Short note: Friis transmission equation

The Friis transmission equation gives the power received by an antenna from a transmitting antenna in free space, in terms of the transmitted power, antenna gains, wavelength and separation.

Pr = Pt·Gt·Gr·(λ / 4πd)²
Pr(dBW) = Pt + Gt + Gr − 20 log(4πd/λ)

Derivation in brief

1. Power density from an antenna of gain Gt at distance d: Pt·Gt/(4πd²).
2. Power picked up by the receiving antenna = density × effective aperture Aer.
3. Aer = Gr·λ²/4π. Substituting gives the equation above.

Assumptions / conditions

• Far-field separation (d ≥ 2D²/λ).
• Free space: no obstacles, reflections or absorption.
• Matched polarisation and impedances; otherwise multiply by the polarisation loss factor and (1 − |Γt|²)(1 − |Γr|²).

Uses

• Link budget of satellite and microwave LOS links.
• Defines the free space loss LFS = 20 log(4πd/λ) = 22 + 20 log(d/λ) dB.
• Antenna gain measurement (two- and three-antenna methods).

Received power falls as 1/d², i.e. 6 dB for every doubling of distance.