Free space propagation and Friis derivation

Free space propagation is radio communication between a transmitter and receiver that have a clear, unobstructed line-of-sight path, with no reflection, refraction, absorption or diffraction from the earth, atmosphere or objects. The only effect on the signal is the spreading of energy over a larger and larger sphere, so the power density falls as 1/d². Satellite links and terrestrial microwave LOS links (with a clear first Fresnel zone) are close to free space.

Assumptions of the free space model

• Antennas are in each other's far field.
• The medium is lossless, homogeneous and isotropic (εr = μr = 1).
• Polarisations are matched and the antennas are matched to their loads.
• No obstacles in the first Fresnel zone, and no ground reflection.

Derivation of the Friis free space 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)

This shows that the received power falls with the square of distance (6 dB per doubling of d) and, for fixed gains, with the square of frequency.

Free space path loss

The free space path loss LFS is the ratio of transmitted to received power when both antennas are isotropic (Gt = Gr = 1), i.e. the loss due to spreading of the wave alone:

LFS = Pt/Pr = (4πd/λ)²

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

Using λ = c/f, with d in km and f in MHz:

LFS(dB) = 32.44 + 20 log d(km) + 20 log f(MHz)

(With f in GHz the constant becomes 92.44.) The loss grows by 6 dB each time the distance or the frequency is doubled.

Example: At f = 10 GHz (λ = 0.03 m) and d = 50 km, LFS = 20 log(4π × 50000/0.03) = 146.4 dB.