## 2 Antenna Array Design
### Modeling of COTS GNSS Antenna  
The first step after picking the antenna was to draw the antenna in CST using both the 3D CAD model and the technical drawing available online. The antenna is composed of five main structures (Figure 2). The antenna's position in the ground plane was chosen to be the best location specified in the datasheet by the manufacturer and corresponds to the center of the 70 \(\times\) 70 \(\times\) 1.5 mm\({}^{3}\) ground plane. To simulate real testing, a coaxial connector was also added, and the feed port was placed at the end of the connector. The Z-axis of the coordinate system (\(\theta\) = 0\({}^{\circ}\)) is perpendicular to the patch's surface. The dimensions and materials available in the datasheet/CAD were used. Table 1 summarizes the antenna parameters, their values, and assigned CST materials. In the case of the dielectric, the manufacturer only specifies that it is a ceramic. Therefore, a sweep of the dielectric constant, \(\epsilon_{\tau}\), was performed to match both the S\({}_{11}\) and axial ratio presented in the datasheet. Figure 3 shows the results obtained with \(\epsilon_{\tau}\) = 20.65, which agree with the results in the antenna's datasheet [20]. Therefore, this was the value adopted for the antenna array simulations. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Figure 2** & **Parameter** & **Value (mm)** & **Material** \\ \hline \multirow{4}{*}{B} & Px & 20 & \multirow{4}{*}{Silver} \\  & Py & 20 & \\  & Ph & 0.01 & \\  & cut & 2 & \\ \hline \multirow{4}{*}{C} & Gx & 70 & \multirow{4}{*}{Copper (annealed)} \\  & Gy & 70 & \\  & Gh & 1.5 & \\ \hline \multirow{4}{*}{D} & Fr & 0.4 & \multirow{4}{*}{Copper (annealed)} \\  & Fl & 7 & \\ \cline{1-1}  & Frs & 2.6 & \\ \cline{1-1}  & Frl & 3.98 & \\ \cline{1-1}  & Frh & 0.69 & \\ \hline \multirow{4}{*}{E} & Sx & 25 & \multirow{4}{*}{C ceramic} \\  & Sy & 25 & \\ \cline{1-1}  & Sh & 4 & \(\epsilon\tau\) = 20.65 \\ \cline{1-1}  & Sd & 0.4 & \\ \hline \multirow{4}{*}{F} & Gix & 24.6 & \multirow{4}{*}{Copper (annealed)} \\  & Gih & 0.2 & \\ \cline{1-1}  & Gid & 4 & \\ \hline \multirow{4}{*}{G} & Cin & 0.64 & Gold \\ \cline{1-1}  & Cis & 2.13 & Vaccum \\ \cline{1-1}  & Co & 2.64 & \\ \cline{1-1}  & Cl & 15 & - \\ \hline \hline \end{tabular}
\end{table} 
Table 1: COTS antenna geometry parameters. Figure 2: COTS antenna geometry. (**A**) Full antenna design; (**B**) Patch layer; (**C**) Large ground layer; (**D**) Feed pin; (**E**) Substrate layer; (**F**) Small ground layer; (**G**) Coaxial feed. The axial ratios (ARs) at 1.57542 GHz in the elevation plane for \(\varphi=0^{\circ}\) and \(\varphi=90^{\circ}\) are presented in Figure 4. In both planes, the axial ratio is slightly above 3 dB, the desired axial ratio upper limit for circular polarized antennas on the radiating side of the patch. Although, from Figure 3, the AR is shown to be below 3 dB, which is only marginally around the frequency of 1.57 GHz, and not at 1.57542 GHz. Note that these results were obtained from the approximated value of \(\varepsilon_{t}\), meaning that they can be slightly off from the actual values. In Figure 5, the simulated realized gain patterns for LHCP and RHCP are presented in both \(\varphi=0^{\circ}\) and \(\varphi=90^{\circ}\) directions. In both directions, the maximum realized gain is \(-2.95\) dBi for LHCP signals (\(-8.9\) dBi if we consider only the top side of the patch). As for the RHCP, the maximum realized gain is 5.15 dBi, which agrees with the datasheet's stated 5.5 dBi gain. Once again, this behavior shows the RHCP affinity of the antenna. Finally, the antenna's apparent phase center was estimated to be at (0.22, 1.70, 1.32) mm, very close to the geometric center of the antenna, which is the reference point (0, 0, 0).