Frequency Dependent Solutions Research Articles

Radiation hydrodynamics in a moving plasma with Compton scattering: Frequency-dependent solutions

Abstract Radiation hydrodynamical equations with Compton scattering are generally difficult to solve analytically, and usually examined numerically, even if in the subrelativistic regime. We examine the equations available in the subrelativistic regime of kBT$/$(mec2) ≲ 0.1, hν$/$(mec2) ≲ 0.1, and v$/$c ≲ 0.1, where T is the electron temperature, ν the photon frequency, and v the fluid bulk velocity. For simplicity, we ignore the induced scattering terms. We then seek and obtain analytical solutions of frequency-dependent radiative moment equations of a hot plasma with bulk motions for several situations in the subrelativistic regime. For example, in the static case of a plane-parallel atmosphere without bulk motions, where equations involve the generalized Kompaneets equation with subrelativistic corrections, we find the Wien-type solution, which reduces to the usual Milne–Eddington solution in the nonrelativistic limit, as well as the power-law-type one, which has a form of [hν$/$(kBT)]−4. In the moving case of an accelerating one-dimensional flow with bulk motions, we also find the Wien-type and the power-law-type solutions affected by the bulk Compton effect. Particularly, in the Wien-type solutions, due to the bulk Compton effect, the radiation fields gain momentum from the hot plasma in the low-frequency regime of hν < 3kBT, while they lose it in the high-frequency regime of hν > 3kBT.

P- and SV-wave dispersion and attenuation in saturated microcracked porous rock with aligned penny-shaped fractures

P- and SV-wave dispersion and attenuation have been extensively investigated in saturated poroelastic media with aligned fractures. However, there are few existing models that incorporate the multiple wave attenuation mechanisms from the microscopic scale to the macroscopic scale. Hence, in this work, we developed a unified model to incorporate the wave attenuation mechanisms at different scales, which includes the microscopic squirt flow between the microcracks and pores, the mesoscopic wave-induced fluid flow between fractures and background (FB-WIFF), and the macroscopic Biot's global flow and elastic scattering (ES) from the fractures. Using Tang's modified Biot's theory and the mixed-boundary conditions, we derived the exact frequency-dependent solutions of the scattering problem for a single penny-shaped fracture with oblique incident P- and SV- waves. We then developed theoretical models for a set of aligned fractures and randomly oriented fractures using the Foldy approximation. The results indicated that microcrack squirt flow considerably influences the dispersion and attenuation of P- and SV-wave velocities. The coupling effects of microcrack squirt flow with the FB-WIFF and ES of fractures cause much higher velocity dispersion and attenuation for P waves than for SV waves. Randomly oriented fractures substantially reduce the attenuation caused by the FB-WIFF and ES, particularly for the ES attenuation of SV waves. Through a comparison with existing models in the limiting cases and previous experimental measurements, we validated our model.

An Analysis of Transient Impedance-like Diagnostic Signals in Batteries

Electrochemical impedance spectroscopy (EIS) is an established technique for monitoring and testing of electrochemical systems, including batteries. Parameters related to electrochemical phenomena such as charge-transfer kinetics and mass transport can be estimated from the impedance response over a range of frequencies1. The classical description for impedance of an electrochemical system, subjected to a periodic input perturbation, depends upon achieving a steady periodic output state2. However, there may be value in analyzing the joint time-frequency response of a dynamic impedance-like output signal under conditions that are not steady periodic3. For example, it has been shown that using a dynamic impedance-like response allows parameter estimation to be carried out in shorter times than waiting for a steady periodic state to be reached4. This may be particularly valuable for accessing low frequency (slow) phenomena such as thermodynamic and diffusion effects. As a result, dynamic impedance-like signals may be a valuable tool for real time diagnostics of battery degradation processes. The objective of this study is to develop time and frequency dependent solutions for existing battery models to further investigate the full dynamic response of the system. Devan et al. has performed transient voltage analysis for a porous electrode model5 and short-time current density response for an intercalation particle model4. A comparison to the steady periodic response for parameter estimation and the choice of optimum frequencies for measurements will be explored. While the analysis will begin with a simple physics-based model, the goal is to develop a fast and efficient tool for porous electrode 2D models. In particular, the focus will be on the gain in estimating parameters with specific confidence intervals, and mechanism validation from using the transient data compared to steady state models. References Yu, B. N. Popov, J. A. Ritter, and R. E. White, Journal of The Electrochemical Society, 146, 8 (1999).Ceder, M. Doyle, P. Arora, and Y. Fuentes, MRS Bulletin, 27, 619–623 (2002).Doyle and J. Newman, in Tutorials in Electrochemical Engineering—Mathematical Modeling, edited by R.F. Savinell, J.M. Fenton, A.C. West, S.L. Scanlon, and J. Weidner (The Electrochemical Society, Seattle, 1999) p. 144.Devan and R. E. White, Journal of The Electrochemical Society, 154(2007).Devan, V. R. Subramanian, and R. E. White, Journal of The Electrochemical Society, 152 (2005).

Effective thermal conductivity of polycrystalline materials with randomly oriented superlattice grains

A model has been established for the effective thermal conductivity of a bulk polycrystal made of randomly oriented superlattice grains with anisotropic thermal conductivity. The in-plane and cross-plane thermal conductivities of each superlattice grain are combined using an analytical averaging rule that is verified using finite element methods. The superlattice conductivities are calculated using frequency dependent solutions of the Boltzmann transport equation, which capture greater thermal conductivity reductions as compared to the simpler gray medium approximation. The model is applied to a PbTe/Sb2Te3 nanobulk material to investigate the effects of period, specularity, and temperature. The calculations show that the effective thermal conductivity of the polycrystal is most sensitive to the in-plane conductivity of each superlattice grain, which is generally four to five times larger than the cross-plane conductivity of a grain. The model is compared to experimental measurements of the same system for periods ranging from 287 to 1590 nm and temperatures from 300 to 500 K. The comparison suggests that the effective specularity increases with increasing annealing temperature and shows that these samples are in a mixed regime where both Umklapp and boundary scattering are important.

Dynamic vertical response of model piles — experimental and analytical investigations

Dynamic response characteristics of reinforced concrete model piles were investigated in the field under varying levels of vertical harmonic excitation. In the investigations single piles and 2 × 2 group piles with length to diameter ratios 10, 15 and 20. For group piles, spacing to diameter ratios of 2, 3 and 4 for each length to diameter ratio were used. In both the cases, two different conditions of pile cap — embedded into soil and above the ground surface, were considered in this investigation. The measured response was compared with the response obtained by different analytical methods. The stiffness and damping of piles under vertical vibration were computed by two different analytical approaches — (i) Novak's frequency independent solutions with static interaction factor for parabolic soil profile and (ii) Novak's complex frequency dependent analytical solutions with dynamic interaction factor approach for layered media. From the comparison of these theories with the experimental data it was found that the Novak's frequency dependent solutions with dynamic interaction factor approach produced reasonable estimates of the experimental results. In this study the influence of excitation intensity, static load on pile and different contact condition of pile cap with soil on the dynamic behaviour of pile foundation are reported.

A validation study on mathematical models of speed and frequency dependence in seakeeping

An extensive theoretical validation exercise is presented into the speed and frequency dependent solutions associated with surface piercing vessels travelling in waves. The basis of the study lies in the formulation of the Green's function satisfying the traditionally posed linearized boundary value problem of an oscillating ship with forward speed and the development/implementation of appropriate numerical schemes of study for solution. Two widely different mathematical models in formulation and numerical algorithm, denoted as methods A and B, are discussed and employed to predict hydrodynamic coefficients, wave loads and responses of a Series 60 form and an NPL monohull. Only a selection of results are shown, but great care was taken in the overall investigation to verify and validate intermediate steps within the separate calculation procedures as well as to compare final predicted values. The extensive qualitative and quantitative agreement of comparable results from methods A and B provides a measure of confidence that the presented findings are solutions to the posed seakeeping problem. The influence of speed and frequency dependence within the context of this study are discussed as well as a preliminary study into their influence in the occurrence of irregular frequencies in the numerical schemes.

1/f noise due to temperature fluctuations in heat conduction in bipolar transistors

Abstract Noise is an important consideration in the reliability of microelectronic circuits and often sets a lower bound on their sensitivity and limits operation. High device temperature which result from high power operation is shown to result in an additional noise source or mechanism which can become important, become a limiting factor on circuit operation, and limit reliability. Power dissipation at high currents and voltages in bipolar transistors results in significant heat generation and heat conduction towards the heat sink. As might be expected the device temperature is only an average value and there are, as a consequence of the diffusion equation for heat flow itself, temperature fluctuations about this average value. It will be shown that these temperature fluctuations can result in 1/ f noise at moderately low frequencies where these frequencies are determined by the physical dimensions over which the heat flows and the diffusion transit time. This physical phenomena is another mechanism which can be explained by the equivalent circuit representations to obtain frequency dependent solutions to the diffusion equation. The results presented here are then related to the shot noise or white noise due to the collector current allowing a determination of the l/ f noise corner frequency.

SPECTRAL FINITE ELEMENT ANALYSIS OF THE VIBRATION OF STRAIGHT FLUID-FILLED PIPES WITH FLANGES

A spectral finite element formulation for the analysis of stationary vibration of straight fluid-filled pipes is introduced. Element formulations for flanges and rigid masses attached to the pipe are also presented. In the spectral finite element formulation, the base functions are frequency-dependent solutions to the local equations of motion. The formulation is valid for arbitrarily long pipes and losses may be distributed in the system and may vary with frequency. The solutions of the equations of motion are expressed in terms of exponential functions, describing propagation in the waveguide, together with corresponding cross-sectional mode shapes. These solutions are found by using an FE discretization of the cross-sectional motion. To increase the numerical efficiency, methods for using FE shape functions with higher order polynomials are developed. The numerical accuracy is investigated by comparisons with results achieved with an exact formulation. It is found that, for frequencies of interest in many engineering problems, pipes may be modelled by using only one element to describe the fluid motion. The vibrations of a simple pipe structure with an infinite pipe, a flange and a small rigid mass are calculated. Just below the cut-on frequency of a shell mode, the stiffness controlled shell mode and the rigid mass may resonate, resulting in high vibration levels concentrated near the mass.

Frequency dependent transmission characteristics of microstrip lines on the finite width substrate or near a substrate edge

An accurate hybrid mode analysis of microstrip lines on the finite width substrate or near a substrate edge including finite metallisation thickness is described. The frequency dependent solutions for the transmission characteristics of the microstrip lines are presented. An approach for avoiding the calculation of higher modes in a rectangular waveguide partially filled with a dielectric is also introduced. The results provide useful information for the design of high packing density millimetre wave MMICs.

Small-signal MOSFET one-dimensional admittance model

A small-signal a.c. admittance ( y-parameter) circuit model is developed by physical analysis of the basic MOSFET structure by using a small signal analysis. The frequency dependent solutions for the derived set of equations for the y-parameters yield results that match experimental data extremely well. The real parts of the y-parameters vary as ω 2, while the imaginary parts vary linearly. In carrying out the analysis it was necessary to add parasitic elements to the physical MOSFET model as well as include the effects of channel length modulation. The y-parameters are expressed as a function of the MOSFET parameters, parasitic elements, frequency and bias conditions.

Analysis of broadside‐coupled strip lines with anisotropic substrate

AbstractTwo analytical approaches are presented for broadside‐coupled strip lines with anisotropic substrates. One is based on the quasi‐static approximation which uses the Fourier transformation and the Ritz procedure. the other is based on the rigorous hybrid‐mode formulation, and the frequency dependent solutions are obtained by using the network analytical methods of electromagnetic fields and Galerkin's procedure. Some numerical results are presented for sapphire and Epsilam‐10 substrates. the effect of the inclination of the optical axis is not so large, and the effect of the shift of central axes of strip lines S/W in the odd modes is larger than that in the even modes. the rigorous frequency dependent solutions converge to quasi‐static values at lower frequencies, and the dispersion of the even modes, which cannot be neglected at higher frequencies.

Critical line shape anomalies of anisotropic ferromagnets

The self-energies derived from the asymptotic frequency-dependent solutions of the mode-coupling equations are utilized to exhibit deviations of the shape functions from the lorentzian line shape for parallel and perpendicular fluctuations of both unixial and planar ferromagnets above T c.