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Examples

Fitting

eisfit.fitting.buildCircuit(circuit, parameters, frequencies)

transforms a circuit, parameters, and frequencies into a string that can be evaluated

Parameters:

circuit : str

parameters : list of floats

frequencies : list of floats

Returns:

eval_string : str

Python expression for calculating the resulting fit

eisfit.fitting.circuit_fit(frequencies, impedances, circuit, initial_guess, algorithm='leastsq', bounds=None)

Main function for fitting an equivalent circuit to data

Parameters:

frequencies : numpy array

Frequencies

impedances : numpy array of dtype ‘complex128’

Impedances

circuit : string

string defining the equivalent circuit to be fit

initial_guess : list of floats

initial guesses for the fit parameters

algorithm: string

Name of algorithm to pass to scipy.optimize.minimize or to instantiate scipy.optimize.leastsq

Returns:

p_values : list of floats

best fit parameters for specified equivalent circuit

p_errors : list of floats

error estimates for fit parameters

Notes

Need to do a better job of handling errors in fitting. Currently, an error of -1 is returned.

eisfit.fitting.computeCircuit(circuit, parameters, frequencies)

evaluates a circuit string for a given set of parameters and frequencies

Parameters:

circuit : string

parameters : list of floats

frequencies : list of floats

Returns:

array of floats

eisfit.fitting.residuals(param, Z, f, circuit)

Calculates the residuals between a given circuit/parameters (fit) and Z/f (data). Minimized by scipy.leastsq()

Parameters:

param : array of floats

parameters for evaluating the circuit

Z : array of complex numbers

impedance data being fit

f : array of floats

frequencies to evaluate

circuit : str

string defining the circuit

Returns:

residual : ndarray

returns array of size 2*len(f) with both real and imaginary residuals

eisfit.fitting.rmse(a, b)

A function which calculates the root mean squared error between two vectors.

Notes

\[RMSE = \sqrt{\frac{1}{n}(a-b)^2}\]

eisfit.fitting.valid(circuit, param)

checks validity of parameters

Parameters:

circuit : string

string defining the circuit

param : list

list of parameter values

Returns:

valid : boolean

Notes

All parameters are considered valid if they are greater than zero – except for E2 (the exponent of CPE) which also must be less than one.

Circuits

class eisfit.circuits.BaseCircuit(initial_guess=None, name=None, algorithm='leastsq', bounds=None)

Base class for equivalent circuit models

Methods

fit(frequencies, impedance)	Fit the circuit model
plot([f_data, Z_data, CI])	a convenience method for plotting Nyquist plots
predict(frequencies)	Predict impedance using a fit equivalent circuit model

fit(frequencies, impedance)

Fit the circuit model

Parameters:

frequencies: numpy array

Frequencies

impedance: numpy array of dtype ‘complex128’

Impedance values to fit

Returns:

self: returns an instance of self

plot(f_data=None, Z_data=None, CI=True)

a convenience method for plotting Nyquist plots

Parameters:

f_data: np.array of type float

Frequencies of input data (for Bode plots)

Z_data: np.array of type complex

Impedance data to plot

CI: boolean

Include bootstrapped confidence intervals in plot

Returns:

ax: matplotlib.axes

axes of the created nyquist plot

predict(frequencies)

Predict impedance using a fit equivalent circuit model

Parameters:

frequencies: numpy array

Frequencies

Returns:

impedance: numpy array of dtype ‘complex128’

Predicted impedance

class eisfit.circuits.DefineCircuit(circuit=None, **kwargs)

Methods

fit(frequencies, impedance)	Fit the circuit model
plot([f_data, Z_data, CI])	a convenience method for plotting Nyquist plots
predict(frequencies)	Predict impedance using a fit equivalent circuit model

class eisfit.circuits.FlexiCircuit(max_elements=None, generations=2, popsize=30, initial_guess=None)

Methods

fit(frequencies, impedances)	Fit the circuit model
plot([f_data, Z_data, CI])	a convenience method for plotting Nyquist plots
predict(frequencies)	Predict impedance using a fit equivalent circuit model

fit(frequencies, impedances)

Fit the circuit model

Parameters:

frequencies: numpy array

Frequencies

impedance: numpy array of dtype ‘complex128’

Impedance values to fit

Returns:

self: returns an instance of self

class eisfit.circuits.Randles(CPE=False, **kwargs)

A Randles circuit model class

Methods

fit(frequencies, impedance)	Fit the circuit model
plot([f_data, Z_data, CI])	a convenience method for plotting Nyquist plots
predict(frequencies)	Predict impedance using a fit equivalent circuit model

eisfit.circuit_elements.A(p, f)

defines a semi-infinite Warburg element

eisfit.circuit_elements.C(p, f)

defines a capacitor

\[Z = \frac{1}{C \times j 2 \pi f}\]

eisfit.circuit_elements.E(p, f)

defines a constant phase element

Notes

\[Z = \frac{1}{Q \times (j 2 \pi f)^\alpha}\]

where \(Q\) = p[0] and \(\alpha\) = p[1].

eisfit.circuit_elements.G(p, f)

defines a Gerischer Element

Notes

\[Z = \frac{1}{Y \times \sqrt{K + j 2 \pi f }}\]

eisfit.circuit_elements.R(p, f)

defines a resistor

Notes

\[Z = R\]

eisfit.circuit_elements.W(p, f)

defines a blocked boundary Finite-length Warburg Element

Notes

\[Z = \frac{R}{\sqrt{ T \times j 2 \pi f}} \coth{\sqrt{T \times j 2 \pi f }} # noqa: E501\]

where \(R\) = p[0] (Ohms) and \(T\) = p[1] (sec) = \(\frac{L^2}{D}\)

eisfit.circuit_elements.p(parallel)

adds elements in parallel

Notes

\[Z = \frac{1}{\frac{1}{Z_1} + \frac{1}{Z_2} + ... + \frac{1}{Z_n}}\]

eisfit.circuit_elements.s(series)

sums elements in series

Notes

\[Z = Z_1 + Z_2 + ... + Z_n\]

Validation

EIS data fundamentally relies on the conditions of linearity,

Measurement Model

Testing your data with the measurement model is straightforward:

import matplotlib.pyplot as plt
import numpy as np
import sys
sys.path.append('../')

from eisfit import validation  # noqa E402

data = np.genfromtxt('./data/exampleData.csv', delimiter=',')
f = data[:, 0]
Z = data[:, 1] + 1j*data[:, 2]

mask = np.imag(Z) < 0

model_list, error_list = validation.measurementModel(f, Z, max_k=25)

fig = plt.figure()
plt.plot(Z.real, -Z.imag, 'o')
for model in model_list:
    Z_fit = model.predict(f)
    plt.plot(Z_fit.real, -Z_fit.imag)

fig2, ax2 = plt.subplots()
ax2.plot(range(1, len(error_list)+1), error_list)
ax2.set_yscale('log')
ax2.set_ylabel('Root Mean Squared Error')
ax2.set_xlabel('Number of RC elements')

plt.show()

eisfit.validation.measurementModel(frequencies, impedances, algorithm='SLSQP', max_k=7, R_guess=0.1, C_guess=10)

Runs a measurement model test for validating impedance data

Iteratively add RC circuit elements until the error converges. If error does not converge, it indicates that the data doesn’t meet standards for linearity.

Notes

\[RMSE = R_0 + \sum_{0}^{k} R_i || C_i\]

frequencies: np.ndarray

A list of frequencies to test

impedances: np.ndarray of complex numbers

A list of values to match to

max_k: int

The maximum number of RC elements to fit

initial_guess: np.ndarray

Initial guesses for R and C elements

eisfit.validation.rmse(a, b)

A function which calculates the root mean squared error between two vectors.

Notes

\[RMSE = \sqrt{\frac{1}{n}(a-b)^2}\]

Preprocessing

Methods for preprocessing impedance data from instrument files

eisfit.preprocessing.readAutolab(filename)

function for reading the .csv file from Autolab

Parameters:

filename: string

Filename of .csv file to extract impedance data from

Returns:

frequencies : np.ndarray

Array of frequencies

impedance : np.ndarray of complex numbers

Array of complex impedances

eisfit.preprocessing.readFile(filename, type=None)

A wrapper for reading in many common types of impedance files

Parameters:

filename: string

Filename to extract impedance data from

type: string

Type of instrument file

Returns:

frequencies : np.ndarray

Array of frequencies

impedance : np.ndarray of complex numbers

Array of complex impedances

eisfit.preprocessing.readGamry(filename)

function for reading the .DTA file from Gamry

Parameters:

filename: string

Filename of .DTA file to extract impedance data from

Returns:

frequencies : np.ndarray

Array of frequencies

impedance : np.ndarray of complex numbers

Array of complex impedances

eisfit.preprocessing.readParstat(filename)

function for reading the .txt file from Parstat

Parameters:

filename: string

Filename of .txt file to extract impedance data from

Returns:

frequencies : np.ndarray

Array of frequencies

impedance : np.ndarray of complex numbers

Array of complex impedances

Genetic Modeling

Indices and tables

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