Nyquist plots

Vibration analysis This ineludes an on-line analysis of the vibration signals, FFT speetral analysis, transient analysis, and diagnosties. A wide variety of displays are available ineluding orbits, easeades, bode and nyquist plots, and transient plots. 

Nyquist Plot a graph of the frequency response of an electrode in which the imaginary component of the impedance is plotted as a function of the real component for a range of frequencies. 

In many cases, the Nyquist plot for SEI electrodes consists of only one, almost perfect, semicircle whose diameter increases with storage time (and a Warburg section at low frequencies). For these cases the following can be concluded the SEI consists of only one sublayer, 7 GT), / GB... 

The technique of AC Impedance Spectroscopy is one of the most commonly used techniques in electrochemistry, both aqueous and solid.49 A small amplitude AC voltage of frequency f is applied between the working and reference electrode, superimposed to the catalyst potential Uwr, and both the real (ZRe) and imaginary (Zim) part of the impedance Z (=dUwR/dI=ZRc+iZim)9 10 are obtained as a function of f (Bode plot, Fig. 5.29a). Upon crossplotting Z m vs ZRe, a Nyquist plot is obtained (Fig. 5.29b). One can also obtain Nyquist plots for various imposed Uwr values as shown in subsequent figures. 

Figures 5.29a and 5.29b show the Bode and Nyquist plot for a resistor, Ro, connected in series with a resistor, Rt, and capacitor, Ci, connected in parallel. This is the simplest model which can be used for a metal-solid electrolyte interface. Note in figure 5.29b how the first intersect of the semicircle with the real axis gives Ro and how the second intersect gives Ro+Rj. Also note how the capacitance, Ct, can be computed from the frequency value, fm, at the top of the semicircle (summit frequency), via C l JifmR .

Figure 5.29. Bode (a) and corresponding Nyquist plot (b) of the circuit shown in inset which is frequently used to model a metal/solid electrolyte interface. Effect (c) of capacitance C2 on the Nyquist plot at fixed R0, R( and R2.

F/gwre 5 JO, (a) Complex impedance spectra (Nyquist plots) of the CH4,02) Pd YSZ system at different Pd catalyst potentials. Open circuit potential U R =-0.13 V. Dependence on catalyst potential of the individual capacitances, C4i (b) and of the corresponding frequencies, fmii, at maximum absolute negative part of impedance (c).54 Reprinted with permission from Elsevier Science. 

NonFaradaic electrochemical modification of catalytic activity, NEMCA, see electrochemical promotion NonFaradaic processes, 2 Nyquist plot, 237... 

Quite often we are face with the task of reducing the order of a transfer function without losing essential dynamic behavior of the system. Many methods have been proposed for model reduction, however quite often with unsatisfactory results. A reliable method has been suggested by Luus (1980) where the deviations between the reduced model and the original one in the Nyquist plot are minimized. 

After the parameters have been estimated, generate the Nyquist plots for the reduced models and the original one. Comment on the result at high frequencies. Is N=100 a wise choice ... 

Nyquist plot Bode plot Nyquist plot is a frequency parametric plot of the magnitude and the argument of the open-loop transfer function in polar coordinates. Bode plot is magnitude vs. frequency and phase angle vs. frequency plotted individually. 

Both the magnitude and the argument are functions of the frequency. The so-named Bode and Nyquist plots are nothing but graphical representations of this functional dependence. 

We can plot the real and imaginary parts of G(jco) on the v-planc with co as the parameter—the so-called Nyquist plot. Since a complex number can be put in polar coordinates, the Nyquist plot is also referred to as the polar plot. 

This plotting format contains the same information as the Bode plot. The polar plot is more compact, but the information on the frequency is not shown explicitly. If we do not have a computer, we theoretically could read numbers off a Bode plot to construct the Nyquist plot. The use of Nyquist plots is more common in multiloop or multivariable analyses. A Bode plot, on the... 

For now, we ll take a look at the construction of Bode and Nyquist plots of transfer functions that we have discussed in Chapters 2 and 3. Keep in mind that these plots contain the same information G(jco). It is important that you run MATLAB with sample numerical values while reading the following examples. Yes, you need to go through MATLAB Session 7 first. 

Example 8.2. What are the Bode and Nyquist plots of a first order transfer function ... 

To make the phase angle plot, we simply use the definition of ZGp(joo). As for the polar (Nyquist) plot, we do a frequency parametric calculation of Gp(jco) and ZGp(joo), or we can simply plot the real part versus the imaginary part of Gptjco).1 To check that a computer program is working properly, we only need to use the high and low frequency asymptotes—the same if we had to do the sketch by hand as in the old days. In the limit of low frequencies,... 

All comments on Nyquist plots are made without the need of formal hand sketching techniques. Strictly speaking, the polar plot is a mapping of the imaginary axis from co = 0+ to You ll see... 

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. 

When we make a Nyquist plot, we usually just map the positive imaginary axis from co = 0 to infinity, as opposed to the entire axis starting from negative infinity. If a system is unstable, the resulting plot will only contribute jt to the (-1,0) point as opposed to 2k—what encirclement really means. However, just mapping the positive imaginary axis is sufficient to observe if the plot may encircle the (-1,0) point. 

Consider first the corrosion of low alloy steel in HC1 per se, i.e. before the addition of organic inhibitors. As shown in Figures 1 and 2 for N80 steel in 15% and 28% HC1 at 65 C, Nyquist plots for steel in concentrated HC1 typically have only one distinct feature a single capacitance loop (a loop above the Z axis) with a hint of a second capacitance loop at lower frequencies. The low-frequency loop is more fully developed in 28% HC1 than in 15% HC1. Mass transport limitations are not evident except under extreme conditions, e.g. above 28% HC1 and 65 C. 

---