Impedance representation

As discussed in Chapter 21, the variances of stochastic errors are equal for real and imaginary parts of the impedance. Thxis, another advantage of presenting real and imaginary parts of the impedance as a function of frequency is that comparison between data and levels of stochastic noise can be easily represented. 

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Remember 16.1 The impedance representation emphasizes values at low frequency and is often used for electrochemical systems for which information is sought regarding mass transfer and reaction kinetics. 

Real The real impedance representation is similarly insensitive to fit quality. 

Imaginary The imaginary impedance representation is modestly sensitive to fit quality. 

The sensitivity of the impedance representation, presented in Figures 20.4(a) and (b) for real and imaginary parts respectively, is somewhat comparable to that seen for the Bode representation. The real part of the impedance is as insensitive to model quality as is the Bode modulus, and the imaginary impedance plots show a discrepancy between model and experiment at intermediate frequencies. 

Fig. 6.5 Schematic diagram of a conductivity cell (a) and its impedance representation as an equivalent circuit (b). Capacitances Ci and C2 are at the electrode I solution...

The impedance plot shown in Figure 2.1a vs. or the Nyquist plot) corresponds to an electrochemical cell (electrode/NaCl solution/electrode) and the equivalent circuit consists of a resistance (R) in parallel with a capacitor (C), which is represented as RQ, while Figure 2.1b shows the variation of the phase angle 4) = arc tan(Zi g/Z,e ) with frequency (4) vs./), but other typical impedance representations correspond to the variation of Z, and -Zj g with frequency (Bode plots), as indicated in Figure 2.1c and 2.1d. This latter representation allows the determination of the interval of frequency associated with a given relaxation process, between KT and 10 Hz, with a maximum frequency around 2 x 10 Hz, for the NaCl solution... 

All these examples show the interest of IS measurements, particularly when the membranes are in contact with electrolyte solutions, since qualitative information on membranes structure can be obtained. Other impedance representations (impedance modulus Z and/or tan 4) = (Zimg/Zreai) versus frequency) as well as different complex magnitudes such as dielectric constant and modulus or dielectric loss can also be determined from IS measurements and they are also commonly used in the literature [47, 48]. 

The usefulness of complex modulus representations in addition to impedance plots is related partially to the composition of the analyzed samples, especially in cases of multicomponent dispersions where the migration-type ionic or particle-based conduction in the bulk sample can be realized by two or more competing processes. As will be shown later, the graphical representation of the modulus often resolves well the resistive differences in the bulk conduction processes, while the impedance representation is preferred to resolve capacitance-related differences [6]. 

Impedance Representation of Bulk Material and Electrode Processes... 

In Chapters 1 through 3 the concept of impedance representation of a complete electrochemical system composed of electrodes, bulk media, and electrode-solution interface was introduced. In this chapter individual electrochemical segments of analyzed experimental or applied system will be separately discussed. Six typical components of the electrochemical process are considered [1, pp. 45 8] ... 

FIGURE 5-5 Warburg Impedance representation as A. Nyquist plot ... 

When the relative porosity effect is large, the penetration depth is small compared to the depth of the pores. De Levie [66] reported that for / > 3Xp the pore behaves as semi-infinite ("deep pore"), and for / < 0.2Xp the impedance behavior is similar to that of a flat electrode ("shallow pore"). For a simplified case where the concentration of electroactive species is high and the concentration gradient in pores is absent but there is a potential gradient, the details of the impedance representation for the porous electrode depend on the pore s... 

Historically, the bulk lubricant has been studied by dielectric spectroscopy and interpreted according to the Debye relaxation theory [3,4]. In impedance terms the system can also be represented according to a theory of colloidal dispersions or polycrystalline media composed of spheres of vastly different conductivities, where the contaminants become a more conductive phase suspended inside the less conductive additive/base oil matrix [6, 34]. Alternatively, when the contaminants are absent, the polar additives can be considered as a conductive discontinuous phase suspended inside insulating continuous base oil. Initially the description of the impedance representation of the fresh, uncontaminated oil will be provided, and then the effects of oxidative degradation and contaminants will be discussed. 

Impedance representation of the combination of a resistor and a constant phase element (a) in series and (b) in parallel. 

It is convenient to display the results of EIS in the complex-plane impedance representation. The X-axis on this plot is ReZ, which is the Ohmic resistance, and the y-axis is -ImZ, which, in the present case, is the capacitive impedance -j/o) C. 

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