---
title: "Friis equation and LFS = 22 + 20 log(d/λ)"
subject: "Propagation and Antenna"
chapter: 5
key: "semester_6/propagation_and_antenna:friis-free-space-loss-22-20log"
source: https://www.bctengineer.com/q/semester_6/propagation_and_antenna/friis-free-space-loss-22-20log
site: BCT Engineering
---

# Friis equation and LFS = 22 + 20 log(d/λ)

Propagation and Antenna · Chapter 5: Propagation between Antennas

- Asked in: 2080 Chaitra, 7+3 marks
- Video: [EC 8701 - Antennas and Microwave Engg: Friis Transmission Formula](https://www.youtube.com/watch?v=fjhamuW8RPU) — Learn through Digital
- 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-free-space-loss-22-20log
- Whole chapter: https://www.bctengineer.com/md/semester_6/propagation_and_antenna/chapter-5

## Question

Derive the Friis transmission equation for the free space propagation. Also find out the free space loss (LFS) equation that equals to 22 + 20 log(d/λ), where d is distance between Tx and Rx antennas and λ is wavelength of the frequency being used.

## Short answer

Friis equation: Pr=Pt Gt Gr λ²/(4πd)². For isotropic antennas Gt=Gr=1, LFS=Pt/Pr=(4πd/λ)²; in dB, LFS=20log(4π)+20log(d/λ)=21.98+20log(d/λ)≈22+20log(d/λ), with d and λ in same units.

## Answer

The **Friis transmission equation** relates the power received by an antenna to the power transmitted by another antenna in free space, through their gains, the wavelength and the distance between them.

#### 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)
```

The equation is valid only in the far field (d ≥ 2D²/λ), for matched polarisation and impedance, and for an unobstructed path. If not, extra factors (polarisation loss factor, reflection loss 1 − |Γ|²) are multiplied in.

#### Free space loss LFS = 22 + 20 log(d/λ)

Free space loss is the ratio Pt/Pr between two **isotropic** antennas (Gt = Gr = 1):

```text
LFS = Pt/Pr = (4πd/λ)²
```

Taking 10 log of both sides:

```text
LFS(dB) = 10 log (4πd/λ)²
        = 20 log(4πd/λ)
        = 20 log(4π) + 20 log(d/λ)
        = 20 × 1.0992 + 20 log(d/λ)
        = 21.98 + 20 log(d/λ)
        ≈ 22 + 20 log(d/λ)  dB
```

Hence proved. Here *d* and λ must be in the same units. In practical units, LFS = 32.44 + 20 log d(km) + 20 log f(MHz) dB.

## Check yourself

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

1. Pr = P_d x Aer where P_d is:
   A. Pt Gt/(4 pi d^2)
   B. Pt/(4 pi d)
   C. Pt Gt d^2
   D. Pt/Gt
   Answer: A. A directive transmitter multiplies the isotropic density by Gt. (Key point: Derivation)

2. LFS = (4 pi d/lambda)^2 assumes:
   A. Gt = Gr = 1 (isotropic antennas)
   B. Gt = Gr = 10
   C. Near-field
   D. Rain present
   Answer: A. LFS is defined between isotropic antennas. (Key point: Isotropic condition)

3. Where does the 22 in 22 + 20 log(d/lambda) come from?
   A. 20 log(4 pi) = 21.98
   B. 10 log(4 pi)
   C. 20 log(2 pi)
   D. log(4 pi)
   Answer: A. 20 x 1.0992 = 21.98, approximately 22 dB. (Key point: Constant 22)

4. The Friis equation is valid in the:
   A. Far field, d >= 2D^2/lambda
   B. Near field
   C. Reactive zone
   D. Any distance
   Answer: A. It requires far field, matched polarisation and impedance, and an unobstructed path. (Key point: Validity)

## Related in Propagation and Antenna

- [Friis equation and free space path loss](https://www.bctengineer.com/question.md?key=semester_6%2Fpropagation_and_antenna%3Afriis-path-loss-free-space) — Propagation and Antenna, chapter 5, asked 2×
- [Free space propagation and Friis derivation](https://www.bctengineer.com/question.md?key=semester_6%2Fpropagation_and_antenna%3Afree-space-friis-complete) — Propagation and Antenna, chapter 5, asked 2×
- [Transmission loss and Friis equation](https://www.bctengineer.com/question.md?key=semester_6%2Fpropagation_and_antenna%3Atransmission-loss-friis) — Propagation and Antenna, chapter 5
- [Free space loss (LFS)](https://www.bctengineer.com/question.md?key=semester_6%2Fpropagation_and_antenna%3Afree-space-loss) — Propagation and Antenna, chapter 5
- [Friis, path loss and LOS distance](https://www.bctengineer.com/question.md?key=semester_6%2Fpropagation_and_antenna%3Afriis-path-loss-los-distance) — Propagation and Antenna, chapter 5
- [10 GHz LOS link loss and Friis derivation](https://www.bctengineer.com/question.md?key=semester_6%2Fpropagation_and_antenna%3Alos-link-10ghz-100km-friis) — Propagation and Antenna, chapter 5

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