Capacitance in parallel

Even in the absence of Faradaic current, ie, in the case of an ideally polarizable electrode, changing the potential of the electrode causes a transient current to flow, charging the double layer. The metal may have an excess charge near its surface to balance the charge of the specifically adsorbed ions. These two planes of charge separated by a small distance are analogous to a capacitor. Thus the electrode is analogous to a double-layer capacitance in parallel with a kinetic resistance. 

The capacitance. The electrical double layer may be regarded as a resistance and capacitance in parallel see Section 20.1), and measurements of the electrical impedance by the imposition of an alternating potential of known frequency can provide information on the nature of a surface. Electrochemical impedance spectroscopy is now well established as a powerful technique for investigating electrochemical and corrosion systems. 

Pore-size-dependent conductances are assigned to individual pores and channels. Three possible types of bonds befween pores exist. The corresponding bond conductances—(T), and o X)—can be established straightforwardly. The model was extended toward calculation of the complex impedance of the membrane by assigning capacitances in parallel to conductances to individual pores. The probability distribution of bonds to have conductivify cr b, <7br/ or O, is... 

First, we analyze the effect of the stray capacitance in parallel with the feedback resistance. Consider a sinusoidal input current with frequency /. From Fig. 11.1, the magnitude of the output voltage is... 

The effectiveness with which a VDR absorbs the very rapid transients that may occur accidentally in a power supply depends on its speed of response. The response time of the voltage-sensitive material is generally adequate, being typically 0.5 ns. The capacitance in parallel with the VDR will partially absorb the transient and delay the rise in voltage, but this will not affect the protective action of the device. The inductance may have an important effect because the steep rise time of a transient results in the presence of very high frequency voltage components so that the impedance due to even a small inductance in series with a VDR becomes significant. The leads to the VDR must therefore be as short as possible. 

The capacitance is due only to the working electrode, whilst the resistance includes the resistive components of the electrode process, of the solution, etc. In some cases a combination of resistance and capacitance in parallel has also been used. In these conditions the analysis is more easily carried out in terms of admittance Y = 1/Z see Appendix 2. 

A2.4 Representation in the complex plane A2.5 Resistance and capacitance in series A2.6 Resistance and capacitance in parallel A2.7 Impedances in series and in parallel A2.8 Admittance... 

Fig. A2.3. Resistance and capacitance in parallel (a) Electrical circuit (b) Diagram showing the vectorial sum of the resistive and capacitive currents for a parallel RC combination (c) Complex-plane impedance plot.

Since the unloaded QCM is an electromechanical transducer, it can be described by the Butterworth-Van Dyke (BVD) equivalent electrical circuit represented in Fig. 12.3 (box) which is formed by a series RLC circuit in parallel with a static capacitance C0. The electrical equivalence to the mechanical model (mass, elastic response and friction losses of the quartz crystal) are represented by the inductance L, the capacitance C and the resistance, R connected in series. The static capacitance in parallel with the series motional RLC arm represents the electrical capacitance of the parallel plate capacitor formed by both metal electrodes that sandwich the thin quartz crystal plus the stray capacitance due to the connectors. However, it is not related with the piezoelectric effect but it influences the QCM resonant frequency. 

In the complex-plane impedance diagram, the Nyquist plot of resistance and capacitance in parallel is an ideal semicircle, as depicted in Figure 4.3b. The diameter equals the value of the resistance, R. The imaginary part of the impedance reaches a maximum at frequency [Pg.146]

Because water can be stored all along the stem, C816 equals the sum of the capacitances for individual parts of the pathway arranged in parallel (0 = J2 for capacitances in parallel). When A1 7 is the same all along the pathway, we can then simply add the component capacitances otherwise, we can let 0 equal d-JA F, where A is the average water potential... 

The electrical circuit providing the impedance response equivalent to the same equation (10.79) is a charge-transfer resistance in series with a Voigt element composed of a capacitance in parallel with a resistance (Figure 10.8(b)). The capacitance has a value of —l/ARj cind the resistance has a value of —ARt /(B - - ARt). 

From this physical model, an electrical model of the interface can be given. Free corrosion is the association of an anodic process (iron dissolution) and a cathodic process (electrolyte reduction). Ther ore, as discussed in Section 9.2.1, the total impedance of the system near the corrosion potential is equivalent to an anodic impedance Za in parallel with a cathodic impedance Zc with a solution resistance Re added in series as shoxvn in Figure 13.13(a). The anodic impedance Za is simply depicted by a double-layer capacitance in parallel with a charge-transfer resistance (Figure 13.13(b)). The cathodic branch is described, following the method of de Levie, by a distributed impedance in space as a transmission line in the conducting macropore (Figure 13.12). The interfacial impedance of the microporous... 

Experiments carried out on monocrystalline Au(lll) and Au(lOO) electrodes in the absence of specific adsorption did not show any fre-quency dispersion. Dispersion was observed, however, in the presence of specific adsorption of halide ions. It was attributed to slow adsorption and diffusion of these ions and phase transitions (reconstructions). In their analysis these authors expressed the electrode impedance as = R, + (jco iJ- where is a complex electrode capacitance. In the case of a simple CPE circuit, this parameter is = T(Jcaif. However, an analysis of the ac impedance spectra in the presence of specific adsorption revealed that the complex plane capacitance plots (C t vs. Cjnt) show the formation of deformed semicircles. Consequently, Pajkossy et al. proposed the electrical equivalent model shown in Fig. 29, in which instead of the CPE there is a double-layer capacitance in parallel with a series connection of the adsorption resistance and capacitance, / ad and Cad, and the semi-infinite Warburg impedance coimected with the diffusion of the adsorbing species. A comparison of the measured and calculated capacitances (using the model in Fig. 29) for Au(lll) in 0.1 M HCIO4 in ths presence of 0.15 mM NaBr is shown in Fig. 30. 

The faradaic impedance is sometimes represented as a resistance and a capacitance in parallel rather... 

Squid axon Hodgkin and Huxley Most models of excitable cells are descendents of this model. The basic circuit equivalent of this model comprises a linear capacitance in parallel with three resistor-battery subcircuits (one each for sodium, potassium, and a nonspecific leakage channel). The HHequations in their original form were ... 

The next step, after all experimental parameters have been given their correct values, is usually a calibration. A dummy cell is used, consisting of electronic components that imitate the behaviour of the real cell as closely as possible. The simplest one, which also is in many cases a completely adequate one, is shown in Fig.5. It consists of a capacitance (double layer capacitance) in parallel with a resistance (charge transfer resistance), and then, in series with this circuit, another resistance (solution resistance). The admittance of the dummy cell is recorded in an ordinary experiment and the transfer function, T(u), of the instrument is set equal to the ratio of the calculated, 0( )5 to the measured, ym( )) admittance of the dummy cell i.e. 

The reason why power sources exhibit transient voltage response and therefore deviate from Ohm s law is the presence of capacitive elements in parallel with resistive ones, for example, a double-layer capacitance in parallel with Re and pseudocapacitance associated with diffusion process in parallel with dififiision resistance. 

CKO The combination of a capacitance in parallel with the KO response model... 

It is fonnd from Figure 16.3 that, the dielectric constants of the composites are non-linearly dependent on volume % of BNN. This shows that the constituent capacitors formed by dielectrics fillers and polymer in the composites are not in parallel combination. From Figure 16.3, it is clear that the inverse of dielectric constant cnrve is not in a harmonic pattern, constituent capacitors formed by dielectrics fillers and polymer in the composites is not in series combinatiom One can choose to model composites as having capacitance in parallel (upper bound) or in series (lower bound). In practice, the answer will lie somewhere between the two. Physically, in composites with (0-3) structures which generally conform to special logarithmic equation, the relation assumes the form of Lichteneker and Rother s (Lich-teneker, 1956) more appropriate to composite stractures where the two-component dielectrics are neither parallel nor perpendicular to the electric field that is, the vahd averages are neither arithmetic nor harmonic. 

If electrostatic and chemisorptive forces are comparable, i.e., if the excess of qj[ is modest, separation of the associated capacitances would seem difficult. If chemisorption, however, is relatively strong, it should result in a substantial change in the interfacial-layer electrolyte composition over the voltage range of interest. The interface capacitance should then be dominated by it and should be simply related to the voltage derivative of the adsorption isotherm of the ion involved. In this case, the adsorptive charging resembles a large capacitance in parallel with other sources of interface capacitance, since the ions involved are free to distribute themselves between the electrolyte bulk and interface sites, irrespective of other capacitance sources. 

The direct measurement of the dielectric constant is straightforward. Indeed, the electrical equipment only requires a variable capacitor (of known capacitance) in parallel with the capacitor whose capacitance is to be determined (i.e., between whose plates is placed a known quantity of material whose dielectric constant is to be found). 

Figure 2-22 shows three types of rectifiers and their output-signal forms. Each uses semiconductor diodes (see Section 2C-2) to block current in one direction while permitting it in the opposite direction. To minimize the current fluctuations shown in Figure 2-22, the output of a rectifier is usually filtered by placing a capacitor with a large capacitance in parallel with the load Ri, as shown in Figure 2-23. The charge and discharge of the capacitor has the effect of decreasing the variations to a relatively small r/pp/c. In some applications, an inductor in series and a capacitor in parallel with the load serve as a filter this type of filter is known as an L section. By suitable choice of capaci-...

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