A Generalized Analytical Solution for the Directivity of Nonuniform Planar Phased Arrays Using a Universal Element Radiation Model

There has of late been a great amount of interest by the antenna design community in the synthesis of optimal phased arrays. It is no wonder, given the increased prevalence of applications such as electronically steered radar, terrestrial multiuser communications (e.g., 5G mobile communications), and space-based shaped beam antennas, among many others. A key figure of merit in the design of a phased array is the directivity pattern, meaning both the peak achievable directivity (<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${D} {_{{0}}}$ </tex-math></inline-formula>) of the array as well as the peak sidelobe level (PSLL) below the main beam. Especially in the case of large element-count aperiodic arrays, these values are quite difficult to accurately determine without first performing computationally intensive numerical integrations of unpredictable radiation patterns. The literature offers several solutions to determine directivity using analytical (i.e., closed form or exact) array factor (AF) methods, but these leave much to be desired in terms of true generality. In this work, we present a new fully generalized analytical directivity solution that may be applied to planar arrays of totally arbitrary topology. The proposed element model is shown to apply to a variety of practical antennas, in addition to enabling rapid array synthesis. Moreover, the solution is shown to be in excellent agreement with prior solutions when considered as special cases (SCs).