Frequency-dependent line model

Most transmission lines in Japan are of double-circuit vertical configuration with two GWs and are thus composed of eight conductors. The use of a frequency-dependent line model is recommended in a numerical simulation, but a distributed line model with fixed propagation ve-locity, attenuation, and surge impedance, that is, the fixed-parameter dis-tributed line model explained in Reference 23, is often used. 

Frequency-Dependent Line Model Constant Frequency Line Model ... 

For a steady-state analysis, a cable can be expressed by a single or a cascaded n-equivalent circuit instead of a distributed parameter line. In the EMTP, even if a cable is represented by a constant-parameter line model (Dommel s line model) or a frequency-dependent line model (Semiyen s or Marti s line model), the distributed parameter line is internally converted into a n-equivalent circuit and is passed to a steady-state analysis routine. 

Even in the case of Pg = 100 Q m, a transient of a 10 ns time region cannot be simulated by Pollaczek s and Carson s impedances [8, 9, 10-11]. It should be noted that most frequency-dependent line models are not applicable in these models because they are based on Pollaczek s and Carson s impedances. 

The results obtained from using a frequency-dependent (distributed) line model and a frequency-independent line model are shown in Figure 2.45 for the case of no flashover of an arc horn. In the latter model, line parameters are calculated at the dominant transient frequency given by... 

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. 

Influence of a tower model on a tower top voltage, (a) Measured results, (b) Frequency-dependent tower model with a resistive-footing impedance, (c) Distributed line tower model with various footing impedances. 

In Equation 1.263, Zq is the characteristic impedance that is frequency dependent. When the frequency dependence of a distributed-parameter line explained in Section 1.5 is to be considered, a frequency-dependent line such as Semiyen s and Marti s line models is prepared as a subroutine in the EMTP. 

In the case of dynamic mechanical relaxation the Zimm model leads to a specific frequency ( ) dependence of the storage [G ( )] and loss [G"(cd)] part of the intrinsic shear modulus [G ( )] [1]. The smallest relaxation rate l/xz [see Eq. (80)], which determines the position of the log G (oi) and log G"(o>) curves on the logarithmic -scale relates to 2Z(Q), if R3/xz is compared with Q(Q)/Q3. The experimental results from dilute PDMS and PS solutions under -conditions [113,114] fit perfectly to the theoretically predicted line shape of the components of the modulus. In addition l/xz is in complete agreement with the theoretical prediction based on the pre-averaged Oseen tensor. 

Fig. 6.8 (a) Calculated frequency-dependent conductivity for a simple dynamic percolation model. Lower line represents the diffusion coefficient without renewal, upper that with renewal. (f ) Frequency-dependent conductivity for pure PEO (bold) and PEO-NaSCN at 22 °C. Only ions are able to diffuse long distances, corresponding to renewal diffusion. 

Fig. 2.17 Frequency dependence of dynamic shear moduli computed using a model for the linear viscoelasticity of a cubic phase based on slip planes, introduced by Jones and McLeish (1995). Dashed line G, solid line G".The bulk modulus is chosen to be G — 105 (arb. units). The calculation is for a slip plane density AT1 - 10 5 and a viscosity ratio rh = = 1, where rjs is the slip plane viscosity and t] is the bulk viscosity. The strain...

Figure 25. Frequency dependence of the absorption coefficient (a) and dielectric loss (b). Liquid fluoromethane CH F at 133 K calculated for that hat-curved model (solid lines). Dashed curve in Fig. (a) refers to the experimental [43] data, vertical line in Fig. (b) marks the experimental position of the maximum dielectric loss. The parameters of the hat-curved model are presented in Table VIII.

Figure 35. Frequency dependence in the submillimeter wavelength region of the real (a, b) and imaginary (c, d) parts of the complex permittivity. Solid lines Calculation for the composite HC-HO model. Dashed lines Experimental data [51]. Dashed-and-dotted lines show the contributions to the calculated quantities due to stretching vibrations of an effective non-rigid dipole. The vertical lines are pertinent to the estimated frequency v b of the second stochastic process. Parts (a) and (c) refer to ordinary water, and parts (b) and (d) refer to heavy water. Temperature 22.2°C.

Figure 37. Absorption-frequency dependence, water H20 at temperature 27°C. Calculation for the HC—HO model (solid line) and for the hybrid-cosine-squared potential model (dashed-and-dotted line). Dahsed curve Experimental data [42], (b) Same as in Fig. 34c but refers to T — 300 K.

Figure 53. Frequency-dependent conductivity of RbAg4ls at 129 K. (More precisely a- represents the real part of the complex conductivity.) As the continuous line shows, the jump-relaxation model in Ref.278-280 can well describe the behavior in the hopping regime.278 Reprinted from K. Funke, Prog. Solid St Chem., 22 (1993) 111-195. Copyright 1993 with permission from Elsevier.

It should be possible to estimate r, the lifetime of the low-frequency Rh-C stretch using a2 and Am in Equation (5). The values for a2 = 1.2 THz and Am = 20 cm-1 yield a value for r of 0.75 ps. For a low-frequency mode that has a number of lower-frequency internal modes and the continuum of solvent modes to relax into, 0.75 ps is not an unreasonable value for the lifetime. A measurement of the Rh-C stretching mode lifetime would provide the necessary information to determine if the proposed dephasing mechanism is valid. In principle, the same mechanism will produce a temperature-dependent absorption line shift. However, other factors, particularly the change in the solvent density with temperature strongly influence the line position. Therefore, temperature-dependent line shifts cannot be used to test the proposed model. 

FIGURE 12.32 Shear moduli and dynamic viscosities measured for silica spheres at = 0.46, a = 28 2nm, O + a = 76 2nm(Mellemaetal. [68]). The broken lines correspond to the infinite shear viscosities (de Kruif et al. [43]) and the solid curves to the frequency dependence predicted by the visco-elastic fluid model of Table 12.4 with the measured values of 170,171 , and Gi. Redrawn from Russel et al. [31]. Reprinted with the permission of Cambridge University Press. 

Figure 1 Real(top) and imaginary (bottom) parts of the frequency dependent dielectric constant for the TIP4P-FQ model (solid lines), compared to experiment (dotted lines).

Fig. 23 column Reduced flow curves (filled squares) for different volume fractions. The solid lines are the results of the schematic model, the dashed line represent the pseudo power law behaviour from [33]. Right column Reduced frequency dependent moduli for different volume fractions. Full symbols/solid lines represent G, hollow symbols/dashed lines represent G". Thick lines ate the results of the schematic model, the thin lines the results of the microscopic MCT. Graphs in one row represent the continuous and dynamic measurements at one volume fraction, (a) and (b) at <]>eff = 0.530, (c) and (d) at (/>eff = 0.595, (e) and (f) at = 0.616, (g) and (h) at < eff = 0.625. and (i) and (j) at = 0.627... 

As for the previous example described in Sect. 3.2.1, the relaxation of the magnetization has been studied using combined ac (Fig. 6a) and dc (Fig. 6b) measurements. In order to extract the relaxation time of the system (r), the obtained frequency dependence of the in-phase x and out-of-phase x" susceptibilities and furthermore the Cole-Cole plots (x" vs. x plot) were fitted simultaneously to a generalized Debye model (solid lines in Figs. 6a and 6b). The fact that the found a parameters of this model are less than 0.06, indicates that the system is close to a pure Debye model with hence a single relaxation time. This indication is confirmed by the quasi-exponential decay of the magnetization observed between 1.8 and 0.8 K (Fig. 6b). 

In the schematic shown in Figure 4.2.10, the RF path is visible between the two signal sources (RF ports) used for extracting the S parameters, and is composed of a length of microstrip transmission line from each port connected to a model for a series-switch plate . Driven by the 6 mechanical wires at each side, which control its position, the switch plate is internally modeled as an equivalent circuit including transmission line, frequency-dependent resistance, and variable capacitance between the conductor on the plate and the underlap of the ends of the microstrip lines separated by the gap for the switch isolation. As with the beams, this model is defined by a complete set of parameters, such as the dimensions and material properties. Parameters can be adjusted quickly to achieve the desired RF performance for different closing states of the electromechanical structure. 

Fig. 2.10. Relation between mean system frequency v and driving frequency i/ (top) and number of locked cycles Ni ck (bottom) of the excitable model as a function of driving frequency v. Lines show dependence as solutions of eqs. (2.36) and (2.37). The + symbols present results from simulations of the discrete system and circles correspond to data from simulations of the FHN system. Parameters for the FHN system eq. (2.11) ao = 0.405, ai = 0.5, = 0.001, D = 10 , Sx(t) = 0, Sy t) = A with A = 0.015. Parameters for the two state model (theory and simulations) from simulations of the interspike interval distribution density (cf. Fig. 2.6) T ss 2620, ro 0.0087 and n 8.3 10- . [15]...

Choosing room temperature as 20.2°C, we depict in Fig. 5a the wideband absorption frequency dependence a(v) of water H20 and in Fig. 6a we depict that of water D20. The fitted parameters of the model are presented in Table II. The total loss spectrum e"(v) is shown in Figs. 5b and 6b, respectively, for OW and HW. The solid lines in Figs. 5a,b and 6a,b mark the results of our calculations. 

Figure 26 The loss frequency dependence, calculated for ice from the composite molecular model (1) and from the empirical formula (72) with the fitting coefficients cyu = 2.35 (line 2) and with cjit = 1 (line 3). The temperature - 7°C. The arrow depicts the point placed at v = 20 cm-1, where the lines 1 and 2 have equal loss values.

We see from Fig. 26a (solid line 1) that the loss spectrum, calculated for our model with the same parameters, as chosen above (Table IX), exhibits resonance lines at the frequencies v < 50 cm-1. At v < 20 cm-1 the calculated solid loss curve 1, becoming nonresonant, coincides with the nonresonant dashed curve 2 calculated from Eqs. (72)-(74) with cfit = 2.35. Both loss s" curves 1 and 2 decrease linearly with v (in the log-log plot) in the interval from 50 to 0.1 cm-1 For further decrease of frequency the empirical dependence (72) exhibits a minimum at v about 0.1 cm-1 (viz, in the millimeter wavelength region). Near this minimum and at lower frequencies, our molecular model should not be applied. 

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