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RF Cascade Workbook

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1996 -
2016

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Kirt Blattenberger,

BSEE
- KB3UON

RF Cafe began life in 1996 as "RF Tools" in an AOL screen name web space totaling 2 MB. Its primary purpose was to provide me with ready access to commonly needed formulas and reference material while performing my work as an RF system and circuit design engineer. The Internet was still largely an unknown entity at the time and not much was available in the form of WYSIWYG ...

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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 them^{1}, 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 paper^{2}
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.

Z_{01} and Z_{02} 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 R_{01}
and R_{02} are the real parts of port impedances Z_{01} and Z_{02}.

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. I_{1} formula corrected to use V_{2}
rather than V_{1}. Thanks to Christoph T. for noticing.

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.

S-Parameters |

Y-Parameters |

Z-Parameters |

h-Parameters |

ABCD-Parameters^{3} |

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.

Z

The values R

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).

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 | |

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. I