.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/control_systems/feedback/plot_03_black_negative_feedback_amplifier.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_feedback_plot_03_black_negative_feedback_amplifier.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_feedback_plot_03_black_negative_feedback_amplifier.py:


Black's negative-feedback amplifier
===================================

Harold Black (1927), commuting on the Lackawanna ferry, sketched an
amplifier that fed a fraction :math:`\beta` of its output back *out of
phase*. With a huge open-loop gain :math:`A`, the closed-loop gain
:math:`A/(1 + A\beta)` (:func:`~engineeringkit.control_systems.feedback`)
is almost exactly :math:`1/\beta`, set by a passive divider rather than by
drifting vacuum tubes. Distortion and gain variations shrink by the factor
:math:`1 + A\beta`, and the bandwidth grows by the same factor.

.. GENERATED FROM PYTHON SOURCE LINES 15-23

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from engineeringkit.control_systems import bode, dc_gain, feedback, poles, transfer_function

    beta = 0.01
    wa = 2 * np.pi * 10.0  # open-loop pole [rad/s]








.. GENERATED FROM PYTHON SOURCE LINES 24-26

Insensitivity to the tube gain
------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 26-35

.. code-block:: Python


    for A0 in [5e4, 1e5, 1.5e5]:
        amp = transfer_function([A0], [1 / wa, 1])
        closed = feedback(amp, beta)
        gain = dc_gain(closed)
        print(f"A0 = {A0:8.0f}: closed-loop gain {gain:.4f}, bandwidth {-poles(closed)[0].real / (2 * np.pi):8.0f} Hz")
        assert np.isclose(-poles(closed)[0].real, wa * (1 + A0 * beta))
        assert np.isclose(gain, A0 / (1 + A0 * beta)) and abs(gain * beta - 1) < 2.5e-3





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

 .. code-block:: none

    A0 =    50000: closed-loop gain 99.8004, bandwidth     5010 Hz
    A0 =   100000: closed-loop gain 99.9001, bandwidth    10010 Hz
    A0 =   150000: closed-loop gain 99.9334, bandwidth    15010 Hz




.. GENERATED FROM PYTHON SOURCE LINES 36-38

Gain traded for bandwidth
-------------------------

.. GENERATED FROM PYTHON SOURCE LINES 38-50

.. code-block:: Python


    amp = transfer_function([1e5], [1 / wa, 1])
    w = np.logspace(0, 8, 400)
    fig, ax = plt.subplots()
    for model, label in [(amp, "open loop $A$"), (feedback(amp, beta), r"closed loop $A/(1+A\beta)$")]:
        _, mag, _ = bode(model, w)
        ax.semilogx(w / (2 * np.pi), mag, label=label)
    ax.set(xlabel="frequency [Hz]", ylabel="gain [dB]", title="Gain-bandwidth exchange")
    ax.grid(True, which="both", alpha=0.3)
    ax.legend()

    plt.show()



.. image-sg:: /api/gallery/control_systems/feedback/images/sphx_glr_plot_03_black_negative_feedback_amplifier_001.png
   :alt: Gain-bandwidth exchange
   :srcset: /api/gallery/control_systems/feedback/images/sphx_glr_plot_03_black_negative_feedback_amplifier_001.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.212 seconds)


.. _sphx_glr_download_api_gallery_control_systems_feedback_plot_03_black_negative_feedback_amplifier.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/feedback/plot_03_black_negative_feedback_amplifier.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_03_black_negative_feedback_amplifier.ipynb <plot_03_black_negative_feedback_amplifier.ipynb>`

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

      :download:`Download Python source code: plot_03_black_negative_feedback_amplifier.py <plot_03_black_negative_feedback_amplifier.py>`

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

      :download:`Download zipped: plot_03_black_negative_feedback_amplifier.zip <plot_03_black_negative_feedback_amplifier.zip>`


.. only:: html

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

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