Dish Antenna Gain and Beam

What is modelled

A parabolic dish turns a plane wave into a spherical wave that converges on the feed, and the same geometry works in reverse for transmitting. The beam it makes is the diffraction pattern of a circular aperture (the Airy pattern). Its width is inversely proportional to the diameter in wavelengths, so a large dish at high frequency has a very narrow beam, a large gain and little tolerance for a rough surface or a pointing error.

Equations

• Gain G = η (πD/λ)2 exp[−(4πε/λ)2] (Ruze).
• Pattern (2 J1(u)/u)2 with u = (πD/λ) sinθ; half-power width 1.029 λ/D.
• Pointing loss ≈ 12 (θe / HPBW)2 dB, here taken directly from the pattern.
• Link Pr = EIRP · Gr (λ/4πR)2, C/N0 = Pr/(k Ts).

Checks run in the page

• The integral of the Airy pattern equal to 2, which fixes the directivity.
• The half-power beamwidth from the root of the pattern.
• The mean phase factor of random surface errors against the Ruze exponential.
• The pointing-loss rule against the pattern.

Honest simplifications

The illumination is uniform with a single efficiency number, so there are no real sidelobe levels for a tapered feed; surface errors are random and uncorrelated across the aperture; the pointing error is a fixed offset; atmosphere, rain, spillover and noise from the sky are folded into one system temperature; and the link has no coding or modulation loss.

• Press 70 m at X band, then 34 m at Ka band and compare the gain and the beamwidth.
• Press Rough surface: at Ka band even a millimetre costs a lot of gain.
• Press Mispointed and watch the yellow line move down the side of the beam.