S-Parameters to voltage transfer function relation

Hello guys,

Simple question whose answer I can't find properly explained anywhere...

What is the relation (in linear scale) between the s-parameters, let's say S21, and the voltage transfer function, Vo/Vi for a 50 Ohms matched system?

I checked many sites and sometimes it is basically that (S21=Vo/Vi), but from another source I get the answer S21= 2*Vo/Vi... and that one would explain a discrepancy I have in my simulation software, but I'd like to understand it first.

Thanks in advance.

Best regards

It really depends what you define as Vi: the input voltage at the device, or the source voltage (unloaded) of a 50 Ohm voltage source.

If Vsrc is the unloaded voltage of a voltage source, with 50 Ohm internal source resistance, the input voltage at the 50 Ohm device is only Vrsc/2. So depending what you specify as Vin, you get a factor of 2.

The pic attached shows the network I'm testing. according to what I've read about the S parameters, S21 would be Vout/Vsrc. My circuit would be the one without the input source and the 50 ohms in series, and also without the final 50 ohms load. What do you mean by unloaded? Wouldn't the source be always loaded by the circuit?

No.

S21 is 2*Vout/Vsrc for your example.

Think of a lossless device (wire) that has 0dB insertion loss. The voltage divider from 50 Ohm internal resistance of your source and 50 Ohm load will already cause Vout=1/2Vsrc.

PS: I am assuming that R5=50 Ohm is part of your 50 Ohm source.

If you wanted this to be part of the device under test, then we found a mistake in your setup: For 50 Ohm S-parameters, both source and and load must have 50 Ohm impedance. And because of these resistances, vload = vin = 1/2*vsrc for a simple through connection.

Got it, the wire example helped a lot!

Thanks!