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Graphical Solution of Array ProblemsAuthor: Edmund A. Laport
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 readymade 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 recomputation 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 threeelement 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 fourelement linear array, with one pair inside the other if S_{1} <> S_{2}. 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 twopair system. A fifth radiator located at the geometric center of the array and at 0 reference phase will provide additional pattern possibilities.


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