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

.. only:: html

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

        Click :ref:`here <sphx_glr_download_examples_example_complex_resonator_model.py>`
        to download the full example code

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

.. _sphx_glr_examples_example_complex_resonator_model.py:


Complex Resonator Model
=======================

This notebook shows how to fit the parameters of a complex resonator,
using `lmfit.Model` and defining a custom `Model` class.

Following Khalil et al. (https://arxiv.org/abs/1108.3117), we can model the
forward transmission of a microwave resonator with total quality factor
:math:`Q`, coupling quality factor :math:`Q_e`, and resonant frequency
:math:`f_0` using:

.. math::

    S_{21}(f) = 1 - \frac{Q Q_e^{-1}}{1+2jQ(f-f_0)/f_0}

:math:`S_{21}` is thus a complex function of a real frequency.

By allowing :math:`Q_e` to be complex, this model can take into account
mismatches in the input and output transmission impedances.

.. GENERATED FROM PYTHON SOURCE LINES 23-28

.. code-block:: default

    import matplotlib.pyplot as plt
    import numpy as np

    import lmfit








.. GENERATED FROM PYTHON SOURCE LINES 29-31

Since ``scipy.optimize`` and ``lmfit`` require real parameters, we represent
:math:`Q_e` as ``Q_e_real + 1j*Q_e_imag``.

.. GENERATED FROM PYTHON SOURCE LINES 31-38

.. code-block:: default



    def linear_resonator(f, f_0, Q, Q_e_real, Q_e_imag):
        Q_e = Q_e_real + 1j*Q_e_imag
        return 1 - (Q * Q_e**-1 / (1 + 2j * Q * (f - f_0) / f_0))









.. GENERATED FROM PYTHON SOURCE LINES 39-40

The standard practice of defining an ``lmfit`` model is as follows:

.. GENERATED FROM PYTHON SOURCE LINES 40-73

.. code-block:: default


    class ResonatorModel(lmfit.model.Model):
        __doc__ = "resonator model" + lmfit.models.COMMON_DOC

        def __init__(self, *args, **kwargs):
            # pass in the defining equation so the user doesn't have to later.
            super().__init__(linear_resonator, *args, **kwargs)

            self.set_param_hint('Q', min=0)  # Enforce Q is positive

        def guess(self, data, f=None, **kwargs):
            verbose = kwargs.pop('verbose', None)
            if f is None:
                return
            argmin_s21 = np.abs(data).argmin()
            fmin = f.min()
            fmax = f.max()
            f_0_guess = f[argmin_s21]  # guess that the resonance is the lowest point
            Q_min = 0.1 * (f_0_guess/(fmax-fmin))  # assume the user isn't trying to fit just a small part of a resonance curve.
            delta_f = np.diff(f)  # assume f is sorted
            min_delta_f = delta_f[delta_f > 0].min()
            Q_max = f_0_guess/min_delta_f  # assume data actually samples the resonance reasonably
            Q_guess = np.sqrt(Q_min*Q_max)  # geometric mean, why not?
            Q_e_real_guess = Q_guess/(1-np.abs(data[argmin_s21]))
            if verbose:
                print("fmin=", fmin, "fmax=", fmax, "f_0_guess=", f_0_guess)
                print("Qmin=", Q_min, "Q_max=", Q_max, "Q_guess=", Q_guess, "Q_e_real_guess=", Q_e_real_guess)
            params = self.make_params(Q=Q_guess, Q_e_real=Q_e_real_guess, Q_e_imag=0, f_0=f_0_guess)
            params['%sQ' % self.prefix].set(min=Q_min, max=Q_max)
            params['%sf_0' % self.prefix].set(min=fmin, max=fmax)
            return lmfit.models.update_param_vals(params, self.prefix, **kwargs)









.. GENERATED FROM PYTHON SOURCE LINES 74-75

Now let's use the model to generate some fake data:

.. GENERATED FROM PYTHON SOURCE LINES 75-91

.. code-block:: default


    resonator = ResonatorModel()
    true_params = resonator.make_params(f_0=100, Q=10000, Q_e_real=9000, Q_e_imag=-9000)

    f = np.linspace(99.95, 100.05, 100)
    true_s21 = resonator.eval(params=true_params, f=f)
    noise_scale = 0.02
    np.random.seed(123)
    measured_s21 = true_s21 + noise_scale*(np.random.randn(100) + 1j*np.random.randn(100))

    plt.figure()
    plt.plot(f, 20*np.log10(np.abs(measured_s21)))
    plt.ylabel('|S21| (dB)')
    plt.xlabel('MHz')
    plt.title('simulated measurement')




.. image-sg:: /examples/images/sphx_glr_example_complex_resonator_model_001.png
   :alt: simulated measurement
   :srcset: /examples/images/sphx_glr_example_complex_resonator_model_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 92-93

Try out the guess method we added:

.. GENERATED FROM PYTHON SOURCE LINES 93-96

.. code-block:: default


    guess = resonator.guess(measured_s21, f=f, verbose=True)





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

 Out:

 .. code-block:: none

    fmin= 99.95 fmax= 100.05 f_0_guess= 100.00353535353536
    Qmin= 100.00353535354105 Q_max= 99003.50000055433 Q_guess= 3146.537781821432 Q_e_real_guess= 5082.2474265369565




.. GENERATED FROM PYTHON SOURCE LINES 97-98

And now fit the data using the guess as a starting point:

.. GENERATED FROM PYTHON SOURCE LINES 98-104

.. code-block:: default


    result = resonator.fit(measured_s21, params=guess, f=f, verbose=True)

    print(result.fit_report() + '\n')
    result.params.pretty_print()





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

 Out:

 .. code-block:: none

    [[Model]]
        Model(linear_resonator)
    [[Fit Statistics]]
        # fitting method   = leastsq
        # function evals   = 36
        # data points      = 200
        # variables        = 4
        chi-square         = 0.08533642
        reduced chi-square = 4.3539e-04
        Akaike info crit   = -1543.89425
        Bayesian info crit = -1530.70099
    [[Variables]]
        f_0:       100.000096 +/- 7.0378e-05 (0.00%) (init = 100.0035)
        Q:         10059.4926 +/- 142.294763 (1.41%) (init = 3146.538)
        Q_e_real:  9180.61941 +/- 133.776876 (1.46%) (init = 5082.247)
        Q_e_imag: -9137.03297 +/- 133.770238 (1.46%) (init = 0)
    [[Correlations]] (unreported correlations are < 0.100)
        C(Q, Q_e_real)   =  0.518
        C(f_0, Q_e_imag) =  0.517
        C(f_0, Q_e_real) =  0.515
        C(Q, Q_e_imag)   = -0.515

    Name         Value      Min      Max   Stderr     Vary     Expr Brute_Step
    Q         1.006e+04      100  9.9e+04    142.3     True     None     None
    Q_e_imag     -9137     -inf      inf    133.8     True     None     None
    Q_e_real      9181     -inf      inf    133.8     True     None     None
    f_0            100    99.95      100 7.038e-05     True     None     None




.. GENERATED FROM PYTHON SOURCE LINES 105-107

Now we'll make some plots of the data and fit. Define a convenience function
for plotting complex quantities:

.. GENERATED FROM PYTHON SOURCE LINES 107-131

.. code-block:: default



    def plot_ri(data, *args, **kwargs):
        plt.plot(data.real, data.imag, *args, **kwargs)


    fit_s21 = resonator.eval(params=result.params, f=f)
    guess_s21 = resonator.eval(params=guess, f=f)

    plt.figure()
    plot_ri(measured_s21, '.')
    plot_ri(fit_s21, 'r.-', label='best fit')
    plot_ri(guess_s21, 'k--', label='inital fit')
    plt.legend(loc='best')
    plt.xlabel('Re(S21)')
    plt.ylabel('Im(S21)')

    plt.figure()
    plt.plot(f, 20*np.log10(np.abs(measured_s21)), '.')
    plt.plot(f, 20*np.log10(np.abs(fit_s21)), 'r.-', label='best fit')
    plt.plot(f, 20*np.log10(np.abs(guess_s21)), 'k--', label='initial fit')
    plt.legend(loc='best')
    plt.ylabel('|S21| (dB)')
    plt.xlabel('MHz')



.. rst-class:: sphx-glr-horizontal


    *

      .. image-sg:: /examples/images/sphx_glr_example_complex_resonator_model_002.png
         :alt: example complex resonator model
         :srcset: /examples/images/sphx_glr_example_complex_resonator_model_002.png
         :class: sphx-glr-multi-img

    *

      .. image-sg:: /examples/images/sphx_glr_example_complex_resonator_model_003.png
         :alt: example complex resonator model
         :srcset: /examples/images/sphx_glr_example_complex_resonator_model_003.png
         :class: sphx-glr-multi-img






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

   **Total running time of the script:** ( 0 minutes  0.451 seconds)


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