Scattering Parameters and different load

Question

I am somewhat confused as to the correct usage of s-parameters. Looking at the problem statement below, would the S-parameters given need to be "transformed" due to the load impedance being different than the reference impedance for calculation purposes?

In other words, can calculation be performed using the given values to find voltages or power at port 2 directly or do the values containing a_2 or b_2 need to be "transformed" to the correct reference impedance?

I hope this makes sense. I have skimmed various literature on the subject but most seem to only deal with matched loads, perhaps someone could clarify or guide me towards literature which deals with such cases.

Answer 1

would the S-parameters given need to be "transformed" due to the load impedance being different than the reference impedance for calculation purposes?

No, the S-parameters don't need to be transformed. Since the load \$Z_L\$ doens't match the reference impedance \$Z_0\$, it will have a nonzero reflection coefficient. This will combine with the part labelled as "two-port network"'s S-parameters to produce a different net reflection looking in to the 2-port than if it were loaded with a matched impedance.

You could, of course, alternatively transform everything to a different reference impedance matching the load.

The result would be the same.

Answer 2

There are different set of matrices of parameters for different purposes on 2 ports.

S-parameters

• Scattering of loss or gain relative to a fixed impedance
  • it is always referenced to a fixed Z eg. Zo = 50 or 75 Ω (video)
• any mismatch in Z causes a loss by reflection.
• it is calibrated with f for 3 impedances Zo=50.00, Short = 0.000 Ω, Open =∞

Y-parameters

• Admittance

H-parameters

• used for Transistors

T-parameters

• used for cascading 2 port parameters easily and supported in Spice

ABCD-parameters

• V/I ratios aka:chain, cascade, for passive elements and transmission line parameters

Answer 3

When all of your ports are specified in terms of S-Parameters, it's simple to combine them using the standard mathematics for that system, Mason's Gain Formula (wikipedia).

You're told the system reference impedance is 50ohms. That means that all the parameters are with respect to 50ohms.

You are only given S-parameters for the two-port.

Fortunately it's trivial to get the S-parameters of the source by inspection, as you are told the output impedance is 50ohms, meaning it matches the system impedance, meaning that S11_G is 0.

The load impedance doesn't match the system, so S22_L will be non-zero. So you do have to transform the load's impedance into an S-parameter for its reflection. I'm not going to do that for you, as I'm sure that grinding through the little bit of algebra to turn impedance to reflection to S-parameter is one of the exercises you will be doing as part of the course, but of course google will be there with a plot-spoiler if you want.

Once generator, two-port and load are all defined in terms of S-parameters, then they can be combined by Mason's.

The other interesting thing you can do is combine them pair-wise. So for instance combining the generator+two-port gives you the output impedance that's driving the load. Conversely combining the two-port and load gives you the effective load that the generator sees.