Circuit calculations series

Figure 5.25c represents an equivalent circuit that consists of resistance and doublelayer capacitance in series. Equation (5.66) is used to calculate the impedance. For an R-C circuit in series, Zl = R and Z = — j/o)C. The phase angle 6 varies between 0° and 90°, depending on the frequency used in the measurement. Equation (5.66) when 1/2... 

In Figs. 4.19-4.25 we display typical results from the SigmaPlot analysis for a range of oxidation potentials of a polypyrrole film in aqueous solution. Some data are plotted in Fig. 4.13. In each case we obtain a very good fit of experimental data with results calculated from the derived parameters. Hence the Case A circuit involving the Randles circuit in series with the transmission line provides an excellent explanation of results at any particular potential. In Table 4.2 we collect results from the analysis. In addition to the Case A circuit, we include an... 

Figure 10 Typical modeling of experimental results (Li electrode in PC-1.5 M LiAsF solution after 24 h) by equivalent circuit of five RC circuits in series. Both the Nyquist and the Bode plots are shown. Dashed line, experimental results. Solid line, calculated response. Reprinted with copyright from The Electrochemical Society Inc. (See [72].)...

A series of calculations for open circuit gas turbines, with realistic a.s.sumptions for various parameters, have been made using a code developed by Young [7], using real gas tables. These illustrate how the analysis developed in this chapter provides an understanding of, and guidance to, the performance of the real practical plants. The subscript G here indicates that the real gas effects have been included. 

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. 

Dielectric losses arise from the direct capacitive coupling of the coil and the sample. Areas of high dielectric loss are associated with the presence of axial electric fields, which exist half way along the length of the solenoid, for example. Dielectric losses can be modeled by the circuit given in Figure 2.5.3. The other major noise source arises from the coil itself, in the form of an equivalent series resistance, Rcoii. Exact calculations of noise in solenoidal coils at high frequencies and small diameters are complex, and involve considerations of the proximity and skin depth effects [23],... 

Now, if R is much greater than R2, we can assume that R2 is zero without compromising the accuracy of the rate calculation. In electric circuits, two resistances applied in series are simply added together in calculating the line resistance. The same is true for resistance to chemical transport. If is 1,000 resistance units and R2 is 1 resistance unit, we can ignore R2 and still be within 99.9% of the correct answer. For most environmental transport and fate computations, it is sufficient to be within 99.9% of the correct answer. 

One theme of this discussion can now be stated as follows when transport processes occur in series, it is the slower transport processes that are important for chemical transport calculations, because the resistance to transport is large, just as the large resistors of a series in an electronic circuit are the most important. 

Knowing the constant current imposed on the circuit and the final potential in the C-D region, the value of the overpotential corresponding to the current can be obtained. Repeating the measurement at a series of constant current densities allows determination of the t0 and the a value of Tafel s equation. If is available, the corresponding rate constant k0 can be calculated from the equation i0 = Fk0%c0, where X is the thickness of the reacting layer. 

In the above example, by changing the capacitor bank to a 500-kVAR unit, the resonance frequency is increased to 490 Hz, or the 8.2 harmonic. This frequency is potentially less troublesome. (The reader is encouraged to work out the calculations.) In addition, the transformer and the capacitor bank may also form a series resonance circuit as viewed from the power source. This condition can cause a large voltage rise on the 480-V bus with unwanted results. Prior to installing a capacitor bank, it is important to perform a harmonic analysis to ensure that resonance frequencies do not coincide with any of the characteristic harmonic frequencies of the power system. 

The open circuit Ecm values for each metal are the entries in the traditional galvanic series. Kinetic information is also available via analysis of the polarization curves. The 4ouPie can be used to calculate the increased corrosion rate of Metal 2. Because of the coupling to Metal 1, the dissoultion rate has increased from 4cathodic kinetics on Metal 1 must now be satisfied. In addition to determining the increase in the corrosion... 

According to the above calculations, a graphical representation of the AC impedance of a series RC circuit is presented in Figure 2.19. As shown in the complex plane of Figure 2.19, the AC impedance of a series RC circuit is a straight vertical line in the fourth quadrant with a constant Z value of R. 

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... 

Figure 11. Plots of log Z and / v.v. log / for a thiol-hexapeptide-coated mercury drop immersed in 5xlO 3M (a), I.3xlO 2M (b), 3.6xlO 2M (c), and 0.1M (d) KC1, as obtained at -1.000 V over the frequency range from 0.1 to 105 Hz. At frequencies <102 Hz all Bode plots coincide hence, only the experimental points for the lower KC1 concentration were reported. The solid curves are least-squares fits to the simple equivalent circuit of inset (1), which consists of the electrolyte resistance Ra, with in series a RSCS mesh representing the self-assembled monolayer and a further RjiCji mesh representing the diffuse layer. Rs = 0.14 Mfi cm2 C, = 11 pF cm-2 Ra = 4.53 (a), 4.17 (b), 1.27 (c) and 0.87 KO cm2 (d). CW 68 (a), 61 (b), 80 (c) and 84 pF cm 2 (d). Inset (2) shows the reciprocal, 1/Cji, of the experimental diffuse-layer capacitance vs. the l/C fajj = 0) value corresponding to the same KC1 concentration, as calculated on the basis of the Gouy-Chapman (GC) theory. The solid curves are 1 /Ca(OM) vs 1 /C,ii(ctm = 0) plots calculated from the GC theory for different charge densities afo on the metal, whose values are reported on each curve. (Reprinted from Ref.114 with permission from the Am. Chem. Soc.)...

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 equivalent circuit for the calculation of the differential electrode capacity is shown in Fig. 2. It consists of a series resistance Rg, which represents the internal ohmic resistance of the germanium disk and of the metal contact. Since... 

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... 

The change in rf voltage at the detector due to a change in the loaded Q may be calculated easily from the equivalent lumped circuit for the resonator, where the resonance absorption may be modeled as a small series resistance 8R. The detector voltage is... 

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. 

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