## 2 Materials and Methods  
As per the transmission-line model, a microstrip antenna is conceptualized with two radiating slots separated by a low-impedance transmission line characterized by dimensions such as length (L), impedance (Z\({}_{\text{o}}\)), width (W), and the height of the antenna substrate (h), ensuring that the length over height ratio (L/h) is significantly greater than 1. The radiated fields extend beyond the physical edges of the rectangular patch, enhancing the antenna's overall performance. The fringing effect, which extends beyond the patch's boundaries, is influenced by the patch size, substrate height, and dielectric constant. Although larger L/h ratios can reduce the impact of fringing, electromagnetic fields still induce slight deviations in the antenna's resonant frequency (f\({}_{\text{r}}\)). The design process commences with determining the dimensions of the patch array. The subsequent set of equations elucidates this calculation. The effective relative permittivity for the substrate can be calculated as per the following formula [32]:  
\[\epsilon_{\text{reff}}=\frac{\epsilon_{\text{r}}+1}{2}+\frac{\epsilon_{\text{ r}}-1}{2}\bigg{[}1+12\frac{\text{h}}{\text{W}}\bigg{]}^{\frac{-1}{2}} \tag{1}\]  
The patch effective length, denoted as L\({}_{\text{eff}}\), is determined by:  
\[\Delta\text{L}=0.412\text{h}\frac{(\epsilon_{\text{reff}}+0.3)\Big{(}\frac{ \text{W}}{\text{h}}+0.264\Big{)}}{(\epsilon_{\text{reff}}-0.254)\Big{(}\frac{ \text{W}}{\text{h}}+0.8\Big{)}} \tag{2}\]The width of a microstrip patch antenna is related to its resonant frequency and the effective dielectric constant of the substrate material. The relationship is derived from the antenna's design and is crucial for determining its operating frequency. The patch width, W, is given by the formula:  
\[\mathrm{W}=\frac{\mathrm{c}}{2\mathrm{f}_{\mathrm{r}}}\sqrt{\frac{2}{\epsilon_{ \mathrm{r}}+1}} \tag{3}\]  
Due to the field fringing effect, the patch length is extended by \(\mathrm{\Delta L}\) on both sides. The effective length \(\mathrm{L_{eff}}\) is given by:  
\[\mathrm{L_{eff}}=\mathrm{L}+2\mathrm{\Delta L} \tag{4}\]  
where L represents the original length of the patch. Thus, the patch length can be calculated as:  
\[\mathrm{L}=\frac{\mathrm{c}}{2\mathrm{f}_{\mathrm{r}}\sqrt{\epsilon_{\mathrm{ ref}}}}-2\mathrm{\Delta L} \tag{5}\]  
At resonance, the input resistance at the edge of the patch (y = 0) can be accurately characterized by the expression:  
\[\mathrm{R_{in}}(0)=\frac{1}{2(\mathrm{G_{1}}+\mathrm{G_{12}})} \tag{6}\]  
where \(\mathrm{G_{1}}\) represents the conductance of a single slot of a rectangular patch that can also be obtained by using the field expression derived by the cavity model, and \(\mathrm{G_{12}}\) accounts for the conductance induced by mutual effects between the two radiating slots of the patch. The conductance terms \(\mathrm{G_{1}}\) and \(\mathrm{G_{12}}\) play crucial roles in determining the overall performance and efficiency of the microstrip patch antenna system, and they can be calculated using:  
\[\mathrm{G_{1}}=\frac{1}{120\pi^{2}}\int\limits_{0}^{\pi}\left[\frac{\sin\left( \frac{\mathrm{k_{0}W}}{2}\cos\theta\right)}{\cos\theta}\right]^{2}\sin^{3} \theta\ \mathrm{d}\theta \tag{7}\]  
\[\mathrm{G_{12}}=\frac{1}{120\pi^{2}}\int\limits_{0}^{\pi}\left[\frac{\sin\left( \frac{\mathrm{k_{0}W}}{2}\cos\theta\right)}{\cos\theta}\right]^{2}\mathrm{J_{0 }}(\mathrm{K_{0}L\sin\theta})\sin^{3}\theta\mathrm{d}\theta \tag{8}\]  
where \(\mathrm{J_{0}}\) is the Bessel function of the first kind of order zero. \(\mathrm{K_{0}}\) is free-space phase constant that is equal to \(2\pi/\lambda_{0}\), where \(\lambda_{0}\) is free-space wavelength.