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The NanoVNA doesn't have a separate Source and receiver ports, so you can't do the external amplifier, coupler approach in the Rackley paper.

Is there a reason you need high power for the test? (that is, can't you just push the mighty milliwatt from the VNA into the antenna)

What you might be able to do is sequentially make S21 measurements, switching the second port (CH1) of the NanoVNA to the various ports of the directional coupler (with attenuators as required). The challenge is that the power amplifier will be "outside the calibration plane", so you'd need to separately characterize it and then do the math to remove its contribution. A complication is that the PA's characteristics probably change with the load impedance.

The techniques in the paper rely on having multiple coherent receivers that are separately accessible, which the NanoVNA doesn't have - two of the three receivers are "dedicated".

Now it's true that you could probably go in and do some surgery on a NanoVNA (they're inexpensive, and you're collecting data into a computer, so you could get the one with the small display) and bring out the different ports you need. (basically, the inputs to the mixers fed from the stimulus, and the bridge on CH0) - you could replace the circuitry with a little board with the same input circuit that CH1 has (which is, I believe, a resistive pad and some coupling capacitors).

You could then use software (Scikit-rf in python for instance, has the libraries) to do the calibrations, etc.; using the modified NanoVNA just as a data collector.

Another approach would be to build up an equivalent system using some inexpensive SDRs all locked to a common reference - A programmable source (the NanoVNA) and 3 or 4 RTL-SDRs might work