Asymptote low-frequency

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,... 

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,... 

A common term used in control engineering is bandwidth, which is defined as the frequency at which any given G(jco) drops to 70.7% of its low frequency asymptotic value (Fig. 8.2). It... 

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. 

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. 

Figure E8.12. Only high and low frequency asymptotes are shown here. Fill in the rest with the help of MATLAB.

Fig. E8.13 is a rough hand sketch with the high and low frequency asymptotes. It is meant to help interpret the MATLAB plots that we will generate next.

The shape of the magnitude plot resembles that of a PI controller, but with an upper limit on the low frequency asymptote. We can infer that the phase-lag compensator could be more stabilizing than a PI controller with very slow systems.1 The notch-shaped phase angle plot of the phase-lag compensator is quite different from that of a PI controller. The phase lag starts at 0° versus -90°... 

Note that the low frequency asymptote of the magnitude plot is not 1 (0 dB). Why That s because the transfer function is not in the time constant form. If we factor the function accordingly, we should expect a low frequency asymptote of 1/6 (-15.6 dB). 

We need to find the frequency m, when the magnitude drops from the low frequency asymptote by 1/V2. From the magnitude equation in Example 8.3, we need to solve... 

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. 

In this case the analytic result for the low frequency asymptotics of the third order polarization operator (see Fig. 3.6) [35] is used... 

Note that the numerical factor before the leading logarithm here is simply the product of the respective numerical factors in the correction to the wave function in (3.65), the low-frequency asymptote of the one-loop polarization — 1/(157t) in (2.5), the factor 47r(Za) in (3.62), and factor 2 which reflects that both wave functions in the matrix element in (3.62) have to be corrected. 

Technically the lower order contributions to HFS are produced by the constant terms in the low-frequency asymptotic expansion of the electron factor. These lower order contributions are connected with integration over external photon momenta of the characteristic atomic scale mZa and the approximation based on the skeleton integrals in (9.9) is inadequate for their calculation. In the skeleton integral approach these previous order contributions arise as the infrared divergences induced by the low-frequency terms in the electron factors. We subtract leading low-frequency terms in the low-frequency as Tnp-totic expansions of the electron factors, when necessary, and thus get rid of the previous order contributions. 

Graphical methods provide a first step toward interpretation and evaluation of impedance data. An outline of graphical methods is presented in Chapter 16 for simple reactive and blocking circuits. The same concepts are applied here for systems that are more typical of practical applications. The graphical techniques presented in this chapter do not depend on any specific model. The approaches, therefore, can provide a qualitative interpretation. Surprisingly, even in the absence of specific models, values of such physically meaningful parameters as the double-layer capacitance can be obtained from high- or low-frequency asymptotes. 

Note. If Kp 1, then as can be seen from eq. (17.6), the low-frequency asymptote shifts vertically by the value log Kp. Equation (17.5) shows that Kp has no effect on the phase shift. 

We can use frequency response techniques (see Chapter 17) to identify experimentally a poorly known process. Do you have any ideas on how you could do it To help you in your thoughts, consider the Bode diagrams of various systems that were examined in Chapter 17. Notice the information provided by characteristics such as the corner frequency (determines the unknown time constant), the level of low-frequency asymptotes (determines the value of static process gains), the slope of high-frequency asymptotes (determines the order of a system), and the behavior of phase lag (keeps increasing for systems with dead time). Note For further details, consult Ref. 11.)... 

Now consider function m,Ci for a three-layer medium and transform it in a way which is convenient for obtaining the low-frequency asymptote. 

The FR method was introduced as a method for macroscopic measurement of diffusion coefficients, but it can also be used for measurement of equilibrium data. The diffusion coefficient is obtained from the locus of the maximum of the out-of-phase function, while the slope of the adsorption isotherm can be obtained from the low-frequency asymptote of the in-phase function [15]. Some applications of the FR methods to investigation of heterogeneous reaction systems have also been reported [46-51]. 

The equilibrium parameters are obtained easily from the low-frequency asymptotes of the particle FRFs. 

In these equations, we use the fact that Op, bpp,. .. and Oq, f qq,. .. are essentially derivatives of two inverse functions (0 and P). (dQ/dP)s and (9 2/9P )s are the first and second derivative of the adsorption isotherm written in the dimensional form [56] at steady state defined by P and Q. It can be shown that the low-frequency asymptotic values of the third- and higher-order functions are proportional to the third- and higher-order derivatives of the adsorption isotherm... 

For the case of nonisothermal adsorption governed by micropore diffusion, treated in Section III.D, the isotherm derivatives with respect to temperature can also be estimated from the low-frequency asymptotes of the Fj functions ... 

The low-frequency asymptotic values of first- and second-order FRFs for the pore-surface diffusion model are ... 

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