## 2 Materials and Methods  
The design of a rectangular antenna starts with determining the length and width of the patch forming the array. The effective relative permittivity for the substrate dielectric material is computed by [32]:  
\[\epsilon_{reff}=\frac{\epsilon_{r}+1}{2}+\frac{\epsilon_{r}-1}{2}\bigg{[}1+12 \frac{h}{W}\bigg{]}^{\frac{-1}{2}} \tag{1}\]  
The effective length of the patch, L\({}_{\text{eff}}\), is calculated by  
\[\Delta L=0.412h\frac{\Big{(}\epsilon_{reff}+0.3\Big{)}\Big{(}\frac{W}{h}+0.264 \Big{)}}{\Big{(}\epsilon_{reff}-0.254\Big{)}\Big{(}\frac{W}{h}+0.8\Big{)}} \tag{2}\]  
A suitable patch width that ensures low loss and therefore higher radiation efficiency is determined by  
\[\text{W}=\frac{c}{2f_{r}}\sqrt{\frac{2}{\epsilon_{r}+1}} \tag{3}\]  
The fringing fields at the edges of a microstrip patch antenna effectively extend its physical length (L) by an additional length (\(\Delta\)L) on each end. The total effective length (L\({}_{\text{eff}}\)) is therefore expressed as follows:  
\[L_{eff}=L+2\Delta L \tag{4}\]Therefore, the length of the patch is calculated as follows:  
\[\text{L}=\frac{c}{2f_{r}\sqrt{c_{reff}}}-2\Delta L \tag{5}\]  
The input resistance (R\({}_{\text{in}}\)) of the antenna at resonance frequency and at the patch edges (at y = 0) is determined by the parallel combination of the conductances of the two radiating slots. This is expressed as follows:  
\[R_{in}(0)=\frac{1}{2(G_{1}+G_{12})} \tag{6}\]  
Here, G\({}_{1}\) denotes conductance of single radiating slot in the patch, G\({}_{12}\) represents the conductance arising from the mutual coupling between the two radiating slots of the patch. These terms can be computed using the methodology outlined in [32]. \[G_{1}=\frac{1}{120\pi^{2}}\int_{0}^{\pi}\left[\frac{sin\left(\frac{k_{0}w}{2} cos\theta\right)}{cos\theta}\right]^{2}sin^{3}\theta d\theta \tag{7}\]  
\[G_{12}=\frac{1}{120\pi^{2}}\int_{0}^{\pi}\left[\frac{sin\left(\frac{k_{0}w}{2} cos\theta\right)}{cos\theta}\right]^{2}J_{0}(k_{0}L\text{sin}\,\theta)sin^{3} \theta d\theta \tag{8}\]  
Here, J\({}_{0}\) represents the first kind of Bessel function of the order zero, and k\({}_{0}\) = 2\(\pi\)/\(\lambda_{0}\) is the free-space phase constant for \(\lambda_{0}\), the free-space wavelength. Input impedance at distance \(y_{0}\) from the edge along the patch length can be expressed as follows [32]:  
\[R_{in}(y_{0})=R_{in}(0)cos^{2}\left(\frac{\pi}{L}y_{0}\right) \tag{9}\]  
Antenna directivity (\(D\)) quantifies the effectiveness of the array in concentrating radiation in a specific direction. It can be calculated using the array factor (AF) and element pattern, providing insights into the array's ability to focus radiation in the desired direction. The antenna gain is directly proportional to its directivity, which quantifies how efficiently the antenna radiates energy compared to an ideal isotropic radiator, and is proportional to its directivity. In other words, higher directivity leads to higher gain, reflecting the antenna's ability to concentrate radiation in a particular direction. The directivity of the patch antenna is calculated as follows [32]:  
\[D=\left(\frac{2\pi w}{\lambda_{0}}\right)^{2}\frac{\pi}{I_{1}} \tag{10}\]  
where  
\[\Delta L=0.412h\frac{\Big{(}\epsilon_{reff}+0.3\Big{)}\Big{(}\frac{W}{h}+0.264 \Big{)}}{\Big{(}\epsilon_{reff}-0.254\Big{)}\Big{(}\frac{W}{h}+0.8\Big{)}} \tag{2}\]0  
When designing a series-fed antenna array, several key design considerations and equations come into play. The spacing between antenna elements is crucial for achieving the desired radiation pattern. The spacing distance (d) between adjacent elements in a uniform linear array is typically set to half the guided wavelength (i.e., \(\lambda_{\text{g}}/2\)) to promote constructive interference in the major lobe direction and enable good radiation efficiency.