The maximum theoretical gain of a dish antenna can be calculated using the following formula:
\[ G = \frac{4 \pi A_e}{\lambda^2} \]
where:
- \( G \) is the antenna gain (in a unitless ratio),
- \( A_e \) is the effective aperture area of the antenna (in square meters),
- \( \lambda \) is the wavelength (in meters).
The effective aperture \( A_e \) can also be related to the physical area of the dish, using:
\[ A_e = \eta A \]
where:
- \( \eta \) is the efficiency of the antenna (typically between 0.5 and 0.7 for a well-designed dish antenna),
- \( A \) is the physical area of the dish.
To find the wavelength \( \lambda \) at a frequency \( f \):
\[ \lambda = \frac{c}{f} \]
where \( c \) is the speed of light (approximately \( 3 \times 10^8 \) m/s).
At 2.4 GHz (which is \( 2.4 \times 10^9 \) Hz):
\[ \lambda = \frac{3 \times 10^8 \ \text{m/s}}{2.4 \times 10^9 \ \text{Hz}} \approx 0.125 \ \text{meters} \]
Now, we need to consider the physical area of a typical dish. Let's take a common consumer dish antenna with a diameter of about 1 meter. The area \( A \) of the dish is given by:
\[ A = \pi \left( \frac{d}{2} \right)^2 = \pi \left( \frac{1}{2} \right)^2 \approx 0.785 \ \text{m}^2 \]
Assuming an efficiency \( \eta \) of 0.6 for a good dish antenna, we get:
\[ A_e \approx 0.6 \times 0.785 \approx 0.471 \ \text{m}^2 \]
Now we can calculate the maximum theoretical gain:
\[ G = \frac{4 \pi A_e}{\lambda^2} = \frac{4 \pi (0.471)}{(0.125)^2} \]
Calculating the denominator:
\[ (0.125)^2 = 0.015625 \]
Then, putting it all into the gain formula:
\[ G \approx \frac{4 \pi (0.471)}{0.015625} \approx \frac{5.91}{0.015625} \approx 378.24 \]
To express this in dBi (decibels relative to an isotropic radiator), we can use:
\[ G_{dBi} = 10 \log_{10}(G) \approx 10 \log_{10}(378.24) \approx 10 \times 2.577 = 25.77 , \text{dBi} \]
Thus, the maximum theoretical gain of a common dish antenna at the 2.4 GHz band is approximately 25.8 dBi.