Ideal component equivalent circuits

Many applications of this strategy are based on extensions of the concepts of impedance developed earlier in this chapter (41-43). However, the excitation waveform is usually an impulse in potential (rather than a periodic perturbation), and a transient current is measured. One records both E t) and i t) as observed functions. Then both are subjected to transformations, and comparisons are made in the frequency domain between E s) and i s). Ratios of the form i s)IE s) are transient impedances, which can be interpreted in terms of equivalent circuits in exactly the fashion we have come to understand. The advantages of this approach are (a) that the analysis of data is often simpler in the frequency domain, (b) that the multiplex advantage applies, and (c) the waveform E(f) does not have to be ideal or even precisely predictable. The last point is especially useful in high-frequency regions, where potentiostat response is far from perfect. Laplace domain analyses have been carried out for frequency components above 10 MHz. 

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... 

A problem with Debye theory and the use of ideal components in the equivalent circuits has been that most dielectrics actually do not follow an exponential discharge curve, but a fractional power discharge curve. This law is called the Curie — von Schweidler s law (Schweidler, 1907). We shall revert to this phenomenon later in Section 9.2.12. 

Consider two materials, each homogeneous, as two dielectric slabs between the capacitor plates. The models with equivalent circuits are shown in Figure 3.9. Let us consider ideal components and therefore the resistors with frequency-independent values right from DC. 

Figure 7.18 Equivalent circuits for the electrode polarization impedance found with one AgCI/wet-gel electrode, (a) with frequency-dependent CPE components (m = 0.47) (b) more simplified version with ideal components, valid around 10 Hz (ECG).

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. 

The bulk of the electrolyte obeys Ohm s law, Eq. (2.2). Accordingly, the bulk electrolyte is modeled as an ideal resistor Rsoiu in series with the electrode components. This is to indicate that bulk electrolytic conductance is considered frequency independent, but dependent on the geometry and possible current constrictional effects. If the bulk electrolyte is replaced by tissue, a more complicated equivalent circuit must replace Rsoi, and we are confronted with the basic problem of division between tissue and electrode contributions. 

We will now discuss the simplest equivalent circuits mimicking the immittance found in tissue measurements. In this section, the R-C components are considered ideal that is, frequency independent and linear. Immittance values are examined with sine waves, relaxation times with step functions. A sine wave excitation results in a sine wave response. A square wave excitation results in a single exponential response with a simple R-C combination. 

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). 

An ideal resistor has only a resistance, but in practice resistors always also have a capacitance and an inductance component. The simplest equivalent circuit for a practical resistor is given in Fig. 2.24. 

Figure 2 shows the equivalent circuit with component overpotentials represented by resistors. The circuit consists of an ideal fuel cell operating at its reversible potential, but non-ideahties in the cell reduce the potential firom Er to E. The resistors symbolize the loss in potential for each component. Each resistor has a voltage drop, which establishes the overpotential for that resistor. Depiction of overpotentials as... 

The Thevenin equivalent circuit is the simplest combination, since it is the association of an ideal voltage source and a resistor connected in series. This is a much more realistic way of modeling a lead-acid battery. Indeed, the resistor illustrates the voltage drop due to the current passing through the components of the battery. In the case of LABs, this instantaneous voltage drop mainly results from the low electrical conductivity of electrolyte and is proportional to the current. But, such a simple combination does not account for the polarization of the electrodes happening later on, when the battery is operated. 

We may cite representative EIS studies of CPs. Firstly, for reference. Fig. 11-6 shows a Cole-Cole (real vs. imaginary impedance, also called Nyquist) plot of an idealized, planar CP film electrode, also showing the equivalent circuit in a situation where the charge-transfer component of the impedance is negligibly small, the... 

An ideal electrode-electrolyte interface with an electron-transfer process can be described using Randle equivalent circuit shown in Fig. 2.7. The Faradaic electron-transfer reaction is represented by a charge transfer resistance and the mass transfer of the electroactive species is described by Warburg element (W). The electrolyte resistance R is in series with the parallel combination of the double-layer capacitance Cdi and an impedance of a Faradaic reaction. However, in practical application, the impedance results for a solid electrode/electrolyte interface often reveal a frequency dispersion that cannot be described by simple Randle circuit and simple electronic components. The interaction of each component in an electrochemical system contributes to the complexity of final impedance spectroscopy results. The FIS results often consist of resistive, capacitive, and inductive components, and all of them can be influenced by analytes and their local environment, corresponding to solvent, electrolyte, electrode condition, and other possible electrochemically active species. It is important to characterize the electrode/electrolyte interface properties by FIS for their real-world applications in sensors and energy storage applications. 

One must always remember that the impedance EC analysis is an attempt to represent a complex phenomenon combining chemical, physical, electrical, and mechanical components in purely electrical terms. Impedance data are frequently fitted with an equivalent circuit made up of circuit elements related to the physical processes in the investigated system. In many cases, ideal circuit elements such as resistors and capacitors can be applied. Mostly, however. 

Equivalent circuit (EC) analysis is relatively simple for a circuit containing ideal elements R, C, and L. It may also be carried out for circuits containing distributed elements that can be described by a closed-form equation, such as CPE, semi-infinite, finite length, or spherical diffusion. Many "ideal" resistances and capacitances chosen to represent a real physicochemical system are really nonideal as any resistor has a capacitive component and vice versa. However, for the broad frequency range utilized by UBEIS it is usually adequate to incorporate "ideal" resistors, capacitors, and inductances [29, p. 87]. The type of electrical components in the model and their interconnections... 

Modeling an electrochemical interface by the equivalent circuit (EC) representation approach has been exceptionally popular in studies of electrodes modified with polymer membranes, although an analytical approach based on transport equations derived from irreversible thermodynamics was also attempted [6,7]. ECs are typically composed of numerous ideal electrical components, which attempt to represent the redox electrochemistry of the polymer itself, its highly developed morphology, the interpenetration of the electrolyte solution and the polymer matrix, and the extended electrochemical double layer established between the solution and the polymer with variable localized properties (degree of oxidation, porosity, conductivity, etc.). 

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