## 3 Presented Method Consideration  
The method presented in this paper involves discretely dividing the surface of a parabolic reflector to analyze radiation characteristics. While it allows simple formulas to be derived for radiation characteristic analysis, errors may occur due to discretization. It is essential to consider an appropriate division interval \(\big{(}g_{\psi},\ g_{\phi}\big{)}\) for the parabolic reflector's surface to reduce errors resulting from discretization. In this paper, the presented method was compared with the results of a radiation characteristic analysis using GRASP by TICRA to analyze the errors associated with the division intervals. The configuration of the parabolic reflector antenna for analysis is given in Table 1.  
The division intervals \(\big{(}g_{\psi},\ g_{\phi}\big{)}\) for comparative analysis varied from 0.1\(\lambda\) to 2.0\(\lambda\). Based on the parabolic reflector antenna configuration given in Table 1, Figure 7 depicts the distribution of charge point sources and the radiation characteristic comparison result (theta-cut at \(\phi=0^{\circ}\)) for division intervals \(\big{(}g_{\psi},\ g_{\phi}\big{)}\) of 0.25\(\lambda\), 0.75\(\lambda\), 1.25\(\lambda\), and 1.75\(\lambda\), respectively. Figure 6: Radiation pattern analysis results: (**a**) contour pattern; (**b**) theta-cut pattern (\(\phi=45^{\circ}\)). \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Value** \\ \hline D & 30\(\lambda\) \\ FD ratio & 0.35 \\ \(q\) (cosine-q model) & 1 \\ Feeder position & (0\(\lambda\), 0\(\lambda\), 0\(\lambda\)) \\ Feeder beam steering angle (\(\theta\), \(\phi\)) & (180\({}^{\circ}\), 0\({}^{\circ}\)) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Configuration of parabolic reflector for analysis. This paper established criteria based on errors in directivity, a \(10^{\circ}\) reference pattern, and a \(45^{\circ}\) reference pattern to facilitate a comparative analysis between the results obtained using the presented method and GRASP. The calculation formulas for each criterion are provided in Equations (18)-(20). \[D_{d}=|F_{m}-G_{m}| \tag{18}\]  
\[PE_{10}=mean\Big{(}|F(\theta)-G(\theta)|^{2}\Big{)},\ (-10^{\circ}\ \leq\ \theta\ \leq 10^{\circ}) \tag{19}\]  
\[PE_{45}=mean\Big{(}|F(\theta)-G(\theta)|^{2}\Big{)},\ (-45^{\circ}\ \leq\ \theta\ \leq 45^{\circ}) \tag{20}\]  
where \(F_{m}\) and \(G_{m}\) denote the maximum directivity corresponding to the results of each radiation pattern analysis. Additionally, \(\theta\) is the observation angle, and \(F(\theta)\) and \(G(\theta)\) represent the radiation patterns (theta-cut at \(\phi=0^{\circ}\)) analyzed using the presented method and GRASP, respectively. The overall error (\(E_{T}\)) is defined in Equation (21) based on the criteria established by Equations (18)-(20). The overall error is a weighted sum based on the importance of each criterion. \[E_{T}=10\times D_{d}+3\times PE_{10}+1\times PE_{45} \tag{21}\]  
Figure 8 shows the errors for each criterion based on Equations (18)-(20) and the overall error defined by Equation (21) for division intervals \(\big{(}g_{\psi},\ g_{\phi}\big{)}\) ranging from \(0.1\lambda\) to \(2.0\lambda\), where \(g_{\psi}\) and \(g_{\phi}\) are set to the same value. As is evident from the results in Figure 8, the division intervals \(\big{(}g_{\psi},\ g_{\phi}\big{)}\) increase sharply at values greater than \(0.75\lambda\). This issue arises from the process of discretizing the continuous effects of the reflector surface for analysis, which can be considered a form of sampling error. As a result of the analysis based on the analysis results of the presented method and GRASP for various division intervals, it is observed that in the technique presented in this paper, when \(\big{(}g_{\psi},\ g_{\phi}\big{)}\) is equal to or less than \((0.5\lambda,\ 0.5\lambda)\), the results of the two radiation pattern analyses are similar.