# Comparison Between Heidler Function And The Pulse Function For Modeling The Lightning Return-Stroke Current

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- Comparison Between Heidler Function And The Pulse Function For Modeling The Lightning Return-Stroke Current

## Abstract

In the past, many functions were considered for simulating the lightning return-stroke current. Some of these functions were found to have problems related to their discontinuities or the discontinuities of their derivatives at onset time. Such problems appear in the double exponential function and its modifications. However, other functions like the Pulse function and Heidler function do not suffer from such problems.One of the main objectives of this work is to simulate the lightning return-stroke current full wave, including the decay part, using either Heidler function or the Pulse function. This work is not only necessary for the evaluation and development of lightning return-stroke modeling, but also for the calculation of the lightning current waveform parameters.Although the lightning return-stroke current, measured at the CN Tower, is simulated using the Pulse function and Heidler function, the simulation of the CN Tower lightning current derivative signal is considered using the derivative of the Pulse and Heidler functions.First, we build a modeling environment for each function, which can be described as parameter estimation system. This system, which represents an automated approach for estimating the analytical parameters of a given function, is capable to best fit the function with the measured data. Using these analytical parameters transforms the discrete data into a continuous signal, from which the current waveform parameters can be estimated.This analytical parameters estimation system is recognized as a curve fitting system. For curve fitting technique, the initial value of each analytical parameter and its feasible region, where the optimal value of this analytical parameter is located, must be specified. The more accurate the initial point is the easier and faster the optimal value can be estimated. On choosing the best approach of the initial condition, which gives the nearest location to the optimal point, applying the estimation system and achieving the analytical model that fits the CN Tower measured current derivative, the current waveform parameters can be easily studied.In order to be sure that the analytical parameter extraction system gives the best fit of a function, it needs to be evaluated. Instead of going through the measured data, we first use artificial digital data as a productive way to evaluate the system. Also, a comparison between both the Pulse and Heidler functions is performed.The described fitting process is applied on 15 flashes, containing 31 return strokes. The calculated current waveform parameters were used to form statistics to determine the probability distribution of the value of each parameter, including the range and the 50% probability level, which is fundamental in building lightning protection systems.