Nyquist diagram

In practice, only the frequencies lu = 0 to+oo are of interest and since in the frequency domain. v = jtu, a simplified Nyquist stability criterion, as shown in Figure 6.18 is A closed-loop system is stable if, and only if, the locus of the G(iLu)H(iuj) function does not enclose the (—l,j0) point as lu is varied from zero to infinity. Enclosing the (—1, jO) point may be interpreted as passing to the left of the point . The G(iLu)H(iLu) locus is referred to as the Nyquist Diagram. 

Fig. 6.18 Nyquist diagram showing stable and unstable contours.

Construet the Nyquist diagram for the eontrol system shown in Figure 6.20 and find the eontroller gain K that... 

Table 6.3 Data for Nyquist diagram for system in Figure 6.20...

Then n in equation (6.62) is the type number of the system and J([ denotes the product of the factors. The system type can be observed from the starting point uj 0) of the Nyquist diagram, and the system order from the finishing point bj oo), see Figure 6.22. 

Fig. 6.22 Relationship between system type classification and the Nyquist diagram. For a step input,...

The M and N circles can be superimposed on a Nyquist diagram (called a Hall chart) to directly obtain closed-loop frequency response information. 

Alternatively, the closed-loop frequency response can be obtained from a Nyquist diagram using the direct construction method shown in Figure 6.25. From equation (6.73)... 

Fig. 6.25 Closed-loop frequency response from Nyquist diagram using the direct construction method.

This tutorial shows how MATLAB can be used to construct all the classical frequency domain plots, i.e. Bode gain and phase diagrams, Nyquist diagrams and Nichols charts. Control system design problems from Chapter 6 are used as examples. 

The Nyquist diagram uj varying from —oo to +oo) is produced by examp64.m where... 

Example 6.4 Nyquist Diagram %Third-order type one system num=[1] ... 

The resulting dependence of Z" on Z (Nyquist diagram) is involved but for values of Rp that are not too small it has the form of a semicircle with diameter Rp which continues as a straight line with a slope of unity at lower frequencies (higher values of Z and Z"). Analysis of the impedance diagram then yields the polarization resistance (and thus also the exchange current), the differential capacity of the electrode and the resistance of the electrolyte. 

The experimental results mainly obtained by EIS supported the mechanism of the build up of an oxide passive layer. They lead to the determination of quantitative parameters related to the change of surface reactivity. The measurements constituted a series of impedance diagrams obtained at successive time intervals. Examples given in Eig. 11 represent [12] the time variation of the Nyquist diagram resulting from the build up of an insulating layer, after reaction ofSClonan initially bare hydrophobic... 

It can be shown061 that, if there are any net encirclements of the point (-1,0) on the Nyquist diagram (i.e. if nE > 0), then the system characteristic equation will have roots lying to the right of the imaginary axis and consequently the system will be unstable (Fig. 7.53). 

Fig. 7.51. Polar plots (Nyquist diagrams) of some common functions...

V comp Phase shift of lead, lag or lag-lead compensator Vpm Angle representing phase margin on Nyquist diagram... 

Idealised Nyquist diagram corresponding to the equivalent circuit of Ershler-Randles (Fig.2.6). 

Figure 2.37 shows an example impedance spectrum of an electrochemical system with two time constants. Figure 2.37a, b, and c are the equivalent circuit, simulated Nyquist diagram, and Bode plot, respectively. 

Figure 5.5. Experimental Nyquist diagrams for the ORR on a Pt nanoparticle catalyst (40% Pt/Pt+C) in alkaline solution (1 M NaOH) using an active layer on an RDE at 25°C (EIS frequency range 5 mHz-105 Hz). Electrode potentials (versus Hg/HgO electrode) ( ) -0.02 V ( ) -0.03 V (A) -0.06 V and ( ) -0.08 V [4], (Reprinted from Electrochimica Acta, 48(25-6), Genies L, Bultel Y, Faure R, Durand R. Impedance study of the oxygen reduction reaction on platinum nanoparticles in alkaline media, 3879-90, 2003, with permission from Elsevier.)...

An impedance response can be interpreted graphically as a vector on the complex plane. The imaginary axis is the out-of-phase response (Z"), and the real axis is the in-phase response (Z ). The magnitude of the impedance response Z is the length of the vector, and the phase angle (]) describes its direction (Fig. 3). Each point on the plane defines an impedance response at a particular frequency. Such representations are commonly referred to as complex plane plots, Nyquist diagrams, or Cole-Cole plots. However, the Cole-Cole plot is actually the complex plane representation of the dielectric response of a material. 

FIG. 7.15 Nyquist diagram for the faradic impedance measurements at the I la/l tth-monok er-modifled Au electrode in the presence of 10 mM [Fe(CN)J (a) 11 b-state, and (b) I la-state nwno-iayer. Applied bias constant potential, 0.6 V. amplitude of the alternating voltage, 10 mV Performed in 0.01 M phosphate buffer, pH 7.0. (Adapted from reference 77, Figure 4. Copyright 1998, American Chemical Society.)... 

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