Equivalent electric circuit, corresponding

Figure 8.7 (a) Equivalent electric circuit corresponding to an oxide ion conductor and the corresponding electrochemical Nyquist diagram, (b) Example of a polycrystalline sample. 

Fig. 5.6 Equivalent electrical circuit of electrochemical cell (top) and corresponding Nyquist plot containing Warburg impedance W (bottom)...

Consider the schematic representation of a coated electrode presented in Figure 9.7. Develop the corresponding equivalent electrical circuit. 

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. 

PDEIS spectra were recorded and analysed in terms of equivalent electric circuits with PDEIS spectrometer [2]. Fig. 1 shows PDEIS spectra and the corresponding equivalent circuits for Cd atomic layer electrodeposition on bulk... 

In the geometric representation of the composition of the membrane, the volume fraction of each component in each layer is described quantitatively by corresponding geometric parameters. In the electrochemical part of the model, each layer is treated as a set of two resistors and all sets, whose number equals the number of layers, are arranged in series forming an equivalent electrical circuit. Summation of the resistances of the layers expressed with appropriate equations leads to the final formula on specific conductivity of the membrane. Calculations based on the model require measurements of the conductivity of the membrane in contact with electrolyte solutions of different concentration. 

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. 

The equivalent electrical circuit, rearranged under the influence of an apphed physical field, is considered as a parallel resonant circuit coupled to another circuit such as an antenna output circuit Thus, in Figure 15.4c, Wj, Cd, La, and Ra correspond to the circuit elements each Wd represents active emitter-coupled oscillator and Cd, Ld, and Rd, represent passive capacitive, inductive, and resistive elements respectively. The subscript d is related to the particular droplet diameter, that is, the droplet under consideration. Now, again the initial electromagnetic oscillation is represented by... 

Correspondence between Components of an Equivalent Electric Circuit and Algebraic Models of (Integral) Admittances and Complex Admittances in the Linear Approximation... 

The electrical impedance of a pyroelectric detector is almost that of a pure capacitance. Hence an output signal only appears when the input radiation is changing. For maximum output the rate of change of the input radiation should be comparable with the electrical (RC) time constant of the element. Figure 3.10 is the equivalent electrical circuit of a pyroelectric detector Putley [3.11, 51]). Assume that the element receives radiation over an area A normal to the polar axis of the material and that this produces a modulated temperature rise 0 (3.4). The corresponding voltage developed across the amplifier input is... 

Fig. 6 presents the system of A1 corrosion in 0.5 M NaCl solution, its frequency impedance characteristic in the form of Nyquist plot and the equivalent electrical circuit. Individual parts of the electric circuit reflect the electrochemical and electrical characteristics of the corrosion systems. In this arrangement, the spectral characteristic of the impedance in the Nyquist plot has the shape of a semicircle, whose intersection with the real axis in the high-frequency range determines the electrolyte solution resistance Rs. Conversely, the intersection of the real axis in the low-frequency range corresponds to the sum of Rs + Rci/ where Ret indicates the charge transfer resistance of the boundary metal/electrolyte, and characterizes the rate of corrosion. On the other hand, Cdi component of the circuit represents capacity of the double layer at the interface metal/electrolyte. 

Further assessment of the characteristics of the impedance at the system boundaries for nickel and its alloy in 0.5M NaCl solution was obtained by approximation of experimental data using equivalent electrical circuits. The equivalent electrical circuits most suitable to represent the measured impedance characteristics of studied systems of nickel with different structures and its alloy in corrosive environment of 0.5M NaCl solution are shown in Figs. 19, 20 and 21. The corresponding resulting impedances are described in the expressions (4) -5- (6). For the analysis of the corrosion of microcrystalline structure nickel with the equivalent electrical circuit a simple layout shown in Fig. 19 was used. 

EIS data are commonly analyzed by fitting them to an equivalent electrical circuit model corresponding to a fuel cell component or components. Most of the circuit elements in the model are common electrical elements such as resistors, capacitors, and inductors. As an example, the electrolyte ohmic resistance can be represented with a resistor. Very few electrochemical cells... 

How does one know when the complete roster of reaction schemes that are consistent with the rate law has been obtained One method is based on an analogy between electrical circuits and reaction mechanisms.13 One constructs an electrical circuit analogous to the reaction scheme. Resistors correspond to transition states, junctions to intermediates, and terminals to reactants and products. The precepts are these (1) any other electrical circuit with the same conductance corresponds to a different but kinetically equivalent reaction scheme, and (2) these circuits correspond to all of the fundamentally different schemes. 

Let us introduce into the titrant one Pt indicator electrode vs. an SCE and maintain in the electric circuit a low constant current + /, as indicated by the broken horizontal line in Fig. 3.71. For this line we shall consider the successive points of its intersection with the voltammetric curves during titration and observe the following phenomena as expressed in the corresponding electrode potentials. Immediately from the beginning of the titration E remains high (nearly 1.44 V), but falls sharply just before the equivalence point (E = 1.107 V), and soon approaches a low E value (below 0.77 V) (see Fig. 3.72, cathodic curve +1). 

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). 

If the process is irreversible, q m spontaneous change, weieo will be less than the best available, which is AG. We will see later that the maximum electrical work available from an electrical cell will be obtained under reversible conditions, where the cell e.m.f. is opposed by an infinitesimally smaller potential. The electrons are made to work their passage around the external circuit to the maximum of their ability. Under these conditions, the electrical work depends on the equilibrium voltage, E, and on the number of electrons made to go through the circuit, corresponding to nF coulombs F is a unit of charge, the Faraday=96 485 coulombs/mol of electrons and / =numbcr of moles of electrons or equivalents . This is expressed as ... 

Fig. 2 Different circuits to be inserted for the load in Fig. 1. The conversion from the physical situation (right) to the equivalent circuits (left) entails a complication because networks are depicted such that the electrical Kirchhoff rules apply. Elements which are placed in series, physically, are represented as parallel circuit elements and vice versa (cf. Fig. 5 in Chap. 2 in this volume). For instance, the forces exerted by the spring and the dashpot in e are additive. In order to let the corresponding voltages in the electrical circuit also be additive, the circuit elements have to be placed in series. In the literature on polymer rheology, networks of springs and dashpots are drawn according to the physical situation (right-hand-side in this figure), which comes down to a different set of Kirchhoff rules...

These network equations differ from the Kirchhoff equations used in electrical circuit theory where the sum in the node Equation 10.5 is zero. O Keeffe (Struct. Bonding 1989, 71, 161-190) has shown that a correct mathematical correspondence requires that Kirchhoff s loop law be equivalenced with Equation 10.5 and the junction law with Equation 10.6. This requires replacing the nodes of the bond network with the loops of the equivalent Kirchhoff network and vice versa. For practical purposes it is simpler to stay with Equations 10.5 and 10.6... 

Electric circuit is needed for a constant Vs output signal to control Ip as a series of operations. Actual Vs output is controlled at a constant voltage of 450 mV. As a result. Vs output is equivalent to the electromotive force generated by oxygen concentration cell at the stoichiometric point as is mentioned in the previous section. In other words, gas detection chamber is always maintained at the stoichiometric air/fuel ratio even though exhaust gas is under any atmosphere. Ip current for the retention corresponds to the equation below. 

Computer Circuits. Shannon showed how logic could be used to design and simplify electric circuits. For example, consider a circuit with switches p and q that can be open or closed, corresponding to the Boolean binary elements, 0 and 1. A series circuit corresponds to a conjunction because both switches must be closed for electric current to flow. A circuit where electricity flows whenever at least one of the switches is closed is a parallel circuit this corresponds to a disjunction. Because the complement for a given switch is a switch in the opposite position, this corresponds to a negation table. When a circuit is represented in symbolic notation, its simplification may use the laws of logic, such as De Morgan s laws. The simplification may also use tables in the same way as the analysis of the equivalence of propositions. 

In an analysis of an electrode process, it is useful to obtain the impedance spectrum —the dependence of the impedance on the frequency in the complex plane, or the dependence of Z" on Z, and to analyse it by using suitable equivalent circuits for the given electrode system and electrode process. Figure 5.21 depicts four basic types of impedance spectra and the corresponding equivalent circuits for the capacity of the electrical double layer alone (A), for the capacity of the electrical double layer when the electrolytic cell has an ohmic resistance RB (B), for an electrode with a double-layer capacity CD and simultaneous electrode reaction with polarization resistance Rp(C) and for the same case as C where the ohmic resistance of the cell RB is also included (D). It is obvious from the diagram that the impedance for case A is... 

Fig. 108a-c. Proposed equivalent circuits for. a an empty and b a semiconductor-particle-coated BLM. Porous structure of the semiconductor particles allowed c the simplification of the equivalent circuit. Rm, RH, and Rsol are resistances due to the membrane, to the Helmholtz electrical double layer, and to the electrolyte solutions, while C and CH are the corresponding capacitances Rf and Cf are the resistance and capacitance due to the particulate semiconductor film R m and Cm are the resistance and capacitance of the parts of the BLM which remained unaltered by the incorporation of the semiconductor particles R and Csc are the space charge resistance and capacitance at the semiconductor particle-BLM interface and Rss and C are the resistance and capacitance due to surface-state on the semiconductor particles in the BLM [652]... 

Figure 2.7 shows a Nyquist plot corresponding to the electrical equivalent circuit of Fig. 2.6. The slope of the impedance can be explained by a circuit, consisting of different resistive and capacitive components37. The...

Here, Ry and ay are the active resistance and the corresponding electric conductance of the circuit fragment between points i and j. Note that in kinetic equation (1.31), thermodynamic rushes fr of the reactant groups behave as electric potentials in the points, while parameter Ey is equivalent to electric conductance ay. 

The equivalent circuits (Figure 3.5) can be used to describe the electrical response of the perturbed device. The lumped-element model. Figure 3.Sb, is most convenient to use. When the resonator has a surface perturbation, the motional impedance increases, as represented by the equivalent-circuit model of Figure 3.7. This model contains the elements C , Li, C, and Ri corresponding to the unperturbed resonator. In addition, the surface perturbation causes an increase in the motional impedance Z(n as described by the complex electrical element Ze in Figure 3.7a. This element is given by [12]... 

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