Electrical Equivalent Models

Double-Layer Capacitor Electrical Equivalent Model.443... 

DOUBLE-LAYER CAPACITOR ELECTRICAL EQUIVALENT MODEL... 

III.l [see also Eq. (17) and Fig. 2], and that in the presence of a faradaic reaction [Section III. 2, Fig. 4(a)] are found experimentally on liquid electrodes (e.g., mercury, amalgams, and indium-gallium). On solid electrodes, deviations from the ideal behavior are often observed. On ideally polarizable solid electrodes, the electrically equivalent model usually cannot be represented (with the exception of monocrystalline electrodes in the absence of adsorption) as a smies connection of the solution resistance and double-layer capacitance. However, on solid electrodes a frequency dispersion is observed that is, the observed impedances cannot be represented by the connection of simple R-C-L elements. The impedance of such systems may be approximated by an infinite series of parallel R-C circuits, that is, a transmission line [see Section VI, Fig. 41(b), ladder circuit]. The impedances may often be represented by an equation without simple electrical representation, through distributed elements. The Warburg impedance is an example of a distributed element. 

Experiments carried out on monocrystalline Au(lll) and Au(lOO) electrodes in the absence of specific adsorption did not show any fre-quency dispersion. Dispersion was observed, however, in the presence of specific adsorption of halide ions. It was attributed to slow adsorption and diffusion of these ions and phase transitions (reconstructions). In their analysis these authors expressed the electrode impedance as = R, + (jco iJ- where is a complex electrode capacitance. In the case of a simple CPE circuit, this parameter is = T(Jcaif. However, an analysis of the ac impedance spectra in the presence of specific adsorption revealed that the complex plane capacitance plots (C t vs. Cjnt) show the formation of deformed semicircles. Consequently, Pajkossy et al. proposed the electrical equivalent model shown in Fig. 29, in which instead of the CPE there is a double-layer capacitance in parallel with a series connection of the adsorption resistance and capacitance, / ad and Cad, and the semi-infinite Warburg impedance coimected with the diffusion of the adsorbing species. A comparison of the measured and calculated capacitances (using the model in Fig. 29) for Au(lll) in 0.1 M HCIO4 in ths presence of 0.15 mM NaBr is shown in Fig. 30. 

For developing the MRR equation for EMM, the following resistances and impedances are to be considered (1) double-layer capacitance (2) Warburg impedance, (3) charge transfer resistance, and (4) electrolyte resistance [12]. Let us consider the double-layer electrical equivalent model circuit for EMM as shown in Fig. 3.6. It consists of an active electrolyte resistance along shorter path, Rshort. in series with the parallel combination of the double-layer capacitance, Cj, and an impedance of a faradaic reaction. The faradaic reaction consists of an active charge transfer resistance R and Warburg resistance Rw-... 

Non-sinusoidal excitation waveforms and several synchronous detectors operating at different frequencies can be used for simultaneous measurements of an entire impedance spectrum [4]. Spectroscopy, i.e., the measurement over a range of frequencies, is required for system identification when the electrical equivalent model contains several components (Fig. 12). 

Fig. 6.1 Electrical equivalent model of faradaic impedance described by Eqs. (6.28) and (6.26)...

Fig. 8.21 Application of DIA to impedance data simulated using electrical equivalent model containing two time constants (a) electrical equivalent model and impedance complex plane plot (b) log of obtained parameters / 2, C, and time constant r = T as functions of log of inverse frequency log Tp = — log (c) and (d) cumulative spectral line s intensity plots as functions of log of values of parameters (From Ref. [392], copyright (2004), with permission from Elsevier)...

The parameters appearing in electrical equivalent models depend on the surface area [473, 474, 479-481]. There are two principal parameters Aq, the geometric surface area of the sample, and Aj, the disbonded surface area under the coating. Initially, only pinhole pores perpendicular to the electrode surface are formed without delamination (Fig. 11.1b), Cc is as defined in Eq. (11.1), and... 

Usually, quite compact layers are obtained. The simplest electrical equivalent model represents the solution resistance in series with the capacitance of a SAM, CsAM (Fig- 12.2a). More detailed analysis reveals that the layers are rarely purely capacitive and their capacitance is in parallel with their resistance, Rsam. leading to a circuit R iCsam Rsam)- Moreover, a diffuse double layer exists at the SAM/solution interface [485,486]. In such a case, the electrical equivalent circuit contains a diffuse-layer capacitance, C, in parallel with the resistance, (Fig. 12.2b). 

However, the SAMs formed are rarely ideal and they contain small defects called pinholes, e.g., bare metal sites or other defects. In such cases, an additional branch must be added in parallel (Fig. 12.2c) consisting of the resistance, R, in series with the parallel connection of the pinhole resistance, Rp, and its capacitance, Cph [488, 489]. The surface coverage of the pinholes is usually very small and does not influence very much the electrode capacitance. The presence of such pinholes can be easily detected using cyclic voltammetry or EIS [483, 490, 491]. The simplified electrical equivalent model of the redox reaction in the presence of pinholes is displayed in Fig. 12.3 [490,492],... 

Fig. 14.7 Electrical equivalent model of mechanism involving one adsorbed species with diffusion (a) and a model with different position of Warburg element (b)...

The experimental data that were checked by the Kramers-Kronig transforms may be used in modeling. First, usually, fit to an electrical equivalent model is carried out. It is important to use a proper weighting procedure and start with the simplest model. Then additional parameters can be added and their importance verified by the appropriate F- and r-tests. The number of adjustable parameters must be kept to a minimum. Additionally, comparison of the experimental and model impedances on complex plane and Bode plots should be carried out. Furthermore, plots of the residuals indicate the correctness of the model used. Next, on the basis of this fit, a physicochemical model might be constructed. One should check how the obtained parameters depend on the potential, concentration, gas pressiue, hydrodynamic conditions, etc. If a strange or unusual dependence is obtained, one should check whether the assumed model is physically correct in the studied case. This is the most difficult part of modeling. 

Exercise 14.2 Find an electrical equivalent model describing data in the file 2.z. Try different weighting techniques. Calculate the sum of squares, S, and reduce the sum of squares, S. Carry out an f-test. 

More sophisticated methods of SoC estimation are based on electrical equivalent models of the cell. Several methods and models are reported in the literature. In general, these methods require more complicated algorithms and computation power than those presented previously. On the other hand, they are capable of SoC estimation from a transient response even when the battery is being used. 

A model for the ac response of real electrodes is the simple electric equivalent circuit consisting of a resistance R and capacitance Q conneeted in series (Fig. 12.12a). It follows from the rules for ac circuits that for this combination... 

The impedance spectroscopy of steel corrosion in concentrated HC1, with and without inhibitors, exhibit relatively straightforward electrochemical phenomenology and can be represented by simple equivalent circuits involving primarily passive electrical elements. Analysis of these circuits for steel corroding in HC1 per se reveals that the heterogeneity of the surface is established rapidly and can be simulated with a simple electrical circuit model. 

To parameterize a Langmuir model, you determine from experimental measurements not only the equilibrium constant K, but the surface s sorption capacity (or exchange capacity). The latter value is a measure of the number of sorbing sites and is commonly reported in moles or electrical equivalents, per gram of dry sediment. Multiplying the sorption capacity in moles by the mass of sediment in a system gives the mole number of sorbing sites, which is... 

The theoretical model that best describes regulation of transepithelial transport is derived from the Ussing-Zerahn equivalent electrical circuit model of ion transport theory [57] (Figure 15.1B). The model predicts that epithelia are organized as a layer(s) of confluent cells, where plasma membranes of neighboring cells come into close contact and functionally occlude the intercellular space. Accordingly, molecules can move across epithelia either through the cells... 

A generalised model of electrical equivalent circuit for painted surfaces has been considered in many of the recent publications. Googan ( 2) used it to study vinyl coatings free of defects and coatings containing defects. Electrocoatings were also evaluated. Muslanl et al (27) in their investigation of mild steel... 

Fig. 6.62. The Helmholtz-Perrin parallel-plate model, (a) A layer of ions on the OHP constitutes the entire excess charge in the solution. (b) The electrical equivalent of such a double layer is a parallel-plate condenser, (c) The corresponding variation of potential is a linear one. (Note The solvation sheaths of the ions and electrode are not shown in this diagram nor in subsequent ones.)...

The output of the model is then compared with the output of the real device and the individual elements are iteratively adjusted. When a good fit is obtained, the model is tested. It is a very important step, because the robustness of this procedure must be characterized by establishing the range of validity of the model, for the frequency and amplitude of the excitation signal, as well as for the range of values of the individual circuit elements. The wider the validity range, the more accurate is the representation of the real device by its model. The flowchart for building the equivalent electrical circuit model is shown in Fig. 4.11, and the equivalent electrical circuit of a QCM harmonic oscillator is shown in Fig. 4.12. Close to its resonance,... 

Rational optimization of performance should be the main goal in development of any chemical sensor. In order to do that, we must have some quantitative tools of determination of key performance parameters. As we have seen already, for electrochemical sensors those parameters are the charge-transfer resistance and the double-layer capacitance. Particularly the former plays a critical role. Here we outline two approaches the Tafel plots, which are simple, inexpensive, but with limited applicability, and the Electrochemical Impedance Spectroscopy (EIS), based on the equivalent electrical circuit model, which is more universal, more accurate, and has a greater didactic value. 

Electrical equivalent circuit representing the model of Ershler-Randles. 

To understand the electrical behaviour of the LAPS-based measurement, the LAPS set-up can be represented by an electrical equivalent circuit (see Fig. 5.2). Vbias represents the voltage source to apply the dc voltage to the LAPS structure. Re is a simple presentation of the reference electrode and the electrolyte resistance followed by a interface capacitance Cinterface (this complex capacitance can be further simulated by different proposed models as they are described, e.g., in Refs. [2,21,22]). In series to the interface capacitance, the insulator capacitance Cj will summarise the capacitances of all insulating layers of the LAPS device. The electrical current due to the photogeneration of electron-hole pairs can be modelled as current source Ip in parallel to the... 

FIGURE 11.9 Basic capacitor electrical equivalent circuit comprising a capacitance, a series inductance, a series resistance, and a parallel resistance. This simple model can fit a DLC behavior in first approximation for a given frequency. 

Since the unloaded QCM is an electromechanical transducer, it can be described by the Butterworth-Van Dyke (BVD) equivalent electrical circuit represented in Fig. 12.3 (box) which is formed by a series RLC circuit in parallel with a static capacitance C0. The electrical equivalence to the mechanical model (mass, elastic response and friction losses of the quartz crystal) are represented by the inductance L, the capacitance C and the resistance, R connected in series. The static capacitance in parallel with the series motional RLC arm represents the electrical capacitance of the parallel plate capacitor formed by both metal electrodes that sandwich the thin quartz crystal plus the stray capacitance due to the connectors. However, it is not related with the piezoelectric effect but it influences the QCM resonant frequency. 

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... 

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-... 

Figure 3 Electrical equivalent circuit model commonly used to represent an electrochemical interface undergoing corrosion. Rp is the polarization resistance, Cd] is the double layer capacitance, Rct is the charge transfer resistance in the absence of mass transport and reaction intermediates, RD is the diffusional resistance, and Rs is the solution resistance, (a) Rp = Rct when there are no mass transport limitations and electrochemical reactions involve no absorbed intermediates and nearly instantaneous charge transfer control prevails, (b) Rp = Rd + Rct in the case of mass transport limitations.

Figure 5 Nyquist, Bode magnitude and Bode phase angle plots for hypothetical corroding interfaces with Rp = 10, 100, or 1,000 ohms, Cd, = 100 tF, and Rs = 10 ohms using the electrical equivalent circuit model of Fig. 3a.

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