Final answer:
The antenna operating at 2.4 GHz with 85% efficiency and a directivity of 40 has a gain of approximately 16.02 dBi or 13.87 dBd. For a 5 GHz array of three antennas, the maximum spacing is 0.03 m, and the maximum radiation is produced broadside to the array axis. Each λ/2 dipole in the array would also be 0.03 m long.
Step-by-step explanation:
Antenna Gain and Array Spacing
To calculate the gain of an antenna at 2.4 GHz with 85% efficiency and a directivity of 40, we use the formula:
Gain (dBi) = 10 × log10(Directivity × Efficiency)
Here, Directivity is 40 (unitless) and Efficiency is 0.85. The Gain (dBi) can be calculated as follows:
Gain (dBi) = 10 × log10(40 × 0.85) ≈ 16.02 dBi
To obtain gain in dBd, subtract 2.15 dB from the dBi value (since dBi is referenced to an isotropic radiator and dBd is referenced to a dipole antenna).
Gain (dBd) ≈ 16.02 dBi - 2.15 dB = 13.87 dBd
For a linear array of three antennas operating at 5 GHz, the maximum spacing to avoid grating lobes is half the wavelength (λ/2). The wavelength can be calculated using the formula λ = c/f, where c is the speed of light (3×10⁸ m/s) and f is the frequency (5 GHz or 5×10⁹ Hz).
λ = 3×10⁸ m/s / 5×10⁹ Hz = 0.06 m
Maximum spacing = λ/2 = 0.06 m / 2 = 0.03 m
The maximum radiation for a phased array is produced in the direction where the phase progression matches the progression of waves in space, usually broadside to the array axis. For an array constructed with λ/2 dipoles, each dipole would be λ/2 in length, which is 0.03 m for the 5 GHz signal.