Graphical Solution of Array Problems

Author: Edmund A. Laport

Once the patterns are calculated for each pair with their correct relative field-strength values for a particular ratio of pair amplitudes, they can be separately plotted in rectangular coordinates on semitransparent graph sheets. They can then be superimposed and examined. If each pattern can be drawn with the axis of each array as zero azimuth reference, the coincidence of these axes would show the possibilities of one limit of the array when the parallelogram is squashed into a single line. By sliding one pattern with respect to the other along the azimuth scale, one is effectively rotating the angle βx between the pair axes. For this purpose, one of the pair patterns should be repeated from 360 to 720 degrees so that there is no break in the one pattern as it is slid along over the other. It is also helpful, in locating null angles, to draw the pattern for the second pair with inverted polarity with respect to the first, because then there will

FIG. 2.48. Example of pattern from two crossed pairs forming a parallelogram array.

be a null indicated directly by the intersection of any point on one pattern with that of the second - points of equal fields but opposite polarity. This graphical method is one way of searching for a possible solution of a given allocation problem. Its advantage is that all possible effects of varying the angle between any two trial pairs can be examined very quickly. One can also form some qualitative idea of the effects of changing the current ratio between the pairs, thus often indicating possible solutions in this direction. This method reduces the number of tries that may be necessary to find the required array specifications for a desired arrangement of nulls and maximums in the final pattern. The person who must frequently solve such problems finds it advantageous to plot all such patterns in a uniform manner and accumulate them over a period of time. Then when a new problem arises, the stock of ready-made patterns leaves only the procedure of superimposing them before a bright light. The inversion of polarity of the lobes can be accomplished by turning a sheet over.

When the graphical method reveals a solution, an accurate recomputa-tion can be made from the indicated array parameters.

Figure 2.48 demonstrates an overlay of two pair patterns from which the resultant pattern can be plotted directly with a pair of dividers, using the distance between the curves for each azimuth. It is seen that zeros occur at 171, 282, 298, and 334 degrees and that very low fields exist between 275 and 336 degrees, not exceeding 2.9 percent of the maximum lobe through this interval of 61 degrees. The polar resultant pattern is given in Fig. 2.49.

This example was chosen to illustrate a further point of importance in making this kind of synthesis: by choosing pair patterns that cancel over a wide range of angles the resulting pattern can be made to suppress radiation over a wide angle.

In general, parallelogram arrays have azimuthal patterns derivable from the following equation:

In this equation A and B are scalar coefficients for the maximum field strength for the pattern of each pair, the other symbols being the same as those previously introduced.

It must now be apparent that a parallelogram array degenerates into a three-element array as the minor axis approaches zero (a single central radiator). On the other hand, as βx approaches zero, the parallelogram again degenerates into a four-element linear array, with one pair inside the other if S₁ <> S₂.

The parallelogram array is composed of only two cocentered pairs. Obviously more than two cocentered pairs could be employed to form still more extensive possibilities for radiation patterns, using the same method of pattern synthesis as developed for the two-pair system. A fifth radiator located at the geometric center of the array and at 0 reference phase will provide additional pattern possibilities.

Fig. 2.49 : Polar resultant pattern redrwan from Fig. 2.48