Frequency response diagrams

Fig. 6.6 Frequency response diagrams for a second-order system.

A sinusoidal input. The frequency of the sinusoidal variation is changed and the steady-state response of the effluent at different input frequencies is determined, thus generating a frequency-response diagram for the system. 

The time variations of the effluent tracer concentration in response to step and pulse inputs and the frequency-response diagram all contain essentially the same information. In principle, any one can be mathematically transformed into the other two. However, since it is easier experimentally to effect a change in input tracer concentration that approximates a step change or an impulse function, and since the measurements associated with sinusoidal variations are much more time consuming and require special equipment, the latter are used much less often in simple reactor studies. Even in the first two cases, one can obtain good experimental results only if the average residence time in the system is relatively long. 

FIGURE 13.80 Typical frequency response diagrams (a) Two-, (b) three-, (c) four-cavity bandpass filters. 

Bode plot A type of frequency response diagram used for analysing the frequency response of a system to a disturbance signal. It is the plot of the logarithm of the amplitude ratio with the logarithm of the phase angle measurements. 

The M and N circles can be superimposed on a Nyquist diagram (called a Hall chart) to directly obtain closed-loop frequency response information. 

Alternatively, the closed-loop frequency response can be obtained from a Nyquist diagram using the direct construction method shown in Figure 6.25. From equation (6.73)... 

Fig. 6.25 Closed-loop frequency response from Nyquist diagram using the direct construction method.

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. 

Running script file fig629.m will produce the closed-loop frequency response gain diagrams shown in Figure 6.29 for Example 6.4 when K = 3.8 and 3.2 (value of K for best flatband response). 

The command cloop is used to find the closed-loop transfer function. The command max is used to find the maximum value of 20 logio (mag), i.e. Mp and the frequency at which it occurs i.e. tUp = uj k). A while loop is used to find the —3 dB point and hence bandwidth = ca (n). Thus, in addition to plotting the closed-loop frequency response gain diagrams,/ gd29.7 will print in the command window ... 

Figure 1. Schematic diagram of frequency response chemisorption apparatus.

Fig. 3. Bode diagram of the closed-loop under the estimation/compensation control approach. Closed-loop system is expected to be non-sensitive to high frequency signais. Arrows indicate how the frequency response of the ciosed-ioop as Kc is increased.

Evidence for the formation of gels from aPS systems is obtained from simple mechanical, (1.4.5) viscoelastic, (7.8) thermodynamic (1.6) and spectroscopic ( ) techniques. Simple tube tilting, falling ball methods and differential scanning calorimetry have been used to determine the phase diagrams for a number of systems. Viscoelastic measurements on the aPS-carbon disulfide system show that the low frequency response indicative of a... 

This heuristic argument forms the basis of the Bode stability criterion(22,24) which states that a control system is unstable if its open-loop frequency response exhibits an AR greater than unity at the frequency for which the phase shift is —180°. This frequency is termed the cross-over frequency (coco) for reasons which become evident when using the Bode diagram (see Example 7.7). Thus if the open-loop AR is unity when i/r = —180°, then the closed-loop control system will oscillate with constant amplitude, i.e. it will be on the verge of instability. The greater the difference between the open-loop AR (< I) at coc and AR = 1, the more stable the closed-loop... 

In order to facilitate the application of the Bode criterion the system frequency response may be represented graphically in the form of a Bode diagram or plot. This consists of two graphs which are normally drawn with the axes ... 

The polar plot is an alternative to the Bode diagram for representing frequency response data and is the locus of all points occupied by the tip of a vector in the complex plane whose magnitude and direction are determined by the amplitude ratio and phase shift, respectively, as the frequency of the forcing function applied to the system is varied from zero to infinity. 

Figure 3.12 Comb filter (clockwise from top-left) flow diagram, time response, frequency response, and pole diagram.

Figure 6.7 (a) Diagram of the use of a network analyzer as used to measure one-port frequency response for a TSM resonator, (b) Measurement of one-port frequency response using a synthesized oscillator source together with a vector voltmeter and a pair of directional couplers. 

Fig 5.13 Block diagram, showing frequency response analyser method of measurement. 

It has to be mentioned that such equivalent circuits as circuits (Cl) or (C2) above, which can represent the kinetic behavior of electrode reactions in terms of the electrical response to a modulation or discontinuity of potential or current, do not necessarily uniquely represent this behavior that is other equivalent circuits with different arrangements and different values of the components can also represent the frequency-response behavior, especially for the cases of more complex multistep reactions, for example, as represented above in circuit (C2). In such cases, it is preferable to make a mathematical or numerical analysis of the frequency response, based on a supposed mechanism of the reaction and its kinetic equations. This was the basis of the important paper of Armstrong and Henderson (108) and later developments by Bai and Conway (113), and by McDonald (114) and MacDonald (115). In these cases, the real (Z ) and imaginary (Z") components of the overall impedance vector (Z) can be evaluated as a function of frequency and are often plotted against one another in a so-called complex-plane or Argand diagram (110). The procedures follow closely those developed earlier for the representation of dielectric relaxation and dielectric loss in dielectric materials and solutions [e.g., the Cole and Cole plots (116) ]. 

Figures like Fig. 4, called Nyquist diagrams, are widely used in frequency response characterization. 

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. 

A further use for the control engineer is that the linear analysis allows him to use frequency response methods for control system design. The fact that the system has been broken down into subsystems also means that any nonlinearities such as deadzone that are known to exist may be introduced easily into the block diagram layout and represented by describing functions. 

The Bode diagrams (in honour of H. W. Bode) constitute a convenient way to represent the frequency response characteristics of a system. As we can see from Eqs. (17.14a) and (17.14b), the amplitude ratio and the phase shift of the ultimate response of a system are functions of the frequency to. The Bode diagrams consist of a pair of plots showing ... 

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