Impedance parameters

Impedance parameters or Z-parameters are properties used in electrical engineering, electronics engineering, and communication systems engineering describe the electrical behavior of linear electrical networks when undergoing various steady state stimuli by small signals. They are members of a family of similar parameters used in electronics engineering, other examples being: S-parameters, [David M. Pozar, "Microwave Engineering", Third Edition, John Wiley & Sons Inc., 2005; pp. 170-174, ISBN 0-471-44878-8.] Y-parameters, [David M. Pozar, 2005 (op. cit); pp 170-174.] H-parameters, T-parameters or ABCD-parameters. [David M. Pozar, 2005 (op. cit); pp 183-186.] [A.H. Morton, " Advanced Electrical Engineering", Pitman Publishing Ltd., 1985; pp 33-72, ISBN 0-273-40172-6.]

The General Z-Parameter Matrix

For a generic multi-port network definition, it is assumed that each of the ports is allocated an integer 'n' ranging from 1 to N, where N is the total number of ports. For port n, the associated Z-parameter definition is in terms of input currents and output voltages, $I\_n,$ and $V\_n,$ respectively.

For all ports the output voltages may be defined in terms of the Z-parameter matrix and the input currents by the following matrix equation:

:$V\; =\; Z\; I,$

where Z is an N x N matrix the elements of which can be indexed using conventional matrix notation. In general the elements of the Z-parameter matrix are complex numbers.

The phase part of a Z-parameter is the "spatial" phase at the test frequency, not the temporal (time-related) phase.

Two-Port Networks

The Z-parameter matrix for the two-port network is probably the most common. In this case the relationship between the input currents, output voltages and the Z-parameter matrix is given by:

:.

where

:

:

For the general case of an n-port network, it can be stated that:

Impedance relations

The input impedance of a two-port network is given by:

:$Z\_\{in\}\; =\; z\_\{11\}\; -\; frac\{z\_\{12\}z\_\{21\{z\_\{22\}+Z\_L\}$

where Z_L is the impedance of the load connected to port two.

Similarly, the output impedance is given by:

:$Z\_\{out\}\; =\; z\_\{22\}\; -\; frac\{z\_\{12\}z\_\{21\{z\_\{11\}+Z\_S\}$

where Z_S is the impedance of the source connected to port one.

Converting Two-Port Parameters

The two-port S-parameters may be obtained from the equivalent two-port Z-parameters by means of the following expressions. [Simon Ramo, John R. Whinnery, Theodore Van Duzer, "Fields and Waves in Communication Electronics", Third Edition, John Wiley & Sons Inc.; 1993, pp. 537-541, ISBN 0-471-58551-3.]

:$S\_\{11\}\; =\; \{(Z\_\{11\}\; -\; Z\_0)\; (Z\_\{22\}\; +\; Z\_0)\; -\; Z\_\{12\}\; Z\_\{21\}\; over\; Delta\}\; ,$

:$S\_\{12\}\; =\; \{2\; Z\_0\; Z\_\{12\}\; over\; Delta\}\; ,$

:$S\_\{21\}\; =\; \{2\; Z\_0\; Z\_\{21\}\; over\; Delta\}\; ,$

:$S\_\{22\}\; =\; \{(Z\_\{11\}\; +\; Z\_0)\; (Z\_\{22\}\; -\; Z\_0)\; -\; Z\_\{12\}\; Z\_\{21\}\; over\; Delta\}\; ,$

Where

:$Delta\; =\; (Z\_\{11\}\; +\; Z\_0)\; (Z\_\{22\}\; +\; Z\_0)\; -\; Z\_\{12\}\; Z\_\{21\}\; ,$

The above expressions will generally use complex numbers for $S\_\{ij\}$ and $Z\_\{ij\}$. Note that the value of $Delta$ can become 0 for specific values of $Z\_\{ij\}$ so the division by $Delta$ in the calculations of $S\_\{ij\}$ may lead to a division by 0.

S-parameter conversions into other matrices by simply multiplying with e.g. $Z\_0\; =\; 50Omega$ are only valid if the characteristic impedance $Z\_0$ is not frequency dependent.

Conversion from Y-parameters to Z-parameters is much simpler, as the Z-parameter matrix is basically the matrix inverse of the Y-parameter matrix. The following expressions show the applicable relations:

:$Z\_\{11\}\; =\; \{Y\_\{22\}\; over\; Delta\_Y\}\; ,$

:$Z\_\{12\}\; =\; \{-Y\_\{12\}\; over\; Delta\_Y\}\; ,$

:$Z\_\{21\}\; =\; \{-Y\_\{21\}\; over\; Delta\_Y\}\; ,$

:$Z\_\{22\}\; =\; \{Y\_\{11\}\; over\; Delta\_Y\}\; ,$

Where

:$Delta\_Y\; =\; Y\_\{11\}\; Y\_\{22\}\; -\; Y\_\{12\}\; Y\_\{21\}\; ,$

In this case $Delta\_Y$ is the determinant of the Y-parameter matrix.

References

Bibliography

*David M. Pozar, "Microwave Engineering", Third Edition, John Wiley & Sons Inc.; ISBN 0-471-44878-8.
*Simon Ramo, John R. Whinnery, Theodore Van Duzer, "Fields and Waves in Communication Electronics", Third Edition, John Wiley & Sons Inc.; ISBN 0-471-58551-3.

ee also

*Scattering parameters
*Admittance parameters
*Two-port network