Admittance characteristic

There are two general interpretations for the admittance characteristics of barrier anodic films (98). The first interpretation is based on film conduction mechanisms, either electronic or ionic, and the influence of solution ions on the oxide film lattice defect structure on conduction behavior. The second is based on the behavior of preexisting defects in anodic films and the effects of attacking or passivating solutions. 

When the propellant burning rate is espressed by Eq. (5b), the parameter p /e can be considered as the transient sensitivity of the burning rate to pressure. This parameter depends on the transient combustion characteristics, and its evaluation depends on the particular model of the combustion process. Thus, the acoustic admittance provides the link between experimental observation and theoretical prediction. 

Horton and Price (H11) have obtained acoustic-admittance data for a series of double-base and composite propellants with different burning-rate characteristics. They examined the effects of pressure at various frequencies... 

The reduction of Cd(II) ions on DME was also investigated in 1 M perchlorate, fluoride and chloride solutions using dc, ac admittance, and demodulation methods [27]. It was found that in the perchlorate supporting electrolyte, the reduction mechanism is also CEE, and that the rate constant of the chemical step is quite close to the value characteristic for fluoride solutions. The theories available at present could not be applied to the Cd(II) reduction in chloride solution because of the inapplicability of the Randles equivalent circuit. 

In studies of these and other items, the impedance method is often invoked because of the diagnostic value of complex impedance or admittance plots, determined in an extremely wide frequency range (typically from 104 Hz down to 10 2 or 10 3 Hz). The data contained in these plots are analyzed by fitting them to equivalent circuits constructed of simple elements like resistances, capacitors, Warburg impedances or transmission line networks [101, 102]. Frequently, the complete equivalent circuit is a network made of sub-circuits, each with its own characteristic relaxation time or its own frequency spectrum. 

Fig. 4.15 Admittance diagram of quartz crystal microbalance (QCM) crystal and the relationship to characteristic frequencies (adapted from Yang and Thompson, 1993)...

As we have already discussed, a lossless FDN results when the feedback matrix is chosen to be unitary. Smith and Rocchesso have shown that the waveguide interpretation leads to a more general class of lossless scattering matrices [Smith and Rocchesso, 1994], This is due to the fact that each waveguide may have a different characteristic admittance. A scattering matrix is lossless if and only if the active complex power is scattering-invariant, i.e., if and only if... 

Figure 5. Dispersion characteristics (Nomenclature (v) applied voltage (peak to peak) (p) relative concentration of NaPSS (Y) admittance of the sample in mho (Y) normalized admittance of the sample (Z ) normalization factor in ohm (6)...

Johannsmann et al. have reported the use of quartz crystal electrical impedance to study the viscoelastic properties of poly(y-methyl-L-glutama-te-co-y-zi-octadecyl-L-glutamate) thin films [65, 66] and Ward studied the viscoelastic characteristics of poly-styrene films with 2-chloro-toluene solvent by measuring the admittance near resonance at 5 MHz of the unloaded quartz resonator and the composite resonator [67]. 

In the event that the film is not rigid, the EQCM response is a function of both the film mass and its rheological characteristics. Application of the Sauerbrey equation under these circumstances is inappropriate it underestimates the mass change, to an extent that is dependent on the viscoelastic properties of the film. Under these circumstances, there are two questions to be addressed first, how does one diagnose film (non-)rigidity and, second, how does one interpret responses from a non-rigid film The answers to both questions can be found from crystal impedance measurements. This is a technique in which one determines the admittance (or impedance) of the loaded crystal as a function of frequency in the vicinity of resonance. Effectively, one determines the shape (width and height) and position (on the frequency axis) of the resonance, rather than just its position (as in the simple EQCM technique). 

During exposure to chromate solutions, defects in the anodic film heal, and there is some thinning of the barrier layer. Under these conditions, the characteristic response in a 1 kHz admittance measurement is a slight increase in the capacitance and dissipation factor. An anodic film with an intentional defect will show an increase in capacitance due to flaw healing. 

In the previous section we considered the conditions under which mechanical resonances would occur in a TSM resonator. In considering only the mechanical properties of the crystal, however, we neglected consideration of how these resonances would actually be excited or detected. The device uses a piezoelectric substrate material in which the electric field generated between electrodes couples to mechanical displacement. This allows electrical excitation and detection of mechanical resonances. In constructing a practical sensor, changes in resonant frequency of the device are measured electrically. The electrical characteristics of the resonator can be described in terms of an equivalent-circuit model that describes the impedance (ratio of applied voltage to current) or admittance (reciprocal of impedance) over a range of frequencies near resonance. 

TSM resonator electrical characteristics are typically described in terms of electrical admittance, defined as the ratio of current flow to applied voltage (the reciprocal of impedance). The total TSM resonator admittance can be determined from inspection of the equivalent circuit model ... 

When a mass layer is added to one side of the TSM resotuitor, the electrical characteristics are changed, as described by the elentent L2. Figure 3.8 shows the effect of mass loading on TSM resonator admittance near restmance. It is apparent that the major effect of the mass layer is to translate the admittance curves... 

This polarizability involves a set of characteristic times jlkT, Id(, illoil, and (IcIIodI), between any two of which the correlation function may take a fairly simple form. However, this example indicates that for a linear system which falls to show normal mode behaviour the frequency-dependent admittance may be much more straightforward than the corresponding correlation function. 

Table 16.1 Summary of complex impedance, admittance, and capacitance characteristics for simple blocking and reactive circuits.

The high-frequency limit is the same as is foimd for the series combination of the resistance Rg and capacitance C shown as equation (16.26). The parallel resistance R does, however, influence the value obtained for the imaginary part of the admittance at the characteristic angular frequency cOg = ReC), i.e.. 

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. 

The maximum value for the imaginary part of the admittance is equal to 1 /2Rg for the blocking system, and the characteristic angular frequency at the maximum is equal to (Vc = 1/ReC. 

Figure 16.7 Real and imaginary parts of the admittance as a function of frequency for Re — 10 Qcm, R = 100 Qcm, and C = 20 piF/cm. The blocking system of Table 16.1(a) is represented by dashed lines, and the reactive system of Table 16.1(b) is represented by solid lines, Characteristic frequencies are noted as /rc == (2n J C) and fc = 2nReC) a) real part of admittance and b) imaginary part of admittance.

The characteristic frequency evident as a peak for the imaginary part of the admittance in Figures 16.7(b) and 16.8(b) has a value corresponding exactly to fc = (2nReC) for the blocking system. As shown by equation (16.31), the presence of a Faradaic process confoimds use of graphical techniques to assess this characteristic frequency. 

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. 

Example 16.1 Admittance of Dielectrics Find an expression for the admittance of the electrical circuit shown in Figure 16.9. Identify the characteristic frequencies. 

Rt,2Ct,2) The characteristic angular frequencies could be easily identified from plots of real and imaginary parts of the admittance as functions of frequency. 

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. 

There are two limits of the impedance (O = 0, Z = / and ro —> o°, Z = 0. The corresponding complex plane and Bode plots for the same values of R and C elements as those used in the series R-C model above, are shown in Fig. 3. The Nyquist plot shows a semicircle of radius RH with the center on the real axis and the frequency at the semicircle maximum equal to (0= RC. The circuit s characteristic breakpoint frequency (the inverse of the characteristic time constant), as observed in the impedance Bode graph, is the same as for the series and the parallel R-C circuits. The complex plane admittance plot represents a straight line parallel to the imaginary axis [Fig. 3(c)], which is similar to the impedance complex plane plot for the series R-C connection. 

For definition of the different characteristic resonance frequencies, the locus of impedance Z = R+jX (a) and admittance Y = l/Z = G + jB (h) of a bare quartz crystal are shown in Fig. 12 (for better distinction Rs has been set to 160 12 here, a typical value for single-side liquid load). The definilions are summarized in Table 2 [34]. 

Fig. 12 Characteristic resonance frequencies of quartz crystal resonators, shown in the locus of impedance, Z = R+jX (a), and admittance, Y = G+jB (b). is the parallel resonant frequency/p at Umax, O is the parallel resonant frequency/a at X = 0, O is the parallel resonant frequency/ at 2 max, i Ih series resonance frequency/s at Gmax, is the series resonant frequency/r at = 0, and is the series resonant frequency/m...

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