## 2 Parabolic Antenna Analysis Method
### Parabolic Reflector Surface Division  
The procedure of parabolic reflector surface division is as follows, where, for the configuration of a parabolic antenna, the focal point is established as the origin of the coordinate system and the aperture plane is defined in the x-y plane:  
1. Considering the x-z plane as a reference, the parabolic curve is uniformly divided at equidistant intervals, and the calculation of \(\psi_{i}\) is based on this division.
2. Considering each \(\psi_{i}\) as a reference for the uniform division of the parabolic surface's circumference, the calculation of \(\phi_{S}^{i}\) is based on circumference division.  
Figure 1 shows a conceptual diagram of the procedure for parabolic reflector surface division.  
The Cartesian coordinates \((x,\,z)\) corresponding to \(\psi_{i}\) for step #1 of the parabolic reflector surface division procedure are given by Equation (1). This is derived based on the formula in [30] for analyzing the structure of a typical parabolic reflector antenna.  
\[(x_{i},\,z_{i})=\left(\frac{F}{\cos^{2}(\psi_{i}/2)}\sin(\psi_{i})\,\ \ \ \ \ -\frac{F}{\cos^{2}(\psi_{i}/2)}\cos(\psi_{i})\right) \tag{1}\]  
where \(F\) is the focal length of the parabolic reflector.  
Let the position vector of the \(i\)-th division point be denoted \(\overrightarrow{r_{i}}\), according to Equation (1); assuming a required uniform division interval of \(g_{\psi}\), \(\psi_{i}\) can be determined by satisfying the condition given in Equation (2).  
\[\left|\overrightarrow{r_{i}}-\overrightarrow{r_{i+1}}\right|\approx\ g_{\psi} \tag{2}\]  
where the value \(\psi_{1}\) required to determine the position vector \(\overrightarrow{r_{1}}\) is given by the maximum angle for the parabolic reflector configuration, which is 2 _arctan_ (0.25 \(D/F\)), in which \(D\) is the diameter of the parabolic aperture. As \(i\) increases, \(\psi\) gradually decreases until it reaches \(0^{\circ}\).  
Step #2 of the parabolic division procedure involves dividing a circumference on the parabolic surface at equidistant intervals, determined by \(\psi_{i}\). When the required interval is \(g_{\phi}\), the value of \(\phi_{s}^{\ \ i}\) resulting from equidistant division with respect to \(\psi_{i}\) can be obtained using Equations (3) and (4).  
\[\phi_{s}^{\ \ i}=2\pi x_{i}\ /\ M^{i} \tag{3}\]  
\[M^{i}=round\left(2\pi x_{i}/g_{\phi}\right) \tag{4}\]  
where \(x_{i}\) is calculated with Equation (1), and \(round(\ )\) represents the rounding function.  
In this paper, a parabolic reflector surface is divided and the resulting points are replaced with charge point sources for the purpose of analyzing the radiation characteristics of a parabolic antenna. Therefore, the positions of each charge point source resulting from the parabolic reflector surface division procedure can be defined through the following equations:  
\[P_{(i,j)}^{x}=R_{i}\sin(\pi-\psi_{i})\ \cos\left(\phi_{s}^{\ \ i}j\right) \tag{5}\]  
\[P_{(i,j)}^{y}=R_{i}\sin(\pi-\psi_{i})\ \sin\left(\phi_{s}^{\ \ i}j\right) \tag{6}\]  
\[P_{(i,j)}^{z}=R_{i}\cos(\pi-\psi_{i}) \tag{7}\]  
where \(j\) is an integer satisfying \(0\leq j\leq M^{i}\).  
Figure 1: Parabolic reflector surface division: **(a)** step #1; **(b)** step #2.  
The positions of the point sources according to Equations (5)-(7) form a ragged matrix. For the convenience of analysis, flattening the matrix yields the following equations:  
\[P_{n}^{x}=R_{n}\sin(\theta_{n})\ \cos(\phi_{n}) \tag{8}\]  
\[P_{n}^{y}=R_{n}\sin(\theta_{n})\ \sin(\phi_{n}) \tag{9}\]  
\[P_{n}^{z}=R_{n}\cos(\theta_{n}) \tag{10}\]  
where \(n\) is a natural number satisfying \(1\ \leq n\ \leq N\), in which \(N\) is equal to \(\sum M^{i}\).  
Figure 2 shows the results of the presented parabolic reflector surface division procedure based on \(\left(g_{\phi},\ g_{\phi}\right)=\left(0.5\lambda,\,0.5\lambda\right)\) and \(\left(1.0\lambda,\,1.0\lambda\right)\). Here, D is set to \(30\lambda\), and the FD ratio \(\left(F/D\right)\) is set to \(0.35\).