Return Stroke Model Research Articles

Reply to comment by Rajeev Thottappillil and Vladimir A. Rakov on “Radio frequency radiation beam pattern of return strokes: A revisit to theoretical analysis”

Reply to comment by Rajeev Thottappillil and Vladimir A. Rakov on “Radio frequency radiation beam pattern of return strokes: A revisit to theoretical analysis”

On the Interpretation of Ground Reflections Observed in Small-Scale Experiments Simulating Lightning Strikes to Towers

Using the finite-difference time-domain (FDTD) method for solving Maxwell's equations, we have simulated small-scale experiments intended to study the interaction of lightning with towers. In these experiments, employing the time-domain reflectometry (TDR), the tower was represented by a conical conductor placed between two horizontal conducting planes, and a relatively high grounding impedance (about 60 /spl Omega/, constant or decreasing with time) of the bottom plane was inferred, based on the assumption that a conical conductor could support propagation of unattenuated waves in either direction. We have shown, using the FDTD simulations, that a current pulse suffers no attenuation when it propagates downward from the apex of the conical conductor to its base, but it attenuates significantly when it propagates upward from the base of the conical conductor to its apex. We show that the current reflection coefficient at the base of the conical conductor is close to 1, so that the equivalent grounding impedance of the conducting plane is close to zero. Our analysis suggests that the relatively high grounding impedance of conducting plane inferred from the small-scale experiments is an engineering approximation to the neglected attenuation of upward propagating waves. When the dependence of cone's waveguiding properties on the direction of propagation is taken into account, the results of small-scale experiments simulating lightning strikes to towers can be interpreted without invoking the fictitious grounding impedance of conducting plane. Representation of a vertical strike object by a uniform transmission line terminated in a fictitious grounding impedance appears to be justified in computing lightning-generated magnetic fields and relatively distant electric fields, but may be inadequate for calculating electric fields in the immediate vicinity of the object. This study was motivated by the growing interest in extending lightning return stroke models to include a tall strike object and calculating associated electric and magnetic fields.

On the use of lumped sources in lightning return stroke models

We consider the use of lumped voltage and current sources in engineering lightning return stroke models with emphasis on those including a tall strike object. If the model is to be used for computing remote electric and magnetic fields, we suggest a representation of the lightning channel as a transmission line energized by a lumped voltage source, with the voltage magnitude being expressed in terms of the lightning short‐circuit current and equivalent impedance of the lightning channel. Such a representation assures appropriate boundary conditions (reflection and transmission coefficients) at the channel attachment point and is equivalent to a distributed‐shunt‐current‐source representation of the lightning channel. This is in contrast with the use of series ideal current source which presents infinitely large impedance to current waves reflected from the ground and/or from discontinuities in the lightning channel, such as the moving return stroke front or branches, and therefore is inadequate when such reflections are involved. If the model is to be used only for injecting lightning current into a grounded object or system, a Norton equivalent circuit (an ideal current source in parallel with the equivalent impedance of the lightning channel) is sufficient to represent the lightning discharge.

Reproduction of Features of Electromagnetic Field Waveforms Associated with Lightning Return Stroke

A lightning return stroke model is indispensable in calculating electromagnetic field associated with a return stroke. A lot of return stroke models have been proposed, however, no model has ever succeeded in reproducing all the features of typical electromagnetic field waveforms at various distances. In this paper, two return stroke models are proposed. One is a Transmission-Line type model, and the other is modification of Diendorfer-Uman model. These models successfully reproduce all the features of typical electromagnetic field observed on ground.

Radio frequency radiation beam pattern of lightning return strokes: A revisit to theoretical analysis

Return stroke current pulses can propagate at speeds approaching the speed of light c. Such a fast‐moving pulse is expected to radiate differently than conventional dipole emitters. In this study, we revisit the theoretical analysis for the high‐speed effect on the radiation beam pattern. Instead of starting with specific return stroke models, as has been done before by other investigators, we start the analysis with a general moving current pulse. Through a simple differential transformation between the retarded time and stationary time/space, the so‐called F factor (1 − v cos θ/c)−1 can be readily obtained. This factor is found to be fundamental and is explicitly associated with the radiation beam pattern but is not limited only to the lossless transmission line (TL) return stroke model. It is demonstrated that different beam pattern factors could be derived from this fundamental factor under different return stroke model assumptions.

Modification of Engineering Return-Stroke Models to Include Dispersion Effects

Recently, Baba and Ishii (see ibid., vol.44, p.476-7, Aug. 2002) have presented a method to modify the transmission-line (TL) type model to incorporate current distortion. In this brief, we generalize their method, as well as an exponential decay return stroke speed model, to simple engineering return-stroke models, and investigate the influence of dispersion effects on associated electromagnetic fields. Results show that the proposed modification to the models eliminates the inherent discontinuity in the traveling current source (TCS) model and the Bruce-Golde (BG) model at the upward-moving front. In addition, it turns out that an increase in the current risetime with height does not have significant effect on magnetic hump, electric ramp, and distant zero crossing, while a rapid decrease in return stroke speed with height could affect these features.

Electromagnetic field radiated by lightning to tall towers: Treatment of the discontinuity at the return stroke wave front

Recently, engineering return stroke models were extended to take into account the presence of an elevated strike object, which was modeled as an ideal uniform transmission line. We show in this paper that the current distribution associated with these extended models exhibits a discontinuity at the return stroke wave front, which cannot be considered to be physically plausible. This discontinuity arises from the fact that the current injected into the tower from its top is reflected back and forth at its ends and portions of it are transmitted into the channel; these transmitted pulses, which are supposed to travel at the speed of light, catch up with the return stroke wave front traveling at a lower speed, and no current is allowed to flow in the leader region above the front. This discontinuity needs to be carefully treated when calculating the radiated electromagnetic field through an additional term in the electromagnetic field equations, the so‐called “turn‐on” term. We additionally derive a general analytical formula describing the turn‐on term associated with this discontinuity for various engineering models. We present simulation results illustrating the effect of the turn‐on term on the radiated electric and magnetic fields for an elevated strike object of height h = 168 m, corresponding to the Peissenberg tower in Germany. Finally, possible modifications to engineering models are suggested in order to eliminate this discontinuity.

On the estimation of lightning peak currents from measured fields using lightning location systems

Although, due to the high variability of key parameters such as the return-stroke speed, it is impossible to determine the lightning current accurately from the remotely measured electric or magnetic field for a given event, we show in this paper that, for an assumed return-stroke model, the statistical estimation (e.g. in terms of mean values and standard deviations) is possible. We show additionally that for the transmission line (TL) model, the equation permitting to infer the mean value of the return-stroke current from the mean value of electric or magnetic field and the mean value of speed has the same functional form as the well-known TL current—far field relationship. This result gives to some extent a theoretical justification to the use of lightning location systems to infer parameters of lightning current statistical distributions from measured fields alone.

Luminosity characteristics of multiple dart leader/return stroke sequences measured with a high‐speed digital image system

Using a newly‐developed high‐speed digital optical system, we measured the luminosity profiles of four leader/return stroke sequences within a flash occurred on a high tower. By analyzing those profiles, we obtained two simple expressions to characterize the light pulse decay features of subsequent return strokes along the channel bottom 210 m. These expressions can be extended to return stroke pulse currents and be easily included in return stroke modeling. In addition, we found that the peak light intensity of a leader and its following return stroke at various heights correlates positively in the channel bottom 210 m.

Test of the transmission line model and the traveling current source model with triggered lightning return strokes at very close range

We test the two simplest and most conceptually different return stroke models, the transmission line model (TLM) and the traveling current source model (TCSM), by comparing the first microsecond of model‐predicted electric and magnetic field wave forms and field derivative wave forms at 15 m and 30 m with the corresponding measured wave forms from triggered lightning return strokes. In the TLM the return stroke process is modeled as a current wave injected at the base of the lightning channel and propagating upward along the channel with neither attenuation nor dispersion and at an assumed constant speed. In the TCSM the return stroke process is modeled as a current source traveling upward at an assumed constant speed and injecting a current wave into the channel, which then propagates downward at the speed of light and is absorbed at ground without reflection. The electric and magnetic fields were calculated from Maxwell's equations given the measured current or current derivative at the channel base, an assumed return stroke speed, and the temporal and spatial distribution of the channel current specified by the return stroke model. Electric and magnetic fields and their derivatives were measured 15 m and 30 m from rocket‐triggered lightning during the summer of 2001 at the International Center for Lightning Research and Testing at Camp Blanding, Florida. We present data from a five‐stroke flash, S0105, and compare the measured fields and field derivatives with the model‐predicted ones for three assumed lightning return stroke speeds, v = 1 × 108 m/s, v = 2 × 108 m/s, and v = 2.99 × 108 m/s (essentially the speed of light). The results presented show that the TLM works reasonably well in predicting the measured electric and magnetic fields (field derivatives) at 15 m and 30 m if return stroke speeds during the first microsecond are chosen to be between 1 × 108 m/s and 2 × 108 m/s (near 2 × 108 m/s). In general, the TLM works better in predicting the measured field derivatives than in predicting the measured fields. The TCSM does not adequately predict either the measured electric fields or the measured electric and magnetic field derivatives at 15 and 30 m during the first microsecond or so.

Correction to "on the concepts used in return stroke models applied in engineering practice"

For original paper see V. Cooray, IEEE Trans. Electromagn. Compat., vol.44, p.101-108, (2003). This paper gives correction to several equations in the aforementioned paper.

Comments on "On the concepts used in return stroke models applied in engineering practice

For original paper see V. Cooray, ibid., vol.45, no.1, p.101-8, (2003). Cooray's paper is an important contribution to the lightning return stroke modeling literature. It shows that any engineering model implying a lumped current source at the lightning channel base - any transmission-line (TL) type model - can be formulated in terms of sources distributed along the channel and progressively activated by the upward-moving return stroke front. Comments are made on Cooray's division of the total charge density into its deposited and transferred components. The charge-density formulation in engineering return-stroke models by other ways is discussed.

Author's reply

For original paper see ibid., vol. 45, no. 1, p. 101-8, (2003). For comment see V.A. Rakov et al., ibid., p. 567. The paper further discusses current generation in lightning return stroke modeling. Rakov et al. have commented on the author's division of the total charge density into its "deposited" and "transferred" components. Any division of the total charge in a given lightning channel element at a given time into deposited and transferred components is model dependent and one cannot define uniquely the deposited charge at a given time. Even though the physical description of the composition of the charge on a given channel element at a given time is different in the current propagation and current generation points of view, both descriptions agree on the total charge in a given channel element at a given time, the total charge deposited on the channel element as time goes to infinity and the total charge transferred across the channel element as time goes to infinity.

On the concepts used in return stroke models applied in engineering practice

In this paper, we investigate the similarities between two basic concepts used in engineering return stroke models. In one type, the current propagation (CP) models, the return stroke channel merely acts as a medium for the current propagation with the driving source being at the ground. In the other type, the current generation (CG) models, the current sources (corona current) are assumed to be distributed along the return stroke channel. Our analysis shows that popular CP models, such as transmission line (TL), the modified transmission-line exponential (MTLE) model , and the modified transmission-line linear (MTLL) can be converted to CG models leaving the temporal and spatial variation of the return stroke current the same. In this alternative representation, the equivalent corona currents of the TL, MTLE, and MTLL models are bipolar, indicating initial deposition and subsequent removal of positive charge from the channel. This knowledge is applied to construct a simple CG model which generates electromagnetic fields identical to those obtained with the TL model at short times, but free from the difficulties associated with the latter at longer times.

Characteristics of electromagnetic return-stroke models

Impedance and associated electromagnetic (EM) fields of an EM return-stroke model are investigated with the help of NEC-2. The impedance of a cylindrical vertical conductor increases with time. The impedance of a vertical conductor having additional distributed inductance is initially 700 to 1000 /spl Omega/ for the current propagation velocity ranging from 170 to 130 m//spl mu/s. This is in good agreement with the apparent impedance of a lightning channel, estimated from lightning-current waveforms measured at the top and bottom of tall towers. A model considering a horizontally bent channel in a thundercloud succeeds in reproducing all of the established features of measured field waveforms except the hump of close magnetic field, while a vertically straight conductor fails to reproduce two of the features.

Analytical formulation of lightning-induced voltages on multiconductor overhead lines above lossy ground

This paper presents analytical expressions for calculation of lightning-induced voltages (LIV) on multiconductor overhead lines above lossy ground. The transmission-line return-stroke model is used and the propagation effect on the electromagnetic field is approximated by the surface impedance method. Assuming a linear return-stroke current, the standard numerical integrations along the lightning channel and the overhead line are avoided, and the result is a fast and stable formulation that contains only a single convolution integral. The described method applies for distances between 100 m and 10 km and is valid the first few microseconds where the maximum LIV usually occurs. A simplified approach to how to include overhead line losses is also outlined. A model is proposed that can be included in the electromagnetic transients program ATP-EMTP for calculation of LIV in larger and practical systems.

Effect of vertically extended strike object on the distribution of current along the lightning channel

On the basis of a distributed‐source representation of the lightning channel, the mathematical formulations of the so‐called engineering lightning return stroke models are generalized to take into account the presence of a vertically extended strike object. The strike object is modeled as a lossless uniform transmission line, and the reflection coefficients are all assumed to be constant. The distribution of current along the lightning channel for each model is expressed in terms of the “undisturbed” current, object height, and current reflection coefficients at the top and the bottom of the object. The undisturbed current is defined as the current that would flow in the channel if the current reflection coefficients at the extremities of the strike object were equal to zero, that is, the characteristic impedances of the lightning channel and the strike object were equal to each other and equal to the grounding impedance of the strike object. The distributed‐source representation of the lightning channel adopted in this study allows for a more general and straightforward formulations of the generalized return‐stroke models than the traditional representations implying a lumped current source at the bottom of the channel, including a self‐consistent treatment of the impedance discontinuity at the tower top.

Lightning return-stroke model incorporating current distortion

A lightning return-stroke model of transmission-line (TL) type, which incorporates both the attenuation and distortion of the waveform of the propagating current, is proposed. With the proposed model, the influence of the distortion of the propagating current on associated electromagnetic field waveforms is investigated and found to be small.

Correlated time derivatives of current, electric field intensity, and magnetic flux density for triggered lightning at 15 m

We present measured current and its time derivative correlated with the corresponding electric field intensity and magnetic flux density and their time derivatives measured at 15 m for two lightning return strokes triggered in 1999 at Camp Blanding, Florida. Lightning was triggered to a vertical 2‐m rod mounted at the center of a 70 × 70 m buried metallic grid. The rocket‐launching system was located underground at the center of the grid. The experiment was designed to minimize any influence of either the strike object or a finite‐conducting Earth (ground surface arcing and propagation effects) on the fields and field derivatives. The measured current derivative waveform and the return stroke portion of the magnetic flux density derivative and electric field intensity derivative waveforms associated with the two strokes are observed to be essentially unipolar pulses that have similar waveshapes for the first 150 ns or so, including the initial rising portion, the peak, and about 50 ns after the peak. The current and magnetic field derivative waveshapes are very similar for their total duration, and both decay to near zero about 200 ns after the peak derivative is reached. The electric field derivative decays more slowly than the current derivative after about 150 ns, taking about 500 ns to decay to near zero. The transmission‐line model, the simplest available and most used return stroke model, is employed to calculate the return stroke field derivatives, given the measured current derivative as a model input, for return stroke speeds of 1 × 108 m s−1, 2 × 108 m s−1, and 3 × 108 m s−1 (the speed of light). A reasonable match between calculated and measured field derivative waveshapes is achieved for both strokes for a return stroke speed between 2 × 108 m s−1 and 3 × 108 m s−1. Although the measured field and current derivatives have similar waveshapes for about 150 ns, which might appear to be consistent with the hypothesis that the radiation field component of the total field derivative is dominant for that time, transmission line model calculations indicate that this is not the case. Further, electric field derivatives measured simultaneously at 15 m and at 30 m for many strokes are observed to have similar waveshapes, which also might appear to be consistent with the hypothesis that the derivatives are dominated by the radiation field component; however, according to transmission line model calculations, while the calculated total field derivative waveshapes are similar at 15 and 30 m, all field components significantly contribute to the waveforms at both distances, and the mix of field components at 15 m and at 30 m is quite different.

Comment on “Return stroke transmission line model for stroke speed near and equal that of light’ by R. Thottappillil, J. Schoene, and M. A. Uman

[1] Electric fields on the ground 500 m, 5 km, and 50 km from the lightning channel base computed using the transmission line (TL) and antenna theory (AT) return-stroke models for a typical subsequent stroke current at the channel base and a return-stroke speed equal to the speed of light (v = c) are compared. The AT model gives a total electric field peak that is about 10% lower than the field peak predicted by the TL model. Optical measurements of the return-stroke speed within the bottom 100 m of the lightning channel are reviewed and found to be not supporting the assumption of v = c.