Nonlinear higher-harmonic impedance analysis

In a linear EIS analysis a small AC perturbation amplitude is typically chosen so that the inherently nonlinear electrochemical system can be approximated by a linear system. In this approximation the measured impedance is independent of the amplitude of the perturbation = V + sin((nt), and there is no higher-harmonic response. Additionally, higher-order harmonics are normally filtered out during EIS measurements as a part of noise reduction. 

Another advantage of nonlinear impedance analysis is that measurement of several harmonics may facilitate extraction of kinetic parameters at a single DC offset potential V [8] not available from small-amplitude fundamental-frequency impedance measurement. NLEIS can be used to calculate all the harmonics of the current response to a sinusoidal potential perturbation -V+ V sin(( t) and derive the nonlinear impedance. Results from a simulation study can be compared with experimental NLEIS data, leading to more accurate quantification and modeling of the impedance data and better interpretation of the electrochemical kinetic processes [8,9,10,11,12,13]. 

When a pure sinusoidal voltage is applied to an electrochemical cell, the waveform of the resulting current is very often distorted due to this nonlinear current-potential relationship unless the excitation voltage amplitude is sufficiently small. The response signal can be described as  

From the separate lines of Eqs. 13-12 and 13-13 real and imaginary components of impedance at first (fundamental), second, third, and fourth harmonics can be calculated from the known voltage signal parameters and measured frequency-dependent current. The values for the Aaracteristic total capacitance C V ) and conductance G(f,j.) of the circuit can be computed. Comparison of the experimental and calculated frequency-dependent data for each harmonic serves as a diagnostic criterion that the system can indeed be represented by a simple parallel G C combination. Poor fit between the experimental and the calculated frequency-dependent impedance or current functions implies that a more complicated kinetic mechanism is responsible for the measured impedance characteristics. 

In conventional EIS experiments the electrode response to a perturbation signal corresponds to a measurement averaged across the whole electrode surface area. However, electrochemical systems show nonuniform current and potential distributions, resulting in CPE behavior. Such distributions can be studied bythe local EIS method, which employs in situ probing of local current density distribution in the vicinity of the working electrode surface. Local EIS (LEIS) relies on the fact that AC current density in the solution very near the working electrode is proportional to the local impedance properties of the electrode [22]. The AC current spreads in the solution as a funchon of the distance from the electrode surface, and as a consequence the LEIS results depend on the distance between the probe and the surface. That allows for spatially resolved LEIS measurements of the surface topography and kinetics at the electrochemical interface. 

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A number of other operational problems exist when using the FFT algorithm. The most important of these, as far as electrochemistry is concerned, is due to the inherently nonlinear nature of the system. When Eq. (56) is used to measure the impedance with an arbitrary time domain input function (i.e. not a single-frequency sinusoidal perturbation), then the Fourier analysis will incorrectly ascribe the harmonic responses due to system nonlinearity, to input signal components which may or may not be present at higher frequencies. As a consequence, the measured impedance spectrum may be seriously in error. 

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