Two-term Approximation Research Articles

О подвижности носителей заряда определенной энергии

AbstractThe quasi-mobility function of charge carriers with a specified energy for describing their dynamics using the kinetic equation is studied in the important case of two-term isotropic approximation. In the stationary case, the quasi-mobility function is independent of the source function of charge carriers and makes it possible to calculate the integral mobility. The correlation between the quasi-mobility and parameters of the system is analyzed. It is proved that this characteristic does not generally describe the contribution of charge carriers with a specified energy to the integral mobility. In the case of almost elastic scattering, the quasi-mobility, as is known, can have a clear physical meaning; however, in the case of the scattering of charge carriers at acoustic phonons in a solid, this quasi-mobility interpretation is found to be incorrect due to the specific features of the collision integral and the form of the quasi-mobility function.

On Mobility of Definite Energy Charge Carriers

The quasi-mobility function of charge carriers with a specified energy for describing their dynamics using the kinetic equation is studied in the important case of two-term isotropic approximation. In the stationary case, the quasi-mobility function is independent of the source function of charge carriers and makes it possible to calculate the integral mobility. The correlation between the quasi-mobility and parameters of the system is analyzed. It is proved that this characteristic does not generally describe the contribution of charge carriers with a specified energy to the integral mobility. In the case of almost elastic scattering, the quasi-mobility, as is known, can have a clear physical meaning; however, in the case of the scattering of charge carriers at acoustic phonons in a solid, this quasi-mobility interpretation is found to be incorrect due to the specific features of the collision integral and the form of the quasi-mobility function.

Electron transport coefficients in binary mixtures of tetramethylsilane gas with Kr, Xe, He and Ne gases

Tetramethylsilane, alone and in combination with inert gases are widely used in various material processing. The electron transport coefficients in binary mixtures tetramethylsilane gas with buffer gases such as Kr, Xe, He, and Ne gases, therefore, was firstly calculated and analyzed by a two-term approximation of the Boltzmann equation in the E/N (ratio of the electric field E to the neutral number density) range of 0.1-1000 Td (Townsend). These results can be considered to use in many industrial applications depending on particular application of gas, especially on plasma polymerization and plasma enhanced chemical vapour deposition.

Convective Stability of Vertical Throughflow of a Non-Newtonian Fluid in a Porous Channel with Soret Effect

The linear stability analysis of vertical throughflow of power law fluid for double-diffusive convection with Soret effect in a porous channel is investigated in this study. The upper and lower boundaries are assumed to be permeable, isothermal and isosolutal. The linear stability of vertical through flow is influenced by the interactions among the non-Newtonian Rayleigh number (Ra), Buoyancy ratio (N), Lewis number (Le), Peclet number (Pe), Soret parameter (Sr) and power law index (n). The results indicate that the Soret parameter has a significant influence on convective instability of power law fluid. It has also been noticed that buoyancy ratio has a dual effect on the instability of fluid flow. Further, it is noticed that the basic temperature and concentration profiles have singularities at \(Pe = 0\) and \(Le = 1\), the convective instability is looked into for the limiting case of \(Pe\rightarrow 0\) and \(Le \rightarrow 1\). For the case of pure thermal convection with no vertical throughflow, the present numerical results coincide with the solution of standard Horton–Rogers–Lapwood problem. The present results for critical Rayleigh number obtained using bvp4c and two-term Galerkin approximation are compared with those available in the literature and are tabulated.

Finite-time H∞ boundedness of discrete-time neural networks normbounded disturbances with time-varying delay

The problem of finite-time H∞ performance of discrete time neural network with norm-bounded disturbances and time varying delays is studied in this paper. By constructing a delay-dependent Lyapunov-Krasovskii functional and using a two-term approximation of the time-varying delay, sufficient conditions of finite-time stability are derived and expressed in terms of linear matrix inequalities (LMIs). The derived stability conditions can be applied into analyzing the finite-time stability and deriving the maximally tolerable delay. Comparision with the existing results given to show that, the derived stability conditions are less conservative. Finally, numerical examples are provided to illustrate the effectiveness of the proposed method.

The critical electric field of gas mixtures over the extended range of cryogenic operating conditions

In this study, we provide the critical electric field and dielectric strength of cryogenic gas mixtures over an extended cryogenic temperature and pressure range that covers most of the operating conditions of superconducting applications. For gas-cooled cryogenic systems, condensation must not occur during operation. To prevent condensation, we determine the maximum allowed mole fractions of gas species consisting cryogenic gas mixtures by accounting for the operating conditions of cryogenic applications. Subsequently, we estimate the dielectric strength of the gas mixtures in terms of the density-reduced critical electric field ((E/N)cr), obtained by solving the Boltzmann equation with the two-term approximation method. Using the values of (E/N)cr, we calculate the critical electric field (Ecr) over the extended cryogenic operation range of 10–100 K at pressures between 1.0 and 2.0 MPa. The results show that the dielectric strength of cryogenic gas mixtures varies as a function of temperature at a constant operating pressure and reaches its minimum at the condensation point of each gas mixture. The results also suggest that (E/N)cr cannot accurately represent the maximum achievable dielectric strength of a gas mixture unless the maximum allowed mole fractions of gas species have been taken into account. Hence, we discuss (E/N)cr values that are derived from the regulated concentration of gas constituents, which will prevent the components of gas mixtures from condensing. This study provides useful recommendations on the suitability of the gas mixtures and useful reference data for the dielectric design of superconducting and cryogenic applications.

1D kinetic simulations of a short glow discharge in helium

This paper presents a 1D model of a direct current glow discharge based on the solution of the kinetic Boltzmann equation in the two-term approximation. The model takes into account electron-electron coulomb collisions, the corresponding collision integral is written in both detailed and simplified forms. The Boltzmann equation for electrons is coupled with continuity equations for ions and metastable atoms and the Poisson equation for electric potential. Simulations are carried out self-consistently for the whole length of discharge in helium (from cathode to anode) for cases p = 1 Torr, L = 3.6 cm and p = 20 Torr, L = 1.8 mm, so that pL = 3.6 cm·Torr in both cases. It is shown that simulations based on the kinetic approach give lower values of electron temperature in plasma than fluid simulations. Peaks in spatial differential flux corresponding to the electrons originating from superelastic collisions and Penning ionization were observed in simulations. Different approaches of taking coulomb collisions into account give significantly different values of electron density and electron temperature in plasma. Analysis showed that using a simplified approach gives a non-zero contribution to the electron energy balance, which is comparable to energy losses on elastic and inelastic collisions and leads to significant errors and thus is not recommended.

Multiphysics Model for Radio-Frequency Gridded Ion Thruster Performance

A multiphysics radio-frequency gridded-ion thruster performance model is presented. The model is composed of various submodels to account for different physics phenomena. A two-dimensional axisymmetric ion extraction submodel is used to estimate the ion optics effective transparency and beam divergence. A two-dimensional axisymmetric and three-dimensional molecular neutral gas submodels are used to calculate the ion optics Clausing factor and the discharge chamber pressure, respectively. An electron transport submodel determines the rate coefficients and the effective collision frequency by solving a two-term approximation Boltzmann equation. The plasma properties are calculated by employing a volume-averaged zero-dimensional plasma submodel. A fully three-dimensional thruster geometry is used for determining electromagnetic thruster and plasma parameters, such as the coupling coefficient, power losses, and field distributions. The submodel solves Maxwell’s equations in the thruster and plasma represented by a complex conductivity. To improve on the uniform plasma assumption, a radial variation in plasma density/conductivity is introduced. The impedance matching and circuit power losses are accounted for by a radio-frequency circuit submodel, in which the thruster is represented by a complex impedance obtained from the electromagnetic submodel. It is shown that the model predicts the Radio-Frequency Ion Thruster 3.5 experimentally measured radio-frequency generator power and current data values (and trends) within 13% error bound for various operational conditions.

Developments in the kinetic theories of ion and electron swarms in the 1960s and 70s

The two decades between 1960 to 1980 saw quite a fantastic development in diverse areas in physics, and so also in the quantitative theoretical treatment and deeper understanding of the behaviour of isolated electrons and ions in gases—that is ‘charged particle swarm physics’. The evolution in swarm theory was strongly correlated with the contemporary advances in computer technology and the emergence of new and accurate experimental methods for finding charged particle transport parameters, as drift velocities, diffusion coefficients and reaction rates, and also with developments in neighbouring fields as plasma physics and the physics of electronic and molecular collisions. In 1960, low energy electron behaviour could already be calculated with reasonable accuracy in the so-called two-term approximation, while ion behaviour could only be treated at weak electric fields. By 1980, reasonably complete theories had been developed for perhaps most cases in interest—which is reflected in a number of reviews, books and journal articles published in the early 1980s. We will present a journey through the developments in this period and the basic theories behind the Boltzmann equation and Maxwell’s transfer equations. We will also indicate how the interaction between different studies of the same basic processes have led to the elimination of shortcomings and a better understanding.

Monte Carlo simulations of electron transport coefficients in low temperature streamer discharge plasmas

Streamer is usually present at the initial stage of atmospheric pressure air discharge, which occurs in nature as a precursor to lightning, transient luminous events in upper atmosphere and has much potential applications in industry, such as the treatment of polluted gases/liquids, assisted combustion, plasma enhanced deposition etc. Streamer is a multi-scale problem both in time and in space, which brings much difficulty to the conventional diagnostic approaches. In past decades, fluid or particle-fluid hybrid models have been frequently used for understanding the mechanisms of streamer discharges because of their high efficiencies of calculations. Accuracies of the electron transport coefficients (including drift/diffusion coefficient, ionization/attachment coefficient, electron mean energy and extra) play a key role in ensuring the correctness of the fluid or hybrid simulations. As far as we know, BOLSIG+ and MAGBOLTZ are two typical tools for obtaining the electron transport coefficients and have been widely utilized in previous models. BOLSIG+ uses two-term approximation which is not sufficient for some molecular gases, MAGBOLTZ cannot calculate the bulk transport coefficients:these data are required for some models. METHES is an additional tool for computing electron transport coefficients, however, specific platform is required which is not very user-friendly. As sorts of drawbacks exist in currently available calculating tools, in the paper, a Monte Carlo model is developed for computing the electron transport coefficients in gases, the model is flexible to choose any type of gas mixture and its accuracy has been validated by comparing with BOLSIG+ and METHES. Furthermore, the influences of N2-O2 mixture and three-body attachment process in high gas pressures on the transport coefficient are investigated. It is worth mentioning that three-body attachment process can significantly change the electron transport properties at a relatively low reduced electric field. Thus, specific attention must be paid to the transport coefficients if simulation is performed at a high pressure. In addition, differences between the bulk and flux coefficients are analyzed which are not distinguished in some previous models. Finally, we further validate the present Monte Carlo model by performing simulation of streamer discharge in atmospheric N2, which shows that the improved electron transport coefficient from our Monte Carlo model can improve the simulated plasma properties, in particular at the interior of the streamer channel. The existence of divergence at the tip of the streamer channel might be due to our local field approximation; if a density gradient term is included in the impact ionization term and local electron energy approximation of the electron transport coefficients is used, the accuracy of the fluid can be improved further.

Normal moveout for long offset in isotropic media using the Padé approximation

The normal moveout correction is important to long-offset observations, especially deep layers. For isotropic media, the conventional two-term approximation of the normal moveout function assumes a small offset-to-depth ratio and thus fails at large offset-to-depth ratios. We approximate the long-offset moveout using the Pade approximation. This method is superior to typical methods and flattens the seismic gathers over a wide range of offsets in multilayered media. For a four-layer model, traditional methods show traveltime errors of about 5 ms for offset-to-depth ratio of 2 and greater than 10 ms for offset-to-depth ratio of 3; in contrast, the maximum traveltime error for the [3, 3]-order Pade approximation is no more than 5 ms at offset-to-depth ratio of 3. For the Cooper Basin model, the maximum offset-to-depth ratio for the [3, 3]-order Pade approximation is typically double of those in typical methods. The [7, 7]-order Pade approximation performs better than the [3, 3]-order Pade approximation.

An approximation to the reflection coefficient of plane longitudinal waves based on the diffusive-viscous wave equation

The frequency-dependent seismic anomalies related to hydrocarbon reservoirs have lately attracted wide interest. The diffusive-viscous model was proposed to explain these anomalies. When an incident diffusive-viscous wave strikes a boundary between two different media, it is reflected and transmitted. The equation for the reflection coefficient is quite complex and laborious, so it does not provide an intuitive understanding of how different amplitude relates to the parameters of the media and how variation of a particular parameter affects the reflection coefficient. In this paper, we firstly derive a two-term (intercept-gradient) and three-term (intercept-gradient-curvature) approximation to the reflection coefficient of the plane diffusive-viscous wave without any assumptions. Then, we study the limitations of the obtained approximations by comparing the approximate value of the reflection coefficient with its exact value. Our results show that the two approximations match well with the exact solutions within the incident angle of 35°. Finally, we analyze the effects of diffusive and viscous attenuation parameters, velocity and density in the diffusive-viscous wave equation on the intercept, gradient and curvature terms in the approximations. The results show that the diffusive attenuation parameter has a big impact on them, while the viscous attenuation parameter is insensitive to them; the velocity and density have a significant influence on the normal reflections and they distinctly affect the intercept, gradient and curvature term at lower acoustic impedance.

Effective ionisation coefficients and critical breakdown electric field of CO2 at elevated temperature: effect of excited states and ion kinetics

Electrical breakdown by the application of an electric field occurs more easily in hot gases than in cold gases because of the extra electron-species interactions that occur as a result of dissociation, ionization and excitation at higher temperature. This paper discusses some overlooked physics and clarifies inaccuracies in the evaluation of the effective ionization coefficients and the critical reduced breakdown electric field of CO2 at elevated temperature, considering the influence of excited states and ion kinetics. The critical reduced breakdown electric field is obtained by balancing electron generation and loss mechanisms using the electron energy distribution function (EEDF) derived from the Boltzmann transport equation under the two-term approximation. The equilibrium compositions of the hot gas mixtures are determined based on Gibbs free energy minimization considering the ground states as well as vibrationally and electronically excited states as independent species, which follow a Boltzmann distribution with a fixed excitation temperature. The interaction cross sections between electrons and the excited species, not reported previously, are properly taken into account. Furthermore, the ion kinetics, including electron–ion recombination, associative electron detachment, charge transfer and ion conversion into stable negative ion clusters, are also considered. Our results indicate that the excited species lead to a greater population of high-energy electrons at higher gas temperature and this affects the Townsend rate coefficients (i.e. of electron impact ionization and attachment), but the critical reduced breakdown electric field strength of CO2 is only affected when also properly accounting for the ion kinetics. Indeed, the latter greatly influences the effective ionization coefficients and hence the critical reduced breakdown electric field at temperatures above 1500 K. The rapid increase of the dissociative electron attachment cross-section of molecular oxygen with rising vibrational quantum number leads to a larger electron loss rate and this enhances the critical reduced breakdown electric field strength in the temperature range where the concentration of molecular oxygen is relatively high. The results obtained in this work show reasonable agreement with experimental results from literature, and are important for the evaluation of the dielectric strength of CO2 in a highly reactive environment at elevated temperature.

Dynamic Stability of a Cylindrical Shell Reinforced by Longitudinal Ribs and a Hollow Cylinder Under the Action of Axial Forces

The dynamic stability of a cylindrical orthotropic shell reinforced by longitudinal ribs and a hollow cylinder under the action of axial forces changing harmonically with time was investigated with regard for the axial contact interaction of the shell with the ribs. A solution of the differential equations defining this process has been obtained in the form of trigonometric series in the angular and time coordinates. A two-term approximation of the Mathieu–Hill equations of motion was used for construction of the main region of instability of the shell. As a result, the problem was reduced to a system of algebraic equations for components of displacements of the shell at the locations of the ribs. The problem for uniformly spaced ribs was solved in the explicit form. A numerical example of this solution is presented.

Trace estimates for unimodal Lévy processes

We give two-term approximation for the trace of the Dirichlet heat kernel of bounded smooth open set for unimodal Levy processes satisfying the weak scaling conditions.

Fault detection and control co-design for discrete-time delayed fuzzy networked control systems subject to quantization and multiple packet dropouts

This paper investigates the problem of fault detection (FD) for discrete-time delayed fuzzy networked control systems with quantization and packet dropouts. Different from existing results for FD, the proposed ones are toward closed-loop design problem, that is, the controller gain, and the fault detection filter (FDF) gains are designed simultaneously. The missing phenomenon is assumed to occur, in the communication links for quantizer-to-FDF and controller-to-physical plant, where the missing probability of packet dropouts is governed by an individual random binary distribution, while the quantization errors are treated as sector-bound uncertainties. The discrete-time fuzzy networked system is first transformed into the form of interconnection of two subsystems by applying an input–output method and a two-term approximation approach, which is employed to approximate the time-varying delay. Our attention is focused on the design of fuzzy fault detection filter (FFDF) such that, for all data missing conditions, and measurement quantization, the residual system is stochastically stable with a guaranteed H∞ performance. Sufficient conditions are first established via introducing some slack matrices to facilitate the FDF design procedure by eliminating the coupling between the Lyapunov matrices and the system matrices. Numerical examples are provided to demonstrate the effectiveness and applicability of the proposed method.

Analysis of Electron Transport Coefficients in Binary Mixtures of TEOS Gas with Kr, Xe, He and Ne Gases for Using in Plasma Assisted Thin-film Deposition

The electron transport coefficients in not only pure atoms and molecules but also in the binary gas mixtures are necessary, especially on understanding quantitatively plasma phenomena and ionized gases. Electron transport coefficients (electron drift velocity, density-normalized longitudinal diffusion coefficient, and density-normalized effective ionization coefficient) in binary mixtures of TEOS gas with buffer gases such as Kr, Xe, He, and Ne gases, therefore, was analyzed and calculated by a two-term approximation of the Boltzmann equation in the E/N range (ratio of the electric field E to the neutral number density N) of 0.1 - 1000 Td (1 Td = 10^−17 V.cm²). These binary gas mixtures can be considered to use as the silicon sources in many industrial applications depending on mixture ratio and particular application of gas, especially on plasma assisted thin-film deposition.

Decentralized $$\mathscr{H}_{\infty }$$ H ∞ Sampled-Data Control for Continuous-Time Large-Scale Networked Nonlinear Systems

The decentralized \({\mathscr{H}}_{\infty }\) sampled-data control problem is investigated for a class of continuous-time large-scale networked nonlinear systems with interconnection. Each nonlinear subsystem in the considered large-scale system is represented by a Takagi–Sugeno model and is closed by a communication channel with transformed time delay. Our objective is to design a decentralized sampled-data fuzzy controller such that the resulting fuzzy control system is asymptotically stable with an \({\mathscr{H}}_{\infty }\) performance. Firstly, using an input delay approach, the sampled-data control system is formulated into the system with time-varying delay, and a two-term approximation method is proposed such that the delayed system is reformulated into an interconnected framework with input and output. Then, we introduce a Lyapunov–Krasovskii functional that all Lyapunov matrices are no longer required to be positive definite. Combined with the scaled small gain theorem, the less conservative solutions to the decentralized \({\mathscr{H}}_{\infty }\) sampled-data control problem for the considered system are derived in the form of linear matrix inequalities. Finally, the effectiveness of the proposed methods is illustrated by two numerical examples.

CF<sub>4</sub>, CH<sub>4</sub>, Ar 혼합기체의 전자 평균에너지

Energy Distribution Function in pure , and mixtures of and Ar, have been analyzed over a range of the reduced electric field strength between 0.1 and 350[Td] by the two-term approximation of the Boltzmann equation (BEq.) method and the Monte Carlo simulation (MCS). The calculations of electron swarm parameters require the knowledge of several collision cross-sections of electron beam. Thus, published momentum transfer, ionization, vibration, attachment, electronic excitation, and dissociation cross-sections of electrons for , and Ar, were used. The differences of the transport coefficients of electrons in , mixtures of and Ar, have been explained by the deduced energy distribution functions for electrons and the complete collision cross-sections for electrons. The results of the Boltzmann equation and the Monte Carlo simulation have been compared with the data presented by several workers. The deduced transport coefficients for electrons agree reasonably well with the experimental and simulation data obtained by Nakamura and Hayashi. The energy distribution function of electrons in -Ar mixtures shows the Maxwellian distribution for energy. That is, has the symmetrical shape whose axis of symmetry is a most probably energy.

Finite-Time $H_{\infty }$ Filtering for T–S Fuzzy Discrete-Time Systems With Time-Varying Delay and Norm-Bounded Uncertainties

In this paper, we investigate the filtering problem of discrete-time Takagi–Sugeno (T–S) fuzzy uncertain systems subject to time-varying delays. A reduced-order filter is designed. With the augmentation technique, a filtering error system with delayed states is obtained. In order to deal with time delays in system states, the filtering error system is first transformed into two interconnected subsystems. By using a two-term approximation for the time-varying delay, sufficient delay-dependent conditions of finite-time boundedness and $H_{\infty }$ performance of the filtering error system are derived with the Lyapunov function. Based on these conditions, the filter design methods are proposed and the filter gain matrices can be obtained by calculating a set of linear matrix inequalities. A numerical example is used to illustrate the effectiveness of the proposed approaches.