eisModel

R2026b

Create fractional-order equivalent circuit model object for analyzing battery impedance data

Since R2025a

Description

Use eisModel to create a fractional-order equivalent circuit model EISModel object. Use this object to analyze or interpret battery or fuel cell impedance data.

You can obtain the impedance data at different frequencies from multiple techniques, including electrochemical impedance spectroscopy (EIS). Fractional-order equivalent circuit models are the equivalent circuit models used to fit EIS or frequency-domain data. This object models a fractional-order equivalent circuit model because the Laplacian in the impedance expression of the electrical elements has a fractional exponent.

The EISModel object allows you to specify the type, amount, and connectivity of the electrical circuit elements that comprise the equivalent circuit by using the CircuitTopology property. You define the circuit topology by using a string value. Use a "+" sign to indicate a series connection. Use a "( , )" sign to indicate a parallel connection. There is no limit to the amount of nested parallel or series connections.

The CircuitTopology property supports these electrical circuit elements:

Circuit Element	Icon	Identifier String	Impedance Value	Mechanism
Resistor		R	$$R$$	Ohmic resistance of the electrolyte, electrodes, and other conductors
Capacitor		C	$$\frac{1}{j\cdot w\cdot C}$$	Double layer at the electrode or electrolyte interface, for example battery or supercapacitor
Inductor		L	$$j\cdot w\cdot L$$	Inductive effects due to leads, conductors, and wirings in the measuring device
Constant Phase Element		CPE	$$\frac{1}{{\left(j\cdot w\right)}^n\text{Q}}$$	Accounts for non-ideal capacitive behavior, often due to surface roughness, inhomogeneity, or porous electrodes
Finite-Space Warburg		FSW	$$\frac{Z}{\sqrt{j\cdot w\cdot t}}\cdot \coth \left(\sqrt{j\cdot w\cdot t}\right)$$	Diffusion processes that occur inside a finite region, for example within a solid-state battery electrolyte
Semi-Infinite Warburg		SIW	$$\frac{\text{W}}{\sqrt{w}}\left(1-j\right)$$	Diffusion processes in an unbounded medium, such as diffusion of redox species in the bulk electrolyte of a redox flow battery
Finite-Length Warburg		FLW	$$\frac{Z}{\sqrt{j\cdot w\cdot t}}\cdot \tanh \left(\sqrt{j\cdot w\cdot t}\right)$$	Diffusion processes in a medium with a defined length, for example in a thin-layer cell

To estimate model parameters over a defined frequency range, use this object and an EISTest object as input to the fitEISModel function. This figure shows the typical workflow to estimate the parameters of a fractional-order equivalent circuit model:

Creation

Syntax

eisfom = eisModel

eisfom = eisModel(CircuitTopology)

eisfom = eisModel(CircuitTopology,PropertyName=Value)

Description

eisfom = eisModel creates a default EISModel object with default property values.

eisfom = eisModel(CircuitTopology) creates an EISModel object with the type and number of electrical circuit elements specified in the CircuitTopology argument.

eisfom = eisModel(CircuitTopology,PropertyName=Value) specifies the eisfom properties using one or more name-value arguments.

Input Arguments

CircuitTopology — Type and number of electrical circuit elements and their interconnectivity

"R0+(L1,R3)+(R1,CPE1)+(R2,CPE2)+CPE3" (default) | string scalar | character vector

Type and number of the electrical circuit elements and their interconnectivity inside the fractional-order equivalent circuit model, specified as a string scalar or character vector.

To define series connections, use the + symbol. To define parallel connections, use the (x,y) syntax, where x and y are one of these circuit elements:

• R — Resistor

• C — Capacitor

• L — Inductor

• CPE — Constant phase element

• SIW — Semi-infinite Warburg impedance

• FSW — Finite-space Warburg impedance

• FLW — Finite-length Warburg impedance

The default value represents this 2-ZARC circuit topology:

This argument sets the CircuitTopology property.

Data Types: char | string

Properties

CircuitTopology — Type and number of electrical circuit elements and their interconnectivity

"R0+(L1,R3)+(R1,CPE1)+(R2,CPE2)+CPE3" (default) | string scalar | character vector

Type and number of the electrical circuit elements and their interconnectivity inside the fractional-order equivalent circuit model, specified as a string scalar or character vector.

To define series connections, use the + symbol. To define parallel connections, use the (x,y) syntax, where x and y are one of these circuit elements:

• R — Resistor

• C — Capacitor

• L — Inductor

• CPE — Constant phase element

• SIW — Semi-infinite Warburg impedance

• FSW — Finite-space Warburg impedance

• FLW — Finite-length Warburg impedance

The default value represents this 2-ZARC circuit topology:

Data Types: char | string

ParameterValues — Parameter values used for frequency-based simulation

[0.0250 1.0000e-06 0.0035 0.8000 1 0.0350 0.8000 10 0.3500 0.8000 100] (default) | vector of nonnegative elements

Parameter values of the electrical circuit elements of the fractional-order model, specified as a vector of nonnegative elements. These values are used for frequency-based simulation when you use the simulateFrequencyResponse function.

Data Types: double

SSE — Sum of squared errors from optimization algorithm

scalar | vector

Sum of the squared errors from the optimization algorithm, returned as a nonnegative scalar or vector.

Data Types: double

OustaloupOrder — Order of approximation for constant phase elements and semi-infinite Warburg impedances

1 (default) | nonnegative scalar

Since R2026b

Order of the Oustaloup approximation for constant phase elements and semi-infinite Warburg impedances.

Note

Use the modifyStateSpaceTransferFunction function to change the Oustaloup and Padé approximation properties and regenerate the corresponding state-space and transfer function representations. Setting the OustaloupOrder, OustaloupFrequencyRange, and PadeApproximant properties directly does not update the system representation.

Data Types: double

OustaloupFrequencyRange — Frequency range used in approximating constant phase elements and semi-infinite Warburg impedances

[10^-2 10^2] (default) | vector of two strictly ascending positive scalars

Since R2026b

Frequency range, in Hertz, used in approximating constant phase elements and semi-infinite Warburg impedances.

Note

Use the modifyStateSpaceTransferFunction function to change the Oustaloup and Padé approximation properties and regenerate the corresponding state-space and transfer function representations. Setting the OustaloupOrder, OustaloupFrequencyRange, and PadeApproximant properties directly does not update the system representation.

Data Types: double

PadeApproximant — Order of approximation for finite-space Warburg and finite-length Warburg impedances

2 (default) | 1 | 3 | 4

Since R2026b

Order of the Pade approximation for finite-space and finite-length Warburg impedances.

Note

Use the modifyStateSpaceTransferFunction function to change the Oustaloup and Padé approximation properties and regenerate the corresponding state-space and transfer function representations. Setting the OustaloupOrder, OustaloupFrequencyRange, and PadeApproximant properties directly does not update the system representation.

Data Types: double

StateSpace — State-space representation generated from fractional-order equivalent circuit model

struct

Since R2026b

State-space representation generated from the fractional-order equivalent circuit model. The value of this property must be a structure with fields A, B, C, and D, representing the state matrix, input matrix, output matrix, and direct transmission matrix of a state-space representation, respectively.

TransferFunction — Transfer function representation generated from fractional-order equivalent circuit model

struct

Since R2026b

Transfer function representation generated from the fractional-order equivalent circuit model. The value of this property must be a structure with fields Numerator and Denominator, representing the numerator and denominator of a transfer function, respectively.

NumParameters — Total number of parameters in fractional-order equivalent circuit model

Read-only: scalar

This property is read-only.

Total number of parameters in the fractional-order equivalent circuit model, returned as a scalar.

Data Types: double

TestParameterTables — Test parameter values over specified dimensions

Read-only: structure

This property is read-only.

Test parameter values over the specified dimensions, returned as a structure.

Data Types: struct

ParameterList — List of parameters in equivalent circuit model

Read-only: string scalar | character vector

This property is read-only.

List of parameters in the fractional-order equivalent circuit model, returned as a string scalar or character vector.

Data Types: char | string

CircuitImpedance — Impedance expression of circuit

Read-only: string scalar | character vector

This property is read-only.

Impedance expression of the circuit, returned as a string scalar or character vector.

Data Types: char | string

ParameterSummary — Summary of parameters

Read-only: table

This property is read-only.

Summary of the parameters and related data, returned as a table.

Object Functions

plot	Plot measured and simulated impedance for profile at specific index
simulateFrequencyResponse	Simulate circuit in frequency domain and obtain real and imaginary impedance
modifyStateSpaceTransferFunction	Modify system representation for state-space and transfer function of fractional-order equivalent circuit model
parameterizeImpedanceBlock	Parameterize Impedance (State-Space) block with EISModel state-space matrices
plotMagnitudeAndPhase	Plot magnitude and phase for profile at specific index
simulateMagnitudeAndPhase	Simulate circuit in frequency domain and obtain magnitude and phase data
visualizeAlgebraicTopologyTree	Visualize tree node of algebraic topology

Version History

Introduced in R2025a

R2026b: Obtain system representation of battery cell impedance in state-space or transfer function form

The EISModel object provides a system representation of the battery cell impedance in state-space or transfer function form. Using this system representation, you can then parameterize an Impedance (State-Space) block to simulate the data in the time domain.

As a result of this change, you can specify the Oustaloup order, the Oustaloup frequency range, and the Padé approximation of the EISModel object by using the OustaloupRange, OustaloupFrequencyRange, and PadeApproximant properties.

Use these additional functions to support this workflow:

• modifyStateSpaceTransferFunction — Modify the system representation that the EISModel object provides.

• parameterizeImpedanceBlock — Parameterize the Impedance (State-Space) block from EISModel state-space matrices.

• plotMagnitudeAndPhase — Plot the magnitude and phase of the circuit impedance.

• simulateMagnitudeAndPhase — Simulate the EISModel object in the frequency domain and obtain the magnitude and phase data.

• visualizeAlgebraicTopologyTree — Plot the algebraic topology tree of the EISModel object.