Resistor/capacitor parallel circuit

The equivalent impedance of the resistor/capacitor parallel circuit in Fig. 6.14(a), Z, must be determined by application of Kirchhoff s rule (i.e., the algebraic sum of the voltages of the voltage sources in any circuit loop must equal the algebraic sum of the voltage drops in the same loop). Thus ... 

In a parallel resistor-capacitor (RC) circuit (R/C), the overall AC impedance of the circuit is denoted as ZR/C. 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 overall admittance (Equation 2.3) for a parallel resistor-capacitor (RC) circuit is given by the sum of the conductance (l/R) and capacitance contributions, where the resistance (R) represents the dissipative component of the dielectric response, while the capacitance (C) describes the storage component. The impedance function for that circuit is... 

The impedance behavior can be described by Debye s formula for a serial-parallel resistor-capacitor (RC) circuit [27] with elements that correspond to the dielectric behavior of different components. The high-temperature impedance behavior can be described by a series of triple parallel RC circuit elements [27]... 

In the parallel configuration, the same potential difference occurs across each and every element with the total current being the algebraic sum of the current flowing through each individual circuit element. Table 2-35 summarizes the equivalent resistance, conductance, capacitance, and inductance of series-parallel configurations of resistors, capacitors, and inductors. 

A more realistic picture of the double-layer has an RC element (that is, a capacitor and resistor in parallel) itself in series with a second resistor Rs (see Figure 8.11(d)). This circuit yields a similar Nyquist plot to that of an RC element... 

Each of these layers behaves just like an RC element (that is, a capacitor and resistor in parallel) within the equivalent circuit (see Figure 8.13). The respective values o/R, and C, will be unique to each RC element since each layer has a distinct value of [H ]. In order to simplify the equivalent circuit, this infinite sum ofRC elements is given the symbol Zw or -W and is termed a Warburg impedance, or just a Warburg . The Warburg in Figure 8.12 extends from about 50 down to 15 Hz. 

It should also be mentioned that capacitors were then added in parallel with the resistors in equivalent circuit elements because the frequency-dependent experimental electrical impedance data had a component that was 90° out of phase with the resistor. 

Commercial impedance analyzers offer equivalent circuit interpretation software that greatly simplifies the interpretation of results. In this Appendix we show two simple steps that were encountered in Chapters 3 and 4 and that illustrate the approach to the solution of equivalent electrical circuits. First is the conversion of parallel to series resistor/capacitor combination (Fig. D.l). This is a very useful procedure that can be used to simplify complex RC networks. Second is the step for separation of real and imaginary parts of the complex equations. 

When a voltage step is applied to the simple RC parallel circuit shown in Fig. 2.54 the response current decays to zero in a manner describable by a single relaxation time. The frequency response of the impedance also yields a semicircle as shown below. Such a circuit can represent a lossy capacitor, and more elaborate combinations of resistors and capacitors correspondingly more electrically complex materials and systems. It is this rather more general approach which is described by impedance spectroscopy . 

Fig. 12L Complex-plane representation of the impedance vector as a function of frequency for a simple circuit, consisting of a capacitor and resistor in parallel.

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. 

Now we apply Kirchhoff s voltage and current laws. For this parallel circuit, the voltage drop across each branch must be equal, and hence all the voltages are equal to V, the voltage across the junction. Hence the current through the capacitor equals CV and the current through the resistor equals V/R. The sum of these currents and the supercurrent sin 0 must equal the bias current I hence... 

Parallel The impedance for a resistor and a capacitor in parallel shows the shape of a semicircle in the complex plane diagram (Fig. 3d). According to KirchhofFs law for a parallel circuit, the potentials across both circuit elements are equal, while the total current can be calculated from the sum of the currents flowing through resistor and capacitor... 

A variety of bridges have been used for immittance measurements. The general principle of an AC bridge is illustrated as an admittance bridge in Figure 8.23, where Yi = 1/Zi is a parallel circuit of a variable resistor and a variable capacitor, and Y2 = I/Z2 is the measured sample. 

Impedance is the preferred parameter characterizing the two resistors, one capacitor series circuit, because it is defined by one unique time constant Xz (Eq. (12.8)). This time constant is independent of R, as if the circuit was current driven. The impedance parameter therefore has the advantage that measured characteristic frequency determining Xz is directly related to the capacitance and parallel conductance (e.g., membrane effects in tissue), undisturbed by an access resistance. The same is not true for the admittance the admittance is dependent both on xz and X2, and therefore on both R and G. 

Both Z and Z can be combined in a single plot A Nyquist plot is obtained by plotting Z on the horizontal axis and Z on the vertical axis. An example of a Nyqnist plot is illustrated in Figure 5. As compared to a Bode plot, a Nyquist plot does not indicate the frequency response of a material directly. A Nyquist plot represents the electrical characteristic of a material. This electrical characteristic can be represented by an equivalent circuit that may consist of a resistor and capacitor, resistor in series with capacitor, resistor in parallel with capacitor, and so oa... 

Another fault scenario may consider the degradation of the capacitor. Reference [19] lists various causes for a failure of an electrolyte capacitor and considers the current ripple which causes internal heating, i.e. an increase of the core temperature which results in a gradual aging of the capacitor. Another possible cause for a failure of the capacitor is a leakage current that may lead to a short circuit. Such a leakage can be accounted for by adding a resistor in parallel to the capacitor. 

Algebraic Equations Modeling Two Electrical Circuits, One with a Capacitor and a Resistor in Parallel (Left Column), the Other with a Self-Inductance in Series with a Resistor (Right Column)... 

The above results show that the = 1 - a parameter which appears in the eARC Cole-Cole function, Eq. (20), associated with a CPE and ideal capacitor in series, and the t/s appearing in the ZARC and YARC functions, Eqs (25) and (27), associated with a CPE and resistor in parallel or in series, may all be interpreted as the t/rof a CPE. The t/r values estimated from fitting with these forms are thus comparable. Although the CPE has sometimes been found in equivalent circuit data fitting to appear separately and not directly in any of the above compound forms (e.g. Macdonald, Hooper, and Lehnen [1982]), its presence as a direct part of the eARC, ZARC, and YARC functions, ones which have long been used in the inter-... 

If a voltage U is applied to one of the X conductors, for example row X2, and one of the Y conductors, say column Y2, the lattice element corresponding to the intersection of X2 and Y2 sees the full voltage U. In the equivalent circuit diagram Fig. 39b, the liquid crystal is represented by a capacitor with a resistor in parallel. As a result of parasitic currents, voltages also appear at the other lattice elements, especially those in row X2 and column Y2, but they are always less than U/2. These elements are therefore also activated for instance to dynamic scattering, although much weaker. 

Then the system in figure 3 was implemented with the Howland current source described before. A RC parallel circuit worked as our load impedance. An 100 nF capacitor was in parallel with a variable resistor. An algorithm to obtain the real part of the impedance was realized in the DSP given the linearity between Vref and the current generated. Figure 7 compares the DSP s estimated real part from the data measured with the codec with theoretical values and values obtained from the data measured with the oscilloscope. The EVM responds correctly to changes in the load impedance. 

Fig. 7 Comparison of real part estimated in EVM, Oscilloscope and theor cal for a RC parallel circuit with 100 nF capacitor and variable resistor.

For example, whilst modelling high frequency noise in nanopore devices, Smeets el al. demonstrated the necessity of accounting for non-ideal capacitive behaviour and it has become common since for the generalised device capacitance to be replaced by either a parallel resistor-capacitor unit or a constant phase element [23]. Chien et al. used the circuit shown in Fig. 14 B, replacing C with a CPE, to determine pore resistance and diameter from Bode plots for Si pores. [42]. Pedone et al. used two parallel resistor-CPE units in series with a solution resistance for characterising the electrical properties of a pore-cavity-pore deviee [43]. 

An actual plot of impedance in the complex plane for the polyethylene oxide NaSCN complex (n = 22, x = 2.0) is shown in Figure 5. This impedance plot represents an equivalent electrical circuit having a capacitor and a resistor in parallel with no resistor in series because the semicircle starts right at the origin. Here, the diameter of the semicircle gives the resistance of the circuit, the capacitance value is given by the formula ... 

Fig. 19 Equivalent circuit models for caibon-based porous electrodes RC circuits for a series and b parallel connections, representing an equivalent circuit (simplest) of a capacitor. R resistor, C capacitor. Equivalent circuits of only one capacitor (Cdl or CP) in parallel to a resistor R and in series to resistor RS (c) and considering both Cdl (in parallel to RE) and CP (in parallel to RE ) in series with RS (d) are also shown. The ac responses to the latter two circuits are shown in (e, f) [33] (Reprinted with permission from Ref. [33] Copyright (2012) by John Wiley and Sons)...

The simplest description of this behaviour is a parallel circuit of a resistor and a capacitor. The conductive and dielectric properties are coupled complex and frequency-dependent properties ... 

The various resistors, capacitors, etc., which are connected together in series and parallel combinations to form the equivalent circuit that corresponds to the test-cell, contribute to the real and imaginary parts in a way that is quite easy to calculate. The frequency-dependent information can be displayed in the form of Z or Z" as functions of the angular frequency, (o, or of log (co). 

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