Rectangular Antenna Pattern Calculator

This calculator computes the half-power beamwidth (HPBW) and first null beamwidth (FNBW) of a rectangular aperture antenna (like a horn or a patch array) in both principal planes. You enter the aperture dimensions and the operating frequency, and it returns the beamwidths in degrees for the E-plane and H-plane, plus the approximate directivity in dBi. The calculation uses the standard uniform-aperture formulas that assume an ideal rectangular aperture with a constant phase and amplitude distribution.

Formula

For a rectangular aperture with uniform illumination, the far-field radiation pattern is a sinc function. The half-power beamwidth (HPBW) is the angular width between the points where the power drops to half (−3 dB) of its maximum. For a rectangular aperture of dimension D (in the plane of interest) and wavelength λ, the HPBW is:

HPBW ≈ 50.8° × (λ / D)

The first null beamwidth (FNBW) is the angular width between the first nulls on either side of the main beam:

FNBW ≈ 114.6° × (λ / D)

Here:

• λ = wavelength = c / f, where c = 3×10⁸ m/s and f is frequency in Hz
• D = aperture dimension in the plane of interest (width for E-plane, height for H-plane), in meters
• Both beamwidths are in degrees

The directivity of a uniform rectangular aperture is:

D₀ = 4π A / λ²

where A = a × b is the physical area of the aperture. The formula assumes 100% aperture efficiency (no tapering, no losses).

Worked Example

Using the default values: frequency f = 10 GHz, width a = 10 cm, height b = 5 cm.

1. Wavelength: λ = 3×10⁸ / (10×10⁹) = 0.03 m (3 cm)
2. E-plane HPBW: 50.8 × (0.03 / 0.10) = 15.24°
3. H-plane HPBW: 50.8 × (0.03 / 0.05) = 30.48°
4. E-plane FNBW: 114.6 × (0.03 / 0.10) = 34.38°
5. H-plane FNBW: 114.6 × (0.03 / 0.05) = 68.76°
6. Directivity: 10 × log10(4π × 0.10 × 0.05 / 0.03²) = 10 × log10(0.0628 / 0.0009) ≈ 10 × log10(69.8) ≈ 18.44 dBi

The calculator rounds these to 15.24°, 30.48°, 34.38°, 68.76°, and 18.44 dBi, matching the example.

When to Use This Calculator

This tool is for rectangular aperture antennas such as horn antennas, rectangular microstrip patch arrays, and open-ended rectangular waveguides. It is most accurate for antennas with nearly uniform amplitude and phase across the aperture. Common applications include:

• Designing feed horns for parabolic reflectors
• Estimating coverage area of a patch array
• Comparing beamwidths of different aperture sizes
• Quickly checking if an antenna meets a link budget’s beamwidth requirement

The calculator gives the ideal pattern. Real antennas have tapered illumination (to reduce sidelobes) which widens the beamwidth and lowers directivity. If your antenna uses a cosine taper (common in horns), the HPBW is about 60–70° × λ/D instead of 50.8°.

How to Read the Results

The E-plane is the plane containing the electric field vector and the direction of propagation. For a rectangular horn, the E-plane is the plane of the wider dimension (often called the H-plane in some textbooks, but we follow the IEEE standard: E-plane is parallel to the electric field). The H-plane is perpendicular to the E-plane.

The HPBW tells you the angular width of the main beam at half power. The FNBW is typically about 2.25 times the HPBW for a uniform aperture. Directivity in dBi is relative to an isotropic radiator. A higher directivity means a narrower beam. For example, a 10 cm × 5 cm aperture at 10 GHz has a directivity of about 18.4 dBi, which is typical for a small horn.

If you need to cover a specific angular sector, compare the HPBW to your required coverage. If you need to minimize sidelobes, you may need a tapered aperture, which will increase the HPBW.

Common Mistakes and Limitations

The most common error is mixing units. Always convert centimeters to meters before using the formula. Another mistake is using the wrong plane: some textbooks label the wide dimension as the H-plane. Check your coordinate system.

Limitations of the uniform-aperture model:

• It ignores edge diffraction, which causes ripples in the pattern and slightly widens the beam.
• It assumes a flat phase front. In a horn, the phase is spherical, which broadens the beam and reduces directivity.
• It does not account for the feed mismatch or ohmic losses.
• For electrically small apertures (D < 2λ), the formulas become inaccurate because the sinc approximation breaks down.

For a real horn, the actual HPBW can be 10–20% wider than the uniform-aperture value. For a patch array with a cosine taper, the HPBW is about 60–70° × λ/D.

How to Choose Aperture Dimensions

If you have a desired beamwidth, you can solve for the aperture dimension: D ≈ 50.8° × λ / HPBW. For example, to get a 10° HPBW at 10 GHz (λ = 3 cm), you need D ≈ 50.8 × 3 / 10 = 15.24 cm. That is the minimum width; a tapered aperture would require a larger dimension.

For a rectangular aperture, the beamwidths in the two planes are independent: you can choose width and height separately to get an elliptical beam. This is useful for satellite dishes that need a different beamwidth in elevation versus azimuth.

Remember that directivity increases with area, but the physical size may be constrained by the platform. A 20 dB directivity requires an aperture of about 10λ², which at 10 GHz is 10 × (0.03)² = 0.009 m², or about 9 cm × 10 cm.

Comparison with Circular Apertures

For a circular aperture of diameter D, the HPBW is approximately 58.4° × λ/D (for uniform illumination), and the first null is at 70.5° × λ/D. The directivity is 4πA/λ², same as rectangular, but the sidelobe structure differs. Circular apertures have slightly wider beamwidths for the same area because the effective area is πD²/4, while a square aperture of side D has area D².

For example, a circular aperture with diameter 10 cm at 10 GHz has HPBW ≈ 58.4 × 0.03/0.10 = 17.5°, compared to 15.2° for a square aperture of the same side. The circular aperture has a lower first sidelobe (−17.6 dB) than the rectangular (−13.3 dB).

Choose a rectangular aperture when you need independent control of the two principal plane beamwidths. Choose a circular aperture for symmetric beams and lower sidelobes.

Frequently Asked Questions

What is the difference between E-plane and H-plane?

The E-plane is the plane that contains the electric field vector and the direction of maximum radiation. The H-plane is the plane that contains the magnetic field vector and the direction of maximum radiation. For a rectangular aperture, the E-plane is the plane of the dimension parallel to the electric field. In a typical rectangular horn, the E-plane is the narrower dimension (height) and the H-plane is the wider dimension (width), but this depends on the feed polarization.

Why is the HPBW formula 50.8° and not something else?

The 50.8° comes from solving the sinc function for the half-power point. For a uniform aperture, the power pattern is (sin x / x)². The half-power point occurs when x = 1.391 radians. The angular width is then 2 × arcsin(1.391 × λ / (πD)) ≈ 50.8° × λ/D for small angles. The approximation is valid for D >> λ.

Can I use this calculator for a patch antenna?

Yes, for a rectangular microstrip patch antenna, the aperture is roughly the physical dimensions of the patch. However, the effective aperture is slightly larger due to fringing fields, so the actual beamwidth is a bit narrower than the physical dimensions suggest. Also, patch antennas have a cosine-like taper, so the HPBW is closer to 60–70° × λ/D. Use the calculator as a rough estimate, then verify with full-wave simulation.

How do I convert beamwidth to gain?

There is an approximate formula: G ≈ 41253 / (θ_E × θ_H) where θ_E and θ_H are the half-power beamwidths in degrees. This gives gain as a ratio (not in dB). For the example above, θ_E = 15.24°, θ_H = 30.48°, so G ≈ 41253 / (15.24 × 30.48) ≈ 88.8, or 19.5 dBi. This is close to the directivity from the area formula (18.4 dBi) but not exact because the approximation assumes a certain pattern shape.

What if my aperture is not uniformly illuminated?

If the illumination tapers toward the edges (common in real antennas to reduce sidelobes), the beamwidth increases and the directivity decreases. A cosine taper gives HPBW ≈ 60–70° × λ/D and directivity about 0.8–0.9 times the uniform aperture. The calculator assumes uniform illumination, so it gives the best-case beamwidth. For a tapered aperture, use a larger dimension or a different formula.

Does the calculator work for any frequency?

The formulas are valid for any frequency as long as the aperture is much larger than a wavelength (D >> λ). At low frequencies where the aperture is small (D < 2λ), the sinc approximation fails, and the beamwidth is wider than predicted. The calculator uses the speed of light constant, so it is valid from DC to light, but practical antennas operate in the RF and microwave range.

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in f | Frequency | GHz | 10 | 0
in a | Aperture width (E-plane) | cm | 10 | 0
in b | Aperture height (H-plane) | cm | 5 | 0
out hp_e | E-plane HPBW | deg | 50.8 * (3e8 / (f * 1e9)) / (a / 100) | 2
out hp_h | H-plane HPBW | deg | 50.8 * (3e8 / (f * 1e9)) / (b / 100) | 2
out fn_e | E-plane FNBW | deg | 114.6 * (3e8 / (f * 1e9)) / (a / 100) | 2
out fn_h | H-plane FNBW | deg | 114.6 * (3e8 / (f * 1e9)) / (b / 100) | 2
out dir | Directivity | dBi | 10 * log10(4 * pi * (a / 100) * (b / 100) / ((3e8 / (f * 1e9))^2)) | 2