Hello,

Question: S-matrix of a transistor is given (for a reference impedance \(Z_0 = 50 Ohm\)). Transistor is unilateral (\(S_{12} = 0\)). Obtain matching networks (source and load) for maximum transducer gain when both source and load have impedances of \(100 Ohm\).

My approach: Since the device is unilateral, the gain is maximum if both gains of input and output matching networks are maximum as well, i. e., \(p_s=(S_{11})*=p_{in}*\) and \(p_L=(S_{22})*=p_{out}*\).
In the design of the networks I don't have problems. My doubt is about the impedances of \(100 Ohm\) since the S-matrix given is for \(Z_0 = 50 Ohm\).

a) Should I design as I described above considering the ports with \(100 Ohm\) (usually is for \(50 Ohm\))?

b) Or before any design I need to re-normalize the S-matrix given to \(Z_0= 100 Ohm\)?

Regards.

 

Normalized s parameters in one impedance base Zo are typically transformed to a new impedance base by a process of transformation to actual Z parameters and then reversing the process by going from actual Z parameters to re-normalized s parameters with a new impedance base Zo'.

I presume this would apply in your example where one is seeking to design matching networks with a different impedance base for source and load.

If you happen by remote chance to use Matlab, the RF toolkit includes functions which can readily provide transformation between arbitrary impedance bases. Otherwise its a case of algebraic manipulations and step-by-step calculation. There is info on the web as to how to do this.

 

Thanks for the answer.
So, for this case the procedure would be:

a) Re-normalize the S-matrix to 100 Ω:
a.1) Calculate the S-parameters of a thru-connection with a mixed port impedance of 100 Ω/50 Ω and 50 Ω/100 Ω.

S(a) - S - S(b)

Where S stands for the original S matrix (50 Ω), S(a) and S(b) for the respective matrices of the impedance transformation. In terms of a circuit S(a) and S(b) could be transformers at the input and output of the transistor described by the original S-matrix.

a.2) Connect all networks in series and calculate the overall S-parameters by utilizing the T-matrix. As result I would get the new S-matrix with reference to 100 Ω.

b) Design the input and output matching networks (for example L/C networks) for the S-matrix obtained in a.2) in order to obtain the matching for the ports of 100 Ω.

Am I correct?

 

I'm not sure of the correctness of your approach in your case as little is revealed of the specifics of the problem at hand - if any. The original question seems very general in nature, which may well be the intent of the person who originally asked it.

My guess would be - do the impedance base transformation and then perform the matching network design (using the transformed s parameters) as per your "usual method" having regard to any other constraints imposed upon you.

That's the best I can offer, unless there a more undisclosed details in the question you are attempting to solve.

 

Yes, it was a general question.

All exercises I've been solving always have a transistor described by a S-matrix with reference to 50 Ohms. Then, there is a question to design matching networks for maximum gain (assuming a unilateral device) where both ports have 50 Ohms.

After solving some of them I was wondering what happens or how to solve when those ports have different impedance (in the case I presented 100 Ohms) than the given S-matrix of the transistor.

Thanks for the help.

 

There are well known techniques for matching impedances at a specfic frequency. No frequency dependent sub-system can be matched for all frequencies at the system base impedance, although one can achieve a useful bandwidth over which matching is acceptable.
Often one is compelled in physical systems to provide tunable matching networks. A typical example is the antenna tuning/matching hardware in mult-band radio transmission with a fixed antenna & feeder line configuration.