Equivalent circuit electrical components

To learn what a Nyquist plot is, and what such a plot looks like for simple electrical components, plus appreciate that the Nyquist plot for an actual electrochemical cell can be mimicked by constructing an equivalent circuit comprising arrangements of various components. 

At the heart of impedance analysis is the concept of an equivalent circuit. We assume that any cell (and its constituent phases, planes and layers) can be approximated to an array of electrical components. This array is termed the equivalent circuit , with a knowledge of its make-up being an extremely powetfitl simulation technique. Basically, we mentally dissect the cell or sample into resistors and capacitors, and then arrange them in such a way that the impedance behaviour in the Nyquist plot is reproduced exactly (see Section 10.2 below on electrochemical simulation). 

Equivalent circuit In impedance analyses, a collection of electrical components used to mimic the frequency behaviour of a cell or electrochemical system. 

It should also be mentioned that capacitors were then added in parallel with the resistors in equivalent circuit elements because the frequency-dependent experimental electrical impedance data had a component that was 90° out of phase with the resistor. 

Basically, the impedance behavior of a porous electrode cannot be described by using only one RC circuit, corresponding to a single time constant RC. In fact, a porous electrode can be described as a succession of series/parallel RC components, when starting from the outer interface in contact with the bulk electrolyte solution, toward the inner distribution of pore channels and pore surfaces [4], This series of RC components leads to different time constant RC that can be seen as the electrical response of the double layer charging in the depth of the electrode. Armed with this evidence, De Levie [27] proposed in 1963 a (simplified) schematic model of a porous electrode (Figure 1.24a) and its related equivalent circuit deduced from the model (Figure 1.24b). 

The second meaning of the word circuit is related to electrochemical impedance spectroscopy. A key point in this spectroscopy is the fact that any -> electrochemical cell can be represented by an equivalent electrical circuit that consists of electronic (resistances, capacitances, and inductances) and mathematical components. The equivalent circuit is a model that more or less correctly reflects the reality of the cell examined. At minimum, the equivalent circuit should contain a capacitor of - capacity Ca representing the -> double layer, the - impedance of the faradaic process Zf, and the uncompensated - resistance Ru (see -> IRU potential drop). The electronic components in the equivalent circuit can be arranged in series (series circuit) and parallel (parallel circuit). An equivalent circuit representing an electrochemical - half-cell or an -> electrode and an uncomplicated electrode process (-> Randles circuit) is shown below. Ic and If in the figure are the -> capacitive current and the -+ faradaic current, respectively. 

Electrochemical reactions consist of electron transfer at the electrode surface. These reactions mainly involve electrolyte resistance, adsorption of electroactive species, charge transfer at the electrode surface, and mass transfer from the bulk solution to the electrode surface. Each process can be considered as an electric component or a simple electric circuit. The whole reaction process can be represented by an electric circuit composed of resistance, capacitors, or constant phase elements combined in parallel or in series. The most popular electric circuit for a simple electrochemical reaction is the Randles-Ershler electric equivalent... 

EIS data analysis is commonly carried out by fitting it to an equivalent electric circuit model. An equivalent circuit model is a combination of resistances, capacitances, and/or inductances, as well as a few specialized electrochemical elements (such as Warburg diffusion elements and constant phase elements), which produces the same response as the electrochemical system does when the same excitation signal is imposed. Equivalent circuit models can be partially or completely empirical. In the model, each circuit component comes from a physical process in the electrochemical cell and has a characteristic impedance behaviour. The shape of the model s impedance spectrum is controlled by the style of electrical elements in the model and the interconnections between them (series or parallel combinations). The size of each feature in the spectrum is controlled by the circuit elements parameters. 

The electrical characteristics of the TSM resonator with a generalized surface perturbation can be described by the equivalent-circuit model of Figure 3.7b. Measurements can be made on a dry TSM resonator to determine C o,L, Ci, and R. Fixing these parameters and fitting the equivalent-circuit model to data measured on an immersed device determines R2 and L2. Equations 3.21 can then be used to determine the components of from L2 and R2. 

Alternatively, an equally powerful visualization of impedance data involves Bode analysis. In this case, the magnitude of the impedance and the phase shift are plotted separately as functions of the frequency of the perturbation. This approach was developed to analyze electric circuits in terms of critical resistive and capacitive elements. A similar approach is taken in impedance spectroscopy, and impedance responses of materials are interpreted in terms of equivalent electric circuits. The individual components of the equivalent circuit are further interpreted in terms of phemonenological responses such as ionic conductivity, dielectric behavior, relaxation times, mobility, and diffusion. 

It has to be mentioned that such equivalent circuits as circuits (Cl) or (C2) above, which can represent the kinetic behavior of electrode reactions in terms of the electrical response to a modulation or discontinuity of potential or current, do not necessarily uniquely represent this behavior that is other equivalent circuits with different arrangements and different values of the components can also represent the frequency-response behavior, especially for the cases of more complex multistep reactions, for example, as represented above in circuit (C2). In such cases, it is preferable to make a mathematical or numerical analysis of the frequency response, based on a supposed mechanism of the reaction and its kinetic equations. This was the basis of the important paper of Armstrong and Henderson (108) and later developments by Bai and Conway (113), and by McDonald (114) and MacDonald (115). In these cases, the real (Z ) and imaginary (Z") components of the overall impedance vector (Z) can be evaluated as a function of frequency and are often plotted against one another in a so-called complex-plane or Argand diagram (110). The procedures follow closely those developed earlier for the representation of dielectric relaxation and dielectric loss in dielectric materials and solutions [e.g., the Cole and Cole plots (116) ]. 

FIGURE 1.10. An equivalent circuit for the electrical components at the semiconductor/electrolyte interface in the absence of an oxide. represents the resistance of the electrolyte Ch is the capacity of the Helmholtz double layer and Rf is the charge transfer resistance 0, and Ru are the capacitance and resistance associated with the space charge layer in the semiconductor C, and are the capacitance and resistance of the surface states. 

The electrical characteristics of biopotential electrodes are generally nonlinear and are a function of the current density at their surface. Thus, having the devices represented by linear models requires that they be operated at low potentials and currents. Under these idealized conditions, electrodes can be represented by an equivalent circuit of the form shown in Figure 4.2. In this circuit, Rj and Q are components... 

The film grown on an electrode surface has a duplex structure with a thin, compact first layer that is directly on the electrode surface and a porous second layer contacting the electrolyte. An equivalent circuit can be used to represent the electrical properties of this film. The components of an equivalent circuit can be determined by impedance spectroscopy. Therefore, this method has become one of the key methods for the characterization of conducting polymers. 

Any electrochemical cell can be represented in terms of an equivalent electrical circuit that comprises a combination of resistances, capacitances or inductances as well as mathematical components. At least the circuit should contain the doublelayer capacity, the impedance of the faradaic or non-faradaic process and the high-frequency resistance. The equivalent circuit has the character of a model, which more or less precisely reflects the reality. The equivalent circuit should not involve too many elements because then the standard errors of the corresponding parameters become too large (see Sect. II.5.7), and the model considered has to be assessed as not determined, i.e. it is not valid. 

Table 7.1 Electrical component values of the equivalent circuit fitted to the measured impedance data during the anodic deposition process at various temperatures. Data are obtained from Ref [64],...

In electrical/electrochemical impedance, it is common practice to represent the system under study by an electrical equivalent circuit, in which a combination of resistors, capacitors, and inductors represents the physical components of the system. For the system as a whole, the impedance is the quotient of the applied potential (the... 

Two possible electrical equivalent circuits are shown in Figure 7.18. Such equivalent circuits are often given in the literamre in the most simplified way with ideal components as shown for 10 Hz in Figure 7.18(b). Important characteristics of an electrode are lost by such a simplification. For two-electrode tissue measurements, the immittance of the equivalent circuit of Figure 7.18 is a source of error physicaUy in series with the tissue, and must either be negligible, or be subtracted as impedance from the measured impedance. 

A possible but rough electrical DC/AC equivalent circuit for the electrode processes is shown in Figure 7.20. The electrodic part consists of three principal current paths in parallel. The elements are Cole-like as discussed in Section 9.2, and some of the used component symbols indicate that their values are nonideal, frequency-dependent. 

He discussed the three-component electric equivalent circuit with two resistors (one ideal, lumped, physically realizable electronic component one frequency-dependent not realizable) and a capacitor (frequency-dependent) in two different configurations. He discussed his model first as a descriptive model, but later discussed Philippson s explanatory interpretation (extra-/intracellular liquids and cell membranes). 

From the expression in Eq. 19, most forms of equivalent circuit models of piezoelectric elements may be found. The Van Dyke circuit [4] is the simplest, using discrete electrical components combined to approximate the piezoelectric... 

Fig. 5.13 shows the equivalent circuit for the electrochemical dynamics. The total current I consists of three components, the double layer charging current Ic, the diffusion current In, and the electric field-induced migration current Im- Here C denotes the double layer capacitance, Ri is the ohmic resistance of the oxidized PPy layer, and i 2 denotes the resistance of the reduced PPy layer, the PVDF (electrol3de) layer, and the contacts. Zd and Zm represent the diffusion dynamics and the migration dynamics, respectively. 

In an equivalent electrical circuit all components are supposed to belong to the same energy variety, as is the case in a dipole assembly in the Formal Graph classification of objects. The question raised when representing a dipole assembly by a Formal Graph is with regard to the number of units... 

The striking difference lies in the mounting, in series for the Maxwell model and in parallel for the equivalent electric circuit. However, this is purely a question of convention. In electricity, and therefore in an equivalent circuit, the current flows through the components and the potential difference develops between their ends. In rheology, it is the opposite, stress circulates through the components and the shear rate develops across them. 

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