Editing Two-port network (section)

==''ABCD''-parameters {{anchor|ABCD-parameters}}== <!-- Other articles link to this section. Fix these links if moving or renaming -->
The {{mvar|ABCD}}-parameters are known variously as chain, cascade, or transmission parameters.  There are a number of definitions given for {{mvar|ABCD}} parameters, the most common is,<ref>Matthaei et al, p. 26.</ref><ref>Ghosh, p. 353.</ref>

:<math> \begin{bmatrix} V_1 \\ I_1 \end{bmatrix} = \begin{bmatrix} A & B \\ C & D \end{bmatrix} \begin{bmatrix} V_2 \\ -I_2 \end{bmatrix} </math>

Note: Some authors chose to reverse the indicated direction of I<sub>2</sub> and suppress the negative sign on I<sub>2</sub>.

where

:<math>\begin{align}
  A &\mathrel{\stackrel{\text{def}}{=}} \left.  \frac{V_1}{V_2} \right|_{I_2 = 0} &
  B &\mathrel{\stackrel{\text{def}}{=}} \left. -\frac{V_1}{I_2} \right|_{V_2 = 0} \\
  C &\mathrel{\stackrel{\text{def}}{=}} \left.  \frac{I_1}{V_2} \right|_{I_2 = 0} &
  D &\mathrel{\stackrel{\text{def}}{=}} \left. -\frac{I_1}{I_2} \right|_{V_2 = 0}
\end{align}</math>

For reciprocal networks {{math|1=''AD'' – ''BC'' = 1}}.  For symmetrical networks {{math|1=''A'' = ''D''}}.  For networks which are reciprocal and lossless, {{mvar|A}} and {{mvar|D}} are purely real while {{mvar|B}} and {{mvar|C}} are purely imaginary.<ref name=Matt27/>

This representation is preferred because when the parameters are used to represent a cascade of two-ports, the matrices are written in the same order that a network diagram would be drawn, that is, left to right.  However, a variant definition is also in use,<ref>A. Chakrabarti, p. 581, {{ISBN|81-7700-000-4}}, Dhanpat Rai & Co pvt. ltd.</ref>

:<math> \begin{bmatrix} V_2 \\ -I_2 \end{bmatrix} = \begin{bmatrix} A' & B' \\ C' & D' \end{bmatrix} \begin{bmatrix} V_1 \\ I_1 \end{bmatrix} </math>

where

:<math>\begin{align}
  A' &\mathrel{\stackrel{\text{def}}{=}} \left.  \frac{V_2}{V_1} \right|_{I_1 = 0} &
  B' &\mathrel{\stackrel{\text{def}}{=}} \left.  \frac{V_2}{I_1} \right|_{V_1 = 0} \\
  C' &\mathrel{\stackrel{\text{def}}{=}} \left. -\frac{I_2}{V_1} \right|_{I_1 = 0} &
  D' &\mathrel{\stackrel{\text{def}}{=}} \left. -\frac{I_2}{I_1} \right|_{V_1 = 0}
\end{align}</math>

The negative sign of {{math|–''I''{{sub|2}}}} arises to make the output current of one cascaded stage (as it appears in the matrix) equal to the input current of the next.  Without the minus sign the two currents would have opposite senses because the positive direction of current, by convention, is taken as the current entering the port.  Consequently, the input voltage/current matrix vector can be directly replaced with the matrix equation of the preceding cascaded stage to form a combined {{mvar| A'B'C'D'}} matrix.

The terminology of representing the {{mvar| ABCD}} parameters as a matrix of elements designated {{math|''a''<sub>11</sub>}} etc. as adopted by some authors<ref>Farago, p. 102.</ref> and the inverse {{mvar| A'B'C'D'}} parameters as a matrix of elements designated {{math|''b''<sub>11</sub>}} etc. is used here for both brevity and to avoid confusion with circuit elements.

:<math>\begin{align}
  \left[\mathbf{a}\right] &= \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix} = \begin{bmatrix} A  & B  \\ C  & D  \end{bmatrix} \\
  \left[\mathbf{b}\right] &= \begin{bmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{bmatrix} = \begin{bmatrix} A' & B' \\ C' & D' \end{bmatrix}
\end{align}</math>

===Table of transmission parameters===
The table below lists {{mvar|ABCD}} and inverse {{mvar|ABCD}} parameters for some simple network elements.

{| class="wikitable" border="1"
|-
! Element
! {{math|['''a''']}} matrix
! {{math|['''b''']}} matrix
! Remarks
|- 
| Series impedance
| align="center" | <math>\begin{bmatrix} 1 & Z \\ 0 & 1 \end{bmatrix} </math>
| align="center" | <math>\begin{bmatrix} 1 & -Z \\ 0 & 1 \end{bmatrix} </math>
| {{mvar|Z}}, impedance<br/>
|- 
| Shunt admittance
| align="center" | <math>\begin{bmatrix} 1 & 0 \\ Y & 1 \end{bmatrix} </math>
| align="center" | <math>\begin{bmatrix} 1 & 0 \\ -Y & 1 \end{bmatrix} </math>
| {{mvar|Y}}, admittance<br/>
|- 
| Series inductor
| align="center" | <math>\begin{bmatrix} 1 & sL \\ 0 & 1 \end{bmatrix} </math>
| align="center" | <math>\begin{bmatrix} 1 & -sL \\ 0 & 1 \end{bmatrix} </math>
| {{mvar|L}}, inductance<br/> {{mvar|s}}, complex angular frequency
|- 
| Shunt inductor
| align="center" | <math>\begin{bmatrix} 1 & 0 \\ {1\over sL} & 1 \end{bmatrix} </math>
| align="center" | <math>\begin{bmatrix} 1 & 0 \\ -\frac{1}{sL} & 1 \end{bmatrix} </math>
| {{mvar|L}}, inductance<br/> {{mvar|s}}, complex angular frequency
|- 
| Series capacitor
| align="center" | <math>\begin{bmatrix} 1 & {1\over sC} \\ 0 & 1 \end{bmatrix} </math>   
| align="center" | <math>\begin{bmatrix} 1 & -\frac{1}{sC} \\ 0 & 1 \end{bmatrix} </math>
| {{mvar|C}}, capacitance<br />{{mvar|s}}, complex angular frequency
|- 
| Shunt capacitor
| align="center" | <math>\begin{bmatrix} 1 & 0 \\ sC & 1 \end{bmatrix} </math>
| align="center" | <math>\begin{bmatrix} 1 & 0 \\ -sC & 1 \end{bmatrix} </math>
| {{mvar|C}}, capacitance<br/>{{mvar|s}}, complex angular frequency
|-

| Transmission line
| align="center" | <math>\begin{bmatrix} \cosh(\gamma l) & Z_0 \sinh(\gamma l) \\ \frac{1}{Z_0} \sinh(\gamma l) & \cosh(\gamma l) \end{bmatrix} </math>
| align="center" | <math>\begin{bmatrix} \cosh(\gamma l) & -Z_0 \sinh(\gamma l) \\ -\frac{1}{Z_0} \sinh\left(\gamma l\right) & \cosh(\gamma l) \end{bmatrix} </math> <ref>Clayton, p. 271.</ref>
| {{math|''Z''<sub>0</sub>}}, [[characteristic impedance]]<br/>{{mvar|γ}}, [[propagation constant]] (<math>\gamma = \alpha +i \beta</math>)<br/>{{mvar|l}}, length of transmission line ({{mvar|m}})
|}