Equivalent circuit method impedance modeling

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. 

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 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-... 

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). 

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. 

The aim of network analysis is the investigation of the amplitude and phase response of a two- or four-port network. Impedance analysis determines the complex impedance or admittance of a device. This method is appropriate for quartz resonators in order to obtain more complete information than is conceivable by merely considering the shift of the resonance frequency. The method especially allows the determination of the equivalent circuit elements (BVD) presented in Fig. 8. Actually many commercial instriunents directly provide this information. Determination of the physical parameters, or their effective values, for accurate modeling of the sensor behavior based on Eq. 5 requires mathematical procedures which fit the calculated curves (e.g., with Eq. 2) to the experimentally measured values. It is recommended to include an external capacitance parallel to Co to accoimt for uncompensated para-... 

FIGURE 25-7 Circuit model of an eteclrochemical cell. is the cell resistance, C j is the double-layer capacitance, and 7, is the faradaic impedance, which may be represented by either of the equivalent circuits shown. is the cel) resistance, C is the so-called pseudocapacitance, and is the Warburg impedance. (Adapted from A. J. Bard and L. R. Faulkner. ElectrochemicSl Methods, 2nd ed, p. 376, New York Wiley, 2001. Reprinted by permission of John Wiley Sons, Inc.)... 

The techniques of impedance spectroscopy, widely used in dielectrics (Jonscher, 1983 MacDonald, 1987) have been applied to magnetic materials. In this method, impedance measurements as a function of frequency are modelled by means of an equivalent circuit and its elements are associated with the physical parameters of the material. The complex permeability, p, is determined from the complex impedance, Z, by ... 

Electric Double Layer and Fractal Structure of Surface Electrochemical impedance spectroscopy (EIS) in a sufficiently broad frequency range is a method well suited for the determination of equilibrium and kinetic parameters (faradaic or non-faradaic) at a given applied potential. The main difficulty in the analysis of impedance spectra of solid electrodes is the frequency dispersion of the impedance values, referred to the constant phase or fractal behavior and modeled in the equivalent circuit by the so-called constant phase element (CPE) [5,15,16, 22, 35, 36]. The frequency dependence is usually attributed to the geometrical nonuniformity and the roughness of PC surfaces having fractal nature with so-called selfsimilarity or self-affinity of the structure resulting in an unusual fractal dimension... 

Many researchers take the view that the transfer function for a given system should be derived from the equations governing the kinetics of the electrochemical reactions involved. This will be demonstrated for a simple charge-transfer reaction in Sect. 2.6.3. A second method for modeling electrochemical processes involves the use of networks of electrical circuit elements, so-called equivalent circuits, which can be selected on the basis of an intuitive understanding of the electrochemical system. It has been shown many times that for simple systems, equivalent circuits can be used to derive useful information from impedance spectra as long as they are based on the physical and chemical properties of the system and do not contain arbitrarily chosen circuit elements. 

Harrington et al. [229-235] proposed a more general method based on a general model for chemical reactions [236] and linear algebra, making it possible to predict the number and nature of parameters and the equivalent circuit for mechanisms involving diffusion and adsorption. Practical information about the stability and complexity of impedance plots and the relation between the reaction mechanism and equivalent circuit may be deduced. Eor example, the model predicts that inductive loops cannot be observed at equilibrium. However, a detailed presentation of this method is beyond the scope of this book. 

Electrochemical methods are well adapted for characterizing the corrosion behavior of coated metals in solution. Because of the high resistance of organic coatings, ac methods are generally more suited than dc polarization methods. In electrochemical impedance spectroscopy (EIC) one measures the response of the coated electrode to a small amplitude ac perturbation as a function of frequency (Chapter 5). The interpretation of the measured frequency response, in principle, requires a physical model. However, for coated metals useful information is more easily obtained by representing the metal-coating-electrolyte interface by an electrical circuit (equivalent circuit). 

Another weakness of the CN LS analysis method is the ambiguity of the equivalent circuits, meaning that identical impedance spectra may be obtained from different circuits [1, 8]. Therefore, it is questionable to propose an a priori model without any knowledge about the real number and physical origin of the polarization processes contributing to the impedance response of the cell. 

AC impedance spectroscopy measurements further confirmed that the measured electron transport is electronic in nature. The necessity to include an additional current path to the ionic resistance shown in the equivalent circuit model of the control setup showed that the biofilm possesses significant electronic conductivity. This method has been employed successMly for mixed conductors [49]. 

The EIS technique is based on a transient response of an equivalent circuit for an electrode/solution interface. The response can be analyzed by transfer functions due to an applied smaU-amphtude potential excitation at varying signals or sweep rates, hi turn, the potential excitation yields current response and vice verse, hi impedance methods, a sine-wave perturbation of small amphtude is employed on a corroding system being modeled as an equivalent circuit (Figure 3.8) for determining the corrosion mechanism and the polarization resistance. Thus, a complex transfer function takes the form... 

A widely used approach toward the interpretation of cell impedance is a method of equivalent circuit (Orazem and Tribollet, 2008). The great advantage of the method is simplicity. However, as discussed in the section Introductory Remarks, the equivalent circuit is not unique, that is, similar spectra could be generated by different circuits. In recent years, there has been a growing interest in the direct physical modeling of the cell impedance. 

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