Lambert W-function Research Articles

Solar cell array parameters using Lambert W-function

Abstract Exact closed-form solution based on Lambert W-function are presented to express the transcendental current–voltage characteristic containing parasitic power consuming parameters like series and shunt resistances for solar cell array. Maple software was used to solve the transcendental equation of solar cell array.

Integrating the LambertW Function to a Tolerance Optimization Problem

This paper explores the integration of the Lambert W function to a tolerance optimization problem with the assessment of costs incurred by both the customer and a manufacturer. By trading off manufacturing and rejection costs, and a quality loss, this paper shows how the Lambert W function, widely used in physics, can be efficiently applied to the tolerance optimization problem, which may be the first attempt in the literature related to tolerance optimization and synthesis. Using the concept of the Lambert W function, a closed-form solution is derived, which may serve as a means for quality practitioners to make a quick decision on their optimal tolerance without resorting to rigorous optimization procedures using numerical methods. A numerical example is illustrated and a sensitivity analysis is performed. Copyright © 2005 John Wiley & Sons, Ltd.

State-dependent impulsive models of integrated pest management (IPM) strategies and their dynamic consequences

A state-dependent impulsive model is proposed for integrated pest management (IPM). IPM involves combining biological, mechanical, and chemical tactics to reduce pest numbers to tolerable levels after a pest population has reached its economic threshold (ET). The complete expression of an orbitally asymptotically stable periodic solution to the model with a maximum value no larger than the given ET is presented, the existence of which implies that pests can be controlled at or below their ET levels. We also prove that there is no periodic solution with order larger than or equal to three, except for one special case, by using the properties of the LambertW function and Poincare map. Moreover, we show that the existence of an order two periodic solution implies the existence of an order one periodic solution. Various positive invariant sets and attractors of this impulsive semi-dynamical system are described and discussed. In particular, several horseshoe-like attractors, whose interiors can simultaneously contain stable order 1 periodic solutions and order 2 periodic solutions, are found and the interior structure of the horseshoe-like attractors is discussed. Finally, the largest invariant set and the sufficient conditions which guarantee the global orbital and asymptotic stability of the order 1 periodic solution in the meaningful domain for the system are given using the Lyapunov function. Our results show that, in theory, a pest can be controlled such that its population size is no larger than its ET by applying effects impulsively once, twice, or at most, a finite number of times, or according to a periodic regime. Moreover, our theoretical work suggests how IPM strategies could be used to alter the levels of the ET in the farmers' favour.

A new approach to study organic solar cell using Lambert W-function

Abstract Organic photovoltaic solar cells bear an important potential of development in the search for low-cost modules for the production of domestic electricity. One of the main differences between inorganic and organic solar cells is that photo-excitation in these materials does not automatically lead to the generation of free charge carriers, but to bind electron–hole pairs (exciton) with a binding energy of about 0.4 eV. Till now various numerical methods using approximations have been reported to study different aspects of organic solar cells. For the first time an accurate method using Lambert W-function is presented to study different parameters of organic solar cells.

A new method to determine the diode ideality factor of real solar cell using Lambert W-function

A new method using Lambert W-function is presented to determine the diode ideality factor of real solar cell.

Study on the Dynamic Range of Delta Modulations With Time-Varying Sampling Periods

The theory of 1-bit adaptive delta modulations (ADM) systems with uniform sampling rate is well known. Analytical study on the influence of sampling rate adaptation on companding capabilities of delta modulations have been carried out in the present paper. The described ADM systems encode the input signal by changing the sampling intervals and/or step size according to it's time-varying slope characteristics. Based on quantitative analysis, a rule that the total companding gain of the doubly adaptive DM system can be expressed, in decibels, as a sum of the companding gain from sampling frequency and step size, has been proved. This rule indicates on possibility obtaining of large dynamic ranges, without considerable variation of both companded parameters. In practice, it is reflected by the increase of not only compression ratio but also the SNR peak value. The dependences of ADM systems companding gain from range changes of both the sampling frequency and the step size, have been found. The relation between the range of sampling frequency changes and companding gain of the variable-rate sampling ADM systems have been described by the LambertW function. The analytical results presented in this study are useful for design the dynamic range of NS-DM and ANS-DM modulations.

Surface changes of temperature and matter due to coupled transport processes through porous media

A new calculation is elaborated for the description of surface changes of temperature and matter due to coupled transport through a porous medium. It is based on a finite-size supposition (Neumann's type boundary condition) and on the solutions of parabolic type partial differential equations combined with Lambert's W-function. The boundary layer phenomena are also incorporated into the description of the general transport. The procedure leads to a direct computer simulation, providing concrete results on the real physical picture of the given problem in good agreement with some experiments analysed.

Modeling of beam width in passively Q-switched end-pumped lasers

An iterative model based on the LambertW function was developed for estimating the fundamental mode parameters of resonators with saturable gain guiding. The process of pulse buildup in passively Qswitched, end-pumped lasers was analyzed. The effective ABCD cavity matrix for consecutive round-trips was calculated, taking into account spatially variable saturated gain in an active medium and absorption bleaching in a saturable absorber. The twofold decrease in beam width, as compared with the fundamental mode of the bare cavity, was demonstrated. The application of such a model for resonators with other nonlinear elements is feasible.

Chebyshev polynomial expansions for simultaneous approximation of two branches of a function with application to the one-dimensional Bratu equation

We introduce two methods for simultaneously approximating both branches of a two-branched function using Chebyshev polynomials. Both schemes remove the pernicious, convergence-wrecking effects of the square root singularity at the limit point where the two branches meet. The “Chebyshev–Shafer” method gives the approximants as the solution to a quadratic equation; the “mapping” algorithm makes a quadratic change of variable. For both, the only input information is the set of values of f( x) at a set of discrete points. There is little to choose between the two schemes in accuracy, but the single expansion/mapping method is more flexible in that it can accommodate different ranges on the two branches. The eigenrelation for the one-dimensional Bratu equation is an interesting example because the upper branch is also singular at another point besides the limit point; this, too, can be removed by subtracting the asymptotic solution, which is the Lambert W-function, from the upper branch only. When the domain of the variable is infinite, the quadratic change of variable can still be applied by substituting rational Chebyshev functions, which are basis functions for an unbounded interval, for Chebyshev polynomials. We illustrate this by approximating the real-valued root of the Brill quintic, u 5− u− λ, λ∈[−∞,∞], which was first solved by Hermite using elliptic modular functions more than a century ago.

Explicit infiltration equations and the Lambert W-function

The Green and Ampt infiltration formula, as well as the Talsma and Parlange formula, are two-parameter equations that are both expressible in terms of Lambert W-functions. These representations are used to derive explicit, simple and accurate approximations for each case. The two infiltration formulas are limiting cases that can be deduced from an existing three-parameter infiltration equation, the third parameter allowing for interpolation between the limiting cases. Besides the limiting cases, there is another case for which the three-parameter infiltration equation yields an exact solution. The three-parameter equation can be solved by fixed-point iteration, a scheme which can be exploited to obtain a sequence of increasingly complex explicit infiltration equations. For routine use, a simple, explicit approximation to the three-parameter infiltration equation is derived. This approximation eliminates the need to iterate for most practical circumstances.

Renormalon-inspired resummations for vector and scalar correlators — estimating the uncertainty in αs( M2τ) and α( M2Z)

We perform an all-orders resummation of the QCD Adler D-function for the vector correlator, in which the portion of perturbative coefficients containing the leading power of b, the first beta-function coefficient, is resummed. To avoid a renormalization scale dependence when we match the resummation to the exactly known next-to-leading order (NLO), and next-NLO (NNLO) results, we employ the Complete Renormalization Group Improvement (CORGI) approach in which all RG-predictable ultraviolet logarithms are resummed to all-orders, removing all dependence on the renormalization scale. We can also obtain fixed-order CORGI results. Including suitable weight-functions we can numerically integrate these results for the D-function in the complex energy plane to obtain so-called “contour-improved” results for the ratio R and its tau decay analogue R τ . We use the difference between the all-orders and fixed-order (NNLO) results to estimate the uncertainty in α s ( M 2 Z ) extracted from R τ measurements, and find α s ( M 2 Z )=0.120±0.002. We also estimate the corresponding uncertainty in α( M 2 Z ) arising from hadronic corrections by considering the uncertainty in R( s), in the low-energy region, and compare with other estimates. Analogous resummations are also given for the scalar correlator. As an adjunct to these studies we show how fixed-order contour-improved results can be obtained analytically in closed form at the two-loop level in terms of the Lambert W-function and hypergeometric functions.

Exact analytical solutions of the forward non-ideal diode equation with series and shunt parasitic resistances

Exact closed form solutions based on the Lambert W-function are presented to express the forward current–voltage characteristics of non-ideal single-exponential diodes containing all possible combinations of series and shunt parasitic resistances. It is shown that these expressions could be useful for carrying out highly accurate computations at speeds almost as fast as those possible when using less precise approximate solutions based on common elementary functions.

Exact analytical solution for current flow through diode with series resistance

A simple analytical expression is presented for the current flow in a diode driven by a voltage source through a series resistance. The proposed solution is based on the Lambert W-function. The new expression leads to an efficient method for extracting series resistance from measured current-voltage data. Experimental results are presented which validate the proposed solution and extraction method.

Bipolar transistor circuit analysis using the Lambert W-function

The generalized diode equation describes conduction in a diode with series resistance. An analytical solution for the generalized diode equation has been elusive; however, one was found based on the transcendental equation w=ln(x/w). The solution of this equation; w=W(x), is traditionally referred to as the Lambert W-function. This function provides a long sought after natural continuity between exponential diode and linear resistor behavior. The W-function also describes more general circuits consisting of a diode or bipolar transistor with local linear negative or positive feedback. The properties of W(x) are reviewed and several iterative methods for its calculation are compared. Three approximations for the W function are derived which can simplify bipolar circuit analysis and design. The practical utility of the proposed solutions are demonstrated in four circuits along with experimental confirmation: a common emitter amplifier with an emitter or collector feedback resistor, Schmitt trigger threshold temperature compensation, bandgap stabilized current source, and a novel current-efficient laser driver.

Global approximations to the principal real-valued branch of the Lambert W-function

W( z) is defined implicitly as the root of W exp( W) = z. It is shown that a simple analytic approximation has a relative error of less than 5% over the whole domain z ϵ [−exp(−1), ∞] of the principle branch—sufficiently accurate so that four Newton iterations refine this approximation to a relative error smaller than 1.E-12. As a second form of global approximation, the W-function is expanded as a series of rational Chebyshev functions TB j in a shifted, logarithmic coordinate with an error that decreases exponentially fast with the series truncation.

Kinetics of the electroporative deformation of lipid vesicles and biological cells in an electric field

AbstractThe electric conductivity of a suspension of unilamellar lipid vesicles of phosphatidylcholine and phosphatidylglycerol, filled with electrolyte, increases after exposure to external field pulses. The electrolyte efflux through the electropores is described in terms of volume decrease under Maxwell stress at constant membrane surface area. Applying the Hagen‐Poiseuille Ansatz, the exact analytical solution for the kinetics of the electromechanical vesicle deformation is obtained in terms of a LambertW function. The initial and final phases of the volume decrease can be approximated by exponential functions with the time constants τ and 2τ, respectively. It is found that τ is proportional to E2·a9, describing the extreme sensitivity of the deformation to the vesicle radius a in electric field of strength E.The kinetic analysis yields the membrane bending rigidity κ = 3.0±0.3·10−20 J. At the field strength E = 1.0 MV m−1 and in the range of pulse duration 5 ≤ tE/ms ≤ 60, the number of water‐permeable electropores is found to be N=35±5 per vesicle of radius rp = 50 nm with mean pore radius rp=0.9±0.1 nm.The kinetic analysis developed here for vesicles is readily applied to cell membranes, aiming at physicalchemical guidelines to optimize the membrane electroporation techniques for the direct transfer of drugs and genes into tissue cells.

Colebrook-White Formula for Pipe Flows

Flow-resistance laws—as used, for example, in water-supply pipe networks—are formulas relating the volume flow rate, q, along a pipe to the pressure-head difference, t, between its ends, such that q=ψ(t), in which ψ is monotonic. The simple Hazen-Williams power law is often used, but in appropriate circumstances the more complicated Colebrook-White law (CW) may better represent aspects of the experimental data. Result 1, the first and easiest-to-state result in the paper, is that ϕCW, the inverse of ψCW, can be expressed in terms of the Lambert W-function (Corless et al. 1993). Result 2 summarizes one use of this, and of related results, in convex optimization problems describing equilibrium flows in pipe networks.

On the LambertW function

The LambertW function is defined to be the multivalued inverse of the functionw →we w . It has many applications in pure and applied mathematics, some of which are briefly described here. We present a new discussion of the complex branches ofW, an asymptotic expansion valid for all branches, an efficient numerical procedure for evaluating the function to arbitrary precision, and a method for the symbolic integration of expressions containingW.