---
title: "Pyramidal horn flare angle and length"
subject: "Propagation and Antenna"
chapter: 3
key: "semester_6/propagation_and_antenna:pyramidal-horn-flare-angle"
source: https://www.bctengineer.com/q/semester_6/propagation_and_antenna/pyramidal-horn-flare-angle
site: BCT Engineering
---

# Pyramidal horn flare angle and length

Propagation and Antenna · Chapter 3: Antenna Classification

- Asked in: 2070 Bhadra, 5 marks
- Video: [Horn Antenna Simplified |Flaring,Flaring Length,Flaring Angle,Flaring Difference| AWP Module - 4 |](https://www.youtube.com/watch?v=A02F9lVN9P0) — THE BACKBENCH ENGINEERING COMMUNITY
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/propagation_and_antenna/pyramidal-horn-flare-angle
- Whole chapter: https://www.bctengineer.com/md/semester_6/propagation_and_antenna/chapter-3

## Question

Derive the relation for flare angle and length of a pyramidal horn antenna.

## Short answer

For a horn, the path difference δ between centre and edge gives L = A²/(8δ), with flare angle θ = 2 tan⁻¹(A/2L) = 2 tan⁻¹(4δ/A). For an optimum pyramidal horn, δ ≤ 0.25λ in E-plane and ≤0.4λ in H-plane, so L_E = B²/(2λ) and L_H = A²/(3.2λ). For an E-plane aperture of 10λ, L = 50λ and θ ≈ 11.4°.

## Answer

A horn is a flared waveguide. Waves from the throat reach the edge of the aperture by a longer path than at the centre, so the aperture field is not in phase. The allowed path difference δ fixes the relation between flare angle θ, axial length *L* and aperture *A*.

```text
                 . P (aperture edge)
              .  |
  L + δ    .     | A/2
        .  θ/2   |
  O  .___________| C (aperture centre)
  (apex)    L
```

**Derivation** (applies to each plane of the pyramidal horn):
From the right triangle OCP, with OC = *L*, CP = *A*/2, OP = *L* + δ:

```text
cos(θ/2) = L / (L + δ)                     ...(1)
tan(θ/2) = (A/2) / L = A / (2L)            ...(2)

(L + δ)² = L² + (A/2)²
L² + 2Lδ + δ² = L² + A²/4
Since δ << L, drop δ²:
2Lδ = A²/4
L = A² / (8δ)                              ...(3)

Flare angle:
θ = 2 tan⁻¹(A / 2L) = 2 cos⁻¹[L / (L + δ)]  ...(4)
```

Putting (3) into (2):
```text
tan(θ/2) = A / (2 · A²/8δ) = 4δ / A
θ = 2 tan⁻¹(4δ / A)
```

**Optimum horn:** for the most directivity with a given length, the path difference is limited to:
- E-plane: δ ≤ 0.25λ (E-field must be nearly in phase)
- H-plane: δ ≤ 0.4λ (field tapers to zero at the edges, so more phase error is allowed)

So for a pyramidal horn with E-plane aperture *B* and H-plane aperture *A*:
```text
L_E = B² / (8 × 0.25λ) = B² / (2λ),   θ_E = 2 tan⁻¹(B / 2L)
L_H = A² / (8 × 0.4λ)  = A² / (3.2λ), θ_H = 2 tan⁻¹(A / 2L)
```
The horn length is the larger of the two; the other flare is then fixed by geometry.

**Example:** E-plane aperture B = 10λ, δ = 0.25λ:
L = 100λ²/(8 × 0.25λ) = 50λ; θ/2 = tan⁻¹(10/100) = 5.71°, so θ ≈ 11.4°.

## Check yourself

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

1. Why is the aperture field of a horn not in phase?
   A. The wave from the throat travels a longer path to the aperture edge than to the centre
   B. The aperture is metallic
   C. The wave slows in the centre only
   D. The throat radiates more power
   Answer: A. The edge point P is at distance L + δ from the apex while the centre is at L, creating a path difference δ. (Key point: Why a path difference arises)

2. From (L + δ)² = L² + (A/2)², neglecting δ², what is the horn length?
   A. L = A²/(8δ)
   B. L = A/(2δ)
   C. L = 8δ/A²
   D. L = A²/(2δ)
   Answer: A. 2Lδ = A²/4 gives L = A²/(8δ). (Key point: Length relation)

3. Which expression gives the flare angle?
   A. θ = 2 tan⁻¹(A/2L)
   B. θ = tan⁻¹(2L/A)
   C. θ = 2 sin⁻¹(L/A)
   D. θ = A/(2L)
   Answer: A. From the right triangle, tan(θ/2) = (A/2)/L, so θ = 2 tan⁻¹(A/2L), which also equals 2 tan⁻¹(4δ/A). (Key point: Flare angle)

4. Why is a larger path difference allowed in the H-plane (0.4λ) than in the E-plane (0.25λ)?
   A. In the H-plane the field tapers to zero at the edges so phase error matters less
   B. The H-plane aperture is smaller
   C. E-plane has no phase error
   D. H-plane fields are stronger at the edges
   Answer: A. The cosine taper in the H-plane makes the edges weak, so more phase error is tolerable; the E-plane field is uniform. (Key point: Optimum horn limits)

## Related in Propagation and Antenna

- [Short note: Pyramidal horn antenna](https://www.bctengineer.com/question.md?key=semester_6%2Fpropagation_and_antenna%3Apyramidal-horn-short-note) — Propagation and Antenna, chapter 3
- [Short note: Horn antenna](https://www.bctengineer.com/question.md?key=semester_6%2Fpropagation_and_antenna%3Ahorn-antenna-short-note) — Propagation and Antenna, chapter 3, asked 2×
- [Aperture antenna advantages and horn types](https://www.bctengineer.com/question.md?key=semester_6%2Fpropagation_and_antenna%3Aaperture-advantages-horn-types) — Propagation and Antenna, chapter 3
- [Aperture antennas and rectangular horn radiation](https://www.bctengineer.com/question.md?key=semester_6%2Fpropagation_and_antenna%3Aaperture-antenna-rectangular-horn) — Propagation and Antenna, chapter 3

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