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 Matrices & Matrix Conversions for 2-Port Nets   Sabato 22 Maggio 2010 19:20

Let's consider the following net: It is possible to represent the net characteristics by one of the following matrices

 Parameters [Z] [Y] [ABCD] [S] [Z] X [Z]→[Y] [Z]→[ABCD] [Z]→[S] [Y] [Y]→[Z] X [Y]→[ABCD] [Y]→[S] [ABCD] [ABCD]→[Z] [ABCD]→[Y] X [ABCD]→[S] [S] [S]→[Z] [S]→[Y] [S]→[ABCD] X

[Z] Parameters:

The open circuit impedance for a 2-port net is the following:

 [Z]= \begin{bmatrix} Z_{11} \; Z_{12} \\ Z_{21} \; Z_{22} \\ \end{bmatrix} (1)

it is possible to show the connection between voltages and currents in a matricial form:

 \begin{bmatrix} V_{1} \\ V_{2} \end{bmatrix} =[Z] \begin{bmatrix} I_{1} \\ I_{2} \end{bmatrix} (2)

by substituting (1) in (2) we obtain (3):

 \left\{\begin{matrix} V_{1}=Z_{11}I_{1}+Z_{12}I_{2}\\ V_{2}=Z_{21}I_{1}+Z_{22}I_{2}\\ \end{matrix}\right. (3)

now it is easy to find:

 Z_{11}=\frac{V_1}{I_1}\left.\begin{matrix} \\ \\ \end{matrix}\right| _{I_2=0} \; \: Z_{12}=\frac{V_1}{I_2}\left.\begin{matrix} \\ \\ \end{matrix}\right| _{I_1=0} \\" border="0" title="Z_{11}=\frac{V_1}{I_1}\left.\begin{matrix} \\ \\ \end{matrix}\right| _{I_2=0} \; \: Z_{12}=\frac{V_1}{I_2}\left.\begin{matrix} \\ \\ \end{matrix}\right| _{I_1=0} \\ Z_{21}=\frac{V_2}{I_1}\left.\begin{matrix} \\ \\ \end{matrix}\right| _{I_2=0} \; \: Z_{22}=\frac{V_2}{I_2}\left.\begin{matrix} \\ \\ \end{matrix}\right| _{I_1=0} \\" border="0" title="Z_{11}=\frac{V_1}{I_1}\left.\begin{matrix} \\ \\ \end{matrix}\right| _{I_2=0} \; \: Z_{12}=\frac{V_1}{I_2}\left.\begin{matrix} \\ \\ \end{matrix}\right| _{I_1=0} \\ (4)

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[Y] Parameters :

The short circuit admittance matrix of a  2-port net is the following:

 [Y]= \begin{bmatrix} Y_{11} \; Y_{12} \\ Y_{21} \; Y_{22} \\ \end{bmatrix} (5)

it is possible to show the connection between currents and voltages and currents in a matricial form:

 \begin{bmatrix} I_{1} \\ I_{2} \end{bmatrix} =[Y] \begin{bmatrix} V_{1} \\ V_{2} \end{bmatrix} (6)

by substituting (5) in (6) we obtain (7):

 \left\{\begin{matrix} I_{1}=Y_{11}V_{1}+Y_{12}V_{2}\\ I_{2}=Y_{21}V_{1}+Y_{22}V_{2}\\ \end{matrix}\right. (7)

now it is easy to find:

 Y_{11}=\frac{I_1}{V_1}\left.\begin{matrix} \\ \\ \end{matrix}\right| _{V_2=0} \; Y_{12}=\frac{I_1}{V_2}\left.\begin{matrix} \\ \\ \end{matrix}\right| _{V_1=0} \\ \\ \\ (8) Y_{21}=\frac{I_2}{V_1}\left.\begin{matrix} \\ \\ \end{matrix}\right| _{V_2=0} \; Y_{22}=\frac{I_2}{V_2}\left.\begin{matrix} \\ \\ \end{matrix}\right| _{V_1=0} \\ \\ \\ (9)

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[ABCD] Parameters:

The [ABCD] matrix is useful when we want to calculate a chain of multiple 2-port nets, the [ABCD] matrix is defined as follows:

 [ABCD]= \begin{bmatrix} A \; B \\ C \; D \\ \end{bmatrix} (10)

it is possible to show the connection between currents and voltages and currents in a matricial form:

 \begin{bmatrix} V_{1} \\ I_{1} \end{bmatrix} =[ABCD] \begin{bmatrix} V_{2} \\ -I_{2} \end{bmatrix} (11)

by substituting (10) in (11) we obtain (12):

 \left\{\begin{matrix} V_{1}=AV_{2}-BI_{2}\\ I_{1}=CV_{2}-DI_{2}\\ \end{matrix}\right. (12)

now it is easy to find:

 A=\frac{V_1}{V_2}\left.\begin{matrix} \\ \\ \end{matrix}\right| _{-I_2=0} \; B=\frac{V_1}{-I_2}\left.\begin{matrix} \\ \\ \end{matrix}\right| _{V_2=0} \\ \\  (13) C=\frac{I_1}{V_2}\left.\begin{matrix} \\ \\ \end{matrix}\right| _{-I_2=0} \; D=\frac{I_1}{-I_2}\left.\begin{matrix} \\ \\ \end{matrix}\right| _{V_2=0} \\ \\ (14)

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[S] Parameters:

The parameters calculated as a ratio between voltages and current with reference to a particular impedance (short circuit or open circuit) applied to the ports are not suitable for high frequency application due to fact that is not easy to make a short circuit or an open circuit at high frequencies. In fact at high frequencies we have the line effect of the connection between the load and the net so an open circuit or a short circuit became a reactive load. Let's consider the following net. Instead of voltages and currents we introduce the wave concept. This waves are so defined:

  \begin{matrix} a_1 = \frac{V_1 + Z_{01} I_1}{2 \sqrt{\mathbf{Re} \left\{ Z_{01} \right\} }} & a_2 = \frac{V_2 + Z_{02} I_2}{2 \sqrt{\mathbf{Re} \left\{ Z_{02} \right\} }} \\ b_1 = \frac{V_1 - Z^*_{01} I_1}{2 \sqrt{\mathbf{Re} \left\{ Z_{01} \right\} }} & b_2 = \frac{V_2 - Z^*_{02} I_2}{2 \sqrt{\mathbf{Re} \left\{ Z_{02} \right\} }}\end{matrix} (15)

The wave unit is $\sqrt{W}$ and are defined with respect to two impedances $Z_{01}$ and $Z_{02}$. In microwave and rf fields these impedances are selected as follows:

 Z_{01}=Z_{02}=Z_{0}=50 \Omega

We define the scattering matrix of a 2-port circuit:

 [S]= \begin{bmatrix} S_{11} \; S_{12} \\ S_{21} \; S_{22} \\ \end{bmatrix} (16)

it is possible to show the connection between reflected and incident waves:

 \begin{bmatrix} b_{1} \\ b_{2} \end{bmatrix} =[S] \begin{bmatrix} a_{1} \\ a_{2} \end{bmatrix} (17)

by substituting (16) in (17) we obtain (18):

 \left\{\begin{matrix} b_{1}=S_{11}a_{1}+S_{12}a_{2}\\ b_{2}=S_{21}a_{1}+S_{22}b_{2}\\ \end{matrix}\right (18)

now it is easy to find:

 S_{11}=\frac{b_1}{a_1}\left.\begin{matrix} \\ \\ \end{matrix}\right| _{a_2=0} \; S_{12}=\frac{b_1}{a_2}\left.\begin{matrix} \\ \\ \end{matrix}\right| _{a_1=0} \\ \\ \\ (19)
 S_{21}=\frac{b_2}{a_1}\left.\begin{matrix} \\ \\ \end{matrix}\right| _{a_2=0} \; S_{22}=\frac{b_2}{a_2}\left.\begin{matrix} \\ \\ \end{matrix}\right| _{a_1=0} \\ \\ \\ (20)

It is important to notice that this time the terminating condition at the ports are not a short circuit or an open circuit but are represented by an impedance of normalization $Z_{0}=50 \Omega$. If we connect this impedance at each port we will have the relfected waves from the load $a_{1}$ or $a_{2}$ are 0. For this reason $Z_{0}=50 \Omega$ is also called a matched load.

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[Z] to [Y] Conversion:

 Y_{11}=\frac{Z_{22}}{\left |Z \right |} \;\;\;\; Y_{12}=\frac{-Z_{12}}{\left |Z \right |} (21)
 Y_{21}=\frac{-Z_{21}}{\left |Z \right |} \;\;\;\; Y_{22}=\frac{Z_{11}}{\left |Z \right |} (22)

where $\left |Z \right |=Z_{11}Z_{22}-Z_{21}Z_{12}$

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[Y] to [Z] Conversion:

 Z_{11}=\frac{Y_{22}}{\left |Y \right |} \;\;\;\; Z_{12}=\frac{-Y_{12}}{\left |Y \right |} (23)
 Z_{21}=\frac{-Y_{21}}{\left |Y \right |} \;\;\;\; Z_{22}=\frac{Y_{11}}{\left |Y \right |} (24)

where $\left |Y \right |=Y_{11}Y_{22}-Y_{21}Y_{12}$

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[Z] to [ABCD] Conversion:

 A=\frac{Z_{11}}{Z_{21}} \;\;\;\; B=\frac{\left |Z \right |}{Z_{21}} (25)
 C=\frac{1}{Z_{21}} \;\;\;\; D=\frac{Z_{22}}{Z_{21}} (26)

where $\left |Z \right |=Z_{11}Z_{22}-Z_{21}Z_{12}$

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[Y] to [ABCD] Convertion:

 A=-\frac{Y_{22}}{Y_{21}} \;\;\;\; B=-\frac{1}{Y_{21}} (27)
 C=-\frac{\left |Y \right |}{Y_{21}} \;\;\;\; D=-\frac{Y_{11}}{Y_{21}} (28)

where $\left |Y \right |=Y_{11}Y_{22}-Y_{21}Y_{12}$

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[ABCD] to [Z] Convertion:

 Z_{11}=\frac{A}{C} \;\;\;\; Z_{12}=\frac{\left |ABCD \right |}{C} (29)
 Z_{21}=\frac{1}{C} \;\;\;\; Z_{22}=\frac{D}{C} (30)

where $\left |ABCD \right |=AD-BC$

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[ABCD] to [Y] Conversion:

 Y_{11}=\frac{D}{B} \;\;\;\; Y_{12}=-\frac{\left |ABCD \right |}{B} (31)
 Y_{21}=-\frac{1}{B} \;\;\;\; Y_{22}=\frac{A}{B} (32)

where $\left |ABCD \right |=AD-BC$

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[S] to [Z] Conversion:

 Z_{11}=Z_{0}\frac{(1+S_{11})(1-S_{22})+S_{12}S_{21}}{(1-S_{11})(1-S_{22})-S_{12}S_{21}} \;\;\;\; Z_{12}=\frac{2S_{12}Z_{0}}{(1-S_{11})(1-S_{22})-S_{12}S_{21}} (33)
 Z_{21}=\frac{2S_{21}Z_{0}}{(1-S_{11})(1-S_{22})-S_{12}S_{21}} \;\;\;\; Z_{22}=Z_{0}\frac{(1-S_{11})(1+S_{22})+S_{12}S_{21}}{(1-S_{11})(1-S_{22})-S_{12}S_{21}} (34)

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[S] to [Y] Conversion

 Y_{11}=Y_{0}\frac{(1-S_{11})(1+S_{22})+S_{12}S_{21}}{(1+S_{11})(1+S_{22})-S_{12}S_{21}} \;\;\;\; Y_{12}=\frac{-2S_{12}Y_{0}}{(1+S_{11})(1+S_{22})-S_{12}S_{21}} (35)
 Y_{21}=\frac{-2S_{21}Y_{0}}{(1+S_{11})(1+S_{22})-S_{12}S_{21}} \;\;\;\; Y_{22}=Y_{0}\frac{(1+S_{11})(1-S_{22})+S_{12}S_{21}}{(1+S_{11})(1+S_{22})-S_{12}S_{21}} (36)

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[Y] to [S] Conversion

 S_{11}=\frac{(1-Y_{11}Z_{0})(1+Y_{22}Z_{0})+Y_{12}Y_{21}Z_{0}^2}{(1+Y_{11}Z_{0})(1+Y_{22}Z_{0})- Y_{12}Y_{21}Z_{0}^2} \;\;\;\; S_{12}=\frac{-2Y_{12}Z_{0}}{(1+Y_{11}Z_{0})(1+Y_{22}Z_{0})- Y_{12}Y_{21}Z_{0}^2} (37)
 S_{21}=\frac{-2Y_{21}Z_{0}}{(1+Y_{11}Z_{0})(1+Y_{22}Z_{0})- Y_{12}Y_{21}Z_{0}^2} \;\;\;\; S_{22}=\frac{(1+Y_{11}Z_{0})(1-Y_{22}Z_{0})+Y_{12}Y_{21}Z_{0}^2}{(1+Y_{11}Z_{0})(1+Y_{22}Z_{0})- Y_{12}Y_{21}Z_{0}^2} (38)

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[Z] to [S] Conversion

 S_{11}=\frac{(Z_{11}Y_{0}-1)(Z_{22}Y_{0}+1)-Z_{12}Z_{21}Y_{0}^2}{(Z_{11}Y_{0}+1)(Z_{22}Y_{0}+1)- Z_{12}Z_{21}Y_{0}^2} \;\;\;\; S_{12}=\frac{2Z_{12}Y_{0}}{(Z_{11}Y_{0}+1)(Z_{22}Y_{0}+1)- Z_{12}Z_{21}Y_{0}^2} (39)
 S_{21}=\frac{2Z_{21}Y_{0}}{(Z_{11}Y_{0}+1)(Z_{22}Y_{0}+1)- Z_{12}Z_{21}Y_{0}^2} \;\;\;\; S_{22}=\frac{(Z_{11}Y_{0}+1)(Z_{22}Y_{0}-1)-Z_{12}Z_{21}Y_{0}^2}{(Z_{11}Y_{0}+1)(Z_{22}Y_{0}+1)- Z_{12}Z_{21}Y_{0}^2} (40)

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[ABCD] to [S] Conversion

 S_{11}=\frac{AZ_{0}+B-CZ_{0}^2-DZ_{0}}{AZ_{0}+B+CZ_{0}^2+DZ_{0}} \;\;\;\; S_{12}=\frac{2 \left| ABCD \right | Z_{0}}{AZ_{0}+B+CZ_{0}^2+DZ_{0}} (41)
 S_{21}=\frac{2Z_{0}}{AZ_{0}+B+CZ_{0}^2+DZ_{0}} \;\;\;\; S_{22}=\frac{-AZ_{0}+B-CZ_{0}^2+DZ_{0}}{AZ_{0}+B+CZ_{0}^2+DZ_{0}} (42)

where $\left |ABCD \right |=AD-BC$

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[S] to [ABCD] Conversion

 A=\frac{(1+S_{11})(1-S_{22})+S_{12}S_{21}}{2S_{21}}\;\;\;\; B=Z_{0}\frac{(1+S_{11})(1+S_{22})-S_{12}S_{21}}{2S_{21}} (43)
 C=Y_{0}\frac{(1-S_{11})(1-S_{22})-S_{12}S_{21}}{2S_{21}} \;\;\;\; D=\frac{(1-S_{11})(1+S_{22})+S_{12}S_{21}}{2S_{21}} (44)

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Ultimo aggiornamento Sabato 03 Settembre 2016 07:49