On Embedded Element Radiation Function and Beam Direction of Wide-Scanning Array Antennas

This paper presents a rigorous new analytical derivation of the theorem on the embedded element radiation function of ideally infinite planar array antennas, along with a formula for accurately calculating the main beam direction of finite-sized array antennas based on the theorem. It validates the previously established formula of the embedded element radiation function, where the amplitude is proportional to $\sqrt {\cos \theta }$ , based on the intuitive reasoning that the effective area of an element should be proportional to its projected area in the direction of interest angle $\theta $ , provided that the array antenna has no grating lobes for the full scan, no surface waves, no losses, and active impedance matched. More importantly, the new analytical derivation can accurately predict the embedded element radiation function in cases where there are grating lobes for the array antenna with the full scan, which the intuitive area projection reasoning cannot provide. The theorem concludes that the array’s active element reflection coefficient and inter-element spacing fully determine the embedded element radiation function in all cases. Utilizing this theorem, a new formula for the element phase progression is derived to accurately steer the main beam in the desired direction of the array with a finite number of elements. Several verification cases of wide-scanning array antennas are presented, and the comparisons between numerical simulations, measurements, theoretical results, and some interesting conclusions are discussed in the paper.