Marquardt Least-squares Method Research Articles

Modeling of artificially triggered lightning currents by multi-peaked analytically extended functions

PurposeThis paper aims to present the approximation of lightning currents waveshapes by the multi-peaked analytically extended function (MP-AEF) for the experimentally measured channel-base currents in the artificially triggered lightning discharges. Modified transmission line model of lightning return strokes having the channel current both linearly decaying and sinusoidally changing with height (MTLSIN) is used to calculate the lightning electromagnetic field.Design/methodology/approachMP-AEF’s parameters for the artificially triggered lightning channel-base currents are calculated by using Marquardt least squares method (MLSM). Lightning electromagnetic fields are calculated based on electromagnetic theory relations, thin-wire antenna model of the vertical lightning channel and the assumption of the perfectly conducting ground. MTLSIN model as an engineering model of lightning strokes is used to obtain the electric field results as these are simultaneously measured in rocket-triggered lightning experiments together with the channel-base currents.FindingsMP-AEF approximates multi-peaked pulse waveshapes. Some important function parameters are chosen prior to the approximation procedure, such as current peaks and the corresponding time moments of those peaks, which presents an advantage in comparison to other functions. The desired accuracy of approximation is obtained by choosing an adequate number of function terms. MLSM is used for the estimation of unknown parameters. Using MTLSIN model, the influence of the channel height and return stroke speed on the lightning electromagnetic field waveshape is analyzed in this paper.Research limitations/implicationsMP-AEF may be used for approximation of various multi-peaked waveshapes. It has no errors in the points of maxima which is important for the lightning protection systems design. MTLSIN model may be validated by using simultaneously measured lightning electromagnetic fields at various distances from the channel and for channel heights estimated in the experiments. It is also possible to approximate measured current derivatives by MP-AEF and use them for further computation.Originality/valueMTLSIN model is proposed in this paper for the evaluation of lightning electromagnetic fields induced by artificially triggered lightning discharges. The procedure is based on the approximation of lightning channel-base currents by the multi-peaked analytically extended function previously proposed by the authors. This function may be used not only for representing lightning currents but also for other waveshapes as current derivatives, electric and magnetic fields and their derivatives, which are all important for the lightning protection design. MTLSIN gives lightning electromagnetic fields results which are in better agreement with measured fields than those obtained by other models from literature.

Electrostatic discharge currents and their derivatives' approximation by piecewise power-exponential functions

An analytically extended function based on power-exponential functions is used in this paper for approximation of electrostatic discharge (ESD) currents and their derivatives. The Marquardt least-squares method (MLSM) is applied for obtaining nonlinear function parameters. IEC 61000-4-2 Standard ESD current is approximated, as well as some measured ESD currents' wave shapes. Power-exponential terms are extended at the local maxima and minima of the represented wave shape, so that this approximation is done from peak to peak. ESD current derivative is approximated using the same procedure in order to obtain the continuous second order derivative of the current, as all piecewise functions are of differentiability class C$^{1}$. Currents and their derivatives are often measured in ESD experiments so that their analytical representation is needed for simulation of ESD phenomena, better definition of standard requirements, and computation of the transient fields and induced effects.

Analytical Representation of Measured Lightning Currents and Its Application to Electromagnetic Field Estimation

Lightning discharge currents waveshapes and their derivatives with multiple peaks are measured in artificially triggered lightning experiments and at instrumented tall towers. Such waveshapes are represented in this paper by the N -peaked analytically extended function ( N P-AEF) and Marquardt least-squares method is applied for the estimation of its nonlinear parameters. Typical channel-base currents of the first negative, subsequent negative strokes, and positive strokes, based on comprehensive measurements by Berger et al. at Monte San Salvatore in Switzerland, are approximated by N P-AEFs and used in the computation of lightning electromagnetic fields. A new attenuation factor introducing nonlinear current attenuation along the channel is applied within the modified transmission line model (MTLSIN). A lightning return stroke is assumed to have a vertical discharge channel at the perfectly conducting ground. In order to validate this model, calculated lightning electromagnetic fields are compared with the typical, as measured by Lin et al. at various distances from the discharges. Lightning currents and their derivatives measured at the tall towers are also approximated by N P-AEFs. For the measured artificially triggered lightning currents, MTLSIN is applied for the calculation of their electromagnetic fields. These are compared with the measured fields of specific lightning strokes, so as to results of other models.

Estimation of Parameters for the Multi-peaked AEF Current Functions

An examination of how the analytically extended function (AEF) can be used to approximate multi-peaked lightning current waveforms, is presented in the paper. A general framework for estimating the parameters of the AEF using the Marquardt least-squares method (MLSM) for a waveform with an arbitrary (finite) number of peaks is presented. This framework is used to find parameters for some common waveforms with a single peak, such as Standard IEC 62305 lightning currents. Illustration of fitting a p-peak AEF to recorded lightning current data is also presented.

Analytically extended function to representation of some measured lightning currents

A multi-peaked form of the analytically extended function (AEF) is used for approximation of lightning current waveforms in this paper. The AEF function's parameters are estimated using the Marquardt least-squares method (MLSM), and the general procedure for fitting the $p$-peaked AEF function to a waveform with an arbitrary (finite) number of peaks is briefly described. This framework is used for obtaining parameters of 2-peaked waveforms typically present when measuring first negative stroke currents. Advantages, disadvantages and possible improvements of the approach are also discussed.

Novel approach to modelling of lightning current derivative

A new approach to mathematical modelling of lightning current derivative is proposed in this paper. It builds on the methodology, previously developed by the authors, for representing lightning currents and electrostatic discharge (ESD) currents waveshapes. It considers usage of a multi-peaked form of the analytically extended function (AEF) for approximation of current derivative waveshapes. The AEF function parameters are estimated using the Marquardt least-squares method (MLSM), and the framework for fitting the multi-peaked AEF to a waveshape with an arbitrary number of peaks is briefly described. This procedure is validated performing a few numerical experiments, including fitting the AEF to single- and multi-peaked waveshapes corresponding to measured current derivatives.

ALCHEMI for coexistent Heusler and half-Heusler phases in TiNi 1.5Sn

Atomic location by channeling-enhanced microanalysis (ALCHEMI) is an effective technique to clarify the atomic configurations in multicomponent compounds. Recent development of the theory on the characteristic X-ray emission has made ALCHEMI a more reliable and expansive technique. In this revised ALCHEMI, the characteristic X-ray intensities are measured at various electron-incidence directions, and are compared with the X-ray intensities calculated from first principles. In the present work, this technique was applied to coexistent Heusler and half-Heusler phases in a thermoelectric material, TiNi 1.5Sn. The X-ray intensities calculated for both phases were significantly dependent on the fitting parameters of the atomic occupancy on each site and the sample thickness. Using the Levenberg–Marquardt least-squares method, the residual errors between the experimental and the calculated X-ray intensities were minimized to determine the fitting parameters quantitatively. In the Heusler phase, the ordered structure was observed because the atoms of each element almost occupy each site. On the other hand, in the half-Heusler phase, the “Heusler-like” structure was observed because the occupation probabilities of Ni atoms on the Ni-site and on the vacancy-site were measured as 0.49±0.01 and 0.49±0.01, respectively, whereas 1 and 0 is expected for the normal half-Huesler structure.

Isomerization of linear alkenes on Amberlyst 15

The kinetics of the isomerization of 1-butene to cis-2-butene and trans-2-butene, of 1-pentene to cis-2-pentene and trans-2-pentene as well as of 1-hexene to cis-2-hexene and trans-2-hexene on sulfonated copolymer Amberlyst 15 in the gas phase was investigated. The Langmuir–Hinshelwood equations were adjusted to the results of the kinetic experiment using the values of adsorption equilibrium constants of the reactants from the adsorption measurements separately taken. The calculations were done by the Levenberg–Marquardt least-squares method using the Origin Microcal program.

Determination of the Langmuir–Hinshelwood kinetic equation of synthesis of ethers

The kinetics of the synthesis of methyl- tert-butyl and ethyl- tert-butyl ethers from isobutene and alcohols (methanol and ethanol) on sulfonated Amberlyst 15 copolymer in the gas phase was investigated. The Langmuir–Hinshelwood equations were adjusted to the results of the kinetic experiment using the values of the adsorption equilibrium constants of the reactants from the adsorption measurements separately taken. The calculations were done by the Levenberg–Marquardt least-squares method accompanied by the Origin Microcal program. A single-stage mechanism of the synthesis of ethers consisting in the surface reaction between the molecule of alcohol adsorbed by one sulfonic group and the molecule of isobutene adsorbed by two sulfonic groups was suggested.

Determination of the Langmuir?Hinshelwood kinetic equation of synthesis of ethers

The kinetics of the synthesis of methyl- tert -butyl and ethyl- tert -butyl ethers from isobutene and alcohols (methanol and ethanol) on sulfonated Amberlyst 15 copolymer in the gas phase was investigated. The Langmuir–Hinshelwood equations were adjusted to the results of the kinetic experiment using the values of the adsorption equilibrium constants of the reactants from the adsorption measurements separately taken. The calculations were done by the Levenberg–Marquardt least-squares method accompanied by the Origin Microcal program. A single-stage mechanism of the synthesis of ethers consisting in the surface reaction between the molecule of alcohol adsorbed by one sulfonic group and the molecule of isobutene adsorbed by two sulfonic groups was suggested.

Evaluation of thermodynamic parameters for highly correlated chemical systems: a spectrophotometric study of the 1 : 1 and 2 : 1 equilibria between I2and 1,1′-methylenebis(3-methyl-4-imidazoline-2-thione)(mbit) and 1,1′-ethylenebis(3-methyl-4-imidazoline-2-thione)(ebit). Crystal and molecular structures of mbit·2I2and ebit·2I2

The reactions of I2 with 1,1′-methylene- and 1,1′-ethylene-bis(3-methyl-4-imidazoline-2-thione)(mbit and ebit) have been investigated in CHCl3 solution at different temperatures by spectrophotometry. The experimental data have been processed using two different procedures of computer analysis: SUPERSPEC based on the conventional Gauss–Newton–Marquardt least-squares method; and POWELSPEC, based on Powell's direct search method for the refinement of ΔH° and ΔS°. The ability of the two methods to identify the involved equilibria correctly was compared. Evidence for the stepwise formation of the 1 : 1 and 1 : 2 adducts has been obtained by the two methods for the mbit case, while POWELSPEC only was able to solve the ebit case satisfactorily. The crystal structures of the adducts mbit·2I2 and ebit·2I2 have been determined. They show that the thionic sulfur atoms co-ordinate two diiodine molecules, with S–I 2.683(2)(mbit·2I2) and 2.642(3)(ebit·2I2)A and with I–I 2.897(1) and 2.903(2)A respectively, the two S–I–I groups beings related by a two-fold axis. The structural features of the S–I–I linear group are in accordance with Fourier-transform IR and Raman spectral data and can be taken into account with a three-centre two-electron axial orbitally deficient bonding scheme.

Numerical analysis reveals complexities of glutamate transport

The uptake of radiolabeled glutamate into cultured human (HeLa S 3) and hamster (CHO-K1) cells was analyzed according to modified Michaelis-Menten models fit by the Marquardt least-squares method. Kinetic parameters not previously reported for these cells were obtained. Some rarely used features available with this fitting method proved to be extremely helpful. Most importantly, a goodness-of-fit measure revealed a significant alteration of glutamate uptake in HeLa cells that was induced by starvation. This apparent regulation, unexpected for glutamate transport, might have been missed if the fit had been judged by eye or by the magnitude of parameter standard deviations. Techniques for analyzing parameter distributions, improving experimental design and performing tests of significance are also described.