Approximation of measured or numerically simulated pulses by double exponential pulses

Abstract

The lightning qualification of parts and equipment is done on the basis of well-defined test waveforms, often defined as double exponential pulses (ED-84A [1], DO-160 [2]). The environmental data determined by coupling measurements and simulations often show not only pulses of different durations, but also pulses with different shape e.g. a lower rise at start, undershoot at start or at the trailing edge. However, measured or simulated results are often approximated with double exponential curves in order to compare such results with standard test pulses. Sometimes the double exponential pulses are converted to standard test pulses showing the same energy content. This second step can be solved fully analytical; a formula has been derived for this purpose. But in a first step, it is necessary to approximate the measured or computed pulse, known only by a series of value pairs, by a double exponential pulse. The half value time (at leading or trailing edge) is often depicted directly from measurement or simulation results as well as the time to peak without considering that they don't belong to a double exponential pulse. With these depicted values one tries to solve a system of non-linear equations in order to find the exponential coefficients α and β for a double exponential pulse covering the test or simulation results. The result can only be found by iteration (e.g. also used in mathematic software), requiring guess values at start, which have to be placed near enough to the solution, otherwise the algorithm shows no convergence or only a trivial but not searched solution as α = β. But this can also be caused by the used half value time, not matching with a double exponential pulse. Therefore the approximation of simulation or coupling measurement results with double exponential pulses requires a play with the depicted parameters in order to find an approximation such that maximum value and at least the same energy content is maintained (worst case approach). When the maximum value and the time to peak value are used as a fix point, the half value time picked up from the leading or the trailing edge of the original pulse has to be varied. The reason is also that two limit values exist for the normalized half value time (normalized to time to peak), where in between no solution exists for the searched exponential coefficients. The paper describes a way, how the approximation can be found without too many trials varying guess values and half value times. Some helpful formulas and diagrams are included. One first practical application has been the proof for limit values for equipment qualification by numerical simulation of lightning attachment scenarios on the aircraft skin. A Mathcad worksheet has been used for this purpose; changing the value for the half value time by hand let you find the best curve fitting by a double exponential pulse.