Asymptotic magnitude

Aj value attains its minimum asymptotic magnitude Aj = 0.213 [12]. TTiis means, that for the indicated values nanoclusters in PC and PAr structure are adopted well to the external influence change A > 0.91). 

The form of that function is shown in Figure 3.2. There are two specific parameters that can be immediately observed from this function. The first is that the maximal asymptote of the function is given solely by the magnitude of A/B. The second is that the location parameter of the function (where it lies along the input axis) is given by C/B. It can be seen that when [Input] equals C/B the output necessarily will be 0.5. Therefore, whatever the function the midpoint of the curve will lie on a point at Input = C/B. These ideas are useful since they describe two essential behaviors of any dmg-receptor model namely, the maximal response (A/B) and the potency (concentration of input required for effect C/B). Many of the complex equations... 

Even with MATLAB, we should still know the expected shape of the curves and its telltale features. This understanding is crucial in developing our problem solving skills. Thus doing a few simple hand constructions is very instructive. When we sketch the Bode plot, we must identify the comer (break) frequencies, slopes of the magnitude asymptotes and the contributions of phase lags at small and large frequencies. We ll pick up the details in the examples. 

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,... 

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. 

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. 

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. 

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). 

In Example 8.12, we used the interacting form of a PID controller. Derive the magnitude and phase angle equations for the ideal non-interacting PID controller. (It is called non-interacting because the three controller modes are simply added together.) See that this function will have the same frequency asymptotes. 

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... 

The time dependence of the various species concentrations will depend on the relative magnitudes of the four rate constants. In some cases the curves will involve a simple exponential rise to an asymptote, as is the case for irreversible reactions. In other cases the possibility of overshoot (exists, as indicated in Figure 5.2. Whether or not this phenomenon will occur depends on the relative magnitudes of the rate constants and the initial conditions. However, the fact that both roots of equation 5.2.34 must be real requires that there be only one maximum in the curve for R(t) or S(t). 

The relationship between the difference spacing and the magnitude of the derivative departs from the degree of proportionality we observe at smaller spacings. As the spacing increases, the maximum value of the computed difference asymptotically approaches the value of unity. 

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