Frequency-Domain Solution Techniques

In the preceding sections of this chapter we assumed that the system transfer function was known. Then the simple substitution s = i o into gave the frequency response of the system. 

These large systems of equations can be solved fairly easily by going into the frequency domain. The procedure is  

A specific numerical value of frequency (o is chosen and substituted into the equations. 

The algebraic equations, which are now in terms of complex variables, are solved numerically to obtain the desired transfer-function relationships. The 

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. 

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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. 

Another variant devised by Martem Yanov et al. [16] ensures strong electrolyte stirring for generating turbulent fluctuations in solution. Assuming a pseudo hydrodynamical white noise, the responding current can be analyzed in the frequency domain to provide the same information as that obtained from any of the techniques mentioned above. 

The introduction of the SWIFT technique (10,14,21,22) makes possible FT/ICR frequency-domain excitation with the same mass resolution as has already been demonstrated for FT/ICR detect ion, provided only that sufficient computer memory is available to store a sufficiently long time-domain waveform. When ejection must be performed with ultrahigh mass resolution over a wide mass range, a simple solution is to use two successive SWIFT waveforms first, a broad-band low-resolution excitation designed to eject ions except over (say) a 1 amu mass range and then a second SWIFT waveform, heterodyned to put 2 8K data points spanning a mass range of 1-2 amu. 

Forced oscillations in torsion are used in the most versatile and accurate technique for measuring the viscoelastic functions, in the frequency domain, of melts and concentration solutions (12). In this case, the second-order differential equation governing the motion is given by... 

The solution of nonlinear evolution equations in the time domain is known analytically only in very simple cases such as reversible redox processes limited by diffusion. For electrochemical nonlinear systems, the treatment of nonsteady-state techniques generally requires calculations that are at least partially numerical. In addition, the solutions found to express the response to a perturbing signal depend specifically on the form of the perturbation. These drawbacks are largely eliminated if the amplitude perturbation is limited to a sufficiently low value to allow the equations to be linearized. In this case, analyses in the frequency domain are very powerful. 

In seismology, linear inversion techniques were proposed to determine the moment tensor component in both time and frequency domains (Stump Johnson 1977) and (Kanamori Given 1981). Although all components of the moment tensor must be determined, the moment tensor inversion with constraints has been normally applied to obtaining stable solutions in seismology (Dziewonski Woodhouse 1981). This is partly because a fault motion of an earthquake is primarily associated with shear motion, corresponding to off-diagonal components in the moment tensor. One application of the moment tensor inversion with constraints is found in rock mechanics (Dai, Labuz et al. 2000). 

The dynamical response of optical properties (modulation depth and phase shift of intensity modulation of the backscattered light) of a tissue in respeet to interval of a chemical agent (solution, gel or oil) administration can be measured using a photon density wave (frequency-domain) technique [23]. When intensity of the light source is modulated at a frequency co, a photon density wave is induced in a seattering medium [2, 3, 6, 30]... 

The application of the technique in the time and frequency domains is illustrated hereafter for the passivation of iron in acidic solutions. 

The more recently developed cryo-TEM technique has started to be used with increasing frequency for block copolymer micelle characterization in aqueous solution, as illustrated by the reports of Esselink and coworkers [49], Lam et al. [50], and Talmon et al. [51]. It has the advantage that it allows for direct observation of micelles in a glassy water phase and accordingly determines the characteristic dimensions of both the core and swollen corona provided that a sufficient electronic contrast is observed between these two domains. Very recent studies on core-shell structure in block copolymer micelles as visualized by the cryo-TEM technique have been reported by Talmon et al. [52] and Forster and coworkers [53]. In a very recent investigation, cryo-TEM was used to characterize aqueous micelles from metallosupramolecular copolymers (see Sect. 7.5 for further details) containing PS and PEO blocks. The results were compared to the covalent PS-PEO counterpart [54]. Figure 5 shows a typical cryo-TEM picture of both types of micelles. 

As you will see, several different approaches are used in this book to analyze the dynamics of systems. Direct solution of the differential equations to give functions of time is a time domain teehnique. Use of Laplace transforms to characterize the dynamics of systems is a Laplace domain technique. Frequency response methods provide another approaeh to the problem. 

This chapter concentrates on the results of DS study of the structure, dynamics, and macroscopic behavior of complex materials. First, we present an introduction to the basic concepts of dielectric polarization in static and time-dependent fields, before the dielectric spectroscopy technique itself is reviewed for both frequency and time domains. This part has three sections, namely, broadband dielectric spectroscopy, time-domain dielectric spectroscopy, and a section where different aspects of data treatment and fitting routines are discussed in detail. Then, some examples of dielectric responses observed in various disordered materials are presented. Finally, we will consider the experimental evidence of non-Debye dielectric responses in several complex disordered systems such as microemulsions, porous glasses, porous silicon, H-bonding liquids, aqueous solutions of polymers, and composite materials. 

Much of the high-frequency work that has been done on aqueous non-electrolyte solutions has yielded data at too few frequencies for an analysis of them to be free from ambiguity. It is to be hoped that the new time-domain techniques will remedy this deficiency. 

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