--- title: "Transmission loss and Friis equation" subject: "Propagation and Antenna" chapter: 5 key: "semester_6/propagation_and_antenna:transmission-loss-friis" source: https://www.bctengineer.com/q/semester_6/propagation_and_antenna/transmission-loss-friis site: BCT Engineering --- # Transmission loss and Friis equation Propagation and Antenna · Chapter 5: Propagation between Antennas - Asked in: 2076 Baisakh, 2+6 marks - 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/transmission-loss-friis - Whole chapter: https://www.bctengineer.com/md/semester_6/propagation_and_antenna/chapter-5 ## Question Define transmission loss. Derive the Friis transmission equation of free space propagation. ## Short answer Transmission loss is L = 10 log(Pt/Pr). Free-space path loss with isotropic antennas is LFS = 20 log(4πd/λ). Friis equation: Pr = Pt Gt Gr (λ/(4πd))^2. In dB: Pr(dBW) = Pt(dBW) + Gt + Gr − 20 log(4πd/λ). ## Answer #### Transmission loss **Transmission loss** is the ratio (in dB) of the power fed to the transmitting antenna to the power available at the receiving antenna terminals: L = 10 log(Pt/Pr). In free space it is the spreading loss reduced by the antenna gains, L = 20 log(4πd/λ) − Gt − Gr. With isotropic antennas it equals the **basic free space loss** LFS = 20 log(4πd/λ). Real links add losses from obstacles, ground, atmosphere, rain and feeders. #### Derivation of the 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) ``` From this, the transmission loss is ```text L = Pt/Pr = (4πd/λ)² / (Gt·Gr) L(dB) = 32.44 + 20 log d(km) + 20 log f(MHz) − Gt − Gr ``` The received power falls as 1/d² (6 dB per doubling of distance). Doubling the frequency adds 6 dB of free space loss, but if dish antennas of fixed size are used, both gains rise by 6 dB, so high-frequency links with dishes actually gain overall. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Transmission loss is the ratio of: A. Antenna gain to path loss B. Noise power to signal power C. Received to transmitted power D. Power fed to the transmitting antenna to power at the receiving antenna terminals Answer: D. L = 10 log(Pt/Pr). (Key point: Transmission loss) 2. In free space the transmission loss with gains is: A. 20 log(4πd/λ) + Gt + Gr B. 20 log(λ/4πd) − Gt − Gr C. 10 log(4πd/λ) − Gt D. 20 log(4πd/λ) − Gt − Gr Answer: D. Antenna gains reduce the spreading loss. (Key point: Loss with gains) 3. The Friis transmission equation in free space is: A. Pr = Pt·Gt·Gr·(4πd/λ)^2 B. Pr = Pt·Gt·Gr·(λ/4πd)^2 C. Pr = Pt·(λ/4πd) D. Pr = Pt·Gt·Gr·λ/(4πd) Answer: B. Substituting Aer into Pt·Gt·Aer/(4πd^2) gives Pr = Pt·Gt·Gr·(λ/4πd)^2. (Key point: Friis result) 4. In deriving the Friis equation, which step introduces the receiving gain Gr? A. Squaring the ratio λ/d B. Dividing Pt by the sphere area 4πd^2 C. Multiplying the power density by Gt D. Replacing the effective aperture Aer by Gr·λ^2/4π Answer: D. The received power is density × Aer, and Aer = Gr·λ^2/(4π) links aperture to gain, which brings Gr into the equation. (Key point: Friis steps) 5. What happens to the free space loss when the distance doubles? A. Increases by 12 dB B. Unchanged C. Increases by 3 dB D. Increases by 6 dB Answer: D. Pr falls as 1/d^2, so loss rises by 20 log 2 = 6 dB. 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