A formalism for the computation of lightning transfer functions by the method of moments, which involves shielding structures that may consist of layered, anisotropically conducting composite materials, is presented in this contribution. The composite materials, being of a type that is widely used in space- and aircraft design, are electrically characterized by an equivalent conductivity. As basis for the quantitative analysis the method of moments is used where shielding surfaces can be treated by a thin layer technique which utilizes analytical solutions inside the layer. Also the effect of an extended lightning channel can be taken into account. The method is applied to geometries that resemble an actual airplane fuselage.

Despite the constantly ongoing progress in the development and application of
numerical methods in electromagnetics it turns out that the computation of
lightning-related effects in the framework of Electromagnetic Compatibility
(EMC) still constitutes a highly challenging task. This is due to a number of
difficulties that can be characterized as follows

In this contribution it is outlined how to numerically calculate LEMP
transfer functions by the method of moments (MoM), taking into account the
main difficulties mentioned above. To this end, in Sect. 2 it is outlined,
based on early work

There are several engineering approaches for the modeling of the electric
current of a lightning channel

In the TL type model

Illustration of the upward moving current front in the lightning channel.

For the current

Lightning current pulse according to Heidler.

Composite materials are widely used in aircraft design for weight reduction and improvement of mechanical strength. In this publication a type of carbon fiber composite is considered which consists of several layers of carbon fibers with different orientations, enclosed in resin.

Instead of modeling single fibers one may assume an anisotropic conductivity
for each layer

A further simplification can be made by replacing the various anisotropically
conducting layers by a single layer with the same overall thickness and an
equivalent isotropic conductivity

Local coordinate system for one layer of carbon fibers, where the

There are several methods to numerically model thin layers. The surface
impedance boundary condition method

Two MoM regions that are separated by a thin finitely conducting layer. Its electromagnetic properties can be described by an analytical formulation.

A priori, the lightning current is formulated in time domain while the MoM and the layer technique are formulated in frequency domain. In this section it is explained how to relate both formulations to calculate the impulse response of a lightning current.

The spectrum

The calculated spectrum consists of values that refer to discrete
frequencies. This leads to a periodic extension of the pulse response in time
domain. The time

Another important point is the maximum frequency that has to be considered to reproduce the steep rise of the pulse. In case of a lightning pulse with a rise time of a few microseconds a maximum frequency in the range of a few MHz is sufficient.

Finally, it should be mentioned that this is a linear formulation, hence non-linear effects, which could result from matter interaction at very high field magnitudes, are not taken into account.

In this section the proposed formalism is illustrated by means of several
examples. In all cases the pulse introduced in Sect.

The first configuration to be considered is a single lightning channel
without a neighboring structure. Then a transmission line structure, which is
loaded by

Dimensions and positions of the transmission line structure and its surrounding cylinder, representing a simple model of an aircraft fuselage.

The considered simulation models where the cylindrical cavity is
positioned at a height of 100 m above perfectly conducting ground. Four
cases are considered: closed finitely conducting cylinder 10 m distant from
the lightning channel

Electric field of the channel at different distances from the
channel at

Magnetic field of the channel at different distances from the
channel at

Electric field of the channel at different distances from the
channel at

Electric field at the center of the transmission line
without

As a prerequisite, in this subsection the electric and magnetic fields of the lightning
channel without a neighboring structure are investigated and compared to the
semi-analytical formulas derived in

The corresponding curves for the electric and magnetic fields are shown in
Figs.

In Fig.

The electric field at the center of the transmission line structure, which
corresponds to the point (25, 0, 100) m, with and without the finitely
conducting cylinder as shield is illustrated in Fig.

Magnetic field at the center of the transmission line, 10 m distant from the lightning channel, with and without conducting cylinder.

Frequency spectrum of the system response for a lightning channel close to the structure without cylinder, with finitely conducting cylinder, and with PEC cylinder with apertures.

Voltage at the termination resistor of the transmission line with
and without conducting cylinder

The magnetic fields for these two cases are plotted in Fig.

The related transfer function

Frequency spectrum of the system response for the cases of a closed conductive cylinder and a PEC cylinder with apertures if a direct lightning strike is applied.

Voltage at the termination resistor of the wire loop inside the conductive cylinder and the PEC cylinder with apertures in case of a direct lightning strike.

Finally, the time-domain voltage at the terminating resistor is shown in
Fig.

In this subsection cases are considered where the lightning channel is
directly attached to the structure, compare Fig.

In Fig.

The time domain response of the voltages are shown in
Fig.

A formalism for the calculation of LEMP transfer functions by means of the MoM has been proposed. Composite materials are modeled by an equivalent conductivity which is applicable for the frequency range of the lightning spectrum. In the examples considered it turned out that diffusion coupling has a larger influence on the transfer function than the aperture coupling. Therefore, the conductivity and thickness of the shell of the fuselage are very important parameters and it clearly is not sufficient to approximate composite materials by PEC material for lightning analysis.

This research was supported in part by the German Research Foundation (Deutsche Forschungsgemeinschaft, DFG). Edited by: F. Sabath Reviewed by: R. Bunger and two anonymous referees