Complex-Capacitance Representation

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

As CO — 0, Cr Csc + Cfj + Ct. As CO oo, Cr Csc- The characteristic angular frequencies are cofi = (Rt iCt,i) and C0f2 = Rt,2Ct ). The characteristic frequencies could be identified easily from plots of real and imaginary parts of the complex capacitance as functions of frequency. 

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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 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... 

Therefore admittance data can also be plotted in the complex plane (V versus F with (o implicit). Some researchers choose to display data in terms of the complex capacitance C( o>) here C( a>) = Y j(o)lj(o. The latter type of representation can be useful when examining the electrochemical response of electronically conducting polymer films. The low-frequency redox pseudocapacitance can be read directly from a plot of C" versus C at low frequency. 

This impedance is plotted in the complex plane representation in Fig. 6. Qualitatively, the impedance appears to be a pure capacitance at low frequencies, where the phase angle tends toward ir/2. At higher frequencies. 

The usefulness of complex modulus representations in addition to impedance plots is related partially to the composition of the analyzed samples, especially in cases of multicomponent dispersions where the migration-type ionic or particle-based conduction in the bulk sample can be realized by two or more competing processes. As will be shown later, the graphical representation of the modulus often resolves well the resistive differences in the bulk conduction processes, while the impedance representation is preferred to resolve capacitance-related differences [6]. 

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... 

Fig. A2.1. Representation in the complex plane of an impedance containing resistive and capacitive components.

The above analysis shows that in the simple case of one adsorbed intermediate (according to Langmuirian adsorption), various complex plane plots may be obtained, depending on the relative values of the system parameters. These plots are described by various equivalent circuits, which are only the electrical representations of the interfacial phenomena. In fact, there are no real capacitances, inductances, or resistances in the circuit (faradaic process). These parameters originate from the behavior of the kinetic equations and are functions of the rate constants, transfer coefficients, potential, diffusion coefficients, concentrations, etc. In addition, all these parameters are highly nonlinear, that is, they depend on the electrode potential. It seems that the electrical representation of the faradaic impedance, however useful it may sound, is not necessary in the description of the system. The systen may be described in a simpler way directly by the equations describing impedances or admittances (see also Section IV). In... 

The results obtained through complex impedance spectroscopy for various samples are presented in Figure 6.52 in two representation modes, that is to say the variations of the resistance Y and the capacitance X as a function of frequency, and the variations of Y according to X at various frequency values (Nyquist diagram). 

Cjj capacitance of the electrode, (h) Representation of the complex impedance as function of frequency, CO in a Nyquist diagram. 

In order to highlight the computing ability of Formal Graphs, Figure 10.17 shows two responses of the previous circuit (Randles circuit) by different techniques. The left plot represents the variation of the current resulting from a linear variation with time of the imposed potential (potential ramp) and the right plot represents the complex impedance in the Cole-Cole (Nyquist) representation. The parallel capacitance Cq is supposed to be sufficiently important for dominating the... 

The composites are analyzed by means of complex impedance spectroscopy as the total Impedance is a function of temperature and frequency. Impedance measurements are often made with a Wheatstone bridge type of apparatus in which the resistance, R , and capacitance, C of the sample are balanced against variable resistors and capacitors. The central problem with this measurement arises over the interpretation of the data. This is because the sample and the electrode arrangement is electrically a black box whose equivalent circuit (i.e. its representation by some... 

The (relative) dielectric permittivity (e,) of a material system can be calculated from impedance capacitance measurements via Eqs. 6.11, 6.13. As Er normally is a complex quantity it is appropriate to consider its real part (e ) and imaginary part (e") separately. An elegant way to do this is to consider the functions s = e (o) = 2tiv), e" = s"(m = 2jtv), which provide a parametric representation of the curve e = 8"(8 ). This was suggested first by K. S. Cole and R. H. Cole. Hence plots of these curves today are called Cole-Cole-plots, an example of which already has been given in Fig. 6.11. 

Fig. 3. Vector diagram of the current in a parallel representation of a dielectric. E = Voltage, V 7 = total curent. A 7c = capacitive component of current 7rp = resistive component of current 6 = phase (power factor) angle S = loss (phase defect) angle f = frequency of applied voltage Cp = parallel capacitance, F and = parallel resistance, electric constant and loss index to give a complex dielectric constant, = - je", of which the dielectric constant, e, is the real part and the loss index, e", the theoretical part.

It is convenient to start the discussion of the fundamentals of impedance data representation with an analysis of very simple systems. If a sinusoidal voltage is applied to a pure resistor of value R, then the measured complex impedance is entirely resistive at all frequencies asZ = R and the impedance magnitude IZI = R. If a sinusoidal voltage is applied across a pure capacitor, the measured impedance can be calculated according to the relationship Z = -jloaCy where C is the capacitance. The magnitude of the impedance for a pure capacitor is IZI = (coC) L This impedance depends on the frequency and is entirely capacitive (see Chapter 3). 

It is convenient to display the results of EIS in the complex-plane impedance representation. The X-axis on this plot is ReZ, which is the Ohmic resistance, and the y-axis is -ImZ, which, in the present case, is the capacitive impedance -j/o) C. 

AKernating current conductivity The alternating current conductivity (trjc) measurements are usrraUy carried out in a frequency range between 10" and 10 Hz for the sandwich-type samples. The equivalent parallel capacitance C co) and conductance G co) are obtained from the experiments and then they are transformed into different representations, like complex resistivity p, complex conductivity a, complex modulus M, and complex permittivity ... 

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