# Are T-parameters the same thing as ABCD parameters?

S-parameters can be converted to either T-parameters or ABCD-parameters, and both are considered a "transfer matrix". The conversion from S-parameters to T- or ABCD-parameters are defined differently, but they seem to be used similarly in the sense that matrix multiplcation can combine different S-parameter component matrixes by converting S->[T1][T2]->S or S->[ABCD1][ABCD2]->S.

T and ABCD parameters are used to chain, so for example you could compose matrixes in such a way that multiplying the matrix creates a parallel circuit. Thus, you could get a 10 nH inductor with two 20nH inductors if you have the S parameters for each converted to T / ABCD matrix (for example from a 0405DC-10N .S2P file from Coilcraft).

What I'm trying to understand is if a T-matrix and ABCD parameters are the same thing, or if they represent different ways of doing the same thing.

@LorenzoMarcantonio provides an appnote from MathWorks in an answer below that states:

The cascadesparams function uses ABCD-parameters. Alternatively, one can use S-parameters and ABCD-parameters (or T-parameters) to cascade S-parameters together by hand (assuming identical frequencies)

but the article doesn't get into the details about T vs ABCD.

This SE answer gets close, but still doesn't address T vs ABCD.

... so are they really the same or just similar?

1. Does $$\T = \begin{pmatrix}A & B\\\ C & D\end{pmatrix}\$$?, such that ABCD are really just the elements of the 2x2 T matrix, or is an ABCD matrix fundamentally different from a T-matrix?

2. If not, then what is the conceptual difference?

• When would you use T-parameters?
• When would you use ABCD-parameters?

Both T parameters and ABCD parameters allow you to analyse a cascade by simply multiplying matrices for each component part. They are different, but in general allow you to achieve the same thing, use whichever is more convenient for the problem at hand. You can convert from one to another:

from: D. A. Frickey, "Conversions between S, Z, Y, H, ABCD, and T parameters which are valid for complex source and load impedances," in IEEE Transactions on Microwave Theory and Techniques, vol. 42, no. 2, pp. 205-211, Feb. 1994

available here:

• That research paper is gold! May 26 at 2:44

As far as I know they are simply a mathematical artifact to help in calculation. They carry the same information as the S parameter, which are important since they are easy to measure (for some definition of ease). However chaining them is easier (as a computation) so that's why they are used. Some more info appnotehere.

I've only used them once or twice so maybe there are more uses for them.

• Interesting, your appnote link says "The cascadesparams function uses ABCD-parameters. Alternatively, one can use S-parameters and ABCD-parameters (or T-parameters) to cascade S-parameters together by hand (assuming identical frequencies)" so either they are similar, related, or the same. Still not sure which, would love a definitive answer to the relation between T vs ABCD. May 25 at 18:48
• I guess it's only a numeric manipulation without a real physic meaning. But S and T depends on the impedence outside the two port network, while ABCD is derived from the network only. tspi.at/2020/07/12/zparameter.html gives some insights May 26 at 6:03

My limited understanding of T-parms is that they represent a way to separate or analyze Power waves from Pseudo waves and Youla? waves.

It may be a useful way to compare with a 4 way impedance analyzer to maximize power point transfer (MPPT) which is essentially conjugate impedance matching.

Like taking a capacitive PV current source matched with an inductive voltage source and a pulsed switch in between in battery charger or matching a grid on a long transmission line or a microwave power amp to an antenna.