An analytical approach to the coverage of a hemisphere by n planar phased arrays

The coverage of a hemisphere by planar phased arrays arranged in pyramids or pyramidal frustra is discussed. Transformations between the spherical coordinates of individual arrays and of the hemisphere of coverage are derived. These results are then used to determine the minimum value of the maximum scan angle required of each array and the positioning of the arrays. Numerical results are given. These transformations are also used to minimize the number of elements needed in a given array aperture. This is shown for an example with both equilateral triangular and rectangular element spacing. It is further demonstrated how the use of an isosceles triangular element spacing affords additional reduction of the number of elements needed.