Two Dimensional Phased Arrays
In this section, we'll look at twodimensional or planar arrays with uniform spacing and with
phased weights. Fortunately,
twodimensional phased arrays are not much more difficult to understand than onedimensional arrays. In onedimensional arrays, the weights are chosen to compensate for the phase propagation of a wave in a desired direction. The extension to twodimensional arrays is straight forward. Lets assume we have an array in the (x,y) plane, with positions given by:
This array is plotted in Figure 1.
Figure 1. Array geometry for twodimensional (planar) array. The phased weights for a twodimensional planar array steered towards are given by:
In the above equation, the weights are designed to steer the array towards the wave vector in the desired direction, written as . To illustrate this, we'll assume the array is steered to broadside  . In this case, all of the weights simply become equal to unity (1.0). The array factor becomes:
Applying the sum formula twice, the above reduces to:
To plot this function, we'll introduce simplifying variables, u and v:
The above variables are often used in plotting twodimensional patterns, and are known as directional cosines. A plot of the AF above is shown in Figure 2:
Figure 2. Magnitude of AF for twodimensional array described above. As expected, the array is maximum when (u, v)=(0,0), which corresponds to the desired direction .

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