## 3 Experimental Setup
### Finite Element Modeling (FEM)  
It is difficult to accurately measure the electric field at every point between the electrodes. Additional challenges should be solved since the electrodes are plunged into pressurized gas in a test chamber. A more practical and economical method of carrying out  
Figure 4: HVDC test setup. Figure 3: HVAC test setup. the measurements is using numerical simulation techniques with computer-aided software. This enables users to skip expensive, intricate, trial-and-error laboratory procedures that are pretty challenging. The electric field utilization factor and field distribution modeling is carried out in this research using the finite element modeling software COMSOL Multiphysics. In FEM software, the geometry of the configuration is constructed, and boundary value conditions are specified [25]. The distribution of the electric field was simulated using the electrostatics module. The analysis necessitates the identification of electrostatic boundary conditions and material properties. An essential element for the simulation is the relative permittivity \(\varepsilon_{\mathrm{r}}\); the \(\varepsilon_{\mathrm{r}}\) values of plexiglass and aluminum are 3.4 and 1, respectively, and the gas permittivity value is also selected as 1. The 0 V voltage is at the ground electrode, while the high-voltage electrode is energized with a voltage of 10 V. The bounding box that reflects the real size of the gas chamber is used to symbolize it with dimensions of 140 mm \(\times\) 80 mm. Similar to the practical experiments, the air gap between electrodes is fixed at 6 mm. The electric field distribution at each point is precisely calculated with the meshing size set to a user-defined extra fine. The design meshing consists of 15,235 domain elements and 384 boundary elements. The electric field potential for uniform fields configured by plane-plane electrode configurations is shown in Figure 5. It can be seen that the potential is highest at the HV electrode surface and decreases gradually until 0 V at the ground electrode. When the boundary conditions are satisfied, calculating the electrostatic field requires [PERSON]'s and [PERSON]'s equations. The equations are stated as  
\[E=-\
abla V \tag{1}\]  
\[\
abla V=-\frac{\rho}{\epsilon_{0}} \tag{2}\]  
Here \(E\) represents the electric field, the applied voltage is represented by \(V\), \(\rho\) shows space charge density, and \(\epsilon_{0}\) represents the dielectric permittivity of free space. The law of charge conservation must be obeyed by the electric field strength:  
\[\
abla.j=0 \tag{3}\]  
\[j=\rho.v=\rho.\mu.E \tag{4}\]  
Figure 5: Electric potential simulation of electrodes in a uniform field. Here \(v\) represents the drift velocity and \(\mu\) represents the ion mobility across electrode gap distance, by the following equation:  
\[\
abla\Big{\{}\Big{(}\
abla^{2}V\Big{)}(\
abla V)\Big{\}}=0 \tag{5}\]  
The FEA software ascertains the behavior of the electric field. The \(E_{max}\) and field utilization factor \(\eta\) for each electrode layout were assessed using the breakdown information from the experimental testing. For the sphere-plane and plane-plane configurations, Figures 6 and 7 illustrate that the \(E_{max}\) occurs in a location of extremely high stress, which measures 1.7475 V/mm on a sphere-plane system and 2.24986 V/mm on a plane-plane system. FEA simulation aims to gather more information about the electric field distribution at every step, which is impossible to observe during practical experiments. As seen in Figures 6 and 7, the sphere-sphere electrode configuration reveals a higher stress point at the center of the electrodes, and the plane-plane electrode configuration shows the highest electric field point at the side edge of the electrode. The field utilization factor \(\eta\) is calculated using Equation (6).