Mathematical Parameters Calculation of Double Exponential Function by a New Numerical Method

Abstract

Background: Lightning electromagnetic pulse (LEMP) and high-altitude electromagnetic pulse (HEMP) are widely described by three physical parameters (rise time tr, full-width at halfmaximum pulse width tw, and maximum electric field strength E0). These pulse shapes are often given by a double exponential form concerning four mathematical parameters, namely α, β, k and Ep. Objective: The transformation from physical parameters into mathematical parameters is necessary in waveform simulation and is traditionally accomplished by linear fit functions regarding the two groups of parameters. However, traditional methods commonly rely on data analysis and calculation. In order to obtain the mathematical parameters more concisely and clearly. Methods: In this paper, a numerical method to calculate the mathematical parameters by solving nonlinear equations with three key constraints is proposed. Firstly, we establish the nonlinear system of equations regarding four variables, namely t1, t2, α and β. Then, three constraints are given to converge the solutions of the equations. Lastly, select the minimal value of the convergent solution of each equation. Results: Comparing the solutions obtained by our proposed method to the iterated ones, the overall relative error is less than 2×10-8. Conclusion: The results show that our proposed method not only simplifies the transformation from physical parameters to mathematical parameters, but also keeps the solutions high accurate.