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Numerical: free-space received power (165 W, 325 MHz)

Short answer

Convert 165 W to 52.17 dBm. Free-space path loss at 325 MHz over 15 km is 106.2 dB. With antenna gains 12 and 6 dBi, received power is 52.17+12+6-106.2=-36.03 dBm, about 0.25 microwatt or 2.5e-7 W.

A wireless communication transmitter has an output power of 165 watts at a carrier frequency of 325 MHz. It is connected to an antenna with a gain of 12 dBi. The receiving antenna is 15 km away and has a gain of 6 dBi. Calculate the power delivered to the receiver, considering free-space propagation. (Assume that there are no other losses or mismatches in the system.)

For free space propagation, the received power in dB form (Friis equation) is:

Pr(dBm) = Pt(dBm) + Gt(dBi) + Gr(dBi) − LFS(dB)
LFS = 20 log(4πd/λ)
    = 32.44 + 20 log f(MHz) + 20 log d(km)

Given: Pt = 165 W, f = 325 MHz, Gt = 12 dBi, Gr = 6 dBi, d = 15 km, no other losses.

Step 1: Transmit power in dBm

Pt = 10 log(165 / 0.001) = 10 log(165 000) = 52.17 dBm

Step 2: Free space path loss

λ   = 3×10⁸ / 325×10⁶ = 0.923 m
LFS = 20 log(4π × 15 000 / 0.923)
    = 106.20 dB
(check: 32.44 + 50.24 + 23.52 = 106.20 dB)

Step 3: Received power

Pr = 52.17 + 12 + 6 − 106.20
   = −36.03 dBm
Pr = 10^(−36.03/10) mW = 2.50 × 10⁻⁴ mW = 0.25 µW

Answer: Power delivered to the receiver ≈ −36.0 dBm, i.e. about 2.5 × 10⁻⁷ W (0.25 µW).

✓ Answer checked by Claude Opus 5.5 (high effort)

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