Scattering matrix to transmission matrix

Knowing the parameters of the scattering matrix, it is possible to determine the parameters of the transmission matrix.

Notes:

1. See Transmission Matrix.

2. See Scattering Matrix.

3. See this site for more information about \(S\)-parameters.

4. See this Wikipedia page for more information about \(S\)-parameters.

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.

Symbol:

Z_S

Latex:

\(Z_\text{S}\)

Dimension:

impedance

input_voltage_reflection_coefficient

Input port voltage reflection_coefficient.

Symbol:

S_ii

Latex:

\(S_\text{ii}\)

Dimension:

dimensionless

reverse_voltage_gain

Reverse voltage circuit_gain.

Symbol:

S_io

Latex:

\(S_\text{io}\)

Dimension:

dimensionless

forward_voltage_gain

Forward voltage circuit_gain.

Symbol:

S_oi

Latex:

\(S_\text{oi}\)

Dimension:

dimensionless

output_voltage_reflection_coefficient

Output port voltage reflection_coefficient.

Symbol:

S_oo

Latex:

\(S_\text{oo}\)

Dimension:

dimensionless

law

[[A, B], [C, D]] = [[((1 + S_ii) * (1 - S_oo) + S_io * S_oi) / (2 * S_oi), Z_S * ((1 + S_ii) * (1 + S_oo) - S_io * S_oi) / (2 * S_oi)], [((1 - S_ii) * (1 - S_oo) - S_io * S_oi) / Z_S / (2 * S_oi), ((1 - S_ii) * (1 + S_oo) + S_io * S_oi) / (2 * S_oi)]]

Latex:

\[\begin{split}\begin{pmatrix} A & B \\ C & D \end{pmatrix} = \begin{pmatrix} \frac{\left(1 + S_\text{ii}\right) \left(1 - S_\text{oo}\right) + S_\text{io} S_\text{oi}}{2 S_\text{oi}} & \frac{Z_\text{S} \left(\left(1 + S_\text{ii}\right) \left(1 + S_\text{oo}\right) - S_\text{io} S_\text{oi}\right)}{2 S_\text{oi}} \\ \frac{\left(1 - S_\text{ii}\right) \left(1 - S_\text{oo}\right) - S_\text{io} S_\text{oi}}{Z_\text{S}} \frac{1}{2 S_\text{oi}} & \frac{\left(1 - S_\text{ii}\right) \left(1 + S_\text{oo}\right) + S_\text{io} S_\text{oi}}{2 S_\text{oi}} \end{pmatrix}\end{split}\]