# 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, 2020 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/>.
import numpy as np
from .elements import Z_CPE, parallel
[docs]def Z_randles(omega, Rct, Rs, Aw, C0):
r"""Randles circuit.
Parameters
-----------
omega: :class:`numpy.ndarray`, double
list of frequencies
Rct: double
Resistance in series with Warburg element, e.g.
charge transfer resistance
Rs: double
Resistance of the DUT, e.g. electrolyte resistance
Aw: double
Warburg coefficient
C0: double
capacitance
Returns
-------
:class:`numpy.ndarray`, complex
Impedance array
Notes
-----
Function holding the Randles equation with capacitor in parallel to resistor in series with Warburg element
and another resistor in series.
Equations for calculations:
Impedance of resistor and Warburg element:
.. math::
Z_\mathrm{RW} = R_0 + A_\mathrm{W} \frac{1 - j}{\sqrt{\omega}}
Impedance of capacitor:
.. math::
Z_C = (j \omega C_0)^{-1}
.. math::
Z_\mathrm{fit} = R_s + \frac{Z_C Z_\mathrm{RW}}{Z_C + Z_\mathrm{RW}}
"""
Z_RW = Rct + Aw * (1. - 1j) / np.sqrt(omega)
Z_C = 1. / (1j * omega * C0)
Z_par = parallel(Z_RW, Z_C)
Z_fit = Rs + Z_par
return Z_fit
[docs]def Z_randles_CPE(omega, Rct, Rs, Aw, k, alpha):
"""Randles circuit with CPE instead of capacitor.
Parameters
-----------
omega: :class:`numpy.ndarray`, double
list of frequencies
Rct: double
Resistance in series with Warburg element, e.g.
charge transfer resistance
Rs: double
Resistance of the DUT, e.g. electrolyte resistance
Aw: double
Warburg coefficient
k: double
CPE coefficient
alpha: double
CPE exponent
Returns
-------
:class:`numpy.ndarray`, complex
Impedance array
See also
--------
:func:`Z_randles`
:func:`impedancefitter.elements.Z_CPE`
"""
Z_RW = Rct + Aw * (1. - 1j) / np.sqrt(omega)
Z_cpe = Z_CPE(omega, k, alpha)
Z_par = parallel(Z_RW, Z_cpe)
Z_fit = Rs + Z_par
return Z_fit