## 4 Parametric Study of S-Band Antenna  
In a stacked microstrip patch antenna fed by a single point, the size of the lower and parasitic patches affects several performance parameters, including the resonant frequency, bandwidth, axial ratio (circular polarization), and radiation pattern [31]. In fact, the lower frequency is determined by the larger patch, which is usually the lower one, while the higher frequency is determined by the smaller patch, which typically corresponds to the parasitic one. Furthermore, it has been demonstrated that altering the dimensions (length and depth) and positioning of the notches in both patches of the antenna can have a substantial impact on the axial ratio bandwidth and, consequently, on the polarization characteristics of the antenna [32; 33; 34; 35]. As a result, a parametric study was conducted to ensure a suitable design was achieved. The final design parameters following optimization are presented in Table 2.  
Figure 5: S-Band QM Antenna Design: (**a**) upper parasitic patch; (**b**) lower patch; (**c**) back part with SMP connectors (**d**) S-band antenna. As demonstrated in Figure 6a, the dimensions of the lower patch (RL) within the stacked microstrip antenna have a significant impact on critical parameters such as resonant frequency, bandwidth, and impedance matching. It is observed that increasing the size of the lower patch leads to a decrease in resonant frequency and a concomitant reduction in bandwidth. Conversely, a reduction in the size of the lower patch results in an increase in resonant frequency and an enhancement in bandwidth. It is also noteworthy that the change in size can impact the antenna's efficiency due to weaker coupling between stacked patches. Conversely, an augmentation in the dimensions of the parasitic patch (RP) has been observed to result in a shift towards lower frequencies of the lower resonant frequency, thereby enhancing the impedance bandwidth through an improvement in coupling between the patches. However, this augmentation can potentially lead to a degradation in the axial ratio (circular polarization), resulting in diminished efficiency of the polarization. Moreover, an inadequate sizing of the parasitic patch can give rise to inefficient coupling and an escalation in return loss. Lastly, changes in patch size can alter the radiation pattern, potentially affecting directivity and side lobes. Therefore, the size of the lower and parasitic patches plays a crucial role in optimizing antenna performances [14; 16; 31]. The separation thickness between two stacked microstrip patches significantly influences the antenna's performance (H3 and H6) [26; 27]. Increasing the separation leads to a reduction in the coupling between the patches, resulting in a decrease in the resonance frequency and an increase in the bandwidth as the effective interaction between the patches becomes weaker. Conversely, although reducing the separation enhances the coupling, increases the resonant frequency, and improves impedance matching, the bandwidth decreases, and it can also lead to greater cross-talk (undesired coupling of signals between the two orthogonal polarizations) or undesirable radiation characteristics. Furthermore, the separation affects the overall efficiency, with optimal spacing ensuring efficient power transfer between the patches without excessive loss. It is, therefore, crucial to fine-tune the separation in order to control the antenna's radiation pattern and optimize its performance since there is always \begin{table}
\begin{tabular}{c c} \hline RP (radius\_parasitic\_patch) & 30.6 mm \\ \hline RL (radius\_lower\_patch) & 30.7 mm \\ \hline LNP (L\_notch\_parasitic) & 20 mm \\ \hline LNL (L\_notch\_low) & 18 mm \\ \hline DNP (D\_notch\_parasitic) & 5.6 mm \\ \hline DNL (L\_notch\_low) & 4.35 mm \\ \hline d\_probe & 20 mm \\ \hline Size & 80 mm \(\times\) 80 mm \\ \hline \end{tabular}
\end{table} 
Table 2: Parameters values of designed antenna. Figure 6: 1S11 as a function of patches size: (a) lower patch; (b) parasitic patch.