Physics courses introduce the concept of electrical resistance via Ohm's law (Figure 1.1). Where is the voltage between two points, is the current that flows between the two points, and is the electrical resistance, represented symbolically by a resistor in a circuit diagram. Conceptually, represents the opposition to the current flowing through an electrical circuit. The larger the , the less current will flow across the resistor, requiring a higher voltage to generate more current.
This description of resistance via Ohm's law applies specifically to a direct current (DC), where a static voltage or current is applied across a resistor. Impedance, by contrast, is a measure of the resistance a circuit experiences related to the passage of an alternating electrical current (AC) signal. In an AC system, the applied signal is no longer static but oscillates as a sinusoidal wave at a given frequency. The equation for impedance is analogous to Ohm's law, however instead of using for resistance we use for impedance.
Additionally, the impedance is not proportional to the voltage and the current, but to the frequency-dependent voltage and frequency-dependent current . Where is the angular frequency of an oscillating sine wave. The definition of impedance comes from electrical circuits, and as a result, voltage is commonly used to define the impedance. However, in electrochemical impedance spectroscopy (EIS), we will switch from using voltage to using electrical potential , because the signal that is applied and measured is an electrochemical potential.
Electrochemical impedance spectroscopy is an electroanalytical technique where a sinusoidal potential (or current) signal is applied to an electrochemical system and the resulting current (or potential) signal is recorded and analyzed (Figure 1.1).
Figure 1.1 Simplified Electrochemical Impedance Spectroscopy Diagram. If the input signal is potential, then the green wave represents the input sinusoidal potential signal and the red wave represents the output sinusoidal current.
If the applied signal is potential and the measured result is current, it is referred to as “potentiostatic EIS.” When the applied signal is current and the measured result is potential, it is referred to as “galvanostatic EIS.” For the case of potentiostatic EIS, a potential is applied with the form shown in Equation 1.4:
Where is the potential sine wave amplitude, is the angular frequency, is the time, and the term represents the phase of the waveform. The angular frequency is a measure of how many cycles per unit time the signal oscillates. The amplitude is a measure of how large or the magnitude of the potential or current signal is. The frequency and amplitude of the input potential signal are tuned by the user, while the output current signal (Equation 1.5) has the same frequency as the input signal but its phase may be shifted by a finite amount, known as a phase shift or phase angle .
Where is the current sine wave amplitude and is the phase shift or phase angle.
Figure 1.2 Simplified Electrochemical Impedance Spectroscopy Diagram with Phase Angle. The teal sinusoidal wave represents input potential signal visually overlapped in time with the output current signal. Phase angle represents the shift is phase when the input and output signals are overlapped in time.
In an EIS experiment, a sequence of sinusoidal potential signals centered around a potential setpoint, are applied to an electrochemical system. The amplitude of each sinusoidal signal remains constant, but the frequency of the input signal will vary. Typically, frequencies of each input signal are equally spaced on a descending logarithmic scale from ~10 kHz - 1 MHz to a lower limit of ~10 mHz - 1 Hz. For each input potential a corresponding output current is measured at a given frequency. Application of these input and output signals is usually performed automatically via a potentiostat/galvanostat.
Figure 1.3 Visualization of EIS experiment. Input signal is applied at different frequencies, measured output signal at the same frequency.
Practically, frequency (f) is reported in units of Hz. However, for mathematical convenience the angular frequency (ω), which has units of rad/s and is equivalent to 2πf, is typically used for calculations instead (e.g., see input and output signal equations in Figure 1.1). Similarly, the phase angle () is typically reported in units of degrees but calculated in units of radians.
There are three conventions often used to define the input (and sometimes output) signal amplitude: peak, peak-to-peak, and RMS. “Peak” refers to the difference between the sine wave set point (i.e., the potential or current at the beginning of the sine wave period) and its maximum or minimum point (i.e., the potential or current at one quarter of the sine wave period). “Peak-to-peak” is simply twice the peak value (see Figure 1.1).
“RMS”, which stands for “root mean square”, is a mathematical quantity used primarily in electrical engineering to compare AC and DC voltages or currents. Though its practical relevance and importance to EIS measurements are somewhat minimal, it is still widely used in the industry to characterize input signal amplitude. Mathematically, it is equivalent to the peak value divided by , or roughly peak times 0.707 (see Figure 1.1).
Below you can find a table of symbols and definitions used in Equations 1.1 and 1.2.
peak potential amplitude
root mean square potential amplitude
peak-to-peak potential amplitude
peak current amplitude
frequency (units of Hz)
angular frequency (units of rad/s)
Table 1.1 Electrochemical Impedance Spectroscopy Input and Output Symbol Definitions
Figure 1.1.1 Diagram of Potentiostatic Electrochemical Impedance Spectroscopy
Monitoring the progress of an EIS experiment can be done by observing the input and output signals on a single current vs. potential graph called a Lissajous plot (see Figure 1.2). Depending on the system under study, as well as the applied frequency and amplitude, the shape of the resulting Lissajous plot may vary. Throughout an EIS experiment, the user can observe the progression and pattern of Lissajous plots as a means of identifying possibly erroneous data.
Figure 1.2 Examples of Typical Lissajous Plots for Stable and Linear Systems
The shape of the current vs. potential Lissajous plot for a stable, linear electrochemical system typically appears as either a tilted oval or straight line that repeatedly traces over itself (see Figure 1.2). The width of the oval is indicative of the magnitude of the output signal phase angle. For example, if the Lissajous plot looks like a perfect circle, it means the output signal is completely out of phase (i.e., +90°) with respect to the input signal. This is also the EIS response experienced by an ideal capacitor or inductor.
Impedance is defined as the measure of difficulty a circuit experiences related to the passage of an applied alternating electrical current. It is analogous to Ohm’s law, but unlike Ohm’s law impedance can change as a function of the frequency of the applied potential or current.
During an EIS experiment, if the applied signal is potential and the measured result is current, it is referred to as “potentiostatic EIS”. When the applied signal is current and the measured result is potential, it is referred to as “galvanostatic EIS”. For the case of potentiostatic EIS, a potential is applied with the form shown in Equation 2.1:
where is the potential sine wave amplitude, is the angular frequency, is the time, and the term represents the phase of the waveform. If is small enough such that the system is linear, the resultant current waveform is also sinusoidal and will have the same frequency as the input signal; however, it may be shifted in phase, as shown in Equation 2.2:
where is the current sine wave amplitude and is the phase angle shift.
A more convenient mathematical rearrangement of the applied and resultant waveforms (Equations 2.1 and 2.2) is needed to clarify the measured impedance. This is accomplished by using complex coordinates via Euler’s formula, which is defined as:
where represents the imaginary unit (rather than the conventional symbol, , to prevent confusion with the symbol for current) and is any real number. If is substituted by an angle with the form similarly to Equations 2.1 and 2.2, where is any given phase angle shift, and all terms are multiplied by a given constant amplitude, , Equation 2.3 is rearranged as follows:
The right-hand side of Equation 2.4 is a complex number while the left-hand side contains only a real quantity. Therefore, by equating the real portions of both sides of the equation, the following expression is obtained:
where is the real portion of a number. The second term on the right-hand side of Equation 2.5 is eliminated because it is entirely imaginary, and the operator is removed from the left-hand side because the cosine of a real number is also a real number:
Equation 2.6 is further simplified as follows:
where is a collection of all time-invariant terms and is equivalent to the following:
Using Equation 2.7, the applied potential and resultant current waveforms shown in Equations 2.1 and 2.2, respectively, are rewritten as:
where the time-invariant terms and are equivalent to:
One condition of validity for impedance is stability (i.e., steady-state or time-invariant - see Section 4.1). Therefore, following Equations 2.9 through 2.12, an expression for impedance, , analogous to Ohm’s law is shown below:
Substituting Equations 2.11 and 2.12 into Equation 2.13 yields the expression:
where is the impedance magnitude and is equivalent to . Using Euler’s formula (Equation 2.3), Equation 2.14 is finally expanded to the form:
This mathematical rearrangement allows the real (, equivalent to ) and imaginary (, equivalent to ) components of the impedance to be calculated separately.
During an EIS experiment, most commercial software packages automatically perform Fourier transform on the input and output signals at each frequency to extract corresponding values for and . These values are then used with Equation 2.15 to calculate and . Hence, the result from a typical EIS experiment is a data table with five columns: , , , , and .
There are two standard types of plots generated from five-column EIS data: Nyquist and Bode plots. A Nyquist plot normally consists of vs. , and this type of plot is most commonly used to identify distinctive patterns and shapes in the data (see Figure 3.1 for examples of Nyquist plots for several different circuit networks).
Figure 3.1 Example Nyquist Plots for Different Circuit Networks
The imaginary impedance values on a Nyquist plot are commonly inverted, or conversely the axis is sometimes displayed in reverse numerical order, due to the fact that almost all values are normally less than zero and it is more convenient to view shapes and patterns primarily in the first quadrant on a Cartesian graph (see Figure 3.1).
Another convention traditionally applied to Nyquist plots is orthonormality, also sometimes called orthogonality, which refers to the usage of a 1:1 ratio for the visual scale of x- and y-axes. Note that this does not necessarily mean the numerical scales of the axes need necessarily be identical (see plots in Figure 3.1 for example). One simple way to think of this principle is that when drawing lines on an orthonormal plot around identical values on both axes, it would always make a perfect square (e.g., connect the points (0,0), (0,100), (100,100), and (100,0), and it will be a perfect square).
Historically, orthonormality was used exclusively on Nyquist plots because some of the ubiquitous shapes (e.g., semicircles and tilted lines) are more easily discerned when viewed on an orthonormal plot. In modern times, however, some of this plotting necessity has become diminished with the advent of circuit fitting software. While non-idealities in data can be observed through distorted semicircles and tilted line angles on an orthonormal Nyquist plot, they are also easily visualized and quantified via circuit fitting algorithms. It can also occasionally be inconvenient to force orthonormality on some Nyquist plots if the data becomes concentrated to a small portion of the plot, leaving large swaths of empty graphical space (e.g., upper right plot on Figure 3.1).
Those points notwithstanding, it is still widely considered standard form to employ orthonormality to Nyquist plots, and most published EIS data use this convention. Therefore, it is generally recommended that the user prepare orthonormal Nyquist plots when presenting their EIS data.
The second standard type of plot used with EIS data is a Bode plot. A Bode plot is a double-axis plot consisting of both vs. (on the primary vertical axis) and vs. (on the secondary vertical axis). Frequency and impedance magnitude are normally plotted on a logarithmic scale, while the phase angle is displayed linearly (see Figure 3.2 for examples of Bode plots for several different circuit networks).
Figure 3.2 Example Bode Plots for Different Circuit Networks
Observing the phase angle on a Bode plot is a quick way to understand the type of circuit behavior a system is experiencing at any particular frequency. For example, a phase angle of 0° corresponds to an ideal resistor, +90° corresponds to an ideal inductor, and -90° corresponds to an ideal capacitor. Values in between may indicate mixed behavior or non-ideality depending on the system under study.
Bode plots also allow easy determination of frequency values, compared with Nyquist plots where frequency values are not plotted. Generally, the lower-leftmost points on a Nyquist plot correspond to the highest frequencies, and following the trace to the right moves from high to low frequency.
Prior to analysis of EIS data, it is critical to consider how potentiostat limits may affect the accuracy of results. All commercial potentiostats have calibrated internal hardware (and often external hardware, like the cell cable, as well), but there are always physical limitations on the range of conditions an instrument can accurately measure. The user should consult the potentiostat’s accuracy contour plot (ACP), which is a half-Bode plot illustrating the accuracy limits of measured over a wide range of frequencies, to evaluate data accuracy, particularly at high frequencies where the accuracy becomes substantially reduced.
In addition to the physical limitations of the potentiostat being used to collect EIS data, there are other factors to consider related to data accuracy and validity. While an AC signal (potential or current waveform) can be physically applied to almost any electrochemical system, it is not guaranteed that the resulting data may be accurately classified as “impedance”. There are three primary conditions that must be met during an EIS experiment for the results to be considered valid impedance data:
Stability - the electrochemical system must not change with respect to time, and it must return to its initial state without further oscillations once the applied signal is terminated
Causality - the resultant signal must only be caused by, and be solely a function of, the applied signal
Linearity - the resultant signal must exhibit a linear response to the applied signal; or, the resultant signal must obey the law of superposition with respect to the input signal; or, the measured impedance of the resultant signal must be independent of the magnitude of the applied signal amplitude
One strategy for maintaining stability is to allow sufficient settling time at the EIS setpoint before applied sinusoidal signals begin. Enough time must be allowed for the system to reach steady-state before collecting EIS data, otherwise there may be drift in the baseline that leads to unsteady and erroneous EIS data. Effect of drift is also more pronounced at lower frequencies due to the extended time required to complete slower sine waves (see Figure 4.1 for effect of drift on sinusoidal signals and Figure 4.2 for effect on Lissajous plots).
Figure 4.1 Drifting Baseline Effect on Sinusoidal EIS Waveform
Figure 4.2 Drifting Baseline Effect on EIS Lissajous Plot
Often, the EIS setpoint is chosen as the open circuit potential. In these cases, the system may not need much extra time to reach a stable condition since it may already be at open circuit during idle periods before running the EIS experiment. When applying sinusoidal signals on top of a potential or current setpoint, however, lack of sufficient settling time can lead to erroneous data. Additionally, since most commercial potentiostats automatically perform an OCP measurement during EIS experiments, switching between setpoint and OCP just prior to applied sinusoidal signals may interrupt the stability of an electrochemical system. When possible, a final settling period placed between the OCP step and applied sinusoidal signals should be added to prevent baseline drift.
Causality is a difficult condition to determine practically. It can be tricky to know definitively whether the electrochemical response during an EIS experiment is solely a result of the applied signal. One way to indirectly infer causality is to observe the system response, if any, once the applied sinusoidal signals have completed (see Figure 4.3 for examples of both potentially causal and non-causal cases). While it is not absolute evidence of a lack of causality, observing continued noise or oscillations after the signal has been discontinued may cast doubt on whether the electrochemical signal was influenced by other sources or general system instability.
Figure 4.3 Illustration of EIS Causality Condition
The condition of linearity is often satisfied by taking precautions with respect to the amplitude of an EIS experiment. For example, even though the average electrochemical system is entirely nonlinear, it may be considered roughly linear over a narrow potential window. It is therefore commonplace for the applied signal amplitude to be very small (around 5 - 20 mV for potentiostatic EIS) to ensure linearity. However, it is also important that the applied signal is large enough to properly induce a measurable response for the potentiostat to monitor. This creates a balancing act to find the optimal range of applied signal amplitudes: large enough to adequately excite the system but small enough to maintain linearity. Typically, trial-and-error is the most effective way to clarify this range and determine the desirable experimental parameters for a given electrochemical system.
As this optimal amplitude is elucidated during experimentation of EIS parameters, another check on linearity can be done by observing the ratio of input and output amplitudes. For example, if the input amplitude is doubled and the output amplitude does not also double, the linearity condition is not likely satisfied.
Finally, perhaps the most obvious indication of a lack of linearity can be quickly determined by the shape of the Lissajous plot. Figure 4.4 shows examples of Lissajous plots for linear systems, and examples of distorted Lissajous plots from nonlinear systems are shown in Figure 4.5. Some commercial software packages display Lissajous plots as they occur live during an EIS experiment. In these cases, the user can rapidly determine if the system is displaying nonlinear behavior and cancel the test if necessary.
Figure 4.4 Examples of Typical Lissajous Plots for Linear Systems
Figure 4.5 Examples of Typical Lissajous Plots for Nonlinear Systems
Figure 5.1 Representative Circuit Used in Kramers-Kronig Fitting
The Kramers-Kronig representative circuit (see Figure 5.1) contains an indeterminate number of Voigt elements (resistor-capacitor-in-parallel). The optimal number of these circuit elements varies depending on a few factors, including the number of data points and decades of frequency. It can therefore be tricky for the user to intuitively know the precise number of elements to select. Too few or too many Voigt elements can lead to either an under-constrained or over-constrained condition, which then may produce a false negative result. It is critical that the user know whether a poor Kramers-Kronig fit is the result of improper circuit design or impedance data that truly does not meet the conditions of validity. For this reason, AfterMath software automatically performs an algorithm on every Kramers-Kronig fit, without requiring user input, to determine the optimal number of circuit elements so that the result may be trusted.
An example EIS validity test using Kramers-Kronig analysis in AfterMath is shown below (see Figure 5.2) for EIS data collected using a dummy cell circuit. The upper pair of plots (Bode and Nyquist) show excellent correlation between experimental results (individual point markers) and the Kramers-Kronig fit (solid lines), with a error statistic of only 0.0004675. The lower pair of plots (Bode and Nyquist) were generated by cascading a baseline drift over the duration of all frequencies, and calculating the effect on resulting EIS data. Since the EIS experiment is conducted from high to low frequency, the greatest impact is observed at low frequencies because the baseline drift is on a similar time scale as the period of sine wave being applied. A deviation between experimental data and Kramers-Kronig fit, as well as a error statistic more than two orders of magnitude higher than the data with no drift, suggests the baseline drift data is not valid impedance.
Figure 5.2 Effect of Drift on EIS Data and Kramers-Kronig Analysis
NOTE: The quantity χ2 is a measure of the “quality of fit” between experimental and fitted data. There are several different mathematical definitions of χ2, which are typically based on a sum of squared residuals. While there is no definitive value of χ2 that is used as a baseline to separate “good” fits from “bad” fits, relative result quality may be determined by comparing χ2 values across similar datasets with circuit fit and/or Kramers-Kronig analyses