# Phased Array Antenna Analysis Workflow Applied to Gateways for LEO Satellite Communications
## 2 Array Theory  
A phased array antenna is a computer-controlled antenna array that produces a beam of radio waves that can be electrically directed in various directions without moving the antennas [9]. The working principle is based on phase shifting and coherent combining signals from different antennas in a process called beamforming, as shown in Figure 2.  
In the case of a ground station antenna, a two-dimensional phased array is chosen. These alter pointing angles in terms of elevation \(\theta\), or scanning angle, and azimuth \(\phi\), or beam direction. Phase differences are defined by (1) and (2), where \(\Delta\Phi x\) is the phase difference between the array \(x\)-axis elements, \(\Delta\Phi y\) is the phase difference between \(y\)-axis elements, \(\lambda\) is the wavelength, \(d_{x}\) is the distance between \(x\)-axis elements and \(d_{y}\) is the distance between \(y\)-axis elements [10].  
\[\Delta\Phi x=\frac{2\pi}{\lambda}d_{x}sent\theta cos\phi \tag{1}\]  
\[\Delta\Phi y=\frac{2\pi}{\lambda}d_{y}sent\theta \tag{2}\]  
For this type of application, antenna arrays are generally composed of patch antennas due to their low profile, low cost, light weight and ease of connection to solid state devices. They consist of a thin metallic layer bonded to a grounded dielectric substrate and can have any geometric shape such as rectangles or circles [11].  
Phased array antennas consist of a large number of radiating elements, in some cases on the order of thousands of elements, and can be distributed in different geometrical  
Figure 2: One-dimensional phased array diagram.  
shapes, such as square, circular or hexagonal (see Figure 3). The geometrical distribution of the patch antennas will determine the radiation pattern of the whole array.  
To design the antenna, some key parameters should be tackled. First is the radiation pattern, which is the evaluation of the far-field amplitude for all possible angular positions of the antenna, and is normalized to the maximum value [9]. Second is the matching bandwidth of the antenna when connected to the transmitter or receiver, which is measured by the input impedance or reflection coefficient [9]. This parameter depends on how the antenna is fed. As shown in Figure 4, there are two methods to feed the patch antenna:  
1. Through a microstrip line (see Figure 4a): This method is used when the circuit that feeds the patch antenna is in the same layer as the antenna. The dimensions of the microstrip line that feeds the antenna are \(W_{f}\) and \(Y_{f}\).
2. Through a point inside the patch antenna itself (see Figure 4b): This method is used when the circuit that feeds the patch antenna is in a different layer as the antenna. This point corresponds to (\(X_{f}\), \(Y_{f}\)).  
The last parameter that is analysed is the gain, which is the maximum of the gain function. The gain function \(G(\theta,\varphi)\) is the ratio between the power radiated by the antenna into a direction \(P(\theta,\varphi)\) and the total accepted power in the same direction by an isotropic radiator \(Pin/4\pi\)[9], as observed in (3). This function does not consider the impedance mismatch on the antenna terminals [9].  
\[G(\theta,\varphi)=\frac{P(\theta,\varphi)}{Pin/4\pi} \tag{3}\]  
Figure 4: Methods to feed patch antennas: (**a**) using microstrip, (**b**) using feeding point.  
Figure 3: (**a**) Square. (**b**) Circular. (**c**) Hexagonal distribution for phased array antenna design.  
In the case of the gateway phased array with thousands of patch antennas, due to size restrictions and the high amount of associated circuitry which is placed in a different layer than the patch antenna, the feed of the patch antenna is usually made by a point inside the patch antenna itself. The use of circular polarization would complicate its implementation, adding an extra \(\lambda/4\) phase shift. This is the reason why this study has been carried out with linear polarization and feeding through a point inside the patch antenna itself.