Why Power Falls with Range

  • Same transmitted power, larger spherical area:
    • Area=4𝜋𝑅2⇒flux∝1𝑅2
  • 𝑅→2𝑅: area ×4, flux ÷4
    • Δ[𝑃𝑟]=−6dB
  • Range ×10 → area ×100 → Δ[𝑃𝑟]=−20dB (see useful decibel relationships)

Calculating Free-Space Path Loss

Free-Space Path Loss

  • [𝐿𝑝]=20log10(4𝜋𝑅𝜆)
  • In km and GHz:
    • [𝐿𝑝]=92.45+20log10(𝑅)+20log10(𝑓)
  • With 𝑅 in m and 𝑓 in GHz the constant is 32.45 instead (used in Class 3); Class 4 rounds 92.45 to 92.4
  • [𝐿𝑝] is a positive loss; subtract it in the power ledger (received-power equation)
  • At one frequency, free-space loss changes only because range changes
  • It also grows with frequency (for a fixed range) — but so does the gain of fixed apertures: Frequency Selection

TERPSat: The Edge Cost

  • 𝑓=8.4GHz
  • Overhead, 𝑅=705km: [𝐿𝑝]=167.9dB
  • At 5° elevation, 𝑅=2574km (slant range):
    • [𝐿𝑝]=92.45+20log10(2574)+20log10(8.4)=179.1dB
  • Edge cost: Δ[𝐿𝑝]=11.2dB — the price of serving the low-elevation edge of the pass
  • Deep-space scale: Voyager 2 at about 2.15×1010 km gives [𝐿𝑝]≈317.6dB (Homework 2)
  • References: ITU-R P.525-4 (2019) and P.525-5 (2024), Calculation of Free-Space Attenuation