Ideal capacitor

The electrical double layer resembles an ordinary (parallel-plate) capacitor. For an ideal capacitor, the charge (q) is directly proportional to the potential difference ... 

The term 47r(qAzA+ qBzB) is a free-charge term, resembling that of an ideal capacitor of surface charge density qA = -qB and interplanar spacing zA - zB. 

The electric field or ionic term corresponds to an ideal parallel-plate capacitor, with potential drop g (ion) = qMd/4ire. Itincludes a contribution from the polarizability of the electrolyte, since the dielectric constant is included in the expression. The distance d between the layers of charge is often taken to be from the outer Helmholtz plane (distance of closest approach of ions in solution to the metal in the absence of specific adsorption) to the position of the image charge in the metal a model for the metal is required to define this position properly. The capacitance per unit area of the ideal capacitor is a constant, e/Aird, often written as Klon. The contribution to 1/C is 1 /Klon this term is much less important in the sum (larger capacitance) than the other two contributions.2... 

The ESR of the decoupling capacitors has to be minimized, as does the ESL. As can be seen from Figure 4-14, everything becomes important. In effect, this is the final search for the ideal capacitor Though there is no ideal capacitor, they certainly are getting better and better, as the next sections reveal. 

The interphase between an electrolyte solution and an electrode has become known as the electrical double layer. It was recognized early that the interphase behaves like a capacitor in its ability to store charge. Helmholtz therefore proposed a simple electrostatic model of the interphase based on charge separation across a constant distance as illustrated in Figure 2.12. This parallel-plate capacitor model survives principally in the use of the term double layer to describe a situation that is quite obviously far more complex. Helmholtz was unable to account for the experimentally observed potential dependence and ionic strength dependence of the capacitance. For an ideal capacitor, Q = CV, and the capacitance C is not a function of V. 

Rule 1. The first rule is the requirement of the closed electrical circuit. This means that at least two electrodes must be present in the electrochemical cell. From a purely electrical point of view, it means that we have a sensor electrode (the working electrode) and a signal return electrode (often called the auxiliary electrode). This requirement does not necessarily mean that a DC electrical current will flow in a closed circuit. Obviously, if we consider an ideal capacitor C in series with a resistor R (Appendix C), a DC voltage will appear across the capacitor, but only as a transient DC current will not flow through it. On the other hand, if an AC voltage is applied to the cell, a continuous displacement charging current will flow. 

From the thermodynamic point of view, this is a multiphase system for which, at equilibrium, the Gibbs equation (A.20) must apply at each interface. Because there is no charge transfer in and out of layer (4) (an ideal insulator) the sandwich of the layers (3)/(4)/(5) also represents an ideal capacitor. It follows from the Gibbs equation that this system will reach electrostatic equilibrium when the switch Sw is closed. On the other hand, if the switch Sw remains open, another capacitor (l)/( )/(6) is formed, thus violating the one-capacitor rule. The signifies the undefined nature of such a capacitor. The open switch situation is equivalent to operation without a reference electrode (or a signal return). Acceptable equilibrium electrostatic conditions would be reached only if the second capacitor had a defined and invariable geometry. 

When an a.c. voltage is applied to a perfect capacitor, no energy is dissipated. However, a real capacitor dissipates energy because of lead and electrode resistances, d.c. leakage resistance and, most importantly, dielectric losses. These account for the capacitor s dissipation factor or loss tangent tan 3. It is sometimes convenient to regard the lossy capacitor as an ideal capacitor shunted by a resistance Rp or in series with a resistance rs, as shown in Fig. 5.5. 

A basic electric equivalent circuit to describe an EDLC is presented in Figures 1.22a and b, which shows the Nyquist (Figure 1.22b) plot of an ideal capacitor C, in series with a resistance... 

Although described here in terms of a capacitor model, these connections between the fluctuation and the dissipative part e" a>) of the susceptibility are general. They hold well beyond the illustrative example of an idealized capacitor used here to describe them. It is not always possible to express fluctuations in so many equivalent... 

The true phase angle (j)w of an ideal capacitor is 90°. Therefore real and imaginary part of Z can be approximated by ... 

Real capacitors have an increased surface charge density at the border of the capacitor plates. They have a bigger capacity than calculated from the formula of an ideal capacitor [5]. The true e is therefore always smaller than calculated. Because it is not possible to determine the real lines of the electric field within the real capacitor it is not possible to correct this error. 

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. 

Cyclic voltammetries (CVs) of CNTs A/CoSi700 modified by electrodepos-ited polypyrrole present the characteristic boxlike shape of an ideal capacitor even at moderate scan rates of 2 mV/sec, as shown in Figure 7.13 (Frackowiak and Beguin, 2002). Values of specific capacitance of nanotubes are significantly enhanced after modifications such as electrodeposition of a thin layer of conducting polymers, because of the contribution of pseudofaradaic properties of the polymer. 

An ideally polarizable electrode behaves as an ideal capacitor because there is no charge transfer across the solution/electrode boundary. In this case, the equivalent electrical model consists of the solution resistance, R, in series with the double-layer capacitance, Cdi. An analysis of such a circuit was presented in Section I.2(i). 

This replacement was necessary to adapt the equivalent circuit to the nonideal behaviour of the aluminium oxide film. The exponent n of the CPE element can be regarded as a measure of the inhomogeneity of the film structure [17]. For an ideal capacitor the exponent n is one. For the calculation of the CPE values, the fitting program in Ref [18] was used. 

This differential capacitance is obviously the slope of the plot of v.y. E at any point. Figure 13.2.4 helps to clarify the definition. One can see there and in Figure 13.2.3 that Q is not constant with potential, as it is for an ideal capacitor. 

---