.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/electrical_engineering/transients/plot_01_kelvin_rc_cable.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_transients_plot_01_kelvin_rc_cable.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_transients_plot_01_kelvin_rc_cable.py:


Kelvin's telegraph cable and the RC time constant (1855)
========================================================

Before the first transatlantic cable was laid, William Thomson (later Lord
Kelvin) analysed how a signal spreads along a submarine cable. In 1855 he
showed that, with negligible inductance, the line obeys a diffusion
equation: a pulse arrives smeared over a time proportional to
:math:`rc\,\ell^2`, the resistance and capacitance per unit length times the
length squared, the *law of squares*. The building block is the single RC
section, whose voltage rises as :math:`1 - e^{-t/RC}`
(:func:`~engineeringkit.electrical_engineering.rc_charging_voltage`). This
example checks the time constant of one section, then chains sections into
a cable and confirms the law of squares.

.. GENERATED FROM PYTHON SOURCE LINES 18-23

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from engineeringkit.electrical_engineering import rc_charging_voltage, rc_time_constant








.. GENERATED FROM PYTHON SOURCE LINES 24-26

One RC section
--------------

.. GENERATED FROM PYTHON SOURCE LINES 26-34

.. code-block:: Python


    R, C = 1e3, 1e-6
    tau = rc_time_constant(R, C)
    t = np.linspace(0, 5 * tau, 501)
    v = rc_charging_voltage(1.0, R, C, t)
    print(f"tau = {tau * 1e3:.1f} ms; v(tau) = {rc_charging_voltage(1.0, R, C, tau):.4f} (1 - 1/e = {1 - np.exp(-1):.4f})")
    assert np.isclose(rc_charging_voltage(1.0, R, C, tau), 1 - np.exp(-1))





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

 .. code-block:: none

    tau = 1.0 ms; v(tau) = 0.6321 (1 - 1/e = 0.6321)




.. GENERATED FROM PYTHON SOURCE LINES 35-41

A cable of N sections
---------------------

A cable of length :math:`\ell` is a ladder of RC sections; a unit step at
the sending end reaches half amplitude at the far end (open circuit) after
a delay that grows as :math:`\ell^2` (here, the number of sections squared).

.. GENERATED FROM PYTHON SOURCE LINES 41-75

.. code-block:: Python



    def far_end_half_time(n):
        """Far-end step response of n unit RC sections, by eigen-decomposition."""
        A = -2 * np.eye(n) + np.eye(n, k=1) + np.eye(n, k=-1)
        A[-1, -1] = -1.0  # the open far end has only one neighbour
        b = np.zeros(n)
        b[0] = 1.0
        lam, Q = np.linalg.eigh(A)  # A is symmetric
        x_final = -np.linalg.solve(A, b)
        times = np.linspace(0, 3 * n * n, 20001)
        far = x_final[-1] - (Q[-1] * (Q.T @ x_final)) @ np.exp(np.outer(lam, times))
        return times[np.argmax(far >= 0.5)], times, far


    fig, axes = plt.subplots(1, 2, figsize=(10, 3.6))
    axes[0].plot(t / tau, v)
    axes[0].axhline(1 - np.exp(-1), color="gray", ls=":")
    axes[0].set(xlabel="time / RC", ylabel="capacitor voltage [V]", title="One RC section")
    sections = [10, 20, 40, 80]
    delays = []
    for n in sections:
        t_half, times, far = far_end_half_time(n)
        delays.append(t_half)
        axes[1].plot(times, far, label=f"{n} sections")
    axes[1].set(xscale="log", xlabel="time / (r c)", ylabel="far-end voltage [V]", title="Signal smearing on a cable")
    axes[1].legend()
    fig.tight_layout()

    exponent = np.polyfit(np.log(sections), np.log(delays), 1)[0]
    print(f"delay grows as length^{exponent:.2f}")
    assert abs(exponent - 2) < 0.05

    plt.show()



.. image-sg:: /api/gallery/electrical_engineering/transients/images/sphx_glr_plot_01_kelvin_rc_cable_001.png
   :alt: One RC section, Signal smearing on a cable
   :srcset: /api/gallery/electrical_engineering/transients/images/sphx_glr_plot_01_kelvin_rc_cable_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    delay grows as length^1.96





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

   **Total running time of the script:** (0 minutes 0.106 seconds)


.. _sphx_glr_download_api_gallery_electrical_engineering_transients_plot_01_kelvin_rc_cable.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/transients/plot_01_kelvin_rc_cable.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_01_kelvin_rc_cable.ipynb <plot_01_kelvin_rc_cable.ipynb>`

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

      :download:`Download Python source code: plot_01_kelvin_rc_cable.py <plot_01_kelvin_rc_cable.py>`

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

      :download:`Download zipped: plot_01_kelvin_rc_cable.zip <plot_01_kelvin_rc_cable.zip>`


.. only:: html

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

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