Analytic solution for the lightning current induced mutually coupled resistive filament wire model

Abstract

When a lightning current flows between the lightning entry and exit points of a structure, the lightning current density varies in different parts of the structure depending on the shape of the structure and material variance. The structure can be discretized into parallel wires, called filament wires, running parallel to the current direction. Furthermore, using the filament wire method, we can calculate the current distribution among the wires. For a structure that has a low resistance material such as aluminum, current distribution can be calculated by considering self-inductance of the wire and mutual-inductance between wires but resistance is not considered. However, in modern aircraft, composite materials are used for parts of the structure because of their strength and weight. These composite materials have high resistance compared to metal, and resistance cannot be ignored. Thus, to solve a system of ordinary differential equations for a filament model, inhomogeneous structure, aperture, and resistance of each wire must be considered to obtain the correct current distribution of each part of the structure. However, the numerical solution of the filament wire model does not reveal the region of convergence and the accuracy of the given mathematical model. It also has high time complexity. This paper presents the analytic solution and stability condition for the mutually coupled resistive filament wire model using eigenvalues of given filament wire matrix model. The stability condition is rigorously calculated and the solution is also consistent with the numerical model.

Keywords

• lightning current
• differential equation
• filament wire model
• matrix equation
• systems of ode