# Parameter Identification Method of a Double-Layer Supercapacitor by Using a Real Voltage Source
## 2 Materials  
### Variable Capacitance Model  
The short-term branch, as depicted in Figure 1b, comprises a resistor, R, of a constant value, connected in series with a capacitance that varies linearly with the internal voltage, \(\mathrm{u}\), according to the following expression:  
\[\mathrm{C(u)}=\mathrm{C_{0}}+\mathrm{C_{1}(u)}=\mathrm{C_{0}}+\mathrm{k_{c}} \cdot\mathrm{u}. \tag{1}\]  
The capacitance of the SC, \(\mathrm{C(u)}\), is equivalent to the parallel association of two others. The first is a constant value, \(\mathrm{C_{0}}\), which is known as the Initial Capacitance. This is the value presented by the SC when it is completely discharged. The second one is \(\mathrm{C_{1}(u)}\), which represents the variable part and grows linearly with the internal voltage. The slope of the linear function \(\mathrm{C(u)}\), \(\mathrm{k_{c}}\), is measured in [F/V]. The rated capacitance, \(\mathrm{C_{N}}\), provided by the manufacturers, is the capacitance of the SC when it is charged up to its rated voltage, \(\mathrm{U_{N}}\), and exhibits the following value:  
\[\mathrm{C_{N}}=\mathrm{C_{0}}+\mathrm{k_{c}}\cdot\mathrm{U_{N}}. \tag{2}\]  
The capacity \(\mathrm{C(u)}\) can be expressed as a function of the rated capacity, \(\mathrm{C_{N}}\), and the rated voltage, \(\mathrm{U_{N}}\), as follows:  
\[\mathrm{C(u)}=\mathrm{C_{0}}+\frac{\mathrm{C_{N}}}{\mathrm{U_{N}}}\cdot \left(1-\frac{\mathrm{C_{0}}}{\mathrm{C_{N}}}\right)\cdot\mathrm{u}= \mathrm{C_{N}}\cdot\mathrm{k_{0}}+\frac{\mathrm{C_{N}}}{\mathrm{U_{N}}} \cdot(1-\mathrm{k_{0}})\cdot\mathrm{u}. \tag{3}\]  
The \(\mathrm{k_{0}}\) value, designated as the Normalized Initial Capacitance, is the quotient between the Initial Capacitance, \(\mathrm{C_{0}}\), and the rated capacitance, \(\mathrm{C_{N}}\). This variable typically ranges between 0.6 and 0.85 in most families of SCs [46]. In the one-branch model (Figure 1b), three parameters must be obtained: R, \(\mathrm{C_{0}}\), and \(\mathrm{k_{c}}\). Although all three components are known to vary slightly with temperature, if the SC is operating between 20 \({}^{\circ}\)C and 65 \({}^{\circ}\)C, the variations they undergo can be considered negligible [47; 48]. Significant changes in the resistance R occur only below 0 \({}^{\circ}\)C, which increases as the viscosity of the electrolyte increases (the capacitance decreases very slightly); therefore, the influence of temperature will not be considered in this study. The proposed method is valid for the determination of the cell parameters at any operating temperature. Over the brief duration of the test, the temperature of the SC will remain constant or will present small variations that are considered negligible.  
In the calculation of the variables of the first branch of the SC, it can be assumed that all the current injected during fast charging will flow into the variable capacitance of the first branch. The secondary branch (\(\mathrm{R_{2}}\), \(\mathrm{C_{2}}\)-Figure 1c) will have a negligible influence because its dynamics are much slower, given that it presents a much larger time constant (R\(\cdot\mathrm{C_{N}}<<\mathrm{R_{2}}\cdot\mathrm{C_{2}}\)). While the voltage in the first branch rises rapidly, it hardly increases in the secondary branch. Once the charging current is canceled, part of the charge stored in the first branch will be slowly redistributed to the second branch, in a new transient state, until the voltages of the capacities of both branches are equalized. It will therefore be considered that, during fast charging, the secondary branch can be neglected [49; 50; 51].