102 / 2021-06-05 21:54:52

Simplified Boundary Element Model for Lightning Current Distribution on the Arbitrary Cross Section of Conductor

Metal structures with a certain extent of height such as lightning rods, wind turbine towers, steel frames of viaducts might be more likely struck by lightning and attract large transient current along them [1]. The current distribution raises concern of researchers in the field of lightning protection and electromagnetic compatibility, but the electromagnetic simulation on such large size and complicated conductor structures often costs too much to get acceptable results. In this paper, the boundary element method (BEM) is involved to simplify the lightning current distribution calculation through some approximation techniques. Firstly, the partial differential equation (PDE) of the current surface density is deduced on the two-dimension cross section Ω. Represented by magnetic vector potential, the PDE from Maxwell’s equation turns to a diffusion equation and a Laplace equation,

\({\nabla ^2}{{\vec A}_1} - j\omega \mu \sigma {{\vec A}_1} = - \mu \vec J,\ {\rm{ in \ \Omega }}\)

\({\nabla ^2}{{\vec A}_2} = 0\ {\rm{ outside \ \Omega }}\)

\({{\vec A}_1} = {{\vec A}_2} \buildrel \Delta \over = g, \ {\rm{ on }}\ \partial {\rm{\Omega }}\)

(1)

where μ and σ are permeability and conductivity of the conductor, ω is the angular frequency and J_e is the surface density of lightning current excitation [2]. If the difference of conductor potential φ per unit length along the current direction is fixed, J_e=-σ▽φ , which is also a constant. Thus, J_e is known and A₁, A₂, g are to be solved from the equation.

Then, by Green's second identity and Gauss divergence theorem, (1) can be transformed into an integral,

\({A_1} = \iint\limits_\Omega {\mu {J_e}{G_1}{\text{d}}S} - \oint\limits_{\partial \Omega } {g\nabla {G_1} \cdot \vec n{\text{d}}l} = \mu {J_e}\oint\limits_{\partial \Omega } {{{\vec F}_1} \cdot \vec n{\text{d}}l} - \oint\limits_{\partial \Omega } {g\nabla {G_1} \cdot \vec n{\text{d}}l}\)

\({A_2} = - \oint\limits_{\partial \Omega } {g\nabla {G_2} \cdot \vec n{\text{d}}l} \)

(2)

which is only valued on the boundary of the cross section.  G₁ and G₂ are the Green’s functions of the boundary value problem of the two PDEs in (1) and ▽·F₁=G₁ , which can be solved separately. Since  A₁=A₂ on the boundary, the equation only contains unknown function g. The point collection method is adopted in order to construct a linear algebraic equation and finally results in an approximation solution of current distribution. The proposed method only needs discrete on the boundary of the cross section of the conductor, which greatly simplifies the calculation progress.