# Voltage Transfer Function determination from S-Parameters

1. Jan 22, 2010

### Seven77

Hello everybody

I'm trying to obtain the transfer function (Vout/Vin) of some circuits employing an E5062A Vector Network Analyzer.

I've been looking for some information on the topic, but I'm quite confused.

Somebody told me it can be obtained simply as S21*sqrt(Zo).

Additionally, I found some documentation in which it is obtained in a more complicated way:

http://cp.literature.agilent.com/litweb/pdf/5952-1087.pdf [Broken] (pages 24, 23)

I would like to know which is the correct way to obtain it.
Additionally, in the expressions found in these last documents, I would like to know if it is correct to assume that when I'm measuring my circuit Zl=Zs=Zo=50Ohm and, thus, the reflection coefficients for the source and the load are equal to zero. This would simplify the expressions to S21/(1+S11) in the first case and S21/2 in the second case.

I'm very confused. ¿Could anybody help me?

Thank you very much

Last edited by a moderator: May 4, 2017
2. Feb 19, 2010

### jsgruszynski

Assuming that your test system Thevenin(s) and loads are matching, you can use the S21*sqrt(Zo) pretty much exactly.

If you don't have source or load matches (pretty much a practical reality), you have to consider the effects of source/load mismatch on the uncertainty of your measurement as well but the same equation is approximately correct with the uncertainty added in. Basically mismatch causes signal to be sloshed back and forth between source and load through your dut but the instrument can't separate original incident and reflected from the mismatch created n-th order incident and reflected signals. An upper bound on degradation of accuracy can be estimated from what your do know about the test system which is what source-mismatch uncertainty is about.

This is what the Agilent AN154 and the ADS page is going into. There is always some mismatch even if you are using the best-of-the-best VNA with a "perfect" calibration. The basic accuracy of a VNA is only 1-5% as it is but with mismatch and calibration error you can multiply this up pretty quick and easily.

You may have an Agilent Uncertainty Calculator. It's simply a nomographic calculator for the same equations in the above links.

The short answer is: to a first order, just use the denormalization formula to characteristic impedance and just be aware that the numbers you get are "worse" than what it says and how much can be estimated with a source-mismatch uncertainty calculation.

3. Feb 19, 2010

### sophiecentaur

"Somebody told me it can be obtained simply as S21*sqrt(Zo)."
Looks right to me. S21 is just the forward power gain with the input and output working matched.
Power is Vsquared over Z so the ratio of volts will be just as you say- for a starting point, at least.
It depends upon how accurate you want to be - jsgruszynski's remarks will be more and more relevant, the better you want your results to be.