Simple equivalent circuit

Figure 1.15 Simple equivalent circuit diagram of a motor...

Nevertheless, we will show that all of the systems studied exhibited relatively straightforward electrochemical phenomenology and could be represented by simple equivalent circuits involving primarily passive electrical elements. 

The impedance spectroscopy of steel corrosion in concentrated HC1, with and without inhibitors, exhibit relatively straightforward electrochemical phenomenology and can be represented by simple equivalent circuits involving primarily passive electrical elements. Analysis of these circuits for steel corroding in HC1 per se reveals that the heterogeneity of the surface is established rapidly and can be simulated with a simple electrical circuit model. 

Figure 5.10 is EIS of marmatite electrode in O.lmol/L KNO3 solution with different pH modifiers at open circuit potential. This EIS is very complicated. Simple equivalent circuit can be treated as the series of electrochemical reaction resistance R with the capacitance impedance Q == (nFr )/(icR ) resulting fi-om adsorbing action, and then parallel with the capacitance Ca of double electric... 

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). 

A simple equivalent circuit diagram for a two-electrode contactless conductivity measurement is shown in Figure 7.8. The impedance is given by... 

In the Cole-Cole (or complex impedance ) plot, one takes the real as ordinate and the Zimag part as abscissa. Each point on the resulting diagram is made up of a Z resolved into two components measured at a chosen frequency. There may be 20-30 points, each at different frequencies. Such plots tend to be semicircles (see Fig. 7.47), but even simple equivalent circuits have some structure (i.e., deviations from the semicircle), and these deviations provide information concerning events at the elec-trode/solution interface. 

Fig 29. A simple equivalent circuit for the artificial permeable membrane. Physical meaning of the elements C, membrane capacitance (dielectric charge displaceme-ment) R, membrane resistance (ion transport across membrane) f pt, Phase transfer resistance (ion transport across interface) Zw, Warburg impedance (diffusion through aqueous phase) Ctt, adsorption capacitance (ion adsorption at membrane side of interface) Cwa, aqueous adsorption capacitance (ion adsorption at water side of interface). From ref. 109. 

If we switch-off the current after the steady state has been reached, the voltage relaxes to the initial zero-level. The electrical behavior can be taken into account by introducing in the simple equivalent circuit (eqc) (Eq. (60)) a capacitor Cl in series to Rioa or to R,.ml (if ions or electrons are blocked) In the language of system theory the equivalent circuit of the bulk represents a PDTi-element and reads3 15 e.g., for cells 3 and 4 Par (Cx. Par (Rm , Ser ( AJrai, Cl))). 

The simple equivalent circuit (eqc) which reads for ion blocking as... 

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 11. Plots of log Z and / v.v. log / for a thiol-hexapeptide-coated mercury drop immersed in 5xlO 3M (a), I.3xlO 2M (b), 3.6xlO 2M (c), and 0.1M (d) KC1, as obtained at -1.000 V over the frequency range from 0.1 to 105 Hz. At frequencies <102 Hz all Bode plots coincide hence, only the experimental points for the lower KC1 concentration were reported. The solid curves are least-squares fits to the simple equivalent circuit of inset (1), which consists of the electrolyte resistance Ra, with in series a RSCS mesh representing the self-assembled monolayer and a further RjiCji mesh representing the diffuse layer. Rs = 0.14 Mfi cm2 C, = 11 pF cm-2 Ra = 4.53 (a), 4.17 (b), 1.27 (c) and 0.87 KO cm2 (d). CW 68 (a), 61 (b), 80 (c) and 84 pF cm 2 (d). Inset (2) shows the reciprocal, 1/Cji, of the experimental diffuse-layer capacitance vs. the l/C fajj = 0) value corresponding to the same KC1 concentration, as calculated on the basis of the Gouy-Chapman (GC) theory. The solid curves are 1 /Ca(OM) vs 1 /C,ii(ctm = 0) plots calculated from the GC theory for different charge densities afo on the metal, whose values are reported on each curve. (Reprinted from Ref.114 with permission from the Am. Chem. Soc.)...

Provided that Ls is large compared with VP, Cb may be neglected and we arrive at the very simple equivalent circuit... 

The first of these can be treated with a simple equivalent circuit of the form shown in Fig. 104. It is normally assumed that processes (b) and (c) are very fast compared with (d) and (e) and that, if a direct bandgap semiconductor is used, (a) can also be minimised. Under these circumstances, the initial condition is... 

To use this simple equivalent circuit, it must be realized that the rate constants of the various steps can be represented by the inverse of the corresponding resistances the higher the resistance, the lower the rate constant, and vice versa. Thus, the overall effective rate constant is given by... 

A result such as shown in Fig. 16L indicates clearly that the system cannot be described correctly by a simple equivalent circuit of the types discussed so far. Sometimes, if the depressed angle a is small (say, less than 10°) the problem may perhaps be ignored, and one may obtain R either as the distance from /I to B or from A to B , which will differ in this case by a few 10%. In any event, changes in the angle of depression or in the radius of the depressed semicircle still can be taken as an indication of variation in the properties of the... 

The operative working condition of a battery can be represented by simple equivalent circuits, which are only valid for charging or discharging operations without considering the current transient phenomena. The discharging equivalent circuit of a battery is shown in Fig. 5.11. 

Among the various models ° " proposed in order to explain two-arc behavior in the Nyquist plot for the insertion electrode, the simple equivalent circuit " depicted in Figure 10(b) was used to analyze the impedance spectra. Flere, / ,. is the sum of the electrolyte and conducting substrate resistances Ream is the resistance associated with the particle-to-particle contact among the oxide particles Ccom is the contact capacitance due to the accumulation of charged species at the surface of the oxide particles Rat is the resistance associated with the absorption reaction of adsorbed lithium into the oxide and Cad represents the capacitance arising from the adsorption of lithium in the near-surface region of the oxide. 

These oscillatory torques are approximately symmetrical about the torque calculated form the simple equivalent circuit and decay to zero as the rotor accelerates. 

Figure 5.30 shows a comparison of Nyquist responses obtained for Zn, Zn-Ni, Cd, Zn-Ni-Cu, nano Ni-Zn, and electroless Ni-Zn-P coatings. The solution resistance remains the same for aU frequencies. This is to be expected because the studies were done under similar conditions in pH 7.0. Rp values can be approximately determined by fitting the Nyquist response to a simple equivalent circuit consisting of ohmic resistance, doublelayer capacitance, and polarization resistance. 

It was shown that an electrochemical cell could be described with a simple equivalent circuit (Fig. 4a) containing the electrolyte resistance, the double-layer capacitance, and the impedance of the faradaic process. Previously, the faradaic process was described in terms of a simple charge-transfer resistance (Sect. 2.6.2.3) neglecting the diffusion of electroactive species. 

This behavior is observed when the surface of an electrode is not perfectly flat. In the simple equivalent circuits for electrochemical cells discussed in Sects. 2.6.2 and 2.6.3, the double-layer capacitance (see Figs. 3c, 4b, and 7) is... 

Fig. 26.1. Different contributions from the protonic conductivity of a powder sample bulk conductivity (1), grain boundary conductivity in series (2) and parallel (3) to the grains. A simple equivalent circuit including simulated impedance spectra for two different cases is shown. The bulk resistance (Rj) can be extracted only from spectrum (b).

It should also be pointed out at the beginning of this chapter that frequent use will be made of rather simple equivalent circuits. These are intended only... 

Everybody can learn to run somebody else s program. Only a minority is capable of making the correct interpretation. But a simple equivalent circuit certainly helps. 

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