Sinusoidal Method

This approach is essentially an extension of the step method. The concentration of hydrogen at the input side of a membrane is varied periodically around a value with an amplitude Y and an angular frequency co. After a few periods, the concentration at the detection side also begins to oscillate, but the amplitude is damped and the phase is shifted with respect to the oscillation at the entry side (Fig. 10). Comparison of the concentration waves at the two sides allows the diffusivity to be calculated from either the phase shift or the ratio of the amplitudes. 

The boundary conditions for the entry and detection sides (taken as jc = L and jc = 0, respectively) are assumed to be given by 

If ff 1, then ax and so the apparent diffusivity can be determined from the dependence of phase shift on frequency according to 

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This group of methods relies on potentiometric measurement at the detection side to determine the hydrogen concentration. Three basic variations of this approach have been described the step, pulse, and sinusoidal methods [97j. In the step method, the hydrogen concentration is initially homogeneous throughout the membrane. The concentration is then increased at the input side and kept constant under potentiostatic control. The change in concentration at the detection side is followed by monitoring the potential. The... 

Examples of the fit of the same G(f) function to both stress relaxation and sinusoidal results appear in Figures 3.3.6 and 3.3.7. Table 3.3.1 gives the relaxation times and moduli used to fit all these data. Figure 3.3.7 illustrates that short time (high frequency) data are more readily obtained with sinusoidal methods. 

Figure 10. Variation in concentration gradient with time for a membrane used in the potentiometric sinusoidal method. The entry side was taken as jc = L. (After Ref. 11. Reprinted with permission from J. Less Common Metals, Elsevier Sequoia S.A.)...

For fluorescent compounds and for times in die range of a tenth of a nanosecond to a hundred microseconds, two very successftd teclmiques have been used. One is die phase-shift teclmique. In this method the fluorescence is excited by light whose intensity is modulated sinusoidally at a frequency / chosen so its period is not too different from die expected lifetime. The fluorescent light is then also modulated at the same frequency but with a time delay. If the fluorescence decays exponentially, its phase is shifted by an angle A([) which is related to the mean life, i, of the excited state. The relationship is... 

Pulsed-force mode AFM (PFM-AFM) is a method introduced for fast mapping of local stiffness and adliesion with lower required data storage than recording force-distance curves at each point on the x-y plane [115]. A sinusoidal or triangular modulation is applied between the tip and sample (either via lever or sample piezo) at a lower frequency than that of either the piezo or cantilever resonance frequency. Tip and sample then come... 

The problem is heated in elementary physical chemishy books (e.g., Atkins, 1998) and leads to a set of eigenvalues (energies) and eigenfunctions (wave functions) as depicted in Fig. 6-1. It is solved by much the same methods as the hamionic oscillator in Chapter 4, and the solutions are sine, cosine, and exponential solutions just as those of the harmonic oscillator are. This gives the wave function in Fig. 6-1 its sinusoidal fonn. 

Orowan (1949) suggested a method for estimating the theoretical tensile fracture strength based on a simple model for the intermolecular potential of a solid. These calculations indicate that the theoretical tensile strength of solids is an appreciable fraction of the elastic modulus of the material. Following these ideas, a theoretical spall strength of Bq/ti, where Bq is the bulk modulus of the material, is derived through an application of the Orowan approach based on a sinusoidal representation of the cohesive force (Lawn and Wilshaw, 1975). 

Quasi-resonant converters are a separate class of switching power supplies that tune the ac power waveforms to reduce or eliminate the switching loss within the supply. This is done by placing resonant tank circuits within the ac current paths to create pseudo-sinusoidal voltage or current waveforms. Because the tank circuits have one resonant frequency, the method of control needs to be modified to a variable frequency control where the resonant period is fixed and the control varies the period of the non-resonant period. The quasi-resonant converters usually operate in the 300 kHz to 2 MHz frequency range. 

An alternative method of studying the molecular motions of a polymeric chain is to measure the complex permitivity of the sample, mounted as dielectric of a capacitor and subjected to a sinusoidal voltage, which produces polarization of the sample macromolecules. The storage and loss factor of the complex permitivity are related to the dipolar orientations and the corresponding motional processes. The application of the dielectric thermal analysis (DETA) is obviously limited to macromolecules possessing heteroatomic dipoles but, on the other hand, it allows a range of frequency measurement much wider than DMTA and its theoretical foundations are better established. 

CALCULATION OF THE ELECTRONIC STRUCTURE OF ANTIFERROMAGNETIC CHROMIUM WITH A SINUSOIDAL SPIN DENSITY WAVE BY THE METHOD OF DIRAC FUNCTION LINEAR COMBINATION... 

When a pure sinusoidal AC current passes across the electrode/solution interface, the cell voltage (a two electrode arrangement is used) shows a sinusoidal perturbation. It contains multiples of the fundamental frequency of the modulation, the first harmonie dominates. The magnitude of the effect is comparable to Faradaie rectification, but experiments may be easier to perform. Measurement and evaluation have been described in detail [60Old, 72Hil2]. (Data obtained with this method are labelled FD.)... 

Theory. If two or more fluorophores with different emission lifetimes contribute to the same broad, unresolved emission spectrum, their separate emission spectra often can be resolved by the technique of phase-resolved fluorometry. In this method the excitation light is modulated sinusoidally, usually in the radio-frequency range, and the emission is analyzed with a phase sensitive detector. The emission appears as a sinusoidally modulated signal, shifted in phase from the excitation modulation and partially demodulated by an amount dependent on the lifetime of the fluorophore excited state (5, Chapter 4). The detector phase can be adjusted to be exactly out-of-phase with the emission from any one fluorophore, so that the contribution to the total spectrum from that fluorophore is suppressed. For a sample with two fluorophores, suppressing the emission from one fluorophore leaves a spectrum caused only by the other, which then can be directly recorded. With more than two flurophores the problem is more complicated but a number of techniques for deconvoluting the complex emission curve have been developed making use of several modulation frequencies and measurement phase angles (79). 

Goresky CA. A linear method for determining liver sinusoidal and extravascular volumes. Am J Physiol 1963 204 626-40. 

In electrochemical measurements, a method often employed is that of applying an alternating current of low amplitude to the electrode. For sinusoidal ac i = sin tof is the amplitude and co the angular frequency), the concentration gradient at the surface varies according to the law... 

Both the subtractive and the phase shift method are valid as long as the RF waveform remains sinusoidal [184]. For asymmetrical discharges the integral method is to be used. 

In order to increase the overall extraction efficiency during SFE sonication has been applied [352]. Ultrasound creates intense sinusoidal variations in density and pressure, which improve solute mass transfer. Development of an SFE method is a time-consuming process. For new methods, analysts should refer the results to a traditional sample preparation method such as Soxhlet or LLE. 

Frequency methods can give us the relative stability (the gain and phase margins). In addition, we could construct the Bode plot with experimental data using a sinusoidal or pulse input, i.e., the subsequent design does not need a (theoretical) model. If we do have a model, the data can be used to verify the model. However, there are systems which have more than one crossover frequency on the Bode plot (the magnitude and phase lag do not decrease monotonically with frequency), and it would be hard to judge which is the appropriate one with the Bode plot alone. 

The Fourier method is not a requirement, and direct sinusoidal fitting procedures are also used to fit the data from a set of images. A number of specialized procedures have been described over the years and it is worth noting that extracting the amplitude and phase may be done as a simple extension to conventional linear regression. 

Sinusoidal fitting is more flexible than Fourier methods, as it does not require evenly spaced phase steps. There is no special convenience associated with sampling of an angle of 2n and estimation of errors in parameters is somewhat more straightforward. 

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