.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/control_systems/root_locus/plot_01_evans_root_locus.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_control_systems_root_locus_plot_01_evans_root_locus.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_control_systems_root_locus_plot_01_evans_root_locus.py:


Evans's root locus
==================

Walter Evans at North American Aviation (1948) showed how to sketch the
paths of the closed-loop poles as a single gain :math:`K` rises from zero,
without solving a polynomial for each gain
(:func:`~engineeringkit.control_systems.root_locus`). For
:math:`K/(s(s+2)(s+4))` the branches leave the poles at 0, -2 and -4. Two of
them meet and break away from the real axis where
:math:`\tfrac{d}{ds}[s(s+2)(s+4)] = 0`, at :math:`s = -2 + 2/\sqrt3`. They
cross the imaginary axis at :math:`\pm j\sqrt8` when :math:`K = 48`.

.. GENERATED FROM PYTHON SOURCE LINES 16-24

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from engineeringkit.control_systems import root_locus
    from engineeringkit.control_systems.visualizers import plot_root_locus

    num, den = [1.0], [1.0, 6.0, 8.0, 0.0]








.. GENERATED FROM PYTHON SOURCE LINES 25-27

Break-away point and imaginary-axis crossing
--------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 27-45

.. code-block:: Python


    s_b = -2 + 2 / np.sqrt(3)
    K_b = -np.polyval(den, s_b)
    pair = np.sort_complex(root_locus(num, den, [K_b])[0])[1:]
    print(f"break-away at s = {s_b:.4f} for K = {K_b:.4f}: roots {np.round(pair, 3)}")
    assert np.allclose(pair, s_b, atol=1e-4)

    crossing = root_locus(num, den, [48.0])[0]
    print("K = 48 roots:", np.round(crossing, 6))
    assert np.isclose(np.abs(crossing.imag).max(), np.sqrt(8)) and abs(crossing.real[np.argmax(crossing.imag)]) < 1e-9

    ax = plot_root_locus(num, den, np.concatenate((np.linspace(0, 5, 400), np.linspace(5, 200, 800))))
    ax.plot([0, 0], [-np.sqrt(8), np.sqrt(8)], "C3o", label="$K = 48$")
    ax.plot(s_b, 0, "C2s", label="break-away")
    ax.set(xlim=(-8, 3), ylim=(-6, 6), title=r"Root locus of $K/(s(s+2)(s+4))$")
    ax.legend()

    plt.show()



.. image-sg:: /api/gallery/control_systems/root_locus/images/sphx_glr_plot_01_evans_root_locus_001.png
   :alt: Root locus of $K/(s(s+2)(s+4))$
   :srcset: /api/gallery/control_systems/root_locus/images/sphx_glr_plot_01_evans_root_locus_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    break-away at s = -0.8453 for K = 3.0792: roots [-0.845+0.j -0.845+0.j]
    K = 48 roots: [-6.+0.j        0.+2.828427j  0.-2.828427j]





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

   **Total running time of the script:** (0 minutes 0.153 seconds)


.. _sphx_glr_download_api_gallery_control_systems_root_locus_plot_01_evans_root_locus.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/control_systems/root_locus/plot_01_evans_root_locus.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_01_evans_root_locus.ipynb <plot_01_evans_root_locus.ipynb>`

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

      :download:`Download Python source code: plot_01_evans_root_locus.py <plot_01_evans_root_locus.py>`

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

      :download:`Download zipped: plot_01_evans_root_locus.zip <plot_01_evans_root_locus.zip>`


.. only:: html

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

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