Series RC circuit

Alternately, for potential-step experiments (e.g., chronoamperometry, see Section 3-1), the charging current is die same as that obtained when a potential step is applied to a series RC circuit ... 

At relatively low applied frequencies, a conductance cell may be represented as the double-layer capacitance Cs in series with the solution resistance R, as shown in Figure 8.8a. When a sinusoidal voltage es is applied to the series RC circuit, the instantaneous current i is the same in every part of the circuit and is given by... 

Figure 8.8 Series RC circuit (a) circuit (b) current-voltage relationships (c) frequency dependence of impedance Z and phase angle .

In a series RC circuit (R-C), according to the primary rules, the overall impedance, ZR C, is expressed as... 

The real and imaginary components Zre, Zim in the AC impedance of the series RC circuit are given by... 

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. 

Figure 2.19. Graphical representation of the AC impedance of a series RC circuit 2.4.2 AC Impedance of a Resistor-Inductor Circuit...

Electrochemical cells can be represented via an equivalent circuit formed by an association of impedances that pass current with the same amplitude and phase angle of the real cell under a given potential input. Thus, for a series RC circuit, the impedance and the phase angle are given by ... 

The variation of the impedance with frequency is often of interest and can be displayed in different ways. In a Bode plot, log Z and are both plotted against log cu. An alternative representation, a Nyquist plot, displays Zi vs. Zrc for different values of cu. Plots for the series RC circuit are shown in Figures 10.1.8 and 10.1.9. Similar plots for a parallel RC circuit are shown in Figures 10.1.10 and 10.1.11. 

Figure 10.1.8 Bode plots for a series RC circuit with / = 100 n and C = 1 fiF.

In the sections that follow, we investigate the res Kmsc of series RC circuits lo a sinusoidal ac voltage signal. rile input signal is descrilx d by Ivquation 2-24 that is. 

Hz, 2 mV rms perturbation with capacitance calculated assuming a series RC circuit, (b) The average fluorescence intensity and selected images of the surface which correspond to the labels in (b). The images A and F were taken before the interaction with the liposomes and are represented with differing gray-scale palettes. 

However, if we excite the same series RC-circuit with a controlled current step and record the voltage across the RC circuit, the voltage will increase linearly with time ad infinitum. The time constant is infinite. Clearly, the time constant is dependent not only on the network itself, but on how it is excited. The time constant of a network is not a parameter uniquely defined by the network itself. Just as immittance must be divided between impedance and admittance dependent on voltage or current driven excitation, there are two time constants dependent on how the circuit is driven. The network may also be a three-or four-terminal network. The time constant is then defined with a step excitation signal at the first port, and the possibly exponential response is recorded at the second port. 

Impedance is the inverse of admittance Z = 1/Y = 1/(G + jwC) = (G — jwC)/ Zp = R + jX. This corresponds to a series RC circuit where both components pass the same current. The voltage is the dependent parameter and divided between the R and C components. The voltage in the capacitor lags the current and the voltage across the resistor. By the same convention, this is defined as a negative phase shift and the voltage and impedance vector is clockwise to the real axis, X = — l/wC and for inductance X = +wL. 

The in-phase and out-of-phase AC current components are measured as a function of the frequency. The data are analyzed in terms of an equivalent circuit (Fig. 8) containing the uncompensated resistance, the interfacial capacitance, and a series RC circuit for the faradaic current component. Laviron has derived expressions for the adsorption resistance Ra and adsorption capacitance Ca, assuming that a is 0.5 [217]. At r] = 0V, the expressions are particularly simple (Eqs 12 and 13) ... 

Figure 4.5 3. Complex-plane or Nyquist plot for the impedance spectrum of a simple series RC circuit.

Fig. 2 Cyclic voltammetry (top panels) and differential capacity (bottom panels) recorded for Au(l 11) (left) and hanging mercury drop electrode (right) in contact with 0.05 M KCIO4 in the absence (dotted line) and in the presence (solid line-voltage scan in the positive direction, dashed Une-voltage scan in the negative direction) of CjsOH spread at the GS interface.CV 20 mV/s, capacity 5 mV rms, 25 Hz for Au(l 11) or 253 Hz for Hg, 5 mV/s sweep rate, a series RC circuit was assumed in the calculation...

Fig. 9 Differential capacity of Au(l 11), without (dotted line) and with adsorbed 12-AS, single, double and triple touch film (as labeled). The positive (solid hne) and negative (dashed line) directions of the voltage sweep are shown in each figure. The capacity was measured using 5 mV rms sine wave perturbation modulated at 25 Hz. A series RC circuit was assumed in determining the capacity...

The s -domain input-output transfer function if (s) relating the output (capacitor voltage) to the input source voltage in Fig. 1.4 is obtained by using voltage division with the generalized impedances (Ciletti, 1988) in the series RC circuit... 

Several important properties of a series RC circuit are illustrated by the results obtained in Example 2-4. First, the sum of the peak voltages for the resistor and the capacitor are not equal to the peak voltage of the source. At the lower frequency, for example, the sum is 21.1 V compared with 20.0 V for the source. This apparent anomaly is understandable when we realize that the peak voltage occurs in the resistor at an earlier time than in the capacitor because of the voltage lag in the latter. At any time, however, the sum of the instantaneous voltages across the two elements equals the instantaneous voltage of the source. 

FIGURE 2-8 (a) A series RC circuit. Time response of circuit when switch 5 is (b) in position 1 and (c) in position 2. 

To describe the properties of a capacitor, consider the series RC circuit shown in Figure 2-8a, which contains a battery V j, a resistor R. and a capacitor C, in series. The capacitor is symbolized by a pair of parallel lines of equal length. 

Impedance in a Series RC Circuit The impedance Z of an KC circuit is made up of two components the resistance of the resistor and the reactance of the capacitor. Because of the phase shift with the latter, however, the two cannot be combined directly but must be added vectorially, as shown in Figure 2-10. Here the phase angle for R is chosen as zero. As we have shown, the phase angle for a pure capacitive element is -90°. Thus, the X - vector is drawn at a right angle to and extends down from the R vector. From the Pythagorean theorem, the quantity Z. called the impedance, is given by... 

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