Parabolic Dish Antenna

What is modelled

A satellite is so far away that its signal reaches the dish as a plane wave. The paraboloid reflects every ray of that wave through one point, the focus, where the feed horn sits, and because all the paths have the same length the reflected waves add in phase. The page computes how well the dish does this: how much of the feed power the dish catches, how evenly it is lit, the gain, the width of the beam, and what happens when the dish is pointed slightly away from the satellite.

Equations

• Reflector z = ρ² / (4 f). For a ray parallel to the axis, (zref - z) + |P - F| = zref + f for every ρ: equal path length, and the surface normal reflects it exactly onto the focus. Off-axis rays are reflected with the same law and miss the focus (coma); the ray picture solves the ray-paraboloid intersection exactly.
• Feed power pattern cos2qψ; the dish edge is seen at ψ0 = 2 atan(D / 4F). Spillover (feed power that misses the dish): ηs = 1 - cos2q+1ψ0. The aperture amplitude is A(ρ) = cosqψ cos²(ψ/2), the second factor being the 1/r spreading of the feed wave; the taper efficiency is ηt = (∫A dA)² / (Aap ∫A² dA).
• Gain G = ηs ηt ηb ηr (π D / λ)², with blockage by the feed and four struts ηb = (1 - Ab/A)² and surface roughness (Ruze) ηr = exp(-(4 π ε / λ)²).
• Far field of the circular aperture: F(u) = ∫01 A(x) J0(u x) x dx / ∫01 A(x) x dx, u = (π D / λ) sinθ. Uniform illumination gives the Airy pattern 2 J1(u) / u with the first null at u = 3.8317, i.e. 1.22 λ/D; tapering widens the beam (about 1.2 to 1.5 λ/D) and lowers the sidelobes.

Simplifications

• Scalar aperture-field (Kirchhoff) theory in the far field: no polarisation, no diffraction from the rim, no near-field region, no spillover radiation into the pattern, and the feed pattern is an idealised rotationally symmetric cos^2q horn. Blockage is a simple area estimate with fixed feed and strut sizes, and the pattern ignores blockage scattering.
• The satellite and the ray lines are drawn close to the dish and not to scale (a real satellite is tens of thousands of kilometres away); the lobe is drawn on a dB scale with angles stretched about 150 times, otherwise it would be a needle thinner than a pencil line. The pointing-error exaggeration option multiplies only the ray picture, never the numbers.

Numerical checks

Four checks run when the page loads (see Model check): the aperture integral against the Airy form and its first null, the spillover closed form and the directivity from integrating the whole far field, the equal-path-length and reflection-law identities for 60 rays, and the Blender dish vertices against the paraboloid equation.

• Start with the home TV dish and watch the amber rays: every parallel ray from the satellite reflects through the feed horn, and the first graph shows the beam the whole aperture makes.
• Raise the pointing error: the rays miss the focus and the received power falls by the pattern value at that angle; tick the exaggeration box to see the coma in the picture.
• Increase the frequency with a rough surface (30 GHz preset): the gain rises with (pi D / lambda)^2 until the surface error eats it (second graph).