Admittance plots

Fig. 17. (a) The components of the faradaic impedance plotted against to. (b) The components of the faradaic admittance plotted against to1/2. System parameters fict = lficm1, p = 0.03. The solid parts of the plots indicate the frequency range that is normally accessible for meaningful analysis of data with the simple theory described in this section... 

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. 

For the complex-plane admittance plot the real and the imaginary parts of the admittance are defined as... 

We shall refer to it as the complex-plane impedance plot, recognizing that the same data can also he represented in the complex-plane capacitance or the complex-plane admittance plots. The terms Cole-Cole plot, Nyqidst plot and Argand plot are also found in the literature. 

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. 

Usually, the properties of a quartz crystal resonator with respect to frequency can be discerned from admittance plots, where the abscissa represents the real part of the admittance (conductance, G) and the ordinate the imaginary component (susceptance, B). Resonance occurs at two frequencies where the admittance locus crosses the real axis, fs and fp, which are the series and parallel resonance frequencies, respectively. If Rt is negligible, the series and parallel frequencies at which resonance occurs are given by eqn.(5) [2],... 

The electrical properties of PEDOT-PSS/PVAc electrospun nanofibers on the indium tin oxide-poly(ethyleneterephthalate) (ITO-PET surface were determined by electrochemical impedance measurements in a monomer-free solution 0.1 M. NaC104 in H2O solution (Table 6.1, Fig. 6.7). The admittance plots of the solutions were in accordance with the conductivity increase (decrease in impedance) by the increase of PEDOT/PSS content... 

The highest value of the semicircle admittance plot decreased with an increase of PEDOT in the composites due to the porous coating of semiconducting material and different morphology of fibers. 

Figure 6.27 (a] Nyquist, (b) Bode magnitude, and (c] Admittance plots of different mole fractions of PANA/PAN composite nanofibers, correlated with the calculated data from the equivalent circuit modeling. (R(Q[R)). Reprinted with permission from Ref. 190, Copyright 2013 Elsevier Ltd. 

More specifically, we show in column 1 what happened in any of the complex planes when we add a pure reactance jX in series with an arbitrary load impedance Zl. As jX varies, we readily see how the series combination in row 1 moves along the vertical line going through Zl. Furthermore, the admittance plot in row 2 is obtained from the case in row 1 by noticing that... 

FIGURE 11.9 Cole-Cole (Nyquist) plots of the impedance (a) and of the admittance (b) of a parallel RC circuit in the linear case. The apex of the semicircle occurs when the angular frequency m equates the kinetic constant Kq (=1/Tc), which corresponds to the point intersecting the bisector in the admittance plot... 

Fig. 2.33 Complex plane impedance. Bode, and complex plane admittance plots for a connection of R and C in parallel (RQ, R = 100 i2, C = 20 pF...

The admittance plot for RC) connections is similar in shape to the impedance plot for an R-C connection in series. The difference is that for circuits containing capacitances the imaginary part of the impedance is negative and that of the admittance positive. 

Fig. 2.35 Complex plane impedance, Bode, and complex plane admittance plots for circuit...

Exercise 2.9 Make complex plane. Bode, and complex admittance plots of an RC coimection in series R = 150 Q, C = 40 pF. 

Fig. 13.1 Complex plane impedance and admittance plots for circuit in Fig. 2.34 with = 10 O,...

Fig. 13.5 Complex plane impedance and admittance plots fin hydrogen-doped LisN monocrystals (a) impedance plot and fit to model indicated in inset, (b) admittance plot and fit to Eq. (13.12) points experimental, line fits (From Ref. [575] Reproduced with pramission of Electrochemical Society)...

Fig. 12 (A) Nynquist plot of pristine graphene before and after anodisation at 2 V for 500 s. (B) Admittance plot for different potentials for pristine graphene before and after anodisation (reprinted with permission from ref 65).

In Figure 16.5 the different ways of presenting the response of the same circuit are compared. The values of the components constituting the equivalent circuit were chosen as Rp = 10 kQ, Rs = 1 kQ and C = 20 p,F. Each way of presentation has its own advantages and disadvantages. From the impedance plot R and R can be read directly and the double-layer capacitance can be calculated, employing Eq. (16.12), The relevant time constant in this case is Xc = Rf x Cdi = 0.20 s, hence 03 = 5 rad s" = 0.80 Hz. The complex-plane admittance plot yields the values of 1 /Rp and 1/Rs but it is rarely used. 

Figure 7.43 shows the plots of other system functions, the complex admittance G = Z S the complex dielectric modulus M = jwZ and the complex capacitance K = M . In particular we can see that when M is used the roles of R and are exchanged compared to the Z plot, so this plot is particularly suitable for analysis of capacitances, although the information content is, in principle , the same for all system functions. The significance of the admittance plot consists in the fact that, for a parallel connection of R and C (and suflBciently high frequencies), the real part is identical with the sample conductance. 

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