What is the Friis transmission equation?

From RF Design Wiki · By the RF Design Wiki editorial team · Published October 21, 2026 · Last reviewed October 21, 2026 · How we research

The Friis transmission equation, published by Harald Friis in 1946, gives the power received between two antennas in free space: P_r = P_tG_tG_r(λ/(4πd))². In decibels, P_r = P_t + G_t + G_r − FSPL.^[1]^[2]

What assumptions does it make?

Far-field separation (d ≫ 2D²/λ).
Free space with no obstructions or reflections.
Antennas impedance- and polarization-matched (otherwise multiply by (1 − |Γ|²) and PLF).
Gains in the direction of the other antenna.

Why does path loss rise with frequency?

The isotropic path loss term (4πd/λ)² rises with frequency because a fixed-gain receiving antenna’s effective aperture A_e = Gλ²/(4π) shrinks. With fixed-aperture antennas at both ends, received power actually increases with frequency: P_r = P_tA_etA_er/(λ²d²).^[1]

What is a worked example?

2.4 GHz, 100 m, P_t = +20 dBm, 2 dBi antennas each: FSPL = 20·log10(4π·100/0.125) ≈ 80.0 dB, so P_r ≈ 20 + 2 + 2 − 80 = −56 dBm in free space. Real indoor links lose much more; see path loss models.

Frequently asked questions

Is Friis valid indoors?

Only as a best-case reference; walls and multipath change received power.

What is the radar version?

The radar equation, with an R⁴ dependence; see radar basics.

Who was Friis?

Harald T. Friis, a Bell Labs engineer who also derived the noise figure cascade formula.

References

Online sources accessed October 21, 2026; printed works cited by edition. See all sources.

“A Note on a Simple Transmission Formula”. H. T. Friis, Proceedings of the IRE, vol. 34, no. 5, pp. 254-256. May 1946.
“Antenna Theory: Analysis and Design, 4th ed.”. C. A. Balanis, Wiley. 2016.

Categories: Antennas and propagation

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