Circuit Randles

This circuit is usually referred to as the Randles circuit and its analysis has been a major feature of AC impedance studies in the last fifty years. In principle, we can measure the impedance of our cell as a function of frequency and then obtain the best values of the parameters Rct,<7,C4i and Rso by a least squares algorithm. The advent of fast micro-computers makes this the normal method nowadays but it is often extremely helpful to represent the AC data graphically since the suitability of a simple model, such as the Randles model, can usually be immediately assessed. The most common graphical representation is the impedance plot in which the real part of the measured impedance (i.e. that in phase with the impressed cell voltage) is plotted against the 90° out-of-phase quadrature or imaginary part of the impedance. 

Further information on this subject can be obtained by frequency response analysis and this technique has proved to be very valuable for studying the kinetics of polymer electrodes. Initially, it has been shown that the overall impedance response of polymer electrodes generally resembles that of intercalation electrodes, such as TiS2 and WO3 (Ho, Raistrick and Huggins, 1980 Naoi, Ueyama, Osaka and Smyrl, 1990). On the other hand this was to be expected since polymer and intercalation electrodes both undergo somewhat similar electrochemical redox reactions, which include the diffusion of ions in the bulk of the host structures. One aspect of this conclusion is that the impedance response of polymer electrodes may be interpreted on the basis of electrical circuits which are representative of the intercalation electrodes, such as the Randles circuit illustrated in Fig. 9.13. The figure also illustrates the idealised response of this circuit in the complex impedance jZ"-Z ) plane. 

A significant step forward in our understanding of Pt was taken by Verkerk and Burgraff, who in 1983 analyzed the impedance of porous sputtered Pt (and Pt gauze) electrodes on YSZ and gadolinia-doped ceria (GDC). As shown in Figure 11, they used a Randles circuit to model the interfacial contributions to the impedance, allowing them to subtract from the data the contributions of uncompensated iR and... 

This circuit is usually referred to as the Randles circuit and its analysis has been a major Feature of AC impedance studies in the last fifty years. In principle, we can measure the impedance of our cell as a Function of Frequency and then obtain the best values of the parameters and by a... 

In a simple case, the electrochemical reaction at the electrode-electrolyte interface of one of the electrodes of the battery can be represented by the so-called Randles circuit (Figure 8.19), which is composed of [129] a double layer capacitor formed by the charge separation at the electrodeelectrolyte interface, in parallel to a polarization resistor and the Warburg impedance connected in series with a resistor, which represents the resistance of the electrolyte. 

The geometrical capacitance Cg is of the order of 10 10 F cm 2 for the majority of films studied, whose thickness is 1 pm by order the corresponding impedance would be an order higher than the film resistance Rs (with exception of very low-doped films) and can be neglected. We then obtain a simpler three-element circuit (Fig. 10b) often called the Randles circuit [65], An essential assumption is that all elements of the circuits in Figs. 10a and b are frequency-independent. 

Fig. 10. Equivalent circuits of electrode (a) general circuit of a thin-film electrode (b) the Randles circuit and (c) circuit with a constant phase element.

For consecutive or parallel electrode reactions it is logical to construct circuits based on the Randles circuit, but with more components. Figure 11.16 shows a simulation of a two-step electrode reaction, with strongly adsorbed intermediate, in the absence of mass transport control. When the combinations are more complex it is indispensable to resort to digital simulation so that the values of the components in the simulation can be optimized, generally using a non-linear least squares method (complex non-linear least squares fitting). 

Figure 28 Randles circuit that serves as an electrical analog of the corroding interface.

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. 

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. 

Figure 4.176 shows the simulated Nyquist plot of a modified adsorption model with an extra CPE in the Randles circuit. This modification of the adsorption model strongly influences the low-fiequency shape of the complex plane impedance diagram. More examples with variations on the CPE exponent are presented in Appendix D (Model D16). 

Figure 4.38. Equivalent circuits of conducting polymers with a Randles circuit [12]. (Reprinted from Journal of Electroanalytical Chemistry, 420, Ren X, Pickup PG. An impedance study of electron transport and electron transfer in composite polypyrrole plus polystyrenesulphonate films, 251-7, 1997 with permission from Elsevier and from the authors.)...

When R2 R, the polymer is more conducting than the pores. The Randles circuit, which is located at the polymer/electrolyte interface (case b), shunts the resistive ionic rail through the polymer. At high frequencies, the equation can be simplified to... 

The Randles circuit is at the electrode/polymer interface (case a). The imaginary component of the impedance is dominated by the double-layer capacitance, and the real one is controlled by the double layer. The capacitance of the transmission line is shunted and the transmission fine is not involved. 

At high frequencies, the plot is linear and the slope is similar to that for a bare Pt electrode. The double-layer capacitance is 28 pF cm-2 for the PPY-PSS film and 23 pF cm-2 for the bare Pt electrode. The plot of real resistance minus the solution resistance versus log(tw) over the frequency range shown in Figure 4.43 is also linear and fits Equation 4.121 well. This indicates that a Randles circuit is at the electrode/polymer interface. 

For the double layer to be studied, the three elements can be grouped Into a Randles circuit (fig. 3.31), which In turn can be connected In series to other elements of the cell. In passing, a Randles circuit may also be drawn for Individual colloidal peuticles. Then, this circuit is isolated. It stands on its own... 

Figure 3.31. Randles circuit, consisting of a pure capacitance C. an ion transfer resistance 0 and a diflusion impedance all counted per unit area.

Raman scattering see electromagnetic radiation scattering Raman spectroscopy 1.7.12 Randles circuit fig. 3.31 random coll 5.3... 

The electrical circuit presented in Figure 10.5 yields the impedance response equivalent to equation (10.49) for a single Faradaic reaction coupled with a mass transfer. This circuit is known as the Randles circuit. Such a circuit may provide a building block for development of circuit models as shown in Chapter 9 for the impedance response of a more complicated system involving, for example, coupled reactions or more complicated 2- or 3-dimensional geometries. 

Figure 17.1 Equivalent circuits used to demonstrate the graphical representation of reactive impedance data a) Randles circuit and b) blocking circuit.

Table 17.1 Randles circuit parameters used for simulations.

A Remdles circuit is used here to demonstrate the graphical representation for reactive (nonblocking) systems. The impedance of the Randles circuit presented in Figure 17.1(a) is given by... 

The Randles circuit provides an example of a class of systems for which, at the zero-frequency or dc limit, the resistance to passage of current is finite, and current can pass. Many electrochemical and electroiuc systems exhibit such nonblocking or reactive behavior. Even though the impedance response of the system presented in this section is relatively simple as compared to that of t37pical electrochemical and electronic systems, the nonblocking systems comprise a broad cross-section of electrochemical and electronic systems. The concepts described in this chapter therefore can be easily adapted to experimental data. 

Figure 17.2 Traditional representation of impedance data for the Randles circuit presented as Figure 17.1(a) with a as a parameter, a) complex-impedance-plane or Nyquist representation (symbols are used to designate decades of frequency.) b) Bode representation of the magnitude of the impedance and c) Bode representation of the phase angle. (Taken from Orazem et al. ° and reproduced with permission of The Electrochemical Society.)...

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