# Phased Array Antenna Analysis Workflow Applied to Gateways for LEO Satellite Communications
## 5 Phased Array Modelling
### Transmission Phased Array Modelling with SystemVue  
The EIRP values specified in Table 3 are the key parameters for estimating the number of transmission gateway antenna elements. A theoretical calculation of \(EIRP\) in dBW can be conducted using (10), where \(N\) is the number of elements of the array, \(G_{e}\) and \(\Gamma\) are the gain in dBi, and the reflection coefficient of each radiating element, and \(P_{in}\) is the radiofrequency (RF) power input to each patch antenna, in dBm. This equation does not consider the electromagnetic interaction between the radiating elements, so we must use Systemvue to better estimate the gateway EIRP. \[EIRP=10 log(N)+G_{e}+10 log(1-|\Gamma|^{2}+Pin-30 \tag{10}\]  
SystemVue schematic for transmission is shown in Figure 13. To simplify the calculations,it was assumed that the components have no insertion loss. A description of each component is summarized below:  
* ArrayPort: it defines the operation frequency (Freq) and the input power of the phased array (\(TxPwrIn\)) in dBm. According to (11), \(TxPwrIn\) depends on the number of array elements (\(N\)), and the power output of each patch antenna (\(P_{patch}\)) in dBm, which we have assumed to be 20 dBm. For example, for a 2x2 array, is calculated a \(TxPwrIn\) of 26 dBm. \[TxPwrIn=P_{patch}+10 logN\] (11)
* ArraySplit: it splits equally \(TxPwrIn\) power into \(N\) radiating elements. * ArrayAttn: with this weighting attenuator we can achieve low peak sidelobes to concentrate all gain in the main lobe. In this case, a Taylor weighting was used [38]. * ArrayPhase: this component is used to apply the phase shifts. * ArrayAnt: this block defines the number of radiation elements and their spatial distribution. The separation between each element (DistanceX and DistanceY) is 0.55\(\lambda\). In this component, the gain pattern of each element was imported using the.UAN file obtained from EMpro. SystemVue simulations were run by increasing the number of elements while keeping simulation time reasonable. From approximately 1K elements, the simulation time was very long. For this reason, to obtain the EIRP for a large number of radiating elements, the simulation data were extrapolated and fitted with (12)-(14) in dBW. These Equations are for circular, hexagonal and square array configurations and they are depicted in Figure 14. For a circular array, we obtain the same EIRP but with fewer radiating elements compared to the square and hexagonal configuration. In terms of saving space, the circular array is the most appropriate. Figure 13: Transmission phased array model. \[EIRP_{circular}=8.4953 ln(N)-2.0311 \tag{12}\]  
\[EIRP_{hexagonal}=8.5584 ln(N)-3.5192 \tag{13}\]  
\[EIRP_{square}=2.6718 ln(N)+23.464 \tag{14}\]  
Table 4 shows the number of radiating elements for each LEO constellation using circular configuration. An RF power input of 20 dBm is assumed for each patch antenna. Theoretical calculations (10) and those obtained through simulations (12) have been included. As expected, there is a difference between the results obtained by (10) and (12). As stated above, this is because, in theoretical calculations, the electromagnetic interaction between the radiating elements is not taken into account. Considering the logarithmic relationship between EIRP and N, Telesat needs a huge number of elements. This number can be reduced by increasing the input power of the signal applied to each patch antenna. A principle of the phased array is that it is electronically steerable by changes on feeding relative phases and focusing all power on the principal beam. To verify this principle, elevation angle (\(\Theta\)) was swept between \(-45^{\circ}\) and \(45^{\circ}\) for a circular shape-phased array with 625 elements. As observed in Figure 15, all the power is concentrated on the principal beam.