For estimating and education only. Results come from the simplified models described below, are provided as is with no warranty, and are not a substitute for engineering analysis, simulation or measurement. Check them before relying on them in a design, purchase or safety decision. See the terms of use.
Equations
Gain
G = \varepsilon_{ap}\,\frac{4\pi}{\lambda^2}\,A Bεap ≈ 0.51, optimum hornOptimum design
\left(\sqrt{2\chi} - \frac{b}{\lambda}\right)^2 (2\chi - 1) = \left(\frac{G_0}{2\pi}\sqrt{\frac{3}{2\pi\chi}} - \frac{a}{\lambda}\right)^2 \left(\frac{G_0^2}{6\pi^3\chi} - 1\right)solve for χ\rho_e = \chi\lambda, \qquad \rho_h = \frac{G_0^2}{8\pi^3\chi}\,\lambdaA = \sqrt{3\lambda\rho_h}, \qquad B = \sqrt{2\lambda\rho_e}p_e = (B - b)\sqrt{\left(\frac{\rho_e}{B}\right)^2 - \frac{1}{4}}, \qquad p_h = (A - a)\sqrt{\left(\frac{\rho_h}{A}\right)^2 - \frac{1}{4}}pe = phExact
D_p = \frac{\pi\lambda^2}{32\,a b}\,D_E D_HDE, DH from Fresnel integralsBeamwidth
\theta_E \approx 56\,\frac{\lambda}{B}, \qquad \theta_H \approx 67\,\frac{\lambda}{A}degreesVariables
| Symbol | Meaning | Unit |
|---|---|---|
a,\ b | Feed waveguide inner width and height | m |
A,\ B | Aperture width (H-plane) and height (E-plane) | m |
G_0 | Target gain, linear | — |
\chi | Normalized E-plane slant length, ρe / λ | — |
\rho_e,\ \rho_h | Slant lengths from apex to aperture edge | m |
p_e,\ p_h | Flare length from waveguide to aperture | m |
\varepsilon_{ap} | Aperture efficiency | — |
Worked example
| Input | Value |
|---|---|
| Target gain | 22.6 dBi |
| Frequency | 11 GHz |
| Feed | WR-90 (0.9 × 0.4 in) |
| Result | Value |
|---|---|
| Aperture width, A (6.44 in) | 163.6 mm |
| Aperture height, B (5.06 in) | 128.5 mm |
| Flare length (10.7 in) | 272.6 mm |
| Slant length ρe (11.9 in) | 302.9 mm |
| Slant length ρh (12.9 in) | 327.3 mm |
| Gain, exact formula | 22.51 dBi |
| Aperture efficiency | 50.1 % |
| Beamwidth, E-plane (≈ 56 λ/B) | 11.9 ° |
| Beamwidth, H-plane (≈ 67 λ/A) | 11.2 ° |
| Far-field distance 2A²/λ | 1.96 m |
This is Balanis’s worked example: A ≈ 6.0 λ, B ≈ 4.71 λ and a flare length of about 10.0 λ. The exact formula gives 22.5 dBi, within 0.1 dB of the target.
Load this example in the calculatorValidity
- Optimum-gain design: the most gain for a given flare length. It is not the shortest or widest-band horn.
- Feed waveguide in its single-mode band; 1.25–1.89 fc is recommended.
- Gain shown is directivity. Wall, flange and transition losses are not included.
- Beamwidth formulas are approximations for optimum horns.
Sources
- C. A. Balanis, Antenna Theory: Analysis and Design, 4th ed., Wiley, 2016. Chapter 13 (pyramidal horn directivity and optimum design).
- J. D. Kraus and R. J. Marhefka, Antennas for All Applications, 3rd ed., McGraw-Hill, 2002 (horn beamwidths).