Scattering matrix equation

The scattering matrix is one of the ways to describe a microwave device. The \(S\)-parameters of the device act as elements of this matrix. The matrix equation relates the waves reflected from the input and output to the waves incident on the input and output.

Notes:

1. For definitions of \(a\) and \(b\) presented below, refer here (Wikipedia).

Links:

1. Wikipedia.

input_reflected_power_wave

Outgoing (“reflected”) power wave at input port.

Symbol:

b_i

Latex:

\(b_\text{i}\)

Dimension:

sqrt(power)

output_reflected_power_wave

Outgoing (“reflected”) power wave at output port.

Symbol:

b_o

Latex:

\(b_\text{o}\)

Dimension:

sqrt(power)

input_incident_power_wave

Incident power wave at input port.

Symbol:

a_i

Latex:

\(a_\text{i}\)

Dimension:

sqrt(power)

output_incident_power_wave

Incident power wave at output port.

Symbol:

a_o

Latex:

\(a_\text{o}\)

Dimension:

sqrt(power)

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

[b_i, b_o] = [[S_ii, S_io], [S_oi, S_oo]]*[a_i, a_o]

Latex:

\[\begin{split}\begin{pmatrix} b_\text{i} \\ b_\text{o} \end{pmatrix} = \begin{pmatrix} S_\text{ii} & S_\text{io} \\ S_\text{oi} & S_\text{oo} \end{pmatrix} \begin{pmatrix} a_\text{i} \\ a_\text{o} \end{pmatrix}\end{split}\]