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Microwave link: 90 km, 10 GHz, 40 dB antennas

Short answer

λ = 0.03 m. Free space loss LFS = 20 log(4πd/λ) = 151.53 dB; net loss after 40 dB gains is 71.53 dB. Received power Pr = 40 dBW + 40 + 40 − 151.53 = −31.53 dBW (0.704 mW).

Microwave link is assumed to be free space condition. The antenna gains are each 40 dB, the frequency is 10 GHz and the path length is 90 km. Calculate the transmission path loss and received power for transmitted power of 10 kW.

Under free space conditions the Friis equation gives the path loss and received power.

Given

  • Gt = Gr = 40 dB, f = 10 GHz, d = 90 km, Pt = 10 kW

Step 1: Wavelength

λ = c/f = 3×10⁸ / 10¹⁰ = 0.03 m

Step 2: Free space (transmission path) loss

4πd/λ = 4π × 9×10⁴ / 0.03 = 3.770×10⁷
LFS   = 20 log(3.770×10⁷) = 151.53 dB

Check: 32.44 + 20 log 90 + 20 log 10000
     = 32.44 + 39.08 + 80 = 151.53 dB

Net loss after the antenna gains: 151.53 − 40 − 40 = 71.53 dB.

Step 3: Received power

Pt = 10 log(10⁴) = 40 dBW
Pr = Pt + Gt + Gr − LFS
   = 40 + 40 + 40 − 151.53
   = −31.53 dBW = −1.53 dBm
Pr = 10^(−3.153) W ≈ 7.04×10⁻⁴ W

Answer: Path loss = 151.53 dB (71.53 dB including antenna gains); received power ≈ −31.53 dBW = 0.704 mW.

✓ Answer checked by Claude Opus 5.5 (high effort)

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