Capacitance in series

Instead of an exact calculation, Gouy and Chapman have assumed that (4) can be approximated by combining the Poisson equation with a Boltzmann factor which contains the mean electrical potential existing in the interface. (This approximation will be rederived below). From this approach the distribution of the potential across the interface can be calculated as the function of a and from (2) we get a differential capacitance Cqc- It has been shown by Grahame that Cqc fits very well the measurements in the case of low ionic concentrations [11]. For higher concentrations another capacitance in series, Q, had to be introduced. It is called the inner layer capacitance and it was first considered by Stern [1,2]. Then the experimental capacitance Cexp is analyzed according to ... 

For the impedance of the resistance and capacitance in series, the current will lead the potential phase angle by... 

Starting Capacitance for a Capacitor Motor. The starting capacitance for a capacitor motor is the total effective capacitance in series with the starting winding under locked-rotor conditions. 

The model more generally accepted for metal/electrolyte interfaces envisages the electrical double layer as split into two parts the inner layer and the diffuse layer, which can be represented by two capacitances in series.1,3-7,10,15,32 Thus, the total differential capacitance C is equal to... 

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 capacitance of the system can be written as that of four capacitances in series, i.e.,... 

Variations of resistance with frequency can also be caused by electrode polarization. A conductance cell can be represented in a simplified way as resistance and capacitance in series, the latter being the double layer capacitance at the electrodes. Only if this capacitance is sufficiently large will the measured resistance be independent of frequency. To accomplish this, electrodes are often covered with platinum black 2>. This is generally unsuitable in nonaqueous solvent studies because of possible catalysis of chemical reactions and because of adsorption problems encountered with dilute solutions required for useful data. The equivalent circuit for a conductance cell is also complicated by impedances due to faradaic processes and the geometric capacity of the cell 2>3( . 

The experimental impedance is always obtained as if it were the result of a resistance and capacitance in series. We have already seen in (11.20) and (11.21) the relation between an RC series combination and the Rct + zw combination. It can be shown for the full Randles equivalent circuit for this simple charge transfer reaction, see Fig. 11.4, on separating the in-phase and out-of-phase components of the impedance, that... 

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... 

We exemplify the use of vectors in the complex plane with a resistance and capacitance in series (Fig. A2.2a). The total potential difference is the sum of the potential differences across the two elements. From Kirchhoff s law the currents have to be equal, that is... 

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 simulated Nyquist plot of resistance and capacitance in series is a vertical line in the complex-plane impedance diagram, as shown in Figure 4.2(b). The effect of the parameter R on the position of the line is presented in Appendix D (Model Dl). 

Some elaborations are given In fig. 3.22. Figure 3.22a Illustrates, in line with [3.6.25], that for two capacitances In series the lower one plays the greater role in determining the overall value. For C o the purely diffuse limit is recovered, identical to the 10 M curve of fig. 3.5. Integration gives the [Pg.309]

Figure 3.30. Relation between the imaginary and real parts of the impedance (left) and admittance (right). The arrows indicate the direction of the increase of for V = R". Figure (b) is circular. Starting from the origin (where oj = 0) the frequency Increases. Solid lines for a resistance and ideal capacitance in series dashed for resistance and a constant phase element in series.

If the circuit can be regarded as consisting of resistance and capacitance in series, then ... 

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... 

Resistance and Capacitance in Series For the serial combination of a resistor and capacitor, the result is shown in Fig. 3(c). According to KirchhofFs law, the currents through both elements are the same. The total potential equals the sum of the potentials across the capacitor and resistor... 

Binding of metal ions to the dendrimer immobilized on the electrode is believed to affect the interfacial properties of the monolayer through ionic interactions. These changes are monitored by changes in the capacitance of the film [65,66]. The total capacitance as described by Eq. 6.3 is composed of a series of capacitances (Cjno oiayer inter ce fo 6d by thc protonatioii/deprotonafion of the COOH group at the pH 7.0) [40,67-69]. For a capacitance in series ... 

The operation of a PEC cell can be influenced by applying an external bias potential to the semiconductor. When the bias is applied with respect to a reference electrode, the potential difference will be distributed over the space charge layer and the Helmholtz layer. These layers act as two capacitances in series [41] ... 

---