Calculation of ring-shaped phase centers of feeds for ring-focus paraboloids

Some primary feeds such as the hat feed radiate around the feed waveguide, which also acts as an axial support tube. Such feed antennas have phase centers in the form of rings rather than points. This paper presents a formula to calculate the location of the ring-shaped phase center. The optimum reflector is a ring-focus paraboloid with the ring focus coinciding with the ring-shaped phase center. The phase center formula is applied to two versions of the hat feed and it is shown that the aperture efficiencies can be improved by up to 0.5 dB when optimum ring-focus paraboloids are used instead of point-focus paraboloids