*3.2. Representation of Dynamic UC Parameters Using Reverse Fourier Transform*

The equivalent circuit and obtained parameters (r and C) can be utilized to determine the voltage and current parameters of UC for different working conditions. For this purpose, a reverse Fourier transformation was applied. Let us consider that current (*Iin*) is applied as input for the equivalent circuit (Figure 5). Using nodal analysis, the voltage (*V*1) is expressed as:

$$V\_1(j\omega \mathcal{C} + \frac{1}{Z}) = I\_{\text{in}} \Rightarrow V\_1 = \frac{I\_{\text{in}}}{j\omega \mathcal{C} + \frac{1}{Z}}\tag{5}$$

where *V*<sup>1</sup> and *Z* are assumed as complex variables. The solution of the following Fourier integral is needed to restore the capacitor voltage [41]:

$$V\_1(t) = \frac{2}{\pi} \int\_0^\infty \frac{R(\omega)\sin(\omega t)}{\omega} d\omega \tag{6}$$

where *R*(ω) is the real part of the Fourier transform of *V*<sup>1</sup> (Equation (4)). The input voltage (*Vin*) can be calculated:

$$V\_{in} = V\_1 + \Delta V\_r = V\_1 + I\_{in}r\tag{7}$$

where Δ*Vr* = *Iinr* is the voltage drop at the resistance *r*. The numerical approaches can only solve the integral (6) because of its irrational form. First, the real part of the impedance is obtained using the specific values of r and C following the expression (4). Second, the real part of Z is substituted to the integral (6), which is numerically solved for the required series of time (t0–tmax) and time resolution (Δt). Using the proposed model and EIS data, the equivalent circuit parameters can be calculated as *r* = 0.1 mΩ, C = 800 F. The voltage of UC for charge-discharge processes was simulated for the constant input current of 20 A. The simulated and measured dynamic characteristics of UC for both charge-discharge cycles are demonstrated in Figure 8. The numerical analysis of Fourier integral provides the expected output for charging-discharging processes. The theoretical analysis was consistent with the experimental one, which approves the transverse Fourier transformation for the restoration of UC dynamic behavior.

**Figure 8.** Representation of dynamic behavior of UC voltage using reverse Fourier transform (**a**) charging process, (**b**) discharging process. The UC characteristics are measured at constant input current of 20 A and temperature of 30 ◦C.

The restoration of parameters can also be achieved by the integral given below:

$$V(t) = \frac{1}{2\pi} \int\_{-\infty}^{\infty} F(\omega) e^{j\omega t} d\omega \tag{8}$$

where *F*(ω) is the Fourier transform of the input current and can be written as:

$$F(\omega) = \int\_{-\infty}^{\infty} V(t)e^{-j\omega t}dt\tag{9}$$

Generally, *F*(ω) is a complex variable and contains real and imaginary parts. The rigorous finite analytical representation of the Fourier transform exists for several functions (e.g., step- and pulse-function). However, an analytical Fourier formulation could not be presented for each mathematical expression.
