How to compute the efficency of an omnidirectional antenna from its gain?

Question

Some antennas have very limited specifications with no stated efficiency. For example, this antenna has the following specifications:

Frequency Band	902-928 MHz
VSWR	<2.0:1
Impedance	50 Ω
Gain	9 dBi
Polarization	Vertical
Radiation Pattern	Omnidirectional

The manufacturer does not mention other radio characteristics such as the efficiency in the datasheet. How to know if the 9dBi gain comes from an excellent efficiency or from an excessive directivity?

Gain is defined as $G=eD$ where $e$ is efficiency and $D$ directivity. As the radiation pattern is omnidirectional, can we assume that the pattern is always the same and use the following formula to substitute $D$?

$$ D \approx 10 \log_{10} \left( 101.5 \over \text{HPBW} - 0.00272\:\text{HPBW}^2 \right) \mathrm{dB} $$

If yes, how to compute $\text{HPBW}$?

Answer 1

There isn't a way to calculate efficiency from gain, in general. If you had the gain for every possible direction then you could integrate over the entire sphere and calculate the total radiated power and compare that to an ideal isotropic antenna. From that you could calculate efficiency since the reference isotropic antenna is 100% efficient by definition.

But the gain figure in an antenna datasheet is just gain in one direction, usually the direction with peak gain. A more professional antenna might give the radiation pattern for two planes in azimuth and elevation, but that still isn't enough information.

Half-power beamwidth (HPBW) can't really be calculated (except perhaps by modeling the antenna), but you might be able to read it from the datasheet if they give you the radiation pattern. It's simply the width of the beam bounded by the points where the power is diminished to half its peak value. But this isn't likely to be accurate enough to calculate an accurate efficiency number since it relies on a number of assumptions that won't be entirely true in practice.

So unfortunately, for a anything but the very most expensive antennas we just have to guess at efficiency. Many resonant antennas are nearly 100% efficient since there aren't any matching networks to introduce much loss. Electrically small antennas on the other hand are much more likely to have worse efficiency, but unless it's specified on the datasheet, there's not a good way to know it without having the antenna tested in a lab.

Answer 2

Given the information supplied, and no other independent measurements, you can't really tell.

You could guess, by looking at the size of the antenna, which is ~3.25 wavelengths long, and figuring that it's probably such-and-such a colinear design, which should have such-and-such a directivity. But that's probably a pointless exercise, because manufacturers that provide such limited information frequently make up their gain numbers anyway — either by providing a gain figure based on a simulation of the antenna design with 100% efficiency, or just by looking at what the competition is selling and copying their figures, or adding a dB or two.

So, sorry, unless you buy one and measure it yourself, or get the information from someone you really trust, you will never really be sure.