Relative operating bandwidth of quarter-wave transformer
========================================================

The relative operating bandwidth of a quarter-wave transformer depends on the reflection
coefficient, the surge impendance of the transmission line to which the transformer is
connected and the load resistance.

..
    TODO: find link

.. py:currentmodule:: symplyphysics.laws.electricity.circuits.couplers.relative_operating_band_of_quarter_wave_transformer

.. py:data:: relative_bandwidth

    Relative operating bandwidth is the ratio of the bandwidth to the center frequency.

Symbol:
    :code:`b`

Latex:
    :math:`b`

Dimension:
    :code:`dimensionless`


.. py:data:: load_resistance

    :attr:`~symplyphysics.symbols.electrodynamics.electrical_resistance` of the load.

Symbol:
    :code:`R_L`

Latex:
    :math:`R_\text{L}`

Dimension:
    :code:`impedance`


.. 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:: reflection_coefficient

    :attr:`~symplyphysics.symbols.electrodynamics.reflection_coefficient`.

Symbol:
    :code:`Gamma`

Latex:
    :math:`\Gamma`

Dimension:
    :code:`dimensionless`


.. py:data:: law

    :code:`b = 2 - 4 / pi * acos(2 * Gamma * sqrt(R_L * Z_S) / (sqrt(1 - Gamma^2) * Abs(R_L - Z_S)))`


    Latex:
        .. math::
            b = 2 - \frac{4}{\pi} \operatorname{acos}{\left(\frac{2 \Gamma \sqrt{R_\text{L} Z_\text{S}}}{\sqrt{1 - \Gamma^{2}} \left|{R_\text{L} - Z_\text{S}}\right|} \right)}