Critical, frequency point

Earlier experiments indicate clearly that a lowered sound pressure level can be an effective measure to reduce the inconvenience reactions due to a ventilation noise, provided that it is targeted at the most critical frequency range from the point of view of influence or that the measure results in a general lowering over the entire spectral range of the ventilation noise. 

However, it can be very useful for getting accurate data at one or two important frequencies. For example, it can be used to get amplitude and phase angle data near the critical -180° point. 

Figure 6-5 shows the evolution of the dynamic moduli for a LM pectin/caldum system in the vicinity of the gel point as a function of the aging time. The evolution of the dynamic moduli was similar to that one observed as a function of the calcium concentration. In the initial period of aging the system showed the typical liquid-like behavior. Then both moduli increased with time, G increasing more rapidly than G" and with lower dependence on frequency. After 15 hr of aging, the system was just above the critical gel point, with a viscoelastic exponent A in the range of0.65-0.68. After the gel point, G passed beyond G", first in the lower frequency range where one can observe the initial formation of the elastic plateau. 

A final point needs to be made regarding taking measurements in time. Suppose we sample a signal y(t) at a sampling rate of A. Nyquist showed that if there exists a critical frequency /c = 1 /(2A) for a system such that measurements are limited to frequencies smaller than fc, then the function y t) is completely determined by these measurements. 

Experiments have been carried out on p-(nitrobenzyloxy)-biphenyl [30] and typical patterns in the conductive range at onset are shown in Fig. 6. At low frequencies disordered rolls without point defects have been observed with a strong zig-zag (ZZ) modulation (see Fig. 6a) which can be interpreted as the isotropic version of oblique rolls. Above a critical frequency, a square pattern is observed which retains the ZZ character because the lines making up the squares are undulated. At onset the structure is disordered however, after a transient period defects are pushed out and the structure relaxes into a nearly defect-free, long-wave modulated, quasi-periodic square pattern (see Fig. 6b). 

The sampling procedure used will obviously depend on the type of sample whether it is liquid or solid fresh, chilled or frozen and the type of container e.g. tinned, bottled). Other major problems are the frequency of sampling and the position on a production line from which a sample is taken. For example, when sampling from a food production line is carried out, an important consideration is whether or not the food has been subjected to sterilisation, or any form of pasteurisation after the point from which the sample was taken. This is considered further in the chapter on food microbiology under the concept of Hazard Analysis Critical Control Point (HACCP). Whatever the form of the sample, it should be collected in a sterile container using aseptic techniques, returned to the laboratory under conditions identical to those from which it was taken, and processed as rapidly as possible. 

The solution can be found conveniently by using Laplace transform technique with respect to time, but the solution takes somewhat complicated form, since the inversion integral includes the branch points corresponding to the critical frequency. The detail of the solution has been described by Toma Morioka [4]. Here we say only that the present result, generalized for arbitrary propagation direction, can be obtained by replacing M in [4] to m = ucos6/c. 

The potential to extend 2-dimensional covariance NMR to higher dimensionality has its foundations in Eq. (5.13). Thus, Snyder et al. [16] laid the basis for the computation of 4D NOESY spectra. In their strategy, the critical entry point consisted of considering a 4D data set an array of 2D data or a plane-of-planes. In order to Ulustrate the calculations, the terms donor and acceptor planes in combination with donor and acceptor pairs were coined. It should be noted that an acceptor plane is associated with each donor pair 01,0)2) at frequencies < i and < 2- Mapping of either the acceptor planes onto the donor planes or vice versa describes the projection of a dimension onto another, which leads to an increase for direct covariance or to a decrease for indirect covariance in dimensionality of resolution. 

In the complex-plane representation of the impedance behavior of a parallel RC circuit, it is convenient to identify the maximum (so-called top point ) in the semicircular plot which is at a critical frequency O) = l/RC, the reciprocal of the time constant for the response of the circnit. A similar sitnation arises for a series RC... 

A filter is a circuit that passes certain frequencies and attenuates or rejects all other frequencies. The passband of a filter is the region of frequencies with which the signals are allowed to pass through the filter with minimum attenuation, usually defined as less than -3 decibels (dB) attenuation. The critical frequency, f, (also called the cutoff frequency) defines the end of the passband and is normally specified at the point where the response drops -3 dB (about 70.7%) from the passband response. 

If Eq. (7.120) is rewritten with a common denominator, it is noticed that the poles and zeros of the results are located on the negative real axis of the s plane, and the lowest critical frequency is a pole (at 5 = —< ). But the poles and zeros mayor may not be interlaced depending on the values chosen. Thus, Z may be an RC or an RLC driving point impedance. Nevertheless, Z falls off more or less uniformly until it levels out at high frequency. At DC, Z is given by... 

Critical frequencies A collective term for the poles and zeros of a transfer function or a driving point function of a network. 

More detailed measurements of the dependences Uth f) in pure and doped MBBA at various temperatures (for sandwich cells) were performed [76, 109]. The results of these measurements are represented in Fig. 5.20. The threshold of the vortical motion was taken as the onset of the circular tion of the solid impurity particles in the electrode plane. The shape of the curves in Fig. 5.20 depends on the electrical conductivity. With a high electrical conductivity the curves have a plateau in the low-frequency region and a characteristic dependence I7th oc at frequencies above the critical frequency. At the transition point to the nematic phase the threshold voltage of the instability does not change. It is shown in [109] that the height of the low-frequency plateau is proportional to and at frequencies of u > 47r(j/e the threshold field does not depend on <7. Moreover, it does not depend on the thickness of the sample, i.e., on the separation between the electrodes. 

The upper value of n corresponds exactly to the critical sampling frequency of two sample points per cycle (i.e., the Nyquist critical frequency). Thus, in general, the discrete Fourier transform maps N complex numbers into N/2 complex numbers [75],... 

For ideal end-linked polymer gels, the critical gel point can be defined as the cross-over point where the elastic and viscous moduli are equal (G =G"). However for non-ideal gels where network defects or physical entanglements are present, the cross-over point dejjends on the applied frequency. The critical gel pwint represents a physical transition from a hquid to a solid and hence should not depend on the measurement parameters. Chamhon and Winter (1987) proposed what is now the definitive criterion for determination of the critical gel point. The critical gel pwint is when the two moduli exhibit a power law dependence on the applied frequency over a wide range of frequencies. Alternately at the critical gel point, the ratio of the shear moduh, tan(5), is independent of frequency (Gupta, 2000)... 

Fig. 8. Determination of critical gel point and network quality from oscillatory rheology. The critical gel point is the time when the curves of tan(5) at various frequencies coincide. The table shows the relaxation exponent n for gels of various equilibrium moduli.

To further illustrate why feedback control can produce sustained oscillations, consider the following thought experiment for the feedback control system shown in Fig. 14.8. Assume that the open-loop system is stable and that no disturbances occur D = 0). Suppose that the set-point is varied sinusoidally at the critical frequency, ysp t)=A m ((x)ct), for a long period of time. Assume that, during this period, the measured output, y, is disconnected, so that the feedback loop is broken before the comparator. After the initial transient dies out, y will oscillate at the excitation frequency coc because the response of a linear system to a sinusoidal input is a sinusoidal output at the same frequency (see Section 14.2). Suppose that two events occur simultaneously (i) the set-point is set to zero, and (ii) y is reconnected. If the feedback control system is marginally stable, the controlled variable y will then exhibit a sustained sinusoidal oscillation with amplitude A and frequency (o. ... 

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