.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/electrical_engineering/networks/plot_01_jacobi_maximum_power.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_networks_plot_01_jacobi_maximum_power.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_networks_plot_01_jacobi_maximum_power.py:


Jacobi's maximum power theorem (c. 1840)
========================================

Around 1840 Moritz von Jacobi, building battery-powered electric motors in
St Petersburg (one drove a boat on the Neva), asked what load draws the
most power from a battery. A source of EMF :math:`V` and internal resistance
:math:`R_s` delivers :math:`P = V^2 R_L/(R_s + R_L)^2` to a load
:math:`R_L`, largest when :math:`R_L = R_s`
(:func:`~engineeringkit.electrical_engineering.maximum_power_transfer`).
At that match the source wastes as much as it delivers, so the efficiency
is only 50%. Engineers later learned to read the theorem the other way:
power grids run far from the match, at high efficiency, while radio
receivers match to capture every last picowatt.

.. GENERATED FROM PYTHON SOURCE LINES 18-23

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from engineeringkit.electrical_engineering import maximum_power_transfer, ohms_law, series_resistance








.. GENERATED FROM PYTHON SOURCE LINES 24-26

Sweeping the load
-----------------

.. GENERATED FROM PYTHON SOURCE LINES 26-48

.. code-block:: Python


    V, Rs = 12.0, 4.0
    RL = np.logspace(-1, 2, 601)
    I = np.array([ohms_law(voltage=V, resistance=series_resistance(Rs, r)) for r in RL])
    P_load = I**2 * RL
    efficiency = RL / (Rs + RL)

    R_match, P_max = maximum_power_transfer(V, Rs)
    best = RL[np.argmax(P_load)]
    print(f"numerical optimum R_L = {best:.3f} ohm, theorem {R_match:.1f} ohm; P_max = {P_max:.1f} W")
    print(f"efficiency at the match {np.interp(R_match, RL, efficiency):.0%}")
    assert np.isclose(best, R_match, rtol=0.01)
    assert np.isclose(P_load.max(), P_max, rtol=1e-4)

    fig, ax = plt.subplots()
    ax.semilogx(RL / Rs, P_load / P_max, label="load power / $P_{max}$")
    ax.semilogx(RL / Rs, efficiency, label="efficiency")
    ax.axvline(1, color="gray", ls="--")
    ax.set(xlabel="$R_L / R_s$", ylabel="fraction", title="Jacobi's maximum power theorem")
    ax.legend()

    plt.show()



.. image-sg:: /api/gallery/electrical_engineering/networks/images/sphx_glr_plot_01_jacobi_maximum_power_001.png
   :alt: Jacobi's maximum power theorem
   :srcset: /api/gallery/electrical_engineering/networks/images/sphx_glr_plot_01_jacobi_maximum_power_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    numerical optimum R_L = 3.981 ohm, theorem 4.0 ohm; P_max = 9.0 W
    efficiency at the match 50%





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

   **Total running time of the script:** (0 minutes 0.036 seconds)


.. _sphx_glr_download_api_gallery_electrical_engineering_networks_plot_01_jacobi_maximum_power.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/networks/plot_01_jacobi_maximum_power.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_01_jacobi_maximum_power.ipynb <plot_01_jacobi_maximum_power.ipynb>`

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

      :download:`Download Python source code: plot_01_jacobi_maximum_power.py <plot_01_jacobi_maximum_power.py>`

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

      :download:`Download zipped: plot_01_jacobi_maximum_power.zip <plot_01_jacobi_maximum_power.zip>`


.. only:: html

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

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