Frequency-Domain Effects

Sampled-data controllers can be designed in the same way continuous controllers are designed. Root locus plots in the z plane or frequency-response plots are made with various types of (z) s (different orders of M and N and different values of the a, and 6, coefficients). This is the same as using different combinations of lead-lag elements in continuous systems. 

These parameters are varied to achieve some desired performance criteria. In the z-plane root locus plots, the specifications of closedloop time constant and damping coefficient are usually used. The roots of the closedloop characteristic equation 1 -I- are modified by changing.  

In the frequency domain, the conventional criteria of phase margin, gain margin, or maximum closedloop log modulus are used. The shape of the or curve is modified by changing The simplest form of a 0,, sampled-data controller is 

Example 20.1. Suppose we use a I),) that approximates a continuous PI controller, as discussed in Chap. 19. 

This pulse transfer function has a zero at z = a and a pole at z = +1. It cannot produce any phase-angle advance since the pole lies to the right of the zero (a is less than 1). The pole at -I-1 is equivalent to integration (pole at s = 0 in continuous systems) which drives the system to zero steadystate error for step disturbances. 

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The Nichols chart shown in Figure 6.26 is a rectangular plot of open-loop phase on the x-axis against open-loop modulus (dB) on the jr-axis. M and N contours are superimposed so that open-loop and closed-loop frequency response characteristics can be evaluated simultaneously. Like the Bode diagram, the effect of increasing the open-loop gain constant K is to move the open-loop frequency response locus in the y-direction. The Nichols chart is one of the most useful tools in frequency domain analysis. 

The spectral width SWi associated with the F, frequency domain may be dehned as F, = SWi. The time increment for the ti domain, which is the effective dwell time, DWi for this period, is related to SW as follows DWi = (V2)SW]. The time increments during [Pg.158]

The first correlation is observed at 40 ppm ( 48 ppm relative to the transmitter position) in the Fi frequency domain at the F2 shifts of the anisochronous Hll methylene protons in Figure 13. Rather than an m/cH transfer across three bonds, instead a 2/ch transfer is observed, probably an effect of the oxygen of the oxepin ring being attached to Cl 2 as shown by 43. The Jcc long-range correlation in this case is to the C7 resonance. 

ESE envelope modulation. In the context of the present paper the nuclear modulation effect in ESE is of particular interest110, mi. Rowan et al.1 1) have shown that the amplitude of the two- and three-pulse echoes1081 does not always decay smoothly as a function of the pulse time interval r. Instead, an oscillation in the envelope of the echo associated with the hf frequencies of nuclei near the unpaired electron is observed. In systems with a large number of interacting nuclei the analysis of this modulated envelope by computer simulation has proved to be difficult in the time domain. However, it has been shown by Mims1121 that the Fourier transform of the modulation data of a three-pulse echo into the frequency domain yields a spectrum similar to that of an ENDOR spectrum. Merks and de Beer1131 have demonstrated that the display in the frequency domain has many advantages over the parameter estimation procedure in the time domain. 

Many interesting phenomena can arise in nonlinear periodic structures that possess the Kerr nonlinearity. For analytic description of such effects, the slowly varying amplitude (or envelope) approximation is usually applied. Alternatively, in order to avoid any approximation, we can use various numerical methods that solve Maxwell s equations or the wave equation directly. Examples of these rigorous methods that were applied to the modelling of nonlinear periodical structures are the finite-difference time-domain method, transmission-line modelling and the finite-element frequency-domain method." ... 

The actual limit of the summation is the extent of the weighting filter. Zero padding is used to ensure that the discretized matrices have sizes which are a power of two so that the computation can be done in the frequency domain using fast fourier transform (FFT) techniques. The effective discretized density, pin, Wj), is then given by... 

Strongly overlapping multiplets may be resolved by two-dimensional J,<5-spectros-copy2" 11G 118, where the first frequency domain (F,) contains coupling and the second frequency domain (F2) chemical shift information. The spectrum in Figure 2 (homonuclear [JH H, 6 ( H)]) demonstrates the use of this technique by showing unperturbed multiplets for ll signals. Second-order effects are principally not eliminated. Heteronuclear experiments [7uc,<5(13C)] are also common. 

By applying the convolution theorem, we see that replication in the x domain has produced a sampling effect in the frequency domain. The wider the replication interval, the finer is the frequency sampling. Sampling in the x domain, on the other hand, appears in Fourier space as replication. Fine sampling in x produces wide spacing between cycles in co. The area under each scaled Dirac function of co may be taken as the numerical value of a sample. 

Figure 4.4 Frequency-domain representation of the dynamically controlled decoherence rate in various limits (Section 4.4). (a) Golden-Rule limit, (b) Anti-Zeno effect (AZE) limit (c) Quantum Zeno effect (QZE) limit. Here, F,( ) and G(w) are the modulation and bath spectra, respectively and F are the interval of change and width of G( ), respectively and is the interruption rate.

The CC pulse train experiments in Refs [63-65] utilize shaped pulses that use a transform-limited (TL) Gaussian pulse its phase is modulated in the frequency domain with a sine function, p ( ) = a sin( -I- c), while keeping the amplitude profile intact. The parameters a, b, and c are further varied to control molecular populations. In Reference [35], the effect of different values of these parameters on the IC dynamics of pyrazine and / -carotene is investigated and the significant role of overlapping resonances is exposed. 

The effects of the circuit in the frequency domain were also characterized. The Fourier transform of the quasi-square waveform in Figure 8.41 was taken and the results shown in Fig. 8.44. Note that the third, fifth, seventh, and ninth harmonics are suppressed by about 40db, while the eleventh and thirteenth harmonics are about 20 dB less. The IsSpice simulation of this circuit was generated using the ICL feature of IsSpice. The format of the FOURIER command is shown below in Table 8.2. The resulting circuit characteristics in the frequency domain (Fig. 8.44) compare favorably to the resulting output from the IsSpice file (Table... 

NMR data i,s usually processed using one or more processing functions, some of which are applied in the time domain, others in the frequency domain. Each processing function in the time domain f(t) also has its counterpart F(f) in the frequency domain and forms a Fourier pair. In principle the same effect in the final spectrum S"(f) may be obtained with a given processing function, applied either in the time or the frequency domain as long as a few important rules are followed when performing the... 

Such calculations could in principle be performed either in the time or the frequency domain. However, it is recommended to do these calculations in the time domain to avoid any loss of spectral quality caused by rounding effects introduced with the Fourier transformation. 

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