# The ImpedanceFitter is a package to fit impedance spectra to
# equivalent-circuit models using open-source software.
#
# Copyright (C) 2018, 2019 Leonard Thiele, leonard.thiele[AT]uni-rostock.de
# Copyright (C) 2018, 2019 Julius Zimmermann,
# julius.zimmermann[AT]uni-rostock.de
#
# This program is free software: you can redistribute it and/or modify
# it under the terms of the GNU General Public License as published by
# the Free Software Foundation, either version 3 of the License, or
# (at your option) any later version.
#
# This program is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
# GNU General Public License for more details.
#
# You should have received a copy of the GNU General Public License
# along with this program. If not, see <https://www.gnu.org/licenses/>.
from numpy import power
from scipy.constants import epsilon_0 as e0
[docs]
def RC_model(omega, Rd, Cd):
"""Simple RC model, resistor in parallel with capacitor.
Parameters
----------
omega: :class:`numpy.ndarray`, double
list of frequencies
Rd: double
Resistance.
Cd: double
Capacitance
Notes
-----
.. warning::
`Cd` is in pF!
Returns
-------
:class:`numpy.ndarray`, complex
Impedance array
"""
Cfit = Cd * 1e-12
Z_fit = Rd / (1.0 + 1j * omega * Cfit * Rd)
return Z_fit
[docs]
def rc_model(omega, c0, kdc, eps):
"""Simple RC model of a lossy dielectric to obtain dielectric properties.
Parameters
----------
omega: :class:`numpy.ndarray`, double
list of frequencies
c0: double
unit capacitance in pF
eps: double
relative permittivity
kdc: double
conductivity
Returns
-------
:class:`numpy.ndarray`, complex
Impedance array
Notes
-----
.. warning::
C0 is in pF!
"""
c0 *= 1e-12
Rd = e0 / (kdc * c0)
# Cd = eps * c0
factor = eps * e0 / kdc # Cd * Rd
Z_fit = Rd / (1.0 + 1j * omega * factor)
return Z_fit
[docs]
def drc_model(omega, RE, tauE, alpha, beta):
"""Distributed RC circuit.
Parameters
----------
omega: :class:`numpy.ndarray`, double
list of frequencies
RE: double
resistance
tauE: double
relaxation time, in ns
alpha: double
Cole-Cole exponent
beta: double
DRC exponent, beta = 1 equals Cole-Cole model
Returns
-------
:class:`numpy.ndarray`, complex
Impedance array
Notes
-----
Described for example in [Emmert2011]_.
.. warning::
The time constant tauE is in ns!
References
----------
.. [Emmert2011] Emmert, S., Wolf, M., Gulich, R., Krohns, S., Kastner, S.,
Lunkenheimer, P., & Loidl, A. (2011).
Electrode polarization effects in broadband dielectric spectroscopy.
European Physical Journal B, 83(2), 157–165.
https://doi.org/10.1140/epjb/e2011-20439-8
"""
tauE *= 1e-9
nom = power(1.0 + power(1j * omega * tauE, 1.0 - alpha), beta)
return RE / nom
[docs]
def rc_tau_model(omega, Rk, tauk):
r"""Compute RC model without explicit capacitance.
Parameters
----------
omega: :class:`numpy.ndarray`, double
list of frequencies
Rk: double
resistance
tauk: double
relaxation time
Returns
-------
:class:`numpy.ndarray`, complex
Impedance array
Notes
-----
Used for example in the Lin-KK test [Schoenleber2014]_.
The impedance reads
.. math::
Z = \frac{R_\mathrm{k}}{1+ j \omega \tau_\mathrm{k}}
"""
return Rk / (1.0 + 1j * omega * tauk)