & 6.08 & 6.37 & 0 \\
600 & 58.32 & 58.53 & 0 & 0 & 5 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Values of the wiring resistances in the tested SCs and initial voltage of the cells. Figure 4: Connection of the measurement instruments. Table 2 presents the most significant data for the SCs under consideration. These include the manufacturer, the rated capacitance, the rated resistance, and the rated voltage. Regarding the values in the datasheets provided by the manufacturers, it is necessary to clarify certain aspects. Firstly, the resistance and capacitance are obtained by assuming that the SC behaves as a constant capacitance RC circuit (Figure 1a), for which standardized IEC 62391-1 [56] and IEC 62391-2 [57] tests are used, without considering the leakage resistance R\({}_{\text{L}}\). Furthermore, it should be noted that there are certain unavoidable tolerances, inherent to the manufacturing process, which may result in discrepancies between the rated capacitance provided by the manufacturer and the real value measured.