Frequency response curves

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... 

FIG. 8-75 Frequency response curves for a pneumatic positioner/actuator (a) input signal to stem travel for a 69-inch spring and diaphragm actuator with a 1.5-inch total travel and. 3-15 psig input pressure (h ) dynamic stiffness for the same positioner/actuator. 

Figure 6.29 (see also Appendix 1, fig629.m) shows the closed-loop modulus frequency response. Curve (a) is the best flatband response, curve (b) is the response with Mp set to 3dB. 

Sanders We did a frequency-response curve in the knockout mice. We were still unable to elicit responses. 

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)...

It will be seen that, as in the case of the LED, control of the bias voltage gives simple modulation of the laser output intensity. This is particularly useful in phase-modulation fluorometry. However, a measure of the late awareness of the advantages of IR techniques in fluorescence is that only recently has this approach been applied to the study of aromatic fluorophores. Thompson et al.(51) have combined modulated diode laser excitation at 670 and 791 nm with a commercial fluorimeter in order to measure the fluorescence lifetimes of some common carbocyanine dyes. Modulation frequencies up to 300 MHz were used in conjunction with a Hamamatsu R928 photomultipler for detecting the fluorescence. Figure 12.18 shows typical phase-modulation data taken from their work, the form of the frequency response curves is as shown in Figure 12.2 which describes the response to a monoexponential fluorescence decay. 

Different processes have different MR and 0 dependence on ft). Since each process is unique, the frequency-response curves are like fingerprints. By merely looking at curves of MR and 0 we can tell the kind of system (order and damping) and the values of parameters (time constants, stcadystate gain, and damping coefficient). 

Commercial software can also be used to generate these frequency response curves (CC, CONSYD, MATRIX-X). 

Another numerical value of m is specified and step 4 is repeated. Picking a number of frequencies over the range of interest for the process gives the complete frequency-response curves. 

Chap. 12, the total frequency-response curve of a complex system is easily obtained on a Bode plot by splitting the system into its simple elements, plotting each of these, and merely adding log moduli and phase angles together. Therefore the graphical generation of the required curve is relatively easy. Of... 

The most useful frequency-domain specification is the maximum closedloop fog modulus. The phase margin and gain margin spedfications can sometimes give poor results when the shape of the frequency-response curve is unusual. 

Both the openloop and the closedloop frequency-response curves can be easily generated on a digital computer by using the complex variables and functions discussed in ( han. 12, Tlie freauencv-resnnnse curves for the closedloon... 

The main disadvantage of direct sine-wave testing is that it can be very time-consuming when applied to typical large time-constant chemical process equipment. The steadystate oscillation must be established at each value of frequency. It can lake days to generate the complete frequency-response curves of a slow process. 

One of the most useful and practical methods for obtaining experimental dynamic data from many chemical engineering processes is pulse testing. It yields reasonably accurate frequency-response curves and requires only a fraction of the time that direct sine-wave testing takes. 

In theory only one pulse input is required to generate the entire frequency-response curve. In practice several pulses arc usually needed to establish the required size and duration of the input pulse. Some tips on the practical aspects of pulse testing are discussed in Sec. 14.3.3. 

A specific numerical value of frequency m is picked. The integrations are performed numerically (see Sec. 14.3.2 below), giving one point on the frequency-response curves. Then frequency is changed and the integrations repeated, using the same experimental time functions and m,) but a new value of frequency m. 

These data can be converted into frequency-response curves, basically by difTerentiating both input and output curves in the frequency domain. The process transfer function G(jj is... 

We are trying to extract a lot of information from one pulse test, i.e., the whole frequency response curve. This is asking a lot from one experiment. 

Instead of converting the step or pulse responses of a system into frequency response curves, it is fairly easy to use classical least-squares methods to solve for the best values of parameters of a model that fit the time-domain data. 

Schroeder, 1987] Schroeder, M. R. (1987). Statistical parameters of the frequency response curves of large rooms . J. Audio Eng. Soc., 35(5) 299-306. English translation of Schroeder (1954). 

Schroeder and Kuttruff, 1962] Schroeder, M. R. and Kuttruff, K. H. (1962). On frequency response curves in rooms. Comparison of experimental, theoretical, and... 

On the basis of this model one expects that if the spatial period of the grating pattern being recorded is increased, the achievable index modulation will drop off when the period becomes larger than the distance over which monomer can diffuse in the time before the fixing exposure. The experimental spatial-frequency response curve in Fig. 18 shows this expected low-spatial-frequency cutoff (37). Measured rates of monomer diffusion in polymer films are also consistent with the basic diffusion model (38). 

The frequency-response curves of three 15-point filters with a = 0.5, 1, and 2 are illustrated in Figure 14. The case of the filter constructed with a = 2 is of particular interest as its frequency response increases beyond zero frequency and then falls off rapidly. The effect of convoluting a spectrum with this function is apparently to enhance resolution. The practical use of such filters should be undertaken with care, however, and they are best used in an interactive mode when the user can visibly assess the effects before proceeding to further data manipulation. 

For the case of the frequency response of a real catalyst in which the number and magnitude of different types of adsorption is not known beforehand, the response may be interpreted by curve fitting to give a distribution of adsorption types versus rate constant. The interpretation of an experimentally determined frequency response curve would not be too dissimilar, in principle, from the interpretation of the output of an infrared spectrum where two or more unknown compounds are to be identified and quantitatively estimated from a single IR scan. 

Both the openloop and the closedloop frequency response curves can be easily generated on a digital computer by using the complex variables and functions in FORTRAN discussed in Chapter 10 or by using MATLAB software. The frequency response curves for the closedloop servo transfer function can also be fairly easily found graphically by using a Nichols chart. This chart was developed many years ago, before computers were available, and was widely used because it greatly facilitated the conversion of openloop frequency response to closedloop frequency response. 

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