AC impedance theory

The simple electrode model in Figure 3-19 illustrates the ac impedance theory following Lamarre and Melville (1992). The capacitive Zc and resistive Zr (due to the electrolyte) impedances are in parallel with the inductance being ignored. The following definitions and limitations together with Equations 3.7 and 3.15 give the complex impedance Z of a liquid-gas probe ... 

Classic techniques for measuring electrical resistivity of miCToporous separators have been described by Falk and Salkind and Robinson and Walker. The resistivity of an electrolyte is more accurately determined by AC methods since DC can polarize the electrodes and cause electrolysis of the solution. Modem AC impedance-measuring systems allow rapid measurements of cell resistance over a wide range or frequencies from which resistance can be calculated free of capacitance effects. Compared to the DC techniques, the equipment required and the theory... 

The electrochemical impedance spectroscopy (EIS) method is very useful in characterizing an electrode corrosion behavior. The electrode characterization includes the determination of the polarization resistance (/J ), corrosion rate (Cfl), and electrochemical mechanism [1,4,6,19-28]. The usefulness of this method permits the analysis of the alternating current (AC) impedance data, which is based on modeling a corrosion process by an electrical circuit. Several review papers address the electrochemical impedance technique based on the AC circuit theory [22-24,29-30]. 

Different transient techniques have also been suggested for the measurement of corrosion rate. Pulse techniques can be used to eliminate from the polarization data the effects of uncompensated solution resistance and mass transport, or to minimize the effect of time-dependent phenomena. However, these techniques must be used with caution because the classical electrode kinetic theory can be used in the data evaluation only if /corrA/<0.9. The square-wave techniqueand ac impedance techniquehave also been used to measure the polarization resistance. The linear potential scan (potentiodynamic) technique has been used to obtain the polarization curve or the polarization resistance (small-amplitude cyclic voltammetry and exponential scan techniques were also proposed to determine the polarization curve. 

D. A. Harrington, B. E. Conway, AC Impedance cf Faradaic reactions involving electrosorbed intermediates—I. kinetic theory, Electrochim. Acta, 1997,32, pp. 1703-1712. 

Low-amplitude perturbation — A potential perturbation (rarely a current perturbation) whose magnitude is small enough to permit linearization of the exponential terms associated with the relevant theory [i]. See for example -> electrochemical impedance spectroscopy where low-amplitude voltage perturbations (usually sinusoidal) are the sole perturbations see also AC -> po-larography where, historically, a small amplitude voltage perturbation was imposed on a DC ramp [ii]. 

In order to understand electrochemical impedance spectroscopy (EIS), we first need to learn and understand the principles of electronics. In this chapter, we will introduce the basic electric circuit theories, including the behaviours of circuit elements in direct current (DC) and alternating current (AC) circuits, complex algebra, electrical impedance, as well as network analysis. These electric circuit theories lay a solid foundation for understanding and practising EIS measurements and data analysis. 

This chapter has provided basic electrical fundamentals, including concepts and definitions for circuit elements, and their relationships within electric circuits. Various basic AC electric circuits were also presented. Following upon primary circuit theories, the concept of electrochemical impedance spectroscopy and basic information about EIS was introduced. This chapter lays a foundation for readers to expand their study of EIS and its applications in PEM fuel cell research and development. 

The experimental situation is inconclusive and sometimes even the same experimental techniques used by different groups give contrary results. Especially for the compounds k-(ET)2Cu(NCS)2 and K-(ET)2Cu[N(CN)2]Br many different techniques have been employed to measure A(T). Evidence for non BCS-like behavior has been obtained by complex ac susceptibility [220], radio-frequency penetration depth [221], muon spin relaxation (//SR) [222], and microwave surface impedance measurements [223]. In contrast, results consistent with conventional BCS theory, sometimes revealing a tendency towards strong coupling, are reported for measurements of the //SR [224], microwave surface impedance [225, 226], and dc magnetization [227]. 

This operation determines the values of R and C that, in series, behave as the cell does at the measurement frequency. The impedance is measured as a function of the frequency of the ac source. The technique where the cell or electrode impedance is plotted V5. frequency is called electrochemical impedance spectroscopy (EIS). In modem practice, the impedance is usually measured with lock-in amplifiers or frequency-response analyzers, which are faster and more convenient than impedance bridges. Such approaches are introduced in Section 10.8. The job of theory is to interpret the equivalent resistance and capacitance values in terms of interfacial phenomena. The mean potential of the working electrode (the dc potential ) is simply the equilibrium potential determined by the ratio of oxidized and reduced forms of the couple. Measurements can be made at other potentials by preparing additional solutions with different concentration ratios. The faradaic impedance method, including EIS, is capable of high precision and is frequently used for the evaluation of heterogeneous charge-transfer parameters and for studies of double-layer structure. 

The porous electrode theory was developed by several authors for dc conditions [185-188], bnt the theory is usually applied in the ac regime [92,100,101,189-199], where mainly small signal frequency-resolved techniques are used, the best example of which are ac theory and impedance spectra representation, introdnced in the previons section. The porous theory was first described by de Levi [92], who assumed that the interfacial impedance is independent of the distance within the pores to obtain an analytical solution. Becanse the dc potential decreases as a fnnction of depth, this corresponds to the assnmption that the faradaic impedance is independent of potential or that the porons model may only be applied in the absence of dc cnrrent. In snch a context, the effect of the transport and reaction phenomena and the capacitance effects on the pores of nanostructured electrodes are equally important, i.e., the effects associated with the capacitance of the ionic donble layer at the electrode/electrolyte-solntion interface. For instance, with regard to energy storage devices, the desirable specifications for energy density and power density, etc., are related to capacitance effects. It is a known fact that energy density decreases as the power density increases. This is true for EDLC or supercapacitors as well as for secondary batteries and fnel cells, particnlarly due to the distributed nature of the pores... 

In conclusion, it appears that few metal-molten salt systems behave in the ideally polarizable sense generally associated with the mercury/aqueous solution interface at 298 K. Possible exceptions include some noble liquid metal/melt systems such as mercury/molten nitrates and lead/molten halides at low temperatures (<773 K), but only when the molten electrolyte is extensively purified. Otherwise, systems need to be analyzed as complex impedances, using ac or pulse techniques, to determine whether the minimum interfacial capacitance is affected by extensive factors, leading to parallel pseudocapacitances and Faradaic components. The range of potentials and measuring frequencies for which the interface approaches ideally polarizable behavior also needs to be established. It now seems clear that the multilayer ionic model of charge distribution at the metal/melt interface is more pertinent to molten media than the familiar double layer associated with aqueous solutions. However, the quantitative theories derived for the former model will have to be revised if it is confirmed that the interfacial capacitance is, indeed, independent of temperature in the ideally polarizable region. 

Electrochemistry in general and the EIS in particular are often used to analyze both bulk sample conduction mechanisms and interfacial processes, where electron transfer, mass transport, and adsorption are often present. EIS analysis has often treated the bulk and interfacial processes separately [4]. The analysis is achieved on the basis of selective responses of bulk and interfacial processes to sampling AC frequencies. The features appearing in the impedance AC frequency spectmm can be described according to the theory of impedance relaxations. Again, as in the case of any other spectroscopy method, the subject of the EIS analysis is the detection and interpretation of these spectrum features. 

As you can see, the LTSpice simulation plots the DC current dependency correctly, but the influence of the AC amplitude is distorted. The AC amplitude has a strong influence on the results, which is to be expected according to the theory [8, 9]. The measured curve with 1 A AC amplitude deviates more from the Simulation with 1 A and the curve with 10 mV amplitude is closer to the simulation, for the case without any superposed DC current. This means a better linearity is achieved by continuously adjusting the current amplitude so a maximal voltage response of 10 mV is met. The asymmetric behavior is clearly visible when comparing the curves with the superposed charging and discharging DC currents. In both cases, the measured impedance is smaller than the impedance without any superposed DC current. 

As is well known, all alternating current (AC) power systems are basically three-phase circuits. This fact makes voltage, current, and impedance a 3-D matrix form. A symmetrical component transformation (i.e., Fortescue and Clarke transformations) is well known to deal with three-phase voltages and currents. However, the transformation cannot diagonalize an n X n impedance/admittance matrix. In general, modal theory is necessary to deal with an untransposed transmission line. In this chapter, modal theory is explained. By adopting modal... 

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