Frequency-dependent effect

It is well known that a current distributes nearby a conductor surface when fhe frequency of the current is high. Under such a condition, the resistance (impedance) of the conductor becomes higher than that at a low frequency, because the resistance is proportional to the cross section of the conductor. This is called frequency dependence of the conductor impedance. As a result, the propagation constant and the characteristic impedance are also frequency dependent. 

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Pre-clinical work and neuroimaging suggest potential frequency-dependent effects of rTMS. Thus, higher frequencies may increase while lower frequencies may decrease brain metabolism (205). Clinically, repetitive, high frequency stimulation (i.e., >1 Hz or HF-rTMS) and repetitive, low frequency stimulation (i.e., 1 Hz or SF-rTMS), have been used. 

The fully general situation of a particle diffusing in an out-of-equilibrium environment is much more difficult to describe. Except for the particular case of a stationary environment, the motion of the diffusing particle cannot be described by the generalized Langevin equation (22). A more general equation of motion has to be used. The fluctuation-dissipation theorems are a fortiori not valid. However, one can try to extend these relations with the help of an age- and frequency-dependent effective temperature, such as proposed and discussed, for instance, in Refs. 5 and 6. 

In an out-of-equilibrium environment, no well-defined thermodynamical temperature does exist. Since, in an out-of-equilibrium regime, even if stationary, the FDTs are not satisfied, one can try to rewrite them in a modified way, and thus to extend linear response theory, with the help of a (frequency-dependent) effective temperature. 

Equation (183) [or Eq. (184)] defines a frequency-dependent effective temperature Teff(co). 

The experimental measures of these molecular electric properties involve oscillating fields. Thus, the frequency-dependence effects should be considered when comparing the experimental results . Currently, there are fewer calculations of the frequency-dependent polarizabilities and hyperpolarizabilities than those of the static properties. Recent advances have enabled one to study the frequency dispersion effects of polyatomic molecules by ab initio methods In particular, the frequency-dependent polarizability a and hyperpolarizability y of short polyenes have been computed by using the time-dependent coupled perturbed Hartree-Fock method. The results obtained show that the dispersion of a increases with the increase in the optical frequency. At a given frequency, a and its relative dispersion increase with the chain length. Also, like a, the hyperpolarizability y values increase with the chain length. While the electronic static polarizability is smaller than the dynamic one, the vibrational contribution is smaller at optical frequencies. ... 

This expansion must be regarded as a shorthand notation if frequency dependent effects are to be treated. For a composite field such as the one acting in an EFISH experiment... 

In compound C the static nuclear relaxation term is nearly seven times greater than the static electronic term, although in the frequency-dependent effects its value is greatly reduced while the electronic effect is expected to get substantially greater. Generally the correlated and HF vibrational values show similar trends the differences for the electronic contributions are much greater. 

The Clausius-Mossotti factor, characterizes the frequency-dependent effective dipole moment. Separating the real and imaginary part of the Clausius-Mossotti factor give a Debye relaxation of the form ... 

Warashina A (1985) Frequency-dependent effects of aconitine and veratridine on membrane currents in the crayfish giant axon. Jap J Physiol 35 463-482 Weigele JB, Barchi RL (1982) Functional reconstitution of purified sodium channel protein from rat sarcolemma. Proc Natl Acad Sci USA 79 3651-3655 Yoshii M, Narahashi T (1984) Patch clamp analysis of veratridine-induced sodium channels (abstract). Biophys J 45 184a... 

Typically, both forms of error occur in a spectrum directly after the FT. The procedure for phase correction is essentially the same on all spectrometers. The zero-order correction is used to adjust the phase of one signal in the spectrum to pure absorption mode, as judged by eye , and the first-order correction is then used to adjust the phase of a signal far away from the first in a similar manner. Ideally, the two chosen resonances should be as far apart in the spectrum as possible to maximise the frequency-dependent effect. Experimentally, this process of phase correction involves mixing of the real and imaginary parts of the spectra produced by the FT process such that the final displayed real spectrum is in pure absorption mode whereas the usually unseen imaginary spectrum is pure dispersion. 

From Fig. 16 it can be seen that stiffnesses were only slightly dependent on orientation, the largest change being an increase of 30 o in Young s modulus in the axial direction for polymethylmethacrylate. Static loading experiments on strips cut at various angles to the stretch direction enabled some check values to be obtained. The trend was similar to that in the ultrasonic measurements, but absolute values of stiffness were lower, presumably due to a frequency dependent effect. The authors comment that despite the small effect of orientation on low strain... 

In all real systems, some deviation from ideal behavior can he observed. If a potential is applied to a macroscopic system, the total current is the sum of a large number of microscopic current filaments, which originate and end at the electrodes. If the electrode surfaces are rough or one or more of the dielectric materials in the system are inhomogeneous, then all these microscopic current filaments would be different. In a response to a small-amplitude excitation signal, this would lead to frequency-dependent effects that can often be modeled with simple distributed circuit elements. One of these elements, which have found widespread use in the modeling of impedance spectra, is the so-called constant phase element (CPE). A CPE is defined as... 

However, the interpretation of /t (o) and m /m obtained from the one-component analysis is more difficult. Obviously, it is tempting to assign the frequency-dependent effective mass and the scattering rate to the real and imaginary parts of the electronic self-energy 2 (ty) = 2 i((y) -f-i22(o)) entering the equation for the spectral function of the electronic excitation A k, (w) ... 

The most accurate compilation of experimental flat-band potentials for many semiconductors, measured for both n- and p-type materials at several pH values and accounting for frequency dependent effects, has been presented by Gomes Cardon (12) the Ufb values... 

The frequency-dependent effect is most noticeable in the propagation constant. Table 1.2 shows the frequency dependences of modal attenuations and velocities for untransposed and transposed lines. Figure 1.25 illustrates a vertical twin-circuit line. Figure 1.26 shows the frequency dependence of the modal propagation constant for the vertical twin-circuit line illustrated in Figure 1.25. 

It is observed that the frequency dependence of A2 in Table 1.4 is less than 10% for the range of frequencies from 100 Hz to 1 MHz. The change is small compared with the parameters explained earlier , thus, the frequency-dependent effect of the transformation matrix in the case of an untransposed horizontal line can be neglected. Then, the following approximation is convenient because it agrees with the traveling-wave transformation of Equation 1.179, explained in Sections 1.4.4.1 and 1.4.4.2 ... 

Table 2.6 shows the maximum voltages calculated by the frequency-dependent Semiyen model and the frequency-independent distributed parameter line model of the EMTP. It is clear from Figure 2.45 and Table 2.6 that the results neglecting the frequency-dependent effect show a mi-nor difference from the results including the effect. Thus, it can be con-cluded that the frequency-dependent model does not have a significant effect on a lightning surge. 

The advantage of the FD method is that any frequency dependent effect is easy to handle as it is based on the frequency response of a transient to be solved. Thus, the frequency-dependent effect of a transmission line or cable, explained in Chapter 1, is very easily included in a simulation. 

On the contrary, a sudden change in the time domain, such as switching, causes a difficulty because the change involves an initial condition problem that requires repeated time/frequency transforms. A nonlinear element (e.g., an arrester) requires a number of time/frequency transforms. Thus, the FD method is often used to check the accuracy of the time-domain method, such as the EMTP, on the frequency-dependent effect. 

A complementary approach is based on empirical relationships between a ground-motion parameter Py measured in the early portion of the P-wave train and the final PGX. This is physically grounded on the first-order approximation that log Py has the same magnitude and distance dependence of log PGX, being differences concentrated only on static and possibly frequency-dependent effects. In such a case, estimation of source parameters is hidden in the common dependence and the micertainty may be... 

Equations (8) and (9) constitute the TDLBA which requires their simultaneous solution. The resulting 6n(x o)) is then combined with (5) and (6) to yield the total photoabsorption. Alternatively, it can be shown that the Golden Rule expression, (4), remains valid if uext(x) is replaced by the complex, frequency dependent effective field... 

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