## 3 Phased Array Comparison of Triangular and Square Lattices
To further emphasize the superior robustness of the antenna elements' impedance variation when using a triangular lattice, the simulated array matching efficiency (\(\eta_{\mathrm{T}}\)) was evaluated for all of the investigated steering angles by, using the following equation:  
\[\eta_{\mathrm{T}}=\frac{\sum\limits_{i=1}^{64}\left(1-\left|S_{ii}\right|^{2} \right)}{64} \tag{8}\]  
The color maps of Figure 13 confirm the advantages of adopting triangular lattices in planar arrays, rather than square ones, in that they provide higher efficiency values for all investigated frequencies (24.25-29.5 GHz). Specifically, the triangular lattice provides a higher matching efficiency value than the square lattice for all of the investigated steering angles, except around \(\phi=90^{\circ}\) plane (H-plane of the array) for \(\theta\) angles greater than \(40^{\circ}\). In particular, the triangular lattice shows a better matching efficiency in approximately 90% of the covered area within the addressed frequency band. Figure 12: Active \(S_{ii}\) CDF for square and equilateral triangular lattice planar arrays composed of 64 elements at 24.25 and 29.5 GHz. Figure 13: _Cont._  
The main reason the triangular lattice's matching efficiency undergoes a value reduction in the principal planes for \(\theta\) angles close to \(60^{\circ}\) at the highest frequency is due to scan blindness onset. This phenomenon can be observed through the active \(S_{1I}\) parameter related to the array's central element at the highest frequency (29.5 GHz), as a function of the beam steering shown in Figure 14. To emphasize the scan blindness onset, the active \(S_{1I}\) parameter was considered during beam steering within the whole visible region. More in detail, the color maps highlight that the triangular lattice does not avoid the scan blindness onset--characterized by a strong antenna element mismatch--but only reduces it by pushing the involved angular sector a bit further [43]. In fact, by considering the simulated phased array in the event of beam steering, the square lattice provides an active \(S_{1I}\) higher than -10 dB within the angular ranges \(|\,\theta_{0}\,|>\,60^{\circ}\), \(\,\theta_{0}\,<\,60^{\circ}\), and \(\,\phi_{0}\,>\,120^{\circ}\) (Figure 14b). On the other hand, the triangular lattice presents a milder mismatch of the central element for \(|\,\theta_{0}\,|>\,60^{\circ}\), \(\,\phi_{0}\,<\,30^{\circ}\), \(\,\phi_{0}\,>\,150^{\circ}\), and \(\,80^{\circ}\,<\,\phi_{0}\,<\,100^{\circ}\) (Figure 14a). Moreover, by considering the angular sector with \(|\,\theta_{0}\,|>\,60^{\circ}\)--hence, outside the desired scan angle area (highlighted by dashed red lines)--the triangular lattice presents an \(S_{1I}\) central array element higher than \(-10\)dB in 40 % of cases, whereas the square lattice does so in 50 % of cases. With the aim of evaluating the impact of the antenna array lattice on the radiation pattern shape, the realized gain as a function of the \(\theta\) angle at a broadside direction is reported in Figure 15.  
Figure 14: Active \(S_{1I}\) parameter related to the central array element as a function of beam steering for a planar array composed of 64 elements with (**a**) a triangular lattice; and (**b**) a square lattice. Figure 13: Matching efficiency (\(\eta_{\Gamma}\)) of a planar array composed of 64 elements (8 \(\times\) 8) as a function of the main beam steering inside the circular angular sector (\(-60^{\circ}\leq\theta_{0}\,\leq\,60^{\circ}\), \(\,0^{\circ}\leq\,\theta_{0}\,\leq\,180^{\circ}\)). Equilateral triangular lattice at (**a**) 24.25 GHz, (**b**) 26.875 GHz, and (**c**) 29.5 GHz; square lattice at (**d**) 24.25 GHz, (**e**) 26.875 GHz, and (**f**) 29.5 GHz. Figure 15 confirms that a triangular lattice provides a narrower main beam in the \(\phi=90^{\circ}\) plane, thus providing a higher maximum gain and better angular resolution.