Circuits

There exist a few predefined circuits that were implemented based on published papers. Usually, those circuits are rather complex and cannot be built by the existing elements or feature parameters in certain ranges or units that are not consistent with the generally chosen unit set.

Names for building the model

To build your equivalent circuit model, choose a model from the following list. The function, which tells you the parameter names for the respective model is linked to each model.

Name	Corresponding function
ColeCole	impedancefitter.cole_cole.cole_cole_model()
ColeCole2	impedancefitter.cole_cole.cole_cole_2_model()
ColeCole2tissue	impedancefitter.cole_cole.cole_cole_2tissue_model()
ColeCole3	impedancefitter.cole_cole.cole_cole_3_model()
ColeCole4	impedancefitter.cole_cole.cole_cole_4_model()
ColeColeR	impedancefitter.cole_cole.cole_cole_R_model()
Randles	impedancefitter.randles.Z_randles()
RandlesCPE	impedancefitter.randles.Z_randles_CPE()
DRC	impedancefitter.RC.drc_model()
RCfull	impedancefitter.RC.RC_model()
RC	impedancefitter.RC.rc_model()
RCtau	impedancefitter.RC.rc_tau_model()
ParticleSuspension	impedancefitter.particle_suspension.particle_model()
ParticleSuspensionBH	impedancefitter.particle_suspension.particle_model_bh()
SingleShell	impedancefitter.single_shell.single_shell_model()
SingleShellBH	impedancefitter.single_shell.single_shell_bh_model()
SingleShellWall	impedancefitter.single_shell_wall.single_shell_wall_model()
SingleShellWallBH	impedancefitter.single_shell_wall.single_shell_wall_bh_model()
DoubleShell	impedancefitter.double_shell.double_shell_model()
DoubleShellBH	impedancefitter.double_shell.double_shell_bh_model()
DoubleShellWall	impedancefitter.double_shell_wall.double_shell_wall_model()
DoubleShellWallBH	impedancefitter.double_shell_wall.double_shell_wall_bh_model()
CPE	impedancefitter.cpe.cpe_model()
CPECT	impedancefitter.cpe.cpe_ct_model()
CPECTW	impedancefitter.cpe.cpe_ct_w_model()
CPETissue	impedancefitter.cpe.cpetissue_model()
CPECTTissue	impedancefitter.cpe.cpe_ct_tissue_model()
CPEonset	impedancefitter.cpe.cpe_onset_model()
LR	impedancefitter.loss.Z_in()
LCR	impedancefitter.loss.Z_loss()
HavriliakNegami	impedancefitter.cole_cole.havriliak_negami()
HavriliakNegamiTissue	impedancefitter.cole_cole.havriliak_negamitissue()

Cole-Cole circuits

impedancefitter.cole_cole.cole_cole_2_model(omega, c0, epsinf, deps1, deps2, tau1, tau2, a1, a2, sigma)

Standard 2-Cole-Cole impedance model.

Parameters

• omega (numpy.ndarray, double) – list of frequencies

• c0 (double) – value for unit capacitance, in pF

• epsinf (double) – value for \(\varepsilon_\infty\)

• deps1 (double) – value for \(\Delta\varepsilon_1\)

• deps2 (double) – value for \(\Delta\varepsilon_2\)

• deps3 (double) – value for \(\Delta\varepsilon_3\)

• tau1 (double) – value for \(\tau_1\), in ps

• tau2 (double) – value for \(\tau_2\), in ns

• a1 (double) – value for \(1 - \alpha_1 = a\)

• a2 (double) – value for \(1 - \alpha_2 = a\)

• sigma (double) – conductivity value

Returns

Impedance array

Return type

numpy.ndarray, complex

Notes

The original model has been described in [Gabriel1996]. Here, two instead of four dispersions are used.

impedancefitter.cole_cole.cole_cole_2tissue_model(omega, c0, epsinf, deps1, deps2, tau1, tau2, a1, a2, sigma)

2-Cole-Cole impedance model for tissues.

Parameters

• omega (numpy.ndarray, double) – list of frequencies

• c0 (double) – value for unit capacitance, in pF

• epsinf (double) – value for \(\varepsilon_\infty\)

• deps1 (double) – value for \(\Delta\varepsilon_1 \cdot 10^3\)

• deps2 (double) – value for \(\Delta\varepsilon_2 \cdot 10^6\)

• tau1 (double) – value for \(\tau_1\), in us

• tau2 (double) – value for \(\tau_2\), in ms

• a1 (double) – value for \(1 - \alpha_1 = a\)

• a2 (double) – value for \(1 - \alpha_2 = a\)

• sigma (double) – conductivity value

Returns

Impedance array

Return type

numpy.ndarray, complex

Notes

The original model has been described in [Gabriel1996]. Here, two instead of four dispersions are used.

impedancefitter.cole_cole.cole_cole_3_model(omega, c0, epsinf, deps1, deps2, deps3, tau1, tau2, tau3, a1, a2, a3, sigma)

Standard 3-Cole-Cole impedance model.

Parameters

• omega (numpy.ndarray, double) – list of frequencies

• c0 (double) – value for unit capacitance, in pF

• epsinf (double) – value for \(\varepsilon_\infty\)

• deps1 (double) – value for \(\Delta\varepsilon_1\)

• deps2 (double) – value for \(\Delta\varepsilon_2\)

• deps3 (double) – value for \(\Delta\varepsilon_3\)

• tau1 (double) – value for \(\tau_1\), in ps

• tau2 (double) – value for \(\tau_2\), in ns

• tau3 (double) – value for \(\tau_3\), in us

• a1 (double) – value for \(1 - \alpha_1 = a\)

• a2 (double) – value for \(1 - \alpha_2 = a\)

• a3 (double) – value for \(1 - \alpha_3 = a\)

• sigma (double) – conductivity value

Returns

Impedance array

Return type

numpy.ndarray, complex

Notes

The original model has been described in [Gabriel1996]. Here, three instead of four dispersions are used.

impedancefitter.cole_cole.cole_cole_4_model(omega, c0, epsinf, deps1, deps2, deps3, deps4, tau1, tau2, tau3, tau4, a1, a2, a3, a4, sigma)

Standard 4-Cole-Cole impedance model.

Parameters

• omega (numpy.ndarray, double) – list of frequencies

• c0 (double) – value for unit capacitance, in pF

• epsinf (double) – value for \(\varepsilon_\infty\)

• deps1 (double) – value for \(\Delta\varepsilon_1\)

• deps2 (double) – value for \(\Delta\varepsilon_2\)

• deps3 (double) – value for \(\Delta\varepsilon_3\)

• deps4 (double) – value for \(\Delta\varepsilon_4\)

• tau1 (double) – value for \(\tau_1\), in ps

• tau2 (double) – value for \(\tau_2\), in ns

• tau3 (double) – value for \(\tau_3\), in us

• tau4 (double) – value for \(\tau_4\), in ms

• a1 (double) – value for \(1 - \alpha_1 = a\)

• a2 (double) – value for \(1 - \alpha_2 = a\)

• a3 (double) – value for \(1 - \alpha_3 = a\)

• a4 (double) – value for \(1 - \alpha_4 = a\)

• sigma (double) – conductivity value

Returns

Impedance array

Return type

numpy.ndarray, complex

Notes

The original model has been described in [Gabriel1996].

References

Gabriel1996(1,2,3,4)

Gabriel, S., Lau, R. W., & Gabriel, C. (1996). The dielectric properties of biological tissues: III. Parametric models for the dielectric spectrum of tissues. Physics in Medicine and Biology, 41(11), 2271–2293. https://doi.org/10.1088/0031-9155/41/11/003

impedancefitter.cole_cole.cole_cole_R_model(omega, Rinf, R0, tau, a)

Standard Cole-Cole circuit for macroscopic quantities.

See for example [Schwan1957] for more information.

Parameters

• omega (numpy.ndarray, double) – list of frequencies

• Rinf (double) – value for \(R_\infty\)

• R0 (double) – value for \(R_0\)

• tau (double) – value for \(\tau\), in ns

• a (double) – value for \(1 - \alpha = a\)

Returns

Impedance array

Return type

numpy.ndarray, complex

Notes

Equation for calculations:

\[Z_\mathrm{Cole} = R_\infty + \frac{R_0-R_\infty}{1+(j\omega \tau)^a}\]

Warning

The time constant tau is in ns!

References

Schwan1957

Schwan, H. P. (1957). Electrical properties of tissue and cell suspensions. Advances in biological and medical physics (Vol. 5). ACADEMIC PRESS INC. https://doi.org/10.1016/b978-1-4832-3111-2.50008-0

impedancefitter.cole_cole.cole_cole_model(omega, c0, epsl, tau, a, sigma, epsinf)

Cole-Cole model for dielectric properties.

The model was implemented as presented in [Sabuncu2012]. You need to provide the unit capacitance of your device to get the dielectric properties of the Cole-Cole model.

Parameters

• omega (numpy.ndarray, double) – list of frequencies

• c0 (double) – value for \(c_0\), unit capacitance in pF

• epsinf (double) – value for \(\varepsilon_\infty\)

• epsl (double) – value for \(\varepsilon_\mathrm{l}\)

• tau (double) – value for \(\tau\), in ns

• sigma (double) – value for \(\sigma_\mathrm{dc}\)

• a (double) – value for \(1 - \alpha = a\)

Returns

Impedance array

Return type

numpy.ndarray, complex

Notes

Warning

The unit capacitance is in pF! The time constant tau is in ns!

Equations for calculations:

\[\varepsilon^\ast = \varepsilon_\infty + \frac{\varepsilon_\mathrm{l}-\varepsilon_\infty}{1+(j \omega \tau)^a} - \frac{j \sigma_\mathrm{dc}}{\omega \varepsilon_\mathrm{0}}\]

\[Z = \frac{1}{j\varepsilon^\ast \omega c_\mathrm{0}}\]

References

Sabuncu2012

Sabuncu, A. C., Zhuang, J., Kolb, J. F., & Beskok, A. (2012). Microfluidic impedance spectroscopy as a tool for quantitative biology and biotechnology. Biomicrofluidics, 6(3). https://doi.org/10.1063/1.4737121

impedancefitter.cole_cole.havriliak_negami(omega, c0, epsinf, deps, tau, a, beta, sigma)

Havriliak-Negami relaxation.

Parameters

• omega (numpy.ndarray, double) – list of frequencies

• c0 (double) – value for \(c_0\), unit capacitance in pF

• epsinf (double) – value for \(\varepsilon_\infty\)

• deps (double) – value for \(\Delta\varepsilon\)

• tau (double) – value for \(\tau\), in ns

• sigma (double) – value for \(\sigma_\mathrm{dc}\)

• a (double) – value for \(1 - \alpha = a\)

• beta (double) – value for \(\beta\)

Returns

Impedance array

Return type

numpy.ndarray, complex

Notes

Warning

The unit capacitance is in pF! The time constant tau is in ns!

Equations for calculations:

\[\varepsilon^\ast = \varepsilon_\infty + \frac{\Delta\varepsilon}{\left(1 + (j \omega \tau)^{a}\right)^\beta} - \frac{j\sigma_{\mathrm{DC}}}{\omega \varepsilon_0} \enspace ,\]

\[Z = \frac{1}{j\varepsilon^\ast \omega c_\mathrm{0}}\]

impedancefitter.cole_cole.havriliak_negamitissue(omega, c0, epsinf, deps, tau, a, beta, sigma)

Havriliak-Negami relaxation.

Parameters

• omega (numpy.ndarray, double) – list of frequencies

• c0 (double) – value for \(c_0\), unit capacitance in pF

• epsinf (double) – value for \(\varepsilon_\infty\)

• deps (double) – value for \(\Delta\varepsilon\)

• tau (double) – value for \(\tau\), in \(\mu\)s

• sigma (double) – value for \(\sigma_\mathrm{dc}\)

• a (double) – value for \(1 - \alpha = a\)

• beta (double) – value for \(\beta\)

Returns

Impedance array

Return type

numpy.ndarray, complex

Notes

Warning

The unit capacitance is in pF! The time constant tau is in \(\mu\)s!

Equations for calculations:

\[\varepsilon^\ast = \varepsilon_\infty + \frac{\Delta\varepsilon}{\left(1 + (j \omega \tau)^{a}\right)^\beta} - \frac{j\sigma_{\mathrm{DC}}}{\omega \varepsilon_0} \enspace ,\]

\[Z = \frac{1}{j\varepsilon^\ast \omega c_\mathrm{0}}\]

impedancefitter.cole_cole.raicu(omega, c0, epsinf, deps, tau, alpha, beta, gamma, sigma)

Raicu relaxation.

Parameters

• omega (numpy.ndarray, double) – list of frequencies

• c0 (double) – value for \(c_0\), unit capacitance in pF

• epsinf (double) – value for \(\varepsilon_\infty\)

• deps (double) – value for \(\Delta\varepsilon\)

• tau (double) – value for \(\tau\), in \(\mu\)s

• sigma (double) – value for \(\sigma_\mathrm{dc}\)

• a (double) – value for \(\alpha\)

• beta (double) – value for \(\beta\)

• gamma (double) – value for \(\gamma\)

Returns

Impedance array

Return type

numpy.ndarray, complex

Notes

Model was described in [Raicu1999].

Warning

The unit capacitance is in pF! The time constant tau is in \(\mu\)s!

Equations for calculations:

\[\varepsilon^\ast = \varepsilon_\infty + \frac{\Delta\varepsilon}{\left[ (j \omega \tau)^\alpha + (j \omega \tau)^{1-\beta}\right])^\gamma} - \frac{j\sigma_{\mathrm{DC}}}{\omega \varepsilon_0} \enspace ,\]

\[Z = \frac{1}{j\varepsilon^\ast \omega c_\mathrm{0}}\]

Note that \(f_c = 2 \pi / \tau\). The formulation using fc has for example been used in [Stoneman2007].

References

Raicu1999

Raicu, V. (1999). Dielectric dispersion of biological matter: Model combining Debye-type and “universal” responses. Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics, 60(4), 4677–4680. https://doi.org/10.1103/PhysRevE.60.4677

Stoneman2007

Stoneman, M. R., Kosempa, M., Gregory, W. D., Gregory, C. W., Marx, J. J., Mikkelson, W., … Raicu, V. (2007). Correction of electrode polarization contributions to the dielectric properties of normal and cancerous breast tissues at audio/radiofrequencies. Physics in Medicine and Biology, 52(22), 6589–6604. https://doi.org/10.1088/0031-9155/52/22/003

Particle suspension model

impedancefitter.particle_suspension.particle_bh_model(omega, ep, kp, kmed, emed, p, c0)

Single particle in Bruggeman-Hanai suspension model

Parameters

• omega (numpy.ndarray, double) – list of frequencies

• c0 (double) – value for \(c_0\), unit capacitance in pF

• ep (double) – particle permittivity, value for \(\varepsilon_\mathrm{p}\)

• kp (double) – particle conductivity, value for \(\sigma_\mathrm{p}\) in \(\mu\)S/m

• emed (double) – medium permittivity, value for \(\varepsilon_\mathrm{med}\)

• kmed (double) – medium conductivity, value for \(\sigma_\mathrm{med}\)

• p (double) – volume fraction

Returns

Impedance array

Return type

numpy.ndarray, complex

Notes

Warning

The unit capacitance is in pF! The particle conductivity is in uF/m!

impedancefitter.particle_suspension.particle_model(omega, ep, kp, kmed, emed, p, c0)

Single particle in suspension model

Parameters

• omega (numpy.ndarray, double) – list of frequencies

• c0 (double) – value for \(c_0\), unit capacitance in pF

• ep (double) – particle permittivity, value for \(\varepsilon_\mathrm{p}\)

• kp (double) – particle conductivity, value for \(\sigma_\mathrm{p}\) in \(\mu\)S/m

• emed (double) – medium permittivity, value for \(\varepsilon_\mathrm{med}\)

• kmed (double) – medium conductivity, value for \(\sigma_\mathrm{med}\)

• p (double) – volume fraction

Returns

Impedance array

Return type

numpy.ndarray, complex

Notes

Warning

The unit capacitance is in pF! The particle conductivity is in uF/m!

Single-Shell model

impedancefitter.single_shell.eps_cell_single_shell(omega, km, em, kcp, ecp, dm, Rc)

Complex permittivity of single shell model

Parameters

• omega (numpy.ndarray, double) – list of frequencies

• em (double) – membrane permittivity, value for \(\varepsilon_\mathrm{m}\)

• km (double) – membrane conductivity, value for \(\sigma_\mathrm{m}\)

• ecp (double) – cytoplasm permittivity, value for \(\varepsilon_\mathrm{cp}\)

• kcp (double) – cytoplasm conductivity, value for \(\sigma_\mathrm{cp}\)

• dm (double) – membrane thickness, value for \(d_\mathrm{m}\)

• Rc (double) – cell radius, value for \(R_\mathrm{c}\)

Returns

Complex permittivity array

Return type

numpy.ndarray, complex

impedancefitter.single_shell.single_shell_bh_model(omega, km, em, kcp, ecp, kmed, emed, p, c0, dm, Rc)

Impedance of single shell model using Bruggeman-Hanai approach

Parameters

• omega (numpy.ndarray, double) – list of frequencies

• c0 (double) – value for \(c_0\), unit capacitance in pF

• em (double) – membrane permittivity, value for \(\varepsilon_\mathrm{m}\)

• km (double) – membrane conductivity, value for \(\sigma_\mathrm{m}\) in \(\mu\)S/m

• ecp (double) – cytoplasm permittivity, value for \(\varepsilon_\mathrm{cp}\)

• kcp (double) – cytoplasm conductivity, value for \(\sigma_\mathrm{cp}\)

• emed (double) – medium permittivity, value for \(\varepsilon_\mathrm{med}\)

• kmed (double) – medium conductivity, value for \(\sigma_\mathrm{med}\)

• p (double) – volume fraction

• dm (double) – membrane thickness, value for \(d_\mathrm{m}\)

• Rc (double) – cell radius, value for \(R_\mathrm{c}\)

Returns

Impedance array

Return type

numpy.ndarray, complex

Notes

Warning

The unit capacitance is in pF!

See also

impedancefitter.single_shell.single_shell_model()

impedancefitter.single_shell.single_shell_model(omega, km, em, kcp, ecp, kmed, emed, p, c0, dm, Rc)

Impedance of single shell model

Parameters

• omega (numpy.ndarray, double) – list of frequencies

• c0 (double) – value for \(c_0\), unit capacitance in pF

• em (double) – membrane permittivity, value for \(\varepsilon_\mathrm{m}\)

• km (double) – membrane conductivity, value for \(\sigma_\mathrm{m}\) in \(\mu\)S/m

• ecp (double) – cytoplasm permittivity, value for \(\varepsilon_\mathrm{cp}\)

• kcp (double) – cytoplasm conductivity, value for \(\sigma_\mathrm{cp}\)

• emed (double) – medium permittivity, value for \(\varepsilon_\mathrm{med}\)

• kmed (double) – medium conductivity, value for \(\sigma_\mathrm{med}\)

• p (double) – volume fraction

• dm (double) – membrane thickness, value for \(d_\mathrm{m}\)

• Rc (double) – cell radius, value for \(R_\mathrm{c}\)

Returns

Impedance array

Return type

numpy.ndarray, complex

Notes

Warning

The unit capacitance is in pF!

Equations for the single-shell-model [Feldman2003]:

\[\nu_1 = \left(1-\frac{d_\mathrm{m}}{R_\mathrm{c}}\right)^3\]

\[\varepsilon_\mathrm{m} = \varepsilon_\mathrm{m} - j \frac{\sigma_\mathrm{m}}{\varepsilon_0 \omega}\]

\[\varepsilon_\mathrm{cp} = \varepsilon_\mathrm{cp} - j \frac{\sigma_\mathrm{cp}}{\varepsilon_0 \omega}\]

\[\varepsilon_\mathrm{cell}^\ast = \varepsilon_\mathrm{m}^\ast \frac{2 (1 - \nu_1) + (1 + 2 \nu_1) E_1}{(2 + \nu_1) + (1 - \nu_1) E_1}\]

\[E_1 = \frac{\varepsilon_\mathrm{cp}^\ast}{\varepsilon_\mathrm{m}^\ast}\]

\[\varepsilon_\mathrm{sus}^\ast = \varepsilon_\mathrm{med}^\ast \frac{(2 \varepsilon_\mathrm{med}^\ast + \varepsilon_\mathrm{cell}^\ast) - 2 p (\varepsilon_\mathrm{med}^\ast - \varepsilon_\mathrm{cell}^\ast)} {(2 \varepsilon_\mathrm{med}^\ast + \varepsilon_\mathrm{cell}^\ast) + p (\varepsilon_\mathrm{med}^\ast - \varepsilon_\mathrm{cell}^\ast)}\]

\[Z = \frac{1}{j \varepsilon_\mathrm{sus}^\ast \omega c_0}\]

References

Feldman2003(1,2)

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

impedancefitter.double_shell.double_shell_model()

impedancefitter.single_shell_wall.eps_cell_single_shell_wall(omega, km, em, kcp, ecp, kw, ew, dm, Rc, dw)

Single shell model with cell wall

Parameters

• omega (numpy.ndarray, double) – list of frequencies

• em (double) – membrane permittivity, value for \(\varepsilon_\mathrm{m}\)

• km (double) – membrane conductivity, value for \(\sigma_\mathrm{m}\)

• ecp (double) – cytoplasm permittivity, value for \(\varepsilon_\mathrm{cp}\)

• kcp (double) – cytoplasm conductivity, value for \(\sigma_\mathrm{cp}\)

• ew (double) – cell wall permittivity, value for \(\varepsilon_\mathrm{w}\)

• kw (double) – cell wall conductivity, value for \(\sigma_\mathrm{w}\)

• dm (double) – membrane thickness, value for \(d_\mathrm{m}\)

• Rc (double) – cell radius, value for \(R_\mathrm{c}\)

• dw (double) – cell wall thickness, value for \(R_\mathrm{c}\)

Returns

Complex permittivity array

Return type

numpy.ndarray, complex

Note

Asami, K. (2002). Characterization of biological cells by dielectric spectroscopy. Journal of Non-Crystalline Solids, 305(1–3), 268–277. https://doi.org/10.1016/S0022-3093(02)01110-9

impedancefitter.single_shell_wall.single_shell_wall_bh_model(omega, km, em, kcp, ecp, kw, ew, kmed, emed, p, c0, dm, Rc, dw)

Impedance of single shell model using Bruggeman-Hanai approach

Parameters

• omega (numpy.ndarray, double) – list of frequencies

• c0 (double) – value for \(c_0\), unit capacitance in pF

• em (double) – membrane permittivity, value for \(\varepsilon_\mathrm{m}\)

• km (double) – membrane conductivity, value for \(\sigma_\mathrm{m}\) in \(\mu\)S/m

• ecp (double) – cytoplasm permittivity, value for \(\varepsilon_\mathrm{cp}\)

• kcp (double) – cytoplasm conductivity, value for \(\sigma_\mathrm{cp}\)

• ew (double) – cell wall permittivity, value for \(\varepsilon_\mathrm{w}\)

• kw (double) – cell wall conductivity, value for \(\sigma_\mathrm{w}\)

• emed (double) – medium permittivity, value for \(\varepsilon_\mathrm{med}\)

• kmed (double) – medium conductivity, value for \(\sigma_\mathrm{med}\)

• p (double) – volume fraction

• dm (double) – membrane thickness, value for \(d_\mathrm{m}\)

• Rc (double) – cell radius, value for \(R_\mathrm{c}\)

• dw (double) – cell wall thickness, value for \(d_\mathrm{w}\)

Returns

Impedance array

Return type

numpy.ndarray, complex

Notes

Warning

The unit capacitance is in pF! The membrane conductivity is in uS/m!

See also

impedancefitter.single_shell.single_shell_wall_model()

impedancefitter.single_shell_wall.single_shell_wall_model(omega, km, em, kcp, ecp, kw, ew, kmed, emed, p, c0, dm, Rc, dw)

Impedance of single shell model

Parameters

• omega (numpy.ndarray, double) – list of frequencies

• c0 (double) – value for \(c_0\), unit capacitance in pF

• em (double) – membrane permittivity, value for \(\varepsilon_\mathrm{m}\)

• km (double) – membrane conductivity, value for \(\sigma_\mathrm{m}\) in \(\mu\)S/m

• ecp (double) – cytoplasm permittivity, value for \(\varepsilon_\mathrm{cp}\)

• kcp (double) – cytoplasm conductivity, value for \(\sigma_\mathrm{cp}\)

• ew (double) – cell wall permittivity, value for \(\varepsilon_\mathrm{w}\)

• kw (double) – cell wall conductivity, value for \(\sigma_\mathrm{w}\)

• emed (double) – medium permittivity, value for \(\varepsilon_\mathrm{med}\)

• kmed (double) – medium conductivity, value for \(\sigma_\mathrm{med}\)

• p (double) – volume fraction

• dm (double) – membrane thickness, value for \(d_\mathrm{m}\)

• Rc (double) – cell radius, value for \(R_\mathrm{c}\)

• dw (double) – cell wall thickness, value for \(d_\mathrm{w}\)

Returns

Impedance array

Return type

numpy.ndarray, complex

Notes

Warning

The unit capacitance is in pF! The membrane conductivity is in uS/m!

See also

impedancefitter.single_shell.single_shell_wall_model()

Double-Shell model

impedancefitter.double_shell.double_shell_bh_model(omega, km, em, kcp, ecp, ene, kne, knp, enp, kmed, emed, p, c0, dm, Rc, dn, Rn)

Impedance of double shell model

Parameters

• omega (numpy.ndarray, double) – list of frequencies

• c0 (double) – value for \(c_0\), unit capacitance in pF

• em (double) – membrane permittivity,membrane permittivity, value for \(\varepsilon_\mathrm{m}\)

• km (double) – membrane conductivity, value for \(\sigma_\mathrm{m}\) in \(\mu\)S/m

• ecp (double) – cytoplasm permittivity, value for \(\varepsilon_\mathrm{cp}\)

• kcp (double) – cytoplasm conductivity, value for \(\sigma_\mathrm{cp}\)

• ene (double) – nuclear envelope permittivity, value for \(\varepsilon_\mathrm{ne}\)

• kne (double) – nuclear envelope conductivity, value for \(\sigma_\mathrm{ne}\) in mS/m

• enp (double) – nucleoplasm permittivity, value for \(\varepsilon_\mathrm{np}\)

• knp (double) – nucleoplasm conductivity, value for \(\sigma_\mathrm{np}\)

• emed (double) – medium permittivity, value for \(\varepsilon_\mathrm{med}\)

• kmed (double) – medium conductivity, value for \(\sigma_\mathrm{med}\)

• p (double) – volume fraction

• dm (double) – membrane thickness, value for \(d_\mathrm{m}\)

• Rc (double) – cell radius, value for \(R_\mathrm{c}\)

• dn (double) – nuclear envelope thickness, value for \(d_\mathrm{n}\)

• Rn (double) – nucleus radius, value for \(R_\mathrm{n}\)

Returns

Impedance array

Return type

numpy.ndarray, complex

Notes

Warning

The unit capacitance is in pF!

Note that here the Bruggeman-Hanai formula is used.

See also

impedancefitter.double_shell.double_shell_model()

impedancefitter.double_shell.double_shell_model(omega, km, em, kcp, ecp, ene, kne, knp, enp, kmed, emed, p, c0, dm, Rc, dn, Rn)

Impedance of double shell model

Parameters

• omega (numpy.ndarray, double) – list of frequencies

• c0 (double) – value for \(c_0\), unit capacitance in pF

• em (double) – membrane permittivity,membrane permittivity, value for \(\varepsilon_\mathrm{m}\)

• km (double) – membrane conductivity, value for \(\sigma_\mathrm{m}\) in \(\mu\)S/m

• ecp (double) – cytoplasm permittivity, value for \(\varepsilon_\mathrm{cp}\)

• kcp (double) – cytoplasm conductivity, value for \(\sigma_\mathrm{cp}\)

• ene (double) – nuclear envelope permittivity, value for \(\varepsilon_\mathrm{ne}\)

• kne (double) – nuclear envelope conductivity, value for \(\sigma_\mathrm{ne}\) in mS/m

• enp (double) – nucleoplasm permittivity, value for \(\varepsilon_\mathrm{np}\)

• knp (double) – nucleoplasm conductivity, value for \(\sigma_\mathrm{np}\)

• emed (double) – medium permittivity, value for \(\varepsilon_\mathrm{med}\)

• kmed (double) – medium conductivity, value for \(\sigma_\mathrm{med}\)

• p (double) – volume fraction

• dm (double) – membrane thickness, value for \(d_\mathrm{m}\)

• Rc (double) – cell radius, value for \(R_\mathrm{c}\)

• dn (double) – nuclear envelope thickness, value for \(d_\mathrm{n}\)

• Rn (double) – nucleus radius, value for \(R_\mathrm{n}\)

Returns

Impedance array

Return type

numpy.ndarray, complex

Notes

Warning

The unit capacitance is in pF!

Equations for the single-shell-model [Feldman2003]:

\[\nu_1 = \left(1-\frac{d_\mathrm{m}}{R_\mathrm{c}}\right)^3\]

\[\nu_\mathrm{2} = \left(\frac{R_\mathrm{n}}{R-d}\right)^3\]

\[\nu_\mathrm{3} = \left(1-\frac{d_\mathrm{n}}{R_\mathrm{n}}\right)^3\]

\[\varepsilon_\mathrm{c}^\ast = \varepsilon_\mathrm{m}^\ast\frac{2(1-\nu_\mathrm{1})+(1+2\nu_\mathrm{1})E_\mathrm{1}}{(2+\nu_\mathrm{1})+(1-\nu_\mathrm{1})E_\mathrm{1}}\]

\[E_\mathrm{1} = \frac{\varepsilon_\mathrm{cp}^\ast}{\varepsilon_\mathrm{m}^\ast} \frac{2(1-\nu_\mathrm{2})+(1+2\nu_\mathrm{2})E_\mathrm{2}}{(2+\nu_\mathrm{2})+(1-\nu_\mathrm{2})E_\mathrm{2}}\]

\[E_\mathrm{2} = \frac{\varepsilon_\mathrm{ne}^\ast}{\varepsilon_\mathrm{cp}^\ast}\frac{2(1-\nu_\mathrm{3})+(1+2\nu_\mathrm{3})E_\mathrm{3}}{(2+\nu_\mathrm{3})+(1-\nu_\mathrm{3})E_\mathrm{3}}\]

\[E_\mathrm{3} = \frac{\varepsilon_\mathrm{np}^\ast}{\varepsilon_\mathrm{ne}^\ast}\]

\[\varepsilon_\mathrm{m} = \varepsilon_\mathrm{m} - j \frac{\sigma_\mathrm{m}}{\varepsilon_0 \omega}\]

\[\varepsilon_\mathrm{cp} = \varepsilon_\mathrm{cp} - j \frac{\sigma_\mathrm{cp}}{\varepsilon_0 \omega}\]

\[\varepsilon_\mathrm{ne} = \varepsilon_\mathrm{ne} - j \frac{\sigma_\mathrm{ne}}{\varepsilon_0 \omega}\]

\[\varepsilon_\mathrm{np} = \varepsilon_\mathrm{np} - j \frac{\sigma_\mathrm{np}}{\varepsilon_0 \omega}\]

\[\varepsilon_\mathrm{cell}^\ast = \varepsilon_\mathrm{m}^\ast \frac{2 (1 - \nu_1) + (1 + 2 \nu_1) E_1}{(2 + \nu_1) + (1 - \nu_1) E_1}\]

\[\varepsilon_\mathrm{sus}^\ast = \varepsilon_\mathrm{med}^\ast \frac{(2 \varepsilon_\mathrm{med}^\ast + \varepsilon_\mathrm{cell}^\ast) - 2 p (\varepsilon_\mathrm{med}^\ast - \varepsilon_\mathrm{cell}^\ast)} {(2 \varepsilon_\mathrm{med}^\ast + \varepsilon_\mathrm{cell}^\ast) + p (\varepsilon_\mathrm{med}^\ast - \varepsilon_\mathrm{cell}^\ast)}\]

\[Z = \frac{1}{j \varepsilon_\mathrm{sus}^\ast \omega c_0}\]

In [Ermolina2000] , there have been reported upper/lower limits for certain parameters. They could act as a first guess for the bounds of the optimization method.

Parameter	lower limit	upper limit
\(\varepsilon_\mathrm{m}\)	1.4	16.8
\(\sigma_\mathrm{m}\)	8e-8	5.6e-5
\(\varepsilon_\mathrm{cp}\)	60	77
\(\sigma_\mathrm{cp}\)	0.033	1.1
\(\varepsilon_\mathrm{ne}\)	6.8	100
\(\sigma_\mathrm{ne}\)	8.3e-5	7e-3
\(\varepsilon_\mathrm{np}\)	32	300
\(\sigma_\mathrm{np}\)	0.25	2.2
R	3.5e-6	10.5e-6
\(R_\mathrm{n}\)	2.95e-6	8.85e-6
d	3.5e-9	10.5e-9
\(d_\mathrm{n}\)	2e-8	6e-8

Ermolina2000

Ermolina, I., Polevaya, Y., & Feldman, Y. (2000). Analysis of dielectric spectra of eukaryotic cells by computer modeling. European Biophysics Journal, 29(2), 141–145. https://doi.org/10.1007/s002490050259

See also

impedancefitter.single_shell.single_shell_model()

impedancefitter.double_shell.eps_cell_double_shell(omega, km, em, kcp, ecp, kne, ene, knp, enp, dm, Rc, dn, Rn)

Complex permittivity of double shell model

Parameters

• omega (numpy.ndarray, double) – list of frequencies value for \(c_0\), unit capacitance in pF

• em (double) – membrane permittivity,membrane permittivity, value for \(\varepsilon_\mathrm{m}\)

• km (double) – membrane conductivity, value for \(\sigma_\mathrm{m}\) in \(\mu\)S/m

• ecp (double) – cytoplasm permittivity, value for \(\varepsilon_\mathrm{cp}\)

• kcp (double) – cytoplasm conductivity, value for \(\sigma_\mathrm{cp}\)

• ene (double) – nuclear envelope permittivity, value for \(\varepsilon_\mathrm{ne}\)

• kne (double) – nuclear envelope conductivity, value for \(\sigma_\mathrm{ne}\) in mS/m

• enp (double) – nucleoplasm permittivity, value for \(\varepsilon_\mathrm{np}\)

• knp (double) – nucleoplasm conductivity, value for \(\sigma_\mathrm{np}\)

• dm (double) – membrane thickness, value for \(d_\mathrm{m}\)

• Rc (double) – cell radius, value for \(R_\mathrm{c}\)

• dn (double) – nuclear envelope thickness, value for \(d_\mathrm{n}\)

• Rn (double) – nucleus radius, value for \(R_\mathrm{n}\)

Returns

Permittivity array

Return type

numpy.ndarray, complex

impedancefitter.double_shell_wall.double_shell_wall_bh_model(omega, km, em, kcp, ecp, kne, ene, knp, enp, kw, ew, kmed, emed, p, c0, dm, Rc, dn, Rn, dw)

Impedance of double shell model with cell wall using Bruggeman-Hanai approach

Parameters

• omega (numpy.ndarray, double) – list of frequencies

• c0 (double) – value for \(c_0\), unit capacitance in pF

• em (double) – membrane permittivity,membrane permittivity, value for \(\varepsilon_\mathrm{m}\)

• km (double) – membrane conductivity, value for \(\sigma_\mathrm{m}\) in \(\mu\)S/m

• ecp (double) – cytoplasm permittivity, value for \(\varepsilon_\mathrm{cp}\)

• kcp (double) – cytoplasm conductivity, value for \(\sigma_\mathrm{cp}\)

• ene (double) – nuclear envelope permittivity, value for \(\varepsilon_\mathrm{ne}\)

• kne (double) – nuclear envelope conductivity, value for \(\sigma_\mathrm{ne}\) in mS/m

• kw (double) – cell wall conductivity, value for \(\sigma_\mathrm{w}\) in S/m

• ew (double) – cell wall permittivity, value for \(\varepsilon_\mathrm{w}\)

• enp (double) – nucleoplasm permittivity, value for \(\varepsilon_\mathrm{np}\)

• knp (double) – nucleoplasm conductivity, value for \(\sigma_\mathrm{np}\)

• emed (double) – medium permittivity, value for \(\varepsilon_\mathrm{med}\)

• kmed (double) – medium conductivity, value for \(\sigma_\mathrm{med}\)

• p (double) – volume fraction

• dm (double) – membrane thickness, value for \(d_\mathrm{m}\)

• Rc (double) – cell radius, value for \(R_\mathrm{c}\)

• dn (double) – nuclear envelope thickness, value for \(d_\mathrm{n}\)

• Rn (double) – nucleus radius, value for \(R_\mathrm{n}\)

• dw (double) – wall thickness

Returns

Impedance array

Return type

numpy.ndarray, complex

Notes

Warning

The unit capacitance is in pF!

Note that here the Bruggeman-Hanai formula is used.

See also

impedancefitter.double_shell.double_shell_model()

impedancefitter.double_shell_wall.double_shell_wall_model(omega, km, em, kcp, ecp, kne, ene, knp, enp, kw, ew, kmed, emed, p, c0, dm, Rc, dn, Rn, dw)

Impedance of double shell model with cell wall

Parameters

• omega (numpy.ndarray, double) – list of frequencies

• c0 (double) – value for \(c_0\), unit capacitance in pF

• em (double) – membrane permittivity,membrane permittivity, value for \(\varepsilon_\mathrm{m}\)

• km (double) – membrane conductivity, value for \(\sigma_\mathrm{m}\) in \(\mu\)S/m

• ecp (double) – cytoplasm permittivity, value for \(\varepsilon_\mathrm{cp}\)

• kcp (double) – cytoplasm conductivity, value for \(\sigma_\mathrm{cp}\)

• ene (double) – nuclear envelope permittivity, value for \(\varepsilon_\mathrm{ne}\)

• kne (double) – nuclear envelope conductivity, value for \(\sigma_\mathrm{ne}\) in mS/m

• enp (double) – nucleoplasm permittivity, value for \(\varepsilon_\mathrm{np}\)

• knp (double) – nucleoplasm conductivity, value for \(\sigma_\mathrm{np}\)

• kw (double) – cell wall conductivity, value for \(\sigma_\mathrm{w}\) in S/m

• ew (double) – cell wall permittivity, value for \(\varepsilon_\mathrm{w}\)

• emed (double) – medium permittivity, value for \(\varepsilon_\mathrm{med}\)

• kmed (double) – medium conductivity, value for \(\sigma_\mathrm{med}\)

• p (double) – volume fraction

• dm (double) – membrane thickness, value for \(d_\mathrm{m}\)

• Rc (double) – cell radius, value for \(R_\mathrm{c}\)

• dn (double) – nuclear envelope thickness, value for \(d_\mathrm{n}\)

• Rn (double) – nucleus radius, value for \(R_\mathrm{n}\)

• dw (double) – cell wall thickness

Returns

Impedance array

Return type

numpy.ndarray, complex

Notes

Warning

The unit capacitance is in pF!

impedancefitter.double_shell_wall.eps_cell_double_shell_wall(omega, km, em, kcp, ecp, kne, ene, knp, enp, kw, ew, dm, Rc, dn, Rn, dw)

Complex permittivity of double shell model with cell wall

Parameters

• omega (numpy.ndarray, double) – list of frequencies

• em (double) – membrane permittivity, value for \(\varepsilon_\mathrm{m}\)

• km (double) – membrane conductivity, value for \(\sigma_\mathrm{m}\)

• ecp (double) – cytoplasm permittivity, value for \(\varepsilon_\mathrm{cp}\)

• kcp (double) – cytoplasm conductivity, value for \(\sigma_\mathrm{cp}\)

• ew (double) – cell wall permittivity, value for \(\varepsilon_\mathrm{w}\)

• kw (double) – cell wall conductivity, value for \(\sigma_\mathrm{w}\)

• dm (double) – membrane thickness, value for \(d_\mathrm{m}\)

• Rc (double) – cell radius, value for \(R_\mathrm{c}\)

• dw (double) – cell wall thickness, value for \(d_\mathrm{w}\)

Returns

Complex permittivity array

Return type

numpy.ndarray, complex

Note

Asami, K. (2002). Characterization of biological cells by dielectric spectroscopy. Journal of Non-Crystalline Solids, 305(1–3), 268–277. https://doi.org/10.1016/S0022-3093(02)01110-9

Inductance circuits

impedancefitter.loss.Z_in(omega, L, R)

Lead inductance of wires connecting DUT.

Described for instance in [Kordzadeh2016].

Parameters

• omega (numpy.ndarray) – List of frequencies.

• L (double) – inductance

• R (double) – resistance

Returns

Impedance array

Return type

numpy.ndarray, complex

References

Kordzadeh2016(1,2,3,4)

Kordzadeh, A., & De Zanche, N. (2016). Permittivity measurement of liquids, powders, and suspensions using a parallel-plate cell. Concepts in Magnetic Resonance Part B: Magnetic Resonance Engineering, 46(1), 19–24. https://doi.org/10.1002/cmr.b.21318

Notes

As mentioned in [Kordzadeh2016], the unit of the inductance is nH.

impedancefitter.loss.Z_loss(omega, L, C, R)

Impedance for high loss materials, where LCR are in parallel.

Described for instance in [Kordzadeh2016].

Parameters

• omega (numpy.ndarray) – List of frequencies.

• L (double) – inductance

• C (double) – capacitance

• R (double) – resistance

Returns

Impedance array

Return type

numpy.ndarray, complex

Notes

As mentioned in [Kordzadeh2016], the unit of the capacitance is pF and the unit of the inductance is nH.

impedancefitter.loss.Z_skin(omega, L, Rb, gamma)

Lead inductance of wires connecting DUT considering skin effect.

Parameters

• omega (numpy.ndarray) – List of frequencies.

• L (double) – inductance

• Rb (double) – resistance of wire, about mOhm

• gamma (double) – exponent (should be about 0.5 probably)

Returns

Impedance array

Return type

numpy.ndarray, complex

References

Levitskaya2000

Levitskaya, T. M., & Sternberg, B. K. (2000). Laboratory measurement of material electrical properties: extending the application of lumped-circuit equivalent models to 1 GHz. Radio Science, 35(2), 371–383. https://doi.org/10.1029/1999RS002186

Notes

Described for instance in [Levitskaya2000]. The idea is to take frequency-dependent resistance.

The unit of the inductance is nH. The wire resistance is about mOhm. The impedance is computed by

\[Z = R_b \omega^{\gamma} + j \omega L\]

Note that the frequency omega in this formula is in MHz.

CPE circuits

impedancefitter.cpe.cpe_ct_model(omega, k, alpha, Rct)

Constant Phase Element in parallel with charge transfer resistance.

Parameters

• omega (numpy.ndarray) – List of frequencies.

• k (double) – CPE factor

• alpha (double) – CPE phase

• Rct (double) – charge transfer resistance

Returns

Impedance array

Return type

numpy.ndarray, complex

See also

impedancefitter.cpe.cpe_model()

impedancefitter.cpe.cpe_ct_tissue_model(omega, k, alpha, Rct)

Constant Phase Element in parallel with charge transfer resistance.

Parameters

• omega (numpy.ndarray) – List of frequencies.

• k (double) – CPE factor

• alpha (double) – CPE phase

• Rct (double) – charge transfer resistance

Returns

Impedance array

Return type

numpy.ndarray, complex

See also

impedancefitter.cpe.cpe_model()

impedancefitter.cpe.cpe_ct_w_model(omega, k, alpha, Rct, Aw)

Constant Phase Element in parallel with

charge transfer resistance, which is in series with Warburg element.

Parameters

• omega (numpy.ndarray) – List of frequencies.

• k (double) – CPE factor

• alpha (double) – CPE phase

• Rct (double) – charge transfer resistance

• Aw (double) – Warburg coefficient

Returns

Impedance array

Return type

numpy.ndarray, complex

See also

impedancefitter.cpe.cpe_model()

impedancefitter.cpe.cpe_model(omega, k, alpha)

Constant Phase Element.

\[Z_\mathrm{CPE} = k (j \omega)^{-\alpha}\]

Parameters

• omega (numpy.ndarray) – List of frequencies.

• k (double) – CPE factor

• alpha (double) – CPE phase

Returns

Impedance array

Return type

numpy.ndarray, complex

impedancefitter.cpe.cpe_onset_model(omega, f0, nu, Z0)

Alternative Constant Phase Element Formulation.

Notes

\[Z_\mathrm{CPE} = Z_0 \left(j \frac{\omega}{2 \pi f_0} \right)^{-\nu}\]

See for example [Ishai2012].

Parameters

• omega (numpy.ndarray) – List of frequencies.

• k (double) – CPE factor

• alpha (double) – CPE phase

Returns

Impedance array

Return type

numpy.ndarray, complex

References

Ishai2012

Ishai, P. Ben, Sobol, Z., Nickels, J. D., Agapov, A. L., & Sokolov, A. P. (2012). An assessment of comparative methods for approaching electrode polarization in dielectric permittivity measurements. Review of Scientific Instruments, 83(8). https://doi.org/10.1063/1.4746992

impedancefitter.cpe.cpetissue_model(omega, k, alpha)

Constant Phase Element.

\[Z_\mathrm{CPE} = k (j \omega)^{-\alpha}\]

Parameters

• omega (numpy.ndarray) – List of frequencies.

• k (double) – CPE factor

• alpha (double) – CPE phase

Returns

Impedance array

Return type

numpy.ndarray, complex

RC circuits

impedancefitter.RC.RC_model(omega, Rd, Cd)

Simple RC model, resistor in parallel with capacitor.

Parameters

• omega (numpy.ndarray, double) – list of frequencies

• Rd (double) – Resistance.

• Cd (double) – Capacitance

Notes

Warning

Cd is in pF!

Returns

Impedance array

Return type

numpy.ndarray, complex

impedancefitter.RC.drc_model(omega, RE, tauE, alpha, beta)

Distributed RC circuit.

Parameters

• omega (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

Impedance array

Return type

numpy.ndarray, complex

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

impedancefitter.RC.rc_model(omega, c0, kdc, eps)

Simple RC model to obtain dielectric properties.

Parameters

• omega (numpy.ndarray, double) – list of frequencies

• c0 (double) – unit capacitance in pF

• eps (double) – relative permittivity

• kdc (double) – conductivity

Returns

Impedance array

Return type

numpy.ndarray, complex

Notes

Warning

C0 is in pF!

impedancefitter.RC.rc_tau_model(omega, Rk, tauk)

Compute RC model without explicit capacitance.

Parameters

• omega (numpy.ndarray, double) – list of frequencies

• Rk (double) – resistance

• tauk (double) – relaxation time

Notes

Returns

Impedance array

Return type

numpy.ndarray, complex

Notes

Used for example in the Lin-KK test [Schoenleber2014].

The impedance reads

\[Z = \frac{R_\mathrm{k}}{1+ j \omega \tau_\mathrm{k}}\]

Randles circuits

impedancefitter.randles.Z_randles(omega, Rct, Rs, Aw, C0)

Randles circuit.

Parameters

• omega (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

Impedance array

Return type

numpy.ndarray, complex

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:

\[Z_\mathrm{RW} = R_0 + A_\mathrm{W} \frac{1 - j}{\sqrt{\omega}}\]

Impedance of capacitor:

\[Z_C = (j \omega C_0)^{-1}\]

\[Z_\mathrm{fit} = R_s + \frac{Z_C Z_\mathrm{RW}}{Z_C + Z_\mathrm{RW}}\]

impedancefitter.randles.Z_randles_CPE(omega, Rct, Rs, Aw, k, alpha)

Randles circuit with CPE instead of capacitor.

Parameters

• omega (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

Impedance array

Return type

numpy.ndarray, complex

See also

Z_randles(), impedancefitter.elements.Z_CPE()