Traveling Current Source Research Articles

Review and Extension of the TCS Model to Consider the Current Reflections at Ground and at the Upper End of the Lightning Channel

This paper gives the overview and review of the traveling current source (TCS) model, which describes the electrical processes during the lightning return stroke. It is shown that the main characteristics of the measured electric and magnetic fields can be reproduced with the TCS model. These are the of the electric and magnetic field for near, intermediate and far distances, the ramp (up to the maximum) of the near electric field, the hump of the near magnetic field after the initial peak, and the zero crossing of the far distant electric and magnetic field. The TCS model is based on the assumption that the lightning current is injected at the top of the increasing return stroke channel represented by a transmission line. The original model is presented and then extended to take into account the current reflections occurring at ground and at the upper end of the return stroke channel. To this end, the ground reflection factor ρ and the top reflection factor R are introduced. Due to the increasing return stroke channel, the top reflection factor is a function of the return stroke velocity. The total current is composed of the source current according to the original TCS model and the reflected currents. An example shows that the ground reflection causes significant variation in the peak values of the channel-base current and of the maximum current derivative. A major effect was also found for the initial peak of the electric and magnetic fields calculated at 3 km distance.

Vertical electric field inside the lightning channel and the channel-core conductivity during discharge – Comparison of different return stroke models

The longitudinal electric field along the axis and the channel core conductivity of a straight, vertical lightning channel above ground has been calculated by using different return stroke models. The lightning channel is modeled by a negatively charged corona sheath that stretches around a thin, very conductive central core. It is commonly held that the majority of the leader charge is located within the corona sheath radius, which is of the order of meters, while the highly conductive channel core, with a diameter assessed at around 1cm, in effect transports the whole of the axial current. An inhomogeneous channel line charge density generates as strong radial as well as vertical electric field inside the corona sheath. Transmission-line-type models and the generalized traveling current source model, representing the “engineering” return stroke models, are used for the calculation of the vertical field and the core conductivity in the channel. For the purpose of the present study only the influence of the charge in the corona sheath on the vertical electric field are taken into account while all other effects are neglected. For comparison purposes, the same channel-base current for all models is assumed. First, we calculated the vertical electric field along the axis on the channel. Knowing the axial current density profile along the channel determined by the particular return stroke model and the assumed channel core diameter, the core conductivity is calculated using simple scalar relationship. The conductivity is compared between the models and with the values found in the existing literature. It is concluded that all considered models give the maximum value of the core conductivity more or less in accordance with the predictions in the literature (of the order of 104S/m). Some discrepancies (negative conductivity) are observed for two transmission-line-type models at the very bottom of the channel. They can be explained by the great amount of injected positive charge in zone 1 of the channel sheath and by the presence of the image charge, small changes in input parameters could diminished or avoid it. Due to the big charge accumulation near the ground the generalized traveling current source model gives greater discrepancies regarding negative conductivity at 25–35m above ground. It is concluded that the removing of these discrepancies requires a more complex approach, the inclusion of the new physical mechanisms during the discharge (for example the magnetic field generated by the core current in the channel), that is a more accurate calculation of the channel discharge function. On the other hand our results are consistent with the 1cm core diameter value found in the literature.

The influence of the breakdown electric field in the configuration of lightning corona sheath on charge distribution in the channel

A model of corona sheath that surrounds the thin core of the lightning channel has been investigated by using a generalized traveling current source return stroke model. The lightning channel is modeled by a charged corona sheath that stretches around a highly conductive central core through which the main current flows. The channel core with the negatively charged outer channel sheath forms a strong electric field, with an overall radial orientation. The return stroke process is modeled as the negative leader charge in the corona sheath being discharged by the positive charge coming from the channel core. Expressions that describe how the corona sheath radius evolves during the return stroke are obtained from the corona sheath model, which predicts charge motion within the sheath. The corona sheath model, set forth by Maslowski and Rakov (2006), Tausanovic et al. (2010), Marjanovic and Cvetic (2009), Cvetic et al. (2011) and Cvetic et al. (2012), divides the sheath onto three zones: zone 1 (surrounding the channel core with net positive charge), zone 2 (surrounding zone 1 with negative charge) and zone 3 (the outer zone, representing uncharged virgin air). In the present study, we have assumed a constant electric field inside zone 1, as suggested by experimental research of corona discharges in coaxial geometry conducted by Cooray (2000). The present investigation builds upon previous studies by Tausanovic et al. (2010) and Cvetic et al. (2012) in several ways. The value of the breakdown electric field has been varied for probing its effect on channel charge distribution prior and during the return stroke. With the aim of investigating initial space charge distribution along the channel, total electric field at the outer surface of the channel corona sheath, just before the return stroke, is calculated and compared for various return stroke models. A self-consistent algorithm is applied to the generalized traveling current source return stroke model, so that the boundary condition for total electric field is fulfilled. The new density of space charge and the new radius of channel corona envelope, immediately before the return stroke stage, are calculated. The obtained results indicate a strong dependence of channel charge distribution on the breakdown electric field value. Among the compared return stroke models, transmission-line-type models have exhibited a good agreement with the predictions of the Gauss' law regarding total breakdown electric field on the corona sheath's outer surface. The generalized lightning traveling current source return stroke model gives similar results if the adjustment of the space charge density inside the corona sheath is performed.

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).

Analytical Expressions for Zero-Crossing Times in Lightning Return-Stroke Engineering Models

One of the characteristic features of the electromagnetic fields radiated by lightning is the far-field inversion of polarity with a zero-crossing occurring in the tens of microseconds range. This feature has been used in several studies to test the ability of return-stroke models to reproduce the observed electromagnetic fields. In this paper, we derive closed-form analytical expressions for the zero-crossing times associated with six engineering models for the lightning return strokes, namely the Bruce-Golde (BG) model, the transmission-line (TL) model, the traveling current source (TCS) model, the modified transmission-line linear (MTLL) model, the modified transmission-line exponential (MTLE) model, and the Diendorfer-Uman (DU) model. For the derivation, the late-time behavior of the current is expressed in terms of a single exponential function. It is shown from the derived expressions that except for the TL model, according to which the zero-crossing time is independent of the current decay constant tau, all the models exhibit greater zero-crossing time as the value of tau increases. This increase is found to be linear for the BG and TCS models, nearly linear for the DU model, and quasi-logarithmic for the MTLL and MTLE models. It is also shown that all the models exhibit a decrease of the zero-crossing time with increasing return-stroke speed, except for the BG model for which the zero-crossing time appears to be independent of v . The DU and TCS models predict almost the same zero-crossing times that vary quasi-linearly with the corresponding running variables. The zero-crossing times predicted by the MTLL and MTLE models are less sensitive to the current decay time and more sensitive to the return-stroke speed, compared to the predictions of the BG, TCS, and DU models. The BG, DU, and TCS models predict, in general, larger values for the zero-crossing times than those predicted by MTLL and MTLE models. The derived expressions are then used to discuss the conditions required for every engineering return-stroke model to reproduce the expected far-field zero-crossing times. The used procedure is to compute the zero-crossing times for each model, starting from channel-base current waveforms typical of the first and subsequent return strokes, and to examine how well the predicted values are in agreement with typical, experimentally observed zero-crossing times. Other adjustable parameters are varied within their typical range of variation. It is shown that only two models, namely MTLL and MTLE, are able to reproduce the typical zero-crossing times.

On return stroke currents and remote electromagnetic fields associated with lightning strikes to tall structures: 2. Experiment and model validation

In this paper, simultaneous GPS time‐stamped measurements of the electric and magnetic fields at three distances and of the return stroke current associated with lightning strikes to the Toronto CN Tower (553 m) during the summer of 2005 are presented. The lightning return stroke current was measured using a Rogowski coil installed at a height of 474 m above ground level (AGL). The vertical component of the electric field and the azimuthal component of the magnetic field were measured simultaneously at distances of 2.0 km, 16.8 km, and 50.9 km from the CN Tower. The propagation path from the CN Tower to the first two stations (2.0 and 16.8 km) was along the soil and through the Toronto city, whereas for the third location (50.9 km) the propagation path was nearly entirely across Lake Ontario. The waveforms of the electric and magnetic fields at 16.8 km and 50.9 km exhibit a first zero crossing about 5 μs after the onset of the return stroke. This early zero crossing is part of a narrow undershoot. For fields at 50.9 km the expected zero crossing at about 40 μs is also observed. Metallic beams and other conducting parts in buildings on which electric and magnetic field sensors were located cause an enhancement effect on the measured fields. Although an enhancement can be identified both on the electric and the magnetic fields, the degree of enhancement is actually more significant for the electric field than for the magnetic field. It is shown that the value of the wave impedance (E‐field peak to H‐field peak ratio) could give an estimate of the enhancement effect of the building on the electric field. Propagation effects (decrease of field amplitude and increase of its risetime) can also be observed in experimental records. It is shown that the fields at 50.9 km are less affected by such attenuation, compared to those at 16.8 km, presumably because the path of propagation is mostly across Lake Ontario. Measured waveforms are compared with theoretical predictions obtained using the five engineering return stroke models extended to include the presence of the strike object, namely, transmission line (TL), modified transmission line (MTLL and MTLE), Bruce‐Golde (BG), and traveling current source (TCS) models. A reasonable agreement is found with all five engineering models for the magnetic field waveforms at the three considered distances, although the peak values of the computed fields are systematically about 25% lower than measured values. None of the models was able to reproduce the early zero crossing and the narrow undershoot. As far as the electric field is concerned, larger differences have been observed between simulations and measurements. This may be due to the fact that the enhancement effect of the building on the electric field is stronger than that on the magnetic field. The expression relating current and field peaks associated with strikes to tall structures is also tested versus obtained sets of experimental data. The overall agreement between the theoretically predicted and the experimentally observed field‐to‐current ratio is reasonable, although the formula of Bermudez et al. (2005) appears also to underestimate the experimentally measured ratio (by about 25%). This may be due, at least in part, to the enhancement effect of the buildings on which the field measurement antennae were installed.

Equivalency of Lightning Return-Stroke Models Employing Lumped and Distributed Current Sources

We show that any engineering return-stroke model can be expressed, using an appropriate continuity equation, in terms of either lumped or distributed current sources with the resultant longitudinal-current distribution along the channel being the same. This property can be viewed as the duality of engineering models. The conversion alters the actual-corona current (if any) of the model. For lumped-source (LS) models the actual-corona current is unipolar and directed radially out of the channel core, while for distributed-source (DS) models it is unipolar and directed into the channel core. For LS models converted to DS models and for the Diendorfer-Uman (DU) model converted to the equivalent LS model, the corona current is the sum of the negated actual-corona current (if any) and a fictitious-corona current, the latter being bipolar. For the transmission-line (TL) model (no longitudinal current attenuation with height) expressed in terms of DSs, there is only a fictitious bipolar corona current component. Conversion of the traveling-current source (TCS) and Bruce-Golde (BG) models to equivalent LS models involves replacement of the actual, unipolar corona current with a fictitious one, the latter current being bipolar near the channel base and unipolar at higher altitudes

Return stroke current profiles and electromagnetic fields associated with lightning strikes to tall towers: Comparison of engineering models

In this paper, we compare five engineering return stroke models, extended to include the presence of an elevated strike object, namely the Bruce–Golde (BG) model, the transmission line (TL) model, the traveling current source (TCS) model, and the two modified TL models (MTLL and MTLE). The model-predicted current profile along the channel and along the strike object, as well as radiated electric and magnetic fields at different distances are presented and discussed. The comparison is made assuming the same ‘undisturbed current’ (that is the idealized current that would be measured at the tower top if the current reflection coefficients at its both extremities were equal to zero) and the same return-stroke speed. Except for the case of very close (50 m) electric field, it is found that the computed electromagnetic fields associated with a strike to a 168-m tall tower are less model-dependent than those corresponding to a strike to flat ground.

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”

Radio frequency radiation beam pattern of lightning return strokes: Inferred from FORTE satellite observations

The fast propagating return stroke is expected to produce a radiation pattern differing from a dipole pattern, with a “correction” of (1 − v cos θ/c)−1 or the so‐called F factor. The FORTE satellite measures lightning Very High Frequency (VHF) radiation at different angles from the up space and offers the first opportunity of examining the F factor. In this report, we studied a group of FORTE‐detected lightning events that were also observed by the National Lightning Detection Network (NLDN). NLDN provided the discharges' geolocations and helped to identify the discharge types. During the summers of 1998 and 1999, 25,721 coincident events were found. Among these, 2092 were found to be narrow (<100 ns), VHF‐intense and highly polarized bursts and were found to be associated with the beginning of return strokes. Through careful statistical analysis regarding the distributions of the event occurrence, we found that the overall ensemble of events can be considered to have an isotropic pattern in the upper half‐space. The subset of the narrow bursts displays a beam pattern that agrees with the F factor at a propagating speed of v = 0.75c. The latter is inferred by comparing FORTE observations to a free‐space transmission line (TL) model. The analysis shows that the ground does not affect the narrow‐burst beam pattern observed from the upper half‐space; the source for the narrow burst needs to be a few tens of meters above the surface of the Earth and is apparently associated with the junction point of the attachment process. The physical size of the corresponding discharge is estimated to be less than 20 m. The analysis also suggests that a single upward current is responsible for the observations, rather than a bidirectional current as suggested by others for the attachment process. Similarly, the traveling current source (TCS) model that consists of a simultaneous downward current is found less suitable for the initiation of return strokes.

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.

Treatment of retardation effects in calculating the radiated electromagnetic fields from the lightning discharge

The analytical treatment of retardation effects in calculating lightning electromagnetic fields far from the source has often involved the use of a so‐called F factor. The literature concerning the F factor in the lightning field equations is often confusing and sometimes in error. The aim of this paper is to clarify, to correct when needed, and to extend previous views of the F factor by considering and consolidating the various situations, both mathematically and physically, in which this factor can occur. The F factor arises because of the retardation effects occurring when the distance to the observer from a point on a propagating current is changing with time. In this paper we (1) discuss the various situations in which the F factor can arise, such as in the determination of the “radiating” channel length “seen” by the observer, in the calculation of fields due to a propagating current wave up or down the discharge channel, and in the calculation of fields due to a propagating current discontinuity extending the channel upward; (2) give a unifying physical interpretation for this factor; and (3) show that the retardation effects in calculating lightning fields can be accounted for without the explicit use of an F factor. Relative to item 2 above, we will show that for simple return‐stroke models like the transmission line (TL) and traveling current source (TCS) models, in which the current at one point on the channel appears at another point at another time, the F factor associated with current behind the front can be interpreted physically as the ratio of the apparent propagation speed (upward or downward) of the current wave “seen” by a distant observer to its actual speed. In the TL model, a current wave moves upward at speed ν, the same speed as that of the front, and the F factor is given by [1‐(ν/c) cos θ]−1, where θ is the angle between the direction of propagation of the source and the line joining the source point and the field point (observer), and c is the speed of light in vacuum. In the TCS model, a current wave moves downward at a speed equal c, and the F factor is given by [1+cosθ]−1. The F factor associated with an upward propagating current discontinuity can always (in any model) be interpreted physically as the ratio of the apparent propagation speed of the discontinuity to its actual speed and is given by the same expression as for the F factor for the upward propagating current wave in the TL model.

Distribution of charge along the lightning channel: Relation to remote electric and magnetic fields and to return‐stroke models

We derive exact expressions for remote electric and magnetic fields as a function of the time‐ and height‐varying charge density on the lightning channel for both leader and return‐stroke processes. Further, we determine the charge density distributions for six return‐stroke models. The charge density during the return‐stroke process is expressed as the sum of two components, one component being associated with the return‐stroke charge transferred through a given channel section and the other component with the charge deposited by the return stroke on this channel section. After the return‐stroke process has been completed, the total charge density on the channel is equal to the deposited charge density component. The charge density distribution along the channel corresponding to the original transmission line (TL) model has only a transferred charge density component so that the charge density is everywhere zero after the wave has traversed the channel. For the Bruce‐Golde (BG) model there is no transferred, only a deposited, charge density component. The total charge density distribution for the version of the modified transmission line model that is characterized by an exponential current decay with height (MTLE) is unrealistically skewed toward the bottom of the channel, as evidenced by field calculations using this distribution that yield (1) a large electric field ramp at ranges of the order of some tens of meters not observed in the measured electric fields from triggered‐lightning return strokes and (2) a ratio of leader‐to‐return‐stroke electric field at far distances that is about 3 times larger than typically observed. The BG model, the traveling current source (TCS) model, the version of the modified transmission line model that is characterized by a linear current decay with height (MTLL), and the Diendorfer‐Uman (DU) model appear to be consistent with the available experimental data on very close electric fields from triggered‐lightning return strokes and predict a distant leader‐to‐return‐stroke electric field ratio not far from unity, in keeping with the observations. In the TCS and DU models the distribution of total charge density along the channel during the return‐stroke process is influenced by the inherent assumption that the current reflection coefficient at ground is equal to zero, the latter condition being invalid for the case of a lightning strike to a well‐grounded object where an appreciable reflection is expected from ground.

Comparison of lightning return‐stroke models

Five return‐stroke models, each allowing the use of measured channel‐base current and return‐stroke speed as inputs for the computation of channel current distribution and remote electric field, are compared and evaluated using 18 sets of three simultaneously measured triggered lightning features: channel‐base current, return‐stroke speed, and electric field at a distance of about 5 km from the channel base. The experimental data were acquired during a triggered lightning experiment at the NASA Kennedy Space Center, Florida, in 1987 and were reported in part by Willett et al. (1989) and Leteinturier et al. (1991). The models compared are the transmission line (TL) model, the modified transmission line (MTL) model, the traveling current source (TCS) model, the Diendorfer‐Uman (DU) model, and the modified Diendorfer‐Uman (MDU) model. The TL, MTL, DU, and MDU models each predict the measured initial electric field peaks with an error whose mean absolute value is about 20%; the TCS model has a mean absolute error about twice that value. For the prediction of overall measured field wave shape, none of the models is clearly preferred, although for the model parameters assumed, the MDU model gave the best wave shape match. Most of the return strokes that exhibited very narrow sharp initial peaks in the measured electric field waveforms had a maximum rate of rise of channel‐base current closer to the peak of the measured channel‐base current waveform than did return strokes not exhibiting these sharp field peaks. The calculated fields from the TL and the MTL models do not have narrow sharp peaks similar to those found in many of the measured fields, while the fields calculated from the TCS, DU, and MDU models had somewhat similar peaks in most of the cases where those peaks were found in the measured fields. On the basis of the comparison of the five models, we recommend the TL model for calculating the peak channel‐base current from the measured initial peak electric field because the TL model provides a similar or better result from a simpler mathematical relation.

Determination of lightning currents from far electromagnetic fields

Expressions relating far electromagnetic fields and return stroke channel base currents have been derived in the literature for various lightning return stroke models. The use of such relations permits the estimation of channel base currents of return strokes, testing of the individual models using sets of simultaneous current and field measurements, and also the determination of nondirectly measurable parameters of the models. In this paper we derive a relation between the far electromagnetic field and the channelbase current for the Diendorfer‐Uman (DU) and the modified Diendorfer‐Uman (MDU) models. Additionally, the current field relations for all models for which such an analytical relation can be derived are summarized. Finally, a numerical solution technique is proposed to solve the equations relating the far field and channel base current of the DU and the traveling current source (TCS) models.

Lightning return stroke current models with specified channel‐base current: A review and comparison

We compare five lightning return stroke current models that exhibit a simple relationship between the current at the return stroke channel‐base and the current along the return stroke channel, namely the Bruce‐Golde (BG) model, the transmission line (TL) model, the Master, Uman, Lin, and Standler (MULS) model, the Traveling Current Source (TCS) model, and the Modified Transmission Line (MTL) model, by assuming a common current wave shape at the channel base and then calculating the channel currents and charges and the resultant electric and magnetic fields. There are basically two characteristics that distinguish the models, namely, (1) the treatment of the return stroke wave front and (2) the spatial and temporal distribution of charge removed from the leader channel. The simpler BG model can be used as an excellent approximation to the TCS model. The latter reduces to the former when the current injected downward by the traveling current source has an infinite speed. The MULS model is equivalent to the MTL model when the MULS uniform current is assumed to be zero. The BG and TCS models produce sharper initial field peaks than do the TL, MTL, and MULS models. The ratio of the peak field derivative to the peak current derivative is near the ratio of the peak field to the peak current for the MULS and MTL models and is equal for the TL model, whereas for the BG and TCS models the ratio of the peak derivatives is about twice the ratio of the peak field to peak current. The TL model is unrealistic for long‐time field calculations due to the fact that no net charge is removed from the channel. The other four models produce overall fields which are reasonable approximations to measured fields from natural lightning even though, for the assumed channel‐base current, the BG and TCS models do not reproduce the observed distant‐field zero crossing and the MTL and MULS models do not reproduce the magnetic “hump” observed after the initial field peak at close range. None of the models can reproduce the fine structure observed in the measured fields.

Induced voltage on an overhead line due to nearby lightning

Voltage waveshapes induced on an overhead line by nearby lightning return strokes are calculated numerically by solving the transmission-line equations using the time-domain approach of A.K. Agrawal et al. (1980) and a finite-difference technique. The traveling current source (TCS) model is used to calculate the return-stroke vertical and horizontal-electric fields (which are the sources for a line voltage) above a perfectly conducting ground plane. The effect of the striking-point distances and the return-stroke current waveshape on the induced voltages are illustrated. The exact location of the maximum voltage along the line depends not only on the closest distance from the striking point to the line but also on lightning-current parameters. In the case of a 2000-m-long line and a striking point a distance of 100 m from the line center, almost 90% of the induced voltage at the line terminations is due to the horizontal electric field coupling.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>