Friis equation and LFS = 22 + 20 log(d/λ)

The Friis transmission equation relates the power received by an antenna to the power transmitted by another antenna in free space, through their gains, the wavelength and the distance between them.

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)

The equation is valid only in the far field (d ≥ 2D²/λ), for matched polarisation and impedance, and for an unobstructed path. If not, extra factors (polarisation loss factor, reflection loss 1 − |Γ|²) are multiplied in.

Free space loss LFS = 22 + 20 log(d/λ)

Free space loss is the ratio Pt/Pr between two isotropic antennas (Gt = Gr = 1):

LFS = Pt/Pr = (4πd/λ)²

Taking 10 log of both sides:

LFS(dB) = 10 log (4πd/λ)²
        = 20 log(4πd/λ)
        = 20 log(4π) + 20 log(d/λ)
        = 20 × 1.0992 + 20 log(d/λ)
        = 21.98 + 20 log(d/λ)
        ≈ 22 + 20 log(d/λ)  dB

Hence proved. Here d and λ must be in the same units. In practical units, LFS = 32.44 + 20 log d(km) + 20 log f(MHz) dB.