Frequency response scales

With these results, we are ready to construct plots used in frequency response analysis. The important message is that we can add up the contributions of individual terms to construct the final curve. The magnitude, of course, would be on the logarithmic scale. 

In spectroscopy we may distinguish two types of process, adiabatic and vertical. Adiabatic excitation energies are by definition thermodynamic ones, and they are usually further defined to refer to at 0° K. In practice, at least for electronic spectroscopy, one is more likely to observe vertical processes, because of the Franck-Condon principle. The simplest principle for understandings solvation effects on vertical electronic transitions is the two-response-time model in which the solvent is assumed to have a fast response time associated with electronic polarization and a slow response time associated with translational, librational, and vibrational motions of the nuclei.92 One assumes that electronic excitation is slow compared with electronic response but fast compared with nuclear response. The latter assumption is quite reasonable, but the former is questionable since the time scale of electronic excitation is quite comparable to solvent electronic polarization (consider, e.g., the excitation of a 4.5 eV n — n carbonyl transition in a solvent whose frequency response is centered at 10 eV the corresponding time scales are 10 15 s and 2 x 10 15 s respectively). A theory that takes account of the similarity of these time scales would be very difficult, involving explicit electron correlation between the solute and the macroscopic solvent. One can, however, treat the limit where the solvent electronic response is fast compared to solute electronic transitions this is called the direct reaction field (DRF). 49,93 The accurate answer must lie somewhere between the SCRF and DRF limits 94 nevertheless one can obtain very useful results with a two-time-scale version of the more manageable SCRF limit, as illustrated by a very successful recent treatment... 

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. 

Furthermore, Boukamp and Adler showed that when the electrodes on opposite sides of a cell are different from each other (as they are in a fuel cell), errors may not only involve a numerical scaling factor but also cross-contamination of anode and cathode frequency response in the measured half-cell impedances. For example. Figure 55a shows the calculated half-cell impedance of the cell idealized in Figure 53a, assuming alignment errors of 1 electrolyte thickness. Significant distortion of the halfcell impedances (Za and Zb) from the actual impedance of the electrodes are apparent, including cross-talk of anode and cathode frequency response (1 and 10 Hz, respectively), as well as a... 

The frequency response measured between a pair of transducers having A<, = 32 ftm and Np = 50 finger pairs is shown in Figure 3.21 (page 79). The amplitude, shown on a log (decibel) scale, shows the characteristic sin(X)/X behavior. The delay line phase shift

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Figure 22 shows the flow pattern when there is sufficient power and low enough viscosity for turbulence to form. Now a velocity probe must be used that can pick up the high frequency response of these turbulent flow patterns, and a chart as shown in Fig. 23 is typical. The shear rate between the small scale velocity fluctuations is called microscale shear rate, while the shear rates between the average velocity at this point are called the macroscale rates. These macroscale shear rates still have the same general form and are determined the same way as shown in Fig. 21. 

A Bode plot (Figure 7.60) displays the impedance Z against the frequency. The absolute value of the amplitude A (o>) and the phase angle rp(co) of the frequency response are separately plotted over the frequency co. A co) and co are usually displayed on a logarithmic scale whereas rp(co) is plotted linearly. The dimension of A(co) is in decibels (dB), and by definition A()dB = 20 log A co). Consequently, the logarithmic representation of A has a linear scale called the magnitude. 

In this study, the relationship between oscillatory heat release and large vortex structure was systematically examined as a function of flow and chemistry scaling. Figure 16.3 shows the experimental setup used for producing vortex flames. A premixed propane-air jet was ignited, and the flame was stabilized at the exit downstream of a sudden-expansion flameholder. A 75-watt compression driver was used to apply controlled disturbance and to produce periodic vortices into a jet flame. The frequency response of actuation was evaluated separately to maintain a similar amplitude of acoustic disturbance at the exit plane. The objectives were to identify the location for active fuel injection in the general case and to establish a scaling criterion. 

Part IV (Chapters 13 through 18) covers the analysis and design of feedback control systems, which represent the control schemes encountered most often in a chemical plant. Emphasis has been placed on understanding the effects which various feedback controllers have on the response of controlled processes, and on the selection of the most appropriate among them. The subject of controller tuning has been deemphasized, and as a consequence, the traditional root-locus techniques and frequency response tuning methods have been scaled down. 

Xia et al. (1992) applied this signal analysis method to study the oscillatory behavior of light output signals in a fast fluidized bed. Figure 4-23 shows the typical power spectral density of optic output signals in the fast fluidized bed. The oscillatory behavior of the optic output signals has no characteristic time scale, or a deterministic frequency response, but forms fractal time characteristics. 

Scale the model. For each disturbance, obtain the frequency response. The range of frequencies to be considered in further analysis corresponds to disturbance transfer function larger than 1 in magnitude. 

When studying a polymer on a large frequency/time scale, the response of a given material under a dynamic stimulus usually exhibits several relaxations. Moreover, the peaks are usually broad and sometimes and are associated with superposed processes. The relaxation rate, shape of the loss peak, and relaxation strength depend on the motion associated with a given relaxation process [41]. In general, the same relaxation/retardation processes are responsible for the mechanical and dielectric dispersion observed in polar materials [40]. In materials with low polarity, the dielectric relaxations are very weak and cannot be easily detected. The main relaxation processes detected in polymeric systems are analyzed next. 

FIGURE 633 Comparison of 3-D model calculations with experimental results of Zhou and coworkers [1994] for amplitude at the place x = 19 mm as a function of frequency. The scales are logarithmic (20 dB is a factor of 10 in amplitude). Case 1 shows a direct comparison with the physical parameters of the experiment, with isotropic BM and viscosity 28 times that of water. Case 2 is computed for the viscosity reduced to that of water. Case 3 is computed for the BM made of transverse fibers. Case 4 shows the effect of active OHC feed forward, with the pressure gain a = 0.21 and feed-forward distance )a25 fim. Thus lower viscosity, BM orthotropy, and active feed forward all contribute to higher amplitude and increased localization of the response. 

In the frequency response technique the volume is periodically increased and decreased in square waves, and the response of the pressure measured and analysed in terms of Fick s laws under equilibrium conditions. It has the twin advantages of being applicable over a wide frequency range and being able to distinguish between independent kinetic processes. As a result, it appears to show close agreement with microscopic processes, because it can resolve the effects on different length scales. 

The radius of the pole, r controls both the amplitude and the bandwidth of the resonance. As the value of oi decreases, its pole moves towards along the real axis towards the origin, and hence the frequency response has a lower amplitude at fi) = 0. From the difference equations, we know that a first order HR filter with oi < 1 is simply a decaying exponential. Small values of ai correspond to slow decay and narrow bandwidth, large values (still < 1) a steeper decay and wider bandwidth. This is exactly what we would e q)ect from the Fourier transform scaling property The standard measure of bandwidth is the 3db down point which is the width of the resonance at 1/ /2 down from the peak. For larger values of r = a, two useful approximations of the bandwidth (in normalised frequency) are ... 

Current accuracy Frequency response 0.2% of full scale (2% for 1-10 nA current ranges)... 

Typical electro-optical responses of the suspensions of spherical particles (CS81), recorded by the scattered light intensity are presented in Fig. 2. They are detected for the crystal state of the systems and for different intensities and frequencies of the applied sine-wave electric pulses. The low-frequency responses are modulated they follow the field frequency at sufficiently low field intensity and exhibit a double frequency modulation at higher field intensity. Two different time scales are involved in the decay of the responses (10 " and 1 s), which can be both exponential and oscillatory. At higher field intensity or frequency the effects cannot be distinguished by the responses of anisotropic colloids. 

Fig. 5.5. (top) Example of frequency response functions of a resonant and a so-called broadband transducer. The frequency response function was measured using a laser-vibrometer (middle) and a face-to-face method (bottom). Amplitudes are on a linear scale. 

The calibration sheets of frequency transfer functions provided by the sensor manufactures are often displayed on a logarithmic scale. Resonances are more difficult to detect in this format and, therefore, it is advised to ask for calibration sheets on both linear and logarithmic scales. The record of the frequency responses should be available in electronic, as well as in paper form, to allow for the application of deconvolution techniques. 

Just as the spectra describe the distribution of variance over different freqnencies or wave numbers, the cospectra describe the distribution of covariances over different frequencies or wave numbers. Vertical fluxes of momentum, heat, moisture, and pollutants are proportional to the co-variances and are important in atmospheric dynamics and thermodynamics. They also have important applications to agricultural problems and to oceanography. In order to measure these quantities, instruments must be designed with the proper frequency response to give correct estimates. Area estimates of surface fluxes can be obtained by using remote-sensing sensors in the atmospheric sm-face layer on a scale of kilometers. 

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