This table of conversion between various forms of 2-port network electrical parameters
is difficult to find, so once I finally located a paper that included them1, I felt it was my duty to publish it for public access.
The paper is available on the IEEE website by subscribers only. Other have published
the full paper without permission of author Frickey. None that I found also include the
correction paper2 published a year later that
address some of the technicalities of the S- and T-parameter translations when complex
impedance reference planes are used. In order to avoid those sticky issues, I have reproduced
only the sets of translations that are unaffected. Many thanks to Mr. Frickey for his
One of the most sought-after sets of conversion is from s-parameters to T-parameters,
and then back to s-parameters. This is because matrix multiplications can be performed
directly on T-parameters in order to calculate cascaded component responses. That is,
s-parameters matrices cannot be multiplied in series to obtain cascaded s-parameters,
but T-parameters can be. So, convert your component s-parameters to T-parameters, multiply
matrices, then convert the result back to s-parameters.
The 2-port network
shown to the left is representative of that implied in the application of these equations.
Basic relationships of voltage and current are given in the table to the right. Many
other sources exist on the particulars of 2-port network analysis, so it will not be
All of the parameter equations make use of complex values for all numbers of impedance
and the resulting matrix parameters, i.e., Z = R ± jX.
Z01 and Z02 are the complex impedances of ports 1 and 2, respectively;
similarly, Z*01 and Z*02 are the complex conjugates of the respective
The values R01 and R02 are the real parts of port impedances
Z01 and Z02.
If you do not already know, here is the meaning of each type of parameter matrix:
S (scattering), Y (admittance), Z (impedance), h (hybrid), ABCD (chain), and T (chain
scattering or chain transfer).
These are all I have, so please do not write to ask if I have others.
1. IEEE Transactions on Microwave Theory and Techniques.
Vol 42, No 2. February 1994.
Conversions Between S, Z, Y, h, ABCD,
and T Parameters which are Valid for Complex Source
and Load Impedances.
By Dean A. Frickey, Member, IEEE
2. IEEE Transactions on Microwave Theory and Techniques.
Vol 43, No 4. April 1995.
A correction was printed by Roger B.
Marks and Dylan F. Williams.
3. I1 formula corrected to use V2
rather than V1. Thanks to Christoph T. for noticing.