|
en>Ser Amantio di Nicolao |
| Line 1: |
Line 1: |
| '''Admittance parameters''' or '''Y-parameters''' (the elements of an '''admittance matrix''' or '''Y-matrix''') are properties used in [[electrical engineering]], [[electronic engineering]], and [[communications system|communication systems]] engineering 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>Pozar, David M. (2005); ''Microwave Engineering, Third Edition'' (Intl. Ed.); John Wiley & Sons, Inc.; pp 170-174. ISBN 0-471-44878-8.</ref> [[Z-parameters]],<ref>Pozar, David M. (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>Pozar, David M. (2005) (op. cit); pp 183-186.</ref><ref>Morton, A. H. (1985); '' Advanced Electrical Engineering'';Pitman Publishing Ltd.; pp 33-72. ISBN 0-273-40172-6</ref>
| | The individual who wrote the post is called Jayson Hirano and he completely digs that title. Distributing manufacturing is where my primary income arrives from and [https://www.machlitim.org.il/subdomain/megila/end/node/12300 accurate psychic readings] it's something I really appreciate. For a whilst I've been in Mississippi but now I'm considering other options. One of the extremely very best things in the globe for him is doing ballet and he'll be starting some thing else along with it.<br><br>My webpage :: free [http://breenq.com/index.php?do=/profile-1144/info/ accurate psychic readings] ([http://165.132.39.93/xe/visitors/372912 39.93]) |
| | |
| ==The Y-parameter matrix==
| |
| A Y-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 Y-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 Y-parameter definition is in terms of the port voltage and port current, <math>V_n\,</math> and <math>I_n\,</math> respectively. | |
| | |
| For all ports the currents may be defined in terms of the Y-parameter matrix and the voltages by the following matrix equation:
| |
| | |
| :<math>I = Y V\,</math>
| |
| | |
| where Y is an ''N'' × ''N'' matrix the elements of which can be indexed using conventional [[matrix (mathematics)|matrix]] notation. In general the elements of the Y-parameter matrix are [[complex number]]s and functions of frequency. For a one-port network, the Y-matrix reduces to a single element, being the ordinary [[admittance]] measured between the two terminals.
| |
| | |
| ==Two-port networks==
| |
| [[Image:Y-parameter two-port.PNG|thumbnail|300px| Figure 5: Y-equivalent two port showing independent variables ''V''<sub>1</sub> and ''V''<sub>2</sub>. Although resistors are shown, general admittances can be used instead.]]
| |
| The Y-parameter matrix for the [[two-port network]] is probably the most common. In this case the relationship between the port voltages, port currents and the Y-parameter matrix is given by:
| |
| | |
| :<math>{I_1 \choose I_2} = \begin{pmatrix} Y_{11} & Y_{12} \\ Y_{21} & Y_{22} \end{pmatrix}{V_1 \choose V_2} </math>.
| |
| | |
| where
| |
| | |
| :<math>Y_{11} = {I_1 \over V_1 } \bigg|_{V_2 = 0} \qquad Y_{12} = {I_1 \over V_2 } \bigg|_{V_1 = 0}</math> | |
| | |
| :<math>Y_{21} = {I_2 \over V_1 } \bigg|_{V_2 = 0} \qquad Y_{22} = {I_2 \over V_2 } \bigg|_{V_1 = 0}</math> | |
| | |
| ===Admittance relations===
| |
| | |
| The input admittance of a two-port network is given by:
| |
| | |
| :<math>Y_{in} = y_{11} - \frac{y_{12}y_{21}}{y_{22}+Y_L}</math>
| |
| | |
| where Y<sub>L</sub> is the admittance of the load connected to port two.
| |
| | |
| Similarly, the output admittance is given by:
| |
| | |
| :<math>Y_{out} = y_{22} - \frac{y_{12}y_{21}}{y_{11}+Y_S}</math>
| |
| | |
| where Y<sub>S</sub> is the admittance of the source connected to port one.
| |
| | |
| ==Relation to S-parameters==
| |
| | |
| The Y-parameters of a network are related to its S-Parameters by<ref name=Russer>{{cite book |title=Electromagnetics, microwave circuit and antenna design for communications engineering |last=Russer |first=Peter |year=2003 |publisher=Artech House |isbn=1-58053-532-1}}</ref>
| |
| | |
| :<math> \begin{align}
| |
| Y &= \sqrt{y} (1_{\!N} - S) (1_{\!N} + S)^{-1} \sqrt{y} \\
| |
| &= \sqrt{y} (1_{\!N} + S)^{-1} (1_{\!N} - S) \sqrt{y} \\
| |
| \end{align} </math>
| |
| | |
| and<ref name=Russer/>
| |
| | |
| :<math> \begin{align}
| |
| S &= (1_{\!N} - \sqrt{z}Y\sqrt{z}) (1_{\!N} + \sqrt{z}Y\sqrt{z})^{-1} \\
| |
| &= (1_{\!N} + \sqrt{z}Y\sqrt{z})^{-1} (1_{\!N} - \sqrt{z}Y\sqrt{z}) \\
| |
| \end{align} </math>
| |
| | |
| where <math>1_{\!N}</math> is the [[identity matrix]], <math>\sqrt{y}</math> is a [[diagonal matrix]] having the square root of the [[characteristic admittance]] (the reciprocal of the [[characteristic impedance]]) at each port as its non-zero elements,
| |
| | |
| <math>\sqrt{y} = \begin{pmatrix}
| |
| \sqrt{y_{01}} & \\
| |
| & \sqrt{y_{02}} \\
| |
| & & \ddots \\
| |
| & & & \sqrt{y_{0N}}
| |
| \end{pmatrix}
| |
| </math>
| |
| | |
| and <math>\sqrt{z} = (\sqrt{y})^{-1}</math> is the corresponding diagonal matrix of square roots of [[characteristic impedance]]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'''''<sup>-1</sup> (since '''''AB'''''<sup>-1</sup> = '''''B'''''<sup>-1</sup>'''''BAB'''''<sup>-1</sup> = '''''B'''''<sup>-1</sup>'''''ABB'''''<sup>-1</sup> = '''''B'''''<sup>-1</sup>'''''A''''')</ref>
| |
| | |
| ===Two port===
| |
| In the special case of a two-port network, with the same characteristic admittance <math>y_{01} = y_{02} = Y_0</math> at each port, the above expressions reduce to
| |
| | |
| :<math>Y_{11} = {((1 - S_{11}) (1 + S_{22}) + S_{12} S_{21}) \over \Delta_S} Y_0 \,</math>
| |
| | |
| :<math>Y_{12} = {-2 S_{12} \over \Delta_S} Y_0 \,</math>
| |
| | |
| :<math>Y_{21} = {-2 S_{21} \over \Delta_S} Y_0 \,</math>
| |
| | |
| :<math>Y_{22} = {((1 + S_{11}) (1 - S_{22}) + S_{12} S_{21}) \over \Delta_S} Y_0 \,</math>
| |
| | |
| Where
| |
| | |
| :<math>\Delta_S = (1 + S_{11}) (1 + S_{22}) - S_{12} S_{21} \,</math>
| |
| | |
| The above expressions will generally use complex numbers for <math>S_{ij}</math> and <math>Y_{ij}</math>. Note that the value of <math>\Delta</math> can become 0 for specific values of <math>S_{ij}</math> so the division by <math>\Delta</math> in the calculations of <math>Y_{ij}</math> may lead to a division by 0.
| |
| | |
| The two-port S-parameters may also be obtained from the equivalent two-port Y-parameters by means of the following expressions.<ref>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.</ref>
| |
| | |
| :<math>S_{11} = {(1 - Z_0 Y_{11}) (1 + Z_0 Y_{22}) + Z^2_0 Y_{12} Y_{21} \over \Delta} \,</math>
| |
| | |
| :<math>S_{12} = {-2 Z_0 Y_{12} \over \Delta} \,</math>
| |
| | |
| :<math>S_{21} = {-2 Z_0 Y_{21} \over \Delta} \,</math>
| |
| | |
| :<math>S_{22} = {(1 + Z_0 Y_{11}) (1 - Z_0 Y_{22}) + Z^2_0 Y_{12} Y_{21} \over \Delta} \,</math>
| |
| | |
| where
| |
| | |
| :<math>\Delta = (1 + Z_0 Y_{11}) (1 + Z_0 Y_{22}) - Z^2_0 Y_{12} Y_{21} \,</math>
| |
| | |
| and <math>Z_0</math> is the [[characteristic impedance]] at each port (assumed the same for the two ports).
| |
| | |
| ==Relation to Z-parameters==
| |
| Conversion from [[Z-parameters]] to Y-parameters is much simpler, as the Y-parameter matrix is just the [[matrix inverse|inverse]] of the Z-parameter matrix. The following expressions show the applicable relations:
| |
| | |
| :<math>Y_{11} = {Z_{22} \over \Delta_Z} \,</math>
| |
| | |
| :<math>Y_{12} = {-Z_{12} \over \Delta_Z} \,</math>
| |
| | |
| :<math>Y_{21} = {-Z_{21} \over \Delta_Z} \,</math>
| |
| | |
| :<math>Y_{22} = {Z_{11} \over \Delta_Z} \,</math>
| |
| | |
| Where
| |
| | |
| :<math>\Delta_Z = Z_{11} Z_{22} - Z_{12} Z_{21} \,</math>
| |
| | |
| In this case <math>\Delta_Z</math> is the [[determinant]] of the Z-parameter matrix.
| |
| | |
| Vice versa the Y-parameters can be used to determine the Z-parameters, essentially using the
| |
| same expressions since
| |
| | |
| :<math>Y = Z^{-1} \,</math>
| |
| | |
| And
| |
| | |
| :<math>Z = Y^{-1} \,</math>
| |
| | |
| ==Notes==
| |
| <references group="note" />
| |
| | |
| ==References==
| |
| <references/>
| |
| | |
| ==See also==
| |
| *[[Scattering parameters]]
| |
| *[[Impedance parameters]]
| |
| *[[Two-port network]]
| |
| *[[Hybrid-pi model]]
| |
| *[[Power gain]]
| |
| | |
| [[Category:Two-port networks]]
| |
| | |
| [[de:Zweitor#Zweitorgleichungen_und_Parameter]]
| |
The individual who wrote the post is called Jayson Hirano and he completely digs that title. Distributing manufacturing is where my primary income arrives from and accurate psychic readings it's something I really appreciate. For a whilst I've been in Mississippi but now I'm considering other options. One of the extremely very best things in the globe for him is doing ballet and he'll be starting some thing else along with it.
My webpage :: free accurate psychic readings (39.93)