---
title: "Path loss in radio wave propagation"
subject: "Propagation and Antenna"
chapter: 5
key: "semester_6/propagation_and_antenna:path-loss-derivation"
source: https://www.bctengineer.com/q/semester_6/propagation_and_antenna/path-loss-derivation
site: BCT Engineering
---

# Path loss in radio wave propagation

Propagation and Antenna · Chapter 5: Propagation between Antennas

- Asked in: 2073 Bhadra, 6 marks; 2069 Bhadra, 12 marks (asked 2 times)
- Video: [6 - Free Space Path Loss](https://www.youtube.com/watch?v=-R4ljyNamlc) — GTHillWiFi
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/propagation_and_antenna/path-loss-derivation
- Whole chapter: https://www.bctengineer.com/md/semester_6/propagation_and_antenna/chapter-5

## Question

Derive the expression for the path loss in case of radio wave propagation.

## Short answer

Path loss is Pt/Pr; free-space loss with isotropic antennas is LFS=(4πd/λ)^2, so LFS(dB)=20log(4πd/λ)=32.44+20log d(km)+20log f(MHz). With antenna gains, L(dB)=20log(4πd/λ)-Gt-Gr. Over a plane reflecting earth loss becomes 40 dB per decade and independent of frequency.

## Answer

**Path loss** is the reduction in power of a radio signal as it travels from the transmitting antenna to the receiving antenna, expressed as the ratio of transmitted power to received power (usually in dB). In free space it is caused only by the spherical spreading of the wave; over real paths, ground reflection, obstacles and the atmosphere add further loss.

#### Basic idea: spreading of power

```text
           .-~~~-.      sphere of area 4πd²
         /         \
   Tx  (  * ------- )--> Rx (aperture Ae)
         \         /
           '-...-.
```

An isotropic antenna radiating *Pt* watts spreads it uniformly over a sphere of radius *d*:

```text
P_iso = Pt / (4πd²)   W/m²
```

#### Effect of antenna gains

A transmitting antenna of gain *Gt* concentrates the power in its main beam:

```text
P_d = Pt·Gt / (4πd²)
```

The receiving antenna extracts power in proportion to its effective aperture:

```text
Pr = P_d · Ae ,   with  Ae = Gr·λ² / 4π

Pr = Pt·Gt·Gr·λ² / (4πd)²        (Friis equation)
```

In dB form:

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

#### Path loss (with antenna gains)

The overall transmission loss between the transmitter output and the receiver input is

```text
L = Pt/Pr = (4πd/λ)² / (Gt·Gr)

L(dB) = 20 log(4πd/λ) − Gt(dB) − Gr(dB)
```

#### Free space (basic) path loss

With isotropic antennas (Gt = Gr = 1) we get the **basic free space loss**, which depends only on distance and wavelength:

```text
LFS = (4πd/λ)²

LFS(dB) = 20 log(4π) + 20 log d − 20 log λ
        = 21.98 + 20 log(d/λ)
```

Putting λ = c/f = 3×10⁸/f:

```text
LFS = 20 log(4π/3×10⁸) + 20 log d(m) + 20 log f(Hz)
    = −147.56 + 20 log d(m) + 20 log f(Hz)

With d in km and f in MHz:
LFS = 32.44 + 20 log d(km) + 20 log f(MHz)  dB
```

#### Path loss over a flat reflecting earth (two-ray model)

For a real LOS link above ground, the receiver gets a direct ray and a ground-reflected ray. For a perfectly reflecting ground (Γ ≈ −1) and d ≫ ht, hr, the path difference is Δ = 2ht·hr/d, and the received power becomes

```text
Pr = Pt·Gt·Gr · (ht·hr)² / d⁴

L(dB) = 40 log d − 20 log ht − 20 log hr − Gt − Gr
```

So over a plane earth the loss rises at **40 dB per decade** (12 dB per doubling of distance), compared with 20 dB per decade in free space, and it no longer depends on frequency.

#### Factors that add to path loss in practice

- Obstacles and diffraction (knife-edge loss, Fresnel zone blockage)
- Atmospheric absorption by oxygen and water vapour (above ~10 GHz)
- Rain attenuation (important above ~10 GHz)
- Multipath fading and shadowing in mobile links

#### Summary

| Model | Pr depends on | Loss slope |
|---|---|---|
| Free space | 1/d², λ² | 20 dB/decade |
| Plane earth (two-ray) | 1/d⁴, (ht·hr)² | 40 dB/decade |

**Example:** f = 6 GHz, d = 10,500 km gives LFS = 20 log(4π × 1.05×10⁷/0.05) = 188.4 dB.

## Check yourself

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

1. Path loss including antenna gains is:
   A. L = (4 pi d/lambda)^2 x Gt x Gr
   B. L = (4 pi d/lambda)^2/(Gt Gr)
   C. L = Gt Gr/(4 pi d/lambda)^2
   D. L = (lambda/4 pi d)^2
   Answer: B. L = Pt/Pr = (4 pi d/lambda)^2/(Gt Gr); gains reduce the loss. (Key point: Path loss with gains)

2. Over flat reflecting earth with d much larger than heights, Pr is proportional to:
   A. 1/d^2
   B. 1/d^4
   C. 1/d
   D. 1/d^3
   Answer: B. Pr = Pt Gt Gr (ht hr)^2/d^4, giving 40 dB/decade. (Key point: Two-ray model)

3. In the two-ray plane-earth model the loss:
   A. Depends strongly on frequency
   B. Does not depend on frequency
   C. Falls with distance
   D. Depends on lambda^2
   Answer: B. The result depends on ht, hr and d only. (Key point: Frequency dependence)

4. Free space loss slope compared with plane earth:
   A. 20 dB/decade versus 40 dB/decade
   B. 40 versus 20
   C. Both 20
   D. Both 40
   Answer: A. Free space loses 20 dB per decade; plane earth 40 dB per decade. (Key point: Slope)

## 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×
- [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
- [Free space path loss: 6 GHz, 10,500 km](https://www.bctengineer.com/question.md?key=semester_6%2Fpropagation_and_antenna%3Afspl-derivation-6ghz-10500km) — Propagation and Antenna, chapter 5
- [Radio wave types and band characteristics](https://www.bctengineer.com/question.md?key=semester_6%2Fpropagation_and_antenna%3Aradio-wave-designation-bands) — Propagation and Antenna, chapter 4, asked 2×
- [Ionospheric wave propagation and layers](https://www.bctengineer.com/question.md?key=semester_6%2Fpropagation_and_antenna%3Aionospheric-propagation-layers) — Propagation and Antenna, chapter 4, asked 3×
- [Short note: Medium wave propagation](https://www.bctengineer.com/question.md?key=semester_6%2Fpropagation_and_antenna%3Amw-propagation-short-note) — Propagation and Antenna, chapter 4, asked 3×

## Related in other subjects

- [Okumura model and median path loss numerical](https://www.bctengineer.com/question.md?key=semester_7%2Fwireless_communication%3Aokumura-model-median-path-loss) — Wireless Communication, chapter 3
- [Large scale path loss and small scale fading](https://www.bctengineer.com/question.md?key=semester_7%2Fwireless_communication%3Alarge-scale-loss-small-fading) — Wireless Communication, chapter 3
- [Radiation pressure of radio wave](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Aradiation-pressure-radio-wave) — Engineering Physics, chapter 9
- [Two-ray model advantages and path loss derivation](https://www.bctengineer.com/question.md?key=semester_7%2Fwireless_communication%3Atwo-ray-advantages-path-loss) — Wireless Communication, chapter 3