---
title: "Gain, efficiency and beamwidth"
subject: "Propagation and Antenna"
chapter: 2
key: "semester_6/propagation_and_antenna:gain-efficiency-beamwidth"
source: https://www.bctengineer.com/q/semester_6/propagation_and_antenna/gain-efficiency-beamwidth
site: BCT Engineering
---

# Gain, efficiency and beamwidth

Propagation and Antenna · Chapter 2: Antenna Parameters and Arrays

- Asked in: 2072 Magh, 6 marks
- Video: [Antenna Beamwidth - HPBW and FNBW](https://www.youtube.com/watch?v=aqSGyrhH0AU) — Padmasri Naban
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/propagation_and_antenna/gain-efficiency-beamwidth
- Whole chapter: https://www.bctengineer.com/md/semester_6/propagation_and_antenna/chapter-2

## Question

Describe antenna gain, antenna efficiency and beam width of antenna with mathematical derivations if necessary.

## Short answer

Gain is G = 4πU/P_in = ηD. Efficiency is η = P_rad/P_in = Rr/(Rr+RL). HPBW is the angle between half-power directions; for a short dipole sin²θ = 1/2 gives θ = 45° and 135°, so HPBW = 90°, while a λ/2 dipole has about 78°.

## Answer

#### Antenna gain
Gain is the ratio of the radiation intensity in a given direction to the radiation intensity that would be obtained if the **input power** were radiated isotropically:

```text
G(θ,φ) = 4π U(θ,φ) / P_in
P_rad = η P_in   →   G = η D
G(dBi) = 10 log10 G
```

It includes losses, so G ≤ D. Example: λ/2 dipole, D = 1.64; if η = 0.97, G = 1.59 (2.0 dBi). For aperture antennas G = 4πAe/λ².

#### Antenna efficiency
Radiation efficiency is the ratio of power radiated to power accepted at the terminals:

```text
η = P_rad / P_in = Rr / (Rr + RL)
```

Rr = radiation resistance, RL = ohmic/dielectric loss resistance. Total efficiency also includes mismatch: e0 = (1 − |Γ|²) · η.

Example: Rr = 73 Ω, RL = 2 Ω → η = 73/75 = 97.3 %.

#### Beamwidth
**HPBW** is the angle between the two directions in the main lobe where power falls to half (field to 0.707) of maximum. **FNBW** is the angle between the first nulls.

**Derivation for a short dipole** (E ∝ sinθ, power ∝ sin²θ):

```text
sin²θ = 1/2  →  sinθ = 1/√2
θ1 = 45°, θ2 = 135°
HPBW = 135° − 45° = 90°
```

For a λ/2 dipole, E = cos[(π/2)cosθ]/sinθ = 0.707 gives θ ≈ 51° and 129° → HPBW ≈ 78°.

```text
        max
       .---.
 0.707/  |  \0.707
     /<HPBW>\
    / <-FNBW->\
```

A narrower beam means higher directivity: D ≈ 41253/(θE° θH°).

## Check yourself

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

1. Antenna gain is referred to which power?
   A. Radiated power
   B. Reflected power
   C. Input power, G = ηD
   D. Average radiated power
   Answer: C. Gain includes losses. (Key point: Gain definition)

2. Which expression gives radiation efficiency?
   A. η = Rr/(Rr + RL)
   B. η = RL/(Rr + RL)
   C. η = Rr/RL
   D. η = Rr + RL
   Answer: A. It is the ratio of radiated to accepted power. (Key point: Efficiency)

3. What is HPBW?
   A. The angle between the first nulls
   B. The angle between the directions in the main lobe where power falls to half (field 0.707) of maximum
   C. The angle of the side lobes
   D. The angle where the field is zero
   Answer: B. FNBW is the first-null angle. (Key point: HPBW definition)

4. For a short dipole sin²θ = 1/2 gives what HPBW?
   A. 45°
   B. 78°
   C. 180°
   D. 90° (from 45° to 135°)
   Answer: D. 135° - 45° = 90°; the λ/2 dipole is about 78°. (Key point: Short dipole HPBW)

## Related in Propagation and Antenna

- [Antenna beamwidth (HPBW)](https://www.bctengineer.com/question.md?key=semester_6%2Fpropagation_and_antenna%3Abeamwidth) — Propagation and Antenna, chapter 2, asked 3×
- [Gain, efficiency and polarization](https://www.bctengineer.com/question.md?key=semester_6%2Fpropagation_and_antenna%3Again-efficiency-polarization) — Propagation and Antenna, chapter 2
- [Gain, efficiency and directivity](https://www.bctengineer.com/question.md?key=semester_6%2Fpropagation_and_antenna%3Again-efficiency-directivity) — Propagation and Antenna, chapter 2
- [Antenna gain](https://www.bctengineer.com/question.md?key=semester_6%2Fpropagation_and_antenna%3Aantenna-gain) — Propagation and Antenna, chapter 2
- [Antenna gain and directivity](https://www.bctengineer.com/question.md?key=semester_6%2Fpropagation_and_antenna%3Aantenna-gain-and-directivity) — Propagation and Antenna, chapter 2
- [Receiving antenna gain](https://www.bctengineer.com/question.md?key=semester_6%2Fpropagation_and_antenna%3Areceiving-antenna-gain) — Propagation and Antenna, chapter 2

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