Quick Reference: Paraboloid Antenna Formulas and Worked Examples

Quick Reference: Paraboloid Antenna Formulas and Worked Examples

Introduction

This quick reference summarizes core formulas for circular-paraboloid (dish) antennas and shows worked examples for common design calculations: aperture area, directivity/gain, beamwidth, focal length and f/D ratio, surface accuracy, and pointing loss. Use SI units throughout.

Key parameters and symbols

• D — dish diameter (m)
• A — physical aperture area (m²)
• λ — wavelength (m)
• f — focal length (m)
• F/D — focal-length-to-diameter ratio (dimensionless)
• η_ap — aperture efficiency (fraction, typical 0.55–0.75)
• G — gain (linear, not dB)
• G_dBi — gain in dBi
• θ_3dB — half-power beamwidth (radians or degrees)
• k — illumination taper factor (used in beamwidth approximations; often 1.02–1.22)
• ε_surface — RMS surface error (m)
• L_pointing — pointing loss (fraction)

Basic formulas

1. Aperture area A = π(D/2)^2

2. Ideal directivity (aperture-based) G = η_ap(4πA / λ^2)

3. Gain in dBi G_dBi = 10 · log10(G)

4. Approximate half-power beamwidth (degrees) θ_3dB ≈ k * (λ / D) * (180/π)

• Use k ≈ 70 for degrees when converting commonly used approximations: θ_3dB(deg) ≈ 70 · λ/D (this is an empirical shortcut for many dish illuminations).

5. Focal length and F/D F/D = f / D

• Typical F/D values: 0.25–0.6; smaller F/D → deeper dish → different feed pattern required.

6. Surface accuracy (Ruze’s formula for gain loss) Gain loss factor due to RMS surface error: L_surface = exp[−(4π ε_surface / λ)^2]

• Effective gain with errors: G_eff = G · L_surface

7. Pointing loss (approximate) L_pointing ≈ exp[−( (2π/λ) · (θ_pointing_rms · D / 2.35) )^2 ]

• θ_pointing_rms in radians; this approximates loss from boresight pointing jitter relative to beamwidth.

8. Illumination taper efficiency (approximate) η_taper depends on feed taper; common values 0.7–0.9. Overall aperture efficiency: η_ap = η_taper · η_spill · η_pol · η_misc (combine feed, spillover, polarization, blockage losses)

Worked examples

Assume: D = 3.0 m, frequency f0 = 10 GHz (λ = 0.03 m), η_ap = 0.65, ε_surface = 0.5 mm = 0.0005 m, F/D = 0.4.

1. Aperture area A = π(3.0/2)^2 = π(1.5)^2 = 7.069 m²

2. Ideal directivity and gain G = 0.65 * (4π * 7.069 / 0.03^2)
   = 0.65 * (4π * 7.069 / 0.0009)
   = 0.65 * (4π * 7854.44) ≈ 0.65 * (98711) ≈ 64162
   G_dBi = 10·log10(64162) ≈ 10·(4.807) ≈ 48.07 dBi

3. Beamwidth θ_3dB ≈ 70 · λ / D = 70 · 0.03 / 3.0 = 0.7°
   (or using radians: θ ≈ 1.22·λ/D = 1.22·0.03/3 = 0.0122 rad = 0.7°)

4. Surface loss (Ruze) L_surface = exp[−(4π·0.0005 / 0.03)^2] = exp[−(0.20944)^2] = exp[−0.04386] ≈ 0.9571
   G_eff ≈ 64162 · 0.9571 ≈ 61439 → G_eff_dBi ≈ 10·log10(61439) ≈ 47.88 dBi

5. Pointing loss (example: pointing RMS = 0.1° = 0.001745 rad) First estimate beam sigma: beamwidth (HW) ≈ 0.0122 rad; approximate beam sigma ≈ HW/2.35 = 0.00519 rad
   Normalized pointing offset ratio = θ_pointing_rms / beam_sigma = 0.001745 / 0.00519 ≈ 0.336
   L_pointing ≈ exp[−(2π/λ · (θ_rms · D / 2.35))^2] — evaluating directly gives small loss; numerically L_pointing ≈ 0.90 (approx).
   Combined effective gain ≈ G_eff · L_pointing ≈ 61439 · 0.90 ≈ 55295 → ≈ 47.42 dBi

Quick reference checklist

• Compute A from D.
• Choose η_ap based on feed and blockage.
• Compute G = η_ap·4πA/λ^2 and convert to dBi.
• Use θ3dB ≈ 70·λ/D (deg) or θ ≈ 1.22·λ/D (rad).
• Apply Ruze: multiply by exp[−(4π·ε/λ)^2] for surface errors.
• Apply pointing loss and other efficiencies multiplicatively.

python
 ffON2NH02oMAcqyoh2UU MQCbz04ET5EljRmK3YpQ CPXAhl7VTkj2dHDyAYAf” data-copycode=“true” role=“button” aria-label=“Copy Code”>Copy CodeCopied
import math 
def paraboloid_gain(D, freq, eta_ap=0.65, eps=0.0005):
    c = 3e8
    lam = c / freq     A = math.pi * (D/2)2
    G = eta_ap  4math.pi*A / lam2
    L_surface = math.exp(-(4math.pieps/lam)*2)
    return G  L_surface 
# Example
D = 3.0
freq = 10e9
G_eff = paraboloid_gain(D, freq)
print(“G_eff_dBi =”, 10*math.log10(G_eff))