Martin A. Uman

The electromagnetic radiation from a finite antenna

Textbooks rarely give time−domain solutions to antenna problems. For the case of a finite linear antenna along which a fixed current waveform propagates, we present analytical time−domain solutions for the electric and magnetic radiation (far) fields. We also give computer solutions for the total (near and far) fields. The current waveform used as an example in the computer calculations approximates that of a lightning return−stroke, a common geophysical example of the type of radiation source under consideration.

A Gated, Wideband Magnetic Direction Finder for Lightning Return Strokes

Abstract A magnetic direction finder has been developed which utilizes only the initial few microseconds of wide-band return stroke waveforms to provide accurate directions to the channel bases of lightning discharges to ground. Bearing errors are minimized because, near the ground, most channels tend to be straight and vertical with no large branches or horizontal sections. Tests on a number of lightning storms at distances of 10 to 100 km indicate the angular resolution is in the range from 1° to 2°, with little or no systematic dependence on azimuth or distance.

New insights into lightning processes gained from triggered-lightning experiments in Florida and Alabama

Analyses of electric and magnetic fields measured at distances from tens to hundreds of meters from the ground strike point of triggered lightning at Camp Blanding, Florida, and at 10 and 20 m at Fort McClellan, Alabama, in conjunction with currents measured at the lightning channel base and with optical observations, allow us to make new inferences on several aspects of the lightning discharge and additionally to verify the recently published “two-wave” mechanism of the lightning M component. At very close ranges (a few tens of meters or less) the time rate of change of the final portion of the dart leader electric field can be comparable to that of the return stroke. The variation of the close dart leader electric field change with distance is somewhat slower than the inverse proportionality predicted by the uniformly charged leader model, perhaps because of a decrease of leader charge density with decreasing height associated with an incomplete development of the corona sheath at the bottom of the channel. There is a positive linear correlation between the leader electric field change at close range and the succeeding return stroke current peak at the channel base. The formation of each step of a dart-stepped leader is associated with a charge of a few millicoulombs and a current of a few kiloamperes. In an altitude-triggered lightning the downward negative leader of the bidirectional leader system and the resulting return stroke serve to provide a relatively low-impedance connection between the upward moving positive leader tip and the ground, the processes that follow likely being similar to those in classical triggered lightning. Lightning appears to be able to reduce, via breakdown processes in the soil and on the ground surface, the grounding impedance which it initially encounters at the strike point, so at the time of channel-base current peak the reduced grounding impedance is always much lower than the equivalent impedance of the channel. At close ranges the measured M-component magnetic fields have waveshapes that are similar to those of the channel-base currents, whereas the measured M-component electric fields have waveforms that appear to be the time derivatives of the channel-base current waveforms, in further confirmation of the “two-wave” M-component mechanism.

Parameters of triggered-lightning flashes in Florida and Alabama

Channel base currents from triggered lightning were measured at the NASA Kennedy Space Center, Florida, during summer 1990 and at Fort McClellan, Alabama, during summer 1991. Additionally, 16-mm cinematic records with 3- or 5-ms resolution were obtained for all flashes, and streak camera records were obtained for three of the Florida flashes. The 17 flashes analyzed here contained 69 strokes, all lowering negative charge from cloud to ground. Statistics on interstroke interval, no-current interstroke interval, total stroke duration, total stroke charge, total stroke action integral (∫ i2dt), return stroke current wave front characteristics, time to half peak value, and return stroke peak current are presented. Return stroke current pulses, characterized by rise times of the order of a few microseconds or less and peak values in the range of 4 to 38 kA, were found not to occur until after any preceding current at the bottom of the lightning channel fell below the noise level of less than 2 A. Current pulses associated with M components, characterized by slower rise times (typically tens to hundreds of microseconds) and peak values generally smaller than those of the return stroke pulses, occurred during established channel current flow of some tens to some hundreds of amperes. A relatively strong positive correlation was found between return stroke current average rate of rise and current peak. There was essentially no correlation between return stroke current peak and 10–90% rise time or between return stroke peak and the width of the current waveform at half of its peak value. Parameters of the lightning flashes triggered in Florida and Alabama are similar to each other but are different from those of triggered lightning recorded in New Mexico during the 1981 Thunderstorm Research International Program. Continuing currents that follow return stroke current peaks and last for more than 10 ms exhibit a variety of wave shapes that we have subdivided into four categories. All such continuing currents appear to start with a current pulse presumably associated with an M component. A brief summary of lightning parameters important for lightning protection, in a form convenient for practical use, is presented in an appendix.

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.

Observed leader and return-stroke propagation characteristics in the bottom 400 m of a rocket-triggered lightning channel

Using a high-speed digital optical system, we determined the propagation characteristics of two leader/return-stroke sequences in the bottom 400 m of the channel of two lightning flashes triggered at Camp Blanding, Florida. One sequence involved a dart leader and the other a dart-stepped leader. The time resolution of the measuring system was 100 ns, and the spatial resolution was about 30 m. The leaders exhibit an increasing speed in propagating downward over the bottom some hundreds of meters, while the return strokes show a decreasing speed when propagating upward over the same distance. Twelve dart-stepped leader luminosity pulses observed in the bottom 200 m of the channel have been analyzed in detail. The luminosity pulses associated with steps have a 10-90% risetime ranging from 0.3 to 0.8 μs with a mean value of 0.5 μs and a half-peak width ranging from 0.9 to 1.9 μs with a mean of 1.3 μs. The interpulse interval ranges from 1.7 to 7.2 μs with a mean value of 4.6 μs. The step luminosity pulses apparently originate in the process of step formation, which is unresolved with our limited spatial resolution of 30 m, and propagate upward over distances from several tens of meters to more than 200 m, beyond which they are undetectable. This finding represents the first experimental evidence that the luminosity pulses associated with the steps of a downward moving leader propagate upward. The upward propagation speeds of the step luminosity pulses range from 1.9×10 7 to 1.0×10 8 m/s with a mean value of 6.7×10 7 m/s. In particular, the last seven pronounced light pulses immediately prior to the return stroke pulse exhibit more or less similar upward speeds, near 8×10 7 m/s, very close to the return-stroke speed over the same portion of the channel. On the basis of this result, we infer that the propagation speed of a pulse traveling along the leader-conditioned channel is primarily determined by the channel characteristics rather than the pulse magnitude. An inspection of four selected step luminosity pulses shows that the pulse peak decreases significantly as the pulse propagates in the upward direction, to about 10% of the original value within the first 50 m. The return-stroke speeds within the bottom 60 m or so of the channel are 1.3×10 8 and 1.5×10 8 m/s for the two events analyzed, with a potential error of less than 20%.

Leader properties determined with triggered lightning techniques

This paper presents current and electric field measurements from two triggered lightning flashes, 9519 and 9516, initiated by the classical and altitude technique, respectively, at Camp Blanding, Florida, in 1995. The current measurement for flash 9519 shows that the upward positive leader, initiated at the top of the grounded wire unreeled by the triggering rocket, propagates in a discontinuous pattern made of successive current pulses of tens to a few hundreds of amperes and separated by intervals of 20-25 μs. The downward negative leader in flash 9516, initiated from the electrically floating conductor, has a velocity greater than 1.3 x 10 5 m s -1 , a stepping interval of 18 μs, and step length of about 3-5 m; the associated peak currents inferred from the electric field steps are at least 600 A.

Methods for calculating the electromagnetic fields from a known source distribution: application to lightning

Two different techniques (monopole and dipole) for calculating the electric and magnetic fields from a distribution of currents and charges are discussed. Both techniques have been used for calculating the fields from lightning. A simple lightning return stroke current model, consisting of a square current pulse traveling up a vertical antenna above a ground plane, is used to compare the two techniques. Analytical expressions are obtained for the fields using each technique. These expressions are shown to be numerically equivalent, but the authors are unable to prove their equivalence analytically. It is concluded that the monopole and the dipole techniques can both be derived from Maxwells equations and hence that both are correct. In attempting to dispel the apparent confusion that has existed regarding the validity of the monopole technique, the authors show that the monopole approach, as discussed in the literature, is applicable only to upward-traveling current waves and hence is not particularly useful in the realistic modeling of lightning return strokes. >

Transient electric and magnetic fields associated with establishing a finite electrostatic dipole

We obtain analytical solutions in the time domain for the electric and magnetic fields associated with establishing a finite electrostatic dipole. We assume that a simple source current distribution, a square pulse of current, produces the dipole, and solve for the fields produced by that source current distribution using Maxwell’s equations. Salient features of the fields are discussed from a physical point of view. We outline a technique to determine in the time domain the electric and magnetic fields produced by any arbitrary time‐varying current propagating along a straight antenna, given the calculated fields due to a short square pulse of current.