Antenna scaling with frequency: A 20 m dish provides a certain gain at 4/6 GHz. What dish diameter is required to obtain the same gain at 20/30 GHz?

Introduction / Context:
Antenna gain for a parabolic dish scales with physical size relative to wavelength. When operating at a higher frequency (shorter wavelength), a smaller dish can achieve the same gain as a larger dish at a lower frequency, assuming similar efficiency.

Given Data / Assumptions:

• Dish 1: D1 = 20 m operating at f1 = 4–6 GHz.
• Dish 2: find D2 at f2 = 20–30 GHz for same gain.
• Assume similar aperture efficiency η and compare uplink 6→30 GHz (or downlink 4→20 GHz), both with the same 5× frequency increase.

Concept / Approach:

Parabolic dish gain G ∝ (D/λ)^2. For constant G and η, D ∝ λ. If frequency increases by 5×, wavelength decreases by 5×, so required diameter decreases by 5×.

Step-by-Step Solution:

Verification / Alternative check:

Back-of-the-envelope: G(dBi) ≈ 10 log10[(πD/λ)^2 η]. Keeping (D/λ) constant preserves gain.

Why Other Options Are Wrong:

• 10 m or 100 m: would dramatically change gain; not required for same gain at higher frequency.
• 1–2 m: too small given only a 5× frequency increase.

Common Pitfalls:

• Mistaking gain scaling as linear with frequency rather than with D/λ.

Final Answer:

4 m