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Suppose that I have a 9-element Yagi-uda Antenna. My goal is to calculate the gain function of the antenna as function of zenith and azimuth angles, i.e. \$G\left(\theta, \psi\right)\$. As far as I read there is no close formula that I can use.

So, what is the way to calculate the \$G\left(\theta, \psi\right)\$? Is it through calibration, i.e. go to the field with a drone and measure the received signal strength and then do what with that?

Or I think it is equivalent to know how to calculate the radiation pattern for each angle.

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What you are referring to is antenna characterization.

If you are a big company with lots of money you can go to some specialized laboratory with an anechoic chamber. They will then study the far field pattern of said antenna.

If you are some person working in you garage your best chance may be to modelize your antenna in a simulator. Run it to obtain again the far field pattern.

With the far field in your possession, you could now proceed to find/extrapolate a corresponding function. Which will give you the corresponding gain for a position.

Note: A Yagi antenna can be seen has made from a lot of discrete antenna (9 in your case + the reflector). Each antenna effect is summed to obtain the directivity. This give a far field pattern with lots of lobes. These lobes depend on a lot of factor, making a direct calculus really complicated and repetitive. (A job for a computer)

Note 2: If you are looking for a simulator, I can recommend you ANSYS HFSS, and 4NEC2. Respectively for the two previous cases.

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