Return Stroke Model Research Articles

Compensation of the Instrumental Decay in Measured Lightning Electric Field Waveforms

The proper characterization of lightning electric and magnetic fields is very important for a number of reasons, such as the study and design of lightning protection and the validation of return stroke models. Also, an accurate late time response of the sensor used to record electric fields radiated by a lightning return stroke is very important to correctly detect, classify, and locate lightning discharges. Lightning fields are commonly obtained by integrating the output of antennas that produce replicas of the derivatives of the wanted quantities. The integration is carried out using either passive or active analog circuits that introduce a distortion in the form of a decay time constant that may affect the late-time response of the sensor. We derive the transfer function required for the reconstruction of signals integrated using a single-pole integrator commonly used in field and current waveform measurements. Making use of the convolution theorem, we propose an equation that changes a signal that resulted from integration with a given time constant, to the signal that would have been obtained if a different time constant had been used. Finally, we propose an equation to fully compensate a measured signal from an insufficient time constant of an analog integrator.

On the behavior of return stroke current and the remotely detected electric field change waveform

After accumulating a large number of remotely recorded negative return stroke electric field change waveforms, a subtle but persistent kink was found following the main return stroke peak by several microseconds. To understand the corresponding return stroke current properties behind the kink and the general return stroke radiation waveform, we analyze strokes occurring in triggered lightning flashes for which have been measured both the channel base current and simultaneous remote electric radiation field. In this study, the channel base current is assumed to propagate along the return stroke channel in a dispersive and lossy manner. The measured channel base current is band‐pass filtered, and the higher‐frequency component is assumed to attenuate faster than the lower‐frequency component. The radiation electric field is computed for such a current behavior and is then propagated to distant sensors. It is found that such a return stroke model is capable of very closely reproducing the measured electric waveforms at multiple stations for the triggered return strokes, and such a model is considered applicable to the common behavior of the natural return stroke as well. On the basis of the analysis, a number of other observables are derived. The time‐evolving current dispersion and attenuation compare well with previously reported optical observations. The observable speed tends to agree with optical and VHF observations. Line charge density that is removed or deposited by the return stroke is derived, and the implication of the charge density distribution on leader channel decay is discussed.

Dynamics of a lightning corona sheath—A constant field approach using the generalized traveling current source return stroke model

A generalized lightning traveling current source return stroke model has been used to examine the characteristics of the lightning channel corona sheath surrounding a thin channel core. A model of the lightning channel consisting of a charged corona sheath and a narrow, highly conducting central core that conducts the main current flow is assumed. Strong electric field, with a predominant radial direction, has been created during the return stroke between the channel core and the outer channel sheath containing the negative charge. The return stroke process is modeled with the positive charge coming from the channel core discharging the negative leader charge in the corona sheath. The corona sheath model that predicts the charge motion in the sheath is used to derive the expressions of the sheath radius vs. time during the return stroke. According to the corona sheath model proposed earlier by Maslowski and Rakov (2006) and Maslowski et al. (2009), it consists of three zones, zone 1 (surrounding channel core with net positive charge), zone 2 (surrounding zone 1 with negative charge) and zone 3 (outer zone representing the virgin air without charges). We adopted the assumption of a constant electric field inside zone 1 of the corona sheath observed in the experimental research of corona discharges in a coaxial geometry by Cooray (2000). This assumption seems to be more realistic than the assumption of a uniform corona space charge density used previously in the study of Maslowski and Rakov (2006), Marjanovic and Cvetic (2009), and Tausanovic et al. (2010). Applying the Gauss’ law on the infinitesimally small cylindrical section of the channel the expressions for time-dependence of the radii of zones 1 and 2 during the return stroke are derived. The calculations have shown that the overall channel dynamics concerning electrical discharge is roughly 50% slower and the maximum radius of zone 1 is about 33% smaller compared to the corresponding values calculated in the study of Tausanovic et al. (2010).

On the Mechanism of Current Pulse Propagation Along Conical Structures: Application to Tall Towers Struck by Lightning

We discuss in this paper the propagation of lightning current pulses along conical tall structures. Although the dominant mode of an infinitely long conical transmission line is transverse electric and magnetic (TEM), such a structure can also support higher order TE and TM modes which display a gradual cutoff frequency variation with height. Recently, Baba and Rakov's finite-difference time domain numerical analysis revealed that for a perfectly conducting conical structure, while the current pulses suffer no attenuation as they travel from the cone's apex to its base, the attenuation is significant when pulses propagate from the base to the apex. Adopting an analysis method using 1) the COMSOL Multiphysics simulation environment based on the finite element method and 2) the partial equivalent element circuit method, we study the same reduced-scale structure analyzed by Baba and Rakov. The obtained results confirm the conclusions drawn by Baba and Rakov. We also perform simulations for the case of a 100-m tall tower considering different tower-base radii. It is shown that the upward current pulses are affected by a strong attenuation resulting from the field scattering near the discontinuity at the tower base, followed by a weaker attenuation resulting from the propagation along the cone from its base to the apex. A simple way to modify the engineering lightning return stroke models to account for the attenuation of the upward current pulses is suggested. Finally, we report on experiments to study the current pulse propagation along a 1/582 reduced scale model of the Toronto, CN, tower. The obtained experimental data support the numerical simulations.

Modeling of Lightning Strokes Using Two-Peaked Channel-Base Currents

Lightning electromagnetic field is obtained by using “engineering” models of lightning return strokes and new channel-base current functions and the results are presented in this paper. Experimentally measured channel-base currents are approximated not only with functions having two-peaked waveshapes but also with the one-peaked function so as usually used in the literature. These functions are simple to be applied in any “engineering” or electromagnetic model as well. For the three “engineering” models: transmission line model (without the peak current decay), transmission line model with linear decay, and transmission line model with exponential decay with height, the comparison of electric and magnetic field components at different distances from the lightning channel-base is presented in the case of a perfectly conducting ground. Different heights of lightning channels are also considered. These results enable analysis of advantages/shortages of the used return stroke models according to the electromagnetic field features to be achieved, as obtained by measurements.

Electromagnetic Environment in the Vicinity of a Tall Tower Struck by Lightning—A Review—

In this paper, an overview of recent theoretical investigations and experimental characterization of the electromagnetic fields in the vicinity of a tower struck by lightning is presented. The presence of an elevated strike object has been included in two classes of return-stroke models, namely the engineering models and the electromagnetic or Antenna-Theory (AT) models. Extensions of engineering models based on distributed-source and lumped-source representations are described. The adequacy of the representation of the tall structure as an ideal uniform transmission line is discussed. Different approaches and assumptions for the computation of electromagnetic fields are summarized. Finally, recent progress in the characterization of electromagnetic fields radiated by lightning to tall structures is presented.

Extension of lightning corona sheath model during return stroke

A generalized lightning traveling current source return stroke model has been used to examine the characteristics of the lightning channel corona sheath. A model of lightning channel consisting of a charged corona sheath and a narrow, high conducting central core is assumed. The return stroke process is modeled with the positive charge coming from the channel core discharging the negative leader charge in the corona sheath. According to the corona sheath model proposed earlier by Maslowski and Rakov, it consists of three zones, zone 1 (inner zone containing net positive charge), zone 2 (surrounding zone 1 with negative charge and outer zone 3 without charge. We adopted the assumption of a constant electric field inside zone 1 of the corona sheath proved in the experimental research of the corona discharges in a coaxial geometry of Cooray. This assumption seems to be more realistic than the assumption of a uniform corona space charge density used previously in the study of Maslowski and Rakov, Marjanovic and Cvetic, and Tausanovic et al. Based on the measurements of the electric field performed by Miki et al, the conductivity of the channel sheath in zone 1 is calculated.

Radiated Fields From Lightning Strikes to Tall Structures: Effect of Upward-Connecting Leader and Reflections at the Return Stroke Wavefront

An extension of the engineering return-stroke models for lightning strikes to tall structures that takes into account the presence of possible reflections at the return-stroke wavefront and the presence of an upward-connecting leader is presented. Based on the approach proposed by Shostak et al., closed-form, iterative solutions for the current distribution along the channel, and the strike object are derived. Simulation results for the magnetic fields are compared with experimental waveforms associated with lightning strikes to the CN Tower (553 m). It is shown that taking into account the reflections at the return-stroke wavefront results in better agreement with the fine structure of the magnetic-field waveforms. Moreover, the obtained results are in better agreement with experimental observations, reproducing both the early narrow undershoot and the far-field zero crossing. The results also suggest that the typical double-peak response of the radiated fields from tall structures might be due to the combined effect of upward-connecting leaders and reflections at the return-stroke wavefront.

Stability and accuracy of the finite difference time domain (FDTD) method to determine transmission line traveling wave voltages and currents: The lightning pulse

The numerical solution of the transmission line wave equation is examined using the transmission line model of a lightning return stroke (LRS) taken as a case study. The LRS is represented by a transverse electromagnetic wave travelling at a velocity close to the velocity of light along a highly ionized lightning leader channel. The leader channel is modeled as a lossy transmission line. Attempts to model the lightning return stroke by an electric circuit in order to determine the currents and radiated electromagnetic fields is categorized as the distributed transmission line model (DTLM). The DTLM method in turn may be subdivided into two categories: Distributed inductance (L), capacitance (C) and resistance (R) model (DLCRM) and the lumped LCR model (LCRM). In this brief communication, we indicate some pitfalls to watch out for in solving for the transmission line wave equationusing numerical methods, and how these may be avoided. Furthermore, we propose tests that may be applied, including the analytical solution of diffusion waves, to ensure the stability and accuracy of the numerical solution DLCRM simulations. Key words: Finite difference time domain method, numerical solution of transmission line wave equation, lightning return stroke model, lightning radiated electromagnetic fields.

Return-Stroke Model Segmentation and its Application to Lightning-Induced Surges Calculation

This paper proposes an analytical formula for calculating lightning-induced voltages on an infinitely long overhead line generated by a lightning stroke departing from an arbitrary altitude. A calculation method of the lightning-induced voltages to consider various engineering return-stroke models that can account for the variation of both current amplitude and velocity in a lightning channel is proposed. The lightning stroke is divided into a number of segments, and the analytical formula of the lightning-induced voltage is applied to each segment. This paper discusses the influence of parameters of return-stroke models on lightning-induced voltages. The proposed method is applied to a case of lightning strike to a tall stack.

The characteristics and simulation of close leader/return stroke field change waveforms

In order to reproduce the V‐shape structure characteristics of the close dart leader/return stroke field change, we employ two existing models, one for the “source charge” leader model and the other for MTLL return stroke model, both based on the assumption of uniform leader charge distribution along the channel and the charges deposited by the dart leader are completely neutralized by the following return stroke process. The simulated results show that the return stroke electric field is inversely related with the return stroke speed at early times (within few tens of microseconds of the beginning of the return stroke), while at later times (after 100 μs or so) the field is dominated by the deposited charge density component, the close electric field is independent of speed and the ratio of the leader field to the corresponding return stroke field tends to be equal to −1. Therefore, at early times (within few tens of microseconds) there is often some uncertainty regarding whether the charges deposited by the dart leader are completely neutralized by the following return stroke process based on the difference between the return stroke and the leader field on the ground. However, although several other return strokes models (TL, MTLE, TCS, BG and DU) exist, the results shown in the paper are strictly valid only for the used MTLL model.

An Analytical Formulation of the Electromagnetic Field Generated by Lightning Return Strokes

The evaluation of the lightning performance of overhead lines requires, mainly for distribution lines, the calculation of the voltages induced by nearby lightning strokes hitting the ground. Generally, when calculating lightning-induced voltages using the time-domain approach and the TEM assumption, the most time-consuming stage is the calculation of the source terms of the coupling equations, which means, in turn, the electromagnetic field radiated by the lightning channel. Hence, simplified approaches are still desirable in order to save computational time. Analytical solution of the lighting-originated electric- and magnetic field is available in literature for the case of an ideal transmission-line (TL) return-stroke model with channel-base current having a stepwise waveshape. This paper describes an analytical approach that applies to the case of a general waveshape of the channel-base current, based on the piecewise straight-line approximation, in cylindrical coordinates, which is functional when using the TL return-stroke current model and the Agrawal et al. coupling model. This paper reports the exact solution for the case of trapezoidal waveshape of the current at the base of the channel and the analytical solution for the case of piecewise linear waveform.

A Channel-Base Current Function for Lightning Return-Stroke Modeling

A mathematical function for the approximation of various lightning channel-base current waveshapes, which can be included in different lightning return-stroke models is presented in this paper. Its derivative, integral, and Fourier transform are determined analytically. The parameters of this function are also calculated according to the waveshapes of the first positive, the first and subsequent negative stroke channel-base currents as given in the International Standard IEC 62305-1.

Engineering Lightning Return Stroke Models Incorporating Current Reflection From Ground and Finitely Conducting Ground Effects

A return stroke model that incorporates a reflected wave from ground without introducing any current discontinuities at the return stroke front is introduced. The incident current is treated using current generation concepts and the reflected current using current dissipation concepts. It is shown that the effect of the reflected current wave is to cause flattening of close electric field waveforms within about 10 μs. Additionally, it is shown how a return stroke model could be utilized to study the effect of ground conductivity on the return stroke current. The results show that the peak time derivative of current in lightning strokes terminating on poorly conducting ground is significantly lower than in the case of highly conducting ground. The model is also used to predict the spatial variation of return stroke velocity. The results show that the return stroke velocity increases initially, reaches a peak, and then decays with increasing height.

A Macroscopic Model for First Return Stroke of Lightning

A lightning return stroke model for a downward flash is proposed. The model includes underlying physical phenomena governing return stroke evolution, namely, electric field due to charge distributed along the leader and cloud, transient enhancement of series channel conductance at the bridging regime, and the nonlinear variation of channel conductance, which supports the return stroke current evolution. Thermal effects of free burning arc at the stroke wave front and its impact on channel conductance are studied. A first-order arc model for determining the dynamic channel conductance along with a field-dependent conductivity for corona sheath is used in the model. The model predicts consistent current propagation along the channel with regard to current amplitude and return stroke velocity. The model is also capable of predicting the remote electromagnetic fields that are consistent with the experimental observations.

Corona current concept in lightning return-stroke models of engineering type

Corona current concept in lightning return-stroke models of engineering typeA role of radial corona current in a lightning discharge is discussed in the paper. It is shown that the corona current concept previously introduced by Cooray for lightning return stroke models of distributed-current-source (DCS) type, and later, by Maslowski and Rakov for lumped-current-source (LCS) type models enables to show duality between these two types of models. Further, it is demonstrated that the corona current is useful during consideration of dynamics of the lightning-channel corona sheath. As an example of application of presented approach a relaxation model of charge motion in the corona sheath is analysed together with plots which show the rate of expansion and shrinkage of the lightning corona sheath on both microsecond and millisecond time scales.

Influence of a Nonlinear Channel Resistance on Lightning-Induced Voltages on Overhead Lines

In this paper, a lightning return-stroke model proposed by the authors is used for investigating the effect of a nonlinear (current-dependent) channel resistance on lightning-induced voltages on an overhead wire. Lightning-induced voltages are shown to behave at early times as if they were calculated by assuming a lossy channel and at later times as if they were calculated by assuming a lossless channel, the transition between early and latter stages being smoothed by the continuous time decay of the channel resistance. Despite such a transitional behavior, it is argued that similar induced-voltage waveforms can be obtained if a channel resistance with constant value between the upper and lower limits associated to the leader and return-stroke channels is suitably selected. It is also shown that in the cases evaluated in this paper simple engineering return-stroke models are able to lead to lightning-induced voltages in relatively good agreement with those predicted by the considered nonlinear return-stroke model. Finally, lightning-induced voltages calculated using the nonlinear return-stroke model are shown to be in good agreement with experimental data.

Dynamics of a Lightning Channel Corona Sheath Using a Generalized Traveling-Current-Source Return Stroke Model—Theory and Calculations

Abstract-A generalized lightning traveling-current-source return stroke model (GTCS) and the measurements of Miki et al. [J. Geophys. Res., vol. 107, no. D16, pp. ACL2.1-ACL2.11, 2002], are used to calculate the dynamics of a lightning channel corona sheath surrounding a thin channel core during the return stroke stage. The channel corona sheath model that predicts the charge motion in the corona sheath is used to determine the corona sheath dynamics. This model can be viewed as the generalization of the corona sheath model proposed by Maslowski and Rakov [J. Geophys. Res., vol. 111, D14110, pp. 1-16, 2006]. According to this model, the corona sheath surrounding the thin channel core consists of two zones containing charge, zone 1 (inner zone containing net positive charge) and zone 2 (zone containing negative charge surrounding zone 1), respectively, and an outer zone 3 surrounding zone 2 without charge. Theoretical expressions for the corona sheath radii and the velocities of both zones are derived. Using a theoretical expression for the radial electric field in the immediate vicinity of the channel core derived for the GTCS model and the measured electric field waveforms of Miki et al. [J. Geophys. Res., vol. 107, no. D16, pp. ACL2.1-ACL2.11, 2002], the channel discharge function versus time is calculated. Based on this function and the measured channel-base current function, the corona sheath radii of both zones and their velocities versus time in the bottom portion of the channel are calculated. It is shown that the maximum radii of zones 1 and 2 at 2 m above ground are less than 1.5 and 6 cm, respectively. Corresponding maximum radial corona sheath velocities are less than 6 × 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4</sup> m/s. Small values of the maximum radii of zones 1 and 2 can be explained by the small value of the channel line charge density of 6.7 μC/m, due to vicinity of perfect ground. Using measured channel-base current and the calculated channel discharge function, the line charge distribution versus height is calculated. The current reflections from the striking point are considered. For the ground current reflection factor R = 1 (the reflection from the perfectly conducting ground) the peak value of the channel line charge density is 0.75 mC/m at the height of about 12 m above ground and for R = 0 (no current reflections) the peak value is 1.3 mC/m, at about 17 m above ground. The corresponding calculated values of the return stroke velocities in the channel bottom are 1.29 × 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">8</sup> m/s (0.43c) and 1.68 × 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">8</sup> m/s (0.56c), respectively. The corona sheath expansion velocity is about three orders of magnitude less than the calculated lightning return stroke velocity. The result concerning the channel line charge distribution is in the agreement with the measurements of Crawford et al. [J. Geophys. Res., vol. 106, pp. 14909-14917, 2001], whereas the calculated return stroke velocities are in a good agreement with the optical measurements of Willet et al. [J. Geophys. Res., vol. 93, pp. 3867-3878, 1988] and [J. Geophys. Res., vol. 94, pp. 1327513286,1989].

A time-domain formula for the horizontal electric field at the earth surface in the vicinity of lightning

This paper presents a time-domain formula for the calculation of the horizontal electric field in the vicinity of the lightning channel. The formula is based on the incident azimuthal magnetic field, so that it does not depend on the assumptions considered in the calculation of the magnetic field, such as return stroke model or propagation effects. The application of the formula with the magnetic field calculated from perfectly conducting earth provided results that agreed very well with results from Sommerfeld integrals, even in the vicinity of the lightning channel, where the Cooray-Rubinstein formula does not hold. The deviations of the results of the formula with respect to the Sommerfeld integrals are on the order of a few percent, which is the same order of magnitude as the deviations involved in the numerical evaluation of Sommerfeld integrals.

HOW LIGHTNING TORTUOSITY AFFECTS THE ELECTROMAGNETIC FIELDS BY AUGMENTING THEIR EFFECTIVE DISTANCE

A novel approach for developing the electromagnetic flelds from a lightning return stroke which follows a tortuous path will be presented. The proposed model is unique in that it recognizes that the symmetrical tortuosity of lightning directly impacts the observable distance r, which in turn, alters the resulting electromagnetic flelds. In the literature, lightning return stroke models typically employ the assumption that the cloud-to-ground path is straight. Although this assumption yields fairly consistent results across an array of varying approaches, it does not account for lightning's natural physical appearance. Furthermore, straight-line models only account for the cloud-to-ground discharges and do not address branching and/or cloud- to-cloud discharges which are far more common. In reality, the \steps which make up the lightning channel's initial descent are staggered or tortuous with respect to each other. Given this fact, the upward traveling current wavefront which follows this prescribed path will exhibit the same characteristics. In doing so, each current segment,