This Julia module allows users to analyse their electrochemical impedance spectroscopy (EIS) data using equivalent electrical circuits. EquivalentCircuits.jl can be used to either fit the parameters of a given equivalent electrical circuit or to get recommendations for an appropriate equivalent electrical circuit configuration. The latter is done by conducting an automatic literature search, where the compatibility of impedance measurements with a variety of equivalent circuits from the EIS literature is evaluated. Alternatively, a gene expression programming-based approach can be used to algorithmically search for compatible circuits.

The package can be installed using the package manager.

] add EquivalentCircuits

Equivalent electrical circuit models are composed of electrical elements, connected in series or in parallel. The four fundamental elements that are most commonly encountered in equivalent electrical circuits are resistors, capacitors, inductors and constant-phase elements. These four elements are represented by the capital letters R, C, L and P, respectively. serially connected elements have dashes between them, whereas parallelly connected elements are placed between square brackets and separated by a comma. Finally, all the elements in a circuit are numbered. Using these notation rules, the circuit R1-[C2,R3-[C4,R5]] corresponds to:

When using this package, the circuit should input as a String:

using EquivalentCircuits

circuitstring = "R1-[C2,R3-[C4,R5]]"

When an appropriate circuit model is available, the parameters can be fitted to experimental data using the parameteroptimisation function which accepts two mandatory arguments:

• circuitstring : the equivalent circuit, provided as a string with the circuit notation displayed above.
• data : the filepath of the electrochemical impedance measurement data.

Furthermore, there are three optional keyword arguments for increased functionality:

• x0: An optional initial parameter set
• weights: A vector of equal length as the frequencies. This can be used to attach more importance to specific areas within the frequency range.
• fixed_params: A tuple with the indices of the parameters that are to be fixed during the optimisation and the corresponding fixed parameter values.

The data should be provided as a CSV file with three columns: imaginary impedance, real impedance and frequency (see example_measurements.csv).

Let's first take a look at what the contents of the example_measurements.csv file look like:

using CSV, DataFrames

#Load the measurement data.

data = "example_measurements.csv"; #This should be the filepath of the example_measurements.csv file.

df = CSV.read("example_measurements.csv",DataFrame,header = false);

#Rename the columns for illustration purposes.

rename_dict = Dict("Column1"=>"Reals","Column2"=>"Imags","Column3"=>"Frequencies");

rename!(df, rename_dict);

println(df)

Next, we can fit the parameters of our example circuit to the example measurement data as follows:

circuitparams = parameteroptimisation(circuitstring,data)

Some users may find it more convenient to directly input the complex-valued impedance measurements and their corresponding frequency values to the function vectors. This is illustrated below.

measurements = [5919.90 - 15.79im, 5919.58 - 32.68im, 5918.18 - 67.58im, 5912.24 - 139.49im, 5887.12 - 285.74im, 5785.04 - 566.88im, 5428.94 - 997.19im, 4640.21 - 1257.83im, 3871.84 - 978.97im, 3537.68 - 564.96im, 3442.94 - 315.40im, 3418.14 - 219.69im, 3405.51 - 242.57im, 3373.90 - 396.07im, 3249.67 - 742.03im, 2808.42 - 1305.92im, 1779.41 - 1698.97im, 701.96 - 1361.47im, 208.29 - 777.65im, 65.93 - 392.51im]

frequencies = [0.10, 0.21, 0.43, 0.89, 1.83, 3.79, 7.85, 16.24, 33.60, 69.52, 143.84, 297.64, 615.85, 1274.27, 2636.65, 5455.59, 11288.38, 23357.21, 48329.30, 100000.00]

circuitparams = parameteroptimisation(circuitstring,measurements,frequencies)

The compatibility of a given set of impedance measurements with circuits from similar applications is evaluated with the circuit_search(data,domain;kwargs) function. Users can finetune the search by restricting the complexity and element composition of the returned circuits. The function returns the compatible circuits along with their Digital Object Identifier (DOI) so that users can examine the circuits' other uses to further evaluate their suitability. An overview of the function's inputs is provided below:

• data : A CSV filepath to the measurements with their frequency information.
• domain : The application. The supported applications are: "Animals", "Plants", "Biosensors" , "Batteries", "Fuel_cells" , "Supercapacitors", and "Materials".
• terminals : (optional) the circuit components that are to be included in the circuit identification.
• max_complexity : a hyperparameter that controls the maximum considered complexity of the circuits.

When only the electrochemical impedance measurements are available, equivalent electrical circuit recommendations can also be obtained using the circuit_evolution(data;kwargs) function. Details on the algorithm can be found in the paper. The data can once again be provided as a CSV file's filepath. A variety of keyword arguments can be adjusted to fine-tune the gene expression programming circuit identification procedure. The possible keyword arguments to tune the cirucit identification are:

• generations : the maximum number of algorithm iterations.
• population_size : the number of individuals in the population during each iteration.
• terminals : the circuit components that are to be included in the circuit identification.
• cutoff: a hyperparameter that controls the circuit complexity by removing redundant components. Lower values lead to more simple circuits, however too low values will lead to circuits that no longer fit the measurements well.
• head : a hyperparameter that controls the maximum considered complexity of the circuits.
• initial_population : the option to provide an initial population of circuits with which the algorithm starts, this initial population can be either a list of Circuit objects (as loaded with the loadpopulation function) or a custom list of circuits which can either be a list of one or more circuit strings or a list of tuples where each tuple has the circuit string as first value and the parameters as second value.

The defaults values are as follows:

As an example, by running the code below you can see if a circuit can be found, consisting of only resistors and capacitors, that is capable of fitting the example measurement data. The data argument is the filepath of the example_measurements.csv file.

circuit_evolution(data,terminals="RC")

Alternatively, this function can also accept the measurements and frequencies as vectors, rather than a CSV file:

circuit_evolution(measurements,frequencies,terminals="RC")

Next, the file Circuitlibrary.csv contains a collection of various circuit topologies. We can allow the algorithm to start from this circuit collection as the initial population as follows:

# Load the population from the CSV file, using the loadpopulation function.
# The input of the loadpopulation should be the filepath of Circuitpopulation.csv.
circuit_library = loadpopulation("Circuitlibrary.csv"); #The input should be the filepath of the Circuitlibrary.csv file.

# Now find a circuit that fits the data, starting from the initial population of circuits
circuit_evolution(data,initial_population = circuit_library)