## 2 Design Methodology  
In microstrip patch antennas, the size and shape of the patch determine the antenna's resonance frequency, bandwidth, return loss, gain, and radiation pattern. Radiation efficiency and resonance frequency are two of the many variables whose values are directly determined by the patch's width and length. The feasible and efficient dimensions (length and width) of a rectangular microstrip antenna can be calculated using the formulae below. Based on the transmission line model, the dimensions of a radiation patch can be calculated using the following equations [33; 34]:  
\[W=\frac{c}{2f_{r}}\sqrt{\frac{2}{\epsilon_{r}+1}} \tag{1}\]  
\[L=\frac{c}{2f_{r}\sqrt{\epsilon_{eff}+1}}-2\Delta L \tag{2}\]  
where  
\[\Delta L=0.412h\frac{\epsilon_{eff}+0.3}{\epsilon_{eff}-0.258}\left[\frac{ \frac{w}{h}+0.264}{\frac{w}{h}+0.8}\right] \tag{3}\]  
\[\epsilon_{eff}=\frac{\epsilon_{r}+1}{2}+\frac{\epsilon_{r}-1}{2}\left[1+12 \frac{h}{w}\right]^{-\frac{1}{2}} \tag{4}\]  
and feedline \(W_{0}\) is evaluated from the input impedance \(Z_{in}\):  
\[Z_{in}=\frac{120\pi}{\sqrt{\epsilon_{r}}\left[\frac{h}{w}+1.393+0.667 ln\left( \frac{h}{w}+1.444\right)\right]} \tag{5}\]  
Here, \(c\) is the speed at which light travels through a vacuum, (\(f_{r}\)) is the resonance frequency, and \(\epsilon_{r}\) is the relative permittivity (also called the dielectric constant) of the dielectric material. In real life, the fields do not just stay on the patch. The fringing field is a field component that goes beyond the patch's physical limits (\(L\times W\)). The effective patch width could be calculated using the effective dielectric constant (\(\epsilon_{ref}\)) for the width of the patch (\(W\)) while considering the influence of the fringing field effect. The effective dielectric constant (\(\epsilon_{ref}\)) is determined by the following equation:\[\epsilon_{eff}=\frac{\epsilon_{r}+1}{2}+\frac{\epsilon_{r}-1}{2}\bigg{[}1+12\frac{h} {w}\bigg{]}^{-\frac{1}{2}} \tag{6}\]  
Here, \(h\) is the thickness of dielectric substrate. To account for the influence of the fringing field given a patch length \(L\), both ends of the line must have a length added to them. \[\frac{\Delta L_{eff}}{h}=0.412h\frac{\left(\epsilon_{eff}+0.3\right)\left( \frac{w}{R}+0.264\right)}{\left(\epsilon_{eff}-0.258\right)\left(\frac{w}{R}+0.8\right)} \tag{7}\]  
The antenna has a total footprint of 28.2 \(\times\) 23.8 \(\times\) 1.6 mm\({}^{3}\) with a patch area of 28.2 \(\times\) 23.8 mm\({}^{2}\). Annealed copper is used as a patch and ground plane material and is matched to the real-world antenna construction material. FR4 is, rightly, the most commonly used material in PCB construction. Boards from FR4 are robust, water resistant, and provide sound insulation between copper layers that minimize interference and support good signal integrity. This research investigated the feasibility of using the FR-4 substrate for microstrip antennas throughout a wide frequency range (8-12 GHz). The purpose of this investigation was to examine the FR-4 substrate as a potential option for designing an X-band microstrip antenna, aiming to achieve a high degree of agreement between simulated and measured results. Due to its inexpensive cost and widespread availability, FR-4 was selected for this research because it can be utilized for prototyping microstrip antennas. A microstrip feed line with a 50 \(\Omega\) impedance was used to feed the input signal to the antenna. CST carries out the preliminary design and performance optimization. The dimensional parameters of the proposed MPA are presented in Table 1.  
The proposed MPA utilizes FR-4 Epoxy dielectric material, which has a relative permittivity of \(\epsilon_{r}=4.4\) and loss tangent of 0.007. The antenna features a rectangle-shaped patch with two inset slot cuts and a partial ground plane with a smaller area than the substrate. The parametric analysis in CST derives the optimal dimensions of insets and ground planes.