Impedance noise perturbation

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. 

White noise, that is, noise consisting of a continuous spectrum of frequencies (or a computer-generated pseudo-random white noise), may be used as a perturbation signal in practical impedance measurements. However, single-frequency components obtained by the FFT have relatively low amplitudes and a long data acquisition time is necessary to... 

Figure 9. FFT analysis of the sum of sine wave perturbation left side, no optimization right side, optimization of phases, (a) Perturbation voltage in the time domain, (b) Perturbation voltage in the frequency domain, (c) Complex plane plots of simulated impedance spectra with 5% noise added to the current response. Solid lines show response without noise.

This relation was proved by Nyquist (10) to be a consequence of basic thermodynamics laws and, except for quantum corrections, was never really challenged. Studies performed with glass microelectrodes (II) and heterogeneous ionic systems (12) showed that for zero ionic gradients and zero applied currents, the measured levels of noise were in agreement with noise levels calculated from the impedance according to eq 1. Hence, a study of electrical noise of a system under equilibrium conditions can be initiated for only two reasons. First, if there is some a priori information that the system is in equilibrium, then measurements of the system impedance or temperature can be performed without external perturbations (quantum effects are not considered here). Second, if impedance and temperature are measured independently by some other techniques, noise measurements can verify that the system under study is in an equilibrium state. 

The effect is normally one of degree. The measurement of a corrosion potential does not influence the surface condition. Electrochemical noise and impedance measurements carried out at the corrosion potential also have little effect as does a polarization resistance measurement if the perturbation is small, although rest potential drift may be a problem if potential control techniques are used. Techruques involving large potential differences will in general modify a surface significantly. 

Fig. 3.9 Complex plane plots of numerictilly simulated impedances, in Ci, for different perturbation waveforms with 1 % noise added (a) rectangular pulse, (b) exponentially decaying perturbation, (c) quasi-random noise, (d) sum of sine waves with constant amplitudes and zero phases (From Ref. [105] with permission of editorial board)...

White noise is a signal that contains a continuous spectrum of frequencies with flat amplitudes [99, 104, 105]. However, single-frequency components have quite low amplitudes, and the response to individual frequencies is also weak. Impedance calculated using a white nose perturbation signal with small 1 % noise added is displayed in Fig. 3.9c. It is obvious that very noisy results are generated, and such an excitation is not recommended for acquiring impedance spectra. 

Consider a rectangular acoustic space occupying a volume V = abd zis shown in Fig. 1. The interior surface of the enclosure is assumed to be covered with absorptive materials for which the impedance characteristics are specified. Noise is generated in the acoustic enclosure through the vibration of the flexible portions of the side-walls, the partitions or the sound sources located in the interior. The perturbation pressure p within the enclosure satisfies the linearized acoustic wave equation... 

As electrochemical measurements are of particular importance for corrosion studies, this chapter will only concentrate on them. However, since many textbooks and monographs discuss the earlier-mentioned simple analysis of current-voltage plots, this discussion will not be covered here. In recent years, more sophisticated techniques have been developed, and these partly overcome the restrictions of conventional electrochemical measurements as they either provide only a small potential perturbation on the corroding system (impedance spectroscopy), use no perturbation at all (electrochemical noise analysis), are able to measure current and potential fluctuations on inhomogeneous corroding surfaces (vibrating electrochemical electrode techniques), or... 

For the simplest case of two WEs with the same impedance (Zj = Z ) the noise impedance is equal to the modulus of the electrode impedance I Z(f) I. This result is valid independently of the origin of the noise signals (localized or uniform corrosion, bubble evolution due to the cathodic reaction) and the shape of the impedance plot, even if the noise levels of the two electrodes are different. In such case, noise measurements are equivalent to impedance measurements for which the external signal perturbation has been replaced by the internal noise generated by the corrosion processes [24]. 

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