Two Dimensional Phased Arrays

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Analysis of Uniform Phased Arrays

In this section, we'll look at two-dimensional or planar arrays with uniform spacing and with phased weights. Fortunately, two-dimensional phased arrays are not much more difficult to understand than one-dimensional arrays.

In one-dimensional arrays, the weights are chosen to compensate for the phase propagation of a wave in a desired direction. The extension to two-dimensional arrays is straight forward. Lets assume we have an array in the (x,y) plane, with positions given by:

antenna positions for 2d array

This array is plotted in Figure 1.

example phased array positions for two dimensional array

Figure 1. Array geometry for two-dimensional (planar) array.

The phased weights for a two-dimensional planar array steered towards direction of scanned array are given by:

weighting vector for 2d phased antenna array

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:

normalized array factor for 2d uniform array

Applying the sum formula twice, the above reduces to:

To plot this function, we'll introduce simplifying variables, u and v:

variables u and v directional cosines

The above variables are often used in plotting two-dimensional patterns, and are known as directional cosines. A plot of the AF above is shown in Figure 2:

magnitude of AF for antenna phased array

Figure 2. Magnitude of AF for two-dimensional array described above.

As expected, the array is maximum when (u, v)=(0,0), which corresponds to the desired direction .

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