Complex-plane admittance

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

Fig. 13L Comparison of a) complex-plane impedance, (b) complex-plane admittance, (c) complex-plane capacitance and (d) Bode magnitude and Bode angle plots for the same equivalent circuit. C =20 uF R = 10 kFl R = I kO.. Values of li) (Rad/s) at which some of the points were calculated are shown.

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. 

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...

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

Figure 16.5 Comparison of (a) complex-plane equivalent circuit. Cd/ = 20 pF / f = 10 kQ impedance, (b) complex-plane admittance, R = 1 kQ. Values of co at which some of the...

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. 

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... 

The - complex plane plot of the -> admittance of a solitary CPE is a straight line which makes an angle of (1 - af) 7t/2 with the Y axis. 

The immittance analysis can be performed using different kinds of plots, including complex plane plots of X vs. R for impedance and B vs. G for admittance. These plots can also be denoted as Z" vs. Z and Y" vs. Y, or Im(Z) vs. Rc(Z), and Im( Y) vs. Re( Y). Another type of general analysis of immittance is based on network analysis utilizing logarithmic Bode plots of impedance or admittance modulus vs. frequency (e.g., log Y vs. logo)) and phase shift vs. frequency ( vs. log co). Other dependencies taking into account specific equivalent circuit behavior, for instance, due to diffusion of reactants in solution, film formation, or electrode porosity are considered in - electrochemical impedance spectroscopy. Refs. [i] Macdonald JR (1987) Impedance spectroscopy. Wiley, New York [ii] Jurczakowski R, Hitz C, Lasia A (2004) J Electroanal Chem 572 355... 

Figure 12.27 shows a schematic IMPS respouse for a mouocrystalhue semiconductor electrode. It is easy to confuse IMPS plots with EIS plots since both contain semicircles. However the quantity displayed in an IMPS plot is not an impedance or an admittance it is the dimensionless transfer function corresponding to the ratio of the elecuon flux to the photon flux. The interpretation of the phase relationship between photocurrent and illumination requires some care. A response in the first upper quadrant indicates that the photocurrent leads the illumination. A response in the lower quadrant indicates that the photocurrent lags behind the illumination. In the schematic response shown in Pig. 12.27, the low-frequency response in the upper quadrant of the complex plane arises from surface... 

The complex plane plots may also be obtained for admittances. The admittance for the series R-C coimection equals... 

It represents a semicircle on the complex plane plot [Fig. 2(c)]. It should be stressed that for capacitive circuits the imaginary impedance is always negative and the imaginary admittance is positive. 

The faradaic admittance of reactions (116) and (117) is described by Eq. (135). Analysis of the complex plane plots in such a case was presented by Cao. Bai and Conway presented three-dimensional plots for such a reaction. Two general cases should be considered, depending on the sign of the parameter B ... 

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... 

Fig. 2 Impedance analysis is based on the conductance curve of the crystal. The central parameters of measiuement are the resonance frequency, /r, and the half-band-half-width, r. The insert shows the admittance diagram in the complex plane of the admittance Y(a>) = G(a>) + iB(a>). The series resonance frequency, /r, corresponds to the peak of the conductance. The frequency corresponding to B = 0 is the zero-phase frequency...

Figure 3. Admittance data from a K +-conducting membrane and curve fits (solid curves) of eqs 2, 3, and 4 with Y /jf,) = 0 plotted in the complex plane [X(f) vs. R(f)] as impedance [Z(jf) = R(f) + jX(f) = Y 1(jf/)] loci (400 frequency points) over the 12.5 5000-Hz frequency range. These data were acquired rapidly as complex admittance data, as illustrated in Figure 1, at premeasurement intervals of 0.1 and 0.5 s after step voltage clamps to each of the indicated membrane potentials from a holding of —65 mV. The near superposition and similarity in shape of the two loci at 0.1 and 0.5 s, at each voltage, indicates that the admittance data reflect a steady state in this interval after step clamps. Axon 86-41 internally perfused with buffered KF and externally perfused in ASW + TTX at 12 °C. The membrane area is 0.045 cm2.

Figure 6. Admittance data from a Na+-conducting membrane and curve Jits (solid curves) of eqs 2, 3, and 5 with YK()f) = 0 plotted in the complex plane [X(f) vs. R(f)] as impedance loci (400 frequency points) over the frequency range 5 to 2000 Hz. Same axon and conditions as in Figure 5.

Fig. II.5.8 Complex plane plot of the frequency normalized admittance. Curve 1 control by the adsorption exchange rate only curve 2 control by diffusion only curve 3 mixed control by the adsorption exchange rate and diffusion...

An important finding by Yamamoto and Yamamoto (1976) was that the impedance of the removed SC layers did not produce a circular arc in the complex impedance plane. This is obvious from Figure 4.19, in which the admittance data from Figure 4.17 have been transformed to impedance values and plotted in the complex plane. 

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... 

Fig. 2.32 Complex plane (a). Bode magnitude (b), Bode phase angle (c), and complex admittance (d) plots for R-C connection in series R = 150 l,C = 40 pF...

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

As can be seen from Eig. 5.3 and the preceding discussion, one simple Eq. (5.51) describing the faradaic admittance in the presence of one adsorbed species, in combination with the double-layer capacitance, can produce many different complex plane plots and electrical equivalent circuits. It should be stressed that the kinetic equations with physically possible rate parameters in Eqs. (5.25) and (5.26) may not give all the behaviors, i.e. impedance plots, found by arbitrarily allocating values to the circuit elements. 

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. 13.23 Complex plane impedance data (continuous lines) and their Kramers-Kronig transformations (symbol +) for simple model with negative resistance (a) transformation of impedance, (b) transformation of admittance (displayed here as impedance) data as in Fig. 13.15b...

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