# Phased Array Antenna Analysis Workflow Applied to Gateways for LEO Satellite Communications
## 5 Phased Array Modelling
### Reception Phased Array Modelling with SystemVue  
The receiver was simulated using SystemVue to obtain its gain versus the number of radiating elements. The gain values specified in [31; 33; 39; 40] are the target of this section. A theoretical calculation of reception gain (\(G_{array}\)) in dB can be conducted using (16), where \(N\) is the number of radiating elements and \(G_{e}\) and \(\Gamma\) are the gain in dBi, and the reflection coefficient of each radiating element. \[G_{array}=10 log(N)+G_{e}+10 log(1-|\Gamma|^{2}) \tag{16}\]  
The schematic used to model the receiver using SystemVue is shown in Figure 17. Most of the components are the same as in the case of transmission except for the RFAMP, which models the RF frontend. A gain of 60 dB and a Noise Figure (NF) of 1.3 dB is assumed for the receiver frontend. ArrayAnt component is configured to 19 GHz. To model the receiver, the sensitivity (\(S\)) in dBm for each constellation is calculated using (17), where \(EIRP_{sat}\) is the effective isotropic radiated power of the satellite in dBm, \(L_{TM}\) is the transmission medium loss dB, \(G_{e}\) is the gain of one radiating element in dBi and \(FSPL\) is the free space path loss in dB, described by (18) (in dB), where \(\lambda\) is the wavelength, and \(R\) is the path length in m. In this case, the working frequency is set to 19 GHz in Telesat and Amazon, and 18.55 GHz in SpaceX and OneWeb. The gain of each radiating element is 6.2 dB. Table 7 shows the sensitivity calculation for each constellation. The OneWeb system requires the minimum sensitivity (\(-114\) dBm). To ensure the reliability of the system, a sensitivity of \(-120\) dBm is assumed in the simulations. \[S=EIRP_{sat}+FSPL-L_{TM}+G_{e} \tag{17}\]  
\[FSPL=10 log(\frac{\lambda}{4\pi R})^{2} \tag{18}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**LEO Constellation** & **Diameter (m)** & **Area (m2)** \\ \hline SpaceX & 1.06 & 0.88 \\ \hline Telesat & 1.56 & 1.92 \\ \hline OneWeb & 0.78 & 0.47 \\ \hline Amazon & 0.75 & 0.44 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Area used by phased array antennas in transmission. Figure 17: Reception phased array model. As in the case of transmission, due to simulation time constraints, SystemVue simulations were run approximately to 1K elements. To obtain the gain for a large number of radiating elements, the simulation data were extrapolated and fitted with (19)-(21) in dB. These equations are for circular, hexagonal, and square array configurations, and they are represented in Figure 18. For a hexagonal distribution, we obtain the lower number of radiated elements for the same gain compared to the circular and square configuration. \[Gain_{circular}=4.3188 ln(N)+4.9112 \tag{19}\]  
\[Gain_{hexagonal}=4.1123 ln(N)+9.535 \tag{20}\]  
\[Gain_{square}=4.2642 ln(N)+4.646 \tag{21}\]  
Table 8 shows the number of radiating elements for each LEO constellation using hexagonal configuration modelled using (20). All reception gains are based in FCC filings of the constellations: SpaceX's gain is based on [39], Telesat's gain is based on [40], OneWeb's gain is based on [31] and Amazon's gain is based on [33]. As expected, there is a difference between the results obtained by (16) and (20). This is because in theoretical calculations, the electromagnetic interaction between the radiating elements is not taken into account. In some cases, a huge gain is required, so a large number of elements is needed. Figure 19 shows the radiation pattern of a 625 elements hexagonal phased array for different elevation (\(\Theta\)) angles. Tables 9 and 10 shows a comparison of the area used by parabolic and phased array antennas. In this case, as the hexagonal distribution was chosen, the area of the phased array \(A_{hexagonal}\) was calculated using (22) and Figure 20, where \(S\) is the length of the side of the array and \(a\) is the apothem, corresponding to \(S_{array}\) and \(a_{array}\) in the Figure 20, respectively. \[A_{hexagonal}=\frac{6\times S\times a}{2} \tag{22}\]