Circuit calculations parallel

Figure 4.7a shows a resistor and an inductor in parallel connection. The impedance of the circuit in parallel is calculated as... 

Since all three generators are operating in parallel, the impedance of each circuit, as calculated above, will fall in parallel and the equivalent impedance will become... 

There will be three such parallel circuits to make it 60 kVAr. To calculate equivalent capacitance and reactance in delta, we may convert it into an equivalent star as shown in Figure 23.24(b) by maintaining the same line parameters as in Figure 23.24(a). If the impedance of each phase in delta is Z, then to maintain the same steady-state line current, 4 , in star also let the equivalent impedance of each phase in star be Z. Then,... 

It is unusual to be able to And one capacitor to handle the entire ripple current of the supply. Typically one should consider paralleling two or more capacitors (n) of I/n the capacitance of the calculated capacitance. This will cut the ripple current into each capacitor by the number of paralleled capacitors. Each capacitor can then operate below its maximum ripple current rating. It is critical that the printed circuit board be laid out with symmetrical traced to each capacitor so that they truly share the current. A ceramic capacitor ( 0.I pF) should also be placed in parallel with the input capacitor(s) to accommodate the high frequency components of the ripple current. 

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 methodology for the calculation of the complex relative permittivity for the dipolar relaxation mechanism is founded on the calculation of the dielectric response function, f(t), for a depolarization produced by the discharge of a previously charged capacitor. In Figure 1.29a, a circuit is shown where a capacitor is inserted in which a dipolar dielectric material is enclosed in the parallel plate capacitor of area, A, and thickness, d, with empty capacitance C0 = Q0/U0 = 0(A/d), and E0 = U0ld. In Figure 1.29b, the corresponding depolarization process is shown. 

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) ...

According to the above calculations, a graphical representation of the AC impedance of a parallel CL circuit is depicted in Figure 2.22. In the complex plane, the AC impedance of the parallel CL circuit is represented by a straight vertical line on the Z"-axis with a constant Z value of zero. 

The circuit elements can be connected in series or in parallel. The basic rule for the calculation of the circuits is for an electric circuit with elements in series connection, the total impedance is the sum of the impedances of the individual elements for an electric circuit with elements in parallel connection, the total... 

We can therefore calculate the relative permittivity and dielectric loss of a material from measured values of either equivalent series or parallel circuit components of a specimen. 

The reciprocal of the specimen resistance in the equivalent parallel circuit for a given frequency is sometimes called the specimen conductance GP. It is a combination of DC conductance, by which we mean any real flow of charge through the sample under the influence of the applied field, and the anomalous conductance due to any time-dependent polarisation processes. The contribution that a true DC conductivity dielectric loss at an angular frequency w can be readily calculated as follows for the material in a parallel-plate capacitor. If the capacitor plates have area A and separation s ... 

The impedance of the skin has been generally modeled by using a parallel resistance/capacitor equivalent circuit (Fig. 4a). The skin s capacitance is mainly attributed to the dielectric properties of the lipid-protein components of the human epidermis [5,8,9,12]. The resistance is associated primarily with the skin s stratum comeum layer [5,8,9,12]. Several extensions to the basic parallel resistor/capacitor circuit model have appeared in the literature [5,8,9,13]. Most involve two modified parallel resistor/capacitor combinations connected in series [5,8,9]. The interpretation of this series combination is that the first parallel resistor/capacitor circuit represents the stratum comeum and the second resistor/capacitor parallel combination represents the deeper tissues [5,8,9]. The modification generally employed is to add another resistance, either in series and/or in parallel with the original parallel resistor/capacitor combination [8,9]. Realize that because all of these circuits contain a capacitance, they will all exhibit a decrease in impedance as the frequency is increased. This is actually what is observed in all impedance measurements of the skin [5,6,8-15]. In addition, note that the capacitance associated with the skin is 10 times less than that calculated for a biological membrane [12]. This... 

After discussing the generation and quantitation of the potential term of the proton circuit, we shall now turn to the proton current, and examine the factors which control the flux of protons around the circuit. Although it is not possible to determine the proton current under steady-state conditions directly, the parameter may be calculated indirectly from the respiratory rate and the stoicheiometry of proton extrusion by the respiratory chain. It is outside the scope of this chapter to discuss the contentious issue of the proton stoicheiometries of the complexes, but the important feature is that, unless the complexities of variable stoicheiometry are invoked, respiration and proton current vary in parallel. 

The objective function, equation (19.3), is presented in Figure 19.1(a) for an RC circuit as a function of parallel resistor and RC-time-constant values. The circuit parameters were R = 1 Ocm, and trc = 1 s (see, e.g.. Figure 4.3(b) and the corresponding Example 4.2). The synthetic data were calculated for frequencies ranging from 1 to 10 Hz at a spacing of 10 points per decade, and the noise was determined by machine precision. 

For any pair of non-adjacent vertices, the resistance distance is the effective resistance calculated according to the two classical Kirchhoff laws for series and parallel electrical circuits some examples of calculations are given in Box R-1 for ethylbenzene. 

The behavior of a resistor in parallel with an ideal capacitor (see above) is recovered when n is 1 (Q = C). When n is close to 1, the CPE resembles a capacitor, but the phase angle is not 90°. The real capacitance can be calculated from Q and n. When n is zero, only a resistive influence is found. For all impedance spectra shown in this work, fitting with a single RC circuit was found to be sufficient, i.e., n was in all cases larger than 0.9. Figure 11.10 shows that a good accordance of measuring data and fit function is evident. 

If we permit charge penetration to the interface, the simplest equivalent circuit diagram is then a parallel circuit composed of R t and Cai in series with Rsuik. Its impedance is calculated as ... 

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 aim of network analysis is the investigation of the amplitude and phase response of a two- or four-port network. Impedance analysis determines the complex impedance or admittance of a device. This method is appropriate for quartz resonators in order to obtain more complete information than is conceivable by merely considering the shift of the resonance frequency. The method especially allows the determination of the equivalent circuit elements (BVD) presented in Fig. 8. Actually many commercial instriunents directly provide this information. Determination of the physical parameters, or their effective values, for accurate modeling of the sensor behavior based on Eq. 5 requires mathematical procedures which fit the calculated curves (e.g., with Eq. 2) to the experimentally measured values. It is recommended to include an external capacitance parallel to Co to accoimt for uncompensated para-... 

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