Electrode equivalent circuit

Finally, it should be noted that in many cases where < 0, is determined by the capacity method uncertainty arises, which is related to the frequency dependence of Mott-Schottky plots. (In particular, the frequency of the measuring current is increased in order to reduce the contribution of surface states to the capacity measured.) As the frequency varies, these plots, as well as the plots of the squared leakage resistance R vs. the potential (in the electrode equivalent circuit, R and C are connected in parallel), are deformed in either of two ways (see Figs. 6a and 6b). In most of the cases, only the slopes of these plots change but their intercepts on the potential axis remain unchanged and are the same for capacity and resistance plots (Fig. 6b). Sometimes, however, not only does the slope vary but the straight line shifts, as a whole, with respect to the potential axis, so that the intercept on this axis depends upon the frequency (Fig. 6a). 

This model is known as the model of the common diffuse layer (CDL) [27]. As shown in Ref. [30], both models can describe only some limiting cases and the exposition for the total capacity of the PC electrode (equivalent circuit) investigated depends on the relationship between the three lengths (1) the characteristic size of the individual planes at a PC electrode surface (2) the effective... 

FTEIS is an experimentally convenient technique for such potentiodynam-ic impedance measurements. In this approach, a series of Nyquist (or Bode) spectra is obtained in parallel with the recording of CV data. DC voltage-dependent electrode equivalent-circuit models can be developed through CNLLS analysis of the impedance spectra. However, in FTEIS postexperiment processing is often cumbersome and inconvenient, and FTEIS limits the amplitude of the excitation signal to imder 10 mV (for aqueous solutions) because of the nonlinearity of the electrochemical system. 

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))...

A value of Rqb for an SEI lOnm thick can be estimated from its values for CPE and CSE by assuming that these solid electrolytes consist of nanometer-sized particles. Thus the expected value for / GB at 30 °C for a lOnm SEI is in the range 10-lOOQcm2, i.e., it cannot be neglected. In some cases it may be larger than the ionic (bulk) resistance of the SEI. This calculation leads us to the conclusion that 7 GB and CGB must be included in the equivalent circuits of the SEI, for both metallic lithium and for LixC6 electrodes. The equivalent circuit for a mosaic-type... 

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... 

But when considered over a wide range of frequencies, the properties of a real electrode do not match those of the equivalent circuits shown in Fig. 12.12 the actual frequency dependence of Z and a does not obey Eq. (12.21) or (12.22). In other words, the actual values of R and or R and are not constant but depend on frequency. In this sense the equivalent circuits described are simplified. In practice they are used only for recording the original experimental data. The values of R and Cj (or R and C ) found experimentally for each frequency are displayed as functions of frequency. In a subsequent analysis of these data, more complex equivalent circuits are explored which might describe the experimental frequency dependence and where the parameters of the individual elements remain constant. It is the task of theory to interpret the circuits obtained and find the physical significance of the individual elements. 

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.)... 

Under this electrochemical configuration, it is commonly accepted that the system can be expressed by the Randles-type equivalent circuit (Fig. 6, inset) [23]. For reactions on the bare Au electrode, mathematical simsulations based on the equivalent circuit satisfactorily reproduced the experimental data. The parameters used for the simulation are as follows solution resistance, = 40 kS2 cm double-layer capacitance, C = 28 /xF cm equivalent resistance of Warburg element, W — R = 1.1 x 10 cm equivalent capacitance of Warburg element, IF—7 =l.lxl0 F cm (

charge-transfer resistance, R = 80 kf2 cm. Note that these equivalent parameters are normalized to the electrode geometrical area. On the other hand, results of the mathematical simulation were unsatisfactory due to the nonideal impedance behavior of the DNA adlayer. This should... 

FIG. 6 Complex impedance plots for the electrode reaction of [Fe(CN)6] on bare (open circle) and DNA-modilied (filled circle) An electrodes. An equivalent circuit for the electrode system is shown in the inset and solid lines represent theoretical responses from the circuit. Parameters used in simulation are cited in the text. Electrode potential, + 205 mV (vs. Ag/AgCl) AC amplitude, 25 mV (p-p). Other conditions are the same as those in Fig. 5. 

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... 

When the rate of the electrode reaction is measurable, being characterized by a definite polarization resistance RP (Eq. 5.2.31), the electrode system can be characterized by the equivalent circuit shown in Fig. 5.22. 

Fig. 5.22 Equivalent circuit of an electrode with diffusing el-ectroactive substances. W is the Warburg impedance (Eq. 5.5.21)...

Ion transport across membranes can be evaluated by using mucosal and serosal electrodes to read transepithelial current (I) and potential difference OP). With these parameters, equivalent circuit analysis can be utilized to account for the relative contributions of transcellular and paracellular pathways. Ionic flux (J) is defined by the Nernst-Planck equation,... 

The distribution of potential in TC is practically the same as that near the flat surface if the electrolyte concentration is about 1 mol/1 [2], So the discharge of TC may be considered as that of a double electric layer formed at the flat electrode surface/electrolyte solution interface, and hence, an equivalent circuit for the TC discharge may be presented as an RC circuit, where C is the double layer capacitance and R is the electrolyte resistance. 

Now an equivalent circuit, which takes into account both the ion transport along the TC and the charge transfer through the carbon electrode material to the current collector, may be represented as in Fig. 2, wherein N = a(c)/4r, Cm and Rm are the total NP capacitance and resistance in a unit electrode volume (defined here as a product of a unit electrode area and the tier thickness), Re is the electrical resistance of an electrode in the same unit... 

Figure 2. An equivalent circuit of a porous carbon electrode in SC.

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 ... 

Very often, the electrode-solution interface can be represented by an equivalent circuit, as shown in Fig. 5.10, where Rs denotes the ohmic resistance of the electrolyte solution, Cdl, the double layer capacitance, Rct the charge (or electron) transfer resistance that exists if a redox probe is present in the electrolyte solution, and Zw the Warburg impedance arising from the diffusion of redox probe ions from the bulk electrolyte to the electrode interface. Note that both Rs and Zw represent bulk properties and are not expected to be affected by an immunocomplex structure on an electrode surface. On the other hand, Cdl and Rct depend on the dielectric and insulating properties of the electrode-electrolyte solution interface. For example, for an electrode surface immobilized with an immunocomplex, the double layer capacitance would consist of a constant capacitance of the bare electrode (Cbare) and a variable capacitance arising from the immunocomplex structure (Cimmun), expressed as in Eq. (4). 

Figure 6.7 Complex impedance of a polycrystalline ceramic sample (a) representation of the equivalent circuit of a component (b) the impedance spectrum of the equivalent circuit in (a) (c) the impedance spectrum of a typical ceramic sample. Each semicircular arc represents one component with an equivalent circuit as in (a) that at the highest frequency corresponds to the repose of the bulk, that at middle frequencies to the grain boundary response, and that at lowest frequencies to the electrodes.

Typically, the frequency lu of the modulation is varied over a considerable range, and an impedance spectrum Z(lS) recorded. Various electrode processes make different contributions to the total impedance. In many cases it is useful to draw an equivalent circuit consisting of... 

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 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( . 

Several factors contribute to the impedance behavior of a silicon electrode. Figure 10.1a shows a proposal for an equivalent circuit that considers most of these factors, which are ... 

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. 

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