Impedance equivalent circuit

Recall that in a purely capacitive circuit element, the phase angle between the current and applied potential is 0 = Jt/2 and Z2 = -jXc, where Xc is the capacitive reactance, in this case normalized with respect to the specimen area (ohm-m2). Xc is equal to 1/toC, where (0 is the angular frequency (radians/s, that is, (0 = 27tf, where f is frequency in cycles/s or Hertz) and C is the normalized capacitance (farad/m2). In a purely resistive circuit element (e g., Rp and Rs), the current is exactly in phase with the applied potential (0 = 0) thus, Z3 = Rp and Z3 = R, where, again, Rp and Rs are normalized with respect to specimen area (ohm-m2). Upon substitution of Eq 6.61 to 6.63 into Eq 6.60, the equivalent circuit impedance is determined to be ... 

With reference to the EIS method, prove that for the electrical-circuit model of Fig. 6.18, the equivalent circuit impedance is given by Eq 6.64. 

FIGURE 6-1 An equivalent circuit impedance model for a realistic experimental system... 

For ease of calculation, let us consider the impedance of the transformer as its leakage reactance, ignoring resistance and draw an equivalent circuit diagram as in Figures 24.25 (b) and (c) Assuming the length of the primary distribution line to be 15 km, the total line parameters will become... 

The equivalent circuit diagram in Fig. 23-4 shows a differential fundamental section and the closed-circuit impedance at the end of the close proximity region. 

Normally, the impedance plots are fitted to an often-complex equivalent circuit. Mathematically, this means searching for a global solution in R". However, problems arise if a complicated equivalent circuit is found which does not allow physical interpretation. Therefore, it is preferable to run a wide variety of experiments with different samples rather than trying to fit in detail the results of a single measurement in order to analyze the resulting impedance plots. 

FIGURE 12.13 Equivalent circuit for the impedance of a galvanic cell. 

The impedance with its components R and C is known as the Warburg diffusion impedance, and constant as the Warburg constant. In the equivalent circuits for electrochemical reactions, a Warburg impedance is represented by the symbol -W- as shown in the lower part of Fig. 12.15b. 

In the case of reactions that are not completely irreversible (or not completely reversible), we must account for both the kinetic factors (e.g., the polarization resistance R and the concentration changes (the Warburg impedance). The simplest equivalent circuit for this case is shown in Fig. 12.15c, while Fig. 12.17c shows the impedance diagram for this circuit (AJS = 10 = 1 Q the other parameters... 

FIG. 7 Simplified equivalent circuit for charge-transfer processes at externally biased ITIES. The parallel arrangement of double layer capacitance (Cdi), impedance of base electrolyte transfer (Zj,) and electron-transfer impedance (Zf) is coupled in series with the uncompensated resistance (R ) between the reference electrodes. (Reprinted from Ref. 74 with permission from Elsevier Science.)... 

Under potentiostatic conditions, the photocurrent dynamics is not only determined by faradaic elements, but also by double layer relaxation. A simplified equivalent circuit for the liquid-liquid junction under illumination at a constant DC potential is shown in Fig. 18. The difference between this case and the one shown in Fig. 7 arises from the type of perturbation introduced to the interface. For impedance measurements, a modulated potential is superimposed on the DC polarization, which induces periodic responses in connection with the ET reaction as well as transfer of the supporting electrolyte. In principle, periodic light intensity perturbations at constant potential do not affect the transfer behavior of the supporting electrolyte, therefore this element does not contribute to the frequency-dependent photocurrent. As further clarified later, the photoinduced ET... 

The impedance data have been usually interpreted in terms of the Randles-type equivalent circuit, which consists of the parallel combination of the capacitance Zq of the ITIES and the faradaic impedances of the charge transfer reactions, with the solution resistance in series [15], cf. Fig. 6. While this is a convenient model in many cases, its limitations have to be always considered. First, it is necessary to justify the validity of the basic model assumption that the charging and faradaic currents are additive. Second, the conditions have to be analyzed, under which the measured impedance of the electrochemical cell can represent the impedance of the ITIES. 

In particular, the coupling between the ion transfer and ion adsorption process has serious consequences for the evaluation of the differential capacity or the kinetic parameters from the impedance data [55]. This is the case, e.g., of the interface between two immiscible electrolyte solutions each containing a transferable ion, which adsorbs specifically on both sides of the interface. In general, the separation of the real and the imaginary terms in the complex impedance of such an ITIES is not straightforward, and the interpretation of the impedance in terms of the Randles-type equivalent circuit is not appropriate [54]. More transparent expressions are obtained when the effect of either the potential difference or the ion concentration on the specific ion adsorption is negli-... 

FIG. 6 Randles equivalent circuit for the ITIES Zq is the interfacial capacitance, Zy)v are the faradaic impedances of the charge transfer reactions, and is the solution resistance. 

Since the ion transfer is a rather fast process, the faradaic impedance Zj can be replaced by the Warburg impedance Zfy corresponding to the diffusion-controlled process. Provided that the Randles equivalent circuit represents the plausible model, the real Z and the imaginary Z" components of the complex impedance Z = Z —jZ " [/ = (—1) ] are given by [60]... 

Under this electrochemical configuration, it is commonly accepted that the system can be expressed by the Randles-type equivalent circuit (Fig. 6, inset) [23]. For reactions on the bare Au electrode, mathematical simsulations based on the equivalent circuit satisfactorily reproduced the experimental data. The parameters used for the simulation are as follows solution resistance, = 40 kS2 cm double-layer capacitance, C = 28 /xF cm equivalent resistance of Warburg element, W — R = 1.1 x 10 cm equivalent capacitance of Warburg element, IF—7 =l.lxl0 F cm (

charge-transfer resistance, R = 80 kf2 cm. Note that these equivalent parameters are normalized to the electrode geometrical area. On the other hand, results of the mathematical simulation were unsatisfactory due to the nonideal impedance behavior of the DNA adlayer. This should... 

FIG. 6 Complex impedance plots for the electrode reaction of [Fe(CN)6] on bare (open circle) and DNA-modilied (filled circle) An electrodes. An equivalent circuit for the electrode system is shown in the inset and solid lines represent theoretical responses from the circuit. Parameters used in simulation are cited in the text. Electrode potential, + 205 mV (vs. Ag/AgCl) AC amplitude, 25 mV (p-p). Other conditions are the same as those in Fig. 5. 

In an analysis of an electrode process, it is useful to obtain the impedance spectrum —the dependence of the impedance on the frequency in the complex plane, or the dependence of Z" on Z, and to analyse it by using suitable equivalent circuits for the given electrode system and electrode process. Figure 5.21 depicts four basic types of impedance spectra and the corresponding equivalent circuits for the capacity of the electrical double layer alone (A), for the capacity of the electrical double layer when the electrolytic cell has an ohmic resistance RB (B), for an electrode with a double-layer capacity CD and simultaneous electrode reaction with polarization resistance Rp(C) and for the same case as C where the ohmic resistance of the cell RB is also included (D). It is obvious from the diagram that the impedance for case A is... 

Fig. 5.21 Basic types of impedance spectra and of corresponding equivalent circuits (Eqs 5.5.16 to 5.5.19). (According to R. D. Armstrong et al)...

Fig. 5.22 Equivalent circuit of an electrode with diffusing el-ectroactive substances. W is the Warburg impedance (Eq. 5.5.21)...

Equivalent Circuit Analysis. IS measurements yield values of V and Z the real and imaginary components of the impedance, as a function of f, the AC frequency. The data are usually displayed as Nvauist plots (Z, vs. Z ) or Bode plots (impedance modulus,... 

The impedance spectroscopy of steel corrosion in concentrated HC1, with and without inhibitors, exhibit relatively straightforward electrochemical phenomenology and can be represented by simple equivalent circuits involving primarily passive electrical elements. Analysis of these circuits for steel corroding in HC1 per se reveals that the heterogeneity of the surface is established rapidly and can be simulated with a simple electrical circuit model. 

There is also a term representing the impedance of the second electrode in the cell and a term representing the geometrical capacitance of the whole cell. These latter two can, however, be minimised by proper choice of cell geometry, but we cannot eliminate the first two in any practical measurement, with the result that our final equivalent circuit for the cell looks like ... 

At very high frequencies, Cdl will have a low impedance (since Z 1 /ojCdl) and will effectively short out RCT and R0 + koL the impedance at high frequencies thus collapses to Rn. At lower frequencies the impedance of Cdl will be high but if the inductance still has a high impedance, the overall impedance will reduce to Rn + RCT, as marked in Figure 2.85(b). At lower frequencies still, a>L R0 and the equivalent circuit will be ... 

Very often, the electrode-solution interface can be represented by an equivalent circuit, as shown in Fig. 5.10, where Rs denotes the ohmic resistance of the electrolyte solution, Cdl, the double layer capacitance, Rct the charge (or electron) transfer resistance that exists if a redox probe is present in the electrolyte solution, and Zw the Warburg impedance arising from the diffusion of redox probe ions from the bulk electrolyte to the electrode interface. Note that both Rs and Zw represent bulk properties and are not expected to be affected by an immunocomplex structure on an electrode surface. On the other hand, Cdl and Rct depend on the dielectric and insulating properties of the electrode-electrolyte solution interface. For example, for an electrode surface immobilized with an immunocomplex, the double layer capacitance would consist of a constant capacitance of the bare electrode (Cbare) and a variable capacitance arising from the immunocomplex structure (Cimmun), expressed as in Eq. (4). 

Figure 6.7 Complex impedance of a polycrystalline ceramic sample (a) representation of the equivalent circuit of a component (b) the impedance spectrum of the equivalent circuit in (a) (c) the impedance spectrum of a typical ceramic sample. Each semicircular arc represents one component with an equivalent circuit as in (a) that at the highest frequency corresponds to the repose of the bulk, that at middle frequencies to the grain boundary response, and that at lowest frequencies to the electrodes.

Typically, the frequency lu of the modulation is varied over a considerable range, and an impedance spectrum Z(lS) recorded. Various electrode processes make different contributions to the total impedance. In many cases it is useful to draw an equivalent circuit consisting of... 

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