Two-term Approximation Research Articles

Einstein Relation for Electrons in an Electric Field

We study the diffusion of electrons moving through a monatomic gas. The two-term Legendre approximation for solving the Boltzmann equation is used to obtain an analytical expression for the isotropic part of the distribution function for hard spheres. We then obtain explicit expressions for the drift velocity, transversal diffusion coefficient, and mean kinetic energy in terms of the confluent hypergeometric function U. The longitudinal diffusion coefficient is obtained by direct numerical integration and the results are used to obtain generalized Einstein relations. The Langevin approach is analyzed and shown to be consistent with the results from the kinetic theory of gases if anisotropic friction is used. An example of the work fluctuation theorem is considered and the differences of the work fluctuation theorem obtained by using the Nernst–Townsend (Einstein) relation and the more accurate results from kinetic theory are calculated.

Investigation of dielectric properties of cold C3F8 mixtures and hot C3F8 gas as Substitutes for SF6

In order to reduce the global warming potential resulting from SF6 widely used as an insulating and arc quenching medium, the substitutes need to be found. This paper focuses on different cold C3F8 mixtures (at room temperature) as an insulating gas and hot C3F8 gas (at temperatures of 300–3500 K) as an arc quenching medium, which seem to be a good replacement of SF6. Firstly, the dielectric properties, including the reduced ionization coefficient α / N, reduced electron attachment coefficient η / N and reduced critical electric field strength (E / N)cr, of the cold C3F8-CF4, C3F8-CO2, C3F8-N2, C3F8-O2 and C3F8-Ar mixtures are calculated numerically using the two-term approximation of the Boltzmann equation. The dependence of such dielectric properties on the buffer gas proportion is investigated. Among the various C3F8 mixtures, the C3F8-N2 mixture has the lowest α / N and the C3F8-CF4 mixture has the largest η / N, and moreover, the C3F8-N2 mixture is the best insulator in terms of breakdown strength because it has the largest (E / N)cr. Secondly, the (E / N)cr of hot C3F8 at temperatures up to 3500 K and various pressures is determined and compared with that of hot SF6 gas. It is found that the hot C3F8 gas has much poorer dielectric performance than hot SF6 because the (E / N)cr of C3F8 decreases significantly above room temperature.

Analysis of Insulating Characteristics of Cl2-He Mixture Gases in Gas Discharges

Insulating characteristics of Cl₂-He mixture gases in gas discharges were analysed to evaluate ability of these gases for using in medium voltage and many industries. These are electron transport coefficients, which are the electron drift velocity, density-normalized longitudinal diffusion coefficient, and density-normalized effective ionization coefficient, in Cl₂-He mixtures. A two-term approximation of the Boltzmann equation was used to calculate the electron transport coefficients for the first time over a wide range of E/N (ratio of the electric field E to the neutral number density N). The limiting field strength values of E/N, (E/N)_lim, for these binary gas mixtures were also derived and compared with those of the pure SF₆ gas.

Bistable solutions for the electron energy distribution function in electron swarms in xenon: a comparison between the results of first-principles particle simulations and conventional Boltzmann equation analysis

At low reduced electric fields the electron energy distribution function in heavy noble gases can take two distinct shapes. This ‘bistability effect’—in which electron–electron (Coulomb) collisions play an essential role—is analyzed here for Xe with a Boltzmann equation approach and with a first principles particle simulation method. The solution of the Boltzmann equation adopts the usual approximations of (i) searching for the distribution function in the form of two terms (‘two-term approximation’), (ii) neglecting the Coulomb part of the collision integral for the anisotropic part of the distribution function, (iii) treating Coulomb collisions as binary events, and (iv) truncating the range of the electron–electron interaction beyond a characteristic distance. The particle-based simulation method avoids these approximations: the many-body interactions within the electron gas with a true (un-truncated) Coulomb potential are described by a molecular dynamics algorithm, while the collisions between electrons and the background gas atoms are treated with Monte Carlo simulation. We find a good general agreement between the results of the two techniques, which confirms, to a certain extent, the approximations used in the solution of the Boltzmann equation. The differences observed between the results are believed to originate from these approximations and from the presence of statistical noise in the particle simulations.

New Results on Decentralized $${\fancyscript{H}}_{\infty }$$ H ∞ Fuzzy Filtering Design for Continuous-Time Large-Scale Nonlinear Systems with Time-Varying Delay

This paper studies the problem of decentralized $${\fancyscript{H}}_{\infty }$$H? fuzzy filtering design for a class of continuous-time large-scale nonlinear systems with time-varying delay. The considered large-scale system consists of several nonlinear subsystems with interconnections and state delays. Each nonlinear subsystem is represented by a Takagi---Sugeno (T---S) model, and the state delay of each subsystem is assumed to be of an interval-like time-varying form. Attention is focused on designing a decentralized fuzzy filter such that the resulting filtering error system is asymptotically stable with a guaranteed $${\fancyscript{H}}_{\infty }$$H? disturbance attenuation level. We firstly propose a two-term approximation method to transform the filtering error system into an interconnected formulation and reformulate the problem of decentralized $${\fancyscript{H}}_{\infty }$$H? fuzzy filtering design in the context of input---output (IO) stability. Then, based on a Lyapunov---Krasovskii functional (LKF) combined with the scaled small gain (SSG) theorem, less conservative results are presented for the decentralized $${\fancyscript{H}}_{\infty }$$H? fuzzy filtering design in terms of linear matrix inequalities (LMIs). Finally, two examples are provided to illustrate the effectiveness of the proposed method.

The role of rotational mechanisms in electron swarm parameters at low reduced electric field in N2, O2 and H2

The homogeneous Boltzmann equation for electrons in N2, O2 and H2 is solved under the classical two-term approximation, for reduced electric fields in the interval 10 − 4 −10 Td where the electron-neutral encounters are limited to elastic, rotational and vibrational collisions. Rotational excitations/de-excitations are described using the following three different approaches: the discrete inelastic/superelastic collisional operator, written for a number of rotational levels that depends on the molecular gas and the specific rotational cross sections considered; the continuous approximation for rotations; a modified version of the continuous approximation for rotations, including a Chapman–Cowling corrective term proportional to the gas temperature. The expression of the rotational collision operator for this latter approach is deduced here and the results show that it bridges the gap between the discrete and the continuous descriptions at low/intermediate reduced electric fields. The calculations are compared with the measurements for the available swarm parameters to assess the validity of the different approaches and cross sections adopted to describe the rotational mechanisms.

Decentralized piecewise [formula omitted] fuzzy filtering design for discrete-time large-scale nonlinear systems with time-varying delay

This paper investigates the problem of decentralized piecewise H∞ filtering design for a class of discrete-time large-scale nonlinear systems with time-varying delay. The considered large-scale system consists of a number of nonlinear subsystems, and each nonlinear subsystem is represented by a Takagi–Sugeno (T–S) model. The time-varying state delay of each subsystem is assumed to be of an interval-like type with lower and upper bounds. The objective is to design a decentralized piecewise filter such that the filtering error system is asymptotically stable with a guaranteed H∞ disturbance attenuation level. A two-term approximation method is proposed to transform the filtering error system into an interconnected formulation, and the decentralized H∞ filtering problem is reformulated in the context of input–output (IO) stability. Based on a piecewise Lyapunov–Krasovskii functional (PLKF) combined with the scaled small gain (SSG) theorem, less conservative results are presented for the decentralized piecewise H∞ filtering design of the large-scale T–S fuzzy system in terms of linear matrix inequalities. Two examples are provided to illustrate the effectiveness of the proposed method.

Finite-time stability analysis and stabilization for uncertain continuous-time system with time-varying delay

The problem of finite-time stability for a class of continuous-time system with norm-bounded uncertainties and time-varying delay is studied in this paper. The original system is firstly transformed into two interconnected subsystems. In order to extract the time-varying term of time delay, a two-term approximation of time-varying delay is used. By using the delay-dependent Lyapunov–Krasovskii-like functional and the method of linear matrix inequality (LMI), sufficient conditions for finite-time stability are derived. The derived conditions can analyze the finite-time stability of system and calculate the upper bound of time delay. In order to stabilize unstable system, the state-feedback and output-feedback controller are respectively designed. Results of numerical examples show the effectiveness of the proposed approach.

ON DOUBLE-DIFFUSIVE HADLEY FLOW IN MAXWELL FLUIDS THROUGH A HORIZONTAL POROUS LAYER

The present study deals with the effect of stress relaxation on the mechanism of instability of the double-diffusive Hadley flow through a shallow horizontal layer of a porous medium induced by thermal and solutal gradients inclined to the vertical by means of linear stability analysis. The Hadley flow which is set up by the horizontal components of these gradients becomes unstable when vertical components are sufficiently large and this instability is analyzed using three-dimensional normal modes. The vertical thermal Rayleigh number is treated as the eigenvalue. The system that constitutes the eigenvalue problem is solved by using a two-term Galerkin approximation for various modes of instability. The detailed analysis is carried out for both the stationary and oscillatory modes. Results are presented for various values of the governing parameters of the flow. It is observed that the stress relaxation parameter has a significant role in convective instability.

Mode-dependent nonrational output feedback control for continuous-time semi-Markovian jump systems with time-varying delay

This paper is concerned with the problem of delay and mode-dependent robust ℋ∞ nonrational dynamic output feedback controller (DOFC) synthesis for a class of continuous-time semi-Markovian jump linear systems (S-MJLSs) with time-varying delay. Due to the relaxed conditions on the stochastic process, the S-MJLSs are with time-varying transition rates and can describe a broader class of dynamical systems than the traditional Markovian jump linear systems. By introducing a two-term approximation for the time-varying delay, the original system is firstly reformulated into a feedback interconnection configuration, which is well-posed in the sense that the scaled small gain (SSG) technique can be applied to the reformulated system to derive robust performance analysis criteria. Then, based on a semi-Markovian Lyapunov–Krasovskii formulation of SSG condition combined with the sojourn-time fractionizing technique, the ℋ∞ performance analysis and mode-dependent nonrational DOFC synthesis conditions for the underlying S-MJLSs are developed, respectively. It is shown that the controller gains can be obtained in terms of linear matrix inequalities. Finally, simulation studies are provided to illustrate the effectiveness and superiority of the proposed design method.

New approach to delay-dependent [formula omitted] control for continuous-time Markovian jump systems with time-varying delay and deficient transition descriptions

This paper proposes an input–output (IO) approach to the delay-dependent stability analysis and H∞ controller synthesis for a class of continuous-time Markovian jump linear systems (MJLSs). The concerned systems are with a time-varying delay in the state and deficient mode information in the Markov stochastic process, which simultaneously involves the exactly known, partially unknown and uncertain transition rates. It is first shown that the original system with time-varying delay can be reformulated by a new IO model through a process of two-term approximation and the stability problem of the original system can be transformed into the scaled small gain (SSG) problem of the IO model. Then, based on a Markovian Lyapunov–Krasovskii formulation of SSG condition together with some convexification techniques, the stability analysis and state-feedback H∞ controller synthesis conditions for the underlying MJLSs are formulated in terms of linear matrix inequalities. Simulation studies are provided to illustrate the effectiveness and superiority of the proposed analysis and design methods.

Fault Detection for Discrete Systems With Network-Induced Nonlinearities

This paper addresses the fault detection filtering issue for a class of discrete system with network-induced nonlinear characteristics. The communication limitations, including measurements quantization, signal transmission delay, and data packet dropout, frequent in many practical networked control systems (NCSs) are considered. By transforming the residual system into an input-output form consisting two interconnected subsystems via a two-term approximation to the state delay variables, sufficient conditions are established. Under these conditions, the residual system is stochastically stable with a prescribed $\mathcal{H}_{\infty}$ level with the help of the scaled small gain (SSG) theorem developed for stochastic systems. Furthermore, the fault detection filter design approach is presented. A numerical example is also provided to demonstrate the effectiveness of the proposed method.

Electron Transport Coefficients and Effective Ionization Coefficients in SF6-O2 and SF6-Air Mixtures Using Boltzmann Analysis

The electron drift velocity, electron energy distribution function (EEDF), density-normalized effective ionization coefficient and density-normalized longitudinal diffusion velocity are calculated in SF6-O2 and SF6-Air mixtures. The experimental results from a pulsed Townsend discharge are plotted for comparison with the numerical results. The reduced field strength varies from 40 Td to 500 Td (1 Townsend=10−17 V·cm2) and the SF6 concentration ranges from 10% to 100%. A Boltzmann equation associated with the two-term spherical harmonic expansion approximation is utilized to gain the swarm parameters in steady-state Townsend. Results show that the accuracy of the Boltzmann solution with a two-term expansion in calculating the electron drift velocity, electron energy distribution function, and density-normalized effective ionization coefficient is acceptable. The effective ionization coefficient presents a distinct relationship with the SF6 content in the mixtures. Moreover, the E/Ncr values in SF6-Air mixtures are higher than those in SF6-O2 mixtures and the calculated value E/Ncr in SF6-O2 and SF6-Air mixtures is lower than the measured value in SF6-N2. Parametric studies conducted on these parameters using the Boltzmann analysis offer substantial insight into the plasma physics, as well as a basis to explore the ozone generation process.

Transport properties of electron swarms in gaseous neon at low values of E/N

A detailed analysis of electron swarm transport through neon gas at applied reduced electric fields of E/N < 2 Td is presented. The root mean square difference of transport parameters calculated from a recent all-order many-body perturbation theory treatment (Cheng et al 2014 Phys. Rev. A 89 012701) with drift velocity measurements by the Australian National University group (Robertson 1972 J. Phys. B 5 648) is less than 1%. Differences of about 3% exist with characteristic energies, DT/μ, (Koizumi et al 1984 J. Phys. B 17 4387) indicating an incompatibility at the 3% level between drift velocity and transverse diffusion coefficient measurements. Multi-term solutions of the Boltzmann equation indicate that the two-term approximation gives transport parameters accurate to better than 0.01%. The diffusion constant at thermal energies is found to be sensitive to the numerical representation of the cross section. A recommended elastic momentum transfer cross section has been constructed that has a maximum difference of 0.5% with all ANU drift velocity data for E/N < 1.6 Td and a root mean square difference that is about a factor of 2 smaller.

Analysis of the insulation characteristics of CF3I gas mixtures with Ar, Xe, He, N2, and CO2 using Boltzmann equation method

The present study is devoted to the calculation of electron swarm parameters, including the reduced effective ionization coefficient, electron mean energy, and electron drift velocity, for the gas mixtures of CF3I with Ar, Xe, He, N2, and CO2. These data are computed by employing the Boltzmann equation method with two-term approximation in the condition of steady-state Townsend (SST) discharge. For the purpose of evaluating the insulation strength of CF3I gas mixtures, values of the limiting field strength (E/N)lim for which the ionization exactly balances the electron attachment are determined from the variation curves of (α − η)/N. The results indicate that mixtures of CF3I–N2 present the greatest insulation strength among all the combinations for CF3I content varied from 20 to 90%. Furthermore, the gas mixture with 70% CF3I can achieve a very similar dielectric strength to that of SF6. The concerned liquefaction issues are also taken into account to fully assess the possibility of applying CF3I gas mixtures in power equipment as an insulation medium.

Input–output method to fault detection for discrete-time fuzzy networked systems with time-varying delay and multiple packet losses

The fault detection (FD) problem for discrete-time fuzzy networked systems with time-varying delay and multiple packet losses is investigated in this paper. The communication links between the plant and the FD filter (FDF) are assumed to be imperfect, and the missing probability is governed by an individual random variable satisfying a certain probabilistic distribution over the interval [0 1]. The discrete-time delayed fuzzy networked system is first transformed into the form of interconnect ion of two subsystems by applying an input–output method and a two-term approximation approach, which are employed to approximate the time-varying delay. Our attention is focused on the design of fuzzy FDF (FFDF) such that, for all data missing conditions, the overall FD dynamics are input–output stable in mean square and preserves a guaranteed performance. Sufficient conditions are first established via H∞ performance analysis for the existence of the desired FFDF; meanwhile, the corresponding solvability conditions for the desired FFDF gains are characterised in terms of the feasibility of a convex optimisation problem. Moreover, we show that the obtained criteria based on the input–output approach can also be established by applying the direct Lyapunov method to the original time-delay systems. Finally, simulation examples are provided to demonstrate the effectiveness of the proposed approaches.

Prediction of the critical reduced electric field strength for carbon dioxide and its mixtures with 50% O2 and 50% H2 from Boltzmann analysis for gas temperatures up to 3500 K at atmospheric pressure

This paper provides theoretical calculations that predict the dielectric breakdown properties of carbon dioxide (CO2) and its mixtures with 50% O2 and 50% H2 for a gas temperature range of 300–3500 K at 0.1 MPa. CO2 is one of the most likely candidates for an environment-friendly arc-quenching medium to replace SF6 in high-voltage circuit breakers. Initially, the electron energy distribution function (EEDF) is derived by solving the Boltzmann equation under the zero-dimensional two-term spherical harmonic approximation. Then the reduced ionization and attachment coefficients are obtained, based on the calculated EEDF. Finally, the critical reduced electric field strength (E/N)cr, which is defined as the value for which total ionization reactions are equal to total attachment reactions, is obtained and analysed. The results demonstrate the superior breakdown properties of a 50% CO2–50% O2 mixture to those of both pure CO2 and 50% CO2–50% H2. Nearly no deviation in (E/N)cr is found in a 50% CO2–50% O2 mixture for gas temperatures up to 2500 K, and although there is clear reduction as the gas temperature is increased further to 3500 K, the value remains higher than that of pure CO2.

Fault detection for T-S fuzzy time-delay systems: delta operator and input-output methods.

This paper focuses on the problem of fault detection for Takagi-Sugeno fuzzy systems with time-varying delays via delta operator approach. By designing a filter to generate a residual signal, the fault detection problem addressed in this paper can be converted into a filtering problem. The time-varying delay is approximated by the two-term approximation method. Fuzzy augmented fault detection system is constructed in δ -domain, and a threshold function is given. By applying the scaled small gain theorem and choosing a Lyapunov-Krasovskii functional in δ -domain, a sufficient condition of asymptotic stability with a prescribed H∞ disturbance attenuation level is derived for the proposed fault detection system. Then, a solvability condition for the designed fault detection filter is established, with which the desired filter can be obtained by solving a convex optimization problem. Finally, an example is given to demonstrate the feasibility and effectiveness of the proposed method.

On CDO Tranche Pricing When Copula is Nearly Comonotone

Pricing of a CDO base tranche is considered when copula is comonotone and approximations are derived for the base tranche expected loss, obtained with a one-factor Gaussian copula model, when correlations are sufficiently close to 100%. Numerical examples are provided showing that a two-term approximation reproduces the true value with no more than approximately 5% relative error, uniformly across detachments and average default probabilities for correlations between 95% and 100%.

Determination of the electron collision cross-section set for the C4F6 molecule by using an electron swarm study

A set of electron collision cross-sections for the C4F6 molecule has been successfully determined for the first time by using an electron swarm method with a two-term approximation of the Boltzmann analysis. The set is evaluated by fitting the calculated electron drift velocities in pure C4F6 and in C4F6-Ar mixtures to the experimental data in the E/N range of 0.35 — 1200 Td (1 Td = 10−17 Vcm2), where E/N is the ratio of the electric field (E) to the neutral number density (N). The set is confirmed to be reliable due to our ability to reproduce satisfactorily the electron drift velocities in pure C4F6 and in 0.1, 0.5, 1, 5, and 10% C4F6-Ar mixtures with deviations of ±5%. The set may be the best available so far for use in the quantitative numerical modeling of a4 plasma discharge for various processing procedures with materials containing C4F6 molecules.