**A. Capacitor Analysis**

Capacitor fundamentals require a clear understanding of the transition from cylindrical trimmers to the simpler model of planar capacitors. An evolution in geometry and calculation methodology has occurred. Capacitors can be analyzed more simply while retaining their fundamental properties. Figure 1 illustrates the simplicity and adaptation of the capacitor model through a visual representation of this transition.

The adaptation from cylindrical to planar models signifies a crucial advancement in capacitor analysis methodology.

Gauss's theorem plays a pivotal role in analyzing capacitors, enabling insight into the distribution of the electric field within various geometries. The electric flux Φ across a surface S is expressed through the surface integral, as denoted by Equation I:

\({\Phi }=\text{Q} {{\epsilon }}_{0}=\iint \text{E}. \text{d}\text{S}\) (I)

Moreover, the electric field *E* is calculated by the charge *Q*, free space permittivity *ε*0, relative permittivity *ε**r*, and geometric parameters such as the radius *r* and length *l* of the capacitor:

\(\text{E}=\frac{\text{Q}}{2\pi {\epsilon }_{0}{\epsilon }_{r}rl}\) (II)

The potential difference *V* between the two electrodes (*V*1 and *V*2), reflecting the radii (*R*1 and *R*2) of the capacitor, is mathematically expressed in Equations (III) to (VI):

\(V={\text{V}}_{2}-{V}_{1}=-{\int }_{{R}_{1}}^{{R}_{2}}\overrightarrow{E}d\overrightarrow{r}\) (III)

\(V={\text{V}}_{2}-{V}_{1}=-\frac{\text{Q}}{2\pi {\epsilon }_{0}{\epsilon }_{r}l}{\int }_{{R}_{1}}^{{R}_{2}}\frac{1}{r}dr\) (IV)

\(V={\text{V}}_{2}-{V}_{1}=-\frac{\text{Q}}{2\pi {\epsilon }_{0}{\epsilon }_{r}l}\text{ln}\left(\frac{{R}_{2}}{{R}_{1}}\right)\) (V)

\(C=\frac{Q}{V}=\frac{2\pi {\epsilon }_{0}{\epsilon }_{r}l}{\text{ln}\left(\frac{{R}_{2}}{{R}_{1}}\right)}\) (VI)

With the potential difference V.

In this analysis, equations (I) to (VI) are used to calculate capacitance in planar capacitors [2]. The equations explain how geometric choices and material characteristics influence capacitance values, emphasizing the crucial role geometry and material characteristics play.

Consequently, we find an expression similar to that of the planar capacitor:

2πR1: the perimeter of a cylindrical conductor 1

: the length of the cylindrical capacitor

2𝜋𝑅1𝑙 = 𝑆: the surface of the electrodes

R2 − R1 : the thickness of the dielectric of the cylindrical capacitor

This equation of capacitance can then be used for a planar capacitor:

\(C=\frac{2\pi {r²\epsilon }_{0}{\epsilon }_{r}}{e}\) (VII)

The profound understanding of geometric and material influences in capacitance emphasizes the critical role these elements play, setting the stage to investigate and address edge effects through the strategic utilization of circular electrodes in planar capacitors [2][3].

**B. Significance of Circular Electrodes in Edge Effect Mitigation**

The electrode shape plays a crucial role in mitigating edge effects in planar capacitors. Circular electrodes, as depicted in Fig. 2, demonstrate distinct advantages in reducing the edge effect by facilitating a more evenly distributed electric field [4]. This section brings together theoretical underpinnings, modeling methodologies, and core principles of capacitors, emphasizing the pivotal role of geometry in curbing edge effects. Figure 2 illustrates the profound impact of geometric shapes on field divergence within planar capacitors.

As well as enhancing our understanding of planar capacitor structures, Equation VII provides a valuable adaptation of the capacitance of cylindrical electrodes. Through Maxwell 3D simulations, we evaluated the electric field distribution within the model. A ceramic electrode with the composition MgTiCa and a dielectric constant of 15 was used with a thickness of 1mm. As depicted in Fig. 3, simulations enabled an electrostatic resolution of Maxwell's equations for a capacitor with metalized surfaces with a ceramic radius of 13mm. This allowed a comprehensive evaluation of capacitance and a visual representation of electric field distribution [6], [7].

In this configuration, the measured capacitance was 76.5 pF. An intensified electric field was observed around the electrodes and their proximity, which resulted in breakdown voltages of 35 kV. Design failure points were clearly highlighted in red, indicating areas of heightened concern.

In this study, circular electrodes play an essential role in shaping field distribution, which affects planar capacitor operational efficiency and safety margins, and sheds light on critical design and optimization considerations for practical electronic applications.

**C. Edge Effects in Planar Capacitors**

The electric potential of a generalized planar capacitor with parallel electrodes can be seen in Fig. 4. In the diagram above, electrodes 1 and 2 are connected to ground and have a 5V potential, respectively.

For an electrode surface S enclosing an inward-facing charge Q_in within a volume V, Green's divergence theorem (Ostrogradsky) is utilized to derive Maxwell-Gauss equations in the air:

\({\iint }_{S}\overrightarrow{E}\left(M\right).d\overrightarrow{S} = {\iiint }_{V}div(\overrightarrow{E})dv = {\iiint }_{V}\frac{\rho }{\epsilon } dv\) (VIII)

\({\iint }_{S}\overrightarrow{E}\left(M\right)·d\overrightarrow{S} =\frac{{Q}_{in}}{\epsilon }\) (IV)

Here, ρ represents the volumetric charge density.

**D. Divergent Electric Fields and Edge Effects**

Divergent electric fields are observed outside the electrodes, producing edge effects, as depicted in Fig. 5.

In Fig. 5, arrow indicators illustrate electric field lines perpendicular to the electrodes, diverging at the electrode ends where a concentrated charge is located.

**E. Evaluating External Field and Potential**

To assess the field and potential outside the electrodes, a symmetrical charge distribution within a planar capacitor is considered, as shown in Fig. 6.

The potential of the dipole NP is calculated as follows:

\(V\left(M\right)= -\frac{q}{4\pi {\epsilon }^{0}NM}+\frac{q}{4\pi {\epsilon }^{0}PM}\) (X)

\(V\left(M\right)=\frac{q}{4\pi {\epsilon }^{0}}\left(-\frac{1}{\sqrt{\frac{{d}^{2}}{4}+{r}^{2}+cos\left(\theta \right).d. r}}+\frac{1}{\sqrt{\frac{{d}^{2}}{4}+{r}^{2}-cos\left(\theta \right).d.r}}\right)\) (XI)

With the above equation, the potential at various points near the electrode boundaries can be calculated and displayed in Fig. 7.

This section establishes an understanding of edge effects and the corresponding electric fields and potentials in planar capacitors.