.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "auto_examples/plot_08_child_langmuir.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_auto_examples_plot_08_child_langmuir.py>`
        to download the full example code.

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_auto_examples_plot_08_child_langmuir.py:


Power law: vacuum diode
=======================

In a vacuum diode, the space-charge limited anode current follows the Child–Langmuir law
I = k · U^(3/2). Fitting a general power law tests whether the exponent really is 3/2.

.. GENERATED FROM PYTHON SOURCE LINES 10-22

.. code-block:: Python

    import numpy as np

    import fitwise as fw

    rng = np.random.default_rng(10)
    voltage = np.linspace(10, 200, 25)  # V
    current = 0.018 * voltage**1.5 * (1 + rng.normal(0, 0.02, voltage.size))  # mA
    dI = 0.02 * current

    result = fw.fit(fw.models.powerlaw, voltage, current, yerr=dI)
    print(result)





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

 .. code-block:: none

    Model:      powerlaw(x; amp, exponent)
    Method:     Trust Region Reflective (scipy least_squares) (auto: nonlinear)
    Converged:  yes (14 evals)
    Data:       25 points, yerr given (absolute)
    Start:      amp, exponent: log-log fit

    Parameter     Value   ± Std.Err    Rel.   Bounds
    amp         0.01786     0.00041   2.3 %   –
    exponent     1.5002      0.0051   0.3 %   –

    χ²/ndf = 0.532 (23 dof, p = 0.97)    R² = 0.99976    AIC = -13.9
    Correlations > 0.5:  amp–exponent (-0.98)
    Warnings:   none




.. GENERATED FROM PYTHON SOURCE LINES 23-24

Is the exponent compatible with 3/2?

.. GENERATED FROM PYTHON SOURCE LINES 24-28

.. code-block:: Python

    n = result.params["exponent"]
    deviation = (n.value - 1.5) / n.stderr
    print(f"exponent = {n.value:.4f} ± {n.stderr:.4f}, deviation from 1.5: {deviation:+.1f} σ")





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

 .. code-block:: none

    exponent = 1.5002 ± 0.0051, deviation from 1.5: +0.0 σ




.. GENERATED FROM PYTHON SOURCE LINES 29-30

If theory fixes the exponent, fix it and fit only the prefactor:

.. GENERATED FROM PYTHON SOURCE LINES 30-33

.. code-block:: Python

    fixed = fw.fit(fw.models.powerlaw, voltage, current, yerr=dI, fixed={"exponent": 1.5})
    print(fixed)





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

 .. code-block:: none

    Model:      powerlaw(x; amp, exponent)
    Method:     Trust Region Reflective (scipy least_squares) (auto: nonlinear)
    Converged:  yes (11 evals)
    Data:       25 points, yerr given (absolute)
    Start:      amp: log-log fit · exponent: fixed

    Parameter      Value   ± Std.Err    Rel.   Bounds
    amp         0.017871    0.000071   0.4 %   –
    exponent         1.5           –       –   fixed

    χ²/ndf = 0.51 (24 dof, p = 0.98)    R² = 0.99976    AIC = -15.9
    Warnings:   none




.. GENERATED FROM PYTHON SOURCE LINES 34-39

.. code-block:: Python

    fig, axes = result.plot()
    axes[0].set_xscale("log")
    axes[0].set_yscale("log")
    axes[0].set_ylabel("I (mA)")
    axes[1].set_xlabel("U (V)")



.. image-sg:: /auto_examples/images/sphx_glr_plot_08_child_langmuir_001.png
   :alt: plot 08 child langmuir
   :srcset: /auto_examples/images/sphx_glr_plot_08_child_langmuir_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none


    Text(0.5, 18.503588541666673, 'U (V)')




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

   **Total running time of the script:** (0 minutes 0.182 seconds)


.. _sphx_glr_download_auto_examples_plot_08_child_langmuir.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

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

      :download:`Download Jupyter notebook: plot_08_child_langmuir.ipynb <plot_08_child_langmuir.ipynb>`

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

      :download:`Download Python source code: plot_08_child_langmuir.py <plot_08_child_langmuir.py>`

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

      :download:`Download zipped: plot_08_child_langmuir.zip <plot_08_child_langmuir.zip>`


.. only:: html

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

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