= Solution
A <phased array> uses fixed <antennas> driven coherently, with adjustable electronic delays or <phases>. Changing those relative <phases> changes the direction of <constructive interference>.
For $N$ identical elements at $\mathbf r_j=jd\hat{\mathbf x}$, take time dependence $e^{-i\omega t}$ and feed <phase> $j\psi$. In a far-field direction at angle $\theta$ from the array normal, the <array factor> is
$$
F(\theta)=\sum_{j=0}^{N-1}e^{ij(\psi-kd\sin\theta)}=e^{i(N-1)\chi/2}\frac{\sin(N\chi/2)}{\sin(\chi/2)},\qquad \chi=\psi-kd\sin\theta.
$$
Multiply $|F|^2$ by the single-element <antenna beam pattern> to obtain the total angular <intensity>. All elements add in <phase> in the selected direction if
$$
\boxed{\psi=kd\sin\theta_0\pmod{2\pi}.}
$$
Thus electronic control of $\psi$ steers the beam to $\theta_0$ without moving an element. <Reciprocity> gives the corresponding directionality in reception.
\Image[/past-exam-of-the-optics-and-electrodynamics-course-of-the-university-of-cambridge/2017-q2-phased-array.png]
{title=An electronically steered linear array and its normalized array factors}
The aperture $Nd$ controls angular resolution: the nearest nulls obey $|\sin\theta-\sin\theta_0|=\lambda/(Nd)$, giving a main-beam width of order $\lambda/(Nd\cos\theta_0)$ away from endfire. Additional equally strong maxima can occur when $\chi=2\pi m$; these are <grating lobes>. Choosing $d<\lambda/2$ avoids them for all steering directions, with $d=\lambda/2$ the usual limiting spacing. Amplitude tapering reduces side lobes at the cost of a wider main beam. For a broad <frequency> band, true time delays preserve steering more accurately than frequency-independent <phase> shifts.
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