## 4 The 3D Array: Methodology and Numerical Results
### The Faceted Linear Array  
A trade-off study on a linear faceted array is conducted on different geometries depicted in Figure 8. Starting from a linear array (\(N_{f}=1\)), Figure 8a, the number of facets \(N_{f}\) incrementally increases by reaching the full conformal aperture in Figure 8e (\(N_{f}=\infty\)). In Figure 8, the elements are represented in red when a rotation of \(180^{\circ}\) is applied for symmetrical steering, compensating for the non-ideal far field of the element pattern. All the geometries involve \(N=60\) elements, which is the simplification of the \(60\times 60\) reference planar array presented in Figure 9a. Figure 8: Geometries of the different faceted arrays considered in this analysis. The geometries are listed for number of facets \(N_{f}\) starting by the (**a**) linear case \(N_{f}=1\), (**b**) triangular \(N_{f}=2\), (**c**) trapezoidal \(N_{f}=3\), (**d**) 4 segment \(N_{f}=4\) and ending with the (**e**) fully conformal scenario \(N_{f}=\infty\). The evaluated performances are depicted in Figure 10a,b, respectively, for the directivity and axial ratio FoV. Directivity and axial ratio are depicted at the steering direction, \(D(\theta=\theta_{s},\varphi=0)\) and \(AR(\theta=\theta_{s},\varphi=0)\). In Table 2, a complete list of the main values is presented. It is worth mentioning that for \(N_{f}=1\) (the linear case), the steering is equivalent to the element pattern which is limiting the FoV to \(\pm 42^{\circ}\). An axial ratio, \(AR(\theta=\theta_{s},\varphi=0)\leq 3.5\) dB, is achieved within the FoV \(\pm 33^{\circ}\). Beyond this range, degraded performance would significantly impact the partial directivity of the array (e.g., LHCP/RHCP components). \begin{table}
\begin{tabular}{c c c c c c} \hline \multirow{2}{*}{**Geometry**} & \multirow{2}{*}{\(N_{f}\)} & \multicolumn{4}{c}{**Field of View (deg)**} & \multirow{2}{*}{**Directivity Loss, \(\Gamma\) (dB)**} \\ \cline{3-3} \cline{5-5}  & & **Directivity** & & **Axial Ratio** & **Total** & \\ \hline Linear & 1 & \(\pm 42\) & \(\pm 33\) & \(\pm 33\) & 0 \\ Triangular & 2 & \(\pm 48\) & \(\pm 64\) & \(\pm 48\) & 2.39 \\ Trapezoidal & 3 & \(\pm 81\) & \(\pm 90\) & \(\pm 81\) & 3.92 \\
4-Segment & 4 & \(\pm 90\) & \(\pm 88\) & \(\pm 88\) & 3.98 \\ Conformal & \(\infty\) & \(\pm 90\) & \(\pm 87\) & \(\pm 87\) & 4.51 \\ \hline \end{tabular}
\end{table} 
Table 2: Summary of the array performance for the faceted linear case. Figure 10: Summary of the array performance for the different 1D faceted arrays considered: (**a**) Directivity vs. steering and (**b**) axial ratio vs. steering. Figure 9: (**a**) Geometry and (**b**) directivity of the reference \(60\times 60\) planar array for the boresight case. The different colors in (**a**) represent the different azimuth orientations of the elements. As the number of facets is increased (\(N_{f}>1\)), jumps in the normalized directivity during steering occur. These jumps result from clusters of elements being turned on when the condition for steering direction, explained in Section 3.3.1, is met. However, directivity FoV significantly improves already at \(N_{f}=2\) (the triangular case) with \(\pm 48^{\circ}\) and axial ratio FoV \(\pm 64^{\circ}\). Due to the tilted position of the elements, a directivity loss of \(\Gamma=2.39\) dB is recorded with respect to the linear array (the reference case). At \(N_{f}=3\), a trapezoidal geometry is considered, expanding directivity FoV to \(\pm 81^{\circ}\) and axial ratio FoV to \(\pm 90^{\circ}\). The overall directivity loss, compared to the linear case, is equivalent to \(\Gamma=3.92\) dB. In the case of a four-segment array at \(N_{f}=4\), the simulated directivity FoV is equivalent to \(\pm 99^{\circ}\) which is more than the required \(\pm 90^{\circ}\). However, since the array is considered installed on a ground metal platform (e.g., the aircraft fuselage), scan range above \(\pm 90^{\circ}\) is not here considered. Additionally,  axial ratio FoV is \(\pm 88^{\circ}\), and a directivity loss of \(\Gamma=3.98\) dB is observed.