Effective capacitance

A more attractive representation of effective capacitance for electrochemical systems may be obtained directly from the imaginary part of the impedance as 

In contrast to the complex capacitance described in Section 16.3, the effective capacitance described in equation (16.41) is defined to be a real number. 

Example 16.3 Evaluation of Double-Layer Capacitance Find the meaning of the effective capacitance obtained using equation (16.41) for the convective-diffusion impedance expressed as equation (11.20), i.e., 

Solution At high-frequencies, all models for convective diffusion to a rotating disk approach the Vfarburg impedance, given as equation (11.52). Thus, the convective diffusion impedance can be expressed as Zo co) = Zd(0)/y/Jcor. Following Example 1.7, which 

After multiplying by the complex conjugate, the real and imaginary contributions are obtained as 

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Two-value capacitor motor. A two-value capacitor motor is a capacitor motor using different values of effective capacitance for the starting and running conditions. 

Starting Capacitance for a Capacitor Motor. The starting capacitance for a capacitor motor is the total effective capacitance in series with the starting winding under locked-rotor conditions. 

Chemical capacitance. When the mechanism involves significant involvement of the bulk, accumulation of reactive intermediates not only involves surface species but oxidation and reduction of the bulk. This can be detected as an anomalously high effective capacitance, often referred to as a chemical (or pseudo) capacitance. This capacitance can be as large as 0.1 — 1 F/cm and thus easily detected by current-interruption or impedance techniques. Thus, capacitance is a strong indicator (independent of resistance) as to what degree the interface, surface, and/or bulk are playing in the... 

Table I shows the details of surface treatment and coating along with the calcxilated values of total resistance R and effective capacitance C. For specimens with Initial mechanical surface preparation, the Nyqulst Impedance plot shows the characteristic semicircular behavior with a resistance of the order of 1800 n cm and a capacitance of about 40 yF cm. As different surface treatments are Incorporated on a sequential basis, the complex plane diagram shows a gradual evolution.

It must be emphasized again that the mid-peak potential is equal to E° for a simple, reversible redox reaction when neither any experimental artifact nor kinetic effect (ohmic drop effect, capacitive current, adsorption side reactions, etc.) occurs, and macroscopic inlaid disc electrodes are used, that is, the thickness of the diffusion layer is much higher than that of the diameter of the electrode. 

Dayhoff [50] suggested that one might measure a rest mass of photon by designing a low-frequency oscillator from an inductor-capacitor (LC) network. The expected frequency can be calculated from Maxwell s equations, and this may be used to give an effective wavelength for photons of that frequency. He claimed that one would have a measure of the dispersion relationship at low frequencies. Williams [51] calculated the effective capacitance of a spherical capacitor using Proca equations. This calculation can then be generalized to any capacitor with the result that a capacitor has an additional term that is quadratic in the area of the plates of the capacitor. However, this term is not exactly the one that Dayhoff referred to. But it seems to be a very close description of it. One can add two identical capacitors C in parallel and obtain the result... 

P. Temple-Boyer, J. Launay, G. Sarrabayrouse and A. Martinez, Amplifying structure for the development of field-effect capacitive sensors, Sens. Actuators B Chem., 86(2-3) (2002) 111-121. 

As an example, consider line A of Table 9.1 for the timescale of the cell charging time due to so called double-layer capacitance. As discussed later in this chapter, effective capacitance values are on the order of 1 Farad m-2. If current densities of a cell are on the order of 1.0 A cm-2, and representative over-potentials at the triple-... 

Volume 21, Part C, is concerned with electronic and transport properties, including investigative techniques employing field effect, capacitance and deep level transient spectroscopy, nuclear and optically detected magnetic resonance, and electron spin resonance. Parameters and phenomena considered include electron densities, carrier mobilities and diffusion lengths, densities of states, surface effects, and the Staebler-Wronski effect. 

Diffusion resistances can occur for Li in the electrode, but also for the salt in the electrolyte (if anion conductivity in the electrolyte is significant). Further effects are due to depletion of carriers at a phase boundary. In such cases, time dependencies of the electrical properties occur (in addition to Rs, effective capacitances Cs also appear). The same is true for impeded nucleation processes. Since any potential step of the electrochemical potential can be connected with current-dependent effective resistances and capacitances, the kinetic description is typically very specific and complex. As the storage processes in Li-based batteries are solid-state processes, the... 

The recommended procedure for determining the dielectric constant of an unknown gas with the cell described here depends on the relationship of changes in the capacitance of the cell to changes produced in the effective capacitance in the tank circuit. These are not the same because other capacitances are present besides that of the capacitor contained in the cell, Cceii. These always include the capacitance of the shielded cable connecting the cell to the oscillator and stray capacitances in the oscillator tank these are in parallel with the cell, and lumped together may be called In addition there is in the circuit described... 

These techniques can also be developed further to deal with Stern layer problems. Note that for Z) the effective capacitance distance D is reduced by a factor (1 —0.62y/alD). If image effects are included, it can be shown that (when the zwitterions are immersed in a high dielectric medium e = 80, adjacent to a hydrocarbon surface s = 2), image effects can virtually double the electrostatic energy so that the two effects may partially cancel out. Nonetheless the observation that is reduced from the intuitive capacitance form is of some interest, as it is known that for ionic micelles use of Debye-Huckel theory (at low salt equivalent to the capacitance form) gives results for the repulsive free energy too large by a factor of 2. 

Figure 8.12 The effective capacitance calculated using equation (16.41) and the imaginary impedance given in Figure 8.11(b),...

As suggested by Figure 14.4, the double-layer capacitance at the bare surface and at a surface covered by the reaction intermediate can be given as Cdi,6 and Cdi,7, respectively. Under the assumption that these double-layer capacitances are different, the effective capacitance can be given by... 

Linear and logarithmic plots of effective capacitance are presented as functions of frequency in Figures 16.14(a) and (b), respectively. The high-frequency asymptote... 

Remember 16.5 The effective capacitance provides a means of quantitatively determining the interfacial capacitance of a system. 

Figure 16.14 Effective capacitance as a function of frequency for Re — 10 flcm, R = 100 flcm, and C = 20 f4F/cm. The blocking system of Table 16.1(a) is represented by dashed lines, and the reactive system of Table 16.1(b) is represented by solid lines. The characteristic frequencies noted are/j c = (2nRC) and fc = (2nReC) a) linear scale and b) logarithmic scale.

Thus, the high-frequency limit of the effective capacitance can be used to obtain the doublelayer capacitance for even quite complicated systems. The reason this approach works is that, at high frequencies, the Faradaic current is blocked, and all current passes through the double-layer capacitor. 

Remember 17.3 Calculation of an effective capacitance or CPE coefficient according to equation (17.7) yields, in the high-frequency limit, properties associated with the electrode under study. 

The conductance of a tunnel junction array is determined by three energy contributions the thermal energy kBT, the electric potential energy eV at bias voltage V and the charging energy ec = e2/2Ceff, where Ceff is the effective capacitance of the array and sc = [(At - 1)/N e2/C with N the number of junctions in series. 

Basically, we are using the gate charge factors to tell us what the effective capacitances are (and the voltage swings from 0 to Vin). We see that the effective input capacitance (Ciss), for... 

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