Reference characteristic impedance in TRL calibration

Scattering parameters once a TRL calibration is performed are referred to the Zc of the micro-strip line.
Determining with measurement Zc (that is not an outcome of the calibration process) is a problem ( myabe a next question to ask about).
However from the theory we know that Zc is in general complex.
Let say nearly real, but with a small imaginary part.
If great accuracy is requested using the measured calibrated data, my question is:
What is the suitable general scattering parameter mathematics to be used when Zc is complex?
I found papers on generalized scattering parameter but they refer to real characteristics impedances.
What are the consequences in term of scattering parameters property and theorems like reciprocity, lossless network, power transmission etc?
Does someane have reference to paper dealing with this matter?
Many thanks for helping

The assumption is wrong. Scattering parameters are usually referred to a nominal, real Z0, in most cases 50 ohm.

TY for replying but I think I have been misunderstood.. I agree that Scattering parameters are usually referred to a nominal real Z0, but I want to clarify my question asking you a thing related to TRL calibration.

Assume somebody gives you coaxial standards for calibrating a VNA with TRL technique, namely two coaxial lines ( the longer is the "line" and the shorter is called the thru) and a coaxial short ( it is the R). (All the 3 standard are made from the same kind of coaxial, let say rigid, cable and are properly assembled with suitable connector in order to be measured on the VNA).
Perform now the TRL calibration so that you are able then to take measurement of other two ports DUTs.
The numbers you get measuring DUTs, namely scattering parameters of the DUT, are normalized to a Zc ... which one?
I would say Zc= characterisitic impedance of the coaxial cable.. ( this is what i understood of TRL calibration, if it is wrong advise).
Now, it could happen that the coaxial cable has a "not real" characteristic impedance (Zc=sqrt( (jwL+R)/(jwC+G)) https://en.wikipedia.org/wiki/Characteristic_impedance
The question is : how should I manage the numbers I got from the measurement after the TRL calibration using that standards?
Are they suitable S parameters? To what Zc are thy referred?
Not to 50 ohms I would say, to the Zc of the cable then.... rigth but it is not real ! so tho which impedance are they referred?
It may be the Real part fot the Zc of the cable... but this is naively not consistent (as different cables can have the same real part of Zc but measuremetns will be different).
Many thanks



Eventually

Yes, a calibration set is referenced to a real Z0 and calibrates the measurement setup to give results referring to this real Z0, although the used cables are lossy and have slightly different (e.g. complex) impedance.

Real calibration sets can have deviations from ideal Z0 or need a small amount of port extension. If so, correction data may me shipped with the calibration set that gives almost ideal Z0 reference after applying it.

Through calibration is a different thing. There's no ideal through calibration and no dedicated through calibration provided by the calibration sets. Instead through calibration is used to compensate properties of your test setup.



After reviewing your post, I think I overlooked a point. Sorry for talking an cross purposes.

Uncorrected TRL calibration is actually referred to the used line standard.

As far as I understand, an instrument will deal with possible deviations in two ways:

- Ignore a small frequency dependent loss term (In this case it would actually refer to a complex Z0)
- Apply a correction table that refers it to an ideal real Z0

So I understand that calibration standard suppliers can provide information (correction table) to take in account the problem in the case of commercial TRL cal kit ( and offering possibility to come back to a real Zo), while uncorrected TRL calibration is actually referred to the used line standard (complex).
My problem however is that I'm used realizing my own "microstrip TRL standards" in the same RF PCB were I design my RF circuit. With the self made microstrip standards I, then, perform "on board" TRL calibration to move the reference planes beyond the connectors on the PCB (normally I use Southwest solderless end launch) and very close to the DUT's pins. This is definitively more accurate then ports extension which is a weaker approximation based on assumptions not always true that don't consider connector discontinuities for example.
In this case the (complex) Zc, which the S parameters result to be normalized to, is the characteristic impedance of the microstrip, often just nominally 50 ohm but in the reality not known.
In this circumstances I've not a suppliers offering correction table and I have to deal and carry on with the raw calibration.
Since theoretically the Zc should have a very small imaginary part, in most of the situation I neglect it and go ahead with a real estimated value.
However it is really frustrating when accuracy is needed, not because of the committed error itself, but because , not being able to measure the actual Zc, I can't therefore estimate the order of magnitude of the error (assessing a confidence on the measure).
The only idea I have was to asses the measurement confidence just using the theoretical Zc. I wonder if the problem was already considered and better solved..