## 3 Numerical Analysis Results  
Figure 2 shows the result of the numerical model to simulate the electric field in insulation air at 0.5 kV. The maximum value of the electric field, according to the air space, was formed at 1.2 \(\upmu\)s, which is similar to the applied standard lightning impulse voltage waveform. The characteristics of the electric field depend on air space between the ionizer conductor and current collector. The results reveal that the electric field strength increases with a decreasing air space. The electric field strength between the ionizer conductor and current collector (0.0005 mm and 0.001 mm air space) are higher than that of the SK-AOR380 tip part by 51% and 17%, respectively. On the other hand, as the air space increased by more than 0.0025 mm, the electric field between the ionizer conductor and the current collector decreased, and the electric field at the tip part of the SK-AOR380 lightning rod was increased. The induced high electric field region due to the charged cloud was formed between the SK-AOR380 ionizer conductor and current collector at 0.0005 mm. In addition, the induced electric field between the ionizer conductor and the current collector was 20.5 kV/m and tip region was 13.5 kV/m, respectively. This means that the corona discharge occurs between the ionizer conductor and current collector, and it could be the \begin{table}
\begin{tabular}{c c} \hline \hline
**Modelling Parts** & **Materials** \\ \hline Ionizer & Aluminum \\ \hline Main body & Copper \\ \hline Current collector & Copper \\ \hline Insulation material & Epoxy resin \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Each material for simulation model. Figure 1: Computer-aided design representation of the numerical model geometry. starting point of the corona discharge inception. From the above results, the air space is determined by the electric field strength and partial discharge inception voltage (PDIV). Figure 3 shows the electron density according to the air space. The trend of electron density is similar to electric field results as a function of time, meaning that the electron density is concentrated where the peak of the electric field is formed. The SK-AOR380 AOR380 between the ionizer conductor and the current collector creates a non-uniform electric field with a large electric field enhancement due to ionization caused by consecutive electron avalanches. Figure 4 shows negative and positive ion density distribution obtained from Equations (1) and (2) including ionization, dissociation, and recombination. The negative ion density showed a similar trend to positive ion density; the maximum value of negative and positive ion density according to the air space was formed at 1.34 \(\upmu\)s and 1.32 \(\upmu\)s with different free electron densities, respectively. The reason or the time delay is that the free electron mobility (\(1\times 10^{-4}\) m\({}^{2}\)s\({}^{-1}\)V\({}^{-1}\)) had a higher mobility than ions (\(1\times 10^{-9}\) m\({}^{2}\)s\({}^{-1}\)V\({}^{-1}\)) [10; 11], and the ion charge carriers with a relatively slow mobility remained in space and then led to the dissociation, recombination, ionization, and electron attachment in this simulation analysis. The slope of negative (\(\Delta\rho_{-}\)) and positive ion density (\(\Delta\rho_{+}\)) dramatically decreased non-linearly as a function of time with different free electron density (\(\Delta\rho_{e}\)) from the peak value to 3.5 \(\upmu\)s, as shown in Figures 3 and 4. This result is influenced by recombination and dissociation terms from Equations (1) and (2). Figure 3: Electron density as a function of time steps. Figure 2: Electric field as a function of time at 0.5 kV (solid line: air space between ionizer conductor and current collector, dash line: tip part of SK-AOR380 lightning rod). Table 3 shows the distribution of the electric field, free electron density, and negative ion density.