# On the Trade-Off between the Main Parameters of Planar Antenna Arrays
## 2 Planar Arrays with Regular Lattices
### Comparing Directivities  
Let us now apply the aforementioned procedure to the synthesis of a specific pattern. We want to radiate a pencil beam with SLL lower than \(-20\) dB for \(w\geq 0.067\), and we want to have the SLL specification met for any beam focused in the conical region \(\theta\leq 50^{\circ}\)(deg). Using the aforementioned rules for the creation of the two lattices, it is possible to achieve a layout with a square lattice of  
Figure 1: Uniform lattices comparison. (**a**) Square, (**b**) triangular. The blue circles describe the radiating element positions, the red curve represents the limit circle of radius \(R_{MAX}\).  
65 elements, and a layout with a triangular lattice of 571 elements (which are the ones depicted in Figure 1).  
In Figure 2, we can see the result of the pattern synthesis with the square lattice (the pink/cyan colormap is chosen to emphasize violation of the SLL constraint). The distance between elements is \(d_{S}\approx 0.54\lambda\), and the directivity of the broadside beam is \(D=29.0\) dBi. The grating lobes are outside the \((u,v)\) plane region \(w\leq 1+\sin(50^{\circ})\approx 1.77\), so they will not appear when scanning the beam up to \(50^{\circ}\)(deg).  
In Figure 3, we can see the result of the synthesis with the triangular lattice. The distance between elements is \(d_{S}\approx 0.62\lambda\), and the directivity of the broadside beam is \(D=28.0\) dBi. Even in this case, the grating lobes are outside the \((u,v)\) plane region \(w\leq 1+\sin(50^{\circ})\approx 1.77\), so they will not appear when scanning the beam up to \(50^{\circ}\)(deg).  
Let us discuss the results of the two lattices. The higher directivity of the former compared to the latter is mostly due to the larger number of radiating elements of the square lattice; it is worth noting that the difference in directivity can be also inferred from the different power pattern imagemaps of Figures 2 and 3, since the former shows slightly darker tones of cyan.  
It is worth underlining that the employed inter-element distances in Equations (4) and (5) are only the maximum ones, to avoid the appearance of grating lobes when focusing the beam in the angular  
Figure 3: Triangular regular lattice array with 665 elements. Left subplot: imagemap with the power pattern of the radiated pattern. Right subplot: some relevant cuts for the normalized AF of the radiated pattern. Negative values of \(w\) are related to the cuts \(\phi^{\prime}=\phi+\pi\).  
Figure 2: Square regular lattice array with 665 elements. Left subplot: imagemap with the power pattern of the radiated pattern. Right subplot: some relevant cuts for the normalized array factor (AF) of the radiated pattern. Negative values of \(w\) are related to the cuts \(\phi^{\prime}=\phi+\pi\).  
region of interest. Smaller distances can be used for achieving a better aperture efficiency and, hence, directivity. This choice would imply using many more radiating elements, so it will be not considered in these notes.  
The use of a regular grid of radiators allows a very high directivity, but the number of employed elements is also huge: for many applications, the number of radiators should be reduced as much as possible to obtain a reasonable system cost, space occupation, and weight. For this reason, it could be convenient to use sparse, non-regular lattices.