Received power: 15.5 km, 3.4 GHz, dish receiver

Maximum power is received when the antennas are matched and aligned in free space, so the Friis equation applies directly. The receiving gain is found from the dish size and efficiency.

Given

• d = 15.5 km, f = 3.4 GHz, Pt = 250 W
• Gt = 25 dB = 10^2.5 = 316.23
• Receiving dish D = 6.4 m, η = 0.75

Step 1: Wavelength

λ = c/f = 3×10⁸ / 3.4×10⁹ = 0.08824 m

Step 2: Effective aperture and gain of the receiving dish

Ae = η·πD²/4 = 0.75 × π × 6.4² / 4 = 24.13 m²
Gr = 4π·Ae/λ² = 4π × 24.13 / 0.08824²
   = 3.894×10⁴  = 45.90 dB

Step 3: Received power (Friis, aperture form)

Pr = Pt·Gt·Ae / (4πd²)
   = 250 × 316.23 × 24.13 / (4π × (15500)²)
   = 1.9075×10⁶ / 3.0191×10⁹
   = 6.32×10⁻⁴ W

Check in dB

LFS = 20 log(4πd/λ) = 126.88 dB
Pt  = 10 log 250    = 23.98 dBW
Pr  = 23.98 + 25 + 45.90 − 126.88 = −31.99 dBW
    = −1.99 dBm ≈ 0.632 mW

Answer: Maximum received power ≈ 6.32×10⁻⁴ W = 0.632 mW (−31.99 dBW, −1.99 dBm).