Relativistic Doppler effect in an extending transmission line: Application to lightning

Abstract

[1] We present in this paper a thorough analysis of current wave propagation with arbitrary speed along an extending transmission line. We derive rigorous analytical equations in the time and frequency domains expressing the reflections of the current wave occurring at the extending end of the line. The derived equations reveal that it is not possible to represent current reflections occurring at the extending end of a transmission line using a constant, frequency-independent reflection coefficient, as previously done in the literature. The reflected wave from the extending end of the line is shown to be affected by the Doppler frequency shift. In other words, the reflected wave from an extending transmission line suffers distortion, the amount of which depends on the incident wave form, its frequency content, and the speed of the extending end of the line. The derived expression is in agreement with the relativistic Doppler effect and is consistent with the Lorentz transformation. Finally, engineering models for return strokes are generalized and closed-form analytical expressions are derived for the spatial-temporal distribution of the current along the channel accounting for reflections at ground and at the return stroke wave front taking into account the Doppler effect.

Lightning strikes to tall objects: A study of wave interactions at the return‐stroke front using a nonlinear transmission line model

“…In general, engineering return-stroke models that include the possibility of current reflections at the upward moving return-stroke front as well as the presence of an upward connecting leader [e.g., Shostak et al, 2000;Mosaddeghi et al, 2011] are believed to reproduce the fine structure of current waveforms and remote electromagnetic fields associated with lightning strikes to tall towers better than models that neglect such factors [e.g., Pavanello et al, 2007aPavanello et al, , 2007b, even though a constant reflection coefficient at the upward moving channel front is usually assumed for that purpose (a time-variant current reflection coefficient should be expected at the end of an extending transmission line as discussed by Shoory et al [2011Shoory et al [ , 2012) and current attenuation and distortion due to channel losses are ultimately disregarded. More recently, an electromagnetic return-stroke model considering corona and nonlinear losses was used to investigate the interaction of lightning with an elevated strike object [Raysaha et al, 2012].…”

Correction to “Relativistic Doppler effect in an extending transmission line: Application to lightning”

“…[9] Note that, in the original paper [Shoory et al, 2011], the above equation (numbered as (31)) was erroneously written in terms of the channel-base current i(0, t À z/v), instead of the undisturbed current i o (t À z/v).…”

“…The derived expressions were found to be in agreement with the relativistic Doppler effect and are consistent with the Lorentz transformation. In the last part of their paper, Shoory et al [2011] generalized the engineering return stroke models to account for reflections at the ground and at the return stroke wavefront taking into account the Doppler effect. Closed-form analytical expressions were derived for the spatial-temporal distribution of the current along the channel.…”

“…In doing so, the return stroke channel is assumed to be a transmission line whose bottom end is fixed at ground level and features a constant, frequency-independent reflection coefficient, r g , for downward current waves and its upper end is extending with speed v, featuring reflections characterized by the Doppler effect for upward propagating current waves formulated in section 2 of Shoory et al [2011]. For the case where z′ > z [see Shoory et al, 2011, Figure 2a], the elemental current seen by an observer at z due to a current source at z′ can be written as…”

“…P(z) and v* for five of the engineering return stroke models to be discussed in what follows are shown in Table 1 of Shoory et al [2011].…”