Polar plots

For a given value of lu, equation (6.9) represents a point in complex space P(lu). When LU is varied from zero to infinity, a locus will be generated in the complex space. This locus, shown in Figure 6.2, is in effect a polar plot, and is sometimes called a harmonic response diagram. An important feature of such a diagram is that its shape is uniquely related to the dynamic characteristics of the system. 

Using equations (6.18) and (6.21), values for the modulus and phase angle may be ealeulated as shown in Table 6.1. The results in Table 6.1 may be represented as a Polar Plot, Figure 6.4(a) or as a reetangular plot. Figures 6.4(b) and (e). Sinee the reetangular plots show the system response as a funetion of frequeney, they are usually referred to as frequeney response diagrams. 

The phase relationship of each point of imbalance is the third factor that must be known. Balancing instruments isolate each point of imbalance and determine their phase relationship. Plotting each point of imbalance on a polar plot does this. In simple terms, a polar plot is a circular display of the shaft end. Each point of imbalance is located on the polar plot as a specific radial, ranging from 0 to 360°. 

Figure 18. Contour plots of the potential energy surfaces of the first three electronic states of H2O. The polar plots depict the movement of one H atom around OH with an OH bond length fixed at 1.07 A. Energies are in electron volts relative to the ground electronic state. The X and B states are degenerate at the conical intersection (denoted by (g)) in the (a) H—OH geometry and (b) H—HO geometry. Reprinted fix)m [75] with permission from the American Association for the Advancement of Science.

The surface energy y hkl) for each surface h,k,l) is plotted in a polar plot such that the length of the vector is proportional to the surface energy. 

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... 

On the magnitude plot, the low frequency (also called zero frequency) asymptote is a horizontal line at Kp. On the phase angle plot, the low frequency asymptote is the 0° line. On the polar plot, the zero frequency limit is represented by the point Kp on the real axis. In the limit of high 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... 

For a process or system that is sufficiently underdamped, Z, < 1/2, the magnitude curve will rise above the low frequency asymptote, or the polar plot will extend beyond the K-radius circle. 

The magnitude and phase angle plots are sort of "upside down" versions of first order lag, with the phase angle increasing from 0° to 90° in the high frequency asymptote. The polar plot, on the other hand, is entirely different. The real part of G(jco) is always 1 and not dependent on frequency. 

When co = ir/0. ZG(jco) = -k. On the polar plot, the dead time function is a unit circle. 

The magnitude log-log plot is a line with slope -1. The phase angle plot is a line at -90°. The polar plot is the negative imaginary axis, approaching from negative infinity with co = 0 to the origin with to —> °°. 

Nyquist stability criterion Given the closed-loop equation 1 + Gol (joi) = 0, if the function G0l(J ) has P open-loop poles and if the polar plot of GOL(](o) encircles the (-1,0) point... 

In this statement, we have used "polar plot of G0l" to replace a mouthful of words. We have added G0L-plane in the wording to emphasize that we are using an analysis based on Eq. (7-2a). The real question lies in what safety margin we should impose on a given system. This question leads to the definitions of gain and phase margins, which constitute the basis of the general relative stability criteria for closed-loop systems. 

Figure 8 Polar plots of the in-phase and quadrature components of the 79Br NMR signal in a powder sample of KBr in a magnetic field of 7 T under MAS at 5.1 kHz. The carrier frequencies were (A) 100.280545 MHz, (B) 100.281545 MHz, (C) 100.282545 MHz, and (D) 100.283545 MHz. The 79Br resonance frequency was 100.282545 MHz.

Redford, G. I. and Clegg, R. M. (2005). Polar plot representation for frequency-domain analysis of fluorescence lifetimes. J. Fluoresc. 15, 805-15. 

Figure 6.2 (a) Orientation of the main magnetic axis in the ground Kramers doublet of [Cp ErCOT], [62]. Colour scheme Er, purple C, yellow H, white, (b) Polar plot of the angular dependence of the single-crystal magnetic susceptibility recorded in the a-b plane, comparison between the results of the... 

Figure 7. Polar plots of the reaction product vector for three surface mechanisms. Key a, simple adsorption-desorption b, series process, kt = kd = k and c, branch...

If a polar plot of the total openloop tranter function of the system B( , wraps around the ( — 1,0) point in the Gf/B plane as frequency co goes from zero to infinity, the system is closedloop unstable. 

The two polar plots sketched in Fig. 13.1a show that system A is closedloop unstable whereas system B is closedloop stable. 

Polar plots showing closedloop Msbility or instability. 

We now let m take on values from 0 to + oo and plot the real and imaginary parts of Gjk(( ) B(( ). This, of course, is just a polar plot of as sketched... 

The Nyquist plot does not encircle the (—1, 0) point if the polar plot of Mreal axis inside the unit circle. The system would then be dosedloop stable. 

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