---
title: "Short note: Friis transmission equation"
subject: "Propagation and Antenna"
chapter: 5
key: "semester_6/propagation_and_antenna:friis-short-note"
source: https://www.bctengineer.com/q/semester_6/propagation_and_antenna/friis-short-note
site: BCT Engineering
---

# Short note: Friis transmission equation

Propagation and Antenna · Chapter 5: Propagation between Antennas

- Asked in: 2073 Bhadra, 3 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/friis-short-note
- Whole chapter: https://www.bctengineer.com/md/semester_6/propagation_and_antenna/chapter-5

## Question

Write a short note on Friis transmission equation.

## Short answer

The Friis transmission equation gives the received power in free space as Pr = Pt Gt Gr (λ/(4πd))^2, or in dB, Pr(dBW) = Pt(dBW) + Gt + Gr − 20 log(4πd/λ). It assumes far-field, matched polarization, and no obstacles. Received power falls 6 dB per doubling of distance.

## Answer

The **Friis transmission equation** gives the power received by an antenna from a transmitting antenna in free space, in terms of the transmitted power, antenna gains, wavelength and separation.

```text
Pr = Pt·Gt·Gr·(λ / 4πd)²
Pr(dBW) = Pt + Gt + Gr − 20 log(4πd/λ)
```

#### Derivation in brief

1. Power density from an antenna of gain Gt at distance d: Pt·Gt/(4πd²).
2. Power picked up by the receiving antenna = density × effective aperture Aer.
3. Aer = Gr·λ²/4π. Substituting gives the equation above.

#### Assumptions / conditions

- Far-field separation (d ≥ 2D²/λ).
- Free space: no obstacles, reflections or absorption.
- Matched polarisation and impedances; otherwise multiply by the polarisation loss factor and (1 − |Γt|²)(1 − |Γr|²).

#### Uses

- Link budget of satellite and microwave LOS links.
- Defines the free space loss LFS = 20 log(4πd/λ) = 22 + 20 log(d/λ) dB.
- Antenna gain measurement (two- and three-antenna methods).

Received power falls as 1/d², i.e. 6 dB for every doubling of distance.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. The Friis transmission equation in free space is:
   A. Pr = Pt·Gt·Gr·(λ/4πd)^2
   B. Pr = Pt·(λ/4πd)
   C. Pr = Pt·Gt·Gr·λ/(4πd)
   D. Pr = Pt·Gt·Gr·(4πd/λ)^2
   Answer: A. Substituting Aer into Pt·Gt·Aer/(4πd^2) gives Pr = Pt·Gt·Gr·(λ/4πd)^2. (Key point: Friis result)

2. Friis equation requires the separation to satisfy:
   A. d ≤ λ
   B. d ≥ 2D^2/λ
   C. d = D
   D. d ≤ D^2
   Answer: B. Far-field distance is d ≥ 2D^2/λ. (Key point: Far field condition)

3. If the distance is doubled, the received power changes by:
   A. +6 dB
   B. −3 dB
   C. −6 dB
   D. −12 dB
   Answer: C. Power falls as 1/d^2. (Key point: Distance dependence)

4. If polarisation or impedance is mismatched, the received power is:
   A. Unchanged
   B. Zero in all cases
   C. Increased
   D. Multiplied by the polarisation loss factor and mismatch factors
   Answer: D. Extra factors (PLF, 1−|Γ|^2) reduce power. (Key point: Mismatch)

## Related in Propagation and Antenna

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