Transmission loss and Friis equation

Transmission loss

Transmission loss is the ratio (in dB) of the power fed to the transmitting antenna to the power available at the receiving antenna terminals: L = 10 log(Pt/Pr). In free space it is the spreading loss reduced by the antenna gains, L = 20 log(4πd/λ) − Gt − Gr. With isotropic antennas it equals the basic free space loss LFS = 20 log(4πd/λ). Real links add losses from obstacles, ground, atmosphere, rain and feeders.

Derivation of the Friis transmission equation

Consider a transmitter of power Pt feeding an antenna of gain Gt, and a receiving antenna of gain Gr (effective aperture Aer) at distance d in free space. Assume far field, matched polarisation, matched loads and no obstacles.

1. An isotropic source spreads Pt uniformly over a sphere of area 4πd², so the power density at distance d is P_iso = Pt / (4πd²) (W/m²)
2. A directive antenna with gain Gt increases the density in its main direction by Gt: P_d = Pt·Gt / (4πd²)
3. The receiving antenna collects the power falling on its effective aperture: Pr = P_d · Aer = Pt·Gt·Aer / (4πd²)
4. Effective aperture and gain are related by Aer = Gr·λ² / (4π). Substituting:

Pr = Pt·Gt·(Gr·λ²/4π) / (4πd²)

Pr = Pt·Gt·Gr·λ² / (4πd)²

Pr/Pt = Gt·Gr·(λ / 4πd)²

This is the Friis transmission equation. In terms of the two effective apertures (using Gt = 4πAet/λ²) it can also be written as Pr/Pt = Aet·Aer / (λ²d²).

In decibels:

Pr(dBW) = Pt(dBW) + Gt(dB) + Gr(dB) − 20 log(4πd/λ)

From this, the transmission loss is

L = Pt/Pr = (4πd/λ)² / (Gt·Gr)
L(dB) = 32.44 + 20 log d(km) + 20 log f(MHz) − Gt − Gr

The received power falls as 1/d² (6 dB per doubling of distance). Doubling the frequency adds 6 dB of free space loss, but if dish antennas of fixed size are used, both gains rise by 6 dB, so high-frequency links with dishes actually gain overall.