.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/electrical_engineering/ac_circuits/plot_02_steinmetz_phasors.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_ac_circuits_plot_02_steinmetz_phasors.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_ac_circuits_plot_02_steinmetz_phasors.py:


Steinmetz's phasors and complex impedance (1893)
================================================

AC circuits obey linear differential equations, and before 1893 engineers
solved them with trigonometry and graphical constructions. At the
International Electrical Congress in Chicago, Charles Proteus Steinmetz
showed that a sinusoid :math:`A\cos(\omega t + \phi)` can be carried as the
complex number :math:`Ae^{j\phi}`. Differentiation becomes multiplication
by :math:`j\omega`, so an inductor becomes the impedance :math:`j\omega L`
(:func:`~engineeringkit.electrical_engineering.inductor_impedance`), a
capacitor :math:`1/j\omega C`
(:func:`~engineeringkit.electrical_engineering.capacitor_impedance`), and AC
circuits are solved by Ohm's law with complex numbers
(:func:`~engineeringkit.electrical_engineering.series_rlc_impedance`). This
example solves a series RLC circuit by phasors and checks the answer
against a time-domain simulation.

.. GENERATED FROM PYTHON SOURCE LINES 21-34

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np
    from scipy.integrate import solve_ivp

    from engineeringkit.electrical_engineering import (
        capacitor_impedance,
        inductor_impedance,
        phasor,
        polar,
        series_rlc_impedance,
    )
    from engineeringkit.electrical_engineering.visualizers import plot_phasors








.. GENERATED FROM PYTHON SOURCE LINES 35-37

Phasor solution
---------------

.. GENERATED FROM PYTHON SOURCE LINES 37-47

.. code-block:: Python


    R, L, C, f, Vm = 30.0, 0.1, 50e-6, 50.0, 100.0
    w = 2 * np.pi * f
    Z = series_rlc_impedance(R, L, C, w)
    I = phasor(Vm, 0.0) / Z
    magnitude, phase = polar(I)
    print(f"Z = {Z:.2f} ohm, I = {magnitude:.3f} A at {np.degrees(phase):.1f} degrees")
    VR, VL, VC = I * R, I * inductor_impedance(L, w), I * capacitor_impedance(C, w)
    assert np.isclose(VR + VL + VC, Vm)  # Kirchhoff's voltage law holds for phasors





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

 .. code-block:: none

    Z = 30.00-32.25j ohm, I = 2.270 A at 47.1 degrees




.. GENERATED FROM PYTHON SOURCE LINES 48-53

Time-domain check
-----------------

Integrate :math:`L\,di/dt + Ri + q/C = V_m\cos\omega t` for 40 cycles and
compare the steady state with :math:`|I|\cos(\omega t + \arg I)`.

.. GENERATED FROM PYTHON SOURCE LINES 53-78

.. code-block:: Python



    def rhs(t, x):
        q, i = x
        return [i, (Vm * np.cos(w * t) - R * i - q / C) / L]


    t_end = 40 / f
    sol = solve_ivp(rhs, (0, t_end), [0, 0], rtol=1e-9, atol=1e-12, dense_output=True)
    t = np.linspace(t_end - 2 / f, t_end, 400)
    i_sim = sol.sol(t)[1]
    i_phasor = magnitude * np.cos(w * t + phase)
    print(f"largest difference after 40 cycles: {np.max(np.abs(i_sim - i_phasor)):.1e} A")
    assert np.allclose(i_sim, i_phasor, atol=1e-4 * magnitude)

    fig, axes = plt.subplots(1, 2, figsize=(10, 4))
    plot_phasors([Vm, VR, VL, VC, I * 50], labels=["$V$", "$V_R$", "$V_L$", "$V_C$", "$I$ (x50)"], ax=axes[0])
    axes[0].set_title("Phasor diagram")
    axes[1].plot(t * 1e3, i_sim, label="ODE simulation")
    axes[1].plot(t * 1e3, i_phasor, "--", label="phasor solution")
    axes[1].set(xlabel="time [ms]", ylabel="current [A]", title="Steady state")
    axes[1].legend()
    fig.tight_layout()

    plt.show()



.. image-sg:: /api/gallery/electrical_engineering/ac_circuits/images/sphx_glr_plot_02_steinmetz_phasors_001.png
   :alt: Phasor diagram, Steady state
   :srcset: /api/gallery/electrical_engineering/ac_circuits/images/sphx_glr_plot_02_steinmetz_phasors_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    largest difference after 40 cycles: 1.4e-09 A





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

   **Total running time of the script:** (0 minutes 0.155 seconds)


.. _sphx_glr_download_api_gallery_electrical_engineering_ac_circuits_plot_02_steinmetz_phasors.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/ac_circuits/plot_02_steinmetz_phasors.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_02_steinmetz_phasors.ipynb <plot_02_steinmetz_phasors.ipynb>`

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

      :download:`Download Python source code: plot_02_steinmetz_phasors.py <plot_02_steinmetz_phasors.py>`

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

      :download:`Download zipped: plot_02_steinmetz_phasors.zip <plot_02_steinmetz_phasors.zip>`


.. only:: html

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

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