Low-frequency limit

Once again, high and low frequency limiting must be introduced on the control IC in order to minimize output ripple voltage and zero switching loss conditions. 

Whichever physical interpretation is chosen, the difference between the high-frequency real axis intercept [Z (high) and the low-frequency limiting real impedance [Z (low)] is one-third of the film s ionic resistance (i.e., R[ = 3[Z (low) - Z (high)]). Ideally, the real component of the... 

In the low-frequency limit, they show that the normalized response of the flame is given by... 

If not otherwise stated the four-component Dirac method was used. The Hartree-Fock (HF) calculations are numerical and contain Breit and QED corrections (self-energy and vacuum polarization). For Au and Rg, the Fock-space coupled cluster (CC) results are taken from Kaldor and co-workers [4, 90], which contains the Breit term in the low-frequency limit. For Cu and Ag, Douglas-Kroll scalar relativistic CCSD(T) results are used from Sadlej and co-workers [6]. Experimental values are from Refs. [91, 92]. 

Tests were run with N80 steel in 15% and 28% HC1 at 25 C with and without octynol for periods extending up to 2 hours. Immediately after injection of octynol into the acid, two phenomena were observed. First, near the low-frequency limit of the tests, a prominent inductive loop (below the Z axis) appeared which then vanished within a few minutes. Secondly, fits of the data above 1 Hz to the Rfl+P/Rfc circuit, i.e. ignoring the inductive loop, gave rise to a higher CPE n-value, which then remained relatively constant for the duration of each experiment. This result is shown in Figure 5. 

Similar analysis of the MDT may be performed for arbitrary initial phase / 7 0. We note that, depending on the initial phase, x( ) may vary significantly (especially in the low-frequency limit). This is due to the fact that the height of the potential barrier at initial instant of time has a large variation (from zero to some maximal value). Because in real experiments the initial phase is usually... 

This result, that the low frequency limit of the in phase component of the viscosity equates to the viscosity of the dashpot, means that for a single Maxwell model it is possible to replace rj by rj(0). Thus far we have concentrated on the description of experimental responses to the application of a strain. Similar constructions can be developed for the application of a stress. For example the application of an oscillating stress to a sample gives rise to an oscillating strain. We can define a complex compliance J which is the ratio of the strain to the stress. We will explore the relationship between different experiments and the resulting models in Section 4.6. 

The expression for the real component of the complex viscosity allows us to express the relaxation times as experimentally realisable parameters. In the low frequency limit we can rewrite Equation (5.88) in terms of the concentration c in gem-3 ... 

However, in such a high concentration regime we can no longer represent the relaxation times (Equation (5.92)) in terms of the intrinsic viscosity. In the low frequency limit, because there is no permanent crosslinking present, the loss modulus divided by the frequency should equate with ... 

Aliphatic amines are characterized by nitrogen NMR signals at the high-field (low-frequency) limit of the normal range of shifts (—50 to +15 ppm referred to Me4N+). The increasing alkyl substitution of the nitrogen atom in the series... 

Often a non-blocking interface will behave like a resistance (/ ct) and capacitance (Q,) in parallel. This leads to a semicircle in the impedance plane which has a high frequency limit at the origin and a low frequency limit at Z = (Fig. 10.4). At the maximum of the semicircle if the angular frequency is then ctQin>max = fro which dl can be evaluated. 

One complication which may be present, when the Helmholtz model is in other respects appropriate, is that of specific adsorption. If one of the mobile species is to some extent chemically bound rather than being simply electrostatically bound to the metal electrode, Cji may show a dependence on the dc bias potential. Indeed this is the normal method of inferring specific adsorption. Another possibility in this case is that dl exhibits different high frequency and low frequency limits because at high frequencies the specific adsorption being an activated process is too slow to follow changes in interface potential. A further complication which is often present in real systems is the presence of an oxide layer on the surface of the metal electrode. Such an oxide layer can generate a potential... 

Notions of high - and low -frequency limiting behavior depend on one s point of view, and the notation reflects this what is low frequency to an ultraviolet spectroscopist may be high frequency to an infrared spectroscopist. For insulating solids the value of c in the near infrared is often denoted as c0 by ultraviolet spectroscopists it refers to frequencies low compared with certain oscillators—electrons in this example—which may, however, be high compared with lattice vibrational frequencies. Consequently, this same limiting value is denoted as by infrared workers. 

The low-frequency limit of c" (9.16) correctly describes the far-infrared (1 /X less than about 100 cm-1) behavior of many crystalline solids because their strong vibrational absorption bands are at higher frequencies. This limiting value for the bulk absorption, coupled with the absorption efficiency in the Rayleigh limit (Section 5.1), gives an to2 dependence for absorption by small particles this is expected to be valid for many particles at far-infrared wavelengths. 

It is not difficult to show that the emissivity of small spherical particles, composed of both insulating and metallic crystalline solids, is expected to vary as 1/A2 in the far infrared. For example, if the low-frequency limit of the dielectric function for a single Lorentz oscillator (9.16) is combined with (5.11), the resulting emissivity is... 

Table 2. Phenomenological dielectric functions e C = dielectric constant and capacitance of reference system (air), JJ = low frequency limiting parallel resistance.

In the low frequency limit this is equivalent to the time-dependent perturbation theory expression [1-4] ... 

If the voltage is high enough, the noise of isolated contacts can be considered as white at frequencies at which the distribution function / fluctuates. This allows us to consider the contacts as independent generators of white noise, whose intensity is determined by the instantaneous distribution function of electrons in the cavity. Based on this time-scale separation, we perform a recursive expansion of higher cumulants of current in terms of its lower cumulants. In the low-frequency limit, the expressions for the third and fourth cumulants coincide with those obtained by quantum-mechanical methods for arbitrary ratio of conductances Gl/Gr and transparencies Pl,r [9]. Very recently, the same recursive relations were obtained as a saddle-point expansion of a stochastic path integral [10]. 

In general, s(0, to), or simply e(oj). is a complex function, but real dielectric constants may be defined for certain regions along the o) axis [12,17,20]. In the low frequency limit, the static dielectric constant, e0 = e (0) corresponds to a medium at full equilibrium... 

One may surmise that the low-frequency limit, introduced while discussing the linear relaxation, would also lead to a reliable simplification in the nonlinear case since the process is governed mainly by the relaxation time xio. As we were tempted by this idea, in Ref. 67 we have supposed that the approximate expression... 

In the low-frequency limit only coii is set to be nonzero while all the higher modes are taken at equilibrium (cox = 0). Thence, when constructing via Eq. (4.182), by adding and subtracting a term with 4 (0), one can present the first-order solution in the form... 

The suppression effect is most pronounced in the adiabatic (low-frequency limit). A typical zero-level curve p(i 0) (see Fig- 4.28) may be, although roughly but reasonably, divided into three characteristic parts the steep ascend with the noise strength (NIR branch), the bend (NIR-FIR crossover), and the noise-independent saturation (FIR branch). To evaluate the parameters of the suppression resonance, namely, the positions of the branchoff points c0/i and the saturation values pt-(oo) of the zero-level curves for particular harmonics, we have obtained simple but rather accurate approximate expressions. 

There are three classes of instruments for the measurement of VCD. The first is based on a dispersive grating monochromator as the source of wavelength discrimination. This was the first kind of VCD instrument to be built and this design was used in the discovery of VCD in 1974 [1]. The early versions of these instruments have been described in detail [3,4,44-47], The low-frequency limit was initially 1900 cm"l, the cut-off of the InSb detector. Using a PbSnTe detector the low-frequency limit was extended to 1550 cm l [46], and subsequently using HgCdTe detectors the limit was lowered to 1250 cm l[48] and then 900 cm l [49], and finally using a Si As detector it was lowered to 650 cm l [50]. 

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