Antenna beamwidth ↔ gain calculator
A directional antenna’s gain follows from how narrow its beam is. Multiply the horizontal and vertical half-power beamwidths, divide about 41 253 by that product, and you have the directivity. Take off the radiation efficiency and that becomes the gain in dBi. This calculator runs the estimate in both directions.
Opening angle of the main lobe in the horizontal plane, measured at −3 dB.
The same angle in the vertical plane, read off the elevation pattern.
The constant stands in for the beam solid angle. The lower it is, the more power is assumed to land outside the main lobe.
Share of the power fed in that is actually radiated. At 100 % gain equals directivity.
The two fans are the main lobe seen from above and from the side. Their opening angles are the half-power beamwidths, drawn at their true size, and the dots mark the −3 dB edges where the beam is defined to end. The bar underneath places the resulting gain on a dBi scale; below 100 % efficiency a dashed D marks the directivity the beam alone would give, and the gap to it is the loss.
Two beamwidths cannot describe a whole radiation pattern, so treat the result as an estimate: about ±1 dB for a clean single-lobe antenna, worse for anything broader than 90° or with prominent sidelobes. Use the measured datasheet figure whenever you have one.
How it works
- 01
Read both half-power beamwidths off the pattern
A datasheet plots two cuts through the main lobe, one horizontal and one vertical. The half-power beamwidth is the angle between the two points where the pattern has fallen 3 dB below the peak. A sector panel might be 65° wide and 7.5° tall, a dish only a couple of degrees in each plane.
- 02
Turn the beam area into directivity
Directivity is D = 4π / ΩA, the full sphere divided by the solid angle the beam occupies. Approximate that solid angle by the product of the two beamwidths, work in degrees, and you get D ≈ 41 253 / (θaz · θel), because the whole sky measures 41 253 square degrees.
- 03
Pick a constant that suits the antenna
41 253 assumes every watt lands inside the main lobe. Real antennas leak into sidelobes and a back lobe, so 32 400 (a flat 1.05 dB less) tracks measured patterns better. ITU-R F.1336 uses 31 000 for sector antennas. The Tai–Pereira form D ≈ 72 815 / (θaz2 + θel2) suits pencil beams whose two widths are similar.
- 04
Convert to dBi and subtract the losses
Directivity in dBi is 10 · log10(D). Gain is that minus whatever the antenna turns into heat, so GdBi = DdBi + 10 · log10(η). At 100 % efficiency the two numbers are identical, which is why datasheets often print gain where they computed directivity.
- 05
Run it backwards from a published gain
One gain figure matches infinitely many beam shapes, so fix the shape first with the aspect r = θel / θaz. Then θaz = K / (D · r) and θel = r · θaz. With the 32 400 constant, an 18 dBi panel at r = 0.115 comes out near 65° × 7.5°, the familiar cellular sector.
Formulas
- D — directivity, a linear power ratio
- θaz, θel — half-power beamwidths, degrees
- K — 41 253 ideal, 32 400 with sidelobes, 31 000 for sector antennas (ITU-R F.1336)
- better for a pencil beam whose two widths are similar
- the sum of squares is dominated by the wider angle, so it under-reads flat fan beams
- GdBi — gain over an isotropic radiator
- η — radiation efficiency, 0 to 1 (100 % gives 0 dB)
- dBd = dBi − 2.15
Worked example
- θaz = 20°, θel = 20°, K = 41 253
- D = 41 253 / (20 · 20) = 103.1
- DdBi = 10 · log10(103.1) = 20.1 dBi
- with sidelobes, K = 32 400: D = 81.0, so 19.1 dBi
- Tai–Pereira: D = 72 815 / (202 + 202) = 91.0, so 19.6 dBi
- → quote roughly 19–20 dBi and expect about ±1 dB against a measurement
FAQ
- Why does the beamwidth estimate ignore sidelobes?
- Because it replaces the entire radiation pattern with one rectangle. The product θaz · θel counts the main lobe only, so every watt that goes into a sidelobe or out the back is credited to the beam and the gain comes out too high. Dropping the constant from 41 253 to 32 400 is a flat 1.05 dB allowance for that leakage.
- What is the difference between dBi and dBd?
- dBi compares the antenna with an isotropic radiator, dBd with a half-wave dipole. A lossless dipole has 2.15 dBi, so dBd = dBi − 2.15. Beamwidth formulas always produce dBi, because the sphere they divide up is the isotropic reference.
- How much gain does a 65° × 7.5° sector antenna have?
- Multiply the beamwidths: 65 × 7.5 = 487.5 square degrees. With the sidelobe constant, D = 32 400 / 487.5 = 66.5, which is 18.2 dBi. Panels of that shape are sold as 17–18 dBi antennas, so the estimate lands within a few tenths of a decibel of the datasheet.
- When does the formula stop working?
- When the pattern is not one clean, narrow lobe. Beams wider than about 90° break the rectangular approximation, and strong sidelobes or several lobes break the assumption behind it. Take a half-wave dipole at 78° × 360°. The product rule gives 1.7 dBi against a true 2.15 dBi. The Tai–Pereira form is worse still on very flat fan beams, because the wider angle dominates the sum of squares.
- Does a datasheet quote directivity or gain?
- Gain, on any serious datasheet. Directivity is pure geometry and says how sharply the radiated power is concentrated. Gain also counts what the antenna loses as heat in its conductors, dielectrics and matching network. For a well-built panel or dish the two sit within a few tenths of a dB of each other. Efficiency well below 90 % shows up mainly in electrically small or heavily loaded antennas.
Gain says how tightly the energy is packed, not where it lands. Waveshed aims the beam over real elevation data and shows the ground it actually reaches, free in your browser.
Put the antenna on the map →Related guides
- Antenna patterns & the sensor panel — how the simulator uses a horizontal and a vertical pattern instead of a single gain figure
- Plan radio coverage (RF / ITM) — where antenna gain sits in a full coverage prediction
Sources & further reading
- Recommendation ITU-R F.1336 — Reference radiation patterns of omnidirectional, sectoral and other antennas ↗ — ties the elevation beamwidth of a sector antenna to its peak gain through the constant 31 000
- C. A. Balanis, Antenna Theory: Analysis and Design, chapter 2 (Directivity) — derives the beam solid angle approximation and both the 41 253 and the Tai–Pereira 72 815 forms
- Directivity — Wikipedia ↗ — states both approximations, including 32 400 as the better fit for planar arrays
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