Scattering matrix to transmission matrix
========================================

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

**Notes:**

#. See :ref:`Transmission Matrix <transmission_matrix_def>`.
#. See :ref:`Scattering Matrix <scattering_matrix_def>`.
#. See `this site <https://www.microwaves101.com/encyclopedias/s-parameters>`__ for more
   information about :math:`S`-parameters.
#. See this `Wikipedia page <https://en.wikipedia.org/wiki/Scattering_parameters#Two-port_S-parameters>`__
   for more information about :math:`S`-parameters.

..
    TODO: find link
    TODO: make laws for the S-parameters, see second link above.

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

.. 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`.

Symbol:
    :code:`Z_S`

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

Dimension:
    :code:`impedance`


.. py:data:: input_voltage_reflection_coefficient

    Input port :attr:`~symplyphysics.symbols.electrodynamics.voltage` :attr:`~symplyphysics.symbols.electrodynamics.reflection_coefficient`.

Symbol:
    :code:`S_ii`

Latex:
    :math:`S_\text{ii}`

Dimension:
    :code:`dimensionless`


.. py:data:: reverse_voltage_gain

    Reverse :attr:`~symplyphysics.symbols.electrodynamics.voltage` :attr:`~symplyphysics.symbols.electrodynamics.circuit_gain`.

Symbol:
    :code:`S_io`

Latex:
    :math:`S_\text{io}`

Dimension:
    :code:`dimensionless`


.. py:data:: forward_voltage_gain

    Forward :attr:`~symplyphysics.symbols.electrodynamics.voltage` :attr:`~symplyphysics.symbols.electrodynamics.circuit_gain`.

Symbol:
    :code:`S_oi`

Latex:
    :math:`S_\text{oi}`

Dimension:
    :code:`dimensionless`


.. py:data:: output_voltage_reflection_coefficient

    Output port :attr:`~symplyphysics.symbols.electrodynamics.voltage` :attr:`~symplyphysics.symbols.electrodynamics.reflection_coefficient`.

Symbol:
    :code:`S_oo`

Latex:
    :math:`S_\text{oo}`

Dimension:
    :code:`dimensionless`


.. py:data:: law

    :code:`[[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:
        .. math::
            \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}