**4. Equivalent Circuit Model with PSIM Software**

The previous section stated that the Fourier reverse transformation could be applied only for a restricted class of mathematical expressions. Hence, simulation of the electrical circuit using appropriate software is a convenient and efficient method. For arbitrary functions, one of the most efficient software is the PSIM simulating package [42].

The equivalent circuit model was developed to simulate the periodic charge-discharge process of UC (Figure 9). The model includes a ladder of ten r–C chains and inductivity of connecting cables. This circuit can be applied for any arbitrary current input function. For example, the galvanostatic measurements can be carried out by applying the AC control signal as the input of a current source. Figure 10 shows the measured and simulated (using proposed model) voltage behavior of the symmetric double layer UC for periodic charging-discharging cycles with a constant current of ±100 A, a period of 3 s, and duty-cycle relation of 0.5. The instantaneous deviations of a voltage over its steady-state magnitude were measured. The simulation study yielded an excellent agreement between the experimental and calculated charge-discharge characteristics of UC. The accuracy of such a model lay inside 5–8% of the relative error between the calculated and experimental data.

**Figure 9.** Model of the equivalent electrical circuit for the galvanostatic measurement. This model comprises ten r–C chains and includes the inductivity of connecting cables.

**Figure 10.** The experimental and simulated characteristics of symmetric double layer UC for periodic charging and discharging cycles.
