Transmission matrix of lossless transmission line

Knowing the length of the transmission line, as well as the surge impedance of the line and the propagation constant of the signal, it is possible to calculate the parameters \(A, B, C, D\) of the transmission matrix of a lossless line.

Notes:

1. See Transmission Matrix.

Conditions:

1. The transmission line is lossless.

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

phase_constant

:symbols`phase_constant`.

Symbol:

beta

Latex:

\(\beta\)

Dimension:

1/length

law

[[A, B], [C, D]] = [[cos(beta * l), I * Z_S * sin(beta * l)], [I / Z_S * sin(beta * l), cos(beta * l)]]

Latex:

\[\begin{split}\begin{pmatrix} A & B \\ C & D \end{pmatrix} = \begin{pmatrix} \cos{\left(\beta l \right)} & i Z_\text{S} \sin{\left(\beta l \right)} \\ \frac{i}{Z_\text{S}} \sin{\left(\beta l \right)} & \cos{\left(\beta l \right)} \end{pmatrix}\end{split}\]