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.

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

The singularity expansion method (SEM) for quantifying the transient electromagnetic (EM) scattering from targets illuminated by pulsed EM radiation is reviewed. SEM representations for both induced currents and scattered fields are presented. Natural-resonance-based target identification schemes, based upon the SEM, are described. Various techniques for the extraction of natural-resonance modes from measured transient response waveforms are reviewed. Particular attention is given to the aspect-independent (extinction) E-pulse and (single-mode) S-pulse discriminant waveforms which, when convolved with the late-time pulse response of a matched target, produce null or mono-mode responses, respectively, through natural-mode annihilation. Extensive experiment results for practical target models are included to validate the E-pulse target discrimination technique. Finally, anticipated future extensions and areas requiring additional research are identified. >

The singularity expansion method and its application to target identification

Compares 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. Two characteristics distinguish the models: the treatment of the return stroke wave front, and the spatial and temporal distribution of charge removed from the leader channel. The TL model is unrealistic for long-time field calculations owing 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

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

This paper is concerned with the approximation of the Maxwell equations by the eddy currents model, which appears as a correction of the quasi-static model. The eddy currents model is obtained by neglecting the displacement currents in the Maxwell equations and exhibits an elliptic character in the time-harmonic formulation. Our main concern in this paper is to show that the eddy currents model approximates the full Maxwell system up to the second order with respect to the frequency if and only if an additional condition on the current source is fulfilled. Otherwise, it is a first-order approximation to the Maxwell equations. We also study the well-posedness of the eddy currents model and investigate the time-dependent case. All our results strongly depend on the topology properties of the domains under consideration. This dependence which is specific to Maxwell's equations does not appear for the two- or the three-dimensional Helmholtz operator.

A justification of eddy currents model for the Maxwell equations

Forty simultaneous, submicrosecond time-resolved measurements of triggered lightning returnstroke current (I), current derivative (dI/dt), and electric field derivative (dE/dt) were made in Florida and France in 1985 and 1986. Peak currents ranged from about 5 to 50 kA, peak dI/dt amplitudes from 60 to 260 kA/μs in 1985 and from 20 to 140 kA/μs in 1986. The mean peak dI/dt values, 111 kA/μs (1985) and 68 kA/μs (1986), are 2–3 times higher than data from instrumented towers, and peak I and dI/dt appear to be positively correlated. The dE/dt and dI/dt waveform pairs have similar shapes, and the peak amplitudes are linearly proportional. Return-stroke velocity, computed from the ratio of peak dE/dt and dI/dt signal amplitudes using an expression derived from the radiation field term of the transmission line model (TLM), averaged 2.9×108 m/s and 3.0×108 m/s in 1985 and 1986, respectively, which is about 2 times higher than most optical measurements. The TLM velocity may be erroneous because (1) the dE/dt measurement was made only 50 m from the lightning channel, where fields other than the radiation field component, that is near fields, may contribute to the total dE/dt and (2) fine structure on the measured E fields was not consistent with a single upwardly propagating return-stroke current wave assumed by the TLM (two waves are more consistent).

Current and electric field derivatives in triggered lightning return strokes

The singularity expansion method (SEM) arose from the observation that the transient response of complex electromagnetic scatterers appeared to be dominated by a small number of damped sinusoids. In the complex frequency plane, these damped sinusoids are poles of the Laplace-transformed response. The question is then one of characterizing the object response (time and frequency domains) in terms of all the singularities (poles, branch cuts, entire functions) in the complex frequency plane (hence singularity expansion method). Building on the older concept of natural frequencies, formulae were developed for the pole terms from an integral-equation formulation of the scattering process. The resulting factoring of the pole terms has important application consequences. Later developments include the eigenmode expansion method (EEM) which diagonalizes the integral-equation kernels and which can be used as an intermediate step in ordering the SEM terms. Additional concepts which have appeared include eigenimpedance synthesis and equivalent electrical networks. Of current interest is the use of the theoretical formulae to efficiently analyze and order experimental data, Related to this is the application of SEM results to target identification. This paper does not delve into the mathematical details; it presents an overview of the history and major concepts and results in SEM and EEM and related matters.

The singularity expansion method: Background and developments

The transmission-line model previously developed for the leader stroke is applied to the return-stroke of a lightning channel. The charge stored in the corona is converted into current, by a discharge wave. The dependence of propagation velocity on coronal charge and radius leads to front steepening and the development of a jump discontinuity or electromagnetic shock in a finite time.

Analytic Return-Stroke Transmission-Line Model

The emerging fundamental basis for analysis and design of complex electromagnetic systems, for protection against undesirable propagation of electromagnetic signals through the system, is known as electromagnetic topology. While this may be viewed as the basis for grounding and shielding, it is more general than that.

Electromagnetic Topology for the Analysis and Design of Complex Electromagnetic Systems

This paper reviews a subject which has developed over the past two decades: the optimal design of sensors for transient (or broadband) electromagnetic measurements and proper installation of the sensors involving concepts from topology and symmetry.

Electromagnetic Sensors and Measurement Techniques

The singularity expansion method (SEM) arose from the observation that the transient response of complex electromagnetic scatterers appeared to be dominated by a small number of damped sinusoids. In the complex frequency plane, these damped sinusoids are poles of the Laplace-transformed response. The question is then one of characterizing the object response (time and frequency domains) in terms of all the singularities (poles, branch cuts, entire functions) in the complex frequency plane (hence singularity expansion method). Building on the older concept of natural frequencies, formulae were developed for the pole terms from an integral-equation formulation of the scattering process. The resulting factoring of the pole terms has important application consequences. Later developments include the eigenmode expansion method (EEM) which diagonalizes the integral-equation kernels and which can be used as an intermediate step in ordering the SEM terms. Additional concepts which have appeared include e...

Carl E. Baum

Papers

The singularity expansion method: Background and developments

Analytic Return-Stroke Transmission-Line Model

Electromagnetic Topology for the Analysis and Design of Complex Electromagnetic Systems

Electromagnetic Sensors and Measurement Techniques

The singularity expansion method: background and developments