When applying electrochemical impedance spectroscopy to a system it is crucial to correctly parameterize the input signal. This requires a careful balance between ensuring the signal is strong enough to be discerned from system noise, and not too large that the system becomes unstable or non-linear. This module will discuss the challenges of signal optimization, how to effectively set your EIS signal parameters and validate your measurement to ensure that the requirements of linearity, causality, and stability are met.
2.0.0.1 Noise in the electrochemical system
When tuning the EIS signal the noise of the electrochemical system is a factor that must be accounted for. Common sources of noise in an electrochemical system include broadband noise, interference from power sources, and noise at nearby unexcited frequencies and harmonics [1]. These noise sources can heavily interfere with EIS measurement quality and fail the requirement of causality if the signal is not properly tuned.
One common method of mitigating noise in the measurement is to average results over multiple data points to counter the randomness of noise. Another method is to increase the signal amplitude to overcome the noise threshold. The amplitude is tuned based on the signal-to-noise ratio, a high signal-to-noise ratio is necessary and should be valid across low and high frequencies. Since an acceptable signal-to-ratio will differ between systems a source of feedback is required to properly tune the signal.
2.0.0.2 Non-linearity
While overcoming system noise is required for EIS, the system must also remain linear and stable. Electrochemical systems are naturally non-linear as explained by the Butler-volmer equations. This equation highlights the relationship between current density and electrode potential. As shown in the figure below higher currents result in greater exponential change in potential and vice versa. To keep our system stable when performing EIS it is best to operate in the pseudo-linear region using sufficiently small perturbations.
To ensure non-linearity the choice of running galvanostatic or potentiostatic EIS must also be considered. When dealing with low impedance systems potentiostatic control may result in large current flows inducing non-linearity following Ohm’s law. For this reason galvanostatic EIS is recommended for lower impedance systems such as batteries [3].
2.0.0.3 Signal Validation
Now that we understand how to combat non-linearity and ensure the signal-to-noise ratio is appropriate, a method of validating the measurement is required. Validation is imperative for effective signal optimization and ensuing the requirements of linearity, causality, and stability are met. One method of validating your data uses the Lissajous plot. The Lissajous plot of current and voltage will appear as a symmetrical ellipsis shape when the signal is optimal [2]. However, the most common method of validation uses the Kramers-Kronig relations. The Kramers Kronig relations are a set of mathematical equations that connect real and imaginary impedance. This allows for theoretical calculation of impedance to compare to your experimental impedance. The Kramers Kronig relations have been simplified to avoid the computation of complex integrals, this is denoted as the linear KK-test. The linear KK-test consists of an equivalent circuit model with N number of RC circuit elements that satisfies the KK relations [4]. The model is fit using linear equations and can be compared to the experimental data via Nyquist plot or residual error [5][6].
In practice we can see that the linear KK-test returns BLANK accuracy metrics, a plot of the Nyquist fit, and the residual error. Using this information you can decipher if the measurement is strong based on a visually good Nyquist fit, error lower than an outlined threshold, and the residual error appearing to be random. If the residual errors appear to be following a function this may be an indication that the model is fitting to noise. The figure below is an example of a valid EIS measurement exhibiting strong accuracy metrics, model fit, and residual error.
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