Equivalent circuits Parallel

According to this model, the SEI is made of ordered or disordered crystals that are thermodynamically stable with respect to lithium. The grain boundaries (parallel to the current lines) of these crystals make a significant contribution to the conduction of ions in the SEI [1, 2], It was suggested that the equivalent circuit for the SEI consists of three parallel RC circuits in series combination (Fig. 12). Later, Thevenin and Muller [29] suggested several modifications to the SEI model ... 

FIGURE 12.12 Equivalent circuits with resistance and capacitance in series (a) and in parallel b). 

FIG. 7 Simplified equivalent circuit for charge-transfer processes at externally biased ITIES. The parallel arrangement of double layer capacitance (Cdi), impedance of base electrolyte transfer (Zj,) and electron-transfer impedance (Zf) is coupled in series with the uncompensated resistance (R ) between the reference electrodes. (Reprinted from Ref. 74 with permission from Elsevier Science.)... 

The impedance data have been usually interpreted in terms of the Randles-type equivalent circuit, which consists of the parallel combination of the capacitance Zq of the ITIES and the faradaic impedances of the charge transfer reactions, with the solution resistance in series [15], cf. Fig. 6. While this is a convenient model in many cases, its limitations have to be always considered. First, it is necessary to justify the validity of the basic model assumption that the charging and faradaic currents are additive. Second, the conditions have to be analyzed, under which the measured impedance of the electrochemical cell can represent the impedance of the ITIES. 

Figure 5.10 is EIS of marmatite electrode in O.lmol/L KNO3 solution with different pH modifiers at open circuit potential. This EIS is very complicated. Simple equivalent circuit can be treated as the series of electrochemical reaction resistance R with the capacitance impedance Q == (nFr )/(icR ) resulting fi-om adsorbing action, and then parallel with the capacitance Ca of double electric... 

We shall now consider how these components of an equivalent circuit behave in combination. There are two types of circuit we need to think about here, i.e. with the components in series and with the components in parallel. 

Each of these layers behaves just like an RC element (that is, a capacitor and resistor in parallel) within the equivalent circuit (see Figure 8.13). The respective values o/R, and C, will be unique to each RC element since each layer has a distinct value of [H ]. In order to simplify the equivalent circuit, this infinite sum ofRC elements is given the symbol Zw or -W and is termed a Warburg impedance, or just a Warburg . The Warburg in Figure 8.12 extends from about 50 down to 15 Hz. 

It should also be mentioned that capacitors were then added in parallel with the resistors in equivalent circuit elements because the frequency-dependent experimental electrical impedance data had a component that was 90° out of phase with the resistor. 

In practice, poor charge mobility, energetic disorder, carrier trapping, and physical aberrations comphcate device characterization. The effects of these nonidealities are often modeled according to an equivalent circuit shown in Fig. 12. Incorporating all specific series resistive elements as R, and all specific parallel resistances as R, one obtains the expression... 

Most often, the electrochemical impedance spectroscopy (EIS) measurements are undertaken with a potentiostat, which maintains the electrode at a precisely constant bias potential. A sinusoidal perturbation of 10 mV in a frequency range from 10 to 10 Hz is superimposed on the electrode, and the response is acquired by an impedance analyzer. In the case of semiconductor/electrolyte interfaces, the equivalent circuit fitting the experimental data is modeled as one and sometimes two loops involving a capacitance imaginary term in parallel with a purely ohmic resistance R. 

CN ions. Anodic dissolution of silver electrode in cyanide solutions and also the behavior of Ag at potentials preceding dissolution have been studied applying electrode impedance measurements [381]. At potentials of anodic dissolution, the process was represented by the equivalent circuit with two parallel branches. 

To a first approximation, the BLM can be considered to behave like a parallel plate capacitor immersed in a conducting electrolyte solution. In reality, even such a thin insulator as the modified BLM (designated by and R, in Fig. 108) could block the specific adsorption of some species from solution and/or modify the electrochemical behavior of the system. Similarly, System C may turn out to be a semiconductor(l)-insulator-semiconductor(2) (SIS ) rather than a semiconductor(l)-semiconductor(2) (SS ) junction. The obtained data, however, did not allow for an unambiguous distinction between these two alternative junctions we have chosen the simpler of the two [652], The equivalent circuit describing the working (Ew), the reference (Eg), and the counter (Ec) electrodes the resistance (Rm) and the capacitance (C of the BLM the resistance (R ) and capacitance (Ch) of the Helmholtz electrical double layer surrounding the BLM as well as the resistance of the electrolyte solution (RSO ) is shown in Fig. 108a [652],... 

Fig. 6.33. (a) The equivalent circuit for an electrified interface is a capacitor and resistor connected in parallel, (b) In the equivalent circuit for an ideally polarizable interface, the resistance tends to infinity, and fora nonpolarizable interface, the resistance tends to zero. 

In drawing an appropriate equivalent circuit, it is clear that the resistance of the solution should be placed first in the intended diagram, but how should the capacitative impedance be coupled with that of the interfacial resistance One simple test decides this issue. We know that electrochemical interfaces pass both dc and ac. It was seen in Eq. (7.103) that for a series arrangement of a capacitor and a resistor, the net resistance is infinite for = 0, i.e., for dc. Our circuit must therefore have its capacitance and resistance in parallel for under these circumstances, for = 0, a direct current can indeed pass the impedance has become entirely resistive.51... 

A more complicated model situation is demanded if one thinks of the equivalent circuit for an electrode covered with an oxide film. One might think of A1 and the protective oxide film that grows upon it during anodic polarization. One has to allow for the resistance of the solution, as before. Then there is an equivalent circuit element to model the metal oxide/solution interface, a capacitance and interfacial resistance in parallel. The electrons that enter the oxide by passing across the interfacial region can be shown to go to certain surface states (Section 6.10.1.8) on the oxide surface, and they must be represented. Finally, on the way to the underlying metal, the electron... 

Figure 5.1 Schematic representation of an electrochemical cell (a) three electrodes (b) equivalent circuit for three-electrode cell (c) equivalent circuit for the working-electrode interphase (d) a solution impedance in series with two parallel surface impedances.

A few comments are in order on the probable validity of conclusions based on this equivalent circuit to real cells. Quite simply stated, real cells that are properly designed will have the same properties as dummy cells of the same values of Rs, Ru, and Cdl. Important design features of a cell are (1) equal resistance between all points on the surface of the working electrode and the auxiliary electrode (2) low-impedance reference electrode and (3) low stray capacitance between electrodes, between leads, and to shields. Spherical symmetry is a good, but somewhat inconvenient, method of meeting the first requirement a parallel arrangement also works with planar electrodes. At the very... 

For more complex current sources, it is necessary to employ Norton s theorem0 which states that any linear network of impedances and voltage sources can be substituted by an equivalent circuit containing a current source iN in parallel with an impedance 2 x, where iN is the current which flows when the output terminals of the network are short-circuited and 2EX is the network impedance with all source voltages put equal to zero and replaced by their internal impedances. 

The Relaxation Spectrum Analysis was carried out for a cell consisting of n-CdSe in a liquid junction configuration with NaOH/S=/S 1 1 1M as the electrolyte. Three parallel RC elements were identified for the equivalent circuit of this cell, and the fastest relaxing capacitive element obeys the Mott-Schottky relation. 

Figure 7. Equivalent circuit for interphase (Raei) resistance of semiconductor (Retec) electrolyte resistance, (Rfar) fara-daic resistance (Csc) space charge capacitance (CDl) double-layer capacitance and (z) parallel impedances associated with surface states, faradaic reactions, etc.

Commercial impedance analyzers offer equivalent circuit interpretation software that greatly simplifies the interpretation of results. In this Appendix we show two simple steps that were encountered in Chapters 3 and 4 and that illustrate the approach to the solution of equivalent electrical circuits. First is the conversion of parallel to series resistor/capacitor combination (Fig. D.l). This is a very useful procedure that can be used to simplify complex RC networks. Second is the step for separation of real and imaginary parts of the complex equations. 

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... 

The electrical behavior of a metal-solution interface is similar to that shown by the equivalent circuit in Fig. 1.6 formed by a capacitor and a resistor connected in parallel. If the resistance is small, the potential changes across the capacitor are... 

This circuit could represent, for instance, a flashlight. Notice that there is only one path that can be taken by the current, thus we call this a series circuit. Parallel circuits offer the current more than one path and will have junctions where wires intersect. Of course, circuits, and hence circuit diagrams, can get very complicated. As such, rules have been developed to help simplify or reduce circuits to equivalent circuits containing fewer components. 

Here, we concentrate on cell 1 and assume negligible electrode effects. If a constant current is switched on, both a faradaic as well as a displacement current flows (cf. Section I). Hence the actual current can be ionic/electronic or capacitive, the relative proportions depending on the electronic (creon) and ionic (crion) conductivities and the dielectric constant. Correspondingly, the elements are, as long as creon and crion are summed locally, in parallel (oo denotes the bulk and / , = ReonRtJ Re(m + 70) and the equivalent circuit is given by (cf. also Eq. (5))... 

Figure 6.24 (a) Electrical equivalent circuit for a conductance cell (b) AC bridge with the cell impedance balanced by a series R-C combination (c) AC bridge with the cell impedance balanced by a parallel R-C combination (see Table 6.7). 

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). 

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