## 3 Low-Power Electronic Converters
### Conventional Parallel-Plate Capacitors  
Parallel-plate capacitors are a system of two conductive bodies--electrodes--separated by a dielectric. The two electrodes have equal magnitude charges with different signs, where \(Q_{1}=Q\) and \(Q_{2}=-Q.\) Thus, the capacitance is defined as the ratio of the stored charge \(Q\) and the voltage \(v\) between the electrodes [67]:  
\[C=\frac{Q}{v}. \tag{31}\]  
For a parallel-plate capacitor, the capacitance depends on the geometric dimensions of the electrodes and on the absolute permittivity. In principle, the structure of a parallel-plate capacitor is given in Figure 27. In it, the deformation of the electric field at the edges is neglected, and the distance between the electrodes is \(d\), the value of which is significantly smaller compared to the transverse dimensions. In this case, the electric field in the interelectrode space has a uniform character with an electric intensity of constant value (\(E=const\)), and outside the capacitor, there is no electric field. Then, for the closed surface, shown in Figure 10 with a dashed line based on [PERSON]'s law of electric intensity, the following is obtained:  
\[E=\frac{Q}{\epsilon A}, \tag{32}\]  
where \(\epsilon\) is the absolute electrical permeability of the insulating material (\(\epsilon=\varepsilon_{0}\times\varepsilon_{r}\), where \(\varepsilon_{0}\) is the vacuum permittivity and is equal to \(8.854187\ldots\times 10^{-12}\)\(F/\)m, and \(\varepsilon_{r}\) is the relative dielectric permittivity (or effective permittivity), indicating how many times the force of interaction between two charged bodies is reduced if the space between them is not a vacuum but is filled with a corresponding material), and \(A\) is the area of the electrodes. Then, the voltage \(v_{1I2}\) between the electrodes (1 and 2) is defined as the circulation of the electric intensity vector \(\overset{\rightarrow}{E}\) between the two points along the path \(l\):\[v_{1l2}=\int\limits_{1I2}^{\rightarrow}\stackrel{{\rightarrow}}{{E}}d \stackrel{{\rightarrow}}{{l}}=E\times l=\frac{Q}{\varepsilon\times A}l, \tag{33}\]  
where the designation \(1l2\) indicates that the circulation of the vector \(\stackrel{{\rightarrow}}{{E}}\) depends not only on the endpoints \(1\) and \(2\) but also on the selected circulation loop. With Formula (31), the capacitance of the parallel-plate capacitor in the presence of geometrical symmetry of both electrodes is determined as follows:  
\[C=\varepsilon_{0}\varepsilon_{r}\frac{A}{l}\left[F\right]. \tag{34}\]  
As can be seen from Formula (34), two of the parameters have a constant value, where \(\varepsilon_{0}\) is the dielectric constant in vacuum and \(\varepsilon_{r}\) is the dielectric constant depending on the material contained between the electrodes. As a result, an increase in capacity can be obtained under the condition of changing the type of material between the electrodes and increasing their area. Also, a larger value of the capacitance of the parallel-plate capacitors can be obtained by reducing the thickness of the electrolytic dielectric between the two electrodes. At a larger value of the capacitance, a larger value of the stored electrical energy \(W_{C}\) can be obtained, i.e.,  
\[W_{C}=\int v\times idt=\frac{Cv^{2}}{2}, \tag{35}\]  
where \(idt=Idt=dQ=Cdv\). The electrical power \(P\) of a capacitor is defined as the energy delivered per unit of time. In order to determine the maximum power that can be delivered to a given load \(R_{L}\), Figure 28 shows the equivalent series diagram of a parallel-plate capacitor with internal resistance \(r_{s}\), which is connected to the load. In this case, the power \(P\) of the load at a voltage resonance (\(X_{C}+X_{L}=0\)) has the form [25]  
\[P=R_{L}i=\frac{R_{L}v_{C}^{2}}{\left(r_{s}+R_{L}\right)^{2}}. \tag{36}\]  
Figure 28: Equivalent series circuit of a parallel-plate capacitor with a given active load \(R_{L}\). Figure 27: Simplified structure of a parallel-plate capacitor.