Asymptote high-frequency

From the structure of the convective diffusion equation it comes out that an asymptotic high frequency solution can be obtained in the form H(jf) f > whatever the geometry, since the convective terms become... 

High frequency (HF) asymptote When uj 3> 1/T, equation (6.18) approximates to... 

This is a rather general conclusion independent of the model of rotational relaxation. It is quite clear from Eq. (2.70) and Eq. (2.16) that the high-frequency asymptotic behaviour of both spectra is determined by the shape of g(co) ... 

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,... 

Accordingly, the phase lag also decreases quickly to the theoretical high frequency asymptote. 

We choose -180° (and not 0°) because we know that there must be a phase lag. On the magnitude log-log plot, the high frequency asymptote has a slope of-2. This asymptote intersects the horizontal K line at co = 1/x. 

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. 

To help understand MATLAB results, a sketch of the low and high frequency asymptotes is provided in Fig. E8.9. A key step is to identify the comer frequencies. In this case, the comer frequency of the first order lead is at 1/5 or 0.2 rad/s, while the two first order lag terms have their comer frequencies at 1/10, and 1/2 rad/s. The final curve is a superimposition of the contributions from each term in the overall transfer function. 

On the magnitude plot, the low frequency asymptote is a horizontal line at Kc. The high frequency... 

By choosing xD < (i.e., comer frequencies l/xD > 1/Xj), the magnitude plot has a notch shape. How sharp it is will depend on the relative values of the comer frequencies. The low frequency asymptote below 1/Xj has a slope of-1. The high frequency asymptote above l/xD has a slope of +1. The phase angle plot starts at -90°, rises to 0° after the frequency l/xIs and finally reaches 90° at the high frequency limit. 

On the Bode plot, the comer frequencies are, in increasing order, l/xp, Zq, and p0. The frequency asymptotes meeting at co = l/xp and p0 are those of a first-order lag. The frequency asymptotes meeting at co = z0 are those of a first-order lead. The largest phase lag of the system is -90° at very high frequencies. The system is always stable as displayed by the root locus plot. 

What are the low and high frequency asymptotes of the minimum phase function (s + z)/(s + p) versus the simplest nonminimum phase function (s - z)/(s + p) in a Bode plot ... 

The Bode plots are shown in Fig. 12.15. One of the most convenient features of Bode plots is that the L curves can be easily sketched by considering the low-and high-frequency asymptotes. As a> goes to zero, L goes to zero. As to becomes very large, Eq. (12.39) reduces to... 

The high-frequency asymptote intersects the L = 0 line at breakpoint frequency. The log modulus is "flat (horizontal) out to this point and then begins to drop off. 

Note that the total phase angle drops down to -180° and the slope of the high-frequency asymptote of the log modulus line is —40 dB/decade since the process is net second-order. 

If a transfer-function model is desired, approximate transfer functions can be fitted to the experimental curves. First the log modulus Bode plot is used. The low-frequency asymptote gives the steadystate gain. The time constants can be found from the breakpoint frequency and the slope of the high-frequency asymptote. The damping coefficient can be found from the resonant peak. 

Where in Fig. 6.3 two curves are drawn with the same type of line, the upper curve describes the free — free transitions of the collisional pair, and the lower one the bound — free contributions. Asymptotically, at high frequencies, the bound — free contributions amount to only about 1% of the free — free components. At the lower frequencies, however, the bound — free components are relatively more significant. In fact, the bound — free components which must be superimposed with the free — free components to obtain the spectral function, Eq. 6.54, affect the shapes of the profiles near the line centers. We note that the quadrupole-induced components 0223, 2023 do not feature a bound — bound spectrum, but the less important overlap components 0221, 2021 do. However, absorption due to this overlap component is insignificant, a few percent of the total absorption. 

The Bode diagram in this case (Fig. 7.46) is distinguished by the fact that f is a parameter which affects both the AR and the yr plots. However, the asymptotes may be determined in the same manner as for the first-order system. It is found that, for all , the AR high frequency asymptote is a straight line of slope -2 passing through the point (1,1) and the LFA is represented by the line AR = 1. The yr plots all tend to zero degrees as tor- 0 and to -180° as When toc= lr, y/--90°... 

For the high frequencies, Sharma [37] obtained an asymptotic solution which can also be expressed in terms of f and g ... 

The temperature-frequency behavior of the phase shift in magnetic SR evaluated by rigorous numerical procedure is presented in Figure 4.16 and with special emphasis on high frequencies,—in Figure 4.17. The asymptotic... 

Caprani et al. [104], defining the cut-off frequency as the intersection of the low and high-frequency asymptotes, as indicated on Fig. 10.18, have given an approximate method to deduce the size of active sites on a partially blocked electrode from the ratio of the two cut-off frequencies ... 

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