The maximum theoretical antenna gain of a dish antenna can be estimated using the formula:
\[ G = \frac{4 \pi A}{\lambda^2} \]
where:
- \( G \) is the gain in linear scale,
- \( A \) is the effective aperture area of the antenna (in square meters),
- \( \lambda \) is the wavelength of the signal (in meters).
To find the gain in decibels (dBi), we use:
\[ G_{dBi} = 10 \log_{10}(G) \]
First, let's calculate the wavelength \( \lambda \) at 5 GHz. The frequency \( f \) is 5 GHz, which is \( 5 \times 10^9 \) Hz. The speed of light \( c \) is approximately \( 3 \times 10^8 \) m/s. The wavelength can be calculated using:
\[ \lambda = \frac{c}{f} \]
\[ \lambda = \frac{3 \times 10^8 , \text{m/s}}{5 \times 10^9 , \text{Hz}} = 0.06 , \text{m} \text{ or } 6 , \text{cm} \]
To calculate the gain, we need the effective aperture area \( A \). The area of a circular dish can be calculated using:
\[ A = \pi \left(\frac{D}{2}\right)^2 \]
where \( D \) is the diameter of the dish. The larger the dish, the greater the gain.
For illustration, let's consider a common size for a dish antenna, such as a 1-meter diameter dish:
\[ A = \pi \left(\frac{1}{2}\right)^2 \approx 0.785 , \text{m}^2 \]
Now, substituting into the gain formula:
\[ G = \frac{4 \pi (0.785)}{(0.06)^2} \]
Calculating it step by step:
- \( 0.06^2 \approx 0.0036 \)
- \( 4 \pi \approx 12.5664 \)
- \( G \approx \frac{12.5664 \times 0.785}{0.0036} \approx \frac{9.8696}{0.0036} \approx 2743.2 \)
Then we convert to dBi:
\[ G_{dBi} = 10 \log_{10}(2743.2) \approx 10 \times 3.438 \approx 34.38 , \text{dBi} \]
Thus, the approximate maximum theoretical gain of a 1-meter diameter dish antenna at the 5 GHz band is around 34.4 dBi.
Larger dishes can achieve even higher gains. For example, a 2-meter dish could yield gains exceeding 40 dBi. The maximum gain is limited by practical factors, including dish surface imperfections and the efficiency of the feed system.