Fitting EIS Data – Diffusion Elements

Posted on July 11, 2014May 29, 2018 by Dr.Bob

Whenever you look at a Complex-Plane Impedance Plot ( Nyquist or Cole-Cole plot) and see a 45° line, or fit data to an equivalent circuit and find a Constant Phase Element (CPE) with an n-value close to 0.5, you should consider diffusion as a possible explanation.

Diffusion Circuit Elements – Warburg

The most common diffusion circuit is the so-called “Warburg” diffusion element, but it is not the only one! A Warburg impedance element can be used to model semi-infinite linear diffusion, that is, unrestricted diffusion to a large planar electrode. This is the simplest diffusion situation because it is only the linear distance from the electrode that matters.
The Warburg impedance is an example of a constant phase element for which the phase angle is a constant 45° and independent of frequency. The magnitude of the Warburg impedance is inversely proportional to the square root of the frequency
as you would expect for a CPE with an n-value of 0.5. The Warburg is unique among CPE’s because the real and imaginary components are equal at all frequencies and both depend upon
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The Constant Phase Element (CPE)

Posted on July 7, 2014May 29, 2018 by Dr.Bob

What is a Constant Phase Element?

The Constant Phase Element (CPE) is a non-intuitive circuit element that was discovered (or invented) while looking at the response of real-world systems. In some systems the Nyquist plot (also called the Cole-Cole plot or Complex Impedance Plane plot) was expected to be a semicircle with the center on the x-axis. However, the observed plot was indeed the arc of a circle, but with the center some distance below the x-axis.
These depressed semicircles have been explained by a number of phenomena, depending on the nature of the system being investigated. However, the common thread among these explanations is that some property of the system is not homogeneous or that there is some distribution (dispersion) of the value of some physical property of the system.

CPE equations

Mathematically, a CPE’s impedance is given by

1 / Z = Y = Q° ( j omega )n

where Q° has the numerical value of the admittance (1/ |Z|) at omega =1 rad/s. The units of Q° are S•sn (ref 1).
A consequence of this simple equation is that the phase angle of the CPE impedance is independent of the frequency and has a value of -(90*n) degrees. This gives the CPE its name.

When n=1, this is the same equation as that for the impedance of a capacitor, where Q° =C.

1 / Z = Y = j omega Q° = j omega C

When n is close to 1.0, the CPE resembles a capacitor, but the phase angle is not 90°. It is constant and somewhat less than 90° at all frequencies. In some cases, the ‘true’ capacitance can be calculated from Q° and n
The Nyquist (Complex Impedance Plane) Plot of a CPE is a simple one. For a solitary CPE (symbolized here by Q), it is just a straight line which makes an angle of (n*90°) with the x-axis as shown in pink in the Figure. The plot for a resistor (symbolized by R) in parallel with a CPE is shown in green. In this case the center of the semicircle is depressed by an angle of (1-n)*90°

What Causes a CPE?

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There is a description of potentiostat stability (written by DK Roe) in the Kissinger & Heineman book (

Fitting EIS Data – Adding Components

Posted on July 2, 2014May 29, 2018 by Dr.Bob

One guideline that I have heard recommended (although I cannot give a reference for it) is that data over a decade range of frequency is required to support each circuit component.

All curve-fitting software should report some measure of the “goodness of fit.” Often this is the chi-squared parameter ( X²) or a value related to it. Boukamp makes the recommendation that the value of X² should decrease by tenfold if a new circuit element is introduced into the circuit model. The tenfold decrease provides the justification for including the new circuit element. If the inclusion of an additional circuit element does not substantially improve the goodness-of-fit (as evidenced by the decrease in the X² value), then based on Occam’s Razor, you should keep the simpler model, or continue your search for an improved one.

The old joke about the ability to “fit an elephant” if you use enough parameters is all too true with impedance data. Each component added to the model should have a physical explanation. Adding components only because they make the fit look better (smaller X²) without a physical interpretation is the equivalent to “fitting an elephant.”

What is X²?

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Electrochemistry Impedance Spectroscopy Basics

Basics of Electrochemical Impedance Spectroscopy

Posted on June 30, 2014May 29, 2018 by Dr.Bob

This application note presents an introduction to Electrochemical Impedance Spectroscopy (EIS) theory and has been kept as free from mathematics and electrical theory as possible. If you still find the material presented here difficult to understand, don’t stop reading. You will get useful information from this application note, even if you don’t follow all of the discussions.

Four major topics are covered in this Application Note.

AC Circuit Theory and Representation of Complex Impedance Values
Physical Electrochemistry and Circuit Elements
Common Equivalent Circuit Models
Extracting Model Parameters from Impedance Data

No prior knowledge of electrical circuit theory or electrochemistry is assumed. Each topic starts out at a quite elementary level, then proceeds to cover more advanced material.

AC Circuit Theory and Representation of Complex Impedance Values

Impedance Definition: Concept of Complex Impedance

(1)

Almost everyone knows about the concept of electrical resistance. It is the ability of a circuit element to resist the flow of electrical current. Ohm’s law (Equation 1) defines resistance in terms of the ratio between voltage, E, and current, I.

While this is a well known relationship, its use is limited to only one circuit element — the ideal resistor. An ideal resistor has several simplifying properties:

It follows Ohm’s Law at all current and voltage levels.
Its resistance value is independent of frequency. AC current and voltage signals though a resistor are in phase with each other.

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Fitting EIS Data to Equivalent Circuits

Posted on June 26, 2014June 25, 2015 by Dr.Bob

“Although the equivalent circuit approach is looked down upon by some, analyzing EIS data by fitting it to equivalent circuit models can be a valid and rewarding approach, particularly in the early stages of an investigation.” When you first begin an electrochemical investigation, very often you may know little or nothing about the process or Read more about Fitting EIS Data to Equivalent Circuits[…]