## 3 Low-Power Electronic Converters
### Thin-Film Ultra-Capacitors (or Supercapacitors)  
To increase the energy density while maintaining a relatively small mass and volume in portable alternative energy sources, capacitors with high and even ultra-high capacitance are used, or ultra-capacitors, sometimes called \"supercapacitors\" [71; 72]. In comparison with parallel-plate capacitors, supercapacitors use the same principle of operation described above; however, they are characterized by a larger area that allows a greater density of charges, and they also have thinner dielectric layers. As a result, a larger capacitance value is achieved for supercapacitors compared to parallel-plate capacitors. The most commonly used supercapacitors (Figure 29) are based on electrochemical double-layer capacitors (EDLCs) [73; 74; 75; 76]. When an AC voltage is applied between the two electrodes, the ions in the bulk or on the electric double layers diffuse through the separator or separators to the oppositely charged electrode. Moreover, the electrodes are designed to limit recombination processes at the surface of the electrodes. For the supercapacitors, the specific capacitance \(C_{S}\) and the areal capacitance \(C_{A}\) can be found as follows [71]:  
\[C_{S}=C/m\,\text{ and } \tag{39}\]  
\[C_{A}=C/S_{A}, \tag{40}\]  
where \(m\) [g] is the total mass of the active material, \(C=\frac{i}{-div/dt}\) is the capacitance (\(dv/dt\) [V/s] is the slope of the discharging curve), and \(S_{A}\) [cm\({}^{2}\)] is the area of the electrodes. Currently, for the realization of thin-film supercapacitors, MXenes are widely used, which are 2D layered transition metal carbides or carbonitrides exfoliated from their parent MAX phases by selective wet chemical etching processes [77]. Furthermore, MXene  
Figure 29: Simplified structure of an electrochemical double-layer capacitor. supercapacitors as electrochemical energy storage devices ensure high power density, high charge/discharge efficiency, and long duration of the real devices [78; 79; 80]. Often, when using thin-film supercapacitors, the capacitance of a single element is not sufficient, and then up to \(n\) number of capacitors are connected in parallel, while the voltage drop across each of them remains the same. In the general case, for \(n\) number of capacitors, the total state of charge has the form  
\[Q_{\alpha q}=Q_{1}+Q_{2}+Q_{3}+\ldots+Q_{n}=C_{1}v_{C}+C_{2}v_{C}+C_{3}v_{C}+ \ldots+C_{n}v_{C}\ [C]. \tag{41}\]  
Then, the equivalent capacitance will be obtained from the individual capacitances connected in parallel, i.e.,  
\[C_{\alpha q}=\frac{Q_{\alpha q}}{v_{C}}=C_{1}+C_{2}+C_{3}+\ldots+C_{n}\ [F]. \tag{42}\]  
For comparison with the parallel-plane capacitors, an effective power density parameter is defined for the supercapacitors. This parameter is defined as the power density per unit of volume or galvanometric power density per unit of mass. The maximum value of the galvanometric power density is determined for the instantaneous power delivered at the full rated voltage \(V_{c,max}\) with an internal equivalent series resistance \(r_{s}\) specified at a certain frequency [71]:  
\[P_{\max}=\frac{v_{C}^{2}}{4r_{s}m}\ [W/k_{S}]. \tag{43}\]  
The analysis of various literature sources shows that for the purposes of the design process, considering the specific structure, the equivalent electric circuits of supercapacitors can contain RC groups (or RC branches) connected in parallel. Moreover, the individual branches, which have an active-capacitive character, have different time constants. The electrical double layers of the individual separators in the multilayer structure create an additional interfacial tension, in which the capacitance between the external electrodes depends on the potential difference. With a certain approximation, in the formula for the capacity, the component is also obtained, which depends linearly on the applied voltage.