S-, h-, T-, Y-, Z-, ABCD- Parameter Conversions

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 unique work.

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 covered here.

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 impedances.

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.

S-Parameters from Z-Parameters
S-Parameters from Y-Parameters
S-Parameters from h-Parameters
S-Parameters from ABCD-Parameters
T-Parameters from Z-Parameters
T-Parameters from Y-Parameters
T-Parameters from h-Parameters
T-Parameters from ABCD-Parameters
Y-Parameters from Z-Parameters
Y-Parameters from S-Parameters
Z-Parameters from T-Parameters
h-Parameters from S-Parameters
h-Parameters from T-Parameters
ABCD-Parameters from S-Parameters
ABCD-Parameters from T-Parameters

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.