Transmission matrix of lossy transmission line

The transmission parameters matrix is one of the ways to describe a microwave device. The \(ABCD\)-parameters of the device act as elements of this matrix. The matrix equation relates the input voltage and input current to the output voltage and output current. Knowing the length and the loss factor of the transmission line, as well as the surge impedance of the line and the constant propagation of signal, it is possible to calculate the parameters \(A, B, C, D\) of the transmission matrix of this line.

Notes:

1. See Transmission Matrix.

voltage_voltage_parameter

Ratio of input voltage to output voltage at idle at the output.

Symbol:

A

Latex:

\(A\)

Dimension:

dimensionless

voltage_current_parameter

Ratio of input voltage to output current in case of a short circuit at the output.

Symbol:

B

Latex:

\(B\)

Dimension:

impedance

current_voltage_parameter

Ratio of input current to output voltage at idle at the output.

Symbol:

C

Latex:

\(C\)

Dimension:

conductance

current_current_parameter

Ratio of input current to output current in case of a short circuit at the output.

Symbol:

D

Latex:

\(D\)

Dimension:

dimensionless

surge_impedance

surge_impedance of the transmission line.

Symbol:

Z_S

Latex:

\(Z_\text{S}\)

Dimension:

impedance

length

length of the transmission line.

Symbol:

l

Latex:

\(l\)

Dimension:

length

propagation_constant

propagation_constant.

Symbol:

gamma

Latex:

\(\gamma\)

Dimension:

1/length

law

[[A, B], [C, D]] = [[cosh(l * gamma), Z_S * sinh(l * gamma)], [sinh(l * gamma) / Z_S, cosh(l * gamma)]]

Latex:

\[\begin{split}\begin{pmatrix} A & B \\ C & D \end{pmatrix} = \begin{pmatrix} \cosh{\left(l \gamma \right)} & Z_\text{S} \sinh{\left(l \gamma \right)} \\ \frac{\sinh{\left(l \gamma \right)}}{Z_\text{S}} & \cosh{\left(l \gamma \right)} \end{pmatrix}\end{split}\]