Final answer:
The effective antenna area of an antenna at 3000 MHz with a gain of 13 dBi is approximately 0.159 square meters, assuming ideal efficiency.
Step-by-step explanation:
To determine the effective antenna area (also known as the aperture) of an antenna at 3000 MHz with a gain of 13 dBi, we utilize the antenna gain formula in terms of wavelength (λ) and efficiency (e). The formula for the effective area Ae when the gain is given in isotropic decibels (dBi) is:
Ae = (λ^2 * G) / (4π * e)
At 3000 MHz (which is the same as 3 GHz), the wavelength λ can be found using the relation c = λ*f where c is the speed of light (approximately 3x10^8 m/s) and f is the frequency. Thus:
λ = c / f = (3x10^8 m/s) / (3x10^9 Hz) = 0.1 m
The gain G in linear terms is derived from the gain in dBi (Gi) using the formula G = 10^(Gi/10). For Gi = 13 dBi:
G = 10^(13/10) = 19.95 (approx.)
Assuming the antenna is perfectly efficient (e=1, which is an idealization), the formula simplifies to:
Ae = (λ^2 * G) / (4π) = (0.1^2 * 19.95) / (4π) = 0.159 m^2 (approx.)
Therefore, the effective antenna area of an antenna at 3000 MHz with 13 dBi gain is approximately 0.159 square meters.