Frequency response analysis diagram

Frequency response analysis is another classical tool that has been used in the analysis and design of process control systems. The Laplace variable s is replaced by jfrequency response is then plotted using an Argand diagram approach. 

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. 

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) ]. 

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 frequency response (although called transfer function) is a conmum function in signal analysis and control engineering when the dynamic behavior of a system must be analyzed. Therefore, the input and output parameters of the system will be compared as a function of frequency. For example, when the system is stimulated with a harmonic input signal of a certain frequency, the system will answer with the same frequency, but with attenuated amplitude and a shifted phase. Since the amplitude attenuation and the phase shift are both functiOTis of the stimulation frequency, it is common to plot them in Bode diagrams, where the amplirnde response and the phase response are displayed separately over the frequency. 

In addition to an examination of the frequency response of series and parallel components of the circuit impedance/admittance, another approach may be particularly valuable. This analysis method involves plotting the real versus imaginary parts of some such complex quantity as admittance or impedance as parametric functions of frequency. Such Argand or "circle diagrams" have been used for many years in electrical engineering when complex dielectric constant is the quantity considered, they are known as Cole-Cole plots. ... 

Frequency response techniques are powerful tools for the design and analysis of feedback control systems. The frequency response characteristics of a process, its amplitude ratio AR and phase angle, characterize the dynamic behavior of the process and can be plotted as functions of frequency in Bode diagrams. The Bode stability criterion provides exact stability results for a... 

Figure 1-3 shows a schematic diagram of a dynamic IR linear dichroism (DIRLD) experiment [20-25] which provided the foundation for the 2D IR analysis of polymers. In DIRLD spectroscopy, a small-amplitude oscillatory strain (ca. 0.1% of the sample dimension) with an acoustic-range frequency is applied to a thin polymer film. The submolecular-level response of individual chemical constituents induced by the applied dynamic strain is then monitored by using a polarized IR probe as a function of deformation frequency and other variables such as temperature. The macroscopic stress response of the system may also be measured simultaneously. In short, a DIRLD experiment may be regarded as a combination of two well-established characterization techniques already used extensively for polymers dynamic mechanical analysis (DMA) [26, 27] and infrared dichroism (IRD) spectroscopy [10, 11]. 

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