Phased array

A phased array combines radiation from spatially separated antennas with controlled relative phases. Changing those phases steers the direction of constructive interference without moving the elements. For an equally spaced linear array with spacing $d$, free-space wavenumber $k$ and phase increment $\psi$, a principal beam at angle ${\theta }_0$ from the array normal satisfies $\psi =kd\sin {\theta }_0$ when the time dependence is $e^{-i\omega t}$ and the element drives are $e^{ij\psi }$.

A grating lobe is an additional principal maximum caused by spatial sampling in a phased array. A regularly spaced linear array steered towards ${\theta }_0$ also has maxima wherever $\sin \theta =\sin {\theta }_0+m\lambda /d$ lies in $[-1,1]$. Taking $d<\lambda /2$ excludes such maxima for all steering angles; $d=\lambda /2$ is the usual limiting choice, with an endfire degeneracy at the extreme scan directions.

The array factor describes the interference between the elements of a phased array, separately from the radiation pattern of each element. For $N$ identical equally spaced elements,At $\chi =2\pi m$ the limiting value has magnitude $N$. The total intensity is proportional to $|F{|}^2$ times the single-element antenna beam pattern.