Equivalent circuit distributed model

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). 

Fig. 13. (a) Sketch of the microelectrode configuration used to investigate the distribution of grain boundary properties, (b) Typical impedance spectrum calculated for a model sample (inset) consisting of 24 cubic grains and two microelectrodes on adjacent grains. An equivalent circuit consisting of two serial RC-elements (inset) can be used to fit the spectrum. 

Quantitatively, we proceed via the use of equivalent circuit models. The most general model is the distributed transmission line model of Fig. 

One can show [42] that, when the surface mechanical impedance is not large, the distributed model in the vicinity of resonance (where we make measurements) can be reduced to the simpler lumped-element model of Fig. 13.8(b). This modified Butterworth-van Dyke (BVD) electrical equivalent circuit comprises parallel static and motional arms. The static... 

Figure 3.5 Equivalent-circuit models to describe the near-resonant electrical characteristics of the resonator (a) distributed model (b) lumped-element model. (Reprinted with permission. See Refs. [7 14J. (a) 1994 American Institute of Physics and (b) 1993 American Chemical Society.)...

The standard model for analyzing QCM data is based on linear mechanics. All forces and stresses are assiuned to be proportional to displacement or speed. Such a linear behavior is a prerequisite for equivalent circuits to apply. Nonlinear behavior, generally speaking, is often found in contact mechanics because of the sharp peaks in the stress distribution. 

Plasma processing reactors normally operate with the wafer biased at radio frequencies, typically in the range 0.1 to 13.56 MHz. Even if the ions injected at the sheath edge were monoenergetic, an lED would result in an RF (time-dependent) sheath, even in the absence of collisions. The literature on RF sheaths is voluminous. Both fluid [170-175] and kinetic (e.g., Monte Carlo) [176-180] simulations have been reported. One of the most important results of such simulations is the lED. The ion angular distribution (IAD) [74, 75] and sheath impedance (for use in equivalent circuit models) [32] are also of importance. 

To discuss the results, the sensor is represented as a lossy capacitor, with both the capacitance C and the resistance R depending on frequency (Fig. 4 the frequency dependence of the equivalent-circuit elements is a consequence of the distributed nature of the processes in the sensor, which cannot be modeled appropriately by only two lumped elements with frequency-independent element values.). That simplifies the recognition of even small changes in the impedance, as changes at low frequencies become easily visible in the representation of the resistance R(f) and changes at higher frequencies become even more visible in the representation of the capacitance C(f). 

Other empirical distributed elements have been described, which can be expressed as a combination of a CPE and one or more ideal circuit elements. Cole and Cole found that frequency dispersion in dielectrics results in an arc in the complex e plane (an alternative form of presentation) with its center below the real axis (Fig. 10a) [16]. They suggested the equivalent circuit shown in Fig. 10(b), which includes a CPE and two capacitors. For ft) —> 0, the model yields capacitance Co and for ft) —> oo the model yields capacitance Coo- The model can be expressed with the following empirical formula for the complex dielectric constant... 

The modeling of an oscillatory movement of fluid is done by mounting in series an inductive dipole and a capacitive dipole that can be represented by an equivalent circuit in series, despite the fact that both are distributed in space. 

Although the CPE and fractal systems give the same impedance in the absence of redox reactions, a comparison of Eq. (8.9) for the CPE model with Eq. (8.17) for a fractal system in the presence of a redox reaction shows that they are structurally different. In fact, they produce different complex plane and Bode plots. This is clearly visible from Fig. (8.9), which can be compared with Fig. 8.4 for the CPE model. With a decrease in the value of , an asymmetry on the complex plane plot occurs that is also visible oti the phase angle Bode plots. This is related to the different topology of the equivalent circuits they are compared in Fig. 8.10. In the CPE model, only the impedance of the double-layer capacitance is taken to the power while in the fractal model the whole electrode impedance is taken to the power (p. The asynunetiy of the complex plane and Bode plots for fractal systems arises from the asymmetric distribution function of time constants in Eq. (8.4) according to the equation [298, 347]... 

Zoltowski [608, 609] proposed that one should first use measurement modeling to determine the number and nature of the circuit elements and parameters describing the studied system. One could use equivalent circuits containing simple R, C, and L parameters, or one could use more complex distributed elements such as the CPE and other analytically described elements such as, for example, mass transfer impedance and a porous model. 

Figure 11. Double-layer model for monomolecular oxide layer with chemisorbed oxygen ions, a, Potential distribution b, charge distribution c, equivalent circuit diagram with regard to adsorption equilibrium of oxygen ions. Symbols defined in text. ...

In general, the impedance of solid electrodes exhibits a more complicated behavior than predicted by the Randles model. Several factors are responsible for this. Firstly, the simple Randles model does not take into account the time constants of adsorption phenomena and the individual reaction steps of the overall charge transfer reaction (Section 5.1). In fact the kinetic impedance may include several time constants, and sometimes one even observes inductive behavior. Secondly, surface roughness or non-uniformly distributed reaction sites lead to a dispersion of the capacitive time constants. As a consequence, in a Nyquist plot the semicircle corresponding to a charge-transfer resistance in parallel to the double-layer capacitance becomes flattened. To account for this effect it has become current practice in corrosion science and engineering to replace the double layer capacitance in the equivalent circuit by a... 

Figure 16.7 Interpretation of EIS data in terms of equivalent circuit models and distribution of relaxation times. Dynamic processes are represented in the distribution by peaks in the case of ideal RC...

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