Equivalent circuit of a cell

FIGURE 30.2 Electrical equivalent circuit of a cell membrane. 

In the three electrode arrangement, we usually, but not always, place the tip of the reference electrode very near the surface of the working electrode. This is done in order to minimize the electric field, which is generated by the IR drop, between the working and reference electrodes and hence get a true reading of potential. The equivalent circuit of a cell is shown in Fig. 44 in... 

Figure 55 shows the equivalent circuit of a cell used for impedance measurements, is not shown and Z is given as the resistance of electron transfer Q and that of diffusion W, The impedance of this circuit is given by the following equation... 

Fig. 7.45 Impedance spectrum and approximate equivalent circuit of a cell with blocking electrodes. (The galvanostatic response was given in Fig. 7.25.) The simplifications apply to the particular frequency ranges. The discussion is analogous to that for Fig. 7.25 Short/long times correspond to high/low frequencies. The approximations axe not valid in the region of the broken line and do not correspond to the accurate calculation (cf. also Fig. 7.44) [431].

FIGURE 1.5. a Three- electrode electrochemical cell, b General equivalent circuit, c equivalent circuit of the cell + potentiostat and current measurer (the symbols are defined in the text). 

Figure 6.7 Evolution of a potentiostat (continued), (a) Equivalent circuit of a three-electrode cell, (b) Addition of current-to-voltage converter.

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. 

The phenomena important in electrolytic conductance have been discussed96 and are represented by the electrical equivalent circuit of a conductance cell shown in Figure 6.24a. 

Figure 4.33. Equivalent circuit of a catalyst layer [8]. (Reproduced by permission of the authors and of ECS—The Electrochemical Society, from Lefebvre MC, Martin RB, Pickup PG. Characterization of ionic conductivity within proton exchange membrane fuel cell gas diffusion electrodes by impedance spectroscopy.)...

Figure 5.17. Equivalent circuit of a PEM fuel cell, with a and b H2/02 gas supply, and c symmetrical gas supply [19]. (With kind permission from Springer Science+Business Media Journal of Applied Electrochemistry, Characterization of membrane electrode assemblies in polymer electrolyte fuel cells using a.c. impedance spectroscopy, 32, 2002, 859-63, Wagner N, Figure 3, 2002 Springer.)...

At a given frequency, the equivalent circuit of the cell can be taken as in Figure 10.1.14, but we measure its impedance as a resistance value R and the capacitance value Cb in series [or equivalently as Zrc = im l/coC ]. One approach to obtaining... 

The function of the AC technique is the following the equivalent circuit of a conductance cell (Fig. 19) is quite complex and the conditions of experiments must be such that the solution resistance R is the principal component that determines the observed cell response. The individual parts of the equivalent circuit in Fig. 19 should be easy to understand on the basis of the description given in section 2.2. 

Fig. 19. Equivalent circuit of a conductometric cell. R is the resistance of interest, Cp ras is a stray capacitance.

Potentiostatic setups control the potential drop between electrode and electrolyte. This requires a probe to measure the potential of the electrolyte, the so-called reference electrode. The probe tip, the sensing point, is positioned somewhere in the electrolyte. This is illustrated in Fig. 1, showing the equivalent circuit of a common electrochemical cell. 

Indeed the degradation of cells by corrosion due to water vapour has been attributed to the growth of the insulating layer or the development of an additional layer Fig. 13.5 gives experimental points marked by crosses on current-voltage curves for p-type Schottky barrier solar cells. They are compared with the theory described in these lectures, using a series resistance of 3.2 (in an equivalent circuit) for a cell area of Icm, and adopting = 6 lO m eV One sees the severe depression in... 

FIGURE 2.29 Equivalent circuit for a cell where the cell impedance is kinetically controlled and is localized at the working electrode by using a large, unpolarized countrelectrode. —nonfaradic capacitance Cj, —faradic components of impedance ... 

Figure 24. Hypothetic equivalent circuit for a cell of one hybrid electrode where the insertion materials and electrochemical double layer capacitor (EDLC) materials are conceptually combined.

The equivalent circuit of a solar cell consists of current source from the incident photons in parallel with a forward-biased diode as shown in Figure 21.20. The circuit current produced by the cell can be expressed as... 

Figure Bl.28.8. Equivalent circuit for a tliree-electrode electrochemical cell. WE, CE and RE represent the working, counter and reference electrodes is the solution resistance, the uncompensated resistance, R the charge-transfer resistance, R the resistance of the reference electrode, the double-layer capacitance and the parasitic loss to tire ground.

Fig. 19.36 Basic circuit for a poiemiostat. (a) Basic circuit for a potentiostat and electrochemical cell, (b) Equivalent circuit, (c) Circuit of a basic potentiostat. A.E. is the auxiliary electrode, R.E. the reference electrode and W.E. the working electrode (6 and c are from Polen-tiostat and its Applications by J. A. von Fraunhofer and C. H. Banks, Butlerworths (1972))...

There is also a term representing the impedance of the second electrode in the cell and a term representing the geometrical capacitance of the whole cell. These latter two can, however, be minimised by proper choice of cell geometry, but we cannot eliminate the first two in any practical measurement, with the result that our final equivalent circuit for the cell looks like ... 

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( . 

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 5.2 Schematic representation of a cell with a power pack (in series) to force electron-transfer reactions to occur also indicated on the circuit are the anode (the positive electrode at which oxidation occurs) and the cathode (the negative electrode at which reduction occurs). Note that this figure would be equivalent to Figure 5.1 if the power pack was to drive the electrons in the opposite direction.

At the heart of impedance analysis is the concept of an equivalent circuit. We assume that any cell (and its constituent phases, planes and layers) can be approximated to an array of electrical components. This array is termed the equivalent circuit , with a knowledge of its make-up being an extremely powetfitl simulation technique. Basically, we mentally dissect the cell or sample into resistors and capacitors, and then arrange them in such a way that the impedance behaviour in the Nyquist plot is reproduced exactly (see Section 10.2 below on electrochemical simulation). 

Figure 8.12 (a) Nyquist plot obtained for the all-solid-state cell, ITOAVO3/PEO-H3PO4/ ITO(H) at 8°C, with the electrolyte being unplasticized. The WO3 layer was 0.3 pm in thickness (as gauged during vacuum evaporation with a thin-film monitor), while the electrolyte thickness was 0.24 mm (achieved by using 0.3 mm spacers of inert plastic placed between the two ITO electrodes), (b) Schematic representation of the equivalent circuit for this cell. 

Equivalent circuit In impedance analyses, a collection of electrical components used to mimic the frequency behaviour of a cell or electrochemical system. 

Figure 7.1 (A) Typical controlled-potential circuit and cell OA1, the control amplifier OA2, the voltage follower (Vr = Er) OA3, the current-to-voltage converter. (B) Equivalent circuit of cell Rc, solution resistance between auxiliary and working electrodes Ru, solution resistance between reference and working electrodes, Rs = Rc + Ru and Cdl, capacitance of interface between solution and working electrode. (C) Equivalent circuit with the addition of faradaic impedance Zf due to charge transfer. Potentials are relative to circuit common, and working electrode is effectively held at circuit common (Ew = 0) by OA3.

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... 

An equivalent circuit of the three-electrode cell discussed in Chapters 6 and 7 is illustrated in Figure 9.1. In this simple model, Rr is the resistance of the reference electrode (including the resistance of a reference electrode probe, i.e., salt bridge), Rc is the resistance between the reference probe tip and the auxiliary electrode (which is compensated for by the potentiostat), Ru is the uncompensated resistance between the reference probe and the working-electrode interphase (Rt is the total cell resistance between the auxiliary and working electrodes and is equal to the sum of Rc and Ru), Cdl is the double-layer capacitance of the working-electrode interface, and Zf is the faradaic impedance of the electrode reaction. 

Figure 9.1 Equivalent circuit of an electrochemical cell. A, Auxiliary electrode R, reference electrode W, working electrode Rc, compensated resistance R , uncompensated resistance Rr, reference electrode impedance Zf, faradaic impedance Cdl, doublelayer capacitance.

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. 

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