Antenna gain relates the intensity of an antenna in a given direction to the intensity that would be produced by a hypothetical ideal antenna that radiates equally in all directions (isotropically) and has no losses. Since the radiation intensity from a lossless isotropic antenna equals the power into the antenna divided by a solid angle of 4π steradians, we can write the following equation:
Although the gain of an antenna is directly related to its directivity, the antenna gain is a measure that also takes into account the efficiency of the antenna, that is, the fraction of the input power dissipated in losses such as resistance. In contrast, directivity is defined as a measure that takes into account only the directional properties of the antenna and therefore it is only influenced by the antenna pattern. However, if we assumed an ideal antenna without losses then antenna gain will equal directivity as the antenna efficiency factor equals 1 (100% efficiency). Therefore, for real antennas, the gain of an antenna is always less than its directivity.
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Relative gain
The formulas above show the relationship between antenna gain and directivity, where ε_{cd} = P_{rad} / P_{in} is the antenna efficiency factor, D the directivity of the antenna and G the antenna gain. In the antenna world, we usually deal with a “relative gain” which is defined as the power gain ratio in a specific direction of the antenna, to the power gain ratio of a reference antenna in the same direction. The input power must be the same for both antennas while performing this type of measurement. The reference antenna is usually a dipole, horn or any other type of antenna whose power gain is already calculated or known.
In the case that the direction of radiation is not stated, the power gain is always calculated in the direction of maximum radiation. The maximum directivity of an actual antenna can vary from 1.76 dB for a short dipole, to as much as 50 dB for a large dish antenna. The maximum gain of a real antenna has no lower bound, and is often 10 dB or less for electrically small antennas^{[1]}.
Efficiency
Taking into consideration the radiation efficiency of an antenna, we can express a relationship between the antenna’s total radiated power and the total power input as:
In the above formula, antenna radiation efficiency only includes conduction efficiency and dielectric efficiency and does not include reflection efficiency as part of the total efficiency factor. Moreover, the IEEE standards state that “gain does not include losses arising from impedance mismatches and polarization mismatches”^{[2]}.
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