Impedance measurement Fourier analysis

Sequential measurement of impedance by Fourier analysis provides good accuracy for stationary systems. The sequence of ftequencies can be arbitrarily selected, and therefore frequency intervals of A///, considered to be the most economical use of frequencies, can be employed. Because the measurements at each frequency are independent of each other, frequencies foimd to be inconsistent with the Kramers-Kronig relations can be deleted. 

Recently, Darowicki [29, 30] has presented a new mode of electrochemical impedance measurements. This method employed a short time Fourier transformation to impedance evaluation. The digital harmonic analysis of cadmium-ion reduction on mercury electrode was presented [31]. A modern concept in nonstationary electrochemical impedance spectroscopy theory and experimental approach was described [32]. The new investigation method allows determination of the dependence of complex impedance versus potential [32] and time [33]. The reduction of cadmium on DM E was chosen to present the possibility of these techniques. Figure 2 illustrates the change of impedance for the Cd(II) reduction on the hanging drop mercury electrode obtained for the scan rate 10 mV s k... 

Equation (7.30) provides the basis for single-frequency Fourier analysis for impedance measurement. 

The magnitude of the stocheistic errors in impedance measurements depends on the selection of experimental parameters as detailed in Chapter 8. The simulation results described by Carson et a 00,25i,255 particular provide insight into differences between commonly used impedance instrumentation, including methods based on Fourier analysis and on phase-sensitive detection. ... 

Some general properties for stochastic errors have been established for impedance measurements through experimental observation and simulations. The results described here correspond to additive time-domain errors. The comparison between simulations and experimental results obtained via Fourier analysis supports the suggestion that the nature of experimental time-domain errors is likely to be additive rather than proportional ... 

Impedance measurements can be made in either the frequency domain with a frequency response analyzer (FRA) or in the time domain using Fourier transformation with a spectrum analyzer. Commercial instrumentation and software is available for these measurements and the analysis of the data. 

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. 

In the present state-of-the-art equipment, it is possible to measure and plot the electrochemical impedance automatically. Electronic circuitry is designed to generate the frequency sweep of a desired resolution over the range of interest. The generator can be programmed to sweep from a maximum to a minimum frequency in a number of required frequency steps. The commonly used modem equipment AC impedance measurement techniques can be subdivided into two main groups - single sine (lock-in amplification and frequency response analysis) and multiple sine techniques such as fast Fourier transforms (FFT). 

The measurement of the electrode impedance has also been ealled Faradaie impedanee method. Since measurements are possible by applying either an electrode potential modulated by an AC voltage of discrete frequeney (which is varied subsequently) or by applying a mix of frequencies (pink noise, white noise) followed by Fourier transform analysis, the former method is sometimes called AC impedance method. The optimization of this method for the use with ultramicroelectrodes has been described [91Barl]. (Data obtained with these methods are labelled IP.)... 

The traditional way is to measure the impedance curve, Z(co), point-after-point, i.e., by measuring the response to each individual sinusoidal perturbation with a frequency, to. Recently, nonconventional approaches to measure the impedance function, Z(a>), have been developed based on the simultaneous imposition of a set of various sinusoidal harmonics, or noise, or a small-amplitude potential step etc, with subsequent Fourier- and Laplace transform data analysis. The self-consistency of the measured spectra is tested with the use of the Kramers-Kronig transformations [iii, iv] whose violation testifies in favor of a non-steady state character of the studied system (e.g., in corrosion). An alternative development is in the area of impedance spectroscopy for nonstationary systems in which the properties of the system change with time. 

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