Electrical circuits models

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. 

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... 

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. 

Figure 3. Electric circuit model for CFT device. Reproduced with permission from reference 18. Copyright 1982 Institute of Electrical and Electronics Engineers.

Numerical calculations using MATHEMATICA software were made based on a theoretical model which assumes flow distribution in circular pipes under laminar conditions as described by the Bernoulli equation and applies an electrical circuit model based on Ohm s law [164],... 

Fig. 6.13. Anodic current vs. potential curves for the process of BH4 ions oxidation on the bulk Cu electrode (curve 1 for comparison see curve 2 registered in the same conditions without BH4 ions), on the initial Ti02 electrodes (curve 7 for Ti02 with Nd = 10 19 cm 3 curve 8 for Ti02 with Nd 1018 cm 3) and on the Ti02 electrodes surface modified with different concentration of Cu (curve 3 - 1018 atoms/cm2, curves 4,5 - 1016 atoms/cm2, curve 6 - 1015 atoms/cm2). The values of Nd for Ti02 were 1018 cm 3 (curve 5) and 1019 cm 3 (curves 3,4,6). Curve 9 was obtained with the use of represented electrical circuit modeling the system Ti02 - Cu particles - electrolyte (D - solid-state Schottky diode R - electrical resistor WE, RE and CE - working, reference and counter electrodes, correspondingly). Electrolyte 0.1 M NaBH4 + 0.1 M NaOH. The potential sweep rate is 5 mV/s.

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. 

Figure 4.1. Nyquist plot showing the impedance spectra of an R/CPE electric circuit model [2], (Reproduced with permission from Research Solutions Resources LLC.)...

It is known that the catalyst layer is far from uniform, especially in the case of a gradient catalyst layer. Thus, profiling properties, such as conductivity, in the catalyst layer are important. Both an electronic conductor (carbon) and an ionic conductor (Nafion ) exist in the catalyst layer, which can be considered a conductive polymer. The conductive polymer electric circuit model has been applied to the catalyst layer, and an ionic conductivity profile was obtained [8], as shown in Figure 4.33. 

Based upon the results of detailed models of these two types, overall models for the performance of fuel cells may be formulated in terms of simple equivalent electric circuit models that parametrise the loss terms and allow calculations of overall efficiencies as a function of such parameters. 

The Kramers-Kronig relations have been applied to electrochemical systems by direct integration of the equations, by experimental observation of stability and linearity, by regression of specific electrical circuit models, and by regression of generalized measurement models. 

Perhaps the major problem with the use of electrical circuit models to determine consistency is that interpretation of a poor fit is ambiguous. A poor fit could... 

For each model system developed in this book, make it a habit to write out the systems description whenever you encounter that model. Tliis includes the kinetic theory of gases, thermodynamic systems, the Born model, the Debye-Htickel model, electric circuit models of electrochemical systems, etc. 

To model the microstructure and evaluate the thermoelectric properties, we used following simple equivalent electric circuit model shown in Figure 2. We considered the two phase composite as a cluster pararrel network circuit. Setting for each cluster the characteristic single phase physical property, and settle the material composition to the cluster number ratio, we can simulate the total thermopower of the system by Millman s theorem of d.c. circuit. 

Figure 2. Schematic diagram of equivalent electrical circuit model of phase A/ B (suffix a andb) thermoelectric composite material.

Electrochemical impedance spectroscopy (EIS) simplest electrical-circuit model... 

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. 

When using the Nyquist data-presentation format in the EIS method (Fig. 6.19) and assuming the simplest equivalent electrical-circuit model of Fig. 6.18, prove that the data points will fit on a semicircle, that the Z value at the -Z" = 0 high-frequency intersection corresponds to Rs, that the Z value at the -Z" = 0 low-frequency intersection corresponds to Rs + Rp, and that C is calculated from C = 1/toRp, where (0 is the angular frequency at the apex of the semicircle. 

Figure 2.12. Conceptual electrical circuit model for clay-electrolyte response when subjected to external potential.

The valves are straightforward and are often implemented as ideal diodes (for electrical circuit models) or as IF-THEN-ELSE statements (for algorithmic models) to keep all flows nonnegative. Defects in the valves can be added to simulate heart defects (e.g., leaky diodes for regurgitation). Other types of heart defects are just as easily simulated. For example, Blackstone et al. [1976] placed an impedance between the atrial chambers to simulate a septal opening. 

Electrochemical micromachining (EMM) is anodic dissolution of metal by the electrochemical reaction while very low voltage is applied between the anode and the cathode separated by a very narrow gap preferably less than 50 pm. Pulsed power supply is applied for better localization of current. In the very narrow gap, in addition to electrolyte resistance other resistances that are not so prominent in conventional ECM have much more influence in EMM. Hence, all these resistance factors within the narrow gap are discussed along with the electrical circuit model of EMM. 

FORMULATION OF MRR BASED ON EQUIVALENT ELECTRICAL CIRCUIT MODEL... 

COMPARISON OF BASIC MODEL AND ELECTRICAL CIRCUIT MODEL OF MRR FOR EMM... 

Theoretical MRR based on equivalent electrical circuit model (Eq. (3.32)) is much more accurate compared to basic MRR model based on Eqn (3.21), as observed from Fig. 3.8. The experimental MRR is much less than the theoretical MRR based on Eqn (3.21). MRR based on equivalent electrical circuit model is perfectly valid over the whole range of frequency. At 2-MHz pulsed frequency, the experimental value of MRR is less than the theoretical MRR based on equivalent electrical circuit model by 4.7 pg and is less than theoretical basic MRR model by 54 pg. At a moderately low frequency of 0.217 MHz, the experimental value of MRR is less than the theoretical MRR based on basic MRR model by 90 pg, which is exceptionally high. This is due to the fact that the charging time constant determines the resolution of machining in low-frequency EMM. 

The equivalent electrical circuit model is perfectly valid at medium range of frequency of pulsed power supply. At high frequencies, the experimental rate of metal removal is less than the theoretical... 

Chapter 3 deals with in depth discussion on basic mechanism of material removal for EMM. Moreover, equivalent electrical circuit, material removal rate (MRR) model, formulation of MRR based on equivalent electrical circuit model as well as comparison of basic model and electrical circuit model of... 

In the theoretical analysis of electrically driven pattern formation in nematics one deals only with the theoretical AC voltage Utheo, which drops over the nematic layer. Utheo differs, however, from the experimental voltage Uexp applied to the whole LC cell and recorded in experiments. Thus a quantitative comparison between experiments and theory is far from trivial as has been emphasized for instance by Krekhov et Typical liquid crystal cells consist of a nematic layer confined between ITO- or Sn02-coated glass plates covered with a thin film of an aligning polymer. As the polymer is a quite good insulator this sandwich has fairly complicated electric properties. In particular, at low frequencies the whole system has to be represented by a complex equivalent electric circuit model. 

The empirical current-duration relationship is in somewhat better accordance with the hyperbolic model than the exponential, but the exponential model is directly derived from the electric circuit model with a current source supplying an ideal resistor and capacitor in parallel. The empirical current-duration relationship is different for myelinated and naked axons. Also, it must be remembered that the excitation process is nonlinear and not easily modeled with ideal electronic components. 

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