# 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  
### Phased Antenna Arrays for LEO Satellite Communications  
Generally, the rectangular planar array radiation pattern can be described by the below Equations (1)-(9) [29]. We will consider a rectangular matrix, having M elements on the Ox axis and N elements on the Oy axis (M \(\times\) N matrix), as represented in Figure 1. The distances between the elements on the Ox axis and those on the Oy were denoted by dx and dy, respectively. Thus, the coordinate of an element of the planar array is (m, n), and its position is indicated by the position vector r\({}_{\text{mn}}\), where 1 \(\leq\) m \(\leq\) M and 1 \(\leq\) n \(\leq\) N. Additionally, since most radiation patterns are represented and studied in the far field (FF), it can be considered that an electromagnetic wave that reaches the array is a plane wave, and its wavefront is perpendicular to its direction of travel, given by R in Figure 1.  
The direction of the wavefront was considered to be from the elements far from the origin of the coordinate system, towards the element considered as the reference and having (1, 1) coordinate. Without introducing additional phase shifts besides the natural ones due to the displacement of the electromagnetic wavefront, it can be deduced that between the array elements, the path traveled by the plane wave between an element of coordinates (m, n) and the element of coordinates (1, 1) chosen as reference, is given by (1), and the phase difference between the element situated in (m, n) position, and the element (1, 1) is given by (2), where \(k_{0}=2\pi/\lambda_{0}\) represents the free space wave number. Thus, the expression of the radiation pattern of a planar antenna array as a function of the elevation angle \(\theta\) and  
Figure 1: Schematic representation of a rectangular planar antenna array having (M \(\times\) N) elements. the azimuth angle \(\varphi\) can be written as in (3) [29]. Equation (4) results from substituting the value of \(\alpha_{mn}\) from (2) into (3). We noted with \(a_{mn}\) the amplitude of the signal at the supply point of the element (m, n). Since we did not intend to work on the amplitudes and, at the same time, keep this parameter constant for each element and normalized to 1 (in theory, the term used is non-weighted array elements), we assume that \(a_{mn}=1\) and (4) changes to Equation (5). It was assumed that the elements were identical, and the effects of the mutual coupling between them were neglected. Equation (5) can also be written as a product of two factors, as can be seen in (6), where \(F_{A}\) is called the Array Factor and is a function of the geometry of the antenna array and the phase of the excitation signals of the radiating elements, in this case, given the normalization to 1 of the elements' signals amplitudes. According to (6), by varying the distances and phase shifts between the array elements, the \(F_{A}\) characteristics and, thereby, the total field of the array, respectively, the radiation pattern, can be controlled. In theory, related to the general case represented in Figure 1, given the sense of the wavefront, which is from the other elements towards the reference element (1, 1), it can be shown that if a phase taper consisting of appropriate values (denoted with \(\beta_{mn}\) in Equation (7)) is applied over the elements, the uniform phased planar antenna array should be able to steer its beam to the desired direction \((\theta_{0},\ \varphi_{0})\).