Admittance Representation

The admittance format 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. When plotted in impedance format, the characteristic time constant is that corresponding to the Faradaic reaction. When plotted in admittance format, the characteristic time constant is that corresponding to the electrol5rte resistance, and that is obtained only approximately when Faradaic reactions are present. 

As shown in Example 16.1, the admittance format is ideally suited for analysis of dielectric systems for which the leading resistance can be neglected entirely. 

Solution The circuit corresponds to the dielectric response of a semiconductor device. The term Csc represents the space-charge capacitance, and the terms Rt,i, Ct,i, Rt,i, and account for the potential-dependent occupancy cfdeep-leoel electronic states, which typically have a small concentration. The term Ri accounts for the leakage current, which would be equal to zero for an ideal dielectric. 

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

If the electrolyte resistance Re is removed from the expression for the admitteince, the admittance is simplified to 

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Remember 16.3 The admittance representation emphasizes values at high frequency and is often used for solid-state systems for which information is sought regarding system capacitance. The admittance format has the advantage that it has a finite value for all frequencies, even for blocking electrodes. 

In the case of blocking electrodes, the impedance increases to infinity as the frequency approaches zero. In such cases, approximation with the Voigt circuit is not appropriate. When a high-frequency impedance is finite, the easiest way to verify the Kramers-Kronig compliancy is to fit the impedances to the admittance representation of the circuit containing a ladder of (RC) element series (Fig. 13.4) [575]. In addition, capacitance, Cq, or inductance can be added in parallel. 

The measurements (Fig. 3) show two areas of dispersion a, /5 in the frequency range 40 Hz-10 MHz. Measured impedance is analyzed by evaluating characteristic points in the impedance and admittance representation and by fitting several impedance models. Characteristic points (Fig. 3) allow analyzing the impedance evolution over time. The characteristic... 

Of the numerous equivalent circuits that have been proposed to describe electrochemical interfaces, only a few really apply in the context of a freely corroding interface at or close to kinetic equilibrium. The first circuit (Fig. 7.25a) corresponds to Eq. (7.2) and to the simplest equivalent circuit that can describe a metal/electrolyte interface. Following Boukamp,25 the term Q has been adopted here to describe the leaky capacitor behavior corresponding to the presence of a constant phase element explained by a fundamental dispersion effect. The admittance representation Y of the CPE behavior with frequency w can be described by Eq. (7.5). For n = (1 - p), Eq. (7.5) describes the behavior of a resistor with R = Yq" and for n = P, that of a capacitor with C = Yo- For n = 0.5, Eq. (7.5) becomes the expression of a Warburg (W) component, and when n = -p, it emulates an inductance with L = Yq-1.25... 

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... 

In an Argand17 diagram representation of Z, called a phasor diagram, 8 is the angle by which the current I(t) lags the voltage V(t). The reciprocal of the impedance Z is the admittance Y ... 

Admittance-plane plots are presented in Figure 16.6 for the series and parallel circuit arrangements shown in Figure 4.3(a). The data are presented as a locus of points, where each data point corresponds to a different measurement frequency. As discussed for the impedance-plane representation (Figure 16.1), the admittance-plane format obscures the frequency dependence. This disadvantage can be mitigated somewhat by labeling some characteristic frequencies. 

Figure 16.6 Admittance-plane representation for Re = 10 flcm, R = 100 flcm, and C = 20 fiF/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 O solid lines.

Figure 3.40 Equivalent circuit representation of an electrode incorporating an adsorbed redox active species (see text for notation). The components responsible for the ER (electroreflectance) response are shown within the dotted frame, where VF is the interfacial admittance, and the subscript F refer to the current and potential associated with the faradaic or interfacial modulation.

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... 

GRAPH 10.14 Formal Graph representations of the admittance of an inductive constant phase element (a) and of a capacitive one (b). The cross near the flow node (current) symbolizes a multiplication of operators (and not an addition) with the respective weights indicated near the arrow head. 

If the representation of a capacitive admittance is extremely simple (a vertical straight line positioned at the value of the conductance on the real axis) and does not need to be plotted, the representation of the capacitive impedance under the form of a semicircle is less straightforward and needs some explanations. 

This is an iterative technique used to solve linear electric networks of the ladder type. Since most radial distribution systems can be represented as ladder circuits, this method is effective in voltage analysis. An example of a distribution feeder and its equivalent ladder representation are shown in Fig. 10.116(a) and Fig. 10.116(b), respectively. It should be mentioned that Fig. 10.116(b) is a linear circuit since the loads are modeled as constant admittances. In such a linear circuit, the analysis starts with an initial guess of the voltage at node n. The current I is calculated as... 

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. 

Instead we want to emphasize that simple electric network models of LPS may include three different elemental systems capacitors, resistances, and inductances [6.12]. The basic physical relations, admittance functions, elements of the representation theorem (6.55) and corresponding static and optical permittivity are collected in Table 6.1 below. These elements can be combined by series or parallel connections in may different ways. For the admittance functions of the electric network generated in this way, the simple rules hold that... 

The optical admittance parameter has been introduced into thin-film optics with one specific aim, namely to visualize optical phenomena occurring within such systems by means of a graphical representation of the optical events known as the admittance diagram. Although this is one of a class of diagrams known collectively as circle diagrams, it is particularly powerful and attractive and therefore it is used extensively in thin-film optics. 

In the AC method, the cell is subjected to an AC source of variable frequency, and the cell response is measured as a function of frequency. Graphical representation involves a plot of negative of the imaginary part of the Impedance, — ImZ(), on the y-axis and real part of impedance, ReZ(imaginary part of the admittance, on the... 

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