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

# Gain, efficiency and polarization

Propagation and Antenna · Chapter 2: Antenna Parameters and Arrays

- Asked in: 2076 Bhadra, 6 marks
- Video: [Antenna efficiency, Power Gain, Relationship between Power gain and Directivity of Antenna](https://www.youtube.com/watch?v=dU9r4YQPcyA) — Prathipati Rakesh Kumar
- 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-polarization
- Whole chapter: https://www.bctengineer.com/md/semester_6/propagation_and_antenna/chapter-2

## Question

Describe the antenna gain, antenna efficiency and antenna polarization with mathematical relation if necessary.

## Short answer

Gain is the ratio of radiation intensity to that of an isotropic radiator fed with the same input power: G = 4πU/P_in = ηD. Efficiency is P_rad/P_in = Rr/(Rr+RL). Polarization is the traced E-field orientation; linear for δ = 0 or π, circular for equal amplitudes with δ = ±90°, otherwise elliptical, with PLF = cos²ψ.

## 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 %.

#### Antenna polarization
Polarization is the orientation of the **E-field vector** traced with time at a fixed far-field point in the direction of maximum radiation. Write the field as two orthogonal components:

```text
Ex = Ex0 cos(ωt − kz)
Ey = Ey0 cos(ωt − kz + δ)
```

- **Linear:** δ = 0 or π (or one component zero).
- **Circular:** Ex0 = Ey0 and δ = ±90° (RHCP / LHCP).
- **Elliptical:** all other cases.

Axial ratio AR = major/minor axis (1 for circular, ∞ for linear). Polarization loss factor PLF = |ρ̂w · ρ̂a|² = cos²ψ.

## Check yourself

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

1. Antenna gain is defined with reference to which power?
   A. Input power, with P_rad = ηP_in so G = ηD
   B. Radiated power only
   C. Reflected power
   D. Average power over a sphere
   Answer: A. Gain includes losses, so G <= D. (Key point: Gain definition)

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

3. Which condition gives circular polarization for Ex = Ex0 cos(ωt - kz), Ey = Ey0 cos(ωt - kz + δ)?
   A. δ = 0 or π
   B. Ex0 = Ey0 and δ = ±90°
   C. Ex0 = 0
   D. Ex0 ≠ Ey0 and δ = 0
   Answer: B. Equal amplitudes and quadrature phase give a rotating vector of constant length. (Key point: Circular polarization condition)

4. What are the axial ratio for linear polarization and the PLF?
   A. AR = 1 for linear; PLF = sin ψ
   B. AR = 0 for linear; PLF = cos ψ
   C. AR = ∞ for circular; PLF = tan²ψ
   D. AR = ∞ for linear (1 for circular); PLF = cos²ψ
   Answer: D. Linear has an infinitely thin ellipse, and PLF is the squared dot product of unit vectors. (Key point: Axial ratio and PLF)

## Related in Propagation and Antenna

- [Gain, efficiency and directivity](https://www.bctengineer.com/question.md?key=semester_6%2Fpropagation_and_antenna%3Again-efficiency-directivity) — Propagation and Antenna, chapter 2
- [Gain, efficiency and beamwidth](https://www.bctengineer.com/question.md?key=semester_6%2Fpropagation_and_antenna%3Again-efficiency-beamwidth) — Propagation and Antenna, chapter 2
- [Antenna polarization](https://www.bctengineer.com/question.md?key=semester_6%2Fpropagation_and_antenna%3Apolarization) — Propagation and Antenna, chapter 2, asked 8×
- [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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