Complex capacitance

To understand the electrical behaviour of the LAPS-based measurement, the LAPS set-up can be represented by an electrical equivalent circuit (see Fig. 5.2). Vbias represents the voltage source to apply the dc voltage to the LAPS structure. Re is a simple presentation of the reference electrode and the electrolyte resistance followed by a interface capacitance Cinterface (this complex capacitance can be further simulated by different proposed models as they are described, e.g., in Refs. [2,21,22]). In series to the interface capacitance, the insulator capacitance Cj will summarise the capacitances of all insulating layers of the LAPS device. The electrical current due to the photogeneration of electron-hole pairs can be modelled as current source Ip in parallel to the... 

This complex capacitance model, even if simplified, gives precious quantitative information about the change of the capacitance of an EDLC device versus the frequency. The knowledge of the ac behavior is indeed important since EDLC, as power devices, are often used in ac modes. Finally, it must be precise that improvements of such approach have been recently developed in a series of papers [35,36], where the De Levie TLM is associated with the complex capacitance model to estimate the porous structure of the carbon electrodes using discrete Fourier transformation. 

Different kinds of plots based on impedance Z, admittance Z 1, modulus icoZ, or complex capacitance (z coZ) 1 can be used to display impedance data. In solid state ionics, particularly plots in the complex impedance plane (real versus imaginary part of Z) and impedance Bode-plots (log(Z) log(co)) are common. A RC element (resistor in parallel with a capacitor) has, for example, an impedance according to... 

A complex capacitance can be expressed as the product of the complex dielectric constant, e, and a geometrical constant, Cq, of the sample ... 

The complex capacitance representation for this type cf system is particularly interesting. See the discussion in Section 16.3 and Example 16.2. 

Remember 16.4 Like the admittance representation, the complex-capacitance representation emphasizes values at high frequency and is often used for solid-state and dielectric systemsfor which information is sought regarding system capacitance. 

The characteristic angular frequency for the blocking circuit is o c = 1/ReC, the same as is found for the admittance of the blocking circuit. At the characteristic angular frequency, the real part of the capacitance is equal to half the double-layer capacitance, and the imaginary part is equal to minus one-half of the double-layer capacitance. The complex-capacitance plot for tire blocking circuit traces a semicircle. 

The complex-capacitance-plane plot is presented in Figure 16.11. The data are presented as a locus of points, where each data point corresponds to a different measurement frequency. As discussed for the impedance- and admittance-plane representations (Figures 16.1 and 16.6, respectively), the complex-capacitance-plane... 

Figure 16.11 Complex-capacitance-plane plots for Re = 10 Qcm, R = 100 flcm, and C = 20 F/cm. The blocking system of Table 16.1(a) is represented by A and dashed lines, and the reactive system of Table 16.1(b) is represented by Q sol d lines.

The characteristic frequency evident as a peak for the imaginary part of the complex-capacitance in Figures 16.12(b) and 16.13(b) has a value corresponding exactly to fc = 27tReC) only for the blocking system. As found for data presentation in admittance format, the presence of a Faradaic process confounds use of graphical techniques to assess this characteristic frequency. Like the admittance format, the complex capacitance is not particularly well suited for analysis of electrochemical and other systems for which identification of Faradaic processes parallel to the capacitance represents the aim of the impedance experiments. It is particularly well suited for analysis of dielectric systems for which the electrolyte resistance can be neglected. 

Example 16.2 Complex Capacitance of Dielectrics Find an expression for the complex capacitance for the electrical circuit shown in Figure 16.9 and discussed in Example 16.1. Identify the limits and characteristic frequencies. 

Solution Insertion cf equation (16.34) into the definition for complex capacitance... 

In contrast to the complex capacitance described in Section 16.3, the effective capacitance described in equation (16.41) is defined to be a real number. 

Example 16.1 Admittance of Dielectrics Example 16.2 Complex Capacitance of Dielectrics Example 16.3 Evaluation of Double-Layer Capacitance... 

Figure 30. Nyquist plot of complex capacitances for Au( 111) in 0.1 M HQ04in the presence of bromide ions Br concentrations given in the figure upper three curves at = 0.3 V lower three curves at = 0.1 V vs. SCE. ...

The connection between the impedance and dielectric measurements can be seen easily for only relatively simple examples. In Figure 7-8c the complex capacitance plot for the same equivalent circuit is similar to typical dielectric response dielectric only in the low-frequency region. The plot of the magnitude of the complex impedance, also shown in this panel, is known as the Bode plot. 

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