Capacitance calculation

Capacitors are often combined in series or parallel, with the resulting circuit capacitance calculated as depicted in Figure 4. An important relationship is the time constant of a capacitor. The time constant is based on the product of the resistance and capacitance and is known as the RC time constant. A capacitor in a dc circuit will charge or discharge 63.2 percent in one RC time constant. The time dependence of a capacitor is shown in the equations. 

The heterogeneity of a metal surface is responsible for the curvature of the Parsons-Zobel (PZ) plot (1/Cvs. 1/Q, where Cis the experimental capacitance and Qthe diffuse layer capacitance calculated on the basis of... 

Following the concepts of H. Helmholtz (1853), the EDL has a rigid structnre, and all excess charges on the solntion side are packed against the interface. Thus, the EDL is likened to a capacitor with plates separated by a distance 5, which is that of the closest approach of an ion s center to the surface. The EDL capacitance depends on 5 and on the value of the dielectric constant s for the medium between the plates. Adopting a value of 5 of 10 to 20 nm and a value of s = 4.5 (the water molecules in the layer between the plates are oriented, and the value of e is much lower than that in the bulk solution), we obtain C = 20 to 40 jjE/cm, which corresponds to the values observed. However, this model has a defect, in that the values of capacitance calculated depend neither on concentration nor on potential, which is at variance with experience (the model disregards thermal motion of the ions). 

FIGURE 10.2 Potential dependence of differential capacitance calculated from Gouy-Chapman theory for z+ = = 1 and various concentrations (1) 10 , (2) 10 , (3) 10 M. 

Figure 13. (a) Nyquist plots of the impedance spectra and (b) plots of reduced capacitance Cred vs. frequency CO experimentally measured on carbon specimens A (-o-), B (- -), and C (-A-) at an applied potential of 0.2 V (vs. SCE) in a 30 wt.% H2S04 solution. Here, the solution resistance was subtracted from the measured impedance spectra. The reduced capacitance in (b) was determined from the normalization of the capacitance with respect to the value of the capacitance calculated from the impedance spectra at 10 Hz. Reprinted from Ref. 17, Copyright (2006), with permission from Elsevier. 

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

The best-fit capacitances on each side of the PZC show the same trend as the capacitances calculated for the entire data sets (cf. Table 5.8), i.e. C increases with the ionic strength. Table 5.11 does not indicate any systematic relationship between C + and C., namely, in two sets Cis greater and in two other sets the C. is greater. The additional adjustable parameter somewhat improves the fit (compare Figs. 5.35-5.38 on the one hand and Figs. 5.21-5.24 on the other). 

Capacitances and resistances calculated for the two maxima at various bias potentials are shown in Table 1. Capacitances calculate from the high frequency maximum are of the order of 0.1 pF cm and increase with increasing cathodic potential. The time constant remains constant in the potential region studied. The value of the capacitance and the potential dependences are close to those expected for the space charge region and hence are assigned to this. 

The surface state capacitance calculated by a similar procedure is shown in Fig. 19 as a function of bias potential. Fig. 20 shows the surface states density as a function of bias potential. Surface states density is an order of magnitude less than that at the CdTe interface under similar conditions. This is consistent with the fact that the photocurrent for GaP is les compared to CdTe for the photoelectrochemical reduction of CO (cf. the model suggested of mediator surface states). The surface state density increases from its minimum at 0.2V NHE, a potential which lies close to the pzc value determined for the system (0.17V NHE). Correspondingly, the form of the surface state density as a function of bias potential resembles an adsorption isotherm. These results support the concept that surface dates are induced by adsorbed ions at the interface. 

Gao et al. [60] reported the CV and electrochemical impedance spectroscopy of self-assembled, stainless steel supported s-BLMs, using a three-electrode system. The membrane resistance and capacitance calculated from the whole impedance spectrum rather than at certain frequencies were close to the conventional BLMs. The time course of impedance under certain applied frequencies was investigated. The results showed that the membrane capacitance of s-BLM fluctuated under low frequency and gradually reached a constant value with the increase in applied frequency. Fullerene-doped s-BLMs were demonstrated to intensify this fluctuation behavior of membrane capacitance. A possible mechanism of this peculiar property of s-BLMs under low frequencies is discussed in Ref. 43. 

Hz, 2 mV rms perturbation with capacitance calculated assuming a series RC circuit, (b) The average fluorescence intensity and selected images of the surface which correspond to the labels in (b). The images A and F were taken before the interaction with the liposomes and are represented with differing gray-scale palettes. 

The parallel capacitance calculated from the data of Figure 7.16 is not a pure double-layer capacitance, it is too frequency-dependent. It must be due to redox or sorption processes at the platinum surface, or the fractal surface of black platinum. Under optimum conditions, it is possible to obtain capacitance values of the order of 50 pF/mm at 20 Hz (Schwan, 1963). 

FIGURE 3.12 Differential capacitance of the interface Hg/0.916 M NaF. The potentials are measured against a normal calomel electrode. (—O—), Experimental results ( —) inner layer capacitance calculated hy suhtracting the diffuse layer capacitance following the GC theory. (Adapted from Grahame, D. C., J. Am. Chem. Soc., 76, 4819-4823, 1954. With... 

IEEE Transactions on Microwave Theory and Techniques (capacitance calculation for different geometric configurations). 

The film thicknesses were corrected using the thermal correction factor of Wilson (Ref. 7). Roughness correction factors as calculated by Patir and Cheng, Ref. 8 were not considered. For the capacitance calculations, each of the loaded rolling element/raceway contacts was divided into five zones the inlet zone, the EHD zone, the outlet zone and... 

Together with the stray capacitance of the system, the total capacitances of the loaded and unloaded zone(s) were added up to give the total bearing capacitance at the particular set of operating conditions. Note that surface roughness effects were not taken into account in the capacitance calculations the bearing surfaces were assumed to be perfectly smooth. 

For the capacitance calculations, the contacts in the loaded zone were divided into five zones the inlet, outlet. Hertz and 2 side zones (see Fig.3). The total capacitance of a loaded IR contact, for example, is now ... 

Fill the cell with a dioxan/water mixture in the range of 10-60 volume %. Equilibrate at the same temperature and measure the capacitance, Calculate the dielectric constant,... 

Figure 25. On-line measurement of heat flow rate and capacitance calculated online as heat flux (—) with off-line measurements of cell density ( ) and changes in the concentrations of glucose ( ), glutamine (A) and IFN-y (+) for a batch culture of CHO 320 cells (Reproduced from Reference [22] with permission).

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