Resistance and capacitance in parallel

The circuit is shown in Fig. A23a. The total current, /tot, is the sum of the two parts, the potential difference across the two components being equal  

We need to calculate the vectorial sum of the currents shown in Fig. A2.3b. Thus 

This is easily separated into real and imaginary parts via multiplication by (1 — itoRC). Thus 

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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. 

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.

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]

FIGURE 12.12 Equivalent circuits with resistance and capacitance in series (a) and in parallel b). 

To find the relation between the values of R and measured experimentally in terms of the circuit of Fig. 12.11a and the parameter values in the circuit of Fig. 12.14a, we must first convert [with the aid of Eq. (12.23)] the parameters of the circuit with parallel elements Ry and Q into the parameters of a circuit with a resistance and capacitance in series, and to the value of resistance obtained we must add R.. As a result, we have... 

The membrane itself can be considered as a resistance and capacitor in parallel, which will be discussed later in this chapter. The resistance and capacitance can be measured using AC impedance, which yields a semicircle from which the values can be extracted. 

In almost all SCP cases of interest, the time constant = RqoC will be appreciably shorter than other time constants present in the system. Furthermore, in most cases, C will also be much smaller than other capacitances present. Undlr these conditions, the parallel combination of C and R may be placed in series with the rest of the circuit to a high degree of approximation. For the special Cases A and B already discussed in connection with Fig. 3, the rest of the circuit may be well approximated by another resistance and capacitance in series, yielding an N = 2 Voigt-model circuit. Finally, whenever one... 

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. 

We see that in this case the phase shift is 7t/4 (45°). This phase shift corresponds to the circuit shown in Eig. I2.l5b, which includes the resistance and a capacitance for which = R hence, tan a = 1 and = 2 - R (it does not matter in this case whether the capacitance and resistance are connected in parallel or in series). It follows from Eq. (12.27) that... 

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. 

The admittance in this region is modeled using either a film capacitance in series with the solution resistance or a parallel resistance and capacitance, which is in series with the solution resistance. This is treated as a parallel R-C combination whose magnitude, expressed as an impedance, is given by... 

Z(a)i,ak) = Zre(a)i,ak) +jZim(cai,ak) is the model function, which can be altered using the adjustable parameters the model function can often be presented by an equivalent circuit, involving such elements as resistance, capacitance, and Warburg in series and/or in parallel ... 

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... 

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. 

Derive formulas for converting a parallel resistance-capacitance network (Rp and Cp in parallel) to a series equivalent (Rg and C in series). 

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

In this model, the polyaniline chain is assimilated to a series network of parallel resistances and capacitances (Figure 8.21). The resistances./ (/) and the capacitances C(0 will correspond to the emeraldine salt part (the conductive part) and will depend on the emeraldine salt segments length /. The resistance / (/ ) and the capacitances C (/ ) correspond to the emeraldine base part (the insulating part) and depend on the emeraldine base segments length /. 

The conductivities of PAn films were also examined with a.c. impedance measurements. Glarum and Marshall [209] reported that the oxidized PAn behaved like a series combination of resistance and capacitance and obtained very similar potential dependencies of equivalent series conductance to those obtained by in situ measurements by Wrighton et al. [28i]. Two different time constants of the R-C circuit were detected in their work, which they related to ionic and electronic conductivities. Rubinstein et al [210], who used a parallel combination of a capacitor with a... 

The leftmost lumped subcircuit (Rcb,p1 bQ) contains the geometrical capacitance, the bulk ionic conductance, and the conduction of ions parallel in the grain boundaries. The next (Rgb,sQgb) contains the resistance and capacitance across grain boundaries, and where g is a constant phase element [37], often with n A reflecting a dispersion of... 

In order to illustrate and simulate such behavior, we can build and calculate the parameters of an electrical circuit made up of two resistances R and r, and a capacitance C (see Figure 6.44). To approach the conditions that characterize a sintered material, we comiect R and C in parallel, while r and the RC circuit are connected in series the impedance of the circuit is calculated in the following way ... 

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. 

Randles model are used to describe the frequency dependence of diffusion and the capacitive impedance observed in the intermediate and low frequency ranges. A dual transmission line model has been proposed by including ionic and electronic resistance rails connected in parallel with a capacitance Cp (Fig. 6.5). The model has been used to define the electrochemical behavior of polyaniline, and the capacitance was explained as a result of oxidation and reduction of the pol5mier. Ionic (i i) and electronic (Rg) resistances are used to describe hindered motion of ions and electrons in the system, respectively. The impedance behavior has been found to be dependent on the ratio of the two resistances. 

Figure 11.4 shows the OLED model used in this Chapter. The model consists of a series resistance and a diode parallel with a capacitor. The capacitor models the total capacitance of the layers, the series resistance models the total resistance of the device and the diode models the rectifying nature of the OLED, the model is based on the... 

Ammeters must be connected In series with the load, and voltmeters in parallel across the load, as shown in Fig. 4.17. The power in a resistive load may be calculated from the readings of voltage and current since P = VI. TTiis will give accurate calculations on both a.c. and d.c. supplies, but when measuring the power of an a.c. circuit which contains inductance or capacitance, a wattmeter must be used because the voltage and cument will be out of phase. 

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