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Rule of Thumb for Gain, Aperture, and Beamwidth
Adapted from a disquisition by Dr. Nathan Cohen, Fractal Antenna Systems, Inc.
Every antenna has an effective aperture (assuming good efficiency) that can be inferred from the gain. The effective aperture can then be used to find the full-width-half-maximum (3dB) beamwidth.
Gain and Diameter
One can determine the gain of an antenna with this formula:
$$ G = \frac{4\pi A_e}{\lambda^2} $$
\( A_e \) = effective aperture (m²)
\( \lambda \) = wavelength (m)
\( \lambda \) = wavelength (m)
Effective aperture can be approximated (using an alternative formula for the area of a circle):
$$ A_e \approx \frac{\pi D^2}{4} $$
Substituting:
$$ G = \frac{\pi^2 D^2}{\lambda^2} $$
Normalized to wavelength:
$$ G = \pi^2 D_\lambda^2 $$
Where \( D_\lambda = \frac{D}{\lambda} \)
Converting to dBi:
$$ G_{dBi} = 10 \log_{10}(\pi^2 D_\lambda^2) $$
Solving for diameter:
$$ D_\lambda \approx 10^{\frac{G - 10}{20}} $$
Diameter and Beamwidth
The FWHM beamwidth is determined by the effective diameter. The Rayleigh Criterion places it as:
$$ \theta(radians) \approx \frac{1.1 \lambda}{D} $$
Normalizing to wavelength and converting to degrees yields:
$$ \theta_{deg} \approx \frac{63}{D_\lambda} $$
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Summary
These simple relations determine gain, aperture and beamwidth of any antenna system, regardless of wavelength.
$$ G_{dBi} \approx 10 \log_{10}(\pi^2 D_\lambda^2) $$
$$ D_\lambda \approx 10^{\frac{G - 10}{20}} $$
$$ \theta_{deg} \approx \frac{63}{D_\lambda} $$
Pretty cool, eh?
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