en>Atepomarus at 14:38, 16 October 2013

2013-10-16T14:38:45Z

New page

'''Impedance parameters''' or '''Z-parameters''' (the elements of an '''impedance matrix''' or '''Z-matrix''') are properties used in [[electrical engineering]], [[electronic engineering]], and [[communications system|communication systems]] engineering to describe the electrical behavior of [[linear]] [[electrical network]]s. They are also used to describe the [[small-signal]] ([[linearization|linearized]]) response of non-linear networks. They are members of a family of similar parameters used in electronic engineering, other examples being: [[S-parameters]],<ref>{{cite book| author = David M. Pozar| title = Microwave Engineering| date = 2004-02-05| publisher = Wiley| isbn = 978-0-471-44878-5| pages = 170-174 }}</ref> [[Y-parameters]],<ref>David M. Pozar, 2005 (op. cit); pp 170-174.</ref> [[Two port#Hybrid parameters (h-parameters)|H-parameters]], [[T-parameters]] or [[Two-port network#ABCD-parameters|ABCD-parameters]].<ref>David M. Pozar, 2005 (op. cit); pp 183-186.</ref><ref>A.H. Morton, '' Advanced Electrical Engineering'', Pitman Publishing Ltd., 1985; pp 33-72, ISBN 0-273-40172-6.</ref>

Z-parameters are also known as ''open-circuit impedance parameters'' as they are calculated under [[open circuit]] conditions. i.e., Ix=0, where x=1,2 refer to input and output currents flowing through the ports (of a [[two-port network]] in this case) respectively.

==The Z-parameter matrix==
A Z-parameter matrix describes the behaviour of any linear electrical network that can be regarded as a [[black box]] with a number of ports. A ''port'' in this context is a pair of [[terminal (electronics)|electrical terminals]] carrying equal and opposite currents into and out-of the network, and having a particular [[voltage]] between them. The Z-matrix gives no information about the behaviour of the network when the currents at any port are not balanced in this way (should this be possible), nor does it give any information about the voltage between terminals not belonging to the same port. Typically, it is intended that each external connection to the network is between the terminals of just one port, so that these limitations are appropriate.

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 the port current and port voltage, <math>I_n\,</math> and <math>V_n\,</math> respectively.

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

:<math>V = Z I\,</math>

where Z is an ''N'' × ''N'' matrix the elements of which can be indexed using conventional [[matrix (mathematics)|matrix]] notation. In general the elements of the Z-parameter matrix are [[complex number]]s and functions of frequency. For a one-port network, the Z-matrix reduces to a single element, being the ordinary [[electrical impedance|impedance]] measured between the two terminals.The Z-parameters are also known as the open circuit parameters as for their determination we need to open circuit one port and analyze the currents and voltages through other ports.

==Two-port networks==
The Z-parameter matrix for the [[two-port network]] is probably the most common. In this case the relationship between the port currents, port voltages and the Z-parameter matrix is given by:

:<math>{V_1 \choose V_2} = \begin{pmatrix} Z_{11} & Z_{12} \\ Z_{21} & Z_{22} \end{pmatrix}{I_1 \choose I_2} </math>.

where

:<math>Z_{11} = {V_1 \over I_1 } \bigg|_{I_2 = 0} \qquad Z_{12} = {V_1 \over I_2 } \bigg|_{I_1 = 0}</math>

:<math>Z_{21} = {V_2 \over I_1 } \bigg|_{I_2 = 0} \qquad Z_{22} = {V_2 \over I_2 } \bigg|_{I_1 = 0}</math>

For the general case of an ''N''-port network,
:<math>Z_{nm} = {V_n \over I_m } \bigg|_{I_k = 0 \text{ for } k \ne m}</math>

===Impedance relations===

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

:<math>Z_{in} = Z_{11} - \frac{Z_{12}Z_{21}}{Z_{22}+Z_L}</math>

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

Similarly, the output impedance is given by:

:<math>Z_{out} = Z_{22} - \frac{Z_{12}Z_{21}}{Z_{11}+Z_S}</math>

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

==Relation to S-parameters==

The Z-parameters of a network are related to its [[S-parameters]] by<ref name="Russer">{{cite book| last = Russer| first = Peter| title = Electromagnetics, microwave circuit and antenna design for communications engineering| year = 2003| publisher = Artech House| isbn = 1-58053-532-1 }}</ref>

:<math> \begin{align}
Z &= \sqrt{z} (1_{\!N} + S) (1_{\!N} - S)^{-1} \sqrt{z} \\
&= \sqrt{z} (1_{\!N} - S)^{-1} (1_{\!N} + S) \sqrt{z} \\
\end{align} </math> 

and<ref name=Russer/>

:<math> \begin{align}
S &= (\sqrt{y}Z\sqrt{y} \,- 1_{\!N}) (\sqrt{y}Z\sqrt{y} \,+ 1_{\!N})^{-1} \\
&= (\sqrt{y}Z\sqrt{y} \,+ 1_{\!N})^{-1} (\sqrt{y}Z\sqrt{y} \,- 1_{\!N}) \\
\end{align} </math> 

where <math>1_{\!N}</math> is the [[identity matrix]], <math>\sqrt{z}</math> is a [[diagonal matrix]] having the square root of the [[characteristic impedance]] at each port as its non-zero elements,

<math>\sqrt{z} = \begin{pmatrix}
\sqrt{z_{01}} & \\
& \sqrt{z_{02}} \\
& & \ddots \\
& & & \sqrt{z_{0N}}
\end{pmatrix}
</math>

and <math>\sqrt{y} = (\sqrt{z})^{-1}</math> is the corresponding diagonal matrix of square roots of [[characteristic admittance]]s. In these expressions the matrices represented by the bracketed factors [[Commuting matrices|commute]] and so, as shown above, may be written in either order.<ref name=Russer/><ref group="note">Any square matrix commutes with itself and with the identity matrix, and if two matrices '''''A''''' and '''''B''''' commute, then so do '''''A''''' and '''''B'''''-1 (since '''''AB'''''-1 = '''''B'''''-1'''''BAB'''''-1 = '''''B'''''-1'''''ABB'''''-1 = '''''B'''''-1'''''A''''')</ref>

===Two port===
In the special case of a two-port network, with the same characteristic impedance <math>z_{01} = z_{02} = Z_0</math> at each port, the above expressions reduce to

:<math>Z_{11} = {((1 + S_{11}) (1 - S_{22}) + S_{12} S_{21}) \over \Delta_S} Z_0 \,</math>

:<math>Z_{12} = {2 S_{12} \over \Delta_S} Z_0 \,</math>

:<math>Z_{21} = {2 S_{21} \over \Delta_S} Z_0 \,</math>

:<math>Z_{22} = {((1 - S_{11}) (1 + S_{22}) + S_{12} S_{21}) \over \Delta_S} Z_0 \,</math>

Where

:<math>\Delta_S = (1 - S_{11}) (1 - S_{22}) - S_{12} S_{21} \,</math>

The two-port S-parameters may be obtained from the equivalent two-port Z-parameters by means of the following expressions<ref>{{cite book| author = Simon Ramo| coauthors = John R. Whinnery, Theodore Van Duzer| title = Fields and Waves in Communication Electronics| date = 1994-02-09| publisher = Wiley| isbn = 978-0-471-58551-0| pages = 537-541 }}</ref>

:<math>S_{11} = {(Z_{11} - Z_0) (Z_{22} + Z_0) - Z_{12} Z_{21} \over \Delta}</math>

:<math>S_{12} = {2 Z_0 Z_{12} \over \Delta} \,</math>

:<math>S_{21} = {2 Z_0 Z_{21} \over \Delta} \,</math>

:<math>S_{22} = {(Z_{11} + Z_0) (Z_{22} - Z_0) - Z_{12} Z_{21} \over \Delta}</math>

where

:<math>\Delta = (Z_{11} + Z_0) (Z_{22} + Z_0) - Z_{12} Z_{21} \, </math>

The above expressions will generally use complex numbers for <math>S_{ij} \, </math> and <math>Z_{ij} \, </math>. Note that the value of <math>\Delta\, </math> can become 0 for specific values of <math>Z_{ij} \, </math> so the division by <math>\Delta \, </math> in the calculations of <math>S_{ij} \,</math> may lead to a division by 0.

==Relation to Y-parameters==

Conversion from [[Y-parameters]] to Z-parameters is much simpler, as the Z-parameter matrix is just the [[matrix inverse|inverse]] of the Y-parameter matrix. For a two-port:

:<math>Z_{11} = {Y_{22} \over \Delta_Y} \,</math>

:<math>Z_{12} = {-Y_{12} \over \Delta_Y} \,</math>

:<math>Z_{21} = {-Y_{21} \over \Delta_Y} \,</math>

:<math>Z_{22} = {Y_{11} \over \Delta_Y} \,</math>

where

:<math>\Delta_Y = Y_{11} Y_{22} - Y_{12} Y_{21} \,</math>

is the [[determinant]] of the Y-parameter matrix.

==Notes==
<references group="note" />

==References==
<references/>

==Bibliography==
* {{cite book| author = David M. Pozar| title = Microwave Engineering| date = 2004-02-05| publisher = Wiley| isbn = 978-0-471-44878-5 }}
* {{cite book| author = Simon Ramo| coauthors = John R. Whinnery, Theodore Van Duzer| title = Fields and Waves in Communication Electronics| date = 1994-02-09| publisher = Wiley| isbn = 978-0-471-58551-0 }}

==See also==
* [[Scattering parameters]]
* [[Admittance parameters]]
* [[Two-port network]]

[[Category:Electronics]]
[[Category:Two-port networks]]

[[de:Zweitor#Zweitorgleichungen_und_Parameter]]

Bidirectional map - Revision history

en>Atepomarus at 14:38, 16 October 2013