Equivalent circuits Series

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). 

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... 

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. 

Variations of resistance with frequency can also be caused by electrode polarization. A conductance cell can be represented in a simplified way as resistance and capacitance in series, the latter being the double layer capacitance at the electrodes. Only if this capacitance is sufficiently large will the measured resistance be independent of frequency. To accomplish this, electrodes are often covered with platinum black 2>. This is generally unsuitable in nonaqueous solvent studies because of possible catalysis of chemical reactions and because of adsorption problems encountered with dilute solutions required for useful data. The equivalent circuit for a conductance cell is also complicated by impedances due to faradaic processes and the geometric capacity of the cell 2>3( . 

Fig. 10.1 Equivalent circuits used to represent the semiconductor-electrolyte interface, (a) A more complete approach taking into account the series resistance (Ry), the depletion layer (Csc, Rsc), an oxide surface film...

To evaluate the magnitude of capacitive currents in an electrochemical experiment, one can consider the equivalent circuit of an electrochemical cell. As illustrated in Figure 24, in a simple description this is composed by a capacitor of capacitance C, representing the electrode/solution double layer, placed in series with a resistance R, representing the solution resistance. 

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. 

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... 

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... 

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.

Figure 25.2 Constant-current source using a battery and series resistor, (a) Dummy (resistor) load (b) Norton s equivalent circuit for part a (c) electrolysis cell as the load.

The equivalent circuit of an electrochemical cell is shown in Fig. 5.6. It can be represented by a capacitive divider consisting of Cw and CAux connected in series. Figure out how the voltage V and charge Q are distributed across this divider when the resistances are (a) finite (b) infinite. 

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... 

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. 

If we switch-off the current after the steady state has been reached, the voltage relaxes to the initial zero-level. The electrical behavior can be taken into account by introducing in the simple equivalent circuit (eqc) (Eq. (60)) a capacitor Cl in series to Rioa or to R,.ml (if ions or electrons are blocked) In the language of system theory the equivalent circuit of the bulk represents a PDTi-element and reads3 15 e.g., for cells 3 and 4 Par (Cx. Par (Rm , Ser ( AJrai, Cl))). 

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). 

A physical insight into eqs. (2)-(5) is gained by considering the equivalent circuit shown in Figure 4, which displays the same frequency response defined in eqs. (2)-(5). The membrane capacitance per unit area Cjj, appears in series with the access impedances p and Pa/2, while the term CTfl (1-1.5p) provides for the conductance of the shunting extracellular fluid. Hence the time constant T which determines the frequency where the impedances l/(jjCmR and (p + Pa/2) are equal is given by eq. (5). 

A basic electric equivalent circuit to describe an EDLC is presented in Figures 1.22a and b, which shows the Nyquist (Figure 1.22b) plot of an ideal capacitor C, in series with a resistance... 

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). 

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. 

FIGURE 11.10 DLC equivalent circuit for capacitance and series resistance dependencies as a function of frequency, voltage, and temperature. 

Consider the situation under potentiostatic conditions. Here, the potential control takes care that the sum of the potential drop across the double layer, DL, and through the electrolyte up to the position of the RE (and possibly additional external series resistances) is constant, i.e. that U = DL + I Rn or / = (U - DL)/Rn. Rn is the sum of the uncompensated cell resistance and possible external resistances and I the total current through the cell. Hence, a perturbation of a state on the NDR branch towards larger values of Dl causes, on the one hand, a decrease of the faradaic current If, and, on the other hand, a decrease of the current through the electrolyte, I. The charge balance through the cell, which can be readily obtained from the general equivalent circuit of an electrochemical cell (Fig. 8), tells us whether the fluctuation is enhanced or decays ... 

Fig. 8. General equivalent circuit of an electrochemical cell. C double layer capacitance qSDL potential drop across the double layer Zp faradaic impedance Rij series resistance (comprising the uncompensated ohmic cell resistance and all external resistances). V is a potentiostatically fixed voltage drop. (It differs from the potentiostatically applied voltage by the constant potential drop across the RE see footnote 3).

Fig. 4.10. Equivalent circuits for solar cells. Left ideal solar cell consisting of a current source — jsc shunted by a diode. Right real solar cell with additional shunt resistor Rp and series resistor Rs...

For practical solar cells, the ideal equivalent circuit will be modified to include the series resistance from Ohmic loss in the two electrodes and the shunt resistance from leakage currents (Fig. 5.36a). The diode current for a realistic setup is then given by... 

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