---
title: "Basic propagation models and far-field distance"
subject: "Wireless Communication"
chapter: 3
key: "semester_7/wireless_communication:propagation-models-far-field"
source: https://www.bctengineer.com/q/semester_7/wireless_communication/propagation-models-far-field
site: BCT Engineering
---

# Basic propagation models and far-field distance

Wireless Communication · Chapter 3: Radio Wave Propagation in Mobile Networks

- Asked in: 2079 Chaitra, 3+3 marks
- Video: [Free Space Propagation Model](https://www.youtube.com/watch?v=-CsPxMjFEOU) — Ekeeda
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_7/wireless_communication/propagation-models-far-field
- Whole chapter: https://www.bctengineer.com/md/semester_7/wireless_communication/chapter-3

## Question

Explain basic propagation models. Compute the far field distance for an antenna with maximum dimension of 1 m and operating frequency of 900 MHz.

## Short answer

Basic models include free space (Friis), two-ray ground reflection, log-distance, and log-normal shadowing. Far field distance df = 2D²/λ; with D=1 m, f=900 MHz (λ=0.333 m), gives df = 6 m.

## Answer

Propagation models predict the average received signal strength at a given distance (**large-scale models**) or its rapid fluctuations over short distances (**small-scale/fading models**).

#### Basic propagation models
**1. Free space model (Friis)**
Used when there is a clear, unobstructed LOS path, e.g. satellite and microwave links.

```text
Pr(d) = Pt·Gt·Gr·λ² / [(4π)²d²L]
PL(dB) = 20 log(4πd/λ)
```
Power falls as 1/d² (20 dB/decade). Valid only in the far field (d ≥ df).

**2. Two-ray ground reflection model**
Considers the direct path plus a ground-reflected path. For large d:

```text
Pr = Pt·Gt·Gr·ht²·hr² / d⁴
```
Power falls as 1/d⁴ (40 dB/decade). More accurate than free space for long distances over flat ground.

**3. Log-distance path loss model**
```text
PL(d) = PL(d0) + 10n log(d/d0)
```
n = 2 free space, 2.7–3.5 urban, 3–5 shadowed urban.

**4. Log-normal shadowing**
Adds a zero-mean Gaussian random variable Xσ (dB) to log-distance loss to account for obstacles: PL(d) = PL(d0) + 10n log(d/d0) + Xσ.

Empirical outdoor models (Okumura, Hata) and indoor models (partition loss, Ericsson) build on these.

#### Far field distance
The far field (Fraunhofer region) begins at

```text
df = 2D²/λ,  with df ≫ D and df ≫ λ
```
Given D = 1 m, f = 900 MHz:

```text
λ  = 3×10⁸ / 900×10⁶ = 0.333 m
df = 2 × 1² / 0.333 = 6 m
Check: 6 m ≫ D (1 m) and 6 m ≫ λ (0.333 m)
```

**Answer: Far field distance df = 6 m.**

## Check yourself

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

1. At large distance, how does received power fall in the two-ray ground reflection model?
   A. As 1/d (10 dB/decade)
   B. As 1/d^2 (20 dB/decade)
   C. As 1/d^4 (40 dB/decade)
   D. As 1/d^3 (30 dB/decade)
   Answer: C. Pr = Pt Gt Gr ht^2 hr^2 / d^4, i.e. 40 dB/decade, versus 20 dB/decade in free space. (Key point: Two-ray power law)

2. What does log-normal shadowing add to the log-distance model?
   A. A fixed 6 dB loss
   B. A deterministic frequency term
   C. A zero-mean Gaussian random variable Xσ in dB for obstacles
   D. A Rayleigh fading term
   Answer: C. PL = PL(d0) + 10n log(d/d0) + Xσ accounts for variation due to obstacles. (Key point: Log-normal shadowing)

3. The far-field (Fraunhofer) distance of an antenna of largest dimension D is:
   A. df = 2D^2/λ
   B. df = D^2/(2λ)
   C. df = 2D/λ
   D. df = λ/(2D^2)
   Answer: A. df = 2D^2/λ, with df much greater than D and λ. (Key point: Far field distance formula)

4. For D = 1 m at 900 MHz, the far-field distance is:
   A. 0.67 m
   B. 3 m
   C. 12 m
   D. 6 m
   Answer: D. λ = 0.333 m, so df = 2 x 1/0.333 = 6 m. (Key point: Far field result)

## Related in Wireless Communication

- [Outdoor propagation models (any two)](https://www.bctengineer.com/question.md?key=semester_7%2Fwireless_communication%3Aoutdoor-propagation-models) — Wireless Communication, chapter 3, asked 3×
- [Indoor propagation models (any two)](https://www.bctengineer.com/question.md?key=semester_7%2Fwireless_communication%3Aindoor-propagation-models) — Wireless Communication, chapter 3
- [Large vs small scale models; propagation mechanisms](https://www.bctengineer.com/question.md?key=semester_7%2Fwireless_communication%3Alarge-small-scale-propagation) — Wireless Communication, chapter 3
- [Numerical: received power, E-field, flux density, voltage](https://www.bctengineer.com/question.md?key=semester_7%2Fwireless_communication%3Afree-space-received-power-e-field) — Wireless Communication, chapter 3, asked 2×
- [Numerical: reverse link distance by link budget](https://www.bctengineer.com/question.md?key=semester_7%2Fwireless_communication%3Alink-budget-reverse-link-distance) — Wireless Communication, chapter 3
- [Numerical: two-ray vs free-space path loss at 800 MHz](https://www.bctengineer.com/question.md?key=semester_7%2Fwireless_communication%3Atwo-ray-path-loss-800mhz) — Wireless Communication, chapter 3

## Related in other subjects

- [BTS radiation fields at 900 and 1800 MHz](https://www.bctengineer.com/question.md?key=semester_7%2Frf_and_microwave_engineering%3Abts-900-1800-radiation-fields) — RF and Microwave Engineering, chapter 7
- [Critical frequency; MUF and skip distance (flat earth)](https://www.bctengineer.com/question.md?key=semester_6%2Fpropagation_and_antenna%3Acritical-frequency-skip-muf-flat) — Propagation and Antenna, chapter 4, asked 2×
- [Define antenna and EMW propagation](https://www.bctengineer.com/question.md?key=semester_2%2Fbasic_electronics_engineering%3Adefine-antenna-emw) — Basic Electronics Engineering, chapter 5
- [Critical frequency and skip distance (flat earth)](https://www.bctengineer.com/question.md?key=semester_6%2Fpropagation_and_antenna%3Acritical-frequency-skip-distance-flat) — Propagation and Antenna, chapter 4, asked 2×