Lightning Return Stroke Models Research Articles

Merging of current generation and current dissipation lightning return stroke models

Current generation and current dissipation return stroke models are engineering models based on the theory associated with the propagation of current pulses along transmission lines undergoing corona. However, neither of these models incorporates the complete theory associated with the phenomenon. One can make the physical scenario complete by combining the current generation concept with the current dissipation concept. In this paper how this can be done is demonstrated by creating a return stroke model which is a combination of these two model types. The new model encompasses the full theory associated with the pulse propagation along transmission lines under corona. The paper provides a full description of the model together with a description of the spatial and temporal variation of the return stroke current and the electric and magnetic fields generated at different distances as predicted by the model.

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.

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.

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.

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,

Why do some lightning return stroke models not reproduce the far‐field zero crossing?

In this paper, we investigate the reasons why some return stroke models do not reproduce one of the characteristic features of the electromagnetic fields radiated by lightning, namely the far‐field inversion of polarity with a zero crossing occurring in the tens of microseconds range. This study leads to physical insights into the behavior of the return stroke attenuation and speed as a function of height. Making reference to the properties of time domain far fields, we show that any one of the existing lightning return stroke channel models, including the electromagnetic models, sometimes known as Antenna Theory models, and the engineering models, should be able to predict a zero crossing in the far field as long as the duration of the return stroke current and the height of the channel are finite. However, the present versions of most of the available engineering and electromagnetic models predict the zero crossing to occur at times that fall well beyond those observed experimentally. We identify three mechanisms responsible for the time of occurrence of the reversal of polarity in the far fields: the current attenuation along the channel, the duration of the return stroke current, and the return stroke speed. The analysis of the MTLE and MTLL engineering models shows that the higher the attenuation of the current along the channel, the earlier the polarity reversal of the vertical electric field. Also, for a given value of the attenuation factor, higher propagation speeds correspond to earlier polarity reversal times. For the TCS model, in which the only adjustable parameter is the return‐stroke speed, we show that the far‐field zero crossing occurs considerably later than the values predicted by both the MTLE and the MTLL models. This is shown to be essentially due to the fact that the decrease of the current wave along the channel according to the TCS model is less pronounced than the current attenuation predicted by the MTLE and MTLL models. Furthermore, it is shown that, in the electromagnetic models, a uniformly distributed resistance along the channel does not lead to a zero crossing in the tens of microseconds range even for large resistance values. This is due to the significant increase of the current pulse duration, as it propagates along the channel as a result of the dispersion effect. The correct zero crossing time, however, can indeed be successfully reproduced if a nonlinear channel resistance is used, which prevents the significant increase of the current pulse duration as it propagates up the channel. Finally, based on the predictions of the MTLE, MTLL, and TCS models, it is shown that the zero crossing time decreases as the observation point moves farther away from the lightning channel, with the amount of variation of less than 10%.

On the Relationship Between the Signature of Close Electric Field and the Equivalent Corona Current in Lightning Return Stroke Models

Engineering return stroke models can be categorized either as current generation (traveling current source type) models or current propagation (transmission line type) models. The current generation models are described among other parameters by a corona current distributed along the channel. Recent studies show that there is equivalence between the models of current generation and current propagation types. Due to this equivalence, any engineering return stroke model of current propagation type can be described in terms of an equivalent corona current per unit channel length. The measurements conducted within 10-500 m from triggered lightning flashes show that the electric field of subsequent return strokes at these distances flattens within 15 mus or so. In this paper, the constraints imposed by this feature on the temporal and spatial variation of the equivalent corona current are investigated. The results show that in order for the close fields to flatten within 15 mus or so, the equivalent corona current, should be bipolar and the corona current wave shape at late times should be identical to that of the longitudinal current time derivative. This is in contrast to most of the engineering models of current generation type, in which the corona current is assumed to be unipolar.

A comparison of different approaches to simulate a nonlinear channel resistance in lightning return stroke models

Different physical models that describe the time variation of the channel resistance are investigated in a lightning return stroke model. Such models consider one of the three following hypotheses: (1) the channel resistance decays exponentially with time; (2) the channel resistance decays with the radial expansion of the channel core, which is assumed to be described by the strong‐shock approximation, or (3) the channel resistance varies with time according to three different arc resistance models (defined by Toepler, Barannik and Kushner et al.). Analyses illustrate the effect of a time‐varying channel resistance on channel currents and corresponding electromagnetic fields. It is shown that the strong‐shock approximation is able to predict typical features of experimentally observed lightning electromagnetic fields and return stroke speed profiles. It is also shown that results predicted by the strong‐shock approximation can be qualitatively reproduced by either using simplified arc resistance equations (such as Toepler's and Barannik's ones) or considering an exponential decay of the channel resistance with attenuation constants linearly increasing with height.

Applications of Electromagnetic Models of the Lightning Return Stroke

Lightning return-stroke models are needed to study lightning effects on various objects and systems, as well as in characterizing the lightning electromagnetic environment. Reviewed here are models based on Maxwell's equations and referred to as electromagnetic models. In contrast to distributed-circuit and so-called engineering models, electromagnetic models of the lightning return stroke allow a self-consistent full-wave solution for both current distribution along the lightning channel and associated electromagnetic fields. In this paper, we review electromagnetic models with an emphasis on their applications.

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

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.

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.

Concepts of lightning return stroke models

Return stroke models have been reviewed according to the assumptions they make and their underlying concepts. We have investigated the roots of the modern concepts used to describe return stroke related phenomena and illustrated their developments and deviations from the original idea. We discuss the return stroke models in two categories in accordance with the direction of the charge transfer along the channel. We show that all the models that we have considered can be described by two general sets of mathematical equations. The validation and comparison of models in reproducing measured lightning parameters are discussed. We also show the different methods employed by return stroke models to estimate the return stroke speed and discuss possible improvements that could be introduced into return stroke models in the future.

Review and evaluation of lightning return stroke models including some aspects of their application

Four classes of models of the lightning return stroke are reviewed. These four classes are: (1) the gas dynamic models; (2) the electromagnetic models; (3) the distributed-circuit models; and (4) the engineering models. Validation of the reviewed models is discussed. For the gas dynamic models, validation is based on observations of the optical power and spectral output from natural lightning. The electromagnetic, distributed-circuit, and engineering models are most conveniently validated using measured electric and magnetic fields from natural and triggered lightning. Based on the entirety of the validation results and on mathematical simplicity, we rank the engineering models in the following descending order: MTLL, DU, MTLE, BG, and TL. When only the initial peak values of the channel-base current and remote electric or magnetic field are concerned, the TL model is preferred. Additionally discussed are several issues in lightning return-stroke modeling that either have been ignored to keep the modeling straightforward or have not been recognized, such as the treatment of the upper, in-cloud portion of the lightning channel, the boundary conditions at the ground, including the presence of a vertically extended strike object, the return-stroke speed at early times, the initial bi-directional extension of the return stroke channel, and the relation between leader and return stroke models. Various aspects of the calculation of lightning electric and magnetic fields in which return stroke models are used to specify the source are considered, including equations for fields and channel-base current, as well as a discussion of channel tortuosity and branches.

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 models

We test the two most commonly used lightning return stroke models, Bruce‐Golde and transmission line, against subsequent stroke electric and magnetic field wave forms measured simultaneously at near and distant stations and show that these models are inadequate to describe the experimental data. We then propose a new return stroke model that is physically plausible and that yields good approximations to the measured two‐station fields. Using the new model, we derive return stroke charge and current statistics for about 100 subsequent strokes.