Circuit Elements¶
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impedance.models.circuits.elements.C(p, f)[source]¶ defines a capacitor
\[Z = \frac{1}{C \times j 2 \pi f}\]
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impedance.models.circuits.elements.CPE(p, f)[source]¶ defines a constant phase element
Notes
\[Z = \frac{1}{Q \times (j 2 \pi f)^\alpha}\]where \(Q\) = p[0] and \(\alpha\) = p[1].
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impedance.models.circuits.elements.G(p, f)[source]¶ defines a Gerischer Element as represented in [1]
Notes
\[Z = \frac{R_G}{\sqrt{1 + j \, 2 \pi f \, t_G}}\]where \(R_G\) = p[0] and \(t_G\) = p[1]
Gerischer impedance is also commonly represented as [2]:
\[Z = \frac{Z_o}{\sqrt{K+ j \, 2 \pi f}}\]where \(Z_o = \frac{R_G}{\sqrt{t_G}}\) and \(K = \frac{1}{t_G}\) with units \(\Omega sec^{1/2}\) and \(sec^{-1}\) , respectively.
[1] Y. Lu, C. Kreller, and S.B. Adler, Journal of The Electrochemical Society, 156, B513-B525 (2009) doi:10.1149/1.3079337.
[2] M. González-Cuenca, W. Zipprich, B.A. Boukamp, G. Pudmich, and F. Tietz, Fuel Cells, 1, 256-264 (2001) doi:10.1016/0013-4686(93)85083-B.
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impedance.models.circuits.elements.Gs(p, f)[source]¶ defines a finite-length Gerischer Element as represented in [1]
Notes
\[Z = \frac{R_G}{\sqrt{1 + j \, 2 \pi f \, t_G} \, tanh(\phi \sqrt{1 + j \, 2 \pi f \, t_G})}\]where \(R_G\) = p[0], \(t_G\) = p[1] and \(\phi\) = p[2]
[1] R.D. Green, C.C Liu, and S.B. Adler, Solid State Ionics, 179, 647-660 (2008) doi:10.1016/j.ssi.2008.04.024.
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impedance.models.circuits.elements.K(p, f)[source]¶ An RC element for use in lin-KK model
Notes
\[Z = \frac{R}{1 + j \omega \tau_k}\]
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impedance.models.circuits.elements.La(p, f)[source]¶ defines a modified inductance element as represented in [1]
Notes
\[Z = L \times (j 2 \pi f)^\alpha\]where \(L\) = p[0] and \(\alpha\) = p[1]
[1] EC-Lab Application Note 42, BioLogic Instruments (2019).
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impedance.models.circuits.elements.T(p, f)[source]¶ A macrohomogeneous porous electrode model from Paasch et al. [1]
Notes
\[Z = A\frac{\coth{\beta}}{\beta} + B\frac{1}{\beta\sinh{\beta}}\]where
\[A = d\frac{\rho_1^2 + \rho_2^2}{\rho_1 + \rho_2} \quad B = d\frac{2 \rho_1 \rho_2}{\rho_1 + \rho_2}\]and
\[\beta = (a + j \omega b)^{1/2} \quad a = \frac{k d^2}{K} \quad b = \frac{d^2}{K}\][1] G. Paasch, K. Micka, and P. Gersdorf, Electrochimica Acta, 38, 2653–2662 (1993) doi: 10.1016/0013-4686(93)85083-B.
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impedance.models.circuits.elements.TLMQ(p, f)[source]¶ Simplified transmission-line model as defined in Eq. 11 of [1]
Notes
\[Z = \sqrt{R_{ion}Z_{S}} \coth \sqrt{\frac{R_{ion}}{Z_{S}}\][1] J. Landesfeind et al., Journal of The Electrochemical Society, 163 (7) A1373-A1387 (2016) doi: 10.1016/10.1149/2.1141607jes.
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impedance.models.circuits.elements.W(p, f)[source]¶ defines a semi-infinite Warburg element
Notes
\[Z = \frac{A_W}{\sqrt{ 2 \pi f}} (1-j)\]
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impedance.models.circuits.elements.Wo(p, f)[source]¶ defines an open (finite-space) Warburg element
Notes
\[Z = \frac{Z_0}{\sqrt{ j \omega \tau }} \coth{\sqrt{j \omega \tau }}\]where \(Z_0\) = p[0] (Ohms) and \(\tau\) = p[1] (sec) = \(\frac{L^2}{D}\)
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impedance.models.circuits.elements.Ws(p, f)[source]¶ defines a short (finite-length) Warburg element
Notes
\[Z = \frac{Z_0}{\sqrt{ j \omega \tau }} \tanh{\sqrt{j \omega \tau }}\]where \(Z_0\) = p[0] (Ohms) and \(\tau\) = p[1] (sec) = \(\frac{L^2}{D}\)
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impedance.models.circuits.elements.element_metadata(num_params, units)[source]¶ decorator to store metadata for a circuit element
Parameters: - num_params : int
number of parameters for an element
- units : list of str
list of units for the element parameters