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Received power: 0.5 km, 1 GHz circuit

Short answer

λ = 0.3 m. Free space loss = 20 log(4π×500/0.3) = 86.42 dB. Transmit power 150 W = 51.76 dBm. Received power Pr = 51.76 + 25 + 20 − 86.42 = 10.34 dBm (10.8 mW).

Find the received power (in dBm) at a distance of 0.5 km over a free space 1 GHz circuit consisting of a transmitting antenna with 25 dB gain and a receiving antenna gain of 20 dB. The power radiated by the transmitting antenna is 150 W.

Using the Friis transmission equation in dB form.

Given

  • d = 0.5 km = 500 m, f = 1 GHz
  • Gt = 25 dB, Gr = 20 dB, Pt = 150 W

Step 1: Wavelength

λ = c/f = 3×10⁸ / 10⁹ = 0.3 m

Step 2: Free space loss

4πd/λ = 4π × 500 / 0.3 = 2.094×10⁴
LFS   = 20 log(2.094×10⁴) = 86.42 dB

Check: 32.44 + 20 log 0.5 + 20 log 1000
     = 32.44 − 6.02 + 60 = 86.42 dB

Step 3: Transmit power in dBm

Pt = 10 log(150 / 1 mW) = 10 log(1.5×10⁵) = 51.76 dBm

Step 4: Received power

Pr = Pt + Gt + Gr − LFS
   = 51.76 + 25 + 20 − 86.42
   = 10.34 dBm
Pr = 10^1.034 mW ≈ 10.8 mW

Answer: Received power ≈ 10.34 dBm (about 10.8 mW).

✓ Answer checked by Claude Opus 5.5 (high effort)

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