# Transistors' Scattering Parameters

I have some doubts about the Scattering Matrix of a transistor seen as a 3 port devices. Precisely, I read (for instance here) that we do not have 9 independent S parameters since there is this relation (6 equations):

So there are only 4 independent S parameters, and this is correct because generally transistors are described as 2 port networks. But I do not understand the proof of these relations. It is shown in these pictures:

First question: is the fact that the sum of current = 0 always verified? Is it a sort of Kirchoff Current Law?

Second question: we are applying equal voltages to the ports, why should this procedure be general, valid for each case?

Third question: why applying equal voltages = each current is 0 ?

• upvoting for the link to the HP memory project. Might try to read some of those some day! – Neil_UK Jun 28 '19 at 13:11
• I forgot about HP memory, must be a senior's moment @Neil_UK but I fondly remember my 1st computer when the HP9825 came out which I used for a dual 9825 SCADA controller. in '78 – Tony Stewart Sunnyskyguy EE75 Jun 28 '19 at 17:22

1. Yes.
2. Because of the assumption to make v1=v2=v3.

• But generally we do not test in this condition.
• We only apply 1 voltage or current to “a” to get the response “b” on all ports
• such that ... $$\S_{xy}=\dfrac{b_x}{a_y}\$$ for each frequency tested or in a continuous display
3. That’s a math problem others can explain better. I think this is to prove conservation of energy and any ratio lost or gained adds up a net power of 0 and a net forward/reverse ratio sum of 1 means there is no net signal lost/generated/reflected as a test to the equation with 0 inputs.

Other:

Three-port S3P files need 19 columns of magnitude and angle data. The data in the Scattering Matrix vectors must be wrapped onto three lines:

For EDA software Insert comments To ensure correct order.
! Freq S11M S11A S21M S21A S12M S12A S22M S22A

S11m S11a S12m S12a S13m S13a
S21m S21a S22m S22a S23m S23a
S31m S31a S32m S32a S33m S33a

is the fact that the sum of current = 0 always verified? Is it a sort of Kirchoff Current Law?

Yes. One version of Kirchhoff's Current law says if you consider any surface that divides the circuit into two parts (such as a ball surrounding the transistor), then the net current through that surface is 0.

we are applying equal voltages to the ports, why should this procedure be general, valid for each case?

We consider this test case to find out some relation between the S-parameters.

But the S-parameters don't change (so long as we operate in a linear regime) when we apply a different set of sources, so the relations between the S-parameters don't change either.

why applying equal voltages = each current is 0 ?

If you apply the same voltage gate, drain, and source, then there is no potential difference between any pair of terminals, so no reason for current to flow.

• Why would you do 3. ? Or even show that in this context . Is it practical? Ie what is the purpose of saying this equation? Which seems obvious. – Tony Stewart Sunnyskyguy EE75 Jun 28 '19 at 16:50
• @SunnyskyguyEE75, you do it as a gedanken experiment to prove the property of the S-parameters that you're trying to prove. – The Photon Jun 28 '19 at 17:08
• YEs of course like Conservation of Energy. But why would anyone want to apply 0V to every port to see if the scattering parameters are still valid when this is not how parameters are measured. Wouldn't it make more sense to prove it with all ports a=0V except one port has a ≠ 0 then validate the sum. What does your Gedankenerfahrung logic say? – Tony Stewart Sunnyskyguy EE75 Jun 28 '19 at 17:16