- Impedance parameters
**Impedance parameters**or**Z-parameters**are properties used inelectrical engineering ,electronics engineering , and communication systems engineering describe the electrical behavior oflinear electrical network s 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 orABCD-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 number s.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::$\{V\_1\; choose\; V\_2\}\; =\; egin\{pmatrix\}\; Z\_\{11\}\; Z\_\{12\}\; \backslash \; Z\_\{21\}\; Z\_\{22\}\; end\{pmatrix\}\{I\_1\; choose\; I\_2\}$.

where

:$Z\_\{11\}\; =\; \{V\_1\; over\; I\_1\; \}\; igg|\_\{I\_2\; =\; 0\}\; qquad\; Z\_\{12\}\; =\; \{V\_1\; over\; I\_2\; \}\; igg|\_\{I\_1\; =\; 0\}$

:$Z\_\{21\}\; =\; \{V\_2\; over\; I\_1\; \}\; igg|\_\{I\_2\; =\; 0\}\; qquad\; Z\_\{22\}\; =\; \{V\_2\; over\; I\_2\; \}\; igg|\_\{I\_1\; =\; 0\}$

For the general case of an n-port network, it can be stated that:$Z\_\{nm\}\; =\; \{V\_n\; over\; I\_m\; \}\; igg|\_\{I\_n\; =\; 0\}$

**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 thematrix 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

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