*3.1. Mathematical Model*

The equivalent electrical circuit for the analysis of UC was developed by employing a ladder of infinite r–C chains. This approach takes into account the real physical nature of charge movement in the electrolyte. Additionally, it provides high precision of the output UC parameters, however, without the application of PDE. Figure 5 shows the schematic of the proposed infinite r–C chains based equivalent circuit model. The resistor (r) models the resistance of diffusion movement, and consequently, the ohmic loss, which is termed as a real part of equivalent impedance. The element 'C' simulates the distribution of space charge in the electrolyte and thus the capacitance of UC. For the mathematical analysis, it was assumed that the r–C chains exhibit similar impedance (Z) as the input one.

**Figure 5.** Infinite r–C chains based equivalent circuit of a capacitor (r: resistor, C: capacitor, Zi: input impedance, *Vin*: input voltage, *Iin*: input current).

In our previous work, the infinite r–C chain-based equivalent circuit of the UCs was proposed and discussed [13]. The resistance (*R*) and reactance (*X*) of this equivalent circuit can be expressed as:

$$R = \frac{1}{2} \Big(r + \sqrt{r^2 + 4X^2} \Big). \tag{1}$$

$$X = -\sqrt{\frac{-r^2 + \sqrt{r^4 + \left(\frac{4r}{a\epsilon\mathcal{C}}\right)^2}}{8}} = -\frac{r}{2\sqrt{2}}\sqrt{\sqrt{1 + \left(\frac{4}{ar\epsilon\mathcal{C}}\right)^2} - 1}.\tag{2}$$

Equations (1) and (2) represent the Fourier transform of internal impedance. If the equivalent capacitance is very large (~100–1000 F), then a relatively small inductivity (~10 nH) of connecting cables and electrodes plays a significant role in the measurement of reactance. Hereafter, the component for the inductivity of cables should be included in the equivalent circuit (Figure 6).

**Figure 6.** Modified equivalent circuit of the ultracapacitor. The component, *L*cc, represents the inductivity of connection wires.

According to the modified equivalent circuit, the total reactance (*XT*) is written as:

$$X\_T = \frac{r}{2} \left( \frac{2\omega L\_{\rm cc}}{r} - \frac{1}{\sqrt{2}} \sqrt{\sqrt{1 + \left(\frac{4}{\omega r \rm C}\right)^2}} - 1\right) \tag{3}$$

Inserting Equation (3) into Equation (1) gives the resistance R:

$$R = \frac{r}{2} \left| 1 + \sqrt{1 + \left(\frac{2\alpha L\_{\rm cc}}{r} - \frac{1}{\sqrt{2}}\sqrt{\sqrt{1 + \left(\frac{4}{\alpha r \rm C}\right)^2} - 1}\right)^2} \right| \tag{4}$$

Figure 7 demonstrates the comparison between experimental and calculated EIS data. Equations (3) and (4) are employed for the simulation, and the least-mean squares approach was applied for the theoretical fitting. The coefficient of determination (χ2) decides the criterion of the proximity between the theoretical and experimental output. This large value of χ<sup>2</sup> (~0.992) proved the accuracy of the proposed method, although this analysis was performed for the EIS data measured at a temperature of 15 ◦C, *V*UC of 2.736 V, and AC current of 1.4 A. However, this model was valid for data measured at different temperatures and voltages (Figure 3).

**Figure 7.** Simulated and measured EIS data of symmetric double-layer UC (**a**) reactance, (**b**) resistance.

The internal resistance and capacitance exhibited relatively small change (±10–15%) for the applicable voltage range (Figure 2). Hence, the constant parameter representation was considered for the calculation of equivalent circuit parameters. The proposed model should be modified accordingly, if the voltage and current parameters are altered significantly.
