--- title: "Friis equation and free space path loss" subject: "Propagation and Antenna" chapter: 5 key: "semester_6/propagation_and_antenna:friis-path-loss-free-space" source: https://www.bctengineer.com/q/semester_6/propagation_and_antenna/friis-path-loss-free-space site: BCT Engineering --- # Friis equation and free space path loss Propagation and Antenna · Chapter 5: Propagation between Antennas - Asked in: 2072 Asoj, 3 marks; 2071 Bhadra, 4+3 marks (asked 2 times) - Video: [Friis Transmission Formula (Basics, Formula & Derivation) Explained](https://www.youtube.com/watch?v=Op7bBz6qbwY) — Engineering Funda - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_6/propagation_and_antenna/friis-path-loss-free-space - Whole chapter: https://www.bctengineer.com/md/semester_6/propagation_and_antenna/chapter-5 ## Question How do you get Friis transmission equation and path loss in case of free space wave propagation? ## Short answer Friis transmission equation: Pr = Pt*Gt*Gr*(lambda/(4*pi*d))^2, derived from power density of isotropic radiator, antenna gain, and effective aperture. Free space path loss LFS = (4*pi*d/lambda)^2 = 32.44 + 20log(d_km) + 20log(f_MHz) dB for isotropic antennas. ## Answer The **Friis transmission equation** gives the power received by one antenna from another in free space in terms of the transmitted power, the two antenna gains, the wavelength and the distance. The **path loss** follows from it by setting both gains to unity. #### 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: ```text 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: ```text Pr(dBW) = Pt(dBW) + Gt(dB) + Gr(dB) − 20 log(4πd/λ) ``` #### Path loss in free space 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: ```text 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: ```text 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. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What is the power density at distance d from an isotropic source of power Pt? A. Pt/(4 pi d) B. Pt/(4 pi d^2) C. Pt/d^2 D. Pt/(2 pi d^2) Answer: B. Power spreads over a sphere of area 4 pi d^2. (Key point: Spreading) 2. The Friis equation is: A. Pr/Pt = Gt Gr (lambda/4 pi d)^2 B. Pr/Pt = Gt Gr (4 pi d/lambda)^2 C. Pr/Pt = Gt Gr lambda/(4 pi d) D. Pr/Pt = (lambda d)^2 Answer: A. Using Aer = Gr lambda^2/4pi gives Pr = Pt Gt Gr lambda^2/(4 pi d)^2. (Key point: Friis result) 3. Free space path loss is obtained by: A. Setting Gt = Gr = 1, giving (4 pi d/lambda)^2 B. Setting d = 0 C. Setting lambda = 1 D. Doubling both gains Answer: A. With isotropic antennas LFS = Pt/Pr = (4 pi d/lambda)^2, rising 6 dB per doubling. 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