Equivalence circuit of an electrochemical cell

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...

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. 

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).

Figure 3.1. a Equivalent circuit of an electrochemical cell b subdivision elements of Zf... 

Figure 3.1 shows a typical equivalent circuit of an electrochemical cell. Rel represents the electrolyte resistance between the working electrode surface and the point of reference electrode Cd is a pure capacitor of the capacity associated with the double layer of the electrode/electrolyte interface and Zf refers to the Faradaic impedance, which corresponds to the impedance of the charge transfer at the electrode/electrolyte interface. The connection of X, and Cd in Figure 3.1 is in parallel. The impedance X, can be subdivided in two equivalent ways, as seen in Figure 3.1 b ... 

As mentioned in the introduction, the electrical nature of a majority of electrochemical oscillators turns out to be decisive for the occurrence of dynamic instahilities. Hence any description of dynamic behavior has to take into consideration all elements of the electric circuit. A useful starting point for investigating the dynamic behavior of electrochemical systems is the equivalent circuit of an electrochemical cell as reproduced in Fig. 1. The parallel connection between the capacitor and the faradaic impedance accounts for the two current pathways through the electrode/electrolyte interface the faradaic and the capacitive routes. The ohmic resistor in series with this interface circuit comprises the electrolyte resistance between working and reference electrodes and possible additional ohmic resistors in the external circuit. The voltage drops across the interface and the series resistance are kept constant, which is generally achieved by means of a potentiostat. 

First, we consider the properties of a microelectrode in contact with a solution of pure electrolyte in the absence of a dissolved or immobilized redox active analyte. The objective is to understand the fundamental behavior of microelectrodes in the absence of an electroactive analyte and to discuss strategies for optimizing the electrode s temporal response. The existence of the double-layer capacitance (see Chapter 1) at the working electrode complicates electrochemical measurements at short timescales. Figure 6.1.1.1 is an equivalent circuit of an electrochemical cell where Zp is the faradaic impedance corresponding to the... 

This chapter is concerned with measurements of kinetic parameters of heterogeneous electron transfer (ET) processes (i.e., standard heterogeneous rate constant k° and transfer coefficient a) and homogeneous rate constants of coupled chemical reactions. A typical electrochemical process comprises at least three consecutive steps diffusion of the reactant to the electrode surface, heterogeneous ET, and diffusion of the product into the bulk solution. The overall kinetics of such a multi-step process is determined by its slow step whose rate can be measured experimentally. The principles of such measurements can be seen from the simplified equivalence circuit of an electrochemical cell (Figure 15.1). 

A brief discussion of cell impedance was given in Section G, Chapter 3 and the equivalent circuit of an electrochemical cell was given in Fig. 18. The following section will be concerned with the way by which impedance... 

Fig. 11.4. Equivalent electrical circuit of an electrochemical cell for a simple electrode process. R is the solution resistance, of the contacts and electrode materials, Zf the impedance of the electrode process, and Cd the double layer...

Fig.II.1.11 (a)RC representation of the simplest equivalent circuit for an electrochemical cell, (b, c) Electrode configuration with dashed lines indicating the flow of current accompanied by potential gradients through the solution phase...

Fig. 11.10 - Equivalent circuit for an electrochemical cell. is the resistance of the reference electrode, and Qef represents parasitic loss to ground in the leads. Rs and are the solution and uncompensated resistances respectively, Cji the double layer capacitance of the working electrode, and / ct the charge transfer resistance.

Almost all users of pH meters are familiar with statib charging phenomena. These can be explained with a modified form of the simplified equivalent circuit of our electrochemical cell (Fig. 39). Here an electrode cable capacitance C is introduced in parallel... 

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.

The use of passive components (resistors and capacitors) to construct an equivalent circuit for the electrochemical cell is carried one step further in Figure... 

Consideration of the equivalent circuit diagram of an electrochemical cell, such as that given in Figure 5.1, reveals the major limitation on the rate at which the potential of an electrode can be varied, namely, the time constant of the electrochemical cell, RuCd. When a potential sweep is applied across the cell, the nonfaradaic charging current that flows is described by [24]... 

The second meaning of the word circuit is related to electrochemical impedance spectroscopy. A key point in this spectroscopy is the fact that any -> electrochemical cell can be represented by an equivalent electrical circuit that consists of electronic (resistances, capacitances, and inductances) and mathematical components. The equivalent circuit is a model that more or less correctly reflects the reality of the cell examined. At minimum, the equivalent circuit should contain a capacitor of - capacity Ca representing the -> double layer, the - impedance of the faradaic process Zf, and the uncompensated - resistance Ru (see -> IRU potential drop). The electronic components in the equivalent circuit can be arranged in series (series circuit) and parallel (parallel circuit). An equivalent circuit representing an electrochemical - half-cell or an -> electrode and an uncomplicated electrode process (-> Randles circuit) is shown below. Ic and If in the figure are the -> capacitive current and the -+ faradaic current, respectively. 

One of the most broadly applied equivalent circuits is shown in Figure 4. It consists of a parallel RpC network in with a resistor R. This particular network is a simple representation of an electrochemical cell. R represents the... 

Usually the impedance is measured as a ftmction of the frequency, and its variation is characteristic of the electrical circuit (where the circuit consists of passive and active circuit elements). An electrochemical cell can be described by an equivalent circuit. Under appropriate conditions, i.e., at well-selected cell geometry, working and auxihary electrodes, etc., the impedance response will be related to the properties of the working electrode and the solution (ohmic) resistance. 

The conventional electrical model of an electrochemical cell that represents the electrode-electrolyte interface (EEI) includes the association of resistances with capacitance as shown in Fig. 1. The parallel elements are related to the total current through the working electrode that is the sum of distinct contributions from the faradaic process and double-layer charging. The double layer capacitance resembles a pure capacitance, represented in the equivalent circuit by the element C and the faradaic process represented by a resistance, R2. The parameters E and Ri represent the equilibrium potential and the electrolyte resistance, respectively. 

Fig. 31 Equivalent circuit for the connection of a potentiostat to a working and reference electrode of an electrochemical cell, with double-layer capacities C . Cp. and faradaic impedances Z , Z of a working and reference electrode, ohmic resistance Re (in most cases electrolyte resistance between the working electrode and end of the Luggin capillary of the reference electrode, and possible external resistance in connection with the working electrode), U applied voltage, i = k + if-...

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

Equivalent electrical circuit for the electrochemical cell described in section 9.2.2 with palladium electrodes of 314mm2, electrically in contact with each other through an electrolyte solution over a distance of 112mm at T=298.0K. 

The simplest and most common model of an electrochemical interface is a Randles circuit. The equivalent circuit and Nyquist and Bode plots for a Randles cell are all shown in Figure 2.39. The circuit includes an electrolyte resistance (sometimes solution resistance), a double-layer capacitance, and a charge-transfer resistance. As seen in Figure 2.39a, Rct is the charge-transfer resistance of the electrode process, Cdl is the capacitance of the double layer, and Rd is the resistance of the electrolyte. The double-layer capacitance is in parallel with the charge-transfer resistance. 

Any electrochemical interface (or cell) can be described in terms of an electric circuit, which is a combination of resistances, capacitances, and complex impedances (and inductances, in the case of very high frequencies). If such an electric circuit produces the same response as the electrochemical interface (or cell) does when the same excitation signal is imposed, it is called the equivalent electric circuit of the electrochemical interface (or cell). The equivalent circuit should be as simple as possible to represent the system targeted. 

While reaction parameters were not identified by regression to impedance data, the simulation presented by Roy et al. demonstrates that side reactions proposed in the literature can account for low-frequency inductive loops. Indeed, the results presented in Figures 23.4 and 23.5 show that both models can account for low-frequency inductive loops. Other models can also account for low-frequency inductive loops so long as they involve potential-dependent adsorbed intermediates. It is generally understood that equivalent circuit models are not unique and have therefore an ambiguous relationship to physical properties of the electrochemical cell. As shown by Roy et al., even models based on physical and chemical processes are ambiguous. In the present case, the ambiguity arises from uncertainty as to which reactions are responsible for the low-frequency inductive features. 

---