I gave some more thoughts on how to do the phasing of the elements. It struck me that the beauty of the 6x6 square array is that the phase delay to form a beam pointing at an arbitrary direction was generated by a (fairly) simple combiner that generated a smooth, linear by-row and by-column phase gradient. The Butler matrix for a linear 8-element array (elements ½ wave apart) generates phase differences between adjacent elements of
The geometry for the various hexagonal grids is more ugly with no such great symmetry. So I have started thing of other geometries. Take a look at the attached "6-spoke" configuration. In this picture we have 3 linear arrays of 13 elements each, with the central "zero phase" element being common to all 3 arrays (i.e. 37 elements -- the array was slightly too large for my scanner, so there are a couple of virtual elements off the right-hand margin).

To point the beam of 6-spoke array, we would resolve the desire pointing direction onto the 3 radial directions, and then apply a linear phase gradient to the element. This means we need only three 13-element progressive power splitters. This could be done with only three 16-port one-dimensional Butler matrices. The problem with this is that the beam pattern has fairly bad diffraction lobes that are smaller than the earth -- i.e. the 13-unit diameter is probably too big.

So I tried another concept. Lets decrease the diameter by 4 (now=9), resulting in a 25 element array. now we fill in the 6 intermediate positions with 3 more elements, also radial from the center. The first 6 elements out from the center are the original hexagon, and outside are 3 more circles with the radii increasing by one unit. This array has 43 elements, as seen in the attached "12_spoke" design. Each of the 12 spokes (arms) can be readily phased by a really elementary 4-port butler matrix (with the 3-element arms having one port terminated).

Once again, comments are solicited.

73, Tom