Transmission matrix of π-type matrix

The π-type circuit consists of the first impedance connected in parallel, the third impedance connected in series, and the second impedance connected in parallel. Knowing the impedances, it is possible to calculate the parameters \(A, B, C, D\) of the transmission matrix of this line.

Notes:

1. See Transmission Matrix.

2. Scheme of the circuit:

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

first_impedance

First electrical_impedance.

Symbol:

Z_1

Latex:

\(Z_{1}\)

Dimension:

impedance

second_impedance

Second electrical_impedance.

Symbol:

Z_2

Latex:

\(Z_{2}\)

Dimension:

impedance

third_impedance

Third electrical_impedance.

Symbol:

Z_3

Latex:

\(Z_{3}\)

Dimension:

impedance

law

[[A, B], [C, D]] = [[1 + Z_3 / Z_2, Z_3], [1 / Z_1 + 1 / Z_2 + Z_3 / (Z_1 * Z_2), 1 + Z_3 / Z_1]]

Latex:

\[\begin{split}\begin{pmatrix} A & B \\ C & D \end{pmatrix} = \begin{pmatrix} 1 + \frac{Z_{3}}{Z_{2}} & Z_{3} \\ \frac{1}{Z_{1}} + \frac{1}{Z_{2}} + \frac{Z_{3}}{Z_{1} Z_{2}} & 1 + \frac{Z_{3}}{Z_{1}} \end{pmatrix}\end{split}\]