# The Design and Implementation of a Phased Antenna Array System for LEO Satellite Communications
## 3 Theoretical Considerations on Phased Antenna Arrays for Leo Satellite Communications
### The Phase Shifting Algorithm  
After many simulations, it was concluded that for the calculation of the phase shifts that must be introduced to each element of the Phased Antenna Array System to orient the beam in the desired direction in the azimuthal plane, it was considered necessary to establish an optimal constant elevation angle (\(\theta_{0}\)), so that functions which theoretically describe these phase shifts to be dependent only on the azimuth angle, \(\varphi_{0}\). The developed numerical evaluations demonstrated that the most accurate orientation towards the desired angle is achieved for the elevation angle kept at the value of 58.5\({}^{\circ}\), for which the antenna array has a maximum gain of approximative 8 dBi without introducing any phase shifts between its elements. In this case, the directivity characteristic is almost omnidirectional, and as will be shown, the gain is reduced when compared to the scenario where the maximum is oriented according to the angle \(\varphi_{0}\). It was found that decreasing the value of \(\theta_{0}\) (lower than 58.5\({}^{\circ}\)) leads to a decrease in gain and an increase in HPBW (Half Power Beamwidth). If the value is increased, the shape of the radiation pattern is distorted by increasing the magnitude of the secondary radiation lobes. Figure 3 shows simulation results corresponding to the topology depicted in Figure 2b, using a negative phase taper (plotted in black in Figure 3) as in (7), positive phase taper (plotted in red in Figure 3) as in (9) and maintaining the same absolute values, corresponding to \(\varphi_{0}=0^{\circ}\) (a) and \(\varphi_{0}=45^{\circ}\) (b). Figure 2: Schematic representation of a square uniform planar antenna array having (2 \(\times\) 2) elements. Direction of the wavefront was considered to be from elements distant from the origin, towards the element considered as the reference (**a**), respectively from the reference element (1, 1) to the other distant elements (**b**). Adapting Equation (9) for the particular case of a phased square uniform planar antenna array, we arrived at the expressions of the phase shifts that must be applied to each element. If we denote by w1, w2, w3, and w4, the phase shifts introduced to the elements with the coordinates (1, 1), (2, 1), (2, 2), respectively (1, 2), according to Figure 2b, for the orientation of the main lobes on the direction \(\varphi_{0}\), the corresponding values are given by:  
\[\text{w1}=0 \tag{10}\]  
\[\text{w2}=\pi\sin(\theta_{0})\text{cos}(\varphi_{0}) \tag{11}\]  
\[\text{w3}=\pi\sin(\theta_{0})\text{[cos}(\varphi_{0})+\sin(\varphi_{0})] \tag{12}\]  
\[\text{w4}=\pi\sin(\theta_{0})\text{sin}(\varphi_{0}) \tag{13}\]  
Additionally, after calculating the constant \(\pi\) sin(\(\theta_{0}\)), in which \(\theta_{0}\) was 58.5\({}^{\circ}\) the resulting value was rounded to 154 and was applied for any beam steering angle \(\varphi_{0}\), in the azimuthal plane. The equations shown in (10)-(13) will be used for programming the phased antenna array for different orientation angles of the main lobe. Referring to the element with coordinates (1, 1), the phase shift introduced to its feeding signal was maintained at w1 = 0 regardless of the main lobe steering angle of the phased antenna array because it was set as the reference element. The orientation angle of the main lobe depends on these values (w1, w2, w3, w4), which basically represent the phase shifts that must be introduced to the feeding signals of each element in order to achieve the orientation in the desired direction (\(\varphi_{0}\)). This orientation will be determined relative to the position of the LEO satellite at a given time, provided that its azimuth angle from the observer's ground location is known at that time.