Frequency response systems

The sharpness of the frequency response of a resonant system is conunonly described by a factor of merit, called the quality factor, Q=v/Av. It may be obtained from a measurement of the frill width at half maxuuum Av, of the resonator frequency response curve obtained from a frequency sweep covering the resonance. The sensitivity of a system (proportional to the inverse of tlie minimum detectable number of paramagnetic centres in an EPR cavity) critically depends on the quality factor... 

In this paper, we discuss semi-implicit/implicit integration methods for highly oscillatory Hamiltonian systems. Such systems arise, for example, in molecular dynamics [1] and in the finite dimensional truncation of Hamiltonian partial differential equations. Classical discretization methods, such as the Verlet method [19], require step-sizes k smaller than the period e of the fast oscillations. Then these methods find pointwise accurate approximate solutions. But the time-step restriction implies an enormous computational burden. Furthermore, in many cases the high-frequency responses are of little or no interest. Consequently, various researchers have considered the use of scini-implicit/implicit methods, e.g. [6, 11, 9, 16, 18, 12, 13, 8, 17, 3]. 

Because of its small size and portabiHty, the hot-wire anemometer is ideally suited to measure gas velocities either continuously or on a troubleshooting basis in systems where excess pressure drop cannot be tolerated. Furnaces, smokestacks, electrostatic precipitators, and air ducts are typical areas of appHcation. Its fast response to velocity or temperature fluctuations in the surrounding gas makes it particularly useful in studying the turbulence characteristics and rapidity of mixing in gas streams. The constant current mode of operation has a wide frequency response and relatively lower noise level, provided a sufficiently small wire can be used. Where a more mgged wire is required, the constant temperature mode is employed because of its insensitivity to sensor heat capacity. In Hquids, hot-film sensors are employed instead of wires. The sensor consists of a thin metallic film mounted on the surface of a thermally and electrically insulated probe. 

Frequency response characteristics of first-order systems... 

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

In general, the complete system frequency response is obtained by summation of the log modulus of the system elements, and also summation of the phase of the system elements. 

An important difference between analysis of stability in the. v-plane and stability in the frequency domain is that, in the former, system models in the form of transfer functions need to be known. In the latter, however, either models or a set of input-output measured open-loop frequency response data from an unknown system may be employed. 

The closer the open-loop frequency response locus G(ja )//(ja ) is to the (—l,j0) point, the nearer the closed-loop system is to instability. In practice, all control... 

Second-order system closed-loop frequency response... 

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

Relationship between frequency response and time response for closed-loop systems... 

An open-loop frequency response test on an unknown system produced the following results ... 

Theoretically, we are making the presumption that we can study and understand the dynamic behavior of a process or system by imposing a sinusoidal input and measuring the frequency response. With chemical systems that cannot be subject to frequency response experiments easily, it is very difficult for a beginner to appreciate what we will go through. So until then, take frequency response as a math problem. 

Frequency response analysis allows us to derive a general relative stability criterion that can easily handle systems with time delay. This property is used in controller design. 

With frequency response analysis, we can derive a general relative stability criterion. The result is apphcable to systems with dead time. The analysis of the closed-loop system can be reduced to using only the open-loop transfer functions in the computation. 

The concept of gain and phase margins derived from the Nyquist criterion provides a general relative stability criterion. Frequency response graphical tools such as Bode, Nyquist and Nichols plots can all be used in ensuring that a control system is stable. As in root locus plots, we can only vary one parameter at a time, and the common practice is to vary the proportional gain. 

With the Routh-Hurwitz analysis in Chapter 7, we should find that to have a stable system, we must keep Kc < 7.5. (You fill in the intermediate steps in the Review Problems. Other techniques such as root locus, direct substitution or frequency response in Chapter 8 should arrive at the same result.)... 

For a given system and closed-loop gain displayed in the root locus plot, we can generate its corresponding time response (step and impulse) and frequency response (Bode, Nyquist, and Nichols) plots. 

We will simply state that the SISO system design tool sisotool, as explained in Session 6, can be used to do frequency response plots. Now, we want to use the default view, so we just need to enter ... 

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. 

The frequency response of the detection system is of low-pass type for characteristog-raphers and band-pass for bridges (see Section 10.4). In both types of measurements the narrowing of the bandwidth corresponds to a longer time of measurement. Depending on the chosen detection system, several problems (true traps) may be encountered in resistance thermometry. 

Fig. 15.3 A typical frequency response curve obtained due to the introduction of air into the resonant cavities. Shown above in pink is a typical absorption profile of the resonator under vacuum, and in blue is the shift of the resonant frequency to a lower value upon introducing air into the system (See Color Plates)...

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