Equivalent Circuit Presentation

Substituting for and using the above expressions in equation 9, we obtain, with dj//dt = juV  

However, since the capacitance C of the equivalent pure ceramic is given by  

Substitution in eqn 12 gives the total current density jJ, with = (y/1)  

Although In this section v e have attempted to separate the e Into Its constituent e or o/ue components, the experimental data presented In the following section do not at first sight distinguish the various component losses and thus the generic terms or tan 6 are used. Closer scrutiny of the experimental data Is needed to be able to differentiate between dipolar and conductivity losses. 

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Electrical parameters of the system under consideration have been estimated using an equivalent circuit presented by Fig. 2. Thus, the anode capacitance can be written as follows ... 

Why such a difference between the real EDLC and the model proposed here In other words, what are the specific features the equivalent circuit does not take into account The oversimplified equivalent circuit presented in Figure 1.22 considers two planar electrodes face to face, with a constant thickness of dielectric material between them. The reality is much more complex since EDLCs use three-dimensional porous electrodes, and the porous electrodes are responsible for the particular shape of the Nyquist plot presented in Figure 1.23. 

In the study of impedance plots, we may observe the depression of semicircles. This is the so-called semicircle rotation of the impedance. This phenomenon is associated with electrode/electrolyte interface double-layer properties. For example, the rough surface of the electrodes or porous electrodes can result in an uneven distribution of the double-layer electric field. This semicircle rotation can be explained using the equivalent circuit presented in Figure 3.10, where R is inversely proportional to the frequency CO (and b is a constant). 

In a first approximation, the equivalent circuit presented in Figure 13.15 can be used to represent the electrode in which the interfacial impedance between the oxide and the electrolyte was eissumed to be negligible as compared to the coating impedance. The impedance of the oxide layer can be considered to correspond to a... 

The equivalent circuit presented in Figure 20.1 was regressed to the impedance data. The mathematical formulation for the model is given as... 

Refined models for mass transfer to a disk electrode are presented in Section 11.6. The equivalent circuit presented in Figure 20.12 was regressed to the impedance data. The mathematical formulation for the model is given as equation (17.1). Four models were considered for the convective-diffusion term Zd (/) ... 

Note that if this is a standard two-stage off-line filter, we need to look at the DM equivalent circuit presented in Figure 10-1 in Chapter 10 to realize what exact values of the various X-capacitors in the figure this C really corresponds to in that schematic. 

At frequencies below 63 Hz, the double-layer capacitance began to dominate the overall impedance of the membrane electrode. The electric potential profile of a bilayer membrane consists of a hydrocarbon core layer and an electrical double layer (49). The dipolar potential, which originates from the lipid bilayer head-group zone and the incorporated protein, partially controls transmembrane ion transport. The model equivalent circuit presented here accounts for the response as a function of frequency of both the hydrocarbon core layer and the double layer at the membrane-water interface. The value of Cdl from the best curve fit for the membrane-coated electrode is lower than that for the bare PtO interface. For the membrane-coated electrode, the model gives a polarization resistance, of 80 kfl compared with 5 kfl for the bare PtO electrode. Formation of the lipid membrane creates a dipolar potential at the interface that results in higher Rdl. The incorporated rhodopsin may also extend the double layer, which makes the layer more diffuse and, therefore, decreases C. ... 

In the simplest case, the equivalent circuit comprised of a capacitance C and a resistance Ri connected in parallel can describe the electrical behavior of the electrodesolution interface (Chapter 3). When a current flows, an ohmic resistance Rq must be added in series to take into account the ohmic drop in the electrolyte between the reference electrode and working electrode. Equations (5.141) and (5.142) express the impedance of the equivalent circuit presented in Figure 5.26. In these equations Zq represents the impedance of the double layer. 

Impedance measurement can be considered a third way to evaluate electrochemical sensors besides potentiometry and amperometry. Electrochemical impedance studies in a narrower sense deal with phenomena at the electrode surface. The overall impedance of a chemosensor also includes effects of charge carrier properties far from the electrode. This was visualized by equivalence circuits presented in Chaps. 2 and 5. By individual experimental design, the study can be focused more on processes at the electrode surface or otherwise on ion properties in homogeneous solution. Even the variation of the dielectric constant in a layer will affect the overall impedance. If impedimetry is designed only to acquire data corresponding to ionic properties or value of the dielectric constant, it is not really an electrochemical method, in a strict sense. 

Figure 13. Schematic presentation of a small segment of polyheteromicrophase SEI (a) and its equivalent circuit (b) A, native oxide film B, LiF or LiCl C, non conducting polymer D, Li2CO, or LiCO, R GB, grain boundary. RA,/ B,RD, ionic resistance of microphase A, B, D. Rc >Rqb charge-transfer resistances at the grain boundary of A to B or A to D, respectively. CA, CB, CD SEI capacitance for each of the particles A to D.

The equivalent circuit of a section of this SEI is presented in Fig. 13(b). It was recently found [123, 124] that at temperatures lower than 90 °C, the grain-boundary resistance of composite polymer electrolytes and composite solid electrolytes based on Lil-A Ojis many times larger than their ionic resistance. At 30 °C / GB is several orders of magnitude larger than (the ionic resistance) and for 100 pm-thick CPE foils or Lil-A Oj pellets it reaches [125] 105-106Qcm2 (depending on CPE composition). 

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. 

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). 

An alternative approach, using semiconductors as light-driven electron donors, has been demonstrated in model systems (Gratzel 1982 Nikhandrov et al. 1988). These are more stable than the photosystems, but show lower photochemical conversion efficiencies owing to short-circuiting of reducing equivalents. The presently used... 

The present study deals with two types of paints applied on phosphated steel, the influence of temperatures upto 90 C and immersion times upto 10 days. The results obtained are analyzed in the light of published information briefly described in this section. An electrical equivalent circuit similar to the one used by Muslanl et al (27) is considered suitable for the analysis. 

The reduction of Cd(II) ions on DME was also investigated in 1 M perchlorate, fluoride and chloride solutions using dc, ac admittance, and demodulation methods [27]. It was found that in the perchlorate supporting electrolyte, the reduction mechanism is also CEE, and that the rate constant of the chemical step is quite close to the value characteristic for fluoride solutions. The theories available at present could not be applied to the Cd(II) reduction in chloride solution because of the inapplicability of the Randles equivalent circuit. 

Noting these uncertainties, we have evaluated the equivalent circuit and present the results in Fig. 3. The potential dependence of the three capacitive elements is shown in Figs. 

Mathematical treatments for equivalent circuit of this type for a single level and a continuum of surface state levels have been presented elsewhere (e.g., 1 0, 20, 21, 22). The admittance of the reduced circuit after removing the influence of R., . 

Finally, it can be seen from Fig. 9.9a that the real impedance does not remain constant at low frequencies for the textile electrode, and this effect is more pronounced at higher electrolyte concentrations. Probably, Zr is influenced by other effects only occurring in the low-frequency range. This effect is frequently observed and described in the literature and is caused by non-uniformity of surfaces at the micro-scale, which in fact is the case for the textile electrodes. It is also not possible to explain this effect by a pure resistor or a pure capacitor in the electrical equivalent circuit. For this purpose, constant-phase elements are implemented as described in the theoretical discussion of electrochemical impedance spectroscopy (presented in Chapter 2, section 2.4). 

To understand the electrical behaviour of the LAPS-based measurement, the LAPS set-up can be represented by an electrical equivalent circuit (see Fig. 5.2). Vbias represents the voltage source to apply the dc voltage to the LAPS structure. Re is a simple presentation of the reference electrode and the electrolyte resistance followed by a interface capacitance Cinterface (this complex capacitance can be further simulated by different proposed models as they are described, e.g., in Refs. [2,21,22]). In series to the interface capacitance, the insulator capacitance Cj will summarise the capacitances of all insulating layers of the LAPS device. The electrical current due to the photogeneration of electron-hole pairs can be modelled as current source Ip in parallel to the... 

Figure 2.23 Faradaic impedance spectra presented in the form of Nyquist plots, along with the electronic equivalent circuit of the electrified interface. (Reproduced with permission from Ref. 87.)...

The reaction of our investigator to the puzzle presented by the black box will differ according to whether he is a mathematician, electrical engineer, physicist or chemist. The mathematician will be satisfied by a description in terms of differential equations and the engineer by an equivalent circuit. However the physicist or chemist will want an interpretation in terms of the structure of the material whose response can be represented by the black box. The materials scientists will often be disappointed. 

A basic electric equivalent circuit to describe an EDLC is presented in Figures 1.22a and b, which shows the Nyquist (Figure 1.22b) plot of an ideal capacitor C, in series with a resistance... 

Rs (Figure 1.22a). The double layer capacitance is represented by the capacitance C, and Rs is the series resistance of the EDLC, also named the equivalent series resistance (ESR). This series resistance shows the nonideal behavior of the system. This resistance is the sum of various ohmic contributions that can be found in the system, such as the electrolyte resistance (ionic contribution), the contact resistance (between the carbon particles, at the current collector/carbon film interface), and the intrinsic resistance of the components (current collectors and carbon). Since the resistivity of the current collectors is low when A1 foils or grids are used, it is generally admitted that the main important contribution to the ESR is the electrolyte resistance (in the bulk and in the porosity of the electrode) and to a smaller extent the current collector/active film contact impedance [25,26], The Nyquist plot related to this simple RC circuit presented in Figure 1.22b shows a vertical line parallel to the imaginary axis. 

The entire diagram illustrating the performance of the electrode processes, corresponding to the simple equivalent circuit diagram of a battery which is shown in Figure 8.18, is presented in Figure 8.23 [6],... 

Figure 18, taken from Ref. 77, describes several models proposed for the Li electrodes in solutions, their equivalent circuit analogs, and the expected impedance spectra (presented as Nyquist plots). Assuming parallel plate geometry for the solid electrolyte interface, as well as knowledge of the surface species involved from spectroscopy (and thus their dielectric constant, which is around 5 for many surface species formed on Li, including R0C02Li, Li2C03, LiF, ROLi, etc. [186]), it is possible to estimate the surface film s thickness from the electrode s capacitance (calculated from the model fitted to the spectra) ...

Figure 18 Various models proposed for the surface films that cover Li electrodes in nonaqueous solutions. The relevant equivalent circuit analog and the expected (theoretical) impedance spectrum (presented as a Nyquist plot) are also shown [77]. (a) A simple, single layer, solid electrolyte interphase (SEI) (b) solid polymer interphase (SPI). Different types of insoluble Li salt products of solution reduction processes are embedded in a polymeric matrix (c) polymeric electrolyte interphase (PEI). The polymer matrix is porous and also contains solution. Note that the PEI and the SPI may be described by a similar equivalent analog. However, the time constants related to SPI film are expected to be poorly separated (compared with a film that behaves like a PEI) [77]. (With copyrights from The Electrochemical Society Inc., 1998.)...

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