Validation

Interpreting EIS data fundamentally relies on the the system conforming to conditions of causality, linearity, and stability. For an example of how the adherence to the Kramers-Kronig relations, see the Validation Example Jupyter Notebook

Lin-KK method

Validating your data with the lin-KK model requires fitting an optimal number of RC-elements and analysis of the residual errors.

impedance.validation.calc_mu(Rs)

Calculates mu for use in LinKK

impedance.validation.eval_linKK(Rs, ts, f)

Builds a circuit of RC elements to be used in LinKK

impedance.validation.fitLinKK(f, ts, M, Z)

Fits the linKK model using scipy.optimize.least_squares

impedance.validation.linKK(f, Z, c=0.85, max_M=50)

A method for implementing the Lin-KK test for validating linearity [1]

Parameters:

f: np.ndarray

measured frequencies

Z: np.ndarray of complex numbers

measured impedances

c: np.float

cutoff for mu

max_M: int

the maximum number of RC elements

Returns:

mu: np.float

under- or over-fitting measure

residuals: np.ndarray of complex numbers

the residuals of the fit at input frequencies

Z_fit: np.ndarray of complex numbers

impedance of fit at input frequencies

Notes

The lin-KK method from Schönleber et al. [1] is a quick test for checking the validity of EIS data. The validity of an impedance spectrum is analyzed by its reproducibility by a Kramers-Kronig (KK) compliant equivalent circuit. In particular, the model used in the lin-KK test is an ohmic resistor, \(R_{Ohm}\), and \(M\) RC elements.

\[\hat Z = R_{Ohm} + \sum_{k=1}^{M} \frac{R_k}{1 + j \omega \tau_k}\]

The \(M\) time constants, \(\tau_k\), are distributed logarithmically,

\[\tau_1 = \frac{1}{\omega_{max}} ; \tau_M = \frac{1}{\omega_{min}} ; \tau_k = 10^{\log{(\tau_{min}) + \frac{k-1}{M-1}\log{{( \frac{\tau_{max}}{\tau_{min}}}})}}\]

and are not fit during the test (only \(R_{Ohm}\) and \(R_{k}\) are free parameters).

In order to prevent under- or over-fitting, Schönleber et al. propose using the ratio of positive resistor mass to negative resistor mass as a metric for finding the optimal number of RC elements.

\[\mu = 1 - \frac{\sum_{R_k \ge 0} |R_k|}{\sum_{R_k < 0} |R_k|}\]

The argument c defines the cutoff value for \(\mu\). The algorithm starts at M = 3 and iterates up to max_M until a \(\mu < c\) is reached. The default of 0.85 is simply a heuristic value based off of the experience of Schönleber et al.

If the argument c is None, then the automatic determination of RC elements is turned off and the solution is calculated for max_M RC elements. This manual mode should be used with caution as under- and over-fitting should be avoided.

[1] Schönleber, M. et al. A Method for Improving the Robustness of linear Kramers-Kronig Validity Tests. Electrochimica Acta 131, 20–27 (2014) doi: 10.1016/j.electacta.2014.01.034.

impedance.validation.residuals_linKK(Rs, ts, Z, f, residuals='real')

Calculates the residual between the data and a LinKK fit