## 4 The 3D Array: Methodology and Numerical Results
### The 3D Array  
In Figure 9a, the reference planar array is shown. As previously mentioned in Section 2, it is composed of \(60\times 60\) elements spaced at \(\lambda/2\) and it achieves a directivity of 42.3 dBiC (LHCP), depicted in Figure 9b. The elements of the array are properly rotated for symmetry in the scan range and the phase is then compensated through the \(\psi_{n}\) term in (12). Figure 11: Directivity FoV, axial ratio FoV and directivity loss \(\Gamma\) vs. the number of facets \(N_{f}\). In red on the left y-axis, the directivity and axial ratio FoV is depicted, while in blue on the right y-axis, the directivity loss \(\Gamma\) is depicted. The selected trapezoidal geometry in Figure 8c is expanded to the 3D case to enable 2D steering, similarly to [37]. The total number of facets is therefore increased to \(N_{f}=5\). In the following, two different geometries, shown in Figure 12, are analyzed. Indeed, for the 3D case, the number of elements for each facets is adapted and this can be obtained either by keeping the number of elements similar to the planar case (configuration T1 in Figure 12a) or by fully using the available footprint of the planar array (configuration T2 in Figure 12b). For T1 configuration, the 3D array is created by a \(3\times 24\) linear array and resulting in 3936 elements in the aperture. If considered the \(3\times 20\) linear array, as for the linear case, the overall number of elements would be much smaller than 3600. When the footprint of the planar reference array is considered (configuration T2), a linear array with \(3\times 30\) elements is considered for the expansion to the 3D case. In this case, the total number of elements in the aperture is 6180. This solution will be latter investigated to compensate for any directivity losses with respect to the planar case. Also in this scenario, the array elements are opportunely rotated for symmetry in the scanning. In Figure 12, the colors of array elements represent the respective rotation which is compensated by the \(\psi_{n}\) term in the phase computation in (12). It is worth to mention that both the geometries, T1 and T2, fit into the footprints foreseen in commercial aircraft fuselages as in [38]. The \(N_{f}=5\) planar PCBs can conveniently host the required amount of beamformer ICs for controlling in amplitude and phase all the elements of the array face. A 5:1 power combiner can be implemented at the bottom of the structure. The steering performance is evaluated once again with an elevation steering resolution of \(1^{\circ}\) in the upper hemisphere \(\theta_{s}\in[0^{\circ},90^{\circ}]\). Regarding the azimuth steering, in Figure 13, the cases for \(\varphi_{s}=\{0^{\circ},22.5^{\circ},45^{\circ}\}\) are shown. Other azimuth angles for \(\varphi_{s}>45^{\circ}\) would be symmetrical for the rotation of the elements. In Table 3, a summary of performances for the 3D arrays (configuration T1 and T2) vs. the reference planar array are listed. \begin{table}
\begin{tabular}{c c c c c c c} \hline \multirow{2}{*}{**Geometry**} & \multirow{2}{*}{\(N_{f}\)} & \multirow{2}{*}{\(N\)} & \multicolumn{3}{c}{**Scan Range (deg)**} & \multirow{2}{*}{**Directity Loss, \(\Gamma\) (dB)**} & \multirow{2}{*}{**Max Directivity (dBiC)**} \\ \cline{3-3} \cline{5-6}  & & & **Directivity** & & & **Axial Ratio** & **Total** \\ \hline Planar & \(\infty\) & 3600 & \(\pm 24\) & \(\pm 31\) & \(\pm 24\) & 0 & 42.3 \\
3D T1 & 5 & 3936 & \(\pm 70\) & \(\pm 76\) & \(\pm 70\) & 4.75 & 37.55 \\
3D T2 & 5 & 6180 & \(\pm 70\) & \(\pm 76\) & \(\pm 70\) & 2.79 & 39.51 \\ \hline \end{tabular}
\end{table} 
Table 3: Summary of performances for the 2 configurations of 3D array vs. the reference planar array. Figure 12: Three-dimensional faceted array based on the trapezoidal geometry for (**a**) T1 and (**b**) T2 configurations. As expected, looking at directivity FoV in Figure 13a for the planar reference case, symmetrical steering limited to the element pattern is achieved (red, green and blue lines).