Admittance parameters

Admittance parameters or Y-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, [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 or ABCD-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 numbers.

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:

:.

where

:

:

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 the matrix 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