Parallel resistance-capacitance

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

Figure 1. The agreement of the spectral density of voltage fluctuations from valinomycin-modified phospholipid bilayers at equilibrium conditions (13, 14) with the Nyquist relation 1. An aqueous 0.01-M KCl solution at 33 °C was used in the experiments. Bilayer direct current resistances and valinomycin solution concentrations were 0.52-Mfl and 1.5 X 10 8 M (l), 0.19 Mfl and 5 X 10 8 M (2), and 0.055 Mi2 and 1.5 X 10 7 M (3). Solid lines are drawn in accordance with relation 1 for the impedance of a parallel resistance-capacitance (RC) circuit using foregoing resistance values and a value of membrane geometrical capacitance.

To investigate the main factors in the low-temperature (LT) power decline, the impedance spectra were first determined at different temperatmes (Figure 20). In the case of the impedance spectra for the cathode, the first and second semicircles, caused by the reactions in the SEl and the interfacial charge transfer combined with double layer charging, respectively, increased in size with lowering temperature while the inclined line due to the solid-state diffusion process became shorter. From the separated semicircles in the impedance spectra for the anode measured at the low temperatures, it is proved that at least four parallel resistive-capacitive elements were needed to properly model the reactions in the anode during battery operation, the basis for which is not yet clearly understood. 

Load inductance Load resistance Load parallel capacitance Load parallel resistance... 

Table 2. Phenomenological dielectric functions e C = dielectric constant and capacitance of reference system (air), JJ = low frequency limiting parallel resistance.

The bipolar pulse technique for measuring solution resistance minimizes the effects of both the series and parallel cell capacitances in a unique way. The instrumentation for this technique is illustrated in Figure 8.15. The technique consists of applying two consecutive voltage pulses of equal magnitude and pulse width but of opposite polarity to a cell and then measuring the cell current precisely at the end of the second pulse [18]. 

Bridge measurements provide the most direct way to compare an unknown impedance with known standard impedances. The impedance may be a pure resistance, capacitance, or inductance, or may be represented by some series or parallel combination. The DC Wheatstone bridge95 provides a simple and... 

Statistical Dispersion of Capacitance, ESR, and Parallel Resistance Values... 

The parameters which affect the spread of both capacitance and parallel resistance values are material and process variability, temperature gradient in the module, aging state, etc. 

The DLC is a physical component which has not only a requested capacitance, but also an unavoidable parasitic inductance due to its geometry, a series resistance caused by the electronic and ionic conductor ohmic resistances and a parallel resistance due to the leakage current between the electrodes. 

FIGURE 11.9 Basic capacitor electrical equivalent circuit comprising a capacitance, a series inductance, a series resistance, and a parallel resistance. This simple model can fit a DLC behavior in first approximation for a given 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... 

The second meaning of the word circuit is related to electrochemical impedance spectroscopy. A key point in this spectroscopy is the fact that any -> electrochemical cell can be represented by an equivalent electrical circuit that consists of electronic (resistances, capacitances, and inductances) and mathematical components. The equivalent circuit is a model that more or less correctly reflects the reality of the cell examined. At minimum, the equivalent circuit should contain a capacitor of - capacity Ca representing the -> double layer, the - impedance of the faradaic process Zf, and the uncompensated - resistance Ru (see -> IRU potential drop). The electronic components in the equivalent circuit can be arranged in series (series circuit) and parallel (parallel circuit). An equivalent circuit representing an electrochemical - half-cell or an -> electrode and an uncomplicated electrode process (-> Randles circuit) is shown below. Ic and If in the figure are the -> capacitive current and the -+ faradaic current, respectively. 

EIS data analysis is commonly carried out by fitting it to an equivalent electric circuit model. An equivalent circuit model is a combination of resistances, capacitances, and/or inductances, as well as a few specialized electrochemical elements (such as Warburg diffusion elements and constant phase elements), which produces the same response as the electrochemical system does when the same excitation signal is imposed. Equivalent circuit models can be partially or completely empirical. In the model, each circuit component comes from a physical process in the electrochemical cell and has a characteristic impedance behaviour. The shape of the model s impedance spectrum is controlled by the style of electrical elements in the model and the interconnections between them (series or parallel combinations). The size of each feature in the spectrum is controlled by the circuit elements parameters. 

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... 

The equivalent circuit of a semiconductor detector operated as a spectrometer is shown in Fig. 5.30. In most cases, effects of high resistance of the reverse-biased junction are negligible. If a zero-electric-field radiation-insensitive region is present in the detector, its impedance (a parallel RC combination) appears in series with the circuit and is indicated in Fig. 5.30 by the impedance Z. The impedance also accounts for any resistance (or resistance-capacitance combination) appearing in series with the contacts. 

The high-frequency limit is the same as is foimd for the series combination of the resistance Rg and capacitance C shown as equation (16.26). The parallel resistance R does, however, influence the value obtained for the imaginary part of the admittance at the characteristic angular frequency cOg = ReC), i.e.. 

The use of a small electrolyte covered resist area around the microelectrode is essential when capacitance measurements are performed, as the resist capacitance is parallel to the electrode capacitance. With a specific resistance of a = 1012 LI cm and a dielectric constant of e = 1.5 for the resist, and assuming a typical electrode capacitance of 10pFcm 2 with a 50 tm electrode, an error of 5% is obtained, if the electrolyte covered surface is 10 3cm2 (/=1000Hz) [88]. Thus, for capacitance measurements, the use of nanoliter droplets is essential. 

Figure 12.1 Simplest small-signal equivalent circuit representing the semiconductor/electrolyte junction under depletion conditions. = total series resistance, t p = parallel resistance due to charge transfer, Csd = space-charge layer capacitance.

AC Impedance of Contaminated Specimens. The ACIS of the contaminated sample under DC bias at 100% RH is consistent with a corroding system (15) in which a fixed number of aqueous pathways have formed, resulting in a constant area of metallization exposed to the electrolyte. In this case, the parallel capacitance corresponds to an electrical double layer of ions on the metallization. The capacitance of the contaminated sample is > 100 times larger than that of the clean sample at 100% RH due to the relatively larger concentrations of ions and water at the IC surface, which overwhelms the oxide capacitance described earlier. The reduction in the parallel resistance with increasing bias arises from the voltage dependent charge transfer process (i.e. electrochemical reaction). 

The revelation that Cp is indeed a novel capacitance requires the inclusion of two capacitances in the equivalent circuit shown in Figure 2 (36, 37). In essence, with the exception of a genuine short-circuit measurement, the macroscopically measured photosignal is a manifestation of the interaction of two circuits in parallel One circuit represents the photochemical event associated with bacteriorhodopsin, and the other circuit represents the resistance-capacitance (RC) relaxation event of the inert supporting structure —the membrane dielectric. 

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 /. 

Capacitance and dissipation factor Capacitance and equivalent series resistance Capacitanc e and equivalent parallel resistance Impedance and dissipation factor Admittance and dissipation factor... 

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