Cell equivalent circuit

The electric circuit of membrane electrode assemblies is a combination of anode and cathode catalyst layers plus the membrane. In general, the anode catalyst layer is considered an electric circuit, the cathode catalyst layer is considered another electric circuit similar to that of the anode but with different RC values, and the membrane is treated as a resistance. These three electric circuits are connected in series to construct a whole-cell equivalent circuit. A typical impedance spectrum is shown in Chapter 1 as Figure 1.16. Since the anode reaction is significantly faster than the cathode, the RC electric circuit of the anode can be disregarded. 

For a fuel cell equivalent circuit, such as the one shown in Figure 8-7, the impedances are ... 

FIGURE 8-7. AC impedance spectroscopy setup and fuel cell equivalent circuit. 

Plotted in a Nyquist diagram (Im Z vs Re Z), the measurements of complex impedance at various frequencies of this simple fuel cell equivalent circuit result in a semicircle (Figure 8-8). At very low frequencies, the resulting impedance is Z = Rr -(- Ract, whereas at very high frequencies, the resulting impedance is Z = Rr. Sometimes, a single measurement at high frequency is used to measure the cell resistance Rr. However, the entire frequency... 

FIGURE 8-8. Resulting complex impedance at various frequencies of the fuel cell equivalent circuit from Figure 8-7. 

FIGURE 8-9. Modified fuel cell equivalent circuit with Warburg impedance representing concentration polarization. 

For the fuel cell equivalent circuit shown in Figure 8-9, derive the equations for real and imaginary parts of the impedance. 

For the fuel cell equivalent circuit shown in Figure 8-7, given with Rr = 0.2mOhm, R ct = 1 mOhm, and C = 5 jJ.F, calculate real and imaginary parts of the impedance, ReZ and ImZ, and plot the Nyquist diagram. 

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))...

FIGURE 12.13 Equivalent circuit for the impedance of a galvanic cell. 

FIGURE 30.2 Electrical equivalent circuit of a cell membrane. 

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. 

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... 

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 ... 

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). 

Once the cell resistance, Ru, or the residual resistance ARu, is known, another possible strategy to handling ohmic drop problems consists of introducing ohmic drop and double-layer charging into the theoretical treatment of the cyclic voltammograms.19 The following relationships, obtained from the equivalent circuits in Figure 1.5, may be used for this purpose. 

FIGURE 2.45. Equivalent circuit for the cell and instrument. WE, RE, and CE, working, reference, and counter electrodes, respectively iph, photocurrent ij/, double-layer charging current Q, double-layer differential capacitance Rc, Ru, cell compensated (by the potentiostat) and uncompensated resistances, respectively Rs, sampling resistance RP, potentiostat resistance E, potential difference imposed by the potentiostat between the reference and working electrodes Vpu, photo-potential as measured across the sampling resistor. Adapted from Figure 1 of reference 51, with permission from Elsevier. 

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 24 The simplest equivalent circuit of an electrochemical cell...

Figure 3 The simplest equivalent circuit of an electrochemical cell. Cd = capacitor emulating the double layer Rn + R c = solution resistance...

The situation is not improved if an equivalent circuit more closely resembling the nature of the electrochemical cell is considered, see Figure 5. 

Figure 5 An equivalent circuit representing the phenomena occurring in an electrochemical cell better than that illustrated in Figure 3...

To learn what a Nyquist plot is, and what such a plot looks like for simple electrical components, plus appreciate that the Nyquist plot for an actual electrochemical cell can be mimicked by constructing an equivalent circuit comprising arrangements of various components. 

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). 

We will now look at a real electrochemical cell, and see how an equivalent circuit yields information about the electron-transfer processes occurring within it. 

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 68. Nyquist plots of a charged lithium ion cell, a lithiated graphite/graphite cell, and a delithiated cathode/ cathode symmetrical cell. The inset is an equivalent circuit used for the interpretation of the impedance spectra. (Reproduced with permission from ref 512 (Figure 3). Copyright 2003 Elsevier.)...

The next set of models examined in this section is impedance models. Impedance is often used to determine parameters and understand how the fuel cell is operating. By applying only a small perturbation during operation, the system can be studied in situ. There are many types of impedance models. They range from very simple analyses to taking a complete fuel-cell model and shifting it to the frequency domain. The very simple models use a simple equivalent circuit just to understand some general aspects (for examples, see refs 302—304). 

While a good equivalent-circuit representation of the transport processes in a fuel cell can lead to an increased understanding, it is not as good as taking a 1-D sandwich model and taking it into the frequency domain. These models typically analyze the cathode side of the fuel cell. °2.3i3 3i4 pj g j ost comprehensive is probably that of Springer et al. °2 The use of impedance models allows for the calculation of parameters, like gas-phase tortuosity, which cannot be determined easily by other means, and can also allow for the separation of diffusion and migra-... 

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