The Frequency Response

The convergence of this series can be shown for any square matrix A. The function A t-to) jg called the transition matrix or transfer matrix of the problem, because for the homogeneous case u = 0) it describes the transition from an initial value xo to the actual value x(t). 

It is rather rare that one has to numerically evaluate the transfer matrix. Computing the numerical solution of a linear system via Eq. (2.4.1) is replaced by modern integration methods which proved to be faster. In general an inhomogeneity u would require to numerically approximate the integral expression in Eq. (2.4.1) anyway. 

For a critical survey of methods for computing the transition matrix we refer the reader to [MvL78]. 

Example 2.5.1 In order to motivate the following we first consider two linear differential equations describing a simple spring/damper system excited by a sinusoidal function 

By standard techniques (e.g. [MK75]) the analytical solution of this system can be determined as 

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Once the probe is set into the target, the acquisitions consist of the peak to peak amplitude, the time of flight and the frequency response of the back-reflected echo. 

The sinc fiinction describes the best possible case, with often a much stronger frequency dependence of power output delivered at the probe-head. (It should be noted here that other excitation schemes are possible such as adiabatic passage [9] and stochastic excitation [fO] but these are only infrequently applied.) The excitation/recording of the NMR signal is further complicated as the pulse is then fed into the probe circuit which itself has a frequency response. As a result, a broad line will not only experience non-unifonn irradiation but also the intensity detected per spin at different frequency offsets will depend on this probe response, which depends on the quality factor (0. The quality factor is a measure of the sharpness of the resonance of the probe circuit and one definition is the resonance frequency/haltwidth of the resonance response of the circuit (also = a L/R where L is the inductance and R is the probe resistance). Flence, the width of the frequency response decreases as Q increases so that, typically, for a 2 of 100, the haltwidth of the frequency response at 100 MFIz is about 1 MFIz. Flence, direct FT-piilse observation of broad spectral lines becomes impractical with pulse teclmiques for linewidths greater than 200 kFIz. For a great majority of... 

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... 

McMorrow D and Lotshaw W T 1990 The frequency response of condensed-phase media to femtosecond optical pulses spectral-filter effects Cham. Phys. Lett. 174 85-94... 

The frequencies responsible for suites IX and X are near the Fj and F2 modes of vibration of thiazole, respectively, and have been assigned to such oscillations. 

The frequency response or switching speed of the bipolar transistor is governed by the same processes which control the speed of thep—n junction, the capacitance associated with the movement of charge into and out of the depletion regions. To achieve high frequencies the dimensions of the active areas and parasitic circuit elements must be reduced. The two critical dimensions are the width of the emitter contact and the base thickness, W. The cutoff frequency,, is the frequency at which = 57 / - b /t > where is the emitter-to-coUector delay time and is the sum of the emitter... 

There are a few obvious relationships between the frequency response and time response of closed-loop systems ... 

Bode Plot a graph of the frequency response see Frequency Response Analysis) of an electrode whereby the magnitude and the phase angle are separately plotted as a function of the frequency. 

The results from EQCM studies on conducting polymer films can be ambiguous because the measured mass change results from a combination of independent ion transport, coupled ion transport (i.e., salt transport), and solvent transport. In addition, changes in the viscoelasticity of the films can cause apparent mass changes. The latter problem can be minimized by checking the frequency response of the EQCM,174 while the various mass transport components can be separated by careful data analysis.175,176... 

To illustrate behaviors of different filters, consider a moving average filter that averages over 11 terms. Such a filter has the frequency response shown in Fig. 8. Note that this filter has a relatively low gain of 0.55 at the break-point frequency of 0.05 cycles per minute. So in the range of... 

We have to wait until the controller chapters to see that this function is the basis of a derivative controller and not till the frequency response chapter to appreciate the terms lead and lag. For now, we take a quick look at its time response. 

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. 

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. 

We also remark that Eq. (5.44) may be decomposed into separate sets of equations for the odd and even ap(t) which are decoupled from each other. Essentially similar differential recurrence relations for a variety of relaxation problems may be derived as described in Refs. 4, 36, and 73-76, where the frequency response and correlation times were determined exactly using scalar or matrix continued fraction methods. Our purpose now is to demonstrate how such differential recurrence relations may be used to calculate mean first passage times by referring to the particular case of Eq. (5.44). 

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. 

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. 

The frequency response may be determined from equations 7.94 and 7.95, Volume 3. 

Q.cm. This suggests the absence of any build-up of a TiC>2 layer between the Ti substrate and the Ru/Ti oxide coating with the onset of anode deactivation. Furthermore, the similarity of the frequency response of a failed electrode to that of freshly prepared low at.% (c. 5-10 at.%) Ru electrodes at low frequencies, supports the conclusion of the absence of the build-up of a TiC>2 layer with failed electrodes. 

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. 

The frequency response of most processes is defined as the steadystate behavior of the system when forced by a sinusoidal input Suppose the input to the process is a sine wave of amplitude m and frequency o) as shown in Fig. 12.1. 

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). 

There are a number of ways to obtain the frequency response of a process. Experimental methods, discussed in Chap. 14, are used when a mathematical model of tbe system is not available. If equations can be developed that adequately describe the system, the frequency response can be obtained directly from the system transfer function. 

As we will show below, the frequency response of a system can be found by simply substituting ift> for s in the system transfer function. Making the substitution s = ift) gives a complex number that has the following ... 

Table 12.1 gives a FORTRAN program that calculates the frequency response of several simple systems. Figure 12.23 gives Bode plots for four difler-ent transfer functions. 

In the preceding sections of this chapter we assumed that the system transfer function was known. Then the simple substitution s = ifrequency response of the system. 

Example 12.6. Let us consider a much more complex system where the advantages of frequencynlomain solution will be apparent. Rippin and Lamb showed how a frequency-domain stepping technique could be used to find the frequency response of a binary, equimolal-overflow distillation column. The column has many trays and therefore the system is of very high order. 

We will show in Sec. 13.1 that chsedhop stability can be determined from the frequency-response plot of the total openloop transfer function of the system (process openloop transfer function and feedback controller This... 

In Chap. 12 we presented three different kinds of graphs that were used to represent the frequency response of a system Nyquist, Bode, and Nichols plots. The Nyquist stability criterion was developed in the previous section for Nyquist or polar plots. The critical point for closedloop stability was shown to be the 1,0) point on the Nyquist plot. 

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. 

J6. The frequency response data given below were obtained by pulse-testing a closed-loop system that contained a proportional-only controller with a proportional band of 25. Controller setpoint was pulsed and the process measurement signal was recorded as the output signal. 

J5. The frequency response Bode plot of the output of a closedloop system for setpoint changes, using a proportional controller with a gain of 10, shows the following features ... 

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