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

# Gain, efficiency and directivity

Propagation and Antenna · Chapter 2: Antenna Parameters and Arrays

- Asked in: 2075 Baisakh, 6 marks
- Video: [Relationship Between Gain, Directivity & Efficiency | Antenna Performance Explained](https://www.youtube.com/watch?v=f-rQ5l6LOuk) — WaveRes
- 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-directivity
- Whole chapter: https://www.bctengineer.com/md/semester_6/propagation_and_antenna/chapter-2

## Question

Describe the antenna gain, antenna efficiency and directivity of antenna with mathematical derivation if necessary.

## Short answer

Gain is G = 4πU/P_in = ηD. Efficiency is η = P_rad/P_in = Rr/(Rr+RL). Directivity is D = 4πU_max/P_rad; for a short dipole U = U0 sin²θ, giving P_rad = 8πU0/3 and D = 1.5 (1.76 dBi).

## 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 %.

#### Directivity
Directivity is the ratio of maximum radiation intensity to average radiation intensity:

```text
U_avg = P_rad / 4π
D = U_max / U_avg = 4π U_max / P_rad
P_rad = ∮ U dΩ = ∫₀²π ∫₀π U(θ,φ) sinθ dθ dφ
```

**Derivation for a short dipole** (U = U0 sin²θ):

```text
P_rad = U0 ∫₀²π ∫₀π sin³θ dθ dφ = U0 · 2π · 4/3 = 8πU0/3
D = 4π U0 / (8πU0/3) = 1.5  (1.76 dBi)
```

Approximation for narrow beams: D ≈ 41253/(θ1° θ2°).

## Check yourself

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

1. How are G, D and η related?
   A. G = D/η, so G >= D
   B. G = D + η
   C. G = ηD, so G <= D
   D. G = D always
   Answer: C. Gain uses input power; P_rad = ηP_in. (Key point: Gain vs directivity)

2. Rr = 73 Ω and RL = 2 Ω. What is η?
   A. 97.3 %
   B. 2.7 %
   C. 50 %
   D. 98.6 %
   Answer: A. η = 73/75. (Key point: Efficiency)

3. Directivity is defined as
   A. P_in divided by P_rad
   B. U_max divided by the average intensity U_avg = P_rad/4π
   C. U_avg divided by U_max
   D. U_max divided by P_in
   Answer: B. D = 4πU_max/P_rad. (Key point: Directivity definition)

4. For U = U0 sin²θ, P_rad = 8πU0/3. What is D?
   A. 1.64
   B. 3
   C. 0.67
   D. 1.5 (1.76 dBi)
   Answer: D. D = 4πU0/(8πU0/3) = 1.5, using ∫ sin³θ dθ = 4/3. (Key point: Short dipole directivity)

## Related in Propagation and Antenna

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- [Antenna directivity](https://www.bctengineer.com/question.md?key=semester_6%2Fpropagation_and_antenna%3Adirectivity) — Propagation and Antenna, chapter 2, asked 3×
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