# 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 scipy.constants import epsilon_0 as e0
[docs]def single_shell_model(omega, em, km, kcp, ecp, kmed, emed, p, c0, dm, Rc):
r"""Single Shell model
Parameters
-----------
omega: :class:`numpy.ndarray`, double
list of frequencies
c0: double
value for :math:`c_0`, unit capacitance in pF
em: double
membrane permittivity, value for :math:`\varepsilon_\mathrm{m}`
km: double
membrane conductivity, value for :math:`\sigma_\mathrm{m}` in :math:`\mu`\ S/m
ecp: double
cytoplasm permittivity, value for :math:`\varepsilon_\mathrm{cp}`
kcp: double
cytoplasm conductivity, value for :math:`\sigma_\mathrm{cp}`
emed: double
medium permittivity, value for :math:`\varepsilon_\mathrm{med}`
kmed: double
medium conductivity, value for :math:`\sigma_\mathrm{med}`
p: double
volume fraction
dm: double
membrane thickness, value for :math:`d_\mathrm{m}`
Rc: double
cell radius, value for :math:`R_\mathrm{c}`
Returns
-------
:class:`numpy.ndarray`, complex
Impedance array
Notes
-----
.. warning::
The unit capacitance is in pF!
Equations for the single-shell-model [Feldman2003]_:
.. math::
\nu_1 = \left(1-\frac{d_\mathrm{m}}{R_\mathrm{c}}\right)^3
.. math::
\varepsilon_\mathrm{m} = \varepsilon_\mathrm{m} - j \frac{\sigma_\mathrm{m}}{\varepsilon_0 \omega}
.. math::
\varepsilon_\mathrm{cp} = \varepsilon_\mathrm{cp} - j \frac{\sigma_\mathrm{cp}}{\varepsilon_0 \omega}
.. math::
\varepsilon_\mathrm{cell}^\ast = \varepsilon_\mathrm{m}^\ast \frac{2 (1 - \nu_1) + (1 + 2 \nu_1) E_1}{(2 + \nu_1) + (1 - \nu_1) E_1}
.. math::
E_1 = \frac{\varepsilon_\mathrm{cp}^\ast}{\varepsilon_\mathrm{m}^\ast}
.. math::
\varepsilon_\mathrm{sus}^\ast = \varepsilon_\mathrm{med}^\ast
\frac{(2 \varepsilon_\mathrm{med}^\ast + \varepsilon_\mathrm{cell}^\ast) - 2 p
(\varepsilon_\mathrm{med}^\ast - \varepsilon_\mathrm{cell}^\ast)}
{(2 \varepsilon_\mathrm{med}^\ast + \varepsilon_\mathrm{cell}^\ast) + p
(\varepsilon_\mathrm{med}^\ast - \varepsilon_\mathrm{cell}^\ast)}
.. math::
Z = \frac{1}{j \varepsilon_\mathrm{sus}^\ast \omega c_0}
References
----------
.. [Feldman2003] Feldman, Y., Ermolina, I., & Hayashi, Y. (2003).
Time domain dielectric spectroscopy study of biological systems.
IEEE Transactions on Dielectrics and Electrical Insulation, 10, 728–753.
https://doi.org/10.1109/TDEI.2003.1237324
See Also
--------
:meth:`impedancefitter.double_shell.double_shell_model`
"""
c0 *= 1e-12 # use pF as unit
km *= 1e-6
v1 = (1. - dm / Rc)**3
epsi_cp = ecp - 1j * kcp / (e0 * omega)
epsi_m = em - 1j * km / (e0 * omega)
epsi_med = emed - 1j * kmed / (e0 * omega)
# model
E1 = epsi_cp / epsi_m
epsi_cell = epsi_m * (2. * (1. - v1) + (1. + 2. * v1) * E1) / ((2. + v1) + (1. - v1) * E1)
# electrode polarization and calculation of Z
esus = epsi_med * (((2. * epsi_med + epsi_cell) - 2. * p * (epsi_med - epsi_cell))
/ ((2. * epsi_med + epsi_cell) + p * (epsi_med - epsi_cell)))
Ys = 1j * esus * omega * c0 # cell suspension admittance spectrum
Z_fit = 1 / Ys
return Z_fit