An expression for free‑space line‑of‑sight link power: the received power equals the transmitted power multiplied by the transmit and receive antenna gains and by the square of the wavelength over (4π times the separation distance) squared, i.e. Pr = Pt·Gt·Gr·(λ/(4πR))^2, valid in the far‑field for matched, polarization‑aligned antennas and no obstructions.

Principle

Received power decays with the square of distance in free space and depends on frequency via the wavelength factor; antenna gains linearly scale the received power in the link budget.

Demonstration

Illustrative calculation → A transmitter radiates Pt = 1 W at frequency f with isotropic antennas (Gt = Gr = 1) separated by R = 100 m. Using λ = c/f, compute Pr = Pt·(λ/(4πR))^2 to obtain the ideal free‑space received power; this serves as the baseline before adding system losses.

Misapplication

Using the Friis formula in near‑field regions (R comparable to antenna dimensions), in obstructed or multipath environments, with mismatched or cross‑polarized antennas, or ignoring additional losses leads to optimistic and incorrect received‑power estimates.

Consequence

Friis provides a simple baseline for link‑budget calculations, antenna sizing and frequency selection under ideal LOS assumptions; real deployments must add diffraction, scattering, fading and implementation losses to obtain realistic budgets.

Reversal

When propagation departs from free‑space — e.g. strong multipath, ground reflection, waveguiding, or near‑field coupling — or when antennas are not in far‑field or not matched/polarized, the equation no longer predicts received power accurately.

Boundary

Valid for: far‑field, line‑of‑sight, free‑space propagation; matched impedances; polarization alignment; narrowband single‑path assumption. Not valid for: near‑field coupling, dense multipath, obstructed terrains, or frequency‑selective channels without modification.

Semantic Tension

Simplicity versus realism — Friis is compact and exact under its assumptions but omits propagation complexity that models (e.g., empirical path‑loss, ray tracing, or stochastic fading) capture for real environments.

Synthesis

Treat Friis as the canonical free‑space reference in link design: it sets an ideal upper‑limit baseline that real‑world link budgets must correct by adding channel, antenna and implementation losses.