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 :math:`A, B, C, D` of the transmission matrix of a lossless line.

**Notes:**

#. See :ref:`Transmission Matrix <transmission_matrix_def>`.

**Conditions:**

#. The transmission line is lossless.

..
    TODO: find link

.. py:currentmodule:: symplyphysics.laws.electricity.circuits.transmission_lines.transmission_matrix_lossless_transmission_line

.. py:data:: voltage_voltage_parameter

    Ratio of input :attr:`~symplyphysics.symbols.electrodynamics.voltage` to output :attr:`~symplyphysics.symbols.electrodynamics.voltage` at idle at the output.

Symbol:
    :code:`A`

Latex:
    :math:`A`

Dimension:
    :code:`dimensionless`


.. py:data:: voltage_current_parameter

    Ratio of input :attr:`~symplyphysics.symbols.electrodynamics.voltage` to output :attr:`~symplyphysics.symbols.electrodynamics.current` in case of a short
    circuit at the output.

Symbol:
    :code:`B`

Latex:
    :math:`B`

Dimension:
    :code:`impedance`


.. py:data:: current_voltage_parameter

    Ratio of input :attr:`~symplyphysics.symbols.electrodynamics.current` to output :attr:`~symplyphysics.symbols.electrodynamics.voltage` at idle at the output.

Symbol:
    :code:`C`

Latex:
    :math:`C`

Dimension:
    :code:`conductance`


.. py:data:: current_current_parameter

    Ratio of input :attr:`~symplyphysics.symbols.electrodynamics.current` to output :attr:`~symplyphysics.symbols.electrodynamics.current` in case of a short
    circuit at the output.

Symbol:
    :code:`D`

Latex:
    :math:`D`

Dimension:
    :code:`dimensionless`


.. py:data:: surge_impedance

    :attr:`~symplyphysics.symbols.electrodynamics.surge_impedance` of the transmission line.

Symbol:
    :code:`Z_S`

Latex:
    :math:`Z_\text{S}`

Dimension:
    :code:`impedance`


.. py:data:: length

    :attr:`~symplyphysics.symbols.classical_mechanics.length` of the transmission line.

Symbol:
    :code:`l`

Latex:
    :math:`l`

Dimension:
    :code:`length`


.. py:data:: phase_constant

    :symbols`phase_constant`.

Symbol:
    :code:`beta`

Latex:
    :math:`\beta`

Dimension:
    :code:`1/length`


.. py:data:: law

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


    Latex:
        .. math::
            \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}