Frequency response characteristics

FIG. 8-64 Pneumatic controller a) example (h) frequency response characteristic,... 

Frequency response characteristics of first-order systems... 

The Nichols chart shown in Figure 6.26 is a rectangular plot of open-loop phase on the x-axis against open-loop modulus (dB) on the jr-axis. M and N contours are superimposed so that open-loop and closed-loop frequency response characteristics can be evaluated simultaneously. Like the Bode diagram, the effect of increasing the open-loop gain constant K is to move the open-loop frequency response locus in the y-direction. The Nichols chart is one of the most useful tools in frequency domain analysis. 

Figure 6.39 shows, for both lead compensator designs, the closed-loop frequency response characteristics for the system. 

FIG. 76. Frequency response characteristics of a conventional, graphite-coaled diaphragm (bar graph) and an o-Si H-coated diaphragm (line graph). 

Recently there has been a growing emphasis on the use of transient methods to study the mechanism and kinetics of catalytic reactions (16, 17, 18). These transient studies gained new impetus with the introduction of computer-controlled catalytic converters for automobile emission control (19) in this large-scale catalytic process the composition of the feedstream is oscillated as a result of a feedback control scheme, and the frequency response characteristics of the catalyst appear to play an important role (20). Preliminary studies (e.g., 15) indicate that the transient response of these catalysts is dominated by the relaxation of surface events, and thus it is necessary to use fast-response, surface-sensitive techniques in order to understand the catalyst s behavior under transient conditions. 

It can be shown(18) that this method may be applied to any system described by a linear differential equation or to a distance-velocity lag in order to obtain the relevant frequency response characteristics. 

In order to determine the frequency response characteristics, substitute s = uo in equation 7.60 ... 

If the frequency response characteristics of the control system are known then it is possible to estimate values of controller parameters which will give specified gain and phase margins. However, this necessitates trial and error procedures. The semi-empirical method of Ziegler and Nichols(26) is more easily applied as follows. 

M. Rosenbaum and D. Race, Frequency-Response Characteristics of Vascular Resistance Vessels, Am. J. Physiol. 215,1397-1402 (1968). 

The frequency range over which accurate data may be obtained is bounded, on the lower end, by the frequency response characteristics of the force gage and accelerometers, and on the... 

Given the importance of mobility in establishing the current gain and frequency-response characteristics of a TTFT, it is worth considering channel mobility assessment. 

The frequency response characteristics of a process element or a group of elements can be computed readily from the corresponding transfer function merely by substituting ju for s, where j is the imaginary number, /— 1, and is the angular velocity. Thus the frequency response characteristics of a simple first order lag are given by... 

By substituting joi for s in Eq. (10), the overall frequency response characteristics of a third order lag are found to be... 

As shown by Cohen and Johnson, Eq. (20) leads to frequency response characteristics which on a Bode diagram exhibit resonances both in the magnitude ratio and phase angle. The first resonance occurs at a period approximating the residence time of a slug of water in the inner pipe. 

Substituting = jw gives the frequency response characteristics, which are that the magnitude ratio is unity for all frequencies, and the phase lag (negative phase angle) increases with increasing frequency without limit. Since the characteristic of unlimited phase angle promotes system instability, time delays are undesirable and should be minimized whenever possible. 

Since the controller output must counteract the measured variable, these two quantities, 0 and 0O, are of opposite sign and hence are inherently 180° out of phase. In commercial proportional controllers this 180° phase shift and also any set gain Kp, are constant for all practical ranges of frequency. Thus the frequency response characteristics of a proportional controller are a magnitude ratio of Kp and a phase lag of 180°. 

From the Laplace transformation of Eq. (25), the transfer function for an integral controller is l/(s2 nt) and by substituting for s the corresponding frequency response characteristics are found to be a phase angle of —90° and a magnitude ratio of l/coTint. 

The transfer function for derivative control is sTd, and the frequency response characteristics are a phase angle of +90° and an amplitude ratio of a)Td. Control stability results from the leading phase angle. 

The frequency response characteristics are sketched in Fig. 6 at low frequencies. Note that the integral action provides infinite gain at zero frequency. 

Fig. 6. Frequency response characteristics of three-mode controller.

Figure 6 gives the frequency response characteristics of ideal threemode control. The chief contributions of the derivative action are to decrease the phase lag of the controller and provide high gain at high frequencies. 

In general, the transient response of stable nonlinear systems cannot be inferred from the frequency response characteristics derived from describing function analysis. 

A generally used set of criteria for good control is that the controlled variable in response to a unit step change in set point (a) overshoot by not more than 20 per cent of the step and (b) damp out with a subsidence ratio of about one-third. This behavior is approximated by many systems if the closed-loop frequency response and the corresponding open-loop frequency response have certain simple characteristics. Since the closed-loop frequency response characteristics can be determined readily from the open-loop frequency response, the latter characteristics of simple control systems can be used as a convenient basis for design. 

Control System Synthesis from Frequency Response Characteristics... 

Generally recommended frequency response characteristics for automatic control systems are given by Oldenburger (03) as follows ... 

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