EISFitpython is a Python package for Electrochemical Impedance Spectroscopy (EIS) analysis, optimized for temperature-dependent experiments and complex circuit models. It applies a unified chi-square optimization to fit multiple impedance datasets at once, preserving inter-parameter relationships and propagating uncertainties across the full temperature range.

Accurate modeling of temperature-dependent impedance demands consistent parameter evolution. EISFitpython delivers:

Standard EIS workflows fit each temperature in isolation, which can produce erratic parameter trends, biased activation-energy estimates, and incomplete error analysis. EISFitpython replaces this with a single global-local optimization:

• Concurrent Dataset Fitting
  Reduces parameter inconsistency by optimizing all temperatures together.
• Global Parameters
  Keeps temperature-independent circuit elements constant across datasets.
• Local Parameters
  Allows temperature-dependent elements to adapt within physical constraints.
• Error Propagation
  Tracks uncertainty through every temperature step.

• Global-Local Parameter Classification
  Automatically labels each circuit element as temperature-independent or temperature-dependent.
• Unified Chi-Square Objective
  Optimizes all datasets against one cost function for consistency.
• Automated Classification
  Detects and assigns global vs. local roles without manual intervention.
• Statistical Validation
  Computes confidence intervals and parameter correlations.

If you use this code in your research, please cite:

S. C. Adediwura, N. Mathew, J. Schmedt auf der Günne, J. Mater. Chem. A 12 (2024) 15847–15857.
https://doi.org/10.1039/d3ta06237f

Electrochemical Impedance Spectroscopy (EIS) is a powerful analytical technique that characterizes the electrical properties of materials and their interfaces with electronically conducting electrodes. The technique involves:

1. Signal Application: $$E(t) = E_0\sin(\omega t)$$

2. System Response: $$I(t) = I_0\sin(\omega t + \phi)$$

3. Impedance Calculation: $$Z(\omega) = \frac{E(t)}{I(t)} = |Z|e^{j\phi} = Z' + jZ''$$

Where:

• $\omega = 2\pi f$ is the angular frequency
• $\phi$ is the phase shift
• $|Z|$ is the impedance magnitude
• $Z'$ and $Z''$ are real and imaginary components

Electrochemical Impedance Spectroscopy (EIS) represents a non-linear system where impedance data must be fitted using complex non-linear least-squares (CNLS) analysis to extract meaningful physical parameters. The CNLS technique operates by minimizing a sum of squares function S, which quantifies the difference between experimental data and the theoretical model.

The sum of squares function for model fitting:

$$S = \sum_{i=1}^{m} w_i[f_i^\mathrm{EXP}(x_i) - f_i^\mathrm{TH}(x, P)]^2$$

For EIS data over frequency range $\omega_i$:

$$S = \sum_{i=1}^{N} {w_i[\mathrm{Re}Z_i^\mathrm{EXP}(\omega_i) - \mathrm{Re}Z_i^\mathrm{TH}(\omega_i, P)]^2 + w_i[\mathrm{Im}Z_i^\mathrm{EXP}(\omega_i) - \mathrm{Im}Z_i^\mathrm{TH}(\omega_i, P)]^2}$$

Where:

• $Z_i^\mathrm{EXP}$ and $Z_i^\mathrm{TH}$ represent experimental and theoretical impedance respectively
• Re and Im denote real and imaginary components
• $w_i$ and $w_i$ are weighting factors for real and imaginary components

The choice of weighting method significantly impacts the fitting quality. EISFitpython implements three established approaches:

1. Unit Weighting $$w_i^\mathrm{RE} = w_i^\mathrm{IM} = 1$$

   • Treats all data points equally
   • May overemphasize larger impedance values
   • Suitable for data with uniform uncertainties

2. Proportional Weighting $$w_i^\mathrm{RE} = \frac{1}{(\mathrm{Re}Z_i)^2}, \quad w_i^\mathrm{IM} = \frac{1}{(\mathrm{Im}Z_i)^2}$$

   • Assumes constant relative errors
   • May overemphasize high and low-frequency regions
   • Useful when uncertainties scale with impedance magnitude

3. Modulus Weighting $$w_i= \frac{1}{|Z_i|^2}$$

   • Balances contributions from small and large impedances
   • Provides equal statistical weight to real and imaginary components
   • Recommended for typical EIS measurements

• Python 3.9 or higher
• NumPy ≥ 1.19.0
• SciPy ≥ 1.6.0
• Matplotlib ≥ 3.3.0
• Pandas ≥ 1.2.0

Download the latest release from the GitHub Releases page, then install it with pip:

python -m pip install eisfitpython-<version>.tar.gz
# or, if you downloaded a wheel:
python -m pip install eisfitpython-<version>-py3-none-any.whl

pip install -r requirements.txt

Core module for defining and evaluating equivalent circuit models.

• R: Resistor (Z = R)
• C: Capacitor (Z = 1/jωC)
• L: Inductor (Z = jωL)
• Q: Constant Phase Element (Z = 1/[Q(jω)ᵅ])
• W: Warburg Element (Z = σω^(-1/2)(1-j))
• F: Finite-length Warburg (Z = σ·tanh(√(jωτ))/√(jωτ))

• Series connections: +
• Parallel connections: |
• Nested circuits: ()

Example: "(R1|Q1)+(R2|Q2)" represents a series combination of two parallel R-Q circuits.

1. compute_impedance(params, circuit_str, freqs)

   • Calculates complex impedance for a given circuit
   • Parameters: element values, circuit string, frequency points

2. Z_curve_fit(circuit_str)

   • Generates fitting function for scipy.optimize.curve_fit
   • Used internally by fitting routines

Handles importing and processing raw EIS data files with sophisticated data validation and preprocessing capabilities.

1. readNEISYS(filename)

   • Reads NEISYS format spectrometer files

2. readTXT(filename) / readCSV(filename)

   • Supports multiple standard data formats:
     • Three-column format: f, Z', Z"

3. trim_data(f, Z, fmin, fmax)

   • Advanced frequency range filtering:
     • Linear or logarithmic range selection
     • Automatic endpoint adjustment
     • Data density preservation
   • Optional noise filtering
   • Returns filtered f, Z arrays
   • Example:

     # Trim to specific frequency range
     f_trim, Z_trim = dt.trim_data(f, Z, 
                                  fmin=1e-1,  # 0.1 Hz
                                  fmax=1e6)   # 1 MHz

4. get_eis_files(base_path, subfolder)

   • Smart file handling:
     • Recursive directory search
     • Pattern matching for EIS data
     • Automatic sorting by temperature
   • Supports multiple file formats
   • Temperature extraction from filenames
   • Example:

     # Get temperature-series files
     
     #if subfolder is not None
      files = dt.get_eis_files(base_path='base folder', subfolder='temp_series')
      #else 
      files = dt.get_eis_files(base_path='basefolder')

Core fitting and analysis functionality.

1. EISFit(f, Z, params, circuit, UB, LB, weight_mtd='M', method='lm', single_chi='No')

   • Main fitting routine
   • Supports multiple weighting methods:
     • 'M': Modulus weighting (|Z|)
     • 'P': Proportional weighting
     • 'U': Unity (no weighting)
   • Returns optimized parameters, errors, and fit statistics

2. full_EIS_report(f, Z, params, circuit, UB, LB, weight_mtd, method, single_chi, plot_type)

   • Comprehensive analysis with parameter values and plots
   • Generates Nyquist and/or Bode plots
   • Calculates statistical metrics (χ², R², AICc)

3. plot_fit(f, Z, Z_fit, plot_type)

   • Visualization of experimental and fitted data
   • Supports Nyquist, Bode, or both plot types
   • Automatic scaling of impedance units (Ω, kΩ, MΩ, GΩ)

Tools for analyzing multiple datasets, especially temperature-dependent studies.

1. Batch_fit(files, params, circuit, Temp, UB, LB, weight_mtd, method)

   • Fits multiple datasets with same circuit model
   • Handles temperature series data
   • Generates combined plots and reports

2. plot_arrhenius(R_values, temp, diameter, thickness)

   • Arrhenius plot generation
   • Calculates activation energies
   • Handles multiple components (bulk, grain boundary, etc.)

3. sigma_report(R, Rerr, l, con, T)

   • Calculates conductivity and its uncertainty
   • Accounts for geometric factors and measurement errors
   • Reports temperature-dependent conductivity values with:
     • Absolute conductivity (S/cm)
     • Standard error
     • Percent error
   • Handles both single and multiple temperature points

4. C_eff(R, R_err, Q, Q_err, n, n_err, T)

   • Calculates effective capacitance from CPE parameters
   • Includes error propagation
   • Useful for analyzing non-ideal capacitive behavior

Specialized tools for complex systems with shared parameters across multiple temperatures.

1. Single_chi_report(f, Z, params, Temp, circuit_str)

   • Advanced fitting with global-local parameter handling:
     • Global parameters remain constant across temperatures
     • Local parameters vary with temperature
   • Temperature-dependent parameter tracking
   • Comprehensive error analysis including:
     • Parameter correlations
     • Standard errors
     • Chi-square statistics

2. flatten_params(params, circuit_str, N_sub)

   • parameter management for complex fits
   • Handles mixed global/local parameters:
     • For global parameters: single value used across all temperatures
     • For local parameters: array of values [T₁, T₂, ..., Tₙ]
   • Automatic parameter validation and structure verification
   • Example usage:

     # Mixed global-local parameter structure
     params = (
         [R1_T1, R1_T2, ..., R1_Tn],  # Local R1 values
         Q1,                           # Global Q1 value
         n1,                           # Global n1 value
         [R2_T1, R2_T2, ..., R2_Tn]   # Local R2 values
     )

3. Single_chi(f, *params, circuit_str, circuit_type)

   • Core computation engine for impedance calculation
   • Handles parameter distribution across temperature points
   • Supports both fitting and prediction modes
   • Automatically manages parameter structure based on temperature points

import EISFit_main as em

# Define a simple RC circuit
circuit = "(R1|C1)"
params = [100, 1e-6]  # R = 100 Ω, C = 1 µF
freqs = np.logspace(-2, 6, 50)  # 0.01 Hz to 1 MHz

# Generate simulated data
Z = em.predict_Z(freqs[0], freqs[-1], len(freqs), params, circuit)

# Import custom EIS modules
from EISFitpython import data_extraction as dt     

# Load EIS data
# NEISYS spectrometer files
f, Z = dt.readNEISYS('my_data.txt')

#.txt file without headers and tab separated, three-column format: f, Z', Z"
f, Z = dt.readTXT('my_data.txt')

#.csv file without headers, three-column format: f, Z', Z"
f, Z = dt.readCSV('my_data.csv')

# Define circuit and initial parameters
circuit = "(R1|Q1)+R2"
params = [1e5, 1e-9, 0.8, 50]  # R1, Q1, n1, R2
UB = []      # Upper bounds
LB = []    # Lower bounds

# Perform fitting
popt, perror = em.full_EIS_report(
    f, Z, params, circuit, 
    UB=UB, LB=LB,
    weight_mtd='M',      # Modulus weighting
    method='lm',         # Levenberg-Marquardt algorithm
    single_chi='No',
    plot_type='both'     # Generate both Nyquist and Bode plots
)

# Import custom EIS modules
from EISFitpython import data_extraction as dt          
from EISFitpython import EIS_Batchfit as ebf 

# Load multiple temperature datasets
#if subfolder is not None
files = dt.get_eis_files(base_path='base folder', subfolder='temp_series')


temps = np.array([25, 50, 75, 100])  # °C

# Equivalent circuit model
circuit_str = "(R1|Q1)+Q2"

# Initial parameter guesses for the circuit model
params = [R1, Q1,n1, Q2, n2]

# Perform batch fitting
fit_params, fit_errors = ebf.Batch_fit(
    files, params, circuit, temps,
    UB, LB, weight_mtd='M', method='lm'
)

# Generate Arrhenius plot
D = 1.3    # Sample diameter (cm)
L = 0.2    # Sample thickness (cm)
ebf.plot_arrhenius(fit_params[:,0], temps, D, L, labels='Bulk')

# Import custom EIS modules
from EISFitpython import data_extraction as dt     
from EISFitpython import singlechi as sc           
from EISFitpython import EISFit_main as em      
from EISFitpython import EIS_Batchfit as ebf 
import numpy as np

#%% Data Processing
# Get EIS data files
filenames=dt.get_eis_files(base_path='../EIS_Data', subfolder='temp_series')

# Extract frequency and impedance data from NEISYS spectrometer files

f, Z = dt.stack_NEISYS_files(filenames)

# Split data at 1MHz frequency point
sublist, _ = dt.split_array(f, Z=None, split_freq=np.max(f))
N_sub = len(sublist)

# Temperature points (in Celsius)
temps = np.array([140, 150, 160, 170, 180])

# Define parameters with temperature dependence
circuit = "(R1|Q1)+(R2|Q2)"
params = (
    R1_values,  # Array of R1 values for each temperature
    Q1,         # Global Q1 (temperature-independent) / local (temperature-dependent, i.e., ferroelectrics )
    n1,         # Global n1 (temperature-independent) / local (temperature-dependent, i.e., ferroelectrics )
    R2_values,  # Array of R2 values for each temperature
    Q2,         # Global/local  Q2
    n2          # Global/local  n2
)

# Perform analysis
fit_params, fit_perror, Z_fit = sc.Single_chi_report(f, Z, params, Temp, circuit_str,weight_mtd='M')

The module handles complex parameter relationships:

• Resistance values (R) typically show temperature dependence
• CPE parameters (Q, n) often remain constant across temperatures
• Automatic handling of parameter interdependencies
• Built-in correlation analysis between parameters

1. Data Preparation:

   • Use consistent file naming
   • Maintain temperature series structure
   • Use trim_data() for focused analysis
   • Check frequency range coverage

2. Circuit Model Selection:

   • Start simple, add complexity as needed
   • Validate with physical meaning
   • Check parameter uncertainties

3. Fitting Strategy:

   • Use reasonable initial guesses
   • Set appropriate parameter bounds
   • Try different weighting methods

4. Temperature Analysis:

   • Ensure consistent sample geometry
   • Use proper temperature units
   • Validate Arrhenius behavior
   • Check activation energy

See Example_scripts/ directory for detailed examples:

• Example1-simulate.py: Circuit simulation
• Example2a-EISfit.py: Single temperature fitting
• Example3-Batchfit.py: Temperature series analysis
• Example4-Singlechi.py: Global-local optimization with one cost function

EISFitpython is distributed under the MIT License.

Sheyi Clement Adediwura (c) 2024