- Admittance parameters
**Admittance parameters**or**Y-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 , [*Pozar, David M. (2005); "Microwave Engineering, Third Edition" (Intl. Ed.); John Wiley & Sons, Inc.; pp 170-174. ISBN 0-471-44878-8.*]Z-parameters , [*Pozar, David M. (2005) (op. cit); pp 170-174.*]H-parameters ,T-parameters orABCD-parameters . [*Pozar, David M. (2005) (op. cit); pp 183-186.*] [*Morton, A. H. (1985); " Advanced Electrical Engineering";Pitman Publishing Ltd.; pp 33-72. ISBN 0-273-40172-6*]**The General Y-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 Y-parameter definition is in terms of input voltages and output currents, $V\_n,$ and $I\_n,$ respectively.

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

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

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

complex number s.The phase part of an Y-parameter is the "spatial" phase at the test frequency, not the temporal (time-related) phase.

**Two-Port Networks**The Y-parameter matrix for the

two-port network is probably the most common. In this case the relationship between the input voltages, output currents and the Y-parameter matrix is given by::$\{I\_1\; choose\; I\_2\}\; =\; egin\{pmatrix\}\; Y\_\{11\}\; Y\_\{12\}\; \backslash \; Y\_\{21\}\; Y\_\{22\}\; end\{pmatrix\}\{V\_1\; choose\; V\_2\}$.

where

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

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

**Admittance relations**The input admittance of a two-port network is given by:

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

where Y

_{L}is the admittance of the load connected to port two.Similarly, the output admittance is given by:

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

where Y

_{S}is the admittance of the source connected to port one.**Converting Two-Port Parameters**The two-port Y-parameters may be obtained from the equivalent two-port

S-parameters by means of the following expressions.:$Y\_\{11\}\; =\; \{((1\; -\; S\_\{11\})\; (1\; +\; S\_\{22\})\; +\; S\_\{12\}\; S\_\{21\})\; over\; Delta\_S\}\; ,$

:$Y\_\{12\}\; =\; \{-2\; S\_\{12\}\; over\; Delta\_S\}\; ,$

:$Y\_\{21\}\; =\; \{-2\; S\_\{21\}\; over\; Delta\_S\}\; ,$

:$Y\_\{22\}\; =\; \{((1\; +\; S\_\{11\})\; (1\; -\; S\_\{22\})\; +\; S\_\{12\}\; S\_\{21\})\; over\; Delta\_S\}\; ,$

Where

:$Delta\_S\; =\; (1\; +\; S\_\{11\})\; (1\; +\; S\_\{22\})\; -\; S\_\{12\}\; S\_\{21\}\; ,$

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

Z-parameters to Y-parameters is much simpler, as the Y-parameter matrix is basically thematrix inverse of the Z-parameter matrix. The following expressions show the applicable relations::$Y\_\{11\}\; =\; \{Z\_\{22\}\; over\; Delta\_Z\}\; ,$

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

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

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

Where

:$Delta\_Z\; =\; Z\_\{11\}\; Z\_\{22\}\; -\; Z\_\{12\}\; Z\_\{21\}\; ,$

In this case $Delta\_Z$ is the

determinant of the Z-parameter matrix.Vice versa the Y-parameters can be used to determine the Z-parameters, essentially using thesame expressions since

:$Y\; =\; Z^\{-1\}\; ,$

And

:$Z\; =\; Y^\{-1\}\; ,$

**References****Bibliography***David M. Pozar, "Microwave Engineering", Third Edition, John Wiley & Sons Inc.; ISBN 0-471-44878-8

**ee also***

Scattering parameters

*Impedance parameters

*Two-port network

*Hybrid-pi model

*Power gain

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