Equivalent circuit method

In a recent study, Harrison et al. [485] used steady-state j-E and Z(co)-E data to characterize the chlorine evolution reaction at Ru02/Ti02 electrodes using a simple redox reaction description of the chlorine evolution process with HOC1 and CR as reactant and product, respectively. The impedance potential data were analyzed by the equivalent circuit method parameter curves such as CiX-E and Rct-E. It has been suggested by the authors [485] that this type of parametric analysis of impedance data can be useful for comparison of the activity of various types of electrodes. 

The theoretical treatments referenced above all suffer from a major deficiency. The nonlinear term of interest in corrosion (the electron transfer process) is contained within a circuit comprising other linear (electrolyte resistance) and nonlinear (double-layer capacitance and diffusional impedance) terms. Since the voltage dropped across nonlinear circuit elements cannot be considered to linearly superimpose, we cannot use the equivalent circuit method to isolate the impedance terms of interest. Properly, one must solve for the system as a whole, including diffusional and double-layer terms, and identify the harmonic components associated with the faradic process of interest. 

However, in order to fit the data accurately in the instances in which biofilms spanned the gap, it was necessary to add an extra resistor, representing an additional conductive pathway, in parallel to the RC model that described the controls without biofilms (Fig. 7.6b dotted line). This equivalent circuit method has been previously employed to measure conductivity of nanostructured films [51, 52] and conducting polymers [53] and serves to separate electronic and ionic conductivity in mixed conductors [49]. The need to include this extra conductance in the circuit model demonstrates that the biofilm is electronically conductive. Additional experiments in the absence of acetate further confirmed that the measured conductance is an intrinsic property of biofilm and does not arise from the charge transfer occurring at the biofilm/anode interface. 

In addition to the equivalent circuit method, the impedance results can also be analyzed using mathematical models based on physicochemical theories. Guo and White developed a steady-state impedance model for the ORR at the PEM fuel cell cathode [15]. They assumed that the electrode consists of flooded ionomer-coated spherical agglomerates surrounded by gas pores. Stefan-Maxwell equations were used to describe the multiphase transport occurring in both the GDL and the catalyst layer. The model predicted a high-frequency loop due to the charge transfer process and a low-frequency loop due to the combined effect of the gas-phase transport resistance and the charge transfer resistance when the cathode is at high current densities. 

The agreement between characteristics predicted by the equivalent circuit methods and those obtained from measurements are illustrated in Figs. 22 and 23. 

A key element of PEM fuel cell is the CCL, which contains a double layer capacitance connected to resistivities due to proton and oxygen transport. To understand measured CCL spectra, many works employ the equivalent circuit method (ECM). This method is based on the construction of an equivalent transmission line, which gives an EIS spectrum close to the spectrum of the system of interest. The components of the resulting circuit are then attributed to CL physical parameters. For example, in a recent work, Nara et al. (2011) used the ECM to study the CCL degradation mechanisms in the PEFC. 

Impedance measurements were performed in different frequency ranges at open circuit potential for an alkyd coating with titanium dioxide as a mineral pigment in 3% sodium chloride. The most probable impedance equivalent circuit method was considered for data analysis. The interpretation of the impedance spectra permitted the determination of water permeation, the formation of blisters, swelling of the coating, and the loss of adhesion. 17 refs. 

The heat run will he equivalent load method. Efficiency at full, and V2 load and power factor at full, /4 and V2 load and breakdown torque will be determined by equivalent circuit calculation (IEEE 112,... 

Electrochemical impedance measurements of the physical adsorption of ssDNA and dsDNA yields useful information about the kinetics and mobihty of the adsorption process. Physical adsorption of DNA is a simple and inexpensive method of immobilization. The ability to detect differences between ssDNA and dsDNA by impedance could be applicable to DNA biosensor technology. EIS measurements were made of the electrical double layer of a hanging drop mercury electrode for both ssDNA and dsDNA [34]. The impedance profiles were modeled by the Debye equivalent circuit for the adsorption and desorption of both ssDNA and dsDNA. Desorption of denatured ssDNA demonstrated greater dielectric loss than desorption of dsDNA. The greater flexibility of the ssDNA compared to dsDNA was proposed to account for this difference. 

The reduction of Cd(II) ions on DME was also investigated in 1 M perchlorate, fluoride and chloride solutions using dc, ac admittance, and demodulation methods [27]. It was found that in the perchlorate supporting electrolyte, the reduction mechanism is also CEE, and that the rate constant of the chemical step is quite close to the value characteristic for fluoride solutions. The theories available at present could not be applied to the Cd(II) reduction in chloride solution because of the inapplicability of the Randles equivalent circuit. 

Capture and PSpice can be used to easily calculate the Norton and Thevenin equivalents of a circuit. The method we will use is the same as if we were going to find the equivalent circuits in the lab. We will make two measurements, the open circuit voltage and the short circuit current. The Thevenin resistance is then the open circuit voltage divided by the short circuit current. This will require us to create two circuits, one to find the open circuit voltage, and the second to find the short circuit current. In this example, we will find the Norton and Thevenin equivalent circuits for a DC circuit. This same procedure can be used to find the equivalent circuits of an AC circuit (a circuit with capacitors or inductors). However, instead of finding the open circuit voltage and short circuit current using the DC Nodal Analysis, we would need to use the AC analysis. 

In studies of these and other items, the impedance method is often invoked because of the diagnostic value of complex impedance or admittance plots, determined in an extremely wide frequency range (typically from 104 Hz down to 10 2 or 10 3 Hz). The data contained in these plots are analyzed by fitting them to equivalent circuits constructed of simple elements like resistances, capacitors, Warburg impedances or transmission line networks [101, 102]. Frequently, the complete equivalent circuit is a network made of sub-circuits, each with its own characteristic relaxation time or its own frequency spectrum. 

From an experimental point of view, it may be worth mentioning that the artificial membrane is reasonably stable, though only for a short time. It is therefore also a clear example of an object that should be studied by means of a quick method. The (small-amplitude) potential step method has been applied successfully [113], as has the FFT method described in Sect. 2.5.5. With the latter, the analysis in terms of an equivalent circuit like in Fig. 29 was demonstrated quite spectacularly [114]. 

A renewal of interest in the other rate-controlling processes started in those groups who were developing the impedance method [49, 53] and the a.c. polarographic method [12, 25], probably because it was found that, in many cases, Randles equivalent circuit did not hold and also because the appropriate mathematics are more tractable in the frequency domain. Still, it is recommended that the a.c. studies are combined with the diagnostic results which can be obtained from steady-state techniques and/or cyclic voltammetry. 

A few comments are in order on the probable validity of conclusions based on this equivalent circuit to real cells. Quite simply stated, real cells that are properly designed will have the same properties as dummy cells of the same values of Rs, Ru, and Cdl. Important design features of a cell are (1) equal resistance between all points on the surface of the working electrode and the auxiliary electrode (2) low-impedance reference electrode and (3) low stray capacitance between electrodes, between leads, and to shields. Spherical symmetry is a good, but somewhat inconvenient, method of meeting the first requirement a parallel arrangement also works with planar electrodes. At the very... 

A solution to this problem is the use of electrical equivalent circuits without constant-phase elements (note that a good numerical simulation of the experiments can be obtained only by inserting constant-phase elements) only pure capacities are used. This method, although not convincing, results in comparable capacities. 

With respect to the equivalent circuit in Figure 3.3, an evaluation of the known methods for hysteresis measurements will be given, in view of the effective parasitic capacitance and the influence of reflection. Well known methods to record the hysteresis loop of ferroelectric capacitors by measuring the current response are Sawyer Tower, Virtual Ground, and Shunt measurement as shown in Figure 3.4. 

Figure 36. Plots of different system functions for the Equivalent Circuit given by Eq. (62) and special cases of it.3,15 Reprinted from J. Maier, Evaluation of Electrochemical Methods in Solid State Research and Their Generalization for Defects with Variable Charges , Z. Phys. Chem. NF, (1984) 191-215. Copyright 1984 with permission from Oldenbourg Verlagsgruppe.223...

Recall that the faradaic resistance can be determined as a low-frequency cut-off at the complex-plane plot of impedance spectrum (compare the equivalent circuit in Fig. 10b). Such plots measured in the Fc(CN)63 /4 solutions of different concentrations are given in Fig. 23a [104] (similar results were obtained in [111]). The plots are (somewhat depressed) semicircles, whose radii decreased with increasing redox couple concentration. Figure 23b shows the line plotted by using the data in Fig. 23a, in accord with Eq. (6). We notice that all three methods yielded similar results. 

The complications and sources of error associated with the polarization resistance method are more readily explained and understood after introducing electrical equivalent circuit parameters to represent and simulate the corroding electrochemical interface (1,16-20). The impedance method is a straightforward approach for analyzing such a circuit. The electrochemical impedance method is conducted in the frequency domain. However, insight is provided into complications with time domain methods given the duality of frequency and time domain phenomena. The simplest form of such a model is shown in Fig. 3a. The three parameters (Rp, Rs, and C d,) that approximate a corroding electrochemical inter-... 

Equivalent circuit analysis is well suited for analysis of EIS measurements of conversion coatings and is the primary method for interpreting EIS spectra from conversion coated metal surfaces. A widely accepted generalized equivalent circuit model for the EIS response of pitted conversion coatings is shown in Fig. 22a (66,67). Several related models discussed below are also shown. In the gener-... 

This method of estimating Rc is useful when it can be applied, since the determination is not based on any presumed model of the corrosion damage process or any of the assumptions that come with assignment of an equivalent circuit model. This method is particularly helpful when there is more than one time constant in the spectrum, or the impedance spectrum is particularly complicated. Caution is warranted however. This method of estimation can be in serious error for samples with large capacitance-dominated low-frequency impedances. As a general rule, for this estimation method to be reasonably accurate, the impedance function must exhibit a clear DC limit, or a diffusional response that can be modeled by a constant phase element in equivalent circuit analysis (75). 

Other methods to simplify the circuit are Thevenin s and Norton s theorems. These two theorems can be used to replace the entire circuit by employing equivalent circuits. For example, Figure 2.34 shows a circuit separated into two parts. Circuit A is linear. Circuit B contains non-linear elements. The essence of Thevenin s and Norton s theorems is that no dependent source in circuit A can be controlled by a voltage or current associated with an element in circuit B, and vice versa. 

The rapid development of computer technology has yielded powerful tools that make it possible for modem EIS analysis software not only to optimize an equivalent circuit, but also to produce much more reliable system parameters. For most EIS data analysis software, a non-linear least squares fitting method, developed by Marquardt and Levenberg, is commonly used. The NLLS Levenberg-Marquardt algorithm has become the basic engine of several data analysis programs. 

Assuming that the Nyquist plot of the impedance does not display an ideal semicircle (e.g., it shows a depressed semicircle or a wide arc), it might be described using two or more discrete time constants or a continuous distribution of time constants. In the former case, the equivalent circuit may involve two or more parallel RCs in series. In the latter case, it may involve one or more parallel CPEs and Rs in series. As mentioned, one solution could be to use several CNLS fittings however, a more direct method would be the deconvolution of the imaginary part of the impedance data. 

Figure 5.8. a Schematic diagram of a two-probe conductivity cell [9], (Reproduced by permission of ECS—The Electrochemical Society, from Xie Z, Song C, Andreaus B, Navessin T, Shi Z, Zhang J, Eloldcroft S. Discrepancies in the measurement of ionic conductivity of PEMs using two- and four-probe AC impedance spectroscopy) b Equivalent circuit of the two-probe method. 

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