Capacitance equivalent circuits

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.

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.

FIGURE 12.12 Equivalent circuits with resistance and capacitance in series (a) and in parallel b). 

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... 

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.)... 

FIG. 18 Simplified equivalent circuit for externally biased ITIES under illumination. The perturbations introduced by the photoreactions in Fig. 11 are contained within the generator term g. Cji and are associated with the interfacial capacitance and the uncompensated resistance. (From Ref. 83. Reproduced by permission of the Royal Society of Chemistry.)... 

The impedance data have been usually interpreted in terms of the Randles-type equivalent circuit, which consists of the parallel combination of the capacitance Zq of the ITIES and the faradaic impedances of the charge transfer reactions, with the solution resistance in series [15], cf. Fig. 6. While this is a convenient model in many cases, its limitations have to be always considered. First, it is necessary to justify the validity of the basic model assumption that the charging and faradaic currents are additive. Second, the conditions have to be analyzed, under which the measured impedance of the electrochemical cell can represent the impedance of the ITIES. 

FIG. 6 Randles equivalent circuit for the ITIES Zq is the interfacial capacitance, Zy)v are the faradaic impedances of the charge transfer reactions, and is the solution resistance. 

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... 

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... 

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 ... 

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 7.3 Simplified equivalent circuit of an original (unmodified) EIS structure (a) and EIS biosensor functionalized with charged macromolecules (b). Cj, Cx and CML are capacitances of the gate insulator, the space-charge region in the semiconductor, and the molecular layer, respectively / u is the resistance of... 

Dielectric constant is directly proportional to the capacitance of a material. Present computer operations are limited by the coupling capacitance between circuit paths and integrated circuits on multilayer boards since the computing speed between integrated circuits is reduced by this capacitance and the power required to operate is increased.11 If the dielectric constant is reduced a thinner dielectric provides equivalent capacitance, and the ground plane can be moved closer to the line, so that additional lines can be accommodated for the same cross-talk. Thus, the effect of a low dielectric constant will be to increase the speed ofthe signal and improve the density of the packaging, and this will result in improved system performance.2... 

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( . 

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 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 impedance response with frequency can be closely simulated by the equivalent circuit shown in Figure 27a, where Re, Ra, Cdi, Rad, and Cad represent the resistance or capacitance for the electrolyte solution, charge-transfer, double layer, and adsorbed layer, respectively. An interesting correlation was found between the passivating ability of various anions and the resistances of the two impedance components R and Rad, which are high for LiPFe-and LiBF4-based electrolytes and low for LiTf- or Lilm-based electrolytes. Using the rationale proposed by the authors, the former component (Ret) is... 

Figure 53. Idealized half-cell response of a thin solid electrolyte cell, (a) Cell geometry including working electrodes A and B and reference electrode (s). (b) Equivalent circuit model for the cell in a, where the electrolyte and two electrodes have area-specific resistances and capacitances as indicated, (c) Total cell and half-cell impedance responses, calculated assuming the reference electrode remains equipotential with a planar surface located somewhere in the middle of the active region, halfway between the two working electrodes, as shown in a.

Figure 6.19. Simplified equivalent circuit for single-electrode reaction [e.g., Eq. (6.6)] Qi, double-layer capacitance of test electrode charge-transfer resistance of electrode reaction.

Most often, the electrochemical impedance spectroscopy (EIS) measurements are undertaken with a potentiostat, which maintains the electrode at a precisely constant bias potential. A sinusoidal perturbation of 10 mV in a frequency range from 10 to 10 Hz is superimposed on the electrode, and the response is acquired by an impedance analyzer. In the case of semiconductor/electrolyte interfaces, the equivalent circuit fitting the experimental data is modeled as one and sometimes two loops involving a capacitance imaginary term in parallel with a purely ohmic resistance R. 

A similar procedure can be used to determine the space charge distribution in n-type Si in the dark with a positive bias polarization so as to generate a depletion layer within the semiconductor substrate. In this case, the situation is somewhat different because the positive polarization in HF results in an anodic etching of the sample with a nonnegligible current density near 7 pA cm . Nevertheless, similar results were obtained, the components of the equivalent circuit were a capacitance of a few 10 F cm , and a resistance term ranging from 1 to 10Mf2cm for a bias potential varying from —0.1 to -1-0.9 V vs. SCE. 

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