In the TL type model a current wave is traveling upwards with propagation speed vf, starting at the channel base, as depicted in Fig. . At positions above z′=vf⋅t the current is zero. The current at any position z′ along the channel and at time t is given by i(z′,t)=i0(t-z′vf)if z′≤vf⋅t0if z′>vf⋅t, where i0(t) is the current pulse at the channel base and vf is the propagation speed, which typically is chosen to be one third of the vacuum speed of light vf=13⋅c.

For the current i0(t) the current pulse according to , which can be found in the standard, is used. The slope of this curve is continuous, which has proven to be consistent with the results of lightning measurements. The entire pulse can be divided into three components that are multiplied by each other and determine the amplitude, the smooth rising edge, and the exponential decay, respectively. The corresponding equation is given by i0(t)=imaxη⋅tTrisen1+tTrisen⋅exp⁡-tThold, where imax is the maximum value of the current, η is the amplitude correction factor, which ensures that the pulse reaches the value imax, Trise is the rise time, and Thold the decay time of the pulse. For the following investigations the parameters imax=100 kA, η=0.986, Trise=1.82 µs, and Thold=285 µs are used. This choice refers to a negative first stroke of threat level “severe”. A pulse with these parameter values is shown in Fig. .

Instead of modeling single fibers one may assume an anisotropic conductivity for each layer , where the direction of the fibers in the layer corresponds to the direction of the highest conductivity in the conductivity tensor.

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 σeq. An analytical investigation of the shielding effectiveness of a plane shield with several anisotropically conducting layers has shown that this simplification is valid up to several tens of MHz . Since lightning is a low frequency phenomenon, the equivalent conductivity can be used to model the shielding properties of the composite material with respect to lightning effects. As a first step to calculate σeq the diagonal entries of the average conductivity tensor are calculated by σxxσyy=1d∑i=1Lσξσψσψσξ⋅cos⁡2(αi)sin⁡2(αi)⋅di, where L is the number of layers. Here it is assumed that each layer of fibers has the conductivity σξ in fiber direction and σψ in cross-fiber direction. The orientation of the fibers is specified by the angle αi. The thickness of layer i is denoted by di and the total thickness of the multilayer material is denoted by d. Then the equivalent conductivity is chosen to be the smaller of the two conductivity values, σeq=min(σxx,σyy), to give a valid value for a worst case approximation. In the following, the conductivities are chosen to be σξ=10 kS m-1 and σξ=100 S m-1 and each layer has thickness di=d/L. For a symmetric layer pattern, e.g., a multiple of four layers with a relative rotation angle of 45∘ between the fibers of adjacent layers, the equivalent isotropic conductivity is given by σeq=5050 S m-1.

There are several methods to numerically model thin layers. The surface impedance boundary condition method , for example, is applicable to scattering problems, but it is not suitable for the calculation of shielding effectiveness, since the coupling between inside and outside region of the shielding geometry has to accurately be taken into account. It might also come to mind to apply a Green's function of layered media , but this typically requires some kind of canonical symmetry of the geometry considered. Therefore it is not immediate to apply this method to arbitrarily shaped three dimensional structures. The thin layer technique that is applied here to efficiently model thin layers of finite size in conjunction with the MoM is based on an analytically formulated coupling matrix. In this case, the layer has to fulfill the requirement to be thin compared to the overall dimension of the body to be modeled and it has to provide a sufficiently high conductivity. If the conductivity is large enough then the wave propagation inside the layer is perpendicular to its surface and can be described by an analytical solution in the form of a coupling matrix which also correctly incorporates all effects related to the wave propagation in a lossy medium. As a consequence, two regions that are separated by a two-dimensional layer, compare Fig. , can be treated by the MoM, where the coupling through the layer is taken into account by a coupling matrix which relates the tangential fields at both sides of the layer to each other. This hybrid-technique, which combines the MoM with an analytical solution, has proven to be stable down to frequencies in the kHz range .

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 . With these formulas the fields at any position on the ground at z=0 can be calculated as a function of the horizontal distance to the channel.

The corresponding curves for the electric and magnetic fields are shown in Figs.  and , respectively. The numerical results obtained from the MoM formalism are in excellent agreement with the semi-analytical results, as exemplified by three observation points at distances 25, 50 and 100 m. For these distances the effect of the finite length of the lightning channel in the used model is negligible.

In Fig.  the same curves for the electric field as in Fig.  are show up to a time of t=1 ms. It can be observed that even for this longer time interval the results of the MoM formalism are still in very good agreement with the results of the semi-analytical method.

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. . In both cases the y component of the field is zero. As expected, the amplitudes of the electric field in the presence of the shield are much smaller if compared to the situation without shield.

The magnetic fields for these two cases are plotted in Fig. . The rise time of the magnetic field inside the finitely conducting cylinder is much larger if compared to the rise time of the magnetic field without the cylinder. This is due to the low pass behavior of the shield with respect to the magnetic field, i.e., the high frequency components of the field are attenuated and therefore the rise time is increased.

The related transfer function G(ω), as defined by the ratio between the voltage U(ω) at the termination resistance and an impressed current of constant amplitude of 1A per frequency, is shown in Fig. . This transfer function includes the following effects: the creation of the fields of the lightning channel, the coupling into the cylindrical cavity by diffusion or aperture coupling, field distribution inside the cavity, and coupling into the transmission line structure. At low frequencies the curves for the case of the transmission line alone and the transmission line inside the finitely conducting cylinder are on top of each other. This shows that in this frequency region the presence of the conducting cylinder is hardly relevant. The magnitude of the transfer function for the case of the PEC cylinder with apertures is lower compared to the other two cases because the magnetic field may only couple through the apertures into the fuselage. This effect is small for the considered wavelengths which are much larger than the aperture dimensions.

Finally, the time-domain voltage at the terminating resistor is shown in Fig.  as a response to the current pulse according to Fig. . A delay time of 1 µs can be observed, which corresponds to the time the pulse needs to travel from the ground to the height of 100 m where the structure is located. The maximum voltage for the cases of no cylinder, finitely conducting cylinder, and PEC cylinder with apertures are 17 kV, 800 V, and 1.5 V, respectively. The main contribution to the voltage is due to the magnetic field. Due to the low pass characteristic of the finitely conducting cylinder the voltage has a decreased rise time, similar to the magnetic field itself.

In this subsection cases are considered where the lightning channel is directly attached to the structure, compare Fig. b and d, such that the lightning current flows directly on the surface of the cylindrical structure.

In Fig.  the spectrum of the transfer functions for both the finitely conducting cylinder and the PEC cylinder with apertures is shown. Two maximum values can be identified in the considered frequency range. The first maximum occurs at a frequency of 372 kHz. This frequency is too low for being a resonance frequency of the cylinder. Analysis shows that this frequency is related to the wire that connects the cylinder to the ground. More precisely, the frequency turns out to be the antiresonance frequency of the wire , i.e., the imaginary part of the impedance of the wire as seen from the current source is zero and the real part of the impedance exhibits a maximum. The second maximum of the transfer function occurs at the frequency 1.6 MHz, where the length of the wire from the cylinder to the ground equals one half of a wavelength.

The time domain response of the voltages are shown in Fig. . Oscillations are clearly visible, where the time period of the oscillation corresponds to the frequency where the first maximum of the transfer function occurs. For the case of the finitely conducting cylinder the voltage is considerably higher if compared to the case of the PEC cylinder. Therefore, in this example, the diffusion coupling is larger if compared to the aperture coupling, at least in the considered frequency range which is relevant for lightning analysis.