Plane, Nyquist

FIGURE 1.19 Complex-plane impedance plot (Nyquist plane) for an electrochemical reaction under kinetic control. 

When the rate determining step of the electrode reaction is the charge transfer process (kinetic control), the faradic impedance ZF in Figure 1.18 can be described as RCJ, the charge transfer resistance [7,8], The impedance plot in the Nyquist plane describes a semicircle, as shown in Figure 1.19. 

We will consider here the case where the mass transfer limitation is due to the diffusion of the species. When the mass transfer becomes predominant in the low-frequency range, experimental plots obtained in the Nyquist plane shift from the semicircular shape. In that case, indeed, the impedance ZFcan no longer be described as only a charge transfer resistance, but as a combination of Rcv with the impedance of diffusion ZD. ZD changes with the frequency it takes into account the relaxation processes inside the diffusion layer. Different cases can be described depending on the diffusion layer thickness. 

FIGURE 1.20 Complex-plane impedance plot (Nyquist plane) for an electrochemical system, with the mass transfer and kinetics (charge transfer) control regions, for an infinite diffusion layer thickness. 

The impedance plot in the Nyquist plane as presented in Figure 1.20 shows two different parts a loop at high frequency and a line at low frequency, also named Warburg line, at 45° angle with the real axis [10]. 

FIGURE 1.21 An example of a complex-plane impedance plot (Nyquist plane) for an electrochemical system under mixed kinetic/diffusion control, with the mass transfer and kinetics (charge transfer) control regions, for a finite thickness 8N of the diffusion layer. Assumption was made that Kf Kh at the bias potential of the measurement, and D0I = Dmd = D, leading to RB = RCT (krb8N/ >). 

By taking into account the double-layer capacity, Q, and the electrolyte resistance, Re, one obtains the Randles equivalent circuit [150] (Fig. 10), where the faradaic impedance Zp is represented by the transfer resistance Rt in series with the Warburg impedance W. It can be shown that the high-frequency part of the impedance diagram plotted in the complex plane (Nyquist plane) is a semicircle representing Rt in parallel with Cd and the low-frequency part is a Warburg impedance. 

Such an expression is represented in the Nyquist plane by a line inclined by 45° as can be seen in Figure 6.45, the 45° angle being a result of the 1/2 pulsation power. This line will be considered as representing an electrode polarization process. 

For various illumination intensities, the diameter of the semicircle fitting the data at high frequencies equals approximately kT/ely pHl [45-47, 49]. In addition, it was shown that upon illumination, a capacitive peak appears in the C versus V plot of the n-GaAsjO.l M H2SO4 interface [45,46, 51], The peak value proved to be a function of the frequency and the photocurrent density as measured in region G [51]. This behavior is markedly different from the purely capacitive impedance (vertical line in the Nyquist plane and straight Mott-Schottky plot) expected for a blocking s/e interface (see Sect. 2.1.3.1). 

In a paper by Kemer and Pajkossy [2002], anion adsorption rates have been measured by impedance spectroscopy at Au (111), for SO4, Cr, Br and T. Cr ion adsorption is very fast, so the equivalent circuit for the process not only has to include a Faradaic pseudocapacitance and its corresponding reaction resistance, but also a Warburg element for the anion diffusion. The interfacial capacitance is then plotted in the Nyquist plane as real vs. imaginary capacitance components, C and... 

As illustrated in this chapter, Eq. (6), in parallel with the double-layer capacitance Qj, generates identifiable shapes on the impedance curves in die Bode or Nyquist plane making possible to determine the number of chemical entities and Cj participating in the reaction mechanism and thus providing information on the reaction pattern. In terms of dissolution-passivation processes, capacitive responses and negative resistances are related to inhibition or passivation whereas inductive behaviors arise from catalytic efiects or activating intermediates [4-8], Acquisition and processing of the transient response of electrochemical systems are easily performed by modem laboratory equipment [5,6,49] and do not deserve special attention in this chapter. 

In order to eneirele any poles or zeros of F s) that lie in the right-hand side of the. v-plane, a Nyquist eontour is eonstrueted as shown in Figure 6.16. To avoid poles at the origin, a small semieirele of radius e, where e 0, is ineluded. 

The Nyquist stability criterion can be stated as A closed-loop control system is stable if, and only if, a contour in the G s)H s) plane describes a number of counterclockwise encirclements of the (—l,jO) point, the number of encirclements being equal to the number of poles of G s)H s) with positive real parts . 

From the Nyquist stability criterion, let N k, G(iuj)) be the net number of clockwise encirclements of a point (k, 0) of the Nyquist contour. Assume that all plants in the family tt, expressed in equation (9.132) have the same number ( ) of right-hand plane (RHP) poles. 

The results are exact—we do not need to make approximations as we had to with root locus or the Routh array. The magnitude plot is the same as the first order function, but the phase lag increases without bound due to the dead time contribution in the second term. We will see that this is a major contribution to instability. On the Nyquist plot, the G(jco) locus starts at Kp on the real axis and then "spirals" into the origin of the s-plane. 

This equation, of course, contains information regarding stability, and as it is written, implies that one may match properties on the LHS with the point (-1,0) on the complex plane. The form in (7-2a) also imphes that in the process of analyzing the closed-loop stability property, the calculation procedures (or computer programs) only require the open-loop transfer functions. For complex problems, this fact eliminates unnecessary algebra. We just state the Nyquist stability criterion here.1... 

Nyquist plots in the G plane, (a) Single point (M complete curve. ... 

The results of the examples above show that adding lags (poles) to the transfer function moves the Nyquist plot clockwise around the origin in the G plane. Adding leads (zeros) moves it counterclockwise. We will return to this generalization in the next chapter when we start designing controllers that shift these curves in the desired way. 

A mathematician would say that a plot of Z" (as y ) against Z (as jc ) forms an Argand diagram (or Argand plane ). As electroanalysts, we will call such a set of axes a Nyquist plot or simply an impedance plot (see Figure 8.9). 

At the heart of impedance analysis is the concept of an equivalent circuit. We assume that any cell (and its constituent phases, planes and layers) can be approximated to an array of electrical components. This array is termed the equivalent circuit , with a knowledge of its make-up being an extremely powetfitl simulation technique. Basically, we mentally dissect the cell or sample into resistors and capacitors, and then arrange them in such a way that the impedance behaviour in the Nyquist plot is reproduced exactly (see Section 10.2 below on electrochemical simulation). 

Figure 10. Impedance complex plane (Nyquist plots) of lithium electrode in (A) 1.0 M LiPFe/EC/PC and (B) 1.0 M LiC104/EC/PC at initial time (0.0 h) and after 24 h. Re and Im stand for the real and imaginary parts of the impedance measured, respectively. Frequency was indicated in the figure for selected data points. Note that the first semicircle corresponds to SEI impedance. (Reproduced with permission from ref 86 (Figure 2). Copyright 1992 The Electrochemical Society.)...

From example 7.6 we know that critical stability occurs for Ac = 1.8, r, = 3.5. Hence, by the Nyquist criterion, when these conditions are applied, the polar plot will pass through the point (-1,0) on the complex plane, i.e. for these values of the controller parameters, 9m (G(i[Pg.631]

It can be shown that there is a Nyquist criterion for sampled data systems which is equivalent to that for continuous systems (see Section 7.10.5) and equation 7.131 can be applied in its comparable r-transformed form(42). In practice it is generally sufficient to ascertain whether the polar plot of G(z) in the complex z-plane encircles the (-1,0) point (as with continuous systems in the j-plane) where 1 + G(r) = 0 is the system z-transformed characteristic equation. The polar plot is constructed from... 

Equations 2.37-2.40 result in the commonly used presentation of the impedance, e.g. the Nyquist and the Bode plots. The first one shows the total impedance vector point for different values of co. The plane of this figure is a complex plane, as shown in the previous section. Electrochemical-related processes and effects result in resistive and capacitive behaviour, so it is common to present the impedance as ... 

The most common graphical representation of experimental impedance is a Nyquist plot (complex-plane diagram), which is more illustrative than a Bode plot. However, a Bode plot sometimes can provide additional information. 

The constant phase element (CPE) is a non-intuitive circuit element that was discovered in the course of investigations into responses from real systems. In general, a Nyquist plot (also called a Cole-Cole plot or complex impedance plane plot) should be a semicircle with the centre on the x-axis. However, the observed plot for some real systems was indeed the arc of a circle but with the centre located somewhere below the x-axis. Figure 4.1 shows the impedance spectra of a circuit of a resistor and a constant phase element connected in parallel. The centre of the semicircle is located at (l-n)x90° below the real axis. 

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