I want to use the Friss transmission equation to measure the gain of a 3D printed horn antenna. I used a VNA and got all of the s-parameters, with the reference antenna at port_1 and the test antenna at port_2. How do I get transmitting and receiving power using these s-Parameters? I'm not getting the expected results even though I took a value from the S21 parameter at the expected frequency as the ratio of receiving and transmitting power on a logarithmic scale. Is this a wrong procedure to measure the antenna gain?
Sheikh Dobir Hossain VNA can be setup to read absolute power at the port. Antenna gain can be measured using gain-transfer method or absolute-gain method - you may want to check these out as well. I think you can use the latter method if you have two AUTs. And make sure that the antennas are copolarized. Since you are testing an antenna with directivity, boresighting (alignment) will be important.
Shiv Prasad Tripathy give good advice.
I will add that you don't need absolute power. You only need relative power. The S21 in dB gives you all the terms in the Frijs equation except the Transmitter power. It doesn't matter if it is in dB or just a ratio of power, as long as you do your working out in the same way, and dB can be obtained easily from the power ratio, and the other way round. When you have the ratio of transmit power to received power then you can easily get the received power when you decide what the transmit power is.
As Shiv Prasad Tripathy points out, the value you get will depend on whether your antennas are pointing at each other properly, if they need to, and if the polarizations are matched. S21 will also depend on the distance between the antennas, so you will need to measure the distance it you want to know what S21 will be at different distances. To be able to do that you will need to measure S21 with the antennas in each other's far field, because if they are closer than that the inverse square law used in the Frijs equations no longer works.
Assuming you ran a loopback calibration (e.g. through or full two-port) on the VNA to remove the cable effects, then the measured S21 is equivalent to the Pr/Pt ratio of the Friis equation. The typical algebraic formulation of the Friis equation is in linear power units, so you'd need to convert everything to linear power ratios from dB. Alternately you can convert all of the products to sums and division to minus and do everything in dB, but that means converting frequency/wavelength to dB as well. Regardless, when you plug everything in you'll only be able to solve for the product (sum in dB) of the two antenna's gains, and can only solve for a single antenna by knowing the gain of the other. Common approaches to address this are to assume the two antennas have the same gain (if they're identical in design) or to use the three antenna method to create three equations and three unknowns and solve for each of the three antennas. As Malcom pointed out, you need to be in the far field for this to be accurate.
Shiv mentioned a gain transfer method, otherwise known as the substitution method, which doesn't require the Friis transmission equation at all. You also don't even have to calibrate your cables. You simply measure the path loss (S21) with an antenna with known gain and then substitute the unknown antenna and measure again. The delta in S21 is the delta in gain between the reference and unknown antenna. Alternately, on the VNA, you can do the through calibration using the reference antenna and then the S21 of the unknown antenna becomes the delta you add to the gain.
Note that all of this discussion assumes a free-space environment so that multipath does not alter the measurement. Rarely does that environment exist except in a fully anechoic chamber (which still isn't perfect), so there are techniques for extrapolating the far-field gain from more complicated measurement processes. Also, mismatch at the antennas will cause standing waves in the cables which will alter your results unless you take steps to mitigate that contribution. This is typically done by applying attenuators to the ends of the cables at the antenna to damp the standing wave reflections.
Equation (13) in this paper:
https://onlinelibrary.wiley.com/share/author/KU6SMKGBPBMGPJ53IY7X?target=10.1002/mmce.22617
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