.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/electrical_engineering/transmission_lines/plot_02_smith_chart.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_api_gallery_electrical_engineering_transmission_lines_plot_02_smith_chart.py>`
        to download the full example code or to run this example in your browser via JupyterLite.

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_api_gallery_electrical_engineering_transmission_lines_plot_02_smith_chart.py:


Smith's chart (1939)
====================

Phillip Hagar Smith, an engineer at Bell Labs, tired of evaluating the
transmission-line equations with a slide rule. His 1939 chart plots the
reflection coefficient :math:`\Gamma = (Z_L - Z_0)/(Z_L + Z_0)`
(:func:`~engineeringkit.electrical_engineering.reflection_coefficient`) in
the unit disc, overlaid with circles of constant normalised resistance and
arcs of constant reactance. Moving along a lossless line just rotates
:math:`\Gamma` around the centre, at constant standing-wave ratio
(:func:`~engineeringkit.electrical_engineering.standing_wave_ratio`), so the
input impedance (:func:`~engineeringkit.electrical_engineering.input_impedance`)
can be read off the grid. This example uses the chart to design a
single-stub match, drawn with
:func:`~engineeringkit.electrical_engineering.visualizers.plot_smith_chart`.

.. GENERATED FROM PYTHON SOURCE LINES 20-27

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np
    from scipy.optimize import brentq

    from engineeringkit.electrical_engineering import input_impedance, reflection_coefficient, standing_wave_ratio
    from engineeringkit.electrical_engineering.visualizers import plot_smith_chart








.. GENERATED FROM PYTHON SOURCE LINES 28-30

Moving along the line
---------------------

.. GENERATED FROM PYTHON SOURCE LINES 30-39

.. code-block:: Python


    z0, zl = 50.0, 25.0 - 30.0j
    gamma_l = reflection_coefficient(zl, z0)
    print(f"load: Gamma = {abs(gamma_l):.3f} at {np.degrees(np.angle(gamma_l)):.1f} deg, VSWR = {standing_wave_ratio(gamma_l):.2f}")
    bl = np.linspace(0, np.pi, 361)  # half a wavelength
    locus = reflection_coefficient(input_impedance(zl, z0, bl), z0)
    assert np.allclose(np.abs(locus), abs(gamma_l))  # a circle about the centre
    assert np.isclose(input_impedance(zl, z0, np.pi), zl)  # half a wavelength repeats the load





.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    load: Gamma = 0.483 at -108.0 deg, VSWR = 2.87




.. GENERATED FROM PYTHON SOURCE LINES 40-46

A shorted shunt stub
--------------------

Move toward the generator until the input conductance is :math:`1/Z_0`
(the unit conductance circle), then cancel the remaining susceptance with
a short-circuited stub in parallel.

.. GENERATED FROM PYTHON SOURCE LINES 46-71

.. code-block:: Python



    def conductance_error(x):
        return (1 / input_impedance(zl, z0, x)).real - 1 / z0


    error = conductance_error(bl)
    k = np.flatnonzero(np.sign(error[:-1]) != np.sign(error[1:]))[0]  # first crossing of the circle
    d = brentq(conductance_error, bl[k], bl[k + 1])
    b = (1 / input_impedance(zl, z0, d)).imag
    stub = brentq(lambda x: (1 / input_impedance(0, z0, x)).imag + b, 0.01, np.pi / 2 - 1e-9)
    y_total = 1 / input_impedance(zl, z0, d) + 1 / input_impedance(0, z0, stub)
    print(f"stub at {d / (2 * np.pi):.4f} wavelengths from the load, stub length {stub / (2 * np.pi):.4f} wavelengths")
    print(f"matched input impedance {1 / y_total:.4f} ohm")
    assert abs(reflection_coefficient(1 / y_total, z0)) < 1e-9

    ax = plot_smith_chart(locus, label="along the line")
    ax.plot(gamma_l.real, gamma_l.imag, "ks", label="load")
    g_d = reflection_coefficient(input_impedance(zl, z0, d), z0)
    ax.plot(g_d.real, g_d.imag, "C3o", label="stub position")
    ax.plot(0, 0, "C2*", ms=12, label="matched")
    ax.legend(loc="upper left", fontsize="small")
    ax.set_title("Single-stub match on the Smith chart")

    plt.show()



.. image-sg:: /api/gallery/electrical_engineering/transmission_lines/images/sphx_glr_plot_02_smith_chart_001.png
   :alt: Single-stub match on the Smith chart
   :srcset: /api/gallery/electrical_engineering/transmission_lines/images/sphx_glr_plot_02_smith_chart_001.png
   :class: sphx-glr-single-img


.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    stub at 0.0151 wavelengths from the load, stub length 0.1171 wavelengths
    matched input impedance 50.0000-0.0000j ohm





.. rst-class:: sphx-glr-timing

   **Total running time of the script:** (0 minutes 0.019 seconds)


.. _sphx_glr_download_api_gallery_electrical_engineering_transmission_lines_plot_02_smith_chart.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

    .. container:: lite-badge

      .. image:: images/jupyterlite_badge_logo.svg
        :target: ../../../../lite/lab/index.html?path=api/gallery/electrical_engineering/transmission_lines/plot_02_smith_chart.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: plot_02_smith_chart.ipynb <plot_02_smith_chart.ipynb>`

    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download Python source code: plot_02_smith_chart.py <plot_02_smith_chart.py>`

    .. container:: sphx-glr-download sphx-glr-download-zip

      :download:`Download zipped: plot_02_smith_chart.zip <plot_02_smith_chart.zip>`


.. only:: html

 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_