How Focused Is an Antenna's Beam?

Two antennas side by side add their waves: in some directions they reinforce, in others they cancel. The field pattern is a polar curve, and the area inside it, ½∫r² dθ, measures the power radiated, so the share of area inside the main beam is the share of power going where you aimed.

How it works

Area with polar coordinates

In one sentence

The area of the region bounded by $r = f(\theta)$ for $\alpha \le \theta \le \beta$ is $A = \frac12\int_\alpha^\beta f(\theta)^2\,d\theta$ .

General formula

{\htmlData{sym=0}{r(\theta)}} = \left|\cos\!\left(\pi {\htmlData{sym=2}{d}}\cos{\htmlData{sym=1}{\theta}} + \tfrac{\htmlData{sym=3}{{\beta}}}{2}\right)\right|, \qquad {\htmlData{sym=4}{P}} = \frac12\int_0^{2\pi} {\htmlData{sym=0}{r(\theta)}}^2\,{\htmlData{sym=2}{d}}{\htmlData{sym=1}{\theta}}

Polar regions are fans of thin wedges, each a sliver of a circle of radius $r$ with area $\frac12r^2\,d\theta$ .

What the symbols mean

$r(\theta)$
Field strength
How strong the combined wave is in direction θ.
$\theta$
Direction
Angle measured from the line through the two antennas.
$d$
Spacing
Distance between the antennas, in wavelengths.
$\beta$
Phase shift
Delay fed to one antenna, which steers the beam.
$P$
Radiated power
Area inside the pattern: total power sent out.

Two elements a distance $d$ apart (in wavelengths) send waves that arrive with a path difference of $d\cos\theta$ in direction $\theta$ .
Adding the two gives a field strength $\left|\cos\left(\pi d\cos\theta + \frac{\beta}{2}\right)\right|$ , where $\beta$ is a phase shift fed to one element.
Power goes like the field squared, so the power radiated is the area inside the polar curve, $\frac12\int r^2\,d\theta$ .
Changing $\beta$ swings the beam without moving anything: that's how phased-array antennas steer.

Learn the math

Area with Polar CoordinatesCalc II · Lesson 22, Parametric Equations and Polar Coordinates Electromagnetic WavesE&M · Lesson 28, Maxwell's Equations and Light Energy, Intensity and the Poynting VectorE&M · Lesson 29, Maxwell's Equations and Light